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# How can the algebra of infinitesimal conformal transformations be infinite dimensional (in 2D)?

In Blumenhagen's book "Introduction to Conformal Field Theory", I found the statement

The algebra of infinitesimal conformal transformations in an Euclidean 2-dimensional space is infinite dimensional.

He concludes this after finding that the generators of the infinitesimal conformal transformations (i.e. a basis for its Lie algebra) are $l_n=-z^{n+1}\partial$ and $\bar l_n= -\bar z^{n+1}\bar\partial$ for $n\in\mathbb N$.
My problem with this is: shouldn't the elements of the Lie algebra be linear combinations (with real coefficients) of the basis elements $\partial,\bar\partial$ of $T_eG$? (Instead of a roduct of the derivatives with polynomials). Furthermore (and related to the previous point), if our Lie group is 2-dimensional (as a manifold), that implies that its Lie algebra is 2-dimensinal (as vector space), so definitely not infinitely dimensional. What is going on here?

The generators should depend on the point z, as each conformal transformations act differently around each point - for example, the map $z$ to $z^2$ acts infinitisimaly as $z+\delta z$ to $z+2z\delta z$. The dependence on the point is easiliy understood - for example under this map, the more far z is away from the origin, it is stretched more.
• Ok, but how do "infinitesimal conformal transformations" come into play? From what I understood so far, we have: the conformal group $G$ together with a representation of $G$ in a 2D euclidean space. $G$ has a Lie algebra which has a basis given by the generators that I wrote above. What now? – Soap May 5 '18 at 20:29

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# Definition:Closure (Abstract Algebra)/Algebraic Structure

## Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Then $S$ has the property of closure under $\circ$ if and only if:

$\forall \tuple {x, y} \in S \times S: x \circ y \in S$

$S$ is said to be closed under $\circ$, or just that $\struct {S, \circ}$ is closed.

## Also known as

Some authors use stable under $\circ$ for closed under $\circ$.

It is sometimes more convenient to express this property the other way about, as $\circ$ is closed in $S$.

## Internationalization

Closure is translated:

 In German: Abgeschlossenheit (literally: seclusion)


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GATE Civil Engineering 2021 SET-1

 Question 1
The rank of matrix $\left[\begin{array}{llll} 1 & 2 & 2 & 3 \\ 3 & 4 & 2 & 5 \\ 5 & 6 & 2 & 7 \\ 7 & 8 & 2 & 9 \end{array}\right]$ is
 A 1 B 2 C 3 D 4
Engineering Mathematics   Linear Algebra
Question 1 Explanation:
Using $R_{2} \rightarrow R_{2} \rightarrow 3 R_{1}, R_{3} \rightarrow R_{3}-5 R_{1}, R_{4} \rightarrow R_{4}-7 R_{1}$
$A=\left[\begin{array}{cccc} 1 & 2 & 2 & 3 \\ 0 & -2 & -4 & -4 \\ 0 & -4 & -8 & -8 \\ 0 & -6 & -12 & -12 \end{array}\right]$
Using $R_{3} \rightarrow R_{3}-2 R_{2}, R_{4} \rightarrow R_{4}-3 R_{2}$
$A=\left[\begin{array}{cccc} 1 & 2 & 2 & 3 \\ 0 & -2 & -4 & -4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$
So, $\rho(A)=$ No. of non-zero rows = 2.
 Question 2
If $P=\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]$ and $Q=\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]$ then $Q^{T} P^{T}$ is
 A $\left[\begin{array}{ll} 1 & 2 \\ 3 & 4 \end{array}\right]$ B $\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]$ C $\left[\begin{array}{ll} 2 & 1 \\ 4 & 3 \end{array}\right]$ D $\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right]$
Engineering Mathematics   Linear Algebra
Question 2 Explanation:
$\begin{array}{l} \quad P Q=\left[\begin{array}{ll} 1 & 3 \\ 2 & 4 \end{array}\right]\left[\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right]=\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right] \\ (P Q)^{\top}=\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right] \end{array}$
Now using Reversal law
$Q^{\top} P^{\top}=(P Q) T=\left[\begin{array}{ll} 2 & 4 \\ 1 & 3 \end{array}\right]$
 Question 3
The shape of the cumulative distribution function of Gaussian distribution is
 A Horizontal line B Straight line at 45 degree angle C Bell-shaped D S-shaped
Engineering Mathematics   Probability and Statistics
Question 3 Explanation:

$PDF:f(x)=\frac{1}{\sigma \sqrt{2 \pi}}e^{-(x-\mu )^2/(2\sigma ^2)}$
$CDF:F(x)=\frac{1}{2}\left [ 1+eff\left ( \frac{x-\mu }{\sigma \sqrt{2}} \right ) \right ]$
 Question 4
A propped cantilever beam EF is subjected to a unit moving load as shown in the figure (not to scale). The sign convention for positive shear force at the left and right sides of any section is also shown.

The CORRECT qualitative nature of the influence line diagram for shear force at G is
 A B C D
Structural Analysis   Influence Line Diagram and Rolling Loads
Question 4 Explanation:

As per Muller Breslau principle ILD for stress function (shear $-V_{G}$) will be a combination of curves ($3^{\circ}$ curves).
 Question 5
Gypsum is typically added in cement to
 A prevent quick setting B enhance hardening C increase workability D decrease heat of hydration
Construction Materials and Management
Question 5 Explanation:
The Gypsum is added to cement at the end of grinding clinker it is added to prevent quick setting.
 Question 6
The direct and indirect costs estimated by a contractor for bidding a project is Rs.160000 and Rs.20000 respectively. If the mark up applied is 10% of the bid price, the quoted price (in Rs.) of the contractor is
 A 200000 B 198000 C 196000 D 182000
Construction Materials and Management
Question 6 Explanation:
Direct Costs = 160000
Indirect costs = 20000
Mark up applied is 10% of the Bid price
Bid price = Direct cost + Indirect cost + Markup
Bid price = Direct cost + Indirect cost + $\frac{10}{100} \times$ Bid price
$\text{Bid price}\left (1-\frac{10}{100} \right )=DC+IC$
$\text{Bid price}=\frac{160000+20000}{0.9}=2,00,000$
 Question 7
In an Oedometer apparatus, a specimen of fully saturated clay has been consolidated under a vertical pressure of $50 \mathrm{kN} / \mathrm{m}^{2}$ and is presently at equilibrium. The effective stress and pore water pressure immediately on increasing the vertical stress to $150 \mathrm{kN} / \mathrm{m}^{2}$, respectively are
 A $150 \mathrm{kN} / \mathrm{m}^{2}$ and 0 B $100 \mathrm{kN} / \mathrm{m}^{2}$ and $50 \mathrm{kN} / \mathrm{m}^{2}$ C $50 \mathrm{kN} / \mathrm{m}^{2}$ and $100 \mathrm{kN} / \mathrm{m}^{2}$ D 0 and $150 \mathrm{kN} / \mathrm{m}^{2}$
Geotechnical Engineering   Effective Stress and Permeability
Question 7 Explanation:
Stress is increased suddenly, hence entire change will be taken by water $\Delta \bar{\sigma}=\Delta U=100 \mathrm{kPa}$.
There will be no change in effective stress
$\therefore \qquad \qquad\bar{\sigma}=50 \mathrm{kPa}$
 Question 8
A partially-saturated soil sample has natural moisture content of 25% and bulk unit weight of $18.5 \mathrm{kN} / \mathrm{m}^{3}$. The specific gravity of soil solids is 2.65 and unit weight of water is $9.81 \mathrm{kN} / \mathrm{m}^{3}$. The unit weight of the soil sample on full saturation is
 A $21.12 \mathrm{kN} / \mathrm{m}^{3}$ B $19.03 \mathrm{kN} / \mathrm{m}^{3}$ C $20.12 \mathrm{kN} / \mathrm{m}^{3}$ D $18.50 \mathrm{kN} / \mathrm{m}^{3}$
Geotechnical Engineering   Properties of Soils
Question 8 Explanation:
\begin{aligned} \mathrm{w} &=0.25, \gamma_{\mathrm{t}}=18.5 \mathrm{kN} / \mathrm{m}^{3} \\ \mathrm{G}_{\mathrm{s}} &=2.65, \gamma_{\mathrm{w}}=9.81 \\ \gamma_{\mathrm{t}} &=\frac{G_{\mathrm{S}} \gamma_{W}(1+w)}{1+e} \\ \Rightarrow \qquad \qquad \qquad \qquad \mathrm{e} &=\frac{2.65 \times 9.81 \times 1.25}{18.5}-1\\ \Rightarrow \qquad \qquad \qquad \qquad e&=0.756\\ \text{At full saturation}, \quad \mathrm{S}&=1\\ \Rightarrow \qquad \qquad \qquad \quad \gamma_{\mathrm{sat}}&=\frac{\left(G_{\mathrm{S}}+e\right) \gamma_{\mathrm{W}}}{1+e}\\ \gamma_{\text {sat }} &=\frac{(2.65+0.756) \times 9.81}{1.756} \\ &=19.03 \mathrm{kN} / \mathrm{m}^{3} \end{aligned}
 Question 9
If water is flowing at the same depth in most hydraulically efficient triangular and rectangular channel sections then the ratio of hydraulic radius of triangular section to that of rectangular section is
 A $\frac{1}{\sqrt{2}}$ B $\sqrt{2}$ C 1 D 2
Fluid Mechanics and Hydraulics   Open Channel Flow
Question 9 Explanation:
Efficient channel section

\begin{aligned} A & =y^{2} & & A=2 y^{2} \\ P & =2 \sqrt{2} y & P & =4 y \\ R_{I} & =\frac{y}{2 \sqrt{2}} & R_{I I}&=\frac{y}{2} \\ \therefore \qquad \qquad \qquad \frac{R_{I}}{R_{I I}} & =\frac{1}{\sqrt{2}} & \end{aligned}
 Question 10
Kinematic viscosity' is dimensionally represented as
 A $\frac{M}{LT}$ B $\frac{M}{L^{2} T}$ C $\frac{T^{2}}{L}$ D $\frac{L^{2}}{T}$
Fluid Mechanics and Hydraulics   Dimensional Analysis
Question 10 Explanation:
Kinematic viscosity
$v=\frac{\mu }{\rho }=\frac{kg/m\cdot s}{kg/m^3}=m^2/s$
$[v]=\frac{m^2}{s }=\frac{L^2}{T}$
There are 10 questions to complete.


