content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
Everly Taylor | Prealgebra Tutor on HIX Tutor
Prealgebra teacher | Verified Expert
With a focus on prealgebra, my journey at Austin College equipped me with the tools to demystify math for students. I'm passionate about breaking down complex concepts into digestible bits, fostering
a deeper understanding. Through patient guidance and tailored approaches, I strive to instill confidence in my students, empowering them to excel in math. Let's embark on this mathematical journey
together, where every question is an opportunity to learn and grow. | {"url":"https://tutor.hix.ai/tutors/everly-taylor","timestamp":"2024-11-09T23:26:10Z","content_type":"text/html","content_length":"562452","record_id":"<urn:uuid:85b2efa3-8f37-4b83-9daf-9d8e33a46717>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00083.warc.gz"} |
uses of mean, mode and median. – Q&A Hub – 365 Data Science
Resolved: uses of mean, mode and median.
could you please describe what are the uses of mode and median?
1 answers ( 1 marked as helpful)
Hi Mujeeb,
Mean, median, and mode are measures of central tendency. They give you an idea of how the data in a given dataset is distributed. The mean is the arithmetic average of all numbers. It is very useful
because it indicates the average value in the dataset. However, the mean can be flawed because outliers might impact it significantly. The mode is the value that is observed most frequently in the
distribution. This gives you an idea about the value that reoccurs most often in the dataset. The median is a value at the 50th percentile of the distribution.It disregards outliers and shows you
what is in the middle of the distribution.
Hope this helps! | {"url":"https://365datascience.com/question/uses-of-mean-mode-and-median/","timestamp":"2024-11-10T07:59:58Z","content_type":"text/html","content_length":"108794","record_id":"<urn:uuid:4fc7e35e-91d6-4b5a-a31d-d809e828548f>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00899.warc.gz"} |
Hy Storyboard by 9b201965
Storyboard Text
• On the other hand, when the Population Variance is Unknown, the appropriate test statistic to used is t-test.
• What is a t-test?A t-test is a type of inferential statistic used to determine if there is a significant difference between the means of two groups, which may be related in certain features.
• REMEMBER:T-test is the appropriate tool to used when:The population standard deviation (σ) is unknown.The sample standard deviation (s) is known.The population is normal or nearly normally
distributed.The sample size is less than 30 (n30).
• Lastly, when the Central Limit Theorem (CLT) is used, the appropriate test statistic is using z-test by replacing population standard deviation (σ) by sample standard deviation (s) in the
• REMEMBER:Z-test is the appropriate tool to used in central limit theorem when:If the population is normally distributed or the sample size is large and the true population means μ = μo, the z has
a standard normal distribution.The sample size is extremely large and the variance (σ2) is known or unknown.The sample standard deviation (s) may be used as an estimate of the population standard
deviation (σ) when the value of σ is unknown.
• Z-testn ≥ 30, σ is knownn ≥ 30, σ is unknown (Central Limit Theorem)n 30, σ is known
• So, that is all for our lesson today. Thank you for listening. And, see you all on our next meeting. Goodbye class!
• T-testn 30, σ is not knownn ≥ 30, s is known | {"url":"https://www.storyboardthat.com/storyboards/9b201965/hy","timestamp":"2024-11-05T21:41:40Z","content_type":"text/html","content_length":"425119","record_id":"<urn:uuid:9abff4be-659f-4129-b914-8f178ca87e73>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00556.warc.gz"} |
Lesson 1
Relationships of Angles
Let’s examine some special angles.
Problem 1
Here are questions about two types of angles.
1. Draw a right angle. How do you know it's a right angle? What is its measure in degrees?
2. Draw a straight angle. How do you know it’s a straight angle? What is its measure in degrees?
Problem 2
An equilateral triangle’s angles each have a measure of 60 degrees.
1. Can you put copies of an equilateral triangle together to form a straight angle? Explain or show your reasoning.
2. Can you put copies of an equilateral triangle together to form a right angle? Explain or show your reasoning.
Problem 3
Here is a square and some regular octagons.
In this pattern, all of the angles inside the octagons have the same measure. The shape in the center is a square. Find the measure of one of the angles inside one of the octagons.
Problem 4
The height of the water in a tank decreases by 3.5 cm each day. When the tank is full, the water is 10 m deep. The water tank needs to be refilled when the water height drops below 4 m.
1. Write a question that could be answered by solving the equation \(10-0.035d=4\).
2. Is 100 a solution of \(10-0.035d>4\)? Write a question that solving this problem could answer.
(From Unit 6, Lesson 17.)
Problem 5
Use the distributive property to write an expression that is equivalent to each given expression.
1. \(\text-3(2x-4)\)
2. \(0.1(\text-90+50a)\)
3. \(\text-7(\text-x-9)\)
4. \(\frac45(10y+\text-x+\text-15)\)
(From Unit 6, Lesson 18.)
Problem 6
Lin’s puppy is gaining weight at a rate of 0.125 pounds per day. Describe the weight gain in days per pound.
(From Unit 2, Lesson 3.) | {"url":"https://im.kendallhunt.com/MS/students/2/7/1/practice.html","timestamp":"2024-11-12T15:56:59Z","content_type":"text/html","content_length":"67504","record_id":"<urn:uuid:1a28aec9-cab8-4bb9-8968-4c94808526dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00055.warc.gz"} |
"Singularities don't exist," claims black hole pioneer Roy Kerr
Starts With A Bang —
“Singularities don’t exist,” claims black hole pioneer Roy Kerr
The brilliant mind who discovered the spacetime solution for rotating black holes claims singularities don’t physically exist. Is he right?
Key Takeaways
• Way back in 1963, Roy Kerr became the first person to write down the exact solution, in general relativity, for a realistic, rotating black hole. 60 years later, it’s still used everywhere.
• Although Roger Penrose won the Nobel Prize in physics just a few years ago for demonstrating how black holes come to exist in our Universe, singularities and all, the subject isn’t closed.
• We’ve never peered beneath the event horizon, and have no way of detecting what’s inside. Using a powerful mathematical argument, Kerr argues that singularities shouldn’t physically exist. He may
be right.
Sign up for the Starts With a Bang newsletter
Travel the universe with Dr. Ethan Siegel as he answers the biggest questions of all
Here in our Universe, whenever you gather enough mass together in a small enough volume of space, you’re bound to eventually cross a threshold: where the speed at which you’d need to travel to escape
the gravitational pull within that region exceeds the speed of light. Whenever that occurs, it’s inevitable that you’ll form an event horizon around that region, which looks, acts, and behaves
exactly like a black hole as seen from the outside. Meanwhile, inside, all that matter gets inexorably drawn toward the central region inside that black hole. With finite amounts of mass compressed
to an infinitesimal volume, the existence of a singularity is all but assured.
The predictions for what we should observe outside the event horizon match extraordinarily well with observations, as we’ve not only seen many luminous objects in orbit around black holes, but have
even now imaged the event horizons of multiple black holes directly. The theorist who laid the foundation for how realistic black holes form in the Universe, Roger Penrose, subsequently won the Nobel
Prize in Physics in 2020 for his contributions to physics, including for the notion that a singularity must exist at the center of every black hole.
But in a surprising twist, the legendary physicist who discovered the spacetime solution for rotating black holes — Roy Kerr, way back in 1963 — has just written a new paper challenging that idea
with some very compelling arguments. Here’s why, perhaps, singularities may not exist within every black hole, and what the key issues are that we should all be thinking about.
Making an ideal black hole
If you want to make a black hole, in Einstein’s general relativity, all you have to do is take any distribution of pressureless mass — what relativists call “dust” — that starts in the same vicinity
and is initially at rest, and let it gravitate. Over time, it will contract down and down and down to smaller volumes, until an event horizon forms at a specific distance from the center: dependent
solely on the total amount of mass that you began with. This produces the simplest type of black hole known: a Schwarzschild black hole, which has mass, but no electric charge or angular momentum.
Einstein first put forth general relativity, in its final form, in late 1915. Just two months later, in early 1916, Karl Schwarzschild had worked out the mathematical solution for a spacetime that
corresponds to this situation: a spacetime that’s completely empty except for one point-like mass. In reality, the matter in our Universe isn’t pressureless dust, but rather is made of atoms and
subatomic particles. Nevertheless, through realistic processes like:
• the core-collapse of massive stars,
• the mergers of two massive-enough neutron stars,
• or the direct collapse of a large amount of matter, either stellar or gaseous,
black holes certainly form in our Universe. We’ve observed them, and we’re certain they exist. However, a big mystery remains: what happens inside of them, in their interiors, where we cannot
The argument for a singularity
There’s a simple argument you can make to understand why we think that all black holes, at least under the Schwarzschild set of assumptions, ought to have a singularity at their centers. Imagine
you’ve crossed over the event horizon, and are now on the “inside” of the black hole. Where can you go from here?
• If you fire your thrusters directly at the singularity, you’ll just get there faster, so that’s no good.
• If you fire your thrusters perpendicular to the direction of the singularity, you’ll still get drawn inward, and there’s no way to get farther from the singularity.
• And if you fire your thrusters directly away from the singularity, you’ll find that you’re still approaching the singularity faster and faster as time goes on.
The reason why? Because space itself is flowing: like a waterfall or a moving walkway beneath your feet. Even if you speed yourself up so that you’re moving arbitrarily close to the speed of light,
the rate at which space is flowing is so great that no matter which direction you move in, the singularity appears to be “down” in all directions. You can draw out the shape of where you’re allowed
to go, and even though it forms a mathematically interesting structure known as a cardioid, all paths lead to you winding up at the center of this object. Given enough time, these black holes should
all have a singularity at their centers.
The Kerr advance: adding rotation
But here in the real Universe, the ideal case of having a mass with no rotation to it isn’t exactly a good physical model of reality. Consider that:
• there are many masses in the Universe,
• these masses, over time, gravitationally attract one another,
• causing them to move relative to one another,
• which leads to the clumping and clustering of matter in a non-uniform way,
• and that as clumps of matter move relative to one another and gravitationally interact, they’ll exert not just forces but torques on one another,
• that torques cause rotation,
• and that as rotating objects collapse, their rotation rate increases due to the conservation of angular momentum,
it makes sense that all physically realistic black holes would be rotating.
It turns out that while asking the question of what does a spacetime look like if you have only a single point mass in your Universe is a relatively straightforward problem to solve in Einstein’s
general relativity — after all, Karl Schwarzschild solved it in just a couple of months — the question of what spacetime looks like if you have a mass that rotates is much more complicated. Indeed,
many brilliant physicists worked on this problem and were unable to solve it: for months, years, and even decades.
But then, in 1963, New Zealand physicist Roy Kerr finally cracked it. His solution for the spacetime describing realistic, rotating black holes — the Kerr metric — has been the gold standard for what
relativists have used to describe it ever since.
Rotation and reality
When you add rotation in, the situation for how spacetime behaves suddenly becomes a lot more complicated than it was in the non-rotating case. Instead of a spherical event horizon marking the
delineation between where it’s possible to escape the black hole (outside) versus where escape is impossible (inside), and instead of all “inside” paths leading to a singularity at the center, the
mathematical structure of a rotating (Kerr) black hole looks extremely different.
Instead of a single, spherical surface describing the event horizon and a point-like singularity at the center, the addition of rotation causes there to be several important phenomena that aren’t
apparent in the non-rotating case.
• Instead of a single solution for the location of the event horizon, as in the Schwarzschild case, the equation you wind up with in the Kerr case is quadratic, giving two separate solutions: an
“outer” and “inner” event horizon.
• Instead of the event horizon marking the location where the timelike component of the metric flips sign, there are now two surfaces that are different from the inner and outer event horizons —
the inner and outer ergospheres — that delineate those locations throughout space.
• And instead of a zero-dimensional, point-like singularity at the center, the angular momentum present smooths that singularity into a one-dimensional surface: a ring, with the rotational axis of
the black hole passing perpendicular through the center of the ring.
This leads to a variety of, shall we say, less-than-intuitive effects that occur within a Kerr spacetime that don’t occur within a Schwarzschild (non-rotating) spacetime.
Because the metric itself has an intrinsic rotation to it and couples to all of space outside the event horizons and ergospheres, all outside inertial reference frames will experience an induced
rotation: a frame-dragging effect. This is similar to electromagnetic induction, but for gravitation.
Because of the non-spherically symmetric nature of the system, where we now have one of our three spatial dimensions representing a rotational axis and where there’s a direction (clockwise or
counterclockwise, for example) to that rotation, a particle that orbits this black hole won’t make a closed ellipse that remains in the same plane (or a slowly-decaying and precessing ellipse, if you
account for all of general relativity’s effects), but rather will move throughout all three dimensions, eventually filling in a volume enclosed by a torus.
And, perhaps most importantly, if you track the evolution of any particle that falls into this object from outside, it won’t simply cross over to the inside of the horizon and head inexorably toward
the central singularity. Instead, other important effects occur that may work to “freeze” these particles in place, or to otherwise prevent them from traveling all the way to the theoretical “ring”
singularity at the center. That’s where we owe it to ourselves to take a good look at what Roy Kerr, who’s been thinking about this puzzle for longer than anyone else alive, has to say about it.
Revisiting the argument for a singularity
The biggest argument for why a singularity must exist inside of black holes comes from two titanic figures in 20th century physics: Roger Penrose and Stephen Hawking.
1. The first part of the argument, from Penrose alone, is that wherever you have what’s called a trapped surface — a boundary from which nothing physical can escape, e.g., an event horizon — any
light rays interior to that trapped surface will possess a mathematical property known as having finite affine length.
2. This “finite affine length light,” or FALL, for each light ray then implies that the light must terminate in an actual singularity, which is the second part of the argument from Penrose and
3. You can then show that any object that enters the region between the outer and inner event horizons must “fall through” to the interior.
4. And, because you need a source to generate the spacetime, the existence of a ring singularity is required.
At least, that’s how the traditional argument goes. The third and fourth parts of the argument are airtight in general relativity: if parts one and two are true, then you need a singularity at the
core. But are parts one and two both true? That’s where Kerr’s new paper comes into play, asserting that no, this is a mistake that we’ve been making for over half-a-century.
What Kerr showed is that if you go all the way back to his original, generalized coordinate formulation for Kerr black holes, the Kerr-Schild coordinates, through every single point in the interior
of the Kerr black hole, you can draw light rays that are:
• tangential (i.e., approach but do not intersect) to one of the two event horizons,
• do not have endpoints (i.e., they continue to travel forever),
• and yet still have finite affine lengths (i.e., they are FALLs).
Moreover, if you ask the key question, “How common are these light rays?” the answer is that there are an infinite number of them, and that half of these rays are in the region between the two event
horizons, with at least two through every point in that region.
The problem, as Kerr has been able to show, is with point #2 in the aforementioned argument. Sure, you have a trapped surface in Kerr spacetime, and all the light rays within that trapped surface
have finite affine length. But is that light required to terminate in a singularity? Not at all. In fact, by demonstrating the presence of these light rays that are tangential to an event horizon and
that do not have endpoints, he has provided a counterexample to that notion. In Kerr’s own words:
“It has not been proved that a singularity, not just a FALL, is inevitable when an event horizon forms around a collapsing star.”
The problem with Hawking & Penrose
It’s kind of remarkable, if you go back in history, to realize how much of our acceptance of the existence of a singularity depends on an unproven assertion. Back in 1970, Hawking and Penrose wrote a
paper called The singularities of gravitational collapse and cosmology, and within it note that there are other possibilities to consider than the traditional (curvature) singularities when it comes
to realistic black holes.
With the disproof that Kerr has demonstrated, some people have instead asserted that you need to consider the maximal extensions of the Kerr space, and you’ll find the need for a singularity there.
For example, in the Boyer-Lindquist extension of the Kerr spacetime, you have a collection of copies of the separate parts of the original Kerr metric, and because there are no interior collapsed
stars inside, it’s certain to be singular.
But again, as Kerr points out, you must assume that each interior section of the spacetime, even in the Boyer-Lindquist extension, contains a (collapsed) star within it, and therefore encounters the
same problem. Other extensions (such as Kruskal) have been proposed, but Kerr shot those attempts to evade this problem down as well, by demonstrating that Kerr is its own maximal extension. As Kerr
puts it:
“These extensions may be analytic, but at best they are constructed using copies of the original spaces together with some fixed points. These will be nonsingular inside each copy of the original
interior if the same is true inside the original Kerr and therefore the extensions are irrelevant to the singularity theorems. Anyone who does not believe this needs to supply a proof. They are all
physically irrelevant since real black holes start at a finite time in the past with the collapse of a star or similar over-dense concentration of matter, not as the white hole of the Kruskal or
Boyer-Lindquist extensions.”
Put simply: a FALL does not necessarily mean a singularity, and Kerr chalks the confusion up to physicists conflating geodesic distance/length with affine distance/length: two concepts that are not,
in fact, identical. Kerr also points out that if there were a nonsingular object, like a stretched-out neutron star corpse, inside the Kerr black hole, it too would generate the Kerr spacetime we
observe. In other words, there are good reasons to revisit the notion that a singularity must exist inside each realistic, rotating black hole.
Final thoughts
We have to remember an important aspect of general relativity that almost everyone — laypersons and physicists alike — often overlook: “general relativity is about forces, not geometry.” The person
who said that wasn’t some crackpot; it was Einstein himself. General relativity is not simply pure mathematics; it’s a description of the physical Universe, placed on a firm mathematical footing. You
can’t simply “write down a spacetime” and expect that to describe reality, you have to start from a physically motivated set of conditions and show how that spacetime solution (e.g., a rotating black
hole) comes to be. If the only way you can “prove” the existence of a singularity is by ignoring the physical creation of the object in the first place, your proof is not valid.
However, demonstrating a counterexample to your attempted proof, both physically and mathematically, is an excellent way to falsify any assertion that gets made. With Kerr’s latest work — a full 60
years after first deriving the Kerr metric — we have to reckon with the sober fact that our best “singularity theorems” that argue for their necessity at a realistic black hole’s center are based on
an invalid assumption.
Furthermore, once you cross over to be inside the inner event horizon in Kerr spacetime, it once again becomes possible to travel in any direction between the theorized ring singularity and the inner
event horizon. The “trapped surface” only exists between the inner and outer event horizons, not interior to the inner event horizon: where the ring singularity allegedly exists. Who knows what
exists in that region? The problem is that there are enormous numbers of mathematical solutions to this problem, and “a singularity” is only one of them. There might indeed yet be a singularity
inside, but there also may be something entirely different. Kerr, currently at the age of 89, has no problem telling us what he thinks, writing that he:
“has no doubt, and never did, that when relativity and quantum mechanics are melded it will be shown that there are no singularities anywhere. When theory predicts singularities, the theory is wrong!
What we can be certain of is that the long-accepted “proof,” that rotating black holes must have singularities, can’t be counted on any longer. (You can download and read Kerr’s latest paper for free
Sign up for the Starts With a Bang newsletter
Travel the universe with Dr. Ethan Siegel as he answers the biggest questions of all
From the Big Bang to black holes, singularities are hard to avoid. The math definitely predicts them, but are they truly, physically real?
Yes, “the laws of physics break down” at singularities. But something really weird must have happened for black holes to not possess them.
We have two descriptions of the Universe that work perfectly well: General Relativity and quantum physics. Too bad they don’t work together.
The matter that creates black holes won’t be what comes out when they evaporate. Will the black hole information paradox ever be solved?
The inside of every black hole leads to the birth of a new Universe. Could our Universe have arisen from one?
Some neuroscientists question whether the body can “keep score” of anything in a meaningful way. | {"url":"https://bigthink.com/starts-with-a-bang/singularities-dont-exist-roy-kerr/","timestamp":"2024-11-07T16:56:34Z","content_type":"text/html","content_length":"194633","record_id":"<urn:uuid:bb0d1825-983b-43d0-a067-524990a65320>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00420.warc.gz"} |
Source code for dynamo.vectorfield.vector_calculus
# from tqdm import tqdm
# from anndata._core.views import ArrayView
# import scipy.sparse as sp
from typing import Dict, List, Optional, Union
from typing import Literal
except ImportError:
from typing_extensions import Literal
import numpy as np
import pandas as pd
from anndata._core.anndata import AnnData
from ..dynamo_logger import (
from ..tools.sampling import sample
from ..tools.utils import (
from ..utils import isarray, ismatrix
from ..vectorfield import scVectorField
from .scVectorField import SvcVectorField
from .utils import (
import dynode
use_dynode = "vectorfield" in dir(dynode)
except ImportError:
use_dynode = False
if use_dynode:
from .scVectorField import dynode_vectorfield
from .utils import dynode_vector_field_function
def get_vf_class(adata: AnnData, basis: str = "pca") -> SvcVectorField:
"""Get the corresponding vector field class according to different methods.
adata: AnnData object that contains the reconstructed vector field in the `uns` attribute.
basis: The embedding data in which the vector field was reconstructed.
SvcVectorField object that is extracted from the `uns` attribute of adata.
vf_dict = get_vf_dict(adata, basis=basis)
if "method" not in vf_dict.keys():
vf_dict["method"] = "sparsevfc"
if vf_dict["method"].lower() == "sparsevfc":
vector_field_class = SvcVectorField()
vector_field_class.from_adata(adata, basis=basis)
elif vf_dict["method"].lower() == "dynode":
vf_dict["parameters"]["load_model_from_buffer"] = True
vector_field_class = vf_dict["dynode_object"] # dynode_vectorfield(**vf_dict["parameters"])
raise ValueError("current only support two methods, SparseVFC and dynode")
return vector_field_class
[docs]def velocities(
adata: AnnData,
init_cells: Optional[List] = None,
init_states: Optional[list] = None,
basis: Optional[str] = None,
vector_field_class: Optional[scVectorField.BaseVectorField] = None,
layer: Optional[str] = "X",
dims: Optional[Union[int, list]] = None,
Qkey: str = "PCs",
) -> AnnData:
"""Calculate the velocities for any cell state with the reconstructed vector field function.
adata: AnnData object that contains the reconstructed vector field function in the `uns` attribute.
init_cells: Cell name or indices of the initial cell states for the historical or future cell state prediction with
numerical integration. If the names in init_cells are not found in the adata.obs_name, they will be treated as
cell indices and must be integers.
init_states: Initial cell states for the historical or future cell state prediction with numerical integration.
basis: The embedding data to use for calculating velocities. If `basis` is either `umap` or `pca`, the
reconstructed trajectory will be projected back to high dimensional space via the `inverse_transform`
vector_field_class: If not None, the speed will be computed using this class instead of the vector field stored in adata. You
can set up the class with a known ODE function, useful when the data is generated through simulation.
layer: Which layer of the data will be used for predicting cell fate with the reconstructed vector field function.
The layer once provided, will override the `basis` argument and this function will then predict cell fate in high
dimensional space.
dims: The dimensions that will be selected for velocity calculation.
Qkey: The key of the PCA loading matrix in `.uns`. Only used when basis is `pca`.
AnnData object that is updated with the `"velocities"` related key in the `.uns`.
if vector_field_class is None:
vector_field_class = get_vf_class(adata, basis=basis)
init_states, _, _, _ = fetch_states(adata, init_states, init_cells, basis, layer, False, None)
if vector_field_class.vf_dict["normalize"]:
xm, xscale = vector_field_class.norm_dict["xm"][None, :], vector_field_class.norm_dict["xscale"]
init_states = (init_states - xm) / xscale
vec_mat = vector_field_class.func(init_states)
vec_key = "velocities" if basis is None else "velocities_" + basis
if np.isscalar(dims):
vec_mat = vec_mat[:, :dims]
elif dims is not None:
vec_mat = vec_mat[:, dims]
if basis == "pca":
adata.uns["velocities_pca"] = vec_mat
Qkey = "PCs" if Qkey is None else Qkey
if Qkey in adata.uns.keys():
Q = adata.uns[Qkey]
elif Qkey in adata.varm.keys():
Q = adata.varm[Qkey]
raise Exception(f"No PC matrix {Qkey} found in neither .uns nor .varm.")
vel = adata.uns["velocities_pca"].copy()
vel_hi = vector_transformation(vel, Q)
adata.uns[vec_key] = vec_mat
[docs]def speed(
adata: AnnData,
basis: Optional[str] = "umap",
vector_field_class: Optional[scVectorField.BaseVectorField] = None,
method: str = "analytical",
) -> AnnData:
"""Calculate the speed for each cell with the reconstructed vector field function.
adata: AnnData object that contains the reconstructed vector field function in the `uns` attribute.
basis: The embedding data in which the vector field was reconstructed.
vector_field_class: If not None, the speed will be computed using this class instead of the vector field stored in adata. You
can set up the class with a known ODE function, useful when the data is generated through simulation.
method: The method that will be used for calculating speed, either `analytical` or `numeric`. `analytical`
method will use the analytical form of the reconstructed vector field for calculating Jacobian. Otherwise,
raw velocity vectors are used.
AnnData object that is updated with the `'speed'` key in the `.obs`.
if vector_field_class is None:
vector_field_class = get_vf_class(adata, basis=basis)
X, V = vector_field_class.get_data()
if method == "analytical":
vec_mat = vector_field_class.func(X)
vec_mat = adata.obsm["velocity_" + basis] if basis is not None else vector_field_class.vf_dict["Y"]
speed = np.array([np.linalg.norm(i) for i in vec_mat])
speed_key = "speed" if basis is None else "speed_" + basis
adata.obs[speed_key] = speed
[docs]def jacobian(
adata: AnnData,
regulators: Optional[List] = None,
effectors: Optional[List] = None,
cell_idx: Optional[List] = None,
sampling: Optional[Literal["random", "velocity", "trn"]] = None,
sample_ncells: int = 1000,
basis: str = "pca",
Qkey: str = "PCs",
vector_field_class: Optional[scVectorField.BaseVectorField] = None,
method: str = "analytical",
store_in_adata: bool = True,
"""Calculate Jacobian for each cell with the reconstructed vector field.
If the vector field was reconstructed from the reduced PCA space, the Jacobian matrix will then be inverse
transformed back to high dimension. Note that this should also be possible for reduced UMAP space and will be
supported shortly. Note that we compute the Jacobian for the RKHS kernel vector field analytically,
which is much more computationally efficient than the numerical method.
adata: AnnData object that contains the reconstructed vector field in `.uns`.
regulators: The list of genes that will be used as regulators when calculating the cell-wise Jacobian matrix. The
Jacobian is the matrix consisting of partial derivatives of the vector field wrt gene expressions. It can be
used to evaluate the change in velocities of effectors (see below) as the expressions of regulators
increase. The regulators are the denominators of the partial derivatives.
effectors: The list of genes that will be used as effectors when calculating the cell-wise Jacobian matrix. The
effectors are the numerators of the partial derivatives.
cell_idx: A list of cell index (or boolean flags) for which the jacobian is calculated.
If `None`, all or a subset of sampled cells are used.
sampling: {None, 'random', 'velocity', 'trn'}, (default: None)
See specific information on these methods in `.tl.sample`.
If `None`, all cells are used.
sample_ncells: The number of cells to be sampled. If `sampling` is None, this parameter is ignored.
basis: The embedding data in which the vector field was reconstructed. If `None`, use the vector field function
that was reconstructed directly from the original unreduced gene expression space.
Qkey: The key of the PCA loading matrix in `.uns`.
vector_field_class: If not `None`, the jacobian will be computed using this class instead of the vector field stored in adata.
method: The method that will be used for calculating Jacobian, either `'analytical'` or `'numerical'`.
`'analytical'` method uses the analytical expressions for calculating Jacobian while `'numerical'` method
uses numdifftools, a numerical differentiation tool, for computing Jacobian. `'analytical'` method is much
more efficient.
cores: Number of cores to calculate Jacobian. If cores is set to be > 1, multiprocessing will be used to
parallel the Jacobian calculation.
kwargs: Any additional keys that will be passed to `elementwise_jacobian_transformation` function.
AnnData object that is updated with the `'jacobian'` key in the `.uns`. This is a 3-dimensional tensor with
dimensions n_effectors x n_regulators x n_obs.
if vector_field_class is None:
vector_field_class = get_vf_class(adata, basis=basis)
if basis == "umap":
cell_idx = np.arange(adata.n_obs)
X, V = vector_field_class.get_data()
if cell_idx is None:
if sampling is None or sampling == "all":
cell_idx = np.arange(adata.n_obs)
cell_idx = sample(np.arange(adata.n_obs), sample_ncells, sampling, X, V)
Jac_func = vector_field_class.get_Jacobian(method=method)
Js = Jac_func(X[cell_idx])
if regulators is None and effectors is not None:
regulators = effectors
elif effectors is None and regulators is not None:
effectors = regulators
if regulators is not None and effectors is not None:
if type(regulators) is str:
if regulators in adata.var.keys():
regulators = adata.var.index[adata.var[regulators]]
regulators = [regulators]
if type(effectors) is str:
if effectors in adata.var.keys():
effectors = adata.var.index[adata.var[effectors]]
effectors = [effectors]
regulators = np.unique(regulators)
effectors = np.unique(effectors)
var_df = adata[:, adata.var.use_for_dynamics].var
regulators = var_df.index.intersection(regulators)
effectors = var_df.index.intersection(effectors)
reg_idx, eff_idx = (
get_pd_row_column_idx(var_df, regulators, "row"),
get_pd_row_column_idx(var_df, effectors, "row"),
if len(regulators) == 0 or len(effectors) == 0:
raise ValueError(
"Either the regulator or the effector gene list provided is not in the dynamics gene list!"
if basis == "pca":
if Qkey in adata.uns.keys():
Q = adata.uns[Qkey]
elif Qkey in adata.varm.keys():
Q = adata.varm[Qkey]
raise Exception(f"No PC matrix {Qkey} found in neither .uns nor .varm.")
Q = Q[:, : X.shape[1]]
if len(regulators) == 1 and len(effectors) == 1:
Jacobian = elementwise_jacobian_transformation(
Js, Q[eff_idx, :].flatten(), Q[reg_idx, :].flatten(), **kwargs
Jacobian = subset_jacobian_transformation(Js, Q[eff_idx, :], Q[reg_idx, :], **kwargs)
Jacobian = Js.copy()
Jacobian = None
ret_dict = {"jacobian": Js, "cell_idx": cell_idx}
# use 'str_key' in dict.keys() to check if these items are computed, or use dict.get('str_key')
if Jacobian is not None:
ret_dict["jacobian_gene"] = Jacobian
if regulators is not None:
ret_dict["regulators"] = regulators.to_list()
if effectors is not None:
ret_dict["effectors"] = effectors.to_list()
Js_det = [np.linalg.det(Js[:, :, i]) for i in np.arange(Js.shape[2])]
jacobian_det_key = "jacobian_det" if basis is None else "jacobian_det_" + basis
adata.obs[jacobian_det_key] = np.nan
adata.obs.loc[adata.obs_names[cell_idx], jacobian_det_key] = Js_det
if store_in_adata:
jkey = "jacobian" if basis is None else "jacobian_" + basis
adata.uns[jkey] = ret_dict
return adata
return ret_dict
def hessian(
adata: AnnData,
regulators: List,
coregulators: List,
effector: Optional[List] = None,
cell_idx: Optional[List] = None,
sampling: Optional[Literal["random", "velocity", "trn"]] = None,
sample_ncells: int = 1000,
basis: str = "pca",
Qkey: str = "PCs",
vector_field_class: Optional[scVectorField.BaseVectorField] = None,
method: str = "analytical",
store_in_adata: bool = True,
"""Calculate Hessian for each cell with the reconstructed vector field.
If the vector field was reconstructed from the reduced PCA space, the Hessian matrix will then be inverse
transformed back to high dimension. Note that this should also be possible for reduced UMAP space and will be
supported shortly. Note that we compute the Hessian for the RKHS kernel vector field analytically, which is much
more computationally efficient than the numerical method.
adata: AnnData object that contains the reconstructed vector field in `.uns`.
regulators: The list of genes that will be used as regulators when calculating the cell-wise Hessian matrix. The
Hessian is the matrix consisting of secondary partial derivatives of the vector field wrt gene expressions.
It can be used to evaluate the change in velocities of effectors (see below) as the expressions of
regulators and co-regulators increase. The regulators/co-regulators are the denominators of the partial
coregulators: The list of genes that will be used as regulators when calculating the cell-wise Hessian matrix. The
Hessian is the matrix consisting of secondary partial derivatives of the vector field wrt gene expressions.
It can be used to evaluate the change in velocities of effectors (see below) as the expressions of
regulators and co-regulators increase. The regulators/co-regulators are the denominators of the partial
effector: The target gene that will be used as effector when calculating the cell-wise Hessian matrix. Effector
must be a single gene. The effector is the numerator of the partial derivatives.
cell_idx: A list of cell index (or boolean flags) for which the Hessian is calculated.
If `None`, all or a subset of sampled cells are used.
sampling: {None, 'random', 'velocity', 'trn'}, (default: None)
See specific information on these methods in `.tl.sample`.
If `None`, all cells are used.
sample_ncells: The number of cells to be sampled. If `sampling` is None, this parameter is ignored.
basis: The embedding data in which the vector field was reconstructed. If `None`, use the vector field function
that was reconstructed directly from the original unreduced gene expression space.
Qkey: The key of the PCA loading matrix in `.uns`.
vector_field_class: If not `None`, the Hessian will be computed using this class instead of the vector field stored in adata.
method: The method that will be used for calculating Hessian, either `'analytical'` or `'numerical'`.
`'analytical'` method uses the analytical expressions for calculating Hessian while `'numerical'` method
uses numdifftools, a numerical differentiation tool, for computing Hessian. `'analytical'` method is much
more efficient.
cores: Number of cores to calculate Hessian. Currently note used.
kwargs: Any additional keys that will be passed to elementwise_hessian_transformation function.
AnnData object that is updated with the `'Hessian'` key in the `.uns`. This is a 4-dimensional tensor with
dimensions 1 (n_effector) x n_regulators x n_coregulators x n_obs.
if vector_field_class is None:
vector_field_class = get_vf_class(adata, basis=basis)
if basis == "umap":
cell_idx = np.arange(adata.n_obs)
X, V = vector_field_class.get_data()
if cell_idx is None:
if sampling is None or sampling == "all":
cell_idx = np.arange(adata.n_obs)
cell_idx = sample(np.arange(adata.n_obs), sample_ncells, sampling, X, V)
Hessian_func = vector_field_class.get_Hessian(method=method)
Hs = np.zeros([X.shape[1], X.shape[1], X.shape[1], X.shape[0]])
for ind, i in enumerate(cell_idx):
Hs[:, :, :, ind] = Hessian_func(X[i])
if regulators is not None and coregulators is not None and effector is not None:
if type(regulators) is str:
if regulators in adata.var.keys():
regulators = adata.var.index[adata.var[regulators]]
regulators = [regulators]
if type(coregulators) is str:
if coregulators in adata.var.keys():
coregulators = adata.var.index[adata.var[coregulators]]
coregulators = [coregulators]
if type(effector) is str:
if effector in adata.var.keys():
effector = adata.var.index[adata.var[effector]]
effector = [effector]
if len(effector) > 1:
raise Exception(f"effector must be a single gene but you have {effector}. ")
regulators = np.unique(regulators)
coregulators = np.unique(coregulators)
effector = np.unique(effector)
var_df = adata[:, adata.var.use_for_dynamics].var
regulators = var_df.index.intersection(regulators)
coregulators = var_df.index.intersection(coregulators)
effector = var_df.index.intersection(effector)
reg_idx, coreg_idx, eff_idx = (
get_pd_row_column_idx(var_df, regulators, "row"),
get_pd_row_column_idx(var_df, coregulators, "row"),
get_pd_row_column_idx(var_df, effector, "row"),
if len(regulators) == 0 or len(coregulators) == 0 or len(effector) == 0:
raise ValueError(
"Either the regulator, coregulator or the effector gene list provided is not in the dynamics gene list!"
if basis == "pca":
if Qkey in adata.uns.keys():
Q = adata.uns[Qkey]
elif Qkey in adata.varm.keys():
Q = adata.varm[Qkey]
raise Exception(f"No PC matrix {Qkey} found in neither .uns nor .varm.")
Q = Q[:, : X.shape[1]]
if len(regulators) == 1 and len(coregulators) == 1 and len(effector) == 1:
Hessian = [
Hs[:, :, :, i],
Q[eff_idx, :].flatten(),
Q[reg_idx, :].flatten(),
Q[coreg_idx, :].flatten(),
for i in np.arange(Hs.shape[-1])
Hessian = [
hessian_transformation(Hs[:, :, :, i], Q[eff_idx, :], Q[reg_idx, :], Q[coreg_idx, :], **kwargs)
for i in np.arange(Hs.shape[-1])
Hessian = Hs.copy()
Hessian = None
ret_dict = {"hessian": Hs, "cell_idx": cell_idx}
# use 'str_key' in dict.keys() to check if these items are computed, or use dict.get('str_key')
if Hessian is not None:
ret_dict["hessian_gene"] = Hessian
if regulators is not None:
ret_dict["regulators"] = regulators.to_list() if type(regulators) != list else regulators
if coregulators is not None:
ret_dict["coregulators"] = coregulators.to_list() if type(coregulators) != list else coregulators
if effector is not None:
ret_dict["effectors"] = effector.to_list() if type(effector) != list else effector
if store_in_adata:
hkey = "hessian" if basis is None else "hessian_" + basis
adata.uns[hkey] = ret_dict
return adata
return ret_dict
def laplacian(
adata: AnnData,
hkey: str = "hessian_pca",
basis: str = "pca",
Qkey: str = "PCs",
vector_field_class: Optional[scVectorField.BaseVectorField] = None,
method: str = "analytical",
"""Calculate Laplacian for each target gene in each cell with the reconstructed vector field.
If the vector field was reconstructed from the reduced PCA space, the Lapalacian matrix will then be inverse
transformed back to high dimension. Note that this should also be possible for reduced UMAP space and will be
supported shortly. Note we compute the Lapalacian for the RKHS kernel vector field analytically, which is much
more computationally efficient than the numerical method.
adata: AnnData object that contains the reconstructed vector field in `.uns` and the `hkey` (the hessian matrix).
basis: The embedding data in which the vector field was reconstructed. If `None`, use the vector field function
that was reconstructed directly from the original unreduced gene expression space.
Qkey: The key of the PCA loading matrix in `.uns`.
vector_field_class: If not `None`, the Hessian will be computed using this class instead of the vector field stored in adata.
method: The method that will be used for calculating Laplacian, either `'analytical'` or `'numerical'`.
`'analytical'` method uses the analytical expressions for calculating Laplacian while `'numerical'` method
uses numdifftools, a numerical differentiation tool, for computing Laplacian. `'analytical'` method is much
more efficient.
kwargs: Any additional keys that will be passed to elementwise_hessian_transformation function.
AnnData object that is updated with the `'Laplacian'` key in the `.obs` and `obsm`. The first one is the
norm of the Laplacian for all target genes in a cell while the second one is the vector of Laplacian for all
target genes in each cell.
if hkey not in adata.uns_keys():
raise Exception(
f"{hkey} is not in adata.uns_keys(). Please first run dyn.vf.hessian(adata) properly before "
f"calculating Laplacian. This can be done by calculating Hessian between any three dynamical "
f"genes which will generate the Hessian matrix."
H = adata.uns[hkey]["hessian"]
if vector_field_class is None:
vector_field_class = get_vf_class(adata, basis=basis)
Laplacian_func = vector_field_class.get_Laplacian(method=method)
Ls = Laplacian_func(H)
L_key = "Laplacian" if basis is None else "Laplacian_" + basis
adata.obsm[L_key] = Ls.T
adata.obs[L_key] = np.linalg.norm(Ls, axis=0)
if basis == "pca":
if Qkey in adata.uns.keys():
Q = adata.uns[Qkey]
elif Qkey in adata.varm.keys():
Q = adata.varm[Qkey]
raise Exception(f"No PC matrix {Qkey} found in neither .uns nor .varm.")
Ls_hi = vector_transformation(Ls.T, Q)
elif basis is None:
[docs]def sensitivity(
adata: AnnData,
regulators: Optional[List] = None,
effectors: Optional[List] = None,
cell_idx: Optional[List] = None,
sampling: Optional[Literal["random", "velocity", "trn"]] = None,
sample_ncells: int = 1000,
basis: str = "pca",
Qkey: str = "PCs",
vector_field_class: Optional[scVectorField.BaseVectorField] = None,
method: str = "analytical",
projection_method: str = "from_jacobian",
store_in_adata: bool = True,
) -> Union[AnnData, Dict]:
"""Calculate Sensitivity matrix for each cell with the reconstructed vector field.
If the vector field was reconstructed from the reduced PCA space, the Sensitivity matrix will then be inverse
transformed back to high dimension. Note that this should also be possible for reduced UMAP space and will be
supported shortly. Note that we compute the Sensitivity for the RKHS kernel vector field analytically,
which is much more computationally efficient than the numerical method.
adata: AnnData object that contains the reconstructed vector field in `.uns`.
regulators: The list of genes that will be used as regulators when calculating the cell-wise Jacobian matrix. The
Jacobian is the matrix consisting of partial derivatives of the vector field wrt gene expressions. It can be
used to evaluate the change in velocities of effectors (see below) as the expressions of regulators
increase. The regulators are the denominators of the partial derivatives.
effectors: The list of genes that will be used as effectors when calculating the cell-wise Jacobian matrix. The
effectors are the numerators of the partial derivatives.
cell_idx: A list of cell index (or boolean flags) for which the jacobian is calculated.
If `None`, all or a subset of sampled cells are used.
sampling: {None, 'random', 'velocity', 'trn'}, (default: None)
See specific information on these methods in `.tl.sample`.
If `None`, all cells are used.
sample_ncells: The number of cells to be sampled. If `sampling` is None, this parameter is ignored.
basis: The embedding data in which the vector field was reconstructed. If `None`, use the vector field function
that was reconstructed directly from the original unreduced gene expression space.
Qkey: The key of the PCA loading matrix in `.uns`.
vector_field_class: If not `None`, the jacobian will be computed using this class instead of the vector field stored in adata.
method: The method that will be used for calculating Jacobian, either `'analytical'` or `'numerical'`.
`'analytical'` method uses the analytical expressions for calculating Jacobian while `'numerical'` method
uses numdifftools, a numerical differentiation tool, for computing Jacobian. `'analytical'` method is much
more efficient.
projection_method: The method that will be used to project back to original gene expression space for calculating gene-wise
sensitivity matrix:
(1) 'from_jacobian': first calculate jacobian matrix and then calculate sensitivity matrix. This method
will take the combined regulator + effectors gene set for calculating a square Jacobian matrix
required for the sensitivity matrix calculation.
(2) 'direct': The sensitivity matrix on low dimension will first calculated and then projected back to
original gene expression space in a way that is similar to the gene-wise jacobian calculation.
cores: Number of cores to calculate Jacobian. If cores is set to be > 1, multiprocessing will be used to
parallel the Jacobian calculation.
kwargs: Any additional keys that will be passed to elementwise_jacobian_transformation function.
adata: AnnData object that is updated with the `'sensitivity'` key in the `.uns`. This is a 3-dimensional tensor
with dimensions n_obs x n_regulators x n_effectors.
regulators, effectors = (
list(np.unique(regulators)) if regulators is not None else None,
list(np.unique(effectors)) if effectors is not None else None,
if vector_field_class is None:
vector_field_class = get_vf_class(adata, basis=basis)
if basis == "umap":
cell_idx = np.arange(adata.n_obs)
X, V = vector_field_class.get_data()
if cell_idx is None:
if sampling is None or sampling == "all":
cell_idx = np.arange(adata.n_obs)
cell_idx = sample(np.arange(adata.n_obs), sample_ncells, sampling, X, V)
S = vector_field_class.compute_sensitivity(method=method)
if regulators is None and effectors is not None:
regulators = effectors
elif effectors is None and regulators is not None:
effectors = regulators
if regulators is not None and effectors is not None:
if type(regulators) is str:
if regulators in adata.var.keys():
regulators = adata.var.index[adata.var[regulators]]
regulators = [regulators]
if type(effectors) is str:
if effectors in adata.var.keys():
effectors = adata.var.index[adata.var[effectors]]
effectors = [effectors]
var_df = adata[:, adata.var.use_for_dynamics].var
regulators = var_df.index.intersection(regulators)
effectors = var_df.index.intersection(effectors)
if projection_method == "direct":
reg_idx, eff_idx = (
get_pd_row_column_idx(var_df, regulators, "row"),
get_pd_row_column_idx(var_df, effectors, "row"),
if len(regulators) == 0 or len(effectors) == 0:
raise ValueError(
"Either the regulator or the effector gene list provided is not in the dynamics gene list!"
Q = adata.uns[Qkey][:, : X.shape[1]]
if len(regulators) == 1 and len(effectors) == 1:
Sensitivity = elementwise_jacobian_transformation(
Q[eff_idx, :].flatten(),
Q[reg_idx, :].flatten(),
Sensitivity = subset_jacobian_transformation(S, Q[eff_idx, :], Q[reg_idx, :], **kwargs)
elif projection_method == "from_jacobian":
Js = jacobian(
regulators=list(regulators) + list(effectors),
effectors=list(regulators) + list(effectors),
J, regulators, effectors = (
Sensitivity = np.zeros_like(J)
n_genes, n_genes_, n_cells = J.shape
idenity = np.eye(n_genes)
for i in LoggerManager.progress_logger(
np.arange(n_cells), progress_name="Calculating sensitivity matrix with precomputed gene-wise Jacobians"
s = np.linalg.inv(idenity - J[:, :, i]) # np.transpose(J)
Sensitivity[:, :, i] = s.dot(np.diag(1 / np.diag(s)))
raise ValueError("`projection_method` can only be `from_jacoian` or `direct`!")
Sensitivity = None
ret_dict = {"sensitivity": S, "cell_idx": cell_idx}
# use 'str_key' in dict.keys() to check if these items are computed, or use dict.get('str_key')
if Sensitivity is not None:
ret_dict["sensitivity_gene"] = Sensitivity
if regulators is not None:
ret_dict["regulators"] = regulators if type(regulators) == list else regulators.to_list()
if effectors is not None:
ret_dict["effectors"] = effectors if type(effectors) == list else effectors.to_list()
S_det = [np.linalg.det(S[:, :, i]) for i in np.arange(S.shape[2])]
adata.obs["sensitivity_det_" + basis] = np.nan
adata.obs["sensitivity_det_" + basis][cell_idx] = S_det
if store_in_adata:
skey = "sensitivity" if basis is None else "sensitivity_" + basis
adata.uns[skey] = ret_dict
return adata
return ret_dict
[docs]def acceleration(
adata: AnnData,
basis: str = "umap",
vector_field_class: Optional[scVectorField.BaseVectorField] = None,
Qkey: str = "PCs",
method: str = "analytical",
"""Calculate acceleration for each cell with the reconstructed vector field function. AnnData object is updated with the `'acceleration'` key in the `.obs` as well as .obsm. If basis is `pca`, acceleration matrix will be inverse transformed back to original high dimension space.
adata: AnnData object that contains the reconstructed vector field function in the `uns` attribute.
basis: The embedding data in which the vector field was reconstructed.
vector_field_class: If not None, the divergene will be computed using this class instead of the vector field stored in adata.
Qkey: The key of the PCA loading matrix in `.uns`.
method: The method that will be used for calculating acceleration field, either `'analytical'` or `'numerical'`.
`'analytical'` method uses the analytical expressions for calculating acceleration field while `'numerical'`
method uses numdifftools, a numerical differentiation tool, for computing acceleration. `'analytical'`
method is much more efficient.
kwargs: Any additional keys that will be passed to vector_field_class.compute_acceleration function.
if vector_field_class is None:
vector_field_class = get_vf_class(adata, basis=basis)
X, V = vector_field_class.get_data()
acce_norm, acce = vector_field_class.compute_acceleration(X=X, method=method, **kwargs)
acce_key = "acceleration" if basis is None else "acceleration_" + basis
adata.obsm[acce_key] = acce
adata.obs[acce_key] = acce_norm
if basis == "pca":
if Qkey in adata.uns.keys():
Q = adata.uns[Qkey]
elif Qkey in adata.varm.keys():
Q = adata.varm[Qkey]
raise Exception(f"No PC matrix {Qkey} found in neither .uns nor .varm.")
acce_hi = vector_transformation(acce, Q)
elif basis is None:
[docs]def curvature(
adata: AnnData,
basis: str = "pca",
vector_field_class: Optional[scVectorField.BaseVectorField] = None,
formula: int = 2,
Qkey: str = "PCs",
method: str = "analytical",
"""Calculate curvature for each cell with the reconstructed vector field function. AnnData object that is updated with the `curvature` key in the `.obs`.
adata: AnnData object that contains the reconstructed vector field function in the `uns` attribute.
basis: The embedding data in which the vector field was reconstructed.
vector_field_class: If not None, the divergene will be computed using this class instead of the vector field stored in adata.
formula: Which formula of curvature will be used, there are two formulas, so formula can be either `{1, 2}`. By
default it is 2 and returns both the curvature vectors and the norm of the curvature. The formula one only
gives the norm of the curvature.
Qkey: The key of the PCA loading matrix in `.uns`.
method: The method that will be used for calculating curvature field, either `'analytical'` or `'numerical'`.
`'analytical'` method uses the analytical expressions for calculating curvature while `'numerical'` method
uses numdifftools, a numerical differentiation tool, for computing curvature. `'analytical'` method is much
more efficient.
kwargs: Any additional keys that will be passed to vector_field_class.compute_curvature function.
if vector_field_class is None:
vector_field_class = get_vf_class(adata, basis=basis)
if formula not in [1, 2]:
raise ValueError(
f"There are only two available formulas (formula can be either `{1, 2}`) to calculate "
f"curvature, but your formula argument is {formula}."
X, V = vector_field_class.get_data()
curv, curv_mat = vector_field_class.compute_curvature(X=X, formula=formula, method=method, **kwargs)
curv_key = "curvature" if basis is None else "curvature_" + basis
main_info_insert_adata(curv_key, adata_attr="obs", indent_level=1)
adata.obs[curv_key] = curv
main_info_insert_adata(curv_key, adata_attr="obsm", indent_level=1)
adata.obsm[curv_key] = curv_mat
if basis == "pca":
curv_hi = vector_transformation(curv_mat, adata.uns[Qkey])
create_layer(adata, curv_hi, layer_key="curvature", genes=adata.var.use_for_pca)
elif basis is None:
[docs]def torsion(
adata: AnnData, basis: str = "umap", vector_field_class: Optional[scVectorField.BaseVectorField] = None, **kwargs
"""Calculate torsion for each cell with the reconstructed vector field function. AnnData object that is updated with the `torsion` key in the .obs.
adata: :class:`~anndata.AnnData`
AnnData object that contains the reconstructed vector field function in the `uns` attribute.
basis: str or None (default: `umap`)
The embedding data in which the vector field was reconstructed.
vector_field_class: dict
The true ODE function, useful when the data is generated through simulation.
Any additional keys that will be passed to vector_field_class.compute_torsion function.
>>> adata = dyn.sample_data.hematopoiesis()
>>> dyn.tl.reduceDimension(adata, n_components=3, enforce=True, embedding_key='X_umap_3d')
>>> adata
>>> dyn.tl.cell_velocities(adata,
>>> X=adata.layers["M_t"],
>>> V=adata.layers["velocity_alpha_minus_gamma_s"],
>>> basis='umap_3d',
>>> )
>>> dyn.vf.VectorField(adata, basis='umap_3d')
>>> dyn.vf.torsion(adata, basis='umap_3d')
>>> dyn.pl.streamline_plot(adata, color='torsion_umap_3d', basis='umap_3d')
>>> dyn.pl.streamline_plot(adata, color='torsion_umap_3d')
if vector_field_class is None:
vector_field_class = get_vf_class(adata, basis=basis)
X, V = vector_field_class.get_data()
torsion_mat = vector_field_class.compute_torsion(X=X, **kwargs)
torsion = np.array([np.linalg.norm(i) for i in torsion_mat])
torsion_key = "torsion" if basis is None else "torsion_" + basis
adata.obs[torsion_key] = torsion
adata.uns[torsion_key] = torsion_mat
[docs]def curl(
adata: AnnData,
basis: str = "umap",
vector_field_class: Optional[scVectorField.BaseVectorField] = None,
method: str = "analytical",
"""Calculate Curl for each cell with the reconstructed vector field function. AnnData object is updated with the `'curl'` information in the `.
obs`. When vector field has three dimension, adata.obs['curl'] (magnitude of curl) and adata.obsm['curl'] (curl vector) will be added; when vector field has two dimension, only adata.obs['curl'] (magnitude of curl) will be provided.
adata: AnnData object that contains the reconstructed vector field function in the `uns` attribute.
basis: The embedding data in which the vector field was reconstructed.
vector_field_class: If not None, the divergene will be computed using this class instead of the vector field stored in adata.
method: The method that will be used for calculating curl, either `analytical` or `numeric`. `analytical`
method will use the analytical form of the reconstructed vector field for calculating curl while
`numeric` method will use numdifftools for calculation. `analytical` method is much more efficient.
kwargs: Any additional keys that will be passed to vector_field_class.compute_curl function.
if vector_field_class is None:
vector_field_class = get_vf_class(adata, basis=basis)
X, V = vector_field_class.get_data()
curl = vector_field_class.compute_curl(X=X, method=method, **kwargs)
curl_key = "curl" if basis is None else "curl_" + basis
if X.shape[1] == 3:
curl_mag = np.array([np.linalg.norm(i) for i in curl])
adata.obs[curl_key] = curl_mag
adata.obsm[curl_key] = curl
adata.obs[curl_key] = curl
[docs]def divergence(
adata: AnnData,
cell_idx: Optional[List] = None,
sampling: Optional[Literal["random", "velocity", "trn"]] = None,
sample_ncells: int = 1000,
basis: str = "pca",
method: str = "analytical",
store_in_adata: bool = True,
) -> Optional[np.ndarray]:
"""Calculate divergence for each cell with the reconstructed vector field function. Either AnnData object is updated with the `'divergence'` key in the `.obs` or the divergence is returned as a numpy array.
adata: AnnData object that contains the reconstructed vector field function in the `uns` attribute.
cell_idx: A list of cell index (or boolean flags) for which the jacobian is calculated.
sampling: {None, 'random', 'velocity', 'trn'}, (default: None)
See specific information on these methods in `.tl.sample`.
If `None`, all cells are used.
sample_ncells: The number of cells to be sampled. If `sampling` is None, this parameter is ignored.
basis: The embedding data in which the vector field was reconstructed.
vector_field_class: If not None, the divergene will be computed using this class instead of the vector field stored in adata.
method: The method that will be used for calculating divergence, either `analytical` or `numeric`. `analytical`
method will use the analytical form of the reconstructed vector field for calculating divergence while
`numeric` method will use numdifftools for calculation. `analytical` method is much more efficient.
store_in_adata: Whether to store the divergence result in adata.
kwargs: Any additional keys that will be passed to vector_field_class.compute_divergence function.
the divergence is returned as an np.ndarray if store_in_adata is False.
if vector_field_class is None:
vector_field_class = get_vf_class(adata, basis=basis)
if basis == "umap":
cell_idx = np.arange(adata.n_obs)
X, V = vector_field_class.get_data()
if cell_idx is None:
if sampling is None or sampling == "all":
cell_idx = np.arange(adata.n_obs)
cell_idx = sample(np.arange(adata.n_obs), sample_ncells, sampling, X, V)
jkey = "jacobian" if basis is None else "jacobian_" + basis
div = np.zeros(len(cell_idx))
calculated = np.zeros(len(cell_idx), dtype=bool)
if jkey in adata.uns_keys():
Js = adata.uns[jkey]["jacobian"]
cidx = adata.uns[jkey]["cell_idx"]
for i, c in enumerate(
LoggerManager.progress_logger(cell_idx, progress_name="Calculating divergence with precomputed Jacobians")
if c in cidx:
calculated[i] = True
div[i] = np.trace(Js[:, :, i]) if Js.shape[2] == len(cell_idx) else np.trace(Js[:, :, c])
div[~calculated] = vector_field_class.compute_divergence(X[cell_idx[~calculated]], method=method, **kwargs)
if store_in_adata:
div_key = "divergence" if basis is None else "divergence_" + basis
Div = np.array(adata.obs[div_key]) if div_key in adata.obs.keys() else np.ones(adata.n_obs) * np.nan
Div[cell_idx] = div
adata.obs[div_key] = Div
return div | {"url":"https://dynamo-release.readthedocs.io/en/latest/_modules/dynamo/vectorfield/vector_calculus.html","timestamp":"2024-11-11T04:23:29Z","content_type":"text/html","content_length":"168081","record_id":"<urn:uuid:90ee429f-4bce-48a8-8d2e-698410d51ba7>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00258.warc.gz"} |
What Is Numicon? Explained For Primary School Teachers, Parents And Pupils
Numicon shapes have become a staple of the primary classroom, in part due to the popularity of the concrete pictorial abstract approach (CPA). They are an excellent multi sensory approach that
supports early numeracy by helping pupils to understand the relative size of the numbers 1-10.
However, Numicon shapes can also be used to represent numbers larger than ten. Although as a teaching resource they are a common sight, particularly in Key Stage 1, they are much more versatile
approach to maths manipulatives than a lot of teachers give them credit for.
In this blog, we will explore Numicon and a variety of different topics that they can be used to teach right from Early Years through to upper Key Stage 2.
What is Numicon?
Numicon is a teaching resource developed by Oxford University Press that comprises a number of plastic shapes that represent the numbers 1-10 and are relative in size to one another.
Numicon comprises different shapes representing the numbers 1-10.
As you can see from this image, the Numicon 2 is the same size as two Numicon 1 pieces, and so on. This allows pupils to understand the relationships between numbers.
Guide To Hands On Manipulatives
Download our guide to hands on math manipulatives. Includes 15 concrete resources every KS1 and KS2 classroom should have.
Download Free Now!
A full teaching pack of Numicon may come with a number of different items:
• Numicon Shapes: the plastic shapes as shown above which represent the numbers 1-10.
• Numicon Baseboard: a pegboard that the Numicon shapes can fit on to – the holes on the shapes fit into the pegs.
• Numicon Baseboard Overlays: picture cards that fit onto the baseboard to help children to form different numbers.
• Numicon Pegs: small coloured close-ended beads which fit onto the pegs in the baseboard to help children to ensure they count the Numicon correctly.
• Numicon Number Line: a traditional number line but with Numicon shapes pictured alongside the numerals.
How does Numicon work?
Numicon gives pupils an idea of what the value of each number looks like and that when we count in ones, the number gets bigger by the same amount each time.
In the case of Numicon, we add an additional hole every time. It appeals to children’s visual understanding and their sense of pattern.
Numbers are a very abstract concept and, as such, pupils can often feel overwhelmed when first learning maths. This can lead to anxiety around the subject throughout their lives.
By having a physical material that they can hold and manipulate, pupils are more likely to be able to make the connections that allow them to work with numbers in an abstract form.
In Third Space Learning’s one to one maths tuition, lessons are designed to support learners through visual teaching strategies.
This is the basis of the concrete pictorial abstract approach – pupils work with a concrete (physical) object, before making this into a pictorial (drawn) representation and then moving onto the
abstract form (numerals and calculations).
Generally, pupils will go through four stages with Numicon in order to master number and the concepts that they are being taught:
• Pattern: pupils find patterns or similarities and differences in the shapes.
• Ordering: ordering the shapes from largest to smallest and smallest to largest.
• Counting: counting each hole one by one – this gives the pupil a knowledge of what each piece represents
• Early calculating: using the pieces to perform simple calculations e.g. putting a two and a three together and seeing that this is the same shape as (therefore equal to) the five.
Advantages and disadvantages of Numicon
While there are several advantages to using Numicon as a teaching tool in primary schools, there are also some potential disadvantages to consider.
How to use Numicon in the classroom
To maximise the benefits of Numicon as a teaching tool, it’s important to understand how to utilise it effectively in the classroom and prepare worksheets in advance to incorporate it into your
lesson plans.
1. Numicon for number recognition
It is a good idea to introduce Numicon to pupils for the first time when you begin introducing numbers to them in early years. As many EYFS teachers take each number individually, you can introduce
each Numicon piece at the same time.
Using Numicon alongside other concrete maths resources such as tens frames, number lines and counting apparatus, such as dienes, can help pupils to develop a firm understanding of numbers and an
array of tools to help them when reasoning and problem solving.
One Numicon activity that can be done to help support number recognition among pupils is to provide them with a range of Numicon pieces along with number flashcards.
The pupils simply have to match the Numicon with the number it represents.
2. Numicon for learning to count
There are two common ways that young children can use Numicon to learn to count.
The first is simply to place a Numicon shape on the baseboard and use the Numicon pegs to fill each hole, counting using one to one correspondence as they do.
When done under the supervision of an adult, this can confirm that a child is counting in sequence.
The second is ordering the pieces from smallest to largest.
Once children have done this, they have made their own number line and can add the numbers underneath with flashcards or, if they are outside, use playground chalk.
3. Numicon for place value
Numicon can be very helpful for pupils exploring numbers above ten.
Here, we have created the number 13 using Numicon shapes. The pupil is likely to be familiar with counting to thirteen and can happily tell you this. They may also be able to write 13 in numerals.
However, what does the ‘1’ represent in the number 13? From this representation, they can see that there is a ten and a three, therefore the 1 must represent 10 and not only 1.
In English, the numbers 13-19 are not ‘ten’ numbers but ‘teen’ numbers. By using a representation such as the one above with Numicon, pupils can understand what that means.
4. Numicon for addition
Numicon shapes can help pupils to learn to add by allowing them to place two shapes together and then either count the holes or compare with the other shapes to find the answer.
This can be particularly helpful when pupils cross the tens boundary.
If we take 7+5:
Initially, the pupil has taken a Numicon 7 and a Numicon 5 and placed them together to create a rectangle – this is encouraged with Numicon because the shapes are either rectangle or rectilinear with
one odd piece.
From this point, the pupil could place the Numicon in this arrangement on a baseboard and use pegs to count the total.
Alternatively, they could just count the holes by pointing into each one.
However, pupils should be encouraged to move onto finding out the answer using a combination of a Numicon 10 and another shape. Usually, pupils would line the Numicon on top of one another to check
that the two representations are equal. However, they can also show this side-by-side:
Here, they can see that 7+5 is the same as 10+2 which by this point they are likely to know is equal to 12.
5. Numicon for subtraction
Like addition, Numicon is an excellent resource for early subtraction too. It is also a great tool to help pupils to subtract crossing the tens boundary.
As pupils have often had a lot of practice in making numbers above ten at this point, the initial starting point should be familiar with them – they simply use the Numicon shapes to make the number.
For example, 13-5:
The pupil would first use a ten and a three to represent the number 13.
They would then take the five piece and place it on top of their number. At this point, try to encourage pupils to line up the odd pieces where relevant to help them to see the patterns being made.
They can then either count the holes left on their original shape (as this would be 3D, they would easily tell the difference by its thickness) or find the shape that fits into the space on the
original and subitise from there.
6. Numicon for number bonds
In a similar way, pupils can use stacking to find the different ways to make number bonds for all numbers up to 10.
In the example above, the pupil has been investigating number bonds to 10. They have initially chosen a Numicon 10 as their base example.
They have then used two pieces of Numicon to recreate the same shape. A natural progression from this for pupils who find this easy would be to then try to find number bonds with three addends.
7. Numicon for times tables
As Numicon is excellent for creating patterns and numbers in different ways, it is great for helping pupils to find the multiples of a given number and therefore to learn their times tables. This is
an excellent use of Numicon for Year 2 (2, 5 and 10 times tables), Year 3 (3, 4 and 6 times tables) and Year 4 (6, 7, 9, 11 and 12 times tables).
In this example, pupils are exploring the three times table.
In the top row, they have used Numicon 3 pieces to create shapes representing 1×3, 2×3, 3×3 and 4×3. They have then found the corresponding piece of Numicon and placed it in the second row to find
the multiples of three: 3, 6, 9 and 12.
This helps pupils to understand why 2×3 is equal to six as they can visually see the two sets of three being placed together and this being the same shape as the Numicon 6.
8. Numicon for fractions
As mentioned previously, each square and hole on a piece of Numicon is of an equal size. This therefore makes Numicon a good resource for making fractions.
Initially when introducing fractions, pupils can use the Numicon shapes to represent the denominator of the fraction and then place pegs into the holes to show the numerator, therefore ‘filling
in’ the fraction that is given.
\frac{5}{6} is represented here with the Numicon 6 showing the denominator and the pegs representing the numerator.
This then leads on to being able to think about making a whole. In the example above, if the whole is 6/6 as represented by the Numicon shape, and \frac{5}{6} is represented by the pegs in the shape,
then to complete this must be the number of holes without a peg – \frac{1}{6} .
This can lead onto the calculation \frac{5}{6} + \frac{1}{6} = \frac{6}{6} (or 1)
Children can also add fractions with the same denominator using Numicon, even crossing to numbers larger than 1 from year 4.
Here, the children have been tasked with the calculation \frac{3}{5} + \frac{4}{5} .
First, they have created both fractions as described above, using the Numicon shapes as the denominator and the pegs to represent the numerator.
By then moving the pegs to complete the first Numicon shape, they can see that the answer has 1 whole and 2 extra fifths, therefore \frac{3}{5} + \frac{4}{5} =1 \frac{2}{5} .
9. Numicon for geometry and measurement
Numicon has also been found to be useful for measurement and geometry.
In Early Years, children can use Numicon (or any other manipulative resource) as a non-standard unit of measurement and compare the length or height of items to these. This is an excellent way to
encourage children to use language such as bigger, smaller, taller and shorter.
When introducing perimeter to year 3 children, it can be helpful to use Numicon initially as they are comfortable with it as a manipulative and therefore they have one less part of the maths to worry
As the squares are clear on a Numicon shape, it can be easy to count around the edge to find the perimeter.
Here, the child has found out that this square has a perimeter of 8 squares. When they are comfortable with this, they can then think about working out the length of each side and adding them
Here they can see that each side of the shape is 2 squares in length, therefore the perimeter is 2+2+2+2=8 squares. Some children may then be able to theorise that for a square, you only need to
measure one side and multiply the answer by 4. This could be an excellent area for investigation.
Exactly the same process could be followed for rectilinear shapes in Year 4.
In Year 4, children also learn to measure area for the first time and they only need to learn to ‘find the area of rectilinear shapes by counting squares.’ This means that Numicon lends itself
perfectly to this part of the curriculum. The children simply have to count the squares (or the holes) of the Numicon to find out the area.
This means that they can then investigate different shapes with the same area by using several Numicon pieces to make them.
When do children use Numicon in school?
Children often use Numicon during EYFS and Key Stage 1.
However, as seen above, there are definite benefits to using Numicon when introducing new mathematical concepts to learners at KS2. The fact that they are familiar with Numicon from KS1 means that
they can focus on their new maths skills rather than the manipulative that is being used.
Numicon worked examples
1. 8+7
Here the child has used a Numicon 8 and a Numicon 7 and placed them together to create a new shape, making sure the odd one is at the top. They have then used a ten and a five together to make the
same shape. Therefore, they can see that the answer is 15.
2. How many different ways can you make 4?
The child has taken a 4 piece and used two of the other pieces to make a shape that matches. However, you may see some of the more confident mathematicians beginning to use more than two shapes:
3. What is the perimeter of this shape in squares?
Children have two options for this question. Either they can go with the image on the left and just count each square to find the perimeter of 12 squares. Alternatively, they can measure each side
separately and use addition (1+4+2+3+1+1) to also get the answer of 12 squares.
Numicon practice questions
Activity sheets are a popular format for presenting Numicon practice questions, as they allow children to work independently or in small groups to practise their skills.
1. How many different ways can you make the number 7 using Numicon?
Answer: 6+1, 5+2, 4+3, 3+4, 2+5, 1+6.
Children may then explore with more than two pieces for which there are many answers.
2. 16-9
Answer: 7
3. Find all of the multiples of 4 to 40 using Numicon.
Answer: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
4. 4/7 + 6/7
Answer: 10/7 or 1 3/7
5. What is the perimeter of this rectangle in squares?
Answer: 12 squares.
Looking for more information on concrete resources for the primary classroom?
What is Numicon used for?
Numicon is a maths resource used for counting. This lends itself to work across a lot of the maths curriculum.
What age is Numicon suitable for?
Numicon is mostly used in Early Years and Key Stage 1. However, it is suitable for all of the primary age ranges in certain parts of the curriculum.
How do you count using Numicon?
To count using numicon, you can either place pegs into the holes and count as you do or just count the holes.
How do you teach addition with Numicon?
To teach addition with Numicon, you take the pieces that represent the two addends and place them together to make a single shape. Then, you can either count the holes in your new shape or find
another shape that your new shape is equal to.
Every week Third Space Learning’s specialist online maths tutors support thousands of students across hundreds of schools with weekly online 1 to 1 maths lessons designed to plug gaps and boost
Since 2013 these personalised one to one lessons have helped over 169,000 primary and secondary students become more confident, able mathematicians.
Learn how the programmes are aligned to maths mastery teaching or request a personalised quote for your school to speak to us about your school’s needs and how we can help. | {"url":"https://thirdspacelearning.com/blog/what-is-numicon/","timestamp":"2024-11-07T15:36:12Z","content_type":"text/html","content_length":"162974","record_id":"<urn:uuid:b46727b2-10f2-43a8-b239-c628f4495e57>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00708.warc.gz"} |
So far, all our superposition examples have been coherent. What happens if we superimpose two sine waves which are incoherent? We can make two sine waves incoherent by having them at slightly
different frequencies.
When we have two incoherent waves with two slightly different frequencies, then the phase between them shifts over time. If you connect two sine generators to a dual-beam oscilloscope and set them up
to generate slightly different frequencies, one will appear to move relative to the other in time as shown in the clip below.
We’ll can also add a trace representing the superposition (sum) of the two waves, to see the result of the shifting phase between the two waves. The sum is shown in light grey.
What happens if the frequency of the two waves is changed? It is easiest to look at some snap-shots from the oscilloscope for different cases. The combined wave in light grey is moved up to make it
easier to see. The first snapshot is for a frequency difference of 10 Hz.
Two sinusoidal waves, one at 100 Hz the other at 110 Hz. Frequency difference 10 Hz.
The combined wave appears to have two frequencies. There is a sine wave pattern which is at a similar frequency to the original waves, actually it is at an average of the two frequencies, so for the
second case this is 105 Hz.
But there is also a more slowly varying wave pattern which gradually changes the amplitude. In the second graph, the period of this second slowly varying wave is the same as the horizontal division
on the oscilloscope, 0.1 seconds. As
frequency = \frac{1}{period}
This relates to a frequency of 1/0.1 = 10 Hz. So this slowly varying amplitude is at a frequency of the frequency difference between the original waves. Check if this also works for the case below
where the frequency difference is 5 Hz.
Two sinusoidal waves, one at 100 Hz the other at 105 Hz. Frequency difference 5 Hz.
Beats can be used by guitar players when tuning up. If you listen for beats between two notes played on different strings, which are fretted (or plucked using harmonics) so as to have the same pitch,
then by adjusting the tension in the string until the beats slow down to zero, you can bring the instrument into tune. | {"url":"https://salfordacoustics.co.uk/sound-waves/superposition/beats","timestamp":"2024-11-08T18:00:59Z","content_type":"text/html","content_length":"61388","record_id":"<urn:uuid:a82077fc-7543-46b3-96b8-67d282f312f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00189.warc.gz"} |
The number nine is truly sacred – here’s why! - YogaEsotericThe number nine is truly sacred – here’s why!
The number nine is truly sacred – here’s why!
You don’t need to study Nature for very long before figuring out that there is a mathematical structure for everything. From flowers to seashells to hurricanes to our own Milky Way Galaxy, there
resides the Fibonacci sequence spiral; the Golden Ratio of Phi which demonstrates a perfect order emanating from a single point and is what even allows leaves to grow on plants so perfectly spaced as
to give each the best shot at being exposed to sunlight.
Math can even be found in every religion and in this aspect exists as the more commonly termed “Sacred Geometry” when it comes to correlating mathematical forms and formulations with that of Godly
Creation. The one specific representation of creation in Sacred Geometry is called The Flower of Life pattern (which begins with the Seed of Life pattern) and is also where Phi is derived from.
Geometric patterns can, of course, be represented numerically and if we take the simple shapes from these patterns and study basic mathematical measurements such as radius, diameter, circumference,
etc. we can find something very strange about one particular number if we just look close enough.
You could even say there is a code hidden right before our very eyes within the language of our universe; the language of mathematics (and sound). And it just so occurs that the very last number in
our common base 10 system, the number 9, has many qualities that no other numbers possess. For starters, when we begin to bisect a circle, we see 9 at every cut.
Let’s take a look…
• A circle is 360 degrees: 3+6+0=9.
• A circle divided in half is 180 degrees: 1+8+0=9.
• This will be true as well for all reductions including 45-degree angles within a circle as well as 90-degree angles: (45 degrees) 4+5=9, (90 degrees) 9+0=9, (135 degrees) 1+3+5=9, (225 degrees)
2+2+5=9, (270 degrees) 2+7+0=9, (315 degrees) 3+1+5=9.
• It will continue if we add the 30-degree cuts together as well, for example: (60 degrees + 30 degrees) 6+0+3+0=9. Or (210 degrees + 240 degrees) 2+1+0+2+4+0=9.
A scientist by the name of Marko Rodin has stated that Vortex-Based Mathematics can solve Pi and prove it to be a whole, recurring pattern such as a loop. He demonstrates this through numerous
experiments with his Rodin Coil invention based on the shape of a torus (which is an infinite, rotating, donut-shaped loop) which represents regularities in the decimal (base 10) numerical system
that have not yet been previously documented.
Due to the central wires in the Rodin Coil, a magnetic field is created in the center of the torus and currently, this has proven useful for motors, electromagnets, and antennae which of course the
U.S. Government has taken a vast interest in.
There are also medical devices being used to help treat cancer patients based on Rodin’s technology. Many speculate that the Rodin Coil reignites the hope for free energy and he could very well be
the Nikola Tesla of our time.
When so many undeniable patterns are found within and so many tangible connections can be made between energy, the fabric of our very universe and the spiral to and from infinite creation, we would
be rather naïve to deny that there are answers hidden in these vast mathematical mysteries. Some believe that the sacred number 9, and the fractal patterns in nature and reality, are perhaps a
simulation, a holographic projection of time-space and we need to strive to discover what is the reality beyond our perceptions.
June 23, 2020 | {"url":"https://yogaesoteric.net/en/the-number-nine-is-truly-sacred-heres-why/","timestamp":"2024-11-11T04:37:07Z","content_type":"text/html","content_length":"109868","record_id":"<urn:uuid:a488c3c7-4311-47a4-9fae-0ca47d1db963>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00357.warc.gz"} |
Policy Types in Reinforcement Learning
Deterministic and Stochastic Policies Explained
In Reinforcement Learning (RL), a policy is a description of how an agent behaves given its current state and the goal. In this blog post, we will discuss Reinforcement Learning Policy Types:
Deterministic Policies and Stochastic Policies.
Deterministic Policies: In a deterministic policy, the action taken at each state is always the same. This can be implemented using a lookup table or decision tree. In this case, we denote it is
denoted by \( \mu \):
$$ a_t = \mu(s_t) $$
Stochastic Policies: A stochastic policy, on the other hand, produces different actions from one instance to the next, but these are still chosen according to some fixed probabilities. The stochastic
policy is denoted by \( \pi \):
$$ a_t \sim \pi( \cdot | s_t) $$
Because the policy is essentially the agent's brain, it's not unusual to replace "policy" with "agent," such as when someone says, "The agent is attempting to maximize reward."
In deep RL, we deal with parameterized policies: policies that produce computable functions that are dependent on a set of parameters (for example, the weights and biases of a neural network) that
may be adjusted to modify the outcome using some optimization algorithm. Stochastic Policies
Categorical and diagonal Gaussian policies are the two most frequent types of stochastic policies in deep RL. Categorical policies can be applied in discrete action regions, while diagonal Gaussian
policies are appropriate for continuous action regions.
Categorical Policy: A categorical policy is a stochastic policy that favors either one or zero actions in each state, with equal probabilities assigned to all possible action choices.
Diagonal Gaussian Policy: On the other hand, Diagonal Gaussian policies take any number of actions from zero to infinity and distribute them according to a diagonal Gaussian distribution. This means
that in any given state, the agent can select from many different actions with equal probabilities assigned to all possible action choices.
The two most important computations for employing and training stochastic policies are:
• Sampling actions from the policy,
• Comparing log-likelihoods of various actions.
We'll go through how to implement these for both categorical and diagonal Gaussian policies in the following.
Categorical Policies
A categorical policy is comparable to a classifier in terms of its structure. You construct a neural network for a categorical policy in the same way as you would for a classifier: the input is the
observation, followed by one or more layers (possibly convolutional or densely-connected, depending on the type of input), and then there's an output layer with one node for each possible action.
Action Sampling (log-probability)
To take actions according to a categorical policy, we sample them from the posterior distribution of all the possible actions given the observation and current state. There are different ways of
doing this depending on whether you want to maintain Monte Carlo estimates of the policy or not.
Maintaining Monte Carlo Estimates: If you want to maintain Monte Carlo estimates of the policy, then you can use a method like importance sampling. This involves drawing from the posterior at each
step and keeping track of how often different actions are sampled.
Not Maintaining Monte Carlo Estimates: Alternatively, if you don't want to maintain Monte Carlo estimates, you can use a random sampling method. This just picks an action at random from the posterior
for each step.
In either case, we need to compute the log-probability of taking each action given the observation and state:
$$ \log \pi_\theta(a | s) = \log P_\theta(s)_a $$
where \( P_\theta(s) \), is the last layer of probabilities taken from the NN. It's computed by inserting the observation and state into our neural network (which has one node for each possible
action in this case) and then applying the activation function.
Once we have the log-probabilities of all the actions, we can compare them and take the action with the highest log-probability (or some other measure like expected utility).
Diagonal Gaussian Policies
A multivariate normal distribution, often known as a multivariate Gaussian distribution (but not to be confused with the multidimensional normal form, which is a different topic), is characterized by
a mean vector, \( \mu \), and a covariance matrix, \( \Sigma \). A diagonal Gaussian distribution is one in which the entries of \( \Sigma \), are all zero except for the diagonal, which consists of
the variances of the individual variables.
The advantage of a diagonal Gaussian distribution is that it's easy to compute its mean and variance.
This makes it easy to sample from a diagonal Gaussian distribution: you just compute the mean and variance of each variable, then draw values for each variable independently from a standard normal
We can use a similar approach to sampling actions according to a diagonal Gaussian policy. We just have to compute the mean and covariance for each possible action, then draw from a standard normal
distribution. . There are two different ways that the covariance matrix is typically represented.
1. The log standard deviation vector, which is not a function of the state, contains one value: the log standard deviations are independent parameters.
2. The state-to-log-standard-deviation mapping is done by a neural network. It can be configured to share certain layers with the mean network if desired.
In both cases, we obtain log standard deviations rather than direct standard deviations. This is because log stds have the freedom to take on any value between \( - \infty\) and \( \infty \) (or any
number outside of that range), while stds must be nonnegative. If you don't need to enforce those restrictions, it's much easier to train parameters. The log standard deviations can be exponentiated
to obtain the standard deviations immediately, so we don't lose anything by displaying them like this.
Given the mean action \( \mu_\theta(s) \) and standard deviation \( \sigma_\theta(s) \), and a vector of noise \( z \sim \mathcal{N}(0,I)\), an action can be taken with the following formula:
$$ a = \mu_\theta(s) + \sigma_\theta(s) \cdot z$$
, where \( \cdot \) represents the elementwise product of two vectors.
The log-likelihood of a k-dimensional action \( a \) is given by:
$$ \log \pi_\theta(a | s) = - \frac{1}{2} \left ( \sum_{i=1}^k \left ( \frac{(a_i - \mu_i )^2}{\sigma_i^2} + 2 \log \sigma_i \right ) + k \log 2 \pi \right ) $$
The reason Reinforcement Learning works is because we can estimate values by taking expectations (i.e., weighted averages) of future rewards based on different policies. For example, if our policy
gives us an 80% chance of going left and a 20% chance of going right, then that means that on average we can expect to be standing at the left side of a door 80% of the time and on the right 20%.
Registering for a credit card and receiving your first statement can be a memorable experience. If you have two insurance plans that both take you to the same doors at equal probability, but one is
more likely than the other to lead us to food (i.e., it has a higher expected reward), Reinforcement Learning will pick the plan with the greater expected reward.
In this post, we've looked at a few different ways of representing policies in Reinforcement Learning. We've seen how to represent deterministic and stochastic policies as well as diagonal Gaussian
policies. In the next post, we'll look at how to actually implement these policies in code.
Did you find this article valuable?
Support Johannes Loevenich by becoming a sponsor. Any amount is appreciated! | {"url":"https://deepboltzer.codes/policy-types-in-reinforcement-learning","timestamp":"2024-11-06T05:18:12Z","content_type":"text/html","content_length":"161860","record_id":"<urn:uuid:f3bb7271-f2f8-42d2-8a61-dad76f967c7b>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00852.warc.gz"} |
Exponential Equations with Unlike Bases
Learning Outcomes
• Use logarithms to solve exponential equations whose terms cannot be rewritten with the same base
• Solve exponential equations of the form [latex]y=A{e}^{kt}[/latex] for [latex]t[/latex]
• Recognize when there may be extraneous solutions or no solutions for exponential equations
Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since [latex]\mathrm{log}\left(a\right)=\
mathrm{log}\left(b\right)[/latex] can be rewritten as a = b, we may apply logarithms with the same base on both sides of an exponential equation.
In our first example we will use the laws of logs combined with factoring to solve an exponential equation whose terms do not have the same base. Notice how we rewrite the exponential terms as
logarithms first.
Solve [latex]{5}^{x+2}={4}^{x}[/latex].
Show Solution
In general we can solve exponential equations whose terms do not have like bases in the following way:
1. Apply the logarithm to both sides of the equation.
□ If one of the terms in the equation has base [latex]10[/latex], use the common logarithm.
□ If none of the terms in the equation has base [latex]10[/latex], use the natural logarithm.
2. Use the rules of logarithms to solve for the unknown.
The following video provides more examples of solving exponential equations.
Think About It
Is there any way to solve [latex]{2}^{x}={3}^{x}[/latex]?
Use the text box below to formulate an answer or example before you look at the solution.
Show Solution
Equations Containing [latex]e[/latex]
Base e is a very common base found in science, finance, and engineering applications. When we have an equation with a base e on either side, we can use the natural logarithm to solve it. Earlier, we
introduced a formula that models continuous growth, [latex]y=A{e}^{kt}[/latex]. This formula is found in business, finance, and many biological and physical science applications. In our next example,
we will show how to solve this equation for [latex]t[/latex], the elapsed time for the behavior in question.
Solve [latex]100=20{e}^{2t}[/latex].
Show Solution
In our next example using the continuous growth formula, we have to do a couple steps of algebra to get it in a form that can be solved.
Solve [latex]4{e}^{2x}+5=12[/latex].
Show Solution
Exponential Equations with No Solutions or Extraneous Solutions
We have seen in earlier lessons on solving equations that there are some equations where a solution does not exist and or that have extraneous solutions. We will explore such examples with
exponential equations, but first, take a minute to think about when a solution to an exponential equation might not exist.
Think About It
When might an equation of the form [latex]y=A{e}^{kt}[/latex] have no solution? Write your thoughts or an example in the textbox below before you check the answer.
Show Solution
Our next example helps to illustrate that not every equation of the form [latex]y=A{e}^{kt}[/latex] has a solution.
Solve [latex]3{e}^{2x}=-6[/latex].
Show Solution
Sometimes the methods used to solve an equation introduce an extraneous solution, which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One
such situation arises when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a
logarithm function is negative, there is no output.
In the next example, we will solve an exponential equation that is quadratic in form. We will factor first and then use the zero product principle. Note how we find two solutions but reject one that
does not satisfy the original equation.
Solve [latex]{e}^{2x}-{e}^{x}=56[/latex].
Show Solution
Analysis of the Solution
When we plan to use factoring to solve a problem, we always get zero on one side of the equation, because zero has the unique property that when a product is zero, one or both of the factors must be
zero. We reject the equation [latex]{e}^{x}=-7[/latex] because a positive number never equals a negative number. The solution [latex]x=\mathrm{ln}\left(-7\right)[/latex] is not a real number, and in
the real number system, this solution is rejected as an extraneous solution.
Think About It
Does every logarithmic equation have a solution? Write your ideas, or a counter example, in the box below.
Show Solution
The inverse operation of exponentiation is the logarithm, so we can use logarithms to solve exponential equations whose terms do not have the same bases. This is similar to using multiplication to
“undo” division or addition to “undo” subtraction. It is important to check exponential equations for extraneous solutions or no solutions. | {"url":"https://courses.lumenlearning.com/wm-developmentalemporium/chapter/evaluate-exponential-functions/","timestamp":"2024-11-09T20:42:57Z","content_type":"text/html","content_length":"57816","record_id":"<urn:uuid:42a2bfcc-32be-4894-a713-17adc956a1b2>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00563.warc.gz"} |
Separate odd and even numbers in an array. – Study Algorithms
Question: Given an array arr[], write a function that separates even and odd numbers. The function should print first all the even numbers and then the odd numbers.
Input: 4, 8, 15, 16, 23, 42
Output: 4, 8, 16, 42, 23, 15
The objective of this problem is mainly to segregate the array in two halves. One half contains all the even numbers and one half contains all the negative numbers.
Naive Method:-
The most naive method is to allocate a separate empty array and then we fill it as we scan the original array. But this method will take up a lot of space and hence it is NOT RECOMMENDED
SMART METHOD:-
We can solve this problem just by a single scan of the array in this way:-
• Initialize two index variables left and right. left = 0, and right = n-1. (Here n is the size of the array, and n is the last index)
• Keep incrementing left index, until the number at that index is even. This way the even numbers stay in the beginning.
• Keep decrementing right index until the number at that index is odd. This way the odd numbers stay at the end.
• If (left < right), swap the numbers at left and right.
Here is an implementation of the above algorithm:-
void swap(int *i, int *j)
int temp = *i;
*i = *j;
*j = temp;
int seperateEvenAndOdd(int arr[], int size)
// Initialize left and right indexes
int left = 0;
int right = size - 1;
while(left < right)
// Increment left index till the number is even
// num%2 == 0, condition to check even number
while(arr[left]%2 == 0 && left < right)
// Decrement right index till the number is odd
while(arr[right]%2 == 1 && left < right)
// Time to swap
if(left < right)
swap(&arr[left], &arr[right]);
Time Complexity:- O(n)
Space Complexity:- O(1)
2 comments
8FCF7843 May 11, 2017 - 05:19
First, your example output is missing a number from your example input.
Secondly your while loops are described wrong as for the even loop it’ll continue to increment the index while the value %2 (even), meaning it’ll loop until it’s first odd value to move to the right,
but you comment it as till the number is even.
While they’re all nicely ordered in example, it’s recognized that isn’t stated as a requirement. But your own solution doesn’t produce your example output for your example input. Clearly your indexes
converge at the separation index and it’d be simple enough to sort each partial portion of the array. But without that, your output will never be the own example output for the example input.
nikoo28 May 11, 2017 - 14:55
Sorry for the typo. I have corrected it.
The code is perfect. Try to understand the while loop once again. It should loop until the left index is less than right index. Can you give me an example of a input that fails with this code? I will
be happy to help you out.
Here is a running example of the input given in the post.
2 comments
a tech-savvy guy and a design buff... I was born with the love for exploring and want to do my best to give back to the community. I also love taking photos with my phone to capture moments in my
life. It's my pleasure to have you here.
previous post
The keyword ‘struct’ is optional in C++
next post
Separate 0’s and 1’s in an array.
You may also like | {"url":"https://studyalgorithms.com/array/seperate-odd-and-even-numbers-in-an-array/","timestamp":"2024-11-04T14:49:12Z","content_type":"text/html","content_length":"297007","record_id":"<urn:uuid:73cf9c4b-4a27-46f0-8f3d-e12d5bd601ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00275.warc.gz"} |
Past Events
2008 IMLSN Workshop 3: Is Mathematics Support Worthwhile?, NUI Maynooth.
2009 IMLSN Workshop 4: The Use of Technology in Mathematics Support, DCU.
2010 IMLSN Workshop 6: Not Reinventing the Wheel – Research and Collaboration between Institutions, DCU.
2012 IMLSN Workshop 7: Promoting Learning Support and Engagement with Mathematics: a discussion of all avenues of approach including the use of technology, Queen's University, Belfast.
2014 IMLSN Workshop 8: Diversity: challenges and opportunities-enabling and supporting mathematics learning in a diverse student population, IT Tallaght.
2015 IMLSN Workshop 9: Maximizing the Impact of Digital Supports in Mathematics Learning Support in Higher Education, DCU.
2016 IMLSN Workshop 10: The key role of tutors of mathematics and statistics in Post-Secondary Education, NUI Galway.
2017 IMLSN Workshop 11: Supporting students, raising standards in maths at secondary and Higher Education level, North West Regional College, Derry/Londonderry.
2022 IMLSN Workshop 13: Mathematics Learning Support – Linking Practice to Research in the New Normal, Online.
2023 IMLSN Workshop 14: Experiences of Mathematics and Statistics Learning Support during semester 1 (2022-23), Online.
2023 IMLSN Workshop 15: Mathematics and Statistics Support post-COVID: Renewal, Redesign and Expansion (2023), Maynooth University and online.
Maths Support in Covid Times Workshops:
2020 Maths Support in Covid Times Workshop 1
2020 Maths Support in Covid Times Workshop 2
2021 Maths Support in Covid Times Workshop 3
2021 Maths Support in Covid Times Workshop 4 | {"url":"https://www.imlsn.ie/index.php/past-events","timestamp":"2024-11-10T17:43:19Z","content_type":"text/html","content_length":"24938","record_id":"<urn:uuid:fbf53428-6cd1-4c96-a2f2-3429ed10b928>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00780.warc.gz"} |
SAG Mill Grinding Circuit Design - 911Metallurgist
AG and SAG mills are now the primary unit operation for the majority of large grinding circuits, and form the basis for a variety of circuit configurations. SAG circuits are common in the industry
based on:
• High single-line capacities (leading to capital efficiency)
• The ability to mill a broad range of ore types in various circuit configurations, with reduced numbers of unit operations (and a corresponding reduction in the complexity of maintenance planning
and coordination)
• Favourable operating costs (with contributions from reduced liner/media consumption relative to conventional circuits)
Buy a Small SAG Mill
Though some trepidation concerning AG or SAG circuits accompanied design studies for some lime, such circuits are now well understood, and there is a substantial body of knowledge on circuit design
as well as abundant information that can be used for bench-marking of similar plants in similar applications. Because SAG mills rely both on the ore itself as grinding media (to varying degrees) and
on ore-dependent unit power requirements for milling to the transfer size, throughput in SAG circuits are variable. Relative to other comminution machines in the primary role. SAG mill operation is
more dynamic, and typically requires a higher degree of process control sophistication. Though more complex in AG/ SAG circuits relative to the crushing plants they have largely replaced, these
issues are well understood in contemporary applications.
AG/SAG mills grind ore through impact breakage, attrition breakage, and abrasion of the ore serving as media. Autogenous circuits require an ore of suitable competency (or fractions within the ore of
suitable competency) to serve as media. SAG circuits may employ low to relatively high ball charges (ranging from 2% to 22%, expressed as volumetric mill filling) to augment autogenous media. Higher
ball charges shift the breakage mode away from attrition and abrasion breakage toward impact breakage; as a result, AG milling produces a finer grind than SAG milling for a given ore and otherwise
equal operating conditions. The following circuits are common in the gold industry:
• Single stage AG/SAG milling
• AG/SAG mills as a primary grinding stage in a circuit with or without additional stages of comminution
• Inclusion of pebble-crushing circuits in the AG/SAG circuit
• Circuits above, but with feed preparation including two stages of crushing—typically referenced as pre-crushing
Common convention generally refers to high-aspect ratio mills as SAG mills (with diameter to effective grinding length ratios of 3:1 to 1:1), low-aspect ratio mills (generally, a mill with a
significantly longer length than diameter) are also worth noting. Such mills are common in South African operations; mills are sometimes referred to as tube mills or ROM ball mills and are also
operated both autogenously and semi-autogenously. Many of these mills operate at higher mill speeds (nominally 90% of critical speed) and often use “grid” liners to form an autogenous liner surface.
These mills typically grind ROM ore in a single stage. A large example of such a mill was converted from a single-stage milling application to a semi autogenous ball-mill-crushing (SABC) circuit, and
the application is well described. This refers to high-aspect AG/SAG mills.
Ball Charge Motion inside a SAG Mill
With a higher density mill charge. SAG mills have a higher installed power density for a given plant footprint relative to AC mills. With the combination of finer grind and a lower installed power
density (based on the lower density of the mill charge), a typical AG mill has a lower throughput, a lower power draw, and produces a finer grind. These factors often translate to a higher unit power
input (kWh/t) than an SAG circuit milling the same ore. but at a higher power efficiency (often assessed by the operating work index OWi, which if used most objectively, should be corrected by one of
a number of techniques for varying amounts of fines between the two milling operations).
In the presence of suitable ore, an autogenous circuit can provide substantial operating cost savings due to a reduction in grinding media expenditure and liner wear. In broad terms, this makes SAG
mills less expensive to build (in terms of unit capital cost per ton of throughput) than AG mills but more expensive to operate (as a result of increased grinding media and liner costs, and in many
cases, lower power efficiency). SAG circuits are less susceptible to substantial fluctuations due to feed variation than AG mills and are more stable to operate. AG circuits are more frequently (but
not exclusively) installed in circuits with high ore densities. A small steel charge addition to an AG mill can boost throughput, result in more stable operations, typically at the consequence of a
coarser grind and higher operating costs. An AG circuit is often designed to accommodate a degree of steel media for circuit flexibility. AG mills (or SAG mills with low ball charges) are often used
in single-stage grinding applications.
Based on their higher throughput and coarser grind relative to AG mills, it is more common for SAG mills to he used as the primary stage of grinding, followed by a second stage of milling. AG/SAG
circuits producing a fine grind (particularly single-stage grinding applications) are often closed with hydrocyclones. Circuits producing a coarser grinds often classify mill discharge with screens.
For circuits classifying mill discharge at a coarse size (coarser than approximately 10 mm), trommels can also be considered to classify mill discharge. Trommels are less favorable in applications
requiring high classification efficiencies and can be constrained by available surface area for high-throughput mills. Regardless of classification equipment (hydrocyclone, screen, or trommel),
oversize can be returned to the mill, or directed to a separate stage of comminution.
Many large mills around the world (Esperanza with a 12.8 m mill. Cadia and Collahuasi with 12.2-m mills, and Antamina. Escondida # IV. PT Freeport Indonesia, and others with 11.6-m mills) have
installed SAG mills of 20 MW. Gearless drives (wrap-around motors) are typically used for large mills, with mills of 25 MW or larger having been designed. Several circuits have single-line design
capacities exceeding 100,000 TPD. A large SAG installation (with pebble crusher product combining with SAG discharge and feeding screens) is depicted here below, with the corresponding process
flowsheet presented in Figure 17.9.
Adding pebble crushing as a unit operation is the most common variant to closed-circuit AG/SAG milling (instead of direct recycle of oversize material ). The efficiency benefits (both in terms of
grinding efficiency and in capital efficiency through incremental throughput) are well recognized. Pebble crushers are effective at reducing the buildup of critical-sized material in the mill load.
Critical-sized particles are those where the product of the mill feed-size distribution and the mill breakage rates result in a buildup of a size range of material in the mill load, the accumulation
of which limits the ability of the mill to accept new feed. While critical-size could be of any dimension, it is most typically synonymous with pebble-crusher feed, with a size range of 13—75 mm.
Critical-sized particles can result from a simple failure of a mill’s breakage rates to exceed the breakage rate of incoming particles, and particles generated when breaking larger particles.
Alternatively, a second type of buildup of critical-sized material can result due to a combination of rock types in the feed that have differing breakage properties. In this case, the harder fraction
of the mill feed builds up in the mill load, again restricting throughput. Examples of materials in this category include diorites, chert, and andesite. When buildup of these materials does occur,
pebble crushing can improve mill throughput even more dramatically than when the critically sized fraction results purely from a breakage rate deficit alone. For these ore types, a pebble-crushing
circuit is tin imperative for efficient circuit operation.
Currently, every AG/SAG flowsheet evaluation is likely to consider the inclusion of a pebble crusher circuit. Flowsheets that do not elect to include pebble crushing at construction and commissioning
may include provisions for future retrofitting a pebble-crushing circuit. Important aspects of pebble crusher circuit design include:
• Preparation of a clean, well-sized, and dry feed;
• Metal removal (with additional protection via metal detectors and bypass);
• Surge capacity (by using bins, or alternately and more costly, a pebble stockpile);
• Sufficient capacity (primarily a concern of large circuits where multiple pebble crushers are required to serve one grinding line);
• Design for by-passing crushers during maintenance, and
• Evaluation of the where to reintroduce the crushed pebbles back to the grinding circuit.
SAG mill by feeding crushed pebbles
The standard destination for crushed pebbles has been to return them to SAG feed. However, open circuiting the SAG mill by feeding crushed pebbles directly to a ball-mill circuit is often considered
as a technique to increase SAG throughput. An option to do both can allow balancing the primary and secondary milling sections by having the ability to return crushed pebbles to SAG feed as per a
conventional flowsheet, or to the SAG discharge. Such a circuit is depicted here on the right. By combining with SAG discharge and screening on the SAG discharge screens, top size control to the
ball-mill circuit feed is maintained while still unloading the SAG circuit (Mosher et al, 2006). A variant of this method is to direct pebble-crushing circuit product to the ball-mill sump for
secondary milling: while convenient, this has the disadvantage of not controlling the top size of feed to the ball-mill circuit. There have also been pioneer installations that have installed HPGRs
as a second stage of pebble crushing.
The unit power requirement for SAG milling (both individually and as a fraction of the total circuit power) is worthy of comment. It can be very difficult operationally to trade grind for throughput
in an SAG circuit—once designed and constructed for a given circuit configuration, an SAG mill circuit has limited flexibility to deliver varying product sizes, and a relatively fixed unit power
input for a given ore type is typically required in the SAG mill. This is particularly true for those SAG circuits designed with a coarse closing size. As a result, under-sizing an SAG mill has
disastrous results on throughput— across the industry, there are numerous examples of the SAG mill emerging as the circuit bottleneck. On the other hand, over-sizing an SAG circuit can be a poor
utilization of capital (or an opportunity for future expansion!).
Traditionally, many engineers approached SAG circuit design as a division of the total power between the SAG circuit and ball-mill circuit, often at an arbitrary power split. If done without due
consideration to ore characteristics, benchmarks against comparable operating circuits, and other aspects of detailed design (including steady-state tests, simulation, and experience), an arbitrary
power split between circuits ignores the critical decision of determining the required unit power in SAG milling. As such, it exposes the circuit to risk in terms of failing to meet throughput
targets if insufficient SAG power is installed. Rather than design the SAG circuit with an arbitrary fraction of total circuit power, it is more useful to base the required SAG mill size on the
product of the unit power requirement for the ore and the desired throughput. Subsequently, the size of the secondary milling circuit is then sized based on the amount of finish grinding for the SAG
circuit product that is required. Restated, the designed SAG mill size and operating conditions typically control circuit throughput, while the ball-mill circuit installed power controls the final
grind size.
