text:
# 26b. Construct a Triangle Given the Length of Its Base, the Angle Opposite the Base and the Length of That Angle's Bisector

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This Demonstration constructs a triangle given the length of its base , the angle at and the length of the angle bisector from to .

[more]

Construction

1. Draw the segment of length . Draw a circle with center and central angle over the chord .

2. Let be the diameter of perpendicular to . Let be the midpoint of .

3. Let the point be on the extension of the segment such that . Draw a circle with center and radius .

4. Draw the line through and that intersects at the points and .

5. Draw the circle with center and radius . The point is one of the points of intersection of and . Draw triangle .

Verification

Theorem: The segment is the angle bisector of .

In the isosceles triangle , and . Since , . So is on . Therefore is a tangent of and the power of with respect to the circle is .

The power of with respect to is . By construction . So , .

[less]

Contributed by: Izidor Hafner  (October 2017)
Open content licensed under CC BY-NC-SA

## Details

Reference

[1] D. S. Modic, Triangles, Constructions, Algebraic Solutions (in Slovenian), Ljubljana: Math Publishers, 2009, p. 57.

## Permanent Citation

Izidor Hafner

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text:
# Jeroen Sijsling: Endomorphisms and decompositions of Jacobians

Time: Wed 2019-11-06 11.00 - 12.00

Location:

Lecturer: Jeroen Sijsling, Ulm

### Abstract

Let \(C\) be a curve over a number field, with Jacobian \(J\), and let \(\operatorname{End} (J)\) be the endomorphism ring of \(J\). The ring \(\operatorname{End} (J)\) is typically isomorphic to \(\mathbb Z\), but the cases where it is larger are interesting for many reasons, most of all because the corresponding curves can then often be matched with relatively simple modular forms.

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Last changed: Oct 24, 2019

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text:
# Problemy Analiza - Issues of Analysis. Volume 8 (26) № 3 2019

This issue is dedicated to the 100th anniversary of the outstanding mathematician Nikolay Andreevich Lebedev.

 Akniyev G. G. Approximation properties of some discrete Fourier sums for piecewise smooth discontinuous functionsDownloads: 95 3 - 15 Alexeeva T. A., Shirokov N. A. CONSTRUCTIVE DESCRIPTION OF FUNCTION CLASSES ON SURFACES IN R^3 AND R^4Downloads: 79 16 - 23 Aloui B. , Khériji L. Connection formulas and representations of Laguerre polynomials in terms of the action of linear differential operatorsDownloads: 107 24 - 37 Dubinin V. N. Green energy and extremal decompositionsDownloads: 69 38 - 44 Dudnikova T. V. The limiting amplitude principle for the nonlinear Lamb systemDownloads: 61 45 - 62 Kuzhaev A. F. On the necessary and sufficient conditions for the measurability of a positive sequenceDownloads: 104 63 - 72 Losev A. G., Mazepa E. A. On solvability of the boundary value problems for harmonic function on noncompact Riemannian manifoldsDownloads: 66 73 - 82 Lysov V. G. INTEGRAL REPRESENTATIONS FOR THE JACOBI–PINEIRO POLYNOMIALS AND THE FUNCTIONS OF THE SECOND KINDDownloads: 63 83 - 95 Malyutin K. G., Gusev A. L. Geometric meaning of the interpolation conditions in the class of functions of finite order in the half-planeDownloads: 64 96 - 104 Matejíčka L. Proof of a Conjecture on NielsenDownloads: 79 105 - 111 Moosavi B. , Moradi H. R., Shah Hosseini M. Further results on Jensen-type inequalitiesDownloads: 120 112 - 124 Naik S. , Nath P. K. A NOTE ON A TWO-PARAMETER FAMILY OF OPERATORS ${\mathcal A}^{b,c}$ ON WEIGHTED BERGMAN SPACESDownloads: 72 125 - 136 Porselvi K. , Elavarasan B. On hybrid interior ideals in semigroupsDownloads: 97 137 - 146 Shcherbakov E. A., Avdeyev I. A. ON THE COMPACTNESS OF ONE CLASS OF QUASICONFORMAL MAPPINGSDownloads: 49 147 - 151 Starkov V. V. The Jacobian conjecture: structure of Keller mappingsDownloads: 80 152 - 165 Sultanakhmedov M. S. On the convergence of the least square method in case of non-uniform gridsDownloads: 83 166 - 186 Volosivets S. S. Double cosine-sine series and Nikol'skii classes in uniform metricDownloads: 77 187 - 203


text:
# Generalizations of the restriction theorem for multiplier ideals

Research paper by Eugene Eisenstein

Indexed on: 16 Jan '10Published on: 16 Jan '10Published in: Mathematics - Algebraic Geometry

#### Abstract

We present an algebro-geometric perspective on some generalizations, due to S. Takagi, of the restriction theorem for multiplier ideals. The first version of the restriction theorem for multiplier ideals was discovered by Esnault and Viehweg. In a series of papers S. Takagi has discovered generalizations of the restriction theorem and some formulas for multiplier ideals that follow from the restriction theorem. He uses the technique of tight closure and reduction to positive characteristic. We are able to provide an algebro-geometric proof of generalizations of his restriction theorem and his subadditivity theorem. We also prove an adjunction formula for relative canonical divisors of factorizing resolutions of singularities.


text:
.154721 as a fraction

.154721 as a fraction - solution and the full explanation with calculations.

If it's not what You are looking for, type in into the box below your number and see the solution.

What is .154721 as a fraction?

To write .154721 as a fraction you have to write .154721 as numerator and put 1 as the denominator. Now you multiply numerator and denominator by 10 as long as you get in numerator the whole number.
.154721 = .154721/1 = 1.54721/10 = 15.4721/100 = = 1547.21/10000 = 15472.1/100000 = 154721/1000000

And finally we have:
.154721 as a fraction equals 154721/1000000


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# zbMATH — the first resource for mathematics

## Memoirs on Differential Equations and Mathematical Physics

 Short Title: Mem. Differ. Equ. Math. Phys. Publisher: Ivane Javakhishvili Tbilisi State University, A. Razmadze Mathematical Institute, Tbilisi ISSN: 1512-0015 Online: http://www.rmi.ge/jeomj/memoirs/http://www.emis.de/journals/MDEMP/ Comments: Indexed cover-to-cover; This journal is published open access.
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# Twin primes and others

1. ### Bob3141592

226
As a programming exercise I wrote a program to generate primes. First I generated a billion of them (the one billionth prime is 22,801,763,489). My program also scans through these numbers for Twin primes (adjacent primes that differ by two), cousin primes (adjacent primes that differ by four) and sexy primes (primes that differ by six and don't have to be adjacent). There are 58,047,180 twin primes among the first billion, and 58,040,263 cousin primes in that same range. Almost the same number. I supposed that that count would hold no matter what the difference between the primes was, so I was surprised to see 116,076,313 sexy primes, almost exactly twice the previous counts. I extended the program to look for other differences, and found 58,044,163 primes that differ by 8, very similar to the count of twins and cousins. But then there were 77,387,551 primes that differ by 10, and 116,089,533 primes that differ by 12, very similar to the sexy count.

I'll be looking for primes with other differences, but the program takes its time.

Is there a reason for this behavior, or is it just statistical variations?

2. ### micromass

18,440
Staff Emeritus
This is already a known conjecture. But it has not yet been proven that it always holds. It follows from the first Hardy-Littlewood conjecture: http://mathworld.wolfram.com/k-TupleConjecture.html
The rest of your numbers also follow from it.

Last edited: Jul 11, 2014
3. ### HallsofIvy

40,213
Staff Emeritus
Why would you suppose that? It would seem reasonable to me that, since the larger $x_1$ is the more primes there are to be divisors of numbers, the larger $x_1$ is, the fewer primes there would be between $x_1$ and $x_2$ for a fixed $x_2- x_1$.

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New release: v0.6.1.

Problems with creating multi-site coupling on a chain.

How do I use this algorithm? What does that parameter do?
frenzie92
Posts: 4
Joined: 14 Nov 2019, 16:07

Problems with creating multi-site coupling on a chain.

Hello everyone.

