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A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet). The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers
are formed by repeated addition.
Characterization by a cutting sequence with a line of slope ${\displaystyle 1/\varphi }$ or ${\displaystyle \varphi -1}$, with ${\displaystyle \varphi }$ the golden ratio.
It is a paradigmatic example of a Sturmian word and specifically, a morphic word.
The name "Fibonacci word" has also been used to refer to the members of a formal language L consisting of strings of zeros and ones with no two repeated ones. Any prefix of the specific Fibonacci
word belongs to L, but so do many other strings. L has a Fibonacci number of members of each possible length.
Let ${\displaystyle S_{0}}$ be "0" and ${\displaystyle S_{1}}$ be "01". Now ${\displaystyle S_{n}=S_{n-1}S_{n-2}}$ (the concatenation of the previous sequence and the one before that).
The infinite Fibonacci word is the limit ${\displaystyle S_{\infty }}$ , that is, the (unique) infinite sequence that contains each ${\displaystyle S_{n}}$ , for finite ${\displaystyle n}$ , as a
Enumerating items from the above definition produces:
${\displaystyle S_{0}}$ 0
${\displaystyle S_{1}}$ 01
${\displaystyle S_{2}}$ 010
${\displaystyle S_{3}}$ 01001
${\displaystyle S_{4}}$ 01001010
${\displaystyle S_{5}}$ 0100101001001
The first few elements of the infinite Fibonacci word are:
0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0,
0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, ... (sequence A003849 in the OEIS)
Closed-form expression for individual digits
The n^th digit of the word is ${\displaystyle 2+\lfloor n\varphi \rfloor -\lfloor (n+1)\varphi \rfloor }$ where ${\displaystyle \varphi }$ is the golden ratio and ${\displaystyle \lfloor \,\ \rfloor
}$ is the floor function (sequence A003849 in the OEIS). As a consequence, the infinite Fibonacci word can be characterized by a cutting sequence of a line of slope ${\displaystyle 1/\varphi }$ or $
{\displaystyle \varphi -1}$ . See the figure above.
Substitution rules
Another way of going from S[n] to S[n+1] is to replace each symbol 0 in S[n] with the pair of consecutive symbols 0,1 in S[n+1], and to replace each symbol 1 in S[n] with the single symbol 0 in S[
Alternatively, one can imagine directly generating the entire infinite Fibonacci word by the following process: start with a cursor pointing to the single digit 0. Then, at each step, if the cursor
is pointing to a 0, append 1,0 to the end of the word, and if the cursor is pointing to a 1, append 0 to the end of the word. In either case, complete the step by moving the cursor one position to
the right.
A similar infinite word, sometimes called the rabbit sequence, is generated by a similar infinite process with a different replacement rule: whenever the cursor is pointing to a 0, append 1, and
whenever the cursor is pointing to a 1, append 0,1. The resulting sequence begins
0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, ...
However this sequence differs from the Fibonacci word only trivially, by swapping 0s for 1s and shifting the positions by one.
A closed form expression for the so-called rabbit sequence:
The n^th digit of the word is ${\displaystyle \lfloor n\varphi \rfloor -\lfloor (n-1)\varphi \rfloor -1.}$
The word is related to the famous sequence of the same name (the Fibonacci sequence) in the sense that addition of integers in the inductive definition is replaced with string concatenation. This
causes the length of S[n] to be F[n+2], the (n+2)nd Fibonacci number. Also the number of 1s in S[n] is F[n] and the number of 0s in S[n] is F[n+1].
Other properties
• The infinite Fibonacci word is not periodic and not ultimately periodic.
• The last two letters of a Fibonacci word are alternately "01" and "10".
• Suppressing the last two letters of a Fibonacci word, or prefixing the complement of the last two letters, creates a palindrome. Example: 01S[4] = 0101001010 is a palindrome. The palindromic
density of the infinite Fibonacci word is thus 1/φ, where φ is the golden ratio: this is the largest possible value for aperiodic words.
• In the infinite Fibonacci word, the ratio (number of letters)/(number of zeroes) is φ, as is the ratio of zeroes to ones.^[4]
• The infinite Fibonacci word is a balanced sequence: Take two factors of the same length anywhere in the Fibonacci word. The difference between their Hamming weights (the number of occurrences of
"1") never exceeds 1.
• The subwords 11 and 000 never occur.^[6]
• The complexity function of the infinite Fibonacci word is n +1: it contains n +1 distinct subwords of length n. Example: There are 4 distinct subwords of length 3: "001", "010", "100" and
"101". Being also non-periodic, it is then of "minimal complexity", and hence a Sturmian word, with slope ${\displaystyle 1/\varphi }$ . The infinite Fibonacci word is the standard word generated
by the directive sequence (1,1,1,....).
• The infinite Fibonacci word is recurrent; that is, every subword occurs infinitely often.
• If ${\displaystyle u}$ is a subword of the infinite Fibonacci word, then so is its reversal, denoted ${\displaystyle u^{R}}$ .
• If ${\displaystyle u}$ is a subword of the infinite Fibonacci word, then the least period of ${\displaystyle u}$ is a Fibonacci number.
• The concatenation of two successive Fibonacci words is "almost commutative". ${\displaystyle S_{n+1}=S_{n}S_{n-1}}$ and ${\displaystyle S_{n-1}S_{n}}$ differ only by their last two letters.
• The number 0.010010100..., whose digits are built with the digits of the infinite Fibonacci word, is transcendental.
• The letters "1" can be found at the positions given by the successive values of the Upper Wythoff sequence (sequence A001950 in the OEIS): ${\displaystyle \lfloor n\varphi ^{2}\rfloor }$
• The letters "0" can be found at the positions given by the successive values of the Lower Wythoff sequence (sequence A000201 in the OEIS): ${\displaystyle \lfloor n\varphi \rfloor }$
• The distribution of ${\displaystyle n=F_{k}}$ points on the unit circle, placed consecutively clockwise by the golden angle ${\displaystyle {\frac {2\pi }{\varphi ^{2}}}}$ , generates a pattern
of two lengths ${\displaystyle {\frac {2\pi }{\varphi ^{k-1}}},{\frac {2\pi }{\varphi ^{k}}}}$ on the unit circle. Although the above generating process of the Fibonacci word does not correspond
directly to the successive division of circle segments, this pattern is ${\displaystyle S_{k-1}}$ if the pattern starts at the point nearest to the first point in clockwise direction, whereupon 0
corresponds to the long distance and 1 to the short distance.
• The infinite Fibonacci word contains repetitions of 3 successive identical subwords, but none of 4. The critical exponent for the infinite Fibonacci word is ${\displaystyle 2+\varphi \approx
3.618}$ . It is the smallest index (or critical exponent) among all Sturmian words.
• The infinite Fibonacci word is often cited as the worst case for algorithms detecting repetitions in a string.
• The infinite Fibonacci word is a morphic word, generated in {0,1}* by the endomorphism 0 → 01, 1 → 0.
• The nth element of a Fibonacci word, ${\displaystyle s_{n}}$ , is 1 if the Zeckendorf representation (the sum of a specific set of Fibonacci numbers) of n includes a 1, and 0 if it does not
include a 1.
• The digits of the Fibonacci word may be obtained by taking the sequence of fibbinary numbers modulo 2.
Fibonacci based constructions are currently used to model physical systems with aperiodic order such as quasicrystals, and in this context the Fibonacci word is also called the Fibonacci quasicrystal
. Crystal growth techniques have been used to grow Fibonacci layered crystals and study their light scattering properties.
See also
External links
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Source code for networkx.generators.trees
"""Functions for generating trees.
The functions sampling trees at random in this module come
in two variants: labeled and unlabeled. The labeled variants
sample from every possible tree with the given number of nodes
uniformly at random. The unlabeled variants sample from every
possible *isomorphism class* of trees with the given number
of nodes uniformly at random.
To understand the difference, consider the following example.
There are two isomorphism classes of trees with four nodes.
One is that of the path graph, the other is that of the
star graph. The unlabeled variant will return a line graph or
a star graph with probability 1/2.
The labeled variant will return the line graph
with probability 3/4 and the star graph with probability 1/4,
because there are more labeled variants of the line graph
than of the star graph. More precisely, the line graph has
an automorphism group of order 2, whereas the star graph has
an automorphism group of order 6, so the line graph has three
times as many labeled variants as the star graph, and thus
three more chances to be drawn.
Additionally, some functions in this module can sample rooted
trees and forests uniformly at random. A rooted tree is a tree
with a designated root node. A rooted forest is a disjoint union
of rooted trees.
import warnings
from collections import Counter, defaultdict
from math import comb, factorial
import networkx as nx
from networkx.utils import py_random_state
__all__ = [
@nx._dispatchable(graphs=None, returns_graph=True)
def prefix_tree(paths):
"""Creates a directed prefix tree from a list of paths.
Usually the paths are described as strings or lists of integers.
A "prefix tree" represents the prefix structure of the strings.
Each node represents a prefix of some string. The root represents
the empty prefix with children for the single letter prefixes which
in turn have children for each double letter prefix starting with
the single letter corresponding to the parent node, and so on.
More generally the prefixes do not need to be strings. A prefix refers
to the start of a sequence. The root has children for each one element
prefix and they have children for each two element prefix that starts
with the one element sequence of the parent, and so on.
Note that this implementation uses integer nodes with an attribute.
Each node has an attribute "source" whose value is the original element
of the path to which this node corresponds. For example, suppose `paths`
consists of one path: "can". Then the nodes `[1, 2, 3]` which represent
this path have "source" values "c", "a" and "n".
All the descendants of a node have a common prefix in the sequence/path
associated with that node. From the returned tree, the prefix for each
node can be constructed by traversing the tree up to the root and
accumulating the "source" values along the way.
The root node is always `0` and has "source" attribute `None`.
The root is the only node with in-degree zero.
The nil node is always `-1` and has "source" attribute `"NIL"`.
The nil node is the only node with out-degree zero.
paths: iterable of paths
An iterable of paths which are themselves sequences.
Matching prefixes among these sequences are identified with
nodes of the prefix tree. One leaf of the tree is associated
with each path. (Identical paths are associated with the same
leaf of the tree.)
tree: DiGraph
A directed graph representing an arborescence consisting of the
prefix tree generated by `paths`. Nodes are directed "downward",
from parent to child. A special "synthetic" root node is added
to be the parent of the first node in each path. A special
"synthetic" leaf node, the "nil" node `-1`, is added to be the child
of all nodes representing the last element in a path. (The
addition of this nil node technically makes this not an
arborescence but a directed acyclic graph; removing the nil node
makes it an arborescence.)
The prefix tree is also known as a *trie*.
Create a prefix tree from a list of strings with common prefixes::
>>> paths = ["ab", "abs", "ad"]
>>> T = nx.prefix_tree(paths)
>>> list(T.edges)
[(0, 1), (1, 2), (1, 4), (2, -1), (2, 3), (3, -1), (4, -1)]
The leaf nodes can be obtained as predecessors of the nil node::
>>> root, NIL = 0, -1
>>> list(T.predecessors(NIL))
[2, 3, 4]
To recover the original paths that generated the prefix tree,
traverse up the tree from the node `-1` to the node `0`::
>>> recovered = []
>>> for v in T.predecessors(NIL):
... prefix = ""
... while v != root:
... prefix = str(T.nodes[v]["source"]) + prefix
... v = next(T.predecessors(v)) # only one predecessor
... recovered.append(prefix)
>>> sorted(recovered)
['ab', 'abs', 'ad']
def get_children(parent, paths):
children = defaultdict(list)
# Populate dictionary with key(s) as the child/children of the root and
# value(s) as the remaining paths of the corresponding child/children
for path in paths:
# If path is empty, we add an edge to the NIL node.
if not path:
tree.add_edge(parent, NIL)
child, *rest = path
# `child` may exist as the head of more than one path in `paths`.
return children
# Initialize the prefix tree with a root node and a nil node.
tree = nx.DiGraph()
root = 0
tree.add_node(root, source=None)
NIL = -1
tree.add_node(NIL, source="NIL")
children = get_children(root, paths)
stack = [(root, iter(children.items()))]
while stack:
parent, remaining_children = stack[-1]
child, remaining_paths = next(remaining_children)
# Pop item off stack if there are no remaining children
except StopIteration:
# We relabel each child with an unused name.
new_name = len(tree) - 1
# The "source" node attribute stores the original node name.
tree.add_node(new_name, source=child)
tree.add_edge(parent, new_name)
children = get_children(new_name, remaining_paths)
stack.append((new_name, iter(children.items())))
return tree
@nx._dispatchable(graphs=None, returns_graph=True)
def prefix_tree_recursive(paths):
"""Recursively creates a directed prefix tree from a list of paths.
The original recursive version of prefix_tree for comparison. It is
the same algorithm but the recursion is unrolled onto a stack.
Usually the paths are described as strings or lists of integers.
A "prefix tree" represents the prefix structure of the strings.
Each node represents a prefix of some string. The root represents
the empty prefix with children for the single letter prefixes which
in turn have children for each double letter prefix starting with
the single letter corresponding to the parent node, and so on.
More generally the prefixes do not need to be strings. A prefix refers
to the start of a sequence. The root has children for each one element
prefix and they have children for each two element prefix that starts
with the one element sequence of the parent, and so on.
Note that this implementation uses integer nodes with an attribute.
Each node has an attribute "source" whose value is the original element
of the path to which this node corresponds. For example, suppose `paths`
consists of one path: "can". Then the nodes `[1, 2, 3]` which represent
this path have "source" values "c", "a" and "n".
All the descendants of a node have a common prefix in the sequence/path
associated with that node. From the returned tree, ehe prefix for each
node can be constructed by traversing the tree up to the root and
accumulating the "source" values along the way.
The root node is always `0` and has "source" attribute `None`.
The root is the only node with in-degree zero.
The nil node is always `-1` and has "source" attribute `"NIL"`.
The nil node is the only node with out-degree zero.
paths: iterable of paths
An iterable of paths which are themselves sequences.
Matching prefixes among these sequences are identified with
nodes of the prefix tree. One leaf of the tree is associated
with each path. (Identical paths are associated with the same
leaf of the tree.)
tree: DiGraph
A directed graph representing an arborescence consisting of the
prefix tree generated by `paths`. Nodes are directed "downward",
from parent to child. A special "synthetic" root node is added
to be the parent of the first node in each path. A special
"synthetic" leaf node, the "nil" node `-1`, is added to be the child
of all nodes representing the last element in a path. (The
addition of this nil node technically makes this not an
arborescence but a directed acyclic graph; removing the nil node
makes it an arborescence.)
The prefix tree is also known as a *trie*.
Create a prefix tree from a list of strings with common prefixes::
>>> paths = ["ab", "abs", "ad"]
>>> T = nx.prefix_tree(paths)
>>> list(T.edges)
[(0, 1), (1, 2), (1, 4), (2, -1), (2, 3), (3, -1), (4, -1)]
The leaf nodes can be obtained as predecessors of the nil node.
>>> root, NIL = 0, -1
>>> list(T.predecessors(NIL))
[2, 3, 4]
To recover the original paths that generated the prefix tree,
traverse up the tree from the node `-1` to the node `0`::
>>> recovered = []
>>> for v in T.predecessors(NIL):
... prefix = ""
... while v != root:
... prefix = str(T.nodes[v]["source"]) + prefix
... v = next(T.predecessors(v)) # only one predecessor
... recovered.append(prefix)
>>> sorted(recovered)
['ab', 'abs', 'ad']
def _helper(paths, root, tree):
"""Recursively create a trie from the given list of paths.
`paths` is a list of paths, each of which is itself a list of
nodes, relative to the given `root` (but not including it). This
list of paths will be interpreted as a tree-like structure, in
which two paths that share a prefix represent two branches of
the tree with the same initial segment.
`root` is the parent of the node at index 0 in each path.
`tree` is the "accumulator", the :class:`networkx.DiGraph`
representing the branching to which the new nodes and edges will
be added.
# For each path, remove the first node and make it a child of root.
# Any remaining paths then get processed recursively.
children = defaultdict(list)
for path in paths:
# If path is empty, we add an edge to the NIL node.
if not path:
tree.add_edge(root, NIL)
child, *rest = path
# `child` may exist as the head of more than one path in `paths`.
# Add a node for each child, connect root, recurse to remaining paths
for child, remaining_paths in children.items():
# We relabel each child with an unused name.
new_name = len(tree) - 1
# The "source" node attribute stores the original node name.
tree.add_node(new_name, source=child)
tree.add_edge(root, new_name)
_helper(remaining_paths, new_name, tree)
# Initialize the prefix tree with a root node and a nil node.
tree = nx.DiGraph()
root = 0
tree.add_node(root, source=None)
NIL = -1
tree.add_node(NIL, source="NIL")
# Populate the tree.
_helper(paths, root, tree)
return tree
@nx._dispatchable(graphs=None, returns_graph=True)
def random_labeled_tree(n, *, seed=None):
"""Returns a labeled tree on `n` nodes chosen uniformly at random.
Generating uniformly distributed random Prüfer sequences and
converting them into the corresponding trees is a straightforward
method of generating uniformly distributed random labeled trees.
This function implements this method.
n : int
The number of nodes, greater than zero.
seed : random_state
Indicator of random number generation state.
See :ref:`Randomness<randomness>`
A `networkx.Graph` with nodes in the set {0, …, *n* - 1}.
If `n` is zero (because the null graph is not a tree).
>>> G = nx.random_labeled_tree(5, seed=42)
>>> nx.is_tree(G)
>>> G.edges
EdgeView([(0, 1), (0, 3), (0, 2), (2, 4)])
A tree with *arbitrarily directed* edges can be created by assigning
generated edges to a ``DiGraph``:
>>> DG = nx.DiGraph()
>>> DG.add_edges_from(G.edges)
>>> nx.is_tree(DG)
>>> DG.edges
OutEdgeView([(0, 1), (0, 3), (0, 2), (2, 4)])
# Cannot create a Prüfer sequence unless `n` is at least two.
if n == 0:
raise nx.NetworkXPointlessConcept("the null graph is not a tree")
if n == 1:
return nx.empty_graph(1)
return nx.from_prufer_sequence([seed.choice(range(n)) for i in range(n - 2)])
@nx._dispatchable(graphs=None, returns_graph=True)
def random_labeled_rooted_tree(n, *, seed=None):
"""Returns a labeled rooted tree with `n` nodes.
The returned tree is chosen uniformly at random from all labeled rooted trees.
n : int
The number of nodes
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
A `networkx.Graph` with integer nodes 0 <= node <= `n` - 1.
The root of the tree is selected uniformly from the nodes.
The "root" graph attribute identifies the root of the tree.
This function returns the result of :func:`random_labeled_tree`
with a randomly selected root.
If `n` is zero (because the null graph is not a tree).
t = random_labeled_tree(n, seed=seed)
t.graph["root"] = seed.randint(0, n - 1)
return t
@nx._dispatchable(graphs=None, returns_graph=True)
def random_labeled_rooted_forest(n, *, seed=None):
"""Returns a labeled rooted forest with `n` nodes.
The returned forest is chosen uniformly at random using a
generalization of Prüfer sequences [1]_ in the form described in [2]_.
n : int
The number of nodes.
seed : random_state
See :ref:`Randomness<randomness>`.
A `networkx.Graph` with integer nodes 0 <= node <= `n` - 1.
The "roots" graph attribute is a set of integers containing the roots.
.. [1] Knuth, Donald E. "Another Enumeration of Trees."
Canadian Journal of Mathematics, 20 (1968): 1077-1086.
.. [2] Rubey, Martin. "Counting Spanning Trees". Diplomarbeit
zur Erlangung des akademischen Grades Magister der
Naturwissenschaften an der Formal- und Naturwissenschaftlichen
Fakultät der Universität Wien. Wien, May 2000.
# Select the number of roots by iterating over the cumulative count of trees
# with at most k roots
def _select_k(n, seed):
r = seed.randint(0, (n + 1) ** (n - 1) - 1)
cum_sum = 0
for k in range(1, n):
cum_sum += (factorial(n - 1) * n ** (n - k)) // (
factorial(k - 1) * factorial(n - k)
if r < cum_sum:
return k
return n
F = nx.empty_graph(n)
if n == 0:
F.graph["roots"] = {}
return F
# Select the number of roots k
k = _select_k(n, seed)
if k == n:
F.graph["roots"] = set(range(n))
return F # Nothing to do
# Select the roots
roots = seed.sample(range(n), k)
# Nonroots
p = set(range(n)).difference(roots)
# Coding sequence
N = [seed.randint(0, n - 1) for i in range(n - k - 1)]
# Multiset of elements in N also in p
degree = Counter([x for x in N if x in p])
# Iterator over the elements of p with degree zero
iterator = iter(x for x in p if degree[x] == 0)
u = last = next(iterator)
# This loop is identical to that for Prüfer sequences,
# except that we can draw nodes only from p
for v in N:
F.add_edge(u, v)
degree[v] -= 1
if v < last and degree[v] == 0:
u = v
last = u = next(iterator)
F.add_edge(u, roots[0])
F.graph["roots"] = set(roots)
return F
# The following functions support generation of unlabeled trees and forests.
def _to_nx(edges, n_nodes, root=None, roots=None):
Converts the (edges, n_nodes) input to a :class:`networkx.Graph`.
The (edges, n_nodes) input is a list of even length, where each pair
of consecutive integers represents an edge, and an integer `n_nodes`.
Integers in the list are elements of `range(n_nodes)`.
edges : list of ints
The flattened list of edges of the graph.
n_nodes : int
The number of nodes of the graph.
root: int (default=None)
If not None, the "root" attribute of the graph will be set to this value.
roots: collection of ints (default=None)
If not None, he "roots" attribute of the graph will be set to this value.
The graph with `n_nodes` nodes and edges given by `edges`.
G = nx.empty_graph(n_nodes)
if root is not None:
G.graph["root"] = root
if roots is not None:
G.graph["roots"] = roots
return G
def _num_rooted_trees(n, cache_trees):
"""Returns the number of unlabeled rooted trees with `n` nodes.
See also https://oeis.org/A000081.
n : int
The number of nodes
cache_trees : list of ints
The $i$-th element is the number of unlabeled rooted trees with $i$ nodes,
which is used as a cache (and is extended to length $n+1$ if needed)
The number of unlabeled rooted trees with `n` nodes.
for n_i in range(len(cache_trees), n + 1):
d * cache_trees[n_i - j * d] * cache_trees[d]
for d in range(1, n_i)
for j in range(1, (n_i - 1) // d + 1)
// (n_i - 1)
return cache_trees[n]
def _select_jd_trees(n, cache_trees, seed):
"""Returns a pair $(j,d)$ with a specific probability
Given $n$, returns a pair of positive integers $(j,d)$ with the probability
specified in formula (5) of Chapter 29 of [1]_.
n : int
The number of nodes
cache_trees : list of ints
Cache for :func:`_num_rooted_trees`.
seed : random_state
See :ref:`Randomness<randomness>`.
(int, int)
A pair of positive integers $(j,d)$ satisfying formula (5) of
Chapter 29 of [1]_.
.. [1] Nijenhuis, Albert, and Wilf, Herbert S.
"Combinatorial algorithms: for computers and calculators."
Academic Press, 1978.
p = seed.randint(0, _num_rooted_trees(n, cache_trees) * (n - 1) - 1)
cumsum = 0
for d in range(n - 1, 0, -1):
for j in range(1, (n - 1) // d + 1):
cumsum += (
* _num_rooted_trees(n - j * d, cache_trees)
* _num_rooted_trees(d, cache_trees)
if p < cumsum:
return (j, d)
def _random_unlabeled_rooted_tree(n, cache_trees, seed):
"""Returns an unlabeled rooted tree with `n` nodes.
Returns an unlabeled rooted tree with `n` nodes chosen uniformly
at random using the "RANRUT" algorithm from [1]_.
The tree is returned in the form: (list_of_edges, number_of_nodes)
n : int
The number of nodes, greater than zero.
cache_trees : list ints
Cache for :func:`_num_rooted_trees`.
seed : random_state
See :ref:`Randomness<randomness>`.
(list_of_edges, number_of_nodes) : list, int
A random unlabeled rooted tree with `n` nodes as a 2-tuple
``(list_of_edges, number_of_nodes)``.
The root is node 0.
.. [1] Nijenhuis, Albert, and Wilf, Herbert S.
"Combinatorial algorithms: for computers and calculators."
Academic Press, 1978.
if n == 1:
edges, n_nodes = [], 1
return edges, n_nodes
if n == 2:
edges, n_nodes = [(0, 1)], 2
return edges, n_nodes
j, d = _select_jd_trees(n, cache_trees, seed)
t1, t1_nodes = _random_unlabeled_rooted_tree(n - j * d, cache_trees, seed)
t2, t2_nodes = _random_unlabeled_rooted_tree(d, cache_trees, seed)
t12 = [(0, t2_nodes * i + t1_nodes) for i in range(j)]
for _ in range(j):
t1.extend((n1 + t1_nodes, n2 + t1_nodes) for n1, n2 in t2)
t1_nodes += t2_nodes
return t1, t1_nodes
@nx._dispatchable(graphs=None, returns_graph=True)
def random_unlabeled_rooted_tree(n, *, number_of_trees=None, seed=None):
"""Returns a number of unlabeled rooted trees uniformly at random
Returns one or more (depending on `number_of_trees`)
unlabeled rooted trees with `n` nodes drawn uniformly
at random.
n : int
The number of nodes
number_of_trees : int or None (default)
If not None, this number of trees is generated and returned.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
:class:`networkx.Graph` or list of :class:`networkx.Graph`
A single `networkx.Graph` (or a list thereof, if `number_of_trees`
is specified) with nodes in the set {0, …, *n* - 1}.
The "root" graph attribute identifies the root of the tree.
The trees are generated using the "RANRUT" algorithm from [1]_.
The algorithm needs to compute some counting functions
that are relatively expensive: in case several trees are needed,
it is advisable to use the `number_of_trees` optional argument
to reuse the counting functions.
If `n` is zero (because the null graph is not a tree).
.. [1] Nijenhuis, Albert, and Wilf, Herbert S.
"Combinatorial algorithms: for computers and calculators."
Academic Press, 1978.
if n == 0:
raise nx.NetworkXPointlessConcept("the null graph is not a tree")
cache_trees = [0, 1] # initial cache of number of rooted trees
if number_of_trees is None:
return _to_nx(*_random_unlabeled_rooted_tree(n, cache_trees, seed), root=0)
return [
_to_nx(*_random_unlabeled_rooted_tree(n, cache_trees, seed), root=0)
for i in range(number_of_trees)
def _num_rooted_forests(n, q, cache_forests):
"""Returns the number of unlabeled rooted forests with `n` nodes, and with
no more than `q` nodes per tree. A recursive formula for this is (2) in
[1]_. This function is implemented using dynamic programming instead of
n : int
The number of nodes.
q : int
The maximum number of nodes for each tree of the forest.
cache_forests : list of ints
The $i$-th element is the number of unlabeled rooted forests with
$i$ nodes, and with no more than `q` nodes per tree; this is used
as a cache (and is extended to length `n` + 1 if needed).
The number of unlabeled rooted forests with `n` nodes with no more than
`q` nodes per tree.
.. [1] Wilf, Herbert S. "The uniform selection of free trees."
Journal of Algorithms 2.2 (1981): 204-207.
for n_i in range(len(cache_forests), n + 1):
q_i = min(n_i, q)
d * cache_forests[n_i - j * d] * cache_forests[d - 1]
for d in range(1, q_i + 1)
for j in range(1, n_i // d + 1)
// n_i
return cache_forests[n]
def _select_jd_forests(n, q, cache_forests, seed):
"""Given `n` and `q`, returns a pair of positive integers $(j,d)$
such that $j\\leq d$, with probability satisfying (F1) of [1]_.
n : int
The number of nodes.
q : int
The maximum number of nodes for each tree of the forest.
cache_forests : list of ints
Cache for :func:`_num_rooted_forests`.
seed : random_state
See :ref:`Randomness<randomness>`.
(int, int)
A pair of positive integers $(j,d)$
.. [1] Wilf, Herbert S. "The uniform selection of free trees."
Journal of Algorithms 2.2 (1981): 204-207.
p = seed.randint(0, _num_rooted_forests(n, q, cache_forests) * n - 1)
cumsum = 0
for d in range(q, 0, -1):
for j in range(1, n // d + 1):
cumsum += (
* _num_rooted_forests(n - j * d, q, cache_forests)
* _num_rooted_forests(d - 1, q, cache_forests)
if p < cumsum:
return (j, d)
def _random_unlabeled_rooted_forest(n, q, cache_trees, cache_forests, seed):
"""Returns an unlabeled rooted forest with `n` nodes, and with no more
than `q` nodes per tree, drawn uniformly at random. It is an implementation
of the algorithm "Forest" of [1]_.
n : int
The number of nodes.
q : int
The maximum number of nodes per tree.
cache_trees :
Cache for :func:`_num_rooted_trees`.
cache_forests :
Cache for :func:`_num_rooted_forests`.
seed : random_state
See :ref:`Randomness<randomness>`.
(edges, n, r) : (list, int, list)
The forest (edges, n) and a list r of root nodes.
.. [1] Wilf, Herbert S. "The uniform selection of free trees."
Journal of Algorithms 2.2 (1981): 204-207.
if n == 0:
return ([], 0, [])
j, d = _select_jd_forests(n, q, cache_forests, seed)
t1, t1_nodes, r1 = _random_unlabeled_rooted_forest(
n - j * d, q, cache_trees, cache_forests, seed
t2, t2_nodes = _random_unlabeled_rooted_tree(d, cache_trees, seed)
for _ in range(j):
t1.extend((n1 + t1_nodes, n2 + t1_nodes) for n1, n2 in t2)
t1_nodes += t2_nodes
return t1, t1_nodes, r1
@nx._dispatchable(graphs=None, returns_graph=True)
def random_unlabeled_rooted_forest(n, *, q=None, number_of_forests=None, seed=None):
"""Returns a forest or list of forests selected at random.
Returns one or more (depending on `number_of_forests`)
unlabeled rooted forests with `n` nodes, and with no more than
`q` nodes per tree, drawn uniformly at random.
The "roots" graph attribute identifies the roots of the forest.
n : int
The number of nodes
q : int or None (default)
The maximum number of nodes per tree.
number_of_forests : int or None (default)
If not None, this number of forests is generated and returned.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
:class:`networkx.Graph` or list of :class:`networkx.Graph`
A single `networkx.Graph` (or a list thereof, if `number_of_forests`
is specified) with nodes in the set {0, …, *n* - 1}.
The "roots" graph attribute is a set containing the roots
of the trees in the forest.
This function implements the algorithm "Forest" of [1]_.
The algorithm needs to compute some counting functions
that are relatively expensive: in case several trees are needed,
it is advisable to use the `number_of_forests` optional argument
to reuse the counting functions.
If `n` is non-zero but `q` is zero.
.. [1] Wilf, Herbert S. "The uniform selection of free trees."
Journal of Algorithms 2.2 (1981): 204-207.
if q is None:
q = n
if q == 0 and n != 0:
raise ValueError("q must be a positive integer if n is positive.")
cache_trees = [0, 1] # initial cache of number of rooted trees
cache_forests = [1] # initial cache of number of rooted forests
if number_of_forests is None:
g, nodes, rs = _random_unlabeled_rooted_forest(
n, q, cache_trees, cache_forests, seed
return _to_nx(g, nodes, roots=set(rs))
res = []
for i in range(number_of_forests):
g, nodes, rs = _random_unlabeled_rooted_forest(
n, q, cache_trees, cache_forests, seed
res.append(_to_nx(g, nodes, roots=set(rs)))
return res
def _num_trees(n, cache_trees):
"""Returns the number of unlabeled trees with `n` nodes.
See also https://oeis.org/A000055.
n : int
The number of nodes.
cache_trees : list of ints
Cache for :func:`_num_rooted_trees`.
The number of unlabeled trees with `n` nodes.
r = _num_rooted_trees(n, cache_trees) - sum(
_num_rooted_trees(j, cache_trees) * _num_rooted_trees(n - j, cache_trees)
for j in range(1, n // 2 + 1)
if n % 2 == 0:
r += comb(_num_rooted_trees(n // 2, cache_trees) + 1, 2)
return r
def _bicenter(n, cache, seed):
"""Returns a bi-centroidal tree on `n` nodes drawn uniformly at random.
This function implements the algorithm Bicenter of [1]_.
n : int
The number of nodes (must be even).
cache : list of ints.
Cache for :func:`_num_rooted_trees`.
seed : random_state
See :ref:`Randomness<randomness>`
(edges, n)
The tree as a list of edges and number of nodes.
.. [1] Wilf, Herbert S. "The uniform selection of free trees."