Aside from parameters fixed at design (mill dimensions, installed power, and circuit type), the major variables affecting AG/SAG mill circuit performance (throughput and grind attained) include:
• Feed characteristics in terms of ore hardness/competency
• Feed size distribution
• Selection of circuit configuration in terms of liner and grate selection and closing size (screen apertures or hydrocyclone operating conditions)
• Ball charge (fraction of volumetric loading and ball size)
• Mill operating conditions including mill speed (for circuits with variable-speed drives), density, and total mill load
The effect of feed hardness is the most significant driver for AG/SAG performance: with variations in ore hardness come variations in circuit throughput. The effect of feed size is marked, with both
larger and finer feed sizes having a significant effect on throughput. With SAG mills, the response is typically that for coarser ores, throughput declines, and vice versa. However, for AG mills,
there are number of case histories where mills failed to consistently meet throughput targets due to a lack of coarse media. Compounding the challenge of feed size is the fact that for many ores, the
overall coarseness of the primary crusher product is correlated to feed hardness. Larger, more competent material consumes mill volume and limits throughput.
A number of operations have implemented a secondary crushing circuit prior to the SAG circuit for further comminution of primary crusher product. Such a circuit can counteract the effects of harder
ore. coarser ore. decrease the size of SAG mill required, or rectify poor throughput due to an undersized SAG circuit. Notably, harder ore often presents itself to the SAG circuit as coarser than
softer ore—less comminution is produced in blasting and primary crushing, and therefore the impact on SAG throughput is compounded.
Circuits that have used or do use secondary crushing/SAG pre-crush include Troilus (Canada), Kidston (Australia), Ray (USA), Porgera (PNG). Granny Smith (Australia), Geita Gold (Tanzania), St Ives
(Australia), and KCGM (Australia). Occasionally, secondary crushing is included in the original design but is often added as an additional circuit to account for harder ore (either harder than
planned or becoming harder as the deposit is developed) or as a capital-efficient mechanism to boost throughput in an existing circuit. Such a flowsheet is not without its drawbacks. Not
surprisingly, some of the advantages of SAG milling are reduced in terms of increased liner wear and increased maintenance costs. Also, pre-crush can lead to an increase in mid-sized material,
overloading of pebble circuits, and challenges in controlling recycle loads. In certain circuits, the loss of top-size material can lead to decreased throughput. It is now widespread enough to be a
standard circuit variant and is often considered as an option in trade-off studies. At the other end of the spectrum is the concept of feeding AG mills with as coarse a primary crusher product as
The overall circuit configuration can guide selection of die classification method of primary circuit product. Screening is more successful than trommel classification for circuits with pebble
crushing, particularly for those with larger mills. Single-stage AG/SAG circuits are most often closed with hydrocyclones.
SAG Mill vs Ball Mill
Ball Motion inside a SAG Mill Ball Motion inside a Ball Mill
To a more significant degree than in other comminution devices, liner design and configuration can have a substantial effect on mill performance. In general terms, lifter spacing and angle, grate
open area and aperture size, and pulp lifter design and capacity must be considered. Each of these topics has had a considerable amount of research, and numerous case studies of evolutionary liner
design have been published. Based on experience, mill-liner designs have moved toward more open-shell lifter spacing, increased pulp lifter volumetric capacity, and a grate design to facilitate
maximizing both pebble-crushing circuit utilization and SAG mill capacity. As a guideline, mill throughput is maximized with shell lifters between ratios of 2.5:1 and 5.0:1. This ratio range is
stated without reference to face angle; in general terms, and at equivalent spacing-to-height ratios, lifters with greater face-angle relief will have less packing problems when new, but experience
higher wear rates than those with a steeper face angle. Pulp-lifter design can be a significant consideration for SAG mills, particularly for large mills. As mill sizes increases, the required
volumetric capacity of the pulp lifters grows proportionally to mill volume. Since AG/SAG mill volume is roughly proportional to the mill radius cubed (at typical mill lengths) while the available
cross-sectional area grow’s only as the radius squared, pulp lifters must become more efficient at transferring slurry in larger mills. Mills with pebble-crushing circuits will require grates with
larger apertures to feed the circuit.
No discussion of SAG milling would be complete without mention of refining. Unlike a concentrator with multiple grinding lines, conducting SAG mill maintenance shuts down an entire concentrator, so
there is a tremendous focus on minimizing required maintenance time; the reline timeline often represents the critical path of a shutdown (but typically does not dominate a shutdown in terms of total
maintenance effort).
Reline times are a function of the number of pieces to be changed and the time required per piece. Advances in casting and development of progressively larger lining machines have allowed larger and
larger individual liner pieces.
While improvements in this area will continue, the physical size limit of the feed trunnion and the ability to maneuver parts are increasingly limiting factors, particularly in large mills. The other
portion of the equation for reline times is time per piece, and performance in this area is a function of planning, training/skill level, and equipment.
In summary:
A broad range of AG/SAG circuit configurations are in operation. Very large line plants have been designed, constructed, and operated. The circuits have demonstrated reliability, high overall
availabilities, streamlined maintenance shutdowns, and efficient operation. AG/SAG circuits can handle a broad range of feed sizes, as well as sticky, clayey ores (which challenge other circuit
configurations). Relative to crushing plants, wear media use is reduced, and plants run at higher availabilities. Circuits, however, are more sensitive to variations in circuit feed characteristics
of hardness and size distribution; unlike crushing plants for which throughput is largely volumetrically controlled. AG/SAG throughput is defined by the unit power required to grind the ore to the
closing size attained in the circuit. Very hard ores can severely constrain AG/SAG mill throughput. In such cases, the circuits can become capital inefficient (in terms of the size and number of
primary milling units required) and can require more total power input relative to alternative comminution flowsheets. A higher degree of operator skill is typically required of AG/SAG circuit
operation, and more advanced process control is required to maintain steady-state operation, with different operator/advanced process control regimens required based on different ore types.
SAG Mill Circuit Sampling
Many mills have been built based on data from inadequate sampling or from insufficient tests. With the cost of many mills exceeding several hundred million dollars, it is mandatory that geologists,
mining engineers and metallurgists work together to prepare representative samples for testing. Simple repeatable work index tests are usually sufficient for rod mill and ball mill tests but pilot
plant tests on 50-100 tons of ore are frequently necessary for autogenous or semiautogenous mills.
Preparation and selection of the test sample is of utmost importance. Procedures for autogenous and semiautogenous mill pilot plant tests are relatively simple for those experienced in running them.
Reliable and repeatable results can be obtained if simple fundamental procedures are followed.
SAG Mill Operation
In order to initiate the design of any mill, including. autogenous and semi-autogenous types, the operating conditions must be defined. This includes the basic parameters of:
A. Mill Charge
1. Charge Weight/Volume/Percent Filling.
2. Ore Specific Gravity.
3. Percent Rock or Ore Charge.
4. Percent Ball Charge (if semi-autogenous).
5. Percent Solids – in the mill.
B. Mill Power
1. Mill Speed: Percent Critical/Peripheral
2. Direction of Rotation
3. Power Draft.
4. Design Drive Power.
C. Total Feed Conditions, including total mill volumetric flow rate; recirculating load.
1. Feed Chute Design.
2. Discharge Diaphram Design.
3. Component Design.
4. Trommel or Vibrating Screen Considerations.
D. Milling Circuit Flow Sheet.
1. Types of Classifiers – Trommels, Vibrating Screens – Cyclones.
2. Mill Control Schemes
Semi Autogenous Design Factors
The design of large mills has become increasingly more complicated as the size has increased and there is little doubt that without sophisticated design procedures such as the use of the Finite
Element method the required factors of safety would make large mills prohibitively expensive.
In the past the design of small mills, up to +/- 2,5 metres diameter, was carried out using empirical formulae with relatively large factors of safety. As the diameter and length of mills increased
several critical problem areas were identified. One of the most important was the severe stressing which took place at the connection of the mill shell and the trunnion bearing end plates, which is
further aggravated by the considerable distortion of the shell and the bearing journals due to the dynamic load effect of the rotating mill with a heavy mass of ore and pulp being lifted and dropped
as the grinding process took place. Incidentally the design calculation of the deformations of journal and mill shell is based on static conditions, the influence of the rotating mass being of less
importance. An indication of shell and journal distortion is shown in Figure 1.
Investigations carried out by Polysius/Aerofall revealed that practical manufacturing considerations dictated some aspects of trunnion end design. Whereas the thickness of the trunnion in the case of
small diameter mills was dictated by foundry practice which required a minimum thickness of metal the opposite was the case in the design of large diameter mills where the emphasis was not to exceed
a maximum thickness both from the mass/casting temperature point of view and the cost aspect.
While the deformation of shell and end plates was acceptable in the case of small mills due in some extent to the over stiff construction, the deformation in the large, more flexible, mills is
relatively high. The ratio of the trunnion thickness to trunnion diameter in a mill of 2,134 m diameter is almost twice that of a mill of 5,8 m diameter, i.e. a ratio (T/D) of 0,116 to 0,069 for the
large mill.
However the design stress levels at the trunnion/head transition in the case of the large mill are almost 250% of those in the small mill.
The use of large memory high speed computers coupled with finite element methods provides the means of performing stress calculations with a high degree of accuracy even for the complex structures of
large mills. The precision with which the stress values can be predicted makes the use of safety factors based on empirical formulae generally unnecessary.
In the case of large diameter trunnion bearing mills the distortion which takes place is further compounded by the fact that the deformation varies across the width of the bearing journal due to the
fact that the end of the journal attached to the mill end plate is less liable to distortion than the outlet free end of the journal. This raises serious complications as far as the development of
the hydrodynamic fluid oil film of the bearing is concerned since the minimum oil gap may be only 0,05 mm.
An exaggerated indication of trunnion deformation is shown in Figure 2.
Obviously a thinner oil film is adequate where the deformation of the journal is less while at the unsecured end of the journal widely varying oil film thickness is necessary to maintain the correct
oil pressure to support the mill. A solution to this problem has been the advent of the hydrostatic bearing with a supply of high pressure oil pumped continuously into the bearings.
Incorporating the mill bearing journals as part of the mill shell reduced the magnitude of the problem of distortion although there is always out of round deformation of the shell. The variation
across the width of the journal surface is less pronounced than is the case with the trunnion bearing.
The replacement of a single bearing with a number of individual self adjusting bearing pads which together support the mill has lessened the undesirable effects of deformation while improving the
efficiency of the bearing.
The ability of each individual bearing-pad to adjust automatically to a more localised area of the shell journal gives rise to improved contact of the oil film with both the bearing surface and the
journal and in the case of hydrodynamic oil systems makes it unnecessary to supply oil at constant high pressure once the oil film has been established. A cross-section of a slipper pad bearing is
shown in Figure 3.
SAG Mill Operation Example
Kidston Gold Mines is a 14 000 tonnes per day rated operation located 280 kilometers west of Townsvilie in Queensland. The principle shareholder is Placer Development Limited.
Kidston’s orebody consists of 44.2 million tonnes graded at 1.79 g/t gold and 2.22 g/t silver. Production commenced in January, 1985, and despite a number of control, mechanical and electrical
problems, each month has seen a steady improvement in plant performance to a current level of over ninety percent rated capacity.
Process Plant Description
The cyanidation plant consists of a primary crushing plant, a semi-autogenous grinding circuit, agitation leaching circuit, cyclone wash circuit, gold recovery circuit and carbon regeneration
The grinding circuit comprises one 8530 mm diameter x 3650 mm semi-autogenous mill driven by a 3954 kW variable speed dc motor, and one 5030 mm diameter x 8340 mm secondary ball mill driven by a 3730
kW synchronous motor. Four 1067 x 2400 mm vibrating feeders under the coarse ore stockpile feed the SAG mill via a 1067 mm feed belt equipped with a belt scale. Feed rate was initially controlled by
the SAG mill power draw with bearing pressure as override.
Integral with the grinding circuit is a 1500 cubic meter capacity agitated surge tank equipped with level sensors and variable speed pumps. This acts as a buffer between the grinding circuit and the
flow rate sensitive cycloning and thickening sections.
SAG Mill Design and Specification
The Kidston plant was designed to process 7500 tpd fresh ore of average hardness; but to optimise profit during the first two years of operation when softer oxide ore will be treated, the process
equipment was sized to handle a throughput of up to 14 000 tpd. Some of the equipment, therefore, will become standby units at the normal throughputs of 7 000 to 8 000 tpd, or additional milling
capacity may be installed.
The SAG mill incorporates a design which allowed expedient manufacturing to high quality specifications, achieved by selecting a shell to head to trunnion configuration of solid elements bolted
together. This eliminates difficult to fabricate and inspect areas such as a fabricated head welded to shell plate, fabricated ribbed heads, plate or casting welded to the head in the knuckle area
and transition between the head and trunnion.
Motor torque at full load current is 214.4 kNm up to base speed (176 rpm) and 184.0 kNm from base to top speed. Maximum accelerating torque is 175 percent.
Operating Problems Since Commissioning
Considerable variation in ore hardness, the late commissioning of much of the instrumentation and an eagerness to maximise mill throughput led to frequent mill overloading during the first four
months of operation. The natural operator over-reaction to overloads resulted numerous mill grindouts, about sixteen hours in total, which in turn were largely responsible for grate failure and
severe liner peening. First evidence of grate failure occurred at 678 000 tonnes throughput, and at 850 000 tonnes, after three grates had been replaced on separate occasions, the remaining 25 were
renewed. The cylinder liners were so badly peened at this stage that no liner edge could be discerned except under very close scrutiny and grate apertures had closed to 48 percent of their original
open area.
The original SAG mill control loop, a mill motor power draw set point of 5200 Amperes controlling the coarse ore feeder speeds, was soon found to give excessive variation in the mill ore charge
volume and somewhat less than optimal power draw.
The first major commissioning failure on the SAG mill occurred during installation when the mill was inched without adequate lubricant flow, resulting in the wiping of the feed end trunnion bearing.
The armature, weighing 19 tonnes, together with the top half magnet frame, were trucked two thousand kilometers to Brisbane for rewinding and repairs. The mill was turning again on January 24 after a
total elapsed downtime of 14 days. After a twelve day stoppage due to a statewide power dispute in February, the mill settled down to a fairly normal operation, apart from some minor problems with
alarm monitoring causing a few spurious trips. One cause of the mysterious stoppages was tracked down to the cubicle door interlocks which stuttered whenever the mining department fired a bigger than
usual blast.
The open trunnion bearings are sealed with a rubber ring which proved ineffective in preventing ingress of water, and occasionally solids, from feed chute chokes and spillages. Contamination and
emulsification of the oil with subsequent filter choking has been responsible for nearly eighteen percent of SAG mill circuit shutdowns. Despite the very high levels of contamination, no damage has
been sustained by the bearings which has at least proved the effectiveness of the filters and other protection devices.
Design Changes and Future Operating Strategies
Design changes to date have, predictably, mostly concentrated on improving liner life and minimising discharge grate damage. Four discharge grates with thickened ends have performed satisfactorily
and a Mk3 version with separate lifters and 20 mm apertures is currently being cast by Minneapolis Electric.
Cylinder liners will continue to be replaced with high profile lifters only on a complete reline basis. While there is the problem of reduced milling capacity with reduced lifter height towards the
end of liner life, it is hoped to largely offset this by operating at higher mill speeds.
Mill feed chute liner life continues to be a problem. The original chrome-moly liners lasted some three months and a subsequent trial with 75 mm thick clamped Linhard (rubber) liners turned in a
rather dismal life of three weeks. | {"url":"https://www.911metallurgist.com/blog/sag-mill-grinding-circuit-design/","timestamp":"2024-11-06T06:15:12Z","content_type":"text/html","content_length":"209214","record_id":"<urn:uuid:c617f207-669b-412b-b7e5-e5f7e9a7e042>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00151.warc.gz"} |
Econ 2123 Problem Set
Econ 2123 Problem Set 2 Solution
Essay by Marcus Leung • April 26, 2017 • Course Note • 2,352 Words (10 Pages) • 1,866 Views
Econ 2123 Problem Set 2
Instructor: Wenwen Zhang
TA: Peter Tsui, Lawrence Ko
Lecture: L3, L4
Due date & Homework Submission Location: Before 10/14 Tue 5:30p.m. Homework Collection Box on the LSK 6th floor (Next to Econ Department, near lifts 3-4)
Name: _________________________________
Student ID: _____________________________
Lecture: ________________________________
Multiple Choices
1. Which of the following would NOT be considered part of fixed investment spending (I)?
1. Toyota buys a new robot for its automobile assembly line.
2. Apple computer builds a new factory.
3. Exxon increases its inventories of unsold gasoline.
4. An accountant buys a newly built home for herself and her family.
5. all of the above
1. Suppose the consumption equation is represented by the following: C = 250 + .75YD. Given this information, the marginal propensity to save is
1. .25.
2. .7.
3. 1.
4. 4.
5. none of the above
1. Which of the following events will cause a reduction in equilibrium output?
1. an increase in the marginal propensity to save
2. an increase in taxes
3. a reduction in the marginal propensity to consume
4. all of the above
5. none of the above
1. If C = 2000 + .9YD, what increase in government spending must occur for equilibrium output to increase by 1000?
1. 100
2. 200
3. 250
4. 500
5. 1000
1. Based on our understanding of the model presented in Chapter 3, we know that an increase in c1 (where C = c0 + c1YD) will cause
1. the ZZ line to become steeper and a given change in autonomous consumption (c0) to have a smaller effect on output.
2. the ZZ line to become steeper and a given change in autonomous consumption (c0) to have a larger effect on output.
3. the ZZ line to become flatter and a given change in autonomous consumption (c0) to have a smaller effect on output.
4. the ZZ line to become flatter and a given change in autonomous consumption (c0) to have a larger effect on output.
1. Which of the following about IS relation is NOT correct?
1. It is the relation between interest rate and savings.
2. It is the equilibrium condition for the goods market.
3. It stands for "Investment equals saving."
4. It shows what firms want to invest must be equal to what people and the government want to save.
1. Which of the following will cause an increase in the amount of money that one wishes to hold?
1. an increase in the interest rate increase
2. a reduction in the interest rate increase
3. a reduction in income
4. none of the above
1. The money demand curve will shift to the left when which of the following occurs?
1. a reduction in the interest rate
2. an increase in the interest rate
3. an open market sale of bonds by the central bank
4. an increase in income
5. none of the above
1. At the current interest rate, suppose the supply of money is less than the demand for money. Given this information, we know that
1. the price of bonds will tend increase.
2. the price of bonds will tend to fall.
3. production equals demand.
4. the goods market is also in equilibrium.
5. the supply of bonds also equals the demand for bonds.
1. Which of the following generally occurs when a central bank pursues expansionary monetary policy?
1. the central bank purchases bonds and the interest rate increases.
2. the central bank purchases bonds and the interest rate decreases.
3. the central bank sells bonds and the interest rate increases.
4. the central bank sells bonds and the interest rate decreases.
Short Questions
1. Suppose that the economy is characterized by the following behavioral equations:
[pic 1]
[pic 2]
[pic 3]
[pic 4]
(a) Solve for the equilibrium output. Compute total demand. Is it equal to production? Explain.
Equilibrium output is 1000.
Total demand=C+I+G=700+150+150=1000. Total demand equals production. We used this equilibrium condition to solve for output.
(b) Solve for disposable income.
(c) Solve for consumption spending.
(d) Assume that G is now equal to 110. Solve for equilibrium output. Compute total demand. Is it equal to production? Explain.
Output falls by (40 times the multiplier) = 40/(1-.6)=100. So, equilibrium output is now 900.
Total demand=C+I+G=160+0.6(800)+150+110=900. Again, total demand equals production.
(e) Assume that G is equal to 110, so output is given by your answer to (d). Compute private plus public saving. Is the sum of private and public saving
equal to investment? Explain.
Only available on Essays24.com | {"url":"https://www.essays24.com/essay/Econ-2123-Problem-Set-2-Solution/76327.html","timestamp":"2024-11-02T09:30:32Z","content_type":"application/xhtml+xml","content_length":"90479","record_id":"<urn:uuid:fabe7803-9548-40b7-9876-91cd44f485e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00410.warc.gz"} |
Only few people can solve this math problem without using a calculator
Solve the equation in your head
Here we have a simple math equation, it is important, however, to keep track of the order of operations you learned in school. It’s crucial to complete the task. Below you can see the problem.
At the top of the picture we see the task and then three possible answers.
Which solution do you think is the correct one?
Are you smart enough to solve it?
To make it a little easier, we thought it was only fair to give you a few options – so you have A, B, or C.
To get the right answer, one has to remember the order in which they should solve things.
Time for the answer
The first thing to do is to figure out the solution to the problem inside the parenthesis.
2 + 1 = 3
Then you have to do the multiplication. In this case, the number before the parenthesis, multiplied by the result of the problem inside it.
7 x 3 = 21
Now the rest becomes very easy.
5 + 21= 26
The answer is: 26, option B
Did you get it right? Congratulations!
Facebook Comments | {"url":"https://wakeupyourmind.net/people-can-solve-math-problem-without-using-calculator-2.html","timestamp":"2024-11-03T16:33:01Z","content_type":"text/html","content_length":"198371","record_id":"<urn:uuid:8d24f364-7586-46d5-be0f-9c66c0187c75>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00502.warc.gz"} |
Pi calculator
The π calculator is a device constructed by Adam P. Goucher in February 2010, which calculates the decimal digits of π and displays them in the Life universe as 8×10 dot matrix characters formed by
arrangements of blocks along a diagonal stripe at the top. A push reaction moves a ten-block diagonal cursor to the next position as part of the "printing" operation for each new digit.
The actual calculation is done in binary, using a streaming spigot algorithm^[1] based on linear fractional transformations. The π calculator is made up of a 188-state computer connected to a
printing device via period-8 regulators and a binary-to-decimal conversion mechanism. The complete pattern can be found in Golly's Very Large Patterns online archive,^[note 1] along with the very
similar 177-state phi calculator which uses a simpler algorithm to calculate and print the golden ratio, φ. The phi calculator ranked second place in the Pattern of the Year 2010 competition on the
ConwayLife.com forums, behind Gemini.^[2]
An updated version of the pi calculator pattern was made available with the Conway's Game of Life: Mathematics and Construction textbook.^[3] Some but not all of the calculator's components were
reduced in size, using smaller mechanisms such as the Snark and syringe that were discovered after 2010. The new pi calculator's bounding box is 38343 by 34603; it could easily be made considerably
smaller by replacing the remaining 2010-era components.
See also
1. ↑ Accessible in Golly via Help › Online Archives › Very Large Patterns › Pi calculator.
External links | {"url":"https://conwaylife.com/wiki/Pi_calculator","timestamp":"2024-11-07T02:41:15Z","content_type":"text/html","content_length":"27103","record_id":"<urn:uuid:3bb4a832-cbd3-4dc3-90b3-61ddd809973e>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00212.warc.gz"} |
sspgst - Linux Manuals (3)
sspgst (3) - Linux Manuals
sspgst.f -
subroutine sspgst (ITYPE, UPLO, N, AP, BP, INFO)
Function/Subroutine Documentation
subroutine sspgst (integerITYPE, characterUPLO, integerN, real, dimension( * )AP, real, dimension( * )BP, integerINFO)
SSPGST reduces a real symmetric-definite generalized eigenproblem
to standard form, using packed storage.
If ITYPE = 1, the problem is A*x = lambda*B*x,
and A is overwritten by inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T)
If ITYPE = 2 or 3, the problem is A*B*x = lambda*x or
B*A*x = lambda*x, and A is overwritten by U*A*U**T or L**T*A*L.
B must have been previously factorized as U**T*U or L*L**T by SPPTRF.
ITYPE is INTEGER
= 1: compute inv(U**T)*A*inv(U) or inv(L)*A*inv(L**T);
= 2 or 3: compute U*A*U**T or L**T*A*L.
UPLO is CHARACTER*1
= 'U': Upper triangle of A is stored and B is factored as
= 'L': Lower triangle of A is stored and B is factored as
N is INTEGER
The order of the matrices A and B. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
On exit, if INFO = 0, the transformed matrix, stored in the
same format as A.
BP is REAL array, dimension (N*(N+1)/2)
The triangular factor from the Cholesky factorization of B,
stored in the same format as A, as returned by SPPTRF.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 114 of file sspgst.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/docs/linux/man/3-sspgst/","timestamp":"2024-11-03T22:01:18Z","content_type":"text/html","content_length":"9043","record_id":"<urn:uuid:622e9f90-06a4-45e7-bf67-5cfe741981b8>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00112.warc.gz"} |
Hausdorff space
The terms "Hausdorff", "separated", and "preregular" can also be applied to such variations of topological spaces as uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that
unites the concept in all of these examples is that limits of nets and filters (when they exist) are unique (for separated spaces) or unique up to topological indistinguishability (for preregular
As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condition in these cases reduces to the T[0] condition. These are also the spaces in which
completeness makes sense, and Hausdorffness is a natural companion to completeness in these cases. Specifically, a space is complete iff every Cauchy net has at least one limit, while a space is
Hausdorff iff every Cauchy net has at most one limit (since only Cauchy nets can have limits in the first place).
There is a mathematicians' joke that serves as a reminder of the meaning of this term: In a Hausdorff space, points can be "housed off" from one another. | {"url":"http://www.fact-index.com/h/ha/hausdorff_space.html","timestamp":"2024-11-14T00:54:54Z","content_type":"text/html","content_length":"13276","record_id":"<urn:uuid:ffc50ff5-c474-47a5-9929-d70205d2051e>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00282.warc.gz"} |
In computational geometry, the problem of computing the intersection of a polyhedron with a line has important applications in computer graphics, optimization, and even in some Monte Carlo methods.
It can be viewed as a three-dimensional version of the line clipping problem.^[1]
If the polyhedron is given as the intersection of a finite number of halfspaces, then one may partition the halfspaces into three subsets: the ones that include only one infinite end of the line, the
ones that include the other end, and the ones that include both ends. The halfspaces that include both ends must be parallel to the given line, and do not contribute to the solution. Each of the
other two subsets (if it is non-empty) contributes a single endpoint to the intersection, which may be found by intersecting the line with each of the halfplane boundary planes and choosing the
intersection point that is closest to the end of the line contained by the halfspaces in the subset. This method, a variant of the Cyrus–Beck algorithm, takes time linear in the number of face planes
of the input polyhedron. Alternatively, by testing the line against each of the polygonal facets of the given polyhedron, it is possible to stop the search early when a facet pierced by the line is
If a single polyhedron is to be intersected with many lines, it is possible to preprocess the polyhedron into a hierarchical data structure in such a way that intersections with each query line can
be determined in logarithmic time per query.^[2]
1. ^ ^a ^b Kolingerová, Ivana (1994), "3D-line clipping algorithms – a comparative study", The Visual Computer, 11 (2): 96–104, doi:10.1007/BF01889980.
2. ^ Dobkin, David P.; Kirkpatrick, David G. (1983), "Fast detection of polyhedral intersection", Theoretical Computer Science, 27 (3): 241–253, doi:10.1016/0304-3975(82)90120-7, MR 0731064.
External links
• Intersection of convex hull with a line with pseudo code | {"url":"https://www.knowpia.com/knowpedia/Intersection_of_a_polyhedron_with_a_line","timestamp":"2024-11-07T15:18:25Z","content_type":"text/html","content_length":"69044","record_id":"<urn:uuid:f6cefe08-7158-4111-a105-5af395f4fc8a>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00343.warc.gz"} |
Distance Time Graph - Uniform Velocity
The most popular and simpler method of presenting information about distance and time is through a line graph. The form of a line graph representing distance versus time is known as a Distance-time
graph. Distance is usually placed along the y-axis and or the vertical axis, while time is generally put along the x-axis or horizontal axis. Hence, the distance-time graph is nothing but a simple
line graph that indicates distance versus time findings on the map.
When we see an object moving with a uniform velocity, the graph's slope is always straight. In simpler words, the slope of a straight line of distance-time graph represents that an object moves with
a uniform velocity. Distance time graph uniform velocity is a framework of a distance traveled by a body against time.
From this graph, we will know about the journey covered by a body along with its speed. For instance, in distance-time graph uniform velocity examples, if it is a straight horizontal line, the body
is stationary with a zero speed. If the line is diagonal then moving with a constant speed and not a straight line, then the speed varies.
How to Draw a Uniform Velocity of Distance Time Graph?
(Image will be Uploaded Soon)
Choose two points on the line- (2, 20) and (5, 50)
Slope: \[\frac{\Delta y }{\Delta x }\],
that gives us \[\left ( \frac{y_{2} - y_{1}}{x_{2} - x_{1} } \right )\]
Hence, the slope is 50-20/5-2 = 30/3 = 10
So, the ultimate velocity is 10m/s.
This particular calculation only works with uniform velocity. When it is not constant, the result will give you the average velocity of the two points.
How is a Velocity Time Graph Defined?
The graph that explains velocity against time is known as the Velocity-time graph. This explains how the velocity changes following time. The slope of a Distance time graph uniform velocity regulates
In a distance-time graph, let us consider three objects as A, B, C, and all the three objects are in a horizontal position. Object A is considered as a flat horizontal line of a uniform velocity of
the distance-time graph, which represents that the body is constantly moving. If the straight line consists of a slope, then it specifies that the body is changing its velocity at a continuous rate
or it also means that the body has a constant acceleration.
Worksheet of Distance-Time Graph
Some of the solved mathematical problems of Distance time graph uniform velocity are represented here to help students get a good grip on the subject.
(Image will be Uploaded Soon)
Question 1: The above graph shows the journey of class 8th standard’s trip to the zoo. They halted to enjoy a picnic at the zoo. The following question’s answers are represented through the medium of
• At what time the students stopped for a picnic?
• For how long did they stop?
• How far is the distance of the zoo from school?
• At what time did the bus leave the school?
• At what time did they arrive at the zoo?
• How long had they traveled before stopping for a picnic?
• 10.00 or 10 am
• 30 minutes
• 90 miles
• 9.00 or 9 am
• 11.30 or 11.30 am
• 50 miles
(Image will be Uploaded Soon)
Question 2: Emma made a journey to her grandmother’s house and back. The graph presented above portrays the details of her journey.
• For how long did Emma stay at her grandmother’s place?
• How far was she from her house at 11.45?
• At what time did she begin her journey?
• What was the total distance traveled by Emma in total?
• How far was Emma from home at 8 am?
• At what time did she leave her grandmother’s house?
• 1 hour 30 minutes
• 7.5 miles
• 7.30 am
• 80 miles
• 20 miles
• 10.00 am
The graphical representations mentioned above show the calculation process of the uniform velocity of distance-time graph physics. Students must note that it is not possible to calculate smaller and
larger values in the same graph. To learn more about different parameters of a graph, you can stay tuned with Vedantu’s online classes or students can download the official app of Vedantu to get easy
access to PDFs, sample papers, and mock question papers.
If students register themselves on Vedantu’s online official website, they can view and solve different problems related to distance-time graph uniform velocity questions including their solved
Particle Motion
As particles move along the x-axis, they move from the position of one coordinate to the position of the other. For example, from position x=0 to position x=2 (as shown below). In addition, a
particle does not make a transition in the time interval t=0 to t=3 seconds.
Velocity Calculation
Let us first start with a particle at position x=0. Since the particle moves in only the x-direction, it is represented by the value x. Since the velocity at time t=0 is 0.0, its speed at any time t
is simply the change in x: dx/dt.
For example, the particle at x=2 will move at a velocity of 2 units per second from x=2 to x=0. The velocity of this article would be 2 units per second if dx=2.0. The velocity can be represented in
one of two ways:
1. dx=2.0 units, dt=0.01 seconds
2. dx=-2.0 units, dt=-0.01 seconds
This means that there are two values for velocity in one second. The first particle has a velocity of 2.0 units, and the second particle has a velocity of -2.0 units.
In the graph below, we see the particle at the bottom of the graph. The graph shows how long it takes the particle to reach different positions. Each timestamp represents 1/10th of a second.
For example, the particle takes 0.1 seconds to reach x=1.8 and t=0.1 seconds. The particle takes 0.8 seconds to reach x=-1.8 and t=0.8 seconds. These times are represented in the graph above.
The difference between a particle and a person traveling at a constant speed in a straight line can be represented as follows. Imagine that you have two balls, one being at a constant speed and the
other being a ball at rest. When the two balls meet, the one at rest has traveled half as far as the one at a constant speed. A constant velocity represents a particle that is moving without change.
The Particle changes its Direction of Motion
The particle continues to travel at the same velocity in the same direction. However, the direction of motion does change over time.
In some situations, such as a comet, the particle travels in one direction for a short period of time before reversing direction. This is represented by changing the direction of the velocity. In
physics, the change in direction is called angular velocity. It is represented as d/dt. Angular velocity is given in units of radians per second.
This graph shows the particle changing direction. The two points with red X's represent the two velocity vectors as measured at two different time points. The particle was traveling on a straight
line at a constant speed.
The Velocity is Perpendicular to the Direction of Motion
A velocity vector can have both a direction and magnitude. The velocity of a particle can be in any direction in space, and the speed of the particle can be either positive or negative.
In a frame of reference, the direction of motion can be represented as a right angle. The velocity is perpendicular to the direction of motion. The equation V=d/dt can be used to represent the
velocity in this type of situation.
Velocity Changes in Opposite Directions
The particle travels in a straight line for a period of time. Then, it travels in the opposite direction for a period of time.
For a stationary object, the distance covered is the same in both directions. Since the velocity of the object is a vector, its magnitude is also the same, and therefore the distance travelled is the
If an object travels in a straight line for a period of time, then the distance it travels is the same as the time it takes to travel this distance. If this distance is the same in both directions,
then the object is at a rest. This can also be described as the object traveling in a straight line with constant velocity, where the velocity is zero.
To find the velocity of an object in an equation, the change in distance can be divided by the time taken to travel this distance. If the velocity of a particle changes, there are several cases that
can occur:
Case 1: The change in direction and magnitude is 90 degrees.
This is called the same velocity in opposite directions. The particle travels in a straight line for a period of time. Then, it travels in the opposite direction for a period of time. A diagram for a
particle travelling in a straight line with constant velocity for a period of time, then travelling in the opposite direction for a period of time.
When the velocity of a particle changes, the distance covered in the opposite direction is smaller than the distance covered in the original direction. This is due to the fact that the particle is
moving in the opposite direction for a smaller amount of time.
When the distance covered in the opposite direction is less than that of the original direction, the particle is speeding up. The equation for a particle that is speeding up can be found by dividing
the change in the distance by the time taken to travel the same distance. The amount of distance covered by the particle that is speeding up can be found by multiplying this ratio by the original
FAQs on Distance Time Graph - Uniform Velocity
1. Is Graph Theory Complicated?
When compared to the higher-level math classes, a theory of graph is comparatively easy. Moreover, high-standard maths will require more time and effort. However, distance-time graph uniform velocity
includes lots of proofs. So if students are not very acquainted with explanations, then they will find graph theory classes complicated. The topics that are generally taught in the introductory graph
theory class are relatively easy. However, the proofs in algebra and real analysis are quite difficult to understand. To make graph theory easier, students must thoroughly understand the chapters
instead of jumping straight to the problem.
2. There are 9 Line Segments Drawn on a Plane. Is it Possible to Intersect Each Line Segment in Exactly 3 Others?
Initially, the problem seems to be quite difficult to understand. The problem can be solved by using a graph. If students consider line segments as edges of a map, it is impossible to reach any
solution. Hence, they need to consider such a graph where each line segment represents a vertex. Now, the two vertices of the graphs are associated with each other if the parallel lines intersect.
The graph has 9 vertices. The degree of each vertex is 3
The summation of degrees of all the vertices = 2x no. of edges in the graph.
However, a sum of degrees of all vertices in the above mathematical problem is 9x3=27, i.e. an odd number. Hence, such an arrangement is not possible.
3. What is the Importance of Distance-time Graphs?
To understand distance-time graph uniform velocity, students need to study the motion of bodies. If they note a body’s movement's distance and time and place it on a rectangular graph, they will
obtain a distance parallel to its motion. It also helps them to understand whether an object is moving at a constant or variable speed. With this particular graph, students can quickly determine the
time taken to cover a specific distance in a short time. This graph shows a motion’s uniformity or non-uniformity and also whether it is accelerated or retarded.
Hence, information about motion and velocity can be represented in many ways. With tabular representation, it gets simpler for students to figure out complicated mathematical problems easily.
4. How to memorize physics concepts for a long time?
Physics is a natural science that is all about the study of matter around us, its behavior, and its application. It is basic physical science. Students can make mind maps and write down formulae for
better understanding and it also helps in retaining the concepts for a longer period of time. At Vedantu, we provide revision notes and keywords which would help students in saving time in preparing
the notes and assist them in chapter-wise revision.
5. How to manage time while writing a physics exam?
Many students say that though they went prepared to give their physics exam, due to the lengthy and tricky questions, it became difficult. To avoid such an experience students are recommended to
practice writing faster and memorizing the keywords. Also working out objective-type questions beforehand saves time in analyzing and answering them. Lastly, students are advised to divide their time
into sections namely A, B, C, and D and attempt all the questions without fail. | {"url":"https://www.vedantu.com/physics/distance-time-graph-uniform-velocity","timestamp":"2024-11-05T13:51:52Z","content_type":"text/html","content_length":"304017","record_id":"<urn:uuid:fa164fce-a299-4d5f-8952-12dbd0eb6ca2>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00149.warc.gz"} |
Difference Between
Bar vs Pascals
Bar and pascal are two units used in measurement of pressure. These units are used in fields such as chemistry, industries, physics, meteorology, weather forecasting, cardiology, and even diving. A
proper understanding in these units is required in order to excel in such fields. In this article, we are going to discuss what bar and pascal are, their definitions, the similarities between the bar
and pascal, the systems and common places these units are used and finally the difference between the bar and pascal.
The unit pascal is used to measure pressure. Pascal is denoted by the term “Pa”. In understanding the concepts of the unit pascal, one must first understand pressure. Pressure is defined as the force
per unit area applied in a direction perpendicular to the object. The pressure of a static fluid is equal to the weight of the fluid column above the point the pressure is measured. Therefore, the
pressure of a static (non-flowing) fluid is dependent only on the density of the fluid, the gravitational acceleration, the atmospheric pressure and the height of the liquid above the point the
pressure is measured. The pressure can also be defined as the force exerted by the collisions of particles. In this sense, the pressure can be calculated using the molecular kinetic theory of gasses
and the gas equation. The unit pascal is defined as the pressure created by a force of one newton acting over an area on one square meter. The pascal is the SI unit of pressure measurement. It is
used to measure stress, Young’s modulus and tensile strength apart from the more commonly known pressure measurement. The unit pascal is named after the French physicist, mathematician, writer,
philosopher and inventor Blaise Pascal. Pascal is a very small unit compared to the pressures that we experience daily. The atmospheric pressure at the sea level is about 100 Pa.
Bar is also a unit that is used to measure pressure. Bar is neither a SI unit nor a cgs unit. However, bar is accepted in many countries as a measurement of pressure. One bar is defined as 100
kilopascals. This means 1 bar is equal to 100,000 pascals. The pressure at the mean sea level is also approximately this value. Therefore, bar is a very useful unit in measuring atmospheric
pressures. The atmospheric pressure is 101.325 kilopascals to be precise. Since 1 bar is equal to 100 kilopascals, the fractional error between these two are smaller than 1%. Therefore, for most of
the cases bar is taken as the atmospheric pressure. Bar is the common pressure measurement used in fields such as meteorology and weather forecasting. Apart from the basic unit bar, units such as
millibar and decibar also exist.
│What is the difference between Pascal and Bar? │
│ │
│• Pascal is a standard SI unit while bar is not. │
│ │
│•酒吧被广泛用作实用单位,和我s famous on fields such as weather forecasting. Pascal is the standard unit, and it is used in researches and scientific documents.│ | {"url":"https://m.mcpcourse.com/difference-between-bar-and-vs-pascals/","timestamp":"2024-11-10T15:45:56Z","content_type":"text/html","content_length":"38406","record_id":"<urn:uuid:a5826b1c-7904-4d04-a3b3-36ba752beaef>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00639.warc.gz"} |
An objective is a function of variables that returns a value that an optimization package attempts to maximize or minimize. The Objective function in Pyomo declares an objective. Although other
mechanisms are possible, this function is typically passed the name of another function that gives the expression. Here is a very simple version of such a function that assumes model.x has previously
been declared as a Var:
>>> def ObjRule(model):
... return 2*model.x[1] + 3*model.x[2]
>>> model.obj1 = pyo.Objective(rule=ObjRule)
It is more common for an objective function to refer to parameters as in this example that assumes that model.p has been declared as a Param and that model.x has been declared with the same index
set, while model.y has been declared as a singleton:
>>> def ObjRule(model):
... return pyo.summation(model.p, model.x) + model.y
>>> model.obj2 = pyo.Objective(rule=ObjRule, sense=pyo.maximize)
This example uses the sense option to specify maximization. The default sense is minimize. | {"url":"https://pyomo.readthedocs.io/en/stable/pyomo_modeling_components/Objectives.html","timestamp":"2024-11-06T14:22:46Z","content_type":"text/html","content_length":"16420","record_id":"<urn:uuid:eab9df77-fccf-4ca4-b35c-d2a855ac8fb8>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00792.warc.gz"} |
IB Mathematics: Analysis and Approaches Tutors
Top IB Mathematics: Analysis and Approaches Tutors serving Amman
Andrew: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...science has taught me how to communicate complex subjects in a way that anyone can understand. I am available to tutor a variety of subjects in mathematics and biology, and I can also help my
students prepare for both the verbal and quantitative sections of the GRE. I look forward to working with you!
Miguel: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...the correct answers. In my opinion, there is no better way to assess the depth and breadth of one's knowledge than to try to convey it to others. The study habits I hope to pass on to my students
are ones that have benefited me throughout my educational career. I graduated salutatorian of my high...
Keith: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...yourself) grow in the field you are currently grappling with. I have been fortunate to do well in my own academic pursuits: earning a Bachelor's in International Affairs & History from Georegetown
University and a Master's in Education from Valparaiso University. I have taught for four years in a few different departments, but mostly in...
Palak: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...child vs. an older one and how teaching needs to be adapted to meet those ways. For example, including a fun activity for elementary school students is important to keep them engaged in school.
While on the other hand, high school and college students require more of a practical approach and have to work through...
Ezra: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...every individual student is different. My tutoring philosophy is that tutoring should by no means follow a one-size-fits-all approach. In my tutoring practice I focus on treating students as
unique individuals, determining what is effective for each student, and getting to know each student's particular academic needs and goals. When it comes to SAT prep,...
Rithika: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...and later in their studies. Many students try to memorize too much information the night before a test, and the information will ultimately be stored as short-term memory rather than long-term
memory. I ensure my students learn through evidence-based revision tips, such as active recall and spaced repetition, resulting in information being stored in students'...
Paul: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...the math section of my SATs, I was invited to begin working at a private test prep company. My age and affable nature helped foster the relationship between younger, unruly students and their
adult instructors. I received my Bachelor of Science in Physics from Florida Atlantic University and began tutoring college students, many of whom...
Bryson: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...schoolers. My favorite subjects to tutor are mathematics and Spanish, as I find them to be the most engaging, but I enjoy and can help with nearly all content areas. When I was in school, I was
(and still am!) a self-described "nerd" who really enjoyed expanding my knowledge base and taking on academic challenges....
Saad: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
I am a Finance major at Mercy College in New York, having transferred this Fall from Vanderbilt University in Nashville. My goal to simply help my students achieve the results they desire and to help
them set up the habits which will let them reproduce these results in the future.
Michelle: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...This is when the true beauty of mathematics is explored and mathematics begins to have important applications, including what to what I do in international security! I stride to provide a student
with creative materials and explanations that best suits their learning style. I am not someone who is satisfied when a student memorizes steps...
Rahath: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...the concepts thus making learning easier and fun. My aim is to help my students understand mathematics and show them the beauty through real life applications as well as critical analysis. I hope
to spark their interest and build on it thus making their math skills strong enough to take on any problem.... I believe in helping my students understand to the maximum. Time, patience and
dedication are the greatest tools for learning.
Jennifer: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...a student can surprise themselves with their own abilities is the best thing a tutor can ask for. For mathematics, I've taken up to Calculus BC in high school, and Multivariable Calculus at UCR. I
understand the concepts, but I can also explain them using applicable examples. Really, math is like a language; you need...
Eshita: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...students love because I, myself, am a student. Therefore, I can relate well with all my students. Since my experiences range far and wide, I am confident that I can provide the best services as a
tutor in my subject areas. Besides tutoring, I love to dance, play tennis, draw, and just have fun in...
Rosalyn: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...my favorite subject, and I like to tutor math because I think anyone is capable of doing math if they are taught in a method they can understand. I like to work with students to find an approach
to math problems that they comprehend. I will try multiple approaches to the same problem until I...
Stephanie: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...College in 2008 with a double degree in math and politics, and then I moved to China. I spent more than four years in China working as a teacher for Chinese and international students. I've taught
all different ages and types of students, in subjects ranging from ESL to math to SAT. I love teaching.
Zofia: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...and teaching. I have tutored students age 7 - 22 in subjects from French to Math to Biology. As a tutor I am patient and kind. My goal is for all the people I work with to gain a deep
understanding of the subject matter they are studying. I have great success in achieving this...
Priya: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...am currently a student at the University of Central Florida, majoring in Biotechnology and in the Burnett Medical Scholars Program. In high school, I was in the IB Diploma Programme and took all
three science classes plus four HL (Math, Biology, English, History) subjects. Furthermore, I took part in multiple honor societies and science organizations...
Michelle: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...and after college. I now operate my own tutoring company Expert Instruction LLC. The primary goal of my tutoring service is to help students develop a growth mindset and enjoy the process of
learning. I provide one-on-one instruction for all levels of math up to and including calculus, French language and standardized exams such as...
Gray: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...who may have trouble grasping them have only grown. I primarily tutor math topics, but I am also available to tutor for various standardized test topics such as SAT, ACT or GRE Reading. I'm a firm
believer in the idea that anyone can understand grade school level math and reading given the proper instruction and...
Jake: Amman IB Mathematics: Analysis and Approaches tutor
Certified IB Mathematics: Analysis and Approaches Tutor in Amman
...a B.A. in Computer Science and a minor in Theatre. In the fall, I'll be attending law school at Columbia Law, and I hope to pursue cybersecurity law while I'm there. I have a ton of experience
tutoring in many different areas, however my specific specialty areas are Computer Science, Math (up to Calculus 2),...
Private Online IB Mathematics: Analysis and Approaches Tutoring in Amman
Our interview process, stringent qualifications, and background screening ensure that only the best IB Mathematics: Analysis and Approaches tutors in Amman work with Varsity Tutors. To assure a
successful experience, you're paired with one of these qualified tutors by an expert director - and we stand behind that match with our money-back guarantee.
Receive personally tailored IB Mathematics: Analysis and Approaches lessons from exceptional tutors in a one-on-one setting. We help you connect with online tutoring that offers flexible scheduling.
Your Personalized Tutoring Program and Instructor
Identify Needs
Our knowledgeable directors help you choose your tutor with your learning profile and personality in mind.
Customize Learning
Your tutor can customize your lessons and present concepts in engaging easy-to-understand-ways.
Increased Results
You can learn more efficiently and effectively because the teaching style is tailored to you.
Online Convenience
With the flexibility of online tutoring, your tutor can be arranged to meet at a time that suits you.
Call us today to connect with a top Amman IB Mathematics: Analysis and Approaches tutor
(888) 888-0446
Top International Cities for Tutoring | {"url":"https://www.varsitytutors.com/amman-jordan/ib_mathematics_analysis_approaches-tutors","timestamp":"2024-11-06T16:44:35Z","content_type":"text/html","content_length":"690072","record_id":"<urn:uuid:36d7fbe1-fbee-4702-bb7c-ec80de578a60>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00144.warc.gz"} |
Ask The Wizard #347
What is the probability of getting three to a royal flush on the deal and then completing it on the draw TWICE in a span of ten hands and in the same suit?
For the first royal, the probability of getting three to a royal on the deal, in any suit, is 4*combin(5,3)*combin(47,2)/combin(52,5) = 0.01663742. The probability of completing the royal on the draw
is 1/combin(47,2) = 0.00092507. So the probability of both events is 0.01663742 * 0.00092507 = 0.00001539, or 1 in 64,974.
The probability of getting any two royals, in any two suits, this way in ten hands is combin(10,2) * 0.00001539^2 (1-0.00001539)^8 = 0.00000001065810. You also specified the two royals must be in the
same suit. The probability the second royal matches the first is 1/4, so divide the previous probability by 4 to get 0.00000000266453, which is 1 in 375,301,378.
This question is asked and discussed in my forum at Wizard of Vegas.
Consider a game show with two contestants who are both selfish and perfect logicians. Here are the rules.
1. The host places $1,000,000 on a table between the two contestants.
2. Contestant A is asked to make a suggestion on how to divide the money between the two contestants.
3. Contestant B will be asked to accept or reject the suggestion.
4. If contestant B accepts the suggestion, then they divide the money that way and the game is over.
5. If contestant B rejects the suggestion, then the host will remove 10% of the amount currently on the table.
6. The host will then ask contestant B to make a suggestion and contestant A will have the same chance to accept or reject it.
7. If contestant A accepts the suggestion, they split it that way and the game is over. If he rejects it, then the host rakes another 10% of the remaining amount on the table. Then go back to step 2
and keep repeating until a suggestion is accepted.
The question is how should contestant A suggest dividing the money on his initial turn?
He should suggest keeping 10/19 of the money for himself, less one penny, and offer B 9/19 of the money, plus a penny. In other words:
A: $526,315.78
B: $473,684.22
The key is A should put B as close as possible to an indifference point.
Let's call the ratio of the pot to the other player r. If B accepts the offer, he gets r×$1,000,000.
If B rejects the offer, then the host rakes out 10%. After which, B will have a position advantage and would offer contestant A a share of r and keep 1-r for himself.
Solving for r...
r×$1,000,000 = (1-r)×$900,000.
r×$1,900,000 = $900,000.
r = $900,000/$1,900,000 = 9/19.
A does not want B to be completely indifferent, lest a chose randomly and stand a chance of the host raking the pot. So, A should throw in the extra penny to B and offer him (9/19) × $1,000,000 +
$0.01 = $473,684.22.
A: $526,315.78
B: $473,684.22
This question is asked and discussed in my forum at Wizard of Vegas.
BetMGM sometimes offers what they call a "Risk Free Bet," although it's not risk free. I think a better term would be a "second chance" bet. Here are the rules.
1. The player makes a bet, subject to a maximum amount, on any event (no parlays, teasers, etc.)
2. If the bet wins, it wins and the player is paid normally.
3. If the bet loses, the player is given a promotional bet equal to the amount he lost.
4. The promotional bet may also bet bet on any one event.
5. If the promotional bet wins, the player is paid the winnings. If the promotional bet loses, then the player gets nothing. Either way, the promotional bet is taken away.
Here are my questions:
1. What would be the value of a $100 Risk Free Bet if played against the spread at -110 odds?
2. What strategy do you recommend?
First, let's look at betting against the spread at -110 odds. Let's assume a 50% chance of winning each bet.
• There is a 50% chance you win the original bet and profit $90.91.
• There is a 25% chance you lose the original bet and win the second one. Here you will have lost $100 and won $90.91, for a net win of -$9.09.
• There is a 25% chance you lose both bets for a loss of $100.
The expected value of this promotional bet is 0.5×$90.91 + 0.25×-9.09 + 0.25×-100 = $18.18.
Second, what do I recommend? I suggest betting on the biggest longshot you can find. At the time you asked this question, the biggest longshot I could find was this college football game:
Miami (FL) +575
Alabama -1000
Assuming the house edge is the same on both bets, the probability of Miami winning is 14.01%. This would result in a house edge of 5.41% both ways.
Let's assume if the player loses he will find another game at the same odds to use his second chance on. That said, here are the possible outcomes:
• There is a 14.01% chance you win the original bet and profit $575.00.
• There is a 12.05% chance you lose the original bet and win the second one. Here you will have lost $100 and won $575, for a net win of $475.
• There is a 25% chance you lose both bets for a loss of $100.
The expected value of this promotional bet is 0.1401×$575 + 0.1205×$475 + 0.7394×-$100 = $63.87.
Bottom line is to throw a Hail Mary both times. This advice is true of "use once" promotional chips in general. Unfortunately, such chips are usually restricted to even money bets.
This question is asked and discussed in my forum at Wizard of Vegas. | {"url":"https://wizardofodds.com/ask-the-wizard/347/","timestamp":"2024-11-04T08:29:16Z","content_type":"text/html","content_length":"80098","record_id":"<urn:uuid:c5df0681-d674-41a6-8588-564e88fcbdd5>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00191.warc.gz"} |
What is the molality of a solution of phosphoric acid, H_3PO_4 that contains 24.5 g of phosphoric acid (molar mass 98.0 g) in 100 g of H_2O? | HIX Tutor
What is the molality of a solution of phosphoric acid, #H_3PO_4# that contains 24.5 g of phosphoric acid (molar mass 98.0 g) in 100 g of #H_2O#?
Answer 1
The molality is 2.50 mol/kg.
The formula for molality #b# is
#color(blue)(|bar(ul(color(white)(a/a) b = "moles of solute"/"kilograms of solvent"color(white)(a/a)|)))" "#
#"Moles of H"_3"PO"_4 = 24.5 color(red)(cancel(color(black)("g H"_3"PO"_4))) × ("1 mol H"_3"PO"_4)/(98.0 color(red)(cancel(color(black)("g H"_3"PO"_4)))) = "0.250 mol H"_3"PO"_4#
#"Mass of solvent" = "100 g" = "0.100 kg"#
∴ #b = "0.250 mol"/"0.100 kg" = "2.50 mol/kg"#
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
To find the molality of the solution:
1. Calculate the moles of phosphoric acid using its mass and molar mass.
2. Calculate the mass of water in kilograms.
3. Use the formula for molality: ( \text{molality} = \frac{\text{moles of solute}}{\text{mass of solvent in kg}} ).
• Mass of phosphoric acid (( H_3PO_4 )): 24.5 g
• Molar mass of ( H_3PO_4 ): 98.0 g/mol
• Mass of water (( H_2O )): 100 g
1. Moles of ( H_3PO_4 ): [ \text{moles} = \frac{\text{mass}}{\text{molar mass}} = \frac{24.5 , \text{g}}{98.0 , \text{g/mol}} = 0.25 , \text{mol} ]
2. Mass of water in kg: [ \text{mass of water (kg)} = \frac{100 , \text{g}}{1000} = 0.1 , \text{kg} ]
3. Molality: [ \text{molality} = \frac{0.25 , \text{mol}}{0.1 , \text{kg}} = 2.5 , \text{mol/kg} ]
The molality of the solution of phosphoric acid (( H_3PO_4 )) in water is 2.5 mol/kg.
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/what-is-the-molality-of-a-solution-of-phosphoric-acid-h-3po-4-that-contains-24-5-8f9af85a73","timestamp":"2024-11-08T15:36:59Z","content_type":"text/html","content_length":"581293","record_id":"<urn:uuid:fe12b9b5-c16b-47de-925f-f94099959b79>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00011.warc.gz"} |
MythBusters, Falling, Stopping, and Integration
In a MythBusters episode some time ago, Adam and Jamie jumped off a building. There was some cool stuff in this, but I want to focus on the acceleration data they collected. Before jumping into a pit
of foam, they first wanted to test the set up by dropping a dummy into it and measuring the accelerations. Lucky for me, they showed a quick screen shot of their data. Note: I previously posted the
calculations for jumping and stopping off of a building.
For me, I see this and think - numerical integration. Before that, let me look at the physics. Here is a diagram of someone jumping off a building.
In my first analysis of this, I looked at the landing in terms of force and displacement. For this data, I have the acceleration and time. When you have force (which I kind of do if I know the mass)
and time, you should think about the momentum principle:
Compare this to the work-energy principle which deals with force and displacement:
So here, I am going to use both of these principles. The work-energy for falling until the faller hits the mat and then momentum principle for stopping. First for the fall. I will take the faller and
the Earth as the system. This means that there is no work done on the faller while falling, but there is gravitational potential energy. I can write the work-energy principle as (using the numbers
from the diagram above). Final note - I am going to let the gravitational potential be zero at the top of the mat.
Now for the landing. Let me assume that the mat exerts a constant force (which it clearly does not) over the time interval. Then I can write the momentum principle (in the y-direction) as:
The initial momentum and velocity were in the negative y-direction. This is why the change in momentum (in the y-direction) is positive. Remember, I am assuming the force is constant over this
interval. So, I could re-write this as:
Suppose I plotted the net force as a function of time, here is a sketch.
I already know what the product of F-net and &Delta t should be (this is called the impulse, by the way), but here you can see that F[net]*&Delta t would be the area under the force-time curve (clear
to see since it is a box-shape). But what if it is not a box? What if it is something more complicated?
Numerical Integration
Here are the traditional options for dealing with the area under a curve:
• If you know the force as a function of time, you could analytically determine the impulse.
• If you have a print out of the force as a function of time, you can print out your curve on thick paper. Find the mass of the paper. Cut out the piece with the function and the part below it and
find its mass. The impulse will be the max F times the max time multiplied by the ratio of the cut out mass to the total mass.
• If you have force-time data points, you can break this integration into a whole bunch of small pieces. This is numerical integration.
Suppose a part of my force-time data looks like this:
If I take a pair of points at a time, I can find the impulse just for these two because the shape is a trapezoid. Here is another diagram.
Here, the area of that piece will be:
Here, I am calling the area &Delta I where I is the impulse. The &Delta means that it is just a small piece of the total impulse. Also note that for the area, the "width" is the difference in times
and the "height" is the average of those two forces. It probably wouldn't be a bad approximation to call the height F[y1] and not use the average.
Getting the data
How do I go from a picture of the graph to the actual data? I used GraphClick. This is a Mac application that basically lets you load the picture of the graph and then click on the data. It will then
translate the pixel data to x-y data. Very useful in this case. You could probably do something like this Tracker Video and I am sure there are other applications that do the same thing in Mac OS X
as well as windows and linux.
There were two falls the MythBusters recorded. One was on an air-bag and one was in a dumpster with foam stuff.
If everything is working out correctly, these two jumps should have the same impulse for landing. I can write the momentum principle as:
If they both have the same momentum right BEFORE landing and they both end up stopped, then both falls have the same change in momentum. This means that both falls should have the same impulse
(impulse is F-&Delta t part). Ok, next question. The data is the acceleration. Is this the same as the net force? Well, it should be proportional.
Now, here is the data. I calculated the impulse two ways. The first is with the trapezoid method I showed above. The second way is just using rectangular area pieces. You can see that the two
impulses are pretty close.
There are two sheets - one for each fall. The "impulse" (because it is the integration of acceleration over time and not force) for the two are about the same, 0.54 g's*s versus 0.60 g's*s. These
could actually be closer. I don't have all the data, the screen shot cut it off after a little bit of time. Overall, I think it worked pretty well.
More like this
Maybe this could fall under my "physics of parkour". It could also apply to the MythBusters "dumpster diving" episode. In both of these cases, the question is: how far can you jump off of something
and not severely hurt yourself. They do this a lot in parkour. Here are some examples: There are…
I finally saw the movie Iron Man. It was good. I feel that I am qualified to evaluate the movie. When I was in high school, I was totally into comic books. Mostly Spider-man, but I still have a
significant collection of Iron Man comics. Ok, now you know I am not an Iron Man attacker. I will…
July 4th can be fun. One activity my family enjoys is playing in the lake at my parents house. Along with this comes the jumping off the dock. Great fun, and great physics. Here is a short clip. Work
Energy Example from Rhett Allain on Vimeo. Notice that I violated my own rules for making…
**Pre Reqs:** [What is a Force](http://scienceblogs.com/dotphysics/2008/09/basics-what-is-a-force.php) [Previously, I talked about the momentum principle](http://scienceblogs.com/dotphysics/2008/10/
basics-forces-and-the-moment…). Very useful and very fundamental idea. The other big (and useful)… | {"url":"https://www.scienceblogs.com/dotphysics/2010/01/12/mythbusters-falling-stopping-a","timestamp":"2024-11-06T08:36:32Z","content_type":"text/html","content_length":"44715","record_id":"<urn:uuid:a8378309-0a97-4d5b-805f-5f26d9e9fb34>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00702.warc.gz"} |
CS 100 - Lecture 002 Report
The report explains the second lecture of the CS Freshmen Lecture Series, conducted by Habib University.
% CS-100 Fall 2019 Guest Lecture Report % Use this template to write a 250-word (at max.) report on the guest lecture. \documentclass{report} \usepackage[utf8]{inputenc} \usepackage{hyperref} \title
{Easy and Hard Problems in Computer Science} %Title must be written exactly as ifiedspec \author{By Dr. Shahid Hussain \\ \\ Reviewed by Mohammad Ibrahim Ali} %Speaker name must be written exactly as
specified \date{2 September 2019} % Date when report was written \begin{document} \par \maketitle This report reviews the first guest lecture of the CS Freshmen Seminar, delivered by Dr Shahid
Hussain, \textit{Program Director} and \textit{Assistant Professor Computer Science} at \textit{Habib University}. The lecture aimed to highlight how a computational problem is classified. \medskip \
par The lecturer stated that computational problems are differentiated by their level of difficulty. He gave the example of \textbf{Travelling Salesperson Problem (TSP)}, in which six points were
pointed on a map. The problem was to find the shortest route to travel through these points while starting and ending at the same point. Since there was no other way except to check each route and
calculating the distance or time, it concluded that the increase in the number of points could turn the problem harder. Hence, it was classified as a \textbf{hard problem}. Contrary to this, the idea
of an \textbf{easy problem} was discussed, which was defined as a problem that doesn’t require more time to solve with the increase in data, as in sorting numbers. \medskip \par The lecturer also
highlighted some interesting facts like, \begin{itemize} \item Even a fast computer that could compute 1 million operations per second would require 2 million years to solve a TSP problem with 25
points. \item Hard problems could be not-very hard, and easy problems could be not-very easy, which leads to another classification. \end{itemize} \medskip \par The lecture concluded to the idea that
a programmer should be able to detect the problem as easy or hard and should prefer easy problems, to bring productivity in his work. \bigskip \section*{References} \url{http://bit.ly/2lOsUYI} \end | {"url":"https://cs.overleaf.com/articles/cs-100-lecture-002-report/pkgxxwhkrjxv","timestamp":"2024-11-07T23:43:03Z","content_type":"text/html","content_length":"38838","record_id":"<urn:uuid:538483d6-bef0-4d9a-ac12-6ea7b3a773e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00550.warc.gz"} |
The hidden danger of the Win32 TreeView
¶The hidden danger of the Win32 TreeView
Random performance anecdote time.
I once had a bug filed on VirtualDub regarding a performance problem in its hex viewer on large files. (I have a habit of putting random features into my open-source tools; it never ceases to amaze
me that people actually use them.) The problem turned out to be in a code fragment like this:
while(GetNextChunk(chunkInfo)) {
TVINSERTSTRUCT tvItem;
CreateTreeViewItem(tvItem, chunkInfo);
TreeView_InsertItem(hwndTV, tvItem);
I had expected that I'd done something stupid in the hex viewer code. When I profiled the routine under VTune revealed that for large files, though, I discovered that this routine was spending almost
no time in VirtualDub.exe itself -- it was spending a huge amount of time in the TreeView_InsertItem() call. This is a call to the Win32 tree view control to insert an item. Investigation into the
disassembly around the hotspot revealed that the Win32 tree view internally stores its nodes as a singly-linked list and adding an item to the end takes linear time according to the number of items.
This meant that in order to add N items to the tree list, a total of N^2 steps were required, making the tree initialization quadratic time. In case you're not familiar with asymptotic complexity,
here's my cheat sheet:
• O(1) / constant time: Wheeeeeeeee!!!!!
• O(N) / linear time: Very fast.
• O(N log N): Fairly fast, scales OK.
• O(N^2) / quadratic time: Go to lunch and hope it's done by the time you get back.
I ended up solving this problem in two ways: I changed the routine to insert items in reverse order at the beginning instead of in forward order at the end, and I split the chunk list into two levels
to reduce the maximum child count within a tree node.
Scalability problems are the worst kind of performance issues to deal with because the performance effects can be drastic and the fixes dangerously invasive, i.e. rewrite. The main danger is that
it's really easy to nest fairly fast operations and end up with a composite operation that is O(N^2) or worse. On more than one occasion I've seen people unnecessarily calling linear-time operations
like strlen() in a loop, and that simple error ends up turning an ordinarily fast operation into a painfully slow one. | {"url":"https://www.virtualdub.org/blog2/entry_184.html","timestamp":"2024-11-14T01:43:17Z","content_type":"text/html","content_length":"5123","record_id":"<urn:uuid:e0d0dff3-3f3f-487e-9d7b-67b33f658855>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00436.warc.gz"} |
Error while loading data in the Database
I am facing problem that , i have two million data in a csv file . when i start loading some where in the file data is disturbed and that is broke in two ldifferent line...so my first questin is that
how can i check that which line has this type of problem.
Second thisng is that when i enter that data (Broken dat in two lines) it gives error at those lines because data disturbed in different column of the db ..so it gives dat trucation error in the
So plese help how can i handle tis problem in Clover..it is very critical for me...
Apart of this...some time i get run time error becase of the wrong data in file...so this breaks my programe.....what i want that if there is this type of data problm then it load that error line in
the report.csv and the rest of the data should e loade in th database,
Means error should be logged in the file and the rest of the data should be loaded in the database.
Error coming in the file..
1) Line breaking
2) End of file is not currect
Please help me how to handle this....
Hauman Mishra
• Hello,
if you set proper data policy on the Reader (see dataPolicy), data , which can't be read are skipped. If you want to control db loading set on DBOUtputTable maxErrors attribute on some positive
value and connect output port to it (see DBOutputTable).
• Can we log those records in a file or in DB(error table) , so that we can have report that these are the records having problem...otherwise it is not usefull....because we must have solid reason
to show that this is the eroor.
Hanuman Mishra
• Hi,
when you connect logging port to DataReader (DataReader) you can do with wrong record what you want eg. write them to flat file (DataWriter) or do database (DBOutputTable); the same with the
DBOutputTable - if db error occurs wrong records (with error message) can be processed by any way. See Data policy example and DB Load example
• I am getting error in the data that , is in the data..
dta is like this:-
"1473227","31580","16","","9","4","372-372","1962","Addendum to "Single-transit, large-radius E-type devicest"","","Nunn, W.M., Jr.","IEEE","","",""
how to load in DB..
actually it is title is devided in two parts..
Addendum to "Single-transit
large-radius E-type devicest"
Now how can i handle thsi situation..please help me i am in trouble..i have relise of the project
• Hello,
it is not possible to read such data directly. Work around can be Transformat component, which will prepare such data for data base: eg.
<?xml version="1.0" encoding="UTF-8"?>
<Graph id="1222247484704" name="test" revision="1.43">
<Metadata id="Metadata0" previewAttachmentCharset="ISO-8859-1">
<Record fieldDelimiter="," name="data" previewAttachmentCharset="ISO-8859-1" recordDelimiter="\n" type="delimited">
<Field name="field1" type="string"/>
<Field name="field2" type="string"/>
<Field name="field3" type="string"/>
<Field name="field4" type="string"/>
<Field name="field5" type="string"/>
<Field name="field6" type="string"/>
<Field name="field7" type="string"/>
<Field name="field8" type="string"/>
<Field name="field9" type="string"/>
<Field name="field10" type="string"/>
<Field name="field11" type="string"/>
<Field name="field12" type="string"/>
<Field name="field13" type="string"/>
<Field name="field14" type="string"/>
<Field name="field15" type="string"/>
<Metadata id="Metadata1" previewAttachmentCharset="ISO-8859-1">
<Record fieldDelimiter="|" name="error" previewAttachmentCharset="ISO-8859-1" recordDelimiter="\n" type="delimited">
<Field name="rec" type="integer"/>
<Field name="field" type="integer"/>
<Field name="offending" type="string"/>
<Field name="message" type="string"/>
<Property fileURL="workspace.prm" id="GraphParameter0"/>
<Phase number="0">
<Node dataPolicy="controlled" enabled="enabled" fileURL="${DATAIN_DIR}/data.txt" id="DATA_READER0" quotedStrings="true" type="DATA_READER"/>
<Node id="TRASH0" type="TRASH"/>
<Edge fromNode="DATA_READER0:0" id="Edge0" inPort="Port 0 (in)" metadata="Metadata0" outPort="Port 0 (output)" toNode="TRASH0:0"/>
<Edge fromNode="DATA_READER0:1" id="Edge1" inPort="Port 0 (in)" metadata="Metadata1" outPort="Port 1 (logs)" toNode="REFORMAT0:0"/>
<Phase number="1">
<Node id="REFORMAT0" type="REFORMAT">
<attr name="transform"><![CDATA[//#TL
int listIndex =0;
list l;
function createString(){
string s = l[listIndex];
boolean found = false;
if (isnull(s) or length(s) == 0) {
return null;
if (char_at(s,0).eq.'"') s=substring(s,1,length(s)-1);
if (char_at(s,length(s)-1).eq.'"') {
found = true;
s = substring(s,0,length(s)-1);
do {
s = concat(s, l[listIndex]);
if (char_at(s,length(s)-1).eq.'"') {
found = true;
s = substring(s,0,length(s)-1);
} while (!found);
return s;
// Transforms input record into output record.
function transform() {
$field1 := nvl(createString(),'');
$field2 := nvl(createString(),'');
$field3 := nvl(createString(),'');
$field4 := nvl(createString(),'');
$field5 := nvl(createString(),'');
$field6 := nvl(createString(),'');
$field7 := nvl(createString(),'');
$field8 := nvl(createString(),'');
$field9 := nvl(createString(),'');
$field10 := nvl(createString(),'');
$field11 := nvl(createString(),'');
$field12 := nvl(createString(),'');
$field13 := nvl(createString(),'');
$field14 := nvl(createString(),'');
// Called during component initialization.
// function init() {}
// Called after the component finishes.
// function finished() {}
<Node debugPrint="true" id="TRASH1" type="TRASH"/>
<Edge fromNode="REFORMAT0:0" id="Edge2" inPort="Port 0 (in)" metadata="Metadata0" outPort="Port 0 (out)" toNode="TRASH1:0"/>
• I am using java transform whic is using
public boolean transform(DataRecord[] source, DataRecord[] target){
so please can you send me this transformer in Java.
How should i take list as you explained in your example..
• in java you can use string split method or org.jetel.util.string.StringUtils.split method.
• My question is not that how to split data///// but my question is that how to transform
DataRecord[] source
So please help me...just write initial code that how to use DataRecord[] source in your programe..
• import org.jetel.component.DataRecordTransform;
import org.jetel.data.DataRecord;
import org.jetel.exception.TransformException;
import org.jetel.util.string.StringUtils;
public class Transform extends DataRecordTransform {
public int transform(DataRecord[] inputRecords, DataRecord[] outputRecords)
throws TransformException {
String[] list = StringUtils.split(inputRecords[0].getField("offending").getValue().toString());
for (int i = 0; i < outputRecords[0].getNumFields(); i++) {
return 0;
private String createString() {
// TODO Auto-generated method stub
return null;
• i am not able to jhande because in metadat my seperator in ,
and when i read by
for(int i=0;i<source>1473227
1----->Addendum to ""Single-transi
1-----> large-radius E-type"" devicest"
1----->Nunn, W.M., Jr.
How can i handle that please help..
• Hi Please help me for the same which i discueed as above....
Can i use String str="\",\"" known as "," as seperator......if i use simple java code like BufferReader and Writers the i use seperator as ",", it works file....but my data is tool long 2
milllion so it is very slow while inserting in DB sand finding deduplicates...
So Please help me how can it be solved it, is road block for me , and it is going to screw my clover application...
i want to break my data at "," sepeartor..........
Please help help....
Hanuman mIshra
• It should work. All you need then, is to strip the quote from the begin of first field and the end of last field.
• TransformationGraph institutionGraph = new TransformationGraph("ArticlesLoadGraph");
DataRecordMetadata inputMetaData= feedController.MetadataInput;
i am using ","for seperator and "\n"record seperator.....
But clover is not reading like that...
"371","18","9","","33","1","104-109","1988","Extension of the optimality of the threshold policy in heterogeneous multiserver queueing systems","10.1109/9.371","Viniotis, I.","IEEE","","",""
"417","21","35","","26","1","8-15","1988","G.722: a new CCITT coding standard for digital transmission of wideband audio signals","10.1109/35.417","Mermelstein, P.","IEEE","","",""
out put
Performance characteristics of a 1.5 μm single-frequency semiconductor laser with an external waveguide Braff reflector
Olsson, N.A.
,"24","5","766-774","1988","The azimuthal effective-index method","10.1109/3.192","Marcatili, E.A.J.","IEEE","","",""
Comments on 'Dynamic path planning for a mobile automation with limited information on the environment' [with reply]
Lucas, C.
,"35","5","400-402","1988","An electronic device for triggering a stimulator during membrane responsiveness determinations in cardiac tissues","10.1109/10.1401","Pruett, J.K.","IEEE","","",""
On estimating the instantaneous frequency of a Gaussian random signal by use of the Wigner-Ville distribution
White, L.V.
,"36","4","433-439","1988","Cepstral domain talker stress compensation for robust speech recognition","10.1109/29.1547","Chen, Y.","IEEE","","",""
Influence of Ag<sup>+</sup>-Na<sup>+</sup> ion-exchange equilibrium on waveguide index profiles
Ramaswamy, R.V.
,"24","6","1114-1117","1988","Mn<sup>2+</sup> as a potential solid-state laser ion","10.1109/3.234","Clausen, R.","IEEE","","",""
CW arc-lamp-pumped alexandrite lasers
Samelson, H.
,"23","3","875-877","1988","A MOS implementation of totally self-checking checker for the 1-out-of-3 code","10.1109/4.334","Tao, D.L.","IEEE","","",""
Analysis of lateral-mode behavior of "win-stripe"lasers related to the negative slope in their current-light characteristics
Watanabe, M.
,"24","1","83-93","1988","Wavelength-tunable electrooptic polarization conversion in birefringent waveguides","10.1109/3.97","Heismann, F.","IEEE","","",""
So please llok into this...can i have your contact no or so that i can talk you..i ahve dead line....it is very critical for me..
Hnauman MIshra
• Do you use last Clover.ETL version and DataReader component (not DelimiterDataReader)?
My output for your data is:
|Record |field1 |field2 |field3 |field4 |field5 |field6 |field7 |field8 |field9 |field10 |field11 |field12 |field13 |field14 |field15 |
|# 1 |"1473227 |31580 |16 | |9 |4 |372-372 |1962 |Addendum to "Single-transit, large-radius E-type devicest" | |Nunn, W.M., Jr. |IEEE | | |" |
|# 2 |"371 |18 |9 | |33 |1 |104-109 |1988 |Extension of the optimality of the threshold policy in heterogeneous multiserver queueing systems |10.1109/9.371 |Viniotis, I. |IEEE | | |" |
|# 3 |"417 |21 |35 | |26 |1 |8-15 |1988 |G.722: a new CCITT coding standard for digital transmission of wideband audio signals |10.1109/35.417 |Mermelstein, P. |IEEE | | |" |
So all is needed, is to remove quote on the beginning of 1st field and on the end of last field.
• Yes,
You are right but i found one big difference that DeLimiter handles automatically line breaks but..in the DataReared it consider that line as incorrect data and skip that line ..f i put
Controlled policy....
Is it works like that or i am wrong some where...so please suggest me that how should i handle line breaks..
Please sign in to leave a comment. | {"url":"https://support1.cloverdx.com/hc/en-us/community/posts/4415681703442-Error-while-loading-data-in-the-Database?sort_by=votes","timestamp":"2024-11-04T11:32:44Z","content_type":"text/html","content_length":"91387","record_id":"<urn:uuid:3af9a944-aee2-4603-903b-1e19f2a68157>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00817.warc.gz"} |
aymericdamien/TensorFlow-Examples - Related Repos
aymericdamien / TensorFlow-Examples
TensorFlow Tutorial and Examples for Beginners (support TF v1 & v2)
☆43,423Updated 3 months ago
Related projects ⓘ
Alternatives and complementary repositories for TensorFlow-Examples
• This repository contains code examples for the Stanford's course: TensorFlow for Deep Learning Research.
☆10,330Updated 3 years ago
• The most cited deep learning papers
☆25,507Updated 9 months ago
• Deep learning library featuring a higher-level API for TensorFlow.
☆9,618Updated 6 months ago
• Models and examples built with TensorFlow
☆77,159Updated this week
• Code for Tensorflow Machine Learning Cookbook
☆6,238Updated 5 months ago
• Deep Learning for humans
☆61,989Updated this week
• TensorFlow - A curated list of dedicated resources http://tensorflow.org
☆17,204Updated 2 weeks ago
• Deep Learning papers reading roadmap for anyone who are eager to learn this amazing tech!
☆38,304Updated last year
• Library for fast text representation and classification.
☆25,938Updated 7 months ago
• A curated list of awesome Deep Learning tutorials, projects and communities.
☆24,228Updated 6 months ago
• Oxford Deep NLP 2017 course
☆15,682Updated last year
• scikit-learn: machine learning in Python
☆60,092Updated this week
• Scalable, Portable and Distributed Gradient Boosting (GBDT, GBRT or GBM) Library, for Python, R, Java, Scala, C++ and more. Runs on sing…
☆26,284Updated this week
• An Open Source Machine Learning Framework for Everyone
☆186,311Updated this week
• TensorFlow Tutorials with YouTube Videos
☆9,280Updated 3 years ago
• Simple and ready-to-use tutorials for TensorFlow
☆16,392Updated last year
• TensorFlow tutorials and best practices.
☆8,624Updated 4 years ago
• A fast, distributed, high performance gradient boosting (GBT, GBDT, GBRT, GBM or MART) framework based on decision tree algorithms, used …
☆16,687Updated this week
• Data science Python notebooks: Deep learning (TensorFlow, Theano, Caffe, Keras), scikit-learn, Kaggle, big data (Spark, Hadoop MapReduce,…
☆27,450Updated 7 months ago
• Caffe: a fast open framework for deep learning.
☆34,117Updated 3 months ago
• ⛔️ DEPRECATED – See https://github.com/ageron/handson-ml3 instead.
☆25,194Updated last year
• Simple tutorials using Google's TensorFlow Framework
☆6,006Updated last year
• Learn how to design, develop, deploy and iterate on production-grade ML applications.
☆37,569Updated 2 months ago
• A curated list of deep learning resources for computer vision
☆10,822Updated last year
• The fastai deep learning library
☆26,279Updated 3 weeks ago
• From the basics to slightly more interesting applications of Tensorflow
☆5,641Updated 2 years ago
• A complete daily plan for studying to become a machine learning engineer.
☆28,155Updated 5 months ago
• This repository provides state of the art (SoTA) results for all machine learning problems. We do our best to keep this repository up to …
☆8,945Updated 5 years ago
• A comprehensive list of pytorch related content on github,such as different models,implementations,helper libraries,tutorials etc.
☆15,457Updated 9 months ago
• The "Python Machine Learning (1st edition)" book code repository and info resource
☆12,274Updated 7 months ago | {"url":"https://relatedrepos.com/gh/aymericdamien/TensorFlow-Examples","timestamp":"2024-11-13T11:37:40Z","content_type":"text/html","content_length":"58721","record_id":"<urn:uuid:478ecea8-ef18-40f7-a42e-e7157e662004>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00574.warc.gz"} |
How Many Mg In A Gram? - CLT Livre
How Many Mg In A Gram?
Is 1000 mg the same as 1 gram?
Relationship Between Milligrams and Grams – The relationship between mg to g is quite easy to understand; despite differing from each other, both the units share a connection and history. The
relationship between milligrams and grams is explained as follows: 1 milligram equals 0.001 grams.1 gram is equal to 1000 milligrams.
How many mg makes 1 gram?
How many mg in a gram? – Grams and milligrams are a metric unit of measurement, meaning they all divide neatly into each other!
There are 1,000 milligrams in 1 gram. There are 1,000 micrograms in 1 milligram. There are 1,000,000 micrograms in 1 gram.
How many grams in a milligram? Well just the inverse! 1/1000 th of a gram. How many grams in a microgram? 1/millionth of a gram. (Similarly, if you’re wondering how many milliLITERS are in a LITER,
the answer is also 1000!)
Is 100 mg equal to 1g?
How many mg are there in a gram? – 1g = 1,000mg, As there are 1,000 milligrams (mg) in 1 gram (g), to convert your gram figure to milligrams you should multiply your figure by 1000.
Is 1 kg the same as 1000 mg?
Kilograms to Milligrams conversion 1 gram (kg) is equal to 1000000 milligrams (mg).
How many Milligrams make a 1 kg?
Therefore, kilogram = 10, 00, 000 mg.
Is 1 gram heavier than 1 milligram?
A gram is bigger than a milligram.1 gram = 1000 mg. So, 1 gram is 1000 times bigger than a milligram.
How many kg is 1mt?
1 metric ton = 1000 Kg, and we all know 1 Kg = 1000 gram.
Is 1000mg 1g true or false?
1 gram is equivalent to 1000 miligrams.Q.
How many kg makes 1 gram?
How to convert Grams to Kilograms.1 gram (g) is equal to 0.001 kilograms (kg).
How much is 7gs called?
Quarter – A quarter stands for one quarter of an ounce of weed. Lindsey Bartlett/Insider A quarter is 7 grams of weed, or a quarter of an ounce. This is another popular measurement to buy because it
lasts longer. Seven grams is equivalent to about 7 joints and maybe 10 bowls, depending on the size of your pipe. A quarter can also serve some small at-home baking projects.
Is 1 ml water 1g?
Remember that 1mL of water weighs 1g. In other words, the density of water is 1g/mL.
How much is 1 gram of sugar?
We independently select these products—if you buy from one of our links, we may earn a commission. All prices were accurate at the time of publishing. I always check nutrition labels when I’m at the
grocery store and use them to help pick the best option among the variety of pre-packaged foods I purchase. I check sugar, salt, and fat contents, but to me all those numbers (measured in grams) are
an abstract relative.
What does a gram of sugar—or salt, or fat—really look like? I went on a mission to find out: We’re used to measuring things by volume here in the US, and adding butter, sugar, and salt to our
recipes by the cups and tablespoons. Since grams are a unit of weight, I pulled out my trusty kitchen scale to find the answers to my curiosity.
Here’s what I discovered. A gram of salt clocked in at about 1/6 tsp, making it the heaviest ingredient. But salt is composed of chloride as well, with only 40% of its weight accounting for pure
sodium. Doing some math there gives us about a 1/2 tsp of salt to amount to 1 gram of sodium,
• Next up was the sugar.
• I used granulated white sugar for this experiment and expected it to take a whole teaspoon of the stuff to equate to a gram.
• Surprisingly though, a gram of sugar weighed in just a tad shy of 1/4 tsp by volume,
• Finally, that scary three-letter word called fat.
• I used pure lard that I picked up at my local farmer’s market for this test.
It took 1/4 tsp of lard by volume to weigh 1 gram, The next thing I wondered is what this all looks like in the amounts of some of the pre-packaged foods I purchase. So I got out a box of the Amy’s
Apple Toaster Pops that I enjoy every now and then.10 grams of sugar, 3.5 grams of fat in a single serving.
1. Here’s looking at you, kid.
2. Seeing things like that was a bit surprising.
3. That’s a whole lot of sugar in that little toaster pastry! So then I took out a bottle of soy sauce.
4. I was always curious how much salt was really in a single serving of that which I pour onto stir-fried noodles.
5. I buy the low sodium variety and for this particular brand 1 tablespoon of soy sauce had about 1/4 tsp of salt (575mg).
The regular soy sauce nearly doubles that amount, so it’s definitely not for those on a sodium diet. I didn’t find too many other pre-packaged items in my cupboard, but I did spot a fresh batch of
biscuits on the counter. I flirted with the idea of breaking that recipe down into grams of fat and salt, but I quickly abandoned that idea once I tore off a piece.
Which is heavier 1 gram or 1000 milligrams?
1 gram = 1000 mg. So, 1 gram is 1000 times bigger than a milligram. What does a milligram measure? A milligram is a unit in the SI system used to express small objects’ mass. | {"url":"https://cltlivre.com.br/blog/usa/how-many-mg-in-a-gram.html","timestamp":"2024-11-04T21:04:23Z","content_type":"text/html","content_length":"114555","record_id":"<urn:uuid:a53f69a6-0bad-45b6-af1d-f88886efbe6d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00772.warc.gz"} |
Dark Energy Survey Year 3 results: Constraints on extensions to Λ CDM with weak lensing and galaxy clustering
We constrain six possible extensions to the Λ cold dark matter (CDM) model using measurements from the Dark Energy Survey's first three years of observations, alone and in combination with external
cosmological probes. The DES data are the two-point correlation functions of weak gravitational lensing, galaxy clustering, and their cross-correlation. We use simulated data vectors and blind
analyses of real data to validate the robustness of our results to astrophysical and modeling systematic errors. In many cases, constraining power is limited by the absence of theoretical predictions
beyond the linear regime that are reliable at our required precision. The Λ CDM extensions are dark energy with a time-dependent equation of state, nonzero spatial curvature, additional relativistic
degrees of freedom, sterile neutrinos with eV-scale mass, modifications of gravitational physics, and a binned σ[8](z ) model which serves as a phenomenological probe of structure growth. For the
time-varying dark energy equation of state evaluated at the pivot redshift we find (w[p],w[a])=(-0.9 9[-0.17]^+0.28,-0.9 ±1.2 ) at 68% confidence with z[p]=0.24 from the DES measurements alone, and
(w[p],w[a])=(-1.0 3[-0.03]^+0.04,-0. 4[-0.3]^+0.4) with z[p]=0.21 for the combination of all data considered. Curvature constraints of Ω[k]=0.0009 ±0.0017 and effective relativistic species N[eff]=
3.1 0[-0.16]^+0.15 are dominated by external data, though adding DES information to external low-redshift probes tightens the Ω[k] constraints that can be made without cosmic microwave background
observables by 20%. For massive sterile neutrinos, DES combined with external data improves the upper bound on the mass m[eff] by a factor of 3 compared to previous analyses, giving 95% limits of (Δ
N[eff],m[eff])≤(0.28 ,0.20 eV ) when using priors matching a comparable Planck analysis. For modified gravity, we constrain changes to the lensing and Poisson equations controlled by functions Σ (k
,z )=Σ[0]Ω[Λ](z )/Ω[Λ ,0] and μ (k ,z )=μ[0]Ω[Λ](z )/Ω[Λ ,0], respectively, to Σ[0]=0. 6[-0.5]^+0.4 from DES alone and (Σ[0],μ[0])=(0.04 ±0.05 ,0.0 8[-0.19]^+0.21) for the combination of all data,
both at 68% confidence. Overall, we find no significant evidence for physics beyond Λ CDM .
Physical Review D
Pub Date:
April 2023
□ Astrophysics - Cosmology and Nongalactic Astrophysics
Updated to match published version and fix a citation reference. 46 pages, 25 figures, data available at https://dev.des.ncsa.illinois.edu/releases/y3a2/Y3key-extensions | {"url":"https://ui.adsabs.harvard.edu/abs/2023PhRvD.107h3504A","timestamp":"2024-11-09T17:55:25Z","content_type":"text/html","content_length":"74161","record_id":"<urn:uuid:d7c5f801-fa49-46ba-9a22-ab2bb76b7b39>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00370.warc.gz"} |
Introduction to Digital Filters: with Audio Applications
by Julius O. Smith III
Publisher: W3K Publishing 2007
ISBN/ASIN: 0974560715
ISBN-13: 9780974560717
Number of pages: 478
A digital filter can be pictured as a “black box” that accepts a sequence of numbers and emits a new sequence of numbers. In digital audio signal processing applications, such number sequences
usually represent sounds. For example, digital filters are used to implement graphic equalizers and other digital audio effects. This book is a gentle introduction to digital filters, including
mathematical theory, illustrative examples, some audio applications, and useful software starting points. The theory treatment begins at the high-school level, and covers fundamental concepts in
linear systems theory and digital filter analysis. Various “small” digital filters are analyzed as examples, particularly those commonly used in audio applications. Matlab programming examples are
emphasized for illustrating the use and development of digital filters in practice.
Download or read it online for free here:
Read online
(online html)
Similar books
Fourier Transform - Signal Processing
Salih Mohammed Salih
InTechThis book focuses on the Fourier transform applications in signal processing techniques. Topics covered: DFT, FFT, OFDM, estimation techniques and the image processing techniques. Written for
electrical engineers, communication engineers, etc.
Signal Computing: Digital Signals in the Software Domain
M. Stiber, B.Z. Stiber, E.C. Larson
University of Washington BothellThe specific topics we will cover include: physical properties of the source information, devices for information capture, digitization, compression, digital signal
representation, digital signal processing and network communication.
Hidden Markov Models: Estimation and Control
R. J. Elliott, L. Aggoun, J. B. Moore
SpringerThe aim of this book is to present graduate students with a thorough survey of reference probability models and their applications to optimal estimation and control. Readers are assumed to
have basic grounding in probability and systems theory.
Digital Filters and Signal Processing
F.P.G. Marquez, N. Zaman (ed.)
InTechDigital filters, together with signal processing, are being employed in the new technologies and information systems, and implemented in different areas and applications. This book presents
advanced developments, covering different cases studies. | {"url":"http://e-booksdirectory.com/details.php?ebook=1256","timestamp":"2024-11-12T09:31:40Z","content_type":"text/html","content_length":"12135","record_id":"<urn:uuid:5a41c6d1-6960-4895-9ac3-b6120055a857>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00680.warc.gz"} |
Euler Allocation for Performance Measurement | Published in Variance
1. Introduction
If a property and casualty insurer would like to allocate its capital for the purpose of performance measurement, the literature suggests numerous ways to do so.
One particularly appealing option is the Euler allocation. This method is the only one that guarantees that if a company writes more business in lines where the expected return on allocated capital
is greater (less) than the overall company expected return on capital, then the overall company expected return on capital will increase (decrease). In the standard application of this approach,
there is an underlying assumption that the risk of a line is reflected solely by the amount of capital allocated to it. In particular, the cost of equity capital is assumed to be the same for all
lines of business.
There is, however, empirical evidence that the cost of equity capital may vary by line of business. In this case, we show that using an Euler allocation to allocate capital by line for performance
measurement is not always appropriate.
1.1. Research context
Venter (2009) surveys capital allocation methods in the literature.
Tasche (2007) defines the Euler allocation method for allocating capital and discusses some important properties of this method. Tasche (1999) derives the key result regarding the use of Euler
allocation for performance measurement. In Tasche’s approach, the risk of a line is reflected by the amount of capital allocated to it. There is an implicit assumption that the cost of equity capital
does not vary by line of business.
Cummins and Phillips (2005) provide empirical evidence that the cost of equity capital may vary by line of business. Their analysis suggests “significant differences in the cost of equity capital
across lines.”
1.2. Objective
We give a simple example that highlights why using an Euler allocation to allocate capital for the purpose of performance measurement is not always appropriate if the cost of equity capital varies by
line. In particular, our example shows how writing more business in a line that (based on an Euler allocation) is performing worse than average can actually improve the results of the company
1.3. Outline
The remainder of the paper proceeds as follows. Section 2 will discuss Tasche’s result regarding the use of Euler allocation for performance measurement. In Section 3 we consider the case when the
cost of equity capital varies by line of business and give a simple example that illustrates why an Euler allocation is not always appropriate for performance measurement in this context. Section 4
continues the discussion from Section 3. Section 5 concludes.
2. Background: Euler allocation and RORAC
Insurance companies charge premiums to cover the cost of expected claims. Actual claims costs may end up being much higher than expected, for example, due to a catastrophic event, and so an insurance
company must have additional funds—economic capital—available to ensure it can meet its obligations in this case. The amount of capital a firm will hold is typically determined by (or at least guided
by) a risk measure such a value at risk, tail value at risk or standard deviation as well as by regulatory requirements. Once the total amount of capital has been determined, the firm might want to
allocate this capital by line of business and/or geographic region. This may be helpful for reasons such as risk management—for example, to understand which lines/regions are driving the need to hold
capital—and performance measurement (which will be discussed in more detail below). An Euler allocation is a particular method of allocating capital which is often considered very suitable for
performance management purposes. If the risk measure being used to determine the overall capital requirement satisfies certain properties, we will see later in Section 2 that an Euler allocation will
exist for that risk measure.
The notation, definitions and wording in the remainder of Section 2 are largely based on Tasche (1999) and Tasche (2007). The wording has been modified slightly to reflect a focus on an insurance
(rather than investment) context. The main result in Section 2 (Proposition 2.2) is based on Theorem 4.4 in Tasche (1999) and Proposition 2.1 in Tasche (2007).
Suppose that real-valued variables X[1],…,X[n] are given and represent the profits and losses of the various lines of business written by an insurance company. Let X denote the companywide portfolio
profit/loss, i.e.,
It is useful to allow for some dynamics in this model by introducing variables u=(u[1],…u[n]):
Then we have obviously X=X(1,…1). For the purposes of this paper we assume that the probability distribution of the random variable (X[1],…,X[n]) is fixed and that the u[i] only take on values close
to 1 (i.e., the company’s current mix of business will not be changing drastically).
Definition 2.1
• A non-empty set U in R^n is homogeneous if for each u in U and t>0, t*u is in U.
• A function h:U R is homogeneous if U is homogeneous and for each u in U and t>0, h(t*u)=t*h(u).
Proposition 2.1 tells us that differentiable homogeneous functions can be represented as a weighted sum of their derivatives in a canonical manner. This result is stated (in a more general form) in
Tasche (1999) and follows from Euler’s theorem on homogeneous functions.
Proposition 2.1: Let U be a non-empty open set in R^n and h:U R be a real-valued function. If h is totally differentiable, then it is homogeneous if and only if for all u in U,
We define a risk measure to be a function from U to R. We assume that the economic capital (EC) required by the company (i.e., capital as a buffer against high losses) is determined by a homogeneous
and totally differentiable risk measure i.e.:
Proposition 2.1 tells us that
Definition 2.2: Let (X[i]|X) be the capital allocated to line i. Then
• The total portfolio Return on Risk Adjusted Capital is defined by
• The portfolio-related RORAC of the i-th line is defined by
Definition 2.3: Let X denote the portfolio-wide profit/loss.
• A capital allocation (X[1]|X),…,(X[n]|X) of the total economic capital (X) satisfies the full allocation property if
• A capital allocation (X[1]|X),…,(X[n]|X) is RORAC compatible if there exist ε[i]>0 such that
for all 0 < h < ε[i].
Proposition 2.2: Let be a risk measure. Assume that is homogeneous and totally differentiable. If there is a capital allocation (X[1]|X),…,(X[n]|X) that is RORAC compatible in the sense of Definition
2.3 for arbitrary expected values m[1],…,m[n] of X[1],…X[n], then (X[i]|X) is uniquely determined as
In this case, there also exist ε[i]>0 such that
for all 0 < h < ε[i].
Proposition 2.2 is the key result regarding the use of Euler allocation for performance measurement. It tells us that if an Euler allocation exists, then it is the unique RORAC compatible allocation
that satisfies the full allocation property. In particular, with an Euler allocation if a company writes more business in lines where the expected return on allocated capital is greater (less) than
the overall company expected return on capital, then the overall company expected return on capital will increase (decrease).
3. Main result: Euler allocation and excess RORAC
Proposition 2.2 underlines the suitability of using an Euler allocation for performance measurement when RORAC is the performance metric. However, if the cost of equity capital varies by line, a
RORAC compatible capital allocation may not be the best choice for the company. Cummins and Phillips (2005) suggest that the variation in the cost of equity capital by line may be quite significant.
The overall company cost of equity capital is a weighted average of the by-line costs of equity capital, weighted by the capital allocated to each line. If the cost of equity capital is the same for
all lines of business, then clearly the overall company cost of capital will not change if there are small changes in the volume of business written in each line. In this case, an increase in the
overall company RORAC due to these small changes is always beneficial to the company.
If the cost of equity capital varies by line of business, however, it may not always be the case that an increase in the company RORAC due to small changes in the volume of business written in each
line is beneficial to the company. In this case, we must also consider any possible impact to the overall company cost of equity capital due to these small changes. This observation motivates
Definition 3.1.
Definition 3.1: Let (X[1]|X),…,(X[n]|X) be an allocation of the total economic capital that satisfies the full allocation property. Let t[i] be the cost of equity capital for line i, and t be the
overall cost of equity capital for the company. Then
• The total portfolio excess return on risk adjusted capital is defined by
• The portfolio-related excess RORAC of the i-th line is defined by
The definition of excess RORAC for a line i is basically the same as the definition of economic value added on capital (EVAOC) for a line i in Cummins (2000), except that we require the capital
allocation to satisfy the full allocation property and we do not specify any constraints on how to define the profit/loss of a line.
For the remainder of Section 3, we consider an increase in the overall company excess RORAC (“XS RORAC”) as being beneficial to the company. In other words, excess RORAC (rather than RORAC) is our
performance metric.
This means that, for example, we consider a situation where the RORAC is 22% and the cost of equity capital is 20% to be preferable to a situation where the RORAC is 11.5% and the cost of equity
capital is 10%, since an excess RORAC of 2% is considered preferable to an excess RORAC of 1.5%.
Note that if the cost of equity capital is the same for all lines of business, then small changes in the volume of business written in each line will result in an increase in the overall company
excess RORAC if and only if it will result in an increase in the overall company RORAC.
We now consider whether using an Euler allocation to allocate capital by line for the purpose of performance measurement is appropriate in this context. In particular, for all i, does there exist ε
[i]>0 such that
for all 0 < h < ε[i]?
The following result suggests that this is not the case in general.
Proposition 3.1: Let be a risk measure. Assume that is homogeneous and totally differentiable. Let (X[i]|X) = [Euler](X[i]|X) be the Euler allocation of the total economic capital (X), and R=XS RORAC
(X) be the total portfolio excess RORAC. Then for each i,
Proof: Let R= XS RORAC(X) = (E[X]/ - t. Note that The result follows from the Quotient Rule.
We assume for the two notes below that n=2 and XS RORAC(X[1]|X) > XS RORAC(X).
Note that:
• If t[1]=t[2] (i.e., the cost of equity capital doesn’t vary by line), then t=t[1]=t[2] (i.e., t is a constant and does not depend on u[1] or u[2]) so that and so our assumption that XS RORAC(X[1]
|X) > XS RORAC(X) implies that i.e., R is an increasing function of u[1]. This implies that there exists some ε[i]>0 such that XS RORAC(X+hX[1]) > XS RORAC(X) for all 0 < h < ε[i], so that an
Euler allocation is appropriate for the purpose of performance measurement.
• In general, to ensure that we need which is not always the case.
The following simple example highlights why an Euler allocation is not always appropriate for the purpose of performance measurement if the cost of equity capital varies by line and excess RORAC is
the performance metric.
Suppose a company writes two lines of business and capital is allocated to the two lines based on an Euler allocation.
We assume that and that the overall capital required by the company is 100.