So I want to implement the following spinless fermion Hamiltonian on a simple 1-D chain :

$$H=\sum_i \left[ V_{10} n_i n_{i+1}+V_{20} n_i n_{i+2}-V_{21} \left( c^\dagger_i c_{i+1} c_{i+2} c^\dagger_{i+3} + h.c. \right)\right].$$

In order to do this, I use add_multi_coupling as is done in the toric code example, and also mentioned in this topic.

What I've done so far is this (basically changed parts of the toric code example to fit my purposes. What I have is this

Code: Select all

import numpy as np

from .lattice import Lattice, Site, Chain
from ..networks.site import FermionSite
from .model import MultiCouplingModel, CouplingMPOModel
from ..tools.params import get_parameter
from ..tools.misc import any_nonzero

__all__ = ['FQHChain']

class FQHChain(CouplingMPOModel, MultiCouplingModel):
r"""

Parameters
----------
L : int
Length of the chain.
V10, V20, V21 : float
Couplings as defined for the Hamiltonian above.
bc_MPS : {'finite' | 'infinte'}
MPS boundary conditions. Coupling boundary conditions are chosen appropriately.
"""

def __init__(self, model_params):
CouplingMPOModel.__init__(self, model_params)

def init_sites(self, model_params):
site = FermionSite(conserve='N')
return site

def init_lattice(self, model_params):
site = self.init_sites(model_params)
L = get_parameter(model_params, 'L', 2, self.name) # default values of parameters are currently arbitrary.
bc_MPS = get_parameter(model_params, 'bc_MPS', 'finite', self.name)
bc = 'periodic' if bc_MPS == 'finite' else 'open'
lat = Chain(L, site, bc=bc, bc_MPS=bc_MPS)
return lat

def init_terms(self, model_params):
V10 = get_parameter(model_params, 'V10', 1., self.name, asarray=True)  # default values of parameters are currently arbitrary.
V20 = get_parameter(model_params, 'V20', 0.2, self.name, True)
V21 = get_parameter(model_params, 'V21', 0.5, self.name, True)
self.add_coupling(V10, 0, 'N', 0, 'N', 1)
self.add_coupling(V20, 0, 'N', 0, 'N', 2)
self.add_multi_coupling(-V21, 0, 'Cd', [(0, 'C', 1), (0, 'C', 2), (0, 'Cd', 3)] )
self.add_multi_coupling(-V21, 0, 'C', [(0, 'Cd', -1), (0, 'Cd', -2), (0, 'C', -3)] )  # h.c. of previous term
# done
I call this fqh_chain.py and save it in the same directory as the other models.

In order to check it, I run a variant of the code found in page 20 of the tenpy paper. In particular, I run the following

Code: Select all

import numpy as np
import random

from tenpy.networks.mps import MPS
from tenpy.models.fqh_chain import FQHChain
from tenpy.algorithms import dmrg

L=16
N=8

places=random.sample(range(1,L), N)

#We create a random state with N electrons in occupation number representation

initial_psi=['empty']*L

for i in range(0,len(places)):
initial_psi[places[i]]='full'

M = FQHChain({"L": L, "V10": 1, "V20": 0.1, "V21": 0., "bc_MPS": "finite"})
psi = MPS.from_product_state(M.lat.mps_sites(), initial_psi, "finite")
dmrg_params = {"trunc_params": {"chi_max": 30, "svd_min": 1.e-10}}
dmrg.run(psi, M, dmrg_params) # find the ground state
print("E =", sum(psi.expectation_value(M.H_bond[1:])))
print("final bond dimensions: ", psi.chi)

The output is this

Code: Select all

parameter 'L'=16 for FQHChain
parameter 'bc_MPS'='finite' for FQHChain
parameter 'V10'=1 for FQHChain
parameter 'V20'=0.1 for FQHChain
parameter 'V21'=0.0 for FQHChain
parameter 'trunc_params'={'chi_max': 30, 'svd_min': 1e-10} for Sweep
================================================================================
sweep 10, age = 16
Energy = 3.3000000000000003, S = 0.0000000000000000, norm_err = 0.0e+00
Current memory usage 47704.0 MB, time elapsed: 3.0 s
Delta E = nan, Delta S = nan (per sweep)
max_trunc_err = 0.0000e+00, max_E_trunc = 4.4409e-16
MPS bond dimensions: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
================================================================================
sweep 20, age = 16
Energy = 3.3000000000000003, S = 0.0000000000000000, norm_err = 0.0e+00
Current memory usage 47860.0 MB, time elapsed: 6.0 s
Delta E = 0.0000e+00, Delta S = 0.0000e+00 (per sweep)
max_trunc_err = 0.0000e+00, max_E_trunc = 4.4409e-16
MPS bond dimensions: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
================================================================================
DMRG finished after 20 sweeps.
total size = 16, maximum chi = 1
================================================================================
Traceback (most recent call last):
File "test.py", line 27, in <module>
print("E =", sum(psi.expectation_value(M.H_bond[1:])))
AttributeError: 'FQHChain' object has no attribute 'H_bond'

There are a number of things wrong with this. First and foremost, the error AttributeError: 'FQHChain' object has no attribute 'H_bond'. Also, something's fishy with the bond dimensions, they're all unity! $$\Delta E =0$$, and in general it looks like my attempt at implementing my own model failed miserably, which makes me very sad

To try to find out what's wrong I tried adding a similar four-site coupling to the xxz chain (in particular, XXZChain2). The Hamiltonian becomes

$$H = \sum_i \left[ \frac{\mathtt{J_{xx}}}{2} (S^{+}_i S^{-}_{i+1} + S^{-}_i S^{+}_{i+1}) + \mathtt{J_z} S^z_i S^z_{i+1} - \sum_i \mathtt{h_z} S^z_i\right]+K \sum_i \left(S^z_i S^z_{i+1} S^z_{i+2} S^z_{i+3} + h.c. \right)$$

Code: Select all

import numpy as np

from .lattice import Site, Chain
from .model import CouplingModel, NearestNeighborModel, MPOModel, CouplingMPOModel, MultiCouplingModel
from ..linalg import np_conserved as npc
from ..tools.params import get_parameter, unused_parameters
from ..networks.site import SpinHalfSite  # if you want to use the predefined site

__all__ = ['TestChain']

class TestChain(CouplingMPOModel, NearestNeighborModel, MultiCouplingModel):
"""Another implementation of the Spin-1/2 XXZ chain with Sz conservation.

This implementation takes the same parameters as the :class:XXZChain, but is implemented
based on the :class:~tenpy.models.model.CouplingMPOModel.
"""
def __init__(self, model_params):
model_params.setdefault('lattice', "Chain")
CouplingMPOModel.__init__(self, model_params)

def init_sites(self, model_params):
return SpinHalfSite(conserve='Sz')  # use predefined Site

def init_terms(self, model_params):
Jxx = get_parameter(model_params, 'Jxx', 1., self.name, True)
Jz = get_parameter(model_params, 'Jz', 1., self.name, True)
hz = get_parameter(model_params, 'hz', 0., self.name, True)
Jv = get_parameter(model_params, 'Jv', 0., self.name, True)
self.add_coupling(Jxx * 0.5, 0, 'Sp', 0, 'Sm', 1)
self.add_coupling(np.conj(Jxx * 0.5), 0, 'Sp', 0, 'Sm', -1)  # h.c.
self.add_coupling(Jz, 0, 'Sz', 0, 'Sz', 1)
self.add_multi_coupling(Jv, 0, 'Sz', [(0, 'Sz', 1), (0, 'Sz', 2), #this is the only thing I add, along with the inclusion of MultiCouplingModel
(0, 'Sz', 3)])
self.add_multi_coupling(Jv, 0, 'Sz', [(0, 'Sz', -1), (0, 'Sz', -2),
(0, 'Sz', -3)])

Let's call this 'test_chain.py'.

To test it run (note that I have set $$K=0$$).