Journal of Algorithms 2.2 (1981): 204-207.
t, t_nodes = _random_unlabeled_rooted_tree(n // 2, cache, seed)
if seed.randint(0, _num_rooted_trees(n // 2, cache)) == 0:
t2, t2_nodes = t, t_nodes
t2, t2_nodes = _random_unlabeled_rooted_tree(n // 2, cache, seed)
t.extend([(n1 + (n // 2), n2 + (n // 2)) for n1, n2 in t2])
t.append((0, n // 2))
return t, t_nodes + t2_nodes
def _random_unlabeled_tree(n, cache_trees, cache_forests, seed):
"""Returns a tree on `n` nodes drawn uniformly at random.
It implements the Wilf's algorithm "Free" of [1]_.
n : int
The number of nodes, greater than zero.
cache_trees : list of ints
Cache for :func:`_num_rooted_trees`.
cache_forests : list of ints
Cache for :func:`_num_rooted_forests`.
seed : random_state
Indicator of random number generation state.
See :ref:`Randomness<randomness>`
(edges, n)
The tree as a list of edges and number of nodes.
.. [1] Wilf, Herbert S. "The uniform selection of free trees."
Journal of Algorithms 2.2 (1981): 204-207.
if n % 2 == 1:
p = 0
p = comb(_num_rooted_trees(n // 2, cache_trees) + 1, 2)
if seed.randint(0, _num_trees(n, cache_trees) - 1) < p:
return _bicenter(n, cache_trees, seed)
f, n_f, r = _random_unlabeled_rooted_forest(
n - 1, (n - 1) // 2, cache_trees, cache_forests, seed
for i in r:
f.append((i, n_f))
return f, n_f + 1
@nx._dispatchable(graphs=None, returns_graph=True)
def random_unlabeled_tree(n, *, number_of_trees=None, seed=None):
"""Returns a tree or list of trees chosen randomly.
Returns one or more (depending on `number_of_trees`)
unlabeled trees with `n` nodes drawn uniformly at random.
n : int
The number of nodes
number_of_trees : int or None (default)
If not None, this number of trees is generated and returned.
seed : integer, random_state, or None (default)
Indicator of random number generation state.
See :ref:`Randomness<randomness>`.
:class:`networkx.Graph` or list of :class:`networkx.Graph`
A single `networkx.Graph` (or a list thereof, if
`number_of_trees` is specified) with nodes in the set {0, …, *n* - 1}.
If `n` is zero (because the null graph is not a tree).
This function generates an unlabeled tree uniformly at random using
Wilf's algorithm "Free" of [1]_. The algorithm needs to
compute some counting functions that are relatively expensive:
in case several trees are needed, it is advisable to use the
`number_of_trees` optional argument to reuse the counting
.. [1] Wilf, Herbert S. "The uniform selection of free trees."
Journal of Algorithms 2.2 (1981): 204-207.
if n == 0:
raise nx.NetworkXPointlessConcept("the null graph is not a tree")
cache_trees = [0, 1] # initial cache of number of rooted trees
cache_forests = [1] # initial cache of number of rooted forests
if number_of_trees is None:
return _to_nx(*_random_unlabeled_tree(n, cache_trees, cache_forests, seed))
return [
_to_nx(*_random_unlabeled_tree(n, cache_trees, cache_forests, seed))
for i in range(number_of_trees)
|
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|
athoms to Kens
Fathoms to Kens Converter
β Switch toKens to Fathoms Converter
How to use this Fathoms to Kens Converter π €
Follow these steps to convert given length from the units of Fathoms to the units of Kens.
1. Enter the input Fathoms value in the text field.
2. The calculator converts the given Fathoms into Kens in realtime β using the conversion formula, and displays under the Kens label. You do not need to click any button. If the input changes,
Kens value is re-calculated, just like that.
3. You may copy the resulting Kens 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 Fathoms to Kens?
The formula to convert given length from Fathoms to Kens is:
Length[(Kens)] = Length[(Fathoms)] / 1.158333333513394
Substitute the given value of length in fathoms, i.e., Length[(Fathoms)] in the above formula and simplify the right-hand side value. The resulting value is the length in kens, i.e., Length[(Kens)].
Calculation will be done after you enter a valid input.
Consider that a ship anchors in water that is 30 fathoms deep.
Convert this depth from fathoms to Kens.
The length in fathoms is:
Length[(Fathoms)] = 30
The formula to convert length from fathoms to kens is:
Length[(Kens)] = Length[(Fathoms)] / 1.158333333513394
Substitute given weight Length[(Fathoms)] = 30 in the above formula.
Length[(Kens)] = 30 / 1.158333333513394
Length[(Kens)] = 25.8993
Final Answer:
Therefore, 30 fath is equal to 25.8993 ken.
The length is 25.8993 ken, in kens.
Consider that a diver descends to a depth of 10 fathoms.
Convert this depth from fathoms to Kens.
The length in fathoms is:
Length[(Fathoms)] = 10
The formula to convert length from fathoms to kens is:
Length[(Kens)] = Length[(Fathoms)] / 1.158333333513394
Substitute given weight Length[(Fathoms)] = 10 in the above formula.
Length[(Kens)] = 10 / 1.158333333513394
Length[(Kens)] = 8.6331
Final Answer:
Therefore, 10 fath is equal to 8.6331 ken.
The length is 8.6331 ken, in kens.
Fathoms to Kens Conversion Table
The following table gives some of the most used conversions from Fathoms to Kens.
Fathoms (fath) Kens (ken)
0 fath 0 ken
1 fath 0.8633 ken
2 fath 1.7266 ken
3 fath 2.5899 ken
4 fath 3.4532 ken
5 fath 4.3165 ken
6 fath 5.1799 ken
7 fath 6.0432 ken
8 fath 6.9065 ken
9 fath 7.7698 ken
10 fath 8.6331 ken
20 fath 17.2662 ken
50 fath 43.1655 ken
100 fath 86.3309 ken
1000 fath 863.3094 ken
10000 fath 8633.0935 ken
100000 fath 86330.9352 ken
A fathom is a unit of length used primarily in maritime contexts to measure water depth. One fathom is equivalent to 6 feet or approximately 1.8288 meters.
The fathom is defined as 6 feet, making it a convenient measurement for nautical and maritime applications, particularly for depth soundings and underwater measurements.
Fathoms are commonly used in navigation, fishing, and marine activities to describe the depth of water. The unit provides a practical measurement for underwater distances and has historical
significance in maritime practices.
A ken is a historical unit of length used in various cultures, particularly in Asia. The length of a ken can vary depending on the region and context. In Japan, one ken is approximately equivalent to
6 feet or about 1.8288 meters.
The ken was traditionally used in architectural and construction measurements, particularly in the design of buildings and layout of spaces.
Ken measurements were utilized in historical architecture and construction practices in Asian cultures. Although not commonly used today, the unit provides historical context for traditional
measurement standards and practices in building and design.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Fathoms to Kens in Length?
The formula to convert Fathoms to Kens in Length is:
Fathoms / 1.158333333513394
2. Is this tool free or paid?
This Length conversion tool, which converts Fathoms to Kens, is completely free to use.
3. How do I convert Length from Fathoms to Kens?
To convert Length from Fathoms to Kens, you can use the following formula:
Fathoms / 1.158333333513394
For example, if you have a value in Fathoms, you substitute that value in place of Fathoms in the above formula, and solve the mathematical expression to get the equivalent value in Kens.
|
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|
4.3 - Binary Calculations - Eduqas GCSE (2020 Spec) | CSNewbs
top of page
4.3: Binary Calculations
Exam Board:
Eduqas / WJEC
2020 +
What are binary calculations?
Binary addition and binary subtraction are methods of adding or subtracting binary values without having to convert them into denary.
How to add binary numbers:
How to subtract binary numbers:
Overflow & Underflow Errors
Overflow and underflow errors occur when there is not enough space to accurately represent a binary number in the bits available.
What is an overflow error?
An overflow error occurs when a binary value is too large to be stored in the bits available.
In technical terms, an overflow error occurs if a carry (remainder) is present on the most significant bit (MSB).
The CPU then sets the overflow flag to true.
The most significant bit (MSB) is the largest bit (always the one furthest to the left) of a binary value (e.g. 128 for an 8 bit value).
A flag is an alert signal. It is either on or off. The overflow flag is turned on by the CPU when an overflow occurs.
What is an underflow error?
An underflow error occurs when a number is too small to be stored in the bits available. The value is too close to 0 to be accurately represented in binary.
Questo's Questions
4.3 - Binary Calculations:
1a. Describe the terms 'most significant bit' and 'flag'. [2]
1b. Using the terms from 1a, explain what an overflow error is. [2]
1c. Describe what is meant by an underflow error. [2]
2. Add together the following binary values. If an overflow error occurs you must state one has occurred.
• a. 010110012 and 010001012 [2]
• b. 110110112 and 010111012 [2]
• c. 001101102 and 011010112 [2]
• d. 110110112 and 010101112 [2]
• e. 011011012 and 110101102 [2]
3. Subtract the following binary values; put the first value on top of the second value:
• a. 100110102 and 000110002 [2]
• b. 110110112 and 010111012 [2]
• c. 011101102 and 011010112 [2]
• d. 110110112 and 010101112 [2]
• e. 111011012 and 110101102 [2]
bottom of page
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|
SAT Tutoring & GMAT Tutoring & GRE Tutoring - Best Math Tutor!
SAT Tutoring & GMAT Tutoring & GRE Tutoring
About Me:
SAT Tutoring
The “SAT” and “SAT Subject Test” are two tests widely used for university admissions offering undergraduate programs in the United States.
SAT Math tests general knowledge of Math and SAT Subject Test Math is designed to measure more specific knowledge in mathematics.
For SAT Math, using my sample questions, I teach my students the techniques and notes for solving the questions of the following topics of SAT:
Arithmetic, Algebra, Geometry, Data Analysis, Counting and Probability, Sequences, Functions, and Their Graphs, Trigonometry, and Complex Numbers.
SAT Subject Test Math Tutoring
In SAT Subject Test Math, we cover the math topics with more details and also new SAT related topics of math including:
Limits of Functions, Parametric Equations, Conic Sections, Polar Coordinates, Matrices, Sequences and Series, and Vectors. ———————————————————-
GMAT Tutoring
GMAT is the abbreviation of Graduate Management Admission. The Math part of GMAT Test is a test to assess certain analytical, and quantitative skills. The GMAT score is used in admission to graduate
management programs, such as MBA.
I have 6 packages for GMAT Tutoring grouped into different topics of GMAT including:
Arithmetic and Number Theory, Venn Diagram and Sets, Ratio, Fractions and Percent, Statistics and Probability, Permutation and Combination, Algebra, 2-D Coordinate System, Geometry, and Data
Sample GMAT Tutoring Questions
Copyright Notice: Individuals may make one copy of the posted pdf files for personal study. No file may be reproduced in any form.
|
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|
Enroll in the Best Math Programme During the March Break
2024 MARCH HOLIDAY PROGRAMME
Unlock your child’s potential this March holiday with Matrix Math’s exclusive holiday programme! Register now to strengthen their math foundations and tackle heuristic problems with ease. Our course
zeroes in on essential concepts and key problem-solving strategies needed for academic success. Join us to simplify complex math and equip your child with confidence and skills to succeed.
Register now and give your child the gift of confidence in math!
Primary Level Lesson Plans
$250 for Matrix Math Students
$260 for non-Matrix Math Students
Number Of Lessons:
3 Lessons x 2 Hours
Math heuristics students should master before the next academic term
LESSON 1
• Comparison of quantity
• Comparison of quantity and units
• Stacking model
• Supposition
LESSON 2
• Interval
• Grouping
• Total concept, regrouping
• Estimation and approximation
LESSON 3
• External transfer
• Internal transfer
• Factors and multiples
• Shortage and surplus
$250 for Matrix Math Students
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Number Of Lessons:
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Math heuristics students should master before transitioning to the next academic term
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• Whole numbers – supposition
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• Fractions as values and units
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• Average
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LESSON 2
• Quantity and value
• Quantity transfer
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• Percentage change
• Percentage as a proportion
• Percentage discount and gst
• Area and perimeter
• Geometry
• Grouping
Secondary Level Lesson Plans
$140 for Matrix Math Students
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Number Of Lessons:
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Skills students should master before transitioning to the next academic term
• Factors and multiples
• Basic algebra and algebraic manipulation
• Simple equations in one variable
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Number Of Lessons:
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Skills students should master before transitioning to the next academic term
• Linear inequalities
• Simultaneous linear equations
• Expansion and factorisation of algebraic expressions
• Algebraic fractions and formulae
$150 for Matrix Math Students
$160 for Non-Matrix Math Students
Number Of Lessons:
1 Lesson X 3 Hours
Skills students should master before transitioning to the next academic term
• Quadratic equations and functions
• Indices
• Inequalities
$150 for Matrix Math Students
$160 for Non-Matrix Math Students
Number Of Lessons:
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Skills students should master before transitioning to the next academic term
• Simultaneous equations
• Quadratic inequalities
• Discriminant
• Quadratic curves
• Surds
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Number Of Lessons:
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Skills students should master before transitioning to the next academic term
• Properties of circles
• Radian measure
• Statistics and probability
$150 for Matrix Math Students
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Number Of Lessons:
1 Lesson X 3 Hours
Skills students should master before transitioning to the next academic term
• Differentiation
• Tangents and normals
• Increasing and decreasing functions
• Rates of change
• Stationary points, maxima and minima
|
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|
How To Calculate Total Interest Paid On A Loan In Excel | SpreadCheaters
How to calculate total interest paid on a loan in Excel
You can watch a video tutorial here.
Excel is frequently used for calculations and supports all basic mathematical operations. Using formulas, you can use Excel to build different types of calculators in which you only need to change
the parameters to get the result. This saves you the problem of doing each step of the calculation repeatedly. For example, you can build a calculator to calculate the total interest paid on a loan
based on the loan amount, interest rate, and period of payment. In Excel, there are 2 ways of doing this.
1. CUMIPMT() function: this returns the cumulative interest between a start and end period
1. Syntax: CUMIPMT(rate, nper, pv, start_period, end_period,[type])
1. rate: the interest rate
2. nper: the total number of payments
3. pv: the principal
4. start_period: the number of the first payment (number of payments)
5. end_period: the number of the last payment
6. [type] optional: specify whether the payment is due at the beginning or end of the loan
2. Use the formula:
1. Total Interest paid = Total amount paid – Principal amount
1. Total amount paid = Monthly payment * Total number of payments
1. Monthly payment is computed using the PMT() function
2. PMT() function: this calculates the monthly payment for a loan
1. Syntax: PMT(rate, nper, pv, [fv], [type])
1. rate: the interest rate
2. nper: the total number of payments
3. pv: the principal
4. [fv] optional: remaining cash balance after the last payment is made, by default this is zero
5. [type] optional: specify whether the payment is due at the beginning or end of the loan
3. The total number of payments is computed using the Period of the loan (years) and the Payment frequency e.g. a 5-year loan with monthly payments will have 60 payments (5*12)
Option 1 – Use the CUMIPMT() function
Step 1 – Find the monthly interest rate
• The payment frequency is ‘Monthly’ so the monthly interest rate is to be computed
• Select the cell for the monthly interest to be displayed
• Type the formula using cell references:
= Interest rate (annual)/12
Step 2 – Find the total number of payments
• Select the cell for the total number of payments to be displayed
• Type the formula using cell references:
= Period of the loan (years) * 12
Because the payment frequency is monthly, the number is multiplied by 12
Step 3 – Create the formula using CUMIPMT()
• Select the cell for the total interest to be displayed
• Type the formula using cell references:
= -CUMIPMT(Monthly interest, Total number of payments, Loan amount,1,120)
Note: There are 120 total payments so the start_period is 1 and the end_period is 120. The function returns a negative number so a minus sign is added at the beginning to make the result positive.
Option 2 – Use the formula
Step 1 – Find the monthly interest rate
• The payment frequency is ‘Monthly’ so the monthly interest rate is to be computed
• Select the cell for the monthly interest to be displayed
• Type the formula using cell references:
= Interest rate (annual)/12
Step 2 – Find the total number of payments
• Select the cell for the total number of payments to be displayed
• Type the formula using cell references:
= Period of the loan (years) * 12
Because the payment frequency is monthly, the number is multiplied by 12
Step 3 – Compute the monthly payment amount
• Select the cell for the monthly payment to be displayed
• Type the formula using cell references:
= PMT(Monthly interest, Total number of payments, – Loan amount)
Note: The PMT() function returns a negative number, so a minus sign is put in front of the ‘Loan amount’ to make the result a positive number.
Step 4 – Calculate the total interest
• Select the cell where the result is to be displayed
• Type the formula using cell references:
= (Monthly payment * Total number of payments) – Loan amount
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Bernoulli Distribution Fitting
Bernoulli Distribution Fitting
In the realm of probability theory and statistics, the Bernoulli distribution stands as a foundational concept, named in honor of the Swiss mathematician Jacob Bernoulli. This distribution
characterizes the discrete probability of a random variable, which assumes the value of 1 with a likelihood denoted as p, and the value of 0 with the complementary probability, q = 1 - p.
To understand it more intuitively, envision a scenario where a single experiment poses a binary question, eliciting a yes or no response. The Bernoulli distribution provides a framework for modeling
such outcomes. These yes-no inquiries yield results that can be represented by a Boolean variable: a single bit of information that holds a value of success, yes, true, or one with a probability of
p, and failure, no, false, or zero with a probability of q.
One common illustration of this distribution lies in its application to coin tosses. Imagine tossing a coin?biased or unbiased?where the outcomes are conventionally labeled as "heads" and "tails."
Here, the Bernoulli distribution offers a means to quantify the probabilities associated with these outcomes. If p represents the probability of the coin landing on heads, then 1 would signify a
head, while 0 would signify a tail. In cases of biased coins, where the probabilities of heads and tails are unequal, p deviates from the standard 1/2.
In summary, the Bernoulli distribution serves as a fundamental tool in modeling binary events, encapsulating the essence of probability in scenarios where outcomes are dichotomous. Whether applied to
coin flips or other yes-no inquiries, its simplicity and versatility render it indispensable in various statistical analyses and decision-making contexts.
The binomial distribution is the discrete probability distribution of the number of successes in a sequence of >n independent yes/no experiments, each of which yields success with probability p. Such
a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial; when n = 1, the binomial distribution is a Bernoulli distribution.
• p is the success probability for each trial
• q is the failure probability for each trial
• f(k,n,p) is probability of k successes in n trials when the success probability is p
How To Cite
We used Accord.Statistics for this calculator
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What is zkVerify | zkVerify Documentation
What is zkVerify
zkVerify Defined
zkVerify is a blockchain designed to provide zero knowledge proof verifications for any project or dApp using zero knowledge proofs. A high performance, public, decentralized, reliable, and secure
blockchain, zkVerify has dedicated proof verification methods written in Rust, and available to be used in a modular and composable way.
Goals of zkVerify
zkVerify is focused on the proof verification aspect of zero knowledge proofs. zk-proofs are used to verify computations without disclosing the underlying data, and can be used to summarize
transactions, for selective disclosure of identity information, and other specific applications like secret ballot voting and hidden cards in games.
There are two essential elements of a complete zero knowledge proof. One is the computation which creates the proof, the other is the verification of the proof. Both are needed to verify the
computation. Many organizations are working diligently to create better, faster, and more compact proofs, and each of these new zk-proof computations needs proof verifiers available in a reliable and
accessible place.
The zkVerify blockchain will accept proofs, verify them, then store both the proof and the verification in the blockchain for future availability
Prohibitive Costs
From a macro cost perspective, the proof verification market is estimated to incur $100+ million in security expenses alone for zkRollups in 2024, extending to $1.5 billion by 2028 when including ZK
On a more granular level, the verification of a single ZK proof on Ethereum can consume upwards of 200,000 to 300,000 gas units, depending on the proof type. Beyond nominal fees today, the
variability of future fees inhibits product adoption. Offloading proof verification from L1s, such as Ethereum, serves to both drastically lower nominal costs, but also to stabilize costs over time
in a way that segregates fees from gas volatility.
For instance, in times of network congestion, gas prices have reached over 100 Gwei, which means that verifying a single proof could cost between $20 to $60 or even more.
Hampering Innovation
Ethereum Improvement Proposals (EIPs) are design documents that outline new features, standards, or processes for the Ethereum blockchain, serving as a primary mechanism for proposing changes to the
network. Two EIPs (EIP-196 and EIP-197) significantly impacted the development of rollups and zkVMs (zero-knowledge virtual machines) by providing essential cryptographic building blocks to perform
ZK proof verification on the Ethereum blockchain.
The choice to standardize around the BN254 curve, while practical at the time of implementation, means that operations involving other elliptic curves are not directly supported and are prohibitively
expensive to execute. This lack of support restricts the variety of cryptographic techniques that can be efficiently employed on the platform, limiting innovation as cryptographic standards evolve.
In general, progressing EIPs forward can be challenging due to the rigorous process involved, its need for widespread consensus via the DAO, and the lack of effective prioritization amongst multiple
competing priorities with different stakeholders.
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Cube Roots | Brilliant Math & Science Wiki (2024)
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Andy Hayes, A Former Brilliant Member, Eli Ross, and
The cube root of a number \(a\) is the answer to the question, "What number, when cubed \((\)raised to the 3\(^\text{rd}\) power\()\), results in \(a?\)" The symbol for cube root is "\(\sqrt[3]{\ }\)
". The cube root of the number \(a\) is written as \(\sqrt[3]{a}\).
What is the cube root of \(64\)?
Ask yourself the question, "What number, when cubed, results in \(64\)?" The answer to that question will be the cube root of \(64\).
\(4^3=64\), so \(\sqrt[3]{64}=4\). \(_\square\)
The cube root is often used to solve cubic equations. In particular, it can be used to solve for the dimensions of a three-dimensional object of a certain volume.
• Definition and Notation
• Basic Calculations
• Cube Roots of Negative Numbers
• Simplifying Cube Roots
• Cube Roots of Complex Numbers
The cube root of a number \(a\), denoted as \(\sqrt[3]{a},\) is the number \(b\) such that
The cube root symbol acts similarly to the square root symbol. It is often called a radical, and the number or expression underneath the top line of the symbol is called the radicand. The cube root
symbol is a grouping symbol, meaning that all operations in the radicand are grouped as if they were in parentheses.
Unlike a square root, the result of a cube root can be any real number: positive, negative, or zero. Also different from a square root is the domain restriction on the radicand: the radicand of a
cube root can be negative while still achieving a real result for the cube root.
To solve for the cube root of any integer, first ask yourself the question, "What integer, when cubed, results in this number?" If none comes to mind, list perfect cubes until a match for the
radicand is found.
What is the value of \(\sqrt[3]{216}\)?
Think of perfect cubes until you find a match for the radicand.
\(6^3=216\), so \(\sqrt[3]{216}=6\). \(_\square\)
The process is similar for the cube roots of fractions. Look for perfect cubes that match the numerator and denominator of the fraction.
What is the value of \(\sqrt[3]{\dfrac{27}{125}}.\)
\(3^3=27\), and \(5^3=125\).
It follows that \(\sqrt[3]{\dfrac{27}{125}}=\dfrac{3}{5}.\) \(_\square\)
Cube Roots of Negative Numbers
Unlike the square root, the cube root has no domain restriction under the real numbers. The radicand can be any real number, and the result of the cube root will be a real number.
What is the value of \(\sqrt[3]{-8}\)?
Similarly to previous examples, the cube root of \(-8\) is the answer to the question, "What number, when cubed, results in \(-8\)?"
\((-2)^3=-8\), so it follows that \(\sqrt[3]{-8}=-2.\ _\square\)
In general, if a cube root operation is done on a negative number, then the result is negative.
Let \(a\) be a real number. Then,
The process to simplify cube roots of non-perfect cubes is like the process to simplify square roots.
Let \(a\) be a non-perfect cube integer.
The simplified radical form of the cube root of \(a\) is
In this form, \(\sqrt[3]{a}=b\sqrt[3]{c}\), \(b\) and \(c\) are integers, and \(c\) is positive with no perfect cube factors other than \(1\).
To simplify a cube root, first look for the largest perfect cube factor of the radicand. Then, apply the following property:
Let \(a\) and \(b\) be real numbers. Then,
\[\sqrt[3]{ab}=\sqrt[3]{a}\times \sqrt[3]{b}.\]
Simplify \(\sqrt[3]{81}.\)
The goal is to find the largest perfect cube factor of \(81\). Since \(27\) is that factor, we have
\[\sqrt[3]{81}&=\sqrt[3]{27\times 3} \\&=\sqrt[3]{27}\times\sqrt[3]{3} \\&=3\sqrt[3]{3}.\ _\square\]
Note: When a number is placed to the left of a cube root symbol, multiplication is implied. Therefore, "\(3\sqrt[3]{3}\)" is read as "\(3\) times the cube root of \(3\)."
Cube Roots of Complex Numbers
The cube root of a complex number is somewhat ambiguous. Non-real complex numbers are neither positive nor negative, so it is not well-defined which cube root is the principal root. Therefore, when a
cube root operation is done on a complex number, the result is interpreted to be all solutions of an equation:
Let \(z\) be a complex number. Then there are up to three values for \(\sqrt[3]{z}\), and they are equal to the solutions of the equation
Note that the cube root operation, when used on complex numbers, is not well-defined in the sense that there is likely more than one result.
The process for finding the cube roots of a complex number is similar to the process for finding the \(3^\text{rd}\) roots of unity.
What are the values of \(\sqrt[3]{i}\)?
First, it is necessary to write the complex number in polar form. For \(i\), \(r=|i|=1\) and \(\theta=\text{arccot}\left(\frac{0}{1}\right)=\frac{\pi}{2}+2k\pi\), where \(k\) is an integer.
\[\begin{array}{c}i & = & e^{i\pi/2} & = & e^{i5\pi/2} & = & e^{i9\pi/2} \\\sqrt[3]{i} & = & i^{1/3} \\\\\sqrt[3]{i} & = & \left(e^{i\pi/2}\right)^{1/3} & = & e^{i\pi/6} & = & \frac{\sqrt{3}}
{2}+\frac{i}{2} \\\sqrt[3]{i} & = & \left(e^{i5\pi/2}\right)^{1/3} & = & e^{i5\pi/6} & = & -\frac{\sqrt{3}}{2}+\frac{i}{2} \\\sqrt[3]{i} & = & \left(e^{i9\pi/2}\right)^{1/3} & = & e^{i3\pi/2}
& = & -i.\end{array}\]
There are three possible results for \(\sqrt[3]{i}\): \(\dfrac{\sqrt{3}}{2}+\dfrac{i}{2}\), \(-\dfrac{\sqrt{3}}{2}+\dfrac{i}{2}\), and \(-i\). \(_\square\)
Cite as: Cube Roots. Brilliant.org. Retrieved from https://brilliant.org/wiki/cube-root/
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To preparation JEE Mains Mathematics in Prayagraj Join Gurukul Classes Mathematics is the most Important subject for JEE entrance. Chemistry, Physics and Mathematics have equal weight in the JEE but
Mathematics has a scoring over the other two as in case of a tie in the overall score of two or more candidates, the score […]
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Silvia Heubach
College of Natural & Social Sciences
Department of Mathematics
Office ST201
I am originally from Germany and came for a one-year student exchange. Life intervened and I decided to stay and do a Ph.D. at USC (University of Southern California in Los Angeles). I graduated in
1992 and then had two one-year visiting appointments, at Colorado College in Colorado Springs and at Humboldt State University in Arcata. I came to Cal State LA in the Fall of 1994 and have been here
ever since. In my free time I like to read, watch foreign and independent movies, hike, camp, dance and stand on my head on an almost daily basis.
I teach courses ranging from general education to graduate level courses, mostly statistics, probability, modeling and the Mathematica course. I use Mathematica in most of my courses to visualize
concepts and algorithms. My most recent focus has been the redesign and teaching of the new mathematics sequence for life sciences majors, Math 1050/1085, Math 2040/41, and Math 2050/51. These
courses were developed with an NIH grant that had the goal to strengthen the quantitative skills of life sciences majors. In these courses, mathematics is taught in the context of life science and
biological applications, and the course topics are those that life science makors will see in the applicaitons in their major courses. Another aspect of the grant was the introducation of a minor in
bioinformatics. I have contributed two chapters on probability and its applications in bioinformatics to a textbook used at Cal State LA, Concepts in Bioinformatics and Genomics, authored by Cal
State LA faculty Drs. Momand and McCurdy, with additional contirbutions by Dr. Warter-Perez.
• I was the faculty learning community coordinator on the First In The World grant on flipped learning. This grant provides funding for faculty from STEM disciplines to try out flipped teaching in
bottle-neck courses. The grant was active from 2014 - 2019.
• I was one of the leads for the course redesign of General Education courses in response to EO 1110. Information about the changes can be found at the math department advisement webpage.
• Recipient of 2018 CSU Chancellor's Office Faculty Innovation and Leadership Award.
• PI, Co-Director and Project Manager on NIH MARC curricular grant (July 2008-June 2014, $1.57 Million)
• NSF travel grant to attend FPSAC conference in Tianjin, China, July 2007, $1150
• Invited Researcher, University of Haifa, Haifa, Israel, supported by the Department of Mathematics and the Caesarea Rothschild Institute, $1500 travel grant + housing, March 2007
• Invited Researcher, University of Haifa, Haifa, Israel, supported by the Department of Mathematics and the Caesarea Rothschild Institute, $1900 travel grant + housing, September 2005
• Invited Researcher, University of Haifa, Haifa, Israel, supported by the Department of Mathematics and the Caesarea Rothschild Institute, $1500 travel grant + housing, May 2005
• 2003/2004 AWM Travel Grant, $1100
• 1999/2000 CSLA Outstanding Professor Award
• 1999/2000 AWM Travel Grant, $500
• 1997-98 Innovative Teaching Award, An Introductory Course in Mathematica, Released Time, Student Assistant and Services $4,600
• NSF-Course and Curriculum Development Grant, An Innovative Approach at the Freshman/Sophomore Level, $121,366 (3/97 - 11/01);
• 1995-96 Proposal Development and Grant-in-Aid Award, Seed Project to develop an NSF Proposal for the Creation of a New Interdisciplinary Modeling Course, Released Time (4 units)
• 1995-96 Discretionary Lottery Funds, Technology Oriented Curriculum for Differential Equation Course, $1000 for purchase of software
• 1995-96 Innovative Instruction Awards, Technology Oriented Curriculum for Differential Equation Course, Released time (4 units), Mini grant $594
• 1995-96 Innovative Instruction Awards, $5000 for purchase of graphing calculat
My research spans a number of areas, ranging from operations research to modeling to combinatorics. My Masters thesis investigated a stochastic model for inventory control (Operations Research),
while in my Ph.D. thesis, I developed a stochastic model for the movement of a white blood cell. With Raj Pamula, I have worked on a model of a system of parallel processes to compare different
methods of error recovery. Work on (enumerative) combinatorics, in particular questions related to tilings and compositions with a number of collaborators has culminated in a book entitled
Combinatorics of Compositions and Words, which is a resource for research in this area. Most recently, I am working on questions related to combinatorial games, an area that lends itself to student
research. If you want to find out more about Combinatorial Game Theory, check out Kyle Burke's blog or the website on combinatorial games.
Listen to my interview with Bonnie Stachowiak of the Teaching in Higher Ed podcast.
I had the pleasure of writing joint research articles with the following individuals (roughly in reverse chronological order): Richard Nowakowski, Craig Tennenhouse, Valdimir Gurvich, Nhan Bao Ho,
Nikolay Chikin, Sharona Krinsky, Kyle Burke, Melissa Huggan, Svenja Huntemann, Urban Larsson, Eric Duchène, Matthieu Dufour, Arnold Knopfmacher, Michael Mays, Sergey Kitaev, Augustine Munagi, N.Y.
Li, Toufik Mansour, Patrick Callahan, Phyllis Chinn, Ralph Grimaldi, Raj Pamula, and Joseph Watkins.
We are looking for integers n whose squares have an average of their digits that is 8.25 or bigger. If you find any, we will post the new record here. To find more about this challenge, read the
article in CRUX Mathematicorum Vol 45(8), October 2020, a special edition in honor of Richard Guy, one of the founders of combinatorial Game theory.