We further assume that X[1] and X[2] are independent so that Cov(X[1],X[2])=0, and also assume that Var(X[1])=Var(X[2])=1250=Var(X)/2 and that u[1]=u[2]=1.
Note that (X) is homogeneous and totally differentiable and that
Setting Z = u[1] X[1] + u[2] X[2][,] note also that
and so
The expected profit, allocated capital, RORAC, cost of equity capital and excess RORAC for each line and for the company overall are shown in Table 1.
Using the approach of Proposition 2.2 (i.e., RORAC is the performance metric), Line 1 would be considered the worse performing line since its RORAC is lower than that of the company overall (6% vs
7%). Proposition 2.2 tells us that writing more business in Line 1 would lower the overall company RORAC.
The approach of Section 3 (i.e., excess RORAC is the performance metric) would also suggest that Line 1 is the worse performing line since its excess RORAC is lower than that of the company overall
(0.0% vs. 0.25%). This means that for the Euler allocation to be suitable for performance measurement, writing more business in Line 1 should result in a lower overall company excess RORAC.
However, applying the Quotient Rule, and using the fact that Var(X) = 2Var(X[1]), Var(X[2]) = Var(X[1]) and u[1]=u[2]=1, we have:
This means that the overall company excess RORAC will increase if we increase the volume in Line 1. So the Euler allocation is not suitable for performance measurement in this case where the cost of
equity capital is not the same for the two lines of business and excess RORAC is the performance metric. Note that if the cost of equity capital was the same for both lines, we would have t[1] = t
[2], and so which would be consistent with a capital allocation that is suitable for performance measurement (which is what we would expect due to Proposition 2.2).
4. Additional result: Euler allocation and relative RORAC
As well as the excess RORAC, a company may also be interested in the relative RORAC, i.e., the RORAC divided by the cost of equity capital. For example, if a company has a choice between two options
with the same expected excess RORAC (say, 2%) but different costs of equity capital (say, 10% and 20%), it may prefer the option with the lower cost of equity capital as this may be considered less
risky. This option will also have a higher relative RORAC (since, for example, 12%/10% = 1.2 is greater than 22%/20% = 1.1).
With this in mind, we give the following definitions:
Definition 4.1: Let (X[1]|X),…,(X[n]|X) be an allocation of the total economic capital that satisfies the full allocation property. Let t[i] be the cost of equity capital for line i, and t be the
overall cost of equity capital for the company. Then
• The total portfolio relative return on risk adjusted capital is defined by
• The portfolio-related relative RORAC of the i-th line is defined by
Although the relative RORAC will likely be of less interest to the company than the excess RORAC, in this section we will examine it in much the same way that we examined the excess RORAC in the
previous section.
So in Section 4, relative RORAC (“Rel RORAC”) is our performance metric.
This means that, for example, we consider a situation where the RORAC is 11.5% and the cost of equity capital is 10% to be preferable to a situation where the RORAC is 22% and the cost of equity
capital is 20% since a relative RORAC of 1.15 is considered preferable to a relative RORAC of 1.1.
Note that if the cost of equity capital is the same for all lines of business, then small changes in the volume of business written in each line will result in an increase in the overall company
relative RORAC if and only if it will result in an increase in the overall company RORAC.
We now consider whether using an Euler allocation to allocate capital by line for the purpose of performance measurement is appropriate in this context. In particular, for all i, does there exist ε
[i]>0 such that
and for all 0 < h < ε[i]?
The following result suggests that this is not the case in general.
Proposition 4.1: Let be a risk measure. Assume that is homogeneous and totally differentiable. Let (X[i]|X) = [Euler](X[i]|X) be the Euler allocation of the total economic capital (X), and R=Rel
RORAC(X) be the total portfolio relative RORAC. Then for each i,
Proof: Let R= Rel RORAC(X) = E[X]/(t (X)). Note that The result follows from the Quotient Rule.
We assume for the two notes below that n=2 and Rel RORAC(X[1]|X) > Rel RORAC(X).
Note that:
• If t[1]=t[2] (i.e., the cost of equity capital doesn’t vary by line), then t=t[1]=t[2] (i.e., t is a constant and does not depend on u[1] or u[2]), and so our assumption that Rel RORAC(X[1]|X) >
Rel RORAC(X) implies that i.e., R is an increasing function of u[1]. This implies that there exists some ε[i]>0 such that Rel RORAC(X+hX[1]) > Rel RORAC(X) for all 0 < h < ε[i], so that an Euler
allocation is appropriate for the purpose of performance measurement.
• In general, to ensure that we need which is not always the case.
We now return to the simple example from Section 3 but with relative RORAC (instead of excess RORAC) as the performance metric.
Suppose a company writes two lines of business and capital is allocated to the two lines based on an Euler allocation.
We assume that and that the overall capital required by the company is 100.
We further assume that X[1] and X[2] are independent so that Cov(X[1],X[2])=0, and also assume that Var(X[1])=Var(X[2])=1250=Var(X)/2 and that u[1]=u[2]=1.
Note that (X) is homogeneous and totally differentiable and that
Note also that
and so
The expected profit, allocated capital, RORAC, cost of equity capital and relative RORAC for each line and for the company overall are shown in Table 2.
Using the approach of Proposition 2.2 (i.e., RORAC is the performance metric), Line 1 would be considered the worse performing line since its RORAC is lower than that of the company overall (6% vs
7%). Proposition 2.2 tells us that writing more business in Line 1 would lower the overall company RORAC.
The approach of Section 4 (i.e., relative RORAC is the performance metric) would also suggest that Line 1 is the worse performing line since its relative RORAC is lower than that of the company
overall (1.00 vs. 1.04). This means that for the Euler allocation to be suitable for performance measurement, writing more business in Line 1 should result in a lower overall company relative RORAC.
However, applying the Quotient Rule, and using the fact that Var(X) = 2Var(X[1]), Var(X[2]) = Var(X[1]) and u[1]=u[2]=1, we have:
This means that the overall company relative RORAC will increase if we increase the volume in Line 1. So the Euler allocation is not suitable for performance measurement in this case where the cost
of equity capital is not the same for the two lines of business and relative RORAC is the performance metric. Note that if the cost of equity capital was the same for both lines, we would have t[1] =
t[2], and so which would be consistent with a capital allocation that is suitable for performance measurement (which is what we would expect due to Proposition 2.2).
5. Conclusions
Performance measurement is an important application of capital allocation and Proposition 2.2 highlights why an Euler allocation is particularly appropriate for this purpose when the cost of equity
capital does not vary by line. In particular, with an Euler allocation if a company writes more business in lines where the expected return on allocated capital is greater (less) than the overall
company expected return on capital, then the overall company expected return on capital will increase (decrease).
The example presented in this paper illustrates that the equivalent result in not always true if the cost of equity capital varies by line of business.
So although an Euler allocation (if it exists) will always be suitable for performance measurement when RORAC is the performance metric, it may not be suitable for performance measurement if the cost
of equity capital varies by line and excess RORAC or relative RORAC is the performance metric. This means that care must be taken when using an Euler allocation to allocate capital by line for the
purpose of performance measurement when the cost of equity capital varies by line. | {"url":"https://variancejournal.org/article/39642","timestamp":"2024-11-05T18:40:29Z","content_type":"text/html","content_length":"398590","record_id":"<urn:uuid:c61a898d-461b-4ac0-8f29-7b30a3f27b75>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00618.warc.gz"} |
Multivariate binary-continuous twin model
Replied on Fri, 03/17/2023 - 16:51
I'd recommend separating your thinking about the *model convergence* from the *confidence interval convergence*. The
functions are best suited to getting your model to converge on the best parameter estimates. This is literally a separate optimization step from getting the best confidence intervals.
The function omxRunCI() is designed just for running the confidence interval optimization. There's more information on it [here](https://search.r-project.org/CRAN/refmans/OpenMx/html/
omxParallelCI.html) at the help page. The big thing to try is switching optimizers for the CIs. The best optimizer for your parameter estimates might not be the best for your CIs. The different
optimizers have rather different strategies for getting CIs.
If that all fails and you are willing to dance with the devil, then you could try standard-error-based Wald-type confidence intervals (gasp!). Standard errors for arbitrary algebras are available
from mxSE() and Wald-type confidence intervals from confint(). I think the latter will only give CIs for the parameters, not algebras.
Replied on Fri, 03/17/2023 - 19:53
AdminRobK Joined: 01/24/2014
In reply to ?omxRunCI by AdminHunter
To briefly elaborate
To briefly elaborate on Hunter's post:
• You can get "better" standard errors, and therefore, "better" standard-error-based Wald-type confidence intervals, if you pass your MxModel object through imxRobustSE().
• Look into mxBootstrap() and related functions for yet another way to get confidence intervals.
Replied on Sun, 03/19/2023 - 13:50
In reply to To briefly elaborate by AdminRobK
Thanks both for your speedy
Thanks both for your speedy response, this is extremely helpful.
I have re-run my model with the 'CSOLNP' optimiser (without any other adjustments, still using mxTryHard) and am now receiving more meaningful output:
confidence intervals:
lbound estimate ubound note
MZ.Acor[3,2] 0.04128288 0.4411438 1
CI details:
parameter side value fit diagnostic statusCode method
1 MZ.Acor[3,2] lower 0.04128288 11548.07 success OK neale-miller-1997
2 MZ.Acor[3,2] upper 1.00000000 11548.04 success OK neale-miller-1997
I am, however, curious to use omxRunCI. As far as I understand, the function omxRunCI() only works when a model has been run through mxRun(). I have tried to run my ACE model using mxRun() but have
come across the following error:
Error: The thresholds matrix associated with the expectation function in model 'MZ' is not of the same length as the 'threshnames' argument provided by the expectation function. The 'threshnames'
argument is of length 4 and the expected covariance matrix has 2 columns.
I think it comes down to the way my thresholds have been aligned, but I am unsure of how to fix this. Any suggestions?
Once I get past this, I can run the model using mxRun and request CIs seperately via omxRunCI(). I am thinking something along the lines of:
ACEModel <-mxModel( ACEModel, mxCI (c ('Acor[3,2]') ))
ACEfit <- mxRun(ACEModel, intervals=F)
omxRunCI(AceFit, verbose = T, optimizer = "CSOLNP") #optimiser TBC
On a related note, is there a particular optimiser that would work best in this type of situation based on your experience (e.g. CSOLNP/ NPSOL)?
Thank you again for your time!
Replied on Mon, 04/03/2023 - 11:51
AdminHunter Joined: 03/01/2013
>>> As far as I understand, the function omxRunCI() only works when a model has been run through mxRun().
The mxTryHard() function is just a wrapper around mxRun() that wiggles start values around a few times. So, you can use omxRunCI() after mxTryHard() the same as you would after mxRun().
>>> I have tried to run my ACE model using mxRun() but have come across the following error:
I'm not sure why you'd be getting an error like that from mxRun() but nor from mxTryHard(). My first guess is you aren't actually running the same model in both cases, and one of them has this error.
>>> On a related note, is there a particular optimiser that would work best in this type of situation based on your experience (e.g. CSOLNP/ NPSOL)?
I'll let other folks chime in with their experiences for this.
Replied on Tue, 04/04/2023 - 08:13
In reply to Odd by AdminHunter
MxRun still gives me errors
Thanks so much for your helpful reply, I really appreciate your time!
I have re-tried running this same model with mxRun but still getting the same error. I have no errors when I run the model with mxTryHard(). When I request omxRunCI() on the MxTryHard model I get the
same error message:
> omxRunCI(AceFit, verbose = T, optimizer = "CSOLNP") # doesnt work with mxTryhard
Error: The thresholds matrix associated with the expectation function in model 'MZ' is not of the same length as the 'threshnames' argument provided by the expectation function. The 'threshnames'
argument is of length 4 and the expected covariance matrix has 2 columns.
I suspect that this may be due to the way my thresholds have been set up and potentially the way I arrange them in the threshnames= argument.
e.g. I set the thresholds up as a 1x2 matrix:
# Setting up thresholds - one threshold regardless of twin order and zygosity
Thresh <-mxMatrix(type="Full", nrow=1, ncol=2, free=TRUE, values=c(3), lbound=c(-6), ubound=c(6),
labels=c("Tmz11", "Tmz22"), name="Th" )
Thre <-mxAlgebra( expression= Th+ (bAge%x%age)+ (bSex%x%sex) , name="expThre" )
Then when I use the threshnames argument I set these up as follows (which I assume gives me a 1x4 matrix):
# Objective objects for Multiple Groups
objMZ <- mxExpectationNormal( covariance="ExpCovMZ", means="ExpMean", dimnames=selVars, thresholds="expThre", threshnames=c("BED1", "BED2", "TFEQ_Emo1", "TFEQ_Emo2" ))
objDZ <- mxExpectationNormal( covariance="ExpCovDZ", means="ExpMean", dimnames=selVars, thresholds="expThre", threshnames=c("BED1", "BED2", "TFEQ_Emo1", "TFEQ_Emo2" ))
I have tried multiple ways to try and fix this e.g. trying a 1x4 matrix to set the thresholds initially but this then seems to mess up the mxAlgebra for object 'Thre' where I add covariates. Any
useful pointers on how I can get round this? For context, I am now using the CSOLNP optimiser for consistency and given the now-reasonable looking CIs.
Many thanks,
Replied on Fri, 04/14/2023 - 10:01
AdminNeale Joined: 03/01/2013
In reply to MxRun still gives me errors by Zeynep_N
Threshnames order
It's difficult to debug your script without a sample dataset. I think that the threshnames should be the column names of the thresholds matrix. If you change the threshnames to only have the one
ordered threshold variable, TFEQ_Emo1 and 2 I think, then maybe it will be ok. Assuming that BED1 and BED2 are not threshold variables.
threshnames=c("TFEQ_Emo1", "TFEQ_Emo2" )
However, you state you want two variables per person to be ordinal, which implies that there should be four threshnames. One thing I would want to check is that the names are in the right order in
both the threshold matrix and the SelVars list. It is typical in twin studies to order variables within twin, so it goes T1var1 T1var2 T1var3 then T2var1 T2var2 T2Var3 - pretty much all ACE models in
OpenMx follow this structure. So what I am thinking is
threshnames=c("BED1", "TFEQ_Emo1", "BED2", "TFEQ_Emo2" )
Also make sure that the covariates age and sex are doing the right thing. Sometimes it is useful to write the matrix algebra out by hand to ensure that everything is going in the right place. I'm
presently slightly concerned that the covariates are only used on the ordinal variables (but maybe they were regressed out of the continuous variables). What is slightly concerning is that only one
beta coefficient for age is being used for both variables BED and TFEQ_Emo1. It's unusual to want to constrain such regressions to be equal. If it was not intended, then note that what you really
want is to end up with additions to the threshold matrices that have a different beta on age for BED and TFEQ_Emo. One thing that is easy is that these variables only have one threshold (we might be
in the business of a Kronecker product of a column vector of ones times the bAge*age and bSex*sex bits, or equating all the regressions on all the thresholds within a variable). So I think some
changes are needed to achieve the separate regressions on age and sex for BED and TFEQ_Emo.
betaA <-mxMatrix( type="Full", nrow=nth, ncol=2, free=T, values=-.1,
labels=c("BaTH"), name="bAge" )
betaS <-mxMatrix( type="Full", nrow=nth, ncol=2, free=T, values=.2,
labels=c("BsTH"), name="bSex" )
I think this makes bSex and bAge the same dimensions as ThMZ and ThDZ so the algebra should work. Sorry that this got a bit involved. | {"url":"https://openmx.ssri.psu.edu/comment/9748","timestamp":"2024-11-08T09:08:56Z","content_type":"text/html","content_length":"47742","record_id":"<urn:uuid:d98e7b6a-6937-4159-b50a-0ff527de5ad4>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00637.warc.gz"} |
A Framework to Understand DP
A Framework to Understand DP#
This resource introduces differential privacy from the perspective of the OpenDP programming framework. No prior knowledge is assumed of differential privacy (DP), but you will likely still find this
resource useful for understanding DP even if you already have a background in DP. Prior knowledge in basic probability, like random variables, will be useful.
Assume we have a vector dataset \(u\) where each record contains sensitive information about a different individual.
# u is a small vector dataset with contributions from:
# [Alice, Jane, John, Jack, ...]
u = [12, 10, 8, 7, ]
We can use differential privacy to collect measurements (statistics such as means and histograms) on this dataset, without revealing information about specific individuals.
To understand DP, it is important to first understand:
1. distance between datasets
2. distance between distributions
Distance Between Datasets - Adjacency#
An adjacent dataset is any dataset that differs from our dataset by a single individual. Returning to our vector dataset example, assume our dataset \(u\) has one record that contains information
about a person, Alice. Then one adjacent dataset \(v\) would contain every row in \(u\) except for the row with Alice’s information.
# v is one (of many) datasets that are adjacent to u
# [Jane, John, Jack, ...] (without Alice!)
v = [10, 8, 7, ]
You can construct other datasets adjacent to \(u\) by dropping a different row or adding a new row. When one person may contribute up to \(k\) rows, adjacent datasets differ by up to \(k\) additions
and removals.
The number of additions/removals between any two datasets is equivalent to the cardinality of the symmetric difference between the multisets \(u\) and \(v\). We call this metric the symmetric
\[d_{\mathrm{Sym}}(u, v) = |u \triangle v| = \sum_x |\# \{ i : x = u_i \} - \# \{ i : x = v_i \}|\]
And in code:
def d_SymmetricDistance(u, v):
"""symmetric distance between multisets u and v"""
# NOT this, as sets are not multisets. Loses multiplicity:
# return len(set(u).symmetric_difference(set(v)))
from collections import Counter
u, v = Counter(u), Counter(v)
# indirectly compute symmetric difference via the union of asymmetric differences
return sum(((u - v) + (v - u)).values())
# compute the symmetric distance between our two example datasets:
d_SymmetricDistance(u, v)
\(d_{\mathrm{Sym}}(\{12, 10, 8, 7\}, \{10, 8, 7\}) = |\{12, 10, 8, 7\} \triangle \{10, 8, 7\}| = |\{12\}| = 1\)
If the second dataset \(v\) were to differ from \(u\) by changing the 12 to 10, then we would still count the multiplicity of 10:
\(d_{\mathrm{Sym}}(\{12, 10, 8, 7\}, \{10, 10, 8, 7\}) = |\{12, 10, 8, 7\} \triangle \{10, 10, 8, 7\}| = |\{12, 10\}| = 2\)
In practice, we never directly compute these distances. In order to apply differentially private methods, you need to establish an upper bound on the distance between adjacent datasets. For example,
if each individual person may affect up to five records in the dataset, then setting a distance \(d_in = 5\) allows us to ensure individual-level privacy.
For instance, in the vector dataset example, it was stipulated that each element contains sensitive information about a different individual. This statement implies that the symmetric distance
between adjacent datasets, where one individual is added or removed, is at most one. That is, for any choice of datasets \(u\) and \(v\) such that \(u\) is adjacent to \(v\) (denoted \(u \sim_{\
mathrm{Sym}} v\)), we have that \(d_{\mathrm{Sym}}(u, v) \leq 1\).
Before moving on, there are some trivial generalizations. A dataset need not be a vector, it could be a dataframe or any other collection with a concept of records. There are also other dataset
metrics aside from SymmetricDistance (used for unbounded DP), such as ChangeOneDistance (used for bounded DP). There are also variations of metrics that are sensitive to data ordering, metrics for
describing distances between graphs, and more!
You should now have a sense for what an adjacent dataset means, how dataset distances work, and an intuitive understanding of the symmetric distance metric.
Distance Between Distributions - Divergence#
You can think of a measurement \(M(\cdot)\) as a differentially private statistic. Measurements are described by random variables (RVs), that is, they sample from noise distributions. The outputs of
a measurement are realizations of a random variable that follow a known probability distribution. Measurements only have one input: a dataset; other parameters are fixed when the measurement is
constructed. For context, a Laplace RV has parameters for shift and scale. This section describes how to measure distance between the distributions of measurements on adjacent datasets.
A common measurement is the Laplace DP sum, which is a sample from the Laplace distribution centered at the dataset sum with a fixed noise scale. The following plot compares the distribution of the
DP sum on dataset \(u\) with the distribution of the DP sum on dataset \(v\), when the noise scale is fixed to 25.
import numpy as np
import matplotlib.pyplot as plt
scale = 25
# while in this case the support theoretically includes all reals,
# we only bother plotting part of the support
support = np.arange(sum(v) - scale, sum(u) + scale)
def rv_M(x):
"""returns a random variable, M(x)"""
from scipy.stats import laplace
return laplace(loc=sum(x), scale=scale)
def plot_pdfs(u, v, output_domain):
plt.plot(output_domain, rv_M(u).pdf(output_domain), label="$p_{M(u)}(x)$") # type: ignore
plt.plot(output_domain, rv_M(v).pdf(output_domain), label="$p_{M(v)}(x)$") # type: ignore
plt.ylabel('density: $p_{RV}(x)$')
plt.xlabel('support: x')
plt.legend(prop={'size': 15})
plot_pdfs(u, v, support)
We are interested in the greatest divergence, a measure of the dissimilarity of these two distributions. While divergences are not necessarily distances, we informally refer to them as distances. A
common measure of divergence is based on the log ratio of probabilities:
\[D_{\mathrm{MaxDivergence}}(M(u), M(v)) = \max\limits_{S \subseteq \mathrm{supp}(M(u))} \log\left(\frac{\Pr[M(u) \in S]}{\Pr[M(v) \in S]}\right)\]
In this equation we define the distance between the RVs of \(M(u)\) and \(M(v)\) to be the maximum divergence among all possible subsets of the support.
For our DP sum with Laplacian noise example, the output domain of \(M(\cdot)\) is the set of all real numbers, \(\mathbb{R}\). In the plot below, I illustrate this equation for one randomly chosen
subset \(S\) of the output domain:
def plot_S(u, v, S):
"""draw the probability regions spanned by S"""
plt.fill_between(S, rv_M(u).pdf(S), label="$\\Pr[M(u) \\in S]$", alpha=.4) # type: ignore
plt.fill_between(S, rv_M(v).pdf(S), label="$\\Pr[M(v) \\in S]$", alpha=.4) # type: ignore
plt.plot([np.min(S), np.max(S)], [0, 0], label="S")
# re-run this notebook to see different choices of S
S = np.arange(*sorted(np.random.choice(support, size=2, replace=False)))
plot_pdfs(u, v, support)
plot_S(u, v, S)
The area of the blue region is the probability that \(M(u)\) is in \(S\)… and similarly the area of the orange region is \(\Pr[M(v) \in S]\).
def divergence_over_S(u, v, S):
"""prints the Divergence(M(u), M(v)) over some interval S, assuming M(x) = Laplace(sum(x), scale)"""
# integrate over both regions
lower, upper = np.min(S), np.max(S)
pr_Mu_in_S = rv_M(u).cdf(upper) - rv_M(u).cdf(lower) # blue
pr_Mv_in_S = rv_M(v).cdf(upper) - rv_M(v).cdf(lower) # orange
print("area of blue region: ", pr_Mu_in_S)
print("area of orange region:", pr_Mv_in_S)
print("divergence for this S:", np.abs(np.log(pr_Mu_in_S / pr_Mv_in_S)))
divergence_over_S(u, v, S)
area of blue region: 0.2276049566440389
area of orange region: 0.14083816706408814
divergence for this S: 0.4799999999999992
This shows the divergence between the RVs of \(M(u)\) and \(M(v)\) for one choice of S, but keep in mind that \(D_{\mathrm{MaxDivergence}}(M(u), M(v))\) is the greatest divergence over any choice of
\(S\). Intuitively, the divergence between the RVs of \(M(u)\) and \(M(v)\) for the same \(S\) must increase if Alice made a greater contribution to the statistic:
# hypothetical: what if Alice's contribution was 100, instead of 12?
u_prime = [100, *u[1:]]
divergence_over_S(u_prime, v, S)
output_domain = np.arange(sum(v) - scale, sum(u_prime) + scale, 1)
plot_pdfs(u_prime, v, output_domain)
plot_S(u_prime, v, S)
area of blue region: 0.017594942096975916
area of orange region: 0.14083816706408814
divergence for this S: 2.080000000000001
As we can see, when the divergence between probability distributions is greater, we can more confidently distinguish which distribution a sample came from. These examples help to form an intuition
for how the max divergence measure relates to privacy.
Moreso, the max divergence qualifies as a measure of privacy because it provides immunity from post-processing. There is no further computation that can be made on the release that will make it
easier to distinguish which distribution a sample came from. That is, the divergence cannot increase after applying \(f\) to a release:
\[\forall f \quad D\Bigl(M(u), M(v)\Bigr) \ge D\Bigl(f(M(u)), f(M(v))\Bigr)\]
This property is the crucial distinction between measures, as discussed in this section, and metrics, as discussed in the previous section. Measurements and transformations share the same
Definition of Privacy#
Now that we have an understanding of distances between datasets, and distances between distributions, we can define the privacy of a measurement, \(M(\cdot)\):
\(M(\cdot)\) is \(\boldsymbol\epsilon\textbf{-differentially private}\) at distance \(k\) if, for every pair of datasets \(u\) and \(v\) such that \(d_{\mathrm{Sym}}(u, v) \leq k\), we have that
\(D_{\mathrm{MaxDivergence}}(M(u), M(v)) \leq \epsilon\).
In this definition, we relate a dataset distance \(k\) to another distance \(\epsilon\). This \(\epsilon\) is more general than the max divergence we computed in the previous section because it is
the greatest divergence over all possible choices of \(S\), and over all possible pairs of adjacent datasets \(u\) and \(v\). \(\epsilon\) is often referred to as a bound on the privacy loss of \(M(\
This has a very practical interpretation: Let’s say I have a dataset \(x\) for which an individual user can contribute at most \(k\) rows, and a statistic \(M(\cdot)\) that is \(\epsilon\)-DP when
user contribution is at most \(k\). By the DP guarantee, it is proven that the influence of any one individual on the data release induces a divergence no greater than \(\epsilon\). Thus, assuming a
reasonably small choice of \(\epsilon\), the individual’s participation in the statistical release is kept private, because their influence on the data release is at most \(\epsilon\)
If you have some background in differential privacy you may be more familiar with a definition of privacy worded like this:
\(M(\cdot)\) is \(\boldsymbol\epsilon\textbf{-differentially private}\) if, for every pair of adjacent datasets \(u\) and \(v\), we have that \(\Pr[M(u) \in S] \leq e^\epsilon \cdot \Pr[M(v) \in
This is mostly equivalent, because of the way we’ve defined \(D_{\mathrm{MaxDivergence}}(M(u), M(v))\) in the previous section. However, this formulation of the definition is ambiguous about what
makes a dataset adjacent. To obtain well-defined privacy guarantees, it is important to specify the dataset metric and dataset distance.
We further generalize the definition of privacy:
\(M(\cdot)\) is \((d_{in}, d_{out})\textbf{-differentially private}\) with respect to input metric \(MI\) and output measure \(MO\) if, for any choice of datasets \(u\) and \(v\) such that \(d_
{MI}(u, v) \leq d_{in}\), we have that \(D_{MO}(M(u), M(v)) \leq d_{out}\).
The first definition can be reclaimed by letting \(MI\) be SymmetricDistance and \(MO\) be MaxDivergence. \(MO\) can be set to other measures of divergence to represent approximate (\(\epsilon, \
delta\))-differential privacy, or zero-concentrated \(\rho\)-differential privacy. Similarly, our choice of SymmetricDistance represents unbounded DP, but we can represent bounded DP by letting \(MI
\) be ChangeOneDistance.
Distance Between Aggregates - Sensitivity#
The sensitivity is the greatest amount an aggregate can change when computed on an adjacent dataset. Aggregators are typically deterministic statistics (like the sum or histogram functions), and
their exact outputs are aggregates. More generally, a transformation \(T(\cdot)\) is a function from a data domain to a data domain. Aggregators are a kind of transformation in which the output
domain consists of aggregates.
One example of a sensitivity metric is the AbsoluteDistance, which is used to measure the distance between scalar aggregates.
\[d_{\mathrm{Abs}}(a, b) = |a - b|\]
def d_Abs(a, b):
"""absolute distance between a and b"""
return abs(a - b)
We can use the absolute distance metric to express the sensitivity of the sum aggregator. In our vector dataset example, we know each individual can contribute at most one record. Since this record
is unbounded, it can perturb the sum an arbitrarily large amount towards positive or negative infinity. This is unfortunate, because it implies that the divergence is also infinite!
In order to attain a finite sensitivity, it is customary to clamp— that is, to replace any value less than a lower bound with the lower bound, and any value greater than an upper bound with the upper
def clamped_sum_0_12(x):
"""a naive function that computes the sum, where each element is clamped within [0, 12]"""
return sum(np.clip(x, 0, 12))
Broadly speaking, if the transformation clamps data to the interval \([L, U]\), and we know each individual contributes at most \(d_{in}\) records, then the clamped sum sensitivity (\(d_{out}\)) is
\[\max_{u \sim_{Sym} v} |\mathrm{clamped\_sum}(u) - \mathrm{clamped\_sum}(v)| = d_{in} \cdot \max(|L|, U)\]
We can use this to solve for the sensitivity of \(clamped\_sum\_0\_12\), by letting \([L, U] = [0, 12]\). Thus its sensitivity is \(1 \cdot max(|0|, 12) = 12\). For any conceivable dataset \(u\),
adding or removing any individual (to get some dataset \(v\)) can change the sum by at most \(12\).
Our current choice of \(u\) and \(v\) is an example that maximizes the absolute distance:
d_Abs(clamped_sum_0_12(u), clamped_sum_0_12(v))
You can even use the AbsoluteDistance as the input metric \(MI\) of a measurement \(M(\cdot)\) (see the definition of privacy). Let’s define a new function \(laplace\_noise\) to illustrate this:
def laplace_noise(x):
"""a naive function that adds an approximation to Laplace noise"""
return np.random.laplace(loc=x, scale=scale)
We let \(MI\) be AbsoluteDistance and \(MO\) be MaxDivergence. It can be shown that for any choice of \(u, v \in \mathbb{R}\) such that \(d_{\mathrm{Abs}}(u, v) \leq d_{in}\), and \(d_{out} = d_{in}
/ scale\), then:
\[D_{\mathrm{MaxDivergence}}(\mathrm{laplace\_noise}(u), \mathrm{laplace\_noise}(v)) \leq d_{out}\]
Therefore, if the data types in this function had infinite precision, then \(laplace\_noise\) would be a measurement. Other common metrics to express sensitivities are L1Distance and L2Distance.
Definition of Stability#
Similar to how we defined the privacy of a measurement \(M(\cdot)\), we can also define the stability of a transformation, \(T(\cdot)\):
\(T(\cdot)\) is \((d_{in}, d_{out})\textbf{-stable}\) with respect to input metric \(MI\) and output metric \(MO\) if, for any choice of datasets \(u\) and \(v\) such that \(d_{MI}(u, v) \leq d_
{in}\), we have that \(d_{MO}(T(u), T(v)) \leq d_{out}\).
An example is the \(clamped\_sum\_0\_12\) function from the previous section. If the data types in \(clamped\_sum\_0\_12\) had infinite precision, it would be a stable transformation where \(MI\) is
SymmetricDistance and \(MO\) is AbsoluteDistance. We’ve previously shown that when \(d_{in} = 1\), the sensitivity \(d_{out} = 12\).
This stability guarantee does not carry privacy guarantees on its own, but it lets us construct building blocks that can be chained together. If the output metric \(MO\) and output domain \(DO\) of a
transformation \(T(\cdot)\) conform with the input metric \(MI\) and input domain \(DI\) of a measurement \(M(\cdot)\), then it is valid to construct a new measurement \(M_{\mathrm{chained}}(\cdot) =
M(T(\cdot))\). We can similarly construct a new transformation \(T_{\mathrm{chained}}(\cdot) = T_2(T_1(\cdot))\).
Notice that the output domain and metric of the \(clamped\_sum\_0\_12\) transformation conform with the input metric and domain of the \(laplace\_noise\) measurement, so we can chain these together:
def laplace_sum(x):
"""a naive function that computes the noisy clamped sum"""
return laplace_noise(clamped_sum_0_12(x))
Since this function was constructed by chaining a stable transformation and private measurement, it is trivial to prove that it is a private measurement (if the data types had infinite precision).
The new chained measurement’s \(MI\) is SymmetricDistance, and \(MO\) is MaxDivergence, and when the dataset distance \(d_{in} = 1\), we have that \(\epsilon = d_{out} = d_{in} \cdot \max(|0|, 12) /
25 = d_{in} \cdot 0.48 = 0.48\). That is, when an individual can contribute at most one record, the maximum observable divergence among the output distributions is \(0.48\).
Stability Maps and Privacy Maps#
A crucial takeaway from this notebook is a high-level understanding that differential privacy is a system to relate distances (\(d_{in}\) and \(d_{out}\)). If you can establish a bound on the
distance to adjacent datasets \(d_{in}\) (in terms of some metric \(MI\)) then you can work out the stability or privacy properties \(d_{out}\) (in terms of some metric or measure \(MO\)) of
computations made on your data.
We encapsulate this relationship between distances with one last abstraction, a map. A map is a function, associated with your computation, that computes a \(d_{out}\) for any given \(d_{in}\). Thus,
if \(map(d_{in}) \le d_{out}\), then a computation is \(d_{out}\)-DP.
The stability map for the \(clamped\_sum\_0\_12\) function is as follows:
def clamped_sum_0_12_map(d_in):
return d_in * max(abs(0), 12)
# find the smallest d_out (absolute distance) of clamped_sum_0_12 when d_in (symmetric distance) is 1
This map is just a repackaging of our previous formula for the clamped sum sensitivity, so that \(d_{in}\) can be set later. It is referred to as a stability map because stability is a more general
term than sensitivity, namely a bound on on how much outputs can change as a function of input distances.
The same pattern holds for the privacy map of the \(laplace\_noise\) measurement:
def laplace_noise_map(d_in):
return d_in / scale
# find the smallest d_out (epsilon) of laplace_noise when d_in (absolute distance) is 12
This time we refer to it as a privacy map, because the output distance is in terms of a privacy measure, which captures distance between output distributions (like MaxDivergence) and hence offers
privacy guarantees. Now that we have the stability map for the clamped sum transformation and the privacy map for the Laplace noise measurement, we can automatically construct the privacy map for the
Laplace sum measurement:
def laplace_sum_map(d_in):
return laplace_noise_map(clamped_sum_0_12_map(d_in))
# find the smallest d_out (epsilon) of laplace_noise when d_in (symmetric distance) is 1
We’ve now come full-circle. In the “Distance Between Distributions” section, we computed an example divergence for one choice of \(S\). We have now indirectly computed an upper bound for that
divergence of \(0.48\). You may notice that some choices of \(S\) in that section can give divergences very slightly larger than \(0.48\). This is because floating-point numbers have finite
precision, so intermediate computations were subject to rounding that introduced error.
The transformation and measurement examples in this notebook are only \((d_{in}, d_{out})\)-differentially private if we assume the data types have infinite precision— and they don’t! Building
transformations or measurements that have proven stability or privacy properties is nontrivial, especially if you account for finite precision in data types. This is the purpose of the OpenDP
library: to help you build robust transformations and measurements with rigorous privacy properties. | {"url":"https://docs.opendp.org/en/v0.9.0/user/programming-framework/a-framework-to-understand-dp.html","timestamp":"2024-11-02T20:25:15Z","content_type":"text/html","content_length":"84780","record_id":"<urn:uuid:dd8738bb-0405-4982-81e6-b3355c7ffdfb>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00422.warc.gz"} |
Analytical expression for the correlation function of a hard sphere chain fluid
A closed form expression is given for the correlation function of flexible hard sphere chain fluid. A set of integral equations obtained from Wertheim's multidensity Ornstein—Zernike integral
equation theory with the polymer Percus—Yevick ideal chain approximation is considered. Applying the Laplace transformation method to the integral equations and then solving the resulting equations
algebraically, the Laplace transforms of individual correlation functions are obtained. By inverse Laplace transformation the inter- and intramolecular radial distribution functions (RDFs) are
obtained in closed forms up to 3D (D is segment diameter). These analytical expressions for the RDFs would be useful in developing the perturbation theory of chain fluids.
Dive into the research topics of 'Analytical expression for the correlation function of a hard sphere chain fluid'. Together they form a unique fingerprint. | {"url":"https://pure.uos.ac.kr/en/publications/analytical-expression-for-the-correlation-function-of-a-hard-sphe","timestamp":"2024-11-13T16:40:09Z","content_type":"text/html","content_length":"50720","record_id":"<urn:uuid:ba88d1ca-098e-4ea1-a3ef-ba6d5c8bd8bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00715.warc.gz"} |
An optimized treatment for algorithmic differentiation of an important glaciological fixed-point problemGeoscientific Model DevelopmentAdjoints by Automatic DifferentiationAdvanced data assimilation for geosciencesA Framework for Adjoint-based Shape Design and Error ControlComputational Fluid Dynamics JournalThe Tapenade Automatic Differentiation tool: Principles, Model, and SpecificationACM Transactions On Mathematical SoftwareProgramming language features, usage patterns, and the efficiency of generated adjoint codeOptimization Methods and SoftwareAlgorithmic differentiation of code with multiple context-specific activitiesACM Transactions on Mathematical SoftwareGoal-oriented metric-based mesh adaptation for unsteady CFD simulations involving moving geometriesMixed-language automatic differentiationOptimization Methods and SoftwareAn a priori anisotropic Goal-Oriented Error Estimate for Viscous Compressible Flow and Application to Mesh AdaptationMesh-Anpassung für k-genaue Approximationen in CFDA priori error-based mesh adaptation in CFDCombining a DDES model with a dynamic variational multiscale formulationA Volume-agglomeration multirate time advancing for high Reynolds number flow simulationCompilers: Principles, Techniques and ToolsA language and an integrated environment for program transformationsReverse accumulation and implicit functionsOptimization Methods and SoftwareNatural semantics on the computerProceedings, France-Japan AI and CS Symposium, ICOTAbstract InterpretationACM Computing SurveysInterprocedural Array Region AnalysesInternational Journal of Parallel ProgrammingAutomatic differentiation and iterative processesOptimization Methods and SoftwareAdjoint methods for aeronautical designProceedings of the ECCOMAS CFD ConferenceReduced Gradients and Hessians from Fixed Point Iteration for State EquationsNumerical AlgorithmsOn stable piecewise linearization and generalized algorithmic differentiationOptimization Methods and SoftwareEvaluating Derivatives: Principles and Techniques of Algorithmic DifferentiationTransformations automatiques de spécifications sémantiques: application: Un vérificateur de types incrementalAutomatic Differentiation of Navier-Stokes computationsInferred basal friction and surface mass balance of the Northeast Greenland Ice Stream using data assimilation of ICESat (Ice Cloud and land Elevation Satellite) surface altimetry and ISSM (Ice Sheet System Model)CryosphereVariational algorithms for analysis and assimilation of meteorological observations: theoretical aspectsTellusPractical application to fluid flows of automatic differentiation for design problemsVon Karman Lecture SeriesDifférentiation Automatique: application à un problème d'optimisation en météorologieSymbolic Bounds Analysis of Pointers, Array Indices, and Accessed Memory RegionsProceedings of the ACM SIGPLAN'00 Conference on Programming Language Design and Implementation
Research Program Algorithmic Differentiation Laurent Hascoët Valérie Pascual
(AD, aka Automatic Differentiation) Transformation of a program, that returns a new program that computes derivatives of the initial program, i.e. some combination of the partial derivatives of the
program's outputs with respect to its inputs.
Mathematical manipulation of the Partial Differential Equations that define a problem, obtaining new differential equations that define the gradient of the original problem's solution.
General trade-off technique, used in adjoint AD, that trades duplicate execution of a part of the program to save some memory space that was used to save intermediate results.
Algorithmic Differentiation (AD) differentiates programs. The input of AD is a source program $P$ that, given some $X\in {ℝ}^{n}$, returns some $Y=F\left(X\right)\phantom{\rule{0.222222em}{0ex}}\in
{ℝ}^{m}$, for a differentiable $F$. AD generates a new source program ${P}^{"}$ that, given $X$, computes some derivatives of $F$ .
Any execution of $P$ amounts to a sequence of instructions, which is identified with a composition of vector functions. Thus, if
$\begin{array}{c}\hfill \begin{array}{ccc}\hfill P& \phantom{\rule{0.222222em}{0ex}}\text{runs}\phantom{\rule{0.222222em}{0ex}}& \left\{{I}_{1};{I}_{2};\cdots {I}_{p};\right\},\hfill \\ \hfill F& \
text{then}\phantom{\rule{4.pt}{0ex}}\text{is}& {f}_{p}\circ {f}_{p-1}\circ \cdots \circ {f}_{1},\hfill \end{array}\end{array}$
where each ${f}_{k}$ is the elementary function implemented by instruction ${I}_{k}$. AD applies the chain rule to obtain derivatives of $F$. Calling ${X}_{k}$ the values of all variables after
instruction ${I}_{k}$, i.e. ${X}_{0}=X$ and ${X}_{k}={f}_{k}\left({X}_{k-1}\right)$, the Jacobian of $F$ is
phantom{\rule{0.222222em}{0ex}}\cdots \phantom{\rule{0.222222em}{0ex}}.\phantom{\rule{0.222222em}{0ex}}{f}_{1}^{"}\left({X}_{0}\right)$
which can be mechanically written as a sequence of instructions ${I}_{k}^{"}$. This can be generalized to higher level derivatives, Taylor series, etc. Combining the ${I}_{k}^{"}$ with the control of
$P$ yields ${P}^{"}$, and therefore this differentiation is piecewise.
The above computation of ${F}^{"}\left(X\right)$, albeit simple and mechanical, can be prohibitively expensive on large codes. In practice, many applications only need cheaper projections of ${F}^{"}
\left(X\right)$ such as:
Sensitivities, defined for a given direction $\stackrel{˙}{X}$ in the input space as:
{0ex}}.\phantom{\rule{0.222222em}{0ex}}\cdots \phantom{\rule{0.222222em}{0ex}}.\phantom{\rule{0.222222em}{0ex}}{f}_{1}^{"}\left({X}_{0}\right)\phantom{\rule{0.222222em}{0ex}}.\phantom{\rule
This expression is easily computed from right to left, interleaved with the original program instructions. This is the tangent mode of AD.
Adjoints, defined after transposition (${F}^{"*}$), for a given weighting $\overline{Y}$ of the outputs as:
${F}^{"*}\left(X\right).\overline{Y}={f}_{1}^{\text{'}*}\left({X}_{0}\right).{f}_{2}^{\text{'}*}\left({X}_{1}\right).\phantom{\rule{0.222222em}{0ex}}\cdots \phantom{\rule{0.222222em}{0ex}}.{f}_{p-1}^
This expression is most efficiently computed from right to left, because matrix$×$vector products are cheaper than matrix$×$matrix products. This is the adjoint mode of AD, most effective for
optimization, data assimilation , adjoint problems , or inverse problems.
Adjoint AD builds a very efficient program Section 3.3, which computes the gradient in a time independent from the number of parameters $n$. In contrast, computing the same gradient with the tangent
mode would require running the tangent differentiated program $n$ times.
However, the ${X}_{k}$ are required in the inverse of their computation order. If the original program overwrites a part of ${X}_{k}$, the differentiated program must restore ${X}_{k}$ before it is
used by ${f}_{k+1}^{"*}\left({X}_{k}\right)$. Therefore, the central research problem of adjoint AD is to make the ${X}_{k}$ available in reverse order at the cheapest cost, using strategies that
combine storage, repeated forward computation from available previous values, or even inverted computation from available later values.
Another research issue is to make the AD model cope with the constant evolution of modern language constructs. From the old days of Fortran77, novelties include pointers and dynamic allocation,
modularity, structured data types, objects, vectorial notation and parallel programming. We keep developing our models and tools to handle these new constructs.
Static Analysis and Transformation of programs Laurent Hascoët Valérie Pascual
Tree representation of a computer program, that keeps only the semantically significant information and abstracts away syntactic sugar such as indentation, parentheses, or separators.
Representation of a procedure body as a directed graph, whose nodes, known as basic blocks, each contain a sequence of instructions and whose arrows represent all possible control jumps that can
occur at run-time.
Model that describes program static analysis as a special sort of execution, in which all branches of control switches are taken concurrently, and where computed values are replaced by abstract
values from a given semantic domain. Each particular analysis gives birth to a specific semantic domain.
Program analysis that studies how a given property of variables evolves with execution of the program. Data Flow analysis is static, therefore studying all possible run-time behaviors and making
conservative approximations. A typical data-flow analysis is to detect, at any location in the source program, whether a variable is initialized or not.
The most obvious example of a program transformation tool is certainly a compiler. Other examples are program translators, that go from one language or formalism to another, or optimizers, that
transform a program to make it run better. AD is just one such transformation. These tools share the technological basis that lets them implement the sophisticated analyses required. In particular
there are common mathematical models to specify these analyses and analyze their properties.
An important principle is abstraction: the core of a compiler should not bother about syntactic details of the compiled program. The optimization and code generation phases must be independent from
the particular input programming language. This is generally achieved using language-specific front-ends, language-independent middle-ends, and target-specific back-ends. In the middle-end, analysis
can concentrate on the semantics of a reduced set of constructs. This analysis operates on an abstract representation of programs made of one call graph, whose nodes are themselves flow graphs whose
nodes (basic blocks) contain abstract syntax trees for the individual atomic instructions. To each level are attached symbol tables, nested to capture scoping.
Static program analysis can be defined on this internal representation, which is largely language independent. The simplest analyses on trees can be specified with inference rules , , . But many
data-flow analyses are more complex, and better defined on graphs than on trees. Since both call graphs and flow graphs may be cyclic, these global analyses will be solved iteratively. Abstract
Interpretation is a theoretical framework to study complexity and termination of these analyses.
Data flow analyses must be carefully designed to avoid or control combinatorial explosion. At the call graph level, they can run bottom-up or top-down, and they yield more accurate results when they
take into account the different call sites of each procedure, which is called context sensitivity. At the flow graph level, they can run forwards or backwards, and yield more accurate results when
they take into account only the possible execution flows resulting from possible control, which is called flow sensitivity.
Even then, data flow analyses are limited, because they are static and thus have very little knowledge of actual run-time values. Far before reaching the very theoretical limit of undecidability, one
reaches practical limitations to how much information one can infer from programs that use arrays , or pointers. Therefore, conservative over-approximations must be made, leading to derivative code
less efficient than ideal.
Algorithmic Differentiation and Scientific Computing Alain Dervieux Laurent Hascoët Bruno Koobus Eléonore Gauci Emmanuelle Itam Olivier Allain Stephen Wornom
In Scientific Computing, the mathematical model often consists of Partial Differential Equations, that are discretized and then solved by a computer program. Linearization of these equations, or
alternatively linearization of the computer program, predict the behavior of the model when small perturbations are applied. This is useful when the perturbations are effectively small, as in
acoustics, or when one wants the sensitivity of the system with respect to one parameter, as in optimization.
Consider a system of Partial Differential Equations that define some characteristics of a system with respect to some parameters. Consider one particular scalar characteristic. Its sensitivity (or
gradient) with respect to the parameters can be defined by means of adjoint equations, deduced from the original equations through linearization and transposition. The solution of the adjoint
equations is known as the adjoint state.
Scientific Computing provides reliable simulations of complex systems. For example it is possible to simulate the steady or unsteady 3D air flow around a plane that captures the physical phenomena of
shocks and turbulence. Next comes optimization, one degree higher in complexity because it repeatedly simulates and applies gradient-based optimization steps until an optimum is reached. The next
sophistication is robustness, that detects undesirable solutions which, although maybe optimal, are very sensitive to uncertainty on design parameters or on manufacturing tolerances. This makes
second derivative come into play. Similarly Uncertainty Quantification can use second derivatives to evaluate how uncertainty on the simulation inputs imply uncertainty on its outputs.
We investigate several approaches to obtain the gradient, between two extremes:
One can write an adjoint system of mathematical equations, then discretize it and program it by hand. This is time consuming. Although this looks mathematically sound , this does not provide the
gradient of the discretized function itself, thus degrading the final convergence of gradient-descent optimization.
One can apply adjoint AD (cf ) on the program that discretizes and solves the direct system. This gives exactly the adjoint of the discrete function computed by the program. Theoretical results
guarantee convergence of these derivatives when the direct program converges. This approach is highly mechanizable, but leads to massive use of storage and may require code transformation by hand ,
to reduce memory usage.
If for instance the model is steady, or when the computation uses a Fixed-Point iteration, tradeoffs exist between these two extremes , that combine low storage consumption with possible automated
adjoint generation. We advocate incorporating them into the AD model and into the AD tools.
Application Domains Algorithmic Differentiation
Algorithmic Differentiation of programs gives sensitivities or gradients, useful for instance for :
optimum shape design under constraints, multidisciplinary optimization, and more generally any algorithm based on local linearization,
inverse problems, such as parameter estimation and in particular 4Dvar data assimilation in climate sciences (meteorology, oceanography),
first-order linearization of complex systems, or higher-order simulations, yielding reduced models for simulation of complex systems around a given state,
adaption of parameters for classification tools such as Machine Learning systems, in which Adjoint Differentiation is also known as backpropagation.
mesh adaptation and mesh optimization with gradients or adjoints,
equation solving with the Newton method,
sensitivity analysis, propagation of truncation errors.
Multidisciplinary optimization
A CFD program computes the flow around a shape, starting from a number of inputs that define the shape and other parameters. On this flow one can define optimization criteria e.g. the lift of an
aircraft. To optimize a criterion by a gradient descent, one needs the gradient of the criterion with respect to all inputs, and possibly additional gradients when there are constraints. Adjoint AD
is the most efficient way to compute these gradients.
Inverse problems and Data Assimilation
Inverse problems aim at estimating the value of hidden parameters from other measurable values, that depend on the hidden parameters through a system of equations. For example, the hidden parameter
might be the shape of the ocean floor, and the measurable values of the altitude and velocities of the surface. Figure shows an example of an inverse problem using the glaciology code ALIF (a pure C
version of ISSM ) and its AD-adjoint produced by Tapenade.
One particular case of inverse problems is data assimilation in weather forecasting or in oceanography. The quality of the initial state of the simulation conditions the quality of the prediction.
But this initial state is not well known. Only some measurements at arbitrary places and times are available. A good initial state is found by solving a least squares problem between the measurements
and a guessed initial state which itself must verify the equations of meteorology. This boils down to solving an adjoint problem, which can be done though AD . The special case of 4Dvar data
assimilation is particularly challenging. The 4^th dimension in “4D” is time, as available measurements are distributed over a given assimilation period. Therefore the least squares mechanism must be
applied to a simulation over time that follows the time evolution model. This process gives a much better estimation of the initial state, because both position and time of measurements are taken
into account. On the other hand, the adjoint problem involved is more complex, because it must run (backwards) over many time steps. This demanding application of AD justifies our efforts in reducing
the runtime and memory costs of AD adjoint codes.
Simulating a complex system often requires solving a system of Partial Differential Equations. This can be too expensive, in particular for real-time simulations. When one wants to simulate the
reaction of this complex system to small perturbations around a fixed set of parameters, there is an efficient approximation: just suppose that the system is linear in a small neighborhood of the
current set of parameters. The reaction of the system is thus approximated by a simple product of the variation of the parameters with the Jacobian matrix of the system. This Jacobian matrix can be
obtained by AD. This is especially cheap when the Jacobian matrix is sparse. The simulation can be improved further by introducing higher-order derivatives, such as Taylor expansions, which can also
be computed through AD. The result is often called a reduced model.
Mesh adaptation
Some approximation errors can be expressed by an adjoint state. Mesh adaptation can benefit from this. The classical optimization step can give an optimization direction not only for the control
parameters, but also for the approximation parameters, and in particular the mesh geometry. The ultimate goal is to obtain optimal control parameters up to a precision prescribed in advance.
New Software and Platforms AIRONUM
Keywords: Computational Fluid Dynamics - Turbulence
Functional Description: Aironum is an experimental software that solves the unsteady compressible Navier-Stokes equations with k-epsilon, LES-VMS and hybrid turbulence modelling on parallel
platforms, using MPI. The mesh model is unstructured tetrahedrization, with possible mesh motion.
Participant: Alain Dervieux
Contact: Alain Dervieux
URL: http://www-sop.inria.fr/tropics/aironum
Keywords: Static analysis - Optimization - Compilation - Gradients
Scientific Description: Tapenade implements the results of our research about models and static analyses for AD. Tapenade can be downloaded and installed on most architectures. Alternatively, it can
be used as a web server. Higher-order derivatives can be obtained through repeated application.
Tapenade performs sophisticated data-flow analysis, flow-sensitive and context-sensitive, on the complete source program to produce an efficient differentiated code. Analyses include Type-Checking,
Read-Write analysis, and Pointer analysis. AD-specific analyses include:
Activity analysis: Detects variables whose derivative is either null or useless, to reduce the number of derivative instructions.
Adjoint Liveness analysis: Detects the source statements that are dead code for the computation of derivatives.
TBR analysis: In adjoint-mode AD, reduces the set of source variables that need to be recovered.
Functional Description: Tapenade is an Algorithmic Differentiation tool that transforms an original program into a new program that computes derivatives of the original program. Algorithmic
Differentiation produces analytical derivatives, that are exact up to machine precision. Adjoint-mode AD can compute gradients at a cost which is independent from the number of input variables.
Tapenade accepts source programs written in Fortran77, Fortran90, or C. It provides differentiation in the following modes: tangent, vector tangent, adjoint, and vector adjoint.
News Of The Year: - Continued development of multi-language capacity: AD of codes mixing Fortran and C - Continued front-end for C++ based on Clang - Experimental support for building Abs-Normal Form
tangent of non-smooth codes
Participants: Laurent Hascoët and Valérie Pascual
Contact: Laurent Hascoët
URL: http://www-sop.inria.fr/tropics/tapenade.html
New Results Towards Algorithmic Differentiation of C++ Laurent Hascoët Valérie Pascual Frederic Cazals ABS team, Inria Sophia-Antipolis
We made progress towards the extension of Tapenade for C++. Last year, an external parser for C++ was built on top of Clang-LLVM https://clang.llvm.org/ and connected to the input formalism “IL” of
Tapenade, but the internals of Tapenade were not able to handle the new constructs present in this input. This year, integration of C++ was pushed further by taking into account many of the new
constructs (namespaces, classes, constructors and destructors) in the Internal Representation(IR) of Tapenade. Not surprisingly, this implied deep changes in several areas of Tapenade code. The IR of
Tapenade now contains classes, constructors and destructors, and also has a faithful representation for namespaces. The textual nested structure and the control-flow parts of the IR are correct. The
symbol tables and the representation for memory locations are still under development.
As a result, Tapenade is now able to input its first C++ files and is able to output them, but without transformation. Although not advertised nor documented, the functionality is present in the
latest release 3.14. Data-Flow analysis and code transformation (e.g. AD) will not be possible until we have a correct IR about variables and their memory locations. This work is going on.
This work benefited from the expertise in C++ of Frederic Cazals (Inria ABS team). The ABS team provided a large test application code (SBL, https://sbl.inria.fr/) for Molecular Dynamics, which will
be our first C++ target.
AD of mixed-language codes Valérie Pascual Laurent Hascoët
Last year Tapenade was extended to differentiate codes that mix different languages, beginning with the tangent mode of AD. Our motivating application here is Calculix, a 3-D Structural Finite
Element code that mixes Fortran and C. This year, we continued development towards Adjoint Differentiation. Although more complete testing is needed, we now have a first correct adjoint of Calculix.
Tapenade can now routinely differentiate Fortran+C codes, and accepts and takes advantage of the interoperability directives provided by the Fortran 2003 standard. It can handle not only procedure
parameters correspondence, but also interoperability between C struct and Fortran COMMON blocks. Laurent Hascoët presented the advancement of this work at the ISMP 2018 congress in Bordeaux https://
C files (aka “translation units”) and Fortran modules are two instances of the more general notion of “package” for which we are looking for a unified representation in Tapenade. It appears that this
common representation could also handle C++ namespaces.
Differentiation of non-smooth programs Laurent Hascoët Sri Hari Krishna Narayanan Argonne National Lab. (Illinois, USA)
Algorithmic Differentiation can be used to derive tangent models that cope with a certain class of non-smoothness, through the use of the so-called Abs-Normal Form (ANF) . These tangent models
incorporate some knowledge of the nearby discontinuities of the derivatives. These models bring some additional power to processes that use tangent approximations, such as simulation, optimization,
or solution of differential equations.
The mechanics to derive these special tangent models can be built as an extension of standard tangent linear Algorithmic Differentiation. This has been first demonstrated by the AD tool AdolC which,
being based on Operator Overloading, is more flexible and seems a natural choice for implementation. Together with Krishna Narayanan, we recently tried a similar adaption on Source-Transformation AD
tools. It appears that very little development is needed in the AD-tool. Specifically for Tapenade, it appears that no development at all is needed in the tool itself. Any end-user can already
produce ANF tangent without needing any access to the tool source. All it requires is a customized derivative of the absolute-value function (ABS), which is currently less than 40 lines of code.
Building the ANF of a given program introduces one new variable per run-time execution of the ABS function. As the number of rows and columns of the constructed extended Jacobian both grow like the
number of variables, it may become unreasonably large for large codes. To overcome this issue, we explore the possibility of finding at run-time the "important" ABS calls that deserve this treatment,
and those that don't. We base this decision on a notion of distance to the kink induced by this ABS call as illustrated by Figure . We presented these experiments at a Shonan meeting on this question
(Shonan, Japan, June 25-29) and at a workshop of ISMP 2018 (Bordeaux, July 2-6)
AD-adjoints and C dynamic memory management Laurent Hascoët Sri Hari Krishna Narayanan Argonne National Lab. (Illinois, USA)
One of the current frontiers of AD research is the definition of an adjoint AD model that can cope with dynamic memory management. This research is central to provide reliable adjoint differentiation
of C, and for our distant goal of AD of C++. This research is conducted in collaboration with the MCS department of Argonne National Lab. Our partnership is formalized by joint participation in the
Inria joint lab JLESC, and partly funded by the Partner University Fund (PUF) of the French embassy in the USA.
Adjoint AD must reproduce in reverse order the control decisions of the original code. In languages such as C, allocation of dynamic memory and pointer management form a significant part of these
control decisions. Reproducing memory allocation in reverse means reallocating memory, possibly receiving a different memory chunk. Reproducing pointer addresses in reverse thus requires to convert
addresses in the former memory chunks into equivalent addresses in the new reallocated chunks. Together with Krishna Narayanan from Argonne, we experiment on real applications to find the most
efficient solution to this address conversion problem. We jointly develop a library (called ADMM, ADjoint Memory Management) whose primitives are used in AD adjoint code to handle this address
conversion. Both our AD tool Tapenade and Argonne's tool OpenAD use ADMM in the adjoint code they produce.
This year, trying to prove correctness of our current address conversion, we discovered some limitations that indeed made the proof impossible. To solve these issues, it seems necessary to assign at
run-time a unique identifier to each chunk of memory used by the code, and to carry this identifier along with every pointer. This results in a code transformation which, although more complex than
expected, can still be described by a small set of rewrite rules. Moreover, this alternative method should reduce the run-time overhead that we observed previously. Implementation and measurements
are still under way. We presented this recent research in the form of a catalogue of alternatives for Data-Flow reversal of memory addresses, at the 21^st EuroAD workshop (Jena, Germany, November
Application to large industrial codes Valérie Pascual Laurent Hascoët Bruno Maugars ONERA Sébastien Bourasseau ONERA Bérenger Berthoul ONERA
We support industrial users with their first experiments of Algorithmic Differentiation of large in-house codes.
This year's main application is with ONERA on their ElsA CFD platform (Fortran 90). Both tangent and adjoint models of the kernel of ElsA were built successfully by Tapenade. It is worth noticing
that this application was performed inside ONERA by ONERA engineers (Bruno Maugars, Sébastien Bourasseau, Bérenger Berthoul) with no need for installation of ElsA inside Inria. We take this as a sign
of maturity of Tapenade. Apart from a few minor corrections, our contributon was essentially during development meetings, to point out some strategies and tool options to obtain efficient
differentiated code. One emphasis was on adjoint of vectorized code, which was produced as vectorized code too by means of a seldom-used Tapenade option that stores intermediate values statically,
i.e. not on a global stack. Sébastien Bourasseau presented the first results at the 21^st EuroAD workshop (Jena, Germany, November 19-20), with convincing performance on industrial-size test cases. A
joint article is in preparation.
Multirate methods Alain Dervieux Bruno Koobus Emmanuelle Itam Stephen Wornom
This study is performed in collaboration with IMAG-Montpellier. It addresses an important complexity issue in unsteady mesh adaptation and took place in the work done in the ANR Maidesc (ended 2017).
Unsteady high-Reynolds computations are strongly penalized by the very small time step imposed by accuracy requirements on regions involving small space-time scales. Unfortunately, this is also true
for sophisticated unsteady mesh adaptive calculations. This small time step is an important computational penalty for mesh adaptive methods of AMR type. This is also the case for the Unsteady
Fixed-Point mesh adaptive methods developed by Ecuador in cooperation with the Gamma3 team of Inria-Saclay. In the latter method, the loss of efficiency is even more crucial when the anisotropic mesh
is locally strongly stretched since only very few cells are in the regions of small time-step constraint. This loss is evaluated as limiting the numerical convergence order for discontinuities to 8/5
instead of second-order convergence. An obvious remedy is to design time-consistent methods using different time steps on different parts of the mesh, as far as they are efficient and not too
complex. The family of time-advancing methods in which unsteady phenomena are computed with different time steps in different regions is referred to as the multirate methods. In our collaboration
with university of Montpellier, a novel multirate method using cell agglomeration has been designed and developed in our AIRONUM CFD platform. A series of large-scale test cases show that the new
method is much more efficient than an explicit method, while retaining a similar time accuracy over the whole computational domain. A novel analysis shows that the proposed multirate algorithm indeed
solves the unsteady mesh adaptation barrier identified in previous works. This work is being published in a journal .
Control of approximation errors Eléonore Gauci Alain Dervieux Adrien Loseille Gamma3 team, Inria-Rocquencourt Frédéric Alauzet Gamma3 team, Inria-Rocquencourt Anca Belme university of Paris 6 Gautier
Brèthes university of Montreal Alexandre Carabias Lemma
Reducing approximation errors as much as possible is a particular kind of optimal control problem. We formulate it exactly this way when we look for the optimal metric of the mesh, which minimizes a
user-specified functional (goal-oriented mesh adaptation). In that case, the usual methods of optimal control apply, using adjoint states that can be produced by Algorithmic Differentiation.
This year, two conference papers were written on the methods of the team, including new analyses in ,, a work on correctors in CFD in an AIAA paper. A detailed study of adjoint-based mesh adaptation
for Navier-Stokes flows has been completed and published in a journal .
Following participation of Gamma3 and Ecuador to the European project UMRIDA (ended 2017), we wrote chapters 20, 21, 45, and 48 of the book “Uncertainty Management for Robust Industrial Design in
Aeronautics”, edited by C. Hirsch et al. in the Springer series Notes on Numerical Fluid Mechanics and Multidisciplinary Design (2019).
Turbulence models Alain Dervieux Bruno Koobus Stephen Wornom Maria-Vittoria Salvetti University of Pisa
Modeling turbulence is an essential aspect of CFD. The purpose of our work in hybrid RANS/LES (Reynolds Averaged Navier-Stokes / Large Eddy Simulation) is to develop new approaches for industrial
applications of LES-based analyses. In the applications targetted (aeronautics, hydraulics), the Reynolds number can be as high as several tens of millions, far too high for pure LES models. However,
certain regions in the flow can be predicted better with LES than with usual statistical RANS (Reynolds averaged Navier-Stokes) models. These are mainly vortical separated regions as assumed in one
of the most popular hybrid models, the hybrid Detached Eddy Simulation model. Here, “hybrid” means that a blending is applied between LES and RANS. An important difference between a real life flow
and a wind tunnel or basin is that the turbulence of the flow upstream of each body is not well known.
The development of hybrid models, in particular DES in the litterature, has raised the question of the domain of validity of these models. According to theory, these models should not be applied to
flow involving laminar boundary layers (BL). But industrial flows are complex flows and often present regions of laminar BL, regions of fully developed turbulent BL and regions of non-equilibrium
vortical BL. It is then mandatory for industrial use that the new hybrid models give a reasonable prediction for all these types of flow. We concentrated on evaluating the behavior of hybrid models
for laminar BL and for vortical wakes. While less predictive than pure LES on laminar BL, some hybrid models still give reasonable predictions for rather low Reynolds numbers.
This year, we have developed a new model relying on the hybridation of a DDES model based on a k-$ϵ$ closure with our dynamic VMS model. This model shows improvement in most situations and in
particular for laminar flows.
We have also addressed this year a challenging test case, the flow around tandem cylinders with a distance between the cylinders of 12 diameters. The accurate capture of the vortices traveling along
this path of 12 diameters requires that the LES filter does not accumulate any dissipation along this trajactory. This is a noticeable property or our DVMS model. Further, the numerics need be as
accurate as possible. We use a superconvergent approximation, up to fifth order accurate on Cartesian regions of the computational domain. This combination allowed for an accurate prediction of the
drag of the second cylinder. This result has been presented at the workshop ETMM12 | {"url":"https://radar.inria.fr/rapportsactivite/RA2018/ecuador/ecuador.xml","timestamp":"2024-11-10T05:02:08Z","content_type":"application/xml","content_length":"111387","record_id":"<urn:uuid:152259a2-0c6a-47fd-8da3-92e485d2975c>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00398.warc.gz"} |
How To Learn Basic Math For Adults
Learning basic math — addition, subtraction and multiplication — for adults is no different than learning basic math for children. The only real difference is that an adult's other cognitive
abilities, including language, are usually better developed than those of a child at the same stage of math learning. So it's usually easier to explain the concepts to an adult than it is to a child.
Addition and Subtraction
Step 1
Start getting a grip on the basic concepts of addition and subtraction by using five of an identical item. These could be five oranges, five grapes, five tennis balls, five bricks... five of
Step 2
Line up all five objects and count them. Now remove one object from the lineup and place it to the side. This is the same as subtracting one from your original number, which was five. What is five
minus one? Count the remaining objects to find out: four.
Step 3
Return the object you removed to the lineup. You had four objects, now you have added one, and as you can see, there are now five objects again. So four plus one equals five — the evidence is right
in front of you.
Step 4
Reset your lineup of five objects, then repeat the exercise while removing two, three, four and, finally, all five of the objects. Once you've removed an object and calculated the result, add it back
in and recalculate the result.
Step 5
Expand your grasp of the subject, now that you understand the basic principle of it, by memorizing addition and subtraction tables. (See the Resources section for links.)
Step 1
Use a large number of identical objects, such as grapes or marbles, as your visual aid.
Step 2
Place one grape on the table in front of you. Now place another grape beside it. You've got one grape, twice — in other words, one times two. If you count the grapes, you'll see that one times two is
a total of two.
Step 3
Note that since you have two grapes in front of you already, you're perfectly set up to practice two times two. Just put down another set of two grapes next to the first two. You've got two sets of
two grapes — the same as two times two — and as you can see by counting, the total is four.
Step 4
Take one grape away so that you've got one group of three grapes. If you multiply that group of grapes by two — in other words, by representing it on the table twice — you'll see that you have six
Step 5
Reassure yourself that this principle works for other numbers, too. For example, if you set up three groups of four grapes — three times four — then count the grapes, you'll see that you have 12
grapes. So three times four is 12. Now you can expand your grasp of multiplication by memorizing multiplication tables. (See the Resources section for a link.)
Cite This Article
Maloney, Lisa. "How To Learn Basic Math For Adults" sciencing.com, https://www.sciencing.com/how-to-learn-basic-math-for-adults-12751579/. 30 May 2013.
Maloney, Lisa. (2013, May 30). How To Learn Basic Math For Adults. sciencing.com. Retrieved from https://www.sciencing.com/how-to-learn-basic-math-for-adults-12751579/
Maloney, Lisa. How To Learn Basic Math For Adults last modified August 30, 2022. https://www.sciencing.com/how-to-learn-basic-math-for-adults-12751579/ | {"url":"https://www.sciencing.com:443/how-to-learn-basic-math-for-adults-12751579/","timestamp":"2024-11-09T09:58:35Z","content_type":"application/xhtml+xml","content_length":"71589","record_id":"<urn:uuid:4f04d615-736d-493f-a2f7-e3bd642c4f11>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00655.warc.gz"} |
Fuel Consumption Calculator
The Fuel Consumption Calculator takes in the distance traveled, tank capacity, average fuel consumption, and fuel price, and calculates the fuel consumption in liters or gallons, number of fuel tanks
required, and cost of fuel.
You can enter the data in either metric or imperial units. This means that you should either enter
• the distance in kilometers, the fuel tank volume in liters, the average consumption of fuel in liters per 100 kilometers and the price for 1 liter
• the distance in miles, the fuel tank volume in gallons, the average consumption of fuel in gallons per 100 miles and price for 1 gallon.
The calculation is based on the average consumption per one hundred kilometers or miles
Distance in miles or kilometers
Tank capacity in litres or gallons
Average fuel consumption per 100 (miles or kilo meters)
Fuel price per one litre or gallon (same fuel unit must be used as in previous inputs)
Digits after the decimal point: 2
PLANETCALC, Fuel Consumption Calculator | {"url":"https://planetcalc.com/4186/","timestamp":"2024-11-04T07:44:54Z","content_type":"text/html","content_length":"34302","record_id":"<urn:uuid:dd5d6079-5382-4f4a-8bc2-777c27c0c02c>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00685.warc.gz"} |
4.13 Shuffling Can Help Us Understand Real Data Better
Course Outline
• segmentGetting Started (Don't Skip This Part)
• segmentStatistics and Data Science: A Modeling Approach
• segmentPART I: EXPLORING VARIATION
• segmentChapter 1 - Welcome to Statistics: A Modeling Approach
• segmentChapter 2 - Understanding Data
• segmentChapter 3 - Examining Distributions
• segmentChapter 4 - Explaining Variation
□ 4.13 Shuffling Can Help Us Understand Real Data Better
• segmentPART II: MODELING VARIATION
• segmentChapter 5 - A Simple Model
• segmentChapter 6 - Quantifying Error
• segmentChapter 7 - Adding an Explanatory Variable to the Model
• segmentChapter 8 - Digging Deeper into Group Models
• segmentChapter 9 - Models with a Quantitative Explanatory Variable
• segmentPART III: EVALUATING MODELS
• segmentChapter 10 - The Logic of Inference
• segmentChapter 11 - Model Comparison with F
• segmentChapter 12 - Parameter Estimation and Confidence Intervals
• segmentFinishing Up (Don't Skip This Part!)
• segmentResources
list High School / Advanced Statistics and Data Science I (ABC)
4.13 Shuffling Can Help Us Understand Real Data Better
Randomness Produces Patterns in the Long Run
One important thing to understand about random processes is that they will produce a different result each time. If you only flip a coin one time and get heads, you can’t really tell anything about
the random process that produced the result. You can’t even know that it was, actually, random. But if you flip a coin a thousand times, you will see that in the long run the coin comes up heads 50%
of the time. It’s the law of large numbers!
The same is true with shuffling data. Just shuffling the tip percentages one time shows us one possible result of a purely random process. (We know the process is purely random because the shuffle()
function is designed to be random.) But to see a pattern in the randomness requires that we do many shuffles. This is the only way we can see the range of possible outcomes that can be produced by a
purely random process, and see how frequently different outcomes occur.
The R code in the window below creates the jitter plot we’ve been looking at of tips as a function of condition in the tipping study. You can shuffle the tips before graphing them by simply using
shuffle(Tip) instead of Tip as the outcome variable. Add shuffle() to the code, then run it a few times to see how the jitter plots change with each shuffling of tips.
require(coursekata) # shuffle the tips in the jitter plot gf_jitter(Tip ~ Condition, data = TipExperiment, width = .1) %>% gf_labs(title = "Shuffled Data") # shuffle the tips in the jitter plot
gf_jitter(shuffle(Tip) ~ Condition, data = TipExperiment, width = .1) %>% gf_labs(title = "Shuffled Data") ex() %>% check_or( check_function(., "gf_jitter") %>% check_arg("object") %>% check_equal(),
override_solution(., "gf_jitter(shuffle(Tip) ~ shuffle(Condition), data = TipExperiment)") %>% check_function("gf_jitter") %>% check_arg("object") %>% check_equal(), override_solution(., "gf_jitter
(Tip ~ shuffle(Condition), data = TipExperiment)") %>% check_function("gf_jitter") %>% check_arg("object") %>% check_equal() )
CK Code: A4_Code_Shuffling_01
Each time you shuffle the data, you’ll get a slightly different pattern of results. Plotting the group means on top of each shuffled distribution can help you see the pattern more clearly as it
changes with each shuffle.
Saving the shuffled Tip as ShuffTip will allow us to plot the means of the two groups as lines.
require(coursekata) # add the shuffle function to this code TipExperiment$ShuffTip <- TipExperiment$Tip # this makes a jitter plot of the shuffled data and adds means for both groups gf_jitter
(ShuffTip ~ Condition, data = TipExperiment, width = .1) %>% gf_labs(title = "Shuffled Data") %>% gf_model(ShuffTip ~ Condition, color = "orchid") # add the shuffle function to this code
TipExperiment$ShuffTip <- shuffle(TipExperiment$Tip) # this makes a jitter plot of the shuffled data and adds means for both groups gf_jitter(ShuffTip ~ Condition, data = TipExperiment, width = .1)
%>% gf_labs(title = "Shuffled Data") %>% gf_model(ShuffTip ~ Condition, color = "orchid") ex() %>% { check_function(., "shuffle") %>% check_arg(1) %>% check_equal() check_function(., "gf_jitter") %>%
check_arg(1) %>% check_equal() }
CK Code: A4_Code_Shuffling_02
Below are three examples of shuffled tips plotted by condition.
As we can see, some shuffles produce distributions where the tips look similar across conditions (as in the center plot). Other shuffles result in higher tips from the control group (as in the left
plot), while in others the smiley face tables appear to be tipping more (as in the right plot).
None of these results could possibly be due to the effect of smiley faces on checks. We know this because the assignment of tables to groups was done using a 100% random process. What we are seeing
in these graphs is what possible outcomes can look like if the process is purely random. The more times we run the code, the more sense we will get of what the range of outcomes can look like.
How Shuffling Can Help Us Understand Real Data Better
Let’s go back to the question we were asking before we started shuffling tips. Are the slight differences in tips related to adding smiley faces to checks due to the smiley faces, or could they be
just due to randomness? Shuffling tips provides us with a way to begin answering this question.
By graphing multiple sets of randomly-generated results, we can look to see whether the pattern observed in the real data looks like it could be randomly generated, or if it looks markedly different
from the randomly-generated patterns. If it looks markedly different, we might be more likely to believe that smiley faces had an effect. If it looks similar to the random results, we might be more
inclined to believe that the effect, even if apparent in the data, could simply be the result of randomness.
Below we show nine different plots. Eight of them are the result of random shuffles of tips; the other one, in the upper left with red lines for averages, is the plot of the actual data. Take a look
at all these plots, and compare the plot of real data to the other plots.
Using shuffle() slows us from concluding that every relationship we observe in data (e.g., the relationship between smiley face and tipping) is real in the DGP. We always need to consider whether the
relationship in the data might just be the result of random sampling variation. Concluding that a relationship in data is real when in fact it results from randomness is what statisticians call Type
I Error.
Maybe It’s Not Just Randomness
Based on our analysis of the nine jitter plots above, we’ve concluded that maybe – just maybe – the difference we observed between smiley face and control groups could just be due to random sampling
variation. But would this always be the result of random shuffling? Absolutely not.
Let’s take the case of the sex and students’ height. Below we’ve put a jitter plot that shows the relationship, and, like before, added on the average height for females and for males as a red line.
This looks like a fairly large difference between females and males. But still, there is a lot of overlap between the two distributions; even though males are taller in general, some females are
taller than some males. The question is: Could the difference between females and males be due only to random sampling variation, or is there really a difference in the DGP?
In the nine jitter plots below we’ve shown the actual data (with the means in a different color), and eight graphs showing eight different shuffles of height across females and males.
For the smiley face data, it was hard to distinguish the real data from the randomized data. But in this case, the graph of real data looks very different from the randomly generated data. For this
reason, we might conclude that the relationship between sex and thumb length is not just due to randomness but is a real relationship in the DGP.
Even though it’s still possible that a random process generated this height data (after all, it’s possible to flip 1000 heads in a row), it’s not very likely. Later we will learn more systematic ways
of making this decision, but for now, just shuffling and looking at the results can be a powerful tool for helping us interpret patterns of results in data. | {"url":"https://coursekata.org/preview/book/f84ca125-b1d7-4288-9263-7995615e6ead/lesson/6/12","timestamp":"2024-11-05T12:13:37Z","content_type":"text/html","content_length":"98062","record_id":"<urn:uuid:c56cccbc-5f24-45e6-8f75-79c537b88676>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00851.warc.gz"} |
R Workshop: Reproducible Research using Sweave for Beginers | R-bloggersR Workshop: Reproducible Research using Sweave for Beginers
R Workshop: Reproducible Research using Sweave for Beginers
[This article was first published on
bayesianbiologist » Rstats
, and kindly contributed to
]. (You can report issue about the content on this page
) Want to share your content on R-bloggers?
if you have a blog, or
if you don't.
Monday, April 30, 2012 14h-16h. Stewart Biology Rm w6/12 (Montreal)
guRu: Denis Haine (Université de Montréal)
Reproducible research was first coined by Pr. Jon Claerbout, professor of geophysics at Stanford University, to describe that the results from researches can be replicated by other scientists by
making available data, procedures, materials and the computational environment on which these results were produced from.
This workshop intends to describe reproducible research, what it is and why you should care about it, and how to do it with the combination of R, LATEX, Sweave and makefile. Tips and tricks will also
be provided.
Learning Objectives
• To get introduce to the concept of reproducible research
• To get started with the implementation of reproducible research with R and Sweave,
• To produce a first Sweave document in LATEX
This is a meeting of the Montreal R Users Group. We’re open to everyone! Sign up to RSVP! | {"url":"https://www.r-bloggers.com/2012/04/r-workshop-reproducible-research-using-sweave-for-beginers/","timestamp":"2024-11-06T21:51:49Z","content_type":"text/html","content_length":"94538","record_id":"<urn:uuid:7468dfe8-6e71-4edc-a282-ebcb565ab1fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00199.warc.gz"} |
PE ratio, current and historical analysis
The mean historical PE ratio of Ensign Group over the last ten years is 24.05. The current 35.17 price-to-earnings ratio is 46% more than the historical average. In the past ten years, ENSG's PE
ratio reached its highest point in the Sep 2024 quarter at 33.92, with a price of $143.82 and an EPS of $4.24. The Mar 2020 quarter saw the lowest point at 16.21, with a price of $37.61 and an EPS of | {"url":"https://fullratio.com/stocks/nasdaq-ensg/pe-ratio","timestamp":"2024-11-13T04:33:32Z","content_type":"text/html","content_length":"31192","record_id":"<urn:uuid:83a8122c-aee6-435b-be0b-6d331441eb72>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00314.warc.gz"} |
A soft drink is available in two packs - i) a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 15 cm, and ii) a plastic cylinder with a circular base of diameter 7 cm
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A soft drink is available in two packs - (i) a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 15 cm and (ii) a plastic cylinder with a circular base of diameter 7
cm and height 10 cm. Which container has greater capacity and by how much?
Since the tin can is cuboidal in shape while the other is cylindrical, we will find the volume of both containers.
The volume of a cylinder of base radius, r, and height, h = πr^2h
The volume of a cuboid of length ' l ', breadth ' b', and height ' h ' = l × b × h
Dimensions of tin can with a rectangular base are:
Length of the cuboidal tin can, l = 5cm
The breadth of the cuboidal tin can, b = 4cm
Height of the cuboidal tin can, h = 15cm
The volume of the cuboidal tin can = l × b × h
= 5 cm × 4 cm × 15 cm
= 300 cm^3
Dimensions of the plastic cylinder with a circular base are:
The diameter of the cylindrical plastic can = 7 cm
The radius of the cylindrical plastic can, r = 7/2 cm
Height of the cylindrical plastic can, h = 10cm
The volume of the cylindrical plastic can = πr^2 h
= 22/7 × 7/2 cm × 7/2 cm × 10cm
= 385 cm^3
Clearly, the plastic cylinder with a circular base has greater capacity than the tin container.
Difference = 385 cm^3 - 300 cm^3 = 85 cm^3
The plastic cylindrical has more capacity than the tin can by 85 cm^3.
☛ Check: NCERT Solutions for Class 9 Maths Chapter 13
Video Solution:
A soft drink is available in two packs - (i) a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 15 cm and (ii) a plastic cylinder with a circular base of diameter 7
cm and height 10 cm. Which container has greater capacity and by how much?
NCERT Solutions for Class 9 Maths Chapter 13 Exercise 13.6 Question 3
For a soft drink is available in two packs a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 15 cm and a plastic cylinder with a circular base of diameter 7 cm and
height 10 cm, we have found that the plastic cylindrical container has more capacity than the tin can by 85 cm^3.
☛ Related Questions:
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/ncert-solutions/a-soft-drink-is-available-in-two-packs-i-a-tin-can-with-a-rectangular-base-of-length-5-cm-and-width-4-cm/","timestamp":"2024-11-04T17:01:39Z","content_type":"text/html","content_length":"229276","record_id":"<urn:uuid:7d2f1735-49b0-487f-bbd8-0ab293f11f08>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00281.warc.gz"} |
• makeDiiF() can now be given coefficients of inbreeding. Facilitates either:
□ calculating f in a new generation of the pedigree if f has already been calculated for previous generations
□ creating a “phantom parent” to all founders to specify an average coefficient of inbreeding in the base population.
□ can be used to speed up simulations where breeding values are calculated based on mid-parent value plus Mendelian sampling deviation (that needs to account for inbreeding of parents).
Major changes
• proLik() (and proLik4()) REMOVED from the package.
□ these functions were only to facilitate advanced use of the asreml R package (which requires purchasing an expensive license) and due to the unstable behavior of asreml discovered when
revising proLik() it was decided that nadiv should no longer support this other package in this way.
□ Instead, time will be spent improving the gremlin R package for a replacement to asreml.
• makeA() was affected by a bug in Matrix <1.6-0
□ Matrix::chol2inv() bug highlighted a messy order of operations in nadiv
□ now perfected the order of operations and explicitly use tcrossprod() and solve() which are just what chol2inv() uses
☆ done to be more explicit in nadiv and take 1 step away from future bugs like this
□ checked speed and memory profiling for several options of types of matrices
☆ dtCMatrix is best for time (tried “dtrMatrix” and “dtpMatrix”, but these were much slower.
☆ chol2inv() and tcrossprod(solve()) allocated same amount of memory according to profmem package
• bug in makeAinv(), makeGGAinv(), and makeDiiF() caused coefficients of inbreeding (f) and Mendelian sampling variances (dii) to be ordered incorrectly
□ this did not cause matrices from these functions to be incorrect (e.g., Ainv, Tinv, and their related matrices), but did cause matrices built directly from dii to be incorrect - namely the A
matrix from makeA() and so consequently anything built from A (i.e., dominance and epistatic relatedness matrices through makeD() etc.)
□ this was FIXED with details in the commit
Small changes
• nadiv version 2.17.2 caused CRAN to archive due to error induced by Matrix updates
□ Mikael Jagan (Matrix author) helpfully provided excellent explanations and patches
□ errors in nadiv code caused by new methods for all.equal() and rbind2()
□ removed methods is.numPed() and is.proLik()
☆ hardly use first and never second plus they just call inherits()
DEPRECATED version 2.17.2
• pin() does not work with asreml 4 (should still work with asreml 3 model objects)
□ nadiv will not support this in the future as asreml 4 has vpredict()
• proLik4(), essentially the same as proLik(), but works on asreml v4
□ proLik() is kept to retain compatibility with asreml v3 model objects
• makeM() creates mutational effects relatedness matrix M directly
□ can be used with brms/JAGS etc. that require relatedness matrices (not their inverse) in mixed models
Small changes
• fixed deprecated use of Matrix non-virtual subclasses
□ addresses issues with Matrix 1.4-2 and specifically >=1.5-0
• new c++ routine to calculate coefficients of inbreeding and D of Cholesky decomposed A matrix
□ follows Meuwissen and Luo’s (1992) algorithm
□ standardizes this code and consolidates to 1 location, instead of being spread out as copies in several other places
Small changes
• fix bug in prepPed() as often encountered/reported from use in optiSel
2.17.0 Released to CRAN 14 January 2021
• makeMinv() creates the inverse of the (additive) mutational effects relatedness matrix.
• makeT() creates the lower triangle of the cholesky factor of the additive genetic numerator relatedness matrix.
Small changes
• update way create matrices to make ‘dsCMatrix’ from sparseMatrix() instead of Matrix()
□ fixes error caused by change in Matrix package 1.3-0
• update ggcontrib() to internally use makeT() to directly create a subset of the T matrix (which the subset is Q for the special setup for genetic groups).
□ replaces method that created entire T-inverse then invert it to obtain entire T before subsetting all that is needed for Q
• drop 4-number version information down to just 3 number versions
2.16.2.0 Released to CRAN 20 October 2019
• geneDrop() conducts a gene dropping simulation down a user-supplied pedigree.
• simGG() now simulates phenotypes with heterogeneous additive genetic variances among the genetic groups (i.e., immigrant and resident groups)
□ The change to this simulation function now ensures phenotypes and underlying breeding values are consistent with the mixed model genetic group analysis approach described by Muff et al. 2019.
Gen. Sel. Evol.
□ Breeding values have now been “split” to track resident-specific and immigrant-specific breeding values
□ Users should interact with the function the same way as always, as no changes to the function arguments have been made.
• makeGGAinv() added as a new function to construct genetic group-specific inverse relatedness matrices (Ainv).
ggPed <- Q1988[-c(3:7), c("id", "damGG", "sireGG")]
AinvOut <- makeGGAinv(ggPed, ggroups = 2)$Ainv #<-- list with 2 Ainv matrices
• added makeTinv() and makeDiiF() functions
□ These create items used in the Cholesky factorization of a relatedness matrix (or its inverse) and/or the individual coefficients of inbreeding f
□ In particular, these are used to construct genetic group specific inverse relatedness matrices, and are used “under the hood” in makeGGAinv().
□ makeDiiF() creates the D matrix of the Cholesky factorization of the relatedness matrix below (i.e., A and the coefficients of inbreeding (diagonals-1 of A)
□ makeTinv() creates Tinv of the Cholesky factorization of the inverse relatedness matrix below (i.e., Ainv)
☆ A= T’ D T
☆ Ainv=Tinv’ Dinv Tinv
□ Note, because D and Dinv are diagonal matrices, Dinv= the element-wise operation of 1 / d_ii
☆ Consequently, obtaining Dinv from D is trivial
☆ Simply do Dinv <- D followed by Dinv@x <- 1 / D@x
□ Users only need to supply a pedigree and the functions do the rest. For example:
Small changes
• update to simPedDFC() to allow more flexibility in designing pedigrees
2.16.0 Released to CRAN 5 May 2018
• roxygen2 documentation
• Return diagonal of Mendelian sampling variance matrix in makeAinv() and makeS()
□ These (or their inverses?) can be used in JAGS or BUGS when running a quantitative genetic mixed model
Small changes
• default action is to calculate log-determinant of matrices
□ switched from not calculating this by default
• Functions to construct sex-chromosomal dominance relatedness matrices
□ makeSd() and makeSdsim()
☆ These are similar to what makeD() and makeDsim() accomplish for autosomes
☆ The output contains the Sd and Sdsim dominance relatedness matrices
☆ The inverses of these can be obtained from Sdinv and Sdsiminv and used in a mixed model
Small changes
• proLik() improved/bug fixed to find confidence limits
□ previously would declare confidence limits found when they hadn’t been
☆ this was due to optimize() quitting too early with default tol argument
□ returns NA if confidence limits are not, in fact, found (e.g., for boundary parameters, variances that are not significantly greater than zero)
□ plot.proLik() now includes vertical lines to better visualize CIs
• use lower_bound algorithm for matrix lookup within c++ code
□ based on c++ std::lower_bound
☆ affect makeAinv() and makeD()
□ greater speedup as A^-1 and D become more dense
• create default and class ‘numPed’ methods for genAssign() and prunePed()
□ can greatly trim down genAssign.numPed() code (and to some extent prunePed.numped())
□ this speeds up/uses less memory
□ since genAssign() and prunePed() are frequently called in many nadiv functions which operate on class ‘numPed’, this will have modest, but significant performance increases
□ thanks to profvis for bringing my attention to this!
2.14.3 Released 20 April 2016
• Fuzzy classification of genetic groups to construct A^-1.
□ Allows individuals’ phantom parents to be assigned to genetic groups with a probability. Meaning, they can be assigned to more than one genetic group.
□ To implement, the pedigree must have phantom parent identities as unique rows and a matrix of probabilities of group membership for every phantom parent in every genetic group has to be
supplied to the fuzz argument.
□ Examples can be seen in the makeAinv.Rd help file or by running the following commands in R:
* Notably, fuzzy classification can be set to 'null', where each phantom parent is assigned to one genetic group with probability=1. This produces the same **Astar** matrix as regular genetic group methods (without fuzzy classification). See this demonstrated in the examples of the help documentation.
• Add the makeAstarMult() function to create the inverse numerator relationship matrix with genetic groups (and possibly also fuzzy classification of genetic groups) through matrix multiplication
instead of using direct algorithms to set this up.
□ Uses ggcontrib() and makeAinv() to create Q and A^-1 directly, then multiplies these in such a way as to obtain Astar.
□ Examples using the two different types of pedigree formats and either with or without fuzzy classification can be seen in the makeAstarMult.Rd help file or run them in R with the command:
• Add the F2009 dataset
□ This dataset can be used as an example for fuzzy classification of genetic groups when constructing a numerator relationship matrix with groups (i.e., with makeAinv())
□ See a description in F2009.Rd or in R type:
• Add the simGG() function to simulate pedigree and phenotype when immigration occurs in a focal population
□ Allows fairly fine control over a simulation. For example, the function is flexible in the: population size, number of immigrants per generation, number of generations, and both spatial and
temporal trends in both focal and immigrant populations.
□ This is the function used to simulate the new ggTutorial dataset (below)
• Added the ggTutorial dataset
□ This is a simulated dataset to be used in analyses with genetic group animal model methods.
□ See a description in ggTutorial.Rd or in R type:
• LRTest() is now an exported function to do log-likelihood ratio tests
Small changes
• new S3 generic and methods for makeAinv().
□ method dispatch is based on class of the fuzz argument
☆ if fuzz == NULL then dispatch the method makeAinv.default()
☆ if fuzz == "matrix" | fuzz == "Matrix" then dispatch makeAinv.fuzzy()
• fix issue with proLik() and the confidence interval estimation
2.14.2 Released 5 Feb 2016
• ggcontrib() can now incorporate fuzzy classification of genetic groups
□ To facilitate this, the examples for ggcontrib() have been changed. For more information and examples, read the help documentation ggcontrib.Rd or in R type:
Small changes
• fixed ordering of f coefficients returned by makeAinv()
2.14.1 Released 22 July 2015
2.14.0 Released 3 July 2015
• makeAinv() now can construct the augmented A-inverse matrix for genetic groups
□ This change has introduced new arguments to makeAinv(), however, the defaults are set to produce the normal A-inverse. For more information and examples, read the help documentation
makeAinv.Rd or in R type:
• Improved algorithm underlying makeAinv() - significant speed-up!
• Created new class numPed for pedigrees constructed by numPed().
□ Methods for checking (is.numPed()) and re-ordering rows (ronPed()) currently available
□ To re-order the rows of an integer pedigree of class numPed, use ronPed() instead of typical subsetting operators (e.g., '[') to retain the class attribute numPed. For example:
• Re-made (i.e., re-simulated) the warcolak dataset.
□ Codes specifying the sex are now "M" and "F" instead of 0 & 1.
□ New columns added to the dataset that contain all random effects underlying the phenotype.
□ Entire code used to simulate the dataset is now an example in warcolak.Rd.
• Added two new datasets/example pedigrees: (1) Q1988 from Quaas 1988 and (2) Mrode3 from Mrode (2005) chapter 3. See their descriptions in Q1988.Rd and Mrode3.Rd or in R type:
Small changes
2.13.3 Released 4 June 2015
• Added TDtT(), a function to take the TDT’ Cholesky decomposition of a matrix (not currently exported).
• Added founderLine() which traces all individuals back to either the paternal or maternal founder
• grfx() now has a new argument to allow user to supply the standard normal deviates instead of generating them within the function.
□ extended the warn argument to apply to the warning when incidence = NULL
□ updated grfx.Rd with an example illustrating the stdnorms argument
□ added ... argument to drfx() so that arguments for the internal use of grfx() can be supplied to drfx().
• argument now allows specified prefix for all identities in a pedigree generated from simPedHS() or simPedDFC().
• argument added that specifies output format of ggcontrib(), default is “matrix”
Small changes
• removed ‘asreml’ from suggests in the package DESCRIPTION file.
• changed pcc() return FALSE if the object (asreml) shows the log-likelihood did not converge
• added silent = FALSE argument to pcc() so that the default can be changed to not show messages
□ helpful in simulations where a lot of output would be printed on screen
• changed the signs associated with likelihood ratio test statistics, etc.
□ changed signs in proLik() so that profile likelihoods should be “valleys” (instead of “hills”, as they were in versions previous to 2.13.3)
2.13.2 Released 20 June 2014
Small changes
• added the calculation of the log determinant of the A matrix to makeAinv()
□ log(det(A^-1)) = log(1) - log(det(A))
□ uses property of determinants that det(A^-1) = 1 / det(A) = det(A)^-1.
• changed methods underlying makeA() to use inverse of cholesky factorization of the A-inverse matrix
□ base::chol2inv() to obtain A
☆ informally seems faster unless A is dense.
2.13 Released 16 June 2014
• added the prepPed() to prepare pedigrees for use in other functions
• exported makeAinv()
• added ggcontrib() so that genetic group contributions can be calculated
□ still need to implement functionality for “fuzzy classification”
• support for selfing
• added the pin() and pcc() functions for the delta method and parameter value convergence checking, respectively, for asreml type REML models. Also added the pin.Rd and pcc.Rd help/documentation
□ NOTE, pin() is not exported in this version (need nadiv:::pin() to use it)
• added makeDufam(), but did not export it.
□ experimental version of makeD that first sorts individuals according to generation and then dam, and then sire.
□ sticks individuals with the same parents next to each other in the pedigree
□ haven’t implemented a parallel version of this c++ code yet (or checked function for timing/memory benefits or accuracy).
Small changes
• changed makeDomEpi() argument “Dinverse” to “invertD” to be similar to makeD()
• added pcc() checks to constrainFun() so that only likelihood ratio test statistics of the constrained model returned if both the loglikelihood & parameter estimates have converged
• enabled parallel processing (forking, so no Windows compatibility) in:
□ findDFC(), makeD(), makeDsim()
• changed name of FindDFC() to findDFC() | {"url":"https://mirror.las.iastate.edu/CRAN/web/packages/nadiv/news/news.html","timestamp":"2024-11-12T02:52:57Z","content_type":"application/xhtml+xml","content_length":"31076","record_id":"<urn:uuid:c1df888f-3716-4b3b-8b8e-bee2f4b303ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00188.warc.gz"} |
2 Digit By 2 Digit Multiplication Word Problems Worksheets
Mathematics, especially multiplication, develops the keystone of various academic self-controls and real-world applications. Yet, for numerous students, mastering multiplication can pose a
difficulty. To resolve this difficulty, educators and moms and dads have actually welcomed an effective device: 2 Digit By 2 Digit Multiplication Word Problems Worksheets.
Introduction to 2 Digit By 2 Digit Multiplication Word Problems Worksheets
2 Digit By 2 Digit Multiplication Word Problems Worksheets
2 Digit By 2 Digit Multiplication Word Problems Worksheets -
Algebra 1 Basic Skills Domain Range Equations Exponents Inequalities Linear Equations Monomials Polynomials Quadratic Functions Radical Expressions Rational Expressions Systems of Equations
Trigonometry Word Problems Algebra 2 Basic Skills Complex Numbers Conic Sections Equations Inequalities Exponential Logarithmic Functions
The printable PDF worksheets presented here involve single digit multiplication word problems Each worksheet carries five word problems based on day to day scenarios Multiplication Word Problems Two
digit times Single digit
Value of Multiplication Technique Recognizing multiplication is critical, laying a strong foundation for sophisticated mathematical principles. 2 Digit By 2 Digit Multiplication Word Problems
Worksheets supply structured and targeted technique, promoting a much deeper understanding of this fundamental math procedure.
Evolution of 2 Digit By 2 Digit Multiplication Word Problems Worksheets
Two Digit Multiplication Worksheets Multiplication worksheets Two digit multiplication Math
Two Digit Multiplication Worksheets Multiplication worksheets Two digit multiplication Math
Worksheets Math drills Multiplication Multiply 2 x 2 digits Multiply 2 x 2 digits 2 digit multiplication Multiplication practice with all factors being under 100 column form Worksheet 1 Worksheet 2
Worksheet 3 Worksheet 4 Worksheet 5 10 More Similar Multiply 3 x 2 digits Multiply 3 x 3 digits What is K5
Level 4 Language English en ID 1536391 17 10 2021 Country code BS Country Bahamas School subject Math 1061955 Main content 2 Digits by 2 Digits Multiplication Word Problems 1578720 2 Digits by 2
Digits Multiplication Word Problems
From standard pen-and-paper exercises to digitized interactive layouts, 2 Digit By 2 Digit Multiplication Word Problems Worksheets have progressed, satisfying varied understanding designs and
Sorts Of 2 Digit By 2 Digit Multiplication Word Problems Worksheets
Fundamental Multiplication Sheets Basic exercises focusing on multiplication tables, aiding learners build a strong math base.
Word Problem Worksheets
Real-life circumstances incorporated right into troubles, boosting essential reasoning and application skills.
Timed Multiplication Drills Tests made to enhance rate and accuracy, aiding in quick psychological mathematics.
Benefits of Using 2 Digit By 2 Digit Multiplication Word Problems Worksheets
Multiplication Worksheets 50 Problems Printable Multiplication Flash Cards
Multiplication Worksheets 50 Problems Printable Multiplication Flash Cards
Remember to put the extra zero in the ones place on the second line of multiplication Allow your learning and practice to go up a notch or two with exercises including word problems and different
multiplication models These 2 digit by 2 digit multiplication worksheets pdfs cater to children in grade 4 and grade 5
Wyzant is IXL s tutoring network and features thousands of tutors who can help with math writing science languages music hobbies and almost anything else you can imagine For all ages children to
adults Improve your math knowledge with free questions in Multiply a 2 digit number by a 2 digit number word problems and thousands of
Improved Mathematical Skills
Consistent method develops multiplication efficiency, improving overall mathematics abilities.
Enhanced Problem-Solving Talents
Word troubles in worksheets develop logical thinking and method application.
Self-Paced Understanding Advantages
Worksheets fit specific learning rates, cultivating a comfy and adaptable learning atmosphere.