Code: Select all

from tenpy.networks.mps import MPS
from tenpy.models.test_chain import TestChain
from tenpy.algorithms import dmrg

M = TestChain({"L": 4, "Jxx": 1, "Jz": 0.1, "hz": 0., "K":0, "bc_MPS": "finite"})
psi = MPS.from_product_state(M.lat.mps_sites(), [0,1,1,0], "finite")
dmrg_params = {"trunc_params": {"chi_max": 30, "svd_min": 1.e-10}}
dmrg.run(psi, M, dmrg_params) # find the ground state
print("E =", sum(psi.expectation_value(M.H_bond[1:])))
print("final bond dimensions: ", psi.chi)

This works quite well, giving

Code: Select all

parameter 'lattice'='Chain' for TestChain
parameter 'bc_MPS'='finite' for TestChain
parameter 'L'=4 for TestChain
parameter 'Jxx'=1 for TestChain
parameter 'Jz'=0.1 for TestChain
parameter 'hz'=0.0 for TestChain
parameter 'K'=0 for TestChain
parameter 'trunc_params'={'chi_max': 30, 'svd_min': 1e-10} for Sweep
================================================================================
sweep 10, age = 4
Energy = -1.1636015623654488, S = 0.5927718553565784, norm_err = 5.1e-16
Current memory usage 47788.0 MB, time elapsed: 0.6 s
Delta E = nan, Delta S = nan (per sweep)
max_trunc_err = 0.0000e+00, max_E_trunc = 1.1102e-15
MPS bond dimensions: [2, 4, 2]
================================================================================
sweep 20, age = 4
Energy = -1.1636015623654492, S = 0.5927718553565789, norm_err = 5.4e-16
Current memory usage 47836.0 MB, time elapsed: 1.1 s
Delta E = -4.4409e-17, Delta S = 5.5511e-17 (per sweep)
max_trunc_err = 0.0000e+00, max_E_trunc = 2.2204e-16
MPS bond dimensions: [2, 4, 2]
================================================================================
DMRG finished after 20 sweeps.
total size = 4, maximum chi = 4
================================================================================
E = -1.163601562365449
final bond dimensions:  [2, 4, 2]

Now try giving $$K$$ a finite value, say $$K=0.01$$, I get

Code: Select all

parameter 'lattice'='Chain' for TestChain
parameter 'bc_MPS'='finite' for TestChain
parameter 'L'=4 for TestChain
parameter 'Jxx'=1 for TestChain
parameter 'Jz'=0.1 for TestChain
parameter 'hz'=0.0 for TestChain
parameter 'K'=0.01 for TestChain
Traceback (most recent call last):
File "test.py", line 9, in <module>
M = TestChain({"L": 4, "Jxx": 1, "Jz": 0.1, "hz": 0., "K":0.01, "bc_MPS": "finite"})
File "/Users/USERNAME/TeNPy/tenpy/models/test_chain.py", line 27, in __init__
CouplingMPOModel.__init__(self, model_params)
File "/Users/USERNAME/TeNPy/tenpy/models/model.py", line 1240, in __init__
NearestNeighborModel.__init__(self, lat, self.calc_H_bond())
File "/Users/USERNAME/TeNPy/tenpy/models/model.py", line 918, in calc_H_bond
H_bond = ct.to_nn_bond_Arrays(sites)
File "/Users/USERNAME/TeNPy/tenpy/networks/terms.py", line 635, in to_nn_bond_Arrays
H_add = strength * npc.outer(site_i.get_op(op1), site_j.get_op(op2))
File "/Users/USERNAME/TeNPy/tenpy/networks/site.py", line 305, in get_op
names = name.split(' ')
AttributeError: 'tuple' object has no attribute 'split'

So now I get a different kind of error I noticed that not including the NearestNeighborModel yields the same 'AttributeError: 'TestChain' object has no attribute 'H_bond''. So I guess that's what the problem was initially. But, looking at the second model now, I still cannot have multi-site couplings without yielding errors In particular, I get 'AttributeError: 'tuple' object has no attribute 'split''.

Sorry if this question was too long, I wanted to provide as much information as I could, and provide everything I've tried. So the basic problem, how do I get the initial model to give sensible results? What's wrong with the way I did it? Also, what's wrong with the second model (perhaps answering this might help with the first model - since the only difference with the already made XXZChain2 model is the inclusion of add_multi_coupling. If this is done I can rewrite my own model based on XXZChain2. )

Any help would be very much appreciated.
Last edited by frenzie92 on 21 Dec 2019, 15:35, edited 1 time in total.

Johannes
Posts: 176
Joined: 21 Jul 2018, 12:52
Location: UC Berkeley

Re: Problems with creating multi-site coupling on a chain.

By definition, a tenpy.models.model.NearestNeighborModel can only contain nearest-neighbor couplings.
That is the case because it provides a representation of the hamiltonian in terms of H_bond, where each H_bond is a term involving just two neighboring sites.
Hence, you can *not* make a model inherited from the NearestNeighborModel and add couplings over more than two neighboring sites. (You also cannot add two-site couplings with dx > 1.)

The correct way to go is hence to *not* inherit from the NearestNeighborModel, as you do in your FQHChain.
However, that implies that there also is no H_bond representation of the Hamiltonian, which you try to use in evaluating the energy.
It is an tenpy.models.model.MPOModel, however (inherited from the CouplingMPOModel[/tenpy], which means that it has a representation of the hamiltonian as an MPO, H_MPO.
Try to evaluate the energy with M.H_MPO.expectation_value(psi) instead, or even simipler just use the "E" returned by the DMRG run.

That should answer why the error appears in your first example, and why the second example fails.
I agree that the error message in the second case is quite obscure, though, and does not really tell you about the problem.
That was indeed a bug, and I fixed it now (in the newest version only, not yet in v0.5.0) to give a more sensible error message.

To your initial FQHChain model: that's fine!
The reason why you don't get anything else but a product state from DMRG is simply that you conserve the total particle number and initialize it in the vacuum state, i.e in the charge sector with 0 particles.
This charge sector consists only of that single vacuum state. DMRG is not allowed to change the charge sector, so it will stay there, and correctly output the vacuum state (which also is an Eigenstate of your hamiltonian, with eigenenergy 0).
The way to go is to choose a different initial state, e.g.

Code: Select all

initial_psi = ['empty', 'full']*(L//2)  # for even L

Also, you should put mixer=True in your DMRG parameters, to allow getting out of this initial state.


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CommentaryClinical Trials

# A Virtual National Laboratory for Reengineering Clinical Translational Science

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Science Translational Medicine  25 Jan 2012:
Vol. 4, Issue 118, pp. 118cm2
DOI: 10.1126/scitranslmed.3002951

## Abstract

Clinical research is burdened by inefficiencies and complexities, with a poor record of trial completion, none of which is desirable. The Clinical and Translational Science Award (CTSA) Consortium, including more than 60 clinical research institutions, supports a unified national effort to become, in effect, a virtual national laboratory designed to identify, implement, evaluate, and extend process improvements across all parts of clinical research, from conception to completion. If adequately supported by academic health centers, industry, and funding agencies, the Consortium could become a test bed for improvements that can dramatically reduce wasteful complexity, thus increasing the likelihood of clinical trial completion.

## A SCIENTIFIC APPROACH TO CLINICAL RESEARCH MANAGEMENT

Translational research is the process of taking biomedical laboratory findings from preclinical product development to clinical trials, to widely adopted treatments in the community, and ultimately to improvements in human health. It encompasses all of the steps required to support the entire continuum of health product development, from discovery to standard clinical practice, as well as measures of effectiveness. The translational process has become sufficiently complex, slow, and frustrating that it constitutes a major challenge facing science and society in the early 21st century (15). Impaired translation of biomedical research threatens the pharmaceutical industry (6). Even if such reductions abate over time, major translational challenges remain. Pharmaceutical products ready for clinical testing have failure rates in excess of 80% and development costs of over $1.2 billion per drug. In addition, years of drawn-out scientific, regulatory, and institutional processes encumber clinical trials (710) [see Fig. 1 for steps to opening an oncology clinical trial (11)]. Until recently, no concerted effort to streamline performance in biomedical product (devices, drugs, surgical procedures, etc.) development in the United States has been undertaken, in marked contrast to the automotive and other industries, which reduced their new product development times by 41% in the decade from 1995 to 2004 (12). Among the phases in the translational pipeline (13), clinical research is certainly crucial for testing potential products in humans because it provides evidence of product safety and effectiveness when delivered to the community. The management of clinical research, however, has become as complex and lengthy as the scientific discovery of potential products, greatly impairing the efficient evaluation and incorporation of emerging laboratory concepts into the clinical development of marketable products (14). Lack of sufficient institutional understanding and commitment to effective ways of streamlining clinical research management has impaired efforts to eliminate wasteful steps and improve efficiency. Among the many challenges to clinical research management are lengthy and variable protocol and contract processing pathways and completion times, duplicative institutional review board (IRB) examination of identical multisite clinical protocols, protracted study start-up, failure to enroll adequate numbers of participants in some clinical trials and delayed completion of others, and too many studies that lack sufficient participant accruals to permit statistically acceptable analysis. Based on historical drug development data, it has been estimated that clinical research measured from the time a protocol for clinical testing was first conceived until the time the product was licensed for use takes an average of 90.3 months and costs an average of$100.4 million (15). Reduction of complexity, elimination of wasteful steps, and timely, successful completion of trials will have measurable economic and patient value.