Current Record for Square with Largest Digit Average, DAve
│ Name & Affiliation │ n │ n^2 │ DA(n) │ Date │
│Matthieu Dufour & Silvia Heubach│707,106,074,079,263,583 │499,998,999,999,788,997,978,888,999,589,997,889 │8.25 │6/23/20 │
│Tomas Rokicki │943,345,110,232,670,883 │889,899,996,999,889,979,488,999,999,795,999,689 │8.25 │1/30/23 │
│Matthieu Dufour & Silvia Heubach│893,241,282,627,485,818,275,387 │797,879,988,989,995,997,899,989,877,988,999,997,998,969,999,769 │8.25 │10/10/20│
│Tomas Rokicki │989,898,919,026,382,208,216,803,790,139,943,167│979,899,869,889,599,999,789,989,999,899,659,988,889,797,488,898,989,989,999,989,949,989,989,889│8.25 │4/20/23 │
│ │ │ │487/59=│ │
│Tomas Rokicki │296,630,559,600,488,563,517,139,286,187 │87,989,688,888,898,997,898,978,579,999,898,999,999,987,799,879,999,888,998,969 │ │4/17/23 │
│ │ │ │8.25424│ │
│ │ │ │487/59=│ │
│Tristrom Cooke, Adelaide │314,610,537,013,606,681,884,298,837,387 │98,979,789,999,989,979,988,999,999,989,499,999,797,975,998,897,999,868,987,769 │ │4/11/23 │
│ │ │ │8.25424│ │
│ │ │ │388/45=│ │
│Tomas Rokicki │312,713,447,088,224,669,275,583 │97,789,699,989,799,889,886,988,697,997,987,998,989,999,989,889 │ │1/31/23 │
│ │ │ │8.25532│ │
│ │ │ │355/43=│ │
│Tomas Rokicki │2,976,388,751,488,907,738,914 │8,858,889,999,989,698,989,999,979,889,997,999,989,899,396 │ │1/31/23 │
│ │ │ │8.25581│ │
│ │ │ │513/62=│ │
│Tomas Rokicki │9984,988,582,817,657,883,693,383,344,833 │99,699,996,998,998,979,989,989,997,788,978,889,798,779,999,999,969,798,987,797,889 │ │4/13/23 │
│ │ │ │8.27419│ │
│Matthieu Dufour & Silvia Heubach│94,180,040,294,109,027,313 │8,869,879,989,799,999,999,898,984,986,998,979,999,969 │8.275 │7/25/20 │
│ │ │ │556/67=│ │
│Tomas Rokicki │2,982,951,558,104,653,129,433,595,167,102,567 │8,897,999,997,998,977,794,997,988,999,999,989,979,999,899,769,898,988,999,997,897,989,489 │ │4/21/23 │
│ │ │ │8.29851│ │
│ │ │ │108/13=│ │
│Tomas Rokicki │264,575,104,459,943,243,164,050,883,010,583 │69,999,985,899,989,878,999,999,889,799,898,899,975,978,988,897,889,989,989,689,999,889 │ │4/14/23 │
│ │ │ │8.30769│ │
• M. Dufour, S. Heubach, and A. Vo, Circular Nim games CN(7, 4). In R. Nowakowski, B. Landman, F. Luca, M. Nathanson, J. Nešetřil & A. Robertson (Ed.), Combinatorial Game Theory: A Special
Collection in Honor of Elwyn Berlekamp, John H. Conway and Richard K. Guy, (2022), pp. 139-156. Berlin, Boston: De Gruyter. https://doi.org/10.1515/9783110755411-009
• M. Dufour, S. Heubach, and A. Vo, Circular Nim games CN(7, 4), INTEGERS, Vol 21B (2021): To the Three Forefathers of Combinatorial Game Theory: The John Conway, Richard Guy, and Elwyn Berlekamp
Memorial Volume, article #A9
• M. Dufour and S. Heubach, Squares with Large Digit Average, Crux Mathematicorum, Vol. 46, No. 8 (2020), pages 384 - 389
• S. Heubach, M. A. Huggan, R.J. Nowakowski, and C. Tennenhouse, Cyclic Subtraction Set Games, Crux Mathematicorum, Vol. 46, No. 8 (2020), pages 413 - 414
• V. Gurvich, S. Heubach, N.B. Ho, N. Chikin, Slow K-Nim, INTEGERS, Vol 20 (2020), article #G3
• K. Burke, S. Heubach, M. Huggan and S. Hunteman, Keeping your Distance is Hard, to appear in Games with No Chance 6. Preprint at arXiv:1605.06801
• M. Dufour, S. Heubach and U. Larsson, (2017) A Misère-Play *-Operator. In: Nathansson M. (Eds) Combinatorial and Additive Numebr Theorey II. CANT 2015, CANT 2016. Springer Proceedings in
Mathematics & Statistics , Vol 220. Springer, Cham. DOI: https://doi.org/10.1007/978-3-319-68032-3_12
• E. Duchene, M. Dufour, S. Heubach and U. Larsson, Building Nim, International Journal of Game Theory, 2016, vol. 45, issue 4, pages 859-873. DOI 10.1007/s00182-015-0489-3
• M. Dufour and S. Heubach, Circular Nim Games, Electronic Journal of Combinatorics, 20:2, (2013) P22 (26 pages
• S. Heubach, Comparison of Recovery Schemes through a Mathematica Simulation, Mathematica in Education and Research, Vol 8, No 3-4, pp.28-36, 1999
• S. Heubach and R.S. Pamula, Implementing an Approximate Probabilistic Algorithm for Error Recovery in Concurrent Processing Systems, AoM/IAoM 1999 Proceedings: Computer Science, Vol 17, No 1, pp.
50 - 55.
• S. Heubach and R. Pamula, Modeling and Simulation of Error Recovery in a Concurrent Processing System, Proceedings of the 2nd IASTED International Conference: European Parallel and Distributed
Systems (Euro-PDS '98), IASTED/ACTA Press, pp 29 - 35, 1998
• S. Heubach and J. Watkins, A Stochastic Model for the Movement of a White Blood Cell, Advances in Applied Probability 27, pp. 443-475, 1995
• Thesis: Lagerhaltung unter Unsicherheit, University of Ulm, Germany, 1986
• S. Heubach and T. Phan-Yamada, Red Light Reaction – A Statistics Project for Real Life Application, to appear in Journal of Statistics and Data Science Education. Preproduction access
• S. Heubach and S. Krinsky, Implementing Mastery-Based Grading at Scale in Introductory Statistics, PRIMUS, January 2020, DOI 10.1080/10511970.2019.1700576
• S. Heubach and E. Torres. A New Mathematics Course Sequence for Life Sciences Majors: A progress report, Proceedings of the Sixth Annual International Symposium on Biomathematics and Ecology:
Education and Research. Web. 23 April 2014, (13 pages)
• Silvia Heubach and Elizabeth Torres. Improving Quantitative Skills of CSULA Life Science Majors. Vision and Change in Undergraduate Biology Education – A view for the 21^st Century.
• S. Heubach, Using The TI-89 To Convey Mathematical Concepts: An Introductory Modeling Course For Non-Science Majors, Proceedings of the 14th International Conference on Technology in Collegiate
Mathematics (ICTCM), Addison Wesley, 2003, pp. 107-111.
• S. Heubach, Using Mathematica To Convey Mathematical Concepts: An Introductory Modeling Course For Non-Science Majors, Proceedings of the 12th International Conference on Technology in Collegiate
Mathematics (ICTCM), Addison Wesley, 2000, pp.160-165.
• S. Heubach, An Innovative Approach to Modeling at the Freshman/Sophomore Level, Proceedings of the 11th International Conference on Technology in Collegiate Mathematics, Addison Wesley, pp. 166 -
170, 1999
• S. Heubach, An Innovative Modeling Approach at the Freshman/Sophomore Level, Proceedings of the 3rd Asian Technology Conference in Mathematics, Springer Verlag, 1998
• CC. Edwards, S. Heubach, V. Howe, and G. Klatt, Floppy Grids: Discovering the Mathematics of Grid Bracing, to appear as a COMAP module.
• S. Heubach, Introducing Laboratories into a Differential Equations Course - How to get started!, Proceedings of the 9th Annual International Conference on Technology in Collegiate Mathematics,
Addison Wesley, pp. 232 - 236, 1997
Expect additional links to some of these presentations soon.
• How to Win in Circular Nim, Mathematics Colloquium West Chester University, September 2024
• How to Win in Slow Exact k-Nim, Combinatorial Game Theory Colloquium IV, Azores, Ponta Delgada, Portugal, January 2023
• Play to Win - How to win in Combinatorial Games, Cal State LA Honor's Colloquium, November 20, 2019
• Combinatorial Games, Lightning Fast Talk at the Natural and Social Sciences College Retreat, Cal State LA, May 25, 2017. The video of the talk starts at time point 42:09.
• New Results on Circular Nim, 48th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 6-10, 2017
• The Game Creation Operator (for general audiences; expanded), Cal State LA Math Club, March 2, 2017
• Keeping your Distance is Hard, Recreational Mathematics Colloquium V and Gathering for Gardener (Europe), January 28 - 31, 2017, Lisbon, Portugal.
• The Game Creation Operator, Combinatorial Game Theory Colloquium II, January 25 - 27, 2017, Lisbon, Portugal.
• The Game Creation Operator (for general audience), invited talk at the SoCal-Nevada MAA Section meeting, October 22, 2016
• The Misère Star Operator, 47th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 7-11, 2016
• Keeping your Distance is Hard, 47th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 7-11, 2016
• The Misère Star Operator, Mathematics Colloquium, Dalhousie University, Halifax, NS, March 4, 2016
• Keeping your Distance is Hard, Discrete Mathematics and Computer Science Seminar, University of Quebec at Montreal, QC, February 19, 2016
• Building Nim, 46th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 2-6, 2015
• Nim on a Tetrahedron, Recreational Mathematics Meeting, Weizman Institute, Rehovot, Israel, June 20, 2014
• Building Nim, Second Joint Meeting of the Israel Mathematical Union and the AMS, Tel Aviv, Israel, June 16 - 19, 2014
• Nim on a Tetrahedron, 45th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 3-7, 2014
• Building Nim, Integers Conference 2013, Carrolton, GA, October 24-27, 2013
• Building Nim, Women in Mathematics Symposium, UCSD, April 21, 2013
• A generalization of Nim, 44th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 4-8, 2013
• Nim, Wythoff and Beyond - Let's Play, Math Club CSU Los Angeles, February 27, 2013
• A Generalization of Nim and Wythoff games, SIAM DM 12, Halifax, Nova Scotia, June 18-22, 2012
• Circular Nim Games, Combinatorics Seminar, UQAM, Montreal, May 24, 2011
• Nim, Wythoff and Beyond - Let's Play , Mathematics Colloquium CSU Long Beach, April 29, 2011
• A Generalization of the Nim and Wythoff games, 42nd Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 7-11, 2011
• Circular (n,k) games, Graduate Seminar, CSU Channel Island, September 8, 2010
• Circular (n,k) games, Math Colloquium, Cal Poly San Luis Obispo, May 7, 2010
• Circular (n,k) games, 41st Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 8-12 2010
• Circular (n,k) games, MAA Mathfest, Portland, OR, August 6-8, 2009
• Analyzing ELLIE - the story of a combinatorial game, San Jose State University, San Jose, CA, May 13, 2009
• Circular (n,k) games, 40th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 2-6, 2009
• Analyzing ELLIE - the story of a combinatorial game, Humboldt State University, Arcata, CA, October 23, 2007
Expect additional links to some of these presentations soon.
• Inversions in compositions of integers, Permutation Patterns 2011, San Luis Obispo, June 20-24, 2011
• Avoiding Substrings in Compositions, 41st Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 8-12, 2010
• Pattern avoidance of type (2,1) multi-permutation patterns in compositions, 40th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 2-6,
• Avoidance of partially ordered patterns in compositions, Dalhousie University, Halifax, NS, August 27, 2007
• Avoidance of partially ordered patterns in compositions, International Conference on Graph Theory and Combinatorics & Fourth Cross-strait Conference on Graph Theory and Combinatorics, National
Taiwan University, Taipei, June 24-29, 2007
• Avoidance of partially ordered patterns in compositions, 38th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 5-9, 2007
• Enumeration of 3-Letter Patterns in Compositions, Integers Conference 2005, University of West Georgia, Carrollton, GA, October 27 – 30, 2005
• Compositions and Multisets Restricted by Patterns of Length 3, Workshop on Permutation Patterns, University of Haifa, Israel, May 29 - June 3, 2005 (invited speaker)
• Tiling with Ls and Squares, 36th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 7-11, 2005
• Counting Rises, Levels and Drops in Compositions, 35th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 8-12, 2004
• Binary Strings Without Odd Runs of Zeros, 34th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 3-7, 2003
• Counting Compositions: Patterns and Combinatorial Proofs, CSU Dominguez Hills, Carson, CA, April 24, 2002
• Counting Compositions with 1s and ks, 33rd Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, March 4-8, 2002
• The Frequency of Summands of Size k in Palindromic Compositions, Fall Meeting of the Southern California Section of the MAA, Los Angeles, CA, October 13, 2001
• Rises, Levels, Drops and "+" Signs in Compositions, 32nd Southeastern International Conference on Combinatorics, Graph Theory and Computing, Baton Rouge, LA, February 26 - March 1, 2001
• Exact and Asymptotic Results for the Number of Tilings of Rectangles with Squares, Joint Meeting of the AMS and MAA, New Orleans, January 10-13, 2001
• Exact and Asymptotic Results for the Number of Tilings of an m-by-n Board with Squares, Mathematical Colloquium, University of Ulm, Germany, November 14, 2000
• Tiling Rectangles with Squares, Mathematical Colloquium, Humboldt State University, Arcata, CA, October 19, 2000
• Patterns Arising From Tiling Rectangles With Squares, 10^th SIAM Conference on Discrete Mathematics, Minneapolis, Minnesota, June 12-15, 2000
• Patterns Arising From Tiling Rectangles With Squares, 31st Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, Florida, March 13-17, 2000
• Tiling an m-by-n Area with Square of Site up to k-by-k (m, 30th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, Florida, March 8-12, 1999
• How many ways are there to tile an n-by-m rectangle using 1-by-1 and 2-by-2 tiles?, MAA (Southern and Northern) Section Meeting, Cal Poly, San Luis Obispo, CA, October 20 - 22, 1995
Expect additional links to some of these presentations soon.
• Do the flip! – Using guided practice for Active Student Engagement, Interactive Workshop, ASEE PSW Conference at Cal State LA, April 2019.
• How to flip Calculus One Lesson at a Time, Lilly Anaheim, March 2, 2019
• Facilitating Flipped Learning: Utilizing Cross-Campus Learning Communities, Lilly Anaheim, March 1, 2019
• Developing a Flipped Lesson Plan: Planning for Active Engagement, Lilly Anaheim, March 1, 2019 (50 minute workshop)
• Mastery-based grading at scale in GE Statistics, CSU Chancellor Office First-Term Reflections: Restructuring First-Year Writing, Mathematics and Quantitative Reasoning, Long Beach, CA, February
1, 2019
• Coordination and Professional Development at Cal State LA, CSU Chancellor Office First-Term Reflections: Restructuring First-Year Writing, Mathematics and Quantitative Reasoning, Long Beach, CA,
February 1, 201
• Flipping Calculus at Cal State LA, AAAS Meeting at Cal Poly Pomona, June 14, 2018
• Facilitating a Culture of Transformative Pedagogical Change in STEM via Focused Faculty Development, AAAS Meeting at Cal Poly Pomona, June 14, 2018
• A New Mathematics Course Sequence for Life Science Majors: A Progress Report, Biomathematics and Ecology: Education and Research 2013, Arlington, VA, October 11 - 13, 2013
• Improving the quantitative skills of life science majors at California State University Los Angeles. Poster presented at the Vision and Change conference, Washington, D.C., August 28-30, 2013
• Improving Quantitative Skills of Life Science Majors at CSULA, Joint Mathematics Meeting, January 9-12, 2013, San Diego, CA
• Improving Quantitative Skills of Life Science Majors at CSULA, CSUPERB Quantitative Biology Network Meeting, January 3, 2013, Anaheim, CA
• Improving Quantitative Skills of Life Science Majors at CSULA, Biomathematics and Ecology: Education and Research 2012, St. Louis, MO, November 9-11, 2012
• Using The TI-89 To Convey Mathematical Concepts: An Introductory Modeling Course For Non-Science Majors, Calculator workshop the 14^th International Conference on Technology in Collegiate
Mathematics (ICTCM), Baltimore, MA, November 1-4, 2001
• An Alternative to College Algebra- An Introductory Modeling Course for Freshman Liberal Arts Majors, Joint Meeting of the AMS and MAA, New Orleans, January 10-13, 2001
• An Innovative Modeling Approach at the Freshman/Sophomore Level (NSF/DUE: 9653262), MAA Poster Session of the NSF DUE CCLI Program, Joint Meeting of the AMS and MAA, New Orleans, January 10-13,
• An Innovative Modeling Course for Freshman Liberal Arts Majors, MAA Mathfest 2000, Los Angeles, California, August 3 - 5, 2000
• Using Mathematica To Convey Mathematical Concepts: An Introductory Modeling Course For Non-Science Majors, Computer workshop at the 12th International Conference on Technology in Collegiate
Mathematics (ICTCM), Burlingame, CA, Nov. 4 - 7, 1999
• An Introductory Modeling Course for Liberal Arts Majors based on Mathematica, Morsels in Math Teaching, California State University Northridge, Northridge, CA, May 18, 1999
• An Innovative Approach to Modeling at the Freshman/Sophomore Level, 11th International Conference on Technology in Collegiate Mathematics (ICTCM), New Orleans, LA, Nov. 19-22, 1998
• An Introductory Modeling Course for Non-Science Majors: Using Mathematica to Convey Mathematical Concepts, Western Regional Meeting of the American Mathematical Society, Tucson, AZ, Nov. 13 - 15,
• A New Introductory Modeling Course for Non-Science Majors, Regional Meeting of the Southern California Section of the Mathematical Association of America (MAA), Pepperdine University, Malibu, CA,
October 19, 1998
• An Innovative Modeling Approach at the Freshman/Sophomore Level, 3^rd Asian Technology Conference in Mathematics (ATCM'), Tsukuba, Japan, August 24 - 28, 1998
• An Introductory Modeling Course Using Mathematica, LACTE Seminar, CSLA, Feb. 19, 1998
• Using Mathematica to bring Research into the Classroom, 10th International Conference on Technology in Collegiate Mathematics (ICTCM), Chicago, IL, Nov. 7 - 9, 1997
• Introducing Laboratories into a Differential Equations Course - How to Get Started!, 9th International Conference on Technology in Collegiate Mathematics, Reno, NV, Nov. 7 -10, 1996
• Highlights and Pitfalls in O.D.E. Reform, 9th International Conference on Technology in Collegiate Mathematics, Reno, NV, Nov. 7 -10, 199
Expect additional links to some of these presentations soon.
• Logarithms - Trick or Treat?, Cal State LA Physics Seminar, October 31, 2019
• Do you Sudoku? Math Club, Cal Poly Pomona, May 4, 2006
• S. Heubach and R.S. Pamula, Implementing an Approximate Probabilistic Algorithm for Error Recovery in Concurrent Processing Systems, 17th International AoM/IAoM Conference, San Diego, CA, August
6-8, 1999
• Modeling and Simulation of Error Recovery in a Concurrent Processing System, 2^nd International IASTED Conference: European Parallel and Distributed Systems (Euro-PDS'98), Vienna, Austria, July 1
- 3, 1998
• Rigid or Not?, Joint Regional Meeting of the AMS/MAA, Claremont McKenna College, Claremont, CA, Oct. 4, 1997
• Optimizing Rollback Schemes for Parallel Processes, 1996 Seminar on Stochastic Processes, Duke University, Durham, NC, March 14 - 16, 1996
• A Stochastic Model for the Movement of a White Blood Cell, Women in Probability, Cornell University, Ithaca, NY, October 16-18, 1994
• A Stochastic Model for the Movement of a White Blood Cell, Combined MAA/AMS Western Section Meeting, University of Oregon, Eugene, OR, June 15-17, 1994
• A Stochastic Model for the Movement of a White Blood Cell, 1st IMS North American New Researchers' Meeting, UC Berkeley, Berkeley, CA, August 4 -7, 1993
• Ph.D in Applied Mathematics, 1992. University of Southern California, Los Angeles, Los Angeles, CA. GPA 4.0. Thesis: A Stochastic Model for the Movement of a White Blood Cell. (Advisor: Dr.
Joseph Watkins)
• M.S. in Mathematics, 1998. University of Southern California, Los Angeles, Los Angeles, CA. GPA 4.0
• Diplom in Wirtschaftsmathematik (Masters in Mathematics and Economics), 1986. University of Ulm, Germany. Thesis: Inventory Control Under Uncertainty. (Advisor: Dr. Ulrich Rieder)
• Vordiplom in Wirtschaftsmathematik (B.A. in Mathematics and Economics), 1986. University of Ulm, Germany
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Deswik White Paper Pseudoflow Explained - PDFCOFFEE.COM
Citation preview
PSEUDOFLOW EXPLAINED A discussion of Deswik Pseudoflow Pit Optimization in comparison to Whittle LG Pit Optimization Julian Poniewierski Senior Mining Consultant FAusIMM (CP)
1. INTRODUCTION The purpose of this document is to inform the user about Deswik Pseudoflow, within the context of the mining industry’s most accepted pit optimization process that uses Whittle
software based on the Lerchs-Grossman (LG) algorithm. In summary, both are variations of network flow algorithms that achieve the same result, with the pseudoflow algorithm being a computationally
more efficient algorithm developed some 35 years after the original Lerchs-Grossman algorithm (1965). It took 20 years from the formulation of the LG algorithm for it to be incorporated in the first
commercially available software (Whittle Three-D), with another 10 years before it became a mainstream approach to open pit optimization.
It should be noted that the Deswik implementation is not constrained (nor is it aided) by the extensive setup parameter tables and inputs provided in the Whittle software for cost and revenue
calculations. For Deswik’s pseudoflow implementation, the user is required to calculate the revenues and costs for each block in the block model used, and is required to do their own block
regularization within the Deswik.CAD environment. The user is thus in full control of how the costs and revenues are calculated and assigned. This does however require the user to be fully familiar
with their block model, cost structures and revenue parameters (which we believe is a “good thing”). This enables the set-up to be as flexible as required by the user (unconstrained by template
set-up dialogs).
In the past 30 years, Whittle and the LG algorithm have become synonymous with the optimization of open pits, and now suffers from having become a generic term for the process of pit optimization–
similar to the genericization of the Hoover brand with vacuuming in the UK (such that in the UK, people hoover their carpets). It is now 15 years since the formulation of the pseudoflow algorithm,
and there are at least three commercial implementations available, including Deswik’s implementation. Just as “hoovering” does not need to be a Hoover vacuum cleaner – indeed the Dyson cyclonic
vacuum is recognized as far more efficient at vacuuming– the pseudoflow algorithm should now be used to replace the LG algorithm in “whittling” your pit optimization.
2. HISTORY OF PIT OPTIMIZATION 2.1. MANUAL PROCESS Prior to the development of computerized methods of pit optimization and pit design, mining engineers used manual interpretation methods with
evaluation on manually drawn cross-sections (on paper, linens, or film), and then a manual pit design. In the manual method, a simple optimization of economic pit depth was usually carried out by
hand calculator (or slide rule) for regular shaped orebodies using incremental cross-sectional areas, for ore and waste, and an overall pit slope. The incremental stripping ratio (the ratio of the
tonnage of waste that must be moved to access the next mass increment of ore that will be accessed) on each crosssection was compared against the break-even stripping ratio for the estimated ore
grade and appropriate revenue and cost inputs.
For this reason, the algorithm fails to consistently give realistic results. Mintec/MineSight (being a US based company and supplier of early solution to Kennecott) were an early implementer of the
floating cone algorithm (and may still offer it in their solution suite).
2.3. LERCHS - GROSSMAN It was also in 1965 that Lerchs and Grossmann published a paper that introduced two modeling approaches to solving the open pit optimization problem. The Lerchs-Grossman (LG)
algorithm is well documented in the technical literature (Lerchs and Grossman, 1965; Zhao and Kim 1992; Seymour, 1995; Hustrulid and Kuchta 2006).
The final pit shell was then produced by drawing increasingly larger pit shells on cross section such that the last increment had a strip ratio equal to the design maximum.
The LG method was based on a mathematical technique which was unusable in practice until a practical optimization program called Whittle Three-D was developed by Jeff Whittle of Whittle Programming
Pty Ltd in the mid-1980s.
This was a very labor intensive approach and could only ever approximate the optimal pit. The design had to be done on a large number of cross sections and was still inaccurate because it treated the
problem in only two dimensions. In cases of highly variable grade the problem became extremely complex, and relied heavily on the “gut feel” of an experienced designer using trial and error.
Two methods to the solution of open pit optimization were detailed by Lerchs and Grossmann, being a Graph Theory algorithm, which is a heuristic approach, and a Dynamic Programming algorithm, which
is an application of an operations research technique. Both methods gave an optimum pit limit for an undiscounted cash flow – based on an economic block model of an ore body and its surrounding
waste, and determined which blocks should be mined to obtain the maximum dollar value from the pit.
2.2. FLOATING CONE Pana (1965) introduced an algorithm called Moving (or Floating) Cone. The method was developed at Kennecott Copper Corporation during the early 1960s (McCarthy, 1993) and was the
first computerized attempt at pit optimization, requiring a three dimensional computerized block model of the mineral deposit. The projected ultimate pit limits are developed by using a technique of
a moving “cone” (or rather a frustum of an inverted cone – that is, the “pointy” end has been cut to a minimum mining area). The cone is moved around in the block model space to generate a series of
interlocking frustum shaped removal increments. However, the shortcoming of this approach is that it creates overlapping cones, and it is incapable of examining all combinations of adjacent blocks.
The LG methods took into account two types of information: i.
The required mining slopes. For each block in the model, the LG method needs details of what other blocks must be removed to uncover it. This information is stored as “arcs” between the blocks
ii. The value in dollars of each block once it has been uncovered. In the case of a waste block this will be negative and will be the cost of blasting, digging and haulage. In the case of an ore
block, the removal cost will be offset by the value of the recovered ore, less any processing, sales, and other associated costs. Any block which can, during mining, be separated into waste and ore
is given a value which reflects this.
Given the block values (positives and negatives) and the structure arcs, the LG method progressively builds up a list of related blocks in the form of branches of a tree (called a “graph” in
mathematics). Branches are flagged as ‘strong’ if the total of their block values is positive. Such branches are worth mining if they are uncovered. Other branches with negative total values are
flagged as ‘weak’. The LG method then searches for structure arcs, which indicate that some part of a strong branch lies below a weak branch. When such a case is found, the two branches are
restructured so as to remove the conflict. This may involve combining the two branches into one (which may be strong or weak) or breaking a ‘twig’ off one branch and adding it to the other branch.
The checking continues until there is no structure arc which goes from a strong branch to a weak branch. At this point the blocks in all the strong branches taken together constitute and define the
optimum pit. The blocks in the weak branches are those which will be left behind when mining is complete. In effect, what the LG algorithm has done is to find the maximum closure of a weighted
directed graph; in this case the vertices represent the blocks in the model, the weights represent the net profit of the block, and the arcs represent the mining (usually slope) constraints. As such
the LG algorithm provides a mathematically optimum solution to the problem of maximizing the pit value (importantly, note that this is for an undiscounted cash flow value). It should be noted that it
is a mathematical solution. Except for the information given by the arcs, the LG algorithm “knows” nothing about the positions of the blocks – nor indeed about mining. The LG algorithm works only
with a list of vertices and a list of arcs. Whether these are laid out in one, two or three dimensions and how many arcs per block are used is immaterial to the logic of the method, which is purely
mathematical. Also note that it took some 20 years between the publication of the LG method (1965, which was also the year that the floating cone method was computerized) and the first commercial
available adoption of the LG method (Whittle’s Three-D). The basic LG algorithm has now been used for over 30 years on many feasibility studies and for many producing mines.
2.4. NETWORK FLOW SOLUTIONS In their 1965 paper, Lerchs and Grossmann indicated that the ultimate-pit problem could be expressed as a maximum closure network flow problem but recommended their direct
approach, possibly due to computer memory constraints at the time. The LG algorithm was therefore a method of solving a special case of a network flow problem. In 1976, Picard provided a mathematical
proof that a “maximum closure” network flow problem (of which the open cut optimization problem is one) were reducible to a “minimum cut” network flow problem, hence solvable by any efficient maximum
flow algorithm. As a consequence, sophisticated network flow algorithms could therefore be used in place of the LG algorithm, and they can calculate identical results in a fraction of the time. One
of the first efficient maximum flow algorithms used in solving the open pit optimization problem was the “pushrelabel” algorithm (Goldberg and Tarjan, 1988; King et al., 1992; Goldfarb and Chen,
1997). Hochbaum and Chen’s study (2000) showed that the pushrelabel algorithm outperformed the LG algorithm in nearly all cases. When the number of vertices is large, greater than a million, network
flow algorithms perform orders of magnitude faster and compute precisely the same results. Numerous authors implemented the push-relabel algorithm, and various heuristics and techniques were
developed to maximize its performance. This was the algorithm that MineMax implemented in their first pit optimizer software offering. Development of more efficient network flow algorithms have
continued. The generally accepted most efficient algorithm currently available are the various pseudoflow algorithms developed by Professor Dorit Hochbaumn and her colleagues at University of
California, Berkeley (Hochbaum, 2002, 2001; Hochbaum and Chen, 2000). Pseudoflow methods give new life to the LG pit optimization. The “highest label” method implementation of the pseudoflow
algorithm in particular is consistently faster than the generic LG methods and is also usually faster than the alternative “lowest label” method implementation of the pseudoflow algorithm. The
increase in speed can be from two to 50 times faster than the LG methods, and theoretically much faster for larger problems (Muir, 2005).
3. ALGORITHM PERFORMANCE COMPARISONS Muir (2005) gave the most comprehensive analysis of the pseudoflow algorithm performance and a practical example of the identical results achieved in comparison
to the LG algorithm in solving a pit optimization. These analyses and results were presented to the mainstream mining industry in the 2005 AusIMM Spectrum series publication: Orebody Modelling and
Strategic Mine Planning. Key results of Muir’s analysis are reproduced herein. It should be noted that the code written by Muir (2005) is the underlying calculation engine that has been implemented
in Deswik Pseudoflow. As a check of the correct implementation of that code, the results for the Deswik implementation were compared against four publicly available test data sets from Minelib1
(Espinoza et al, 2012). The specific data sets the results were checked against were for Marvin, McLaughlin, KD and P4HD. The Pseudoflow results were identical to the published results at Minelib.
Table 1 (from Muir, 2005) shows the relative run-times for several variants of both the LG and pseudoflow algorithms. It can be seen from these results that the “highest label pseudoflow priority
queue” (HLPQ) implementation took just under 2% of the time it took for the standard LG algorithm to solve a 38 bench pit optimization problem. Table 2 (from Muir, 2005) shows that the number of
blocks and the profit value for the HLPQ solution was identical to the LG solution of the same 38 bench pit optimization problem. The relative solution times shown in Table 1 are shown plotted in
Figure 1. In addition to Muir’s paper, there are a couple of other known examples of published comparisons between the LG algorithm and flow network solutions to the pit optimization problem.
and was found to produce “the same results for the actual optimal pit calculations” (to within less than 0.01% - with the differences appearing to be due to block coarseness and slopes). Table 1 –
Optimization times (seconds) to various pit levels for 220 x 119 x 38 profit matrix
Legend: LG LGS LLP LLPS LLPQ HLPQ
Normal Lerchs-Grossmann Subset Lerchs-Grossmann Lowest Label Pseudoflow (no priority queue) Subset Lowest Label Pseudoflow (no priority queue) Lowest Label Pseudoflow (priority queue) Highest Label
Pseudoflow (priority queue) (after Muir, 2005)
Table 2 – Statistics for level 38 for 220 x 119 x 38 profit matrix
Blocks removed
Blocks remaining
Branches relinked
Branches pruned
Profit value
Time (seconds)
Figure 1 – Solution times for four pit optimization algorithms for different bench number pit problems
Jiang (2015) stated that the final pit limits from using a pseudoflow algorithm implementation versus the Whittle LG implementation have always been found to be materially the same, with any minor
differences that are observed always being due to how the various implementations compute the slope angle constraints. The push-relabel algorithm implemented by MineMax was compared to the LG
algorithm by SRK (Kentwell, 2002) 1
4. MODELING ISSUES TO NOTE Having shown that it has been proved that the pseudoflow algorithm will give identical results to the LG algorithm, it is appropriate to also point out that no algorithmic
solution will provide the exact “true” optimization solution. There are a large number of approximations inbuilt into the algorithmic solution to the pit optimization problem as well as a number of
common errors and uncertain assumptions used in the process. The huge efforts dedicated to the development of sophisticated optimization algorithms is usually not matched by similar attention being
paid to improving the correctness and reliability of the data used in the modeling exercise, as well as the correct use of the results of the modeling. Some of the numerous sources of error,
uncertainty and approximations in the process of pit optimization that need to be recognized are discussed below. In summary, be aware that the process of pit optimization is based on coarse and
uncertain estimated input parameters. Deswik therefore recommend that the user concentrate on the overall picture and getting as accurate as possible estimates of the “big ticket” items. And
remember: “don’t sweat the small stuff”. Deswik also advise to design for risk minimization of the downside in your assumptions, as per the scenario strategies advocated by Hall (2014) and Whittle
(2009), but to also check the optimistic case upside scenario to determine infrastructure boundaries.