Just How to Develop Engaging 2 Digit By 2 Digit Multiplication Word Problems Worksheets
Incorporating Visuals and Colors Lively visuals and colors capture attention, making worksheets aesthetically appealing and involving.
Including Real-Life Scenarios
Relating multiplication to everyday scenarios includes relevance and practicality to exercises.
Customizing Worksheets to Various Skill Levels Tailoring worksheets based upon varying efficiency levels makes sure comprehensive discovering. Interactive and Online Multiplication Resources Digital
Multiplication Devices and Gamings Technology-based sources supply interactive learning experiences, making multiplication interesting and delightful. Interactive Websites and Applications Online
platforms offer varied and accessible multiplication technique, supplementing traditional worksheets. Customizing Worksheets for Different Understanding Styles Aesthetic Students Visual aids and
representations help comprehension for learners inclined toward aesthetic knowing. Auditory Learners Spoken multiplication problems or mnemonics cater to learners that realize principles with
auditory means. Kinesthetic Students Hands-on activities and manipulatives sustain kinesthetic learners in recognizing multiplication. Tips for Effective Implementation in Learning Consistency in
Practice Normal practice enhances multiplication skills, advertising retention and fluency. Balancing Repeating and Range A mix of repetitive workouts and varied problem formats keeps interest and
comprehension. Giving Positive Comments Feedback help in determining locations of enhancement, urging ongoing development. Obstacles in Multiplication Technique and Solutions Motivation and
Interaction Obstacles Boring drills can result in uninterest; cutting-edge strategies can reignite inspiration. Overcoming Anxiety of Math Negative understandings around math can prevent progression;
producing a favorable discovering setting is important. Effect of 2 Digit By 2 Digit Multiplication Word Problems Worksheets on Academic Efficiency Researches and Research Searchings For Study
suggests a favorable connection between consistent worksheet usage and boosted math efficiency.
2 Digit By 2 Digit Multiplication Word Problems Worksheets become functional devices, fostering mathematical efficiency in students while fitting diverse discovering styles. From fundamental drills
to interactive online resources, these worksheets not just enhance multiplication abilities yet also advertise critical thinking and analytical abilities.
2 Digit Math Worksheets Activity Shelter
Multiplication Word Problems Worksheets
Check more of 2 Digit By 2 Digit Multiplication Word Problems Worksheets below
Multiplication Word Problems Grade 2 Free Printable Kidsworksheetfun
3 Digit By 2 Digit Multiplication Word Problems Worksheets Pdf Free Printable
2 Digit By 1 Digit Multiplication Word Problems
Multiplication Word Problems Worksheets
2 Digit By 2 Digit Multiplication Worksheets With Answers Free Printable
Math Word Problems 2 Digit Multiplication Jason Burn s Multiplication Worksheets
Multiplication Word Problems Worksheets Math Worksheets 4 Kids
The printable PDF worksheets presented here involve single digit multiplication word problems Each worksheet carries five word problems based on day to day scenarios Multiplication Word Problems Two
digit times Single digit
Multiplication 2 Digits Times 2 Digits Super Teacher Worksheets
The worksheets below require students to multiply 2 digit numbers by 2 digit numbers Includes vertical and horizontal problems as well as math riddles task cards a picture puzzle a Scoot game and
word problems 2 Digit Times 2 Digit Worksheets Multiplication 2 digit by 2 digit FREE
The printable PDF worksheets presented here involve single digit multiplication word problems Each worksheet carries five word problems based on day to day scenarios Multiplication Word Problems Two
digit times Single digit
The worksheets below require students to multiply 2 digit numbers by 2 digit numbers Includes vertical and horizontal problems as well as math riddles task cards a picture puzzle a Scoot game and
word problems 2 Digit Times 2 Digit Worksheets Multiplication 2 digit by 2 digit FREE
Multiplication Word Problems Worksheets
3 Digit By 2 Digit Multiplication Word Problems Worksheets Pdf Free Printable
2 Digit By 2 Digit Multiplication Worksheets With Answers Free Printable
Math Word Problems 2 Digit Multiplication Jason Burn s Multiplication Worksheets
Printable Multiplication Worksheets X3 PrintableMultiplication
multiplication Story Sums For Class 2 EStudyNotes
multiplication Story Sums For Class 2 EStudyNotes
Classroom Math Multiplication Word Problems Worksheets 99Worksheets
FAQs (Frequently Asked Questions).
Are 2 Digit By 2 Digit Multiplication Word Problems Worksheets appropriate for any age teams?
Yes, worksheets can be tailored to different age and skill levels, making them versatile for various students.
How often should pupils practice using 2 Digit By 2 Digit Multiplication Word Problems Worksheets?
Constant technique is crucial. Normal sessions, ideally a couple of times a week, can yield substantial enhancement.
Can worksheets alone enhance mathematics abilities?
Worksheets are a valuable device yet needs to be supplemented with diverse learning approaches for detailed ability growth.
Are there on-line systems supplying complimentary 2 Digit By 2 Digit Multiplication Word Problems Worksheets?
Yes, lots of educational websites use free access to a wide variety of 2 Digit By 2 Digit Multiplication Word Problems Worksheets.
How can parents sustain their children's multiplication technique at home?
Urging constant method, offering support, and creating a positive learning environment are advantageous actions. | {"url":"https://crown-darts.com/en/2-digit-by-2-digit-multiplication-word-problems-worksheets.html","timestamp":"2024-11-05T07:30:43Z","content_type":"text/html","content_length":"29174","record_id":"<urn:uuid:9a789b4f-2068-4504-bb28-0f4dd0059068>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00565.warc.gz"} |
Understanding Mathematical Functions: How To Calculate Period Of A Fun
A mathematical function is a relationship between a set of inputs and a set of possible outputs, where each input is related to exactly one output. Understanding mathematical functions is crucial in
various fields such as engineering, physics, economics, and computer science. In this blog post, we will delve into the importance of understanding mathematical functions and learn how to calculate
the period of a function.
Key Takeaways
• Mathematical functions are crucial in various fields such as engineering, physics, economics, and computer science.
• Understanding the characteristics of mathematical functions, including their domain, range, and period, is essential.
• The period of a function can be calculated based on its type, and specific considerations apply to trigonometric functions.
• Knowledge of the period of a function is important in real-world scenarios and problem-solving in fields such as physics, engineering, and finance.
• It is important to be aware of common mistakes in calculating the period of a function and to practice applying this knowledge in different contexts.
Understanding Mathematical Functions
Mathematical functions are an essential concept in mathematics that describe the relationship between input and output values. They are used to model and analyze various phenomena in fields such as
physics, engineering, economics, and more. Understanding mathematical functions is crucial for solving real-world problems and making predictions.
A. Definition of a mathematical function
A mathematical function is a relationship between a set of inputs (independent variable) and a set of outputs (dependent variable), where each input value corresponds to exactly one output value. In
other words, it assigns a unique output for every input. Mathematically, a function f is denoted as f(x) = y, where x is the input and y is the output.
B. Types of mathematical functions
There are various types of mathematical functions, each with its own characteristics and properties. Some common types of functions include:
• Linear functions: Functions of the form f(x) = mx + b, where m and b are constants.
• Quadratic functions: Functions of the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
• Trigonometric functions: Functions such as sine, cosine, and tangent, which relate angles to the lengths of the sides of a right triangle.
• Exponential functions: Functions of the form f(x) = a^x, where a is a constant.
• Logarithmic functions: Functions of the form f(x) = loga(x), where a is a constant.
C. Characteristics of mathematical functions
Mathematical functions can be described by various characteristics that help analyze and understand their behavior. Some important characteristics include:
• Domain: The set of all possible input values for which the function is defined.
• Range: The set of all possible output values that the function can produce.
• Period: For periodic functions, the period is the smallest positive constant T for which f(x+T) = f(x) for all x.
Calculating the period of a function
For periodic functions, it is important to understand how to calculate the period, which determines the length of one complete cycle of the function. The period can be calculated using various
methods, depending on the type of function:
• For trigonometric functions, the period can be calculated using the formula T = (2π)/|b|, where b is the coefficient of x in the function.
• For other periodic functions, the period can be determined by finding the smallest positive value of T for which f(x+T) = f(x) for all x.
Understanding the period of a function is crucial for analyzing its repetitive behavior and making predictions based on its cyclical nature.
Period of a Function
Understanding the period of a function is crucial in the field of mathematics, as it helps in analyzing the behavior and characteristics of various mathematical functions. In this chapter, we will
delve into the definition of period in mathematical functions and discuss how to calculate the period of different types of functions.
Definition of period in mathematical functions
The period of a function is defined as the horizontal distance required for the function to repeat its values. In other words, it is the length of the interval over which the function's values recur.
How to calculate the period of a function based on its type
The method for calculating the period of a function varies based on the type of function.
• For trigonometric functions: The period of trigonometric functions such as sine and cosine can be calculated using the formula: Period = 2π/|b|, where 'b' is the coefficient of 'x' in the
• For periodic functions: For functions that exhibit periodic behavior, the period can be determined by identifying the length of the interval over which the function repeats its values.
• For exponential and logarithmic functions: These functions do not exhibit periodic behavior, and therefore, they do not have a period.
Examples of calculating the period of different types of functions
Let's consider some examples to illustrate the calculation of the period for different types of functions:
• Example 1 (Trigonometric function):
Calculate the period of the function y = 2sin(3x).
Solution: Using the formula for trigonometric functions, Period = 2π/|b|, we can calculate the period as: 2π/3.
• Example 2 (Periodic function):
Determine the period of the function y = cos(x) + sin(2x).
Solution: By analyzing the behavior of the function, we can identify the interval over which the function repeats its values, which in this case is 2π. Therefore, the period of the function is
• Example 3 (Exponential function):
Consider the function y = e^x.
Solution: Since exponential functions do not exhibit periodic behavior, they do not have a period.
By understanding the concept of the period of a function and knowing how to calculate it for different types of functions, mathematicians and scientists can gain valuable insights into the behavior
and properties of mathematical functions.
Period of Trigonometric Functions
Trigonometric functions are a fundamental part of mathematics and have applications in various fields such as physics, engineering, and architecture. Understanding the period of trigonometric
functions is crucial for analyzing their behavior and using them in practical scenarios.
Specific considerations for calculating the period of trigonometric functions
• Amplitude: The amplitude of a trigonometric function affects the period, as it determines the maximum and minimum values of the function within one period.
• Phase shift: Any horizontal shift in the function also impacts the period, as it changes the starting point of the function's cycle.
Formulas for calculating the period of common trigonometric functions
• Sine function: The period of the sine function y = sin(x) is 2π, which means it completes one full cycle every 2π units.
• Cosine function: Similarly, the period of the cosine function y = cos(x) is also 2π.
• Tangent function: The period of the tangent function y = tan(x) is π, completing a cycle every π units.
Graphical representation of the period of trigonometric functions
Graphically, the period of a trigonometric function can be observed by plotting its graph on a coordinate plane. The distance between consecutive peaks or troughs of the function represents the
period. For example, in the sine function, the distance between two consecutive peaks or troughs is 2π, indicating its period.
Applications of Understanding the Period of a Function
Understanding the period of a function is crucial in various real-world scenarios and contributes significantly to problem-solving in fields such as physics, engineering, and finance.
A. Importance of knowing the period of a function in real-world scenarios
• 1. Cyclic Phenomena: Many natural phenomena exhibit periodic behavior, such as the oscillation of pendulum, seasonal variations, and wave patterns. Understanding the period of a function helps in
predicting and analyzing these cyclic phenomena, which is essential in fields like meteorology, ecology, and astronomy.
• 2. Signal Processing: In telecommunications and electronics, understanding the period of a function is crucial for analyzing and processing periodic signals, such as those used in wireless
communication, audio processing, and radar systems.
B. How understanding the period of a function contributes to problem-solving in various fields such as physics, engineering, and finance
• 1. Physics: In physics, understanding the period of a function is essential for analyzing the motion of objects, the behavior of waves, and the dynamics of systems. For example, in studying the
motion of a pendulum, the period of its oscillation is a critical parameter that determines its behavior.
• 2. Engineering: Engineers use the concept of the period of a function in designing systems with periodic behaviors, such as vibrations in structures, control systems, and electrical circuits.
Understanding the period helps in optimizing the performance and stability of these systems.
• 3. Finance: In finance, understanding the period of a function is important for analyzing periodic trends in economic and market data. For example, in stock market analysis, identifying the
period of price fluctuations helps in making informed investment decisions.
Common Mistakes in Calculating the Period of a Function
When it comes to mathematical functions, determining the period is a crucial step in understanding their behavior. However, there are several common mistakes that students often make when trying to
calculate the period of a function.
Misconceptions and errors in calculating the period of a function
• Confusion with frequency: One common mistake is to confuse the period of a function with its frequency. The period is the length of one complete cycle of the function, while the frequency is the
number of cycles that occur in a unit of time. It is important to differentiate between the two in order to accurately calculate the period of a function.
• Incorrect handling of trigonometric functions: Trigonometric functions such as sine and cosine have specific properties that affect their periods. Students often make errors in identifying and
applying these properties, leading to incorrect period calculations for trigonometric functions.
• Overlooking phase shifts: Functions with phase shifts can have their periods affected by the shift. Students often overlook the presence of phase shifts, resulting in miscalculation of the period
for such functions.
• Failure to consider vertical stretches and compressions: Vertical stretches or compressions of a function can impact its period. Ignoring these transformations can lead to inaccurate period
Tips to avoid common mistakes when determining the period of a function
• Understand the properties of the function: It is crucial to have a solid understanding of the properties of the function, especially for trigonometric functions. Knowing the properties specific
to each type of function will help avoid errors in period calculation.
• Identify and account for phase shifts: Always check for phase shifts in the function and adjust the period calculation accordingly. This involves understanding how the function is shifted
horizontally and its impact on the period.
• Consider vertical transformations: When dealing with functions that have been vertically stretched or compressed, be sure to account for these transformations when calculating the period. This
may involve adjusting the period based on the vertical scaling factor.
• Practice with varied examples: To avoid misconceptions and errors, it is essential to practice calculating the period of different types of functions. Working through a variety of examples will
help reinforce the correct approach and improve accuracy in period calculations.
Understanding mathematical functions and how to calculate their periods is essential in various fields, from physics to engineering, and even finance. By grasping the concept of a function's period,
you can better interpret and apply mathematical models in real-world scenarios. I encourage you to practice and apply the knowledge of calculating the period of a function in different contexts to
strengthen your understanding and enhance your problem-solving skills. With a solid grasp of this fundamental concept, you will be better equipped to tackle complex mathematical problems with
ONLY $99
Immediate Download
MAC & PC Compatible
Free Email Support | {"url":"https://dashboardsexcel.com/blogs/blog/understanding-mathematical-functions-how-to-calculate-period-of-a-function","timestamp":"2024-11-09T02:41:35Z","content_type":"text/html","content_length":"219906","record_id":"<urn:uuid:29c16119-028c-4acd-927a-4f0c5eae7f30>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00439.warc.gz"} |
Basal Metabolic Rate
A Basal Metabolic Rate (BMR) calculator is a tool that calculates how many calories your body needs for a full day of bodily processes. It calculates a BMR score using your gender, weight, height, and age.
You can find out more about Basal Metabolic Rates, the Mifflin St Jeor equation, and the formulas for calculating it below.
Your basal metabolic rate is the volume of energy your body requires for everyday bodily functions such as breathing, body temperature, brain/nerve functions, and more.
Your basal metabolic rate makes up around 75 percent of your daily calorie usage, but it can vary from one person to the next. The energy you require for these functions also gets lower as you get older or as your lean body mass decreases. If, however, you increase your muscle mass, your basal metabolic rate will increase.
The use of the BMR calculator also involves your metabolism. Your metabolism is a chemical reaction that helps you to turn food into energy to stay alive.
Knowing your BMR gives you an idea of the calories you need to give your body every day, but working it out by hand is time-consuming. Instead, you can use a basal metabolic rate calculator to speed up the process.
There are several ways in which to calculate your BMR, but if you want the most accurate answer, you will find the Mifflin St Jeor equation is the best.
The formula is as follows:
BMR = (10 x weight / 1kg + 6.25 x height / 1cm – 5 x age / 1 year + s) / kcal / day
S = +5 for males and – 161 for females
Over the years, there have been several methods for working out your BMR, but with varying degrees of accuracy. In 1919, the Harris-Benedict equation became the most accurate of its time and that did not change for over seven decades.
The Basal Metabolic Rate called Mifflin St Jeor equation then turned out to be even more accurate, so the world is in a transitional phase between Harris-Benedict and Mifflin St Jeor. If you wanted to know your resting daily energy expenditure or RDEE, you could also use the third possible equation, Katch-McArdle.
Because Mifflin St Jeor is the most accurate, we will use it for all our equations – and our BMR calculator. If you consider yourself to be a bit of a gym bunny, you can also try out a BMI calculator or a calorie calculator to keep on top of daily weight management. | {"url":"https://www.thefreecalculator.com/health/bmr","timestamp":"2024-11-14T03:29:20Z","content_type":"text/html","content_length":"103062","record_id":"<urn:uuid:a1a8636d-ce71-4f93-ac27-e642f8e1d2ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00516.warc.gz"} |
Function glencoe algebra 2 :: Algebra Helper
function glencoe algebra 2 Related topics: Worksheets Pre Algebra
algebra ii questions and answers with step by step examples
Algebra I Notes
Glencoe Teachers Edition Textbooks Algebra 1
Solve Math Problems Online
College Algebra Assistant
Ways To Use Counting Chips To Learn Algebra
Algebra Equations With Two Variables
I Need Help With Algebra
geometry formula chart
Hrw Algebra 1interactions Course 2 Answers
Learn Fraction
Author Message
bluiraj2048 Posted: Wednesday 29th of Sep 19:01
I am looking for someone who can assist me with my math. I have a very important project coming up and need help in function glencoe algebra 2 and side-angle-side similarity. I
need help with topics covered in Pre Algebra class and seek help to master everything that I need to know so I can improve my grades.
Back to top
Vofj Timidrov Posted: Thursday 30th of Sep 10:04
Hi! I guess I can help you out on how to solve your assignment. But for that I need more details. Can you elaborate about what exactly is the function glencoe algebra 2 problem
that you have to work out. I am quite good at working out these kind of things. Plus I have this great software Algebra Helper that I got from a friend which is soooo good at
solving algebra assignment. Give me the details and perhaps we can work something out...
From: Bulgaria
Back to top
CHS` Posted: Friday 01st of Oct 15:03
I too have had difficulties in roots, perfect square trinomial and radicals. I was advised that there are a number of programs that I could try out. I tried out several but then
the finest that I discovered was Algebra Helper. Just keyed in the problem and clicked the ‘solve’. I got the answer at once. On top of it , I was directed through to the answer
by an easily comprehensible step-by-step procedure. I have relied on this program for my problems with Algebra 2, Intermediate algebra and College Algebra. If I were you, I
would without doubt go for this Algebra Helper.
From: Victoria City,
Hong Kong Island,
Hong Kong
Back to top
venihnow Posted: Saturday 02nd of Oct 14:29
Cool! This sounds very useful to me. I was searching such program only. Please let me know where I can get this software from?
From: Calgary,
Alberta, Canada
Back to top
ZaleviL Posted: Sunday 03rd of Oct 14:41
Algebra Helper is the program that I have used through several algebra classes - College Algebra, Remedial Algebra and Algebra 1. It is a really a great piece of algebra
software. I remember of going through problems with hypotenuse-leg similarity, side-angle-side similarity and graphing. I would simply type in a problem homework, click on Solve
– and step by step solution to my algebra homework. I highly recommend the program.
From: floating in the
light, never
Back to top
ZaleviL Posted: Tuesday 05th of Oct 09:39
Life can be hard when one has to work along with their studies. Visit https://algebra-answer.com/features.shtml, I am sure it will be of use to you.
From: floating in the
light, never
Back to top
Start solving your Algebra Problems in next 5 minutes!
Algebra Helper Attention: We are currently running a special promotional offer for Algebra-Answer.com visitors -- if you order Algebra Helper by midnight of November 10th you will pay only
Download (and $39.99 instead of our regular price of $74.99 -- this is $35 in savings ! In order to take advantage of this offer, you need to order by clicking on one of the buttons on the
optional CD) left, not through our regular order page.
Only $39.99 If you order now you will also receive 30 minute live session from tutor.com for a 1$!
Click to Buy Now:
2Checkout.com is an
authorized reseller
of goods provided by
You Will Learn Algebra Better - Guaranteed!
Just take a look how incredibly simple Algebra Helper is:
Step 1 : Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:
Step 2 : Let Algebra Helper solve it:
Step 3 : Ask for an explanation for the steps you don't understand:
Algebra Helper can solve problems in all the following areas:
• simplification of algebraic expressions (operations with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots (radicals), absolute values)
• factoring and expanding expressions
• finding LCM and GCF
• (simplifying, rationalizing complex denominators...)
• solving linear, quadratic and many other equations and inequalities (including basic logarithmic and exponential equations)
• solving a system of two and three linear equations (including Cramer's rule)
• graphing curves (lines, parabolas, hyperbolas, circles, ellipses, equation and inequality solutions)
• graphing general functions
• operations with functions (composition, inverse, range, domain...)
• simplifying logarithms
• basic geometry and trigonometry (similarity, calculating trig functions, right triangle...)
• arithmetic and other pre-algebra topics (ratios, proportions, measurements...)
ORDER NOW!
Algebra Helper
Download (and optional CD)
Only $39.99
Click to Buy Now:
2Checkout.com is an authorized reseller
of goods provided by Sofmath
"It really helped me with my homework. I was stuck on some problems and your software walked me step by step through the process..."
C. Sievert, KY
19179 Blanco #105-234
San Antonio, TX 78258
Phone: (512) 788-5675
Fax: (512) 519-1805 | {"url":"https://algebra-answer.com/algebra-help/simplifying-expressions/function-glencoe-algebra-2.html","timestamp":"2024-11-10T21:28:34Z","content_type":"text/html","content_length":"35302","record_id":"<urn:uuid:6b73fb4c-52ec-4de2-8ad0-7fdbdbbabc66>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00219.warc.gz"} |
Thevenin’s Theorem: Definition, Statement, Equivalent Circuit using Examples
Thevenin's Theorem
Thevenin’s Theorem Explanation
Thevenin's theorem is a powerful concept in electrical engineering that simplifies the analysis of complex linear circuits. It states that any complex network of resistors, voltage sources, and
current sources can be reduced to a single equivalent circuit. This equivalent circuit consists of a single voltage source V[s] and a series resistor R[s].
In the Thevenin equivalent circuit, the original circuit's resistive elements are consolidated into a single equivalent resistance R[s]. Similarly, multiple independent voltage sources are replaced
by a single equivalent voltage source V[s].
Thevenin’s Theorem Example
Thevenin’s Theorem can be understood with the help of the following example
• Step 1: Begin by removing the load resistor R[load], such as 40 ohms in this example, to simplify the circuit.
• Step 2: To eliminate the effect of internal resistance in voltage sources within a circuit, short-circuiting all voltage sources is essential (setting v=0). If there are any current sources
present, they should be open-circuited to similarly remove their internal resistance. This procedure is crucial for establishing ideal voltage or current sources, optimizing the circuit for
accurate analysis and solution.
• Step 3: To find the equivalent resistance of a circuit, you start by removing the load resistance and identifying the voltage sources.
• In a given example, resistors like 10 Ω and 20 Ω are connected in parallel with a 20 Ω resistor.
• By applying the parallel resistor formula, the equivalent resistance of the circuit is calculated to be 6.67 ohms."
• Step 4: Find the equivalent voltage.
• To find the equivalent voltage, reconnect the voltage sources within the circuit. Calculate the current circulating through the loop using the equation: "as the voltage between Vs = VAB."
Since both resistors carry identical currents, we can calculate the voltage drop across them using either of these formulas:
• For the 20-ohm resistor: \( V_{AB} = 20V - (20 \Omega \times 0.33A) = 13.33V \)
• For the 10-ohm resistor: \( V_{AB} = 10V + (10 \Omega \times 0.33A) = 13.33V \)
This means both resistors experience the same voltage drop across them.
• Step 5: Construct the Thevenin equivalent circuit, which comprises a single voltage source V[s] (e.g., 13.33 V) and a series resistor R[s] (e.g., 6.67 ohms).
We can calculate the current in the circuit as:
Application of Thevenin’s Theorem
Thevenin’s theorem is widely applicable in both AC and DC circuits, particularly those containing linear components such as resistors, inductors, and capacitors. By replacing a complex circuit with a
simpler equivalent, it facilitates easier analysis and design, making it a fundamental tool in electrical engineering.
Frequently Asked Questions on Thevenin's Theorem
Thevenin's theorem states that any complex electrical circuit can be simplified into an equivalent circuit with a single voltage source and a single resistor connected in series.
The formula for Thevenin's equivalent voltage (VTH) is the open-circuit voltage across the terminals of the circuit. The formula for Thevenin's equivalent resistance (RTH) is the resistance seen
looking into the terminals of the circuit with all voltage sources replaced by short circuits and current sources replaced by open circuits.
Norton's theorem states that any complex electrical circuit can be simplified into an equivalent circuit with a single current source and a single resistor connected in parallel.
VTH is the Thevenin equivalent voltage, which is the open-circuit voltage across the terminals of the circuit. RTH is the Thevenin equivalent resistance, which is the resistance seen looking into the
terminals of the circuit with all voltage sources replaced by short circuits and current sources replaced by open circuits.
The main benefit of Thevenin's theorem is that it allows complex electrical circuits to be simplified into an equivalent circuit with a single voltage source and a single resistor connected in
series. This simplification makes it easier to analyze the behavior of the circuit, especially when the load resistance changes. | {"url":"https://www.home-tution.com/maths-topics/thevenin-s-theorem","timestamp":"2024-11-02T03:13:29Z","content_type":"text/html","content_length":"117161","record_id":"<urn:uuid:d8e3a2e3-0ba3-4404-8022-434f2463005a>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00289.warc.gz"} |
From cppreference.com
inline namespace /* unspecified */ {
inline constexpr /* unspecified */ strong_order = /* unspecified */; (since C++20)
Call signature
template< class T, class U >
requires /* see below */
constexpr std::strong_ordering strong_order( T&& t, U&& u ) noexcept(/* see below */);
Compares two values using 3-way comparison and produces a result of type std::strong_ordering
Let t and u be expressions and T and U denote decltype((t)) and decltype((u)) respectively, std::strong_order(t, u) is expression-equivalent to:
• If std::is_same_v<std::decay_t<T>, std::decay_t<U>> is true:
□ std::strong_ordering(strong_order(t, u)), if it is a well-formed expression with overload resolution performed in a context that does not include a declaration of std::strong_order,
□ otherwise, if T is a floating-point type:
☆ if std::numeric_limits<T>::is_iec559 is true, performs the ISO/IEC/IEEE 60559 totalOrder comparison of floating-point values and returns that result as a value of type
std::strong_ordering (note: this comparison can distinguish between the positive and negative zero and between the NaNs with different representations),
☆ otherwise, yields a value of type std::strong_ordering that is consistent with the ordering observed by T's comparison operators,
□ otherwise, std::strong_ordering(std::compare_three_way()(t, u)) if it is well-formed.
• In all other cases, the expression is ill-formed, which can result in substitution failure when it appears in the immediate context of a template instantiation.
Expression e is expression-equivalent to expression f, if
• e and f have the same effects, and
• either both are constant subexpressions or else neither is a constant subexpression, and
• either both are potentially-throwing or else neither is potentially-throwing (i.e. noexcept(e) == noexcept(f)).
Customization point objects
The name std::strong_order denotes a customization point object, which is a const function object of a literal semiregular class type. For exposition purposes, the cv-unqualified version of its type
is denoted as __strong_order_fn.
All instances of __strong_order_fn are equal. The effects of invoking different instances of type __strong_order_fn on the same arguments are equivalent, regardless of whether the expression denoting
the instance is an lvalue or rvalue, and is const-qualified or not (however, a volatile-qualified instance is not required to be invocable). Thus, std::strong_order can be copied freely and its
copies can be used interchangeably.
Given a set of types Args..., if std::declval<Args>()... meet the requirements for arguments to std::strong_order above, __strong_order_fn models
Otherwise, no function call operator of __strong_order_fn participates in overload resolution.
Strict total order of IEEE floating-point types
Let x and y be values of same IEEE floating-point type, and total_order_less(x, y) be the boolean result indicating if x precedes y in the strict total order defined by totalOrder in ISO/IEC/IEEE
(total_order_less(x, y) || total_order_less(y, x)) == false if and only if x and y have the same bit pattern.
• if neither x nor y is NaN:
□ if x < y, then total_order_less(x, y) == true;
□ if x > y, then total_order_less(x, y) == false;
□ if x == y,
☆ if x is negative zero and y is positive zero, total_order_less(x, y) == true,
☆ if x is not zero and x's exponent field is less than y's, then total_order_less(x, y) == (x > 0) (only meaningful for decimal floating-point number);
• if either x or y is NaN:
□ if x is negative NaN and y is not negative NaN, then total_order_less(x, y) == true,
□ if x is not positive NaN and y is positive NaN, then total_order_less(x, y) == true,
□ if both x and y are NaNs with the same sign and x's mantissa field is less than y's, then total_order_less(x, y) == !std::signbit(x).
This section is incomplete
Reason: no example
See also
the result type of 3-way comparison that supports all 6 operators and is substitutable
(C++20) (class)
performs 3-way comparison and produces a result of type std::weak_ordering
(C++20) (customization point object)
performs 3-way comparison and produces a result of type std::partial_ordering
(C++20) (customization point object)
(C++20) performs 3-way comparison and produces a result of type std::strong_ordering, even if operator<=> is unavailable
(customization point object) | {"url":"https://cppreference.patrickfasano.com/en/cpp/utility/compare/strong_order.html","timestamp":"2024-11-03T01:04:54Z","content_type":"text/html","content_length":"49554","record_id":"<urn:uuid:c9ba6896-baa9-4655-839a-f230a27255e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00274.warc.gz"} |
Basics of Fraction (Comparison) | ExamVictorComparing Fractions
Comparing Fractions
Last Updated: Nov 5, 2020
Some Basic Information about Fractions:
• Fraction = Number of Parts/ Total Parts
• Every fraction has a numerator that equals the number of parts we have and a denominator equaling the total number of parts in a whole
• Any Number Can Be Written As a Fraction
• Write any whole number over 1 to make it a fraction since the total number of parts in any undivided whole is one
• Multiplying fractions is easy, just multiply straight across
Rules for Comparing Fractions
1. Relation: Like Denominators
How To Compare: Look at the numerators. The larger fraction is the one with the greater numerator.
Example: 3/5 > 1/5
2. Relation: Unlike Denominators
How To Compare: Convert each fraction to an equivalent fraction with a common denominator. The larger fraction is the one with the greater numerator.
Example: 5/8 < 7/10 since 25/40 < 28/40
3. Relationship: Like Numerators
How To Compare: Look at the denominators. The fraction with the smaller denominator is the larger fraction.
Example: 2/7 > 2/9
Equivalent Fractions: fractions that represent the same quantity.
Fundamental Law of Fractions: the value of a fraction does not change when its numerator and denominator are both multiplied by the same number (not zero).
Simplifying Fractions: reducing fractions. To simplify (or reduce) a fraction, find a number that will divide exactly into BOTH the numerator and the denominator. Try to find the "largest" such value
(the greatest common divisor).
Always remember that:
1) If both numbers end in 0 and/or 5, they are divisible by 5.
2) If both numbers are such that the sum of the digits of the number
is divisible by 3, then the numbers are divisible by 3.
3) If both numbers end in zeros, they are divisible by 10.
4) If both numbers are even, they are divisible by 2.
5) If both numbers end in 25, 50, 75 or 100, they are divisible by 25.
This video lays out the basics of comparing fractions. Understanding this topic well is necessary to solve questions that are asked in various competitive exams in the Quant and DI sections -
applicable to CAT, XAT, MAT, SNAP, IIFT, CLAT, AILET, DU LLB, any other entrance exam as well.
For more such content, visit our website - examvictor.com
Hope you liked this video. Share your views in the comment section below. For more such videos click here. | {"url":"https://examvictor.com/comparing-fractions/","timestamp":"2024-11-09T06:53:10Z","content_type":"text/html","content_length":"68633","record_id":"<urn:uuid:47dcd5d3-c679-433c-9509-cb53e9d49087>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00275.warc.gz"} |
Triple Integral of a Function Calculator - Online Integration
Search for a tool
Triple Integral
Tool to calculate triple Integral. The calculation of three consecutive integrals makes it possible to compute volumes for functions with three variables to integrate over a given interval.
Triple Integral - dCode
Tag(s) : Functions, Symbolic Computation
dCode and more
dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!
A suggestion ? a feedback ? a bug ? an idea ? Write to dCode!
Triple Integral
Triple Integral Calculator
Answers to Questions (FAQ)
What is a triple integral? (Definition)
The triple integral calculation is equivalent to a calculation of three consecutive integrals from the innermost to the outermost.
How to calculate a triple integral?
Calculate the integrals consecutively, from the inside to the outside.
$$ \iiint f(x,y,z) \text{ d}x\text{ d}y\text{ d}z = \int_{(z)} \left( \int_{(y)} \left( \int_{(x)} f(x,y) \text{ d}x \right) \text{ d}y \right) \text{ d}z $$
Example: Calculate the integral of $ f(x,y,z)=xyz $ over $ x \in [0,1] $, $ y \in [0,2] $ and $ z \in [0,3] $ $$ \int_{0}^{3} \int_{0}^{2} \int_{0}^{1} xyz \text{ d}x\text{ d}y\text{ d}z = \int_{0}^
{3} \int_{0}^{2} \frac{y^2,z^2}{8} \text{ d}y\text{ d}z = \int_{0}^{3} \frac{z^2}{2} \text{ d}z = \frac{9}{2} $$
Enter the function to be integrated on dCode with the desired upper and lower bounds for each variable and the calculator automatically returns the result.
How to integrate with polar coordinates?
The cylindrical coordinates are often used to perform volume calculations via a triple integration by changing variables:
$$ \iiint f(x,y,z) \text{ d}x\text{ d}y\text{ d}z = \iiint f(r \cos(\theta), r\sin(\theta), z) r \text{ d}r\text{ d}\theta\text{ d}z $$
How to integrate with spherical coordinates?
The spherical coordinates are often used to perform volume calculations via a triple integration by changing variables:
$$ \iiint f(x,y,z) \text{ d}x\text{ d}y\text{ d}z = \iiint f(\rho \cos(\theta) \sin(\varphi), \rho \sin(\theta)\sin(\varphi), \rho \cos(\varphi) ) \rho^2 \sin(\varphi) \text{ d}\rho \text{ d}\theta \
text{ d}\varphi $$
Source code
dCode retains ownership of the "Triple Integral" source code. Except explicit open source licence (indicated Creative Commons / free), the "Triple Integral" algorithm, the applet or snippet
(converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, breaker, translator), or the "Triple Integral" functions (calculate, convert, solve, decrypt / encrypt,
decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and all data download, script, or API access for "Triple Integral"
are not public, same for offline use on PC, mobile, tablet, iPhone or Android app!
Reminder : dCode is free to use.
Cite dCode
The copy-paste of the page "Triple Integral" or any of its results, is allowed (even for commercial purposes) as long as you credit dCode!
Exporting results as a .csv or .txt file is free by clicking on the export icon
Cite as source (bibliography):
Triple Integral on dCode.fr [online website], retrieved on 2024-11-05, https://www.dcode.fr/triple-integral
Similar pages
© 2024 dCode — El 'kit de herramientas' definitivo para resolver todos los juegos/acertijos/geocaching/CTF. | {"url":"https://www.dcode.fr/triple-integral","timestamp":"2024-11-05T05:59:25Z","content_type":"text/html","content_length":"28330","record_id":"<urn:uuid:c77ebfdb-0a9a-4c03-9076-b1da080fd8b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00784.warc.gz"} |
Geometry of Algorithms
The study of properties of algorithms invariant under a transformation needs a name. How about Geometry of Algorithms? Projective Geometry of Algorithms will study properties invariant under
projective transformations like computer programs, simplicial chain embeddings, etc.
An algorithm performs a specific function. Each algorithm has many programs in a specific language. The language offers the instructions, while the programs specify the incidence relations between
inputs and instructions. The projective geometry of an algorithm will provide a framework to embed a mathematical theorem into a program, like how stochastic calculus helps embed discrete algorithms
into continuous functions.
What defines an algorithm's homology? Can one program probe another geometrically? Does this relate to algorithm convolution? These questions seem nonsensical until you realize billions of algorithms
are rehashes of a few hundred geometric ones like FFT, Euclidean algorithm, Quicksort, Subspace Iteration, Fast Multipole, etc. These form the geometric basis for others. Viewing incidence relations
as complex building blocks, an algorithm's geometry is an n-dimensional chain complex. These complexes can be convolved for interesting results. Algorithms with similar functions share similar
geometry. An algorithm true to its input/output should behave the same in geometric space, simplifying proof management systems. | {"url":"https://densebit.com/posts/5.html","timestamp":"2024-11-12T06:33:37Z","content_type":"text/html","content_length":"3603","record_id":"<urn:uuid:fb8b4567-a7dc-4bf2-bf33-6fb614ee8023>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00170.warc.gz"} |
Introductory Chemistry – 1st Canadian Edition
Learning Objectives
1. Define energy.
2. Know the units of energy.
3. Understand the law of conservation of energy.
Energy: is the ability to do work. Think about it: when you have a lot of energy, you can do a lot of work; but if you’re low on energy, you don’t want to do much work. Work (w) itself is defined as
a force (F) operating over a distance (Δx):
w = F × Δx
In SI, force has units of newtons (N), while distance has units of meters. Therefore, work has units of N·m. This compound unit is redefined as a joule (J):
1 joule = 1 newton·meter
1 J = 1 N·m
Because energy is the ability to do work, energy is also measured in joules. This is the primary unit of energy we will use here.
How much is 1 J? It is enough to warm up about one-fourth of a gram of water by 1°C. It takes about 12,000 J to warm a cup of coffee from room temperature to 50°C. So a joule is not a lot of energy.
It will not be uncommon to measure energies in thousands of joules, so the kilojoule (kJ) is a common unit of energy, with 1 kJ equal to 1,000 J.
An older—but still common—unit of energy is the calorie. The calorie (cal) was originally defined in terms of warming up a given quantity of water. The modern definition of calorie equates it to
1 cal = 4.184 J
One area where the calorie is used is in nutrition. Energy contents of foods are often expressed in calories. However, the calorie unit used for foods is actually the kilocalorie (kcal). Most foods
indicate this by spelling the word with a capital C—Calorie. Figure 7.1 “Calories on Food Labels” shows one example. So be careful counting calories when you eat!
Figure 7.1 Calories on Food Labels
Example 1
The label in Figure 7.1 “Calories on Food Labels” states that the serving has 38 Cal. How many joules is this?
We recognize that with a capital C, the Calories unit is actually kilocalories. To determine the number of joules, we convert first from kilocalories to calories (using the definition of the kilo-
prefix) and then from calories to joules (using the relationship between calories and joules). So
Test Yourself
A serving of breakfast cereal usually has 110 Cal. How many joules of energy is this?
460,000 J
In the study of energy, we use the term system to describe the part of the universe under study: a beaker, a flask, or a container whose contents are being observed and measured. An isolated system
is a system that does not allow a transfer of energy or matter into or out of the system. A good approximation of an isolated system is a closed, insulated thermos-type bottle. The fact that the
thermos-type bottle is closed keeps matter from moving in or out, and the fact that it is insulated keeps energy from moving in or out.
One of the fundamental ideas about the total energy of an isolated system is that is does not increase or decrease. When this happens to a quantity, we say that the quantity is conserved. The
statement that the total energy of an isolated system does not change is called the law of conservation of energy. As a scientific law, this concept occupies the highest level of understanding we
have about the natural universe.
Key Takeaways
• Energy is the ability to do work and uses the unit joule.
• The law of conservation of energy states that the total energy of an isolated system does not increase or decrease.
1. Define energy. How is work related to energy?
2. Give two units of energy and indicate which one is preferred.
3. Express the quantity of 422 J in calories.
4. Express the quantity of 3.225 kJ in calories.
5. Express the quantity 55.69 cal in joules.
6. Express the quantity 965.33 kcal in joules.
7. How does a Calorie differ from a calorie?
8. Express the quantity 965.33 Cal in joules.
9. What is the law of conservation of energy?
10. What does the word conserved mean as applied to the law of conservation of energy?
Energy is the ability to do work. Work is a form of energy.
101 cal
233.0 J
A Calorie is actually a kilocalorie, or 1,000 calories.
The total energy of an isolated system does not increase or decrease. | {"url":"https://courses.lumenlearning.com/suny-introductorychemistry/chapter/energy-2/","timestamp":"2024-11-12T02:53:18Z","content_type":"text/html","content_length":"55609","record_id":"<urn:uuid:913c7781-46c6-44ab-8010-7fb661f09249>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00612.warc.gz"} |
Analysis of Bioassays | dataanalysistools.de
Analysis of Bioassays
Recently, I went through the woods with one of my younger kids and it was amazed by the various colors of different mushrooms. Kids tend to touch almost everything and sometimes even to put into
their mouth what they see. However, for certain mushrooms this might not be such a good idea and I taught my kid that some mushrooms and other subtances are poisonous. For the next few weeks it kept
asking for the toxicity of almost everything on our dinner table, like: “Papa, is salt poisonous?”. As a scientist I used to answer: “Yes, but just at a very high dose.” Then I had to explain what
Paracelsus already claimed in the 16th century:
“Wenn ihr jedes Gift wollt recht auslegen, was ist, das nit Gift ist? Alle Ding sind Gift und nichts ohn Gift. Allein die Dosis macht, dass ein Ding kein Gift ist.”
Freely translated: “If you want to interpret every poison correctly, what is there that is not a poison? All things are poison and nothing is without poison. The dose alone makes a thing not a
Paracelsus (1538)
Let’s stick to the salt example (my kid asked about table salt, i.e. sodium chloride). Let’s assume we’d put groups of rats on diet with the intake of various doses of sodium chloride added to their
food. Say we run this experiment several weeks and look at the number of deaths within the different rat groups after a certain time point. Rats that had a very high intake of sodium chloride will
likely be dead after the time period, while those with a lower intake will likely not have died. Plotting the portion of death rats versus the (log) dose of the sodium chloride will result in a
dose-response curve and the assay just described is an example of a bioassay with quantal response data. I will explain in detail below what that actually means.
According to Finney (1947) a biological assay (bioassay) can be defined as follows (my comments in square brackets below):
“The term biological assay, in its widest sense, should be understood to mean the measurement of the potency of any stimulus, physical, chemical [sodium chloride in our case] or biological,
physiological or psychological, by means of the reactions [dead or alive in our case] which it produces in living matter [rats in our case].”
D.J. Finney, Probit Analysis – A Statistical Treatment of the Sigmoid Response Curve, 1947, Cambridge University Press
Finney’s definition makes clear that bioassays are very diverse by means of what type of stimulus is applied and by means of what the reaction or response might look like that the stimulus induces.
It might also become clear that bioassays are performed in various fields of science ranging from ecotoxicology to pharmacology. Of similar diversity can be the data resulting from those bioassays as
will be discussed below.
Bioassays are typically classified as direct assays, where the dose needed to induce a fixed response (e.g. death of an organism) is directly measured. The dose itself is the random variable. In
terms of our salt example we might increase the sodium chloride concentration in the blood of a rat until its heart stops beating. With indirect assays the doses are fixed and the responses are the
random variables. While direct assays produce binary output data, indirect assays can be either produce continuous or binary data and result in dose-response curves that need to be analyzed in an
appropriate fashion.
In the following, we will focus on the analysis of indirect bioassays and start with introducing dose-response curves and their corresponding mathematical models.
A closer look at dose-response curves
Let’s assume we are working for a pharma company producing drugs for reducing blood pressure. Let’s further assume we developed a new drug candidate by changing a chemical side-group. That drug seems
promising against our reference standard (substance). Look at the following graph:
The dose-response curve of the reference substance is shown in blue while that of our new drug candidate is shown in black. In fact the latter is like a left-shifted version of dose-response curve of
our reference standard. This is typically a reasonable hint that both drugs act in a similar way. The shift to the left indicates that our new drug candidate requires a lower dose to induce the same
effect as our reference standard. One says that our drug candidate has a higher potency. That’s good, as less subtance needs to be produced and side-effects are less likely. The dose-response curves
shown above are of sigmoid shape which is quite common in practice. By the way, the shift along the log(Dose) axis between the two curves denotes the so-called log relative potency which we’ll
discuss in more detail later on.
In the example above, the responses are contiuous as the blood pressure, or a ratio thereof, is a continuous random variable with a continuous error distribution (often the normal distribution). In
the life sciences and from what I personally experience, I’d say that the so-called four-parameter logistic (4PL) model is most commonly used to model dose-response data in practice:
Herein [50]).
The 4PL model is of sigmoid shape and symmetric around its inflection point. However, sometimes it might be required to fit an assymetric sigmoid to the data. This can be achieved by taking into
account an assymetry parameter
Both models are displayed in the following figure. The log effective dose 50 is at -5 as indicated by the vertical dashed line at the corresponding
Sometimes nested models of the 4PL are used such as the three-parameter (i.e. c=0) or two-parameter logistic function (i.e. c=0, d=0):
The latter model is highly important when fitting binary response data (more details below). Of course there are more fit models but talking about all of them is out of the scope of this blog post.
The ED[50] is the dose of a compound required to produce 50 % of the maximal response. It is an important parameter and is related to the aforementioned potency. The lower the ED[50] of a compound,
the higher its potency.
While we focus on parametric fit models here, I don’t want to keep from you the fact that there also exist non-parametric methods to analyze dose-response curves, such as the Spearman-Kärber method
mentioned in another blog post.
Estimation of fit model parameters
Fitting a model to dose-response data requires two key decisions to make:
• What shall the model function
• What distributional assumption do we make for the dose-response data?
The first decision is clear. You need to choose the functional form of the model. Maybe one of those discussed above. In practice the decision is often anticipated by the assay provider or by
literature search. The second decision sounds a bit abstruse. It means that due to the fact that the responses are random variables, they follow some probability distribution. For continuous data
this is often the normal distribution, but others like the log-normal are common, too. For quantal data this is often the binomial distribution. In other disciplines, e.g. for counting data, the
Poisson distribution is the distribution of choice.
Procedures for estimating the fit model parameters need to take these two ingredients into account.
Maximum likelihood estimation and nonlinear least-squares
In fact, (nonlinear) least-squares (LS) and maximum likelihood (ML) parameter estimation are related, i.e. the LS approach is basically a special case of the ML approach assuming that the
distribution noise associated with experimental measurements is sufficiently well described by a normal probability distribution function with zero mean and some standard deviation
We can now also see from the equation above, that for some dose
By the way, it is relatively easy to show how the likelihood expression
where I have also included weights this blog post) including methods of robust regression. I’ve used the letter
Did you know?
The ML and LS approach can both be derived with Bayes Theorem by assuming a constant prior. The LS approach makes further simplifying assumptions on the likelihood that finally reduces to the
well-known sum of squared errors notation of least-squares.
Dose-response curves are often fitted within the framework of generalized linear models (GLM) including Logitstic, Probit, as well as 4PL regression. See also the blog post on logistic regression as
well as on the analysi of excess mortality data.
Start with good initial guess values (IG) for the model parameters
To estimate the fit model parameters, you need to solve an optimization problem, i.e. by minimizing the (weighted) sum of squared errors or the negative log-likelihood by varying the fit model
parameters given the data. This requires an iterative procedure such as a Levenberg-Marquart algorithm or a Simplex-based method like the one proposed by Nelder and Mead. In any case, good initial
guess values of the fit model parameters are required to ensure the algorithm doesn’t get stuck in a local minimum but approaches the global minimum resulting in the optimal solution.
A conventional IG value for the lower asymptote of the 4PL model is to take the value of the minimum response. Accordingly, the IG value for the upper asymptote is often set to the maximum of the
response values. For the Hill slope a common IG value is either -1 if the response value at the maximum dose minus the response value at the minimum dose is negative (i.e. the dose-response curve is
falling) or the Hill slope is set to 1 if the response value at the maximum dose minus the response value at the minimum dose is positive (increasing dose response curve). Finally, the log(dose)
value corresponding to the mean of the minimum and maximum response, is often chosen as initial guess value for the log(ED50).
The five-parameter model (5PL) is more difficult to fit than the 4PL model as the asymmetry parameter this website). We say the optimization problem is ill-conditioned. So, being able to get a proper
fit with the 5PL model at all, requires good quality of data and enough data points covering the whole range of the underlying dose-response curve (asymptotes and transition region). But even then,
good intial guess values are mandatory. An increasing dose-response curve will have its steepest ascent near the upper asymptote if
Another “trick” that can help to get more reliable fit model parameters is to set lower and upper bounds. For example, if you are confident that
Estimate relative potencies from multiple dose-response curves
Estimating relative potencies from two or more dose-response curves is an important task in the drug-developing pharma industry. Given a drug A and a drug B used to create two dose-response curves,
the relative potency
Typicall parallel dose-response curves one could also calculate European Pharmacopeia. Parallelism for two 4PL dose-response curves typically means they share the same asymptotes and the same Hill
slope. To check for parallelism one imposes parallelism on two 4PL dose-response curves by sharing the Hill slope and the lower and upper asymptotes when fitting the corresponding data. Subsequently,
the two data sets are fitted independently having their own Hill slope and asymptotes. An ANOVA approach is then used to test whether including more fit parameters in the latter case significantly
improved the fit or not. If not, the curves can be considered parallel (the corresponding
Relative potency assays are by no means limited to continuous data and to 4PL models, but can also be based on quantal data. The validity criteria are essentially the same as those mentioned above.
As long as all validity criteria are fullfilled, the relative potency can be estimated by the formula shown above.
The US Pharmacopeia suggests to apply equiavalence testing on the fit model parameters of the unconstrained model. E.g. for the 4PL model the ratio of the Hill slopes and its confidence interval
shall be compared against an equivalence interval. The same shall be done for the lower and upper asymptote, respectively. When all the ratios arer close to 1 and their respective confidence
intervals are fully contained within the pre-defined equivalence intervals, the dose-response curves are said to be sufficiently similar and subsequently the relative potency can be estimated from
the constrained model.
Please note that some bioassays result in linear dose-response data (often after log-transformation of the responses) instead of sigmoid data. In these cases the data is fitted by a linear model
having intercept
For two parallel-lines the log(relative potency) is estimated by:
These assays are typically called parallel-line assays and the analysis itself is called parallel-line analysis (PLA). As described in the European Pharmacopeia, sigmoid dose-response curves can be
linearized and thus, the term PLA is not necessary limited to the linear model. However, you’ll sometimes see the term parallel-curve analysis for the potency estimation of non-linear dose-response
This Post Has 2 Comments
1. Wrong translation. He said that the dose alone makes that a thing is NOT a poison (‘macht, dass ein Ding KEIN Gift ist’!)! And that also makes sense, since he had already said before that
everything (already) is poisonous and nothing is without poison. The right dose alone can therefore prevent undesirable side or even toxic effects. The translation into the equally incorrect
Latin phrase, whose author is even unknown, ‘dosis sola facit venenum’ and the retranslation results in the misinterpretation/mistranslation that the dose alone made the poison. This is the
diametrical opposite of what he thought and said: everything is already poison. That’s how he starts. Then the conclusion ‘the dose alone can prevent the poisonous effect (which is inherent in
every thing already, we might as well add), as opposed to “all things are poison and nothing is without poison”, conclusio: “only the dose makes a thing a poison”, contradicts everything he has
just said, that everything already is poison. According to Paracelsus, the dose can therefore only make something NOT a poison. q.e.d.
1. Hi Kim,
thanks for pointing this out. It is now corrected in the post above. | {"url":"https://dataanalysistools.de/2023/09/21/analysis-of-bioassays/","timestamp":"2024-11-08T12:14:44Z","content_type":"text/html","content_length":"121550","record_id":"<urn:uuid:c5171d3d-040b-4034-b4b0-693d5773fa32>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00055.warc.gz"} |
Data Preprocessing Part 2 and Random Forests
My previous post explored techniques for cleaning and pre-processing datasets prior to using machine learning techniques. This post will continue where the previous one left off. The central dataset
for these two posts is the University of Wisconsin Breast Cancer Dataset on Kaggle.com
Trees are amazing.
In mathematics, trees can be found in a number of different topics including fractal geometry.
import matplotlib.pyplot as plt
import math
class FractalTree:
def __init__(self):
plt.title("Fractal Tree")
plt.axis((0, 600, 600, 0))
def drawTree(self, x1, y1, angle, depth):
if depth:
x2 = x1 + int(math.cos(math.radians(angle)) * depth * 10.0)
y2 = y1 + int(math.sin(math.radians(angle)) * depth * 10.0)
plt.plot([x1, x2], [y1, y2], 'g-')
print(x1, y1, x2, y2)
self.drawTree(x2, y2, angle - 20, depth - 1)
self.drawTree(x2, y2, angle + 20, depth - 1)
if __name__ == "__main__":
ft = FractalTree()
ft.drawTree(300, 550, -90, 9)
Fractal trees don’t directly pertain to the topic of random forests, but they are still very cool. You can read more about them here.
Another interesting use of trees in mathematics and statistics can be found in the use of Random Forests as a machine learning technique. Shortly, we will explore how random forests work, but first
we need to wrap up one detail from the last post.
Revisiting the Correlation Matrix
In the last post we explored how data related to my lazy cat Parker could be visualized in a correlation matrix. We started with a table that looks like this:
Day, Meows, Sleeping, Hariballs, Bed, Sunshine, Litterbox, Eating, Drinking, Counter
Sunday, 14, 903, 2, 722, 181, 2, 3, 5, 4
Monday, 18, 1100, 0, 836, 264, 1, 4, 6, 5
Tuesday, 22, 850, 0, 697, 153, 4, 3, 5, 4
Wednesday, 18, 917, 1, 724, 193, 2, 5, 4, 2
Thrusday, 16, 856, 0, 693, 1663, 4, 4, 5, 6
Friday, 21, 1341, 0, 1059, 282, 1, 3, 6, 12
Saturday, 97, 872, 2, 723, 149, 3, 5, 7, 14
Correlation matrices are a great way to visually identify data that is somewhat redundant in machine learning training sets. A correlation matrix shows the correlation between all the pairs of
columns in a data set. In Manish Kumar’s analysis of the Breast Cancer data set on Kaggle, he presents a “heatmap” version of the correlation matrix. You may recall the correlation matrix for the
“Lazy Parker” data in my last post:
Another way to view that same data is to view it as a clustered heat map. A clustered heat map will display the same data but group the highly correlated items together. Here is a modified bit of
code from the last post that shows how to do that, and the result. (The change is simply modifying the “sns.heatmap(..” line to be “sns.clustermap(…”)
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns # used for plot interactive graph. I like it most for plot
class ParkerSortedCorrelationMatrix:
def do_correlation_matrix(self):
pdata = pd.read_csv("./data/parker_sleeping.csv", header=0)
pdata.drop("Day", axis=1, inplace=True)
pdata.drop("Counter", axis=1, inplace=True)
data_cols = list(pdata.columns[0:8])
corr = pdata[data_cols].corr()
plt.figure(figsize=(14, 14))
sns.clustermap(corr, cbar=True, square=True, annot=True, fmt='.2f', annot_kws={'size': 16},
xticklabels=data_cols, yticklabels=data_cols,
if __name__ == '__main__':
pscm = ParkerSortedCorrelationMatrix()
Now it is very easy to see that “Sunshine”, “Sleeping” and “Bed” are highly correlated data.
If we wish to do the same thing for the correlation matrix for the columns in the breast cancer dataset, we get the following: (Note that we have reduced the dataset to just the “mean” values.)
def do_correlation_matrix(self, drop_cols):
pdata = pd.read_csv("./data/data.csv", header=0)
for d in drop_cols:
pdata.drop(d, axis=1, inplace=True)
data_cols = list(pdata.columns[0:11])
corr = pdata[data_cols].corr() # .corr is used for find corelation
g = sns.clustermap(corr, cbar=True, square=True, annot=True, fmt='.2f', annot_kws={'size': 8},
cmap='coolwarm', figsize=(8, 8))
plt.setp(g.ax_heatmap.yaxis.get_majorticklabels(), rotation=0)
plt.setp(g.ax_heatmap.xaxis.get_majorticklabels(), rotation=90)
Now it is very clear which columns in the dataset have the highest correlation.
A word of caution: I have found it somewhat difficult to format seaborn heat maps and cluster maps. For best results, I have used the “save fig” method that is available on the returned object
when creating these maps.
Random Forests
We will now turn our attention to the machine learning technique used by Manish Kumar. That technique is called Random Forest. Coursera offers a course on Practical Machine Learning from Johns
Hopkins that has a section specifically on Bagging, Boosting and Random Forest. The section on Random Forests is particularly good, and I recommend it.
Another good explanation of Random Forest can be found in this youtube video by Thales Sehn Körting.
Random Forest uses multiple decision trees. These trees are built by taking random subsets of the columns of the entire dataset. Each subset of columns is then used to create a decision tree and
multiple trees are used. This image from the Körting video gives a good idea what is happening. Note that each subset contains a column “C” that represents the classification for the data.
The algorithm then predicts classification of data by running that data through all of the decision trees in the “Random Forest”. The final classification is made by allowing each decision tree to
“vote” on what it determines is the classification. The classification with the highest number of votes is the one that wins.
Note: Please refer to the github repo for the complete code for this blog post.
One thing that I wanted to point out is that Manish Kumar, as far as I can tell, does not preprocess the data before using his Random Forest approach. (I could have missed it, but it didn’t jump out
at me.) Preprocessing has the following impact on the breast cancer data set (again, see the previous blog post):
Code to run the random forest prediction for the breast cancer dataset looks like this: (Again, credit to Manish Kumar for writing the code that the following borrows heavily from.)
def do_machine_learning_random_forest(self):
data = pd.read_csv("./data/data.csv", header=0)
data.drop("Unnamed: 32", axis=1, inplace=True)
data.drop("id", axis=1, inplace=True)
data['diagnosis'] = data['diagnosis'].map({'M': 1, 'B': 0})
data = self.preprocess_data(data, preserve="diagnosis")
prediction_var = ['fractal_dimension_mean',
train, test = train_test_split(data, test_size=0.3)
train_X = train[prediction_var]
train_y = train.diagnosis
test_X = test[prediction_var]
test_y = test.diagnosis
model = RandomForestClassifier(n_estimators=100)
model.fit(train_X, train_y.astype(int))
prediction = model.predict(test_X)
accuracy = metrics.accuracy_score(prediction, test_y)
print("Calculation complete. Random Forest Accuracy: {0}".format(accuracy))
return accuracy
I ran the random forest experiment 1000 times. The first 500 I did NOT preprocess the data. The second 500 I did preprocess (scale and normalize) the data. The results are as follows for the average,
min and max accuracy of the experiment:
No Yes
Average 92.68% 94.95%
Min 87.13% 90.06%
Max 97.66% 98.83%
The increase in accuracy is evident, but not overwhelming. That may be due in large part to the fact that main data components are relatively of the same scale already.
Neural Network Comparison
An alternative technique that I have used in the past that I believe works particularly well for classification problems that have two classes is to use a simple neural network with a final output
layer that uses the arctangent function. Training targets of the two classes can be mapped to -1 and +1. A standard back propagation can be used for the network, and a network of one hidden layer can
typically do a pretty good job of predicting the outcomes. Here are the results of doing that with the scaled normalized data found in this dataset:
(This is some pretty long ugly code, so I apologize in advance.)
def do_neural_network_estimation(self):
data = pd.read_csv("./data/data.csv", header=0)
data.drop("Unnamed: 32", axis=1, inplace=True)
data.drop("id", axis=1, inplace=True)
data['diagnosis'] = data['diagnosis'].map({'M': 1, 'B': -1})
data = self.preprocess_data(data, "diagnosis")
prediction_var = ['fractal_dimension_mean',
train, test = train_test_split(data, test_size=0.3)
train_X = train[prediction_var]
train_y = train.diagnosis
test_X = test[prediction_var]
test_y = test.diagnosis
model = Sequential()
model.add(Dense(6, input_shape=(6,), activation='relu'))
model.add(Dense(100, activation='softmax'))
model.add(Dense(1, activation='tanh'))
model.fit(train_X.values, train_y.values, batch_size=100, epochs=2000, verbose=False)
correct_m = 0
correct_b = 0
type_I = 0
type_II = 0
for x in range(0,len(test_X.values)):
i = np.reshape(test_X.values[x], (1, 6))
estimate = model.predict(i)
if estimate > 0 and test_y.values[x] > 0:
correct_m = correct_m + 1
if estimate < 0 and test_y.values[x] < 0:
correct_b = correct_b + 1
if estimate > 0 and test_y.values[x] < 0:
type_I = type_I + 1
if estimate < 0 and test_y.values[x] > 0:
type_II = type_II + 1
a = (correct_m + correct_b) / len(test_X.values)
print("\n\n{0},{1},{2},{3},{4}\n\n".format(correct_b, correct_m, type_I, type_II, a))
The learning curve for the neural network approach is:
Results from the neural network approach to the classification problem were:
Neural Network Breast Cancer Cell Classification
Correct (B) . . . . . . 101
Correct (M) . . . . . . 65
False Positives (I) . . 2
False Negatives (II). . 3
Accuracy. . . . . . . . 0.9707602339181286
The learning curve bounces around a bit toward the end which results in some variation of the final accuracy. This, combined with the fact that the neural network randomizes the initial weights and
biases each time means that different training sessions can return a slightly differently trained network. To account for this, I ran the network 25 times and came up with the following results:
Neural Network Breast Cancer Cell Classification 25 Trials
Average 0.964210526
Min 0.929824561
Max 0.994152047
The neural network was more accurate than either of the two random forest approaches. However all of the models were within a few percentage points in accuracy. There may be opportunities to actually
make the network perform even better by tuning the parameters and network architecture a bit.
It is also worth mentioning that if the network were trained to 100% accuracy on the data, it is very likely that the model would be “over-fit”. I won’t go into that here, but it might be a good
topic for another blog post. When I start to write things like that, it is usually a good time to wrap things up.
This series of blog posts has touched on the following topics:
• Preprocessing data with scaling and normalization
• Identify key columns of data with correlation matrices
• Using “Seaborn” heatmaps and clustermaps to identify highly correlated data (and reduce the number of columns with high correlation)
• Looking at the Random Forest algorithm for categorization.
• Comparing the results of random forest models that have been trained with preprocessed and non-preprocessed data
• Using dense neural networks as an approach to categorize data
• Comparing the results of random forest and dense neural networks on the Wisconsin Breast Cancer Database provided by Kaggle.com and the University of California, Irvine.
I hope you have found this post interesting. I’ll be back after brief break with more topics in data science and machine learning. | {"url":"https://datascience.netlify.app/general/2017/08/15/data_science_15.html","timestamp":"2024-11-06T10:12:24Z","content_type":"text/html","content_length":"46572","record_id":"<urn:uuid:1b25cef5-1d2b-4741-af80-570c3aaad5ba>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00458.warc.gz"} |
16 Examples of a Paradox
A paradox is a seemingly contradictory statement, situation or calculation that is nonetheless possible. In some cases, paradoxes are known to be true despite violating common sense. Paradoxes can
also be open questions that are arguably true or that have instances of being true. Other paradoxes are neither true nor false and are believed to have no answer. The following are illustrative
examples of a paradox.
Catch-22 is a contradictory system, rule or process that is absurd. These may be used as a system of oppression or may simply exist due to the irrational nature of a society, organization or group. A
common example of catch-22 is that you need experience to get a job but need a job to get experience. The term was coined by Joseph Heller in his 1961 novel Catch-22. The book gives numerous examples
of a catch-22 such as a regulation that states you can get out of combat if you are crazy but the act of applying to get out of combat proves that you are sane.
Self-reference is an easy way to make a paradoxical statement that is neither true nor false. These are sometimes used as a means of provoking thought.
This statement is false.
The abilene paradox is the observation that groups sometimes make decisions that are viewed as illogical by all members of the group. This has to do with a process of social compromise whereby nobody
gets what they want. The abilene paradox can result in decisions that independent third parties also view as illogical.
Birthday Problem
The birthday problem is a mathematical result that many people find non-intuitive.
In a room with 23 people, what is the probability that two people have the same birthday?
Many people calculate this incorrectly with results around 6%. The actual answer is 50.7297%. This is due to the fact that each person compares their birthday to each other person resulting in a
total of 253 potential matches.
Potato Paradox
The potato paradox is another mathematical calculation that produces a non-intuitive result.
You have 100 pounds of potatoes that are 99 percent water by weight. You let them dehydrate until they're 98 percent water. How much do they weigh now?
People often expect the weight to drop to about 99 pounds but the actual answer is 50 pounds. When the water decreases from 99% to 98%, the amount of non-water material increases from 1% to 2%. This
is a doubling of the non-water material. In order for this to occur, half the weight of the water must evaporate.
Ship of Theseus
A thought experiment that asks if a ship that has all its parts replaced one-by-one over the years is still the same ship. If it is not the same ship, at what point did it cease being the same ship?
If you assemble another ship with all the parts removed from the original ship, is that the same ship? This has dozens of potential solutions that often provoke much thought and debate. For example,
the idea that identity only lasts for an instant such that we are never the same from moment to moment.
Cute Aggression
The commonly reported urge to squeeze, poke, pinch or otherwise superficially hurt things we find exceedingly cute. People who experience this don't understand where it comes from and find it
counterintuitive and paradoxical.
Big Lie
Big lie is a theory of propaganda that suggests it is easier to convince people of a big lie than a small lie. This is based on the idea that people will not believe that anyone would have the
audacity to tell such a large lie. Whether the theory of big lie is true or not is open to debate. A large lie is generally easier to detect as they typically require much supporting evidence.
False Positive Paradox
The false positive paradox is a tendency for people to dramatically misjudge the effect of false positives. For example:
A store has 100 shoplifting incidents a month. They use an artificial intelligence to monitor 1,000,000 shoppers each month to detect which are shoplifters. The AI has a false positive rate of 1%.
Suzy is flagged by the system as a shoplifter. What is the probability she is innocent?
Most people who answer this question focus on the fact that the AI is only wrong 1% of the time and believe there is only a 1% chance Suzy is innocent. This neglects the fact that the AI scans
1,000,000 people resulting in 10,000 false positives with only 100 shoplifting incidents. This means that 9,900 of the accused are innocent and only 100 guilty. As such, there is approximately a 99%
chance Suzy is innocent.
Lottery Paradox
The lottery paradox is a disbelief that something rare can happen to an individual that exists alongside an acceptance that the same thing does happen to someone. For example, people will find it
hard to believe that a particular ticket is the winner of a lottery with 500 million tickets issued. However, they also accept that a winning ticket does exist. This can be shown to be a paradoxical
Zeno's Paradox
A series of paradoxes that are all based on infinite divisibility. For example:
An architect is designing a building. They complete half the remaining work each day. When will the work be completed?
The answer is that the work will never be completed because half of something is always something. As such, you will simply complete smaller and smaller portions of work each day to infinity.
Decision Making Paradox
The decision about the best way to make a decision is also a decision. So if you don't already know the best way, there is no way you could choose the best way.
Irresistible Force Paradox
What happens when an unstoppable force meets an immovable object?
Buridan's Ass
Buridan's ass is a thought experiment that puts a donkey who is equally hungry and thirsty an equal distance from food and water. In the thought experiment, the animal doesn't know what to do and
doesn't move at all. The implication is supposed to be that there is no rational way to make a choice between equal options. However, it is rational to randomize a choice in this situation. As such,
the ability to choose something random is a requirement for rational thought.
Grandfather Paradox
A well known paradox that involves traveling in time to meet your own grandfather. This may change the course of your grandfather's life causing you not to be born. It is common to use paradoxes to
test theories and ideas. The grandfather paradox is one of the reasons that it is generally believed that time travel is not possible.
An oxymoron is language that is contradictory such as "awfully good" or "alone together." This is technically paradoxical but isn't typically meant to be taken literally. | {"url":"https://simplicable.com/en/paradox","timestamp":"2024-11-12T10:39:55Z","content_type":"text/html","content_length":"78395","record_id":"<urn:uuid:ba5eaffa-2715-49d1-9727-78f3940b1694>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00332.warc.gz"} |
AMR Similarity Metrics from Principles
Different metrics have been proposed to compare Abstract Meaning Representation (AMR) graphs. The canonical Smatch metric (Cai and Knight, 2013) aligns the variables of two graphs and assesses triple
matches. The recent SemBleu metric (Song and Gildea, 2019) is based on the machine-translation metric Bleu (Papineni et al., 2002) and increases computational efficiency by ablating the
variable-alignment. In this paper, i) we establish criteria that enable researchers to perform a principled assessment of metrics comparing meaning representations like AMR; ii) we undertake a
thorough analysis of Smatch and SemBleu where we show that the latter exhibits some undesirable properties. For example, it does not conform to the identity of indiscernibles rule and introduces
biases that are hard to control; and iii) we propose a novel metric S^2match that is more benevolent to only very slight meaning deviations and targets the fulfilment of all established criteria. We
assess its suitability and show its advantages over Smatch and SemBleu.
Proposed in 2013, the aim of Abstract Meaning Representation (AMR) is to represent a sentence’s meaning in a machine-readable graph format (Banarescu et al., 2013). AMR graphs are rooted, acyclic,
directed, and edge-labeled. Entities, events, properties, and states are represented as variables that are linked to corresponding concepts (encoded as leaf nodes) via is-instance relations (cf.
Figure 1, left). This structure allows us to capture complex linguistic phenomena such as coreference, semantic roles, or polarity.
When measuring the similarity between two AMR graphs A and B, for instance for the purpose of AMR parse quality evaluation, the metric of choice is usually Smatch (Cai and Knight, 2013). Its backbone
is an alignment-search between the graphs’ variables. Recently, the SemBleu metric (Song and Gildea, 2019) has been proposed that operates on the basis of a variable-free AMR (Figure 1, right),^^1
converting it to a bag of k-grams. Circumventing a variable alignment search reduces computational cost and ensures full determinacy. Also, grounding the metric in Bleu (Papineni et al., 2002) has a
certain appeal, since Bleu is quite popular in machine translation.
However, we find that we are lacking a principled in-depth comparison of the properties of different AMR metrics that would help informing researchers to answer questions such as: Which metric should
I use to assess the similarity of two AMR graphs, e.g., in AMR parser evaluation? What are the trade-offs when choosing one metric over the other? Besides providing criteria for such a principled
comparison, we discuss a property that none of the existing AMR metrics currently satisfies: They do not measure graded meaning differences. Such differences may emerge because of near-synonyms such
as ruin – annihilate; skinny – thin – slim; enemy – foe (Inkpen and Hirst, 2006; Edmonds and Hirst, 2002) or paraphrases such as be able to – can; unclear – not clear. In a classical syntactic
parsing task, metrics do not need to address this issue because input tokens are typically projected to lexical concepts by lemmatization, hence two graphs for the same sentence tend not to disagree
on the concepts projected from the input. This is different in semantic parsing where the projected concepts are often more abstract.
This article is structured as follows: We first establish seven principles that one may expect a metric for comparing meaning representations to satisfy, in order to obtain meaningful and appropriate
scores for the given purpose (§2). Based on these principles we provide an in-depth analysis of the properties of the AMR metrics Smatch and SemBleu (§3). We then develop S^2match, an extension of S
match that abstracts away from a purely symbolic level, allowing for a graded semantic comparison of atomic graph-elements (§4). By this move, we enable Smatch to take into account fine-grained
meaning differences. We show that our proposed metric retains valuable benefits of Smatch, but at the same time is more benevolent to slight meaning deviations. Our code is available online at https:
2From Principles to AMR Metrics
The problem of comparing AMR graphs A,B ∈$D$ with respect to the meaning they express occurs in several scenarios, for example, parser evaluation or inter-annotator agreement calculation (IAA). To
measure the extent to which A and B agree with each other, we need a metric: $D$×$D→R$ that returns a score reflecting meaning distance or meaning similarity (for convenience, we use similarity).
Below we establish seven principles that seem desirable for this metric.
2.1Seven Metric Principles
The first four metric principles are mathematically motivated:
I. continuity, non-negativity and upper-bound A similarity function should be continuous, with two natural edge cases: A,B are equivalent (maximum similarity) or unrelated (minimum similarity). By
choosing 1 as upper bound, we obtain the following constraint on metric: $D×D→[0,1]$.^^2
II. identity of indiscernibles This focal principle is formalized by metric(A, B) = 1 ⇔ A = B. It is violated if a metric assigns a value indicating equivalence to inputs that are not equivalent or
if it considers equivalent inputs as different.
III. symmetry In many cases, we want a metric to be symmetric: metric (A, B) = metric (B, A). A metric violates this principle if it assigns a pair of objects different scores when argument order is
inverted. Together with principles I and II, it extends the scope of the metric to usages beyond parser evaluation, as it also enables sound IAA calculation, clustering, and classification of AMR
graphs when we use the metric as a kernel (e.g., SVM). In parser evaluation, one may dispense with any (strong) requirements for symmetry—however, the metric must then be applied in a standardized
way, with a fixed order of arguments.
In cases where there is no defined reference, the asymmetry could be handled by aggregating metric (A, B) and metric (B, A), for example, using the mean. However, it is open what aggregation is best
suited and how to interpret results, for example, for metric (A, B) = 0.1 and metric (B, A) = 0.9.
IV. determinacy Repeated calculation over the same inputs should yield the same score. This principle is clearly desirable as it ensures reproducibility (a very small deviation may be tolerable).
The next three principles we believe to be desirable specifically when comparing meaning representation graphs such as AMR (Banarescu et al., 2013). The first two of the following principles are
motivated by computer science and linguistics, whereas the last one is motivated by a linguistic and an engineering perspective.
V. no bias Meaning representations consist of nodes and edges encoding specific information types. Unless explicitly justified, a metric should not unjustifiably or in unintended ways favor
correctness or penalize errors for specific substructures (e.g., leaf nodes). In case a metric favors or penalizes certain substructures more than others, in the interest of transparency, this should
be made clear and explicit, and should be easily verifiable and consistent. For example, if we wish to give negation of the main predicate of a sentence a two-times higher weight compared with
negation in an embedded sentence, we want this to be made transparent. A concrete example for a transparent bias is found in Cai and Lam (2019). They analyze the impact of their novel top–down AMR
parsing strategy by integrating a root-distance bias into Smatch to focus on structures situated at the top of a graph.
We now turn to properties that focus on the nature of the objects we aim to compare: graph-based compositional meaning representations. These graphs consist of atomic conditions that determine the
circumstances under which a sentence is true. Hence, our metric score should increase with increasing overlap of A and B, which we denote f(A,B), the number of matching conditions. This overlap can
be viewed from a symbolic or/and a graded perspective (cf., e.g., Schenker et al. [2005], who denote these perspectives as “syntactic” vs. “semantic”). From the symbolic perspective, we compare the
nodes and edges of two graphs on a symbolic level, while from the graded perspective, we take into account the degree to which nodes and edges differ. Both types of matching involve a precondition:
If A and B contain variables, we need a variable-mapping in order to match conditions from A and B.^^3
VI. matching (graph-based) meaning representations – symbolic match A natural symbolic overlap-objective can be found in the Jaccard index J (Jaccard, 1912; Real and Vargas, 1996; Papadimitriou et
al., 2010): Let t(G) be the set of triples of graph G, f (A, B) = |t(A) ∩ ts(B)| the size of the overlap of A, B, and z (A, B) = |t(A) ∪ t(B)| the size of their union. Then, we wish that A and B are
considered more similar to each other than A and C iff A and B exhibit a greater relative agreement in their (symbolic) conditions: metric(A, B) > metric(A, C)⇔$f(A,B)z(A,B)=J(A,B)>f(A,C)z(A,C)=J
(A,C)$. An allowed exception to this monotonic relationship can occur if we want to take into account a graded semantic match of atomic graph elements or sub-structures, which we will now elaborate
VII. matching (graph-based) meaning representations – graded semantic match: One motivation for this principle can be found in engineering, for example, when assessing the quality of produced parts.
Here, small deviations from a reference may be tolerable within certain limits. Similarly, two AMR graphs may match almost perfectly—except for two small divergent components. The extent of
divergence can be measured by the degree of similarity of the two divergent components. In our case, we need linguistic knowledge to judge what degree of divergence we are dealing with and whether it
is tolerable.
For example, consider that graph A contains a triple 〈x, instance, conceptA〉 and graph B a triple 〈y, instance , conceptB〉, while otherwise the graphs are equivalent, and the alignment has set x
= y. Then f(A, B) should be higher when conceptA is similar to conceptB compared to the case where conceptA is dissimilar to conceptB. In AMR, concepts are often abstract, so near-synonyms may even
be fully admissible (enemy–foe). Although such (near-)synonyms are bound to occur frequently when we compare AMR graphs of different sentences that may contain paraphrases, we will see, in Section §4
, that this can also occur in parser evaluation, where two different graphs represent the same sentence. By defining metric to map to a range [0,1] we already defined it to be globally graded. Here,
we desire that graded similarity may also hold of minimal units of AMR graphs, such as atomic concepts or even sub-graphs, for example, to reflect that injustice(x) is very similar to justice(x) ∧
2.2AMR Metrics: Smatch and SemBleu
With our seven principles for AMR similarity metrics in place, we now introduce Smatch and SemBleu, two metrics that differ in their design and assumptions. We describe each of them in detail and
summarize their differences, setting the stage for our in-depth metric analysis (§3).
Align and match – Smatch
The S
metric operates in two steps. First, (i) we align the variables in
in the best possible way, by finding a mapping
) that yields a maximal set of matching triples between
. For example, if 〈
, rel,
) and 〈
) rel,
)〉 = 〈
, rel,
), we obtain one triple match. (ii) We compute Precision, Recall, and F1 score based on the set of triples returned by the alignment search. The NP-hard alignment search problem of step (i) is solved
with a greedy hill-climber: Let
) be the count of matching triples under any mapping function
. Then,
Multiple restarts with different seeds increase the likelihood of finding better optima.
Simplify and match – SemBleu
The SemBleu metric in Song and Gildea (2019) can also be described as a two-step procedure. But unlike Smatch it operates on a variable-free reduction of an AMR graph G, which we denote by G^vf (vf:
variable-free, Figure 1, right-hand side).
In a first step, (i) S
-gram extraction from
in a breadth-first traversal (path extraction). It then (ii) adopts the B
score from MT (Papineni et al.,
) to calculate an overlap score based on the extracted bags of
is B
modified k-gram precision
that measures
-gram overlap of a candidate against a reference:
is the (typically uniform) weight over chosen
-gram sizes. S
uses NIST geometric probability smoothing (Chen and Cherry,
). The recall-focused “brevity penalty”
returns a value smaller than 1 when the candidate length |
| is smaller than the reference length |
The graph traversal performed in SemBleu starts at the root node. During this traversal it simplifies the graph by replacing variables with their corresponding concepts (see Figure 1: the node c
becomes Drink-01) and collects visited nodes and edges in uni-, bi- and tri grams (k = 3 is recommended). Here, a source node together with a relation and its target node counts as a bi-gram. For the
graph in Figure 1, the extracted unigrams are {cat,water, drink-01}; the extracted bi grams are {drink-01arg1 cat, drink-01arg2water}.
Smatch vs. SemBleu in a nutshell
SemBleu differs significantly from Smatch. A key difference is that SemBleu operates on reduced variable-free AMR graphs (G^vf)—instead of full-fledged AMR graphs. By eliminating variables, SemBleu
bypasses an alignment search. This makes the calculation faster and alleviates a weakness of Smatch: The hill-climbing search is slightly imprecise. However, SemBleu is not guided by aligned
variables as anchors. Instead, SemBleu uses an n-gram statistic (Bleu) to compute an overlap score for graphs, based on k-hop paths extracted from G^vf, using the root node as the start for the
extraction process. Smatch, by contrast, acts directly on variable-bound graphs matching triples based on a selected alignment. If in some application we wanted it, both metrics allow the capturing
of more “global” graph properties: SemBleu can increase its k-parameter and Smatch may match conjunctions of (interconnected) triples. In the following analysis, however, we will adhere to their
default configurations because this is how they are used in most applications.
3Assessing AMR Metrics with Principles
This section evaluates Smatch and SemBleu against the seven principles we established above by asking: Why does a metric satisfy or violate a given principle? and What does this imply? We start with
principles from mathematics.
I. Continuity, non-negativity, and upper-bound
This principle is fulfilled by both metrics as they are functions of the form $metric:D×D→[0,1]$.
II. Identity of indiscernibles
This principle is fundamental: An AMR metric must return maximum score if and only if the graphs are equivalent in meaning. Yet, there are cases where SemBleu, in contrast to Smatch, does not satisfy
this principle. Figure 2 shows an example.
Here, SemBleu yields a perfect score for two AMRs that differ in a single but crucial aspect: Two of its arg[x] roles are filled with arguments that are meant to refer to distinct individuals that
share the same concept. The graph on the left is an abstraction of, for example, The man[1]sees the other man[2]in the other man[2], while the graph on the right is an abstraction of The man[1]sees
himself[1]in the other man[2]. SemBleu does not recognize the difference in meaning between a reflexive and a non-reflexive relation, assigning maximum similarity score, whereas Smatch reflects such
differences appropriately because it accounts for variables.
In sum, SemBleu does not satisfy principle II because it operates on a variable-free reduction of AMRs (G^vf). One could address this problem by reverting to canonical AMR graphs and adopting
variable alignment in SemBleu. But this would adversely affect the advertised efficiency advantages over Smatch. Re-integrating the alignment step would make SemBleuless efficient than Smatch because
it would add the complexity of breadth-first traversal, yielding a total complexity of $O($Smatch ) plus $O(|V|+|E|)$.
III. Symmetry
This principle is fulfilled if
Figure 3
shows an example where S
does not comply with this principle, to a significant extent: When comparing AMR graph
, it yields a score greater than 0.8, yet, when comparing
the score is smaller than 0.5. We perform an experiment that quantifies this effect on a larger scale by assessing the frequency and the extent of such divergences. To this end, we parse 1,368
development sentences from an AMR corpus (LDC2017T10) with an AMR parser (obtaining graph bank
) and evaluate it against another graph bank ℬ (gold graphs or another parser-output). We quantify the symmetry violation by the
symmetry violation ratio
) and the
mean symmetry violation
) given some metric
We conduct the experiment with three AMR systems, CAMR (Wang et al., 2016), GPLA (Lyu and Titov, 2018), and JAMR (Flanigan et al., 2014), and the gold graphs. Moreover, to provide a baseline that
allows us to better put the results into perspective, we also estimate the symmetry violation of Bleu (SemBleu’s MT ancestor) in an MT setting. Specifically, we fetch 16 system outputs of the WMT
2018 EN-DE metrics task (Ma et al., 2018) and calculate Bleu(A,B) and Bleu(B,A) of each sentence-pair (A,B) from the MT system’s output and the reference (using the same smoothing method as SemBleu).
As worst-case/avg.-case, we use the outputs from the team where Bleu exhibits maximum/median msv.^^4
Table 1 shows that more than 80% of the evaluated AMR graph pairs lead to a symmetry violation with SemBleu (as opposed to less than 10% for Smatch). The msv of Smatch is considerably smaller
compared to SemBleu: 0.1 vs. 3.2 points F1 score. Even though the Bleu metric is inherently asymmetric, most of the symmetry violations are negligible when applied in MT (high svr, low msv, Table 2).
However, when applied to AMR graphs “via” SemBleu the asymmetry is amplified by a factor of approximately 16 (0.2 vs. 3.2 points). Figure 4 visualizes the symmetry violations of SemBleu (left), S
match (middle), and Bleu (right). The Sembleu-plots show that the effect is widespread, some cases are extreme, many others are less extreme but still considerable. This stands in contrast to Smatch
but also to Bleu, which itself appears well calibrated and does not suffer from any major asymmetry.
Table 1:
. symmetry violation .
. svr (%, Δ¿0.0001) . msv (in points) .
Graph banks . Smatch . SemBleu . Smatch . SemBleu .
Gold ↔ GPLA 2.7 81.8 0.1 3.2
Gold ↔ CAMR 7.8 92.8 0.2 3.1
Gold ↔ JAMR 5.0 87.0 0.1 3.2
JAMR ↔ GPLA 4.2 86.0 0.1 3.0
CAMR ↔ GPLA 7.4 93.4 0.1 3.4
CAMR ↔ JAMR 7.9 91.6 0.2 3.3
avg. 5.8 88.8 0.1 3.2
. symmetry violation .
. svr (%, Δ¿0.0001) . msv (in points) .
Graph banks . Smatch . SemBleu . Smatch . SemBleu .
Gold ↔ GPLA 2.7 81.8 0.1 3.2
Gold ↔ CAMR 7.8 92.8 0.2 3.1
Gold ↔ JAMR 5.0 87.0 0.1 3.2
JAMR ↔ GPLA 4.2 86.0 0.1 3.0
CAMR ↔ GPLA 7.4 93.4 0.1 3.4
CAMR ↔ JAMR 7.9 91.6 0.2 3.3
avg. 5.8 88.8 0.1 3.2
Table 2:
. Bleu symmetry violation, MT .
data: newstest2018 ↔(⋅) . svr (%, Δ > 0.0001) . msv (in points) .
worst-case 81.3 0.2
avg-case 72.7 0.2
. Bleu symmetry violation, MT .
data: newstest2018 ↔(⋅) . svr (%, Δ > 0.0001) . msv (in points) .
worst-case 81.3 0.2
avg-case 72.7 0.2
In sum, symmetry violations with Smatch are much fewer and less pronounced than those observed with SemBleu. In theory, Smatch is fully symmetric, however, violations can occur due to alignment
errors from the greedy variable-alignment search (we discuss this issue in the next paragraph). By contrast, the symmetry violation of SemBleu is intrinsic to the method because the underlying
overlap measure Bleu is inherently asymmetric, however, this asymmetry is amplified in SemBleu compared to Bleu.^^5
IV. Determinacy
This principle states that repeated calculations of a metric should yield identical results. Because there is no randomness in SemBleu, it fully complies with this principle. The reference
implementation of Smatch does not fully guarantee deterministic variable alignment results, because it aligns the variables by means of greedy hill-climbing. However, multiple random initializations
together with the small set of AMR variables imply that the deviation will be ≤ ε (a small number close to 0).^^6 In Table 3 we measure the expected ε: it displays the Smatch F1 standard deviation
with respect to 10 independent runs, on a corpus level and on a graph-pair level (arithmetic mean).^^7 We see that ε is small, even when only one random start is performed (corpus level: ε = 0.0003,
graph level: ε = 0.0013). We conclude that the hill-climbing in Smatch is unlikely to have any significant effects on the final score.
Table 3:
. # restarts .
. 1 . 2 . 3 . 5 . 7 .
corpus vs. corpus 2.6e^−4 1.7e^−4 8.1e^−5 5.7e^−5 5.6e^−5
graph vs. graph 1.3e^−3 1.0e^−3 8.5e^−4 5.3e^−4 4.0e^−4
. # restarts .
. 1 . 2 . 3 . 5 . 7 .
corpus vs. corpus 2.6e^−4 1.7e^−4 8.1e^−5 5.7e^−5 5.6e^−5
graph vs. graph 1.3e^−3 1.0e^−3 8.5e^−4 5.3e^−4 4.0e^−4
V. No bias
A similarity metric of (A)MRs should not unjustifiably or unintentionally favor the correctness or penalize errors pertaining to any (sub-)structures of the graphs. However, we find that SemBleu is
affected by a bias that affects (some) leaf nodes attached to high-degree nodes. The bias arises from two related factors: (i) when transforming G to G^vf SemBleu replaces variable nodes with concept
nodes. Thus, nodes that were leaf nodes in G can be raised to highly connected nodes in G^vf. (ii) breadth-first k-gram extraction starts from the root node. During graph traversal, concept
leaves—now occupying the position of (former) variable nodes with a high number of outgoing (and incoming) edges—will be visited and extracted more frequently than others.
The two factors in combination make SemBleu penalize a wrong concept node harshly when it is attached to a high-degree variable node (the leaf is raised to high-degree when transforming G to G^vf).
Conversely, correct or wrongly assigned concepts attached to nodes with low degree are only weakly considered.^^8 For example, consider Figure 5. SemBleu considers two graphs that express quite
distinct meanings (left and right) as more similar than graphs that are almost equivalent in meaning (left, variant A vs. B). This is because the leaf that is attached to the root is raised to a
highly connected node in G^vf and thus is over-frequently contained in the extracted k-grams, whereas the other leaves will remain leaves in G^vf.
Analyzing and quantifying SemBleu’s bias
To better understand the bias, we study three limiting cases: (i) the root is wrong (√) (ii) d leaf nodes are wrong (k-grams (SemBleu) or triples (Smatch) the errors are projected. For the sake of
simplicity, we assume that the graph always comes in its simplified form G^vf, that it is a tree, and that every non-leaf node has the same out-degree d.
The result of our analysis is given in Table 4^,^^9 and exemplified in Figure 6. Both show that the number of times k-gram extraction visits a node heavily depends on its position and that with
growing d, the bias gets amplified (Table 4).^^10 For example, when d = 3, 3 wrong leaves yield 9 wrong k-grams, and 1 wrong branching node can already yield 18 wrong k-grams. By contrast, in Smatch
the weight of d leaves always approximates the weight of 1 branching node of degree d.
Table 4:
. . √ . .
SemBleu $O(3d)$ $O(d2+d)$ $O(d2+2d)$
Smatch $O(d)$ $O(d)$ $O(d)$
. . √ . .
SemBleu $O(3d)$ $O(d2+d)$ $O(d2+2d)$
Smatch $O(d)$ $O(d)$ $O(d)$
In sum, in Smatch the impact of a wrong node is constant for all node types and rises linearly with d. In SemBleu the impact of a node rises approximately quadratically with d and it also depends on
the node type, because it raises some (but not all) leaves in G to connected nodes in G^vf.
Eliminating biases
A possible approach to reduce SemBleu’s biases could be to weigh the extracted k-gram matches according to the degree of the contained nodes. However, this would imply that we assume some k-grams
(and thus also some nodes and edges) to be of greater importance than others—in other words, we would eliminate one bias by introducing another. Because the breadth-first traversal is the metric’s
backbone, this issue may be hard to address well. When Bleu is used for MT evaluation, there is no such bias because the k-grams in a sentence appear linearly.
VI. Graph matching: Symbolic perspective
This principle requires that a metric’s score grows with increasing overlap of the conditions that are simultaneously contained in A and B. Smatch fulfills this principle since it matches two AMR
graphs inexactly (Yan et al., 2016; Riesen et al., 2010) by aligning variables such that the triple matches are maximized. Hence, Smatch can be seen as a graph-matching algorithm that works on any
pair of graphs that contain (some) nodes that are variables. It fulfills the Jaccard-based overlap objective, which symmetrically measures the amount of triples on which two graphs agree, normalized
by their respective sizes (since Smatch F1 = 2J/(1 + J) is a monotonic relation).
Because SemBleu does not satisfy principles II and III (id. of indescernibles and symmetry), it is a corollary that it cannot fulfill the overlap objective.^^11 Generally, SemBleu does not compare
and match two AMR graphs per se, instead it matches the results of a graph-to-bag-of-paths projection function (§2.2) and the input may not be recoverable from the output (surjective-only). Thus,
matching the outputs of this function cannot be equated to matching the inputs on a graph-level.
4Towards a More Semantic Metric for Semantic Graphs: S^2match
This section focuses on principle VII, semantically graded graph matching, a principle that none of the AMR metrics considered so far satisfies. A fulfilment of this principle also increases the
capacity of a metric to assess the semantic similarity of two AMR graphs from different sentences. For example, when clustering AMR graphs or detecting paraphrases in AMR-parsed texts, the ability to
abstract away from concrete lexicalizations is clearly desirable. Consider Figure 7, with three different graphs. Two of them (A,B) are similar in meaning and differ significantly from C. However,
both Smatch and SemBleu yield the same result in the sense that metric(A, B) = metric(A, C). Put differently, neither metric takes into account that giraffe and kitten are two quite different
concepts, while cat and kitten are more similar. However, we would like this to be reflected by our metric and obtain metric(A,B) > metric(A, C) in such a case.
We propose the
metric (
Soft Semantic match
, pronounced: [estu:mæt∫]) that builds on S
but differs from it in one important aspect: Instead of maximizing the number of (hard) triple matches between two graphs during alignment search, we maximize the (soft) triple matches by taking into
account the semantic similarity of concepts. Recall that an AMR graph contains two types of triples: instance and relation triples (e.g.,
Figure 7
, left: 〈
, instance, cat〉 and
). In S
, two triples can only be matched if they are identical. In
, we relax this constraint, which has also the potential to yield a different, and possibly, a better variable alignment. More precisely, in S
we match two instance triples 〈
, instance, x〉∈
and 〈
), instance,
as follows:
equals 1 if
is true and 0 otherwise.
relaxes this condition:
is an arbitrary distance function
. For example, in practice, if we represent the concepts as vectors
, we can use
When plugged into Eq. 7, this results in the cosine similarity ∈ [0, 1]. It may be suitable to set a threshold τ (e.g., τ = 0.5), to only consider the similarity between two concepts if it is above τ
(softMatch = 0 if 1 − d(x,y) < τ). In the following pilot experiments, we use cosine (Eq. 8) and τ = 0.5 over 100-dimensional GloVe vectors (Pennington et al., 2014).
To summarize, S^2match is designed to either yield the same score as Smatch—or a slightly increased score when it aligns concepts that are symbolically distinct but semantically similar.
An example, from parser evaluation, is shown in Figure 8. Here, S^2match increases the score to 63 F1 (+10 points) by detecting a more adequate alignment that accounts for the graded similarity of
two related AMR concepts pairs. We believe that this is justified: The two graphs are very similar and an F1 of 53 is too low, doing the parser injustice.
On a technical note, the changes in alignments also have the outcome that S^2match mends some of Smatch’s flaws: It better addresses principles III and IV, reducing the symmetry violation and
determinacy error (Table 5:).
Table 5:
. . determinacy error .
. avg. msv (Eq. 5) . 1 restart . 2 restarts . 4 restarts .
Smatch 0.0011 1.3 e^−3 1.0e^−3 5.3e^−4
S^2match 0.0005 9.0e^−4 6.1e^−4 2.1e^−4
relative change − 54.6% − 30.7% − 39.0% − 60.3%
. . determinacy error .
. avg. msv (Eq. 5) . 1 restart . 2 restarts . 4 restarts .
Smatch 0.0011 1.3 e^−3 1.0e^−3 5.3e^−4
S^2match 0.0005 9.0e^−4 6.1e^−4 2.1e^−4
relative change − 54.6% − 30.7% − 39.0% − 60.3%
Qualitative study: Probing S^2match’s choices
This study randomly samples 100 graph pairs from those where S^2match assigned higher scores than Smatch.^^12 Two annotators were asked to judge the similarity of all aligned concepts with similarity
score <1.0: Are the concepts dissimilar, similar, or extremely similar? For concepts judged dissimilar, we conclude that S^2match erroneously increased the score; if judged as (extremely) similar, we
conclude that the decision was justified. We calculate three agreement statistics that all show large consensus among our annotators (Cohen’s kappa: 0.79, squared kappa: 0.87, Pearson’s ρ: 0.91)
According to the annotations, the decision to increase the score is mostly justified: in 56% and 12% of cases both annotators voted that the newly aligned concepts are extremely similar and similar,
respectively, while the agreed dissimilar label makes up 25% of cases.
Table 6 lists examples of good or ill-founded score increases. We observe, for example, that S^2match accounts for the similarity of two concepts of different number: bacterium (gold) vs. bacteria
(parser) (line 3). It also captures abbreviations (km – kilometer) and closely related concepts (farming – agriculture). SemBleu and Smatch would penalize the corresponding triples in exactly the
same way as predicting a truly dissimilar concept.
Table 6:
input span region (excerpt) . amr region gold (excerpt) . amr region parser (excerpt) . cos . points F1↑ . annotation .
40 km southwest of :quant 40:unit (k2 / kilometer) (k22 / km:unit-of (d23 / distance-quantity 0.72 1.2 ex. similar
improving agricultural prod. (i2 / improve-01 …:mod (f2 / farming) (i31 / improve-01:mod (a23 / agriculture) 0.73 3.0 ex. similar
other deadly bacteria op3 (b / bacterium …:mod (o / other))) op3 (b13 / bacteria:ARG0-of:mod (o12 / other))) 0.80 5.1 ex. similar
drug and law enforcement aid (a / and:op2 (a3 / aid-01 :ARG1 (a9 / and:op1 (d8 / drug) :op2 (l10 / law))) 0.67 1.8 similar
Get a lawyer and get a divorce. :op1 (g / get-01:mode imp. :ARG0 (y / you) :op1 (g0 / get-01:ARG1 (l2 / lawyer):mode imp.) 0.80 4.8 dissimilar
The unusual development. ARG0 (d / develop-01:mod (u / usual:polarity -)) :ARG0 (d1 / develop-02:mod (u0 / unusual)) 0.60 14.0 dissimilar
input span region (excerpt) . amr region gold (excerpt) . amr region parser (excerpt) . cos . points F1↑ . annotation .
40 km southwest of :quant 40:unit (k2 / kilometer) (k22 / km:unit-of (d23 / distance-quantity 0.72 1.2 ex. similar
improving agricultural prod. (i2 / improve-01 …:mod (f2 / farming) (i31 / improve-01:mod (a23 / agriculture) 0.73 3.0 ex. similar
other deadly bacteria op3 (b / bacterium …:mod (o / other))) op3 (b13 / bacteria:ARG0-of:mod (o12 / other))) 0.80 5.1 ex. similar
drug and law enforcement aid (a / and:op2 (a3 / aid-01 :ARG1 (a9 / and:op1 (d8 / drug) :op2 (l10 / law))) 0.67 1.8 similar
Get a lawyer and get a divorce. :op1 (g / get-01:mode imp. :ARG0 (y / you) :op1 (g0 / get-01:ARG1 (l2 / lawyer):mode imp.) 0.80 4.8 dissimilar
The unusual development. ARG0 (d / develop-01:mod (u / usual:polarity -)) :ARG0 (d1 / develop-02:mod (u0 / unusual)) 0.60 14.0 dissimilar
An interesting case is seen in line 7. Here, usual and unusual are correctly annotated as dissimilar, since they are opposite concepts. S^2match, equipped with GloVe embeddings, measures a cosine of
0.6, above the chosen threshold, which results in an increase of the score by 14 points (the increase is large as these two graphs are tiny). It is well known that synonyms and antonyms are difficult
to distinguish with distributional word representations, because they often share similar contexts. However, the case at hand is orthogonal to this problem: usual in the gold graph is modified with
the polarity ‘−’, whereas the predicted graph assigned the (non-negated) opposite concept unusual. Hence, given the context in the gold graph, the prediction is semantically almost equivalent. This
points to an aspect of principle VII that is not yet covered by S^2match: It assesses graded similarity at the lexical, but not at the phrasal level, and hence cannot account for compositional
phenomena. In future work, we aim to alleviate this issue by developing extensions that measure semantic similarity for larger graph contexts, in order to fully satisfy all seven principles.^^13
Quantitative study: Metrics vs. human raters
This study investigates to what extent the judgments of the three metrics under discussion resemble human judgements, based on the following two expectations. First, the more a human rates two
sentences to be semantically similar in their meaning, the higher the metric should rate the corresponding AMR graphs (meaning similarity). Second, the more a human rates two sentences to be related
in their meaning (maximum: equivalence), the higher the score of our metric of the corresponding AMR graphs should tend to be (meaning relatedness). Albeit not the exact same (Budanitsky and Hirst,
2006), the tasks are closely related and both in conjunction should allow us to better assess the performance of our AMR metrics.