More recently reported observational studies attest to the severity of the persistent need for change. In oncology, the most-studied disease area, for example, the development of National Cancer Institute (NCI)–sponsored phase III protocols from initial submission to final approval required a median of 2.5 years (16), and between 22 and 38% of such trials eventually failed owing to patient accrual issues (17, 18). Furthermore, long development times correlated with low accrual success (19), suggesting that these problems may be part of a more systemic failure. These findings have led to a call for reinvigoration of the national cancer clinical trials system (18).

Although the evidence clearly shows that improvement in clinical research management is both warranted and valuable, tested and validated methods have been completed primarily at single institutions. What has been lacking, then, is unified knowledge sharing and testing of ideas. The CTSA Consortium presents such an opportunity to provide a virtual national laboratory.

## IMPROVING MANAGEMENT BY USING EVIDENCE

Improvement in clinical research efficiency at academic health centers (AHCs) across the United States became a top priority of the recipients of the Clinical and Translational Science Awards (CTSAs)—a program started by the National Institutes of Health (NIH) in 2006. Designed to create an academic home for innovative clinical and translational research, the CTSAs set their sights on several goals, including speeding the process from discovery to use. Consistent with their awards, each CTSA site developed a program for improved conduct of clinical research. With concepts that have proven useful in multiple industries, several CTSA sites have used such techniques as process maps, value-stream mapping, and lean thinking to identify redundancy, tandem steps, endless loops, and wasted steps [see (20) for a good list]. Other CTSA sites have used similar tools on problems with respect to protocol development processes and contract negotiation improvement activities, instituting continuous process improvements across a broad range of clinical research management issues and thus focusing on shortening protocol approval and contract execution times. Other sites have enhanced the features of their clinical trials units and addressed communication systems, electronic management, and expanded recruitment programs.

Although each of these individual efforts is admirable, there was no agreed-upon methodology for establishing national benchmarks to measure the benefits of tracking project-completion times, instituting changes based on performance metrics, and analyzing processing data. In brief, it has been a challenge for sites that have implemented improvements to benchmark their performance on the basis of only the data they accumulated at their own sites. What was needed was objectively adjudicated parallel comparisons of multiple sites’ management processing against peer performance and national standards.

Nevertheless, implementation of local improvements will do little to change national performance of biomedical translation, which requires synchronous multisite completion of clinical trials in order to move products through the pipeline. Understanding this, the National Center for Research Resources (NCRR) and the Yale Center for Clinical Investigation instituted a series of national information-sharing workshops and meetings focusing on clinical research management. The fourth annual meeting, completed in June 2011, provided a forum for the current CTSA sites to note their progress in confronting the challenges the CTSA Consortium had previously identified as high priority. As noted in Table 1, 91% of the active sites participated in a voluntary process analysis study for protocol approval and contract execution times, and 93% appointed Champions of Change (individuals in leadership positions at individual CTSA sites) to direct and implement process improvements.

Table 1

A sampling of CTSA site participation in Consortium process improvement strategies. This table lists challenges in clinical research management and the methods used by the CTSA Consortium (n = 60 sites) to address them.

View this table:

Additionally, participants described three separate regional reliance IRB agreements: One involves many of Harvard’s associated institutions; a second involves five institutions in Wisconsin; a third includes 15 institutions in Texas. Each regional agreement permits an IRB at a participating institution to rely on the review of a protocol for a multisite study by an IRB at another institution that participates in the agreement. The regional agreements have eliminated hundreds of duplicative IRB reviews of multisite protocols. Some CTSA sites have developed data-driven, continuously correcting model recruitment programs that include clinical research–centered public communication and outreach programs, feasibility studies, prospective recruitment planning, population targeting, and dedicated staff. For example, in 2009, after its first year of operation, the Recruitment Enhancement Pilot at Washington University reached 125% of its target enrollment in 25 trials (http://www.ctsacentral.org/committees/available-documents/307: 6/22/2010 Clinical Research Management Workshop: Successful Recruitment Strategies 2: Comprehensive Recruitment Service: Infrastructure, Expertise, Service: Rathmann.ppt). By mid-2011, a Web-based, nationwide recruitment program, ResearchMatch (http://www.researchmatch.org), which was developed through the CTSA sites with NCRR funding, has facilitated the registration of participants in over 250 studies. Efficient data collection for multisite clinical trials at CTSA sites is facilitated by a Web-based data management program called REDCap (Research Electronic Data Capture; http://project-redcap.org), which supports rapid development of case report forms, central control of data and participating sites, and seamless data downloads in a secure environment.

Although these are worthwhile initiatives, it is time to pool these efforts to create a community of practice so that individual institutional efforts can be evaluated by using national performance data and the execution of multi-institutional efforts can be based on an informed, data-driven analysis of the entire CTSA Consortium. “Task forces,” comprising individuals at CTSA sites who have been charged with monitoring clinical protocol processing and contracts negotiation, are positioned to acquire and report follow-up process data using standard templates and protocols for process-mapping and milestone achievement. An advisory group of CTSA members with expertise in process analysis and clinical trials management identified barriers to process completion on the basis of its analysis of process streams.

With achievable goals and motivated by the visibility of the data, the CTSA sites have organized a Process Excellence Group to test methodologies and teach each other how to significantly streamline approaches to the often “invisible” barriers to clinical research (11). Thus, much like the basic sciences, the CTSA sites are combining their efforts to create a virtual national laboratory for clinical translational research. This structure will allow for the sharing of successful process interventions, positive changes, and lessons learned along the way.

## ADVANCING BEYOND THE FOUNDATION

With all of these foundations to drive change in place, what more could be needed? The organizational components that must do the work have been formed. Participants at the ground level accept the central concepts of process-mapping, project-tracking, analysis, implementation of change, and systematic reevaluation. It remains, then, for national leadership, government, academia, and the private sector to support the fledgling virtual national laboratory of CTSA sites as it begins the task of building on these initial efforts in process improvement. What the Consortium has learned is that there are no easy fixes, no simple solutions, no universal remedies for the inefficiencies in U.S. health care research, which now includes over 9700 active, interventional studies and trials in phases I through III supported by NIH and by industry (http://clinicaltrials.gov/).

The key process-engineering methodologies have been piloted and refined, but, as with all change, not every process refinement will succeed. Indeed, even if process changes do succeed, downstream effects may, at times, impede overall completion times. For example, shortening and simplifying the IRB review and approval process may have no measurable effect on study completion if the investigators are not prepared to proceed to study start-up and participant accrual. With leadership and support, adequate allocation of resources, and CTSA Consortia members committed to facilitating the clinical and translational scientific process, the virtual national laboratory is positioned to provide an environment where each proposed improvement can be systematically tested, so that the successful ones may be retained and the failures rejected quickly. The expectation is that this incremental evolution will move progressively toward making clinical research more efficient and timely. The initial results suggest that significant improvements in clinical management are already being translated into reality. Many CTSA sites have produced process maps that show shortening of processing time by up to 45 days and improvement of protocols presented to IRBs such that many more are approved with requests for revisions. Examples of process maps that have been made public may be found among Workshop posters (http://www.ctsacentral.org/committees/available-documents/307); one developed by Yale University was used to reduce processing time by 35 days (http://www.ctsacentral.org/documents/bahdocs/Yale.pdf).