4.1. SOLUTION APPROXIMATION “ERRORS” a. The effect of using blocks with vertical sides to represent a solution (a pit design) that has nonvertical sides. It is possible to output a smoothed shell
through the block centroids, but note that this will not give the same tonnes and grade result as the block based optimization when the surface is cut against the resource model blocks. b. Slope
accuracy representation. The accuracy of the overall slope created in the modeling process with
respect to slope desired to be modeled will depend upon the height and number of dependencies (arcs) used to define the slope. This will always need to be checked for suitability. Larger blocks will
generally give less slope accuracy, and more, smaller blocks that allow greater accuracy will require more modeled arcs (block precedencies) and will slow the processing down. An accuracy tolerance
of around 1° average error is usually considered acceptable. c. Changes in converting a shell to a pit design. A difference of 5% in tonnes is quite common during this process. This is due to the
approximation of the overall slope with respect to the actual design and effects of placement of haul roads on that overall slope. d. Effect of minimum mining width on the bottom of a shell. Many pit
optimizations are undertaken without consideration of the minimum mining width at the bottom of each shell – even when the package used provides such a facility. This will change the value of the
selected shell used for design. At present Deswik’s implementation of Pseudoflow does not have a tool to consider minimum mining width – but this is in the future development plans. e. Effect of
stockpiling. The pit optimization algorithms – both Whittle LG and Deswik Pseudoflow assume the value generated is the value that occurs at time of mining, and stockpiling delays the recovery of that
value. Stockpiling for 10 or more years will mean that the time value of the block of ore stockpiled can be a fraction of the value used in the pit optimization. Mines with significant amounts of
marginal stockpiled ore will suffer a significant oversize effect from the difference in when the algorithm values the block and when the value is actually generated. If an elevated cut-off grade
policy is used in the scheduling of the pit early in the pit’s life, as a means of maximizing the NPV (Lane, 1988), then the tonnage stockpiled will be increased, and the time related differences in
value between when the pit optimization assigns the value and when the value is actually realized in the plan increases further.
4.2. COMMON INPUT/OUTPUT ERRORS AND ISSUES a. Errors in block model regularization and the assumed Smallest Mining Unit (SMU). If a block model is used that features grade estimated blocks at smaller
than the SMU size, then unrealistic mining selectivity will be built into the result. If a model is regularized to a larger than SMU size for purposes of processing speed, then the ore/waste tonnage
classifications and grades at the SMU size need to be maintained and not smoothed out to the larger regularized block size. Not considering the over-selectivity can easily result in pits with an
expectation of double the value of a pit selected from a block model with an appropriately sized SMU. b. Using Revenue Factor (RF) =1 shells for final pit design. The pit limits that maximize the
undiscounted cashflow for a given project will not maximize the NPV of the project. As discussed by Whittle (2009) when the time value of money is taken into account, the outer shells of the RF = 1
pit can be shown to reduce value, due to the fact that the cost of waste stripping precedes the margins derived from ore ultimately obtained. The effect of discounted cash flow means the discounted
costs outweigh the more heavily discounted revenues. The optimal pit from a Net Present Value (NPV) viewpoint can be between revenue factor 0.65 and 0.95, depending on the deposit’s structure and the
mining constraints (minimum mining width, maximum vertical advancement per year, and limit on total movement) and processing capacity. This can be seen where the peak of the discounted cash flow of
the specified case is at a lower overall tonnage than the peak of the undiscounted total cash curve. Despite the fact that this aspect is well discussed in the technical literature, the selection of
RF=1 shell is still commonly seen in the industry for ore reserves work and project feasibility studies.
c. Processing Plant Performance Parameters. Aside from price, the other big factor with significant uncertainty and used in the calculation of revenue received for a block of ore is the processing
plant recovery. Variations in recovery for grade, mineralogy and hardness can be expected compared to the recovery used in the model. The commonly used constant recovery will almost always be wrong
(either because it is optimistically overestimated, or because there is a fixed tail component not be taken into account). Additionally, it should also be noted that project value can often be
increased by sacrificing metal recovery to pursue lower cost, higher throughput – as discussed by Wooller (1999). d. Cut-Off Grade. If blocks with extremely small values (cents per tonne of positive
value) are left within the block model used (effectively the use of a marginal cut-off grade value of zero), then a lot of ore will be processed in the project for very little value. Effectively, a
significant percentage of the ore is being mined and processed for little more than practice – as discussed in Poniewierski (2016). Deswik suggest that in order to avoid this situation that a value
cut-off greater than zero be applied. It is suggested that a suitable value would be the minimum desired percentage margin on the processing and sales costs employed. Such blocks would have their
revenue value set to zero, so they do not influence the optimal shell selection. Once the final shell has been selected and the ultimate pit designed, the marginal material in that pit can be
reconsidered for inclusion in ore reserves and stockpiling if so desired. It should also be noted that for NPV maximization, a variable cut-off grade or cut-off value policy should be adopted (as per
Lane 1988).
Additionally, the curve of discounted cash value versus tonnage tends to be flat at the top. For example, it is common for the last third of the life-of-mine to be quite marginal. Whilst it is worth
maintaining the option to operate during this period and in this part of the deposit in case prices, costs, or technology improve, this part of the resource should not be regarded as a core part of
and driver of a project (Whittle 2009).
4.3. INPUT UNCERTAINTIES a. Geological uncertainty. This is one of the biggest sources of error in a pit optimization, as the pit optimization results ultimately depend on the accuracy of the model
and the competence of the geologist interpreting all the available geological data. The block model has been created from sparse imperfect data that makes assumptions and estimations on
mineralization limits, mineralization grades modeling, fault interpretation and lithology interpretation. In the author’s experience, many resource models have contained metal errors of at least 10%
or more (model over-call) and up to 30% has been seen. Cases of under-call do also occur, and will predominate in the literature as no-one likes to discuss the bad outcomes publically. In the authors
experience 70 to 80% of all resource models suffer from overcall to some degree. In addition to the grade uncertainty, there is also density uncertainty and in-situ moisture uncertainty. b. Effect of
Inferred resources. Should these be included or not included? If included, these can easily be in error by 50% or more. If not included, the design will change when these are converted to Indicated
or Measured status.
e. Economic uncertainty. This is also one of the major sources of “error” with pit optimization. In the analysis of costs and revenues, we have to make assumptions about the macro-economic
environment such as commodity prices, exchange rates, interest rates, inflation, fuel and power costs, sustaining capital costs, contractor costs and labor costs. For the commodity price in
particular, we can confidently state that the price used will be 100% wrong for the life-of-the mine (it will never be one static value). f.
Costs. Except for operating mines with a good understanding of their detailed cost driver history, there is usually a great deal of uncertainty on the costs being used in the pit optimization. Many
parameters used to estimate costs such as equipment selection, annual production rate, plant capacity and requirements, etc. are just estimates. There is usually an imperfect understanding of fixed
and variable costs that do not truly reflect the changes in costs as the pits being assessed change in size. In addition, it needs to be noted that fixed costs (or time period costs) need to be
applied on the basis of the mine/mill system bottle-neck. As a general rule this is often the SAG mill (with power rather than tonnage being the limit).
c. Geotechnical uncertainty. While a lot of focus can be spent on ensuring the desired overall angles are modeled accurately, in many cases the slopes provided for use may be little more than a
geotechnical engineer’s guesstimate based on very little rock mass quality data, sparse and imperfect knowledge of faulting, jointing, bedding and hydrology. Even in operating pits, the geotechnical
conditions can change quickly from that currently being used. d. Dilution and loss are nearly always “guesses” – except for sites with a number of years of operating experience and a good
reconciliation system that allows for assessment of the dilution and loss (which is not all that common).
5. SUMMARY Both the Lerchs-Grossmann and pseudoflow algorithms are variations of network flow algorithms that achieve the same result. Pseudoflow is however a computationally more efficient algorithm
developed some 35 years after the original Lerchs-Grossman algorithm (1965), and has been available for use for some 15 years, with the first implementation for mining discussed in 2005 (Muir, 2005).
If differences are seen between a Whittle LG result and a Deswik Pseudoflow result, it will be a difference in the set-up used. There are numerous set-up factors and parameters that can cause
differences in pit optimization results, and the user should be aware of all of these to avoid falling into common error traps. It should be noted that the Deswik implementation is not constrained
(nor is it aided) by the pre-defined template inputs provided in the Whittle software for cost and revenue calculations (these templates can be restrictive for both very simple set-ups or complex
set-ups not catered for). For Deswik’s Pseudoflow implementation, the user is required to calculate the revenues and costs for each block in the block model used, and is required to do their own
block regularization within the Deswik.CAD environment. The user is thus in full control of how the costs and revenues are calculated and assigned, but this does require the user to be fully familiar
with their block model, cost structures and revenue parameters (which we believe is a “good thing”). This enables the cost and revenue calculations to be as simple or complex as required by the user
(unconstrained by template set-up dialogs).
REFERENCES Alford C G and Whittle J, 1986. Application of Lerchs– Grossmann pit optimization to the design of open pit mines, In Large Open Pit Mining Conference, AusIMM–IE Aust Newman Combined
Group, 1986, 201–207.
Kentwell, D, 2002. MineMax Planner vs Whittle Four-X - an open pit optimization software evaluation and comparison, MineMax Whitepaper available from https://www.minemax. com/downloads/
Carlson, T R; Erickson, J D, O’Brain D T and Pana, M T, 1966. Computer techniques in mine planning, Mining Engineering, Vol. 18, No. 5, p.p. 53-56.
Kim, Y C, 1978. Ultimate pit design methodologies using computer models the state of the art, Mining Engineering, Vol. 30, pp. 1454 1459.
Chandran, B G and Hochbaum, D S, 2009. A computational study of the pseudoflow and push-relabel algorithms for the maximum flow problem, Operations Research, 57(2): 358-376.
King, V, Rao, S and Tarjan, R, 1992. A faster deterministic maximum flow algorithm, Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms. Academic Press, Orlando, FL, USA,
Dagdelen, K, 2005. Open pit optimization — strategies for improving economics of mining projects through mine planning, in Orebody Modelling and Strategic Mine Planning, Spectrum Series No 14, pp
125-128 (The Australasian Institute of Mining and Metallurgy: Melbourne).
Lane, K F, 1988. The Economic Definition of Ore: Cut-off Grades in Theory and Practice (Mining Journal Books: London).
Espinoza, D, Goycoolea, M, Moreno, E and Newman, A N, 2012. MineLib: A Library of Open Pit Mining Problems, Ann. Oper. Res. 206(1), 91-114.
McCarthy, P L, 1993. “Pit Optimization” internal paper for AMC and whitepaper on the AMC website, available at http:// www.amcconsultants.com.au/library
Francois-Bongarcon, D M and Guibal, D, 1984. Parameterization of optimal design of an open pit -beginning of a new phase of research, Transactions of Society of Mining Engineers, AIME, Vol. 274, pp.
Muir, D C W, 2005. Pseudoflow, new life for Lerchs-Grossmann pit optimization, in Orebody Modelling and Strategic Mine Planning, Spectrum Series No 14, pp 97-104 (The Australasian Institute of Mining
and Metallurgy: Melbourne).
Goldberg, A and Tarjan, R E, 1988. A new approach to the maximum flow problem, Journal of the Association for Computing Machinery, 35, 921-940.
Pana, M, 1965. The simulation approach to open-pit design, In J. Dotson and W. Peters, editors, Short Course and Symposium on Computers and Computer Applications in Mining and Exploration, College of
Mines, University of Arizona, Tuscon, Arizona. pp. ZZ–1 – ZZ–24.
Goldfarb, D and Chen, W, 1997. On strongly polynomial dual algorithms for the maximum flow problem, Special Issue of Mathematical Programming B, 78(2):159-168. Hall, B, 2014. Cut-off Grades and
Optimising the Strategic Mine Plan, Spectrum Series 20, 311 p (The Australasian Institute of Mining and Metallurgy: Melbourne).
Lerchs, H and Grossmann, I F, 1965. Optimum design of open pit mines, The Canadian Mining and Metallurgical Bulletin, Vol. 58, January, pp.47-54.
Picard, J, 1976. Maximal closure of a graph and applications to combinatorial problems, Management Science, Vol. 22, No. 11, pp. 1268–1272.
Hochbaum, D S, 2008. A pseudoflow algorithm: a new algorithm for the maximum-flow problem, Operations Research, 56(4): 992-1009.
Poniewierski, J, 2016, Negatively geared ore reserves – a major peril of the break-even cut-off grade, Proc. AusIMM Project Evaluation Conference, Adelaide, 8-9 March 2016. pp236247.
Hochbaum, D S and Chen, A, 2000. Performance analysis and best implementations of old and new algorithms for the open-pit mining problem, Operations Research, 48(6): 894914.
Whittle, G, 2009. Misguided objectives that destroy value, in Proceedings Orebody Modelling and Strategic Mine Planning, pp 97-101 (The Australasian Institute of Mining and Metallurgy: Melbourne).
Hochbaum, D S, 2001. A new-old algorithm for minimum-cut and maximum-flow in closure graphs, Networks, 37(4): 171193.
Wooller, R, 1999. Cut-off grades beyond the mine – optimising mill throughput, in Proceedings Third Biennial Conference on Strategic Mine Planning. pp 217-230 (Whittle Programming: Melbourne)
Jiang, Y, D. 2015, “Impact of Reblocking on Pit Optimization” https://www.linkedin.com/pulse/effect-reblocking-pitoptimization-yaohong-d-jiang
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Bayes@Lund 2019
Thank you all who participated in Bayes@Lund 2019 and who made it such a successful and enjoyable event! :)
All best, The Organizers Rasmus Bååth, Alex Holmes, and Ullrika Sahlin
Below you will find the original webpage of Bayes@Lund 2019.
You are welcome to participate in the sixth edition of Bayes@Lund!
The purpose of this conference is to bring together researchers and professionals working with or interested in Bayesian methods. Bayes@Lund aims at being accessible to researchers with little
experience of Bayesian methods while still being relevant to experienced practitioners. The focus is on how Bayesian methods are used in research and in the industry, what advantages Bayesian methods
have over classical alternatives, and how the use and teaching of Bayesian methods can be encouraged. (see last year's conference for what to expect).
The conference will take place at Lund University, Sweden on the 7th of May 2019 starting at 9.00 and ending at 17.00. It will include contributed talks and invited presentations. Please register for
the conference here.
The program is now finalized! For a list of all the speakers, and abstracts for all talks, do check out the book of abstracts:
Some of the speakers have agreed on sharing slides and information regarding their presentations which you'll find here:
Maggie is an astrophysics research fellow working at the European Space Agency in Madrid. Her main research involves modelling the mass distribution of clusters of galaxies to understand the nature
of dark matter and dark energy in our Universe. Maggie's talk is Hierarchical models and their applications in astronomy; how hierarchical models can be a powerful tool for inference.
Robert Grant is a medical statistician, turned freelance trainer, coach and writer in Bayesian models and data visualisation. His book Data Visualisation: charts maps and interactive graphics is
published by CRC Press. His talk Visualisation for refining and communicating Bayesian analyses will review relevant general principles of effective visualisation, recent work on Bayesian workflow,
and the role of interactive graphics.
Are you interested in Bayesian statistics and want to get up to speed? Then join the pre-conference Bayesian tutorial. This 3h tutorial will be given by Rasmus Bååth and will go through the
fundamentals of Bayesian statistics using R. It will be based on the online course of the same name and requires no prior knowledge of Bayesian statistics but basic knowledge of the R programming
The tutorial is free of charge and takes place on the 6th of May 14.00 - 17.00 at Lund University, Sweden. Please register here.
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Chebyshev’s Inequality - Finance Train
Chebyshev’s Inequality
Chebyshev’s Inequality is used to describe the percentage of values in a distribution within an interval centered at the mean.
It states that for a distribution, the percentage of observations that lie within k standard deviations is atleast 1 – 1/k2
This is illustrated below:
The following table shows the minimum number of observations that lie within a certain number of standard deviations of the mean.
Standard Deviations % of observations
1.5 56%
2 75%
3 89%
4 94%
An important feature of Chebyshev’s Inequality is that it works with any kind of distribution.
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Rendering a torus in Tachyon
Rendering a torus in Tachyon
I'm trying to figure out how to render a torus using Tachyon. The problem is that a solid torus requires a 2 variable parametric equation, and it seems Tachyon only likes to use single variable
parametric equations. Is there a way around this?
2 Answers
Sort by ยป oldest newest most voted
The following is working for me in Sage Math Cloud.
var('s t')
The downside is that you will have to transform the torus in order to change the view. I don't think you can change the viewpoint using the show or parametric_plot3d commands yet.
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I think you're right. I've been able to use the tachyon viewer, but I'm trying to animate a series of nested tori, and was hoping to control tachyon directly. I've got the animation without it, but
it would be nice to change the view and lighting.
Jeff Ford ( 2015-04-07 02:01:49 +0100 )edit
Depending on how hard you want to work, you might be interested in extracting tachyon data from Graphics3d objects. This is the data rendered with the option viewer='tachyon', but you can modify
lighting, camera, etc. before rendering.
This has been a long-time side goal of mine, but currently there's no totally automatic ways of doing it. Here are some things you can use to do it yourself though:
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PLSHADES(3plplot) PLplot API PLSHADES(3plplot)
plshades - Shade regions on the basis of value
plshades(a, nx, ny, defined, xmin, xmax, ymin, ymax, clevel, nlevel, fill_width, cont_color, cont_width, fill, rectangular, pltr, pltr_data)
Shade regions on the basis of value. This is the high-level routine for making continuous color shaded plots with cmap1 while plshade(3plplot) should be used to plot individual shaded regions using
either cmap0 or cmap1. examples/;<language>/x16* shows how to use plshades(3plplot) for each of our supported languages.
Redacted form: General: plshades(a, defined, xmin, xmax, ymin, ymax, clevel, fill_width, cont_color, cont_width, fill, rectangular, pltr, pltr_data)
This function is used in examples 16, 21, and 22.
A matrix containing function values to plot. Should have dimensions of nx by ny.
First dimension of matrix "a".
Second dimension of matrix "a".
Callback function specifying the region that should be plotted in the shade plot. This function accepts x and y coordinates as input arguments and must return 1 if the point is to be included in
the shade plot and 0 otherwise. If you want to plot the entire shade plot (the usual case), this argument should be set to NULL.
See the discussion of pltr below for how these arguments are used (only for the special case when the callback function pltr is not supplied).
A vector containing the data levels corresponding to the edges of each shaded region that will be plotted by this function. To work properly the levels should be monotonic.
Number of shades plus 1 (i.e., the number of shade edge values in clevel).
Defines the line width used by the fill pattern.
Defines cmap0 pen color used for contours defining edges of shaded regions. The pen color is only temporary set for the contour drawing. Set this value to zero or less if no shade edge contours
are wanted.
Defines line width used for contours defining edges of shaded regions. This value may not be honored by all drivers. The pen width is only temporary set for the contour drawing. Set this value to
zero or less if no shade edge contours are wanted.
Callback routine used to fill the region. Use plfill(3plplot) for this purpose.
Set rectangular to true if rectangles map to rectangles after coordinate transformation with pltrl. Otherwise, set rectangular to false. If rectangular is set to true, plshade tries to save time
by filling large rectangles. This optimization fails if the coordinate transformation distorts the shape of rectangles. For example a plot in polar coordinates has to have rectangular set to
A callback function that defines the transformation between the zero-based indices of the matrix a and world coordinates. If pltr is not supplied (e.g., is set to NULL in the C case), then the x
indices of a are mapped to the range xmin through xmax and the y indices of a are mapped to the range ymin through ymax.For the C case, transformation functions are provided in the PLplot
library: pltr0(3plplot) for the identity mapping, and pltr1(3plplot) and pltr2(3plplot) for arbitrary mappings respectively defined by vectors and matrices. In addition, C callback routines for
the transformation can be supplied by the user such as the mypltr function in examples/c/x09c.c which provides a general linear transformation between index coordinates and world coordinates.For
languages other than C you should consult the PLplot documentation for the details concerning how PLTRANSFORM_callback(3plplot) arguments are interfaced. However, in general, a particular pattern
of callback-associated arguments such as a tr vector with 6 elements; xg and yg vectors; or xg and yg matrices are respectively interfaced to a linear-transformation routine similar to the above
mypltr function; pltr1(3plplot); and pltr2(3plplot). Furthermore, some of our more sophisticated bindings (see, e.g., the PLplot documentation) support native language callbacks for handling
index to world-coordinate transformations. Examples of these various approaches are given in examples/<language>x09*, examples/<language>x16*, examples/<language>x20*, examples/<language>x21*,
and examples/<language>x22*, for all our supported languages.
Extra parameter to help pass information to pltr0(3plplot), pltr1(3plplot), pltr2(3plplot), or whatever routine that is externally supplied.
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19 [recommendation system 11] combination of FM and deep learning model
A typical CTR process is shown in the figure below:
As shown in the figure above, it mainly includes two parts: offline part and online part. The goal of offline part is to train available models, while the online part considers that the performance
of the model may decline over time after it is online. In this case, you can choose to use online learning to update the model online:
Offline part:
• Data collection: it mainly collects business-related data. Usually, special colleagues will bury points at the app location to get business data;
• Preprocessing: remove the dirt and duplicate of the business data obtained at the buried point;
• Construct data set: for preprocessed business data, construct data sets. When segmenting training, testing and verification sets, it should be segmented reasonably according to business logic;
• Feature Engineering: basic feature processing of original data, including removing features with large correlation, one-hot of discrete variables, discretization of continuous features, etc;
• Model selection: select a reasonable machine learning model to complete the corresponding work. The principle is from simplicity to depth, find the baseline first, and then optimize step by step;
• Hyperparameter selection: use gridsearch, random search or hyperopt to select the hyperparameter combination with the best performance in the offline data set;
• Online A/B Test: select the optimized model and the original model (such as baseline) for A/B Test. If the performance is improved, replace the original model;
Online part
• Cache & Logic: set simple filtering rules to filter abnormal data;
• Model update: when cache & Logic collects data of appropriate size, pre train + finetuning the model. If the performance of the test set is higher than that of the original model, update the
model parameters of the model server;
• Model Server: accepts data requests and returns forecast results;
1. Foreword
Predicting user response, such as CTR and CVR (conversion rate), plays an important role in web search, personalized recommendation and online advertising; Advertising prediction CTR depends on
whether a specific user thinks the advertisement is relevant and gives the probability of users clicking in a specific scene.
Different from the field of image and audio, the input characteristics of web space are usually discrete and category, and the dependence is basically unknown; To predict user response: linear model
(under fitting) or manual design of high-order interactive features (large amount of calculation); The linear model is simple and effective, but the performance deviation should be unable to learn
the relationship between features. The ability of nonlinear model can be improved by the combination of features.
The effective category feature interaction mode is automatically learned through DNN. In order to make DNN work effectively, three feature conversion methods are used: FM (factorization machines),
RBM (restricted Boltzmann machine) and DAE (noise reduction self encoder).
The previous section introduced the evolution process of FM model family in detail. After entering the era of deep learning, the evolution process of FM has not stopped. The FNN, DeepFM and NFM
models introduced in this section have applied or improved the FM model in different ways, integrated into the deep learning model, and continue to give full play to their advantages in feature
2. FNN -- initialize the Embedding layer with the implicit vector of FM
FNN was proposed by researchers from University College London in 2016. The structure of its model (as shown in Figure 1) is a classical depth neural network similar to Deep Crossing model. The
conversion process from sparse input vector to dense vector is also the structure of classical Embedding layer. So, where is FNN model combined with FM model?
chart 1 : F N N model type Yes junction structure chart Figure 1: structure diagram of FNN model Figure 1: structure diagram of FNN model
The key to the problem lies in the improvement of the Embedding layer. In the process of parameter initialization of neural network, random initialization, which does not contain any prior
information, is often used. Because the input of Embedding layer is extremely sparse, the convergence speed of Embedding layer is very slow. In addition, the number of parameters in the Embedding
layer often accounts for more than half of the parameters of the whole neural network, so the convergence speed of the model is limited by the Embedding layer.
Basics - why does the Embedding layer tend to converge slowly?
In deep learning networks, the function of Embedding layer is to convert sparse input vectors into dense vectors, but the existence of Embedding layer often slows down the convergence speed of the
whole neural network for two reasons:
1. The number of parameters in the Embedding layer is huge. Here you can do a simple calculation. Assuming that the dimension of the input layer is 100000, the output dimension of the Embedding
layer is 32, the upper layer plus five 32-dimensional fully connected layers, and the dimension of the final output layer is 10, the number of parameters from the input layer to the Embedding
layer is 32 x 100000 = 3200000, and the total number of parameters of all other layers is (32x32)x4+32x10=4416. Then, the proportion of the total weight of the Embedding layer is 3200000 /
(3200000 + 4416) = 99.86%. In other words, the weight of the Embedding layer accounts for the vast majority of the weight of the whole network. Then, it can be imagined that most of the training
time and computing overhead are occupied by the Embedding layer.
2. Because the input vector is too sparse, in the process of random gradient descent, only the weight of Embedding layer connected with non-zero features will be updated (please refer to the
parameter update formula and solution of random gradient descent), which further reduces the convergence speed of Embedding layer.
Aiming at the problem of convergence speed of Embedding layer, the solution idea of FNN model is to initialize the parameters of Embedding layer with each feature hidden vector trained by FM model,
which is equivalent to introducing valuable a priori information when initializing the parameters of neural network. In other words, the starting point of neural network training is closer to the
target best advantage, which naturally accelerates the convergence process of the whole neural network.
Here we will review the mathematical form of FM, as shown in equation (1).
y F M ( x ) : = sigmoid ( w 0 + ∑ i = 1 N w i x i + ∑ i = 1 N ∑ j = i + 1 N ⟨ v i , v j ⟩ x i x j ) (1) y_{\mathrm{FM}}(x):=\operatorname{sigmoid}\left(w_{0}+\sum_{i=1}^{N} w_{i} x_{i}+\sum_{i=1}^
{N} \sum_{j=i+1}^{N}\left\langle v_{i}, v_{j}\right\rangle x_{i} x_{j}\right) \tag{1} yFM(x):=sigmoid(w0+i=1∑Nwixi+i=1∑Nj=i+1∑N⟨vi,vj⟩xixj)(1)
The parameters mainly include constant bias w 0 w_{0} w0, first order parameter part w i w_{i} wi # and second order implicit vector part v i v_{i} vi.
In Figure 1, the bottom layer of the model embeds the input data encoded by one hot binary with FM, maps the sparse binary eigenvector to the dense real layer, and then models the dense real layer as
the input variable. This method successfully avoids the computational complexity of high-dimensional binary input data.
Since the input layer is the deny real layer and each field is of one hot type, so:
z i = W 0 i ⋅ x [ start i : e n d i ] = ( w i , v i 1 , v i 2 , ... , v i K ) (2) \boldsymbol{z}_{i}=\boldsymbol{W}_{0}^{i} \cdot \boldsymbol{x}\left[\operatorname{start}_{i}:\right. end \left._{i}
\right]=\left(w_{i}, v_{i}^{1}, v_{i}^{2}, \ldots, v_{i}^{K}\right) \tag{2} zi=W0i⋅x[starti:endi]=(wi,vi1,vi2,...,viK)(2)
The corresponding relationship between FM parameters and parameters of Embedding layer in FNN is shown in the following figure (as shown in Figure 2).
chart 2 : benefit use F M first beginning turn E m b e d d i n g layer Yes too Course Figure 2: process of initializing the Embedding layer with FM Figure 2: process of initializing the Embedding
layer with FM
Through this method, the above neural network can learn more effectively from the representation of FM. We can learn more potential data patterns and get better results.
After initialization of FM layer and other layers, finetune through supervised learning, using the loss function of cross entropy:
L ( y , y ^ ) = − y log y ^ − ( 1 − y ) log ( 1 − y ^ ) (3) L(y, \hat{y})=-y \log \hat{y}-(1-y) \log (1-\hat{y}) \tag{3} L(y,y^)=−ylogy^−(1−y)log(1−y^)(3)
FM layer weight update method:
∂ L ( y , y ^ ) ∂ W 0 i = ∂ L ( y , y ^ ) ∂ z i ∂ z i ∂ W 0 i = ∂ L ( y , y ^ ) ∂ z i x [ start i : end i ] W 0 i ← W 0 i − η ⋅ ∂ L ( y , y ^ ) ∂ z i x [ start i : end i ] (4) \begin{aligned}
\frac{\partial L(y, \hat{y})}{\partial \boldsymbol{W}_{0}^{i}} &=\frac{\partial L(y, \hat{y})}{\partial \boldsymbol{z}_{i}} \frac{\partial \boldsymbol{z}_{i}}{\partial \boldsymbol{W}_{0}^{i}}=\frac{\
partial L(y, \hat{y})}{\partial \boldsymbol{z}_{i}} \boldsymbol{x}\left[\operatorname{start}_{i}: \text { end }_{i}\right] \\ \boldsymbol{W}_{0}^{i} & \leftarrow \boldsymbol{W}_{0}^{i}-\eta \cdot \
frac{\partial L(y, \hat{y})}{\partial \boldsymbol{z}_{i}} \boldsymbol{x}\left[\operatorname{start}_{i}: \text { end }_{i}\right] \end{aligned}\tag{4} ∂W0i∂L(y,y^)W0i=∂zi∂L(y,y^)∂W0i∂zi=
∂zi∂L(y,y^)x[starti: end i]←W0i−η⋅∂zi∂L(y,y^)x[starti: end i](4)
It should be noted that although in Figure 2 F M \mathrm{FM} The parameters in FM point to the neurons in the Embedding layer, but their specific significance is to initialize the connection weight
between the Embedding neuron and the input neuron. Suppose the dimension of FM implicit vector is m m m. The first i i i characteristic fields i i i) section of k k The implicit vector of
k-dimensional feature is v i , k = ( v i , k 1 , v i , k 2 , ... , v i , k l , ... , v i , k m ) v_{i, k}=\left(v_{i, k}^{1}, v_{i, k}^{2}, \ldots, v_{i, k}^{l}, \ldots, v_{i, k}^{m}\right) vi,k =
(vi,k1, vi,k2,..., vi,kl,..., vi,km), then the second dimension of the implicit vector l l l dimension v i , k l v_{i, k}^{l} vi,kl# will become connected input neurons k k k and Embedding neurons l
l l initial value of connection weight between.
It should be noted that during training F M \mathrm{FM} In the process of FM, the feature domain is not distinguished, but F N N \mathrm{FNN} In FNN model, features are divided into different feature
domains, so each feature domain has a corresponding Embedding layer, and the Embedding dimension of each feature domain should be consistent with the FM implicit vector dimension.