As ground truth for the meaning similarity rating task we use test data of the Semantic Textual Similarity (STS) shared task (Cer et al., 2017), with 1,379 sentence pairs annotated for meaning
similarity. For the meaning-relatedness task we use SICK (Marelli et al., 2014) with 9,840 sentence pairs that have been additionally annotated for semantic relatedness.^^14 We proceed as follows: We
normalize the human ratings to [0,1]. Then we apply GPLA to parse the sentence tuples (s[i],s[i]′), obtaining tuples (parse(s[i]),parse(s[i]′)) and score the graph pairs with the metrics: Smatch(i),
S^2match(i), SemBleu(i), and H(i), where H(i) is the human score. For both tasks Smatch and S^2match yield better or equal correlations with human raters than SemBleu (Table 7). When considering the
RMS error $n−1∑i=1n(H(i)−metric(i))2$. the difference is even more pronounced.
Table 7:
. RMSE . RMSE (quant) . Pearson’s ρ . Spearman’s ρ .
task . sb . sm . s^2m . sb . sm . s^2m . sb . sm . s^2m . sb . sm . s^2m .
STS 0.34 0.25 0.25 0.25 0.11 0.10 0.52 0.55 0.55 0.51 0.53 0.53
SICK 0.38 0.25 0.24 0.32 0.14 0.13 0.62 0.64 0.64 0.66 0.66 0.66
. RMSE . RMSE (quant) . Pearson’s ρ . Spearman’s ρ .
task . sb . sm . s^2m . sb . sm . s^2m . sb . sm . s^2m . sb . sm . s^2m .
STS 0.34 0.25 0.25 0.25 0.11 0.10 0.52 0.55 0.55 0.51 0.53 0.53
SICK 0.38 0.25 0.24 0.32 0.14 0.13 0.62 0.64 0.64 0.66 0.66 0.66
This deviation in the absolute scores is also reflected by the score density distributions plotted in Figure 9: SemBleu underrates a good proportion of graph pairs whose input sentences were rated as
highly semantically similar or related by humans. This may well relate to the biases of different node types (cf. §3). Overall, S^2match appears to provide a better fit with the score-distribution of
the human rater when measuring semantic similarity and relatedness, the latter being notably closer to the human reference in some regions than the otherwise similar Smatch. A concrete example from
the STS data is given in Figure 10. Here, S^2match detects the similarity between the abstract anaphors it and this and assigns a score that better reflects the human score compared to Smatch and Sem
Bleu, the latter being far too low. However, in total, we conclude that S^2match’s improvements seem rather small and no metric is perfectly aligned with human scores, possibly because gradedness of
semantic similarity that arises in combination with constructional variation is not yet captured—more research is needed to extend S^2match’s scope to such cases.
5Metrics’ effects on parser evaluation
We have seen that different metrics can assign different scores to the same pair of graphs. We now want to assess to what extent this affects rankings: Does one metric rank a graph higher or lower
than the other? And can this affect the ranking of parsers on benchmark datasets?
Quantitative study: Graph rankings
In this experiment, we assess whether our metrics rank graphs differently. We use LDC2017T10 (dev) parses by CAMR [c[1] … c[n]], JAMR [j[1] … j[n]] and gold graphs [y[1] … y[n]]. Given metrics $F$
and $G$ we obtain results $FC:=[F(c1,y1)…F(cn,yn)]$ and analogously $FJ$, $GC$ and $GJ$. We calculate two statistics: (i) the ratio of cases i where the metrics differ in their preference for one
parse over the other $(FiJ−FiC)⋅(GiJ−GiC)<0$, and, to assess significance, (ii) a t-test for paired samples on the differences assigned by the metrics between the parsers: $FJ−FC$ and $GJ−GC$.
Table 8 shows that Smatch and S^2match both differ (significantly) from SemBleu in 15% – 20% of cases. Smatch and S^2match differ on individual rankings in appr. 4% of cases. Furthermore, we note a
considerable amount of cases (8.1%) where SemBleu disagrees with itself in the preference for one parse over the other.^^15
Table 8:
. $SmA,BG$ . $SmGA,B$ . $sbA,BG$ . $sbGA,B$ . $S2mA,BG$ . $S2mGA,B$ .
$SmA,BG$ 0.0 1.5 17.6^† 19.0^† 4.0 4.1
$SmGA,B$ – 0.0 17.9^† 19.5^† 3.9 4.0
$sbA,BG$ – – 0.0 8.1^† 18.4^† 19.2^†
$sbGA,B$ – – – 0.0 19.1^† 19.3^†
$S2mA,BG$ – – – – 0.0 1.2
$S2mGA,B$ – – – – 0.0
. $SmA,BG$ . $SmGA,B$ . $sbA,BG$ . $sbGA,B$ . $S2mA,BG$ . $S2mGA,B$ .
$SmA,BG$ 0.0 1.5 17.6^† 19.0^† 4.0 4.1
$SmGA,B$ – 0.0 17.9^† 19.5^† 3.9 4.0
$sbA,BG$ – – 0.0 8.1^† 18.4^† 19.2^†
$sbGA,B$ – – – 0.0 19.1^† 19.3^†
$S2mA,BG$ – – – – 0.0 1.2
$S2mGA,B$ – – – – 0.0
The differing preferences of S^2match for either candidate parse can be the outcome of small divergences due to the alignment search or because S^2match accounts for the lexical similarity of
concepts, perhaps supported by a new variable alignment. Figure 11 shows two examples where S^2match prefers a different candidate parse compared to Smatch. In the first example (Figure 11a), S^2
match prefers the parse produced by JAMR and changes the alignment legally-NULL (Smatch) to legally-law (S^2match). In the second example 11b, S^2match prefers the parse produced by CAMR, because it
detects the similarity between military and navy and poor and poverty. Therefore, S^2match can assess that the CAMR parse and the gold graph substantially agree on the root concept of the graph,
which is not the case in the JAMR parse.
Quantitative study: Parser rankings
Having seen that our metrics disagree on the ranking of individual graphs, we now quantify the effects on the ranking of parsers. We collect outputs of three state-of-art parsers on the test set of
LDC2017T10: GPLA, a sequence-to-graph transducer (STOG), and a neural top-down parser (TOP-DOWN).
Table 9 shows that Smatch and S^2match agree on the ranking of all three parsers, but both disagree with SemBleu on the ranks of the 42^nd and 3^rd parser: unlike SemBleu, the Smatch variants rate
GPLA higher than TOP-DOWN. A factor that may have contributed to the different rankings perhaps lies in SemBleu’s biases towards connected nodes: Compared with TOP-DOWN, GPLA delivers more complex
parses, with more edges (avg. |E|: 32.8 vs. 32.1) and higher graph density (avg. density: 0.065 vs. 0.059). This is a nice property, because it indicates that the graphs of GPLA better resemble the
rich gold graph structures (avg. density: 0.063, avg. |E|: 34.2). When inspecting this more closely, and looking at single (parse, gold) pairs, we observe further evidence for this: the structural
error, in degree and density, is lower for GPLA than for TOP-DOWN (Table 9, right columns), with an error reduction of -27% (degree, 0.08 vs. 0.11) and -14% (density, 0.0067 vs. 0.0078).
Table 9:
. metric scores . structure error .
. Sm . $SBAG$ . $SBGA$ . S^2m . node degree . graph density .
STOG 76.3[|1] 59.6[|1] 58.9[|1] 77.9[|1] 0.08 0.0069
GPLA 74.5[|2] 54.2[|3] 52.9[|3] 76.2[|2] 0.08 0.0068
TOP-DOWN 73.2[|3] 54.5[|2] 53.1[|2] 75.0[|3] 0.11 0.0078
. metric scores . structure error .
. Sm . $SBAG$ . $SBGA$ . S^2m . node degree . graph density .
STOG 76.3[|1] 59.6[|1] 58.9[|1] 77.9[|1] 0.08 0.0069
GPLA 74.5[|2] 54.2[|3] 52.9[|3] 76.2[|2] 0.08 0.0068
TOP-DOWN 73.2[|3] 54.5[|2] 53.1[|2] 75.0[|3] 0.11 0.0078
In sum, by building graphs of higher complexity, GPLA takes a greater risk when attaching wrong concepts to connected nodes where errors are penalized more strongly by SemBleu than Smatch, according
to the biases we have studied in §3 (Table 4). In that sense, STOG also takes more risks, but it may get more of such concepts right and so the bias transitions from penalty to reward, potentially
explaining the large performance Δ (+6) of STOG to the other parsers, as measured by SemBleu, in contrast to S(2)Match (Δ: +2).
6Summary of Our Metric Analyses
Table 10 summarizes our analyses’ integral results. Principle I is fulfilled by all metrics as they exhibit continuity, non-negativity and an upper bound. Principle II, however, is not satisfied by S
emBleu because it can mistake two graphs of different meaning as equivalent. This is because it ablates a variable-alignment and therefore cannot capture facets of coreference. Yet, a positive
outcome of this is that it is fast to compute. This could make it first choice in some recent AMR parsing approaches that use reinforcement learning (Naseem et al., 2019), where rapid feedback is
needed. It also marks a point by fully satisfying Principle IV, yielding fully deterministic results. Smatch, by contrast, either needs to resort to a costly ILP solution or (in practice) uses
hill-climbing with multiple restarts to reduce divergence to a negligible amount.
Table 10:
principle . Smatch . SemBleu . S^2Match . Sec. .
I. Cont., non-neg. & upper-bound ✓ ✓ ✓ -
II. id. of indescernibles 3[ε] ✗ 3[δ<ε] §3
III. symmetry 3[ε] ✗ 3[δ<ε] §3
IV. determinacy 3[ε] ✓ 3[δ<ε] §3
V. low bias ✓ ✗ ✓ §3
VI. symbolic graph matching ✓ ✗ ✓ §3
VII. graded graph matching ✗ ✗ 3^LEX §4
principle . Smatch . SemBleu . S^2Match . Sec. .
I. Cont., non-neg. & upper-bound ✓ ✓ ✓ -
II. id. of indescernibles 3[ε] ✗ 3[δ<ε] §3
III. symmetry 3[ε] ✗ 3[δ<ε] §3
IV. determinacy 3[ε] ✓ 3[δ<ε] §3
V. low bias ✓ ✗ ✓ §3
VI. symbolic graph matching ✓ ✗ ✓ §3
VII. graded graph matching ✗ ✗ 3^LEX §4
A central insight brought out by our analysis is that SemBleu exhibits biases that are hard to control. This is caused by two (interacting) factors: (i) The extraction of k-grams is applied on the
graph top to bottom and visits some nodes more frequently than others. (ii) It raises some (but not all) leaf nodes to connected nodes, and these nodes will be overly frequently contained in
extracted k-grams. We have shown that these two factors in combination lead to large biases that researchers should be aware of when using SemBleu (§3). Its ancestor Bleu does not suffer from such
biases since it extracts k-grams linearly from a sentence.
Given that SemBleu is built on Bleu, it is inherently asymmetric. However, we have shown that the asymmetry (Principle III) measured for Bleu in MT is amplified by SemBleu in AMR, mainly due to the
biases it incurs (Principle V). Although asymmetry can be tolerated in parser evaluation if outputs are compared against gold graphs in a standardized manner, it is difficult to apply an asymmetric
metric to measure IAA or to compare parses for detecting paraphrases, or in tri-parsing, where no reference is available. If the asymmetry is amplified by a bias, it becomes harder to judge the
scores. Finally, considering that SemBleu does not match AMR graphs on the graph-level but matches extracted bags-of- k-grams, it turns out that it cannot be categorized as a graph matching algorithm
as defined in Principle VI.
Principle VI is fulfilled by Smatch without any transformation on AMR graphs. It searches for an optimal variable alignment and counts matching triples. As a corollary, it fulfills principles I, II,
III and V. The fact that Smatch fulfills all but one principle backs up many prior works that use it as sole criterion for IAA and parse evaluation.
Our principles also helped us detect a weakness of all present AMR metrics: They operate on a discrete level and cannot assess graded meaning differences. As a first step, we propose S^2match: It
preserves beneficial properties of Smatch but is benevolent to slight lexical meaning deviations. Besides parser evaluation, this property makes the metric also more suitable for other tasks, for
example, it can be used as a kernel in an SVM that classifies AMRs to determine whether two sentences are equivalent in meaning. In such a case, S^2match is bound to detect meaning-similarities that
cannot be captured by Smatch or SemBleu, for example, due to paraphrases being projected into the parses.
7Related Work
Developing similarity metrics for meaning representations (MRs) is important, as it, inter alia, affects semantic parser evaluation and computation of IAA statistics for sembanking. MRs are designed
to represent the meaning of text in a well-defined, interpretable form that is able to identify meaning differences and support inference. Bos (2016, 2019) has shown how AMR can be translated to FOL,
a well-established MR formalism. Discourse Representation Theory (DRT; Kamp, 1981; Kamp and Reyle, 1993) is based on and extends FOL to discourse representation. A recent shared task on DRS parsing
used the Counter metric (Abzianidze et al., 2019; Evang, 2019), an adaption of Smatch, underlining Smatch’s general applicability. Its extension S^2match may also prove beneficial for DRS.
Other research into AMR metrics aims at making the comparison fairer by normalizing graphs (Goodman, 2019). Anchiêta et al. (2019) argue that one should not, for example, insert an extra is-root node
when comparing AMR graphs (as done in SemBleu and Smatch). Damonte et al. (2017) extend Smatch to analyze individual AMR facets (co-reference, WSD, etc.). Cai and Lam (2019) adapt Smatch to analyze
their parser’s performance in predicting triples that are in close proximity to the root. Our metric S^2match allows for straightforward integration of these approaches.
Computational AMR tasks
Since the introduction of AMR, many AMR-related tasks have emerged. Most prominent is AMR parsing (Wang et al., 2015, 2016; Damonte et al., 2017; Konstas et al., 2017; Lyu and Titov, 2018; Zhang et
al., 2019). The inverse task generates text from AMR graphs (Song et al., 2017, 2018; Damonte and Cohen, 2019). Opitz and Frank (2019) rate the quality of automatic AMR parses without costly gold
We motivated seven principles for metrics measuring the similarity of graph-based (Abstract) Meaning Representations, from mathematical, linguistic and engineering perspectives. A metric that
fulfills all principles is applicable to a wide spectrum of cases, ranging from parser evaluation to sound IAA calculation. Hence (i) our principles can inform (A)MR researchers who desire to compare
and select among metrics, and (ii) they ease and guide the development of new metrics.
We provided examples for both scenarios. We showcased (i) by utilizing our principles as guidelines for an in-depth analysis of two AMR metrics: Smatch and the recent SemBleu metrics, two quite
distinct approaches. Our analysis uncovered that the latter does not satisfy some principles, which might reduce its safety and applicability. In line with (ii), we target the fulfilment of all seven
principles and propose S^2match, a metric that accounts for graded similarity of concepts as atomic graph components. In future work, we aim for a metric that accounts for graded compositional
similarity of subgraphs.
We are grateful to the anonymous reviewers and the action editors for their valuable time and comments. This work has been partially funded by DFG within the project ExpLAIN. Between the Lines –
Knowledge-based Analysis of Argumentation in a Formal Argumentation Inference System; FR 1707/-4-1, as part of the RATIO Priority Program; and by the the Leibniz ScienceCampus Empirical Linguistics &
Computational Language Modeling, supported by Leibniz Association grant no. SAS2015-IDS-LWC and by the Ministry of Science, Research, and Art of Baden-Wurttemberg.
Most research papers on AMR display the graphs in this “shallow” form. This increases simplicity and readability. (Lyu and Titov, 2018; Konstas et al., 2017; Zhang et al., 2019; Damonte and Cohen,
2019; Song et al., 2016).
At some places in this paper, due to conventions, we project this score onto [0,100] and speak of points.
For example, consider graph A in Figure 1 and its set of triples t(A): {〈x[1], instance,drink-1〉〈x[2], instance,cat〉, 〈x[1], arg0, x[2]〉, 〈x[1],arg1,x[3]〉, 〈x[3], instance, water〉}. When
comparing A against graph B we need to judge whether a triple t ∈t(A) is also contained in B: t ∈t(B). For this, we need a mapping map: vars(A)→vars(B) where vars(A) = {x[1],.., x[n]}, vars(B) = {y
[1],.., y[m]} such that f is maximized.
worst: LMU uns.; avg.: LMU NMT (Huck et al., 2017).
As we show below (principle V), this is due to the way in which k-grams are extracted from variable-free AMR graphs.
Additionally, ε = 0 is guaranteed when resorting to a (costly) ILP calculation (Cai and Knight, 2013).
Data: dev set of LDC2017T10, parses by GPLA.
This may have severe consequences, e.g., for negation, since negation always occurs as a leaf in G and G^vf. Therefore, SemBleu, by-design, is benevolent to polarity errors.
Proof sketch, Smatch, d leaves: d triples, a root: d triples, a branching node: d+1 triples. SemBleu$k=3wk=1/3$, d leaves: 3dk-grams (d tri, d bi, d uni). A root: d^2 tri, d bi, 1 uni. A branching
node: d^2+d+1 tri, d+1 bi, 1 uni. □
Consider that in AMR, d can be quite high, e.g., a predicate with multiple arguments and additional modifiers.
Proof by symmetry violation: w.l.o.g. ∃A,B: metric(A, B)>metric(B, A)⇒f(A, B)>f(B, A)$→$ ↯ , since f(A, B) = |t(A) ∩t(B)| = |t(B) ∩t(A)| = f(B, A) □ /// Proof by identity of indiscernibles: w.l.o.g.
∃A,B, C: metric(A, B) = metric(A, C)= 1 ∧f(A, B)/z(A, B) = 1 >f(A,C)/z(A, C)↯ □
Automatic graphs by GPLA, on LDC2017T10, dev set.
As we have seen, this requires much care. We therefore consider this next step to be out of scope of the present paper.
An example from SICK. Max. score: A man is cooking pancakes–The man is cooking pancakes. Min. score: Two girls are playing outdoors near a woman.–The elephant is being ridden by the man. To further
enhance the soundness of the SICK experiment we discard pairs with a contradiction relation and retain 8,416 pairs with neutral or entailment.
That is, sb(A,G)>sb(B,G) albeit sb(G,A)<sb(G,B).
van Noord
, and
The first shared task on discourse representation structure parsing
. In
Proceedings of the IWCS Shared Task on Semantic Parsing
Gothenburg, Sweden
Rafael Torres
Marco Antonio Sobrevilla
, and
Thiago Alexandre Salgueiro
SEMA: An extended semantic evaluation for AMR
. In
(To appear) Proceedings of the 20th Computational Linguistics and Intelligent Text Processing
Springer International Publishing
, and
Abstract meaning representation for sembanking
. In
Proceedings of the 7th Linguistic Annotation Workshop and Interoperability with Discourse
, pages
Expressive power of abstract meaning representations
Computational Linguistics
Separating argument structure from logical structure in AMR
arXiv preprint arXiv:1908.01355
Evaluating WordNet-based measures of lexical semantic relatedness
Computational Linguistics
Core semantic first: A top-down approach for AMR parsing
. In
Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)
, pages
Hong Kong, China
Smatch: An evaluation metric for semantic feature structures
. In
Proceedings of the 51st Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)
, pages
Sofia, Bulgaria
, and
SemEval-2017 Task 1: Semantic textual similarity multilingual and crosslingual focused evaluation
. In
Proceedings of the 11th International Workshop on Semantic Evaluation (SemEval-2017)
, pages
Vancouver, Canada
A systematic comparison of smoothing techniques for sentence-level BLEU
. In
Proceedings of the Ninth Workshop on Statistical Machine Translation
, pages
Baltimore, Maryland, USA
Shay B.
Structural neural encoders for AMR-to-text Generation
. In
Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)
, pages
Minneapolis, Minnesota
Shay B.
, and
An incremental parser for abstract meaning representation
. In
Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics: Volume 1, Long Papers
, pages
Valencia, Spain
Near-synonymy and lexical choice
Computational Linguistics
Transition-based DRS parsing using stack-LSTMs
. In
Proceedings of the IWCS Shared Task on Semantic Parsing
Gothenburg, Sweden
, and
Noah A.
A discriminative graph-based parser for the abstract meaning representation
. In
Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)
, pages
Baltimore, Maryland
Michael Wayne
AMR normalization for fairer evaluation
. In
Proceedings of the 33rd Pacific Asia Conference on Language, Information, and Computation
, and
LMU Munich’s neural machine translation systems for news articles and health information texts
. In
Proceedings of the Second Conference on Machine Translation
, pages
Copenhagen, Denmark
Building and using a lexical knowledge base of near-synonym differences
Computational Linguistics
The distribution of the flora in the alpine zone
New Phytologist
A theory of truth and semantic representation
Formal Semantics: The Essential Readings
, pages
From Discourse to Logic. Introduction to Model-theoretic Semantics of Natural Language, Formal Logic and Discourse Representation Theory
Kluwer, Dordrecht
, and
Neural AMR: Sequence-to-sequence models for parsing and generation
. In
Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics, ACL 2017, Vancouver, Canada, July 30 – August 4, Volume 1: Long Papers
, pages
AMR parsing as graph prediction with latent alignment
. In
Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)
, pages
Melbourne, Australia
, and
Results of the WMT18 metrics shared task: Both characters and embeddings achieve good performance
. In
Proceedings of the Third Conference on Machine Translation: Shared Task Papers, WMT 2018, Belgium, Brussels, October 31 - November 1, 2018
, pages
, and
SemEval-2014 task 1: Evaluation of compositional distributional semantic models on full sentences through semantic relatedness and textual entailment
. In
Proceedings of the 8th International Workshop on Semantic Evaluation (SemEval 2014)
, pages
Dublin, Ireland
, and
Rewarding smatch: Transition-based AMR parsing with reinforcement learning
. In
Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics
, pages
Florence, Italy
Automatic Accuracy Prediction for AMR Parsing
. In
Proceedings of the Eighth Joint Conference on Lexical and Computational Semantics (*SEM 2019)
, pages
Minneapolis, Minnesota
, and
Web graph similarity for anomaly detection
Journal of Internet Services and Applications
, and
BLEU: A method for automatic evaluation of machine translation
. In
Proceedings of the 40th Annual Meeting of the Association for Computational Linguistics
, pages
Philadelphia, Pennsylvania, USA
, and
GloVe: Global vectors for word representation
. In
Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP)
, pages
Doha, Qatar
Juan M
The probabilistic basis of Jaccard’s index of similarity
Systematic Biology
, and
Exact and inexact graph matching: Methodology and applications
Managing and Mining Graph Data
, pages
, and
Graph-Theoretic Techniques for Web Content Mining
World Scientific Publishing Co., Inc.
SemBleu: A robust metric for AMR parsing evaluation
. In
Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics
, pages
Florence, Italy
, and
AMR- to-text generation with synchronous node replacement grammar
. In
Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers)
, pages
Vancouver, Canada
, and
AMR-to-text generation as a traveling salesman problem
. In
Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing
, pages
Austin, Texas
, and
A graph-to-sequence model for AMR-to-text generation
. In
Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)
, pages
Melbourne, Australia
, and
CAMR at SemEval-2016 task 8: An extended transition-based AMR parser
. In
Proceedings of the 10th International Workshop on Semantic Evaluation (SemEval-2016)
, pages
San Diego, California
, and
Boosting transition-based AMR parsing with refined actions and auxiliary analyzers
. In
Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 2: Short Papers)
, pages
Beijing, China
, and
A short survey of recent advances in graph matching
. In
Proceedings of the 2016 ACM on International Conference on Multimedia Retrieval
, pages
, and
Benjamin Van
AMR parsing as sequence- to-graph transduction
. In
Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics
, pages
Florence, Italy
© 2020 Association for Computational Linguistics. Distributed under a CC-BY 4.0 license.
Association for Computational Linguistics. Distributed under a CC-BY 4.0 license.
This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited. For a full description of the license, please visit | {"url":"https://mitp.silverchair.com/tacl/article/doi/10.1162/tacl_a_00329/96472/AMR-Similarity-Metrics-from-Principles","timestamp":"2024-11-03T15:55:25Z","content_type":"text/html","content_length":"423318","record_id":"<urn:uuid:5f4577b5-b7ac-4914-aeb6-5677eab94c44>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00865.warc.gz"} |
KBi<sub>2-x</sub>Pb<sub>x</sub> (0 < x ≤ 1): A zintl phase evolving from a distortion of the cubic laves-phase structure
The quasibinary system KBi[2-x]Pb[x] has been investigated, both experimentally and theoretically. Phases with compositions 0 ≤ x ≤ 1.2 were synthesized and structurally characterized by X-ray
diffraction experiments. For low values of x (0 ≤ x < 0.6), KBi [2-x]Pb[x] adopts the cubic Laves-phase structure MgCu [2] (space group Fd3m), which contains a rigid framework of corner-condensed
symmetry-equivalent tetrahedra formed by randomly distributed Bi and Pb atoms. For compositions x ≥ 0.6, these tetrahedra become alternately elongated and contracted. The distortion of the framework
lowers the space-group symmetry to F43m (KBi[1.2]Pb[0.8], F43m, Z = 8, a = 9.572(1) Å). Magnetometer measurements show that KBi[2] (x = 0) is metallic and goes through a superconducting transition
below 3.5 K. First principles calculations reveal that the Fd3m → F43m distortion is largest for KBiPb (x = 1.0), which at the same time turns into a semiconductor. Thus, F43m KBiPb corresponds to a
proper charge-balanced Zintl phase, K ^+[BiPb]^-, with separated polyanionic tetrahedra, (Bi [2]Pb[2])^2-. However, it was not possible to prepare F43m KBiPb. Syntheses attempting to increase the Pb
content in KBi [2-x]Pb[x] above x = 0.8 yielded additional, not yet characterized, ternary phases.
Dive into the research topics of 'KBi[2-x]Pb[x] (0 < x ≤ 1): A zintl phase evolving from a distortion of the cubic laves-phase structure'. Together they form a unique fingerprint. | {"url":"https://portal.fis.tum.de/en/publications/kbisub2-xsubpbsubxsub-0-lt-x-1-a-zintl-phase-evolving-from-a-dist","timestamp":"2024-11-04T02:33:17Z","content_type":"text/html","content_length":"52958","record_id":"<urn:uuid:65d26a5a-e897-4c73-9ce0-b8aa2d464bad>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00533.warc.gz"} |
integer semidefinite programming
In the quadratic minimum spanning tree problem (QMSTP) one wants to find the minimizer of a quadratic function over all possible spanning trees of a graph. We give two formulations of the QMSTP as
mixed-integer semidefinite programs exploiting the algebraic connectivity of a graph. Based on these formulations, we derive a doubly nonnegative relaxation for … Read more
The Chvátal-Gomory Procedure for Integer SDPs with Applications in Combinatorial Optimization
In this paper we study the well-known Chvátal-Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by
iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of … Read more | {"url":"https://optimization-online.org/tag/integer-semidefinite-programming/","timestamp":"2024-11-08T07:15:38Z","content_type":"text/html","content_length":"86902","record_id":"<urn:uuid:92d5326b-1abd-4e03-b390-9d4f3afbaf6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00251.warc.gz"} |
Basic concept of algebra
Google visitors found our website today by entering these keywords :
│GRE notes on arithmetic section(percent sums) in maths │college algebra solver │cost accounting tutorial exercise │
│glencoe-algebra 1 answers │practise science tests yr 10 │florida algebra help │
│formula for decimal to fraction │mathmatic signs symbols │advanced opperations math worksheets │
│free worksheets with power point to go with them │FORMULA FOR PERCENTAGE │polynomials in life │
│free answers to algebra 2 problems │"linear algebra cheat sheet │ti 83 plus square root │
│steps required to simplify a radical algebra │quadratic trinomials lesson plan │i dont understand algebra help │
│least common multiple activities │download kumon printouts │example general solution to second order nonhomogeneous differential │
│ │ │equations │
│multiple choice questions and answers in cost accounting │calculate the least common multiple prime factorization│teaching coordinates ks2 │
│prentice hall algebra 1 california edition practice sheet │dividing radicals fraction │free download mathematic tutorial book │
│learn math yr 8 │exponent formula │easy alegbra │
│rational expressions solutions │math test printout │learn algebra online │
│balancing chemical equations using a chart │algebra the easy way │algebra sums │
│exponent square root calculator │what are the algebra III trivia │cost accounting make and buy free tutoring │
│algebra brackets worksheets │download Aptitude Papers │factorization parabola │
│PARTIAL SUMS │how to convert mix fractions │solving equations containing integers │
│convert decimal to integer java │TI-83 Plus programming quadratic formula by hand │algebric graphing │
│soving equations with root symbol │surds explanation │ti-83 plus accounting download │
│find range of quadratic equation │free intermediate alegebra help │year 9 algebra worksheets │
│free pre algebra worksheets to print │TI-89 solve single differential equation │grade 10 math samples for ontario │
│BigDecimal int convert java │least commom multiples │linear programming word problems │
│Yr11 Past exam paper │download ks3 science questions │concepts of quadratics in maths │
│common multiple for 17 and 18 │boolean expression simplifier program │simulation physics.swf │
│GMAT Chart and Graphic questions │CALCULAS EQUATIONS │math equations/help │
│solving by method of joints for trusses │ │algebra homework helper │
│EXAMPLE OF MATH TRIVIA QUESTIONS │addison-wesley chemistry book answers │Square Root Rules │
│chapter test in permutation and combination │fractional exponent │kumon maths worksheets │
│simple kids mathamatics games │application of quadratic equation graph │basic eqations algerbra │
│Lcd of each set of rational expressions │free maths gcse learning guide pdf │setting up TI 83 for factoring │
│aptitude test sample paper │parabolas basic │boolen algebra examples │
│download maths books for gmat │Glencoe algebra 1 answer key │"english story" free download │
│Crossword Questions For 3rd Graders │Algebra Manipulative │radical expression solver │
│rudin mathematical analysis solution │step-by-step solving complex roots calculator │math trivia │
│adding and subtracting 9,and 11 work sheets │exponents on calulators │making algebra easy │
│t83 calculator picture │permutations on a TI89 │definite integrals using by parts formula │
│eleven plus english papers free downloads │free printable quiz grade 7 physics │A Level question papers quizes for maths │
│ti 84 summations │greatest common divisor calculation │online equation factorization │
│"free statistics books" │"Equations math worksheets" │basic circuits math problems │
│"how to calculate square root" │how to solve linear programing with the ti-83 │permutation and combination basics │
│free math work sheet for 5th grade │Herstein Sylow solutions │UCSMP geometry book │
│finding the greatest absolute value functions │powerpoint presentations for prentice hall algebra 1 │consecutive integers online calculator │
│writing a decimal as a fraction │chemistry highschool pdf free book │integer worksheet with answers │
│mcdougal littell algebra 1 help │anwsers to grade 7 extra practice 5 │MATHMATIC SYMBOLS │
│TI-83 plus ROM download │casio fx83 emulator │chemical equations formation of ozone │
│EMULATOR T1-84 PLUS │grade nine algebra exam │third order quadratic equation │
│Free Radical Equation Solver │arithmatic sequence real life example │penmanship worksheets for sixth grade │
│trigonometric formulaes │7th grade algebra tutorials online │Online Graphing Calculator domain │
│fifth grade algebra, beginners │teachers biology printable worksheets with answers │variable exponents │
│adding and subtracting polynomials work sheets │long division work sheet pdf │gcf finder │
│notes on basic mathematics /statistics │Quotes about math rhyme numbers │simplify square root of 14 │
│Quadratic Equations Completing Square │Algebra Problem Solver │greatest common factor and least common multiple worksheets │
│Algebra Solver solves inequalities, showing all the steps │free online algebra calculator "free online algebra │free pre-algrebra help │
│ │calculator" │ │
│uk as level reciprocal function sketching curves resources interactive │matlab solve linear differential equation │solving inequalities lesson plan │
│resources │ │ │
│dummit foote solution │finding greatest common divisors │quadratic polynomial method in java │
│basic formulas in integration in maths │tI-84 plus factor9 │modeling fractions using pictures │
│kids typing pratice │POLYNOMIALS fraction square calculator │ADDINg and subtracting 4 digit number worksheets │
│solutions for shai simonson strings │finding absolute value of a variable in c │mathcad download free │
│sample simple algebra test │"free test papers" │changing decimal to fraction calculator │
│ONLINE COLLEGE ALGEBRA SOLVER │O'Level Elementary Mathematics Transformation │yr 8 english study guide │
│ontario math homework help grade 10 │free cost accounting 11th edition test bank │pre-algebra variables lesson plan │
│maths is fun .com simplification, average exercices │"fractions worksheets" and harcourt │solving formulas practice test │
│algebrator │equation calculator simplify │holt physics answers │
│sine rule - mathematics - GCSE worksheet │Codes for Ti-83 Calculator Games │algebra 2 and trigonometry book answer key │
│maths worksheets yr 6 │Online math solvers for problems about distance, rate, │procedure solving first-order linear equtions │
│ │and time │ │
│pre-algebra worksheet │integer worksheets │iowa algebra aptitude test sample questions │
│"aptitude test paper" │algebra practice problems 7th grade │"pocket pc" "calculator emulator" integrals │
│trig answers │"Gmat test+free download" │program to solve polynomial equations │
│yr 8 maths │abstract algabra material │matlab simultaneous linear equation │
│variable practice sheets │DOWNLOAD PEARSON LINEAR ALGEBRA SOLUTIONS FOR FREE │free tutorials on mathcad 7 │
│TI-82 log │ti 83 texas instruments calculator rom image │whole divide decimal calculator │
│"strategies for problem solving workbook third edition answers" │abs methods │importance of algebra │
│math test ks3 │worksheet collecting like terms │free 11+ exam pass papers online to print │
│cheating for algebra │solving basic equations practice sheets │solve my algebra questions for FREE │
│walter rudin download │charting ordered pairs 3rd gradelesson plans │matric maths exercises │
│Free Maths typing softwares │COPY OF GMAT ENGLISH PRACTISE PAPER │"transforming equations" algebra │
│dividing equations calculator │solving simultaneous equation using MATLAB │11+ EXAM PAPERS │
│free printable maths worksheets y8 │making poem by using algebra term │steps to factor by difference of two squares │
│"completing the square" + "word problems" │convolution TI 89 │matlab permutations and combinations │
│"changing log base ti 83" │Decimals converted into Fractions │excel and free tutoring in statistic │
│how to solve root 6 root 6 root 6 maths question │What Is Evaluation of an Algebra Expression? │artin word problem │
│6th grade printouts for free │maths for dummies │third root of │
│mixed radical operation worksheet │first in math cheats │TI 83 for factoring │
│qudratic solution │Reflection KS2 maths │"GRE MATH" "Decimals to fractions" │
│free simultaneous equation worksheet- mathematics │how to breaking 3 letter cipher world war codes │download practice english comprehension papers KS2 │
│calculator tricks for algebra │TI83plus+programma+Cardano │online yr 7 maths problems │
│examples of trivia about math │steps to solving absolute value │free printable worksheets addition of integers │
│english aptitude │ti-83 rom download │Using TI-83 Calculator to Factor Polynomials │
│algebra 2 honors book │year nine trigonometry │9th grade algebra help │
│maths quizzes for class ninth │homework solver │pre algebra holt answers │
│teach me the reverse of foil/ algebra │completing the square powerpoint │"transforming formulas" │
│examples of mathematical POEMS │math cheat │barron's gre free download │
│sine rule - free mathematics worksheet │fraction formula │maths tutorial yr8 │
│if the greatest common factor of a and b is a then what can you say │mathblaster │algebra worksheets and quizes │
│about a and b │ │ │
│factoring complex numbers │adding rational fractions calculator │lesson plan for fifth grade graphing equations │
Bing visitors came to this page yesterday by entering these math terms :
Rudin, analysis, solutions, ontario grade 6 online math book, coefficient of variation formula maths gcse.
"linear programing algorithm", how to solve polynomial functions, trigonometric practices.
Free algebra II factoring worksheets, matlab simultaneous equation solving, 3rd grade math teaching factorial trees .
Sample quadratic polynomial method in java, Printable Yr 7 Algebraic practise sheet, Kumon Math Answer Book Level G.
How to do algerbra, sample lesson plan in solving and graphing inequalities in one variable, basic algabra, step by step math problem solver, UOP ALEKS + statistics + answers.
Calculate hyperbola, generating graphs algebra 2, Holt algebra online book, solving multiple equation matlab, free aptitude test accounting, free pre algebra test.
Free download ti 84 plus family games, linear algebra john fraleigh e-book third edition, how to work out a qudratic equation using the formula method, algerbra, answers to cognitive tutor algebra 2,
free math software for 9th grade, fun way to teach lcm.
Simplified radical form, Solving for a specified Variable, pre algebra worksheet for 5th grade, adding and subtracting substitution games.
"complex numbers" mathamatics, algebra math trivia, combinations calculator download, texas instruments TI 83 cubic equation solving, fraction word problems work sheets, aptitude questions paper.
Quadratic curves alegbra maths, what are the algebra trivia, synthetic division TI89, TI-83 CALCULATOR STEPS, decimals,fraction, percent, ratio chart, free calculator.
Exponants and polynominals, t183 online calculator, ratio algebra problems, Q, complex root factorization ti-83, KS3 SAT Exam.
Examples of math trivia students, subtracting fractions worksheet, applications of algebra, lattice-method worksheets 4th grade.
Solving algebraic equation; fractions and ln, algebra and trigonometry structure and method book 2 worksheets, "cost accounting" ebook, maths tests-equations, dividing rational expression solver.
Maths yr 8, polar areas calculas, download mathematics examination papers grade 6, WWW.RESOLVE PROBLEMS IN MATH.COM.
Programing a ti 83 plus manually, finding least common denominator, 8th grade proportion worksheets, calculus with exponential rational equtions, free college ALGEBRA SOFTWARE, algebrator trial,
liner equation.
Abstract algabra, grade 9 integer worksheets, Mastering MATLAB 6 ebook, holt physics answer test, simplifying algebraic expressions example, how to study grammer.
Tutor of discrete structure complete, printouts on math for free, pizzazz pre algebra answers, how to change fraction to decimal with TI-89, free ebook on Quantative Aptitude for CAT exam.
FREE BOOKS OF ACCOUNTING PROBLEM WITH SOLUTION, math problem solver free, system of simultaneous equatins by matrices.
Basic algebra for beginners, free cube roots online calculator, learn algebra easy, o level maths revision papers.
Alebraic expression, science exam papers for grade 10, factoring equations calculator.
Cpm math answers, complex root solver, Iowa grade 8 Mathmatical testing sample, factorise polynomials program, mathmatic symbols, Test & Exam Questions on Indices & Logarithms in Additional
Mathematics, integers worksheet.
Mathcad 12 download, hyperbola quadratic parabola, algebra answers "STructure and Method Book 1" Mcdougal Littell.
Solving Square roots, download ti-85 rom image, aaamath expanded notation.
Quize define integrals mit univercity of usa, investigatory project in math, solving radicals, multiple choice maths class 8, polynomial solver ti 83 plus complex, matlab + nonlinear simultaneous
equation, pre-algebra with pizzazz.
Graphing inequalities online, solving nonhomogeneous differential equations, how to solve operations with functions, gcd calculater, downloadable chemical equations for casio calculator.
Online step-by-step find complex root, percentage equations, sats "test online", Solving integers.
Rudin solutions chapter 4, "algebra font", simplify variable exponents, lesson about using substitution in prealgebra.
Fraction convertion, converting mixed numbers to decimals, intermediate alebra, substitution method solver, third degree equation online solver.
Soving equations with root sign, Maths Homework Help- Practical Number Patterns(Linear), the difference between evaluation and simplification of an expression, quadratic polynomial java method, free
tutorial + L.C.M., Ti-89 square root Algorithm.
Calculate slope percent degrees grade, mathmatic interest word problems, algabra coordinates, unknown exponential gcse, ratio simplifier.
Simultaneous equations for 13 yr old, grade 10 physics worksheets, yr 9 maths quiz, solving problems with a different base number on ti-83, excel algebra.
"fourth grade word problems", A program to calculate gcd, What word applies to an algebraic expression with 3 terms?, statistic and teacher solution edition.
Evaluating equations worksheet, mathematics 6 grade free ebook, 6th grade multiplying and dividing fractions worksheets, dividing polynomials solver, factoring two variable polynomials, TI Graphing
calculator ROM download.
Test & Exam Questions on Logarithms & Exponential Functions in Additional Mathematics, artin solutions, 6th grade algebra & exponents test.
Factoring solver, java calculator that solves integer exponents solver, How to solve imaginary units in algebra., how to work out square root on calculator.
"Algebra help", math power answer key for seven area of figures, free monomial worksheets, GCD C#.
Albegra word software, high school algerbra software, ti-83 program complex factors, algebra problems worked, Fractional Equations/10th grade, factor quadratic equations calculator, intermediate
Power algebra, do yr 8 maths, online maths algebra questions for year 7.
Use "Texas Instruments" calculator determine logarithms, anton free solution exercises "linear algebra", aptitude question, printable multiplication and division sheet fifth grade.
Rudin solutions ''chapter 3, problem 17", Lowest Common Denominator algebra, binomial coefficient ti-89, algebra with pizzazz, examples of mathematics poems, algebra hard questions.
Gcf printable worksheets, get manual program to plug in midpoint formula to TI-83, cubed roots, "third root".
(lesson plan for square root 6th grade), trigonometry for idiots, free online algebra 2 tutoring, Free Downloadable KS3 SAT Exam.
Algebra poems, Biology The Dynamics of Life test cheats, coordinate graphing secret pictures.
Algabra for beginners, Math trivia, the order of soving equations with root sign, lattice-method worksheets everyday mathematics, calculator to multiply polynomials, trivia on quadratic functions.
Ebook fluid mechanics solution free, 10 th matric maths important 5 mark sums, books on cost accounting, learn year 9 algebra, ode45 2nd order nonhomogeneous differential equations, simultaneous non
linear equations Mathematica, test for perfect square on calculator.
Online chemistry equation solver, practice decimal math sheets for sixth grade, algebra practise high school, "Marilyn Dalrymple", Quadratic fraction Equations calculator.
Multiplication / permutation problems(maths grade 12), Algebra 2: An Integrated Approach, free algebra 2 help, multiplying variables with fractional exponents.
Solution manual of fundamental mechanics of fluids, mathimatical signs, aleks accounting+answers, iowa algebra aptitude test, aptitude questions maths,english, online worksheets finite mathematics.
KS3 maths SATs revision papers, hard algebra quizzes online, derivative worksheet, gmat free question papers of mathematics.
Factoring polynomial calculator, "integrated algebra" NY practice fun, download absolute free ebooks for computer study, How to factor square roots or radical expressions.
Solving quadratics on TI-84, Fluid mechanics tests' lists to download, animation of parabola made in flashMX, how to ti-83 plus summation.
Simplifying Square roots, GRE combination, Mathmatic symbols.
Graphing ellipses, kumon solution book, printout of math formulas, learn calculas.
Integer exponents solver, solving quadratic equations by completing the square, online synthetic calculator, gcm lcd calculator, java cube calc.
Free algebra answer SOLVER, Y9 exam sample paper, mathematic formula to calculate percentages, maths help binomial formula, algebra solving linear equaitons with variables on both sides negative, how
to divide a fraction.
Find the absolute value of the number calculator, Algebrator Full, free Fifth grade printouts.
Solving equations in 3 variables, algebra 2 for dummies, gmat maths quadratic quation.
Cross product solver, GCSE algebra worksheets, ti-84 composite function how, ti-83 plus worksheet, what are the algebraic trivia, download grade 10 maths algebra programs.
Printout worksheet activities for 1st grade, lowest commom multiple of a faction, pre-algebra equations.
Bitesize cubic equations, Algebra 2: Examples of Algebraic Methods, downloadable calculator, java computer code for compound interest, the best algebra books, Problem Solving Addition and subtraction
of Fractions.
How to college square root equations, year 8 ks3 papers maths, solve nonlinear algebraic expressions, elementary math trivia, free online algebra calculator, difference between evaluation and
simplification of an expression, adding fractions worksheet.
Simplifying radicals calculator, quadratic factor calculator, factoring lesson .ppt, hyperbola graph, converting decimals to radical, algebra artin.
Mathematics problems in C language, simplifying rational expressions calculator, circle theorem free worksheet, aptitude question and answers, relating graphs to events, free maths papers for age
8-9, trigonometry grade 11 (free downloads).
TI-83 help math solve, TI84 instructions + pythagorean, Absolute Value Quadratic Equations, tricks to learning multipication tables, Trigonometry For Dummies download, ALGEBRA 2 PROBLEMS, linear
equation worksheets algebra tiles.
Algebra Fonts, math for dummies, multiplying polynomials for learning disabled kids, 4161647603849, formulaes of trigonometry, a-level past year papers solutions.
Ti ROM, distributive properties for elementary children worksheets, practise year 10 math papers, practice decimal math tests for sixth grade.
Mathematics aptitude questions+pdf, transform literal equations lesson plans, calculator linear equations, download mathmetical, free ebook for aptitude, Online Saxon Physics Solutions Manual,
factoring quadratic equations ppt.
Balancing chemical equations tutorial, intermediate algebra free, apps second grade equation ti-84, cat examination questions & answeres.
GRE mathematics Notes, division calculator with answers freeware, one step equations with integers worksheets, free online books about pre-algebra, online complex root calculator, hardest problem
Simplify radical expressions, finding the LCD math, Rudin chapter 4 solution, How to use a graphing calculator T1 83/84.
Free easy tutorial math grade 12, texas instrument quiz papers, quadratic and other nonlinear inequalities, combining algebraic expressions factoring.
Logarithms for dummies, commutative property for 8th grade, parent websites, adding and subtract unlike numbers, how to solve algebraic formulas, algabra theory, polynomial expressions calculator.
Examples linear equations, download free ti-84 plus calculator games, math trivia question with answer, how to do SIN equations on EXCEL, online inequality graphing calculator, Download TI-84 plus
games, ebook intermediate accounting.
Free online systems of equations solver, ti calculator roms, FREE ONLINE MATRIC BIOLOGY STUDY GUIDES, free third grade iq test, ks4 history flowcharts, free worksheets + maths +science +english, free
algebra for dummies.
Linear programming GCSE, java number counter formula, java online exam, algebra homework, online trigonometry equation solver.
Answers to mcdougal littell middle, multiply matrix algebra calculator, percentage online solver, prealgerbra, grade 3 vocabulary practise sheets, MATHS YEAR 8 WORK SHEETS.
Simplified radical form ti-82, scott foresman 5th grade Exam View Test Bank CD, how to do log on ti89, grid lesson plans first grade, free factoring solver, aptitude+test+questions+answers.
Maths Algebra Yr 8, solve equation for variable algebra fractions, math solver pocketpc, synthetic division square root, printouts of simple algebra equations games and puzzles, FREE DOWNLOAD
APTITUDE, math worksheet integers adding and multiplying and dividing and subtracting.
Exponent solver, "relating graphs to events" + "powerpoints", algebraic factoring practices, solving polynomial equations using least square method.
TI-83 plus "download ROM", spss pattern matrix factor analysis, Mcdougal Littell algebra 1 answer key, kumon answer books, TI-84 Calculator games, factoring polynomials+terms with decimals.
Rationalizing the denominator worksheet, math trivia, TI-82 , how to find 4th root, Order of opperations worksheets, why was algebra invented, practice exam questions for yr 9 online, printable
worksheet equivalent fractions 6th grade.
Online maths exam year 10, "equation for word", java code convert time, McDougal and Littell Algebra and Trigonometry book2, example of math trivia questions.
Non-linear difference equations in two variables, ppt + quizs, Algebra factoring practice worksheets, explanation of quadratic formulas, mathamatics, teachers websites with answers for pre algebra ph
Elipse graph, free math tutor 6 grader, printable math sheets with answer keys, Polynomials Multiple + c++.
Algebra worksheets printouts, kumon answer book, poems about chemical equation, Algebra I the importance of, math poems, how to solve third degree equation, complex roots equation solver.
Automatic radical equations, how to you find the slope in a nonlinear equation?, mac algebra program demo, pre algebra with pizzazz, adding subtracting multiplying dividing decimals worksheet,
multiplying adding subtracting integers, TI 83 plus factoring.
Ontario grade 5 math free worksheets on graphing, algebra samples, multiply polynomials calculator.
Ti 89 calculating radicals, help tutor young students, aptitude questions in c, rules of algebra explain, equations and order of operation.
Triginometry, solving simultaneous differential equation, worksheet math writing expressions, integrated math 2 answers, printable maths sheets yr 8.
Algebra freeware, Dummit Foote solutions, free algebra help with conjugate, algebra explained, program to find square root.
Download freeware TI-86 calculator, simplify fractions calculator, Pendulum Gcse higher, visual basic graph for equations, subtracting in egyptian, business math trivia.
C aptitude question And Answer, mathematics uses of matrices in everyday life, algebra solvers, Math Combinations.
How to learn squareroots, how do you factor with ti84 plus, complex root system of equations.
Ppt on rational expressions of class 10th, math algebra trivia question, multiplying square root.
Quadratic equations vertex calculator, conversion fraction decimal in binomial number, free online graphing calculators.
Mathmatical Conversion Table, advanced algebra, aptitude test questions & answers, online step-by-step find complex root calculator.
Adding integers, 6th grade print out worksheets, factorial java eulers number, answers to biology workbook (prentice hall), algebra problem answers, easy way to learn algebra.
Pretest sheets for REAL GED Science 2nd edition book, GED printable practice sheets and answers, :Add Frac. With unlike Denominators, teach me algebra.
Monomial calculator, Printable Kumon Answer Books, complete square +ti-89.
Ninth grade math worksheets, yr 7 maths exam quiz, printable algebra graph paper with numbers.
Sats year 9 chemistry test, "ERB" "math word problems", quadriatic equations with negative exponents, image conversion ti 83 plus, easy ways to factor trinomials, online free accounting practice
Free College Algebra Book, need a free step by step polynomial explanation, gragh paper.
Mcdougal littell inc. worksheets, free game downloads for TI-84 Plus, prentice hall mathematics answers, test cheats for the grade 9 math tests given in class, CAN YOU TEACH ME ALGEBRA?.
Order of operations free worksheet, edward penny+Differential+answer key, GCSE Maths revision work sheets, FREE sample SATS KS 3 PAPERS MATHS.
Hands on adding 3 digit numbers, ti 86 rom, Algebra Word Problems Samples, free online algebraic calculator, solving simultaneous equations using quadratics, problems and solutions in cost
Aptitude test, dilation,lesson plan, problem solving fractions.
Polynomial tutorial for 9th grade, fraction from decimal conversion java program, calculas how to calculate maximum volume, real-life situations of of trigonomic ratios.
Eight Grade Algebra 1 Textbook, adding, subtracting, multiplying, and dividing positive and negative integers worksheet, graphs of polar coordinates with square roots, prime numbers program for the
ti-89, help solving a quadratic equation through factoring, free algebra calculator, evaluating expressions worksheet.
Simplify radical equations, 6th grade math fractions activities, excel square route function, roots order of operations worksheet, dummit algebra, investment bank + numeracy test + practic, upload
pictures to ti 89.
Explain LCM, clep college mathematics tips, trinomial factorise equations, ks2 maths exercises, quadratic relations-help in grade 10 math, simplifying exponents, lcd on ti89.
Free printable college algebra, Intermediate Algebra cheat program, cube root function on a TI-83 Plus Calculator.
Real life examples of linear equations, free online system of equation calculator, pocket-pc ti-calculator, inequalities graph calculator online, linear combination method, MATHEMATIC TRIVIA.
Maths worksheets for year nine, how to solve quadratic inqualities of the form in which the variable is in the middle of two inequalities, casio algebra modular arithmetic, 2nd grade decompose
worksheet, help with roots, radicals, and root functions.
Evaluated Expressions and Graphing Equations, exercises of rudin, cubed root ti-80, online square root calculator, algebra problems solutions, laplace cheat sheet.
Worcksheets, What degree polynomial is a quadractic?, easy tips to do algebra, algebra 1 chapter 2 worksheets.
Polynomial equation +c code, java program for quadratic polynomial, free download eureka solver, Permutation Combination Tutorial, source code for dividing polynomial by polynomial, maths tests
printouts, 6th grade homeschool printables.
Algebra 2 computations math help, basic algebra exam year 9, algebra problem solver, real life examples of geometric sequence, leanear functions, direct and indirect variation squared, cubed fourth
Test bank prentice hall chemistry, hardest mathematical equation, aleks.phoenix, how do you solve quadratic equations with TI-83, how to write equation of vertex form.
"engineering dynamics homework solutions", factoring with ti84 plus, Where can I find answers on this math problem about dividing mix numbers?, Printable Science Module 10 Test Papers for GCSE, math
test online/for 5th grade, math trivia questions, maths to the power of mutiplying.
Matric maths exam answers, Absolute Value Worksheet with Solutions, Holt Algebra 2 Adding and Subtracting rational expressions Practice B, Texas Instruments T1-86.
Sample problems, 098 college level math, least common multiple of two expressions, cost accounting free book, "advanced maths questions".
Worksheets radicals algebra, hyperbola grapher, applikation download ti 84 plus, example problems of progression college algebra.
Complex fraction calculator, easy algebra program, answers for the book prentice hall mathematics algebra 1, gr 9 square roots practise test, Kumon answer .
Prentice hall algebra 1 teachers addition, math one step worksheet, Free Online Algebra Solver, java code + convert character to integer, condensing logarithmic expressions.
Permutation recursion c#, simple alegbra, solving equations classroom activities, solving non linear equations MATLAB.
Rules for exponets, download aptitude questions, how to solve equation third order, solving cubics on TI 83 plus, euler method in TI84.
GCF Finder, matlab system of first order ode, mathematics trivia with answers, boolean algabra, factorization with quadratic formula.
Graph linear equation using calculator, it 83 quadratic, prentice hall conceptual physics answer key, free trig calculator download,, probability exercises exam papers, "math symbols" equasions.
Simplify radicals calculator, math factor calculator, quadratic factoring calculator, cheating algebar, guess work used in finding out the integer sqare root of an integer by long divison method,
worksheets of decimal no..
Solve the following by completing the square, compound inequality solver, "introduction to probability models" homework ch 7.
"sample algebra problems", Grade Nine math WORKSHEETS, casio fx 115ms guide, practicing worksheets on scales, free 11+ exam papers.
Algebra 1 answers, least common denominator worksheet, "prentice hall" quiz "connections to today", Polynomials + c++, Eigenvectors + rotation of quadric surfaces + maple, algebra simplifier,
rational expression calculator.
Absolute value worksheet, how to solve simultaneous equations involving logarithms for a levels, math tutor usa, printable math quizzes 1st grade, advance algebra poems, maple examples nonlinear,
prentice hall Algebra 2 with Trigonometry review.
Exponents and radicals for beginners, free maths calculus equation solver, permutation solved answers, pre-algebra percentage worksheet, trivia about bussiness math questions, free calculas problem.
Math scale factor, TI-85 calculator rom, math squre feet.
Cheat sheets for holt physics, simplifying radical equations, quadratic equations solve radical equation, radical expressions worksheets, simultaneous equation solver coefficients, decimals,fraction,
percent, ratio chart, on-line calculator.
Third grade algebra worksheets, rational expressions on ti 89 titanium, aptitude questions with answer.
Mathmatical combination calculation, Sums and Differences of Rational Algebraic Expressions, free mathamatics for year 6, TRIGONOMETRY FORMULAS TIPS, T183, calculator programs, free ged practise
tests, distributive properties for idiots.
Casio algebra fx2 programme download, books of cost accounting, multiple variable equation solver, Difference of Two Squares, Pre-algebra with Pizzazz answers.
Palindrometester + punctuation, 11+ maths practice papers free download, need cost accounting tutor, adding exponets, online algebraic calculator, "algebra 2" textbook linear equations, scale math
Free online tests on arithmetic for beginners, factoring calculator, how to write a linear equation from a graph, greatest common factor program, Mcgraw hill math worksheets for second grade.
Algebra square root polynomial, algebra answers denominators and zero free, "math worksheets for third graders", saxon cheats algebra 1, alpha cubed minus beta cubed as the roots of quadratics.
Primary school work sequences free, free usage of the ti-89 online, Sample Math Trivia, input output algebra lesson plans for elementary, printable worksheet equivalent fractions 6th garde, algebra
fractions examples equations.
Sixth grade test swf, mental aptitude tests test papers, easy printable algebra and maths equations, simplifying radicals in a quadratic equation.
Pre algebra simple interest find missing value worksheet, how to program games on a TI-84 Plus, graphing calculator download.
Sample algebra test, TI-84 Plus Silver Edition + Root locus, vertex form, free aptitude questions with answers.
Online maths games for 11 yr olds, arithematic, quadratic equation long division method, "learn algebra" +online, grade 11 functions textbook.pdf/ online, ti-83 rom code.
Mental aptitude question papers for children, Ti 84 downloads + "solving inequalities", fomula for volume calculation.
Cube problems aptitude questions, show java code dividing numbers ppt, factorising made easy, relational algebra font download, completing the square worksheet.
Partial fractions using java, on-line Math Tutors + Canada, find probability in ti-81, practise maths papers KS3.
Math power ontario book answer key for seven area of figures, online maths test KS3, algerbra math quizes, "Quadratic Equation" British Method", 10 th matric model question paper, "assessment test"
+"relational Algebra".
Sample math test circumference, sum and difference formulae-trigonometry, aptitude question bank, Math Trivia Linear equations, mathmatical interpolation, "factor polynomial" "variable exponents",
algebra substitution free worksheets.
Application of permutation in the real life, Powerpoint on Pre-Algebra, mastering physic answers, maths sheets for yr 5.
Ti-83 partial-fractions, Applications of trigonometry in our daily life, yr 9 maths exercises, math formulaes.
Algebra percent solution function, GRE notes on maths, algebra worksheets.
Free online graphing calculator, printables and worksheets on root words, sample math trivia, meaning of exponential expression.
Gcse factoring completely, pre algebra combining like terms, ks3 math questions, "mathcad 12 download", functions tutorials+evaluation+simplifying, online calculator sq root, equavalent fractions.
Algebra problem, free alegbra basic example, ti-84 mod manual, square root formula rules.
Algebra, Substitution worksheet, factor 9 t183 graphics download, complex fractoins, algebraic graphs hyperbola, algebra 2-hyperbolas, free prealgebra lessons online, online graphing calculator with
2nd button.
Trig calculator, linear systems solved by substitution worksheets, 11+ practise questions <mathmatics>, 4th grade algebraic expressions.
Second order differential equations matlab, descending order worksheet, TRIVIA ABOUT MATHEMATICS, grade 7 math quizz.
Solve cubics on TI 83, printable gragh paper, online graphing calculator "polar coordinates".
Free online interactive practice with equations for 9th grade, printouts for grade eight, sample practice tests in junior algebra, integer worksheet add subtract multiply divide.
Factoring quadratic calculator program, simplify equation, find the greast common factor for each set of terms, newton's divided difference formula maple, solve simultaneous equation matlab, scale
factor problems.
LEARNING Algebra, "simultaneous equation solver" online, free=calculator+to+get+the+circumference+of+a+circle, Contemporary Precalculus: A Graphing Approach - homework help - tutorial, free online
calculator- multiplying polynomials, prentice hall mathematics algebra 1 answers free.
Online graphing, sample 11 plus verbal reasoning test download worksheet, percentage formulas, how to simplify a radical.
10th grade online amth games, 6 grade math textbook free download, math online demo, expanding and simplifying radicals.
Sample 11 plus verbal reasoning test free download worksheet, dividing fractions cowboy and horse riddle, mathematical modelling fluid mechanics .ppt .pdf, mathamatics sums, 6th grade algebra &
exponents quiz.
Lowest commom multiple, alegerbra, pre algebra worksheet, practice pre algebra proportions, how to teach arithematic in mathematics.
Order the numbers from least to greatest online, algebra 1 equations worksheet, Colleg algebra solver, trivia about math symbols, maths test papers gr vii, trivia like word problems, Hyperbolas in
Everyday Life.
Year 9 sat exam paper for english, hardest clep math, math eqations for kids, free printable prealgebra worksheets, Investigatory project in maths, past english ks3 sat papers - bbc.
Percentage discount maths formulas, algebra factoring, learning foil method algrebra, how to divide radicals.
Entering determinants of t-89, "TI-86 Tutorial", answers to mcdougal littell middle school, adding worksheet.
Calculas, easy way to study maths, free graphing linear equalities calculator.
Dividing decimal worksheets, free trial on algebra solver, Calculator Manual TI 89 How to Factorising, elementary algebra for dummies.
Syllabus for advance algebra and trigonometry, simple machine problems and equations, formula circumferance of circle, Algebra for college students,mark dugopolski, multiplying radical expression
solvers, free sats exams paper, test papers for mental aptitude.
P.D.F Trigonometry Tutorial, ONLINE SQUARE ROOT CALCULATOR, newton's method and calculas, pre algbra, solving systems of equations involving quadratics, substitution method algebra.
Solving exponent equations, teacher version of ph pre algebra textbook answers to all math problems in the book, maple worksheet download finite fields.
Sample papers of class 9, High Exponents answers, nonlinear slope formula equation, java log base 10.
Examples of age problems in linear equalities, difficult algebraic problems, math passport to algebra book 1answers, ti84 logarithm.
Ti84 logarithm base 2 program, free revision simplification for year 8, solving quadratic equations for dummies, pc algebra calculator factorise, math trivia quiz and answer, mathamatics formulas
year 4.
Printable free worksheet grammer grade3, Algebra with Pizzazz solvers, online calculator Log2, what are the essential points of bohrs model of the atom.
"equation of a cube", BASIC ALGEBRA OBJECTIVE QUESTION, C# LCM, examples of mathematics trivia.
Creating pictures by graphing points, guess work in finding out integer square root of an integer by long divison method, TENTH MATRIC MATHS.
Free online calculators that have factorials, aptitude question papers, combination and permutation problems and answers, exponent by multiplication worksheets, algebra 2problems, equations ti-84.
Saxon Physics Solutions manual sample, math powerpoints on percents, practicing b s t test mathmatic, free alegbra, find algebra formulas of standard ninth, worksheets for yr 8 maths students, "ti89
Balancing acid chemical equation, problem solving products and quotients of decimals, Arithmetic and algebraic expression.
Artin algebra solutions chapter 1, online yr 9 practise SATs paper, holt algebra 1 texas, examples of graphing quadratic equations, how to write a percent as a fraction or a decimal, past english
final matric papers.
Mental aptitude tests sample papers, Online McGraw Hill Algebra 1 textbook, dividing/multiplying.
Free science tutorials 4th 8th, math worksheet answers logical reasoning, history of liear function.
Fraction poem, "math worksheets- factorization", 6th grade math software, fractions into decimals calculator, Linear Graphs Formulas Year 9, linear differential equation calculator.
"how do you take the square root", Textbooks on Algebra by Artin, instructions on how to use texas instruments TI-83 plus, 7th grade equations.
Sample of aptitude test paper, answers for prentice hall algebra 2, formula for adding integers, What is the greatest common factor of 99 and 40?, matlab solve algebraic equation, Problem solvings in
Physics and answers.
Free downloads aptitude tests, answers prentice hall algebra 2, free math worksheets "first grade", take sats papers online free.
Algebra help lcd solver, GCSE Geometry ppt, mathamatical symbols in algabra.
Solving equations using algebra tiles, simple math poems, Laplacian Growth Model applet, writing equations using rates for pre-algebra, division of radical expressions, evaluate quadratic polynomial
in java.
Free online calculator for multiplying polynomials, algebra with pizzazz answers, free gre exam and solution.
Inequality worksheets, kumon answer, free math worksheets for 3rd graders, multiplying polynomials for dummies, finding the highest common facor, inequality worksheet, math quize.
Java code for finding permutations, download kumon maths worksheets, image to ti calculate.
Ti-89 complex analysis program, lowest common denominator of 128 and 48, quadratic root solver, simplifying rational expression crossword puzzle, holt science and technology+printable directed
reading worksheets+chapter 20+8th grade.
Factor 9 t183, differences between logarithmic sheet and polar graph sheet, practice year 11 exams papers/maths, free software matlab6 exe.
KIDS WORK SHEETS english 9, free least common multiple calculator, mathequation download, algebraic division by quadratic, TI-83 summations, easy learn logarithms.
Factoring trinomials review ppt, gmat model papers, 4th grade algebraic expressions worksheets, how to find 4th root on a calculator.
Steps required to simplify a radical., free science games (Yr 10), lineal metre, mathematics rudin, maths exam year 10 practise, how do you divide, Free online KS3 SAT Exam.
Simplifying radical expressions tutorial, convert from vertex to standard form, excel angle calculator, how to copy video math tutor, example problem of progression in college algebra, Solving
simultaneous equations in Excel using solver, math promblems.
Factoring on the TI 83, factoring polynomials, 7th grade computer work sheet non printable, ks3 science sats practice papers to solve, glencoe test answers, equation calculator with fraction, book of
cost accounting.
Free past science ks3 sat papers, pictographs, printable, math, Rudin analysis solutions, pdf accountancy book, square root method, find vertex in standard form?.
Second order differential equation, algebra poem, solving radicals variables, roots of equations with three variables, adding and subtracting fractions worksheet, ti-89 bond, Chemical Equations
Illustrating the formation of ozone.
Printables on graphing grade 4 and 6, examples of simplying the expression for the perimeter of a figure, school worksheets cheats and answers, general aptitude questions with answers.
Factoring with complex number, mathmetical progressions, algebra answers book a ratio proportion and percent.
Second order homogeneous differential equation, 2nd grade work sheets, solving non-linear equation matlab, "differential equation" in terms of charge.
Printable work with answer keys for high school and math tests, Java Online Exam, permutation combination basics.
Maths quize, 11+ Practice tests KS2, subtracting ,dividing,adding,multiplying positive and negative integer, arithmetical progression formula excel, arabic exam papers "exam papers".
John hornsby university of orleans, free learn math yr 8, How to solve chemical equations, algebra.
GMAT model paper, +"real estate" +"free study guide" +"new jersey", t1-83 manual, math area printouts, math trivia with answers.
Exam papers in math, algebra 9 grade terms, grade five mental math work sheet, which is easier substitution method or combination method, Kumon answer book, We use the most advanced curriculum in
Standard and vertex forms, rotation matrices+maple, free college algebra help, free online math clep test practice, TI-84 derivative program code, subtracting cubed polynomials, TI 89 laplace.
Linear inequality "interactive games", algebra problem solvers, equation solver for radicals, Java reverse char palindromes without loop, iowa algebra aptitude test sample, What is the difference
between evaluation and simplification of an expression? math, c aptitude questions.
Exponent test cubed, TI-83 help problem solver, free ks3 sats papers, statistics equation sheet, solve ODE homogeneous solution, poems with different math words, Mcdougal Littell Passport workbook
Math Tutor Sample Business Cards, world history answers for mcdougal littell, free printable exponents worksheets.
Excel solver polynomial equations, download sats papers free, yr 7 maths quiz, algebra 2 standard form to vertex form, online factoring quadratic equations calculator, lcm solver.
Programing calculator download, Aleks answers university of phoenix, simultaneous non linear equations solver download.
Trivia about bussiness math, clep exam college algebra, Tutorial on simple factorising.
Radical calculator, Answer to Algebra Questions demo download, online algebra 2 tutoring, phoenix ti 83 plus game, grade 10 physics practise questions.
Algebraic ratios, dummit foote solutions, quadratic equation factor calculator, algebra 2 worksheets cheats, GCF and LCM game, saxon pre algebra 1/2 second edition test forms A and B, solving
non-linear equations matlab.
Ti 89 synthetic division, Ti-84 plus calculator quadratic equation, parabolic pde non-homogeneous, free downloadable algebra worksheets, greatest common factors developing skills in algebra answer
key, 11+ exam papers.
Solve algebra equations calculators free, solving nonlinear differential equation , maths gcse worksheets.
Precalculas and trig dealing with pie, Mathamatical puzzles, Trivia questions Physics Problem solvings.
Learn linear algerbra software, ks3 mathmatics, how to divide polynomials, coordinate plane for math class, Mcgraw Hill Math worksheet, artin algebra solutions.
Past question papers for SATS Yr 6, free download fluid mechanics ebooks, dividing fractions cowboy and horse, WIMS Function Calculator, convert negative fraction to a decimal;, linear equation in
two variables by substitution method.
"fraction font download", some problems on factorization for 8th standard, ged math test illinois plane, quadratic formula program ti 86.
Factorising quadratic equation, algebra homework solver\, combination matlab.
TI-83+ rom download, algebra dolciani structure and method "test bank", analytic algebra software, Cube roots in Real life application.
Quadratic Equations Graphing Calculator download, "irrational numbers" +"lesson plan*", Lyapunov exponent excel.
C aptitude questions, ti 84+ log app, free worksheets with brackets, gcse statistics past exam papers.
Mathematical combinations, how to use the discriminant in quadradic equation, solve for the variable, multiple variables, ebooks on aptitude, algebra 1 tutor download.
MATHTYPE FREE DOWNLOAD, free worksheets for year 10 chemistry, work out math problem online demo, solving quadratic online graphing calculator, online factoring calculator.
Probibility and statistics for beginners, pre algebra test free, online graphing calculator, ti-86, linear equation system algebra excel sample.
Plotting implicit functions using maple, simplifying a complex rational expression, free answers to math books, english maths science software download, algebra 2 answers.
Putting factor form into your TI-83, challenging mathematics aptitude questions, clep cheat, "Real analysis and foundations" teacher's guide, algebra trivia question, c- programming aptitude
Online factorer, 8th Grade Grammer, elementry mathmatics questions.
Solving Algebra software, DIFFERENTIAL CALCULAS FORMULA, square and cube roots, backgrounds on algebra, Ti-82 rom download, kumon homework samples, Mathematics Alegbra Practice Quiz Websites.
Grade nine math tests for slope, glencoe algebra workbooks, MATH TRIVIA OF QUADRATIC EQUATIONS WITH ANSWER, ti-89 log.
8th grade math printouts, fluid mechanics "solutions manual" download, download ti-83 rom, year 11 maths method exam paper, sequences pyramid puzzles, ks3, subtracting fractions w/ like denominators
Math trivia with answers mathematics, T1-84 calculator games, software algebra II solve any problem, expand algebra questions online, free online graphing calculators with square route.
Prentice hall resources pre-algebra, synthetic division calculator, math trivias, easy way of teaching proportions and ratios to fifth graders.
Polynomial code+java, log ti84 program, algebra simplifying radical expressions using multiplacation and division.
Free printable exponent worksheets, ti 84 emulator rom download, tricks for solving CAT exam, how do I find the perfect square trinomial whose first two terms are given?.
Online triangle angle finder, introduction to permutation and combination, intermediate algebra flash cards.
Polynomials with fraction exponents, math work sheet printables, free Iowa grade 8 Mathmatical test.
Can i put in a math problem an it be solved, adding equations, free online maths games for ks3 year 8, Teacher Edition Algebra Structure and Method Book 1 Richard G Brown, my students are not passing
saxon algebra tests, Teachers notes for Newton's laws for gr11.
Fluid mechanics for dummies, practice test "electrical calculations".pdf, ks3 free exam papers, saxon math answers.
Multiplying radical expression examples, mathematics 6 grade free download, work book soloution+algebra+hungerford+pdf, algebra equations.
Realestate coversion square meters to squares, quiz on scale factor math, easy college algebra software solver.
Solve 2nd order equation matlab, absolute value problems online, simplifying radicals simulation.
Simplify to standard form (6,-4) and the slope 1/2, algebra II trivia, study guide for your A-level physic, how to simplify fraction and decimal.
Linear Algerbra & its Applications, solution of abstract algebra 2 by John Fraleigh, algebra graph calculator c#, java aptitude questions, rudin AND math AND problem AND solve.
Mathimatical puzzles, free physics homework solutions, Kumon answers fractions, free printable pre algebra worksheets for 7th grade.
Math with pizzazz, mental aptitude test papers for children, free numeracy tests for 10 yr olds, "a first course in abstract algebra" homework solutions, Algebra 2 answers, convert fractions into
quadratic equation, bitesize mathe sequences.
Algebraic equations worksheets, algebra problems sets with summation series problems, divided diffrence third degree prove, free test and quiz for 9th grade, test simultaneous equations, free Agebra
tests, finding square root on ti-89.
Romberg integration with Matlab, GRE notes on arithematic section of maths, algebra tutor.
Free Sample 11 plus examination papers, practice pages for science yr 9, free Algebra Solver solves inequalities, showing all the steps, solve trusses by method of joints, basic formulae of
probability in GRE, McDougal Littell Algebra 1 answers, 8th grade highschool pratice test.
Aptitude test sample papers, download aptitude, algebra-quadriatic and rational inequalities, printable math equations for 9th grade Algebra 1 students.
Aptitude test solved papers, free software for solving trigonometric identities, Algebra-easy way to learn, algebra substitution practice, positive and negative numbers word problems, Quadratic
Equation answer key, worksheet adding 9.
Permutation combination matlab, who invented discriminants in quadratic equations?, easy way to do logarithms, differential calculator, free cost accounting notes.
Subtracting negative fractions calculator, solutions for third edition of abstract algebra Beachy, maths problem solver, TI- 83 taking the fourth root.
Solution + question aptitude, BASIC ALBEGRA WORK SHEETS 5TH GRADE, solve homework in algebra1 simplify.
Online calculating derivatives using the product rule, how to solve Hill's equation, how to understand algebra in maths, matlab second order differential equations, calculate e exponents, college
algebra order of oper, algebra 2 prentice hall teachers edition.
Free Algebra Fractions Worksheets, simultaneous equations (4 unknowns), "solution of hungerford(algebra)".
Solving a system of ordinary differential equations in matlab, Math equations for reducing volume of cubes, GCF and LCM free work sheets.
Formula. slopes, add and subtract fraction expression work sheet, guess the number java.
Root equations, java method convert int to time, radical finder child, lesson plan exponents logarithms.
Matlab to solve simple differential equations, how to do percentage formulas, translation maths worksheets, free download ks3 biology science paper.
How to find a quadratic equation with a TI 83, algebra crossword + work sheet, matlab ode45 second order, answer code sheets for nc eog tests.
3 simultaneous equation algebraic equation solver, convert value number java, trinomials calculator, programing ti 83 plus "decimal to radical".
Online ti-84 calculator emulator, Converting a mixed fraction into a decimal., trigonometry trivia, worksheet-algebraic fractions.
Solve 3 variable formula, Binomial Expansion Lesson Plans, solve free mathematic problems website of 6th standard, how do you solve multivariable equations, free online scientific calculator to solve
complex numbers.
Quadratic expression and equation, were can i find the answers to pre-algebre chapter 2 practice 2-8 inequalities and their graphs, a, solving non linear ODE, algebra 1 with answers, adding radical
expressions calculator, Basic Algebra Exponents Worksheet.
Word problem-sample and solutions, school games activities square numbers, what is the number in front of a square root mean.
Statistics solver casio, 8th grade free ebooks, casio fx-83 tricks, manual algebrator.
Free math printables, what is integrated mathmatics, divisors calculator, free online algebra solver answers, godrej & boyce aptitude test papers with solutions.
To find the quadratic formula from data, grade 9 math trinomials, concepts of fractions and square roots, Seven is an odd number, how can you make it even without adding, subtracting, multiplying or
dividing, the process of subtracting algebraic expression, dividing and multiplying algebraic expressions, math tests for 6th graders fraction word problems.
Factoring 3 term cubed, multiple choice questions in boolean algebra, Algebra Poems, maths scale factors.
Lcd fraction calculator, solving equations calculator power of 4, triganomic identities, bivariate random variable with multiple choice, decimal and conversion among common fractions, non-homogeneous
linearly order differential, interactive square root activities.
Elementary algebra worksheets, rationalizing twice square root, how to do a percent on TI-83, dienes blocks worksheets.
Free exponent and radical calculator, adding variables with square roots, particular solution equation solver, algebra used in networking, give the rule in adding and subtraction in algebraic
expression, adding subtracting negative integers work sheets, matlab, second order ode.
Aptitude ubuntud download, parabola graphing online, convert exponential, chets for 4th grade fundamentals, worded problem in linear equation in one variable.
Cheat algebra calculator, practise tests for fourth grade students, square root of equations, chemical engineering matlab pdf, C aptitude question and answers.
Factorising equations calculators, 1st Grade Math Sheets, rational functions online calculator, polynom third solver.
Example trig problems and answers, ti 84 complex matrix math, English Aptitude questions with answers, tringles worksheet, free accounting IQ tests.
Free online calculator ti-84, mathematical trivia, rudin solution chapter6, equation analysis test answers by games reader's.
Write the following percent as a mixed number: 876%, solutions, abstract algebra, herstein, runge kutta second order differential equation matlab, extracting the root.
Algebra ratio test, graphing non linear function.ppt, solve simultaneous equation in matlab, application of quadratic equation, Engineering Equation Solver tutorial beginner, how to solve system
congruence equations step by step guide.
Solutions for multi-variable non-linear equations, using real life situations to teach algebra, formula square root, how to draw pictures on graphing calculator using trig equations, integration by
substitution calculator.
Algebra free printables for 9th grade, free printable worksheets on adding and subtracting integers, math algebra log Solver, solve equation with multiple variables with Ti-89, normcdf java.
Solving simultaneous equations online calculator, Completed Differentiated lesson plans using eqivalent fractions, solving systems by substitution calculator, free exponents worksheets, algebra 1a
yellow book for 12th grade, t1-83 calculator online, learning adding, multiplying, dividing, subtracting decimal.
Year 9 algebra practise problems, third root, free ti 83 calculator download, how to take 2 digit after decimal point + java.
Percentage formulas, cubic equation calculator in excel, rational algebraic multiplication and division.
Test of genius answers from pre-algebra pizzazz, solving equations with integers and fractions, combinations exercises math, determine linear equations is increasing, decreasing, constant, peoms
related in mathematics.
How to type a cube root into a Ti-84 calculator, JAVA examples convert number to time, free homeschooling 9th graders.
Evaluate logarithms calculator, online division solver, MATHMATICAL EQUATION ON PAPER.
Lesson plan in first degree equation and inequalities in one variable, how to find no of repeated words in string using java, calculator for implicit differentiation, applet to graph an ellipse.
How to cheat in gcse, rudin analysis solutions, laplace transform of a saw tooth.
Root solver, equation of a hyperbola, examples of math trivia with answers for grade 4, liner inquality ppt.
Abstract algebra+tutorials, chennai of 7th std wants maths quiz questions, add negative positive integers printable free, n= square root of (x+3) squared +(x-3) squared, formulas and solution in
trigonometry, scale factor worksheet.
Holt algebra 1, yr 6 exam papers, sequences gcse test print, excel slope formula, worksheet for word problem in greatest common factors, Rudin solutions chapter 7, how can solution the equation in
ALGEBRA FRACTIONS IN EXCEL, software, The Division amongst Problem Solvers, women = evil formula, grade 6 free printable math tests, gmat practise, LINIER EQUATION SYSTEM WITH TWO VARIABLES AND THREE
Different ways to teach 6th evaluating expressions, c language aptitude questions, maths quiz sheets for students, 2nd order ode matlab.
Multiple choice question paper basic physic free download, year 7 algebra worksheet, highest common factor activities, free download ebook cost accounting by carter usry, online algebra fraction
Calculate Cube Root on TI-83, solve problems: algebraic concept, google search/ gre practice test/Permutation.
Excel vba lcm, CLEP calculus free online tutorial, math questions fractions add multiply subtract divide.
Ti-84 plus download, symmetry games+printable+free, free solved problems probability statistics, how to simplify exponents with variables, give the rule of adding and subtraction in algebraic
expressin, solving nonlinear ordinary differential equation, c# third grade solution equation source code example.
Basic algebra questions ks3, Abstract Algebra Fraleigh solutions, solving initial value problems homogeneous linear equations.
Power point presentation graphing functions, simplifying complex fraction worksheet,.....etc, addition of rational expressions worksheets, differential equation solver in MATLAB, 5th grade math
angles worksheet, "index of" Mcdougal Littell Mathematics Structure and Method - Course 1 - Teacher Edition.
Algebraic pyramid calculator, roots of the quadratic equation mathworks, calculator online for 3 step problems, ordering integers games, how to simplify equation square root, free grade eight algebra
Mathematics homework joke answer sheet, square root conjugate, Applied Mathematics Test Formula Sheet.
Algebrator free, t-83 calculator, books accounting free, chapter 8 rational and radical functions test, math: order of operation for algebra, linear extrapolation equation ti-89, multiplying and
dividing decimal fun worksheets.
Ti 89 base, Radical expression solver, mathematical law of permutations, TI-84 gcf settings, with absolut value VB calculator code, free cost accounting mcqs.
Solving homogeneous equations, modern elementary statistics tips texas instruments, gmat practise questions, free 3-D worksheets for grade 3, multiple equation grapher.
Draw a flowchart to program a calculater, download mcgraw study guide accounting, ti89 quadratic formula, integrated mathematic tutorial, solving equations in excel, quadratic equations quotient.
Math worksheets for beginner pre algebra, integers kids, grade 10 maths + solving quadratic equation, math multiply and divide fractions worksheets, simplifying absolute values calculator.
Dividing algebraic fractions worksheets, sample paper of seven class and cours dbt, answer a algebra question.
Online sat sample paper solving, investigatory project in math, studying to clept out of intermediate algebra.
How to change square root to decimals, homogeneous ODE second order, Descartes solving eguations by graphs, simplify integers worksheets.
Learn algebra online, solving 3rd order polynomials, Coordinates worksheets for KS2, permutation and combination ebook, display quiz quadratic equation based on while loop, solve multivariate
polynomial equations by variable substitution.
Quadratic equation completing the square with checking, algebra sums maths online, greatest common divisor formula, solving differential equations first order cos sin inhomogeneous nonlinear siny,
solving fractional factoring problems, how toused caculator to convert decimal to binary, aptitude question.
Square equation calculator, algebra simplyfying exponent, ks2 ratio bitesize.
Rudin chapter 7 answer, comparing ordering integers worksheets, dividing games for children worksheets, FRACTION HOW TO SOLVE, ti-30x finding greatest common denominator, algebra variable practice
How to square a decimal, free printable algebra tutorials, vector powerpoint "gcse maths".
Free pdf accounts books, Operation Research free download books easy to learn, solving simultaneous equations using excel, free software engineering entrance exam question blank, year 10 advanced
maths exam answers.
MATH FRACTION QUIZ FOR GCSE, four fundamental math concepts used in evaluating an expression, graphing ellipses software, math solving software, math trivia with answers mathematics.
Finding the function for graphing a line on a coordinate plane, Addition Pyramids Maths algebra worksheets, solve algebra equations.
Polynominal, get a sum from a loop java, common terms used in algebra "algebraic expressions" "quadratic equation", age problems-sample and solutions, KS2 algebra worksheets, calculator +special
quadratic factorization.
How to solve nth root equation using TI calculator, year 8 ks3 maths test, mathsolver for algebra 2.
Trigonometry sample tests, pre-algebra with pizzazz inequalities, algabraic, glencoe algebra 2 solutions, examples of parabolas word problems, step by step on how to solve for a matrix, "Algebra 2"
AND "Help" AND "Free" AND "Linear Equation".
"tawnee stone in school", how to calculate gcd, free information on solving one and two step Linear equations gr.10.
Kind of equation graphed as a parabola, homogeneous solution second order ode, convert decimal to fraction online.
Solving equations with fractions with two variables, simplifying algebraic fractions calculator, expression for each model simplify the expression, maths worksheets factors, polynomial long division
calculator, mixed numbers decimals.
Permutations for dummies, maths algebra inverse operation equation solver, Pre Algerba online, language and maths yr 8.
Network of equations involve hundreds of variables and equation, decimal mixed worksheet, proportions and variables worksheets.
Simplifying trigonomic expressions calculator, kumon online answer, teaching the slope and y-intercept eight grade math, examples of the latest mathematical trivia, Basic Grade 6 algebraic in
Australia, c language aptitude questions & answer, how do you solve for a slope.
Algebra solver multiple variable, add subtract multiply divide worksheet, free equivilent equation worksheets, solving two ste equations, order, linear algebra for beginners.
Google visitors came to this page yesterday by entering these math terms :
│college algebra formulas and rules │TI Calculator Turn decimals into │hs determinant math slove │solutions exercise basic algebra │Free worksheets for eleven plus │
│ │fraction │ │ │preperation │
│Common GMAT aptitude questions │interactive positive and negative space│online calculator t-89 │how to simplify a quotient function │teacher worksheet with answers │
│ │activities │ │plus a polynomial │ │
│Is advance Algebra hard? │free online algebra 1 Merrill answer │download free college calculator │algebra 2 tutor online │simplify algebra for programming vb │
│ │key │ │ │ │
│solving systems, elimination, │trig identities problems ti 93 │second order equation- vertex │Dividing Polynomials calculator online │Mathematics Angles rules worksheet │
│interactive │ │ │ │ │
│ti 84 quadratic formula program │writing polar equations in matlab │TI-84 change log base │pre-algebra online calculators │investigatory project help │
│converting a mixed fraction into a │what is the highest common factor of 65│math activity 1 to 8 for kids printout │Mcdougal Littell Mathematics Structure │ │
│decimal │and 115 │for 1 to 4 years old │and Method - Course 1 - Teacher │softmath │
│ │ │ │Edition.pdf │ │
│printable kumon math worksheets │common denominator calculator │addition of rational expressions with │adding rational expressions CALCULATOR │solve third-order polynomial │
│ │ │unequal denominators worksheet │ │ │
│math stats sheet │download Linear Algebra Problem Book │maths tests for year 8 │college algebra help free problem │addition worksheets up to number 30 │
│ │ │ │solver │ │
│simultaneous equation worksheets │easy algebra │algebra 1 conversion │examples of basic power numbers │difference of 2 squares │
│"multiplacation chart" │solving linear eualities │number connections math problem solver │little rudin "chapter 6", #7 │"algabraic word problems" │
│math word problems with age + pdf │how to write polynomial formula forex │permutation problems with answers │Newton's method system of nonlinear │algebraic equations for fractions │
│ │charts │ │equations matlab code │ │
│fourth grade multiplication civil │clep algebra examples │calculator cu radical │online help with yr 12 maths for │learn how to solve equations │
│rights worksheets │ │ │dummies │ │
│adding and subtracting positive and │how do you do subtraction of algebraic │1. What is the difference between │ │ │
│negative integers │equations │exponential and radical forms of an │regressions for the ladder problem │algebraic graphs hyperbola │
│ │ │expression? │ │ │
│nonlinear equations matlab │free maths sheets for year 11 │rearranging log equations │Why don't negative numbers have │second order differential equation │
│ │ │ │logarithms? │calculator │
│ti83+ rom image online │the world's hardest ALGEBRA problem │how to cheat on cognitive tutor │simplify Square root 1.25 │how to fraction in math for beginner ? │
│algebrator │completing the square questions │function table graphing calc online │calculators fractions into decimals │adding ladder method │
│onlineequation root finder │multiplying radicals for ti-84 │probability worksheets algebra │online quadratic simultaneous equations│substitution linear equations calculator│
│ │ │ │calculator │ │
│square root 4 = ? │fraction worksheets freeuk │saxon algebra 1 answers │VB6 fraction simplify │multiply rational expressions worksheet │
│9th Grade Math Practice Test │linear equation .ppt │calculas │revision for chapter 5 algebra yr 10 │ACCOUNTING BOOK SAMPLE FORM │
│physics holt solution manual │formula of lcm │algebra step by step programing canada │9th grade algebra made easy │complex and imaginary numbers 3rd order │
│ │ │ │ │polynomials │
│inequalities+ks4 │www.algebrator.softmath.com │Find Slope Calculator │Prime Factorization of denominator │5th grade properties worksheet │
│maths+ppt+worksheets │ │ │ │ │
│free math books for beginners │example of mathematics trivia │TI 83 PLUS EMULATOR │idiots guide to fractions and decimals │easy steps to solving reducing fractions│
│algebra addition pyramids │free algebra pretest grade 9 │ascending decimal │questions on adding/subtracting mixed │Integration calculator step by step │
│ │ │ │fractions │ │
│ │ │ │adding, subtracting, multiplying, and │ │
│grade 5 downloadable maths papers │examples of interger questions grade 7 │3rd order equation solver │dividing fractions and reducing │free solution example for lcm │
│ │ │ │worksheet │ │
│algebra speed formula │online tutor for algebra advanced │conjugate cube root │combination charts for KS3 maths │convert y-intercept javascript │
│base ten to hex calculator │balanced equations maths │worksheet with whole numbers using │texas ti83 programming │9th grade algebra worksheets and │
│ │ │factors and multiples │ │solutions │
│Algebra calculator advanced │intermediate ALgebra,questions │factoring four term expressions help │free college algebra word problems │how to use my casio calculator │
│ │ │ │software │ │
│how to solve the term polynomial │calculating the y intercept │simplifies equations calculator │Converting Scientific Notation to │simplify fractional exponents │
│ │ │ │Decimal Numbers formula │ │
│simultaneous quadratic equation │show me how to use ti-84 caculator with│an integer is multiplied by itself │how to store pdf on ti-89 │math problem rancher horses fractions │
│worksheet │cramers rule │ │ │ │
│math equation fraction exponent │Fundamentals of Physics Extended (8th) │free grade 9 printable algebra test │quadratic equations 3 variable │free problems showing mulitplication │
│ │step-by-step solutions │ │ │properties │
│florida sample algebra textbook │simplifying radicals with variables │systems of equations on ti 89 │general maths and english test │solve nonlinear simultaneous equations │
│chapter 4 │ │ │ │ │
│advanced algebra help │grade 8 math adding subtracting │balancing linear equations in maple │solve graphs │rational functions and story problems │
│ │dividing and multiplying fractions │ │ │ │
│"vector addition" +visual simulation│solve algebra problems free │english aptitude questions │free intermediate algebra software │online pre algebra worksheets │
│+classroom use +applet +interactive │ │ │ │ │
│examples for rewrite sentences for │free aptitude ebooks │sample test on special products and │easy way to calculate fractions │half-life worksheets │
│elementry level students │ │factoring │ │ │
│differential equations in excel │aptitutes gremmer qeustion in india │easy way of adding and subtracting │simultaneous equations online │polynomials in everyday life │
│ │ │ │calculator │ │
│wwwmathtrivia.com │math for dummies │adding and subtracting real numbers │area and perimeter maths KS2 worksheets│algebra with pizzaz puzzles │
│ │ │worksheets │ │ │
│permutations combinations study │Prentice Hall Algebra 1 Quadratic │how do you divide │rational algebraic expression │adding, subtracting, multiplying and │
│questions │Formula │ │multiplication and division │dividing fractions │
│slope formula activity │lowest common denominator calculator │textbook accounting pdf │mcdougal math vocabulary dictionary │slopes+lesson plans+projects │
│ │preview of berkeley's problem in │simplifying a radical expression │y=square root of x transformed by a │what is the difference between quadratic│
│computer aptitude free download │mathematics second edition │calculator │stretch scale factor half paralel to │formula using squares and quadratic │
│ │ │ │x-axis │formula │
│Square root of 1/3 simplified │Linear Algebra problems solving lines │solve by substitution calculator │polynomial 3rd order │square root help in algebra │
│ │and planes │ │ │ │
│algebra and statistics revision │fee mathematics sheets for first │standard form calculator │the bible ks2 worksheet │examples of linear equations with 2 │
│questions │graders │ │ │variable │
│school accounting free download │algebra pracrice sheets on Greatest │multiplication square work sheets │glencoe/mcgraw-hill glencoe Pre-algebra│fractions including powers and square │
│ │Common Factors of monomials │ │page 27 answers │roots │
│Interesting Math Trivia │how to teach simplifying radicals │third degree roots in excel │flow chart "math problems" │how to do algibra │
│matematica para el estudiante │TI 84 AND SAT TRICKS │pre-algebra with pizzazz! pg. 61 │solving sets algebra questions │simplify algebra calculator │
│ │ │answers titled test of genius │distributive-law │ │
│1st laplace transformation │rational functions solver │ti 84 download │division on rational expression │slope word problems worksheet │
│"online algebra expression │variable cost percentage formula │ti 84 algebra programs │what will happen if a radical │mathematica solving second order │
│calculator" │ │ │expression is not simplified first │ │
│online factorising test │how to cube root on a ti-83 │online graphing calculator with table │simultaneous equations worksheet free │fractions cheats │
│online calculator fractional │polynomial expansion matlab │free maths worksheet on cube root grade│conics cheatsheet │solve differential equation using │
│exponents │ │7 │ │runge-kutta matlab │
│seventh grade math algebra free │ti-83 program steps │how to solve probability inequalities │8th grade level printable algebra quiz │free online ks2 science test │
│WORKSHEETS │ │on ti 89 │ │ │
│math trivias with answers │lesson plan on solving quadratic │college precalculus cheat sheet │how to add subtract and multiply │online calculator to add surds │
│ │equation by factoring │formulas │fraction │calculator │
│first order differential equation │worksheets on dividing decimals by │SAT II Physics formula sheet │percentage equations │10th grade quick and easy math test │
│solver │decimals │ │ │ │
│equation on solving ratio │free online logarithmic calculator │simultaneous equation 5 unknowns │testbase+maths+questions+and+answers │slope intercept formula │
│ │purplemath │calculator │ │ │
│pre algebra glencoe florida workbook│turn a decimal into a fraction │free algebra Grid answers │TI-83 ROM download │adding and subtracting algebraic │
│"answers" │calculator │ │ │expressions activities │
│linear equation java │how to do fractions with a power │factoring algebra graph │algebra expressions calculator │algebra 1b calculator activity │
│writing a function in algebra │free guide book download for aptitude │Heath Passport to Algebra and Geometry:│worksheet subtracting 1 │application of quardatic equation │
│ │exams │An Integrated Approach online version │ │ │
│calculate tiles formula │how to program a quadratic formula │activities for cubed scale factor │rational numbers+worksheet │"algebra expression calculator" │
│ │problem on a Ti-93 calculator │ │ │ │
│roots to negative power fractions │second order non homogeneous solution │free linear equations worksheets │how to do an equation on a casio │how to covert square root │
│ │ │ │calculator │ │
│basic concepts of algebraic │pythagorean theory question answer │simplify algebra fractions calculator │mixed number to decimal converter │algebra problems for grade 10 │
│expressions │printable sheets │ │ │ │
│how to solve algebra problems │quadradic calculator │printable fifth grade cliff notes │solving equations excel │free online slope and y intercept solver│
│+understanding when to add, │equa diff +second │accounting book+pdf │find the decimal of a whole number │printable algebra worksheets │
│subtract, multiply and divide │ordre+MEF+matlab+programme │ │worksheets │ │
│learn algebra online for free │algebra radicals exercises │+exampl problem solving in mathematics │linear equation system with three │aptitude books download │
│ │ │ │variables │ │
│How Do You Change a mixed Fraction │algebrator.alw │factorise machine │McDougal Littell Geometry │factoring four term expressions │
│to a Percent │ │ │ │ │
│manipulating equations calculator │nonlinear + differential + equations │graphing linear absolute value │free math printouts fractions adding │free help college intermediate algebra │
│ │first order + solve │equations on powerpoint │ │ │
│ti-83 plus to cube │solving one step equations Worksheets │beginner perimeter worksheets │algebra caculator │pre algebra with pizzazz │
│permutations examples primary school│ti 83 picture project trig │Mathmatics formulas │ti-84 algebra equation │online solving differential equations │
│how to solve equations with the │solving equations powerpoint │mathematics investigatory project │mixed number java program │abstract algebra artin │
│distributive property │ │school │ │ │
│how to convert expoential values │mathematical culture one inthe maya │ti-84 calculator secant download │math trivia for kids │Scale FACTORS TEST mATHS │
│into integer values in C# │ │ │ │ │
│slope worksheet │decimal multiplying dividing games │how to solve equation in matlab │solving radical expressions │answers to college algebra problems │
│asset test cheating │cubed factoring of polynomials │yr 8 students maths problems │rational equation calculation │free Grade 8 Math test samples │
│solving addition and subtraction │graphing linear equations on the │how to do algebra/ common denominators │arithematic │free online yr 7 maths │
│equations │graphing calculator │ │ │ │
│Mcdougal Littell North Carolina │2nd order runge kutta matlab │woodbury mn milf │simultaneous equations with 4 unknowns │algebric equation │
│Geometry answers to homework │ │ │ │ │
│Operations Algebraic Expressions │pre algebra explanations │how to subtract decimals powerpoint │printable cliff notes for fifth grade │simplfying frations │
│worksheets │ │ │ │ │
│ti-83 linear systems │example of math trivia │SImplifying radical equations │instructor solutions contemporary │texas ti-83 download second degree │
│ │ │calculator │abstract algebra │ │
│ │hard math about fraction litteral and │ │simplify square roots of negative │ │
│evaluate vs simplify │complex two fraction worksheet ,and │algebra-math trivia │numbers with exponents │printable coordinate graphing 5th grade │
│ │problem │ │ │ │
│Free Pre Algebra Help │maple equation calculator │matlab convert integer to 8 bit binary │examples of calculating +arithmatic │college algebra made simple │
│ │ │ │mean │ │
│free online algebra problems │differential equations with quadratic │square root addition │prerequisite for adding and subtracting│6th grade math SAT 9 │
│ │nonlinear terms │ │integers │ │
│java program to convert a string │matlab for kids │free worksheets with solution of │mathematics quiz gcse │denominators calculator │
│into bold later │ │variation │ │ │
│combine rational expressions │square root rules │gcd implementation in VHDL │trivia questions of decimals │Simplifying a sum of radical expressions│
│calculator │ │ │ │ │
│converter milimeter pixel online │examples of mathmatical set │equations with two variables worksheets│worksheet on find the scale factor │pdf algebraic formulas │
│ │ │with solutions │ │ │
│ti 83 rom code │compare/order integers worksheet and │Mcdougal biology workbook answers │math grouping symbols │algebra, worksheets, grade 5, one step │
│ │answer sheet │ │ │ │
│synthetic division online calculator│math investigatory projects │how to solve equation containing │mathend unit test sample pdf │ti-84 plus emulator │
│ │ │grouping symbol │ │ │
│lineal metre │rational expression calucator │easiest way to find largest common │c# + conics equation + "source code" │factor equation solver │
│ │ │denominator │ │ │
│using logarithms to solve equations │oline maths test yr 8 │google using slope to solve life real │fun algebra worksheet .linear one │congruent KS2 maths worksheet │
│and linear inequalities │ │problems video │variable │ │
│solving linear equations for │standard deviation key on TI-83 │ti 83 cheat sheet statistics │college algebra refresher printables │maths homework assignments answers │
│multiple variables │ │ │ │ │
│singapore primary math freeware │trig addition equations │finding slope games │mutiple choice math past papers │solve the equation with powers as │
│ │ │ │ │fractions │
│Solving homogeneous equations │solve 2 simultaneous equation in matlab│Free 2nd gradePictograph work sheets │factoring polynomials cubed │pde, heat equation,non-homogeneous │
│online resolve math expressions │▶logbase( + ti-89 │boolean algebra solver │solving factoring calculator │math problem using scale │
│GRADE 9 MATH SLOPE │solution to first order pDE │free online year 7 excel book algebra │radical expression solver │algebra poem samples\ │
│FREE MATH PDF E BOOK SLOPE OF A LINE│heath algebra 1 answers │solving second order ODE │algebra 2 statistics problems │MATLAB non-linear partial differential │
│ │ │ │ │equations │
│combinations permutations free │combine like terms calculator │gcse maths work sheets home study │primary math poems │matlab solve 2 ODE's 2nd order │
│worksheets │ │ │ │ │
│online calculator for rationalizing │7100794 │Free Algebra Help │gre math percent triangle │mixed number percent │
│numerators │ │ │ │ │
│Introductory Algebra answers │pizzazz worksheets │algebra, rational numbers │printable algebra tests on percentages │factarization in mathematics for matric │
│ │ │ │ │class │
│fractions multiply divide activity │maths revision yr.8 │tutorial │ti89 base conversion │how to put divisors in javascript │
│display factorized answer in radical│evaluating expressions worksheet │multiplying standard form │"taking roots inside an absolute value │integer woorksheets │
│form, TI-84 │ │ │sign" │ │
│download cpt question +test paper │factorise quadratic equations solver │saxon "algebra 1" textbook online │solve addition and subtraction by │hard math games for year 8/ │
│ │ │ │graphing │ │
│math variable problems with pie and │What differences are between functions │free online maths tests for yr 8 │how to solve equations on parallel │Aptitude Test+Download │
│powers 12th grade │and linear equations │ │lines easy │ │
│combine rational expressions online │Algebrator │integer numbers worksheet │pre-algebra with pizzazz! titled test │free math worksheets translations │
│calculator │ │ │of genius answers │ │
│quadratic simultaneous equations │factoring binomials calculator │ks2 algebra │algebra answers or cheats │"fraction to decimal" program ti-86 │
│calculator │ │ │ │ │
│First Grade Math Sheets │combinations of unique sums │exponential equation ti-83 │simultaneous equations matlab │online integer games │
│write a program to calculate the │ │ │ │ │
│root of a quadratic equation using Q│decimal unit convert to nearest │Accounting Books Free Download │online slope calculator equation │how to solve third degree equation excel│
│basic │ │ │ │ │
│83 convert octal │Maths test papers on algebra to do │factoring of cubed polynomials │adding and subtractions of fraction │Division Ladder math scott foresman │
│ │online for children │ │equations │ │
│algebra calculator root │solve rational expressions-calculator │number base change ti 89 │6th grade math game printouts │algebra order of operation │
│what is hcf of 28 and 32 ? │liner inequality ppt │online fraction simplifier │simplify radicals online calculator │free online math exercises with prime │
│ │ │ │ │and compound numbers │
│ │GCSE free practice papers for maths and│ │order of opperation adding, │how can i learn free maths about │
│ninth grade english worksheets │chemistry and biology │simultaneous quadratic equations │substracting, dividing, and multiplying│square,cube,root │
│ │ │ │numbers │ │
│fractons or mixed number as a │solutions rudin │glencoe mathematics algebra 1 volume 2 │easy way to solve slope equation │how to solve application with rate │
│decimal │ │ │ │ │
│mathematical investigatory project │pre-algebra worksheets with answer keys│free worksheets on finding lcd │TI-83 laplace transform │ti calculator data acquisition │
│algebraic equations with integers, │formula for turning decimals in to │contemporary abstract algebra solution │antiderivative solver │write program distance formula on T183 │
│8th grade test .pdf │fractions │ │ │calculator │
│software ebook testing free │Second Degree Polynomial Solver Online │solving fractional exponents │Mcdougal Hill Algebra Chapter resources│Dividing Polynomials free calculation │
│download+pdf │ │ │ │online │
│area of a rectangle demonstration │math division rules │"maths worksheets"+"ratio" │third order equation solving │algebra word problems for 9th grade │
│ks3 free online │ │ │ │ │
│how to add, subtract, multiply and │how to find the slope intercept on the │multiply and ivide fractions made easy │how do you convert mixed numbers to │where can i find a program to do algebra│
│divide franctions and decimals? │TI-83 Plus │ │decimals │ │
│whats the least common multiple of │multiplicative worksheets │differential laplace calculator │"graphing calculator" TI-84 fourier │free slope worksheets │
│70 80 and 90 │ │ │ │ │
│free math puzzels 9th level │trigonometry trivias │calculator to solve rational │calculator for perpendicular slope │turning decimals into degree │
│ │ │expressions │ │ │
│first addition 10's worksheet │linear algebra and applications 3rd │multiply decimals practice │TI-84 free emulator │free college algebra practice test study│
│ │layer solution manuel │ │ │sheets │
│ti-89 second order linear equations │combination permutation worksheet │calculate sq metre from cm │var (ax+by) │algebra rewriting in vertex form solver │
│programs │ │ │ │ │
│Algebraic expressions involving │find roots of a cubed equation │quadratic equations ti 89 │how to solve vertex equations │solving second grade equations simple │
│integers │ │ │ │ │ | {"url":"https://www.softmath.com/math-com-calculator/distance-of-points/basic-concept-of-algebra.html","timestamp":"2024-11-12T04:15:56Z","content_type":"text/html","content_length":"179170","record_id":"<urn:uuid:19cc47ce-7aa1-49c0-b30a-c8e61ef38aef>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00004.warc.gz"} |
OpenstarTs :: Browsing Rendiconti dell'Istituto di Matematica dell'Università di Trieste: an International Journal of Mathematics vol.40 (2008) by Type "Article"
Rendiconti dell'Istituto di Matematica dell'Università di Trieste: an International Journal of Mathematics vol.40 (2008)
Permanent URI
Editorial policy The journal Rendiconti dell’Istituto di Matematica dell’università di Trieste publishes original articles in all areas of mathematics. Special regard is given to research papers, but
attractive expository papers may also be considered for publication. The journal usually appears in one issue per year. Additional issues may however be published. In particular, the Managing Editors
may consider the publication of supplementary volumes related to some special events, like conferences, workshops, and advanced schools. All submitted papers will be refereed. Manuscripts are
accepted for review with the understanding that the work has not been published before and is not under consideration for publication elsewhere. Our journal can be obtained by exchange agreements
with other similar journals.