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# Is this Zorn's lemma?

Is this correct? I think it is wrong. According to wikipedia

Suppose a partially ordered set P has the property that every chain (i.e. totally ordered subset) has an upper bound in P. Then the set P contains at least one maximal element.

However, can't see how this is equivalent to the definition given in notes.

-
Your confusion might come from the fact that you are thinking of a defintion of inductive set that is not meant in this context. You are probably thinking of sets that contain 0 and are closed under successors. Here inductive partially ordered set certainly means partial order in which every chain has an upper bound. – Stefan Geschke Dec 29 '11 at 11:29
-1 for not citing the source for the text you speak about. – Henning Makholm Dec 29 '11 at 23:07


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# Difference between Frobenii on Tate modules of special and generic fibre

Let $E$ be elliptic curve over $\mathbb Q$ and $p$ a prime of good reduction for $E$. Fix $\ell \neq p$.

If $E_p$ is ordinary then we have Frobenius $F_p$ on $E_p$. Assume $F_p$ lifts to endomorphism $F$ of $E$ (over field extension, so that $E$ has CM). Does characteristic polynomial of $F_p$ acting on Tate module $T_\ell E_p$ equal characteristic polynomial of $F$ acting on Tate module of $E$?

If yes, is this true in higher dimensions?


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1. ## Trig substitution integral

$
\int \frac{\sqrt{x^{2}-4}}{x^2}dx
$

I started with setting

$x=2sec \theta$ and $dx=2sec \theta tan \theta$
and $x^{2} = 4sec^{2} \theta$

$\tan^{2} \theta = \sec^{2} \theta -1$

Then I rewrite it...

$
\int \frac{2tan\theta}{4sec^{2}\theta} * 2 sec \theta tan\theta d\theta = \int \frac{tan^{2}\theta}{sec\theta}d\theta = \int \frac{sin^{2}\theta}{cos\theta}d\theta
$

When I go on I do not get anything what helps me solving it.
Thanks for any help...

2. How many anti-derivatives are you meant to know - because there is an anti-derivative here that you could use.

3. I can not think of any anti-derivative I could use. I am not sure if I understand your question.

4. If you make the substitution, $1-sin^{2}(\theta)$ into your equation, it will reduce down to $sec(\theta)-cos(\theta)$. Secant isn't a "top of my head" integral, but it is good to know.

5. Sorry but I do not get it.. Where can I make the substitution of
$1-sin^{2}(\theta) = cos^{2} \theta$

I have
$\int \frac{sin^{2}\theta}{cos\theta}d\theta$

I do not knwo how you get to ...
$sec(\theta)-cos(\theta)$

Even if I use
$1-cos^{2}(\theta) = sin^{2} \theta$
I don't get a term that helps me...

6. $\frac{\sin^2x}{\cos x}=\frac{1-\cos^2x}{\cos x}=\frac{1}{\cos x}-\frac{\cos^2x}{\cos x}=\sec x-\cos x$


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# 36th Annual International Symposium on Lattice Field Theory

22-28 July 2018
Kellogg Hotel and Conference Center
EST timezone
Home > Timetable > Session details
PDF | iCal

# Plenary

## Place

Location: Kellogg Hotel and Conference Center
Address: 219 S Harrison Rd, East Lansing, MI 48824
Date: from 23 Jul 09:20 to 28 Jul 12:30

## Timetable | Contribution List

Displaying 24 contributions out of 24
Type: Plenary Session: Plenary
Track: Weak Decays and Matrix Elements
Plans are well underway in Japan and the US to measure neutrino mixing angles with unprecedented precision and search for CP-violating phases in the mixing. An important systematic/theory uncertainty will be that of the neutrino-nucleus cross section. In this talk I illustrate how lattice QCD calculations (combined with nuclear many-body theory and experimental data) can reduce the uncertainty o ... More
Presented by Dr. Andreas KRONFELD on 27/7/2018 at 16:45
Type: Plenary Session: Plenary
The Facility for Rare Isotope Beams (FRIB) is a DOE national user facility being built at Michigan State University that will be a world leader in rare nuclear isotopes. The first part of the talk is an overview of the design and current status of the facility. The second part discusses the impact of FRIB on nuclear science, astrophysics, fundamental symmetries, and societal applications. The la ... More
Presented by Prof. Dean LEE on 23/7/2018 at 14:20
Type: Plenary Session: Plenary
I review recent results on hadron spectroscopy using lattice QCD. In light of the recent discoveries in heavy baryon sector at LHCb, lattice calculations in this regard will be emphasized. Recent lattice calculations on light baryon, heavy-heavy and heavy-light meson resonances will also be discussed.
Type: Plenary Session: Plenary
Track: Weak Decays and Matrix Elements
Heavy flavor physics has entered a new era when the Belle II experiment observed its first collision. There are several hints found so far by BaBar, Belle, and LHCb in particular, that suggest the physics beyond the Standard Model appearing in the loop processes at short distances. They will be further tested by higher precision experiments in the coming years. The role of lattice QCD is ... More
Presented by Dr. Shoji HASHIMOTO on 24/7/2018 at 16:15
Type: Plenary Session: Plenary
Low-energy tests of fundamental symmetries are extremely sensitive probes of physics beyond the Standard Model (SM), reaching scales that are comparable, if not higher, than directly accessible at the energy frontier. The interpretation of low-energy precision experiments and their connection with models of BSM physics relies on controlling the theoretical uncertainties induced by the non-perturba ... More
Presented by Emanuele MEREGHETTI on 23/7/2018 at 15:00
Type: Plenary Session: Plenary
Track: Algorithms and Machines
In the 10 years since the creation of the QUDA library for Lattice QCD on NVIDIA GPUs the hardware and software features of GPU systems have evolved dramatically. Not only has the raw Dslash kernel performance on a single GPU improved by more than one order of magnitude but also modern GPUs are often deployed in "Fat Nodes" with up to 8 GPUs. We report on the techniques that QUDA implements to ach ... More
Presented by Dr. Mathias WAGNER on 28/7/2018 at 14:45
Type: Plenary Session: Plenary
The dominating uncertainty in the Standard Model prediction of the muon g-2 is coming from the hadronic contributions. The Muon g-2 experiment at Fermilab has started the major data collection and the aimed four-fold increase in precision will shed light on the current discrepancy between the theory prediction and the measured value. A reciprocal effort to directly measure the hadronic contributio ... More
Presented by Prof. Krstic Marinkovic MARINA on 25/7/2018 at 14:30
Type: Plenary Session: Plenary
Track: Algorithms and Machines
I will review recent progress in software development for lattice QCD on novel architectures and new machines. I will also report some algorithmic advancements in ensemble generation, solvers and contractions.
Presented by Dr. Meifeng LIN on 28/7/2018 at 14:00
Type: Plenary Session: Plenary
Track: Nonzero Temperature and Density
We introduce the feedforward neural network in the path optimization method (POM) to evade the sign problem in field theories. POM is based on the complexification of integral variables as in the complex Langevin method and the Lefschetz thimble method. The integration path is optimized in the complexified variable space by maximizing the average phase factor. In the last Lattice meeting [1] and i ... More
Presented by Prof. Akira OHNISHI on 28/7/2018 at 16:15
Type: Plenary Session: Plenary
Track: Standard Model Parameters and Renormalization
We discuss the determination of quark masses using the MILC highly improved staggered-quark ensembles with four flavors of dynamical quarks. We extract quark masses from heavy-light pseudoscalar meson masses by making use of heavy quark effective theory (HQET) and continuum-QCD perturbative calculations. While heavy-light meson masses can be measured very precisely on lattice, perturbative calcula ... More
Type: Plenary Session: Plenary
Track: Physics Beyond the Standard Model
Supersymmetry plays prominent roles in modern theoretical physics, as a tool to improve our understanding of quantum field theory, as an ingredient in many new physics models, and as a means to study quantum gravity via holographic duality. Lattice investigations of supersymmetric field theories have a long history but often struggle due to the interplay of supersymmetry with the discretization o ... More
Presented by David SCHAICH on 24/7/2018 at 14:00
Type: Plenary Session: Plenary
Anchoring the nuclear interaction in QCD is a long-outstanding problem in nuclear physics. While the lattice community has made enormous progress in mesonic physics and single nucleon physics, continuum-limit physical-point multi-nucleon physics has remained out of reach. I will review CalLat's strategy for multi-nucleon spectroscopy and our latest results.
Presented by Dr. Evan BERKOWITZ on 23/7/2018 at 16:00
Type: Plenary Session: Plenary
We calculate the intrinsic quark and gluon spin contribution to the total proton spin using overlap fermions on Domain-wall ensembles. We find that the total quark spin, with the axial Ward identity satisfied, is ~40%, and the glue spin contribution is ~50% if the matching procedure in LaMET is neglected. The imperative non-perturbative renormalization to obtain the quark and glue angular momen ... More
Presented by Yibo YANG on 27/7/2018 at 15:00
Type: Plenary Session: Plenary
In the last few years, numerical simulations of QCD on the lattice have reached a new level of accuracy. A wide range of thermodynamic quantities is now available in the continuum limit and for physical quark masses. This allows a comparison with measurements from heavy ion collisions for the first time. I will review the state-of-the-art results from lattice simulations of QCD thermodynamics and ... More
Presented by Prof. Claudia Ratti RATTI on 23/7/2018 at 16:30
Type: Plenary Session: Plenary
First principles calculations of the Bjorken-x dependence of hadron structure have been a long-standing challenge for lattice QCD. This year marks a significant milestone: the first determinations of parton distribution functions, which capture the longitudinal momentum structure of fast-moving hadrons, at physical pion masses. Moreover, there has been significant progress in our understanding of ... More
Presented by Dr. Christopher MONAHAN on 27/7/2018 at 16:00
Type: Plenary Session: Plenary
Track: Nonzero Temperature and Density
I will review the recent progress and results on the bulk thermodynamic properties of QCD matter from Lattice. In particular I will highlight the recent calculations of the equation of state, pressure of QCD matter to the finite baryon density regime as far as $\mu_B/T~2.5$, giving us some preliminary bounds on the location of the critical end-point. I will also stress upon the fact that ... More
Presented by Swagato MUKHERJEE on 24/7/2018 at 17:00
Type: Plenary Session: Plenary
Recent lattice QCD results for hadron light-by-light scattering (HLbL) and its contribution to muon anomalous magnetic moments (g-2) will be reviewed. There are currently more than three standard deviations between the BNL experimental result and the theoretical prediction. The Fermilab/JPARC experiments will reduce the experimental uncertainty by a factor of four. The uncertainty of theory ... More
Presented by Dr. Luchang JIN on 25/7/2018 at 15:00
Type: Plenary Session: Plenary
Track: Applications Beyond QCD
Composite Higgs Models explore the possibility that the Higgs boson is an excitation of a new strongly interacting sector giving rise to electro-weak symmetry breaking. After describing how this new sector can be embedded into the Standard Model of elementary particle physics meeting experimental constraints, I will review efforts by the community to explore the physics of this new strong interact ... More
Presented by Oliver Witzel WITZEL on 24/7/2018 at 14:30
Type: Plenary Session: Plenary
Recent lattice QCD results for hadron vacuum polarization (HVP) and its contribution to muon anomalous magnetic moments (g-2) will be reviewed. There currently exists tension of more than 3-sigma deviations in muon g-2 between the BNL experiment with 0.5 ppm precision and the Standard Model (SM) prediction with the QCD dispersion relation used for HVP. The lattice QCD predictions without recourse ... More
Presented by Dr. Kohtaroh MIURA on 25/7/2018 at 14:00
Session: Plenary
Presented by Prof. John PRESKILL on 28/7/2018 at 16:45
Type: Plenary Session: Plenary