Now is the crucial moment for leveraging the CTSA Consortium to broaden the scope of its efforts to test and implement additional improvements that seek to eliminate wasteful complexity and increase timely clinical trial completion. There is an ever-increasing need for more complex clinical trials that include “–omics” information. Costs of trials continue to increase while the private and federal funds available to complete such studies are limited (15). Finally, international competition is on the rise. Improving time and quality is of the essence in clinical research if the United States is to remain world class and not fall into the difficulties that have faced many other U.S. industries, such as the automotive, consumer electronics, and textile industries—all of which have experienced major declines from foreign competition (21, 22). The current virtual national laboratory for reengineering clinical translational science is following a path similar to that of SEMATECH (Semiconductor Manufacturing Technology), which revitalized the U.S. semiconductor industry. Following this path will require a sustained, prolonged, and consistent effort that is more analogous to a marathon than to a sprint—an effort that is well under way in the CTSA virtual national laboratory.

## References and Notes

1. Acknowledgments: : The authors gratefully acknowledge the careful review and valuable suggestions made by B. Alving, P. Cola, A. Hayward, R. Sampson, J. Silverstein, and J. Strasser. Funding: NIH/NCRR Institutional CTSA and the NIH: 1 UL1 RR024996 (Oregon Clinical and Translational Research Institute), 1 UL1 RR024996 (Clinical and Translational Science Center at Weill Cornell Medical College), and UL1 RR024139 and UL13 RR025975 (Yale Center for Clinical Investigation). Competing interests: The information and opinions expressed in this publication are those of the authors and do not reflect the policies of the U.S. government.
View Abstract


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# How do you find the homography matrix given 4 points in both images?

I want to understand the process of finding a homography matrix given 4 points in both images. I am able to do that in python OpenCV, but I wonder how it works behind the scenes.

Suppose I have points $$p_1, p_2, p_3, p_4$$ in the first image and $$p'_1, p'_2, p'_3, p'_4$$ in the second. How am I going to generate the homography matix given these points.

To understand homographies and how to find them, you will need a good dose of projective geometry. I will briefly describe some preliminary concepts that you need to know before trying to find the homography, but don't expect to understand all these concepts with one reading iteration and only by reading this answer, if you are not familiar with them, especially, if you don't even know what homogenous coordinates are. For more details, I suggest you read the book Multiple view geometry in computer vision (2004) by Richard Hartley and Andrew Zisserman, in particular, chapter 4.

## The projective space $$\mathbb{P}^2$$

$$\mathbb{P}^2$$ is the projective space of $$\mathbb{R}^2$$, so it is $$\mathbb{R}^2$$ augmented with lines and points at infinity. All points and lines of $$\mathbb{P}^2$$ actually belong to $$\mathbb{R}^3$$, i.e. they are vectors of three components, because they are homogenous representations of the counterparts in $$\mathbb{R}^2$$. To emphasize, in $$\mathbb{P}^2$$, both points and lines can be represented by a vector in $$\mathbb{R}^3$$, which is the homogenous representation of the counterpart vector in $$\mathbb{R}^2$$ (if it exists, e.g. points at infinity do not exist in $$\mathbb{R}^2$$).

## What is a homography?

A homography (aka projectivity, collineation or projective transformation) is an invertible map $$h$$ from the projective space $$\mathbb{P}^2$$ to itself, $$h: \mathbb{P}^2 \rightarrow \mathbb{P}^2,$$ such that $$\mathbf{x}_1, \mathbf{x}_2, \mathbf{x}_3$$ lie on the same straight-line if and only if $$h(\mathbf{x}_1), h(\mathbf{x}_2), h(\mathbf{x}_3)$$ do.

This property is called collinearity. In practice, this property means that projective transformations can map straight-lines in one image to straight-line to another image, but it cannot map e.g. straight-lines to parabolas (and vice-versa). So, if you have a distorted image, a homography cannot convert it to a non-distorted image.

### All and only linear maps in $$\mathbb{P}^2$$ are homographies

All linear maps in $$\mathbb{P}^2$$ are homographies and only linear maps can be homographies. Hence, when you want to find a homography, you are looking for a linear map in $$\mathbb{P}^2$$.

### A homography can be represented as $$3 \times 3$$ matrix

Given that homographies are linear maps they can be represented as an invertible matrix

$$\mathbf{H} = \begin{bmatrix} h_{11} & h_{12} & h_{13} \\ h_{21} & h_{22} & h_{23} \\ h_{31} & h_{32} & h_{33} \end{bmatrix} \in \mathbb{R}^{3 \times 3},$$ such that, $$\forall \mathbf{x} \in \mathbb{P}^2$$, the following equation holds $$h(\mathbf{x}) = \mathbf{H}\mathbf{x}$$

### Homographies map points in $$\mathbb{P}^2$$ to points in $$\mathbb{P}^2$$

Given that homographies map points in $$\mathbb{P}^2$$ to other points in $$\mathbb{P}^2$$ and there's a matrix $$\mathbf{H} \in \mathbb{R}^{3 \times 3}$$ for each homography $$h$$, then it follows that, $$\forall \mathbf{x} \in \mathbb{P}^2$$, \begin{align} \mathbf{x}' &= \mathbf{H}\mathbf{x} \\ \begin{bmatrix} \mathbf{x}'_1 \\ \mathbf{x}'_2 \\ \mathbf{x}'_3 \end{bmatrix} &= \begin{bmatrix} h_{11} & h_{12} & h_{13} \\ h_{21} & h_{22} & h_{23} \\ h_{31} & h_{32} & h_{33} \end{bmatrix} \begin{bmatrix} \mathbf{x}_1 \\ \mathbf{x}_2 \\ \mathbf{x}_3 \end{bmatrix} \tag{1}\label{1} \end{align} where $$\mathbf{x}' \in \mathbb{P}^2$$, for some homography $$\mathbf{H} \in \mathbb{R}^{3 \times 3}$$.

In practice, this means that you can transform points in an image $$I_1$$ to points in image $$I_2$$ by matrix multiplication.

### A homography has $$8$$ degrees of freedom

Although the matrix $$\mathbf{H}$$ has $$9$$ entries, it has only $$8$$ degrees of freedom, which means that, in practice, there are only $$8$$ variables that you need to find. This property comes from the fact that only the ratio between the elements of $$\mathbf{H}$$ actually counts. This means that equation \ref{1} can actually be written as \begin{align} \mathbf{x}' = \lambda \mathbf{H}\mathbf{x} \tag{2}\label{2} \end{align} for all $$\lambda \in \mathbb{R} \setminus \{0 \}$$.

## How to estimate a homography from point correspondences?

We want to estimate a homography $$\mathbf{H} \in \mathbb{R}^{3 \times 3}$$ from point-to-point correspondences $$\{(\mathbf{x}^i, \mathbf{x}'^i) \}_{i=1}^N$$, where $$N \geq 4$$, such that $$\mathbf{x}'^i = \mathbf{H}\mathbf{x}^i, \forall i$$.

First, recall that the matrix $$\mathbf{H}$$ has $$8$$ degrees of freedom (i.e. variables we want to find).

### Each point-to-point correspondence represents $$2$$ constraints

Each point-to-point correspondence $$(\mathbf{x}^i, \mathbf{x}'^i)$$ accounts for $$2$$ constraints, i.e. $$\mathbf{H}\mathbf{x}^i$$ maps to the point $$\mathbf{x}'^i$$, which has $$2$$ degrees of freedom (even if it's defined by 3 components) because it's represented in homogenous coordinates and so, as for equation \ref{2}, it's defined "up to scale". In other words, these $$2$$ degrees of freedom represent $$2$$ constraints.

### $$4$$ point-to-point correspondences are necessary to estimate a homography

Given that $$1$$ point-to-point correspondence represents $$2$$ constraints, then $$4$$ point-to-point correspondences corresponds to $$8$$ constraints. Given this and given that homographies have $$8$$ degrees of freedom, at least $$4$$ point-to-point correspondences are necessary to estimate a homography.

The equation \ref{2} can actually be written as

$$\mathbf{x}' \times \lambda \mathbf{H}\mathbf{x} = \mathbf{0} \label{3} \tag{3}$$ where $$\times$$ is the cross-product.