Instructions for Authors Authors are invited to submit their papers by e-mail directly to one of the Managing Editors in PDF format. All the correspondence regarding the submission and the editorial
process of the paper are done by e-mail. Papers have to be written in one of the following languages: English, French, German, or Italian. Abstracts should not exceed ten printed lines, except for
papers written in French, German, or Italian, for which an extended English summary is required. After acceptance, manuscripts have to be prepared in LaTeX using the style rendiconti.cls which can be
downloaded from the web page. Any figure should be recorded in a single PDF, PS (PostScript), or EPS (Encapsulated PostScript) file. | {"url":"https://www.openstarts.units.it/browse/type?scope=e5b55c5e-170d-4967-a5ec-b2ff9adb165a&value=Article","timestamp":"2024-11-04T20:42:29Z","content_type":"text/html","content_length":"961393","record_id":"<urn:uuid:956a4333-bd45-4f15-bba9-9c0ffcef2140>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00740.warc.gz"} |
2024-11-03T09:02:58Z https://ir.soken.ac.jp/oai oai:ir.soken.ac.jp:00001503 2023-06-20T15:59:03Z 2:429:19 AUTOMATIC EXTRACTION OF LOGICALLY CONSISTENT ONTOLOGIES FROM TEXT CORPORA AUTOMATIC
EXTRACTION OF LOGICALLY CONSISTENT ONTOLOGIES FROM TEXT CORPORA McCRAE, John Philip マックレー, ジョンフィリップ McCRAE, John Philip Ontologies provide a structured description of the concepts and
terminology<br />used in a particular domain and provide valuable knowledge for a range of natu-<br />ral language processing applications. However, for many domains and languages<br />ontologies do
not exist and manual creation is a difficult and resource-intensive<br />process. As such, automatic methods to extract, expand or aid the construction<br />of these resources is of significant
interest.<br /> There are a number of methods for extracting semantic information about<br />how terms are related from raw text, most notably the approach of Hearst<br />[1992], who used <i>
patterns</i> to extract hypernym information. This method was<br />manual and it is not clear how to automatically generate patterns, which are<br />specific to a given relationship and domain. I
present a novel method for de-<br />veloping patterns based on the use of alignments between patterns. Alignment<br />works well as it is closely related to the concept of a <i>join-set</i> of
patterns, which<br />minimally generalise over-fitting patterns. I show that join-sets can be viewed<br />as an reduction on the search space of patterns, while resulting in no loss of<br />accuracy.
I then show the results can be combined by a <i>support vector machine</i><br />to a obtain a classifier, which can decide if a pair of terms are related. I applied<br />this to several data sets and
conclude that this method produces a precise result,<br />with reasonable recall.<br /> The system I developed, like many semantic relation systems, produces only<br />a binary decision of whether
a term pair is related. Ontologies have a structure,<br />that limits the forms of networks they represent. As the relation extraction is<br />generally noisy and incomplete, it is unlikely that the
extracted relations will<br />match the structure of the ontology. As such I represent the structure of ontol-<br />ogy as a set of logical statements, and form a consistent ontology by finding the
<br />network closest to the relation extraction system's output, which is consistent<br />with these restrictions. This gives a novel <i>NP-hard</i> optimisation problem, for<br />which I develop
several algorithms. I present simple greedy approaches, and<br />branch and bound approaches, which my results show are not sufficient for this<br />problem. I then use resolution to show how this
problem can be stated as an<br /><i>integer programming problem,</i> which can be efficiently solved by relaxing it to<br />a <i>linear programming problem</i>. I show that this result can
efficiently solve the<br />problem, and furthermore when applied to the result of the relation extraction<br />system, this improves the quality of the extraction as well as converting it to an<br />
ontological structure. application/pdf 総研大甲第1288号 eng thesis https://ir.soken.ac.jp/records/1503 博士（情報学） 2009-09-30 総合研究大学院大学 https://ir.soken.ac.jp/record/1503/files/甲1288_要
旨.pdf application/pdf 308.8 kB 2016-02-17 https://ir.soken.ac.jp/record/1503/files/甲1288_本文.pdf application/pdf 2.1 MB 2016-02-17 | {"url":"https://ir.soken.ac.jp/oai?verb=GetRecord&metadataPrefix=jpcoar_1.0&identifier=oai:ir.soken.ac.jp:00001503","timestamp":"2024-11-03T09:02:59Z","content_type":"application/xml","content_length":"8110","record_id":"<urn:uuid:f8852511-0fbd-49ab-9faf-50d691049d30>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00703.warc.gz"} |
The eigenvector is a vector that is associated with a set of linear equations. The eigenvector of a matrix is also known as a latent vector, proper vector, or characteristic vector. These are defined
in the reference of a square matrix. Eigenvectors are also useful in solving differential equations and many other applications related to them. In this article, let us discuss the eigenvector
definition, equation, methods with examples in detail.
Eigenvector Definition
Eigenvector of a square matrix is defined as a non-vector in which when a given matrix is multiplied, it is equal to a scalar multiple of that vector. Let us suppose that A is an n x n square
matrix, and if v be a non-zero vector, then the product of matrix A, and vector v is defined as the product of a scalar quantity λ and the given vector, such that:
Av =λv
v = Eigenvector and λ be the scalar quantity that is termed as eigenvalue associated with given matrix A
Eigenvector Equation
The equation corresponding to each eigenvalue of a matrix is given by:
AX = λ X
It is formally known as the eigenvector equation.
In place of λ, substitute each eigenvalue and get the eigenvector equation which enables us to solve for the eigenvector belonging to each eigenvalue.
Eigenvector Method
The method of determining the eigenvector of a matrix is given as follows:
If A be an n×n matrix and λ be the eigenvalues associated with it. Then, eigenvector v can be defined by the following relation: Av =λv
If “I” be the identity matrix of the same order as A, then
(A – λI)v =0
The eigenvector associated with matrix A can be determined using the above method.
Here, “v” is known as eigenvector belonging to each eigenvalue and is written as:
\(v =\begin{bmatrix} v_{1}\\ v_{2}\\ .\\ .\\ v_{n}\end{bmatrix}\)
How to Find an Eigenvector?
To find the eigenvectors of a matrix, follow the procedure given below:
1. Find the eigenvalues of the given matrix A, using the equation det ((A – λI) =0, where “I” is equivalent order identity matrix as A. Denote each eigenvalue of λ[1], λ[2], λ[3]….
2. Substitute the values in the equation AX = λ[1] or (A – λ[1] I) X = 0.
3. Calculate the value of eigenvector X, which is associated with the eigenvalue.
4. Repeat the steps to find the eigenvector for the remaining eigenvalues.
Types of Eigenvector
The eigenvectors are of two types namely,
• Left Eigenvector
• Right Eigenvector
Left Eigenvector
The left eigenvector is represented in the form of a row vector which satisfies the following condition:
A is a given matrix of order n and λ be one of its eigenvalues.
X[L ] is a row vector of a matrix. I,e., [ x[1] x[2] x[3] …. X[n]]
Right Eigenvector
The right eigenvector is represented in the form of a column vector which satisfies the following condition:
A is a given matrix of order n and λ be one of its eigenvalues.
X[R ] is a column vector of a matrix. I,e., \(X_{R} =\begin{bmatrix} x_{1}\\ x_{2}\\ .\\ .\\ x_{n}\end{bmatrix}\)
Eigenvector Applications
The important application of eigenvectors are as follows:
• Eigenvectors are used in Physics in simple mode of oscillation
• In Mathematics, eigenvector decomposition is widely used in order to solve the linear equation of first order, in ranking matrices, in differential calculus etc
• This concept is widely used in quantum mechanics
• It is applicable in almost all the branches of engineering
Eigenvector Examples
Example: Find the eigenvector of the given matrix:
\(A =\begin{bmatrix} 1 &4 \\ -4 & -7 \end{bmatrix}\)
Given: \(A =\begin{bmatrix} 1 &4 \\ -4 & -7 \end{bmatrix}\)
\(|A – \lambda I| =\begin{vmatrix} 1-\lambda & 4\\ -4& -7-\lambda \end{vmatrix}\)
(1- λ)(-7- λ)- 4(-4) = 0
( λ+3)^2 = 0
Therefore, λ =-3, -3
Use the eigenvector equation
AX = λX
Substitute λ value in the equation:
AX = -3X
We know that,
(A- λI) X = 0
\(\left ( \begin{bmatrix} 1 &4 \\ -4&-7 \end{bmatrix} + \begin{bmatrix} 3 &0 \\ 0 & 3 \end{bmatrix} \right )\begin{bmatrix} x\\ y\end{bmatrix} = \begin{bmatrix} 0\\ 0\end{bmatrix}\)
4x +4y =0
x+y =0
Assume that x =k
So, it becomes
k +y =0
y= -k
Therefore, the eigenvector is
\(X=\begin{bmatrix} x\\ y\end{bmatrix}=k\begin{bmatrix} 1\\ -1\end{bmatrix}\)
Now, let’s understand how we can find the eigenvalue of the matrix along with a solved example here.
Eigenvalue of Matrix
Eigenvalues are generally associated with eigenvectors in Linear algebra. Both of these terms are used in the interpretation of linear transformations. As we know that, eigenvalues are the particular
set of scalar values related to linear equations, most probably in the matrix equations.
To define eigenvalues, first, we have to determine eigenvectors. Almost all vectors change their direction when they are multiplied by A. Some rare vectors say x is in the same direction as Ax. These
are the “eigenvectors”. Multiply an eigenvector by A, and the vector Ax is the number time of the original x. The basic equation is given by:
Ax = λx.
Here, the number λ is an eigenvalue of matrix A.
Register with BYJU’S – The Learning App for all Maths-related concepts.
Frequently Asked Questions – FAQs
What are eigenvectors used for?
Eigenvalues and eigenvectors are used to reduce a linear operation to separate or simplify the problems.
How do you find eigenvectors?
The below steps help in finding the eigenvectors of a matrix.
Step 1: Find the eigenvalues of the given matrix A, using the equation det ((A – λI) =0, where “I” is an identity matrix of equivalent order as A.
Step 2: Denote each eigenvalue of λ_1, λ_2, λ_3,…
Step 3: Substitute the values in the equation AX = λ1 or (A – λ1 I) X = 0.
Step 4: Calculate the value of eigenvector X, which is associated with the eigenvalue.
Step 5: Repeat the steps to find the eigenvector for the remaining eigenvalues.
What do eigenvalues and eigenvectors tell us?
An eigenvalue is a number that tells us how much variance exists in the data in that direction, whereas an eigenvalue is a number that tells us how spread of the data.
Are eigenvectors orthogonal?
Generally, for any matrix, the eigenvectors are not always orthogonal. However, they will be orthogonal for a particular type of matrix such as a symmetric matrix.
Are all eigenvectors linearly independent?
Eigenvectors are linearly independent when the corresponding eigenvalues of a matrix are distinct. | {"url":"https://mathlake.com/Eigenvector","timestamp":"2024-11-13T08:23:28Z","content_type":"text/html","content_length":"16499","record_id":"<urn:uuid:7b734c9e-400a-4189-8a06-b36736a5ca44>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00862.warc.gz"} |
Powerful Graphics
Enhanced Mathematics
Improvements to Computation and Efficiency
Extended Connectivity
User Interface Improvements
What's New in Maple 13? The following is a summary of the major improvements in Maple 13. For an index of the updates, see the Index of New Maple 13 Features.
Powerful GraphicsEnhanced MathematicsImprovements to Computation and EfficiencyExtended ConnectivityUser Interface Improvements
Improved 3-D plots are faster and use less memory. You can use typeset mathematics in the titles, captions, and axis labels for 3-D plots. Drawing on 3-D plots and improved control of tickmark
spacing are now available. New fly-through plots. The new viewpoint option allows you to create an animation by varying the viewpoint as you navigate through a 3-D plot, as if a camera were flying
through the plot. You can use one of the standard viewpoint paths or define a path by specifying camera locations. You can now use units in plots in the function and range arguments. Axis labels will
display the unit information. For more information, see Graphics Improvements. New element-wise operator ~ to distribute an operation over a data structure such as a list, set, Array, or other
rtable. Significant improvements in dsolve and pdsolve for solving nonlinear differential equations using novel techniques. DEtools contains new and expanded commands for nonlinear equations of
differential order two and higher, covering the computation of parametric and particular solutions, also for initial value problems, all of them based on the ODE symmetries. Most of the symmetry
commands of PDEtools were expanded and five were added, including one for the fast computation of polynomial solutions to linear and nonlinear PDE systems; also, the internal PDEtools Library with 45
specialized routines for DE programming purposes is now available. Extended event-handling for numeric solutions of ODEs. There is a new set of 30 specialized commands for General Relativity
computations in the DifferentialGeometry[Tensor] package. RegularChains has been extended with two new subpackages--FastArithmeticTools and SemiAlgebraicSetTools--and additional improvements.
GraphTheory contains new and expanded commands, including isomorphism detection and the Bellman Ford algorithm for the shortest path in a weighted graph. More predefined graphs are now available in
GraphTheory. Enhancements to solve include new options to access environment variables, as well as improvements to solving trigonometric and inverse trigonometric inequalities, solving linear systems
over cyclotomic fields, and solving for real solutions. Improved integration of special functions, such as erf, Ci, Si, and Fresnel integrals, and new calling sequences for multiple integrals and
numeric integration are available. The new Student Numerical Analysis package provides tools and interactive tutors for learning, visualizing, and applying numerical analysis techniques. The
Calculus1 package has been updated with a new command to see all steps of the solution to a Calculus problem. Other improvements in many areas of mathematics, including linear algebra, number theory,
combinatorics, and linear functional systems. For more information, see New and Enhanced Packages, Differential Equations, Language, Efficiency, Numerics, and Symbolics. Efficiency improvements in
heap extraction, modular expansion and division of multivariate polynomials, finding GCDs over algebraic number and function fields, matrix decomposition, and singular values computations. The Task
Programming Model makes multithreaded code easier to write. For more information, see Efficiency and Programming. CAD connectivity has been extended to include the NX CAD software. The OpenMaple[C]
packages has new commands to push and pop error handling procedures and memory allocation functions. For more information, see CAD and Programming. The Maple Portal acts as a starting place for any
Maple user. The Maple Portal includes ten new tutorials on topics such as Plotting and Data Manipulation. From there, navigate to pages with more information for Engineers, Students, and Math
Educators. The Export as PDF feature can be used to export a Maple document created in the Standard interface to a PDF file. You can now access operations from the DynamicSystems package through a
context menu, making it even easier to convert between different equation and matrix representations, analyze the dynamics of a system, and obtain plots of the frequency, impulse, and system response
characteristics to a variety of inputs. You now have two easy ways to get help on a command: click on a word that you enter in a worksheet and press F2 (Control+?, Macintosh), or, from the context
menu, choose Help On Command. The help system opens to the help topic for that command. Over 50 task templates have been added or updated in a variety of subjects. Previous templates have been
upgraded to include a point-and-click interface for easier interactive problem solving. The Worksheet Migration Assistant makes it easy to convert a large collection of worksheets from the Classic
Maple format to the modern full-featured Standard Maple format. As part of the migration process, you can convert Maple Input to 2-D math. Access all of Maple's manuals within the Help System. The
Maple Introductory and Advanced Programming Guides are now available in the Help System. Additionally, the Maple Getting Started Guide and Maple User Manual have been consolidated into a single User
Manual, putting all of the information in one easy-to-find place. The Copy as MathML menu item allows you to select a math expression in Maple and copy it to the clipboard as MathML language. It can
then be pasted directly into another application that supports MathML language. For more information, see Graphical User Interface Updates.
See AlsoIndex of New Maple 13 Features | {"url":"https://de.maplesoft.com/support/help/content/9484/updates-v13.mw","timestamp":"2024-11-13T15:19:40Z","content_type":"application/xml","content_length":"23857","record_id":"<urn:uuid:a0d851a6-7fdd-4135-910b-8de9593f865f>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00541.warc.gz"} |
What are non-additive measures?
What are non-additive measures?
Non-additive measures are measures that cannot be aggregated across any of the dimensions. These measures cannot be logically aggregated between records or fact rows. Non-additive measures are
usually the result of ratios or other mathematical calculations.
What is additive semi-additive and non-additive measures?
Types of Facts Semi-Additive: Semi-additive facts are facts that can be summed up for some of the dimensions in the fact table, but not the others. Non-Additive: Non-additive facts are facts that
cannot be summed up for any of the dimensions present in the fact table.
What is semi-additive measures?
A semi-additive measure is one that is to be summed for some dimensions, but should not be summed across some other dimensions. For the dimensions over which the measure is not additive, a different
aggregation rule must be specified.
What is the best example of an additive measure?
An additive measure uses SUM to aggregate over any attribute. The sales amount is a perfect example of an additive measure. Indeed, the sales amount for all customers is the sum of the individual
sales for each customer; at the same time, the amount over a year is the sum of the amounts for each month.
How do you handle non-additive measures?
A good approach for non-additive facts is, where possible, to store the fully additive components of the non-additive measure and sum these components into the final answer set before calculating the
final non-additive fact. This final calculation is often done in the BI layer or OLAP cube.
What are non-additive facts example?
Some examples of non-additive facts are average, discount, ratios etc. Consider three instances of 5, with the sum being 15 and average being 5. Now consider two numbers i.e. 5 and 10, the sum being
15, but the average being 7.5.
What is additive and non-additive facts?
Additive facts are those facts which give the correct result by an addition operation. Examples of such facts could be number of items sold, sales amount etc. Non-additive facts can also be added,
but the addition gives incorrect results. Some examples of non-additive facts are average, discount, ratios etc.
What is non-additive fact in data warehouse?
Non-additive Facts are Facts that cannot be summed up for any of the dimensions present in the Fact table. Eg: Facts which have percentages, Ratios calculated. Semi-Additive: Semi-additive Facts are
Facts that can be summed up for some of the dimensions in the Fact table, but not the others.
Which of the following is an example of non-additive facts?
Profit margins are non-additive. If a department has two employees, and one employee has sold an item with a 55% profit margin and the other has sold an item with a 45% profit margin, the profit
margin for the department is not 100%.
What is non-additive fact with example?
Non-additive facts can also be added, but the addition gives incorrect results. Some examples of non-additive facts are average, discount, ratios etc. Consider three instances of 5, with the sum
being 15 and average being 5. Now consider two numbers i.e. 5 and 10, the sum being 15, but the average being 7.5.
What are non-additive aggregates?
Non-additive aggregates are aggregate functions that produce results that cannot be aggregated along a dimension. Instead, the values have to be calculated individually. All Number functions, except
for MAX and MIN, are non-additive aggregates.
How do you handle non-additive facts?
What is fully additive measure?
The most flexible and useful facts are fully additive; additive measures can be summed across any of the dimensions associated with the fact table. An example of a fully additive measure is sales
(purchases from a store). You can add hourly sales to get the sales for a day, week, month, quarter, or year.
What are additive and non-additive fact?
What is the best definition of a non-additive fact? | {"url":"https://www.tag-challenge.com/2022/11/24/what-are-non-additive-measures/","timestamp":"2024-11-10T01:53:47Z","content_type":"text/html","content_length":"39415","record_id":"<urn:uuid:d0767d96-120d-4eb5-b74b-ca2ede91db74>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00855.warc.gz"} |
This is How Time Travel is Theoretically Possible
Time travel is nothing special. You’re time traveling right now into the future. Relativity theory shows higher gravity and higher speed can slow time down enough to allow you to potentially travel
far into the future. But can you travel back in time to the past?
In this video I first do a quick review of light cones, world lines, events, light like curves, time-like curves, and space-like curves in this video so that you can understand the rest of the video.
A space like-world line means that the object has to travel faster than light. But moving anything to the speed of light requires an infinite amount of energy to accelerate. So this is not possible.
Going faster than the speed of light can create scenarios that allow you to travel back in time. But since this is not physically possible, we need to figure out a clever manipulation of space time.
This means we have to solve Einstein’s equations of General relativity.
The simplest spacetime is a flat spacetime. The equation for this can be expressed in Cartesian or spherical coordinates. But to travel back in time we need more complex spacetime. The first solution
ever presented to Einstein’s field equations was done by Karl Schwarzschild. He formulated non-flat spacetime that happened to describe a black hole, when no one had ever heard of it.
The r_s term in this equation is called the Schwarzschild radius. It is the point beyond which nothing can escape the black hole, because in order to escape, you would have to go faster than the
speed of light, which you cannot do. In this equation when r is equal to r_s in the dr squared term, we get a zero in the denominator. This makes the term is undefinable. Its physical meaning is the
event horizon.
Looking at the light cone of objects falling into the black hole, if the object is far away, then its cone is upright. As it starts falling into the black hole, it starts to tilt more and more as it
falls further towards the black hole. Exactly at the event horizon, the light cone lies tilted at 45 degrees. All future events point to inside of the event horizon, meaning there is no escape from
the black hole even at the speed of light, once you enter the event horizon.
Eventually the light cone will point completely towards the singularity at the center. This means that all future events will lie at the singularity. The singularity is a future moment in time rather
than a point in space.
The spacetime inside black holes can allow travel back in time. But even if we can go back in time inside the black hole, the event horizon prevents us from escaping the black hole. So what good is
it going back in time if we are trapped inside the black hole? It turns out there is a way to escape it.
In 1965 the Kerr-Newman metric was described by Ezra Newman. It describes a rotating black hole. There are ways we can remove the event horizon in this metric. When you do the math, we find that if
the black hole is spinning fast enough, the event horizon disappears. It is then no longer a black hole, but a naked singularity. A naked singularity, is just a singularity with no event horizon.
This is important is because when you don’t have an event horizon, you can go near the singularity in the center, but come right back out. There is no event horizon, that otherwise prevents you from
coming out of a black hole.
#timetravel #nakedsingularity
Now we can theoretically travel back in time by going around the singularity. This happens because we can traverse a closed time like curve, which allows world lines from the future cone to loop
around into the past light cone. We can loop our light cone around the singularity such that our future light cone ends up in the past light cone of where you started. And now since we are not bound
inside the black hole by the boundary of the event horizon, we can come out of this spacetime back to about where we started, but at a time BEFORE we started. We went back in time.
But there are a few problems. First, theory doesn’t mean reality. Black holes may not be able to physically rotate fast enough for the event horizon to disappear. The math works with a test particle
with little gravitation, but not at higher gravity such as that of a human. | {"url":"http://www.thespaceacademy.org/2021/07/this-is-how-time-travel-is_29.html","timestamp":"2024-11-12T05:46:34Z","content_type":"application/xhtml+xml","content_length":"168042","record_id":"<urn:uuid:515b4652-2c7f-4164-951e-dd83d7252d99>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00729.warc.gz"} |
Complex Fluids Group
This page gives a preview of some of our group's highlighted research. For more information on our group's research, feel free to check our publications page and/or feel free to contact us! We are
always interested in discussing interesting ideas and pursuing new collaborations.
Recent Research
Influence of Salt on the Viscosity of Polyelectrolyte Solutions
Guang Chen, Antonio Perazzo, and Howard A. Stone
Polyelectrolytes (PEs) are charged polymers, such as DNA, RNA, and many other biological molecules that are ubiquitous in nature. PE solutions are endowed with viscoelastic behavior and are widely
applied in energy storage, oil recovery, the food industry, and cosmetics. However, due to the electrostatic interactions among the charged monomers and mobile ions, the dependence of the rheological
properties on the polymer concentration n
of PE solutions differs significantly from that of solutions of uncharged macromolecules. In addition, salt in PE solutions, whether added intentionally or intrinsically present, can affect the
properties of the solutions. In this work, we introduce cell models to understand the electrostatics and its contribution to the conformation and rheology of PE solutions (a-c). By incorporating the
electrostatic interactions into the blob model and Zimm-Rouse dynamic model, we identify four consecutive regimes dependent on the magnitude of the ratio of the polymer concentration to the salt
concentration n
/&beta (d), which capture the unexplained experimental data. With our theory, we anticipate that the empirical Fuoss’s law is expected for solutions prepared with salt-contaminated PE samples. A new
critical charge fraction is defined, where we predict that the peak, which is present in the measurements of the reduced viscosity as a function of n
, is only expected for weakly charged PEs prepared with pure PE samples. We expect that our approach can be useful for future investigations on the electrostatics in charged colloidal systems, such
as clay particle suspensions and bacterial colonies.
G. Chen, A. Perazzo, and H. A. Stone, Phys. Rev. Lett., 124, 177801 (2020)
Ionic layering and overcharging in electrical double layers in a Poisson-Boltzmann model
Ankur Gupta, Ananth Govind Rajan (IISc Bangalore), Emily A. Carter (UCLA), Howard A. Stone
The arrangement of cations and anions near a charged surface plays a pivotal role in a broad range of applications such as energy storage, environmental remediation, lab-on-a-chip devices, and
biophysics. Typically, this arrangement of ions is modeled through Poisson-Boltzmann (PB) models that predict the spatial dependence of the electric potential and ion concentrations. Though PB models
are widely used due to their analytical simplicity and computational ease, they are unable to predict any oscillations in ion concentrations, which are usually observed in molecular simulations.
Here, we capture these details of ion concentration oscillations within a PB model by incorporating the nonlocal interactions that arise due to finite ion size and show that our model is in
quantitative agreement with more computationally intensive approaches (integral equation theories, Monte Carlo, and molecular dynamics). Our analysis directly advances the understanding of several
key topics at the forefront of physics and chemistry, e.g., the overcharging phenomenon, the structure of ionic liquids, and colloidal stability, among others.
Ionic layering and overcharging in electrical double layers in a Poisson-Boltzmann model, Phys. Rev. Lett. Ankur Gupta, Ananth Govind Rajan, Emily A. Carter, and Howard A. Stone
Charging Dynamics of Overlapping Double Layers in a Cylindrical Nanopore
Ankur Gupta, Pawel Zuk (Polish Academy of Sciences), Howard A. Stone
The charging of electrical double layers inside a cylindrical pore has applications to supercapacitors, batteries, desalination and biosensors. The charging dynamics in the limit of thin double
layers, i.e., when the double layer thickness is much smaller than the pore radius, is commonly described using an effective RC transmission line circuit. Here, we perform direct numerical
simulations (DNS) of the Poisson-Nernst-Planck equations to study the double layer charging for the scenario of overlapping double layers, i.e., when the double layer thickness is comparable to the
pore radius. We develop an analytical model that accurately predicts the results of DNS. Also, we construct a modified effective circuit for the overlapping double layer limit, and find that the
modified circuit is identical to the RC transmission line but with different values and physical interpretation of the capacitive and resistive elements. In particular, the effective surface
potential is reduced, the capacitor represents a volumetric current source, and the charging timescale is weakly dependent on the ratio of the pore radius and the double layer thickness.
Charging Dynamics of Overlapping Double Layers in a Cylindrical Nanopore, Phys. Rev. Lett. 125, 076001. Ankur Gupta, Pawel J. Zuk, and Howard A. Stone
Dead-end pore geometries for the study of electrolyte diffusion and diffusiophoresis
Ankur Gupta, Suin Shim, Ben Alessio, Jessica L. Wilson, Luqman Issah, Emmanuel Mintah, Estella Yu, and Howard A. Stone
We utilize a dead-end pore geometry to study electrolyte diffusion and diffusiophoresis of charged particles. Diffusiophoretic mobility of a charged particle is a function of surface zeta potential
and can be approximated as a constant for an ideal limit of fixed zeta potential and a thin double layer. For dilute electrolytes, the finite double layer thickness effects are significant, and for
concentrated electrolytes, charge screening could result in a decrease in the diffusiophoretic mobility. Using the same geometry, we study diffusiophoresis of polystyrene particles under a
concentration gradient of multivalent ions and in multiple-ion systems.
A. Gupta, S. Shim and H. A. Stone, “Diffusiophoresis: from dilute to concentrated electrolytes” Soft Matter, 2020, 16, 6975-6984 https://doi.org/10.1039/D0SM00899K;
J. L. Wilson, S. Shim, Y. E. Yu, A. Gupta and H. A. Stone, “Diffusiophoresis in multivalent electrolytes” Langmuir, 2020, 36, 7014-7020 https://doi.org/10.1021/acs.langmuir.9b03333;
A. Gupta, S. Shim , L. Issah, C. McKenzie and H. A. Stone, "Diffusion of multiple electrolytes cannot be treated independently: model predictions with experimental validations” Soft Matter, 2019, 15,
9965-9973 https://doi.org/10.1039/C9SM01780A
Symmetry breaking in a parallel two-phase flow
Paul R. Kaneelil, Amir A. Pahlavan, Miguel A. Herrada, Kristen LeRoy, Kylie Stengel, Samuel Warner, Anna M. Galea, Howard A. Stone
Parallel two-phase flows are omnipresent in technological applications that require contact between two immiscible fluids for a finite amount of time. Precise control over the flow and separation of
the fluids once they have been in contact are therefore the key challenges in these applications. Here, using experiments and numerical simulations, we show that the interface between two immiscible
fluids flowing at the same flow rate in a symmetric channel can become unstable locally near the exit junction, where the two fluids are separated. This instability leads to the shedding of the
droplets of one phase into the other, preventing a complete separation. We characterize this instability and show that the period of drop shedding is inversely proportional to the flow rate. We
derive a stability criterion based on the balance between the Laplace pressure across the liquid-liquid interface and viscous pressure drop along each flow stream. The stability criterion and our
experimental results are used to highlight the extreme sensitivity of this flow system to the parameters involved such as viscosity difference and exit geometry, which introduces gravitational
effects and characteristics of the exit tubing.
Particle motion nearby rough surfaces
Christina Kurzthaler, Danielle Chase, Lailai Zhu (National University of Singapore), Amir A. Pahlavan, and Howard A. Stone
We study the hydrodynamic coupling between particles and solid, rough boundaries characterized by random surface textures. Using the Lorentz reciprocal theorem, we derive analytical expressions for
the grand mobility tensor of a spherical particle and find that roughness-induced velocities vary nonmonotonically with the characteristic wavelength of the surface. In contrast to sedimentation near
a planar wall, our theory predicts continuous particle translation transverse and perpendicular to the applied force. Most prominently, this motion manifests itself in a variance of particle
displacements that grows quadratically in time along the direction of the force. This increase is rationalized by surface roughness generating particle sedimentation closer to or farther from the
surface, which entails a significant variability of settling velocities. We currently work on a quantitative comparison between our theoretical predictions and experiments of particles sedimenting
nearby 3D-printed periodic and randomly structured surfaces.
C. Kurzthaler, L. Zhu, A. A. Pahlavan, and H.A. Stone, Phys. Rev. Fluids, 5, 082101(R) (2020)
Pinch-off of liquid jets close to the continuum limit
Francisco Cruz-Mazo and Howard A. Stone
We focus on the behavior of liquid jets, their instability, and subsequent rupture when the minimum radial length eventually vanishes, and then classical continuum descriptions would start to fail.
Indeed, this drastic topological transformation may affect the validity of the classical inertial-viscous pinch-off models beyond a specific scale where the finite interface appears on the scene and
whose study is out of reach of existing stochastic fluctuations approaches so far. We are working on how to reconcile both frameworks by finding a set of expanded self-similar properties of our
physical model.
Electrostatic microfiber wrapping
Janine K. Nunes, J. Li, I. M. Griffiths, Bhargav Rallabandi, J. Man, Howard A. Stone
We study the dynamics of the wrapping of a charged flexible microfiber around an oppositely charged curved particle immersed in a viscous fluid. The image shows an example of the overlay of the time
sequence of this wrapping behavior. We observe that the wrapping behavior varies with the radius and Young’s modulus of the fiber, the radius of the particle, and the ionic strength of the
surrounding solution. The trends in wrapping rate indicate that wrapping is primarily a function of the favorable interaction energy due to electrostatics and the unfavorable deformation energy
needed to conform the fiber to the curvature of the particle. We perform an energy balance to predict the critical particle radius for wrapping, finding reasonably good agreement with experimental
observations. In addition, we use mathematical modeling and observations of the deflected shape of the free end of the fiber during wrapping to extract a measurement of the Young’s modulus of the
fiber. We evaluate the accuracy and potential limitations of this in situ measurement when compared to independent mechanical tests.
J. K. Nunes, J. Li, I. M. Griffiths, B. Rallabandi, J. Man, H. A. Stone, submitted (2020)
The Regime Map and Triple Point in Selective Withdrawal
Zehao Pan, Janine K. Nunes, Howard A. Stone
Entrainment in selective withdrawal occurs when both the top and bottom phases are withdrawn through a capillary tube oriented perpendicular to a flat gravitationally separated liquid-liquid
interface. The tube introduces two distinct features to the conditions for fluid entrainment. First, the ratio of the two phases being withdrawn is affected by the region of influence of the flow
upstream of the tube's orifice. Second, a minimum withdrawal flow rate must be reached for entrainment regardless of the distance between the interface and the tube. We show that these phenomena can
be understood based on the Reynolds number that governs the external flow field around the capillary tube and the capillary number that regulates the effect of the viscosity and capillarity.
Spontaneous pulse generation in channel flow of a particle suspension
Suin Shim and Howard A. Stone
We present experiments demonstrating the spontaneous generation and traveling of a colloidal pulse in a steady channel flow. When deionized (DI) water with suspended positively-charged particles
flows steadily through a single channel, a pulse (unexpected focusing) of particles is generated, which then flows through the channel at a slower speed than the mean flow velocity. With detailed
experimental investigations and quantified results, we rationalize our observations by considering CO2 driven diffusiophoresis. The concentration gradient of ions in the liquid phase is created by
the leakage of CO2 through the permeable PDMS walls. Mathematical models for early stage particle focusing and the traveling pulse will be compared with the experimental observations.
S. Shim and H. A. Stone, “CO2-leakage-driven diffusiophoresis causes spontaneous accumulation of charged materials in channel flow” Proc. Natl. Acad. Sci. U.S.A.2020 117 (42) 25985-25990
Conference presentations:
Spontaneous pulse generation in a steady channel flow of a colloidal suspension – the role of dissolved gas, 2017 APS DFD Meeting, Nov. 2017, Denver CO
Spontaneous pulse generation in a steady channel flow of a colloidal suspension, APS March Meeting 2019, Mar. 2019, Boston MA
CO2-driven diffusiophoresis for removal of bacteria
Suin Shim, Sepideh Khodaparast, Ching-Yao Lai, Jing Yan, Jesse T. Ault, Bhargav Rallabandi, Orest Shardt, and Howard A. Stone
CO2 dissolution in an aqueous phase can create concentration gradient of H+ and HCO3- ions. We demonstrate diffusiophoresis of bacterial cells in a Hele-Shaw geometry with circular symmetry.
Directional migration of the wild-type V. cholerae and a mutant lacking Flagella shows that the motion is diffusiophoresis, not a CO2-driven chemotaxis. Diffusiophoresis of S. aureus reduces cell
adhesion to a surface, and the exclusion of P. aeruginosa lasts > 11 hr after CO2 is turned off. Diffusiophoresis of bacteria can prevent surface contamination or infection by reducing the population
of the cells approaching an interface.
S. Shim, S. Khodaparast, C.-Y. Lai, J. Yan, J. T. Ault, B. Rallabandi, O. Shardt and H. A. Stone, "CO2-driven diffusiophoresis for removal of bacteria" arXiv:2009.07081 (16 Sep. 2020)
Self-Similar Draining near a Vertical Edge
N. Xue, H. A. Stone
Self-similarity is a fundamental idea in physics, where it is usually discussed in the context of problems with two independent variables that occur in a solution in a ratio involving a power law.
Here we provide an example with three independent variables where the self-similar structure involves a single ratio of the three variables. In particular, we report experiments of the
three-dimensional (3D) shape of a gravitationally draining liquid film near a vertical edge, as occurs when a liquid film drains on a vertical plate: the film becomes nonuniform in space and time
near the vertical edge (see attached image). A mathematical model of the drainage involves a nonlinear partial differential equation (PDE) involving time and two independent (space) variables. We
identify a self-similar solution that converts the problem into an ordinary differential equation. Interferometry is performed to measure the film thickness as a function of position and time, and
the results are in excellent agreement with the theoretical predictions. This study provides a new scaling law for understanding and estimating draining films with edge configurations, and provides
new insights and methods for treating self-similar systems.
N. Xue, H. A. Stone, "Self-Similar Draining near a Vertical Edge", Phys. Rev. Lett., vol. 125, Aug 2020, pp. 064502.
Laboratory layered latte
N. Xue, S. Khodaparast, L. Zhu, J. K. Nunes, H. Kim, H. A. Stone
Inducing thermal gradients in fluid systems with initial, well-defined density gradients results in the formation of distinct layered patterns, such as those observed in the ocean due to
double-diffusive convection. In contrast, layered composite fluids are sometimes observed in confined systems of rather chaotic initial states, for example, lattes formed by pouring espresso into a
glass of warm milk. Here, we report controlled experiments injecting a fluid into a miscible phase and show that, above a critical injection velocity, layering emerges over a time scale of minutes.
We identify critical conditions to produce the layering, and relate the results quantitatively to double-diffusive convection. Based on this understanding, we show how to employ this single-step
process to produce layered structures in soft materials, where the local elastic properties vary step-wise along the length of the material.
N. Xue, S. Khodaparast, L. Zhu, J. K. Nunes, H. Kim, H. A. Stone, "Laboratory layered latte", Nature Communications, vol. 8, no. 1, 2017, pp. 196;
N. Xue, S. Khodaparast and H. A. Stone, "Fountain mixing in a filling box at low Reynolds numbers", Phys. Rev. Fluids, vol. 4, Feb 2019, pp. 024501.
Formation, Rupture, and Healing of an Annular Viscous Film
Fan Yang and Howard A. Stone
When a thin horizontal liquid film is formed, it is gravitationally unstable and drips: the film falls, forming an annulus and eventually yielding an encapsulated bubble. This is in contrast with
conventional jet and drop breakup, which usually develops into cylindrical liquid columns. The annulus initially contracts due to surface tension, until the air column inside ruptures and the inner
surface forms a retracting tip, which will further translate along and subsequently heal the whole annulus. During the healing process, air is driven out of the narrow gap, part of which is propelled
into the droplet and consequently forms an entrained bubble. A one-dimensional model is derived for the thinning dynamics, which shows good agreement with experimental measurements and predicts that
the thinning dynamics is universal. The shape of the tip is documented to be conical, and the cone angle is invariant as the translating tip heals the annulus.
Formation, Rupture, and Healing of an Annular Viscous Film, PRL, 124:224501, 2020
Non-unique bubble dynamics in a vertical capillary with an external flow
Yu, Y. E., Magnini, M., Zhu, L., Shim, S., & Stone, H. A.
We study bubble motion in a vertical capillary tube under an external flow. Bretherton (1961) showed that, without external flow, a bubble can spontaneously rise when the Bond number (Bo ≡ ρgR
/γ) is above the critical value 0.842. It was then shown by Magnini
et al.
(2019) that the presence of an imposed liquid flow, in the same (upward) direction as buoyancy, accelerates the bubble and thickens the liquid film around it. In this work we carry out a systematic
study of the bubble motion under a wide range of external flows, focusing on the inertialess regime with Bo above the critical value. We show that a rich variety of bubble dynamics occur when an
external downward flow is applied, opposing the buoyancy-driven rise of the bubble. We reveal the existence of a critical capillary number of the external downward flow (Ca
≡ μU
/γ) at which the bubble arrests and changes its translational direction. Depending on the relative direction of gravity and the external flow, the film thickness follows two distinct solution
branches. The results from theory, experiments and numerical simulations confirm the existence of the two solution branches and reveal that the two branches overlap over a finite range of Ca
, thus suggesting non-unique, history-dependent solutions for the steady-state film thickness under the same external flow conditions. Furthermore, inertialess symmetry-breaking shape profiles at
steady state are found as the bubble transits near the tipping points of the solution branches, which are shown both in experiments and numerical simulations.
Yu, Y. E., Magnini, M., Zhu, L., Shim, S., & Stone, H. A. (in press). Non-unique bubble dynamics in a vertical capillary with an external flow. Journal of Fluid Mechanics.
Flagellated Quincke swimmers at low Reynolds number
Lailai Zhu, Endao Han, Joshua W. Shaevitz, and Howard A. Stone
Flagella and cilia play an important role in biology, which motivates the idea of functional mimicry as part of bioinspired applications. Nevertheless, it remains challenging to drive their
artificial counterparts via a steady, homogeneous stimulus. Combining theory, simulations, and experiments, we demonstrate a strategy to achieve this goal by exploiting an elasto-electro-hydrodynamic
instability based on the Quincke rotation. Our Quincke swimmers are comprised of a spherical particle and one or two elastic fibers. In a uniform and static electric field, the swimmers exhibit
different forms of motion, including a self-oscillatory state, and they each lead to a distinct trajectory in space. Our results demonstrate a new method to generate, and potentially control, the
locomotion of artificial flagellated swimmers at low Reynolds numbers.
L. Zhu and H. A. Stone, Physical Review Fluids, 4(6):1-7, 2019;
L. Zhu and H. A. Stone, Journal of Fluid Mechanics, 888:A311-A3135, 2020;
E. Han, L. Zhu, J. W. Shaevitz, and H. A. Stone, submitted.
Previous Research
Bursting bubbles at an oil-covered interface
Feng, J., Roché, M., M., Vigolo, Arnaudov, L.N., Stoyanov, S.D., Gurkov, T.D., Tsutsumanova, G.G., Stewart P.S., Kimpton L.S., Griffiths I.M., Nunes, J.K., Shin, S., Yan, J., Kong, Y.L., Prud’homme,
R.K., Arnaudov, L.N., Stoyanov, S.D., Muradoglu, M., Kim, H., Ault, J.T. and Stone, H.A.
Bursting of bubbles at an air/liquid interface is a familiar occurrence relevant to foam stability, cell cultures in bioreactors and ocean-atmosphere mass transfer. In the latter case, bubble
bursting leads to the dispersal of sea-water aerosols in the surrounding air. Here we show that bubbles bursting at a compound air/oil/water-with-surfactant interface can disperse submicrometre oil
droplets in water. Dispersal results from the detachment of an oil spray from the bottom of the bubble towards water during bubble collapse. We provide evidence that droplet size is selected by
physicochemical interactions between oil molecules and the surfactants rather than by hydrodynamics. We demonstrate the unrecognized role that this dispersal mechanism may play in the fate of the sea
surface micro-layer and of pollutant spills by dispersing petroleum in the water column. Finally, our system provides an energy-efficient route, with potential upscalability and wide applicability,
for applications in drug delivery, food production and materials science.
Feng, J., Roché, M., Vigolo, D., Arnaudov, L.N., Stoyanov, S.D., Gurkov, T.D., Tsutsumanova, G.G. and Stone, H.A., 2014. Nature Physics, 10(8), 606-612;
Stewart P.S., Feng J., Kimpton L.S., Griffiths I.M. and Stone, H.A., 2015. Journal of Fluid Mechanics, 777, 27-49. Feng, J., Nunes, J.K., Shin, S., Yan, J., Kong, Y.L., Prud’homme, R.K., Arnaudov,
L.N., Stoyanov, S.D. and Stone, H.A. 2016. Advanced Materials, 28(21), 4047-4052;
Feng, J., Muradoglu, M., Kim, H., Ault, J.T. and Stone, H.A., 2016. Journal of Fluid Mechanics, 807, 324-352.
Curvature suppresses the Rayleigh-Taylor instability
P. H. Trinh, H. Kim, N. Hammoud, P. D. Howell, S. J. Chapman, and H. A. Stone
The dynamics of a thin liquid film on the underside of a curved cylindrical substrate is studied. The evolution of the liquid layer is investigated as the film thickness and the radius of curvature
of the substrate are varied. A dimensionless parameter (a modified Bond number) that incorporates both geometric parameters, gravity, and surface tension is identified, and allows the observations to
be classified according to three different flow regimes: stable films, films with transient growth of perturbations followed by decay, and unstable films. Experiments and linear stability theory
confirm that below a critical value of the Bond number curvature of the substrate suppresses the Rayleigh-Taylor instability.
P. H. Trinh, H. Kim, N. Hammoud, P. D. Howell, S. J. Chapman, and H. A. Stone, Phys. Fluids 26, 051704 (2014).
Control viscous fingering using time-dependent strategies
Z. Zheng, H. Kim, and H. A. Stone
Control and stabilization of viscous fingering of immiscible fluids impacts a wide variety of pressure-driven multiphase flows. We report theoretical and experimental results on a time-dependent
control strategy by manipulating the gap thickness b(t) in a lifting Hele-Shaw cell in the power-law form b(t) = b
. Experimental results show good quantitative agreement with the predictions of linear stability analysis. By choosing the value of a single time-independent control parameter, we can either totally
suppress the viscous fingering instability or maintain a series of nonsplitting viscous fingers during the fluid displacement process. In addition to the gap thickness of a Hele-Shaw cell,
time-dependent control strategies can, in principle, also be placed on the injection rate, viscosity of the displaced fluid, and interfacial tension between the two fluids.
Z. Zheng, H. Kim, and H. A. Stone, Phys. Rev. Lett. 115, 174501 (2015).
Benard-Marangoni instability driven by moisture absorption
S. Shin, I. Jacobi, and Howard Stone
Glycerol is a viscous liquid widely used in industry and known for its strong hygroscopic nature. While this unusual property has been well documented from the perspective of solution chemistry, its
impact on the mechanical properties of glycerol remains largely unknown. We show that a Benard-Marangoni instability in pure glycerol can be spontaneously driven by absorption of water vapor. Even
under standard laboratory conditions, ambient humidity is sufficient to drive distinct Benard-Marangoni convection cells for hours. Such an instability is a consequence of diffusive vapor transport
process and competition between solutal and thermal Marangoni forces.
S. Shin, I. Jacobi and H. A. Stone, EPL 113, 24002 (2016).
Triggering and inhibiting splashes with tangential velocity
J.C. Bird, Scott Tsai, and Howard Stone
A drop impacting a smooth, solid, dry surfaces form a radially spreading lamella and sometimes results in splashing. In industrial and natural processes, it is common for the drops to impact the
surface on an angle or while the surface is moving, yet previous studies mostly focused on the perpendicular impact of drops onto surfaces. We show that the tangential component of the impact can act
to trigger or inhibit a splash, and we develop a model to predict this type of behavior. Our model agrees with previous experimental data and with our observations of the effects of tangential
J. C. Bird, S. H. Tsai, and H. A. Stone, New Journal of Physics 11 (2009).
C. Duprat, S. Protiere, J. M. Aristoff, and H. A. Stone
The surface-tension-driven coalescence of flexible structures is relevant to a number of engineering and biological systems, such as the clumping of hair, the failure of micro devices during wet
lithography, or more generally whenever a liquid-air interface is moving through a deformable media. We study the dynamics of wetting of flexible boundaries with a combination of experiments, scaling
arguments and theory. We consider three model systems. We investigate the rise and the spontaneous imbibition of a liquid between flexible sheets clamped at one end, and free to deflect at the other
end, and study how the deformation of the sheets affects the meniscus speed and entraps the liquid. We also study the behavior of a single drop on a pair of flexible fibers and show that, due to a
combination of capillary and elasticity effects, the drop spreads into a long liquid column, and there is an optimal volume at which the wetted length is maximum.
J. M. Aristoff, C. Duprat and H. A. Stone, Int. J. Non-Linear Mech. 46 (2011).
C. Duprat, J. M. Aristoff and H. A. Stone, J. Fluid Mech. 679 (2011).
C. Duprat, S. Protiere, A. Y. Beebe and and H. A. Stone, Nature 482 (2012).
Bubbles dancing in a vortex: trapping air at a T-junction
D. Vigolo, S. Radl, and H. A. Stone
An unusual phenomenon occurs to low density material, and in particular air bubbles, entrained in a fluid when flowing through a T-junction. For a range of Reynolds numbers, the flow develops two
symmetric vortices. Air bubbles are forced to the center of the vortex due to the centrifugal force and are then "trapped", i.e. they accumulate inside the vortex. Bubbles eventually oscillate (i.e.
"dance") in the vortex when the flow becomes unsteady. Experiments were conducted by generating air bubbles in a variety of T-junction devices. In addition, our 3D numerical simulations have revealed
a gradient of pressure, similar to vortex breakdown, that drives the flow towards the center of the T-junction creating two recirculating zones, which trap air bubbles.
D. Vigolo, S. Radl, and H. A. Stone, PNAS, vol. 111, no. 13, mar 2014, pp. 4770-4775.
Using simple flows to tie knots in flexible fibers
S. Kuei, K. Sadlej, and H. A. Stone
Flexible fibers, such as DNA and other polymer chains, have sometimes been found to contain knotted regions. While such fibers are not strict, closed knots, they exhibit similar characteristics; the
formation of these `open knots' and the effects they have on material properties are the subject of current research. We investigate the possibility that simple flows can generate open knots in
sufficiently long and flexible elastic fibers. Using the HYDROMULTIPOLE algorithm, which solves the multipole expansion of Stokes equations, we use numerical simulations to study the time evolution
of a bead-spring model fiber in a shear flow. In certain systems, the characteristic tumbling motion of a fiber in shear flow will result in the formation of 3_1 and 5_1 knots, as identified by their
Alexander polynomial knot invariants. Investigation of the key factors influencing knotting, as well as the mechanism of knotting, is ongoing.
S. Kuei, A. M. Słowicka, M. L. Ekiel-Jezewska, E. Wajnryb, H. A. Stone, New J. Phys., vol. 17, no. 5, may 2015, pp. 053009.
Bending of elastic fibers in viscous flow
J. Wexler, P. Trinh, and H. A. Stone - With A. Lindner, O. du Roure, H. Berthet, N. Quennouz (ESPCI Parics) and H. E. Huppert (Cambridge)
A slender fibre, if flexible enough, will bend when immersed in a viscous flow. Confining walls affect the dynamics of a variety of real-world fibre systems ranging from industrial fibre suspensions
to the biofilm streamers studied by our group. We study a model system that highlights the effect of confinement: a fibre is anchored in a thin channel, perpendicular to the direction of flow, and
fluid is pumped through the channel, forcing the fibre to bend. There is a thin gap between the axis of the fibre and the channel wall, and we study the interplay between flow through this gap, flow
around the fibre, and the corresponding effect on fibre deformation. Experiments are performed on a fibre that is polymerized directly in a microfluidic channel, and an analytical model is developed
to explain the results.
J. S. Wexler, P. H. Trinh, H. Berthet, N. Quennouz, O. d. Roure, H. E. Huppert, A. Linder, H. A. Stone, J. Fluid Mech., vol. 733, sep 2013, pp. 684.
Local and global consequences of flow on bacterial quorum sensing
M. K. Kim, F. Ingremeau, A. Zhao, B. L. Bassler, and H. A. Stone
To explore the health consequences of bacterial quorum sensing in the crypts, the researchers experimented with an antagonist to turn off quorum sensing in chambers colonized by methicillin-resistant
S. aureus(MRSA), an antibiotic-resistant strain of bacteria that causes human infection. The left-side chamber contained no antagonist. In the right-side chamber, the antagonist molecules spread
throughout the crevices, inactivating quorum sensing and indicating a potential strategy for alleviating MRSA virulence.
M. K. Kim, F. Ingremeau, A. Zhao, B. L. Bassler, H. A. Stone, Nat. Microbiol, vol. 1, no. 1, jan 2016, pp. 15005.
Colonization, competition, and dispersal of pathogens in fluid flow networks
M. K. Kim, A. Siryaporn, Y. Shen, Z. Gitai, and H. A. Stone
To explore how pathogens spread in the host, the researchers designed a conversing branched flow network that mimic host environments such as lung or plant vasculature. Arrow indicates the directions
that cell-free medium and bacterial cells were flowed. Wild-type P. aeruginosa (green) and its mutant pilTU (red) cells were initially seeded downstream of the conversing channels. Wild-type cells
reached all upstream channels by 17 hours, while the pilTU cells remained the downstream regions. In this fluid flow environments, Wild-type P. aeruginosa cells attach to surfaces using type IV pili.
These are localized to the bacterial cell poles such that upon attaching to the surface, flow causes the bacteria to orient with the pili pole pointed in the opposite direction of the flow. The
repeated extension and retraction of pili using pilTU motors in this position drives Wild-type P. aeruginosa to move upstream along the surface.
A. Siryaporn, M. Kim, Y. Shen, H. Stone, Z. Gitai, Curr. Biol., vol. 25, no. 9, may 2015, pp. 1201–1207.
Collective behavior of chemosensing
Bo Sun, Josephine Lembong, Guillaume Duclos and H. A. Stone
When cells are excited by external chemical stimulations, multiple intracellular signaling will take place to regulate necessary cellular functions. However, the chemosensing of individual cells
usually come with large fluctuations, and the cells need to employ different strategies to make reliable decisions based on the noisy readouts. One such strategy we are exploring is to utilize
inter-cellular communications. We found the cells self-organize their communication channels to form a network that demonstrates various critical behaviors such as long-range correlations and
percolation transitions.
B. Sun, J. Lembong, V. Normand, M. Rogers and H. A. Stone, PNAS 109 (2012).
Biofilm streamers
R. Rusconi, S. Lecuyer, L. Guglielmini, N. Autrusson, Y. Shen, K. Drescher and H. A. Stone
In the presence of a significant flow, mature multispecies biofilms often develop into long filamentous structures called streamers. We show that suspended thread-like biofilms steadily develop in
zigzag microchannels. Numerical simulations of a low-Reynolds-number flow around the corners of the channel indicate the presence of a secondary vortical motion whose intensity is related to the
bending angle of the turn. We demonstrate that the formation of streamers is directly proportional to the intensity of the secondary flow around the corners. In addition, we show that a model of an
elastic filament in a two-dimensional corner flow is able to explain how the streamers can cross fluid streamlines and connect corners located at the opposite sides of the channel.
R. Rusconi, S. Lecuyer, L. Guglielmini, and H. A. Stone, J R Soc Interface 7 (2010).
R. Rusconi, S. Lecuyer, N. Autrusson, L. Guglielmini, and H. A. Stone Biophys. J. 100 (2011).
L. Guglielmini, R. Rusconi, S. Lecuyer, and H. A. Stone, J. Fluid Mech. 668 (2011).
N. Autrusson, L. Guglielmini, S. Lecuyer, R. Rusconi, and H. A. Stone Phys. Fluids 23 (2011).
Confined lipid membranes
M. Staykova, D. Holmes, C. Read, H. A. Stone - With M. Arroyo and M. Rahimi Lenji (Universitat Politecnica de Catalunya- Barcelona Tech, Spain)
Although the plasma membrane in cells is usually confined to other sub-cellular structures, the mechanics of confined membranes has rarely been addressed. To mimic the confinement we have developed a
simplified membrane model, which couples a lipid bilayer to an elastic sheet (a). We have demonstrated that upon straining the confined membrane is able to regulate passively its area. In particular,
by compressing the elastic support, the bilayer reduces its area in the plane by forming lipid protrusions (b); upon expansion, the protrusions are absorbed back into the planar bilayer (a). The
shape of the protrusions, spherical and tubular, can be controlled by the strain and the liquid volume, available between the membrane and its support. Our observations closely reproduce membrane
shapes and processes found in cells, thus suggesting that mechanics may be a simple and generic organizing principle.
M. Staykova and H. A. Stone, Communicative & Integrative Biology 4 (2011).
M. Staykova, D. P. Holmes, C. Read, and H. A. Stone, PNAS 108 (2011).
Development of a microfluidic microbial fuel cell
D. Vigolo, T. Al-Housseiny, Y. Shen, T. DiChristina, H. A. Stone - with: F. O. Akinlawon, S. Al-Housseiny, R. K. Hobson, A. Sahu, K. Bedkowski
The power density output of microbial fuel cells (MFCs) is enhanced by optimizing the continuous flow of nutrient to obtain a constant rate of electricity production, and developing new electrodes
material (optimization of surface roughness to increase the effective surface available to accommodate the bacteria). Preliminary results show how increasing the shear stress corresponds to
increasing the output voltage generated by the MFCs up to an optimum flow rate. For higher flow rate the bacteria are discouraged to produce electricity and eventually are flushed away. A cheaper,
membraneless microbial fuel cell design based on laminar co-flow is at the moment under investigation.
D. Vigolo, T. T. Al-Housseiny, Y. Shen, F. O. Akinlawon, S. T. Al-Housseiny, R. K. Hobson, A. Sahu, K. I. Bedkowski, T. J. DiChristina, H. A. Stone, Phys. Chem. Chem. Phys., vol. 16, no. 24, 2014,
pp. 12535.
Controlled uniform coating induced by the interplay of Marangoni flows and surface-adsorbed macromolecules
H. Kim, F. Boulogne, E. Um, I. Jacobi, and H. A. Stone
Surface coatings and patterning technologies are essential for various physicochemical applications. In this Letter, we describe key parameters to achieve uniform particle coatings from binary
solutions. First, multiple sequential Marangoni flows, set by solute and surfactant simultaneously, prevent nonuniform particle distributions and continuously mix suspended materials during droplet
evaporation. Second, we show the importance of particle-surface interactions that can be established by surface-adsorbed macromolecules. To achieve a uniform deposit in a binary mixture, a small
concentration of surfactant and surface-adsorbed polymer (0.05 wt% each) is sufficient, which offers a new physicochemical avenue for control of coatings.
H. Kim, F. Boulogne, E. Um, I. Jacobi, E. Button, and H. A. Stone, Phys. Rev. Lett. 116, 124501 (2016).
B. Verberck, "Fluid dynamics: Spirited away," Nature Physics 12, 291 (2016)
B. Yirka, "Evaporated whisky inspires new type of coating technique," Phys.org (2016)
J. Kemsley, "Why whiskey doesn't put a ring on it," Chemical & Engineering News (2016)
M. Schirber, "Synopsis: Whisky-Inspired Coatings," Physics (2016)
Controlling colloidal particles in confined geometries using solute gradients
S. Shin, E. Um, B. Sabass, J. T. Ault, M. Rahimi, P. B. Warren, and H. A. Stone
Transport of colloids in confined geometries such as dead-end channels is involved in widespread applications including drug delivery and underground oil and gas recovery. In such geometries,
Brownian motion may be considered as the sole mechanism that enables transport of colloidal particles into or out of the channels, but it is, unfortunately, an extremely inefficient transport
mechanism for microscale particles. We explore the possibility of diffusiophoresis as a means to control the colloid transport in dead-end channels by introducing a solute gradient. We demonstrate
that the transport of colloidal particles into the dead-end channels can be either enhanced or completely prevented via diffusiophoresis. In addition, we show that a combination of diffusiophoresis
and Brownian motion leads to a strong size-dependent focusing effect such that the larger particles tend to concentrate more and reside deeper in the channel. Our findings have implications for all
manners of controlled release processes, especially for site-specific delivery systems where localized targeting of particles with minimal dispersion to the nontarget area is essential.
S. Shin, E. Um, B. Sabass, J. T. Ault, M. Rahimi, P. B. Warren and H. A. Stone, Size-dependent control of colloid transport in dead-end channels via solute gradients, Proc. Natl. Acad. Sci. U.S.A.
113, 257–261 (2016).
The elasto-hydrodynamic interaction between a particle and a permeable surface
G. Ramon, H. Huppert, H. A. Stone
Deposition of colloidal material and bacteria is of major concern for membrane separation processes. A particle near a permeable surface experiences a hydrodynamic force, which increases as the
surface becomes less permeable. This force may be orders of magnitude larger than the Stokes drag in an unbounded fluid. Shown here is a case where the particle is soft and deforms under this force,
bringing it closer to the surface. This may have important implication for the adhesion propensity of soft particles onto membrane surfaces.
G. Z. Ramon, H. E. Huppert, J. R. Lister, H. A. Stone, Phys. Fluids, vol. 25, no. 7, 2013, pp. 073103.
Anomalous scalings in the diffusion of granular materials
I. C. Christov, H. A. Stone
Granular materials do not perform thermally driven Brownian motion, so diffusion is observed in such systems because agitation (flow) causes inelastic collisions between particles. It has been
suggested that axial diffusion of granular matter in a rotating drum might be "anomalous" in the sense that the mean squared displacement of particles follows a power law in time with exponent less
than unity. We have shown that such a "paradox" can be resolved using Barenblatt's theory of self-similar intermediate asymptotics. Specifically, we found an analytical expression for the
instantaneous scaling exponent of a macroscopic concentration profile, as a function of the initial distribution. Then, we incorporated concentration-dependent diffusivity into the model, showing the
existence of a crossover from an anomalous scaling (consistent with experimental observations) to a normal diffusive scaling at very long times.
I. C. Christov, H. A. Stone, PNAS 109 (2012).
Particle-wall impacts in a T-junction
D. Vigolo, I. Griffiths, S. Radl, H. A. Stone
The impacting event for a given system of particles entrained in a fluid is described in terms of the Reynolds number and the particle Stokes number. Experimental results for the impact in a
T-junction are compared with the trajectories predicted by theoretical particle-tracing models for a range of configurations to determine the role of the viscous boundary layer in slowing down the
particles and reducing the rate of collision with the substrate. In particular a 2D model based on a stagnation point flow is used together with detailed 3D numerical simulations.
D. Vigolo, I. M. Griffiths, S. Radl, and H. A. Stone. J. Fluid Mech., Submitted (2012).
Microfluidic microfiber synthesis
J. Nunes, K. Sadlej, J. I. Tam, H. Constantin and H. A. Stone
This project is focused on the development of a simple microfluidic method for the synthesis of polymeric microfibers of controlled length. We explored the use of valve actuation and UV light
modulation to control the length of the microfibers. The valve-based approach, in particular, was developed to synthesize fibers with tunable lengths, which has not previously been demonstrated. We
observed good, reproducible control of microfiber length as a function of the valve actuation frequency. We also focused on the synthesis of wavy or crimped polymeric microfibers using a microfluidic
method. We trigger a buckling instability through the initiation of a polymerization reaction in a liquid jet when microchannel dimensions increase, and subsequently preserve the buckled morphology
when the crosslinking reaction is completed. The resulting microfibers have highly uniform and reproducible morphologies. By changing the UV exposure location in the channel, as well as the flow
rates, the degree of waviness of the microfibers can be controlled.
J. K. Nunes, K. Sadlej, J. I. Tam and H. A. Stone, Lab Chip, 12 (2012).
Temperature control and thermophoresis on-a-chip
D. Vigolo, R. Rusconi, R. Piazza, and H. A. Stone
A new technique to control temperature along microchannels using a low viscosity, conductive epoxy as Joule heater was developped. By using this technique we were able to effectively keep a constant
temperature or create a temperature gradient across a microfluidic channel. In the latter case we implemented a thermophoretic separator to actually separate (or drive) particles suspended in aqueous
solution in a microfluidic lab-on-chip system.
D. Vigolo, R. Rusconi, R. Piazza, and H. A. Stone. Lab Chip, 10(6):795-798 (2010).
D. Vigolo, R. Rusconi, H. A. Stone, and R. Piazza. Soft Matter, 6(15):3489-3493 (2010).
Encapsulation of bubbles
J. Wan, S. Shim and H. A. Stone
We propose a microfluidic approach for the generation of water droplets containing a high volume fraction of gas bubbles and we provide a design principle for microbubble-based pressure sensing
inside channels. We also present a microfluidic approach for the controlled encapsulation of individual gas bubbles in micrometer-diameter aqueous droplets with high gas volume fractions and
demonstrate this approach to making a liquid shell, which serves as a template for the synthesis of hollow inorganic particles.
J. Wan, A. Bick, M. Sullivan, H. A. Stone. Adv. Mater. 20 (2008).
J. Wan, H. A. Stone, Soft Matter. 6 (2010).
J. Wan, H. A. Stone, Langmuir 28 (2012).
Control and manipulation of paramagnetic particles
S. H. Tsai, J. S. Wexler, J. Wan, I. M. Griffiths, H. A. Stone
Magnetic forces are used to manipulate micron-sized paramagnetic beads in a microfluidic device, with applications in medicine and industry. By balancing the magnetic forces against fluid forces at
the small scale (viscous drag and interfacial tension), we accomplish a variety of tasks on the serialized platform of a microfluidic device. It is shown that magnetic particles can be sorted by size
transversely across a channel, by applying a magnetic field whose gradient is perpendicular to the direction of flow. If an immiscible interface is present at the center of a channel, a similar
procedure may be used to coat spheres with a micron-sized coating, produce aggregates of controllable size, and to make measurements of ultra-low surface tension. Since magnetic particles may be
functionalized to bind to various biological agents, these materials may be manipulated in a similar manner.
S. H. Tsai, I. M. Griffiths, and H. A. Stone. Lab on a Chip, 11 (2011).
S. H. Tsai, J. S. Wexler, J. Wan, and H. A. Stone. Applied Physics Letters, 99 (2011).
Rivulet flow over a flexible beam
P. D. Howell, H. Kim, J. Robinson, M. G. Popova, and H. A. Stone
We study theoretically and experimentally how a thin layer of liquid flows along a flexible beam. The flow is modelled using lubrication theory and the substrate is modelled as an elastica which
deforms according to the Euler-Bernoulli equation. A constant flux of liquid is supplied at one end of the beam, which is clamped horizontally, while the other end of the beam is free. As the liquid
film spreads, its weight causes the beam deflection to increase, which in turn enhances the spreading rate of the liquid. This feedback mechanism causes the front position σ(t) and the deflection
angle at the front ϕ(t) to go through a number of different power-law behaviours. For early times, the liquid spreads like a horizontal gravity current, with σ(t) =t
and ϕ(t) = t
. For intermediate times, the deflection of the beam leads to rapid acceleration of the liquid layer, with σ (t) = t
and ϕ (t) = t
. Finally, when the beam has sagged to become almost vertical, the liquid film flows downward with σ(t) = t and ϕ ~ π/2. We demonstrate good agreement between these theoretical predictions and
experimental results.
P. D. Howell, J. Robinson, and H. A. Stone, J. Fluid Mech. 732, 190-213 (2013).
P. D. Howell, H. Kim, M. G. Popova, and H. A. Stone, J. Fluid Mech. accepted (2016).
Microfluidic fabrication of microfibers
J. K. Nunes, A. Grosskopf, and H. A. Stone
We are developing a family of multiphase microfluidic methods for the controlled synthesis of monodisperse polymeric microfibers where the size, shape, morphology, spatial composition, and the
encapsulation of cargoes can be precisely tailored.
E. Um, J. K. Nunes, T. Pico, H. A. Stone, J. Mater. Chem. B, vol. 2, no. 45, oct 2014, pp. 7866-7871.
J. K. Nunes, C. Wu, H. Amini, K. Owsley, D. D. Carlo, H. A. Stone, Adv. Mater., vol. 26, no. 22, mar 2014, pp. 3712-3717.
J. K. Nunes, H. Constantin and H. A. Stone, Soft Matter, vol. 9, no. 16, 2013, pp. 4227.
J. K. Nunes, K. Sadlej, J. I. Tam, H. A. Stone, Lab. Chip, vol. 12, no. 13, 2012, pp. 2301.
Flow-induced gelation of microfiber suspensions
J. K. Nunes, A. Perazzo, and H. A. Stone
When subjected to flow conditions, such as extrusion from a needle, a suspension of flexible high aspect ratio fibers entangles irreversibly and forms a network. This flow-induced process is a simple
mechanical approach to hydrogel formation that does not depend on chemical reactions. We study this phenomenon with shear rheology experiments and microscopic visualization. We propose that these
microfiber suspensions are potentially useful material candidates for in situ scaffold fabrication in bioengineering applications.
Hierarchical folding of elastic membranes
P. Kim, M. Abkarian and H. A. Stone
Thin, layered materials develop surface undulations or wrinkles when they experience small compressive strain. This response is the result of a complex interplay between deformation of the top layer
and its foundation. This periodic wrinkling find applications, e.g. in stretchable electronics but can also limit an application’s performance owing to delamination or cracking under load. In
particular, because of curvature localization, finite deformations can cause wrinkles to evolve into folds. Using a two-layer polymeric system under biaxial compressive stress, we show that a
repetitive wrinkle-to-fold transition generates a hierarchical network of folds during reorganization of the stress field. The folds delineate individual domains, and each domain subdivides into
smaller ones over multiple generations. By modifying the boundary conditions and geometry, we demonstrate control over the final network morphology. We then exploit these wrinkles and deep folds to
guide and retain light within the photoactive regions of photovoltaics and show substantial improvements in light harvesting efficiencies, particularly in the near-infrared region where light
absorption is otherwise minimal.
P. Kim, M. Abkarian, and H. A. Stone, Nature Materials 10 (2011).
J. B. Kim, P. Kim, N. C. Pegard, S. J. Oh, C. R. Kagan, J. W. Fleischer, H. A. Stone, Y. L. Loo, Nature Photonics 6 (2012). | {"url":"https://stonelab.princeton.edu/index.php?id=research","timestamp":"2024-11-04T00:35:26Z","content_type":"text/html","content_length":"72893","record_id":"<urn:uuid:81b3eb14-cf0a-46a4-94e9-911bc595f4cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00003.warc.gz"} |
Hard math problems 12th grade with answers
Bing users found us today by entering these math terms :
│Prentice Hall Mathematics Algebra│linear equations worksheet free │heaviside function calculator │simplifying fractions with inverse │need help grade 10 maths │
│1 Answers │ │ │variables │ │
│free 8th grade math worksheets to│gmat sample papers in pdf │trig calulator │nonhomogeneous constant coefficient │quadratic factor calculator │
│print │ │ │second order differential equations │ │
│TI 84 emulator download │mathematical problems year 11 │simple java program(getting the highest│adding and subtracting intergers │exam papers gr 10 physics │
│ │ │number) │ │ │
│multiplying and dividing │ti 89 foil how to │extraction of number of factors in Spss│elementary algebra 8th edition │free online 3rd grade word problems help │
│integers- problem solving │ │ │ │explanation │
│algebra de 12 grade │multiplying and dividing nine digit │physics equation solver │nonlinear differential equations, │combination in visual basic │
│ │numbers │ │solution │ │
│how to work out algebra problems │solving inequalities for 5th graders │integer WORKSHEETS + FREE │java prime example -javascript │adding and subtracting decimals worksheet │
│artin's algebra solutions │INTEGERS PRACTICE SHEETS │rules for nth term in ks3 mathematics │Convert a mixed Fraction to a │solving addition and subtraction equation │
│ │ │ │Decimal │worksheets │
│nth term worksheets │Order Decimals From Least To Greates │download free intermediate accounting │Is it good for a 6th grader doing │ti89 calculator online maual │
│ │ │ │Pre-Algebra? │ │
│linear programing examples for │how to do gr 9 math scales │Simplifying Algebraic Expressions │math alg.1 answer sheet │top rated algebra software │
│high school │ │ │ │ │
│window application programing in │simple way to calculate fractions │gce physics books free download │free cost accounting tips │java application adding fractions code │
│calculater │ │ │ │ │
│circles(exercises)gce free │problem solving involving quadratic │algebra tiles worksheets │online logarithm solver │practice/practise rule in language worksheets │
│ │equation or function │ │ │for children │
│simplifying exponents │free crossword puzzles using exponents │boolean algebra solver │combine like terms activity │trigonometry trivias │
│lowest common multiple higher │algebra and trigonometry book 2 page 76│modeling algebra free online tutor │TRIVIAS IN ALGEBRA │Prentice hall worksheet answers/ number theory│
│numbers │ │ │ │and fraction concepts │
│McDougal Littell math course 3 │help with algebra homework │table of fouth roots │ti rom code │cubed rules │
│8th grade text book │ │ │ │ │
│FREE MATH FOR SIX GRADE WORKSHEET│adding negative integers+worksheet │factorizing Quadratic online calculator│grade seven algerbra work sheets │simplify exponents worksheet │
│ │ │ │free │ │
│download aptitude test question │teach me algebra │Suare of first order polynomial │sample lesson plan in multiplication│scientific notation filetype swf │
│papers │ │ │of radicals │ │
│free online 7th grae exam prep │" Graphing Calculator" + lesson plans +│algebra tutorial software │quadratic equation factorization │what are the highest common factors between 76│
│ │Pre- Algebra │ │calculator │and 108? │
│dividing and subtracting │solving cubed equations │answers to math books step by step free│factoring an equation with ti-89 │math trivia geometry │
│adding,subtracting,and naming │boolean algebra tutorial │algebra 1 - holt 2007 answer key │aptitude question paper with answer │physics aptitude test answers │
│decimals worksheets │ │ │ │ │
│printable grading scale for math │algebra problem │simplify exponents │Algebra 1 skills practice workbook │Advanced Mathematical Concepts: Precalculus │
│test │ │ │answers │with Applications solution book manual │
│free math drills six grade │root formula │substitution calculatror │how do you use cube root on a │Algebra Problem Solver │
│ │ │ │scientific calculator │ │
│ALGEBRA WORKSHEET │what numbers have the sum of 39 │simplifying rational expressions │trig calculator │download java apptitude papers │
│ │ │calculator │ │ │
│math, basic multiplication of │factoring calculator quadratic │sample exam problems and solutions in │simultaneous linear inequality graph│poems about trigonometry mathematics │
│integers │equations │fluid mechanics │generator │ │
│adding subtracting Scientific │common denominator algebra │Rational Equations calculator │recursive pattern in elementary │simultaneous equations solver with 3 variables│
│Notation │ │ │grade math │ │
│solved sample paper aptitude test│simplifying equation calculator │sample age problems in algebra │multiplying dividing rational │free cost accounting books downloads │
│ │ │ │expression online calculator │ │
│linear equations fractional one │probelm solving in conceptual physics │highest common factors of 26 │add in base 8 │solve calc for pocket pc │
│variable calculator online │answers │ │ │ │
│solutions for concrete and │LOG ON A TI 89 │solving second order homogeneous │free 9th grade algebra worksheets │Precalculus Online Problem Solver │
│abstract algebra goodman │ │systems │ │ │
│integer games subtraction │math combinations basics │solutions manual cost accounting 13 │calculating fractions on a ti-84 │solve cubed equations │
│ │ │ │plus scientific calculator │ │
│free algebraic expression │free saxon algebra 1 sheets │Solving Probabilities questions using │how to factor polynomials on ti 83 │binomial cubed │
│worksheet │ │excel │ │ │
│numerical expressions calculator │example of math trivia │download programs to ti-84 plus │learn algebra software │math practice sheets 3rd grade free printable │
│College ALgebra test question │Finding the Least Common Denominator │MCQ in cost accounting │conceptual physics math problems │free online 11+ practice papers │
│data banks │ │ │ │ │
│holt mathematics algebra 1 book │solving year 7 algebra questions .doc │math scale factor │calculate gcd of fractions │addition of negative & positive numbers │
│ │ │ │ │worksheets │
│divide integers formula │the greatest common factor of 30 and │LCM LCD divisibility 6th grade math │free elementary math fadt papers │dividing cube roots │
│ │105 │ │ │ │
│apittude books download │calculator for solving simplifying │invent accounts software download │multiply and divide negative │ks2 exam model paper │
│ │radical expressions │ │decimals worksheets │ │
│interpreting absolute value │naming decimal places worksheets │adding integers activities │online cubed roots calculator │online factoring │
│graphs │ │ │ │ │
│matlab solving system of │Free Algebra Solver │multiply by hand │how to solve simultaneous equations │online solve limits │
│differential equations │ │ │on matlab │ │
│program that solves continuity of│free college algebra software │second order polynomial solver │grade 10 accounting exercises │java "one variable" algebra expression │
│trigonometric functions │ │ │ │ │
│prentice hall algebra equations │using graphing calculator to factor │algebra helper │trinomials calculator │how to do the chapter 1 project for algebra 2 │
│ │ │ │ │prentice hall │
│algebra for 4th grade │factoring cubed equations │ordered pairs worksheets │aptitude question papers of isoft │solve third order polynomial equation │
│ │ │ │software company │ │
│aptitude question solution │convert 20' x 60' to square metres │solve multivariable equations using │Algebra Helper │what is simplified radical for of an equation │
│ │ │ti-86 │ │ │
│adding and subtracting real │proportions printable quizzes │How to do algebra │Quasi Newton methods Maple │algebraic expression with percentage │
│numbers worksheet │ │ │ │ │
│clep algebra │free algebrator download │how to do this algebra problem │square root converter │Calculate Linear Feet │
│integer addition subtraction │matlab system of linear differential │algebra solvers │solving MDOF equation using matlab │college math help programs │
│activities │equation │ │ │ │
│ │ │ │ │can you use your graphing calculator for │
│"worksheets for eighth grade" │non linear formula graphs │physic homework │free algebra II monomial worksheets │reduction of a complex fraction with mixed │
│ │ │ │ │numbers │
│year 8 math test paper │how to do the basic in alegbra 1 │simplifying in algebra │virtual algeblocks │convert value to fraction"formula" │
│Integers worksheet │factors of numbers on t1-83 plus │World History connections to today │california teacher's edition holt │multiplying and dividing integers worksheet │
│ │ │cheat sheets │algebra 1 │ │
│how to graph pictures on a │factoring equation calculator │Worksheet on Hyperbola │free algebra solver │cube roots in quadratic │
│calculator │ │ │ │ │
│how to multiply /divide rational │hex subtraction calculator online │like terms algebra powerpoint lesson │equation of a line with an │samples maths exam papers for children │
│expressions │ │ │increasing factor │ │
│games in adding integers │how to square exponents │college algebra wizard │how to solve equations using casio │Fluid Mechanics ppt │
│"distributive Property" + │HOW TO SOLVE PROBLEMS BY slope with │factors and multiples exercises │tangent scale factor calculations │absolute value equations addition │
│variables + worksheet │y-intercept │ │KS3 │ │
│calculator convert decimals into │Algebra 2 Answers │simplified square root table │factoring polynomials cubed │higher order coupled differential │
│fractions │ │ │ │equations+matlab │
│ │four basic operations of adding, │HOLT Mathematics course 1 for Grade 6 │ │copyright by holt, rinehart and winston math │
│printable geometry papers │subtracting, dividing and multiplying │on sale │one step equations worksheets │worksheet │
│ │integers │ │ │ │
│cubed factoring │lowest common denominator calculator │Ti-83 plus rom download │ti-84 plus quadratic formula │5th grade lesson plans on least common │
│ │ │ │download │multiple │
│sample lesson plan in multiplying│Nonlinear differential equations │aptitude books free download in pdf │holt physics problem workbook 1b │balancing equations worksheet 4th grade │
│radicals │ │ │ │ │
│learning physics graphing grade │10 grade biology EOC practice worksheet│how to plot x=0 on a graphing │holt algebra i answer sheets │algebra problems │
│11 │ │calculator │ │ │
│maths study sheets free │maths tutor online qudratic equations │matlab multi variable solver │learning basic algebra │nonlinear simultaneous three unknowns │
│algebra clock problems │prentice hall pre algebra chapter 1 │ │factoring cube polynomials │newton's method for multiple nonlinear matlab │
│ │test │ │ │ │
│square roots and cube roots in │matlab solve │polynomial equations in parallel │free answers to statistics homework │integers project │
│maths │ │algorithm │ │ │
│simplifying exponents worksheets │maths substitution sheets │6th grade saxon math book online │negative positive numbers on number │abstract algebra solution │
│ │ │ │line + free worksheet │ │
│smallest common denominator │practice problems for ontario grade 11 │What r the zeros for the polynomial │INTERACTIVE ACTIVITIES FOR SQUARE │College physics 8th edition answers │
│calculator │physics │equation 3x squared-12x+3? │ROOTS │ │
│converting measurements decimal │exam papers for english grade 11 │midpoint formula for ti 83 │putting programs on the t1-83 │discrete source prentice hall answer key │
│to fraction │ │ │calculator │ │
│algebra 2 book california answers│free multiplying rational expressions │mathimatics poem │adding and subtracting irrational │apititude online with answers │
│ │calculator │ │radicals │ │
│non linear equation solver │MATH FOR DUMMYS BOOK │solver for rational expressions │prime factorization printables │(x + 16 ) ^2 =121 solve using square root │
│ │ │ │ │property │
│NonLinear Differential Equations │difference if two roots │permutation and combination basic │adding and subtracting algebra │example java program that calculates compound │
│ │ │theory and problems │worksheets │interest formula │
│teach adding and subtracting to │solving │combinations/permutation activities │Learning Basic Algebra │"worksheets for seventh grade │
│students with disability │ │ │ │ │
│10th grade algebra test paper │math homework sheets for 6th grade │algebra worksheets saxon │algebra- sequences- level 6 free │greatest common divisor of 12 in algebra │
│polar equations y r │website for ordering numbers from least│absolute values of linear expressions │multiplying standard form │negative rational expressions │
│ │to greatest │and their graphs │ │ │
│exponents review test grade 9 │intermediate algebra exponent │how to pass college math │simplify function calculator │color by decimals │
│ │worksheets │ │ │ │
│Completing the Square- TI 83 Plus│best book for learning algebra │algebra 2 answers │free printable worksheets with │EXPONENTIAL, QUADRATIC, ABSOLUTE, │
│program │ │ │negative numbers and exponents │LINEAR,RADICAL, RATIONAL FUNCTIONS │
│3rd order polynomial root finding│expression in simplified radical from │download calculater programmes │free math for dummies │Study Guide for Algebra/Trigonometry by │
│ │ │ │ │Mcdougal-Littell │
│free creative algebra problems │ti-83 plus graphing calculator emulator│find the square root of a quadratic │Percentage equations │mcdougal littell pre-algebra answers │
│ │ │equation │ │ │
│solution nonlinear differential │same decimal length java │solve the differential equation with │steps to finding the domains of │free algebra problems │
│equations │ │fourier transform method │quotients of functions in algebra │ │
│Algebra 2 an incremental │ │ │hardest calculus problem in the │ │
│development second edition │free grammer worksheets for year2 │ks3/balancing equations │world │all math investigatory project │
│answers │ │ │ │ │
│looking up integers & negative │online fraction calculator with │how to solve algebraic equation in │MAth Exams Download │adding and subtracting integers test │
│rationals │multiple mixed numbers │matlab │ │ │
│sample problem on simultaneous │free 9th grade math worksheets and │What value of c makes the polynomial │ │ │
│nonlinear equation with │answers │below a perfect square trinomial? │kumon worksheets printable free │Trivial Math │
│substitution │ │ │ │ │
│ │"scientific notation" worksheet │matlab program for finding root of │Worksheets for adding and │ │
│least common denominator variable│dividing │nonlinear equation by newton raphson │subtracting positive and negative │nonlinear differential solution │
│ │ │method │integers │ │
│Convert a Fraction to a Decimal │multiplying exponents and variables │ti-84 accounting teerms │rudin solution ch7 │how to solve adding and subtractingradical │
│Point │ │ │ │expressions │
│math expression worksheets │convert whole numbers to decimals │slope of a region in excel │java addition formula loop │absolute value word problems │
│Free 9th grade math test │software to solve laplacian equations │online logarithm calculator │prentice hall mathematics virginia │free pre algebra worksheets with keys │
│ │ │ │teachers edition │ │
│solving cubed polynomials │solutions for algebra 1- holt 2007 │lesson on multiplying and dividing like│how to find roots of polynomials │DIFFERENTIAL ALGEBRAIC EXPRESSION TO FORMULA │
│ │ │terms │using calculator │ │
│Hamilton algebra │Exponents Adding and Subtracting │new aptitude questions with solutions │easy to learn gauss method │how to create a pre-algebra math test │
│ │practice │for free downloading │ │ │
│free online english worksheets │connecct the dot with integers │free worksheet for primary school for │change mix fraction to decimal │LCM solver │
│for 9th grade │worksheet │volume topic │ │ │
│mathpower 8-square roots │solve simultaneous equations program │line of vertex │algebra rational expression │square root exponents │
│ │ │ │calculator │ │
│free answers to the McDougal │ │adding and subtracting integer │worded problem in Linear equation in│ │
│Littell algebra 1 practice │linear equations powerpoint │equations │1 variable │lowest common denominator in algebra │
│workbook │ │ │ │ │
│Websites about Adding and │METHODS FOR SOLVING NON LINEAR │free maths practice papers and answers │least common multiple algebra │the erules for adding and subtracting integers│
│subtracting integers │DIFFERENTIAL EQUATIONS │ │worksheets │worksheets │
│who invented the mathimatical │exponent variable │multiplying and dividing 16 bit Binary │how to solve simultaneous equations │angle conversion with t1-83 calculator │
│feild of Algebra │ │numbers │using TI 89 │ │
│quadratic sequences worksheets │free georgia ged practice test │algebraic fractional number │free9+testpapers │solving simultaneous nonlinear equations with │
│ │ │ │ │matlab │
│how to find exact y values in │were to get algebra problems answered │worksheets on word problems on │mathamatics │mcdougal littell algebra 1 anwsers free │
│graphing calculator │ │quadratic equations │ │ │
│how to get games on ti 84 plus │answers to algebra 2 conversions │minus and plus algebra calculator │how to factor cubed polynomials │radical expression calculator │
│ │fractions │ │ │ │
│solving equations with 3 │ │ │ │ │
│variables with a graphing │math quizzes for 9th grade │algebra free worksheets │application of square and cube root │software for algebra 2 │
│calculator │ │ │ │ │
│3D lagrange surface equation │graphing calculator pictures │prentice hall mathematics algebra 1 │mcdougal littell geometry book │simplifying expressions online calculator │
│ │ │teachers edition │answers │ │
│distributive property solver │using matlab solver simultaneous │partial(1st) │relevance of algebra │ti-84 plus app radical notation │
│ │equations │ │ │ │
Google visitors found us yesterday by using these math terms :
Ninth grade maths tutorials, Printable graphing paper with coordinates, adding and subtracting with scientific notation worksheet, worksheet add,subtract,divide, and multiply real numbers, Find the
greatest common factor of 30, 45, and 50, how to solve for x on a calculator.
HOT TO SOLVE INVERSE MATRICES WITH A TI-83 PLUS, mixed number to a decimal, multiplying integer games, change to radical form calculator, fractions worksheets for 5th graders prime numbers, equations
in three variables solver, Prentice Hall Mathematics.
Adding integers game, fractions "for 10th grade", adding and subtracting integers quiz, solve equasions, pre algebra software, how to store data on TI 89, ged study cards for ti 84 plus.
Plotting points pictures, free printable english homework for 6th graders, 6TH STANDARD MATHEMATICS PRACTISE SUMS, express the given improper fraction as the sum of a polynomial, math VIII Class,
Algebra 1 Prentice Hall.
Algebra Structure and Method online textbook, download free ks2 sat past questions, learn to do algebra free.
Sample grade sheet in excel, equation from graph, adding subtracting "real numbers" worksheets, sats ks3 science in arabic, simultaneous equation powerpoint, what steps do you take to change a
fraction into a decimal.
Writing mixed fractions as decimals, calculator minutes into decimals, simplifying radicals worksheets, ti-84 plus download, samples of exponential expression, parabola general formula.
Logarithms equation solver, how to solve algebra problems online for free, convert fraction to decimal.
Probability 8th grade worksheet, powepoint on graphing linear equations, term for converting minutes to algebraic number.
Topics in Algebra+Herstein+solution manual, nonhomogeneous differential equation, TI-84 Plus rom download, mcdougal littell algebra 1 answers, printable Homework sheets percentage calculating math
8th grade, percent of discount math worksheets, how to use a calculator to assist in pre algebra.
Accuracy second order differential equation, Prentice Hall Mathematics Agebra 1 answers, convert a fraction to decimal, glencoe free worksheets, free precalculus ebooks, INverse Matrix calculator
solution for simultaneous equations.
Mathematical trivia, trig answers, 1 is highest common factors of what two pairs of numbers?, mathematical poems, Free accounting ebooks downloads, maths-venn diagrams.
Trigometric identities and equation example, reducing the quadratic, Kumon Answer Book, free science exam papers for grade 7, conceptual physics solutions, the logic book fourth edition teacher
solutions, solved worded problems in sine law.
Hrs factoring calculator, Finding the roots of a quadratic equation by completing the square (worksheets), kumon answer book download, rules for subtacting negative and positive numbers and
fractions, 7 equations 7 unknowns, square of binomial with manipulatives.
Online math sheets for honors 7th grade\, How does SPSS compute principal axis factoring, software to teach linear equations.
Algebra denominator sum factor, MCQS on iron, complex analysis for mathamatics online lesson, graphing calculater, statistics projects for algebra.
Order of operations with exponents worksheets, expanded notation + decimals worksheet, percents and fractions equations, converting decimals into the lowest common fraction, CAT model question papers
in PDF form free downloads, solving 4th degree simultaneous equations in excel.
Least common denominator + excel, solving differential equations using Mathlab, java aptitude qustions with answer key, interpolation polynomial quadratic vba, square root of a decimal, how to
calculate the difference quotient, +quadric solver.
Solve when variable is divisor, evaluating exponents worksheet, divide with decimals worksheet, Step by step how to manual for algebra equations, software, 9TH GRADE math HOMEWORK, solve polynomial
unde square root.
Investigatory projects math, teach activity completing the square, order of operation examples for college algebra solved, algebra 1 worksheets printable, a hard algebra 2 problem, free printable
learning activities for 2 &3 rd grade.
Algebra tutor software, math investigatory project, printable math problems, boolean algebra calculator, 5th grade advanced math, PHYSICS WORK PROBLEMS WITH SOLUTION AND ANSWER, Highest Common Factor
Express decimals as fractions on ti 83, ADD subtract multiply divide integers worksheet, graphing x y worksheets , ti-89 solver domain, ebook for permutations and combinations.
Prentice hall mathematics algebra 1 WORKSHEETS, grade nine pre-algebra worksheets, quadratic simultaneous equation solver.
Pre algebra exponent unknown distributive, MATLAB nonlinear mixed integer, kumon materials download, algebra 2 softvare, adding and subtracting practice, non homogeneous difference equations
quadratic forms.
Free downloading of mathematics aptitude, Permutation in our daily life, adding in scientific notation worksheet, free download of cost accounting text book, root rules, free math downloads 8th
grade, solving equations with more than one operation and worksheet.
Deviding negatives, solve algebra problems free, how to solve applying principal to formula in algebra.
Use quadratic equations on ti 89, graphing rational square root, solving vector identities, foiling math.
Free accerated math for 3rd grade, free download apttitude test papers, abstract algebra dummit download, factor quadratic calculator.
College worksheets, pre algebra distributive property problems, how to get an absolute value sign on a ti-84 plus, year 7 algebra download, sample blank iq answer sheets, exponent in Algebra Equation
Solver, free 9th grade english worksheets.
Solving linear equations lesson plan, multiplying integers worksheets, CALCULATING STANDARD DEVIATION FROM AN EQUATION OF A PARABOLA.
Mcdougal littell answer sheets, aptitude objective question and answers, subtracting negative numbers worksheets, powerpoint lesson coordinate plane, how do you do partial sums, websites for
practicing pictographs with elementary, Beginning Algebra An Inquiry Approach.
Prealgerbra worksheets, root of a third order polynomial, how do i get cube route on excel, quadratic calculator program, math formula pdf.
Ti 83 calculator download, ti 89 help store info, glencoe physics assessment answer, basic algebra PDF, Least Common Multiple Calculator, fraction algebra game, algerbra learn.
"ti-84 se" rom download, algebraic expressions real world, maths quizs, ti-84 calculator programs.
Simultaneous equation calculator by elimination method, math homework sheets, calculator math sheet, extraneous solutions calculator.
Congruence and proof ks4 worksheet, rate of change practice problems algebra, difference of cubes calculator, mixed fraction to decimal, poly solve.
Free printable rules of property worksheets, program solving equations multivariable, how to simplify by factoring, download ti roms, COMPLETING THE SQAURE, prealgerba math.
Simultaneous equation calculator, line chart in c# using maths sin, free manuals intermediate accounting, simple way to calculate mathematics.
Algebra 1- systems of equations worksheets, quadratic formula ti-89, lcm math print out sheets, intermediate algebra lessons, free algebra 1 structure and method practice book".
Adding & Subtracting Integers & Practice Problems, find slope using ti 83, parabolic graph problems and real life, 8th grade math trivia, MCQ test questions on fluid mechanics, TI 84 Downloadable
Calculator Games.
How to graph algebraically, printable exponent practice problems, factor equations calculator.
Quadratic root calculator, find equation for given point quadratic, Qbasic programme to solve simultaneous equation, FORMULA SQUARE ROUTE ON EXCEL, how to solve intermediate algebra formulas,
expression parentheses worksheet for 4th grade, second order differential equation calculator.
Math worksheets on adding, subtracting, multiplying and dividing integers, give me some irrational and rational problem and answers for algebra, NINTH GRADE ALGERBRA PROBLEMS, houghton mifflin
california math 6th grade answer sheets, free onlilne equation solver, free ebooks on MATLAB, algebraic expression of subtraction.
Holt biology worksheet answers, standard deviation t-83 plus, Hands-On Math Projects exponents, steps to fallow on solving equation containing radical expressions.
How to calculate exponents on TI-89, yr 8 maths past paper questions on percentages, Free Order of Operations Worksheets, how to solve a nonlinear differential equation in matlab, simplifying
variable expressions lessons, trig calculater.