Layer construction:
import tensorflow as tf
from tensorflow.keras.layers import Layer, Dense, Dropout
class FM_layer(Layer):
def __init__(self, k, w_reg=1e-4, v_reg=1e-4):
self.k = k
self.w_reg = w_reg
self.v_reg = v_reg
def build(self, input_shape):
self.w0 = self.add_weight(name='w0', shape=(1,),
self.w1 = self.add_weight(name='w1', shape=(input_shape[-1], 1),
self.v = self.add_weight(name='v', shape=(input_shape[-1], self.k),
def call(self, inputs, **kwargs):
linear_part = tf.matmul(inputs, self.w1) + self.w0
inter_part1 = tf.pow(tf.matmul(inputs, self.v), 2)
inter_part2 = tf.matmul(tf.pow(inputs, 2), tf.pow(self.v, 2))
inter_part = tf.reduce_sum(inter_part1-inter_part2, axis=-1, keepdims=True) / 2
output = linear_part + inter_part
return tf.nn.sigmoid(output)
class DNN_layer(Layer):
def __init__(self, hidden_units, output_dim, activation='relu', dropout=0.2):
self.hidden_layers = [Dense(i, activation=activation) for i in hidden_units]
self.output_layer = Dense(output_dim, activation=None)
self.dropout_layer = Dropout(dropout)
def call(self, inputs, **kwargs):
x = inputs
for layer in self.hidden_layers:
x = layer(x)
x = self.dropout_layer(x)
output = self.output_layer(x)
return tf.nn.sigmoid(output)
Model setup:
FNN is a two-stage training mode, so there are two separate models: FM and DNN.
from layer import FM_layer, DNN_layer
from tensorflow.keras.models import Model
class FM(Model):
def __init__(self, k, w_reg=1e-4, v_reg=1e-4):
self.fm = FM_layer(k, w_reg, v_reg)
def call(self, inputs, training=None, mask=None):
output = self.fm(inputs)
return output
class DNN(Model):
def __init__(self, hidden_units, output_dim, activation='relu'):
self.dnn = DNN_layer(hidden_units, output_dim, activation)
def call(self, inputs, training=None, mask=None):
output = self.dnn(inputs)
return output
Model training code:
When the code is implemented, I write the training process of the two models together, which can complete the training end-to-end.
from model import FM, DNN
from utils import create_criteo_dataset
import tensorflow as tf
from tensorflow.keras import optimizers, losses, metrics
from sklearn.metrics import accuracy_score
if __name__=='__main__':
file_path = 'criteo_sample.txt'
(X_train, y_train), (X_test, y_test) = create_criteo_dataset(file_path, test_size=0.5)
k = 8
#**************** Statement 1 of Training *****************#
model = FM(k)
optimizer = optimizers.SGD(0.01)
train_dataset = tf.data.Dataset.from_tensor_slices((X_train, y_train))
train_dataset = train_dataset.batch(32).prefetch(tf.data.experimental.AUTOTUNE)
model.compile(optimizer=optimizer, loss='binary_crossentropy', metrics=['accuracy'])
model.fit(train_dataset, epochs=200)
#**************** Statement 2 of Training *****************#
# Obtain the implicit vector matrix self.v obtained from FM training
v = model.variables[2] # [None, onehot_dim, k]
X_train = tf.cast(tf.expand_dims(X_train, -1), tf.float32) #[None, onehot_dim, 1]
# Multiply the original input by the implicit vector matrix self.v
X_train = tf.reshape(tf.multiply(X_train, v), shape=(-1, v.shape[0]*v.shape[1])) #[None, onehot_dim*k]
hidden_units = [256, 128, 64]
model = DNN(hidden_units, 1, 'relu')
optimizer = optimizers.SGD(0.0001)
train_dataset = tf.data.Dataset.from_tensor_slices((X_train, y_train))
train_dataset = train_dataset.batch(32).prefetch(tf.data.experimental.AUTOTUNE)
model.compile(optimizer=optimizer, loss='binary_crossentropy', metrics=['accuracy'])
model.fit(train_dataset, epochs=50)
3.DeepFM - replace the Wide part with FM
FNN and PNN models still have an obvious unsolved disadvantage: little is learned about the low-order combined features, which is mainly caused by the serial mode of FM and DNN. That is, although FM
learns the low-order feature combination, the full connection structure of DNN leads to the poor performance of the low-order features at the output of DNN. It seems that we have found the problem.
Improving the serial mode to the parallel mode can better solve this problem. So Google proposed the wide & deep model, but if we deeply explore the composition of wide & deep, although the structure
of the whole model is adjusted to a parallel structure, in actual use, some parts of the Wide Module need more sophisticated feature engineering, In other words, manual processing has a great impact
on the effect of the model (this can be verified in the wide & deep model part).
DeepFM integrates the model structure of FM with the wide & deep model. The deep network on the right and FM on the left are shown in Figure 3.
chart 3 : D e e p F M model type junction structure chart Figure 3: structure diagram of DeepFM model Figure 3: structure diagram of DeepFM model
The improvement of DeepFM to wide & Deep model is that it replaces the original wide part with FM, and strengthens the ability of feature combination of shallow network. As shown in Figure 3, the FM
part on the left and the Deep neural network part on the right share the same Embedding layer. The FM part on the left intersects the Embedding of different feature domains, that is, the Embedding
vector is regarded as the feature hidden vector in the original FM. Finally, the output of FM and the output of Deep part are input into the final output layer to participate in the final target
The mathematical form of FM in equation 1, FM second-order intersection:
Finally, we can get:
y ^ ( X ) = ω 0 + ∑ i = 1 n ω i x i + 1 2 ∑ f = 1 k [ ( ∑ i = 1 n v i , f x i ) 2 − ∑ i = 1 n v i , f 2 x i 2 ] (5) \hat{y}(X)=\omega_{0}+\sum_{i=1}^{n} \omega_{i} x_{i}+\frac{1}{2} \sum_{f=1}^{k}\
left[\left(\sum_{i=1}^{n} v_{i, f} x_{i}\right)^{2}-\sum_{i=1}^{n} v_{i, f}^{2} x_{i}^{2}\right]\tag{5} y^(X)=ω0+i=1∑nωixi+21f=1∑k⎣⎡(i=1∑nvi,fxi)2−i=1∑nvi,f2xi2⎦⎤(5)
ω 0 \omega_{0} ω 0 = global offset;
ω i \omega_{i} ω i ， is the second part of the model i i Weight of i variables;
ω i j = < v i , v j > \omega_{i j}=<v_{i}, v_{j}> ω ij = < VI, vj > characteristic i i i and j j Cross weight of j;
v i v_{i} vi ， is the implicit vector of the second dimensional feature;
< ⋅ , ⋅ > < \cdot , \cdot > < ⋅, ⋅ > represents vector dot product;
So, parameters ω 0 \omega_{0} ω The gradient of 0 is 1, ω i \omega_{i} ω The gradient of i + x i , v i , f x_{i}, v_{i, f} The gradient of xi, vi,f + ∑ f = 1 k ∑ i = 1 n v i , f x i − v i , f x i 2 \
sum_{f=1}^{k} \sum_{i=1}^{n} v_{i, f} x_{i}-v_{i, f} x_{i}^{2} ∑f=1k∑i=1nvi,fxi−vi,fxi2
Calculate FM first:
• The continuous field can be implemented with the Dense(1) layer
• Embedded (n, 1) is used for single valued discrete field. N is the number of median values in the classification
• Multivalued discrete field can take multiple eigenvalues at the same time. In order to batch training, zero padding must be performed on the sample. It can also be implemented with Embedding,
because there are multiple Embedding, and the average value can be taken down.
from keras import backend as K
from keras.engine.topology import Layer
import tensorflow as tf
class MyMeanPool(Layer):
def __init__(self, axis, **kwargs):
self.supports_masking = True
self.axis = axis
super(MyMeanPool, self).__init__(**kwargs)
def compute_mask(self, input, input_mask=None):
# need not to pass the mask to next layers
return None
def call(self, x, mask=None):
if mask is not None:
if K.ndim(x)!=K.ndim(mask):
mask = K.repeat(mask, x.shape[-1])
mask = tf.transpose(mask, [0,2,1])
mask = K.cast(mask, K.floatx())
x = x * mask
return K.sum(x, axis=self.axis) / K.sum(mask, axis=self.axis)
return K.mean(x, axis=self.axis)
def compute_output_shape(self, input_shape):
output_shape = []
for i in range(len(input_shape)):
if i!=self.axis:
return tuple(output_shape)
# coding:utf-8
from keras.layers import *
from keras.models import Model
from MyMeanPooling import MyMeanPool
from keras.utils import plot_model
'''Input Layers'''
# numeric fields
in_score = Input(shape=[1], name="score") # None*1
in_sales = Input(shape=[1], name="sales") # None*1
# single value categorical fields
in_gender = Input(shape=[1], name="gender") # None*1
in_age = Input(shape=[1], name="age") # None*1
# multiple value categorical fields
in_interest = Input(shape=[3], name="interest") # None*3, maximum length 3
in_topic = Input(shape=[4], name="topic") # None*4, maximum length 4
'''First Order Embeddings'''
numeric = Concatenate()([in_score, in_sales]) # None*2
dense_numeric = Dense(1)(numeric) # None*1
emb_gender_1d = Reshape([1])(Embedding(3, 1)(in_gender)) # None*1, 3 gender values
emb_age_1d = Reshape([1])(Embedding(10, 1)(in_age)) # None*1, 10 ages
emb_interest_1d = Embedding(11, 1, mask_zero=True)(in_interest) # None*3*1
emb_interest_1d = MyMeanPool(axis=1)(emb_interest_1d) # None*1
emb_topic_1d = Embedding(22, 1, mask_zero=True)(in_topic) # None*4*1
emb_topic_1d = MyMeanPool(axis=1)(emb_topic_1d) # None*1
'''compute first order'''
y_first_order = Add()([dense_numeric,
emb_topic_1d]) # None*1
'''define model'''
model = Model(inputs=[in_score, in_sales,
in_gender, in_age,
in_interest, in_topic],
'''plot model'''
plot_model(model, 'model.png', show_shapes=True)
Realize FM quadratic term
from keras import backend as K
from keras.engine.topology import Layer
import tensorflow as tf
class MySumLayer(Layer):
def __init__(self, axis, **kwargs):
self.supports_masking = True
self.axis = axis
super(MySumLayer, self).__init__(**kwargs)
def compute_mask(self, input, input_mask=None):
# do not pass the mask to the next layers
return None
def call(self, x, mask=None):
if mask is not None:
# mask (batch, time)
mask = K.cast(mask, K.floatx())
if K.ndim(x)!=K.ndim(mask):
mask = K.repeat(mask, x.shape[-1])
mask = tf.transpose(mask, [0,2,1])
x = x * mask
if K.ndim(x)==2:
x = K.expand_dims(x)
return K.sum(x, axis=self.axis)
if K.ndim(x)==2:
x = K.expand_dims(x)
return K.sum(x, axis=self.axis)
def compute_output_shape(self, input_shape):
output_shape = []
for i in range(len(input_shape)):
if i!=self.axis:
if len(output_shape)==1:
return tuple(output_shape)
# coding:utf-8
from keras.layers import *
from keras.models import Model
from MyMeanPooling import MyMeanPool
from MySumLayer import MySumLayer
from keras.utils import plot_model
'''Input Layers'''
# numeric fields
in_score = Input(shape=[1], name="score") # None*1
in_sales = Input(shape=[1], name="sales") # None*1
# single value categorical fields
in_gender = Input(shape=[1], name="gender") # None*1
in_age = Input(shape=[1], name="age") # None*1
# multiple value categorical fields
in_interest = Input(shape=[3], name="interest") # None*3, maximum length 3
in_topic = Input(shape=[4], name="topic") # None*4, maximum length 4
latent = 8
'''Second Order Embeddings'''
emb_score_Kd = RepeatVector(1)(Dense(latent)(in_score)) # None * 1 * K
emb_sales_Kd = RepeatVector(1)(Dense(latent)(in_sales)) # None * 1 * K
emb_gender_Kd = Embedding(3, latent)(in_gender)
emb_age_Kd = Embedding(10, latent)(in_age)
emb_interest_Kd = Embedding(11, latent, mask_zero=True)(in_interest) # None * 3 * K
emb_interest_Kd = RepeatVector(1)(MyMeanPool(axis=1)(emb_interest_Kd)) # None * 1 * K
emb_topic_Kd = Embedding(22, latent, mask_zero=True)(in_topic) # None * 4 * K
emb_topic_Kd = RepeatVector(1)(MyMeanPool(axis=1)(emb_topic_Kd)) # None * 1 * K
emb = Concatenate(axis=1)([emb_score_Kd,
emb_topic_Kd]) # None * 9 * K
summed_features_emb = MySumLayer(axis=1)(emb) # None * K
summed_features_emb_square = Multiply()([summed_features_emb,summed_features_emb]) # None * K
squared_features_emb = Multiply()([emb, emb]) # None * 6 * K
squared_sum_features_emb = MySumLayer(axis=1)(squared_features_emb) # Non * K
sub = Subtract()([summed_features_emb_square, squared_sum_features_emb]) # None * K
sub = Lambda(lambda x:x*0.5)(sub) # None * K
y_second_order = MySumLayer(axis=1)(sub) # None,
model = Model(inputs=[in_score, in_sales,
in_gender, in_age,
in_interest, in_topic],
plot_model(model, 'model.png', show_shapes=True)
Implement a DNN:
This part is a simple fully connected network, but the input used in it: Dense Embeddings, is on the same layer as the FM part above. DNN shares the sense embedding with FM, which is also the biggest
difference between DeepFM and wide & deep models.
In FM part, sense embedding is used to calculate second-order cross feature information, but in DNN part, sense embedding is used to provide input for higher-order cross feature information (it is
generally considered that DNN can learn high-order feature combination information). Therefore, Dense Embedding encodes the information required by low-order combined features and high-order combined
features at the same time. The author believes that this method is more conducive to the improvement of model performance. Moreover, this way of sharing parameters does not need additional feature
engineering and saves the time of model construction.
The mathematical definition is as follows:
a ( 0 ) = [ e 1 , e 2 , ... , e m ] (6) a^{(0)}=\left[e_{1}, e_{2}, \ldots, e_{m}\right]\tag{6} a(0)=[e1,e2,...,em](6)
among a ( 0 ) a^{(0)} a(0) indicates the output of Dense Embedding, e i e_{i} ei = second i i i eigenvectors corresponding to fields.
a ( l + 1 ) = σ ( W ( l ) a ( l ) + b ( l ) ) (7) a^{(l+1)}=\sigma\left(W^{(l)} a^{(l)}+b^{(l)}\right)\tag{7} a(l+1)=σ(W(l)a(l)+b(l))(7)
among a ( l ) , W ( l ) , b ( l ) a^{(l)}, W^{(l)}, b^{(l)} A (L), w (L) and B (L) are the third l l l layer output, parameters and bias direct terms, σ \sigma σ Is the activation function. Then, the
output of DNN part can be expressed as:
y D N N = σ ( W ( H ) a ( H ) + b ( H ) ) (8) y_{D N N}=\sigma\left(W^{(H)} a^{(H)}+b^{(H)}\right)\tag{8} yDNN=σ(W(H)a(H)+b(H))(8)
H H H indicates the number of hidden layers.
from keras import backend as K
from keras.engine.topology import Layer
import tensorflow as tf
import numpy as np
class MyFlatten(Layer):
def __init__(self, **kwargs):
self.supports_masking = True
super(MyFlatten, self).__init__(**kwargs)
def compute_mask(self, inputs, mask=None):
if mask==None:
return mask
return K.batch_flatten(mask)
def call(self, inputs, mask=None):
return K.batch_flatten(inputs)
def compute_output_shape(self, input_shape):
return (input_shape[0], np.prod(input_shape[1:]))
'''deep parts'''
y_deep = MyFlatten()(emb) # None*(6*K)
y_deep = Dropout(0.5)(Dense(128, activation='relu')(y_deep))
y_deep = Dropout(0.5)(Dense(64, activation='relu')(y_deep))
y_deep = Dropout(0.5)(Dense(32, activation='relu')(y_deep))
y_deep = Dropout(0.5)(Dense(1, activation='relu')(y_deep))
model = Model(inputs=[in_score, in_sales,
in_gender, in_age,
in_interest, in_topic],
plot_model(model, 'model.png', show_shapes=True)
Complete DeepFM
So far, we have completed the analysis of the two modules (really simple, the avenue is simple), and simply combined formula (6) and formula (8) to obtain the initial formula definition:
y ^ = sigmoid ( y F M + y D N N ) (9) \hat{y}=\operatorname{sigmoid}\left(y_{F M}+y_{D N N}\right)\tag{9} y^=sigmoid(yFM+yDNN)(9)
y = Concatenate(axis=1)([y_first_order, y_second_order,y_deep])
y = Dense(1, activation='sigmoid')(y)
model = Model(inputs=[in_score, in_sales,
in_gender, in_age,
in_interest, in_topic],
plot_model(model, 'model.png', show_shapes=True)
4. NFM-FM neural network attempt
introduce F M \mathrm{FM} The limitations of FM have been mentioned by the author: whether it is F M \mathrm{FM} FM, or its improved model F F M \mathrm{FFM} FFM, in the final analysis, is a
second-order feature crossover model. Troubled by the combinatorial explosion problem, FM can hardly be extended to more than third order, which inevitably limits the expression ability of FM model.
So, is it possible to use the stronger expression ability of deep neural network to improve FM model?
In terms of mathematical form, the main idea of NFM model is to replace the inner product of the second-order implicit vector in the original FM with a function with stronger expression ability (as
shown in 9).
y ^ F M ( x ) = w 0 + ∑ i = 1 N w i x i + ∑ i = 1 N ∑ j = i + 1 N v i T v j ⋅ x i x j \begin{aligned} &\hat{y}_{\mathrm{FM}}(x)=w_{0}+\sum_{i=1}^{N} w_{i} x_{i}+\sum_{i=1}^{N} \sum_{j=i+1}^{N} v_{i}^
{\mathrm{T}} v_{j} \cdot x_{i} x_{j} \\ \end{aligned} y^FM(x)=w0+i=1∑Nwixi+i=1∑Nj=i+1∑NviTvj⋅xixj
y ^ N F M ( x ) = w 0 + ∑ i = 1 N w i x i + f ( x ) (9) \hat{y}_{\mathrm{NFM}}(x)=w_{0}+\sum_{i=1}^{N} w_{i} x_{i}+f(x)\tag{9} y^NFM(x)=w0+i=1∑Nwixi+f(x)(9)
If we design with the idea of traditional machine learning N F M \mathrm{NFM} Functions in NFM model f ( x ) f(x) f(x), then it is bound to construct a more expressive function through a series of
mathematical derivation. However, after entering the era of deep learning, because the deep learning network has the ability to fit any complex function in theory, f ( x ) f(x) The construction of f
(x) can be completed by a deep learning network and learned by gradient back propagation. In the NFM model, the neural network structure used to replace the second-order part of FM is shown in Figure
Note that this section does not cover the first-order term and bias, and the complete NFM covers the three.
chart 6 : N F M Yes junction structure chart Figure 6: structure diagram of NFM Figure 6: structure diagram of NFM
The calculation of the embedding vector is consistent with the model described earlier and can be obtained through the lookup table. The final input eigenvector is composed of input eigenvalues x i
x_i xi) and embedding vector v i v_i vi) multiply to obtain, i.e V x = { x 1 v 1 , ... , x n v n } V_x=\{x_1v_1,\dots,x_nv_n\} Vx={x1v1,...,xnvn} .
The feature of NFM network architecture is very obvious, that is, a feature cross pooling layer (. Bi interaction pooling layer) is added between the Embedding layer and multi-layer neural network.
The specific operation of the characteristic cross pool layer is shown in equation (10).
f B I ( V x ) = ∑ i = 1 n ∑ j = i + 1 n x i v i ⊙ x j v j (10) \begin{aligned} f_{BI}(V_x)=\sum_{i=1}^n\sum_{j=i+1}^nx_iv_i \odot x_jv_j \end{aligned}\tag{10} fBI(Vx)=i=1∑nj=i+1∑nxivi⊙xjvj
Among them, ⊙ \odot ⊙ means that the corresponding elements of two vectors are multiplied, and the result is a vector. Therefore, the Bi interaction layer intersects the embedded vectors in pairs ⊙ \
odot ⊙ operation, then sum all vectors with corresponding elements, and finally f B I ( V x ) f_{BI}(V_x) fBI (Vx) is a vector after pooling.
The calculation time complexity of formula (10) is O ( k n 2 ) O(kn^2) O(kn2) ， k k k is the embedding vector dimension, similar to FM. Formula (10) can be further rewritten as:
f B I ( V x ) = 1 2 [ ( ∑ i = 1 n x i v i ) 2 − ∑ i = 1 n ( x i v i ) 2 ] (11) \begin{aligned} f_{BI}(V_x)=\frac{1}{2}\left[\left(\sum_{i=1}^nx_iv_i\right)^2-\sum_{i=1}^n\left(x_iv_i\right)^2\right]
\end{aligned} \tag{11} fBI(Vx)=21⎣⎡(i=1∑nxivi)2−i=1∑n(xivi)2⎦⎤(11)
The time complexity after rewriting is O ( k N x ) O(kN_x) O(kNx), where N x N_x Nx is the input characteristic X X Number of non-zero elements of X. Compared with the second-order cross term in FM,
Bi interaction layer does not introduce additional parameters, and can also be trained with linear time complexity, which is a very good property.
DNN is defined as follows:
z 1 = σ 1 ( W 1 f B I ( V x ) + b 1 ) z 2 = σ 2 ( W 2 z 1 + b 2 ) ... ... z L = σ L ( W L z L − 1 + b L ) (12) \begin{aligned} z1={}&\sigma_1(W_1f_{BI}(V_x)+b_1) \\ z2={}&\sigma_2(W_2z_1+b_2) \\ &\
dots \dots \\ z_L={}&\sigma_L(W_Lz_{L-1}+b_L) \\ \end{aligned}\tag{12} z1=z2=zL=σ1(W1fBI(Vx)+b1)σ2(W2z1+b2)......σL(WLzL−1+bL)(12)
The last hidden layer plus a linear transformation is output as the result, that is:
f ( X ) = h T z L (13) f(X)=h^Tz_L\tag{13} f(X)=hTzL(13)
Finally, formula (9) can be expressed as formula (14)
y ^ N F M ( X ) = w 0 + ∑ i = 1 n w i x i + f ( X ) = w 0 + ∑ i = 1 n w i x i + h T σ L ( W L ( ... σ 1 ( W 1 f B I ( V x ) + b 1 ) ... ) + b L ) (14) \begin{aligned} \hat{y}_{NFM}(X)={}&w_0+\sum_
{i=1}^{n}w_ix_i+f(X) \\ ={}&w_0+\sum_{i=1}^{n}w_ix_i \\ +&h^T\sigma_L(W_L(\dots \sigma_1(W_1f_{BI}(V_x)+b_1)\dots)+b_L) \end{aligned}\tag{14} y^NFM(X)==+w0+i=1∑nwixi+f(X)w0+i=1∑nwixihTσL
h h h does not enhance the characterization ability of FM, because this parameter can be absorbed into the embedding vector of the feature. In other words, even if h h h is not a vector with all 1,
and we can also regard it as the equivalent model of FM. Carefully observe formula (14). The structural diagram of Figure 6 above only corresponds to the structural diagram in formula f ( X ) f(X)
Item f(X). If global offset w 0 w_0 w0} and first order terms ∑ i = 1 n w i x i \sum_{i=1}^nw_ix_i Considering Σ i=1n wi xi, in fact, the structure diagram of NFM is very similar to wide & deep, but
the second-order term in NFM and DNN are in series structure. The left side of NFM can also be regarded as an LR model, but unlike wide & deep, the LR model on the left side of NFM only inputs single
features and does not send combined features into the LR model, so there is no need for additional feature engineering.
reference resources
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Understanding Factorials
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This approach can be especially tempting in high school mathematics courses, which can be consistently frustrating for many students as they find the dizzying array of variables, numbers, and
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Once you fall behind in a high school math course, it can be nearly impossible to get caught up. High school mathematics courses are usually dense, introducing new content at a fast page;
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Boosting the participation ratio during our explanations
How can we ensure more students listen, think and understand during our explanations and worked examples?
A couple of newsletters ago, I shared an exercise I often do with the maths departments I support to encourage them to reflect upon the participation ratio at various stages in their lessons. As a
reminder, a good way to reflect on the participation ratio is to ask yourself:
How easy would it be for a student in this phase of the lesson to be either not listening, not thinking, or not understanding, and you not pick up on it?
I asked teachers who read the post to choose the phase of their lesson where they felt they needed to boost the participation ratio the most. The results suggest we have a clear winner: the
explanation and worked example phase.
This is also the phase of the lesson where I need to boost the participation ratio, so I have been thinking about it a lot. So, in this post, I am going to share five ideas for boosting the
participation ratio during explanations and worked examples.
Thanks for reading Eedi Newsletter! Subscribe for free to receive new posts and support my work.
What not to do
An obvious way to boost the participation ratio is to ask students lots of questions during an explanation or worked example. But we need to be careful what we ask them questions about. If we
question students about a new idea that they know little about (What do you think we do first? Why can't we do this?) there is a danger that our explanation or worked example descends into a game of
Guess what is in my head. I have both seen and delivered explanations like this. They take ages and are inherently confusing to students as wrong answers fly around the room until someone - usually
by a process of elimination - stumbles upon the correct answer.
Idea 1: Check for listening
So, if we don't ask our students questions about new knowledge, then what could we ask them questions about? Well, first we could simply check if they are listening. If our students are not paying
attention, then it does not matter how clear our explanation is as they will not understand it. The simplest check for listening is to ask students to repeat something you have just said. This could
be done using Cold Call (What did I just say the subject of this formula is... Holly?) or Call and Respond (The first thing we do to both sides of the equation is... 3, 2, 1... MULTIPLY BY 3!).
Once students realise you will be regularly checking for listening, they have an added incentive to be paying attention.
I discuss the importance of high-frequency checking for listening during explanations with Science teacher Pritesh Raichura on an upcoming episode of my Mr Barton Maths Podcast (due for release 28th
June). As a sneak preview of our discussion, check out these graphs of two different classrooms from Pritesh’s wonderful blog post:
Idea 2: Ask questions about the prerequisite knowledge you have just assessed
Another thing we could question our students about during an explanation or worked example is skills they have met before which are key to this new idea - in other words, the prerequisite knowledge.
Now, we only want to do this if we have recently assessed that prerequisite knowledge and have evidence that it is secure. If we don't, then we run the risk that we ask students a question that we
assume they know the answer to, but it turns out they don't, and then we are forced to pause our explanation or worked example and intervene.
Again, assessing prerequisite knowledge during an explanation or worked example works well via Cold Call or Call and Respond. But you can also use mini-whiteboards:
So, we have seen we need to multiply both sides by 3. On your mini-whiteboards, please write down what the left-hand side of the equation will look like after we multiply by 3.
Idea 3: Use Silent Teacher
Whilst I like to check for listening and ask questions about prerequisite knowledge during an explanation, when it comes to modelling a worked example I like to start with my Silent Teacher approach.
For those not familiar, this is where I model the worked example in silence, pausing at key points to challenge my students to consider: What has he just done? What do I think he will do next?
Now, of course, whilst I am creating the optimal conditions for students to focus and think hard by removing any distractions, it would be very easy for a student to not participate at all during
Silent Teacher. They could simply stare at the board whilst their mind wanders elsewhere. That is why it is important I have two whole-class checks for understanding coming up next. It is also
important that my students are aware of this so they have the incentive to engage.
Idea 4: Check for understanding using Step by Step
The first of these whole-class checks for understanding following the worked example is called Step by Step. Here, I give my students a problem to solve that is of a similar difficulty to the worked
example, but first I ask them to just write the first step on their mini-whiteboards. For example, if the problem I wanted them to solve is:
I would say:
On your mini-whiteboards, write down the first operation we are going to do to both sides of the equation.
Students would hover their boards face-down to indicate they are ready, and then when I say would hold their boards up so I could see in 3, 2, 1...
If all students have got this correct, I would move on to the next step:
Good. So, we subtract b-squared from both sides of the equation. On your mini-whiteboards, write down what the left-hand side of the equation would look like after we subtract b-squared.
And so on.
Step by Step may also be suited to assessment via Call and Respond. For example, I could ask the following to ensure students have read the question correctly:
We are going to do a Call and Respond. What letter do we want to make the subject of the formula… (wait)… 3, 2, 1…
There are two things I like about this stage of the worked example process:
1. All students are involved, so the participation ratio is high
2. If something goes wrong, then I know the exact part of the process where it happens, and I can pause and intervene accordingly
Idea 5: Check for understanding using the Tick Trick
Following Step by Step, I want to see if students can solve a similar problem on their own from start to finish. I ask them to do this on their mini-whiteboards and show me when I ask. There is
likely to be too much information on each student's board for this to be a thorough check for understanding, but I certainly get a sense of whether all my students are participating. I can also be
tactical and pay particular attention to the boards of the weaker students in my class.
The way I assess this second example is to use Adam Boxer's Tick Trick. I ask students to put their boards down and watch me. I then write the first step of the solution on my board, and say to the
If you have written exactly what I have written, then give yourself a tick.
I ask students to hold their boards up again to show me their tick so I can see if everyone is participating and pick up on any students who have done something different.
Then I do the next step of the solution, and again ask students to tick if their step matches mine.
Here is what I love about the Tick Trick:
1. Again, everyone is involved, so the participation ratio is high
2. Again, I can pick up the exact stage of the process where students are struggling
3. It forces students to look carefully at their working out, and not just the final answer
4. If there is a particular way I want students to set out a solution, then this is a great way of making sure they are doing it
So, here are five ways we could boost the participation ratio during explanations and worked examples:
1. Check for listening
2. Ask questions about prerequisite knowledge you have just assessed
3. Use Silent Teacher
4. Check for understanding using Step by Step
5. Check for understanding using the Tick Trick
Here are some questions to consider:
• Which of these do you already do?
• Which of these could you try?
• What would you need to change to make them work for you?
• What other things do you do that help boost the participation ratio during the explanation or worked example?
Please let me know in the comments.
Three final things from Craig
1. Have you listened to my 3.5+ hour epic podcast on How to lead a maths department?
2. You can see the back catalogue of all my Eedi newsletters and Tips for Teachers newsletters here
3. Have you checked out my Tips for Teachers book, with over 400 ideas to try out the very next time you step into a classroom?
If you found this newsletter useful, subscribe (for free!) so you never miss an edition, share it with one of your colleagues, or let me know your thoughts by leaving a comment.
Thanks so much for taking the time to read this, and have a great week!
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Cherry Picking: Consumer Choices in Swarm Dynamics, Considering Price and Quality of Goods
CIEM CONICET and FaMAF, National University of Cordoba, 5000 Córdoba, Argentina
School of Science of Nature, University of Torino, 10100 Torino, Italy
School of Economics, University of Torino, 10100 Torino, Italy
Fondazione Collegio Carlo Alberto, 10100 Torino, Italy
Author to whom correspondence should be addressed.
Submission received: 23 October 2020 / Revised: 17 November 2020 / Accepted: 18 November 2020 / Published: 20 November 2020
This paper proposes a further development of the mathematical theory of swarms to behavioral dynamics of social and economic systems, with an application to the modeling of price series in a market.
The complexity features of the system are properly described by modeling the asymmetric interactions between buyers and sellers, specifically considering the so-called cherry picking phenomenon, by
which not only prices but also qualities are considered when buying a good. Finally, numerical simulations are performed to depict the predictive ability of the model and to show interesting emerging
behaviors, as the coordination of buyers and their division in endogenous clusters.
1. Objectives and Plan of the Paper
This paper is devoted to develop an approach based on a suitable development of the theory of swarms, arguably initiated by the celebrated paper by Cucker and Smale [
], applied to behavioral dynamics of social and economic systems with particular focus on the study of price sequences. The behavioral swarm theory approach to the dynamics of prices was recently
introduced in [
] and this present paper aims at providing a deeper insight in the role of cherry picking and asymmetric interactions.
The modeling and simulations of large systems of interacting behavioral entities have been developed by methods of the so-called kinetic theory of active particles, in short KTAP, as well as by
recent developments of the theory of swarms. For additional details, the interested reader is addressed to the KTAP approach [
], to the kinetic theory approach by mean field and Fokker-Plank models [
], and to the mathematical theory of behavioral swarms [
]. All approaches refer to large systems of interacting living entities, called active particles, whose state at the microscopic scale, or shortly micro-scale, includes well defined social and/or
economic variables which are heterogeneously distributed over active particles.
The common feature of all different kinetic theory methods is that the overall state of the system is described by a distribution function over the micro-state. This distribution accounts for the
overall heterogeneity of the system. In the case of behavioral swarms, the overall state of the system is delivered by a whole set of micro-states. An additional common feature is that mathematical
models are obtained by inserting into general mathematical structures, which differ for each of the aforementioned approaches, models of the dynamics of interactions. Indeed, this rationale is
followed also in our paper.
The KTAP has been applied to model several socio-economic phenomena, among others, propagation of opinion formation and credit risk over networks [
], idiosyncratic learning [
], opinion dynamics [
] and social inequality [
], while additional bibliography is provided in [
]. Another example about the use of kinetic theory to explain the market mechanisms is in [
] where the microscopic description leads to a system of linear Boltzmann-type equations.