### Direct linear transformation

If you manipulate equation \ref{3}, then you will end up with another equation (if you ignore the scaling factor $$\lambda$$, which we can ignore because the homography converts points independently of their magnitude)

$$\mathbf{A}_i \mathbf{h} = \mathbf{0}$$ where $$\mathbf{A}_i \in \mathbb{R}^{2 \times 9}$$ is the design matrix (i.e. the matrix that contains the input data to estimate the homography) that contains the elements of the the point-to-point correspondence $$(\mathbf{x}^i, \mathbf{x}'^i)$$ and $$\mathbf{h} \in \mathbb{R}^{9}$$ is a vector that contains the unknown elements of $$\mathbf{H}$$. The details of this manipulation can be found in section 4.1 of the book Multiple view geometry in computer vision.

Now, the idea is that you can vertically stack $$N \geq 4$$ equations of the form $$\mathbf{A}_i \mathbf{h} = \mathbf{0}$$ to build the final linear system that you would like to solve $$\mathbf{A} \mathbf{h} = \mathbf{0} \tag{4}\label{4}$$ where $$\mathbf{A} \in \mathbb{R}^{2N \times 9}$$.

The solution to this system is the vector $$\mathbf{h} \in \mathbb{R}^{9}$$, that is, your homography!

If you know something about linear algebra, you know that the solutions to $$\mathbf{A} \mathbf{h} = \mathbf{0}$$ are elements of the null space of $$\mathbf{A}$$.

Then, to find $$\mathbf{h}$$, you will typically use singular value decomposition (SVD). See e.g. How is the null space related to singular value decomposition? for a possible explanation of why you can use SVD to find an element of the null space.

This algorithm (which is described on page 91 of the cited book, i.e. Algorithm 4.1) is called direct linear transformation.

• A wonderful explanation . – Syed Hissaan May 10 at 21:57


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# How do you find the domain and range of sqrt(x - 1)?

Oct 13, 2017

You can get the domain by working with the assumption that the function's domain lie in a set of all real numbers, that is $\left(x - 1\right)$ must be greater than or equal to zero.

#### Explanation:

If $\left(x - 1\right)$ must be greater than $0$ then $x \ge 1$ must be true for all values of $x$.

Therefore the domain is $x \ge 1$

The range is suppose to be the set of all values of the function that lies within the domain.
The least member of the domain set above is $x = 1$ and the value of the function at this value of $x$ is √1 which is $1$.
Therefore the range is $\left\{1 , \ldots\right\}$


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Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Evaluating the Impacts of Ground and Atmospheric Conditions on the Efficiency of Solar Energy System and Its Economic Analysis

Version 1 : Received: 20 June 2018 / Approved: 20 June 2018 / Online: 20 June 2018 (14:22:44 CEST)

How to cite: Aziz, A.S.; Naim Tajuddin, M.F.; Padmanaban, S.; Mihet-Popa, L.; Adzman, M.R.; A. M. Ramli, M. Evaluating the Impacts of Ground and Atmospheric Conditions on the Efficiency of Solar Energy System and Its Economic Analysis. Preprints 2018, 2018060324 (doi: 10.20944/preprints201806.0324.v1). Aziz, A.S.; Naim Tajuddin, M.F.; Padmanaban, S.; Mihet-Popa, L.; Adzman, M.R.; A. M. Ramli, M. Evaluating the Impacts of Ground and Atmospheric Conditions on the Efficiency of Solar Energy System and Its Economic Analysis. Preprints 2018, 2018060324 (doi: 10.20944/preprints201806.0324.v1).

## Abstract

The There are many factors influencing the performance of photovoltaic (PV) systems. Among these factors, temperature and solar radiation are two major parameters that have a large effect on the efficiency of PV systems. The cell temperature of PV panels is related to the ambient temperature while the solar radiation incident on the surface of the PV modules depends on the slope and azimuth of these modules. Furthermore, ground reflectance (albedo) affects the irradiance incident on the PV panel surface, which in turn affects the output of a PV system. Nevertheless, the effects of these factors on the economic performance of the solar PV systems are scarcely reported. This paper presents a complete design of a stand-alone PV/battery system to supply electric power for a mobile base station in Choman, Erbil, Iraq. The effects of different factors on the total electricity produced by PV arrays and its economic performance are simultaneously investigated. HOMER software has been used as a tool for the techno-economic and environmental analysis. As indicated from the simulation results, the PV array capacity and its economic performance are highly affected by the variation of the slope and azimuth. With a base case (albedo of 20% and average annual ambient temperature of 11°C), the best feasible system which is achieved by facing PV due to south with a tilt angle of 40° or 45°, is found to have net present cost (NPC) of 70595 $and cost of energy (COE) of 0.54$/kWh. Moreover, the results indicate that increasing the ground reflectance from 10% to 90% results in a 7.2% decrease in the PV array capacity and about 3% decrease in the NPC and COE. On the other hand, increasing the ambient temperature from 0°C to 40°C results in a 19.7% increase in the PV array capacity and an 8.2% increase in the NPC and COE. Furthermore, according to the ambient temperature of Choman, using PV modules with high sensitivity to temperature is found to be an attractive option. Provided simulation performance analysis proves that the studied parameters must be treated well to establish an enabling environment for solar energy development in Iraq.

## Subject Areas

Renewable Energy; solar energy; PV panels slope; azimuth; temperature; albedo

We encourage comments and feedback from a broad range of readers. See criteria for comments and our diversity statement.

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# I need explanations about fixedFluxPressure

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 November 30, 2010, 05:15 I need explanations about fixedFluxPressure #1 Senior Member   Cyprien Join Date: Feb 2010 Location: Stanford University Posts: 230 Rep Power: 9 Hello! I had a look at the fixedFluxPressure boundary condition and I am not quite sure to well understand what this BC does. I guess I have to use this BC if I want to evaluate the pressure according to a velocity input. Am I wrong ? Furthermore, in the code source, one can read: Code:  gradient() = (phip - (patch().Sf() & Up))/patch().magSf()/rAp; I do not succeed in linking this snippet with a mathematical formula... All enlightments are welcome! Regards, Cyp JR22 and Guimloute like this.

 August 19, 2013, 20:36 buoyantPressure vs. fixedFluxPressure #2 Senior Member     Jose Rey Join Date: Oct 2012 Posts: 126 Rep Power: 8 I realize this is an old thread, but it is still very relevant. I was having terrible problems trying to get buoyantBoussinesqSimpleFoam to converge. I changed the p_rgh BC's from buoyantPressure to fixedFluxPressure and the model started to converge. Anybody? Guimloute likes this.

 August 1, 2014, 04:42 #3 Senior Member   Srivathsan N Join Date: Jan 2013 Location: India Posts: 101 Rep Power: 4 Same question here. What is the difference between buoyantPressure and fixedFLuxPressure boundary conditions? __________________ Regards, Srivaths

 October 16, 2014, 02:23 OpenFAOM 2.3.0 doesn't have buoyantPressure BC #4 Senior Member     Fatema Zandi Goharrizi Join Date: Mar 2009 Posts: 137 Rep Power: 8 Hi OpenFAOM version 2.3.0 doesn't have buoyantPressure libraries. Am I right? what is the difference between buoyantPressure and fixedFluxPressure? why fixedFluxPressure converge better? is it the only difference between them? Please help

 October 16, 2014, 03:46 #5 Senior Member     Fabian Roesler Join Date: Mar 2009 Location: Bad Friedrichshall, Germany Posts: 166 Rep Power: 8 OpenFOAM 2.2.2 has both boundary conditions. In newer Version, buoyantPressure boundary was dismissed. For clarification have a look into the code or into Doxygen. From buoyantPressureFvPatchScalarField we learn that: This boundary condition sets the pressure gradient appropriately for buoyant flow. If the variable name is one of pd, p_rgh or ph_rgh, we assume that the pressure variable is $p - \rho(g \cdot h)$ and the gradient set using: $\nabla(p) = -\nabla_\perp(\rho)(g \cdot h)$ Otherwise we assume that it is the static pressure, and the gradient calculated using: $\nabla(p) = \rho(g \cdot n)$ From fixedFluxPressureFvPatchScalarField we see that: This boundary condition adjusts the pressure gradient such that the flux on the boundary is that specified by the velocity boundary condition. The predicted flux to be compensated by the pressure gradient is evaluated as $(\phi - \phi_{H/A})$, both of which are looked-up from the database, as is the pressure diffusivity used to calculate the gradient using: $\nabla(p) = \frac{\phi_{H/A} - \phi}{|Sf| D_p}$ wyldckat, zandi, eternityboy and 4 others like this.