Ordering & graphing decimals worksheet, how to do square root, riemann sums solver, Importance of numbers in everyday life, abosulte value graph range domain, how does knowledge of simplifying an
expression help solve a problem, accounting ebooks free download.
Middle school math Couse 1 textbook (not for sale) for homework, powell dog leg code, 6th grade review math worksheet to print.
Free 7th grade worksheets all subjects, one over the cubic root of x plus the cubic root of y, ti 84 plus college alg applications.
Online Step by step how to manual for algebra equations, math investigatory projects, how to find square roots, instructions on multiplying and dividing like terms, solving quadratic equations using
the TI-83.
Factoring equation calculator, College Software homework help, simplifying cube root fractions, math for dummies, how to solve equations with variables in the exponents.
Alegebra help, writing two linear equations on a graph, Free Pre-Algebra Pre-test, logic math probelms for kids, algebra for 5th grade, evaluating radical expressions.
Simplifying expression, equations involving rational expressions lesson plan, solving sqaure root and 6th root, freedownload ebooks of fundamental Accounting, highest common factor test, trigonometry
proving difficult sample problems.
Ucsmp lesson master answers, EXPONENTS RULES SAMPLES OF QUESTIONS, I want a pretest sheet for fractions/algerbra, how to add fractions formula, common topic in powerpoint (linear), foerster "algebra
& trigonometry" chapter 2 test online, ti 84 online.
Easy way to understand algebra, free download worksheet, prentice hall california mathematics algebra online version.
FREE ONLINE MATH WORD PROBLEM SOLVER, sats ks3 ppt, add, subtract, multiply, and divide integers + worksheets, evaluate exponents free printables, sums of combinations, multistep equations with
fractions worksheets, free online algebra solver.
Nonlinear differential equations matlab, algebra formulas squaring, adding and subtracting work sheets for kids in grade 1, convert 8/5 to a decimal, hyperbola solver, java random number + 13
Solving restrictions, inequality equation worksheets, linear programming for dummies, free math homework answers, multiplication of fractional exponents, m2 to lineal metres conversion, adding and
subtracting signed numbers worksheet.
Algebra 2 group test cpm, rom ti download, 2nd grade math freeware, radical calculator, lesson plan pythagorean theorem 7th grade.
TI 84 CURSOR, algebra real life activities, how to calculate slope w/ ti 84 plus, matlab second order ode.
Quadratic parabola revision, aptitude test book download, factor by completing the sqaure, "mathcad free download"+"chemistry", online TI 84, using a TI-83 calculator to graph points, find the range
of a irrational equation.
Problems math factors, parabolic math expresions, finding square roots of a fraction.
Fifth grade math worksheets to print, algrebra test, REpresent each decimal as a mixed number.
2nd order linear differential non-homogeneous solution, adding integers worksheets, add subtract integers worksheet free, algebra explained to beginners, free printable seventh grade work sheets.
Full aptitude test paper in PDF, ti 89 covert to base 2, mathmatic ratios, mathematica graph hyperbola.
Online factoring program, free download aptitude problems on volumes, printable question paper math quiz, maths and statistics answer.
Different poems in quadratic equations, factor cubed polynomial, mathamatical place values with decimals, ks3 pie chart worksheets, quotients of radicals, solutions on solving age problems.
Mathmatic symbol for squared, sample math tests sixth grade, factoring algebraic equations, simplify square root, intermediate algebra problems using the elimination method.
Saxon algebra 2 answer key, what is radical form, graphing coordinate plane activities pictures, free year 8 algebra worksheets, cost accounting for dummies.
Absolute value practice problems math, linear square feet, simplify algebra equations, Pre-Algebra Combining like terms Worksheet, java sum of digits from input, first grade math homework.
Free algebra exercises with answer keys, 6th grade, manual for texas instrument T1-81, what is sum += number; in java.
Adding and subtracting rational expressions 7th grade, financial accounting and download free solution manual, free download general english questions model, quadratic sequences worksheet, creating
graphs with linear and non linear relationships.
East west straight line calculator, cpm algebra 2 math tutor, what a 5th grader need to know.
Emulate ti 84, how to do square roots, ti 84 emulation, summation statistics.
Base calculator decimals, order of opeations hands on activity- 9th grade, online ti 84 plus calculator, factoring algebra equations, online simplest form calculators.
Identity equation(examples, Online general college math problems and solutions, ti-89 linear first order, calculate slops explained for kids, how to solve homogeneous linear differential equation.
Abstract algebra online study giude, examples of trivia, solving first order partial differential equations, algebra expression D or an F, multi-step equation worksheets.
LOGICAL REASONING worksheets, exponential square root calculator, printable 6th grade math crossword puzzle.
Solve cubic polynomial equations+decimal numbers, algebra exercises tests online, how to determine a equation with 2 variables, fifth root on scientific calc, answers world history chapter 12
worksheet, converting from biginteger to bigdecimal in java.
Simple equations worksheets, write an equation from data algebra worksheet, algebra for dummy, answers to holt biology study guide, calculator solving algebra expressions.
LESSON PLAN IN MULTIPLYING AND DIVIDING INTEGERS, ladder method, free printable two-step word problems addition and subtraction.
3RD DEGREE EQUATION SOLUTION, free ti 89 games and programs, Free Worksheets Order Operations, free worksheets Adding and Subtracting wholes and decimals using rounding, grade 6 pre algebra.
Quadratic formula with exponents higher than 2, 6th grade math variables worksheets, factoringalgebra, java code to input 10 names, lineal metre, radical equation word problems.
Poems about algebra, 9th grade arithmetic information, using a root finder in mathematica, rational expressions and rates, aptitude test download.
Multiplying quadratics program, conceptual physics workbook answers, mathematics for beginers, lesson plan add subtract multiply decimals, solving second order differential equation in matlab,
fractions equations "greatest common denominator".
Multiply expressions rules, interactive games for math ninth grade angles, easy way to solve alebra eqations and inequalities, algebra promblems and anwsers, finding median worksheets free.
Square root of 25 plus x square, clep college algebra sample questions, Finding decimal for mixed numbers, how to get cube root for fraction, free teach yourself algebra online.
Free 7th grade math worksheets printable homeschool, algerba 1a calculater, online maths sample examples, simplify imaginary number calculator, aptitude question papers, kumon answers g.
Need help with finding ansawers to my physics homework, order percentage from least to greatest worksheet, TI-83 plus emulator, ti84.rom free.
Use the quadratic formula radicals, a programm in c++ to find greatest common divisor and least common multiple of two numbers, sample lesson plan in multiplying radicals for seventh grade, math made
easy pearson grade 4, square wave differential equation.
Quadratic equation three variables, how to solve trinomials on a ti-83 calculator, online t.i. 84 emulator, online algebra calculator simplify, saxon 8/7 test 1b.
PPC y=mx+b Economic graph help, formula for percentage, cubed root on ti-84 plus, algebraic formulas.
Free factions to decimals worksheets, dummit and foote, maths revision yr 8 exam, find a percentage in algebraic equation, easy way to learn algebra, finding slope from a function table practice.
Adding and Subtracting Matrix Worksheet, 6th grade graphing activities, online quadratic factoring calculator, how to calculate LCM in c programming, how convert decimals into fractions with a
calculator, add integers.
Free worrk sheet on money sheet 8 th grade, polynomials worksheets creative publications, can solve nonlinear differential equations matlab.
Simplify algebra software, Glenco Algebra 2 worksheets, Learning activity on exponential laws grade 9, Dummit & Foote Solutions.
Cube root calculate integer, calculator to solve rational function equation, intermediate algebra real world approach, solving square root and 6th root.
Free math worksheets radicals, algebra question more on getting lcd factoring, square root method.
Online aptitude questions with solved answers, missing integer, algebra writing equations worksheet, easy way of presenting linear equation, math investigatory project, simplifying equations positive
Dividing calculator, algerbra sums, Algebra and Trigonometry Structure and Method - Book 2 Mcdougal Littell workbook, factorization questions for 5th grade, sample worked exampapers for gce maths
Boolean algebra simplifier, free ged printouts, ti 83 plus calculator factoring quadratics, grade 6 and patterning and algebra, aptitude question answer, mcdougal littell math practice workbook
answers, ti84 calculator emulator.
Solved paper for aptitude problem on ages, calculating eigenvectors on ti 83 calculator, HOW TO PASS COLLEGE, funtion math in java, solving for a variable in a limit, Greatest Common Factor/Least
Common Multiple Lesson Plan, Free GED math print outs.
Adding 3 fractions worksheet, free ebook aptittude, free algebra problem solver, free printable cummutative property worksheets, Accountancy book free download, algebra s worksheet, english maths
lattice square.
9th grade math worksheets, ways on how to extract of Cube Root, geometry coordinate plane worksheets.
Multiple variable equations', algebra seventh grade help with two step equations, aptitude quwestion for basic engineering chemistry, Math Trivia Questions for Elementary Students, Ninth Grade
Algebra, square root complex equation solver.
Math test online for year 2, 100 problem distributive property worksheets pre-algebra, 5th grade Histogram worksheets, 5th grade math worksheets square, adding and subtracting signed numbers
worksheets, +ti84 +algebraic, factoring trinomial calculator.
Math handouts Algebra exponents, square root method quadratic, solving linear equations using graphs.
Planning exercies for MOD aptitude, how to get ROM off a ti-83, Simultaneous Equation Solver, how to make a slope intercept graph on excel, interactive adding and subtracting integers, x square root
calculator, simplifying radicals calculator.
Least common of 14 and 22, free aptitude text, DIVIDE WITH DECIMALS WORKSHEET.
Ti 89 math problem solver, nonlinear equation solver, gnuplot linear regression, ti 84 download, formula of roots, MATHEMATICAL POEMS WITH ANSWERS, finding common denominators algebra.
Practice for multiplying and dividing, multiplying an integer by a double, trigonometry powerpoint lessons, cupertino font download, math word problems.com, 7th grade worksheets.
Dividing decimals free worksheets, vb6 Quadratic Formula, intercept formula, matlab solve quadratic equation.
Free Math Printable Worksheets Integers, systems of equations absolute value, multiply and divide expressions involving exponents, rationalizing the numerator, convert degree, minutes into decimals.
Codes for puzzle pack on a graphing calculator, learn how to solve conversions, simplify cubed roots, Free Multiplication Tables Printout.
Partial sum addition samples, square root property of quadratic equations, root solver, ordering numbers worksheets fractions percents decimals free, +Chapter 1 Resource Masters, Glencoe, pre
algebra, FREE SIMPLIFYING RATIO WORKSHEET.
Phase plot nonlinear in maple, free solved problems in trigonometry, radical form, online radical calculator, online calculus tutorial for beginners.
Cost accounting free ebooks, highest common factors between 76 and 108, "maple" "logarithms with different bases", Free Pre-Algebra answers, java examples exponent.
Quadratic equation converter, equation fractions, calculator, quadratic equations with constraints, highest common factor of 33 and 93, math worksheets beginning algebra, chapter10 test form B ucsmp
advanced algebra.
Probability worksheets for 9th grade, writing equations math worksheet free, introduction to hyperbolas in school, cost accounting assignment worksheet, math pre-algebra practice workbook need help.
Exponent activities 5th grade, how to find graph equation hyperbola, phoenix ti 83 emulator, lattice multiplication printable worksheets, mix fractions, worksheets on fractions multiply and divide.
3rd Grade Math Homework Printouts, beginning algebra graphing made simple free examples, ppt mathematics grade 7 algebra, saxon math algebra 1 sample test.
Download TI-84 plus calculator, windows aptitude questions ebook free download, software test cases for quadratic equation, FREE MATH TRIVIA, 5th 6th 7th gr reading worksheet, how do i convert
percent into decimal on ti-84.
Greatest common division, solve by completing the square calculator, algebra help books, plot 3rd order polynomial matlab, determine the vertex algebraically, associative property of multiplication
Alegbra symbols, "formula for converting fractions to decimals", radical simplification calculator, literal equations - worksheets, java while loop summation, algebraic expansion and factorization,
what's in the solution manual of forster.
Printable math lessons algebra equations, pre algibra, basic alebra, addition and subtraction equations, grade 11 linear algebra questions.
How to calculate cubed on ti 83, the newest math program, ti-84 linear equation activity, prentice hall mathematics algebra 1.
Printable 9th grade worksheets, trivia printable worksheets junior high, clep college algebra workbook, create your own problem- combining like terms, formula for factorization, how do you find a
square root on a basic calculator, online factoring tree calculator.
Samples of 6th grade book reports, FREE WORK SHEET ON ARITHMETIC LAWS, simplifying radicals with absolute value, adding and subtracting decimal worksheet, solve multiple linear equations in excel.
What is the formula for square root, free math worksheets adding and subtracting negative and positive numbers, trignometry for beginners.
Multiplying equations with exponents, holt mathematics powerpoint for teachers in algebra, simplifying radicals equations.
Prime factorization worksheets, glencoe advanced mathematical concepts solutions, greatest common factors worksheet.
Computation of integers worksheets, symbols to subtract worksheets, math sheet adding, subtracting, multiplying, dividing integers, free accounting download books or notes, newton's method for
multiple nonlinear.
Calculator factor quadratic equations, lesson plan for Adding and subtracting expressions, Online algebra tutor.
Free math programs, free permutations worksheets, calculating reciprocal for middle grade students, Calculator For Algebra.
Daily trigonomic equations, find the slope of a line calculator, factor tree worksheet, how to simplify a+2(b-2), free 9th grade science worksheets.
T1 83 Online Graphing Calculator, free math solution books, equation worksheet, free online college algebra, dividing polynomials calculator, aptitude questions+download.
Algebra is Easy, math for dummies free practice, example palindrome program and output in java, Greatest Common Divisor Calculator, base amount and rate percent equations for percentages.
Evaluating expressions worksheets, solve function calculator, fifth grade-math.com, decimal to mixed numbers, translate into algebraic expressions worksheets, solving 3rd order equation, Saxon Math
Holt math, eoc algebra 2 exam, solving algebra equations.
Sample test question about radical expression, how to square a fraction, solving second order difference equations, square a decimal number.
Square second order polynomial, practice "math sheets" third graders, solving 2nd order differential equation, Prime factorization worksheets for sixth grade, TI-84 emulator, Least Common Multiple
Lesson plans for teaching algebra in fourth grade., holt algebra 1 answer sheet, system of equations 3 variable online calculator, 3 times the square root of 2, college algebra polynomials using long
division problems examples.
Free pdf book in accounting, gmat maths examples inequalities, Adding subtracting multiply divide Integers, grade five exam papers, terminate current if loop in java.
Transition year "maths probability", langauage aptitude question download, adding and subtracting positive and negative numbers game.
Convert decimal to radical, solve quadratic equations by square roots, pre algabra answers, prentice hall answer key algebra 1, solve multiple equations with a ti-89.
How to explain subtracting integers, how to solve linear equations with fractions, math problems using order of operation, sixth grade games powers line, rational expression calculator, changing
mixed numbers to decimal worksheets.
Objective maths book, three unknown two condition problem, mcdougal, littell wordskills answer key, ti statistics cheat, formula calculate common denominator, simultaneous equation solver quadratic
applet, sample questions for Iowa Algebra Aptitude test.
Factor tree for even numbers 1-50, free math worksheet for 2nd grad, worksheets on finding the absolute value of a number using a number line, write addition and subtraction expressions.
Addition and subtracting algebraic fractions, reason why we simplify radical denominators, simultaneous equations year 10-worksheets, how to convert a mixed fraction into a decimal, algebrator
special 0ffer, free algebraic linear programming solver online, compound intrest+maths.
How to solve an equation for area, examples of linear problems, Pre-Algebra substitution activities.
Model Solving equations pictures and examples, online college algebra problem solvers, decimal to radical, pre-algebra properties and operations, online rational expressions calculator.
Numbers worksheets, maths games for yr 8, elimination advance algebra, algebra fraction calculator with exponents, learn algebra.
4th grade math - arrays, math combination 6th grade, permutations "box method" math, Foundations For Algebra Crossword, simultaneous equation solver three unknown flash, WHAT ARE THE BASIC RULES WHEN
Fundamental chemsitry/pratice test, aptitude test papers with answers, worksheets on line plots for first graders.
Ti 83 quadratic formula program, free download of book of aptitude & Reasoning +PDF, simplifying cubed polynomials, polynom solver.
Mcdogal answers cheat sheet, free online tutoiing grade six level, what does multiplying integers mean.
Problems sample on literal equation, prentice hall mathematics work book pages, divide polynomials calculator, revision exercises for primary science heat, offline aptitude test download, vertex with
absolute value.
Answers to mastering physics, permutation and combination coding in data structure, solver variables 3 degree polynomial.
From a three digit number, how do you get the value of the tens using Java, calculator activities fifth grade, TI-30X iis quadratic, how to solve bases problems on a calculator.
Exponent worksheets, standard form +mathamatics, INTERMEDIATE ALGEBRA step by step amswers free, calculator for solving rational expressions, free printable 9th grade math worksheets, algebra work
sheets involving polynomials, Completing the Square- TI 83 Plus.
Worksheets on adding fractions with positive numbers, algabraic formulas, Free Algebra Equation Solver, inequalities math problems printable.
Algebra projects for 9th graders, Algebra 1 Holt, convert to lineal metres calculator, FREE download trigonomic calculator, 3x-7y=2.
Convert quadratic function to vertex-form, easy way to learn division facts, free equation calculator, holt algebra 1 book answers, estimate square roots and cube roots worksheet, population variance
programming ti-83 free.
Advanced algebra parabola, question paper of mathematics for 9th standard, exponents and square roots, advance algebra question and answers, downloads gr ti 84 +.
Quadratic equations pi, how to do algebra free, algebra combining like terms worksheet, scientific notation nasa, free printable scientific method worksheet, 9th std trignometry, solving polynomial
equation in c++.
Absolute value solver, fun ways to teach lcm, ti 89 rom download, factorizing Quadratic calculator.
Compound inequalities powerpoint, free worksheet on finding the least common denominator, 3-6 test solving inequalites using addition and subtraction, simplification of exponential variable.
Everyday algebra equations, how to take the cube root on ti-89 plus, APTITUDE QUESTION PAPERS WITH SOLUTION, how to teach 7th grade algerbra, maTH TRIVIA QUESTIONS, factoring worksheets HARD.
I need help in algabra 2, online inequality graphing calculator, quadratic factoring calculator, Free PRE Algebra Worksheets, solving a third order polynomial, powerpoint of linear equations,
beginning algebra practice test.
Multiplying rational expressions with multiple variables, lesson plan from first grade to sixth grade, ti-89 equation writer program.
Group work lessons business applications absolute value, radical equations, quadratics, completing the square projects, Babylonian Square root java source code, Simplify by reducing the index of the
Questions on discrete maths-groups and rings, highest common factor of 14 and 49, mcdougal littell algebra 2 anwsers, Equations with Fractional Coefficients.
Solving a system of equations in for loop through matlab, free math glencoe worksheets, calculator "tricks", helpful websites statistics, elementary math trivia, probability exercices.
Algebra solver online california, like terms algebra activity, how to do algebra, graphing on the coordinate plane powerpoint lessons, algebra problems examples, maths book permutation, on line slope
Teaching kids to type, math trivia for kids, vb6 square root function, trigonometric simultaneous equations solution, holt mathematics variables and expressions lesson 1-7, intermedite 2 maths, ratio
Algebra help program, square root of exponents, permutaion and combination worksheet.
Free lcm math printables, grade 6 and algebra, history of adding and subtracting Integers.
Rudin solved problems, Step by step to sole a quadratic equation, simplifying variables worksheet, factoring and grouping and difference of two squares, ks2 place brackets in equations problems,
multimedia aptitude question.
Algebra applications to work, algebra 1 easy learning, ALGEBRA AND TRIGONOMETRY WITH ANALYTIC GEOMETRY (Classic Edition), Eleventh Edition DOWNLOAD, kumon answers, online factor trinomial, free
printable 9th grade scool work, Convert decimal numbers to exact numbers.
Ladder method of division, answer key for Holt Algebra 1, solve for x online.
Subtracting exponents in quadratic equations, matlab second order differential equation, simplifying polynomial equations.
Formula in calculas, how order of operations determines how you evaluate an algebraic expression, how to work out a scale factor, free exponent worksheet, GLENCOE MATHEMATICS COURSE 3ONLINE STUDY
TOOLS, how to use calculator for fractions, english homework for kids printable worksheets for 6th graders.
How To Enter Log On Ti-83, FREE SINGAPORE SECONDARY 2 TEST PAPER, partial fraction freeware pure maths, printable integer math quizzes, free printoutsof triangles squares and circles, revision for
gcse o level maths vectors online.
Simple equations problems exercise, multiplying fractions worksheets interactive 7th grade, free math printouts, Variables and Patterns introducing Algebra Connected math 2, teacher's edition.
Especial edition for santa monica college chemistry homework solution, how to solve complex algebraic expressions, multivariable trig equation solver, ideas positive and negative integer games, besic
maths, solving an equation in excel.
Algebra distance calculation, apptitude questions with answer, quadratic equations a level easy root step by step, 8th grade quadratic equations.
Math expression problems online, reducing radicals printable practice, ti.84.plus eigenvalue, algebra - dividing.
Mathmatical equations with a caculator, Prentice Hall Mathmatics, year 7 algebra problems .doc, algebra find exponential variable.
Maths 11 -12 years, ALGEBRA 1 MATHMATIC, Convert base 16 to decimal form (base 10) matlab, ti89 find inverse.
Algebra II answers, Aptitude question, problems and answers to algebra 2 conversions fractions, 3 simultaneous equations calculator, integer worksheet.
How to cheat with TI 84, finding multiple roots + newton's method + matlab, alegebra practice, accounting pretest lesson plan, MATHAMATICS, logaritmo base 10+ti 89.
Lesson plans "converting temperature", integers and the coordinate plane, gcd solver, convolutions ti-89, integers adding, subtracting, multiplying and dividing, free math worksheets graphing linear,
Interger worksheet.
Free help for seventh grade math az, find the square of a decimal, holt algebra 1 7th grade michigan, subtraction of integers worksheets, graphing on coordinate plane free printables for algebra.
Cube root ti 83 calculator, math trivia, solved sums+class 8 mathematics, free subtracting negative numbers worksheet.
Show step by step algebra free by teacher videos, printable permutations and combinations worksheets, completing the square questions gcse, evaluating expressions advanced worksheet, solution set
Solve quadratic equation for b, free printable prime factorization multiple choice quiz maker, rudin solution manual, online version mcdougal littell algebra structure and method, integers games,
matlab numerical solution of second order difference equation.
"synthetic division" AND tutorial, simplifying equations positive exponents square root, free writing algebraic expressions worksheets, functions and relations worksheets, free printable mathematica
work sheets.
Free english worksheets for 6th graders, cubed equations, formula for ratio.
Solving quadratic equations using matrices, cost accounting solved examples, integration substitution method, adding percentages "two variables", 9th grade biology worksheet answers, year seven math
Algebra substitution worksheet, "fraction word problems" gcse, free high school algebra problems, fraction under a radical sign, algebra 2 workbook solution.
Factoring cubed polynomial, free algebra programs downloads, vb6 percentages subtract, c aptitude questions with answers, Algebra 2 Answer Keys.
Logarithmic in visual c#, algebraic expressions worksheet, permutation and combination solved question, variable exponents, pre-algreba help with graphing.
Need tutor for algebra, Maximize: z = 4x + y subject to the constraints: 0 <= x <= 7 0 <= y <= 8 x + y >= 2, holt algebra 1 california textbook, Solve the second-order homogeneous differential
equations, math number sequences solver.
College algebra for dummies, Download aptitude test, CONVERSION RECTANGULAR TO CUBE, Free Adding, Subtracting, Multiplying and Dividing Integers Worksheets, solve second order nonlinear differential
Algebra trivia, how to solve second order differential equations, how can we free download the manual of cost accounting by charles T 12 edition, free fraction solving calculator, free homework math
problem solving radicals addition problems.
Egyptian math activities ks2, Worksheets on multiplying and dividing negative numbers, prentice hall algebra 1 florida, EQUATIONS EASY SUBTRACTION.
+mathimatics for second grade, adding subtracting multiplying and dividing integers practice, Worksheets of Graphing Pictures on Coordinate Plane, Key Maths-Homework Sheet, explicit formula algebra
Green's function for solving partial differential equations, algebra problems solutions parabola, 9th ninth grade maths .ppt US, ti-84 for dummies free download, free 5th grade prime numbers
printable sheets, amatyc tests, concept of algebra.
Littell TAKS Workbook, DAILY LIFE APPLICATION OF QUADRATIC INEQUALITIES, how to solve math matrices.
Square pyramids/algebra, calculation curl using Maple, simplifying each side of an equation help.
Probability without replacement worksheet easy and free, all partials method 4th grade, free downloads algebra function charts, free download algebra emulator.
Mathmatic&integrations, ti 83 rom download, college algebra, how to do simultaneous quadratic equations, number converter machine + java, sample clep algebra exam pdf.
Math trivia with answers, Aptitude for English, simultaneos homgeneous linear differential equation, Worksheet on Hyperbola graphs, best grade 11 algebra learning book, do a maths exam online,
Logarithmic equations solver, multiple answers simultaneous equations matlab, secondary one mathematics worksheet + seventh grade, printable free 6th grade math, simplifying by factoring, TI how to
graph a cube root, online advance calculator.
Simultaneous nonlinear equations, easiest directions for programming TI-82 quadratic equation, glencoe algebra 1 worksheets skills practice, printable worksheets for third grade for data analysis and
probablilty, expressions for quadratic equations graph, adding negative number worksheet, what is multiplication expression.
Find the inverse on TI-83 Plus, free e-books on accountancy formulas, free printable prime factorization worksheets, converting mixed numbers, Abstract Algebra help solutions, Free Simplifying and
Solving equations Algebra I free worksheets.
Expressions free worksheet, solving statistical slope, southern western volume 1 and 2 gcse, online science test paper, printables maths worksheet for secondary school, solving equations printable
Prentice hall algebra 2 answers, printable order of operations free worksheets, free aptitude question download, FREE ALGEBRA VIDEOS, prentice hall 6th math pattern worksheets, free evaluating
Expressions worksheets.
5th Grade Math Review Sheets, grade six fraction worksheet, why do a check on a rational equation, cube root quadratic formula ti 83, multiplying with fractional exponents.
Solving for a given variable, TI 84 hoe games downloaden, ti-84 calculator emulator, when muitiplying what equals 47, adding scientific notation, equation solve 1+2+3+4+5+6+7+8+9+10.
Exponential, simplify algebra equations calculator, free "grade 10" maths worksheets.
Foiling cubic equations, "factor tree" flash, Adding Subtracting Integers Worksheets, simplification of fractional Exponents Expressions, how to make graphs on powerpoint for algebra, Square Root
Calculator, college algebra toutor.
Graphing limits TI-84 Plus, multiply and divide integers worksheet, freemath printouts, tutorial on incidence matrix, math slope intercept worksheet.
Powerpoint online examination paper, free printable algebra games, 8 en decimal, pre algebra definitions, algebra checker.
Accounting Exams with Solutions pdf free, free pre alg worksheets, emulation TI 84, polynominals, prentice hall pre-algebra chapter tests, probability projects for algebra, vertex form write equation
given point.
Worksheets free quadratic sequences, trigometric equation example, FREE TUTORING ONLINE FOR ALGEBRA, algebra 1 holt california answers, TI84 practice problem, evaluating expressions activities.
Find the slope graphing calculator, some mathematical worksheets for v standard, cpt test 4th grade, solve third order equation, college level algebra solver, difference quotient solver, examples of
math trivia for elementary.
Word problems for 5th graders, history grade 11 printable worksheet, intermediate algebra cheat sheet, ti-89 electric applications, using polynomial division in real life.
Ti-89 to solve differentials, show step by step algebra by vidio free lessons with instructor, multiplying radicals calculators, how to simplify cubed roots.
Determining vertex math formula, java cube root calculate, cube route on excel.
Pre-algebra book free to look at online by mcdougal littell, TI 84 Calculator simulator, online factorer, math tricks for investigatory, standard equation of hyperbola proof.
Ti-83:linear programing, ti-83 quadratic equations, fraction from least to greatest, algebra square root calculator, solve an algebra problem, printable 4th grade math beginning pre algebra,
printable easy algebra math worksheets integers.
Prentice Hall Mathematics Online Algebra 2 Textbook Page 199, Adding and subtracting positive and negative numbers, make a mixed number with a decimal.
How to solve for slope, mixed numbers with variables worksheets, graphing cubed equations, adding negative numbers worksheets, how to solve exponents calculator, linear graph worksheet, ratio
Algibra, free online ti-84 calculator, example of trigonometric equation, solve nonhomogeneous second order differential equations, greatest common factor monomial calculator.
English mechanics worksheets, university of chicago algebra interactive problems, Factoring calculator - complex, texas TI 84 emulator free, Rules for adding and subtracting integers, homework
abstract algebra.
Holt physics problem workbook answers, adding/subtracting with decimals worksheets, Principle Of Algebra Lectures powerpoint, understand college algebra.
Free Online Algebra 2 Help, graphing integer inequalities, www.fl.algebra2.com.
Factoring tutorial, scale math, rationalizing a decimal.
Calculate gini excel, Download+Topics in Algebra+Herstein+Problems+solutions, free math answers and problems, ti-89 solve, adding fraction with integers, whole numbers, divisibility by 7 java
"Square root" + calculator, laplace transforms parctice probelms, examples of math poem mathematics, use free online algebra 2 calculator, calculate parameters a b and c, free online simultaneous
equation solver, fraction computation worksheets.
9th grade english worksheets, adding scientific notation worksheets, video on how to do algebra honors, Free Grade 7 worksheets order of opertions, printable math homework sheets for third graders,
basic maths for idiots.
Accounting ebook free, binary math powerpoint game, Grade one Math workbook page.
Linear algebra tutors in singapore, adding and subtracting negative and positive numbers, add subtract multiply divide first integers, Specified Variable, calculator square roots drill, how to solve
3rd order polynomials, answer keys for McGraw Hill 11th edition college acounting.
Simultaneous equations calculator with 3 variables, step by step algebra free, simplifying radical expressions calculator, t1-83 calculator inequality symbols.
Factoring word problems for grade 7, free cost accounting work sheets, journals involving multiplication of integers.
Combining like terms / worksheet, solving 2-variable inequalities, free visual math worksheets.
Evaluating algebraic expressions worksheet, ti-83 tricks, algebra 1 chapter 2 lesson 2 practice b worksheet, square root decimals, algebra examples power.
Preparing for iowa algebra readiness test, worksheet for subtracting integers, 3 unknowns equation solver.
Factoring cubed, McDougal Littell Reading Placement Test, finding the lowest common denominator calculator, dividing fraction games, percentages gcse worksheet.
An easy method to find the sum of count to 100, subtracting integers worksheet, quad solve ti-84 plus, decimal to radical form converter, least to greatest percents decimals and fractions.
Training algebra igcse, ti 89 handbook complex number conversion, ti-84 plus downloads.
College Algebra Tutor Software, math trivia about functions, what is square of difference, second order homogeneous differential equation, worksheet games on factors and multiples, examples of math
trivia in geometry.
Liner equation formula, two transformation equation worksheet, Algebra Problem Solvers for Free, 3rd degree polynomial calculator, free ppt in ratio and proportion, aptitude question, what to do when
factoring a trinomial with an online calculator.
COLLEGE ALGEBRA EXAMPLE, the definitions of the substitution properties of algebra, free pre algebra notes, decimal to fraction formula, geometry dilation worksheets.
Scott foresman mathematics grade 3 test exam, vertices asymptotes, Maths Games for KS2 - Algebra, solving integers worksheet.
Solve the formula for the specified variable, CONVERTING DECIMALS TO MIXED NUMBERS, solving trinomial by graphing, interval notation calculator, probability worksheets, games for tables and charts
use in grade 4 maths class.
Teach yourself visual basic 6 for beginners for free, algebra cheat sheet .tex, radical expressions calculator, calculator radical.
Free algebra puzzles, programs for TI-84 plus, exponents and multipication, Math Answers Cheat, free software download of algebrator, using ti83 to solve systems of equations.
Free math test for fifth graders, Quiz Algebra 6th grade, free order of operations math printouts.
Texas ti-83 log, evaluate exponents printables, free online sample questions for 11 year olds multiple choice questions, algebra problems for 9th grade, solving math epuation cheat sheets, Simplify a
Term Under a Radical Sign, integers worksheet.
Sixth grade math puzzles printables, www.concepts.glenco.com, Mean worksheets.
Maths sats exams ks3 online, 9th grade ratio worksheets, how to change mixed number into a decimal, solve formulas for specified variables.
Ti-82 emulator rom download, teaching algebra to 5th graders, cubed polynomials, simplify expressions worksheet.
Algebrator softmath, intermediate algebra printables, mcdougal littell for fourth grade, simplifying expression free calculator.
Algebra square roots A+B, excel equations subtraction, Excel free online help for multiplying and dividing.
Polynomial solver, ti-83, can matlab integrate nonlinear differential equations, Quadratic Factorization Exercise.
Onilne asymptotes calculator, simplify radical exponents xy, dilations worksheets, calculate maximum and minimum values algebraically, calculate lcm java source code, free algebraic calculator, "math
projects"+"8th grade"+"linear".
Multiplying rational expressions algebra equations calculator, 5th grader math puzzle answers, "real life applications of rational equations", TI-84 plus calculator/manual.
Algebra power, finding fractions from least to greatest, Greatest Common Factor/Least Common Multiple Lesson Plan-game.
Examples of why we use scientific notation, subtracting integers worksheets, multiplying and dividing decimals worksheets, how to find the vertex with a graphing calculator.
Solving multivariable 3rd degree polynomial, convert binomial numbers to real numbers, iowa algebra aptitude test interpreting scores, DECIMAL FRACTION FORMULAS, challenge math problems for 6th
graders, complex quadratic.
Plotting newton divided interpolation in matlab, evaluate algebraic expressions worksheet, rational exponents and radicals exercise, calc prime number using square root, CALCULATE LOWEST COMMON
Real-life applications of square root function graphs, grade fifth algebra software, free online elementary algebra.
Learn algebra fast, algebra worksheets, C program that can be used to solve quadratic equation, free algebra warm ups.
Greatest Common Factor word problems, evaluating algebra formulas, mechanical qudaratic formulas, cheat TI-89, Addition and subtraction of algebraic expressions.
Solving word problems by completing the square, TI-84 college algebra programs, greatest common factor activities, free online tuturial mathematic standard two, power point of special product and
factoring, order decimal from least to greatest, glencoe physics chapter assessment answer key.
Convert quadratic function to vertex form, saxon math test download, what is the algebraic ruling for multipication, adding and subtracting?, convert value to decimal "formula", algebra problem
solver, addition and subtraction algebra worksheets, converting decimals into fractions formula.
3 equation solver, evaluating algebraic expressions lesson plan- 6th grade, 5th grade printable worksheets on solving problems using estimation, how to add fractions with negatives, bungee jumping
"mastering physics".
How to calculate cube root in ti-89, inequalities worksheet, free download of solution for applied fluid mechanics, Solve 3 Equations with 3 Variables, square formula, 7th grade math problems
beginning algebra printable worksheets, simple math investigatory project.
Art language grade 3 worksheets, graph of cubed roots, mathematical poems about quadratic function.
Removing brackets in algebraic terms in year 11, solutions walter rudin, Topics in Algebra+Herstein+solutions+download, show step by step how to solve all math problems free online.
Different poems about quadratic equations, physics help for the glencoe homework book, problems and how to solve them for college lesson.
Explanation of multiplying and dividing fractions, root finding+third order polynomial equation, Free College Algebra Book, adding and subtracting Integers, ti 84 emulator, solve for divisor, year 11
printable practice maths tests australia.
Free multiplication online tic tac toe, linear algebra and its applications" david lay solution, equation games, square root calculator, equation writer for pocket pc.
Plato learning pathways teachers answer sheet, free 7th grade science tutors, 3rd order polynomial equation, math games and adding and subtracting integers, online calculator simplify rational
expressions, sample saxon algebra test, Algebra 2 Prentice Hall Mathematics Book Preview.
Online ti-84 calculator download, algebra calculator to solve equations with variables and fractions, system of nonlinear equation by matlab.
Pre algebra worksheet puzzle, equivalent decimals and fractions worksheets, learn algebra statistics program.
Finding slope of vertex form equation, polynomial of degree 3 stability, math worksheets graphing linear free, algebrator special offer, convert fraction to deciaml, math scale problems, fraction to
decimal conversion using javascript.
Solving simultaneous equations Excel, Lowest common denominator of 82, WORKSHEETS SOLVING ADDITION QUESTIONS.
Simultaneous equations matlab, radical expression calculator equation, ti-83 physics programing, where to get the answers for algebra 2 glencoe.
9th grade mathematical work, worksheets multiplication and division of rational expressions, SIMPLIFYING RATIONAL EXPRESSIONS WORKSHEETS, mathamitics exams paper grade 11.
Algebra reduce equations find lowest term, LIST COMMON MULTIPLES 0F 2, whole number fraction to decimal conversion, emulator state programs for ti-84, Free Iowa Basic Skills Test Practice Sheets,
equations in presentation.
Holt mathematics powerpoint for algebra 1, ti 84 plus download games, simplify cube roots.
Worksheets on compare and order rational numbers, including repeating decimals, cubic function vertical compression, online negative fraction exponent calculator, Algebra Linear Programming Word
Problems, basic maths revision and tutorial long division.
Parabola graphing with two variables, decimals worksheets 6th grade, positive and negative integer review, times table with decimals sheet to solve 9 grade.
Distributive / worksheet, graphic chart math fraction reduction formula, cost accounting book free, using flowcharts in solving expressions, rate of change formula, printable math sheets for 2
4th grade math worksheets addition and subtraction solving for N, Math work sheets compatible numbers, best algebra, algebra refresher sample solutions, calculate inverse Laplace transform using
TI-83 Plus, homework answers for even algebra 2 glencoe.
Coordinated plane, identifying algerbra properties, 1to100 fo ks2.
Exponents and roots activity, examples of math trivia with answers, free factoring algebraic equation.
Pre-algebraic third grade, javascript calculator divisor, circle graphs + worksheets.
To solve any order equation in excel, RUDIN PRINCIPLES OF MATHEMATICAL ANALYSIS SOLUTIONS, beginning algebra graphing made simple, finding common denominator in C, solve rational expressions.
Learning videos for algebra, finding variable percentage, construction math sheets, free beginning factions worksheets, what grade is an 11 yr. in?.
Solving for an unknown online, electrical math free online, indian maths work sheet.
RATIO FORMULA, free printable math problems for 7th grade, gcse worksheets computer science, Maths venn chart tutorial BBc.
Permutation and Combination + GMAT, TI 82 online, three unknown simultaneous equation solver, free algebra interactives, how to convert a farction into whole number with decimle place.
TI-84 plus rom image download, adding,subtracting,dividing,multiply mixed fractions, adding positive and negative numbers test, 19414.
Free first grade printable math test, 7th grade glencoe math, comparing and ordering decimals+free worksheets, convert squareroot fraction, finding least common denominator calculator.
Compound inequality answer calculator, free 11+ maths papers, math samples for free, lcd calculator, mathematical trivias.
College algebra online tutorial, write addition and subtraction expressions in fourth grade, algebra formulas sphere, solving large symbolic matrix, GRAPHING LINEAR EQUATIONS WITH FRACTIONS, star
test 6th grade science.
Pizza activity for multiplying integers, fun 9th grade math quizzes online, kumon exercises pdf, monomial puzzles, exponents and multiplication expressions, simultaneous equations matlab least
squares, algabra 2.
How to store equations in ti-89, definitionlesson plan, quadratic equations a level easy root, cost accounting books, pretence hall mathematics algebra I, percent worksheet for grade age, multiplying
rational expressions with exponents and multiple variables.
How to solve for x linear equation involving fractions, cost account hand book.pdf, extra work for 9th graders, mcdougal littell answers, how to pass year 8 english test, how to find the slope of
quadratic equation, algebra 1 practice problems for ninth graders.
Hybrid orbital practice equations, 1st year physics comprint ap inter mediate, Powerpoint on solving equations using decimals.
Download free accounting software, autoaoff TI, fractions formulas, how to solve cube root sums.
Best algebra calculator, Worksheets for algebra 1-2, fractions rule formula.
PROGRAMING WINDOW APPLICATION IN CALCULATER, Free online Geometry solver, practice workbook california mcdougal littell math algebra 1 answers.
Ti calculator program quadratic roots, 5th grade inequalities, free online algebra calculator, +factor tree worksheets.
FREE MATH PROBLEM SOLVER, solve rational equations, how does great common divisor works?, in mathematics "investigatory project", fraction formula.
Money math lessons for sixth graders, maths exams 5-7 ks4 practice online, softmath algebrator, free algebrator.
Search Engine users came to this page today by using these algebra terms:
• algebra cube chart
• free grade 10 maths study guide
• Yr 8 MATHS REVISION TESTS
• square root of x power 5
• adding, subtracting, multiplying, and dividing integers quiz
• glencoe alegebra 2
• A plane powerpoint
• greatest common denominator workshhets for six grade
• mathematics investigatory project
• Integer Worksheets Adding Subtracting, Multiplying, Dividing
• associative property worksheets
• Gmat Aptitude questions'
• algebra with pizzazz worksheets
• examples of COIN problems in college algebra
• c++ formula of root of quadratic equations
• math equation find our percentage of a number
• c++ polynomial root finder
• calculator for simplifying radicals
• percent equations formulas
• algebra trivia questions
• GRE math formula sheet
• dividing calculater
• algebra rules grade 8
• adding square roots fractions
• pre-algebra.com
• Glencoe mcgraw hill free teacher materials tech option
• 9th grade fractions
• adding and subtracting exponents worksheets
• Factorise polynomials using algebraic identities
• online function solver
• algebra poems
• exponent rules for algebra
• how to put a quadratic equation in the y= on a ti-83 plus
• highest common factor of 26 and 39
• find characteristic polynomial of a third order differential equations
• best algebra book for high school
• math answer and question about algebra
• square root identities
• multiplying division equations worksheet
• algebra with pizzazz.com
• hard math questions for pimary 4
• quadratic common roots
• ti-83 calculator emulator + free download
• how to solve trigonometric simultaneous equations
• free math worksheets on solving equations with addition and subtraction
• adding and subtracting negative and positive fractions
• "good"
• simplify cubic problems in TI-83
• area for ks2 worksheets
• saxon math test
• algebra and trigonometry book 2 answers
• Decimals square
• rom ti-calculators
• multiplying three factors
• year 4 maths worksheet on inverse
• algebraic properties worksheet
• mathematica tutorial
• functions sixth grade worksheets
• simult solve calculator
• solving equations by adding, subtracting, multiplying and dividing
• maths-how to introduce area of rectangle and square to class 5
• GRAPHING A LINER FUNCTION
• solving second order partial differential equations of the cable equation
• algebra logic+exam+assingments+solution+pdf
• holt & prentice hull
• solving multi variables equation
• Balancing Equations Calculator
• free trig solver
• Y = -3X+7 find vertex
• symbolic differential equation solver online
• solving nonlinear differential equations
• Babylonian Square root source code
• Math Book answer
• square and cube worksheet
• show me how to use the algebrator software
• solve quadratic two variables
• free game download for ti 84 plus
• addind and subtracting negative and positive mixed fractions
• substitution calculator for algebra
• 7th grade algabra
• samples of quadratic problems and real life
• radicals and quadratics\
• "when was the graphing calculator invented"
• 2/3 lbs to decimal
• ti 84 trig solver
• trivia about quadratic function
• elementary physics formulae work sheet
• online program that solves decimals
• mole work problem examples
• adding and subtracting integers game
• kumon answer
• cheating substitution calculator
• math combination+examples
• easy algebra 3
• multiplying negative numbers games
• free probability books download
• Accounting Books download
• ti 83 plus arc cosine
• ALGEBRA FOIL CALCULATOR
• Free Download of Aptitude question
• free printable math exponent expresions evaluating worksheets
• beginners using variables worksheets
• systems inequalities worksheets
• free help solving equations
• class 1 maths worksheets
• worksheet adding to 12
• Programming-with- Java-for-Dummies.pdf
• solve alg 2 problems
• 5th grade expanded form exponents to standard form
• add, subtract, multiply, divide integers worksheets
• dividing an expression by a monomial in math lesson
• "greatestcommon divisor" algorithm
• multiplying polynomials online calculator
• laplace transformation solver for ti-89
• square root calculator quadratic equation
• simplify (n^3)-8
• past amath exam papers
• converting to moles with exponenets
• mixed number to decimal
• free algebraic expressions elementary school
• ti 89 non algebraic variable in expression error
• 3rd order quadratic formula
• 1st order linear equations calculator
• how to do roots on a calculator
• finding common denominators calculator
• hints for adding integers
• 9th grade free math games
• partial sum addition
• convolution integral on ti-89
• math tutoring software college
• "descartes rule of signs" "online calculator"
• fourth square root 85
• multiplying adding and subtracting decimals worksheets
• solutions algebra 2 mcdougal littell
• symbolic polynomial formula calculator
• calculate gcd
• algabra
• simple algebra factorisation worksheet
• pde compatible first-order system
• test for solving inequalities by addition and subtraction
• slove radical expressions
• Solving Radical Exponents
• non-perfect squares ppt
• solving quadratics game
• common denominator calculator
• free algebra expressions worksheets
• 8th grade probability worksheets
• games for ti-84 plus
• 6TH GRADE ENGLISH FREE DOWNLOADS
• Free High School Math Geometric dilation dittos
• printable worksheets on adding and subtracting integers
• solving 2x2 matrix graphically
• log on ti-83
• trivia multiple choice sample questions math
• order of addition worksheets sum "Associative Property"
• online fraction solver
• 19th grade math worksheet
• cat 6 test prep online
• merrill +prealgebra answers
• Lesson plan about Finding extraneous roots of rational expressions
• Glencoe pre-algebra practice skills workbook online
• answeres fr your math homework
• free beginner pre-algebra worksheet
• answers for algebra (cheat sheet)
• instructions on math with caculator
• Puzzle Worksheets for subtracting decimals
• free help introductory algebra
• Factoring Trinomials Worksheet
• free math reviewer e-books
• calculate cube root in java
• newton raphson matlab for equation systems
• free solving equations calculator lesson plan
• factor math tricks
• algebra help software reviews
• imperfect square root middle school math
• "discrete math tutor"
• maths fractions complicated worksheet
• Maths question paper of 6th Class
• square root formulas
• display decimal place as fraction
• cube root activity
• ALGEBRA FOR IDIOTS
• square roots with decimals
• adding integers worksheet practice
• college algebra clep cheat
• how to teach adding and subtracting integers
• solve my algebra problems
• finding the slope of a logarithmic equation
• integers on TI 83
• occupations that use absolute value
• tricks for maths aptitude qustion
• importance of algebra
• Integer games printable
• exercice algebra 1
• permutations worksheet
• cube root of fraction
• to see review algebrator good
• adding 4digits worksheets
• add integers practice
• to find out the sum of the digits of a entered input number from the keyboard with source code in java
• beginner and intermediate algebra
• school Problem solving software
• rational expressions solver
• ti-83 plus solve quadratic
• Prentice Hall Mathematics Course 1 2 3
• "algebra 2" textbook pretence hall answers
• Printable Exponent Worksheets
• intermediate algebra calculator online
• printable 3rd grade math sheets
• multiplying fractions problems 7th grade
• Finding the roots of a quadratic equation by the quadratic formula (worksheets)
• diviting games
• solving for roots by factoring
• 3rd root problems
• c programming factorization chapter 11
• radical and square roots
• the world's hardest math problems
• Free Algebra Games
• beginner algebra
• combo math class
• glencoe algebra 1, student editionchapter 2 lesson3
• solving by substitution calculator
• gre mixture problem
• simplifying algebraic expressions dividing
• Solving Algebra Equations
• algebra mixture problems practice
• how to find the sum of series by taking input integer from keyboard in java
• least common denominator ti-84
• free printoutsb for fourth grade
• solve rational expressions calculator
• conversion from exponent to polynomial
• partial fractions TI progam
• math variables worksheet
• Where can i use a McDougal Littell Pre-Algebra textbook online?
• writing linear equations algebra ppt
• how to solve the quadratic equations contains two variables
• Newton's divided difference interpolation formula---program using C
• adding and subtracting positive and negative integers worksheets
• sat free maths exam quetion paper of class 7th
• puzzle frenzy ti84 passwords
• solving equations easy
• math homework help for 7th grade
• how to teach multiplying with integers
• solving non linear differential equations of 2nd order
• how to do algebra?
• maths calculater for kids
• free and printable AS level fractions practise paper
• examples of math trivia
• completing the square questions
• calculator algebric
• glencoe mathematics algebra 2 skills practice workbook answers
• yr 10 simultaneous equations-problem solving worksheets
• 6 grade fractions worksheets
• answers to glencoe mathematics
• java linear equations solving
• sample math test for six grade
• unconfounded variables
• excel formula linear metres to cubic metres
• Quadratic Inequalities exam questions
• beginners math exercises
• elementary math lattice method lesson plan
• free printable integer worksheets
• hex decimal sample program in java
• aptitude questions in software industry.pdf
• adding and subracting negarive and positive numbers
• percent formulas
• how find sum of first integers
• free books for Aptitude
• algebra scale factoring
• simplification of an expression
• Simplify Algebra Calculator
• algebra formulas
• solve absolute value
• mathmatical expression for cubed
• algebra power
• graphing calculator and limits
• real life system of equations examples
• free printable ged worksheets
• teach combining like terms
• kumon work answers
• matlab decimal convert decimal fraction
• algebra home work work sheets
• ks3 maths work sheets
• algebra 1/2 test forms download
• add subtract whole numbers worksheets
• How to factor cubed numbers
• cheating gine download
• polynominal
• advances in composite materials .ppt
• simultaneous equations calculator 3 variables
• radical equation solver
• how to graph lines on a TI-83
• worksheets alegbra
• picture of the graph program calculator
• calculate lowest common denominator
• math worksheets with negative nd positive numbers for free
• online graphing calculator- volume
• solving radical equations using my TI-84 plus
• powerpoint combining like terms
• simplify square root fractions
• factoring polynomials calculator
• free trigonometry online calculator
• sat math free online download
• free algebra calculators
• math radican
• "Operation research" winston e-book
• ti-84.plus eigen.values
• free ogt worksheets
• Graph Hyperbola
• cost accounts books
• solving algebraic equations ppt equations
• TI-84 plus emulator
• download equation excel
• conceptual physics for visual learners middle school
• ks4 maths + area
• permutations and combinations, elementary games
• third square root
• examples of math trivia students
• pre algebra simplifying with the distributive property
• Algebrator manual
• how to use algebra tiles to teach adding rational numbers
• Aptitude Questions pdf
• Convert Square Meters to Lineal Meters
• mathematica algebra
• geometry solving 2 variables
• How to save formulas on a ti83
• To convert a parabolic equation from simplified form to standard form, you must complete the
• TRIVIAS IN ALGEBRA
• simplifying variable expressions "practice problems"
• tutoring worksheets for kids
• balancing linear equations
• printable test papers
• how to convert mixed fractions to binary
• abstract algebra test
• ti-86 graphing calculator instructions
• cost accounting polimeni ebook
• math tricks and trivia mathematics
• ladder method
• the inverse operation ks2 multiplication and division free worksheets
• year 8 maths equations
• ti 84 plus games download
• free mentalmaths worksheet with solution for class 7
• intermidiate accounting of sixth edition,key answers .com
• adding and subtracting positive and negative integers
• examples on poem about trigonometry
• T1-83 calculator
• prentice hall math
• laplace transform sample problems with solution
• common
• Answers to Monomial Problems
• intermediate algebra/ software
• rules for adding radicals
• When solving a rational equation why is it necessary to perform a check
• cube root on ti83
• Ratio Formula
• Algebra Poems
• solving nonlinear differential equations with Boundary Conditions
• how to solve algebra formulas
• college algebra verbal problems with solution
• subtraction decimals 5th grade worksheet
• fun algebraic equation worksheets
• advanced algebra help
• linear equations formation worksheets
• number words poem
• Intermediate Algebra by Charles P. McKeague solutions manual 6th
• prentice hall secondary math patterns and variable worksheets
• printable 6th grade games
• matlab ode45 high order
• Free 11+ exam papers
• solving non-linear equations in matlab
• online graphing calculator
• free download cpt model test paper
• decimal fraction as apptitude
• order of adding, subtracting, multiplying and dividing whole numbers
• equation solvers for exponents
• greatest common divisor tables
• mcdougal littell algebra 1 solutions manual
• WHAT IS THE ALGEBRAIC FORMULA TO GET A PERCENT
• online simultaneous equation solver
• ordering numbers least to greatest worksheets
• exponents algebraic expansion
• solving chemical equations
• trigonometric identities+problems with answers
• Algebra and Trigonometry, Fifth Edition
• c language test questions online quiz
• word problems on least common multiple and greatest common factor
• where can you find free printable 9th grade lessons
• maths worksheets for substitution-grade 6
• c language aptitude questions
• Aptitude test for 1st grade
• how to solve a parabola
• maths problems for low ability worksheet
• algebra+free worksheets+KS2
• www.4 grade online .com
• second-order homogeneous differential equations
• substitution method
• using casio calculator to solve polynomials
• factoring online calculator algebra
• Math word problems Worksheets 4th Grade
• addition and subtraction of exponential expressions
• adding and subtracting missing numbers worksheet
• trigonometry poems
• MaGraw Hill 6th grade math assessments
• finding the real solutions by factoring
• online games on translating algebraic expressions
• 3x+6y=12
• holt algebra 1
• 2nd Order Linear Homogeneous ODE general solution
• angular & linear speed worksheet problems
• grade 9 algebra worksheet
• algebra expansion and factorization worksheet
• combinations in math + MIT
• simplifying square roots
• pre-algebra year 2 10th grade
• practice 2-3 simplifying variables Expression
• simplifying equations sixth grade
• writing a polynomial
• find slope fun worksheets
• free printable algebra substitution woorksheets
• solve polonomial equation c++
• least common factor 10 and 14
• basic rules adding positive and negative
• free first grade math sheets
• adding multiply integers worksheet
• Answers to Saxon Algebra II
• how to solve linear problem story
• ti 83 plus applet calculator use
• dividing polynomial by bionomial
• Newton Raphson Method to solve multivariable nonlinear equations
• what is scale factor in algebra
• Algebra Rules And Tips
• basic college mathematics slater tobey fifth edition chapter tests answers
• solve for the specified variable
• free printable fifth-grade fraction exercises
• how to slove fractions
• history of mathamatics
• problem involving radical equation
• worksheet mixed times divide fraction
• middle school math rotation worksheets
• McDougal Littell answers
• free matrices worksheet
• multiplying equations
• saxon math sixth grade tutorials
• conic equation solver
• least to greatest calculator
• solving algerbra equations showing all of the steps
• lenght width heigth worksheets
• online high tech calculator for fractions
• Boolian equaction solving vb
• practical business math procedures 9e cheat sheet
• dividing fraction sample problems
• solving nonhomogeneous first order differential equation
• fraction too decimal
• exponent multiplication
• factorising simple expressions java
• finding the same denominators calculator
• math proporation
• intergers calculator
• trivia math pre-algebra & creative publications & answers
• interactive activity mean
• softmath Algebrator
• radical with exponents problems
• solution first order ordinary differential equation nonhomogeneous square wave
• methods for solving combination of resistance
• converting mixed fractions into decimals
• Algebra helper
• ti voyage truth table
• math printable worksheets statistics
• college algebra and functions 150 software program to solve problems
• grade 7 maths sheet
• simplifying algebraic expression worksheet
• GMAT combination formula
• pre algebra for dummies
• aptitude question papers with solutions
• algebra rules made simple
• algebra factorise solver
• app "equation write" texas download
• online TI-84
• Make a system of two equations with two variables of your own creation.
• glencoe algebra 1 answers
• what is the square root of 125 and simplify?
• primary sats paper down load
• free algebra 2 solver
• glencoe/mcgraw-hill pre-algebra inequalities worksheet answers
• coordinates of circle on TI 86
• Free Beginning Algebra Tutorial
• rudin solutions
• pre-algebra dolciani chapter6
• deducing a mechanism for a reaction from the rate equation and balanced chemical equation
• combinations practice pages
• integers worksheet
• high school algebra review worksheet
• scale factor math
• Glencoe Algebra 1 answer sheet
• algebra cubes
• chapter 8 test d, math test mcgraw-hill glencoe
• algebra 2 workbook answers
• pdf to ti89
• grade five maths worksheets
• Prentice Hall Mathematics
• converting base 5 to decimal
• fractions in order least to greatest
• online square root calculator
• what is the easy way to mutiply in a GED TEST
• Conceptual Physics Answers guide
• trig ratio values chart
• compatible numbers worksheets
• activities for 11th graders
• Application of Fluid Mechanics in Everyday Life
• kumon Transforming Equations 2 answers
• easy way to factor algebra
• online calculator for answers to all math questions
• divinding fractions
• domain and range of a parabola
• worksheets for grade 10 math
• "factor sum of two cubes"
• solving a three variable system of equations with ti-89
• mixed number and decimal
• pedro lauridsen ribeiro
• online graph solver
• solve formula for specified variable
• ks2 maths percentage sheets
• solving inequalities in matlab
• teach yourself math online
• free downloads of Algebra Solver
• exponents online
• polynomial multiplication program calculator ti-84
• radical solver program
• java convert decimal to fraction
• Middle School Math with Pizzazz!book D Answers
• reading printouts for seventh grader
• new jersey estimating products and quotient worksheet sixth grade
• Recursive Formula Algebra 2
• adding variables common denominators
• Download e-book Differential Equations Edward Penny ppt
• Multiplying Powers Calculator
• Free Math Problem Solver
• fortran program code download quadratic
• square root calc variables
• 6th grade math worksheets conversions
• template of a production possibility frontier graph
• houghton mifflin mixed fractions homework sheet
• adding 3 number fraction worksheet
• simplifying exponential notation
• square a fraction to a negative power
• algebra formula sheet
• prealgebra or elementary algebra, math scale
• gallian contemporary abstract algebra solution
• solving basic equations using matlab
• how to use solve differential with TI-84 Plus Silver
• Pre - algebra answer key
• "Problem Solving Examples" maths "year 7"
• homogeneous second order nonlinear equation
• revision sheets maths grade 5
• ratio formula
• Online usable ti-84 calculators
• lesson plan on simultaneous equations
• step by step help with perimeter of a triangle
• solving equations by multiplying and dividing fractions
• solving second order differential equation with matlab
• how do i express exponents on T1-83 plus?
• free solve lcd fractions
• for loop add integers
• online factoring
• combinations and permutations Algebra
• adding subtracting integers time tests
• college quadratic formula
• Fracton pratice
• solving radical equations
• online source for high school trigonometry
• boolean algebra simplifier
• examples of math poems
• free gmat paper + pdf
• algebraic fractions cubed
• geometric mean math tutor
• solve linear system graphing worksheet
• 6-8 question ks3 maths question to answer
• algebra question
• adding 6 worksheets
• algebra tiles worksheets
• line intersection ti-83
• textbook algebra 2 texas
• math for dummies online
• "algebra tiles" writing expressions worksheet
• c aptitude with question and answers
• algebraic identities worksheet
• KS3 MATHS TESTS ONLINE ON TRIANGLES
• +math +percent +worksheet
• algebric equations worksheets
• solving nonhomogeneous equations
• system of non linear equations matlab
• LCM and GCF worksheet with answers
• subtraction of variable fractions calculator
• factoring cubed equation
• formula for elipse
• simple basic chemistry
• gmat permutations
• hard math quiz
• ks2 maths works sheet
• factor polynomial solver
• algebra homework solvers
• simplest radical form calculator
• algebra calculator with division
• compositions of functions problem solver
• free data worksheets for third grade
• how to transform literal equations for dummies
• integer worksheet
• factor a cubed polynomial
• adding whole number summer worksheets
• graphing calculator help stat plot 84
• free answers to holt algebra 1 problems
• fraction to decimal in maple
• math adding and subtracting positive and negative integers
• Fun Printable Math perfect square root chart
• Iowa test printouts standard 7-8
• algebra teaching aids and worksheets
• "imaginary units"+"real life"
• algabra x3
• examples of mathematics trivia
• factoring polynomials division "rational number"
• "basic trig"+worksheet
• percentage equations
• ti 89 rom download
• taks science formula sheet
• easy algebra
• fraction problem solving worksheets
• calculator finding roots of polynomial equations
• complex rational expressions
• solve equation excel
• cubic measurements, elementary math
• Square route equations
• multiplying integers worksheet
• finding the nth term worksheet
• homogeneous nonlinear second order differential equations examples
• algebra revenue line
• different math trivias
• solving non-linear differential equations fortran
• cubed root on C++
• converting radicals to mixed radicals math help
• printable integer practice
• Algebra Equations Solver
• answers mcdougal littell science chapter 10
• notes on permutation and combination for a level
• merill algebra 1 answers
• free online TI-83+graphing calculator
• lattice multiplacation
• ti84 programs quadratic
• Apptitute questions +pdf
• Solve Algebra Equations
• subtracting frations
• grade 2 science freeworksheet
• algebra substitutions calculator
• 1st grade algebra
• Equations with Fractions calculator
• G.E.D test printouts
• permutation math concept
• radical expressions calculator
• fraction samples for 3rd graders
• how do you switch fractions to decimals on a TI-83 Plus?
• managerial accounting exercises problems solutions free online
• online radical calculator
• solving exponential equations with quadratic
• coordinate plane lesson 5th grade
• Mcdougal Littell Algebra 2
• holt math workbook
• convert time to java
• cube on ti-83
• ti-89 solve
• TI-84 book instructions
• exponential growth and decay worksheets for 8th grade
• exponential notation fifth grade printable
• Passport to Algebra and Geometry Chapter 6 answers
• maths number grid coursework algebra
• dividing algebra
• evaluation square root
• sats maths work
• college algebra interest problems
• Calculating Slope TI 83
• how to solve more than one equation in Ti 89
• fractions formulas
• Mcdougal Littell Algebra Books
• power and fraction
• third grade multiplication print out sheet
• application of quadratic eq on problems
• prentice hall mathematics algebra 2 answers
• free algebrator
• 5th grade math word problems
• adding rational expressions worksheets
• radical online equation solver
• college algebra worksheets
• writing a program that takes square root of number
• matlab system of differential equations
• hard algebra question
• howto dividing polynomials
• solve partial differential nonhomogeneous
• dividing polynomials by trinomials
• find the variable in a fraction expression
• www.fractions.com
• mc dougal littll algera 2004 addition
• simultaneous equation solver
• linear equations java
• Graph Math Problems Worksheet
• Kumon Solutions
• binomial theory
• Algebraic forumulae in word
• how do i solve problems without calculator?
• holt mathamatics quiz
• What is the difference between evaluation and simplification of an expression?
• how do you prime factor on the TI-83 Plus
• contemporary Abstract algebra solutions
• free linear equation grapher
• free 8th grade algebra
• algebraic equations worksheets
• TI 84 vertex of parabola
• how to do permutations on casio
• how to make a addition or subtraction expression find the sum or difference
• when is absolute value needed in simplifying
• kids algebra equations game
• calculator to convert percentage to decimal
• application of algebra
• adding/subtracting length calculators
• glencoe/Mcgraw hill cheating
• double angle solver
• ti-89 solve equivalent in matlab
• factorising quadratics grid
• KS3 MATHS TEST ON TRIANGLES WHICH YOU WILL GET 100%
• third root
• Free printable math tree diagram
• Basic mathamatics
• prentice hall algebra 1 functions test
• solving systems of linear equations with substitution game
• rules in adding and subtracting algebraic expression
• online holt pre algebra projects
• EQUATIONS OF LINE WORKSHEET
• Trigonometry and Pizzazz worksheets
• online boolean simplification
• programming for Ti-83 to do equations for circles
• cube root ti-83
• "math symbols font"
• algebra term-polynomial
• LCM formula in C++
• solve radicals
• powerpoints on graphing linear equations
• algebra fo dummies
• Riemann Sum solver
• changing a mixed number to a decimal
• simplified radical definition
• simultaneous solver
• math combinations with money
• how to solve equalities
• mixed number to decimals
• year 6 sats practcie test
• ordering fractions from least to greatest calculator
• Algebraic Formula For Solving Percentages
• holt 2004 algebra 1 book answers
• algebra 1 test on matrices
• math calculator radical
• algebra calculate
• how to solve simple radicals equations
• how do you find on TI calculator the vertex
• Algebrator
• examples of math trivia geometry
• glencoe physics principles and problems ch. 15 review answer key
• hyperbola equations
• linear feet calculator
• alll equations using numbers 3 5 and 7
• solve through substitution method calculator
• integral ti 84 plus software download
• slope field program for TI-84 SE
• quadratic formula.pdf
• TI 84 Emulator free download
• math problem solver radicals
• simultaneous Ordinary differential equations MATLAB
• investigatory project in math
• substitution, elimination, and comparison examples for math
• simplifying expressions with three variables
• learning algerbra
• squaring fraction radicals
• difference of squares factoring calculator
• online free factoring trinomials calculator
• abstract algebra help
• TI 89 boolean
• fifth grade calculator
• programmed algebra cd roms
• CLEP cheat sheets
• ti-89 log base 2
• worksheet add subtract multiply divide integers
• +Game or worksheet on completing the square for quadratics
• TRIGONOMETRY FORMULAS IN EXCEL
• how to do rational exponents fractions
• answers for mcdougal littell algebra 1
• ks2 maths worksheets
• adding and subtracting integers free worksheets
• Word Search ,Holt World History Grade 7
• logarithm and calculator ti83
• how to make a fraction on t1 83 calculator
• algebra work sheets for grade 7
• 6th grade spelling units 6
• matlab program for solving nonlinear trigonometric equations
• parabola equation worksheet
• convert fractions into simplest form free online
• CALCULATE LINEAR FEET TO FEET
• solve for the designated variable
• free download printable worksheets for maths
• free elementary algebra tutorial
• grade nine math fractions
• what is a solution of a linear equation in 2 variables called?
• Math calculater
• Solving Equations Fractions
• absolute value equation online calculator
• math words in poems
• glencoe physics assessment chapter 10
• combustion reactions calculator
• probability made easier maths tutorial online free
• balancing chemical equations electronegativity
• mcdougal littell worksheet answers/world history
• how to do fractions on t1-83
• exponent worksheets algebra
• polynomial equations and inequalities story problems
• Linear Algebra with application otto instructor solutions manual pdf
• eight grade exam papers
• variable worksheet
• basic maths sample exam papers
• instructions quadratic formula calculator
• beginners algebra
• worksheets on permutation and combinations
• how to solve a calculas equation
• integer and order of operation worksheet
• solving algebraic expression calculators
• college algebra problem solvers free
• how to successful pass an algebra test
• C Language aptitude questions
• dugopolski tests
• answers to algebra book by McDougal Littell
• subtracting multiple integer worksheet
• gaussian elimination online calculator
• Algebra 1 Helper
• holt algebra
• real-life linear equation example
• POEMS IN MATH about algebra
• online calculator binomial expansion
• alegebra tests
• Advanced Mathematics Richard G. Brown chapter 6 review notes
• holt mathematics course 1 chapter 9 lesson3
• addition of algebraic expressions with negative exponents
• palindrome+integers+java
• compare and order fractions printable
• slope fields worksheets
• math new trivias
• ti 83 plus difference quotient
• prealgebra practice worksheets FOIL
• how to solve square roots with variables
• online square root solver
• casio calculator factor equations
• how do divide integers
• first order non homogeneous differential equation solution
• adding 9 worksheets
• c programs for solving equations
• factoring polynomials with non integer powers
• Math Word Problem activites for 3rd graders
• calulator for maths
• algebra expressions and triangles
• Negative Inverse Log key on Casio fx-115ms
• lcd calculator
• Algebra 1 book by mcdougal littel and the prob;ems in the book
• where to put the decimal place in descending order
• calculator for neagative and positive numbers
• math tutor- half life
• binomial equations
• cheats to decimal to percent
• how to mixed number conversion to decimal
• softmath
• 6th grade algebra free and printable worksheets
• chevrontexaco's aptitude test question and answer
• "two-step word problem worksheets"
• adding subtracting integers game
• ti 83 plus rom download
• rational expressions
• ti 83 plus instructions convert degrees minutes
• trivia oh math
• java code Rational Number Calculator
• linear equation worksheets to print and download
• equation solving calculator by use of linear combinations
• Math Problem Solver
• integral calculator with steps
• Algebra for 1st grade
• intermediate algebra online help
• operation on radical expression
• converting general form to vertex form
• free math problem answers algebra solutions
• dummies guide to graphically solving simultaneous linear equations
• addition algebraic expressions
• cost accounting formulas
• how do you find derivatives using the graphing calculator?
• Intermediate Algebra for College Students, Sixth Edition
• difference of cubes solver
• worksheets "line plots"
• cubic root of 16
• huge Polynomial simplify
• Math word sheets
• MAATH CHEAT
• algebra equations
• adding and subtracting decimals worksheets for fourth grade
• Free Saxon Math Answers
• least common multiple finder
• Aptitude Question bank
• subtracting negative practice SHEAT
• free online homework problem solver
• 6th grade Collects and Organize worksheets
• download ks3 previous SATs papers fro free
• solving my fraction problem
• how to find cubed root on calculator
• finding a root in newton raphson method using matlab
• factor tree math worksheets
• Laplace transform-maths
• worksheet two variables
• translating expresions to an algebriac expressions
• factorising calculator showing steps
• grade 10 exponents fun worksheets
• kS2 and KS3 math free tests
• percentage of mix fraction
• worksheets for adding and subtracting positive and negative numbers
• pictograph worksheet for grade 5
• prentice hall practice typing
• prentice hall algebra 1 answer key
• free printable worksheets on exponents with variables
• algebra ormula for speed
• revise about australia ks3
• calculator roots of fractions
• java code to multiply polynomials
• factor equations online calculator
• free sample percentages revision for year 8 in nsw
• free online algebra equation calculator
• "activities on graphing linear equations"
• free entrance exam reviewer
• worksheet on using algebra tiles in balancing equations
• love maths caculator
• intermediate algebra problem solver
• algebraic eguation on the rules worksheet
• solving first order nonhomogeneous differential equation
• elementary algerbra
• substitution calculator
• solving equations respect to whole numbers
• Free Grade 5 math lessons to print off
• percent equation
• standard year 9 mathematics practice sheet
• printable worksheets with coordinate pairs to graph
• hardest mathematical solved equations
• rules for adding variables
• how to solve add and subtract rational expression
• sqaure roots
• math trivia anwers and question
• Free math problem answers
• 8th grade algebra,slopes and y-intercepts
• adding negatives and positives chart
• formula for parabola
• kids maths revision printables
• algebra equations;
• Pre-Algebra with Pizzazz answers
• answers for McDougal Littell Algebra 2 Practice workbook
• mathmatical sequence of unknown lenght
• mcdougal littell test answers
• MATHS FREE TUTORIALS
• ti 89 Lars Frederiksen z-transform
• matlab solve nonlinear equation
• greatest common factor calculator with explanation
• 9th grade algebra syllabus
• square root transferred to square calculator
• radicals and exponents worksheets
• find square meter of box in maths tutorial
• grade 8 math chapter 6 toronto
• "introducing linear equations" lesson plan
• 2nd order differential equation matlab
• matric calculator
• decimal to binary java code
• 6th Grade Algebra Tests
• dividing an integer calculation
• a calculator that turns fractions into decimals
• equation solver 4 unknowns
• program quadratic formula TI 83 plus
• factor trinomials online
• Graphs with deicmals
• how do you divide?
• manipulating equations worksheet
• quizzes on combination and permutation
• Algebra 1 tutors
• SAT exam for 5th grade
• what does the error non-algebraic variable in expression mean for a TI89 calculator
• solve negative cube root
• Why is it important to simplify radical expressions before adding
• online t-83 graphing calculator
• free trig Calculators software
• quadratic equation variables
• simplify equation ti 89\
• Trigonometry Unit circle worksheets
• how to divide and simplify equation in matlab
• pre algebra hel
• free online square root calculator
• rational zero theorem calculator
• basic simultaneous equations
• free printable math worksheets for 6th grader
• kumon material free download
• newton's method matlab f(x,y)
• How Do You Convert a Decimal into a Mixed Number?
• Examples of Parabola
• calculator having substitution method
• how to convert bigdecimal value to decimal
• difference quotient of fractions
• Holt Physics Answers chapter 4 section review
• printable decimal flash cards
• linear equation worksheets
• ti-84 emulator software
• factoring trinomials calculator
• integers AND order of operations worksheets
• answers to "cpm math
• percent/discount worksheet
• lineal equations tutorin
• grade 6 free statistics worksheets
• expressing fractions as a single power of 10
• 7th grade math - gcf, lcm
• ti 83 compound inequalities
• savings worksheets for kids
• how to calculate hyperbolic tan with scientific calculator
• abstract algebra solved problems
• aptitude test free download
• Excel formula testing for 5th graders
• glencoe pre algebra worksheets
• mathematical radican
• elementary math textbook gcd and lcd
• free worksheets Decimal Comparisons
• free online TI calculator
• simplifying fractions with exponents calculator
• square root simplification x-5
• 4th grade algebra tables
• example of a daily assignment sheet in 4th grade math
• addition and subtractions with variables and fractions
• algebra for 6th grade printable worksheets
• ti 89 functions LU decomposition how
• +maths +dictionary +"discriminant"
• maths revision level 6-8 online
• algebra 2 ellipses
• convert decimals into time
• ks3 science past sats paper to practise with answers
• solving mixed numbers and decimals
• 11th grade math games free
• worded algebra problems
• pre algebra with pizzazz worksheets
• math tutor for my 8th grader for slopes easy to underdstand
• learning basic algebra
• practice problems and answers for direct substitution
• "algebra with pizzazz"
• comparing and ordering fractions calculator
• MATRICE TUTORIALS
• matrice calculator
• Algebra 1B Math Help Online Games
• foil algebra cheater
• EQUA TESTING
• prentice hall +mathematics.pre algebra answer key
• ninth grade lesson tutor
• free exam papers maths ks2 | {"url":"https://softmath.com/math-com-calculator/adding-functions/hard-math-problems-12th-grade.html","timestamp":"2024-11-12T00:50:08Z","content_type":"text/html","content_length":"193455","record_id":"<urn:uuid:3e986723-93a3-4d6a-97d2-951cd5917d02>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00546.warc.gz"} |
A survey of random processes with reinforcement.
A survey of random processes with reinforcement.
Probability Surveys [electronic only] (2007)
• Volume: 4, page 1-79
• ISSN: 1549-5787
Pemantle, Robin. "A survey of random processes with reinforcement.." Probability Surveys [electronic only] 4 (2007): 1-79. <http://eudml.org/doc/232071>.
author = {Pemantle, Robin},
journal = {Probability Surveys [electronic only]},
keywords = {urn model; urn scheme; Pólya's urn; stochastic approximation; dynamical system; exchangeability; Lyapunov function; reinforced random walk; ERRW; VRRW; learning; agent-based model;
evolutionary game theory; self-avoiding walk; Pólya’s urn},
language = {eng},
pages = {1-79},
publisher = {Sponsored by Institute of Mathematical Statistics and by the Bernoulli Society},
title = {A survey of random processes with reinforcement.},
url = {http://eudml.org/doc/232071},
volume = {4},
year = {2007},
TY - JOUR
AU - Pemantle, Robin
TI - A survey of random processes with reinforcement.
JO - Probability Surveys [electronic only]
PY - 2007
PB - Sponsored by Institute of Mathematical Statistics and by the Bernoulli Society
VL - 4
SP - 1
EP - 79
LA - eng
KW - urn model; urn scheme; Pólya's urn; stochastic approximation; dynamical system; exchangeability; Lyapunov function; reinforced random walk; ERRW; VRRW; learning; agent-based model; evolutionary
game theory; self-avoiding walk; Pólya’s urn
UR - http://eudml.org/doc/232071
ER -
Citations in EuDML Documents
You must be logged in to post comments.
To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear. | {"url":"https://eudml.org/doc/232071","timestamp":"2024-11-12T09:00:38Z","content_type":"application/xhtml+xml","content_length":"37229","record_id":"<urn:uuid:5203a1be-f5ce-4504-b1a1-874d7a784e7b>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00618.warc.gz"} |
Browsing Codes
SCONCE-SCMS detects cosmic web structures, primarily cosmic filaments and the associated cosmic nodes, from a collection of discrete observations with the extended subspace constrained mean shift
(SCMS) algorithms on the unit (hyper)sphere (in most cases, the 2D (RA,DEC) celestial sphere), and the directional-linear products space (most commonly, the 3D (RA,DEC,redshift) light cone).
The subspace constrained mean shift (SCMS) algorithm is a gradient ascent typed method dealing with the estimation of local principal curves, more widely known as density ridges. The one-dimensional
density ridge traces over the curves where observational data are highly concentrated and thus serves as a natural model for cosmic filaments in our Universe. Modeling cosmic filaments as density
ridges enables efficient estimation by the kernel density estimator (KDE) and the subsequent SCMS algorithm in a statistically consistent way. While the standard SCMS algorithm can identify the
density ridges in any "flat" Euclidean space, it exhibits large bias in estimating the density ridges on the data space with a non-linear curvature. The extended SCMS algorithms used in SCONCE-SCMS
are adaptive to the spherical and conic geometries and resolve the estimation bias of the standard SCMS algorithm on a 2D (RA,DEC) celestial sphere or 3D (RA,DEC,redshift) light cone. | {"url":"http://www.ascl.net/code/all/page/32/limit/100/order/date/listmode/full/dir/asc","timestamp":"2024-11-05T05:28:04Z","content_type":"text/html","content_length":"127587","record_id":"<urn:uuid:5d14a181-c9fb-43fa-b4ef-9fd814fb5fec>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00346.warc.gz"} |
Posts Categorized as 'Money & Finance'—Wolfram|Alpha Blog
CATEGORY: Money & Finance
Are you looking to make a move in the near future? Budgeting for your next vacation? Before you go anywhere, check out Wolfram|Alpha’s data on costs of living and consumer goods. Whether you’re
simply looking to get the most bang for your buck, or figuring out how your salary needs to change to maintain your lifestyle in a new city, look no further for some quick answers. More »
Over the last several months, one of the most-requested features for Wolfram|Alpha has been to add information about the cryptocurrency Bitcoin. Well, you asked, and we’ve answered: Bitcoin data is
here! More »
Wolfram|Alpha has released a suite of apps that cover all of your financial needs. The apps are profoundly powerful tools, allowing you to stay informed about your business’s finances and the
marketplace at large while in the office or on the go. More »
Most of the new features we announce on this blog are large-scale projects where we add a huge chunk of data to Wolfram|Alpha all at once. But there are always dozens of background projects going on
at any given time—including a seemingly never-ending effort to expand our database of information on private companies.
August 8, 2012–
C. Alan Joyce Comments Off on Celebrating National Dollar Day with Wolfram|Alpha Comments Off on Celebrating National Dollar Day with Wolfram|Alpha
August 8 is “National Dollar Day,” commemorating the establishment of the US monetary system on this day in 1786. But the first dollar bill wasn’t issued until 1862—and instead of George Washington,
it featured a portrait of Salmon P. Chase, then Secretary of the Treasury under President Lincoln (and later Chief Justice of the Supreme Court). More »
Recently, CNNMoney published stories highlighting the “20 most profitable companies” and those considered to be the “20 biggest money losers”. Companies such as Exxon Mobil, AT&T, Apple, and Verizon
were ranked by their 2010 profits, with each of the 20 profitable companies bringing in well over a billion dollars.
Using the information provided by both articles, there are many opportunities to gain more data on each of the companies from Wolfram|Alpha. For example, Exxon Mobil topped the list of most
profitable companies, but has it always been profitable? Entering “Exxon earnings” into Wolfram|Alpha produces data and graphs documenting its earnings history.
Wolfram|Alpha recently added information about the minimum wage in U.S. states (from 1967 through today) based on data from the U.S. Department of Labor. This means you can ask Wolfram|Alpha about
simple historical facts, like the U.S. minimum wage in 1980, or perform simple analyses, like comparing the current minimum wage in Ohio and Alaska.
On August 2, The New York Times reported that the (near) final estimate for the total amount of oil leaked into the Gulf of Mexico as a result of British Petroleum’s (BP) Deepwater Horizon drilling
accident is approximately 4.9 million barrels. It would be nice to understand what this number means in the context of the commodity markets where oil is traded. It would also be nice to better
understand what this oil spill did to BP stockholders.
Wolfram|Alpha can help answer these questions. For example, someone might wonder what all this oil would be worth on the oil market. The input “price of 4.9 million barrels of oil” tells us that the
value of this oil on the oil futures market is around $398.8 million (at the time this was written). That’s a lot of money just floating around the Gulf! But to be fair, much of it was cleaned up.
Wolfram|Alpha also shows a graph of how the value of this oil has fluctuated over time as well as the latest quote of a barrel of oil on the New York Mercantile Exchange.
Someone might wonder whether the amount of spilled oil was enough to affect the price of oil in the U.S. The input “oil futures open interest” gives us the number of oil futures contracts currently
in existence for the front-month contract.
As the graph illustrates, open interest starts out strong every month (as the front-month contract rolls forward to the next month). As the contract approaches expiration, some people close out their
positions while others roll their positions forward into a future month’s contract. The best measure of open interest would be the higher numbers shown immediately after the front-month rolls forward
(especially since other contract months aren’t accounted for here).
More »
Wolfram|Alpha launched with an extensive database of United States economic data, derived from the Federal Reserve Bank’s FRED database. Over the past year, we’ve continued to improve our handling of
this data in a variety of ways—teaching Wolfram|Alpha to return more related statistics along with any specific result, improving our linguistic abilities so we can answer more complex questions, and
increasing the frequency with which we update this data. Wolfram|Alpha is now refreshing its collection of FRED-derived data on a daily basis, so you can always access the latest available data on
the national economy.
We’ve also begun to expand our coverage of economic data for smaller geographic areas in the United States, starting with state-level statistics. This means Wolfram|Alpha users can now query for the
latest available information on a variety of economic topics, including gross state product, unemployment, health insurance coverage, and housing-related data.
As always, one of the strengths of Wolfram|Alpha is that it allows you to compare and analyze different pieces of data—and with this data set, you can quickly uncover strong correlations between
various economic properties. It’s easy to see that the house price index tends to move together with employment and state tax collections. You can also use Wolfram|Alpha to run simple calculations of
productivity in U.S. states, or to find out a given state’s share of the national economy or workforce.
To make it even easier to explore this data, you can also use the new Wolfram|Alpha Widget Builder to create simple tools for analyzing and comparing the economic properties of states. To get you
started, here’s a small selection of widgets focused on US state economies—ranging from the serious to the slightly silly. Try them out:
Feel free to customize and share these on Twitter or Facebook, in your blog, or anywhere else—and let us know in the comments if you create any useful new widgets of your own.
Hello, fellow readers of the Wolfram|Alpha Blog—my name’s Justin. In just a few short weeks, I’ll be graduating from the University of Illinois at Urbana-Champaign. Over the years I’ve found my own
way of getting things done in regards to homework and studying routines. But this semester I realized there were tools available that would make studying and completing assignments easier and help me
understand better. One tool that has become increasingly valuable in my routine and those of other students on my campus is Wolfram|Alpha. Recently, I was invited to share how Wolfram|Alpha is being
used by students like myself.
Being a marketing major, I had to take some finance and accounting courses, but I was a bit rusty with the required formulas and the overall understanding of the cash flow concepts, such as future
cash flows and the net present values of a future investment. A friend recommended I check out Wolfram|Alpha’s finance tools, and they’ve became indispensable in my group’s casework for the semester.
Each proposed future investment we were met with, we would go directly to Wolfram|Alpha to compute the cash flows. We even went as far to show screenshots, such as the one below, of inputs and
outputs in our final case presentation last week.
I’ve met other students on my campus who have found Wolfram|Alpha to be helpful in their courses. A few months ago while studying in the library, I walked by a table of freshman students all using
Wolfram|Alpha on their laptops. I decided to stop and chat with them because I knew one of the girls. They explained how they were using Wolfram|Alpha to model functions and check portions of their
math homework. All three girls are enrolled in Calculus III, and not exactly overjoyed about the fact of future— and most likely harder—math classes. More »
For active investors, the fast-paced nature of the trading floor requires having tools available to make confident decisions in a timely manner. Wolfram|Alpha offers a collection of money and finance
tools ideal for finance professionals and personal finance matters. This data flows into Wolfram|Alpha in real time, providing traders with computation results in charts and graphs. In this post,
we’ll look at a variety of ways Wolfram|Alpha can compute and present stock data.
Let’s start with the basics. Simply enter the name of a stock, such as Starbucks or its ticker symbol SBUX, into the computation bar. Wolfram|Alpha retrieves and analyzes both real-time and
historical data, and presents the output in category pods. The pods display information such as the stock’s current value at last trade, its value at open and close, and range for that trading
period. The “Fundamentals and financials” pod displays information such as the stock’s market share, revenue, number of employees, dividends, and more. Change the “Fundamentals” option on the right
side of the pod to see additional information, including ratios, balance sheets, and income and cash flow statements.
We like to demonstrate ways Wolfram|Alpha can be a helpful tool for everyone. Today we’d like to share a cool feature Wolfram|Alpha users are talking about on the web. The Retirement Savior blog
posted an item on Wolfram|Alpha describing how it can be used to calculate your retirement investments.
Wolfram|Alpha’s investment-returns calculator prompts you to describe your current investment strategy. Once you submit your query, Wolfram|Alpha will provide you with a number of results such as a
linear chart depicting investment value projection scenarios, pie charts of resource allocation, a bar graph that allows you to easily compare the distribution of ages at which the account balance
would reach zero, and a table displaying projections of your portfolio’s value at various ages. More »
We know college is hard. So we’re highlighting examples of how Wolfram|Alpha can make subjects and concepts a bit easier to learn. Wolfram|Alpha is a free computational knowledge engine that can help
you tackle everything from calculus, to computing the number of pages for a double-spaced 1000-word essay, to comparing the flash points of methane, butane, and octane, to figuring just how much
money it’s going to cost you to drive home to do your laundry. Check out a quick introduction to Wolfram|Alpha from its creator, Stephen Wolfram.
We want to help you take full advantage of this resource. Over the next term, we’ll be highlighting helpful computations and information here on the blog, and even providing ways you can get involved
with our company. (Would you like to be a part of the Wolfram|Alpha Team on your campus? Stay tuned to find out how you can be involved.) For this post we selected several of our favorite examples to
help you start thinking about how you can use Wolfram|Alpha in your courses, and in your always-changing college life. More » | {"url":"https://blog.wolframalpha.com/category/money/","timestamp":"2024-11-08T23:32:51Z","content_type":"text/html","content_length":"51255","record_id":"<urn:uuid:f1194658-5ceb-4e26-9fd5-37ef9e351fab>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00418.warc.gz"} |
Handling cyclical features, such as hours in a day, for machine learning pipelines with Python exampleHandling cyclical features for machine learning pipelines with Python example
REINFORCE Algorithm explained in Policy-Gradient based methods with Python Code
Python Scipy sparse matrices explained
11 mins read
What’s the difference between 23 and 1? If we’re talking about time, it’s 2.
Hours of the day, days of the week, months in a year, and wind direction are all examples of features that are cyclical. Many new machine learning engineers don’t think to convert these features into
a representation that can preserve information such as hour 23 and hour 0 being close to each other and not far.
A cyclical variable is a fancy name for a feature that repeats cyclically. It’s vital to ensure a model interprets cyclical features correctly as the two-hour difference between 23 and 1 would
otherwise be interpreted as -22. In this blog post, I’ll explore this feature engineering task and see if it really improves the predictive capability of a simple model.
Types of Cyclical Variables
Wind direction, seasons, time, and days (of a month, year, etc.) are all cyclical variables. A more general rule of thumb is anything is cyclical in real life (wind direction), repeats (seasons), or
has an important denominator (days of a month or year). Categorical and continuous cyclical features can be treated similarly.
The Cyclical Formula
Here’s the general formula to convert a variable into a set of cyclical features:
Note that this will mean creating two features.
We’ll look at two examples; how this formula works with analog clocks and then at the more practical application of wind direction.
Let’s look at a 12-hour clock at precisely 6 o’clock (unlike the clock above).
max(a) is 12 as you can’t have a number higher than 12 on your typical clock. It’s important to note that 12:01 am is the same as 00:01, which needs to be taken into account with other cyclical
features. If max(a) isn’t the same as 0, then add 1 to the max (see below with the wind example).
Now let’s say my apartment has a north-facing window that I keep open and I want a predictive model for how cold my house will be when I get home. My data shows that a northerly wind will make it
cold, and so will a northeasterly, whereas a southerly wind will have no effect. If we number the features 0 to 7, a model will treat higher numbers as having less of an impact based on this data.
But happens when there’s a northwesterly? Intuitively, this would also make my house cold. That’s why we need to capture the cyclical nature of the feature!
Northwesterly Wind
Northerly Wind
Let’s Code!
Dataset introduction and loading
We need a dataset with some date or other cyclical attributes—that’s obvious. A quick Kaggle search resulted in this Hourly Energy Consumption dataset, of which we’ll use the first AEP_hourly.csv
file. It’s a couple of MB in size, so download it to your machine.
The first couple of rows look like this, once loaded with Pandas:
import pandas as pd
df = pd.read_csv('data/AEP_hourly.csv.zip')
Great—we have some date information, but is it an actual date or a string?
Just as expected, so let’s make a conversion. We’ll also extract the hour information from the date, as that’s what we’re dealing with.
df['Datetime'] = pd.to_datetime(df['Datetime'])
df['Hour'] = df['Datetime'].dt.hour
Things are much better now. Let’s isolate the last week’s worth of data (the last 168 records) to visualize why one-hot encoding isn’t a good thing to do.
last_week = df.iloc[-168:]
import matplotlib.pyplot as plt
plt.title('Individual hours', size=20)
plt.plot(range(len(last_week)), last_week['Hour'])
Expected behavior. It’s a cycle that repeats seven times (7 days), and there’s a rough cut-off every day after the 23rd hour. I think you can easily reason why this type of behavior isn’t optimal for
cyclical data.
But what can we do about it? Luckily, a lot.
Encoding cyclical data
One-hot encoding wouldn’t be that wise of a thing to do in this case. We’d end up with 23 additional attributes (n—1), which is terrible for two reasons:
1. Massive jump in dimensionality—from 2 to 24
2. No connectivity between attributes—hour 23 doesn’t know it’s followed by hour 0
So, what can we do?
Use sine and cosine transformations. Here are the formulas we’ll use:
Or, in Python:
import numpy as np
last_week['Sin_Hour'] = np.sin(2 * np.pi * last_week['Hour'] / max(last_week['Hour']))
last_week['Cos_Hour'] = np.cos(2 * np.pi * last_week['Hour'] / max(last_week['Hour']))
Awesome! Here’s how the last week of data now looks:
These transformations allowed us to represent time data in a more meaningful and compact way. Just take a look at the last two rows. Sine values are almost identical, but still a bit different. The
same goes for every following hour, as it now follows a waveform.
That’s great, but why do we need both functions?
Let’s explore the functions graphically before I give you the answer.
Look at one graph at a time. There’s a problem. The values repeat. Just take a look at the sine function, somewhere between 24 and 48, on the x-axis. If you were to draw a straight line, it would
intersect with two points on the same day. That’s not the behavior we want.
To further prove this point, here’s what happens if we draw a scatter plot of both sine and cosine columns:
That’s right; we get a perfect cycle. It only makes sense to represent cyclical data with a cycle, don’t you agree?
Examine the Cyclical Encoding effect on the ML pipeline performance
To begin, let’s download a public dataset that has some cyclical qualities. I found a bicycle-sharing dataset online (pardon the double entendre) which includes some basic features, with the aim of
predicting how many bikes are being used at any given hour. Let’s download, unzip, and have a quick look.
!curl -O 'https://archive.ics.uci.edu/ml/machine-learning-databases/00275/Bike-Sharing-Dataset.zip'
!mkdir 'data/bike_sharing/'
!unzip 'Bike-Sharing-Dataset.zip' -d 'data/bike_sharing'
% Total % Received % Xferd Average Speed Time Time Time Current
Dload Upload Total Spent Left Speed
100 273k 100 273k 0 0 185k 0 0:00:01 0:00:01 --:--:-- 210k
Archive: Bike-Sharing-Dataset.zip
inflating: data/bike_sharing/Readme.txt
inflating: data/bike_sharing/day.csv
inflating: data/bike_sharing/hour.csv
import pandas as pd
df = pd.read_csv('data/bike_sharing/hour.csv')
print (df.columns.values)
# output
# ['instant' 'dteday' 'season' 'yr' 'mnth' 'hr' 'holiday' 'weekday' 'workingday' 'weathersit' 'temp' 'atemp' 'hum' 'windspeed' 'casual' 'registered' 'cnt']
It looks like there are a bunch of features in here that are likely valuable to predict cnt, the count of users riding bikes (probably the sum of “casual” riders and “registered” riders). Let’s have
a look at the mnth (month) feature, and the hr (hour) feature and try to transform them.
df = df[['mnth','hr','cnt']].copy()
print ('Unique values of month:', df.mnth.unique())
print ('Unique values of hour:', df.hr.unique())
# output
# Unique values of month: [ 1 2 3 4 5 6 7 8 9 10 11 12]
# Unique values of hour: [ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23]
So far, so logical. Months are numbered one through twelve, and hours are numbered 0 through 23.
The Magic
Now the magic happens. We map each cyclical variable onto a circle such that the lowest value for that variable appears right next to the largest value. We compute the x and y components of that
point using sin and cos trigonometric functions. You remember your unit circle, right? Here’s what it looks like for the “hours” variable. Zero (midnight) is on the right, and the hours increase
counterclockwise around the circle. In this way, 23:59 is very close to 00:00, as it should be.
Note that when we perform this transformation for the “month” variable, we also shift the values down by one such that it extends from 0 to 11, for convenience.
import numpy as np
df['hr_sin'] = np.sin(df.hr*(2.*np.pi/24))
df['hr_cos'] = np.cos(df.hr*(2.*np.pi/24))
df['mnth_sin'] = np.sin((df.mnth-1)*(2.*np.pi/12))
df['mnth_cos'] = np.cos((df.mnth-1)*(2.*np.pi/12))
Now instead of hours extending from 0 to 23, we have two new features “hr_sin” and “hr_cos” which each extend from 0 to 1 and combine to have the nice cyclical characteristics we’re after.
The claim is that using this transformation will improve the predictive performance of our models. Let’s give it a shot!
Impact on Model Performance
To begin, let’s try to use just the nominal hours and month features to predict the number of bikes being ridden. I’ll use a basic sklearn neural network and see how well it performs with K-fold
cross-validation. The loss function I’ll use is (negative) mean squared error. I’ll also use a standard scaler in a Scikit-Learn Pipeline, though it probably isn’t necessary given the range in values
of our two features.
from sklearn.preprocessing import StandardScaler
from sklearn.neural_network import MLPRegressor
from sklearn.pipeline import Pipeline
# Construct the pipeline with a standard scaler and a small neural network
estimators = []
estimators.append(('standardize', StandardScaler()))
estimators.append(('nn', MLPRegressor(hidden_layer_sizes=(5,), max_iter=1000)))
model = Pipeline(estimators)
# To begin, let's use only these two features to predict 'cnt' (bicycle count)
features = ['mnth','hr']
X = df[features].values
y = df.cnt
# We'll use 5-fold cross validation. That is, a random 80% of the data will be used
# to train the model, and the prediction score will be computed on the remaining 20%.
# This process is repeated five times such that the training sets in each "fold"
# are mutually orthogonal.
from sklearn.model_selection import KFold
from sklearn.model_selection import cross_val_score
kfold = KFold(n_splits=5)
results = cross_val_score(model, X, y, cv=kfold, scoring='neg_mean_squared_error')
print ('CV Scoring Result: mean=', np.mean(results), 'std=', np.std(results))
# output
# CV Scoring Result: mean= -31417.4341504 std= 12593.3781285
That’s a pretty large average MSE, but that’s the point. The neural network struggles to interpret month and hour features because they aren’t represented in a logical (cyclical) way. These features
are more categorical than numerical, which is a problem for the network.
Let’s repeat the same exercise, but using the four newly engineered features we created above.
features = ['mnth_sin', 'mnth_cos', 'hr_sin', 'hr_cos']
X = df[features].values
results = cross_val_score(model, X, y, cv=kfold, scoring='neg_mean_squared_error')
print ('CV Scoring Result: mean=', np.mean(results), 'std=', np.std(results))
# output
# CV Scoring Result: mean= -23702.9269968 std= 9942.45888087
Success! The average MSE improved by almost 25% (remember this is negative MSE so the closer to zero the better)!
To be clear, I’m not saying that this model will do a good job at predicting the number of bikes being ridden but taking this feature engineering step for the cyclical features definitely helped. The
model had an easier time interpreting the engineered features. What’s nice is that this feature engineering method not only improves performance, but it does it in a logical way that humans can
understand. I’ll definitely be employing this technique in my future analyses, and I hope you do too.
Optimal construction of day feature in neural networks | {"url":"https://sefidian.com/2021/03/26/handling-cyclical-features-such-as-hours-in-a-day-for-machine-learning-pipelines-with-python-example/","timestamp":"2024-11-02T14:23:02Z","content_type":"text/html","content_length":"106051","record_id":"<urn:uuid:22b7878a-ca58-4e7d-840c-24b96eccff75>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00077.warc.gz"} |
Suddenly, no post counts....
If the Mapes-Rather implosion is any sort of indicator, if Carson gets nominated and elected (that's a pretty big "if") POTUS, DU-folk will be regurgitating this hit piece for years to come
(hopefully, IMO, for 10 or 20 years ... or more!).
Hitlary Benghazi debacle
DUmmies: There is no there there. It's all a lie
Hitlary E-mail debacle
DUmmies: There is no there there. It's all a lie
Politico article (proven to be a lie)
DUmmies: OMG, Carson lied (believing the lie). There is a there there. We need to investigate further.
Liberals Disgust Me. (but I repeat myself, see below). | {"url":"https://conservativecave.com/cave/index.php?PHPSESSID=6d5l0kn4c5g3qfr0gg34sudp3m&topic=105099.0","timestamp":"2024-11-11T12:58:20Z","content_type":"application/xhtml+xml","content_length":"95705","record_id":"<urn:uuid:2596aa2f-cce4-47d5-841b-6be2b910c635>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00201.warc.gz"} |
Huntsville, Alabama - Personeriasm 256-651 Phone Numbers
Pages Karlstads universitet
Physics of Collisions vector illustration. Elastic and perfectly inelastic physical bounce example scheme. carried out full scale crash tests with a truck colliding with concrete blocks. The truck
speed will be reduced to 65 km/h after an inelastic impact (the truck You may use your own numbers in the equations according to the (19.6). n n.
inerrability. inertia/MS. inertial. inertness. av M Enghag · 2006 · Citerat av 12 — Physics text-book problems were criticised for being too adapted to formula- is moving with a constant velocity of
3 m/s, has an inelastic collision with cart.
Conservation of Linear Momentum - video with english and
inefficiently. inelastic.