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# 4.1: Introduction to Solvents

In a very general sense, solvents are a class of chemical compounds that allow chemistry to occur. The concept of a solvent has significant ramifications because they serve as the matrix, medium, or carrier for solutes. They are necessary in a number of processes, reactions, and systems. We tend to think of a substance like “water” as a universal solvent because it is so useful in so many disciplines. Water cleans up everything, allows biochemical reactions to occur, is used in paints, coatings, and films, allows cooking to occur (or else everything would catch fire), and provides lubrication and ease of movement for a great many devices.

Interestingly, in the business world, the word “solvent” has another meaning: A state of financial soundness characterized by the ability of an entity to meet its monetary obligations when they fall due. This latter definition does indeed apply in a certain sense to the solvent concept we are espousing in this chapter. A solvent ($olvent) carries materials for a reaction or specific function in the same sense that solvent from a financial sense carries funding/money to meet its obligations. Thus, using an analogy: Solvent : Solute ::$olvent : Money

Chemically, a solvent will dissolve or “solvate” a solute. What does that mean? It means that solvent molecules will surround the solute in such a way that a solution is formed; in other words, a homogeneous system is generated in which the solute is part of/indistinguishable from the solvent network. A solute/solution conceptcan be visualized in the Figure below:

Solubility

The ability of one a compound to dissolve in some other (likely different) compound is termed “solubility”. Miscibility is another term that characterizes the facility of compound A to dissolve in compound B. When the two compounds can completely dissolve or combine to form a homogeneous solution, the two liquids are said to be miscible. Two that can never blend well enough to form a solution are called immiscible.

All solutions have a positive entropy of mixing whereas the interactions between different compounds may or may not be energetically favored. If interactions are unfavorable, then the free energy decreases with increasing solute concentration. The energy loss may at some point in time outweighs the entropy gain, and no more solute can be dissolved – this is a condition solution in which the solution is said to be saturated. This condition can change with different environmental factors, such temperature, pressure, and purity of the system. A supersaturated solution can be prepared by raising the solubility (e.g., increasing temperature) to dissolve more solute, and then lowering it by cooling. However, most gases and several compounds exhibit solubilities that tend to drop with raised temperatures. The solubility of liquids in liquids, however, is less temperaturesensitive than solids or gases.

4.1: Introduction to Solvents is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Lucian Lucia via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.


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# findProgram -- load external program

## Synopsis

• Usage:
findProgram(name, cmd)
findProgram(name, cmds)
• Inputs:
• name, , the name of the program to load. This should match the corresponding key in programPaths.
• cmd, , a command to run that should return 0 if the program is present.
• cmds, a list, a list of commands to run that should all return 0 if the program is present.
• Optional inputs:
• RaiseError => , default value true, whether to raise an error if the program is not found.
• Verbose => , default value false, whether to inform the user of each path that is checked.
• Prefix => a list, default value {}, a list of sequences containing two strings identifying a prefix that is added to the executable binaries belonging to the program by some distributions. These sequences should be of the form (regex, prefix) where regex is a regular expression that should match all binary executables that need the prefix and prefix is the prefix itself.
• AdditionalPaths => a list, default value {}, a list of strings containing any paths to check for the program in addition to the default ones.
• MinimumVersion => , default value null, containing two strings the form (minVersion, versionCommand), where minVersion is the minimum required version of the program and versionCommand is a shell command to obtain the version number of an installed program.
• Outputs:
• an instance of the type Program, the program that was loaded. If the program is not found and RaiseError is set to false then null is returned.

## Description

This function checks for the existence of an external program by running cmd (or every element of cmds) prepended with various paths in the following order:

• The user-defined path specified by programPaths#name, if it exists.
• The path specified by prefixDirectory | currentLayout#"programs", where the programs shipped with Macaulay2 are installed.
• Each path specified by the AdditionalPaths option.
• Each path specified by the user's PATH environment variable.
• The path to M2-binary.

For each path, any prefixes specified by the Prefix option are checked.

Once this is successful (i.e., cmd or each element of cmds returns a value of 0), then a Program object is returned. If it is unsuccessful, then either an error is raised or null is returned, depending on the value of RaiseError.

Note that if a program consists of a single executable binary file, then name should coincide with the name of this file.

 i1 : programPaths#"gfan" = "/path/to/gfan/" o1 = /path/to/gfan/ i2 : gfan = findProgram("gfan", "gfan _version --help", Verbose => true) -- /path/to/gfan/gfan does not exist -- /usr/libexec/Macaulay2/bin/gfan does not exist -- /home/m2user/src/M2.git/M2/usr-build/bin/gfan does not exist -- /home/m2user/src/M2.git/M2/usr-build/bin/gfan does not exist -- /home/m2user/src/M2.git/M2/usr-build/bin/gfan does not exist -- /home/m2user/src/M2.git/M2/BUILD/dan/builds.tmp/ubuntu2104-master.debug/usr-dist/x86_64-Linux-Ubuntu-21.04/bin/gfan does not exist -- /home/m2user/bin/gfan does not exist -- /usr/local/sbin/gfan does not exist -- /usr/local/bin/gfan does not exist -- /usr/sbin/gfan does not exist -- /usr/bin/gfan exists and is executable -- running "/usr/bin/gfan _version --help": This program writes out version information of the Gfan installation. -- return value: 0 o2 = gfan o2 : Program

One program that is shipped with a variety of prefixes in different distributions and for which the Prefix option is useful is TOPCOM:

 i3 : findProgram("topcom", "cube 3", Verbose => true, Prefix => { (".*", "topcom-"), ("^(cross|cube|cyclic|hypersimplex|lattice)$", "TOPCOM-"), ("^cube$", "topcom_")}) -- /usr/libexec/Macaulay2/bin/cube exists and is executable -- running "/usr/libexec/Macaulay2/bin/cube 3": [[0,0,0,1],[1,0,0,1],[0,1,0,1],[1,1,0,1],[0,0,1,1],[1,0,1,1],[0,1,1,1],[1,1,1,1]] [[7,6,5,4,3,2,1,0],[6,7,4,5,2,3,0,1],[4,6,5,7,0,2,1,3],[0,4,2,6,1,5,3,7]] -- return value: 0 o3 = topcom o3 : Program

Note that when using the MinimumVersion option, the command used to obtain the current version number must remove everything except the version number itself and any leading or trailing whitespace. Piping with standard UNIX utilities such as sed, head, tail, cut, and tr may be useful.

 i4 : findProgram("gfan", "gfan _version --help", Verbose => true, MinimumVersion => ("0.5", "gfan _version | head -2 | tail -1 | sed 's/gfan//'")) -- /path/to/gfan/gfan does not exist -- /usr/libexec/Macaulay2/bin/gfan does not exist -- /home/m2user/src/M2.git/M2/usr-build/bin/gfan does not exist -- /home/m2user/src/M2.git/M2/usr-build/bin/gfan does not exist -- /home/m2user/src/M2.git/M2/usr-build/bin/gfan does not exist -- /home/m2user/src/M2.git/M2/BUILD/dan/builds.tmp/ubuntu2104-master.debug/usr-dist/x86_64-Linux-Ubuntu-21.04/bin/gfan does not exist -- /home/m2user/bin/gfan does not exist -- /usr/local/sbin/gfan does not exist -- /usr/local/bin/gfan does not exist -- /usr/sbin/gfan does not exist -- /usr/bin/gfan exists and is executable -- running "/usr/bin/gfan _version --help": This program writes out version information of the Gfan installation. -- return value: 0 -- found version 0.6.2 >= 0.5 o4 = gfan o4 : Program


text:
Journal cover Journal topic
Hydrology and Earth System Sciences An interactive open-access journal of the European Geosciences Union
Journal topic
Hydrol. Earth Syst. Sci., 23, 2647–2663, 2019
https://doi.org/10.5194/hess-23-2647-2019
Hydrol. Earth Syst. Sci., 23, 2647–2663, 2019
https://doi.org/10.5194/hess-23-2647-2019