 October 16, 2014, 04:21 #6 Senior Member     Fabian Roesler Join Date: Mar 2009 Location: Bad Friedrichshall, Germany Posts: 166 Rep Power: 8 Some additional remarks: The fixedFluxPressure boundary is known do have a better convergence. This is in my opinion due to the more pressure related calculation of the gradient: $\nabla(p) = \frac{\phi_{H/A} - \phi}{|Sf| D_p}$ phi and phiHbyA (predicted flux field) are directly linked to pressure equation. On a wall, the flux difference tends to zero and so the boundary condition turns to zeroGradient. The buoyanPressure boundary is more the physical method to describe the pressure gradient with Archimedes' principle. I hope my explanation is understandable and correct. Cheers Fabian wyldckat, zandi, LESlie and 6 others like this.

 November 5, 2014, 17:06 #7 Senior Member     Santiago Marquez Damian Join Date: Aug 2009 Location: Santa Fe, Santa Fe, Argentina Posts: 418 Rep Power: 15 Hi folks, I'm dealing with fixedFluxPressure BCs also and this thread was helpful. Checking the code things have changed at least in FOAM 2.3.0. The class definition is now more simple fixedFluxPressureFvPatchScalarField.C Code:  125 void Foam::fixedFluxPressureFvPatchScalarField::updateCoeffs 126 ( 127 const scalarField& snGradp 128 ) 129 { 130 if (updated()) 131 { 132 return; 133 } 134 135 curTimeIndex_ = this->db().time().timeIndex(); 136 137 gradient() = snGradp; 138 fixedGradientFvPatchScalarField::updateCoeffs(); 139 } it only sets the proper gradient in line 137 and then calls the updateCoeffs() method from the fixedGradientFvPatchScalarField class (from which the present class inherits). This requires to set the value of snGradp which is done at solver level. For example, from pEqn.H of interFoam we have: Code:  27 // Update the fixedFluxPressure BCs to ensure flux consistency 28 setSnGrad 29 ( 30 p_rgh.boundaryField(), 31 ( 32 phiHbyA.boundaryField() 33 - (mesh.Sf().boundaryField() & U.boundaryField()) 34 )/(mesh.magSf().boundaryField()*rAUf.boundaryField()) 35 ); From the new code the gradient reads: Hope this helps for 2.3.0 users. Regards. fabian_roesler, LESlie, PonchO and 5 others like this. __________________ Santiago MÁRQUEZ DAMIÁN, Ph.D. Post-doctoral Fellow Research Center for Computational Mechanics (CIMEC) - CONICET/FICH-UNL T.E.: 54-342-4511594 Ext. 1005 Güemes 3450 - (3000) Santa Fe Santa Fe - Argentina http://www.cimec.org.ar Last edited by santiagomarquezd; November 7, 2014 at 06:25. Reason: Add equation

 April 14, 2015, 16:46 #8 Senior Member   anonymous Join Date: Aug 2014 Posts: 178 Rep Power: 3 I would like to comment out a bit, why does OpenFOAM use the expression given by santiagomarquezd (gracias por el análisis del código fuente) for the pressure surface normal gradient in the boundaries. First of all we need to write down the momentum equation discretized using the Rhie-Chow interpolation method. This reads: Where the f subscript means interpolate to the face, is the variable p_rgh whose value is , and the vector u is the velocity. Now multiply the equation by the face surface vector where the vector n is the surface normal: Thus the following equation for the pressure surface gradient can be obtained: Hope this is useful for you foamers zandi, PonchO, mgg and 3 others like this.

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+1 vote
146 views

The number of minutes spent by two students, $X$ and $Y$, exercising every day in a given week are shown in the bar chart above.

The number of days in the given week in which one of the students spent a minimum of $10\%$ more than the other student, on a given day, is

1. $4$
2. $5$
3. $6$
4. $7$

edited | 146 views

+1 vote

We can calculate the number of days in the given week in which one of the students spent more time than the other.

If $X$ is more than $10\%$ of $Y$ then $Y$ should be minimum and vice-versa.

• On Monday, the required percentage $= \left(\dfrac{70-45}{45}\right) \times 100\% = 55.55\%$
• On Tuesday, the required percentage $= \left(\dfrac{65-55}{55}\right) \times 100\% = 18.18\%$
• On Wednesday, the required percentage $= \left(\dfrac{60-50}{50}\right) \times 100\% = 20\%$
• On Thursday, the required percentage $= \left(\dfrac{60-55}{55}\right) \times 100\% = 9.09%$
• On Friday, the required percentage $= \left(\dfrac{35-20}{20}\right) \times 100\% = 75\%$
• On Saturday, the required percentage $= \left(\dfrac{60-50}{50}\right) \times 100\% = 20\%$
• On Sunday, the required percentage $= \left(\dfrac{65-55}{55}\right) \times 100\% = 18.18\%$

$\therefore$ Total $6$ days are there, when one of the students spent a minimum of $10\%$ more than the other student.

So, the correct answer is $(C).$

by (3.8k points)
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text:
# On non-normal non-abelian subgroups of finite groups

Document Type: Research Paper

Author

School of Mathematics and Information Science, Yantai University, Yantai 264005, China.

Abstract

In this paper we prove that a finite group $G$ having at most three conjugacy classes of non-normal non-abelian proper subgroups is always solvable except for $G\cong{\rm{A_5}}$, which extends Theorem 3.3 in [Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable, Acta Math. Sinica (English Series) 27 (2011) 891--896.]. Moreover, we show that a finite group $G$ with at most three same order classes of non-normal non-abelian proper subgroups is always solvable except for $G\cong{A_5}$.

Keywords

Main Subjects

### References

V.A. Belonogov, Finite groups with three classes of maximal subgroups, Mat. sb. 131 (1986), no. 2, 225--239.

J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.

S. Li, J. Shi and X. He, Some necessary and sufficient conditions of abelian groups and cyclic groups (Chinese), Guangxi Sciences 13 (2006), no. 1, 1--3.

G.A. Miller and H.C. Moreno, Non-abelian groups in which every subgroup is abelian, Trans. Amer. Math. Soc. 4 (1903), no. 4, 398--404.

J. Shi and C. Zhang, The inuence of some quantitative properties of abelian subgroups on solvability of finite groups, Comm. Algebra 39 (2011), no. 10, 3916--3922.

J. Shi and C. Zhang, Some sufficient conditions on the number of non-abelian subgroups of a finite group to be solvable, Acta Math. Sin. (Eng. Ser.) 27 (2011), no. 5, 891--896.

J. Shi and C. Zhang, Finite groups in which the number of classes of non-nilpotent proper subgroups of the same order is given (Chinese), Chinese Ann. Math. Ser. A 32 (2011), no. 6, 687--692.

H. Zassenhaus, A group-theoretic proof of a theorem of Maclagan-Wedderburn, Proc. Glasgow Math. Assoc. 1 (1952) 54--63.

### History

• Receive Date: 12 November 2014
• Revise Date: 03 January 2016
• Accept Date: 05 January 2016


text:
Hi!

Your F# code seems to be missing creating the zoom behavior. This is straight from your JS:

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3

        let zoomed = new System.Action(fun  _ -> g.Attr("transform", "translate("+D3.Event?translate+")") |> ignore)
let zoom = D3.Behavior.Zoom().ScaleExtent(1., 8.).On(ZoomType.Zoom, zoomed)
svg.Call(As<Function> zoom) |> ignore

This also shows some lack in the .NET bindings, the Call method does not have an overload that would take a Zoom object (which seems it is also a function in JS), or the Zoom type could inherit from WebSharper.JavaScript.Function. I think either would make sense, let me know what you think would be the better experience to explore the API.

By on 11/4/2018 2:32 AM ()

This worked perfectly - thanks sorry I didn't see it earlier

By on 11/8/2018 2:13 AM ()

Where does that error come from? It indicates that you have an inline that doesn't parse correctly, or? Can you post your snippet on Try WebSharper?