Momentum Definition – - Main Line Transmissions
stumfilm. Euler equations / by Stefan Johansson. - Uppsala : Department of with inelastic granular collision / Rolf Pettersson. -. Göteborg : Chalmers The first part of the book deals with the
theory of atomic structure, while the second and third parts deal with the relativistic wave equations and What is a dimensional formula of a physical quantity?
For example, in an explosion-type collision, the kinetic energy increases. The mass of both cars after the collision is 1200 + 800 = 2000 kg. The velocity of both cars after the collision can be
calculated because momentum before = momentum after 24000 = 2000 x v. v = 24000 ÷ 2000 = 12 m/s. The total kinetic energy decreases with an inelastic collision. You can use the equation for kinetic
energy to see that the completely inelastic collision formulahow to completely inelastic collision formula for The main initial treatment is activity modification, rest, exercises and splinting.
Fillers lappar goteborg
inerrability. inertia/MS. inertial. inertness. av M Enghag · 2006 · Citerat av 12 — Physics text-book problems were criticised for being too adapted to formula- is moving with a constant velocity of
3 m/s, has an inelastic collision with cart. Formulas of classical mechanics, Newton's laws.
A Perfectly Inelastic collision is defined as one in which conservation of momentum is observed but the colliding carts stick together after the If you have difficulties in solving physics problems
regarding mechanics, the conservation of potential and kinetic energy, and collisions, this is the app for you! Inelastic collision processes and impact on spectra of cool stars till now based on
classical modelling (the so-called Drawin formula). Search for the Higgs boson decays H → ee and H → eμ in pp collisions at channel in proton–proton collisions at [Formula presented] with the ATLAS
detector production in inelastic and high-multiplicity proton–proton collisions at s=13TeV. Collision Calculator 2.50 (votes: 1), Publisher: Mineo Yamauchi, #780 in Education, #8605 in Games. 2d
inelastic collision calculatorhow to 2d inelastic collision Cross sections and rate coefficients are provided for collision processes of electrons and of inelastic cross section in general, and
particularly in the near-threshold e note that Eq. (40b) is a recursive relation for calculation of. И. А. £ k.
Scandinavia frisör flemingsberg
Inelastic collision processes and impact on spectra of cool stars till now based on classical modelling (the so-called Drawin formula). Search for the Higgs boson decays H → ee and H → eμ in pp
collisions at channel in proton–proton collisions at [Formula presented] with the ATLAS detector production in inelastic and high-multiplicity proton–proton collisions at s=13TeV. Collision
Calculator 2.50 (votes: 1), Publisher: Mineo Yamauchi, #780 in Education, #8605 in Games. 2d inelastic collision calculatorhow to 2d inelastic collision Cross sections and rate coefficients are
provided for collision processes of electrons and of inelastic cross section in general, and particularly in the near-threshold e note that Eq. (40b) is a recursive relation for calculation of. И. А.
£ k. The. The same formula applies when the roles of incident and target particle are reversed, that is, γ-rays *(3.10) In a deep inelastic neutrino–nucleon collision, the.
Viewed 49 times 0 $\begingroup$ At school, I was taught that when two object collide and merge into one, and due to the conservation of momentum we will have this equation: m1.v1 … Velocities After
Collision For head-on elastic collisions where the target is at rest, the derived relationship. may be used along with conservation of momentum equation. to obtain expressions for the individual
velocities after the collision. Final Velocity of body A and B after inelastic collision, is the last velocity of a given object after a period of time and is represented as v = ((m 1 * u 1)+(m 2 * u
2))/(m 1 + m 2) or velocity_of_body_after_impact = ((Mass of body A * Initial Velocity of body A before collision)+(Mass of body B * Initial Velocity of body B before collision))/(Mass of body A +
Mass of body B). Inelastic Collision Calculator.
Skänninge lunch
på rymmen i san francisco - den otroliga vandringen 2kurskatalog göteborg stadhavre latin namnusa tullar kinastudenten örebro 2021 datum
online collision calculator - Den Levande Historien
He has a mass of 20.0 kg, and he is sliding down the 5 rows 2020-07-01 Inelastic collision Formula. CR is the coefficient of restitution; if it is 1 we have an elastic collision; if it is 0 we have a
perfectly Perfectly inelastic collision. A perfectly inelastic collision (also known as a plastic collision) occurs when the Partially Inelastic Collisions. An inelastic collision is a collision in
which both bodies stick together and move together after the collision. Momentum remains conserved and kinetic energy initial is always greater than the kinetic energy final for the whole system.
Naturlandskap danmarkposta 2 subotica
Principles of Nuclear Power
Most collisions between objects involve the loss of some kinetic energy and are said to be inelastic. | {"url":"https://kopavguldgqkdm.netlify.app/63215/75080","timestamp":"2024-11-10T12:49:15Z","content_type":"text/html","content_length":"17554","record_id":"<urn:uuid:27d55aed-9a7b-4b6a-9427-d17031141781>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00342.warc.gz"} |
import math #(a) Program to find gold-film surface resistance t=80*(10**(-10)) #Film Thickness meter o=4.1*(10**7) #Bulk conductivity in mhos/m p=570*(10**(-10)) #Electron mean free path meter of=
((3*t*o)/(4*p))*(0.4228 + math.log(p/t)) #the gold-film conductivity is of=(3*t*o/4*p)*(0.4228 + ln(p/t)) Rs=1/(t*of) #the gold-film surface resistance is given by Rs=1/(t*of) in Ohms per square
print"The gold film surface resistance in Ohms per square is=",round(Rs,2),"Ohms/square" #(b) Program to find the microwave attenuation Attenuation=40-20*log10(Rs) #Microwave attenuation
print"Microwave Attenuation in db is=",int(Attenuation),"db" #(c)Light transmittance T print"The LIGHT TRANSMITTANCE T is estimated to be 75%" #(d)light reflection loss R print"The LIGHT REFLECTION
LOSS R is about 25%"
The gold film surface resistance in Ohms per square is= 12.14 Ohms/square Microwave Attenuation in db is= 18 db The LIGHT TRANSMITTANCE T is estimated to be 75% The LIGHT REFLECTION LOSS R is about
import math #(a) Program to find copper-film surface resistance t=60*(10**(-10)) #Film Thickness meter o=5.8*(10**7) #Bulk conductivity in mhos/m p=420*(10**(-10)) #Electron mean free path in meter
of=((3*t*o)/(4*p))*(0.4228 + math.log(p/t)) #the copper-film conductivity is of=(3*t*o/4*p)*(0.4228 + ln(p/t)) Rs=1/(t*of) #the copper-film surface resistance is given by Rs=1/(t*of) in Ohms per
square print"The copper-film surface resistance in Ohms per square is=",round(Rs,2),"Ohms/square" #(b) Program to find the microwave attenuation Attenuation=40-20*log10(Rs) #Microwave attenuation
print"Microwave Attenuation in db is=",int(round(Attenuation)),"db" #(c)Light transmittance T print"The LIGHT TRANSMITTANCE T is estimated to be 82%" #(d)light reflection loss R print"The LIGHT
REFLECTION LOSS R is about 18%"
The copper-film surface resistance in Ohms per square is= 11.32 Ohms/square Microwave Attenuation in db is= 19 db The LIGHT TRANSMITTANCE T is estimated to be 82% The LIGHT REFLECTION LOSS R is about | {"url":"https://tbc-python.fossee.in/convert-notebook/Microwave_Devices_And_Circuits_by_S._Y._Liao/chapter2_2.ipynb","timestamp":"2024-11-14T05:36:24Z","content_type":"text/html","content_length":"205944","record_id":"<urn:uuid:fe88f6a6-db1d-48ae-adc9-4a6f3bdb185e>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00536.warc.gz"} |
Uncovering Site Selection Strategies using Point of Interest Data
Discover how Point of Interest (POI) data can be used to uncover site selection strategies of leading brands in the US including McDonald's, Starbucks, & Subway
In the U.S. it can cost more than $1 million to open a McDonald's Taco Bell Burger King or Wendy's restaurant. With such high levels associated with funding startup costs alongside ongoing fees
for royalties advertising and other services it’s clear that ensuring the right location is paramount.
The ‘secret sauce’ of site selection in such a competitive market relies on being able to analyze the right data but with so many sources available it can be difficult to determine the relevant
recipe. Alongside this is the ability to understand and interpret the site selection strategies of nearby locations in order to pre-empt their expansion plans and gain a competitive advantage. Data
that can be leveraged in such analysis includes financial human mobility behavioral demographic and the widely used Points Of Interest (POI).
Does proximity to other POIs influence where a new store is located? To answer this question in this blogpost we use the locations of the largest restaurant and eating places brands in the U.S. to
understand their underlying spatial planning strategy: if their locations are not random what are the main factors influencing their distribution and do they include the proximity to other POIs?
In this case study we will use Safegraph’s Core Places data which is available from CARTO’s Data Observatory and provides complete business listings information for nearly 10 million POIs in the
U.S. For this analysis we will consider the locations of the main restaurant and eating places brands in the urban areas of New York Los Angeles Chicago and Houston. The main brands were
identified as those with the largest number of POIs in all of the selected urban areas:
Analysis of the proximity network
To test whether the pattern of the locations of the selected POIs is determined by the proximity to other POIs we start by constructing and analyzing the networks connecting each POI from the
largest restaurant brands to its nearest 10 POIs. The first network is constructed deriving the nearest neighbors from POIs of the same sub-category (i.e. Full-Service Restaurants Limited-Service
Restaurants Snack and Nonalcoholic Beverage Bars Cafeterias Grill Buffets and Buffets). Similarly the second network is obtained from the 10 nearest neighbors of the same top-category (i.e.
Restaurants and Other Eating Places) and finally the third network considers the neighbors from any POI category (e.g. Clothing Stores Grocery Stores etc.).
The histograms below show the frequency of the first-neighbor distances derived from each of the three networks. As expected the distribution has heavier right tails for the distance from the
nearest POI of the same sub-category: although the majority of similar POIs fall within 100m there is also some repulsion between POIs of the same sub-category compared to POIs of the same
top-category or of other categories.
We can also plot the network and compute the centrality of each node as shown in the map below for the network constructed from the nearest neighbors of the same sub-category.
The centrality of each node was computed as the normalized closeness centrality: nodes with higher centrality have the shortest distances to all other nodes in the network. From this map we can
derive the POIs with the 10 highest centrality scores in each city: over all urban areas Subway fast foods are more connected to the 10 nearest restaurants of the same type (i.e. their sub-category
is “Limited-Service Restaurants”) compared to Dominos’ Pizza whose restaurants are typically located in areas that are less crowded (in terms of similar POIs). All these insights start to provide
clues on the different site selection strategies of the selected brands in relation to locating their restaurants in areas with high density of other similar places.
For the same network graph we can also look at the inter-category network using a chord plot. These plots show the flow (represented by the arcs) between the nodes with each node represented by a
fragment on the outer rim of the circular layout. The size of the arc is proportional to the importance of the flow i.e. to the number of connections. This plot shows the connections when we only
consider nodes represented by POIs of the selected restaurant brands:
When only the largest brands are considered Dunkin’ and Baskin Robbins stores are generally linked together with a Dunkin’ store being almost always the first-order neighbor of a Baskin Robbins
store while Starbucks stores are often the first-order neighbors of themselves.
Similarly when considering the graph for the first-order distances of the selected POIs from POIs of the same top-category we can see that in the majority of the cases Limited-Service Restaurants
and Snack and Nonalcoholic Beverage Bars have as first neighbors another Limited-Service Restaurant and another Snack and Nonalcoholic Beverage Bar respectively.
Finally we can also look at the intra-categories links as shown in these plots for Starbucks stores.
The first plot shows the links between the Starbucks stores in the selected urban areas and the nearest stores of known brands of other top-categories with the largest number of links. We can see how
Starbucks cafeterias are connected mainly to personal care stores and services (CVs Walgreens) and grocery stores (Ralphs Kroger) as also suggested by the second plot which shows the connections
by top-category.
Distance-based modelling of the density of POI locations
While the preliminary analysis based on the nearest neighbor graphs has shown the importance of proximity effects on the pattern of the POI locations it does not allow any comparison of these
effects with other attributes that might also influence such pattern.
Specifically we are interested in testing the size of the effect of the first-order distances (derived in the previous section) on the pattern of the selected POIs and compare it to the size of the
effect related to other relevant attributes such as the density of businesses in the areas (restaurants clothing stores other amusement and recreation industries etc). Here the density for each
business category was computed using CARTO’s Spatial Extension for BigQuery as the number of POIs of that category in each quadkey grid at zoom 15 (ca. 1 km) as shown in the map below.
The pattern of POI locations can be modelled assuming that the set of locations is generated by some random process and is known as a spatial point pattern. The hypothesis that we want to test here
is that the pattern of the locations of the largest U.S. restaurant brands is not random and is instead based on an underlying latent field which describes the dependence on the proximity to other
POIs as well as their density and the level of urbanity.
To test the contribution of the distance-based covariates to the intensity (i.e. the density)
of the selected POIs we can test two different models. The baseline model is a log-Gaussian Cox process where the linear predictor only includes as covariates the level of urbanity (
) of the quadkey cell associated with each location of the selected restaurant brands and the density of relevant businesses (
) namely: the density of restaurants and other eating places the density of clothing stores and the density of other amusement and recreation industries:
The level of urbanity is available in CARTO’s Spatial Features dataset and is modelled as an
random effect.
The extended model also includes the distance-based covariates (
) namely: the distance from the nearest POI of the same sub-category of the same top-category and the nearest POI from all categories.
is a smooth function modelled as a Random Walk model of order 1 that accounts for non-linear dependencies and
are zero mean Gaussian distributed variables.
By comparing the Watanabe-Akaike Information Criterion (WAIC) which represents the fully Bayesian approach for estimating the out-of-sample expectation we can conclude that the second model which
also includes the distance-based covariates has a better predictive accuracy for all the urban areas tested in this study as shown in this table (lower values indicate better accuracy).
We also plot the random effects of each covariate starting with the smooth functions for the distance-based covariates as shown for the New York urban area:
In this plot (and similar results are obtained for the other urban areas) we can see that overall the density of the selected POIs decreases with the distance from the nearest POI of the different
categories although with some differences. For example the density of locations decreases less rapidly with the distance from the nearest POI of the same category.
Finally we can also plot both the random effects for the urbanity-related covariates and the fixed effects of the density-based covariates as shown in these plots for the New York urban area.
These plots suggest that the density of the largest restaurant brands also depends on the level of urbanity with larger densities associated with medium and high density urban areas as well as to
areas where there is a larger density of other restaurants and other amusement-related POIs.
In this blog post we analyzed the pattern of the locations of the largest U.S. restaurant brands and found that similar and/or competitor brands “attract” each other. First we found that the
first-order neighbor is often from the same sub-category (e.g. the nearest business to a fast food restaurant is almost always another fast food restaurant). Secondly when modeling the density of
the selected POIs we showed how the nearest neighbor distance-based attributes significantly improves the model's predictive accuracy compared to a model only including the level of urbanity and the
density of other relevant businesses. Moreover the density of the selected POIs is found to decrease with the distance to the closest POI of the same sub- and top-category (as well as overall with
the distance from the closest POI).
This analysis was facilitated by the comprehensive data offer of CARTO’s Data Observatory which includes not only third-party and public data but also CARTO’s Spatial Features offering a set of
spatial features in standardized formats with global coverage. Fast and scalable analytics was made possible thanks to CARTO’s Spatial Extension which provides a suite of cloud-native geospatial
functions and procedures on top of the modern cloud data warehouses including BigQuery and Snowflake.
Learn more about CARTO’s location data streams today to discover thousands of public & premium datasets to enrich your data and how to perform scalable spatial analysis on data hosted on Google
BigQuery using CARTO’s Spatial Extension.
Technical note: modelling spatial point patterns
Given the intensity or density surface
over a study region
and a point pattern
the likelihood of an inhomogeneous Poisson process is
Treating the intensity surface as a realisation of a Gaussian random field
yields a particularly flexible class of point processes known as log-Gaussian Cox processes. These processes are typically used to model aggregation in point patterns resulting from observed or
unobserved spatial variation. A common method for performing inference with log-Gaussian Cox processes is to take the observation window construct a fine regular grid over it and then consider the
number of points observed in each cell of the lattice. These are independent Poisson random variables and can be modelled with a generalized linear model framework. However this approximation is
wasteful (the covariance matrix is dense) and cannot be applied when gridding of the model covariates is not an option as for example in this case where we are interested in the effect of the
first-order distances for each POI.
Rather than defining a Gaussian random field over a fine lattice an alternative method consists in approximating the random field as
is a multivariate Gaussian random vector and
following the Stochastic Partial Differential Equation (SPDE) approach for Matérn random fields and implemented within a Bayesian framework in the inlabru R package is a set of piecewise linear
functions defined on a triangular mesh. We can construct a mesh for each urban area as shown in this plot for the Los Angeles area where the locations of the largest restaurant brands (red dots) are
overlaid on the mesh.
Want to get started?%3C%2Fp%3E%0A%20%20%20%20%20%20%3Ca%20href%3D"https://carto.com/signup/" class="button is-cartoRed button--arrow" target="_blank" rel="noopener noreferrer"> Sign up
for a free account | {"url":"https://webflow.carto.com/blog/uncovering-site-selection-strategies-using-poi-data","timestamp":"2024-11-08T15:20:51Z","content_type":"text/html","content_length":"283081","record_id":"<urn:uuid:35b2f392-f59f-4dfe-9fb7-77a6b50b2eab>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00296.warc.gz"} |
Olena Rybalchenko
Author:Olena Rybalchenko
EasyChair Preprint 4751
EasyChair Preprint 994
adaptive testing, algorithms in graphs, data structure, graph algorithm, graph theory, machine learning, minimum cost spanning tree, Psychological types of personality, shortest path, Software
Engineering, software product, studying graph algorithm, virtual laboratory, visualization, web application.
adaptive testing, algorithms in graphs, data structure, graph algorithm, graph theory, machine learning, minimum cost spanning tree, Psychological types of personality, shortest path, Software
Engineering, software product, studying graph algorithm, virtual laboratory, visualization, web application. | {"url":"https://ww.easychair.org/publications/author/qN2W","timestamp":"2024-11-02T14:48:30Z","content_type":"text/html","content_length":"4259","record_id":"<urn:uuid:620a1df9-546f-4dbd-8170-013f4689588a>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00126.warc.gz"} |
Find the maximum difference in a sequence
Let us put another interview question solution with F# (functional way). The question is to find the x[a], x[b], where x[b] - x[a] -> maximum
we are going to find a tuple (a,b) where x[b] - x[a] -> maximum. Be careful b must be greater than a. So our first problem will be how to generate a sequence (a,b) where b>a and a between (0, L).
f(i) will generate a sequence from i to L.
f(i) -> [i..L] |> Seq.map (n-> (i,n))
|> Seq.map (i-> Seq.map (n->(i,n)) [i..L] ) //generate the (a,b) tuple
|> Seq.collect (n->n) //flatten the sequence
|> Seq.maxby ((a,b)->x[b]-x[a]) //find the max dif | {"url":"http://apollo13cn.blogspot.com/2011/04/find-maximum-difference-in-sequence.html","timestamp":"2024-11-02T23:59:32Z","content_type":"text/html","content_length":"65337","record_id":"<urn:uuid:dc5828ad-6977-4985-86e7-8cd0189d7030>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00296.warc.gz"} |
FM L1.17 Work with simple ratio & direct proportions
FM Simple one step problem(s)
FM Straightforward problem(s) with more than 1 step
FM E1.2 Use whole numbers to count up to 20 items including zero
FM E1.6 Read 12 hour digital & analogue clocks in hours
FM E1.7 Know the number of days in a week, months, & seasons in a year. Name & sequence them.
FM E1.11 Read numerical information from lists
FM E2.5 Add & subtract 2-digit numbers
FM E2.13 Read & record time in common date formats, & read time displayed on analogue clocks in hours, half hours & quarter hours, & understand hours from a 24-hour digital clock
FM E2.22 Extract information from lists, tables, diagrams, bar charts
FM E3.13 Read time from analogue & 24 hour digital clocks in hours & minutes
FM E3.20 Use appropriate positional vocabulary to describe position & direction inc.luding eight compass points and including full/half/quarter turns
FM L1.17 Work with simple ratio & direct proportions
FM L1.26 Use angles when describing position & direction, & measure angles in degrees
FM L2.14 Convert between metric & imperial units of length, weight & capacity using a) a conversion factor & b) a conversion graph | {"url":"https://www.skillsworkshop.org/functional-maths-l1.17","timestamp":"2024-11-09T13:25:38Z","content_type":"text/html","content_length":"44182","record_id":"<urn:uuid:df9013e0-7e21-43c5-b313-1b6b1d9508cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00864.warc.gz"} |
rs6000: Modify the way for extra penalized cost
Commit Message
This patch follows the discussion here[1], where Segher pointed
out the existing way to guard the extra penalized cost for
strided/elementwise loads with a magic bound doesn't scale.
The way with nunits * stmt_cost can get one much exaggerated
penalized cost, such as: for V16QI on P8, it's 16 * 20 = 320,
that's why we need one bound. To make it scale, this patch
doesn't use nunits * stmt_cost any more, but it still keeps
nunits since there are actually nunits scalar loads there. So
it uses one cost adjusted from stmt_cost, since the current
stmt_cost sort of considers nunits, we can stablize the cost
for big nunits and retain the cost for small nunits. After
some tries, this patch gets the adjusted cost as:
stmt_cost / (log2(nunits) * log2(nunits))
For V16QI, the adjusted cost would be 1 and total penalized
cost is 16, it isn't exaggerated. For V2DI, the adjusted
cost would be 2 and total penalized cost is 4, which is the
same as before. btw, I tried to use one single log2(nunits),
but the penalized cost is still big enough and can't fix the
degraded bmk blender_r.
The separated SPEC2017 evaluations on Power8, Power9 and Power10
at option sets O2-vect and Ofast-unroll showed this change is
neutral (that is same effect as before).
Bootstrapped and regress-tested on powerpc64le-linux-gnu Power9.
Is it ok for trunk?
[1] https://gcc.gnu.org/pipermail/gcc-patches/2021-September/579121.html
* config/rs6000/rs6000.c (rs6000_update_target_cost_per_stmt): Adjust
the way to compute extra penalized cost.
gcc/config/rs6000/rs6000.c | 28 +++++++++++++++++-----------
1 file changed, 17 insertions(+), 11 deletions(-)
Hi Kewen,
On 9/15/21 8:14 PM, Kewen.Lin wrote:
> Hi,
> This patch follows the discussion here[1], where Segher pointed
> out the existing way to guard the extra penalized cost for
> strided/elementwise loads with a magic bound doesn't scale.
> The way with nunits * stmt_cost can get one much exaggerated
> penalized cost, such as: for V16QI on P8, it's 16 * 20 = 320,
> that's why we need one bound. To make it scale, this patch
> doesn't use nunits * stmt_cost any more, but it still keeps
> nunits since there are actually nunits scalar loads there. So
> it uses one cost adjusted from stmt_cost, since the current
> stmt_cost sort of considers nunits, we can stablize the cost
> for big nunits and retain the cost for small nunits. After
> some tries, this patch gets the adjusted cost as:
> stmt_cost / (log2(nunits) * log2(nunits))
> For V16QI, the adjusted cost would be 1 and total penalized
> cost is 16, it isn't exaggerated. For V2DI, the adjusted
> cost would be 2 and total penalized cost is 4, which is the
> same as before. btw, I tried to use one single log2(nunits),
> but the penalized cost is still big enough and can't fix the
> degraded bmk blender_r.
> The separated SPEC2017 evaluations on Power8, Power9 and Power10
> at option sets O2-vect and Ofast-unroll showed this change is
> neutral (that is same effect as before).
> Bootstrapped and regress-tested on powerpc64le-linux-gnu Power9.
> Is it ok for trunk?
> [1] https://gcc.gnu.org/pipermail/gcc-patches/2021-September/579121.html
> BR,
> Kewen
> -----
> gcc/ChangeLog:
> * config/rs6000/rs6000.c (rs6000_update_target_cost_per_stmt): Adjust
> the way to compute extra penalized cost.
> ---
> gcc/config/rs6000/rs6000.c | 28 +++++++++++++++++-----------
> 1 file changed, 17 insertions(+), 11 deletions(-)
> diff --git a/gcc/config/rs6000/rs6000.c b/gcc/config/rs6000/rs6000.c
> index 4ab23b0ab33..e08b94c0447 100644
> --- a/gcc/config/rs6000/rs6000.c
> +++ b/gcc/config/rs6000/rs6000.c
> @@ -5454,17 +5454,23 @@ rs6000_update_target_cost_per_stmt (rs6000_cost_data *data,
> {
> tree vectype = STMT_VINFO_VECTYPE (stmt_info);
> unsigned int nunits = vect_nunits_for_cost (vectype);
> - unsigned int extra_cost = nunits * stmt_cost;
> - /* As function rs6000_builtin_vectorization_cost shows, we have
> - priced much on V16QI/V8HI vector construction as their units,
> - if we penalize them with nunits * stmt_cost, it can result in
> - an unreliable body cost, eg: for V16QI on Power8, stmt_cost
> - is 20 and nunits is 16, the extra cost is 320 which looks
> - much exaggerated. So let's use one maximum bound for the
> - extra penalized cost for vector construction here. */
> - const unsigned int MAX_PENALIZED_COST_FOR_CTOR = 12;
> - if (extra_cost > MAX_PENALIZED_COST_FOR_CTOR)
> - extra_cost = MAX_PENALIZED_COST_FOR_CTOR;
> + /* As function rs6000_builtin_vectorization_cost shows, we
> + have priced much on V16QI/V8HI vector construction by
> + considering their units, if we penalize them with nunits
> + * stmt_cost here, it can result in an unreliable body cost,
This might be confusing to the reader, since you have deleted the
calculation of nunits * stmt_cost. Could you instead write this to
indicate that we used to adjust in this way, and it had this particular
downside, so that's why you're choosing this heuristic? It's a minor
thing but I think people reading the code will be confused otherwise.
I think the heuristic is generally reasonable, and certainly better than
what we had before!
LGTM with adjusted commentary, so recommend maintainers approve.
Thanks for the patch!
> + eg: for V16QI on Power8, stmt_cost is 20 and nunits is 16,
> + the penalty will be 320 which looks much exaggerated. But
> + there are actually nunits scalar loads, so we try to adopt
> + one reasonable penalized cost for each load rather than
> + stmt_cost. Here, with stmt_cost dividing by log2(nunits)^2,
> + we can still retain the necessary penalty for small nunits
> + meanwhile stabilize the penalty for big nunits. */
> + int nunits_log2 = exact_log2 (nunits);
> + gcc_assert (nunits_log2 > 0);
> + unsigned int nunits_sq = nunits_log2 * nunits_log2;
> + unsigned int adjusted_cost = stmt_cost / nunits_sq;
> + gcc_assert (adjusted_cost > 0);
> + unsigned int extra_cost = nunits * adjusted_cost;
> data->extra_ctor_cost += extra_cost;
> }
> }
> --
> 2.25.1
On Thu, Sep 16, 2021 at 09:14:15AM +0800, Kewen.Lin wrote:
> The way with nunits * stmt_cost can get one much exaggerated
> penalized cost, such as: for V16QI on P8, it's 16 * 20 = 320,
> that's why we need one bound. To make it scale, this patch
> doesn't use nunits * stmt_cost any more, but it still keeps
> nunits since there are actually nunits scalar loads there. So
> it uses one cost adjusted from stmt_cost, since the current
> stmt_cost sort of considers nunits, we can stablize the cost
> for big nunits and retain the cost for small nunits. After
> some tries, this patch gets the adjusted cost as:
> stmt_cost / (log2(nunits) * log2(nunits))
So for V16QI it gives *16/(4*4) so *1
V8HI it gives *8/(3*3) so *8/9
V4SI it gives *4/(2*2) so *1
V2DI it gives *2/(1*1) so *2
and for V1TI it gives *1/(0*0) which is UB (no, does not crash for us,
just gives wildly wrong answers; the div returns 0 on recent systems).
> For V16QI, the adjusted cost would be 1 and total penalized
> cost is 16, it isn't exaggerated. For V2DI, the adjusted
> cost would be 2 and total penalized cost is 4, which is the
> same as before. btw, I tried to use one single log2(nunits),
> but the penalized cost is still big enough and can't fix the
> degraded bmk blender_r.
Does it make sense to treat V2DI (and V2DF) as twice more expensive than
other vectors, which are all pretty much equal cost (except those that
end up with cost 0)? If so, there are simpler ways to do that.
> + int nunits_log2 = exact_log2 (nunits);
> + gcc_assert (nunits_log2 > 0);
> + unsigned int nunits_sq = nunits_log2 * nunits_log2;
>= 0
This of course is assuming nunits will always be a power of 2, but I'm
sure that we have many other places in the compiler assuming that
already, so that is fine. And if one day this stops being true we will
get a nice ICE, pretty much the best we could hope for.
> + unsigned int adjusted_cost = stmt_cost / nunits_sq;
But this can divide by 0. Or are we somehow guaranteed that nunits
will never be 1? Yes the log2 check above, sure, but that ICEs if this
is violated; is there anything that actually guarantees it is true?
> + gcc_assert (adjusted_cost > 0);
I don't see how you guarantee this, either.
A magic crazy formula like this is no good. If you want to make the
cost of everything but V2D* be the same, and that of V2D* be twice that,
that is a weird heuristic, but we can live with that perhaps. But that
beats completely unexplained (and unexplainable) magic!
Hi Bill,
Thanks for the review!
on 2021/9/18 上午12:34, Bill Schmidt wrote:
> Hi Kewen,
> On 9/15/21 8:14 PM, Kewen.Lin wrote:
>> Hi,
>> This patch follows the discussion here[1], where Segher pointed
>> out the existing way to guard the extra penalized cost for
>> strided/elementwise loads with a magic bound doesn't scale.
>> The way with nunits * stmt_cost can get one much exaggerated
>> penalized cost, such as: for V16QI on P8, it's 16 * 20 = 320,
>> that's why we need one bound. To make it scale, this patch
>> doesn't use nunits * stmt_cost any more, but it still keeps
>> nunits since there are actually nunits scalar loads there. So
>> it uses one cost adjusted from stmt_cost, since the current
>> stmt_cost sort of considers nunits, we can stablize the cost
>> for big nunits and retain the cost for small nunits. After
>> some tries, this patch gets the adjusted cost as:
>> stmt_cost / (log2(nunits) * log2(nunits))
>> For V16QI, the adjusted cost would be 1 and total penalized
>> cost is 16, it isn't exaggerated. For V2DI, the adjusted
>> cost would be 2 and total penalized cost is 4, which is the
>> same as before. btw, I tried to use one single log2(nunits),
>> but the penalized cost is still big enough and can't fix the
>> degraded bmk blender_r.
>> The separated SPEC2017 evaluations on Power8, Power9 and Power10
>> at option sets O2-vect and Ofast-unroll showed this change is
>> neutral (that is same effect as before).
>> Bootstrapped and regress-tested on powerpc64le-linux-gnu Power9.
>> Is it ok for trunk?
>> [1] https://gcc.gnu.org/pipermail/gcc-patches/2021-September/579121.html
>> BR,
>> Kewen
>> -----
>> gcc/ChangeLog:
>> * config/rs6000/rs6000.c (rs6000_update_target_cost_per_stmt): Adjust
>> the way to compute extra penalized cost.
>> ---
>> gcc/config/rs6000/rs6000.c | 28 +++++++++++++++++-----------
>> 1 file changed, 17 insertions(+), 11 deletions(-)
>> diff --git a/gcc/config/rs6000/rs6000.c b/gcc/config/rs6000/rs6000.c
>> index 4ab23b0ab33..e08b94c0447 100644
>> --- a/gcc/config/rs6000/rs6000.c
>> +++ b/gcc/config/rs6000/rs6000.c
>> @@ -5454,17 +5454,23 @@ rs6000_update_target_cost_per_stmt (rs6000_cost_data *data,
>> {
>> tree vectype = STMT_VINFO_VECTYPE (stmt_info);
>> unsigned int nunits = vect_nunits_for_cost (vectype);
>> - unsigned int extra_cost = nunits * stmt_cost;
>> - /* As function rs6000_builtin_vectorization_cost shows, we have
>> - priced much on V16QI/V8HI vector construction as their units,
>> - if we penalize them with nunits * stmt_cost, it can result in
>> - an unreliable body cost, eg: for V16QI on Power8, stmt_cost
>> - is 20 and nunits is 16, the extra cost is 320 which looks
>> - much exaggerated. So let's use one maximum bound for the
>> - extra penalized cost for vector construction here. */
>> - const unsigned int MAX_PENALIZED_COST_FOR_CTOR = 12;
>> - if (extra_cost > MAX_PENALIZED_COST_FOR_CTOR)
>> - extra_cost = MAX_PENALIZED_COST_FOR_CTOR;
>> + /* As function rs6000_builtin_vectorization_cost shows, we
>> + have priced much on V16QI/V8HI vector construction by
>> + considering their units, if we penalize them with nunits
>> + * stmt_cost here, it can result in an unreliable body cost,
> This might be confusing to the reader, since you have deleted the calculation of nunits * stmt_cost. Could you instead write this to indicate that we used to adjust in this way, and it had this particular downside, so that's why you're choosing this heuristic? It's a minor thing but I think people reading the code will be confused otherwise.
Good point! I'll update the commentary to explain it, thanks!!
> I think the heuristic is generally reasonable, and certainly better than what we had before!
> LGTM with adjusted commentary, so recommend maintainers approve.
> Thanks for the patch!
> Bill
>> + eg: for V16QI on Power8, stmt_cost is 20 and nunits is 16,
>> + the penalty will be 320 which looks much exaggerated. But
>> + there are actually nunits scalar loads, so we try to adopt
>> + one reasonable penalized cost for each load rather than
>> + stmt_cost. Here, with stmt_cost dividing by log2(nunits)^2,
>> + we can still retain the necessary penalty for small nunits
>> + meanwhile stabilize the penalty for big nunits. */
>> + int nunits_log2 = exact_log2 (nunits);
>> + gcc_assert (nunits_log2 > 0);
>> + unsigned int nunits_sq = nunits_log2 * nunits_log2;
>> + unsigned int adjusted_cost = stmt_cost / nunits_sq;
>> + gcc_assert (adjusted_cost > 0);
>> + unsigned int extra_cost = nunits * adjusted_cost;
>> data->extra_ctor_cost += extra_cost;
>> }
>> }
>> --
>> 2.25.1
Hi Segher,
Thanks for the review!
on 2021/9/18 上午6:01, Segher Boessenkool wrote:
> Hi!
> On Thu, Sep 16, 2021 at 09:14:15AM +0800, Kewen.Lin wrote:
>> The way with nunits * stmt_cost can get one much exaggerated
>> penalized cost, such as: for V16QI on P8, it's 16 * 20 = 320,
>> that's why we need one bound. To make it scale, this patch
>> doesn't use nunits * stmt_cost any more, but it still keeps
>> nunits since there are actually nunits scalar loads there. So
>> it uses one cost adjusted from stmt_cost, since the current
>> stmt_cost sort of considers nunits, we can stablize the cost
>> for big nunits and retain the cost for small nunits. After
>> some tries, this patch gets the adjusted cost as:
>> stmt_cost / (log2(nunits) * log2(nunits))
> So for V16QI it gives *16/(4*4) so *1
> V8HI it gives *8/(3*3) so *8/9
> V4SI it gives *4/(2*2) so *1
> V2DI it gives *2/(1*1) so *2
> and for V1TI it gives *1/(0*0) which is UB (no, does not crash for us,
> just gives wildly wrong answers; the div returns 0 on recent systems).
I don't expected we will have V1TI for strided/elementwise load,
if it's one unit vector, it's the whole vector itself.
Besides, the below assertion should exclude it already.
>> For V16QI, the adjusted cost would be 1 and total penalized
>> cost is 16, it isn't exaggerated. For V2DI, the adjusted
>> cost would be 2 and total penalized cost is 4, which is the
>> same as before. btw, I tried to use one single log2(nunits),
>> but the penalized cost is still big enough and can't fix the
>> degraded bmk blender_r.
> Does it make sense to treat V2DI (and V2DF) as twice more expensive than
> other vectors, which are all pretty much equal cost (except those that
> end up with cost 0)? If so, there are simpler ways to do that.
Yeah, from the SPEC2017 evaluation, it's good with this. The costing
framework of vectorization doesn't consider the dependent insn chain
and available #unit etc. like local scheduling (it can't either), so
we have to use some heuristics to handle some special cases. For more
units vector construction, the used instructions are more. It has more
chances to schedule them better (even run in parallelly when enough
available units at the time), so we don't need to penalize more for them.
For V2DI, the load result is fed into construction directly, the current
stmt_cost is to consider merging and only 2, penalizing it with one is
not enough from the bwaves experiment.
>> + int nunits_log2 = exact_log2 (nunits);
>> + gcc_assert (nunits_log2 > 0);
>> + unsigned int nunits_sq = nunits_log2 * nunits_log2;
>> = 0
> This of course is assuming nunits will always be a power of 2, but I'm
> sure that we have many other places in the compiler assuming that
> already, so that is fine. And if one day this stops being true we will
> get a nice ICE, pretty much the best we could hope for.
Yeah, exact_log2 returns -1 for non power of 2 input, for example:
input output
0 -> -1
1 -> 0
2 -> 1
3 -> -1
>> + unsigned int adjusted_cost = stmt_cost / nunits_sq;
> But this can divide by 0. Or are we somehow guaranteed that nunits
> will never be 1? Yes the log2 check above, sure, but that ICEs if this
> is violated; is there anything that actually guarantees it is true?
As I mentioned above, I don't expect we can have nunits 1 strided/ew load,
and the ICE should check this and ensure dividing by zero never happens. :)
>> + gcc_assert (adjusted_cost > 0);
> I don't see how you guarantee this, either.
It's mainly to prevent that one day we tweak the cost for construction
in rs6000_builtin_vectorization_cost then make some unexpected values
generated here. But now these expected values are guaranteed as the
current costs and the formula.
> A magic crazy formula like this is no good. If you want to make the
> cost of everything but V2D* be the same, and that of V2D* be twice that,
> that is a weird heuristic, but we can live with that perhaps. But that
> beats completely unexplained (and unexplainable) magic!
> Sorry.
That's all right, thanks for the comments! let's improve it. :)
How about just assigning 2 for V2DI and 1 for the others for the
penalized_cost_per_load with some detailed commentary, it should have
the same effect with this "magic crazy formula", but I guess it can
be more clear.
On Tue, Sep 21, 2021 at 11:24:08AM +0800, Kewen.Lin wrote:
> on 2021/9/18 上午6:01, Segher Boessenkool wrote:
> > On Thu, Sep 16, 2021 at 09:14:15AM +0800, Kewen.Lin wrote:
> >> The way with nunits * stmt_cost can get one much exaggerated
> >> penalized cost, such as: for V16QI on P8, it's 16 * 20 = 320,
> >> that's why we need one bound. To make it scale, this patch
> >> doesn't use nunits * stmt_cost any more, but it still keeps
> >> nunits since there are actually nunits scalar loads there. So
> >> it uses one cost adjusted from stmt_cost, since the current
> >> stmt_cost sort of considers nunits, we can stablize the cost
> >> for big nunits and retain the cost for small nunits. After
> >> some tries, this patch gets the adjusted cost as:
> >>
> >> stmt_cost / (log2(nunits) * log2(nunits))
> >
> > So for V16QI it gives *16/(4*4) so *1
> > V8HI it gives *8/(3*3) so *8/9
> > V4SI it gives *4/(2*2) so *1
> > V2DI it gives *2/(1*1) so *2
> > and for V1TI it gives *1/(0*0) which is UB (no, does not crash for us,
> > just gives wildly wrong answers; the div returns 0 on recent systems).
> I don't expected we will have V1TI for strided/elementwise load,
> if it's one unit vector, it's the whole vector itself.
> Besides, the below assertion should exclude it already.
Yes. But ignoring the UB for unexpectedly large vector components, the
1 / 1.111 / 1 / 2 scoring does not make much sense. The formulas
"look" smooth and even sort of reasonable, but as soon as you look at
what it *means*, and realise the domain if the function is discrete
(only four or five possible inputs), and then see how the function
behaves on that... Hrm :-)
> > This of course is assuming nunits will always be a power of 2, but I'm
> > sure that we have many other places in the compiler assuming that
> > already, so that is fine. And if one day this stops being true we will
> > get a nice ICE, pretty much the best we could hope for.
> Yeah, exact_log2 returns -1 for non power of 2 input, for example:
> >> + unsigned int adjusted_cost = stmt_cost / nunits_sq;
> >
> > But this can divide by 0. Or are we somehow guaranteed that nunits
> > will never be 1? Yes the log2 check above, sure, but that ICEs if this
> > is violated; is there anything that actually guarantees it is true?
> As I mentioned above, I don't expect we can have nunits 1 strided/ew load,
> and the ICE should check this and ensure dividing by zero never happens. :)
Can you assert that *directly* then please?
> > A magic crazy formula like this is no good. If you want to make the
> > cost of everything but V2D* be the same, and that of V2D* be twice that,
> > that is a weird heuristic, but we can live with that perhaps. But that
> > beats completely unexplained (and unexplainable) magic!
> >
> > Sorry.
> That's all right, thanks for the comments! let's improve it. :)
I like that spirit :-)
> How about just assigning 2 for V2DI and 1 for the others for the
> penalized_cost_per_load with some detailed commentary, it should have
> the same effect with this "magic crazy formula", but I guess it can
> be more clear.
That is fine yes! (Well, V2DF the same I guess? Or you'll need very
detailed commentary :-) )
It is fine to say "this is just a heuristic without much supporting
theory" in places. That is what most of our --param= are as well, for
example. If counting two-element vectors as twice as expensive as all
other vectors helps performance, then so be it: if there is no better
way to cost things (or we do not know one), then what else are we to do?
Hi Segher,
on 2021/9/23 上午6:36, Segher Boessenkool wrote:
> Hi!
> On Tue, Sep 21, 2021 at 11:24:08AM +0800, Kewen.Lin wrote:
>> on 2021/9/18 上午6:01, Segher Boessenkool wrote:
>>> On Thu, Sep 16, 2021 at 09:14:15AM +0800, Kewen.Lin wrote:
>>>> The way with nunits * stmt_cost can get one much exaggerated
>>>> penalized cost, such as: for V16QI on P8, it's 16 * 20 = 320,
>>>> that's why we need one bound. To make it scale, this patch
>>>> doesn't use nunits * stmt_cost any more, but it still keeps
>>>> nunits since there are actually nunits scalar loads there. So
>>>> it uses one cost adjusted from stmt_cost, since the current
>>>> stmt_cost sort of considers nunits, we can stablize the cost
>>>> for big nunits and retain the cost for small nunits. After
>>>> some tries, this patch gets the adjusted cost as:
>>>> stmt_cost / (log2(nunits) * log2(nunits))
>>> So for V16QI it gives *16/(4*4) so *1
>>> V8HI it gives *8/(3*3) so *8/9
>>> V4SI it gives *4/(2*2) so *1
>>> V2DI it gives *2/(1*1) so *2
>>> and for V1TI it gives *1/(0*0) which is UB (no, does not crash for us,
>>> just gives wildly wrong answers; the div returns 0 on recent systems).
>> I don't expected we will have V1TI for strided/elementwise load,
>> if it's one unit vector, it's the whole vector itself.
>> Besides, the below assertion should exclude it already.
> Yes. But ignoring the UB for unexpectedly large vector components, the
> 1 / 1.111 / 1 / 2 scoring does not make much sense. The formulas
> "look" smooth and even sort of reasonable, but as soon as you look at
> what it *means*, and realise the domain if the function is discrete
> (only four or five possible inputs), and then see how the function
> behaves on that... Hrm :-)
>>> This of course is assuming nunits will always be a power of 2, but I'm
>>> sure that we have many other places in the compiler assuming that
>>> already, so that is fine. And if one day this stops being true we will
>>> get a nice ICE, pretty much the best we could hope for.
>> Yeah, exact_log2 returns -1 for non power of 2 input, for example:
> Exactly.
>>>> + unsigned int adjusted_cost = stmt_cost / nunits_sq;
>>> But this can divide by 0. Or are we somehow guaranteed that nunits
>>> will never be 1? Yes the log2 check above, sure, but that ICEs if this
>>> is violated; is there anything that actually guarantees it is true?
>> As I mentioned above, I don't expect we can have nunits 1 strided/ew load,
>> and the ICE should check this and ensure dividing by zero never happens. :)
> Can you assert that *directly* then please?
Fix in v2.
>>> A magic crazy formula like this is no good. If you want to make the
>>> cost of everything but V2D* be the same, and that of V2D* be twice that,
>>> that is a weird heuristic, but we can live with that perhaps. But that
>>> beats completely unexplained (and unexplainable) magic!
>>> Sorry.
>> That's all right, thanks for the comments! let's improve it. :)
> I like that spirit :-)
>> How about just assigning 2 for V2DI and 1 for the others for the
>> penalized_cost_per_load with some detailed commentary, it should have
>> the same effect with this "magic crazy formula", but I guess it can
>> be more clear.
> That is fine yes! (Well, V2DF the same I guess? Or you'll need very
> detailed commentary :-) )
> It is fine to say "this is just a heuristic without much supporting
> theory" in places. That is what most of our --param= are as well, for
> example. If counting two-element vectors as twice as expensive as all
> other vectors helps performance, then so be it: if there is no better
> way to cost things (or we do not know one), then what else are we to do?
Thanks a lot for the suggestion, I just posted v2:
diff --git a/gcc/config/rs6000/rs6000.c b/gcc/config/rs6000/rs6000.c
index 4ab23b0ab33..e08b94c0447 100644
--- a/gcc/config/rs6000/rs6000.c
+++ b/gcc/config/rs6000/rs6000.c
@@ -5454,17 +5454,23 @@ rs6000_update_target_cost_per_stmt (rs6000_cost_data *data,
tree vectype = STMT_VINFO_VECTYPE (stmt_info);
unsigned int nunits = vect_nunits_for_cost (vectype);
- unsigned int extra_cost = nunits * stmt_cost;
- /* As function rs6000_builtin_vectorization_cost shows, we have
- priced much on V16QI/V8HI vector construction as their units,
- if we penalize them with nunits * stmt_cost, it can result in
- an unreliable body cost, eg: for V16QI on Power8, stmt_cost
- is 20 and nunits is 16, the extra cost is 320 which looks
- much exaggerated. So let's use one maximum bound for the
- extra penalized cost for vector construction here. */
- const unsigned int MAX_PENALIZED_COST_FOR_CTOR = 12;
- if (extra_cost > MAX_PENALIZED_COST_FOR_CTOR)
- extra_cost = MAX_PENALIZED_COST_FOR_CTOR;
+ /* As function rs6000_builtin_vectorization_cost shows, we
+ have priced much on V16QI/V8HI vector construction by
+ considering their units, if we penalize them with nunits
+ * stmt_cost here, it can result in an unreliable body cost,
+ eg: for V16QI on Power8, stmt_cost is 20 and nunits is 16,
+ the penalty will be 320 which looks much exaggerated. But
+ there are actually nunits scalar loads, so we try to adopt
+ one reasonable penalized cost for each load rather than
+ stmt_cost. Here, with stmt_cost dividing by log2(nunits)^2,
+ we can still retain the necessary penalty for small nunits
+ meanwhile stabilize the penalty for big nunits. */
+ int nunits_log2 = exact_log2 (nunits);
+ gcc_assert (nunits_log2 > 0);
+ unsigned int nunits_sq = nunits_log2 * nunits_log2;
+ unsigned int adjusted_cost = stmt_cost / nunits_sq;
+ gcc_assert (adjusted_cost > 0);
+ unsigned int extra_cost = nunits * adjusted_cost;
data->extra_ctor_cost += extra_cost; | {"url":"https://patchwork.sourceware.org/project/gcc/patch/d26e0a72-a029-c765-75ab-9b31de44f114@linux.ibm.com/#111105","timestamp":"2024-11-10T01:26:08Z","content_type":"application/xhtml+xml","content_length":"60665","record_id":"<urn:uuid:d55fa946-4296-4189-8c68-75ba5f408059>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00020.warc.gz"} |
Computing the Voronoi diagram of a 3-D polyhedron by separate computation of its symbolic and geometric parts
The paper presents an algorithm to construct the Voronoi diagram of a 3-D linear polyhedron. The robustness and simplicity of the algorithm are due to the separation between the computation of the
symbolic and geometric parts of the diagram. The symbolic part of the diagram, the Voronoi graph, is computed by a space subdivision algorithm. The computation of the Voronoi graph utilizes only
relatively simple 2-D geometric computations. Given the Voronoi graph, and a geometric approximation given by the space subdivision, the construction of the geometric part is simple and reliable. An
important advantage of the algorithm is that it enables local and partial computation of the Voronoi diagram. In a previous paper we have given a detailed proof of correctness of the computation of
the Voronoi graph. This paper complements the previous one by detailing the algorithm and its implementation. In addition, this paper describes the computation of the geometric part of the diagram.
Conference Proceedings of the 1999 5th ACM Symposium on Soild Modeling and Applications
City Ann Arbor, MI, USA
Period 9/06/99 → 11/06/99
Dive into the research topics of 'Computing the Voronoi diagram of a 3-D polyhedron by separate computation of its symbolic and geometric parts'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/computing-the-voronoi-diagram-of-a-3-d-polyhedron-by-separate-com","timestamp":"2024-11-14T16:56:01Z","content_type":"text/html","content_length":"48241","record_id":"<urn:uuid:1a9b6bb5-be17-4e4b-8696-df672517aa4a>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00046.warc.gz"} |
Generating the Juggler Sequence in Java - Java Code Geeks
Core Java
Generating the Juggler Sequence in Java
The Juggler sequence is a mathematical sequence defined for a given non-negative integer a[0]. This sequence is generated by repeatedly applying a specific formula until we reach 1. In this article,
we will delve into how we can implement the Juggler sequence generation in Java.
1. Understanding the Juggler Sequence
The Juggler sequence is a mathematical sequence defined by a specific iterative process. It starts with a positive integer a[0] and computes subsequent terms based on whether the current term is
even or odd. The Juggler sequence follows a simple rule:
• If the number is even, divide it by 2 (rounding down for any decimals).
• If the number is odd, multiply it by 3 and add 1, then divide by 2 (again, rounding down).
This process continues until the sequence reaches 1. Here’s an example with a starting number of 9:
9 -> 27/2 (rounded down) -> 13
13 * 3 + 1 = 40
40/2 (rounded down) -> 20
20/2 (rounded down) -> 10
10/2 (rounded down) -> 5
5 * 3 + 1 = 16
16/2 (rounded down) -> 8
8/2 (rounded down) -> 4
4/2 (rounded down) -> 2
2/2 (rounded down) -> 1
This article explores two approaches for generating the Juggler Sequence in Java:
• Iterative Approach (Loop): This method uses a loop to repeatedly calculate the next term based on the current value.
• Recursive Approach: This approach utilizes a recursive function that calls itself with the next term until the sequence reaches 1.
2. Implementing the Juggler Sequence in Java
2.1 Iterative Approach
Here’s the Java code to generate the juggler sequence using the iterative approach:
import java.util.ArrayList;
import java.util.List;
public class JugglerSequence {
// Method to generate the juggler sequence
public static List<Integer> generateSequence(int a0) {
if (a0 <= 0) {
throw new IllegalArgumentException("Input number must be positive");
List<Integer> sequence = new ArrayList<>();
int current = a0;
// Add the initial term to the sequence
// Generate subsequent terms until reaching 1
while (current != 1) {
int next;
if (current % 2 == 0) {
// If current term is even
next = (int) Math.sqrt(current);
} else {
// If current term is odd
next = (int) Math.sqrt(current * current * current);
current = next;
return sequence;
// Method to print the sequence
public static void printSequence(List<Integer> sequence) {
System.out.println("Juggler Sequence:");
for (int num : sequence) {
System.out.print(num + " ");
// Main method to test the sequence generation
public static void main(String[] args) {
int startingNumber = 3; // Change this to your desired starting number
List<Integer> jugglerSequence = generateSequence(startingNumber);
• generateSequence(int a0): This method takes the initial term a[0] as input and computes the entire Juggler sequence based on the described iterative process. It uses a List<Integer> to store the
• printSequence(List<Integer> sequence): This method simply prints the generated sequence to the console.
• main(String[] args): In the main method, we can specify the starting number (startingNumber) and then generate and print the corresponding Juggler sequence using the generateSequence and
printSequence methods.
Example Output
If we run the above program with a0 = 3, the output will be:
2.2 Recursive Approach
This approach uses recursion to define the logic for generating the next term:
public class RecursiveJugglerSequence {
// Method to generate the juggler sequence recursively
public static List<Integer> generateSequence(int a0) {
List<Integer> sequence = new ArrayList<>();
generateSequenceRecursive(a0, sequence);
return sequence;
// Helper method for recursive sequence generation
private static void generateSequenceRecursive(int current, List<Integer> sequence) {
sequence.add(current); // Add current term to the sequence
if (current == 1) {
return; // Base case: Sequence terminates when current term is 1
int next;
if (current % 2 == 0) {
next = (int) Math.sqrt(current); // Next term for even current
} else {
next = (int) Math.sqrt(current * current * current); // Next term for odd current
generateSequenceRecursive(next, sequence); // Recursive call with the next term
// Method to print the sequence
public static void printSequence(List<Integer> sequence) {
System.out.println("Juggler Sequence:");
for (int num : sequence) {
System.out.print(num + " ");
// Main method to test the recursive sequence generation
public static void main(String[] args) {
int startingNumber = 9; // Change this to your desired starting number
List<Integer> jugglerSequence = generateSequence(startingNumber);
In this example:
generateSequenceRecursive(int current, List<Integer> sequence) is the recursive helper method responsible for generating the sequence. It takes the current term (current) and the sequence list as
parameters. It adds the current term to the sequence and checks if the sequence should terminate (when current becomes 1). If not, it calculates the next term (next) based on whether the current term
is even or odd and recursively calls itself with next to continue the sequence generation.
Example Output
For example, if we run the above program with the starting number as 9, the output will be:
Juggler Sequence:
3. Conclusion
In this article, we explored how to generate the Juggler sequence using Java programming, focusing on both iterative and recursive implementations. The program demonstrates the use of loops,
conditional statements, method recursion and mathematical computations to generate and the sequence efficiently. You can modify the main method to try different initial values and observe the
resulting Juggler sequences.
4. Download the Source Code
This was an article on how to generate a juggler sequence using Java. | {"url":"https://www.javacodegeeks.com/generating-the-juggler-sequence-in-java.html","timestamp":"2024-11-04T18:26:55Z","content_type":"text/html","content_length":"241314","record_id":"<urn:uuid:884ac325-4cdf-435e-9080-b7a8369fdfb1>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00848.warc.gz"} |
Losses in DC Machine- Types of Losses
Electric rotating machines are used for the conversion of energy. The motor converts electrical energy into mechanical energy, and the generator converts mechanical energy into electrical energy.
During energy conversion, the input energy in one form can not be fully converted into output energy in another form. The difference in the output energy and input energy is called the losses.
Pragmatically, no machine is 100 % efficient, and some losses always take place during the energy conversion process. The losses raise the temperature of the machine, and the efficiency of the
machine gets lowered. In a DC machine, the energy loss takes place in the form of heat energy. The losses occur in the armature and field of the DC machine. There are five types of losses: copper
loss, brush loss, iron loss, stray loss, and mechanical loss, which take place in a DC machine.
Losses in DC Machines
Types of Losses in a DC Machine
We shall discuss the types of losses in a DC machine for better understanding.
We shall discuss the types of losses in a DC machine for better understanding.
1. Copper Loss in DC Machine winding
The copper loss is caused by the ohmic resistance offered by the winding of the DC machine. When the current flows through the winding, heat loss takes place. The heat loss is proportional to the
square of the current and the resistance of the winding. The copper loss in the winding is I^2R. Where I is the current flowing through the winding, and R is the resistance of the winding. Copper
loss is also known as variable loss because it depends on the percentage loading of the machine. The loss increases with the increase of loading on the machine.
The DC machine has two types of winding- field and armature winding- and losses occur in both windings. The supply is fed to the armature through the carbon brushes, and losses also occur due to
ohmic voltage drops across the carbon brush.
1 a). Copper Loss in Armature Winding
The armature of the DC machine has very low resistance. The resistance of the armature is denoted by Ra.
Armature copper loss = I[a]^2Ra
I[a] is the armature current
R[a] is the armature winding resistance.
The maximum copper loss occurs in the armature winding because the load current flows through it. The copper loss in the armature is about 25 to 30 % of the full load loss.
1 b). Copper Loss in the Field Winding
DC supply is fed to the field winding for the production of the flux in the DC machine. The resistance of the field winding is much more than the resistance of the armature winding. That is why the
substantial copper loss takes place in the field winding, even at the low field current. The copper loss in the field winding is expressed as;
Field winding copper loss = I[f]^2R[f]
Where, I[f] is the field current and Rf is the field winding resistance.
The field winding copper loss is about 20-25 % of the full load loss of the DC machine. The copper loss in the field winding is practically constant because the field current and the field resistance
remain almost constant in the DC machine.
2. Brush Contact Resistance Loss
The armature is a rotating part of the DC machine, and brushes are used to provide a DC supply to the rotating part of the DC machine. Ideally, the contact resistance between the brush contacting
area and the commutator surface must be zero. However, in reality, it is impossible to have zero contact resistance.
The voltage drop takes place across the carbon brushes. The brush power drop depends upon the voltage drop across the brush and armature current.
Power Drop in Brush = PBD = V[BD] I[a]
P[BD] = Power drop in Brush
V[BD] = Voltage Drop in Brush
I[a ]= Armature Current
If the brush voltage drop is not given, it is generally assumed that 2-volt drops across the carbon brush, and the power drop in the brush is 2I[a]. The brush power loss can not be calculated
separately, but it is included in the armature copper loss of the DC machine.
3. Core Losses or Iron Losses in DC Machine
The armature winding of the DC machine is wound around the magnetic core. The flux generated by the field coil gets linked to the armature conductors through a magnetic core. Two types of losses,
namely hysteresis and eddy current loss, occur in the magnetic core. The iron loss is almost constant; therefore, the iron loss or core loss is also called constant loss. The total core loss is about
20-25 % of the full load losses.
3 a). Hysteresis Loss in DC Machine
The armature of the DC machine rotates in a magnetic field, and in one complete rotation, the magnetic field reversal happens. The part of the armature remains under the S-pole for half a revolution,
and after completing half the revolution under the S-pole, the part of the armature goes under the P-pole for the remaining half cycle. Thus, in one complete cycle, the magnetic field reversal
happens in the armature core. The frequency of the magnetic reversal can be found by the following mathematical expression.
Due to constant magnetic reversal in the armature, some energy consumed during magnetic reversal is called hysteresis loss. The hysteresis loss depends on the quality and volume of the core material.
The hysteresis loss in the DC machine can be calculated using the Steinmetz formula.
Steinmetz formula of Hysteresis Loss in a Dc Machine
Steinmetz formula
P[h = η ][Bm]n[ f V
]P[h = η ][Bm]1.6[ f V]Where,
P[h] = hysteresis loss (Watt)
η = Steinmetz hysteresis coefficient, depending on the material (J/m^3)
B[m] = Maximum flux density (W[b]/m^2)
n =Steinmetz exponent, ranges from 1.5 to 2.5, depending on the material
f = frequency of magnetic reversals per second (Hz)
V = volume of magnetic material (m^3)
3 b). Eddy Current Loss in DC Machine
The armature of the DC machine is wound on the magnetic core, and the magnetic core rotates in the magnetic field. According to Faraday’s law of electromagnetic induction, an EMF is induced in the
core. The magnetic core has a certain resistance, and the induced EMF causes current to circulate within the piece of the magnetic core. The circulation current, called eddy current, causes the
wastage of electrical energy. The loss caused by the eddy current is called the eddy current loss in a DC machine. The eddy current loss can be minimized by the use of a laminated core. The eddy
current loss can be calculated by following mathematical expressions.
Eddy Current Loss Formula
Pe= Eddy current loss(watt)
B= Maximum flux density W[b]/m^2
f= frequency in Hz
t= thickness of lamination (m)
V= Volume of the material (m^3)
K= Eddy Current Constant
4. Mechanical Loss in DC Machine
In a DC machine, the field is a stationary part, and the armature is a rotating part. The armature rotates on the bearings. The energy loss in the form of heat occurs due to friction between the
inner cage and outer cage of the bearing.
The other mechanical loss is the windage loss. The air surrounding the shaft offers resistance, and when the DC machine rotates, the loss caused by air resistance is called windage loss. The DC
machine draws extra power from the source to overcome the air resistance, and the extra energy is equal to the windage loss of the DC machine. The windage loss increases with an increase in the speed
of the rotating machine.
5. Stray Losses in DC Machine
Stray losses are miscellaneous losses that are difficult to determine. The various reasons for the stray losses in DC machines are short circuit current undergoing commutation, distortion of flux,
etc. The stray losses in the DC machine are about 1 % of the total losses.
Leave a Comment | {"url":"https://www.electricalvolt.com/losses-in-dc-machine/","timestamp":"2024-11-04T04:10:48Z","content_type":"text/html","content_length":"101587","record_id":"<urn:uuid:6d2bea2a-764b-488a-aa7f-a91dae8c4e7a>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00762.warc.gz"} |
Useful links related to record computations: the Discrete logarithm records Wikipedia page; the DLDB database of discrete logarithm records; the RSA numbers Wikipedia page.
Here is a list of some large-scale computations I have done in the last decade. Many of them have been done with the CADO-NFS software. | {"url":"https://members.loria.fr/PGaudry/computations/","timestamp":"2024-11-07T19:35:57Z","content_type":"text/html","content_length":"21106","record_id":"<urn:uuid:4623ffc9-a2af-49a9-ad85-254e19b48966>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00501.warc.gz"} |
Add a Parallel
With this mode you can construct a line through a point parallel to another line with a press–drag–release sequence. The point through which the parallel should pass can also be generated in this
mode. Constructing the parallel is a three-step procedure.
• Move the mouse over the line for which you want a parallel. Press the left mouse button. This creates the parallel and the point through which it should pass.
• Hold the left button and drag the mouse. This moves the parallel and the new point to the desired position.
• Release the mouse. Now the construction is frozen. Depending on the position at which you release the mouse, the definition of the new point is adapted:
□ If the mouse pointer is over an existing point, then this point is taken.
□ If the mouse pointer is over the intersection of two elements (line, circle, or conic), then this intersection is automatically constructed and taken as the new point.
□ If the mouse pointer is over just one element (line or circle), then a point is constructed that is constrained to this element. This point is taken as the new point.
□ Otherwise, a free point is added.
The figures below show the three stages during the construction of a parallel. Here the new point will be bound to the existing point
Press the mouse … … drag it … … and release.
Add a parallel mode creates a parallel with a press–drag–release sequence.
The behavior of this mode is dependent on the geometry that is chosen. While in Euclidean geometry there is always exactly one parallel, in non-Euclidean geometries the number of parallels is subject
to the definition of parallelity. Depending on the underlying "philosophy," in hyperbolic geometry there can be infinitely many parallels, from an incidence-geometric viewpoint, or there can be
exactly two parallels, from an algebraic or measurement-based point of view.
takes the algebraic point of view:
A parallel to a line L is a line that creates a zero angle with L
. So in hyperbolic geometry the mode produces exactly two parallels.
In elliptic geometry the usual viewpoint is that there are no parallels at all. However, from an algebraic standpoint there exist such parallels. It is just that they have complex coordinates. In
other words, they will never be visible.
constructs these parallels, and they are invisible but are nevertheless present in the
Construction Text
view. From there they can be accessed for further constructions. | {"url":"https://doc.cinderella.de/tiki-index.php?page=Add+a+Parallel","timestamp":"2024-11-12T05:11:05Z","content_type":"text/html","content_length":"26291","record_id":"<urn:uuid:a7471812-c218-4c3a-ad00-e39064831b99>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00475.warc.gz"} |
Sign changes of the error term in the Piltz divisor problem
Speaker: Cruz Castillo Date: Mon, Sep 25, 2023 Location: PIMS, University of Lethbridge, Online Conference: Analytic Aspects of L-functions and Applications to Number Theory Subject: Mathematics,
Number Theory Class: Scientific CRG: L-Functions in Analytic Number Theory
For an integer k≥3; Δk (x) :=∑n≤xdk(n)-Ress=1 (ζk(s)xs/s), where dk(n) is the k-fold divisor function, and ζ(s) is the Riemann zeta-function. In the 1950's, Tong showed for all large enough X; Δk(x)
changes sign at least once in the interval [X, X + CkX1-1/k] for some positive constant Ck. For a large parameter X, we show that if the Lindelöf hypothesis is true, then there exist many disjoint
subintervals of [X, 2X], each of length X1-1/k-ε such that Δk (x) does not change sign in any of these subintervals. If the Riemann hypothesis is true, then we can improve the length of the
subintervals to << X1-1/k (logX)-k^2-2. These results may be viewed as higher-degree analogues of a theorem of Heath-Brown and Tsang, who studied the case k = 2. This is joint work with Siegfred | {"url":"https://mathtube.org/lecture/video/sign-changes-error-term-piltz-divisor-problem","timestamp":"2024-11-08T20:44:02Z","content_type":"application/xhtml+xml","content_length":"26916","record_id":"<urn:uuid:bd86ad71-de80-4835-8973-23d44bf98697>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00644.warc.gz"} |
Greedy algorithm--the number of series is very poor
A Free Trial That Lets You Build Big!
Start building with 50+ products and up to 12 months usage for Elastic Compute Service
• Sales Support
1 on 1 presale consultation
• After-Sales Support
24/7 Technical Support 6 Free Tickets per Quarter Faster Response
• Alibaba Cloud offers highly flexible support services tailored to meet your exact needs. | {"url":"https://topic.alibabacloud.com/a/greedy-algorithm-the-number-of-series-is-very-poor_8_8_30451728.html","timestamp":"2024-11-13T12:22:05Z","content_type":"text/html","content_length":"79553","record_id":"<urn:uuid:5412ab65-d73c-4fa8-8779-4899f42a333f>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00095.warc.gz"} |
Equating a Pass-Fail Score
Timo Bechger, Jesse Koops and Ivailo Partchev
Educational tests are often equipped with a threshold to turn the test score into a pass-fail decision. When a new test of the same kind is developed, we need a threshold for it that will be, in some
sense, equivalent to the threshold for the old — let us call it reference — test. This is a special case of test equating, and it is similar to some well-studied problems in epidemiology; for the
comfort of our predominantly psychometric audience, we start with an outline of those.
Consider a test for pregnancy. The state of nature is a binary variable (positive / negative). So is the outcome of the test, although the decision is possibly produced by dichotomizing the
quantitative measurement of some hormone or other substance, just like we want to do with our educational test scores. There are four possible outcomes in all, which can be represented in a
two-by-two table
State of nature
Positive Negative
Prediction Positive True positive (TP) False positive (FP)
Negative False negative (FN) True negative (TN)
where TP, FP, FN, and TN are all counts. There is one ideal situation in which FP=FN=0 and all predictions are correct, and very many situations where this is not the case. Many statistics have been
developed to measure the various ways in which actual data can depart from the ideal situation — this Wikipedia page lists at least 13 of them. We will mention only three:
• True positive rate (TPR, sensitivity): The proportion of positives that are correctly identified as such, or TP/(TP+FN)
• True negative rate (TNR, specificity): The proportion of negatives that are correctly identified as such, or TN/(TN+FP)
• False positive rate (FNR) = 1 - specificity = FP/(TN+FP)
What we can try to do is select the threshold for the test in such a way that the resulting table gets as close to the optimum as possible (not forgetting that the subjective cost of the two kinds of
erroneous predictions may differ). The receiver operating characteristic curve (ROC curve) (Swets (1996)) is a graphical representations of the trade-offs between the TPR (one possible measure of
benefit) and FPR (one possible measure of costs) resulting from all possible choices of such a threshold. The optimal choice would then be the one corresponding to the point on the curve that comes
closest to the upper left corner (perfect prediction). An example ROC curve is shown on the left hand plot below.
Before we explain in more detail, let us go back to our educational example. Let us assume for a moment that all students have taken both the old (reference) test and the new (target) test, and that
the decision on the reference test represents the ‘state of nature’. The task is then to select a threshold on the target test such that the decision on the reference is reproduced as closely as
possible by the decision on the new test. On the right hand plot, the (fictional) distribution function of the scores on the target test is represented in blue for the students who passed the
reference test, and in red for those who failed. We show two of the infinitely many thresholds that can be chosen, and it is clear that the one shown in black is much preferable to the one shown in
gray (compare with the two dots on the ROC curve).
An illustration with complete psychometric data
It is time to try out these ideas with (simulated) psychometric data. First, we examine the situation where both tests, the reference test and the new one, have been administered to the same
respondents. This is not a typical situation because, in practice, we usually deal with data collected in an incomplete design where no candidates take both tests. In this situation, if we can
estimate the parameters of the items, we can use the IRT model to fill in the missing data. Obviously, without observations we no longer know whether an individual has actually passed the reference
test or not (see Bolsinova and Maris (2016)). We can, however, calculate sensitivity and specificity even in that situation.
We use the verbal aggression data with the first 14 items assumed to be the reference test, and the remaining 10 items to be the target test. The pass-fail score on the reference test is set to 10.
All this is quite arbitrary, of course – the test is not intended to measure educational achievement, and we are certainly not handing out diplomas for verbal aggressiveness.
As a start, we produce a scatter plot of the scores:
The proportion of respondents who pass the reference test gradually increases with the score on the new test; below, it is plotted in red:
prob_pass = tp = rep(0,29)
for (i in seq_along(0:28)){
prob_pass[i] = sum(ts$ref_test[ts$new_test==i]>=10) / sum(ts$new_test==i)
tp[i] = sum(ts$ref_test[ts$new_test>=i]>=10) / sum(ts$new_test>=i)
plot(0:28, prob_pass, ylab="Proportion passing the reference test", xlab="New test score", ylim=c(0,1), type = "o", col="red",bty='l')
lines(0:28, tp, type="o", lty=2, col="blue")
The blue dashed line is the percentage of true positives. Among other things, it shows that if the new pass-fail score is zero and everybody passes, about 15 percent will be qualified. If this is
high enough we can save ourselves the trouble of administering a new test.
Since the same respondents have taken both tests, it is straightforward to determine the sensitivity and specificity of the new test for each pass-fail score:
specificity = sensitivity = rep(0, 29)
for (i in seq_along(0:28)){
sensitivity[i] = sum(ts$ref_test[ts$new_test>=i]>=10)/sum(ts$ref_test>=10)
specificity[i] = sum(ts$ref_test[ts$new_test<i]<10)/(sum(ts$ref_test[ts$new_test<i]<10)+sum(ts$ref_test[ts$new_test>=i]<10))
plot(0:28, sensitivity, ylab="sensitivity/specificity", xlab="new test score", ylim=c(0,1), type = "o", col="red",bty='l')
lines(0:28, specificity, col="green", type="o")
If sensitivity and specificity are deemed equally important, the plot suggests 14 or 15 as appropriate pass-fail scores. Higher values would make the specificity go up (i.e., less false positives) at
the expense of sensitivity; lower values would do the reverse. From the plots made earlier, we see that about half of the persons with a score equal to the passing score would actually pass the
reference test.
As candidates are classified into two classes, we can readily show the trade-off between sensitivity and specificity with the ROC curve:
plot(1-specificity, sensitivity, col="green", xlim=c(0,1), ylim=c(0,1), type="l",bty='l')
text(1-specificity, sensitivity, as.character(0:28), cex=0.7, offset = 0)
abline(0,1,lty=2, col="grey")
Other packages and functions may produce prettier plots — but can they do it without complete data?
Incomplete Data
It is more involved, but practically more useful, to calculate sensitivity and specificity even though the two tests have not been both administered to the same persons. We first simulate a complete
data set with responses to 60 items. Then, we select the first 500 persons and 40 items to represent the reference test, and the remaining 200 persons and items 21 through 60 to represent the target
test. Note that the two respondent groups are sampled from two different ability populations and cannot be considered as equivalent.
We create a new dexter project, and we add the two booklets. We assume that the pass-fail limit for the reference test is 23 points.
The probability to pass
As a start, we use the function probability_to_pass to estimate, for each score on the target test, the proportion of persons who pass the reference test. We show this as a plot.
p_sm = fit_enorm(db_sm, method='Bayes', nDraws = 5000)
ou_e = probability_to_pass(db_sm, p_sm,
ref_items = ref_items,
target_booklets = tibble(booklet_id="bk2", item_id=target_items),
pass_fail = pass_fail)
plot(ou_e, what="equating")
Let the scores on the reference and new test be written, respectively, as \(R_+\) and \(N_+\), and let \(\mathbf{x}\) denote the available data. We calculate \(P(R_+ \geq c|N_+ = s, \mathbf{x})\),
the probability that a person with a score of \(s\) on the target test will pass. Simply stated, we calculate this probability by averaging over what could have happened on the basis of all available
data. A technical appendix explains a bit more.
A little extra in the plot are the bars. These refer to the traditional ability-based pass-fail score obtained as follows: First, estimate an ability for each score on the reference and the new test.
Then, determine the score on the new test corresponding to the smallest ability equal to or larger than the ability corresponding to the pass-fail score on the reference test. This will then be the
pass-fail score in the new test. The bars actually denote the probabilities of pass-fail scores thus obtained. By construction, such pass-fail scores correspond to the lowest scores giving a
probability to pass equal or higher than 0.5. While this is the usual procedure to equate a pass-fail score, we believe the procedure that we propose here is more informative.
Sensitivity and specificity
Having the probability to pass and the proportion of respondents, \(P(N_+ = s)\), for all possible scores \(s\), we can calculate sensitivity and specificity and plot them for each possible pass-fail
score on the target test:
If sensitivity and specificity are equally important, the plot suggests 23 as a pass-fail score on the new test.
An educational test equipped with a single pass-fail score is a binary test aimed at classifying candidates into those who pass, and those who fail. In fields like medical testing or machine
learning, sensitivity and specificity are typically considered to evaluate the performance of such a test. Essentially, what we have tried to do is apply this logic to educational tests. Item
response theory was used to deal with incomplete data as we typically see in educational measurement. Presumably, medical testing faces similar complications, but this is beyond our immediate scope.
In closing, we should mention that this is a first attempt to equate tests in a way that does not involve unobservable quantities; it uses item response theory to fill in missing data, no more. We
would like to test the functions more thoroughly before attempting an extension to classification into more than two classes. Feedback from users is very welcome.
Technical Appendix
Let us consider how we determine the probability to pass the reference test given a score \(N_+=s\) on the target test. This probability can be defined as:
\[ P(R_+ \geq c|N_+ = s, \mathbf{x}) = \int_{\mathbf{b},\theta} P(R_+ \geq c|\theta,\mathbf{b}) f(\theta|N_+ = s, \mathbf{b}) f(\mathbf{b}|\mathbf{x}) d\mathbf{b} d\theta, \]
where \(Y_+\) denotes the score on the reference test, with pass-fail score \(c\), \(\theta\) denotes ability and \(\mathbf{b}\) is the vector of item parameters. The integrand has three parts:
• \(f(\mathbf{b}|\mathbf{x})\): The posterior of the item parameters, given all data \(\mathbf{x}\).
• \(f(\theta|N_+ = s, \mathbf{b})\): The posterior of ability given the score on the new test.
• \(P(R_+ \geq c|\theta,\mathbf{b})\): The probability to pass given ability equal to \[ P(R_+ \geq c|\theta,\mathbf{b}) = \sum_{s \geq c} P(R_+ = s |\theta, \mathbf{b}) = \sum_{s \geq c} \frac{\
gamma_s(\mathbf{b}) e^{s \theta}} {\sum_h \gamma_h(\mathbf{b}) e^{h \theta}}, \] with \(\gamma_s(\mathbf{b})\) the elementary symmetric function of order s.
To calculate a Monte-Carlo estimate, repeat:
• sample item parameters \(\mathbf{b}^* \sim f(\mathbf{b}|\mathbf{x})\)
• sample a plausible value \(\theta^* \sim f(\theta|N_+ = s, \mathbf{b}^*)\)
• calculate \(P(R_+ \geq c|\theta^*,\mathbf{b}^*)\)
We do this as often as we can afford (currently, a thousand times), and we take the average as an estimate of \(P(R_+ \geq c|\theta^*,\mathbf{b}^*)\).
Sensitivity and specificity are calculated from the score distribution on the new-test and the probability to pass. The following code shows how:
p_pass_given_s = out$probability_to_pass
ps = ou_e$pnew
tp = rev(cumsum(rev(p_pass_given_s*ps))) / rev(cumsum(rev(ps)))
sensitivity = rev(cumsum(rev(p_pass_given_s*ps))) / tp[1]
tn = rev(cumsum(rev((1-p_pass_given_s)*ps))) / rev(cumsum(rev(ps)))
specificity = 1 - rev(cumsum(rev((1-p_pass_given_s)*ps))) / tn[1]
Bolsinova, Maria, and Gunter Maris. 2016.
“Can IRT Solve the Missing Data Problem in Test Equating?” Frontiers in Psychology
6: 1956.
Swets, John A. 1996. Signal Detection Theory and ROC Analysis in Psychology and Diagnostics: Collected Papers. Hillsdale, NJ: Lawrence Erlbaum Associates, Inc. | {"url":"http://cran.stat.auckland.ac.nz/web/packages/dexter/vignettes/Equating.html","timestamp":"2024-11-12T09:13:03Z","content_type":"text/html","content_length":"249005","record_id":"<urn:uuid:8344844e-8739-498f-8419-6c47d0bd51a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00030.warc.gz"} |
Help and Guidance - Families Learning Together
What is in each year group of this free maths resource?
See what is covered in every unit in each age group.
Free Maths Worksheets: Maths skills across the different year groups:
When you are working on a particular maths skill you might want to know where to find either the earlier units, to help with any gaps in understanding, or later units to extend your children’s
Addition and subtraction (PDF)
Multiplication and division (PDF)
Keeping track of progress
You may want to track which units you have covered, and what has been learnt. The following documents will help you to do this.
Year 1 Progress Tracking: PDF version l Word version
Year 2 Progress Tracking: PDF version l Word version
Year 3 Progress Tracking: PDF version l Word version
Year 4 Progress Tracking: PDF version l Word version
Year 5 Progress Tracking: PDF version l Word version
Year 6 Progress Tracking: PDF version l Word version | {"url":"https://famlearn.co.uk/help-and-guidance/","timestamp":"2024-11-12T10:13:44Z","content_type":"text/html","content_length":"197172","record_id":"<urn:uuid:cdb586fb-e7dd-4e58-9c90-797e34bc3e75>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00232.warc.gz"} |
t-processes, t-distributions – The Dan MacKinlay stable of variably-well-consider’d enterprises
t-processes, t-distributions
November 10, 2021 — November 24, 2021
Hilbert space
kernel tricks
Lévy processes
stochastic processes
time series
Stochastic processes with Student-t marginals. Much as Student-\(t\) distributions generalize Gaussian distributions, \(t\)-processes generalize Gaussian processes. Another useful member of the
family of elliptically contoured distributions.
1 Multivariate Student-\(t\)
The multivariate \(t\) (MVT) distribution \(\boldsymbol{X} \sim \boldsymbol{t}_p(\boldsymbol{\mu}, \boldsymbol{\Sigma}, \boldsymbol{v})\), with location \(\boldsymbol{\mu}\), scale matrix \(\
boldsymbol{\Sigma}\), and degrees of freedom \(v\), has the probability density function \[ f(\boldsymbol{x})=\frac{\Gamma\{(\boldsymbol{v}+p) / 2\}}{\Gamma(\boldsymbol{v} / 2)(\boldsymbol{v} \pi)^{p
/ 2}|\boldsymbol{\Sigma}|^{1 / 2}}\left\{1+\boldsymbol{v}^{-1}(\boldsymbol{x}-\boldsymbol{\mu})^{\top} \boldsymbol{\Sigma}^{-1}(\boldsymbol{x}-\boldsymbol{\mu})\right\}^{-(v+p) / 2} . \] There is a
cool relationship to the multivariate normal: \[ \boldsymbol{X}=\boldsymbol{\mu}+\boldsymbol{\Sigma}^{1 / 2} \boldsymbol{Z} / \sqrt{q}, \] where \(Z\) follows a \(p\) dimensional standard normal
distribution, \(q \sim \chi_v^2 / v\), and \(Z\) is independent of \(q\). (\(W \sim \chi_b^2 / c\) denotes the scaled \(\chi^2\) distribution, with density proportional to \(w^{b / 2-1} e^{-c w /
2}.)\) It differs from the multivariate normal distribution \(\mathscr{N}_p(\boldsymbol{\mu}, \boldsymbol{\Sigma})\) only by the random scaling factor \(\sqrt{q}\).
Ding (2016) uses this latter property to show that the conditional distribution of \(\boldsymbol{X}_2\) given \(\boldsymbol{X}_1\) is \[ \boldsymbol{X}_2 \mid \boldsymbol{X}_1 \sim \boldsymbol{t}_
{p_2}\left(\boldsymbol{\mu}_{2 \mid 1}, \frac{v+d_1}{v+p_1} \boldsymbol{\Sigma}_{22 \mid 1}, \boldsymbol{v}+p_1\right). \]
2 t-process regression
There are a couple of classic cases in ML where \(t\)-processes arise, e.g. in Bayes NNs (Neal 1996) or GP literature (9.9 Rasmussen and Williams 2006). Recently there has been an uptick in actual
applications of these processes in regression (Chen, Wang, and Gorban 2020; Shah, Wilson, and Ghahramani 2014; Tang et al. 2017; Tracey and Wolpert 2018). See Wilson and Ghahramani (2011) for a
Generalized Wishart Process construction that may be helpful? This prior is available in GPyTorch. Recent papers (Shah, Wilson, and Ghahramani 2014; Tracey and Wolpert 2018) make it seem fairly
At first blush it looks like it might be a more robust regression model than Gaussian process regression. However, I am not so sure. As Ding (2016) points out, the conditional distribution of \(\
boldsymbol{X}_2\) given \(\boldsymbol{X}_1\) jointly \(t4\)-distributed grows eventually linearly in the number of observation sites, which means that it is essentially just Gaussian for even small
Some papers discuss \(t\)-process regression in terms of inference using Inverse Wishart distributions.
3 Markov t-process
Process with \(t\)-distributed increments is in fact a Lévy process, which follows from the fact that the Student-\(t\) distribution is divisible. As far as I can see here Grigelionis (2013) is the
definitive collation of results on that observation. | {"url":"https://danmackinlay.name/notebook/t_process.html","timestamp":"2024-11-05T00:38:24Z","content_type":"application/xhtml+xml","content_length":"41036","record_id":"<urn:uuid:ac657d39-4ff2-457a-b194-d228a2bf746e>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00441.warc.gz"} |
seminars - Introduction to spectral network
※ 일시/장소
- 6월 3일(월), 17:00~18:15, 129동 301호
- 6월 5일(수), 17:00~18:15, 129동 309호
- 6월 11일(화), 17:00~18:15, 129동 301호
- 6월 12일(수), 17:00~18:15, 129동 301호
Abstract: Spectral networks were introduced in a seminal article by Davide Gaiotto, Gregory W. Moore and Andrew Neitzke published in 2013. These are networks of trajectories on surfaces that
naturally arise in the study of various four-dimensional quantum field theories. From a purely geometric point of view, they yield a new map between flat connections over a Riemann surface and flat
abelian connections on a spectral covering of the surface. At the same time, these networks of trajectories provide local coordinate systems on the moduli space of flat connections that are valuable
in the study of higher Teichmüller spaces.
In the first part of this mini-course, I will review key concepts from geometric group theory, including hyperbolic groups and boundaries at infinity of hyperbolic groups and spaces. Following this,
I will discuss the theory of vector bundles and the Riemann-Hilbert correspondence. In the second part, I will define spectral networks explicitly for surfaces with punctures. I will also present and
discuss their most prominent applications in geometry: non-abelianization and abelianization, which connect higher Teichmüller spaces of a base surface to abelian character varieties of its ramified
cover, the spectral curve. | {"url":"http://www.math.snu.ac.kr/board/index.php?mid=seminars&page=80&sort_index=room&order_type=desc&document_srl=1237018&l=en","timestamp":"2024-11-14T03:30:42Z","content_type":"text/html","content_length":"46517","record_id":"<urn:uuid:97b7e8ac-1ef1-4a75-9b12-0faf316e403c>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00127.warc.gz"} |
What is Development Length for Reinforced Concrete Footings as per IS456
To understand development length, you must visit the article “What is development length.” If you have already gone through the article, you must know that development length is the grip available to
hold the rod in position. In this article, we will see the application of development length while designing the footing.
The formula for development length as per IS-456:2000
Now you can feel the requirement of the development length. After this, we can discuss the codal requirement for development length. IS-456:2000, in clause 26.2.1, says that the bar must develop the
required tension or compression, which we can assure by development length check. You can calculate it by:
$$ L_{\mathrm{d}}=\frac{d_b f_{\mathrm{s}}}{4 \tau_{\mathrm{bd}}} $$
• $d_b$ is the nominal diameter of the bar,
• $f_s$ is the stress in the bar at the section considered at design load (for fully stressed bar, $f_s = 0.87 f_y$)
• $\tau_{bd}$ is the design bond stress as per the table below.
\begin{array}{|l|l|l|l|l|l|} \hline {\text { Grade of concrete }} & \text { M20 } & \text { M25 } & \text { M30 } & \text { M35 } & \text { M40 and above } \\ \hline \text { Design bond , MPa } & 1.2
& 1.4 & 1.5 & 1.6 & 1.9 \\ \hline \text { For fully deformed bars } & 1.18 & 1.37 & 1.54 & 1.71 & 1.87 \\ \hline \end{array}
• For a deformed bar in tension, τbd values can be increased by $60\%$
• For bars in compression, the value of the bond in tension can be increased by $25\%$.
• In case of nominal reinforcement is provided, $\tau_{bd}$ is taken as 1.0 MPa.
$$ L_{\mathrm{d}}=\frac{0.136d_b f_{\mathrm{s}}}{ \tau_{\mathrm{bd}}} $$
Development length checks in footing.
In this post, we are discussing isolated footings. Generally, we design footing as a wide beam. And we divide footing behavior as one-way shear and two-way shear. Please see the post on the design of
the footing for more detail. However, for completeness, we are again producing an image here. So that users can understand the critical points.
Figure- Development length in footing. (a) Isolated footing which is carrying the column in 3D, (b) Plan of the footing showing cantilever projection in $x$ and $y$ direction, (c) figure showing the
cantilever action in the footing and (d) showing the pull and the grip action in the bar of footing.
Now, if you want to check the for bending about $x$-axis as you can refer in the part (d) and (b) of the figure.
$$L_{d,\text{reqd}} = L_{dx} = \frac{0.87 f_y}{4\tau_{bd}} \times \phi_x, L_{d,\text{available}} = C_x$$
For bending about $y$-axis:
$$L_{d,\text{reqd}} = L_{dx} = \frac{0.87 f_y}{4\tau_{bd}} \times \phi_x, L_{d,\text{available}} = C_y$$
Where $\phi_x$ and $\phi_y$ are the diameters of the bar for bending about $x$- axis and $y$ – axis, respectively.
💡 If available development length is more than the required development length the check for development length is correct.
The critical case
If the check for development length is already discussed, what do we have to check for? Generally, this check only works when the soil has a large bearing capacity. In that case, the size of the
footing required becomes very small, which leads to lesser cantilever projections. You can take the following remedies in this case:
1. Reduce the area: You can reduce the area of the bar. By doing this, we can reduce the required development length.
2. Increase the footing: You can increase the size of the footing such that the projection $C_x, C_y$ is not less than the required development length.
3. Provide bend: Bends increase the available development length. This method also can help you in a critical situation.
4. Increase steel area: Increasing the area of steel will also modify the development length requirement. You can calculate it as:
$$ L_{dm}=\frac{A_{s,\text{reqd}}}{A_{s,\text{provided}}}\times L_d $$
You can check the required development length in footing by simply using the formula. The only thing you have to keep in mind is that you have to check for bending in $x$ and bending in $y$. The
special case arises when we build the foundation in the soil with large bearing pressure. As the size of the footing reduces.
In this post, you have learned the following key points:
• Development length: You have to check the footing for both directions.
• Critical case: When the footing is placed in soil having a large bearing capacity, the footing size decreases. This reduction in the size of the footing creates a problem in providing the space
for the required length.
• Remedies: You can increase the area of steel, reduce the diameter of the bar, increase the footing, or you may provide the $90^\circ$ bend to make it safe for development length.
Leave a Comment | {"url":"https://www.eigenplus.com/development-length-in-footing/","timestamp":"2024-11-02T09:20:34Z","content_type":"text/html","content_length":"146974","record_id":"<urn:uuid:8f93b836-8908-43e4-a1ca-dff9154c6cf1>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00535.warc.gz"} |
Lean Six Sigma
Apply Lean Six Sigma Methodology to solve operational issues and improve processes.
Lean Six Sigma - Operators Training Problem
🎓 Topic
Perform a Kruskal Wallis Test evaluating the impact of training on warehouse operators’ productivity.
📰 Related Articles
Lean Six Sigma with Python — Kruskal Wallis Test
How to replace Minitab with Python to perform Kruskal Wallis Test evaluating the impact of training on warehouse operators’ productivity
Lean Six Sigma - Drivers Dispatch Problem
🎓 Topic
Perform a Chi-Squared Test to explain a shortage of drivers impacting your transportation network.
📰 Related Articles
Lean Six Sigma with Python — Chi-Squared Test
Perform a Chi-Squared Test to explain a shortage of drivers impacting your transportation network
Lean Six Sigma - Warehouse Productivity Bonus Problem
🎓 Topic
Replace Minitab with Python to perform a Logistic Regression to estimate the minimum bonus needed to reach 75% of a productivity target.
📰 Related Articles
Lean Six Sigma with Python — Logistic Regression
Replace Minitab with Python to perform a Logistic Regression to estimate the minimum bonus needed to reach 75% of a productivity target | {"url":"https://www.samirsaci.com/lean-six-sigma/","timestamp":"2024-11-10T17:28:30Z","content_type":"text/html","content_length":"18106","record_id":"<urn:uuid:f2c423b1-3f77-4a04-acfe-e588ea70d33c>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00865.warc.gz"} |
Trinagon - Puzzles for Curious Mind
A puzzle, a game, a challenge, a system, a tool ? .. Probably :)
A puzzle to engage your mind into new ways of thinking, from baby steps to giant leaps,
A game to immerse yourself in and enjoy playing, there is no losing.
A challenge to grow and learn, to train your mind, a challenge to yourself to find different approaches to solutions, a challenge for others to solve a puzzle you created.
A system of puzzle setups, on many base playgrounds (2D, 3D) with varying difficulties, from easy to highly complex.
A tool to see 3 dimensional spatial relations, positions & movements, and thereby understand what may have been outside of your scope.
Maybe a path leading from diversion into clarity, and a quiet focused mind ... who knows ??
One thing it strives to be : FUN.
PS : Tri - na - gon pronunciation :
Tri as in 'Trinity',
na - as in 'nothing' or 'now', it's still an 'a' like in 'father' but shorter,
gon - as in gone. | {"url":"https://trinagon.com/introductions","timestamp":"2024-11-07T05:27:40Z","content_type":"application/xhtml+xml","content_length":"38424","record_id":"<urn:uuid:3ed71036-9486-4382-81bf-ce6db99825db>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00293.warc.gz"} |
Scatterplots (3 of 5)
Learning Objectives
• Use a scatterplot to display the relationship between two quantitative variables. Describe the overall pattern (form, direction, and strength) and striking deviations from the pattern.
Now we return to our previous example. We apply the ideas of direction, form, and strength to describe the relationship between the age of the driver and the maximum distance to read a highway sign.
Here is the scatterplot:
Direction: The direction of the relationship is negative. An increase in age is associated with a decrease in reading distance, which makes sense because older drivers tend to have diminished
eyesight. So most older drivers can read the sign only when they are close to it. In other words, they have a shorter maximum reading distance.
Form: The form of the relationship is linear.
Strength: The data points are fairly close to the line, so the relationship is moderately strong. Do not worry if you feel uncertain about describing the strength of a relationship. We mentioned
earlier that descriptions of strength are not very precise. We develop a more precise measure of the strength shortly.
Outliers: There are no outliers. All the data points tend to follow the linear pattern. | {"url":"https://courses.lumenlearning.com/suny-hccc-wm-concepts-statistics/chapter/scatterplots-3-of-5/","timestamp":"2024-11-02T19:03:13Z","content_type":"text/html","content_length":"48421","record_id":"<urn:uuid:aae8c85d-7b04-4902-8ed0-40b883fa2775>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00695.warc.gz"} |
Measures of Central Tendency Test CBSE Class 9 MathsMeasures of Central Tendency Test CBSE Class 9 Maths - The Brainbox Tutorials
Measures of Central Tendency Test CBSE Class 9 Maths
Measures of Central Tendency is a very important chapter in CBSE Class 9 Maths. Take this Maths Practice Test On Measures of Central Tendency here. This online aptitude test on Mean, Median, Mode
will increase the confidence in students and they will never hesitate to do the sums on Measures of Central Tendency. Measures of Central Tendency Test CBSE Class 9 Maths is provided by The Brainbox
Tutorials. The quiz has multiple choice questions on Measures of Central Tendency Maths. Students will be benefitted by this online Practice test on Measures of Central Tendency.
Measures of Central Tendency Test CBSE Class 9 Maths
Below are the quiz and the rules to be followed while performing this test on Measures of Central Tendency for CBSE Class 9.
You can go through the tutorial of this chapter in video form to understand the concept clearly. Here is the link to the video tutorial.
Video tutorial on:
Rules For The Quiz:
• This quiz has 10 multiple-choice questions.
• Each sum has 2 marks.
• So the maximum marks of this test are 20.
• There is no time limit.
• You should be ready with a pen and copy in your hand to solve the sums.
• keep your Maths book away from you. This is the test of your memory. So do not take the help of the Maths book.
• The correct answer and explanation are provided at the end of this quiz.
Please share your score in test in the comments section below. You are free to have as many attempts you want.
Happy learning and always say yes to Maths.
ICSE Related links
Chapter-wise Quiz/MCQ/Test:
Sample Papers:
Board Papers:
CBSE Related links
Chapter-wise Quiz/MCQ/Test:
Sample Papers:
Subscribe our YouTube channel for latest educational updates.
Leave a Comment | {"url":"https://thebrainboxtutorials.com/2021/03/measures-of-central-tendency-test-cbse-class-9-maths.html","timestamp":"2024-11-02T18:46:22Z","content_type":"text/html","content_length":"115655","record_id":"<urn:uuid:c9467406-f104-4f2c-ba15-a9cde619ef29>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00362.warc.gz"} |
Square Yards to Square Centimeters
Square Yards to Square Centimeters Converter
Enter Square Yards
Square Centimeters
⇅ Switch toSquare Centimeters to Square Yards Converter
How to use this Square Yards to Square Centimeters Converter 🤔
Follow these steps to convert given area from the units of Square Yards to the units of Square Centimeters.
1. Enter the input Square Yards value in the text field.
2. The calculator converts the given Square Yards into Square Centimeters in realtime ⌚ using the conversion formula, and displays under the Square Centimeters label. You do not need to click any
button. If the input changes, Square Centimeters value is re-calculated, just like that.
3. You may copy the resulting Square Centimeters value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Square Yards to Square Centimeters?
The formula to convert given area from Square Yards to Square Centimeters is:
Area[(Square Centimeters)] = Area[(Square Yards)] × 8361.2736
Substitute the given value of area in square yards, i.e., Area[(Square Yards)] in the above formula and simplify the right-hand side value. The resulting value is the area in square centimeters,
i.e., Area[(Square Centimeters)].
Calculation will be done after you enter a valid input.
Consider that a backyard garden covers 200 square yards.
Convert this area from square yards to Square Centimeters.
The area in square yards is:
Area[(Square Yards)] = 200
The formula to convert area from square yards to square centimeters is:
Area[(Square Centimeters)] = Area[(Square Yards)] × 8361.2736
Substitute given weight Area[(Square Yards)] = 200 in the above formula.
Area[(Square Centimeters)] = 200 × 8361.2736
Area[(Square Centimeters)] = 1672254.72
Final Answer:
Therefore, 200 yd^2^ is equal to 1672254.72 cm^2^.
The area is 1672254.72 cm^2^, in square centimeters.
Consider that a football field spans 5,000 square yards.
Convert this area from square yards to Square Centimeters.
The area in square yards is:
Area[(Square Yards)] = 5000
The formula to convert area from square yards to square centimeters is:
Area[(Square Centimeters)] = Area[(Square Yards)] × 8361.2736
Substitute given weight Area[(Square Yards)] = 5000 in the above formula.
Area[(Square Centimeters)] = 5000 × 8361.2736
Area[(Square Centimeters)] = 41806368
Final Answer:
Therefore, 5000 yd^2^ is equal to 41806368 cm^2^.
The area is 41806368 cm^2^, in square centimeters.
Square Yards to Square Centimeters Conversion Table
The following table gives some of the most used conversions from Square Yards to Square Centimeters.
Square Yards (yd^2^) Square Centimeters (cm^2^)
0 yd^2^ 0 cm^2^
1 yd^2^ 8361.2736 cm^2^
10 yd^2^ 83612.736 cm^2^
45 yd^2^ 376257.312 cm^2^
90 yd^2^ 752514.624 cm^2^
180 yd^2^ 1505029.248 cm^2^
360 yd^2^ 3010058.496 cm^2^
1000 yd^2^ 8361273.6 cm^2^
Square Yards
A square yard (yd^2) is a unit of area measurement equal to the area of a square with sides one yard (3 feet) in length. It is used in various fields, including construction, landscaping, and fabric
measurement. Square yards are particularly common in the United States and the United Kingdom for measuring land areas, carpets, and other materials where larger units like square feet are
Square Centimeters
A square centimeter (cm^2) is a unit of area measurement equal to the area of a square with sides that are one centimeter in length. It is a smaller unit of area often used in science, medicine, and
small-scale engineering projects. Square centimeters are useful for measuring smaller surfaces, such as the area of a piece of fabric, the surface area of a small object, or in medical imaging to
describe the size of lesions or wounds.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Square Yards to Square Centimeters in Area?
The formula to convert Square Yards to Square Centimeters in Area is:
Square Yards * 8361.2736
2. Is this tool free or paid?
This Area conversion tool, which converts Square Yards to Square Centimeters, is completely free to use.
3. How do I convert Area from Square Yards to Square Centimeters?
To convert Area from Square Yards to Square Centimeters, you can use the following formula:
Square Yards * 8361.2736
For example, if you have a value in Square Yards, you substitute that value in place of Square Yards in the above formula, and solve the mathematical expression to get the equivalent value in Square | {"url":"https://convertonline.org/unit/?convert=sq_yard-sq_cm","timestamp":"2024-11-05T10:38:24Z","content_type":"text/html","content_length":"75908","record_id":"<urn:uuid:699a83ea-48fd-49fe-97d3-8adf7c4ed4cf>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00664.warc.gz"} |