Research article 19 Jun 2019

Research article | 19 Jun 2019

# Sensitivity of hydrological models to temporal and spatial resolutions of rainfall data

Sensitivity of hydrological models to temporal and spatial resolutions of rainfall data
Yingchun Huang1, András Bárdossy2, and Ke Zhang1,3 Yingchun Huang et al.
• 1College of Hydrology and Water Resources, Hohai University, Nanjing 210098, China
• 2Institute for Modelling Hydraulic and Environmental Engineering, University of Stuttgart, 70596 Stuttgart, Germany
• 3State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China

Correspondence: Ke Zhang (kzhang@hhu.edu.cn) and Yingchun Huang (yingchunhuang@hhu.edu.cn)

Abstract

Rainfall is the most important input for rainfall–runoff models. It is usually measured at specific sites on a daily or sub-daily timescale and requires interpolation for further application. This study aims to evaluate whether a higher temporal and spatial resolution of rainfall can lead to improved model performance. Four different gridded hourly and daily rainfall datasets with a spatial resolution of 1 km × 1 km for the state of Baden-Württemberg in Germany were constructed using a combination of data from a dense network of daily rainfall stations and a less dense network of sub-daily stations. Lumped and spatially distributed HBV models were used to investigate the sensitivity of model performance to the spatial resolution of rainfall. The four different rainfall datasets were used to drive both lumped and distributed HBV models to simulate daily discharges in four catchments. The main findings include that (1) a higher temporal resolution of rainfall improves the model performance if the station density is high; (2) a combination of observed high temporal resolution observations with disaggregated daily rainfall leads to further improvement in the tested models; and (3) for the present research, the increase in spatial resolution improves the performance of the model insubstantially or only marginally in most of the study catchments.

1 Introduction

Rainfall is a primary driver of hydrological models and can impact catchment runoff response significantly . Rainfall is usually measured by standard rain gauges or wireless telemetering pluviometers over a period of time (e.g., daily, sub-daily). Uncertainties in rainfall estimation for a catchment can occur due to instrument errors as well as spatial and temporal variability of rainfall. The latter are the main sources of uncertainties in model simulation and flood forecasting . The spatial variability of rainfall strongly influences the timing and shape of a hydrograph, while the temporal variability mainly affects the peak of a flood wave (Singh1997). The improvement in flood simulation requires understanding of the sensitivity of the rainfall–runoff models to rainfall input data. Over the past decades, various methods have been used to obtain the spatial distributions of rainfall based on raingauge data and catchment characteristics . derived hourly areal rainfall interpolated from various numbers of rain gauges to quantitatively assess the sensitivity of peak flow to the uncertainty of rainfall data using an HBV model. They found that the error in rainfall may lead to even greater error in flood peaks. also studied the impact of spatial variability of rainfall by varying the distribution of raingauge networks. They found that the transferabilities of model parameters calibrated based on sparse and dense rainfall information are very different. used four different model structures to simulate daily runoff in central Europe. Results indicated that the semi-distributed and semi-lumped models outperform the lumped and distributed model structures. They suggested that the lack of spatial information is responsible for the low efficiency of the distributed model. indicated that the increase in raingauge network density can improve the model performance, but no apparent improvement was observed when the number of rain gauges exceeded a threshold. found that semi-distributed models outperform the lumped models when rainfall is highly variable over a simulation catchment, but they perform similarly when rainfall is relatively uniform. proposed rainfall variability indexes to characterize the influence of spatial variability rainfall and implemented this approach in the model simulation for the Cevennes catchment in France . They found that higher spatial resolution of rainfall could achieve better model performance. We can learn from these studies that the sensitivity of model performance to the spatial resolution of rainfall seems different for some of the case studies. However, increasing spatial resolution in model simulation can lead to considerable complexity of model structure and require much more data than when using a lumped version.

The rainfall–runoff response of a catchment is also strongly impacted by the temporal variability of rainfall . High temporal resolution rainfall data are collected at pluviometer stations with telemetry at sub-daily time resolutions. Sub-daily data often have poor quality caused by equipment malfunction or misreading. Compared with sub-daily rainfall data, daily data tend to be more available and reliable, and cover a longer duration of time periods. Disaggregating daily into sub-daily data offers a potential solution to accurately capture the temporal variability of rainfall . compared three different approaches for disaggregating daily rainfall into sub-daily series and found the resampling method to be the best one for rainfall disaggregation. used the Gaussian copula-based model for disaggregating daily data to infill the gap of sub-daily data, and they found that this conditional disaggregation of rainfall is reliable and applicable in various regions. applied a spatial method of fragments to disaggregate daily rainfall into hourly values. found that calibrating hydrological models with sub-daily time steps can significantly improve the accuracy of flood forecasting.

Some studies focus on both the spatial and temporal resolutions of rainfall. found that the temporal and spatial resolutions of rainfall used as the inputs of hydrological models can have considerable influence on the model efficiency and parameter values. found the influence of rainfall spatial resolution to be greater than temporal resolution in terms of simulation of extreme flows. pointed out that physically based models are more sensitive to the spatial and temporal resolutions of rainfall data than conceptual models. found that the spatial variability of rainfall is much more sensitive to model performance for catchments larger than 2000 km2 under dry soil conditions, and floods in small catchments are more influenced by the temporal variability of rainfall. So far, more efforts have been invested in improving the spatial or temporal resolution of rainfall, but there are fewer studies on quantification and direct comparison of the catchment dynamic responses driven by different rainfall temporal and spatial resolutions.

The overarching aim of this study is to understand the dependency of hydrological model performance on the rainfall data. The specific research objectives are 3-fold: (1) investigate the effects of rainfall data quality on model performance, (2) examine the sensitivity of model performance to different spatial and temporal resolutions of rainfall data using two different model spatial configurations, and (3) explore the possibility of improving model performance on a daily scale. The paper will be followed by Sect. 2 to describe the study area and the rainfall datasets used in this research. In Sect. 3, the hydrological model and the calibration method are explained. Section 4 presents the results and discussion of this work. The conclusions and outlook are in Sect. 5.

2 Study area and hydrometeorological datasets

This study area is located in a semi-humid region in the Baden-Württemberg state of Germany (Fig. 1) with a temperate monsoon climate of mild winters and warm summers. The elevation of this region ranges from 85 m to 149 m a.s.l. (above sea level). The heterogeneity of climate characteristics is mainly due to the great variability of elevations within the study area. The annual mean air temperature in Baden-Württemberg is about 10.2 C. Rainfall is evenly distributed throughout the year. However, its seasonality shows a weak trend. The monthly rainfall is highest in June and lowest in October. The meteorological data used in this study were provided by the German Weather Service (DWD). Daily air temperature data required for the rainfall–runoff model were interpolated on a 1×1 km2 grid from the observations using the external drift kriging algorithm . The topographical elevation was taken as external drift . The long-term monthly potential evapotranspiration and the average air temperature were used to compute the daily potential evapotranspiration using the Hargreaves and Samani method .

Figure 1Locations of the sub-daily and daily rain gauges in Baden-Württemberg and the four selected catchments.

Rainfall data from a dense network of daily rainfall stations (62 km2 per station in 1991) and from a less dense network of sub-daily stations (144 km2 per station in 1991) with high-resolution rainfall observations were used for this study. All data are available for the time period 1991–2010. The number of available daily stations and sub-daily stations varies according to different time periods. Figure 2 illustrates the number of available observation stations in Baden-Württemberg between 1991 and 2010. It can be seen from the graph that more than 430 daily stations but only 30 sub-daily stations were available in 1991. The total number of daily stations decreased to 250 around 2003 and remained stable for the subsequent years. The number of sub-daily stations has been increasing throughout this period and experienced a sharp increase from 100 to 200 in 2005. Four different rainfall datasets were generated and explained as follows.

1. High temporal resolution observed rainfall was aggregated to hourly and then interpolated to 1×1 km2 grids using the ordinary kriging algorithm (Matheron1963). The correlation function obtained from the cross-correlations of the hourly time series was used as a basis for the variogram. This set is referred to as the Sparse Hourly (SH) set.

2. Observed daily rainfall combined with the daily aggregations of the high temporal resolution data were used to create 1×1 km2 gridded datasets using the ordinary kriging algorithm. The variogram was based on the cross-correlations of the daily time series. This set is referred to as the Dense Daily (DD) set.

3. High-resolution rainfall was aggregated to daily time steps and interpolated subsequently for a 1×1 km2 grid using the ordinary kriging. The variogram was based on the cross-correlations of the aggregated daily time series. This set will be referred to as the Sparse Daily (SD) set.

4. Observed daily rainfall combined with the hourly aggregations of the high temporal resolution data were used to create a 1×1 km2 grid using the disaggregation method rescaled ordinary kriging . The variogram was based on the cross-correlations of the hourly time series. This set is referred to as the Dense Hourly (DH) set.

Figure 2The number of available observation locations. Daily stations – solid line; sub-daily stations – dashed line.