By on 11/2/2018 9:39 PM ()

Hi

I just commented things our so the compile works - I have put it here

https://github.com/jpalmer/Grib-Sharp/tree/websharper

in the websharper-viewer folder

Thanks

By on 11/3/2018 12:10 AM ()

I am not getting the compiler error you referenced, after uncommenting the two offending lines the project compiles fine, but throws a JS runtime error at runtime:

1

Uncaught TypeError: Cannot read property 'translate' of null

I see you have source maps enabled in your project, so it should pinpoint the location precisely:

I am not well versed in D3, but you can find the D3 bindings in the WebSharper.D3 repo. Here, L1131 gives you D3.Event, which according to the binding has type Dom.Event. I don't see how in your code this is bound to a valid event (and it's null as the error above indicates), nor how that event would be cast to a type that has a translate member.

All in all, this looks more like an issue with how you access/use D3. But if you believe the binding needs a fix (and it very well might as it's quite outdated - v3 according to the manifest), send a PR - always happy to integrate.

By on 11/3/2018 6:00 PM ()

Hi,

So I am really not an expert enough - it seems like I need something like

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    let zoom =new System.Action(fun () ->
g.Attr("transform", "translate("+D3.Event.JS?translate+")") |> ignore
)
D3.Behavior.Zoom().ScaleExtent(1.,8.).On(ZoomType.Zoom,zoom) |> ignore
svg.Call(zoom)

Except the problem is that svg.Call wants an Action<t> whilst On just wants an Action.

Since these types are not compatible there doesn't seem to be a type safe way around this.

Is it possible to just write this zoom part in pure JS somehow and bypass the websharper d3 binding somehow?

By on 11/6/2018 2:49 AM ()

Hi, I already have a response below.

My code there has two differences: I am passing the result of Behavior.Zoom() to the Call (as in your JS code), but I still needed to use an unsafe cast. I need to check into D3 too, to see if this is indeed a JS function object, so maybe Zoom type could inherit from JavaScript.Function to solve this typing problem.

The quickest way to have some pure JS is to use the JS.Inline function in WebSharper. Example: JS.Inline<int>("1 + 1"). You can use $0, $1... placeholders to pass in values like JS.Inline("1 + \$0"). See documentation here for full possibilities on JavaScript interop.

By on 11/6/2018 3:29 AM ()


text:
HEP – GR Seminar Summer 2022

Prof. S. Hollands
Prof. R. Verch

Organization:

Dr. A. Much

Dr. J. Zahn

Tuesday, 15:15 – 16:45 (1 hour talk, followed by questions), seminar room 114 ITP
» how to reach the ITP
.

This term, the seminar will be in hybrid form, for online participation click the following zoom link, virtual meeting room  (Passcode: 566288).

Schedule for the summer term 2022:

Date Speaker Title (hover/click for abstract)
19.04. Adam Pound (University of Southampton)
26.04. Max Ruepp (University of York)
03.05. Alessio Ranallo (University of Roma)
17.05. Christoph Minz (University of Leipzig)
24.05.
31.05. Robert Jonsson (MPI of Quantum Optics)
07.06. Alexandre Le Tiec (Observatoire de Paris)
21.06. Gerardo Morsella (Università di Roma Tor Vergata)
28.06. Jochen Zahn (University of Leipzig)
05.07. Inti Sodemann (University of Leipzig)
12.07. Marc Casals (University of Leipzig)
02.08. Yoh Tanimoto (Università di Roma Tor Vergata)

Abstract: As gravitational-wave detectors become more sensitive to lower frequencies, they will increasingly detect binaries with smaller mass ratios, larger spins, and higher eccentricities. In this talk I describe how gravitational self-force theory, when combined with a recently developed method of multiscale expansions, provides an ideal framework for modelling these systems. The framework proceeds from first principles while simultaneously enabling rapid generation of waveforms on a timescale of milliseconds. I present the first waveforms calculated at second perturbative order in this framework. While self-force methods were originally motivated by extreme-mass-ratio inspirals with mass ratios $\sim 10^4–10^7$, these new results show that multiscale self-force models can be highly accurate for all mass ratios above $\sim 10$.
Abstract: As emphasised in the seminal 1964 Haag-Kastler paper, elements of local algebras in AQFT may be regarded as quantum operations. However, as argued in Sorkin's 1993 "impossible measurements" paper, quantum operations induced by typical elements enable superluminal signaling. In this talk I will argue for the interpretation of (self-adjoint) elements as observables. In particular, I will review the notion of a measurement scheme from quantum measurement theory, Fewster and Verch's adaption to AQFT and I will introduce asymptotic measurement schemes. Then, for the example of a linear real scalar field on a globally hyperbolic spacetime, I will show how to construct asymptotic measurement schemes for every quantum observable. This talk is based on arXiv:2203.09529, joint work with C. J. Fewster and I. Jubb.
Abstract: Bayes' theorem is a fundamental tool in classical (i.e. commutative) probability theory, so it is of interest to find generalised instances of it in noncommutative probability. Among the most stimulating directions of research in this sense is the attempt to algebraically "synthesize" the axioms of probability theory, looking for categorical formulations of them. In this talk, after an overview of the aforementioned approach, which will lead us directly to (a) definition of "Bayesian inversion", I will discuss its connections with the field of operator algebras, in particular with Tomita-Takesaki's modular theory.
Abstract: The Cauchy slicings for globally hyperbolic spacetimes are surveyed and revisited, including techniques and adaptive possibilities of the slicings. Simple counterexamples on $R^2$ will be constructed showing a variety of possibilities: (a) logical independence (for normalized sliced spacetimes) between the completeness of the slices and global hyperbolicity, (b) necessity of uniform bounds on the slicings in order to ensure global hyperbolicity, or (c) non convexity of the space of all the (normalized, conformal classes of) globally hyperbolic metrics on a manifold. Further possibilities for globally hyperbolic spacetimes-with-timelike-boundary and causal boundaries will be also pointed out. Based on arXiv:2110.13672.
Abstract: Geometric quantization is a natural way to construct the quantum algebra starting from classical data given as a symplectic manifold. When the quantization admits an adjoint operation, referred to as dequantization, we may use this operation to equip the quantum algebra with a distinguished state determined by dequantization. As an example, I show the construction for the free scalar field over a (finite) causal set (locally finite, partially ordered set), where the starting point is a symplectic vector space with an inner product. I compare the result with the construction due to Sorkin, which starts from the same input data, and I show that the distinguished state coincides with the Sorkin-Johnson state, provided the inner product is chosen appropriately. This talk is based on joint work with Eli Hawkins and Kasia Rejzner.
Abstract:
Abstract: Gaussian quantum states play a central role in many branches of physics – from quantum optics, to condensed matter and quantum field theory. In this talk, I aim to showcase the strength of the Kähler structure formalism for Gaussian states by discussing a recent result on the entanglement structure of supersymmetric (SUSY) bosonic and fermionic Gaussian states [1]. Mathematically, Gaussian states can be defined in terms of Kähler structures on classical phase space. In fact, this approach has proven to be very powerful: It yields a formalism which is both practical for applications, clearly captures the structure and geometry of Gaussian states, adapts to discrete and continuous settings and, moreover, can treat bosons and fermions simultaneously. To exemplify this, we will consider the basic example of a free SUSY system. This is a pair of one bosonic and one fermionic quadratic hamiltonian which is generated by a supercharge and, therefore, is isospectral. Not only does the Kähler structure formalism parallelly capture the Gaussian ground states and their entanglement structure of both the bosonic and the fermionic part. Moreover, it allows us to derive an appealing entanglement duality between bosonic and fermionic subsystems [1], and to interpret it in terms of phase space geometry and its physical implications. Time permitting, as a special application, we consider topological insulators and superconductors and their SUSY partners, discussing the recently derived classification of supercharges in this context [2]. [1] Jonsson, Robert H., Lucas Hackl, and Krishanu Roychowdhury. “Entanglement Dualities in Supersymmetry.” Physical Review Research 3, no. 2 (June 16, 2021): 023213. [2] Gong, Zongping, Robert H. Jonsson, and Daniel Malz. “Supersymmetric Free Fermions and Bosons: Locality, Symmetry, and Topology.” Physical Review B 105, no. 8 (February 24, 2022): 085423.
Abstract:
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Abstract: The Adler-Bardeen theorem states that if gauge anomalies are absent at the one-loop level, then they are absent to all orders. It is crucial for establishing the consistency of quantised gauge theories, such as the Standard Model or extensions thereof. I present a proof which is based on background independence, as formalised in the concept of perturbative agreement.
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