An alternative method is the so-called agent-based modeling (ABM). Although not mathematically fully grounded, it is useful to understand complex dynamics. The agents are commonly implemented in
software as objects, with internal rules. With the model, it is possible to instantiate a population agent, observing what is emerging by the actions and interactions that produce following their
rules. From the formal point of view, ABMs are close to a narrative of reality, thanks to their flexibility. Still, they are also close to a mathematical structure if they adhere to a rigorous
representation via computer code [
As stated before, this paper proposes a development and a new vision of the approach proposed in the second part of [
], where price dynamics within a market was studied by a swarm theory approach. The scientific literature on swarms has been arguably initiated by physicists, while the interest of mathematicians has
been boosted by the visionary paper [
]. An overview of the vast literature on this topic is far beyond the aims of our paper which consists of understanding how the structure delivered by the classical mathematical theory of swarms [
] can be modified towards the modeling of social and economic systems. The first step is to set the variables that can describe the individual state of the interacting entities. If particles are
viewed as agents which carry a certain social variable, for instance a social or political opinion, an analogous structure has been used to study and control the dynamics of the collective behavior
of one or more populations [
]. The concept of topological interactions in swarms has been introduced in [
], by which interactions occur with a fixed number of individuals rather than with those belonging to an influence domain, and the mathematical formalization was further developed in [
The main objective of our paper consists of a detailed analysis of the role of the asymmetry in the interactions with respect to symmetric interactions, issue that has been recently treated in [
]. The aforementioned model presented [
] is enriched with a proper description of sticky prices and the so-called cherry picking, which assumes that—in addition to prices—quality is also an important factor to be considered. In [
] a useful economic discussion on market coordination by prices is provided. Other useful references are [
]. An important asymmetry of our work is represented by the stickiness of seller prices, which allows the market prices to stay stable. Stickiness is a crucial characteristic for seller prices,
especially when we consider the realistic phenomenon of
cherry picking,
as shown in [
]. The use of this novel modeling structure for micro-economic analyses of the markets has simultaneously two goals: (i) to observe the agent coordination on the two sides of the market and (ii)
investigate the effects of the information presence on the quality. Instead, in a classical model, we would have aggregate demand and offer curves related to a unique good in each market.
The presentation is as follows:
Section 2
provides a qualitative description of the behavioral economic systems object of the modeling approach. More in details, of the dynamics of prices under asymmetric interactions.
Section 3
first introduces the concept of cherry picking that is applied to derive two specific models focusing on specific types of asymmetries. Finally, is devoted to simulations and interpretation of the
computational results referring to the framework of economic sciences.
Section 4
looks ahead to possible research perspectives induced by a critical analysis of the contents of our paper.
2. Behavioral Dynamics of Prices
Let us consider a market in which
sellers and
buyers trade a specific good. According to the kinetic theory of active particles [
], sellers and buyers can be regarded as functional subsystems (FS). Within each FS, particles (i.e., sellers or buyers) express an activity which is heterogeneously distributed among them. In this
specific case, activity variables for sellers and buyers are the
price assigned by each seller
for this good and the
price that each buyer accepts to pay
for the good, respectively [
]. We introduce the following notation:
$u s$, $s = 1 , … , N$ corresponds to the first functional subsystem (sellers), where each s-firm expresses the price $u s$ of the product (good) offered for sale.
$w b$, $b = 1 , … , M$ corresponds the second functional subsystem (buyers), where each b-buyer expresses the price $w b$ that he/she accepts to pay.
The variables which define the activities within each FS are given by the vectors
$u = ( u 1 , … , u s , … , u N ) and w = ( w 1 , … , w b , … , w M ) ,$
while their corresponding speeds of change are
$v = ( v 1 , … , v s , … , v N ) and z = ( z 1 , … , z b , … , z M ) ,$
where if both prices and related speeds are normalized with respect to their highest value at initial time
$t = 0$
, we can assume that
$u 0 , v 0 ∈ [ 0 , 1 ] N$
$w 0 , z 0 ∈ [ 0 , 1 ] M$
. The dynamics can, however, generate values which do not belong to these intervals for larger times.
According to this representation,
-order moments within each FS can be computed by
$E s m = 1 N ∑ s = 1 N u s m and E b m = 1 M ∑ b = 1 M u b m ,$
where first (
$m = 1$
) and second (
$m = 2$
) order moments provide the expected (or mean) price and variance, respectively, while higher order moments give information on the distortion.
Following the reasonings exposed in [
], we assume that
Micro-micro interactions take place only across FSs, but not within the same FS. By these interactions, firms and customers adjust the price by direct contacts.
Macro-micro interactions take place within the same FS, but not across different ones. By these interactions, each seller adjusts her/his price according to the mean stream of sellers, while
customers adjust the price accounting for the mean stream of buyers.
Let us now introduce the following quantities deemed to model interactions among particles and between particles and FSs:
$η s b ( u s , w b )$ models the rate at which a seller s interacts with a buyer b;
$η b s ( w b , u s )$ models the rate at which a buyer b interacts with a seller s;
$μ s ( u s , E s )$ models the micro-macro interaction rate between a seller s and her/his own FS;
$μ b ( w b , E b )$ models the micro-macro interaction rate between a buyer b and her/his own FS;
$φ s b ( u s , w b , v s , z b )$ denotes the micro-micro action, which occurs with rate $η s b$, of a buyer b over a seller s;
$φ b s ( w b , u s , z b , v s )$ denotes the micro-micro action, which occurs with rate $η b s$, of a seller s over a buyer b;
$ψ s ( u s , E s )$ denotes the micro-macro action, which occurs with rate $μ s$ of the FS of sellers over a seller s;
$ψ b ( w b , E b )$ denotes the micro-macro action, which occurs with rate $μ b$ of the FS of buyers over a buyer b.
Accordingly, the mathematical structure corresponding to the setting given by Equation (
) in [
] is as follows:
$d u s d t = v s , d w b d t = z b , d v s d t = 1 M ∑ q = 1 M η s q ( u s , w q ) φ s q ( u s , w q , v s , z q ) + μ s ( u s , E s ) ψ s ( u s , E s ) , d z b d t = 1 N ∑ q = 1 N η b q ( w b , u q
) φ b q ( w b , u q , z b , v q ) + μ b ( w b , E b ) ψ b ( u b , E b ) ,$
$s = 1 , … N$
$b = 1 , … , M$
. This provides the framework to derive specific models by inserting into Equation (
) a detailed description of the interactions.
As remarked in [
], the system presents asymmetries, since the seller prices are public (e.g., advertised price tags), while buyer prices are unknown to the sellers. This feature is taken into account to properly
model the interactions terms:
• The interaction rates for both micro-micro and macro-micro interactions asymmetrically decay with the distance between the interacting entities starting from the same rates
$η 0$
$μ 0$
. In addition, when
increases with respect to
the interaction rates
both decrease by the so-called
sticking effect
$η s b ≃ η s = η 0 exp − ρ ε u s , η b s = η 0 exp − 1 ε | u s − w b | w b ,$
$ε = N / M$
is a non-negative parameter, and
$μ s = μ 0 , μ b = μ 0 exp − 1 ε | w b − E b | w b .$
• The actions
correspond to a dynamics of consensus driven by the difference between the seller and buyer prices, in the micro-micro interaction, and between the local price and the global one, in the
micro-macro interaction. The following model of interaction is proposed
$φ s b = α u s sign ( w b − u s ) , φ b s = β ( u s − w b ) ,$
$ψ s = γ ( E s − u s ) , ψ b = κ ( E b − u b ) ,$
$α , β , γ , κ$
are non-negative parameters.
If the interactions terms introduced in Equations (
) are replaced into the general structure (
), we get a system of ODEs that the describe the whole dynamics. This will be specified in the next section accounting for cherry picking.
3. Cherry Picking
3.1. Modeling Consumers as Cherry Pickers
cherry picking
to the model introduced in
Section 2
means that an agent chooses a specific other agent to interact with, under some conditions. In this scenario, each buyer chooses a specific seller basing her/his choice on the offered price and/or
quality of the good.
Assume that a level of quality of the product, denoted by $c s$, is assigned to each seller s. This quantity will remain constant during the whole process, which means that we are looking at it in
the short term and thus the agent is not able to improve or worsen the product quality. The buyers are now seen as “cherry pickers” because we start from a world in which each of them is aware of the
seller price, but not vice versa. That means that the buyer has more information than the seller (like in a mall or online shopping), so that it is difficult for the seller to know the buyer
“reservation quality” and “reservation price”.
After the buyer makes her/his choice, the price dynamics will remain: the buyer decides to buy if her/his reservation price is higher or equal to the one of seller she/he has chosen. To choose the
seller, the buyer must “visit” her/his shop (or online shopping site) to check the quality and/or the price of the product offered by every seller and compare them. So the buyer reservation price is
not changed by every seller price, because the buyer is just looking for the condition she/he made and then compares her/his price only with the price of the seller she/he has chosen. On the other
hand, the seller is aware of the visit of every buyer, and if she/he is not chosen then her/his price will go down.
Summarizing the above reasonings, the whole dynamics can be described as follows:
• Each buyer looks for the right seller (under the above-mentioned conditions), visiting and comparing the prices and quality offered by all of them.
• After choosing the right one, the buyer will compare their prices and buy if her/his reservation price is higher or equal than seller price (or not if it is not).
• If the buyer effectively makes the transaction, then her/his reservation price will go down (if not it will go up).
• Each seller is visited by every buyer. If they buy, then she/he will increase the price of the product (if not she/he will decrease it).
3.2. Derivation of Model 1
Let us first consider a scenario in which the buyer choice is conditioned by both features: quality and price. To make the model near the most to reality, we introduce three types of buyers:
• Type of buyer $B 1$, numbered from 1 to $a 1$, who always chooses the seller offering the highest quality product.
• Type of buyer $B 2$, numbered from $a 1 + 1$ to $a 2$, who always chooses the seller with the highest quality-price ratio.
• Type of buyer $B 3$, numbered from $a 2 + 1$ to M, who always chooses the seller with the lowest price.
Remark 1.
Another kind of buyer could have been the one choosing the seller with the highest price (for example in the case of a luxury good), but it is not present in every market and, most of all, it is a
low percentage of it. For the size we are reproducing now, it is negligible, so we will not consider it.
Remark 2.
Buyers belonging to each group $B 1$, $B 2$ and $B 3$ will act in different ways in the micro-micro interactions depending on their type. However, in the macro-micro interactions all buyers will
behave in the same way.
Let us now define the following quantities:
$s c m a x = arg max s ∈ { 1 , … , N } c s$ (for the sake of simplicity $s c$) is the seller offering the highest quality.
$s r m a x = arg max s ∈ { 1 , … , N } c s w s$ (for the sake of simplicity $s r$) is the seller with the highest quality-price ratio.
$s w m i n = arg min s ∈ { 1 , … , N } w s$ (for the sake of simplicity $s w$) is the seller offering the lowest price.
Introducing the above defined types of buyers and sellers in Equation (
), the system describing the dynamics under this scenario is:
$d u s d t = v s , d w b d t = z b , d v s d t = ( 1 a 1 ∑ q = 1 a 1 ( δ s s c η s q ( u s , w q ) φ s q ( u s , w q , v s , z q ) + δ s s c − 1 ) ( η s b α u s ) + + 1 a 2 − a 1 ∑ q = a 1 + 1 a 2 (
δ s s r η s q ( u s , w q ) φ s q ( u s , w q , v s , z q ) + δ s s r − 1 ) ( η s b α u s ) + + 1 M − a 2 ∑ q = a 2 + 1 M δ s s w η s q ( u s , w q ) φ s q ( u s , w q , v s , z q ) + δ s s w − 1 ) (
η s b α u s ) + + μ s ( u s , E s ) ψ s ( u s , E s ) , d z b 1 d t = η b 1 s c ( w b 1 , u s c ) φ b 1 s c ( w b 1 , u s c , z b 1 , v s c ) + μ b ( w b , E b ) ψ b ( w b , E b ) , d z b 2 d t = η b
2 s r ( w b 2 , u s r ) φ b 2 s r ( w b 2 , u s r , z b 2 , v s r ) + μ b ( w b , E b ) ψ b ( w b , E b ) , d z b 3 d t = η b 3 s w ( w b 3 , u s w ) φ b 3 s w ( w b 3 , u s w , z b 3 , v s w ) + μ b
( w b , E b ) ψ b ( w b , E b ) ,$
$δ x y$
denotes a delta Kronecker function, namely
$δ x y = 1$
$x = y$
$δ x y = 0$
Remark 3.
The functions used are the same used in the original model
, except for $η s b$, the one of the sellers. It is
$η s b ≃ η s = η 0 exp − ρ ε u s ,$
where ρ is a parameter and $ε = N M$. In this way we make the price of the sellers more “sticking", because cherry picking creates a sticking effect on the price of the picker (in this case the
buyer). Therefore, the aim is to balance this not wanted effect.
Remark 4.
The Kronecker function δ aims to classify the seller: if she/he is the chosen one by the buyer, then the price dynamics is the same as in the original model without cherry picking. If not, the term $
( δ s s c − 1 ) ( η s b α u s )$ will make the seller price go down for every buyer of that type, following the same lower-price rule introduced in
Remark 5.
Notice that the maximum (resp. minimum) value in the definitions of $c s$ and $s r$ (resp. $w s$), above can be eventually reached by more than one seller. If this is the case, a random seller will
be picked at random among those who attain the maximum (resp. minimum) value.
3.3. Derivation of Model 2
In this case, let us consider the “reservation quality” of the buyer, which is the minimum level of quality that she/he is willing to accept. We denote it as
$c b$
. The
cherry picking
consists of choosing the seller offering the lowest price among those with
$c s ≥ c b$
(so, among sellers with quality high at least as her/his own “reservation quality”, the buyer
will choose the one with lower price ). We denote the chosen seller as:
$s b m i n = arg min s ∈ { 1 , … , N } { w s | c s ≥ c b } ,$
and for the sake of simplicity we will refer to her/him as
$s b$
. This time, the seller will not be the same for all the buyers of the same type (as in the previous case). In principle, it could be different for every buyer and that is why it depends on
The overall dynamics are then described by the following system:
$d u s d t = v s , d w b d t = z b , d v s d t = ( 1 M ∑ q = 1 M δ s s q η s q ( u s , w q ) φ s q ( u s , w q , v s , z q ) + δ s s q − 1 ) ( η 0 α u s ) + + μ s ( u s , E s ) ψ s ( u s , E s ) , d
z b d t = η b s b ( w b , u s b ) φ b s b ( w b , u s b , z b , v s b ) + μ b ( w b , E b ) ψ b ( w b , E b ) ,$
where all the interaction functions are the same than in Model 1.
3.4. Numerical Results
In the following we perform numerical simulations for Models 1 and 2, which are based on some essential premises also assumed in [
• Prices are assumed to be ordered numbers.
• Productive factors do not change (capital and labor, here represented by the number of sellers).
• It is assumed the absence of new seller entries or existent seller exits in or from the market.
• The Statements (2) and (3) consequence is that any automatic price control mechanism is missing; instead, allowing the entry and exit mechanism, if prices go too high new sellers (firms) enter in
the market increasing the offer side and lowering the prices, and vice versa.
• Both in our construction and in reality—when price are exposed by the sellers (e.g., in the mall)—, buyers coordination is easier than that of the sellers, which ignore the reservation prices of
the buyers (the max price that a buyer accepts to pay); sellers blindly react step by step to their successes (made a sale) or failures (no sale) in dealing.
• Consistently with (5), buyers very well coordinate their reservation prices because they see all the set of the sellers, which on turn receive the reactions of all the other buyers; sellers
instead have to act on the basis of information collected observing buyer decisions without seeing their internal reservation prices; certainly, they have micro-macro (mean field) interactions
with the other sellers.
3.5. Numerical Results for Model 1
Let us first perform some numerical experiments by solving Equations (
) with
$N = 10$
sellers and
$M = 50$
buyers. In order to set initial conditions we consider that the initial prices, both for sellers and buyers, are taken randomly following a uniform distribution in the interval
$[ 1000 , 1005 ]$
while initial speeds are assumed to be all equal to 0.
Figure 1
shows the temporal evolution of the system taking
$η 0 = μ 0 = α = 1$
, and
$β = γ = 0.1$
$ρ = 2$
for a short term of
$T = 1000$
time steps. We can see that prices trend is made of regular waves maintaining same frequency and amplitude for each price, especially for the seller prices. The same behavior is observed in the price
variances, both for buyers and sellers, as shown in
Figure 2
Figure 3
represents the corresponding Pareto market efficiencies for short and long terms, which are calculated as follows:
Seller Pareto market efficiency is the sum, at every time t, of $P s − I c$ calculated at every exchange at a selling price $P s$ and for every seller with initial cost $I c$, fixed from the
beginning as $1 10$ of seller price.
Buyer Pareto market efficiency is the sum, at every time t, of $R p − P s$ calculated at every exchange at a selling price $P s$ and for every buyer with reservation price $R p$.
The total Pareto market efficiency is the sum of the two above.
Notice that Pareto market efficiencies show a sort of regular and cyclical trend, where the benefits of the market are practically all on sellers, because of the sticking prices that were introduced
for them and also for the choice made about the initial cost.
In addition, we aim to investigate the influence of some of the model parameters on the overall dynamics. For instance,
Figure 4
shows the trend for two values of
$η 0$
, namely
$η 0 = 1$
$η 0 = 0.1$
, while the other parameters keep the same value. Notice that there is a change in the ratio between the frequencies of the prices of the two different types of agents.
Both seller coordination and buyer differentiation in type are crucial. In particular, when we decrease seller coordination through
(which goes from
), buyer prices begin to change the amplitude of their waves during time and macro-waves appear in the long term, as shown in
Figure 5
Taking a closer look to macro-waves, we can see that they are well differentiated depending on buyer type. That means that a lower seller coordination, brings both to the formation of clusters in the
buyer functional subsystem, depending on their type, and to the aforementioned macro-waves. Recall that buyers
$B 1$
only seek the best quality, buyers
$B 2$
seek for the best quality-price ratio, while buyers
$B 3$
always choose the lowest price.
Figure 6
shows the dynamics for each type of buyer for different time intervals. In particular we use green for type
$B 1$
, purple for
$B 2$
and yellow for
$B 3$
. Although parameter
was reduced to
, all the other parameters keep the initially stated values. Three macro-waves emerge according to the type of buyer.
The stickiness of seller prices do not allow a visible change [for them] in their amplitude, as shown in
Figure 7
However, even a small change in seller prices trend (which are more free to adapt to buyer ones due to a lower coordination) brings to an amplified effect on buyer prices, creating three different
markets. Both the split and the macro-waves are a way for the buyer to reach (also creating it) the market they prefer. For example, macro-waves allow more often type
$B 1$
to have higher probability of grabbing the best quality. In this way, they also reach a higher Pareto market efficiency, as shown in
Figure 8
. A similar result deriving from buyer coordination is also in [
3.6. Numerical Results for Model 2
Recall that Model 2 assumes that each buyer has a reservation quality, which is the minimum level of quality that she/he is willing to accept for the product. Among those sellers satisfying the
quality requirement, the buyer will choose the option with lowest price.
Consider the same initial conditions than in the previous case.
Figure 9
shows the dynamics for the short term of individual and mean prices.
As in the first case, when the value of
is changed, we can see macro-waves and a the formation of clusters depending on their (this time) reservation quality.
Figure 10
shows the case in which the 50 buyers are divided into six reservation qualities that, ordered from larger to lower, will be represented in black, red, cyan, yellow, green and magenta. It is clear
that macro-waves emerge according to the reservation quality and this becomes especially clear for large times.
Notice that, as it usually happens in the simulations, in the short term (
Figure 10
a) there are only three different trends for the six individuated groups, analogously to the first case (
Figure 6
). But taking a look at the red and the black trends, if at the beginning (
Figure 10
a) they stay together, in the medium term (
Figure 10
c) we can see a slight differentiation in the frequency and, in the long term we can see that the red has got its own macrocycle (in
Figure 10
d in the second half). Therefore, we end up with four different clusters
If the aforementioned trend is the most common, the split of the trends can also change depending on the (random) initial conditions of prices. Indeed, we can also see a fewer clusters and a unique
cluster, as shown in
Figure 11
. That means that the formation of clusters is an endogenous effect. In this sense, we may state that the second model is a generalization of the first one, in the sense that buyers can organize both
in three clusters as in the first case, but also in more or less as it is more convenient for them.
4. Conclusions
In this paper, using [
] as a base for our work, we developed the study of price dynamics, applying theory of swarms to describe the interactions of the particles living in our world. We introduced variables that can be
seen as economic features. They are carried by particles that represent the agents divided in two different types: buyers and sellers. We showed a system in which the asymmetry between behaviors of
the two types is a fundamental characteristic and a crucial aspect for the obtained results. We study the dynamics of prices in a perfect competitive market where also the parameter of quality is
crucial. We used the idea of
cherry picking
performed by buyers, which creates a more realistic behavior of our agents. In this context, the model explains the realistic behavior of markets besides the limits of the classical microeconomics
models, with a unique price and a unique good; in the classical framework, goods with quality differences generate multiple markets. From that perspective, it is impossible to analyze the buyer
behavior in the face of quality differences. Considering micro-transactions with prices exposed by the sellers (so-called adhesion contracts), we can instead investigate the effects of the consumer
control about quality, e.g., in food and beverage markets, while cherry picking the products. The relevance of quality is related to goods with a limited range of prices. If the range is enormous,
the quality usually is consistent with the price levels.
Section 2
we present the basis of our world, that will be completed in
Section 3
. We set agent variables which are the price of the offered product for sellers and the reservation price for buyers. Price dynamics is based on the interaction (which affects the acceleration of
prices) between two agents of the opposite type (
interaction) and the interaction between an agent and the whole group it is part of (the
Section 3
we add the main characteristics of our model: the quality variable and
cherry picking
(buyers choose seller to interact with, basing their choice on seller prices and qualities). We develop two models. In the first one we add quality as seller parameter and we distinguish three types
of buyers on the different ways they choose sellers, every type basing its choice on different variables. In the second model we add also buyer reservation quality and every buyer chooses seller
basing both on seller quality, with respect to its reservation one, and seller price. In this further development, the asymmetry consists of
cherry picking
, in the
of sellers prices and in the idea that the buyer knows seller price and quality, but the seller does not know the buyer reservation price and quality (and this is reason it is the buyer to make the
Computational results are also shown in
Section 3
. If seller
interaction is set sufficiently high, we obtain a regular oscillating trend for both seller and buyer prices. Wave trend has a length of few interactions and the amplitude remains always constant in
time. Otherwise, if we lower the interaction among sellers, we see a more interesting behavior, which is the main result of our work. Seller prices do not seem to have an important change, while the
buyer ones show a change in the amplitude of price waves during time that in the long term, creates macro-waves (with long wavelength). Moreover, every price follows a different macro-cycle depending
on buyer type (for the first model) and buyer reservation quality (for second model). In this sense, the second case appear to be a generalization of the first one. We can explain this trend saying
that a higher freedom for sellers, not bounded by the medium seller price, creates a little change in their prices, which brings to an acceleration in buyer ones. However, to understand better the
economic reason behind this trend, we can see the effects on Pareto market efficiency, noticing that macro-waves are not only a way for buyer to “create” different markets to reach the best choice
for their condition, but also a way to increase their own Pareto market efficiency. Our results also suggest a concrete consequence on reality, especially considering the increasing of markets where
the competition and the number of relevant sellers are getting lower: when sellers create a sort of “agreement” about their prices (in the model, when they have a high interaction among them), buyers
suffer of a drawback. On the other hand, a more free market means a gain for buyers, without a significant loss for sellers.
Author Contributions
Conceptualization, D.K. and P.T.; Formal analysis, D.K. and V.S.; Methodology, V.S.; Software, V.S.; Supervision, D.K.; Writing—original draft, D.K. and V.S.; Writing—review and editing, D.K. All
authors read and approved the final manuscript.
D.K. was partially funded by CONICET Grant Number PIP 11220150100500CO and Secretaría de Ciencia y Técnica (UNC) Grant Number 33620180100326CB.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1. Seller and buyer prices and mean prices for a short term $T = 1000$. (a) Buyer prices, (b) seller prices, (c) buyer mean price and (d) seller mean price.
Figure 3. Red: buyers; blue: sellers; purple: total Pareto market efficiency. (a) Pareto market efficiency with $ρ = 0.1$, $γ = 0.1$, short term. (b) Buyer Pareto market efficiency with $ρ = 0.1$, $γ
= 0.1$, short term. (c) Pareto market efficiency with $ρ = 0.1$, $γ = 0.1$, long term.
Figure 5. Buyer prices trends for (a) $γ = 0.1$ and (b) $γ = 0.01$. In the first case prices range remains constant, in second case we can see macro-waves appearing.
Figure 6. Dynamics of buyer prices for different times intervals: (a) [0,1500], (b) [49,000,50,000], (c) [0,150,000]. Each color represents a buyer type, namely green $B 1$, purple $B 2$, yellow $B
3$. Blue in (c) is for seller prices that remain in the same constant interval, as in the previous case.
Figure 7. Comparing sellers prices trend for (a) $γ = 0.1$ and (b) $γ = 0.01$. Here with $ρ = 0.1$ and $η = 1$.
Figure 8. Red: buyers; blue: sellers; purple: total Pareto market efficiency. (a) Pareto market efficiency with $ρ = 0.1$, $γ = 0.01$, medium term. (b) Buyer Pareto market efficiency with $ρ = 0.1$,
$γ = 0.01$, medium term.
Figure 9. Sellers (blue) and buyers (red) prices and mean prices for a short term $T = 1000$, with $ρ = 2$. (a) individual prices (b) buyer mean price and (c) seller mean price.
Figure 10. Evolution of buyer prices for different time intervals: (a) [0,1500], (b) [49,000,50,000], (c) [0,150,000]. Buyers are divided into 6 reservation qualities that, ordered from larger to
lower, will be represented in black, red, cyan, yellow, green and magenta. Here, $ρ = 0.5$, $γ = 0.01$, $η = 1$.
Figure 11. Two different simulations showing evolution of buyer prices in which a unique cluster appears in the medium term. Here with $ρ = 0.5$, $γ = 0.01$, $η = 1$. (a) buyers can organize both in
three clusters. (b) generalization of (a).
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Knopoff, D.; Secchini, V.; Terna, P. Cherry Picking: Consumer Choices in Swarm Dynamics, Considering Price and Quality of Goods. Symmetry 2020, 12, 1912. https://doi.org/10.3390/sym12111912
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Knopoff D, Secchini V, Terna P. Cherry Picking: Consumer Choices in Swarm Dynamics, Considering Price and Quality of Goods. Symmetry. 2020; 12(11):1912. https://doi.org/10.3390/sym12111912
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Knopoff, Damian, Valeria Secchini, and Pietro Terna. 2020. "Cherry Picking: Consumer Choices in Swarm Dynamics, Considering Price and Quality of Goods" Symmetry 12, no. 11: 1912. https://doi.org/
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Who provides assistance with discrete event simulation and stochastic processes assignments? | Hire Someone To Do My Assignment
Who provides assistance with discrete event simulation and stochastic processes assignments? – Andrew Post, Maria de Waldeberg and David Petke Abstract The method to simulate two and three
dimensional real-time crowds [1] with several time-of-day conditions being applied is developed in order to show that any one or two-step time-dependent Poisson processes can be mapped into two-step
Poisson processes via the Poisson formula defined as the sum of the moments of probability of a process in the idealized case and that for any second-order Poisson process is in fact a non-null
probability density. The algorithm is based on a three-dimensional subspace based approach being defined for an abstract example problem discussed in the original paper [2]. It considers, for
simplicity, only the first block of time-of-day conditions. 1 Introduction The common standard in the field of stochastic processes is the Poisson process [1] and the method developed in the context
of data-dependent non-autotoxic models is based on the sum of the moments of probability of a Poisson process with one environmental variable, a time-of-day or user-specified fraction of time-step to
which the environmental variables are not a dependent variable, and an additional disturbance of the environment. In order to convert two or three-dimensional model problems be solved in such a way
that each of these may potentially change at will dig this to a total of the same number of problems, the time-dependent Poisson process was used to estimate the true Poisson rate (number of possible
paths available from the model on the test-bed data data) as one would represent as an unbiased distribution of the true Poisson time-dependent rate (rate of decrease in total number of possible
paths). Similar to the main arguments for the time-dependent Poisson process, the problem has two classical features [2]: first, two-step Poisson processes, which are simultaneously a solution to a
poisson equation and a random walk problem, etc; second, the three-dimensional subspace problem on the model in the case of either time-of-day or user-specified fraction of time-step of the
environmental variables being a conservative estimate of that estimate, and thus only for a single random walk problem: the estimation of the Poisson rate is even more powerful than for the process
of the same equation. A problem in this context is that of counting the probable number of paths available on the test-bed data due to the time-step and the environmental variables being a
conservative estimate; this is due to the random particle (or pseudo drift potential). It would be unwise, however, to attempt such a count if the number of known times that can be identified from
the data in the case of the time-step is zero, rather than from the environment and the number of paths exists, and also if there is no measure of (relative) accuracy about the estimates taken
(satisfying the system, or exact probability, or any number of factors to the contrary in the sense described in [2]) and the total number of paths available for determining all possible paths
available (how often), even if the number of observations to be taken is two or three. An alternative method proposed in this paper is a different approach [1]. It considers neither the solution to
the poisson equation nor the problem of counting the (expected) number of possible paths that can occur on the test-bed data (as one would consider any of these means, e.g. one for calculating the
rate and a sum of the moments on a single test-bed data unit) whilst simultaneously counting the probable number of paths (which amounts to calculating the likelihood function) prior to any path out
of which the probability is zero in the special case for which the study of the Poisson problem is feasible, without use of a pseudo-summation methodology. Its main objective is to find the solution
to the Poisson equation limiting to a subset of the probability distributions whose expected time-evolution can be solved before any other probability distributions become an equalization product of
Poisson’s equations, ignoring the requirement on sample size: if equalization can be achieved, but time-evolutional properties must become irrelevant, the Poisson equation becomes a non-equidominated
(e.g. sub-critical) Poisson probability in non-equidominated probability distributions, as can be seen by comparing the Poisson equation to an ordinary Poisson equation on a real-time test-phone
(i.e., the time-step). The method can be easily applied if two components with an important difference are estimated. Thereby, for an abstract example problem, the algorithm of the method allows for
a procedure to calculate all the potential paths if different distributions are sampled simultaneously; the main goal is to find, for any given configuration of the test-bed data that under cover is
the initial guess of the PoissonWho provides assistance with discrete event simulation and stochastic processes assignments? By its highly detailed description, here, we find that several games, for
instance Flash games, are exactly capable of setting some of the parameters of discrete-events system even when real environment or human parameter settings are not included! At each step, like every
other simulated data, we try to analyze the resulting probability distribution function, whose interpretation is very interesting for many reasons: it can give information about probabilities of
adding or otherwise changing some parameter of some one particular system! There are several ways of obtaining the numerical results for discrete events simulation, but I suppose this list should be
helpful and we can discuss several different ways of describing discrete events simulation procedure with a view to more sophisticated work! 3 ways that we work on discrete event simulation, but it
is not recommended to work with real environment parameters. 1) If “*the world’s starting place*, *what the possible values do*, and what they happen (but not where *you* get everything right*!) are
different from each other that could be treated more as the natural thing to do, like when you ask different humans for their attention, or what we would be seeing in the world but we would see more
of us.
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I don’t mean it to be called “big bangs”. If you’re finding it slightly chaotic than we can solve completely different problems, but if I only want in on it a “big bang”, let me include some random
information or “big” in that? For instance, let’s say that we can find out by looking in a specific box, “C” is a box with “2*(1+4C)” and “1+4*1000” being exactly the list of numbers that we know in
the world. For example, if we know that “100*1000” happens at the first box above, then there is some random number’s in the box, and then there’s some random numbers. What that means is that we
could try to find all “1+4*3” value from the box. Once that is done, we try to get more information on “1+4*3” values up to some point. Such a problem is sometimes quite ugly and has a few
interesting consequences, depending on how far you want to go in finding the “1+4*3” values. For instance, when trying to find out something about changing a local variable, like a shape change,
we’re probably trying to find in a “meagherian” way the factor that may change the shape of the world’s current size, and then try to update it by changing it a random number. A further consequence
is that our problem could even be a “scratch” problem, as “1+4*3” values the same way every time, and then we have something like a “like this, does 1+4*1000” list of numbers based on our “1+4*3”
values. In either case, some errors or imperfections have spread to try to solve, and while we’ve heard many possible solutions, we can come across some we could not find the answer to the problem!
2) “*what you could do with the world’s movement*” / you could try to find something about this movement going right way down. There are many people who are trying to solve this puzzle, and so have
got a lot of different methods for doing it, but the idea would be to find out how the world’s movement spreads. (I’ll skip the details, this is the method I guess I’d actually try.) I would add an
extension to this idea that considers everything we do in the world moves in a straight up linear way, and so the original idea would simply be to find the “vector” at which everything for all that
is “3×3” would move. This could be done by analyzing any random thing we might have to do on a time cycle, for illustration purposes. This is what the following method would look like, and **where
the time period is short enough (but not too short), for example, meaning that we start at a 3×3 position, stay there for 20 min, and increase to a last place after this. To compute the time
evolution, if you read the right way up prior to the world, it would basically be finding the vector of the 3-point displacements of the world at that 5*9 position, and then moving the last 4*9
positions back to that position for 20 min until there are 3-points to move at that last position. If the current world isWho provides assistance with discrete event simulation and stochastic
processes assignments? Background: According to the current policy on distributed artificial intelligence and machine learning by S. F. Hartl, there are a high number of state information of a
document that has to exist during the document, and this makes it unstructured of all existing documents. Also, we have a number of systems to do this: [1] a memory management system [2] a library
for machine learning and visual engineers [3] a training tool for solving a problem [4] a problem finding machine-learning problems [5] a network based intelligence language [6] a database to find
optimal solutions (see below) [7] an association model [8] a generic pattern-based approach to data alignment (K. I.