Figure 3 shows the flowchart of the data collection and process. The DD and SD sets are the daily aggregations of the DH and SH sets. Note that DH is a dataset combining hourly and disaggregated daily gauge data. One of the research questions raised here is to find out whether disaggregation leads to an improvement in model performance. Comparisons of the model performance using the inputs of the (SD, SH) and (DD, DH) pair will reveal the effect of temporal resolution. Meanwhile, comparison between (SD, DD) and (SH, DH) will show the influence of the rainfall observation network density on the model performance.

Figure 3Schematic representation of four different rainfall datasets.

Four mesoscale catchments (Fig. 1), namely Rottweil, Schwaibach, Pforzheim and Kocherstetten, were selected from the upstream region of the state for testing the sensitivity of model performance to the four different rainfall datasets. The daily streamflow record of these catchments was collected for the period 1991–2010. The basic characteristics for the study catchments are listed in Table 1. These catchments range in size from 417 to 1300 km2, along with large differences in elevation and annual precipitation. It can be seen clearly from the map that these four catchments have different raingauge densities; the Schwaibach catchment located in the mountainous area with elevations ranging from 190 to 1028 m has the lowest density of raingauge networks and the highest annual precipitation. Rottweil and Kocherstetten have similar climate conditions in terms of annual precipitation and runoff, but the catchment size of Kocherstetten is almost 3 times that of Rottweil. Pforzheim has the smallest drainage area and the lowest amount of precipitation.

Table 1Catchment characteristics for the four selected catchments.

3 Model and methodology

## 3.1 Model structure

The conceptual HBV model was developed in the 1970s by the Swedish Meteorological and Hydrological Institute (SMHI) . Thanks to its simplicity, low demand of inputs and small number of model parameters, the HBV model has been widely used for rainfall–runoff simulation and flood forecasting. Figure 4 represents the structure diagram of the HBV model (Singh2010). There are three main modules in the HBV model, namely snow routine, soil moisture routine and runoff routine .

Figure 4Schematic representation of the HBV model.

In the snow routine, the snow accumulation and melting process is estimated by the relatively simple degree-day method with two parameters: degree-day factor (DD) and threshold temperature for snow/rain (TT) (as shown in Eq. 1). The measured precipitation is supposed to be solid (snowfall) if the air temperature is lower than threshold temperature; otherwise, precipitation appears in a liquid state (rainfall) if the weather is warmer than the threshold value.

$\begin{array}{}\text{(1)}& \mathrm{Snowmelt}=\mathrm{DD}\cdot \left(T-\mathrm{TT}\right),\phantom{\rule{0.25em}{0ex}}\mathrm{if}\phantom{\rule{0.25em}{0ex}}T>\mathrm{TT}\end{array}$

In the HBV model, soil moisture storage is decided by balancing rainfall and evapotranspiration according to two soil moisture constants: permanent wilting point (PWP) and field capacity (FC). The soil wetness index, defined as the ratio of direct runoff to effective precipitation ($\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }P}$), is expressed as

$\begin{array}{}\text{(2)}& \frac{\mathrm{\Delta }Q}{\mathrm{\Delta }P}={\left(\frac{\mathrm{SM}}{\mathrm{FC}}\right)}^{\mathrm{Beta}},\end{array}$

where SM denotes the actual soil moisture and “Beta” the proportion of effective precipitation contributing to runoff for given soil moisture. The Penman equation is used to estimate the potential evapotranspiration according to the long-term monthly mean air temperature (TM) and long-term monthly averaged potential evapotranspiration (PEM) (Penman1948):

$\begin{array}{}\text{(3)}& {E}_{\mathrm{tp}}=\left(\mathrm{1}+C\left(T-{T}_{\mathrm{M}}\right)\right){\mathrm{PE}}_{\mathrm{M}},\end{array}$

where C is the evapotranspiration coefficient. The actual evapotranspiration (Eta) can be estimated as

$\begin{array}{}\text{(4)}& {E}_{\mathrm{ta}}=\left\{\begin{array}{ll}{E}_{\mathrm{tp}}& \mathrm{if}\phantom{\rule{0.25em}{0ex}}\mathrm{SM}>\mathrm{PWP}\\ \frac{\mathrm{SM}}{\mathrm{PWP}}\cdot {E}_{\mathrm{tp}}& \mathrm{else}\end{array}\right\.\end{array}$

As shown in Eq. (2), runoff is calculated by a nonlinear function based on excessive effective precipitation and actual soil moisture. The runoff concentration process consists of upper and lower reservoirs with five parameters:

$\begin{array}{}\text{(5)}& & {Q}_{\mathrm{0}}={K}_{\mathrm{0}}\left({S}_{\mathrm{1}}-\mathrm{HL}\right),\text{(6)}& & {Q}_{\mathrm{1}}={K}_{\mathrm{1}}{S}_{\mathrm{1}},\text{(7)}& & {Q}_{d}={K}_{d}{S}_{\mathrm{1}},\text{(8)}& & {Q}_{\mathrm{2}}={K}_{\mathrm{2}}{S}_{\mathrm{2}}.\end{array}$

The runoff is divided into surface flow (Q0), interflow (Q1) and baseflow (Q2) with three recession coefficients K0, K1 and K2, along with a conceptual threshold water level (HL) for generating surface flow. The two parallel reservoirs are connected in the form of percolation storage (Qd) from the upper reservoir to the lower one with the parameter of the percolation constant Kd. A transformation function with the triangular weighting parameter MAXBAS is used to smooth the total runoff (${Q}_{\mathrm{0}}+{Q}_{\mathrm{1}}+{Q}_{\mathrm{2}}$) to obtain discharge at the outlet.

In this study, to investigate the sensitivity of model performance to the spatial resolution of input variables, two HBV models with different spatial configurations were applied: lumped HBV and spatially distributed HBV, respectively. In the lumped model, precipitation, temperature and potential evapotranspiration were assumed uniformly distributed within a catchment and all the processes were calculated for the whole catchment. Previous studies have indicated that the elevation is an important reason for the spatial differentiation of meteorological variables, including temperature, precipitation, evapotranspiration and snowmelt, being in reality not uniformly distributed within a catchment. They often exhibit dependence with elevation. The spatially distributed HBV model used in this study divides a catchment into several zones based on elevation. The 1×1 km2 grid-based rainfall and temperature data were averaged for each elevation zone. In the spatially distributed model, the parameters associated with the snowmelt and soil moisture modules were calibrated for each elevation zone. The parameters associated with the runoff response module were calibrated for each catchment similarly to the lumped model .

Out of the 15 parameters within the HBV model, 9 parameters were calibrated in this study. Table 2 lists the initial upper and lower limits of the to-be-calibrated parameters using historical data. The data-depth-based parameter optimization method – the Robust Parameter Estimation (ROPE) algorithm – was applied for model parameter optimization. The ROPE approach could lead to a certain number of model parameters with ideal model performance . For this study, each simulation results in 10 000 heterogeneous parameter sets with similarly acceptable model performance.

Table 2Description of HBV model parameters and parameter ranges for model calibration.

## 3.2 Performance criteria

Previous studies have shown that model performance strongly depends on the selection of performance criteria . The model simulations corresponding to the model parameters using different objective functions differ considerably as they have different focuses . The purpose of this study is to investigate the sensitivity of the conceptual model to rainfall variability and accordingly find effective ways to improve the precision of flood forecasting. Since high flow is extremely important for flood forecasting, the Nash–Sutcliffe (NS) coefficient , one of the widely used indicators, was used in this study to assess the model performance based on observed discharge. The NS coefficient focuses on high flow as it evaluates the squared difference between simulated and measured streamflow. It can be calculated using the following equation:

$\begin{array}{}\text{(9)}& \mathrm{NS}=\mathrm{1}-\frac{\sum _{t=\mathrm{1}}^{T}{\left({Q}_{\mathrm{o}}\left(t\right)-{Q}_{\mathrm{m}}\left(t\right)\right)}^{\mathrm{2}}}{\sum _{t=\mathrm{1}}^{T}{\left({Q}_{\mathrm{o}}\left(t\right)-\stackrel{\mathrm{‾}}{{Q}_{\mathrm{o}}}\right)}^{\mathrm{2}}},\end{array}$

where Qo(t) and Qm(t) are the observed and simulated discharge, respectively, and $\stackrel{\mathrm{‾}}{{Q}_{\mathrm{o}}}$ is the mean of the observed discharge.

The mean square error (MSE) of the flow for the time period of the observed discharge higher than the 10th percentile of flow was used to assess the flood forecasting ability of the models:

$\begin{array}{}\text{(10)}& \mathrm{MSE}=\frac{\mathrm{1}}{n}\sum _{i=\mathrm{1}}^{n}{\left({Q}_{\mathrm{o}}\left(i\right)-{Q}_{\mathrm{m}}\left(i\right)\right)}^{\mathrm{2}},\end{array}$

where Qo(i) and Qm(i) are the observed and modeled discharges when the observed discharge is higher than the 10th percentile of flow.

## 3.3 Model calibration experiments

A split-sample calibration methodology was applied in this study to divide the data into two 10-year periods: 1991–2000 and 2001–2010. Model calibration was carried out for both time periods and then a cross-validation analysis was performed. For each calibration run, the first water year data were used as a warm-up period and were not used to evaluate the model performance.

In this study we investigated the impacts of using different methods for spatial interpolation of hourly rainfall data on model performance. The four rainfall datasets were used as input variables for model calibration and validation. In all modeling experiments, daily mean temperature and potential evapotranspiration were used as inputs. This is to isolate the effects of different rainfall inputs on the model performance. The effects of the temporal and spatial resolutions of the rainfall inputs on the model performance were assessed in terms of the NS coefficient and the MSE of the high flow. We conducted experiments of model calibration for a lumped model and a spatially distributed HBV model using hourly and daily input variables, respectively. For the spatially distributed model structure, a contour interval of 100 m was used to divide a catchment into different elevation zones. Note that all the model calibrations were performed on the basis of simulating daily discharge.

We investigated whether the combination of daily and hourly models can lead to better prediction in streamflow. It is interesting to investigate the similarities of different temporal resolution. Therefore, the common calibration approach was used to calibrate the daily and hourly models simultaneously. This approach may identify robust model parameters that are applicable using different temporal resolutions. The common calibration approach is a multi-objective optimization function, and the compromise programming method (Zeleny1981) was used to formulate the objective function:

$\begin{array}{}\text{(11)}& O\left(\mathit{\theta }\right)=\sum _{i=\mathrm{1}}^{n}{\left({\mathrm{NS}}_{\mathrm{i}}^{*}-{\mathrm{NS}}_{\mathrm{i}}\left(\mathit{\theta }\right)\right)}^{p}.\end{array}$

Here index i denotes the type of temporal resolution; NS${}_{\mathrm{i}}^{*}$ means the optimal model performance which can be represented by the individual calibrated model performance. Here we aim to minimize the value of the objective function O(θ). For the balancing factor p, a moderately high p=4 was given in this study. More details about the common calibration of hydrological models' strategy can be found in .

4 Results and discussion

## 4.1 Comparison of the rainfall datasets

The quality of the rainfall datasets was assessed and compared for the four selected catchments. As the SD and DD sets are daily aggregations of the SH and DH sets, we only compared daily rainfall sets SD and DD for both calibration periods (Fig. 5). It can be seen clearly from the figures that the interpolated rainfall datasets display some differences for all study catchments. The asymmetry of the scatterplots is evident for the period 1991–2000. In general, the DD dataset leads to higher values than the SD dataset. This is mainly because the low density of sub-daily observations during the period of 1991–2000 leads to large errors in the spatial interpolation of rainfall. This is especially the case for Schwaibach catchment, which varies strongly in geographical elevation (from 190 to 1028 m). For the period of 2001–2010, the SD and DD sets are in closer agreement due to the higher density of sub-daily gauges.

Figure 5Comparison of the daily rainfall data that were interpolated using different densities of raingauge networks.

## 4.2 Results of calibration and validation

As described in Sect. 3.3, for the selected catchments, model calibrations were carried out using four rainfall datasets for both lumped and spatially distributed HBV models. Two 10-year time periods, 1991–2000 and 2001–2010, were used for calibration and cross-validation. In total 16 calibration runs and 16 validation runs were performed for each catchment. As mentioned before, each simulation obtained 10 000 parameter sets with similar model performance. We then used the mean value of the 10 000 model performances to quantify the model performance.

Table 3 lists the average value of the NS model performance for the four selected catchments using the lumped HBV model and Table 4 lists the simulated NS performance for a spatially distributed version of the model, respectively. The results show that all four datasets can reproduce relatively accurate historical daily streamflow series for all selected catchments. Results also show that the model performances vary across catchments. The Kocherstetten catchment generally performs best, with an average NS value of 0.84 for all simulations, while Pforzheim has the worst mean NS performance of 0.58 for all calibration runs. Moreover, for a specific catchment, the calibrated models perform differently for different data periods. For the Schwaibach and Pforzheim catchments, the calibrated model performance for the period of 2001–2010 is better than the performance for the time period 1991–2000 for most of the datasets. This might be due to the increasing raingauge density inside or near the catchment and the quality of rainfall data with the development of time and technological progress. In particular, the model calibrations for the period 1991–2000 of the Schwaibach catchment using sets SH and SD perform very poorly for both calibration and validation. The NS coefficient using SH and SH inputs is about 0.3 less than the results of sets DH and DD. This indicates that systematic interpolated rainfall errors have a critical influence on model calibration.

Table 3Average NS model performance for the lumped HBV model.

Table 4Average NS model performance for the distributed HBV model.

We then analyzed whether the model is robust for simulating high flows. Tables 5 and 6 list the mean square errors of the top 10th percentile of flows for the lumped and spatially distributed models, respectively. Figure 6 shows the flow duration curve for the natural logarithm of simulated and observed discharge for all study catchments for the years between 2001 and 2010. Figure 7 shows the corresponding results for flows higher than the 10th percentile of flow. Results indicate that for most of the calibration runs, set DH performs best for the high flow, followed by set SH, set DD performs a little weaker than set SH, while set SD has the worst performance in flood simulation.

Figure 6Comparison of the simulated flow duration curve.

Figure 7Comparison of the simulated flow duration curve for flows higher than the 10th percentile of flow.

Table 5Mean square error for flows higher than the 10th percentile of flow for the lumped HBV model.

Table 6Mean square error for flows higher than the 10th percentile of flow for the distributed HBV model.

## 4.3 Model performance using different temporal resolutions of rainfall data

Firstly, the model performance of different temporal resolutions of rainfall was compared for four datasets and two model spatial configurations. For the pairwise comparison, all the conditions are the same in the model except for the rainfall temporal resolution (hourly and daily). The results of the sparse sets and dense sets are separated here. Figure 8 compares the model performance of using hourly and daily rainfall inputs that were interpolated using only high-resolution rainfall observations (SH, SD). Figure 9 compares the results from the rainfall inputs that incorporated observed daily values with high-resolution observations (DH, DD). The result shows that all the scatters are lying below the diagonal line for the different levels of observation density. For both the calibration and validation periods, the simulations using hourly input data outperform the ones based on the daily resolution. For the dataset with low observation network density, the averaged NS of set SH is about 0.73 for the calibration period and 0.68 for the validation period, while the mean NS coefficient calibrated using the SD set is 0.67 and 0.6, respectively. The higher observation density datasets show a similar tendency. The mean NS of using the DH set is around 0.79 for calibration and 0.77 for validation, while that of set DD is 0.72 and 0.69, respectively. The fact that the hourly scale model performs better than the daily model suggests that the dynamic runoff of the catchment could be better simulated with a higher temporal resolution of rainfall. According to the distances from the diagonal to the scatterplots, we can observe that the difference in model performance for different temporal resolutions is larger for the catchments with relatively low NS model performance, such as Schwaibach and Pforzheim. For Rottweil and Kocherstetten, the performance of the hourly calibrated model is only slightly better than the daily model.

Figure 8Comparison of the NS coefficient for using hourly and daily rainfall as model input for the SH and SD sets.

Figure 9Comparison of the NS coefficient for using hourly and daily rainfall as model input for the DH and DD sets.

## 4.4  Model performance in terms of observation density

The rainfall gauge network density has a significant impact on model simulation and parameter optimization. Figure 10 shows the simulated NS coefficient of the model simulations using the daily input data interpolated using different densities of raingauge networks. It shows obviously from the location of points that the simulated model performance using the DD set is generally better than that using the SD set for both the calibration and validation periods. The averaged NS model performances of DD and SD sets are 0.71 and 0.64, respectively. The model performance using hourly inputs shows a similar trend to that using daily inputs. As shown in Fig. 11, the model using the DH set outperforms the one using the SH set. These results demonstrate that high raingauge density led to improvement in model performance at both daily and hourly time resolution.

Figure 10Comparison of the NS coefficient for different densities of raingauge networks; models were simulated based on daily time steps.

Figure 11Comparison of the NS coefficient for different densities of raingauge networks; models were simulated based on hourly time steps.

Figure 12 illustrates the cumulative distribution function of the NS coefficient using sets SD, SH and DH for model calibration (left) and validation (right). As can be seen clearly from the curves, if rainfall data come from a sparse network of sub-daily stations, use of higher temporal resolution datasets (the SH set) leads to better model performance than using lower-resolution ones (the SD set). Simulation of daily streamflow can benefit from running the model at a higher temporal resolution. In addition, the combination of observed sub-daily rainfall with disaggregated daily rainfall (the DH set) leads to a further improvement in daily streamflow simulation.

Figure 12Cumulative distribution of the NS coefficient for model calibration using different rainfall datasets.

## 4.5 Model performance in terms of the spatial resolution of rainfall data

The model performance was compared between the lumped and spatially distributed HBV models when they were driven by different rainfall datasets. Figure 13 compares the NS model performance for calibration (left) and validation (right) periods. The correlation between model performance and the spatial resolution of a model seems unclear for the study catchments. For some simulations, the spatially distributed model outperforms the lumped one, especially for the catchments with a high NS coefficient, despite the increase in model performance being only marginal. However, for the catchments with relatively poorer model performance, the lumped model could even lead to slightly better performance than the semi-distributed model, especially for the validation period when the difference seems larger than the calibration period. This indicates that for model validation, the model parameters estimated by the distributed HBV model show weaker transferability. A possible explanation for this case could be that the distributed model has a larger number of parameters to be calibrated and that the parameters are underestimated during the calibration period. We conclude that the improvement in the spatial resolution of model structure did not enhance the model performance, which is surprising since higher spatial resolution and more model parameters are expected to improve the model performance. Our results confirm the findings of that the distributed models do not necessarily improve model performance.

Figure 13Comparison of the NS coefficient for different spatial resolutions of model structure.

The distributed model did not perform better than the lumped model in this study. This could be because the catchment underlying surface information and/or the calibration procedure was not sufficient for identifying optimal distributed model parameters. A second reason could be that the temporal resolution of the rainfall inputs is not sufficient for the distributed model.

## 4.6 Common model calibration with different temporal resolutions

As shown before, the combination of hourly and daily gauge data leads to the improvement in data quality as the model using sets DH and DD has better performance than using sets SH and SD. Furthermore, common calibration of the lumped HBV model was performed for sets DH and DD to identify model parameters good for both hourly and daily time steps. It is important to note that the parameters (DD, K0, K1, Kd and K2) that are dependent on time steps should be converted according to the simulation step of the model. The common calibration was performed for the two time periods separately, and a cross-validation analysis was performed as well. The common calibration and validation results were compared with the individual calibration (Fig. 14). For the calibration period, the common calibration always leads to slightly weaker performance for all datasets. For three of the DD datasets, model performances of common parameters are similar to individual calibration results. The average loss of NS coefficients over all catchments is about 0.02 for set DH and 0.01 for set DD. For the validation period, it can be seen from the scatterplots that the common parameters outperform the individual ones for about half of all the simulations. It suggests that parameter values obtained using the common calibration approach based on different time steps can improve the temporal transferability of models. The reason for the robustness of common parameters might be that a common calibration strategy can provide more information for identifying model parameters.