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Krammers on Artificial Intelligence [9] and [10] applications). [1] [2] [3] [4] [5] [6] [7] Introduction This page notes various common non-general features of discrete event model. It notes that
some of this feature is used for feature extraction from documents to understand the structure of the event. Also a relatively simple model like the state information is used in the model. The author
has reviewed his papers on Event Analysis and Process LSTM and has also created generalizations to other work. The book covers the material in the book about the structure of formalized applications
of Event Analysis and Process LSTM. [3] [5][new] The most important of these are: [1]. A tool for event analysis on document identification. [2] [6] Event analysis (for this kind of examples mainly
using fuzzy logic) – detecting performance of a control procedure of a system. [7] Algorithm based control. [8] Event-based processing system. [9] Data in a document. [10] Document generator for
machine learning tasks [11] a random topic generator (a.k.a. network generators) to find optimal solutions. [12] Automated data maintenance (AD) for humans to make this the last option. [13] Network
based computational function analysis. [14] Event-perception with different dynamic attributes. [15] Event-perception with classifiers.
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[16] Event-based learning. [17] Machine learning with machine-learning methods. [18] Manual algorithm for learning systems using different methods (linear kernel activation method, maximum entropy
learning algorithm, some fixed learning rates…) [19] Text processing. [20] Temporal logic. [21] Event processing techniques with non-portable image recognition. [22] Event-analysis machine tool. [23]
Linking machine learning methods to [24] The recent interest in machine learning and Event Analysis was related to the field of Artificial Natural Language Processing [(ANNLP)] and related to
artificial intelligence[25], where a sequence of classes should be attached to the presentation to provide information about the types of an object in the object list. The AOTP, for their work on
machine learning or image recognition algorithms, relies mainly on different models for machine learning and other computer intelligence tasks. However, the structure of the model includes many more
classes of documents. Also, the machine-learning algorithms may work different to the traditional training/feature extraction methods, which is what I was referring to in the book. A summary of some
simple details: An event is represented by a sequence of discrete event system inputs. For instance, the input sequence of a document generated by a document generator is given to the same
classifiers, such as fuzzy neural networks, which in the context of machine learning algorithms have discrete examples. Many approaches for this kind of problem applied to historical data showed that
the model can be approximated by neural networks (e.g. graph theory models). Also, although the event sequence has time-dispersed information, this information doesn’t possess the phase-vari
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Is Friday the 13th Unlucky?
Many people live in fear of the number 13. The technical term for this is Triskaidekaphobia. Yes, it
a real word - check out the
Wikipedia entry
if you don't believe me.
The word for those who are afraid of Friday the Thirteenth is
. It's a not-uncommon fear.
But is it all just superstition or is there some valid basis behind this fear?
There are many articles and experts who can help you draw conclusions regarding the origin of such concerns, but the history is at best sketchy. Is it, then, part of some retained ancient wisdom that
is able to highlight the days on which evil could triumph? Or is it rather the foolish man's misunderstanding of various unrelated stories fashioned into popular myth?
The biggest problems with Friday the 13th lie with two fundamental issues.
Firstly, the number of fingers that we have.
And secondly, the absence of better telescopes.
Let me explain.
Firstly, the fingers.
Human kind has ten fingers. Because of this we work with what is called a base-10 numbering system. In other words, once we get to nine, we then start afresh from zero with a one (1) before it all.
Thirteen, therefore is represented as 10 plus 3 -> 13. This counting system is quite significant. Computers use binary (0, 1, 10, 11, 100 etc) for example.
But if we had only 8 fingers, then once we got to seven we would then introduce that one (1) and restart from zero again. Consider the following table which shows the numbers from one to thirteen in
both base-10 as well as base-8:
│Number │Base-10│Base-8│Base-2 (Binary) │
│one │1 │1 │1 │
│two │2 │2 │10 │
│three │3 │3 │11 │
│four │4 │4 │100 │
│five │5 │5 │101 │
│six │6 │6 │110 │
│seven │7 │7 │111 │
│eight │8 │10 │1000 │
│nine │9 │11 │1001 │
│ten │10 │12 │1010 │
│eleven │11 │13 │1011 │
│twelve │12 │14 │1100 │
│thirteen │13 │15 │1101 │
As you see, with base-8 we would get to '13' quickly as the written equivalent of eleven. Similarly, thirteen would be written as '15'.
Of course, in base-2 i.e. binary, thirteen must be written as 1101 and there is in fact no meaning in the word "13" since only ones (1) and zeroes (0) are used.
Now that throws numerology and other systems off somewhat.
So if we had been born with 8 fingers (or in fact 9 or 11 etc) then our understanding of 13 would be massively different.
Then, there is the problem of the absence of better telescopes.
The main reason that the seven day week was chosen was in part (at least) due to the significance of the number seven, and specifically because there were seven known planets.
Sun, Mercury, Mars, Venus, Moon, Jupiter, Saturn
Poor old Pluto and Neptune had not been discovered!
If they had then we might well have had 9 day weeks which, apart from giving us either a very long working week (or a very long weekend!) would also have affected the days on which our current Friday
the Thirteenth occurs. In other words, because the weeks would be 9 days long, the day which we currently associate with Friday the Thirteenth would probably be a Monday instead. Or a Saturday. Or a
Gooday or a Gnomeday..... or whatever they chose to call the extra two days.
The practical upshot of all this is that the whole association that we have with a day being a Friday and the 13th of a month is completely arbitrary and totally unrelated to fate or evil or
So therefore, there can be no sensible way in which we can assume that any Friday the Thirteenth is unlucky!
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Math 7512: Topology II
Homework Assignments:
Course Content:
The following is a collection of topics that I hope to cover in this course.
• Simplicial and cell complexes
• Singular Homology
• Chain complexes and basic homological algebra
• Singular Cohomology
• Künneth Theorem
• Poincaré-Lefshetz Duality
Text: We won’t follow one book exclusively, but Allen Hatcher’s Algebraic Topology is the best single resource for this course. Other resources include Munkres’sElements of Algebraic Topology, and
Spanier’s Algebraic Topology.
Classroom: Lockett 232
Time: TTh 9:00 – 10:20
Office Hours: Monday 1:30 – 2:30 and Thursday 12:00 – 1:00
Homework and Grades:
Homework Policy: Our grader for this course is to be determined. Homework is to be scanned and turned in via email by 4:00 PM on its due date. If you know in advance you will be unable to turn in
homework when it’s due, you should plan to turn it in ahead of time.
Homework must be neat, well-organized, and legible. If your handwriting is difficult to read, type your homework. If you tend to scratch out or erase incorrect parts of solutions, do a rough draft or
type your homework. Write in paragraphs, sentences, and English words. Use punctuation and conjunctions to indicate your flow of thought rather than arrows or telepathy. Shoot for lucidity rather
than terseness.
Final Course Grades: Your grade will be determined by your performance on homework assignments and your overall classroom engagement:
• Homework problems 50%
• Midterm problems 20%
• Final exam 30%
Academic Dishonesty: While copying your written work from somebody else or from any other source is considered cheating, you are strongly encouraged to talk to others about the material you are
learning — this includes fellow students, me and anyone else in the department. The more engaged you are with the material the better. That said, the work you hand in must be your own. The best way
to avoid any suspicion of copying is to write out your homework solutions on your own, not whilst working with a friend. Cheating on exams is unacceptable. Any cheating during midterms or finals will
result in you failing the exam.
Disability Support: Students who may need accommodations because of a documented disability should meet with me privately within the first week of classes. In addition, students with disabilities
should also contact the Office of Disability Services.
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Numbered Graph Paper
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Best Time to Buy and Sell Stock with K transaction
Best Time to Buy and Sell Stock with K transaction
The original question can be found here.
Lets start from an elegant the solution :
class Solution {
public int maxProfit(int k,int[] prices) {
if (prices.length<2 || k==0) return 0;
int[] profits = new int[k+1];
int[] balance = new int[k];
for (int p : prices) {
for (int i=0;i<k balance math.max profits return>
This solution would run in O(n*k). if we dont mind getting our hands dirty, it can be better implemented and then explained like that:
class Solution {
public int maxProfit(int k,int[] prices) {
if (prices.length<2 || k==0) return 0;
int[] profits = new int[k+1];
int[] balance = new int[k];
int transactions=1;
for (int p : prices) {
if (transactions<k profits>profits[transactions-1]) {
for (int i=0;i<transactions balance math.max profits return>
Now it would run in O(n*m) where is the trade count in the optimal solution. commonly m<<k. Lets begin with the explanation, We’ll keep track of the following loop invariants:
• profits[i] will hold the max profit we can get for i efficient full transactions.
profits[0] is always 0 since that’s the profit when we make no transactions, profits[1] is the maximal profit we could have with 1 efficient full transaction and so on.
• balance[i] will hold the optimal balance we would have if we do i efficient full transactions and then an efficient buy.
since we have only have a buy for balance[0], it will eventually hold the negate of the minimal price, balance[1] will be the best balance after doing one full transaction efficiently and followed by
a buy, and so on.
Whenever we find that profits[i]>profits[i-1] it would mean that we can increase our profits by doing one more full transaction after i-1 efficient full transactions, so we search for this to hold,
and when it does we start to work on profits for next i.
We try to maximize our balance[i] which consists of the profits[i] which is the profit we made so far, and paying for our following buy. for every new price we examine if we improve our balance when
buying at that price, if we don’t we stay with the previous balance.
Then for profits[i+1] we take the optimal balance[i] and we need to add the price we get if we sell to complete the i+1 trade transaction, we will not do the sell if we previously had a better setup.
Eventually we return profits[transaction] which is the max profit for the optimal count of transactions.
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Swarm algorithms
I wrote a book about about genetic algorithms and machine learning. You can buy it
Apart from genetic algorithms and other aspects of machine learning, it includes some swarm algorithms. Where a genetic algorithm mixes up potential solutions, by merging some together, and
periodically mutates some values, swarm algorithms can be regarded as individual agents collaborating, each representing a solution to a problem, They can work together in various ways, giving rise
to a variety of swarm algorithms.
The so-called particle swarm algorithm can be used to find optimal solutions to problems. It's commonly referred to as a particle swarm optimisation, or PSO for short. PSO is often claimed to be
based on the flocking behaviour of birds. Indeed, if you get the parameters right, you might see something similar to a flock of birds. PSO are similar to colony algorithms, which are also nature
inspired, and also having agents collaborating to solve a problem.
Suppose you have some particles in a paper bag, say somewhere near the bottom. If they move about at random, some might get out of the bag in the end. If they follow each other, they might escape,
but more likely than not, they'll hang round together in a gang. By providing a fitness function to encourage them, they can learn, for some definition of learn. Each particle can assess where it is,
and remember the better places. The whole swarm will have a global best too. To escape a paper bag, we want the particles to go up. By inspecting the current (x, y) position, the fitness score can be
the y-value. The bigger, the better. For real world problems, there can be many more than two dimensions, and the fitness function will require some thought.
The algorithms is as follows:
Choose n
Initialize n particles randomly
For a while:
Update best global position
Move particles
Update each particle's best position and velocity
The particles' personal bests and the overall global best give the whole swarm a memory, of sorts. Initially, this is the starting position for each particle. In addition to the current position,
each particle has a velocity, initialised with random numbers. Since we're doing this in two dimensions, the velocity has an x component, and a y component. To move a particle, update each of these
by adding the velocity, v, in that direction to the current position:
x[t+1] = x[t] + v[x,t]
y[t+1] = y[t] + y[x,t]
[Since the velocity starts at random, the particles move in various different directions to begin with. The trick comes in when we update the velocity. There are several ways to do this. The standard
way adds a fraction of the distance between the personal best and global best position for each particle and a proportion of the current velocity, kinda remembering where it was heading. This gives
each a momentum along a trajectory, making it veer towards somewhere between its best spot and the global best spot. You'll need to pick the fractions. Using w, for weight, since we're doing a
weighted sum, and c1 and c2 for the other proportions, we have:]
v[x,t+1] = wv[t] + c[1](p[t]-x[t])+c[2](g[t]-x[t])
If you draw the particles moving around you will see them swarm, in this case out of the paper bag.
[This is one of many ways to code your way out of a paper bag covered in my book. When a particle solves a problem, here being out of the bag, inspecting the x and y values gives a solution to the
problem. PSO can be used for a variety of numerical problems. It's usually described as a stochastic optimisation algorithm. That means it does something random (stochastic) to find the best
(optimal) solution to a problem. You can read more here. ]
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Little guidance needed with meshesi
I decided to dive into an area where I have absolutely no experience. Meshes.
For start of my journey I have chosen to create myself a 3D Convex Hull.
Having a set of random points, I am able to create the initial tetrahedron from the first four points in the list. Now, I can evaluate the points and exclude the ones that are already inside the
initial mesh. Question is when iterating over the ones that outside. How do I choose to which two/three points from the mesh it has to connect? I assume I have to look for the closest point. But how
to look for the second and third closest?
What is the best approach?
• creating expaning sphere?
• creating lines and measuring distances?
• iterating over all points and trying to create mesh checking the result if it is closed?
I don’t yet know what properties I can extract from the mesh, or face, or vertex.
Are the mesh faces (1,2,3), (3,2,1), (1,3,2), (2,1,3), (2,3,1) and (3,1,2) the same mesh?
Thanks in advance.
You could transform your points into a point cloud and recursively check for closest points. At each level/iteration a found closest point gets stored somewhere and removed from the point cloud. This
runs until your desired number of closest points has been found.
Using the rTree algorithm could be more efficient, especially if your dealing with a huge amount of points, but you’d need a search radius, which you don’t have.
The vertices are organised in flat list. The order is up to you. The face vertices are tuples of vertex indices. Each index refers to a vertex inside the vertices list. The list of vertices and the
list of face vertices represent the mesh.
The face vertices of a mesh face are commonly organised in a counterclockwise way. The faces (1,2,3) and (3,2,1) refer to the same vertices, but have an opposed direction.
Furthermore, you should use quad meshes, since they are easier to understand from a human perspective. They have 4 vertex indices per face!
vertices = [ v0, v1, v2, v3, v4, v5, v6, v7, v8 ]
face_vertices = [ (0,1,4,3), (1,2,6,4), (4,5,8,7), (3,4,7,6) ]
1 Like
Thanks for the hints @diff-arch,
Any particular reason why?
I cannot understand why would that be. I prefer triangles because they are strongly inside a single plane.
1 Like
I guess that’s just common practice as far as I know.
No problem, do triangles then.
As far as I’m concerned convex-hull algorithms are not easy to implement. So you either have to read up on some algorithms that are out there or you can try something very grasshoppery:
Maybe try to solve the problem using kangaroo for grasshopper, it’s a physics-engine:
Create a Sphere around your point-cloud and let it shirk until it fit’s your geometry well, the connections between your spheres vertices act like springs, it would result in a convex shape. Do it
like Christev in the video but don’t specify any anchors.
Flexhopper is another engine that might works well.
Take care.
I would say your best bet for finding the closest points would be to use the RTree methods.
They’re extremely fast and deal well with these neighborhood type problems. You’ll need to test some of the functions out but I think there’s some will output a list of points by proximity.
RTree.Point3dKNeighbors seems about right. If you want to just dip your toe, LunchBox tools has a few simple RTree tools in Grasshopper that I’ve started using quite regularly.
As for the clockwise/counter clockwise vertex naming, it’s used to tell if the normal is pointing inward or outward.
I don’t know if you’re working in C# but @LongNguyen 's Grasshopper development class is really awesome and I can’t recommend it enough. https://icd.uni-stuttgart.de/?p=22773 The final exercise
is a mesh problem and he does a good job of explaining the basics and also integrates RTree.
This is what I’m gonna do.
Kangaroo is not grasshoppery. It’s 3rd-party.
And I will not use Kangaroo, unless I have no other choice.
This thread is in the GH Developer section, for a reason. I don’t want to use any plugins but ghpython
Sure it is! Since, v6 it’s an integral part of GH!
Well, it’s a interactive physics/contraints solver, so calling it a physics engine is probably fine.
Yep, me too. I always try to solve everything with GHPython!
2 Likes
As far as algorithms go, I’d start from the outside of the point cloud, that approach would be faster I think, not sure though.
1. Get the outermost point by measuring the distance of all points to the average of the cloud.
2. place a plane with it’s origin at that outermost point and look for points closest to that plane, if you found the closest point, save that in a variable, closest_pt
3. iterate by creating another plane at closest_pt find closest point other than the ones you already checked
You could also create a cone at the outermost point and make it smaller at certain increments, until you intersect another point and repeat that.
You cannot do this. There’s no point cloud, first of all. Second, you can get a point from a randomized list.
No plane should be involved as well. You only work with points and their coordinates, until you get the starting tetrahedron. From there you begin looking for closest points that are not inside this
starting mesh. And here come the algorithms.
Your approach is wrong, you assume you have everything available at start.
1. My approach isn’t wrong, it’s just different and more simple, because I don’t calculate for inclusion. If you have a library that does it for you, good!
2. I assume to only have points in R3, that is simply a point-cloud
“You only work with points and their coordinates”, that’s exactly what I did.
3. I can easily generate planes from normal-vectors very fast. In fact, planes are at the core of vector-math. The cartesian-coordinate-system is based on three of them!
If you don’t want alternative ideas fine, I assume you want to stick to the nearest-neighbor-inclusion-method. It will work fine.
On another note, You can extract the vertices, triangles, UVs, tangents and nomals of a mesh, but you don’t need those for a convex-hull, you only need the vertices, which is, again, a point-cloud.
Two of the best known algorithms for tackling this are Jarvis march aka gift wrapping, and Quickhull.
The first is similar to what @lesan describes, starting from a plane on an outside point (found by sorting the points in 1d), then wrapping around with successive planes.
Quickhull first builds a tetrahedron (after also sorting in 1d to get the extremes) and works outwards, which sounds more like what @ivelin.peychev is thinking of.
Both are valid approaches, and maybe it’s not such a good idea to be so dismissive of people who take the time to answer your questions.
5 Likes
Yes, I read about the algorithms a little. I realized that gift wrapping has really bad performance, iterating through all points in each step.
I assumed the logical approach will be to create a closed mesh at start to remove the points that are inside, thus reducing the number of iterations. Later I saw that Quickhull is also proposing
The issue I have is what happens next.
• say I found the nearest point to one of the vertices of the tetrahedron. How do I define which this nearest point should be added to the mesh, how do I pick the indices? Should that happen again
trial/error until I get a closed volume again?
• Perhaps it is faster to pick a random point instead of the nearest one, then I can enclose points that I’ll later exclude. You know pseudo-genetic approach.
What does that mean?
I don’t dismiss people I dismiss proposals. @lesan’s R-tree proposal was very good suggestion. I didn’t even know what that is before he suggested it, but talking about point cloud and cone,
when you don’t know anything about how points are spread, makes no sense.
Sorting in 1D means the points are sorted by their X,Y and or Z-Values.
1 Like
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Introduction to Signals and Systems - Electronic Engineering (MCQ) questions & answers
1) A system is said to be shift invariant only if______
a. a shift in the input signal also results in the corresponding shift in the output
b. a shift in the input signal does not exhibit the corresponding shift in the output
c. a shifting level does not vary in an input as well as output
d. a shifting at input does not affect the output
Answer Explanation
ANSWER: a shift in the input signal also results in the corresponding shift in the output
No explanation is available for this question!
2) Which among the below specified conditions/cases of discrete time in terms of real constant 'a', represents the double-sided decaying exponential signal?
a. a > 1
b. 0 < a < 1
c. a < -1
d. -1 < a < 0
Answer Explanation
ANSWER: -1 < a < 0
No explanation is available for this question!
3) An equalizer used to compensate the distortion in the communication system by faithful recovery of an original signal is nothing but an illustration of _________
a. Static system
b. Dynamic system
c. Invertible system
d. None of the above
Answer Explanation
ANSWER: Invertible system
No explanation is available for this question!
4) Which among the following are the stable discrete time systems?
1. y(n) = x(4n)
2. y(n) = x(-n)
3. y(n) = ax(n) + 8
4. y(n) = cos x(n)
a. 1 & 3
b. 2 & 4
c. 1, 3 & 4
d. 1, 2, 3 & 4
Answer Explanation
ANSWER: 1, 2, 3 & 4
No explanation is available for this question!
5) Under which conditions does an initially relaxed system become unstable?
a. only if bounded input generates unbounded output
b. only if bounded input generates bounded output
c. only if unbounded input generates unbounded output
d. only if unbounded input generates bounded output
Answer Explanation
ANSWER: only if bounded input generates unbounded output
No explanation is available for this question!
6) Which condition determines the causality of the LTI system in terms of its impulse response?
a. Only if the value of an impulse response is zero for all negative values of time
b. Only if the value of an impulse response is unity for all negative values of time
c. Only if the value of an impulse response is infinity for all negative values of time
d. Only if the value of an impulse response is negative for all negative values of time
Answer Explanation
ANSWER: Only if the value of an impulse response is zero for all negative values of time
No explanation is available for this question!
7) An amplitude of sinc function that passes through zero at multiple values of an independent variable 'x' ______
a. Decreases with an increase in the magnitude of an independent variable (x)
b. Increases with an increase in the magnitude of an independent variable (x)
c. Always remains constant irrespective of variation in magnitude of 'x'
d. Cannot be defined
Answer Explanation
ANSWER: Decreases with an increase in the magnitude of an independent variable (x)
No explanation is available for this question!
8) Damped sinusoids are _____
a. sinusoid signals multiplied by growing exponentials
b. sinusoid signals divided by growing exponentials
c. sinusoid signals multiplied by decaying exponentials
d. sinusoid signals divided by decaying exponentials
Answer Explanation
ANSWER: sinusoid signals multiplied by decaying exponentials
No explanation is available for this question!
9) Which property of delta function indicates the equality between the area under the product of function with shifted impulse and the value of function located at unit impulse instant?
a. Replication
b. Sampling
c. Scaling
d. Product
Answer Explanation
ANSWER: Sampling
No explanation is available for this question!
10) Which mathematical notation specifies the condition of periodicity for a continuous time signal?
a. x(t) = x(t +T[0])
b. x(n) = x(n+ N)
c. x(t) = e^-αt
d. None of the above
Answer Explanation
ANSWER: x(t) = x(t +T[0])
No explanation is available for this question!
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Linear Dissipative Force as a Result of Discrete Time Collisions
2020, v.26, Issue 2, 315-337
We consider two models: one describes the particle movement under the influence of external force and friction, and another one describes the movement of a particle, which is acted upon by the same
external force but it additionally collides with other particles of much lighter masses.
We establish conditions for these two
models to be equivalent in some sense. We also
considered deterministic and stochastic models for collisions, in first case assuming that the time intervals between the collisions are constant, and in another case when these intervals are random
independent random variables.
For various examples of the external force
find parameters which yield asymptotic equivalence of
the velocities of the particle in different models. We also provide conditions whe
n the trajectories of the particles in different models are close to each other in the Chebyshev norm over a certain finite period of time.
Our results confirm that a linear dissipative force such as e.g., friction, can well be
modelled by the
collisions with external light particles if their masses and the time-intervals between the collisions satisfy certain condition. The latter is proved here to be universal for
different forms of the external force.
Keywords: dissipative force, friction, particles collision, Euler's approximation method
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6. (10 points total) An insurance policy pays $1000 per day for up to
6. (10 points total) An insurance policy pays $1000 per day for up to 3 days of hospitalization and$500 per day for each day of hospitalization thereafter. The number of
days of hospitalization,X, is a discrete random variable with probability function p_{X}(k)=\left\{\begin{array}{cc} \frac{6-k}{15} & \text { for } k=1,2,3,4,5 \\ 0 & \text { otherwise } \end{array}\
right. (a) (5 points) Determine the expected payment for hospitalization under this policy. (b) (5 points) Determine the variance of the payment for hospitalization under this policy.
Fig: 1
Fig: 2
Fig: 3
Fig: 4
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CAT Average - Quantifiers
Ready to master your way to CAT exam success? At Quantifiers, we’ll make the world of averages not only understandable but also as easy as pie! Join us on this journey of mastering the Average
concept with a sprinkle of wit and a dash of fun. This section will explore the idea of Average, how important it is to the CAT exam’s Quantitative Aptitude section, and how you can succeed with our
experienced assistance.
At Quantifiers, we’re not just about numbers; we’re about making them work for you. Master the art of averages with us, and you’ll not only understand this concept but also learn to appreciate its
importance in the CAT exam.
Onion is sold for 5 consecutive months at the rate of Rs 10, 20, 25, 25, and 50 per kg, respectively. A family spends a fixed amount of money on onion for each of the first three months, and then
spends half that amount on onion for each of the next two months. The average expense for onion, in rupees per kg, for the family over these 5 months is closest to (CAT 2021)
In a football tournament, a player has played a certain number of matches and 10 more matches are to be played. If he scores a total of one goal over the next 10 matches, his overall average will be
0.15 goals per match. On the other hand, if he scores a total of two goals over the next 10 matches, his overall average will be 0.2 goals per match. The number of matches he has played is: (CAT
The arithmetic mean of scores of 25 students in an examination is 50. Five of these students top the examination with the same score. If the scores of the other students are distinct integers with
the lowest being 30, then the maximum possible score of the toppers is: (CAT 2021)
Dick is thrice as old as Tom and Harry is twice as old as Dick. If Dick’s age is 1 year less than the average age of all three, then Harry’s age, in years, is: (CAT – 2020)
A batsman played n + 2 innings and got out on all occasions. His average score in these n + 2 innings was 29 runs and he scored 38 and 15 runs in the last two innings. The batsman scored less than 38
runs in each of the first n innings. In these n innings, his average score was 30 runs and lowest score was x runs. The smallest possible value of x is (CAT 2020)
There are three categories of jobs P.R.T., T.G.T. and P.G.T. The average salary of the teachers who got the job in P.R.T and T.G.T. categories is 26 lakhs per annum. The average salary of the
teachers who got the job in T.G.T. and P.G.T. categories is 44 lakhs per annum and the average salary of those teachers who got the job of P.R.T and P.G.T categories is 34 lakhs per annum. The most
appropriate range of average salary (in lakhs per annum) of all the three categories (if it is known that each teacher gets only one category of job i.e. P.R.T. or T.G.T. or P.G.T.):
Aman and eight of his friends took a test of 100 marks. Each of them got a different integer score and the average of their scores was 86. The score of Aman was 90 and it was more than that of
exactly three of his friends. What could have been the maximum possible absolute difference between the scores of two of his friends?
Subscribe to our newsletter and stay updated with all the latest news, tips, and information about the CAT exam delivered directly to your inbox.
We at Quantifiers understand and deliver on the personal attention each of our students requires. Whether it is through our pedagogy that enables non-engineers or non-math background students, our
constant effort to proactively provide solutions, or our focus on our student’s goals.
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The travelling salesman problem (TSP) asks the following question:
"Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city and returns to the origin city?"
It is an NP-hard problem in combinatorial optimization, important in operations research and theoretical computer science.
• Click on the map to add a new route point.
• Click on an existing point to remove it from the route.
• Click the reset button to restore the route.
Algorithm in use: {{ max_reached ? '2-opt heuristic' : 'linear programming' }}
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In this comprehensive guide, we will explore the HYPGEOM.DIST formula in Excel. The HYPGEOM.DIST function is a statistical function that calculates the probability of a given number of successes in a
sample drawn from a finite population without replacement. This function is particularly useful in situations where you need to analyze the likelihood of a specific outcome in a sample, such as
quality control, market research, or survey analysis.
HYPGEOM.DIST Syntax
The syntax for the HYPGEOM.DIST function in Excel is as follows:
HYPGEOM.DIST(sample_s, number_sample, population_s, number_population, cumulative)
• sample_s is the number of successes in the sample.
• number_sample is the size of the sample.
• population_s is the number of successes in the population.
• number_population is the size of the population.
• cumulative is a logical value that determines the form of the function. If TRUE, the function returns the cumulative distribution function; if FALSE, it returns the probability mass function.
HYPGEOM.DIST Examples
Let’s look at some examples of how to use the HYPGEOM.DIST function in Excel.
Example 1: Suppose you have a batch of 100 light bulbs, and 10 of them are defective. You randomly select 20 light bulbs from the batch. What is the probability of finding exactly 3 defective light
bulbs in the sample?
=HYPGEOM.DIST(3, 20, 10, 100, FALSE)
In this example, the function returns the probability mass function, which calculates the probability of finding exactly 3 defective light bulbs in the sample.
Example 2: Using the same batch of 100 light bulbs with 10 defective ones, what is the probability of finding 3 or fewer defective light bulbs in a sample of 20?
=HYPGEOM.DIST(3, 20, 10, 100, TRUE)
In this example, the function returns the cumulative distribution function, which calculates the probability of finding 3 or fewer defective light bulbs in the sample.
HYPGEOM.DIST Tips & Tricks
Here are some tips and tricks to help you effectively use the HYPGEOM.DIST function in Excel:
1. Remember that the HYPGEOM.DIST function assumes that the sampling is done without replacement. This means that once an item is selected from the population, it is not returned to the population.
2. Ensure that the values for sample_s, number_sample, population_s, and number_population are non-negative integers. Otherwise, the function will return a #NUM! error.
3. When using the cumulative distribution function (cumulative = TRUE), the result will include the probability of the specified number of successes as well as all smaller numbers of successes.
4. If you need to calculate the probability for a range of successes, you can use the HYPGEOM.DIST function in combination with the SUM function. For example, to calculate the probability of finding
between 2 and 4 defective light bulbs in the previous example, you can use the following formula:
=SUM(HYPGEOM.DIST(2, 20, 10, 100, FALSE), HYPGEOM.DIST(3, 20, 10, 100, FALSE), HYPGEOM.DIST(4, 20, 10, 100, FALSE))
Common Mistakes When Using HYPGEOM.DIST
Here are some common mistakes to avoid when using the HYPGEOM.DIST function:
1. Using decimal values for sample_s, number_sample, population_s, or number_population. These arguments must be non-negative integers.
2. Forgetting to specify the cumulative argument. If omitted, Excel will return a #NUM! error.
3. Using the HYPGEOM.DIST function for situations where sampling is done with replacement. In such cases, consider using the BINOM.DIST function instead.
Why Isn’t My HYPGEOM.DIST Working?
If you encounter issues when using the HYPGEOM.DIST function, consider the following troubleshooting steps:
1. Check the values of sample_s, number_sample, population_s, and number_population. Ensure they are non-negative integers.
2. Verify that the cumulative argument is specified as either TRUE or FALSE.
3. Ensure that the sampling scenario is appropriate for the HYPGEOM.DIST function (i.e., sampling without replacement).
4. Double-check the formula for any typos or incorrect references to cell ranges.
HYPGEOM.DIST: Related Formulae
Here are some related formulae that you might find useful when working with the HYPGEOM.DIST function:
1. BINOM.DIST: Calculates the probability of a given number of successes in a fixed number of trials with a constant probability of success, assuming sampling with replacement.
2. POISSON.DIST: Calculates the probability of a given number of events occurring in a fixed interval, assuming a constant average rate of occurrence.
3. NORM.DIST: Calculates the probability of a given value in a normal distribution, assuming a specified mean and standard deviation.
4. CHISQ.DIST: Calculates the probability of a given value in a chi-square distribution, assuming a specified number of degrees of freedom.
5. F.DIST: Calculates the probability of a given value in an F-distribution, assuming specified numerator and denominator degrees of freedom.
By understanding the HYPGEOM.DIST function and its related formulae, you can effectively analyze and interpret statistical data in Excel. With practice, you’ll be able to confidently apply this
function to a wide range of real-world scenarios.
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ing T
You can see objects moving all around you. See the picture below. A boy is travelling on a train and looking out of the window at his surroundings.
Is the boy on the train moving or stationary?
Moves / Stands
Both answers are correct.
According to the seat and the passengers on the train, the boy is sitting still – he is at rest. Compared to the houses, trees and people sitting in front of the house outside the train, the boy on
the train is moving. So, when we want to tell whether the observed body (in our case, the boy) is moving or stationary, we first need to choose its surroundings.
We describe the motion of bodies in terms of their stationary surroundings.
A boy is moving when it changes position relative to its chosen surroundings. So, the boy moves from one place to another
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Magnetohydrodynamic flow computations in three dimensions
A complete three-dimensional numerical simulation of steady laminar MHD incompressible flow was carried out. A mathematical model of magnetohydrodynamics in which the electric field vector is
eliminated from the Maxwell equations is described. The numerical method for solving the system of governing equations is presented.