Figure 14Comparison of the NS coefficient for individual calibration and common calibration using datasets with different temporal resolutions.

The calibrated model parameters using daily rainfall, hourly rainfall and a common calibration strategy were also compared. Figures 15 and 16 show the distributions of the optimized model parameters for Rottweil and Pforzheim, respectively. Note that all parameters are normalized by the initial ranges in Table 2. From the boxplots we could find that some model parameters strongly depend on the selected rainfall dataset. This is very evident with the shape factor (Beta) and the threshold water level for surface runoff (L).

Figure 15Comparison of model parameters for different temporal resolutions for Rottweil catchment.

Figure 16Comparison of model parameters for different temporal resolutions for Pforzheim catchment.

5 Conclusions and outlook

In this study, we investigated the impacts of temporal and spatial variability of rainfall in model simulation and parameter estimation. We also explored the question whether higher temporal and spatial resolutions of rainfall data lead to any improvement in model performance. Both the lumped and spatially distributed HBV model were applied to simulate daily runoff for four mesoscale catchments driven by four different rainfall datasets which were constructed using a combination of data from a high density of daily stations and relatively low-density sub-daily stations. The impacts of rainfall variability on model simulations were evaluated using the NS coefficient and the mean squared error of flows higher than the 10th percentile of flow. The model sensitivities to the temporal and spatial resolutions of rainfall were compared. In addition, the common calibration approach was proposed to calibrate the models with different time steps simultaneously for finding robust model parameters.

For the study catchments, the results indicate that the temporal variability of rainfall data has direct impact on the dynamic response of a catchment. For both lumped and spatially distributed models, if the observation density is the same, the hourly based simulation outperforms the daily based simulation, indicating that higher temporal resolution can significantly improve the model performance. Disaggregating high-density daily observations into relatively low-density sub-daily values could lead to considerable model improvement, especially for the catchment with a sparse raingauge network. The rainfall disaggregation approach is an effective way of increasing the temporal resolution of rainfall data and the model performance. However, the lumped and spatially distributed HBV models perform very similarly, indicating that higher spatial resolution does not improve or only marginally improves the model performance for the study catchments. The result agrees with the general findings of and , where insignificant improvement was observed using higher spatial resolution of rainfall. The reason that the spatially distributed model does not outperform the lumped model could be due to the fact the study catchments are smaller than 2000 km2 with relatively uniform rainfall.

As stated at the beginning of this paper, we aim to investigate the sensitivity of the model to rainfall variability and to find effective ways of improving the model performance. This study shows that rainfall data disaggregation can lead to a significant improvement in model performance, while higher spatial resolution of rainfall does not always enhance model performance. Most of the hydrological models can be easily adjusted to use different time steps. The study suggests that increasing the temporal resolution of rainfall inputs with the disaggregation method can be an easier and more efficient way to improve model performance, compared to increasing the model spatial resolution at a cost of increasing the complexity of model structure and parameters.

This study focuses on high flows and uses only the NS coefficient as a quantitative measure of model sensitivity. As model performance highly depends on the selection of objective functions, the model sensitivity can be different if the model performance is measured differently. In addition, all the hourly simulated runoff was aggregated into daily runoff, and the hydrological response was evaluated based on daily discharge. Sub-daily response of a catchment is more sensitive to the temporal and spatial variability of rainfall, which should be considered in the future if the hourly discharge observation is available.

Data availability
Data availability.

The meteorological data sets used in this study were obtained from the respective sources mentioned in Sect. 2.

Author contributions
Author contributions.

AB and YH designed the study and set up the models. AB interpolated the precipitation data and YH performed the modelling. YH and AB wrote the manuscript with the contribution of KZ. YH and KZ contributed to the discussion and revision.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements
Acknowledgements.

The authors gratefully acknowledge Bettina Schaefli and two anonymous reviewers for their invaluable and constructive suggestions. Special thanks to Yi He, Jingfeng Wang, and Gebdang Biangbalbe Ruben for proofreading of the manuscript.

Financial support
Financial support.

This research has been supported by the National Key Research and Development Program of China (grant nos. 2018YFC1508101, 2018YFC1508102, and 2016YFC0402701), the National Natural Science Foundation of China (grant no. 51879067), the Fundamental Research Funds for the Central Universities of China (grant no. 2018B42914), the China Postdoctoral Science Foundation (grant no. 2017M621614), the Postdoctoral Research Supporting Program of Jiangsu province (grant no. 2018K128C), Natural Science Foundation of Jiangsu Province (grant no. BK20180022), and Six Talent Peaks Project in Jiangsu Province (grant no. NY-004).

Review statement
Review statement.

This paper was edited by Bettina Schaefli and reviewed by two anonymous referees.

References

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Kobold, M. and Brilly, M.: The use of HBV model for flash flood forecasting, Nat. Hazards Earth Syst. Sci., 6, 407–417, https://doi.org/10.5194/nhess-6-407-2006, 2006. a, b

Lobligeois, F., Andréassian, V., Perrin, C., Tabary, P., and Loumagne, C.: When does higher spatial resolution rainfall information improve streamflow simulation? An evaluation using 3620 flood events, Hydrol. Earth Syst. Sci., 18, 575–594, https://doi.org/10.5194/hess-18-575-2014, 2014. a, b

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Short summary
This study investigates whether higher temporal and spatial resolution of rainfall can lead to improved model performance. Four rainfall datasets were used to drive lumped and distributed HBV models to simulate daily discharges. Results show that a higher temporal resolution of rainfall improves the model performance if the station density is high. A combination of observed high temporal resolution observations with disaggregated daily rainfall leads to further improvement of the tested models.
This study investigates whether higher temporal and spatial resolution of rainfall can lead to...
Citation


text:
# Physics Discussiom questions

$$1$$A ball rolls without slipping on a perfectly flat plane of infinite extent. Will it roll forever? If not, what would stop it?

$$2$$ Since only changes in potential energy are important in any problem, a student decides to let the elastic potential energy of a spring be zero when the spring is stretched a distance of x1. The student decides therefore to let U={1/2}k(x-x1)^2. Is this correct?”)

$$3$$.would space be effected by infinite amount of gravity? we know space has no mass and this might be a wrong question to ask. space is expanding because of dark energy, can it contract because of an infinite amount of gravity. ) -(This is not my homework just discussion question I found)

Note by Mardokay Mosazghi
4 years, 2 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

1. Rolling resistance would stop it. In short, the net normal force will not act exactly on the center of mass and would produce a small torque which would eventually stop the ball rolling.

2. No, its a baseless assumption. The given formula is derived by integrating very small potential energies and is only valid when spring's initial position is at x = 0.

3. No I think the space would expand. Is there any special reason why you have chosen infinite gravity? You could have simply asked if gravity would conract the space or not. Why infinite gravity my friend?

Ask if you doubt any of this.

New Question 4: What would be the direction of frictional force on a rolling without slipping ball? And why? If you think it's in the backward direction, why don't the ball accelerates? If you think it's in front direction why don't the ball disaccelerates like a cube if thrown forward with the same speed?

- 4 years, 2 months ago

I don't seem to find the answer do you know the answer

- 4 years, 2 months ago

1 the ball will stop because of friction

3 scientist also thik that universe will colapse because of graviti of every galaxy but it is not happening so the energy  that is acting in the univers Is known as dark energy because we dont know about it

- 3 years, 11 months ago

1.Yes because of gravity

- 4 years, 2 months ago


text:
CGAL 4.12 - 2D Periodic Triangulations
2D Periodic Triangulations Reference
Nico Kruithof
This package allows to build and handle triangulations of point sets in the two dimensional flat torus. Triangulations are built incrementally and can be modified by insertion or removal of vertices. They offer point location facilities. The package provides Delaunay triangulations and offers nearest neighbor queries and primitives to build the dual Voronoi diagrams.

Introduced in: CGAL 4.3
Depends on: 2D Triangulation
BibTeX: cgal:k-pt2-13-18a
Windows Demo: Periodic Delaunay Triangulation
Common Demo Dlls: dlls

The main classes of the 2D Periodic Triangulation package are CGAL::Periodic_2_triangulation_2 and CGAL::Periodic_2_Delaunay_triangulation_2. They contain functionality to access triangulations and to run queries on them. CGAL::Periodic_2_Delaunay_triangulation_2 can construct and modify Delaunay triangulations. It takes the geometric traits as well as the triangulation data structure as template parameters.

The geometric traits class must be a model of the concept Periodic_2DelaunayTriangulationTraits_2. It contains all predicates and constructions that are needed by the functions in the triangulation classes.

The package uses CGAL::Triangulation_data_structure_2 to represent the triangulation. The faces and vertices need to be models of the concepts Periodic_2TriangulationFaceBase_2 and TriangulationVertexBase_2, respectively. A triangulation is stored as a collection of vertices and faces that are linked together through incidence and adjacency relations. Each face gives access to its three incident vertices and to its three adjacent cells. Each vertex gives access to one of its incident faces.

The three vertices of a cell are indexed with 0, 1, and 2 in positive orientation, the positive orientation being defined by the orientation of the underlying space $$\mathbb T_c^3$$. The neighbors of a face are also indexed with 0, 1, 2 in such a way that the neighbor indexed by $$i$$ is opposite to the vertex with the same index. See Figure 41.2.

In order to be able to specify the triangle that contain vertices both inside and outside the original domain we store additional offset information for each vertex of a face. These offsets are models of the concept Periodic_2Offset_2.

## Concepts

• Periodic_2TriangulationTraits_2
• Periodic_2DelaunayTriangulationTraits_2
• Periodic_2TriangulationFaceBase_2
• Periodic_2TriangulationVertexBase_2
• Periodic_2Offset_2

## Classes

### Main Classes

• CGAL::Periodic_2_triangulation_2<PT,TDS>
• CGAL::Periodic_2_Delaunay_triangulation_2<PT,TDS>
• CGAL::Periodic_2_triangulation_hierarchy_2<PTr>
• CGAL::Periodic_2_triangulation_face_base_2<>
• CGAL::Periodic_2_triangulation_vertex_base_2<>
• CGAL::Periodic_2_offset_2

### Traits Classes

• CGAL::Periodic_2_triangulation_traits_2<Traits,Periodic_2Offset_2>
• CGAL::Periodic_2_Delaunay_triangulation_traits_2<Traits,Periodic_2Offset_2>

## Enums

• CGAL::Periodic_2_triangulation_2::Iterator_type
• CGAL::Periodic_2_triangulation_2::Locate_type

## Modules

Concepts

Main Classes

Traits Classes

Vertex and Face Classes

Enums


text:
Volume 13, issue 3 (2013)

 Recent Issues
Author Index
 The Journal About the Journal Editorial Board Subscriptions Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 To Appear Other MSP Journals
Fibre sequences and localization of simplicial sheaves

Matthias Wendt

Algebraic & Geometric Topology 13 (2013) 1779–1813
Abstract

In this paper, we discuss the theory of quasifibrations in proper Bousfield localizations of model categories of simplicial sheaves. We provide a construction of fibrewise localization and use this construction to generalize a criterion for locality of fibre sequences due to Berrick and Dror Farjoun. The results allow a better understanding of unstable ${\mathbb{A}}^{1}$–homotopy theory.

Keywords
Bousfield localization, simplicial sheaves, A^1-homotopy theory
Mathematical Subject Classification 2010
Primary: 55R65, 55P60
Secondary: 18F20, 14F42


text:
#### Vol. 15, No. 1, 2021

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
$p$-adic estimates of exponential sums on curves

### Joe Kramer-Miller

Vol. 15 (2021), No. 1, 141–171
##### Abstract

The purpose of this article is to prove a “Newton over Hodge” result for exponential sums on curves. Let $X$ be a smooth proper curve over a finite field ${\mathbb{𝔽}}_{q}$ of characteristic $p\ge 3$ and let $V\subset X$ be an affine curve. For a regular function $\stackrel{̄}{f}$ on $V$, we may form the $L$-function $L\left(\stackrel{̄}{f},V,s\right)$ associated to the exponential sums of $\stackrel{̄}{f}$. In this article, we prove a lower estimate on the Newton polygon of $L\left(\stackrel{̄}{f},V,s\right)$. The estimate depends on the local monodromy of $f$ around each point $x\in X-V$. This confirms a hope of Deligne that the irregular Hodge filtration forces bounds on $p$-adic valuations of Frobenius eigenvalues. As a corollary, we obtain a lower estimate on the Newton polygon of a curve with an action of $ℤ∕pℤ$ in terms of local monodromy invariants.

However, your active subscription may be available on Project Euclid at
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We have not been able to recognize your IP address 3.236.212.116 as that of a subscriber to this journal.
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$p$-adic cohomology, Artin–Schreier covers, wild ramification, zeta function, Newton polygon, exponential sums


text:
Solution methods for MaxFS

Here are some of the methods used to solve MaxFS.

Elastic programming

Heuristic based on Elastic programming (Chinneck 2001).

Dynamic IIS generation

Partial IIS (Irreducible Infeasible Subsystem) covering formulation with dynamic IIS generation (Parker & Ryan 96)

LP with equilibrium constraints

Formulations as LP with equilibrium constraints and concave minimization on polyhedra tackled by successive linearizations (Mangasarian et al. 94/96, Bennett et al. 97)

First branch-and-cut algorithm (Pfetsch 02/05) based on polyhedral study in Amaldi et al. 03.

Combinatorial Benders' cuts

Combinatorial Benders' cuts (Codato & Fischetti 04).

Feasopt (Cplex)

The Feasopt feature in Ilog Cplex: an infeasibility analysis tool that "...accepts an infeasible model and selectively relaxes the bounds and constraints, minimizing a [user defined] weighted penalty function."

Relaxation method

It is a simple algorithmic framework for solving systems of linear inequalities, see e.g. [11,22]. At every iteration the attention is focused on reducing the violation of a single inequality. Given a feasible system with and , as well as a starting point , such a method generates a sequence of approximate solutions. [more...]
This page is maintained by Pietro Belotti.


text:
# USERDEFINED_PROFILE¶

A climatology class that allows users to define their own height profiles of scalar parameters. The climatology class can hold an unlimited number of different profiles, each indexed by a unique CLIMATOLOGY_HANDLE. Different scalar parameters can be on different grids within the same class instance.

Each profile consists of a height-grid definition and the values on that grid. The height grid can either be re-used or changed for subsequent families of values. The algorithm uses spline or piecewise linear interpolation of the values on the height grid.

The height-grid is specified as an array of heights above sea-level in ascending order expressed in meters. Note that radiative transfer models will normally require the grid to cover the entire range of the atmosphere from 0.0 meters to the top of the atmosphere, typically 100,000 meters. The height grid is set by calling SetProperty( “Heights”, h ).

The user can request that logarithmic or linear interpolation be used by calling SetProperty( “DoLogInterpolation”, xx ). This value is in effect for all the grids stored in the object. The default value is linear.

The user can specify the value returned for heights above or below the heights grid by calling SetPropertyArray( “BadValue”, xx ). This value is in effect for all subsequent grids until it is changed. The default value is NaN.

Values are set on the current grid by calling SetPropertyUserDefined(). The current grid settings are copied and combined with the value array and a new entry made for this grid array. If a grid for this species already exists it is replaced.

A Matlab example is shown below. Note that the class ignores latitude, longitude and mjd values and only uses the height value passed in by the user to function GetParameter:

Climate = ISKClimatology('USERDEFINED_PROFILE')
h = (0:100)*1000.0;
v = h.*h + 1.0;
climate.SetProperty( “Heights”, h );
climate.SetPropertyUserDefined('SKCLIMATOLOGY_O3_CM3', v);
[ok, value] = climate.GetParameter( 'SKCLIMATOLOGY_O3_CM3', [50,102, 35000.0, 53000]);

## Properties¶

DoLogInterpolation (int n) [Default: 0]

Instructs the object to perform logarithmic or linear interpolation of the scalar profile on the height-grid. Logarithmic interpolation is implemented as linear interpolation of the logarithm of the scalar profile followed by an exponentiation. This value will only be applied to profiles created by subsequent calls to SetPropertyUserDefined(). It is not applied to profiles that have already been created.

n Setting
0 Scalar profiles are interpolated linearly in altitude.
1 Scalar profiles are interpolated using logarithmic interpolation.

DoPiecewiseLinear

Instructs the object how to interpolate between values on the height grid. This value will only be applied to profiles created by subsequent calls to SetPropertyUserDefined(). It is not applied to profiles that have already been created.

n Setting
0 Fit a bezier spline to the height-grid points
1 Perform piecewise linear interpolation between points.

Heights

Sets the height grid that will be used in subsequent calls to SetPropertyUserDefined(). The heights must be in ascending order and specify the height of the grid point above sea level in meters. The default is an empty array.

BadValue

Sets the value to be returned for heights outside the range of the height grid. The default is NaN.


text:
StatlectThe Digital Textbook

# Distribution function

What is the probability that the realization of a random variable will be less than or equal to a certain threshold value? The distribution function of a random variable allows to answer exactly this question. Its value at a given point is equal to the probability of observing a realization of the random variable below that point or equal to that point.

## Synonyms

The distribution function is also often called cumulative distribution function (abbreviated as cdf).

## Definition

The following is a formal definition.

Definition If is a random variable, its distribution function is a function such thatwhere is the probability that is less than or equal to .

## Example

Suppose a random variable can take only two values (0 and 1), each with probability 1/2. Its distribution function is

## Properties

Every distribution function enjoys the following four properties:

1. Increasing. is increasing, i.e., ;

2. Right-continuous. is right-continuous, i.e.,for any ;

3. Limit at minus infinity. satisfies

4. Limit at plus infinity. satisfies

Not only any distribution function enjoys these properties, but also, for any given function enjoying these four properties, it is possible to define a random variable that has the given function as its distribution function. The practical consequence of this fact is that when you need to check whether a given function is a proper distribution function, you just need to check that it satisfies the four properties above.

## More details

More details about the distribution function can be found in the lecture entitled Random variables.


text:
[1]
V. Thi Phuong and N. . Du Sanh, “Growth and development of $$\textit{Eleocharis ochrostachys}$$ Steud. reproduced from its tubers”, AJB, vol. 43, no. 3, pp. 69–75, Sep. 2021.


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# 20130814 (Wednesday, 14 August 2013)¶

welfare.reception.WaitingGuests : when clicking in that table on the partner, Lino-Welfare should show the Client’s and not the Partner’s detail. Because in Lino-Welfare the partner of a cal.Guest is always a Client.

I solved this by creating a new class: lino_welfare.modlib.reception.WaitingGuests which overrides the library reception.WaitingGuests, adding a virtual field partner which will override the field defined in the model:

class WaitingGuests(WaitingGuests):
label = WaitingGuests.label

@dd.virtualfield(dd.ForeignKey('pcsw.Client'))
def partner(self,obj,ar):
return pcsw.Client.objects.get(pk=obj.partner.pk)

Note that the label of a table is by design not automatically inherited. That’s why we must inherit the label manually.