29th AIAA Aerospace Sciences Meeting
Pub Date:
January 1991
□ Computational Fluid Dynamics;
□ Incompressible Flow;
□ Laminar Flow;
□ Magnetohydrodynamic Flow;
□ Mathematical Models;
□ Three Dimensional Flow;
□ Boussinesq Approximation;
□ Electric Fields;
□ Finite Difference Theory;
□ Heat Transfer;
□ Joule-Thomson Effect;
□ Magnetic Fields;
□ Maxwell Equation;
□ Runge-Kutta Method;
□ Steady Flow;
□ Plasma Physics
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How many 24 ounces equals a gallon?
When it comes to liquid measurements, gallons and ounces are two of the most common units used. But how do they relate to each other? Specifically, how many 24-ounce containers does it take to equal
one gallon? Let’s take a closer look at the conversion between gallons and ounces to find the answer.
Quick Answer
It takes four 24-ounce containers to equal one gallon. A gallon contains 128 fluid ounces. Since each 24-ounce container holds 24 ounces, you need four of them to reach the 128 ounces in a gallon.
Breaking Down Gallons and Ounces
Before diving into the specific conversion, let’s break down some key details about gallons and ounces:
• A gallon is a unit of volume used to measure liquids. One gallon equals 128 fluid ounces.
• An ounce is a smaller unit of volume. There are 128 ounces in one gallon.
• When measuring liquid volumes, fluid ounces are used rather than weight ounces.
• Common containers like soda or water bottles often hold 24 fluid ounces of liquid.
Understanding these basic facts helps illustrate the relationship between gallons and ounces. A gallon is a much larger amount of liquid than an ounce. But you can measure out one gallon by summing
up smaller ounce measurements.
Converting Gallons to Ounces
Since a gallon equals 128 fluid ounces, converting between the two units is straightforward:
• 1 gallon = 128 ounces
• To convert gallons to ounces, multiply the gallon amount by 128
• For example: 2 gallons x 128 oz/gal = 256 oz
This gallon to ounce conversion shows that any amount of gallons can be converted to ounces by multiplying by 128. This conversion factor comes from the definition of a gallon equalling 128 fluid
Converting Ounces to Gallons
Going the other direction, from ounces to gallons, the conversion works like this:
• 128 ounces = 1 gallon
• To convert ounces to gallons, divide the ounce amount by 128
• For example: 384 oz / 128 oz/gal = 3 gallons
So to convert any amount of ounces to gallons, you simply divide the total ounces by 128. This gives you the number of gallons, since there are 128 ounces per every 1 gallon.
How Many 24 Ounce Containers Equal a Gallon?
Okay, now we can answer the original question: How many 24-ounce containers does it take to equal 1 gallon? Let’s think step-by-step:
1. There are 128 ounces in 1 gallon
2. Each 24 ounce container holds 24 ounces
3. So to reach 128 ounces total, we need 128 / 24 = 5.3 containers
4. Since we can’t have a partial container, round down to the nearest whole number
5. Therefore, it takes 4 containers of 24 fluid ounces to equal 1 gallon
To summarize:
• 1 gallon = 128 fluid ounces
• Each 24 oz container holds 24 fluid ounces
• So 4 containers x 24 oz/container = 96 oz
• 96 oz is the same as 3 quarts or 0.75 gallons
• It takes 4 full 24-oz containers to reach 128 oz total, which is 1 gallon
Conversion Table
Here is a table summarizing some key equivalent measurements related to this gallon and ounce conversion:
Gallons Ounces 24 oz containers
1 gallon 128 ounces 4 containers
2 gallons 256 ounces 8 containers
3 gallons 384 ounces 12 containers
4 gallons 512 ounces 16 containers
This table shows the number of 24 ounce containers needed for 1, 2, 3, and 4 gallons. You can see the consistent pattern that for any number of gallons, you multiply by 4 containers to get the
24-ounce equivalent.
Practical Examples
Here are some practical examples of how this gallon to 24-ounce conversion might be used:
• Cooking: A recipe calls for 2 gallons of water. You’ll need 8 containers of 24 ounces (oz) each to measure out the full amount.
• Mixing solutions: A cleaning solution is made by combining 2 gallons vinegar and 1 gallon water. You’ll need 8 containers of vinegar (24 oz each) and 4 containers of water (24 oz each) to mix up
the full 3 gallons.
• Buying beverages: You’re having a party and estimate you’ll need 3 gallons of iced tea. So when buying containers of tea at the store, you should get 12 containers that are 24 ounces each.
Any application involving fluid gallons can be converted to 24-ounce containers using the conversions discussed here. This can make measurements and purchasing easier when you need to scale up or
Other Common Gallon Conversions
In addition to 24-ounce containers, there are some other handy conversions between gallons and common liquid amounts:
• 1 gallon = 4 quarts
• 1 gallon = 8 pints
• 1 gallon = 16 cups
• 1 gallon = 32 gills (a gill is 4 ounces)
Here are some examples of converting gallons to these other units:
• 2 gallons = 8 quarts
• 5 gallons = 40 cups
• 0.5 gallons (1/2 gallon) = 4 pints
And going the other direction:
• 12 cups = 3 gallons
• 10 pints = 1.25 gallons
• 6 quarts = 1.5 gallons
Knowing these conversions allows you to move easily between gallons, quarts, pints, cups, and ounces when dealing with liquid volumes.
Why Gallon Conversions Are Useful
Being able to convert between gallons, ounces, and other liquid measures comes in handy for all kinds of situations including:
• Cooking and baking recipes
• Mixing household solutions like cleaners, garden chemicals, etc.
• Measuring for crafts and DIY projects
• Calculating needs for events, parties, restaurants
• Portioning out beverages like juice, milk, or water
• Following instructions for medications, supplements, etc.
Any time you need to measure or divide up liquid amounts, converting gallons to more practical volumes can save time and minimize waste. Some common items like water, juice, soda, and milk are
available in gallon jugs. But for ease of use, it’s good to know how to portion those gallons out into smaller amounts as needed.
Tips for Converting Gallons to Ounces
Here are some helpful tips to keep in mind when converting between gallons and ounces:
• Memorize that 1 gallon = 128 fluid ounces. This makes the math a lot quicker.
• When converting gallons to ounces, multiply by 128. When going ounces to gallons, divide by 128.
• Use round numbers whenever possible. For example, 4 containers of 24 oz instead of 5.3.
• Remember fluid ounces are a volume measurement, not weight ounces.
• Keep a gallon/ounce conversion chart handy for easy reference.
• Use visuals like liquid measuring cups to better understand the amounts.
• Practice with examples from recipes or practical situations.
The Takeaway
Now you know the answer to “How many 24-ounce containers equal a gallon?” The simple math shows that it takes 4 full 24-ounce containers to equal 1 gallon. This useful conversion allows you to
measure gallons in practical volumes for cooking, mixing solutions, purchasing beverages, and any application involving liquid amounts. Understanding gallon to ounce conversions ensures you get the
right volumes for any situation.
So the next time you come across a measurement in gallons, you can quickly convert it to ounces or 24-ounce containers. Thisgallon and ounce equivalency provides flexibility to portion out any liquid
amount as needed.
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Note: this page contains legacy resources that are no longer supported. You are free to continue using these materials but we can only support our current worksheets, available as part of our
membership offering.
The “lessons” below could also be described as animations or even as slide shows. In fact, we are not exactly sure what to call them! They illustrate concepts and steps required to carry out
different operations. They have audio commentary although they can be viewed with just the text. The lessons developed so far include six on fractions, three on percent, and one on long division.
The links below will open the lessons in a new window. There is a navigation bar at the bottom each lesson that will allow you to:
• Pause and play the lesson
• Skip through different sections
• Switch the audio on and off
• View the lesson in full-screen mode
• See thumbnails of each screen and skip around the lesson
We would really like to hear what you think about these lessons. We will use what we learn to make them better.
Fractions Lessons
#1: Introducing Fractions (2:42 min.)
This lesson introduces the concept of fractions as being part
of one whole unit. It illustrates how numerators and
denominators are used when writing and saying fractions.
#2: Equivalent Fractions (3:41 min.)
The concept of equivalency of fractions is explored with
examples of finding equivalent fractions by multiplying and
dividing the numerator and denominator by the same number.
#3: Common Denominators (4:10 min.)
The need for and use of common denominators are discussed
in this brief lesson. Guidance on how to find common
denominators using least common multiples is shown.
#4: Adding and Subtracting Fractions (3:46 min.)
Examples are illustrated showing how to add and
how to subtract fractions. These include adding and subtracting
fractions with both the same denominator
and with different denominators.
#5: Simplifying Fractions (3:12 min.)
Two different methods of how to simplify fractions are shown.
One involves dividing the numerator and denominator
by their greatest common factor. The other uses more of a
“keep dividing until you can’t divide anymore” method.
#6: Mixed Numbers and Improper Fractions (4:12 min.)
The final lesson in the series covers when to use mixed
numbers and when to use improper fractions. It shows
the steps to convert between both formats.
You will also find more text-based guidance on how to work with fractions here. There are also a number of fraction worksheets as well as some fraction games.
Percent Lessons
#1: Introduction
This lesson looks at the concept of percent and illustrates
the importance of the number 100 in any percent calculation.
#2: Calculating with Percent e.g. 40% of 150 is ?
The first of three exercise types is covered in this lesson.
It shows how the part can be found given the percent and
the whole amount. e.g. what is 60% of 80?
#3: Calculating with Percent e.g. 12 out of 20 is
what % & 80 is 12% of ?
Two exercise types are covered in this lesson;
finding the percent when the part and whole are known
and finding the whole amount when the part and percent
are known. As in lesson 2, this is illustrated using a double number line.
You will find more here on calculating with percent. There are also a number of printable worksheets here that can be used for practice.
Division Lessons
Long Division: Step-by-step
A step-by-step animation showing long division as
sharing alongside the algorithmic calculation steps.
There is a long division worksheet generator here that provides limitless questions to practice the steps shown in the mini-lesson above.
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Welcome to Quickmath Solvers!
Enter an expression, enter the variable or variables to integrate with respect to (with their limits, if required) and click the Integrate button.
Integration by Parts
Unlike differentiating, there is no systematic procedure for finding antiderivatives. The rule of substitution discussed in the previous section is effective if we can find a substitution u (x) for
which a term u '(x) appears in the integral. Substitution may not work, but another powerful technique is available based on the product rule for derivatives. Integration by parts is given by the
following formula.
To see this, note that
from the product rule for derivatives. Also,
this statement about antiderivatives is true simply because the derivative of
Since the antiderivative on the right side already involves a constant of integration,c is redundant and can be omitted. This gives us the formula for integrating by parts.
Remark: The g (x) in the integral on the right side of Formula (1) is any antiderivative of g '(x) in the integral on the left side; therefore, the constant in g (x) on the right side may be chosen
to be any convenient value. It is usually chosen to be 0, although occasionally a different choice is useful.
Example 1 Find
To write this integral in the form
in order to apply Formula (1), we must determine how to choose f and g. The rule of thumb is to choose for g '(x) the "most difficult part" of the integrand which can be integrated. In this example,
the rule suggests that we let g'(x) = e^x, and thus f(x) = x. With this choice,
Clearly, f'(x) = I and g(x) = e^x. (Since g(x) is any antiderivative, we choose c = 0 in the general antiderivative e^x + c .) That is,
Example 2 Find
We have a choice to make: either let f(x) = ln(x) and g'(x) = 1 or let f(x) = 1 and g'(x) = ln(x). If we choose g'(x) = ln(x) and f(x) = l, we must then find g(x) by integrating ln(x), which is,
unfortunately, exactly our problem in this example.
Hence, let f(x) = ln(x) and g'(x) = 1. Then f '(x) = 1/x and g(x) = x. We now have
Example 3 Find
We could let g '(x) = ln(x) and use the result of Example 2, but then g (x) would be fairly complicated. Instead we let f(x) = ln(x) and g'(x) = x^1/2, in which case f '(x) = l/x and g(x) = 2/3x^3/
The next example illustrates how we can use integration by parts to evaluate a definite integral. The fundamental theorem of calculus is used together with the fact that the integration by parts
formula changes one antiderivative into another. The integration by parts formula is
Hence, the corresponding formula for the definite integral is the following:
Example 4 Find
We begin by using the conventional techniques for integrating by parts. As in Example 1, let f(x) = x and g'(x) = e^x. Then f '(x) = 1 and g(x) = e^x, and hence, by Formula (3)
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Problem 1065 - TheMathWorld
Problem 1065
Determine the size of the matrix.
To find the size of the matrix, count the rows and the columns.
How many rows does the matrix above have?
How many columns does the matrix above have?
The size of the matrix is 3 × 2.
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Correlation of electrons in a narrow s band for Physical Review
Physical Review
Correlation of electrons in a narrow s band
View publication
The ground-state wave function for the electrons in a narrow s band is investigated for arbitrary density of electrons and arbitrary strength of interaction. An approximation is proposed which limits
all the calculations to counting certain types of configurations and attaching the proper weights. The expectation values of the one-particle and two-particle density matrix are computed for the
ferromagnetic and for the non-ferromagnetic case. The ground-state energy is obtained under the assumption that only the intra-atomic Coulomb interaction is of importance. Ferromagnetism is found to
occur if the density of states is large at the band edges rather than in the center, and if the intra-atomic Coulomb repulsion is sufficiently strong. The relation of this approximation to certain
exact results for one-dimensional models is discussed. © 1965 The American Physical Society.
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Safe Haskell None
Language Haskell2010
Provides NonDetC, a carrier for NonDet effects providing choice and failure.
Under the hood, it uses a Church-encoded structure and a binary tree to prevent the problems associated with a naïve list-based implementation.
Since: 1.0.0.0
NonDet carrier
runNonDet Source #
:: (m b -> m b -> m b) Handles choice (<|>)
-> (a -> m b) Handles embedding results (pure)
-> m b Handles failure (empty)
-> NonDetC m a A nondeterministic computation to execute
-> m b
Run a NonDet effect, using the provided functions to interpret choice, leaf results, and failure.
runNonDet fork leaf nil (pure a <|> empty) = leaf a `fork` nil
Since: 1.0.0.0
runNonDetA :: (Alternative f, Applicative m) => NonDetC m a -> m (f a) Source #
Run a NonDet effect, collecting all branches’ results into an Alternative functor.
Using [] as the Alternative functor will produce all results, while Maybe will return only the first. However, unless used with cull, this will still enumerate the entire search space before
returning, meaning that it will diverge for infinite search spaces, even when using Maybe.
runNonDetA (pure a) = pure [a]
runNonDetA (pure a) = pure (Just a)
Since: 1.0.0.0
newtype NonDetC m a Source #
NonDetC (forall b. (m b -> m b -> m b) -> (a -> m b) -> m b -> m b)
MonadTrans NonDetC Source #
Defined in Control.Carrier.NonDet.Church
Monad (NonDetC m) Source #
Defined in Control.Carrier.NonDet.Church
Functor (NonDetC m) Source #
Defined in Control.Carrier.NonDet.Church
MonadFix m => MonadFix (NonDetC m) Source # Separate fixpoints are computed for each branch.
Defined in Control.Carrier.NonDet.Church
MonadFail m => MonadFail (NonDetC m) Source #
Defined in Control.Carrier.NonDet.Church
Applicative (NonDetC m) Source #
Defined in Control.Carrier.NonDet.Church
MonadIO m => MonadIO (NonDetC m) Source #
Defined in Control.Carrier.NonDet.Church
Alternative (NonDetC m) Source #
Defined in Control.Carrier.NonDet.Church
MonadPlus (NonDetC m) Source #
Defined in Control.Carrier.NonDet.Church
Algebra sig m => Algebra (NonDet :+: sig) (NonDetC m) Source #
Defined in Control.Carrier.NonDet.Church
NonDet effects
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Vizly: AI-powered data analysisVizly: Analyzing Academic Performance by Test Preparation Course Status
T-Test Results for Academic Performance by Test Preparation Course Status
• Math Scores:
□ P-value: $1.54 \times 10^{-8}$
• Reading Scores:
□ P-value: $9.08 \times 10^{-15}$
• Writing Scores:
□ P-value: $3.69 \times 10^{-24}$
Conclusion: The p-values for math, reading, and writing scores are significantly lower than the common alpha level of 0.05, suggesting that we reject the null hypothesis. There is a statistically
significant difference in academic performance between students who completed a test preparation course and those who did not.
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Summarize within | ArcGIS GeoAnalytics Engine | Esri Developer
Summarize Within calculates statistics in areas where geometries from the input DataFrame are within or overlap specified boundaries. The boundaries can be a polygon geometry column or you can
summarize within hexagonal or square bins.
Examples of summarizing points within polygons (first row), linestrings within polygons (second row), and polygons within polygons (third row).
Usage notes
• Summarize Within takes a DataFrame and a boundary, and stacks them on top of each other. After stacking, you can look down through the stack and count the number of summarized input records that
fall within the input boundaries. You can also calculate statistics about the attributes of the input summary records, such as minimum, maximum, sum, mean, weighted mean, and so on.
• There are two ways to specify the boundaries:
□ Use a polygon DataFrame by specifying setSummaryPolygons().
□ Use a square or hexagonal bin of a specified size that is generated when the analysis is run by specifying setSummaryBins().
• The bin size specifies how large the bins are. If you are summarizing into hexagons, the size is the height of each hexagon, and the radius of the resulting hexagon will be the height divided by
the square root of three. If you are aggregating into squares, the bin size is the height of the square, which is equal to the width.
• Use Summarize Within to calculate standard statistics as well as geographically weighted statistics. Standard statistics summarize the statistical values without weighting. Weighted statistics
calculate values using the geographically weighted attributes of lines within a polygon, or the attributes of polygons within a polygon. Weighted statistics do not apply to points within
• Standard statistics and geographically weighted statistics can be calculated for attributes that represent either counts or rates. These are defined as follows:
□ Counts—Attributes that represent a sum or quantity of an entity at a point location, along a line, or within a polygon. Examples of count-type attributes include the population of a country,
the number of taxi pickups in a census block, and the number of dams along a river. For line and polygon features, counts are proportioned before calculating standard or weighted statistics.
□ Rates—Attributes that represent a ratio or index at a point location, along a line, or within a polygon. Examples of rate-type attributes include the population density of a country, the
speed limit of a road, or the walkability score of a neighborhood. Rates are never proportioned.
For count-type attributes, values are proportioned according to the amount of the line within a polygon or the amount of the polygon within another polygon prior to calculating statistics.
Statistics are calculated the same way for count-type and rate-type attributes when the summary features are points.
• You can calculate the lengths and areas of the summarized geometries within each polygon using the options in the table below. Options are based on the geometry of the summarized DataFrame.
Input geometry Description Option
Points The count of summarized points within each boundary. None
• Miles
• Yards
Linestrings The length of summarized linestrings within or intersecting each boundary. • Feet
• Kilometers
• Meters
• Square Miles
• Square Yards
• Square Feet
Polygons The area of summarized polygons within or intersecting each boundary. • Square Kilometers
• Square Meters
• Hectares
• Acres
• For standard statistics, there are ten options: count, sum, mean, minimum, maximum, range, standard deviation, variance, first, and last. Count and sum will not be calculated for rate-type
attributes. There are four options for string statistics: count, any, first, and last.
• For weighted statistics, there are three options: mean, standard deviation, and variance. Weighted statistics are not calculated for string data.
• To calculate first or last, time needs to be enabled on the input DataFrame.
• Analysis with binning requires that your input DataFrame's geometry has a projected coordinate system. If your data is not in a projected coordinate system, the tool will transform your input to
a World Cylindrical Equal Area (SRID: 54034) projection. You can transform your data to a projected coordinate system by using ST_Transform.
• Optionally, specify a field name using setGroupBy() so statistics are calculated separately for each unique field value. When a group by field value is specified, a summary table listing each
record and statistic is also created.
• The options include_minor_major_fields and include_group_percentages are part of the group by option (setGroupBy()). The minority and majority will be the least and most dominant value from the
group field, respectively, where dominance is determined using the count of points, total length, or total area of each value.
• When the specified value for include_minor_major_fields is True, two fields will be added to the result DataFrame. The fields will list the values from the group field that are the minority and
majority for each result.
• The include_group_percentages option can only be used when you specify a value of True for include_minor_major_fields. When the value specified for include_group_percentages is True, two fields
will be added to the result DataFrame listing the percentage of the count of points, total length, or total area that belong to the minority and majority values for each input record. A
percentage field will also be added to the result table listing the percentage of the count of points, total length, or total area that belong to all values from the group by field for each input
• The output DataFrame always contains polygons. Only polygons that intersect the summarized geometries will be returned. Other polygons will be completely removed from the result.
• You can only calculate statistics on the records that intersect your boundary.
The following fields are included in the output polygon DataFrame:
Field Description
bin_geometry The result bin geometries.
count The count of summarized records that intersect each boundary.
sum_length_<linearunit The total length of linestrings within or intersecting the boundary or total area of summarized polygons within or intersecting a polygon. These values are returned when you
>, or sum_area_ specify a value of True for includeShapeSummary() and are returned in the specified unit.
<statistic>_<fieldname> Specified standard statistics will each create a field named in the following format: <statistic>_<fieldname>. For example, the maximum and standard deviation of the field id
is MAX_id and SD_id.
p<statistic>_<fieldname Specified weighted statistics will each create a field named in the following format: p<statistic>_<fieldname>. For example, the mean and standard deviation of the field pop
> is pMEAN_pop and pSD_pop.
This value is returned when you create a group-by table and specify minority and majority calculations. This represents the values for the specified field that is the minority
minority_<fieldname> in each polygon. For example, there are five points within a polygon with a field called color and values of red, blue, blue, green, green. If you create a group by the color
field, the value for the minority_color field is red.
This value is returned when you create a group-by table and specify minority and majority calculations. This represents the values for the specified field that is the majority
majority_<fieldname> in each polygon. For example, there are five points within a polygon with a field called color and values of red, blue, blue, green, green. If you create a group by the color
field, the value for the majority_color field is blue;green.
minority_<fieldname> This value is returned when you create a group-by table and specify percent shapes. This represents the percentages of the count for the specified field that is the minority
_percent in each polygon. For example, there are five points within a polygon with a field called color and values of red, blue, blue, green, green. If you create a group by the color
field, the value for the minority_color_percent field is 20 (calculated as 1/5).
majority_<fieldname> This value is returned when you create a group-by table and specify percent shapes. This represents the percentages of the count for the specified field that is the majority
_percent in each polygon. For example, there are five points within a polygon with a field called color and values of red, blue, blue, green, green. If you create a group by the color
field, the value for the majority_color_percent field is 40 (calculated as 2/5).
join_id This value is returned when you create a group-by table. This is an ID to link records to the group-by table. Every join_id field corresponds to one or more records in the
group-by table.
The following fields are included in the output group-by DataFrame:
Field Description
This is an ID to link records to the polygon DataFrame. Each polygon will have one or more records with the same ID that represent all of the group-by values. For example, there are five
join_id points within a polygon with a field called color and values of red, blue, blue, green, green. The group-by table will have three records representing that polygon (same join ID), one
for each of the colors red, blue, and green.
count The count of the specified group within the joined polygon. For example, red is 1 for the selected polygon.
<statistic>_ Any specified statistic calculated for each group.
percentcount The percentage each group contributes to the total count in the polygon. Using the above example, red contributes 1/5 = 20, blue contributes 2/5 = 40, and green contributes 2/5 = 20.
Performance notes
Improve the performance of Summarize Within by doing one or more of the following:
• Only analyze the records in your area of interest. You can pick the records of interest by using one of the following SQL functions:
• If you are using bins, larger bins will perform better than smaller bins. If you are unsure which size to use, start with a larger bin to prototype.
Similar capabilities
For more details, go to the GeoAnalytics Engine API reference for summarize within.
Setter Description Required
Adds a field in the input DataFrame to a list of fields that represent rates, indices, or ratios. Examples of rate-type attributes
addRateField(rate_field) include the population density of a country, the speed limit of a road, or the walkability score of a neighborhood. Rates are never No
proportioned. By default all fields are assumed to represent counts or amounts and will be proportioned.
addStandardSummaryField(summary_field, Adds a standard summary statistic of a field in the input DataFrame to the result DataFrame. Statistics for numeric fields include
statistic, alias=None) Count, Sum, Mean, Max, Min, Range, Stddev, Var, First, Last, or Any. Count and sum will not be calculated for rate-type fields. There No
are four options for string statistics: Count, Any, First, and Last.
addWeightedSummaryField(summary_field, Adds a weighted summary statistic of a field in the input DataFrame to the result DataFrame. Statistics include Mean, Stddev, and Var. No
statistic, alias=None) Weighted statistics are not calculated for string fields.
includeShapeSummary(include=True, Sets to the tool to calculate statistics based on the geometry type of the primary geometry column in the input DataFrame, such as the No
units=None) length of lines or areas of polygons within each summary polygon.
run(dataframe) Runs the Summarize Within tool using the provided DataFrame. Yes
setGroupBy(group_by_field, include Sets a field from the input DataFrame that will be used to calculate statistics for each unique value. When setGroupBy() is called,
_minor_major_fields=True, include the tool will return a DataFrame containing the grouped statistics in addition to a DataFrame containing the summaries. No
setSummaryBins(bin_size, bin_size One of setSummaryBins()
_unit, bin_type='square') Sets the size and shape of bins that the input DataFrame will be summarized into. or setSummaryPolygons()
is required.
One of setSummaryBins()
setSummaryPolygons(summary_polygons) Sets the DataFrame containing a column of polygons that the input DataFrame will be summarized into. or setSummaryPolygons()
is required.
Run Summarize Within
Use dark colors for code blocksCopy
# Log in
import geoanalytics
geoanalytics.auth(username="myusername", password="mypassword")
# Imports
from geoanalytics.tools import SummarizeWithin
from geoanalytics.tools import ReconstructTracks
from geoanalytics.sql import functions as ST
# Path to the hurricane tracks dataset
hurricanes_data_path = r"https://services2.arcgis.com/FiaPA4ga0iQKduv3/arcgis/rest/" \
# Create a hurricanes tracks DataFrame and filter to a smaller extennt of area
hurricanes_df = spark.read.format("feature-service").load(hurricanes_data_path) \
.withColumn("bbox_intersects", ST.bbox_intersects("shape",-10512137.72, -9527997.38,3278846.39,4303954.46)) \
.where("bbox_intersects == 'true'") \
.where("BASIN == 'NA'")
# Use Reconstruct Tracks to create hurricane paths
rt_result = ReconstructTracks() \
.setTrackFields("NAME") \
.setDistanceMethod(distance_method="Planar") \
# Use Summarize Within to summarize hurricane tracks into bins to
# visualize a track heat map
result = SummarizeWithin() \
.setSummaryBins(bin_size=200, bin_size_unit="Kilometers", bin_type='hexagon') \
.includeShapeSummary(include=True, units="Kilometers") \
Plot results
Use dark colors for code blocksCopy
# Plot the summarized result with shorelines (continent outlines) near Florida
continents_path = "https://services.arcgis.com/P3ePLMYs2RVChkJx/ArcGIS/rest/" \
shoreline_df = spark.read.format("feature-service").load(continents_path)
result_plot = result.output.st.plot(cmap_values="COUNT",
shoreline_plot = shoreline_df.st.plot(edgecolors="black",
result_plot.set_title("Hurricane track heat map near Florida and the Gulf of Mexico")
result_plot.set_xlabel("X (Meters)")
result_plot.set_ylabel("Y (Meters)");
result_plot.set_xlim(left=-10512137, right=-8027997)
result_plot.set_ylim(bottom=2688846, top=4303954)
Version table
Release Notes
1.0.0 Tool introduced
Links to helpful information
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Subtraction Color By Number
These subtraction coloring worksheets require students to solve simple math facts to find the right color to shade in to reveal a picture of their own creation. You'll find a growing set of holiday
and seasonal themed pages that I'll be adding to over time... Please check back often for updates, or if you have a suggestion send me a note at the contact link below!
Practice Subtraction Facts with these Color by Number Worksheets!
First grade and second grade students who are learning their subtraction facts will have a great time completing these fun coloring pages! They also make for a fun art activity for students in later
This collection of worksheets is growing, and I'll continue adding more coloring worksheets for various holidays and seasons... If you like these, be sure to check out the other pages for
color-by-number, addition, multiplication and division problems that also feature coloring solutions.
Be sure to check out the subtraction worksheets at the link below! You'll find printable worksheets that start with subtraction facts and progress through multi-digit subtraction problems with and
without regrouping...
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One Way A
Struggling to Run a One Way ANOVA Test in Excel?
QI Macros can run Anova tests & interpret the results for you!
Run One-Way ANOVA using QI Macros
1. Select your data.
2. Click on QI Macros menu > Statistical Tools > ANOVA > ANOVA Single Factor
3. QI Macros will do the math and analysis for you.
You Don't Have to be a Expert to Run a One-Way ANOVA test
• Does the thought of performing complicated statistical analysis intimidate you?
• Have you struggled with the awkward interface of Excel's Data Analysis Toolpak?
• Have you tried to learn another more complicated statistics program?
One-Way ANOVA Step-by-Step Example
Imagine you manufacture paper bags and you want to improve the tensile strength of the bag. You suspect that changing the concentration of hardwood in the bag will change the tensile strength. You
measure the tensile strength in pounds per square inch (PSI). So, you decide to test this at 5%, 10%, 15% and 20% hardwood concentration levels. These "levels" are also called "treatments."
To perform One-Way ANOVA in Excel using QI Macros follow these steps:
1. Click and drag over your data to select it:
2. Now, click on QI Macros menu > Statistical Tools > ANOVA > ANOVA Single Factor:
3. QI Macros will prompt you for the significance level. The default is 0.05 (95% confident).
4. QI Macros will perform the calculations and analyze the results for you:
QI Macros is Smart Enough to Interpret the Results for You
QI Macros built in code compares the p-value (it calculates) to the significance level (you input) to tell you what the results mean. You will see one of two results:
• Reject the null hypothesis - Means are different/Means are not the same.
• Cannot Reject the null hypothesis (Accept the null hypothesis) - Means are the same/Means are not different
In this example, QI Macros compares the p-value (0.000) to the significance (0.05) and tells you to "Reject the Null Hypothesis because p<0.05 (Means are Different)."
After a one-way ANOVA finds a significant difference in means, Post Hoc testing helps identify which of the differences are significant.
QI Macros Will Even Draw a Chart to Help You Visualize the Results
In this example, QI Macros draws a values plot. Compare the difference in Means using the line graph and the variation using the height of the dots.
Stop Struggling with One-Way ANOVA Tests!
Start conducting One-Way ANOVA Tests in just minutes.
QI Macros can draw these charts too!
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Compound Inequalities Worksheet Answer Key
Compound Inequalities Worksheet Answer Key
Then graph the solution set on a number line. X 3 3 or x 3 3 9.
Matching Activity Solving Inequality Word Problems Algebra Inequality Word Problems Solving Inequalities Word Problems
K 2 12 and k 2.
Compound inequalities worksheet answer key. 2 3b 7 13 8. These printable worksheets are exclusively designed for high school students free handouts are also. Then graph the solution set.
Each section except the first section contains three levels of compound inequalities worksheets based on either solving or graphing or both. All numbers between 1 5 and 1 5 including 1 5 and 1 5 i t
i 5 write an absolute value inequality for each graph. This sort reviews solving 1 variable compound inequalities both and and or this sort has 4 parts compound inequalities 10 problems simplified
inequalities graph interval notationalso included is student answer sheet and answer key this sort can be used several ways.
2s 3 7 or 3s 5 26 write a compound inequality for each problem. K 2 12 and k 2. M z2 y0r152 w vklu ot2a z ysnovf 8tcw za zr re 0 tl ul 1c a l q la hlxlk ir pimgkhptis f 0r2e 9s9e5rtvue wdu 3 g umna4d
iem jw 6izt shp di fnsf oisnbi4t 0e5 qaplkgie kb irja m 92k 6 worksheet by kuta software llc kuta software infinite algebra 2 name compound inequalities date period.
All numbers greater than 4 or less than 4 n 4 2. 15 x 8 4 6. Solve the compound inequality.
Free printable worksheet with answer key on the discriminant in quadratic equations and the nature and number of roots. Than less than worksheets from solving and graphing inequalities worksheet
answer key source. Number of years is 2.
1 m or m. A l 1mda9d ken 6wsi rt 4hw hinnbf ti7n nipt ie2 uajlagte 8b 0r4al y1e e worksheet by kuta software llc kuta software infinite algebra 1 name compound inequalities date period solve each
compound inequality and graph its solution. This site has tons of worksheets and activities he has all of from solving and graphing inequalities worksheet answer key source.
The levels are classified based on the number of steps that required solving compound inequalities. 12 4n 28 7. Solving compound inequalities worksheet.
Solve each compound inequality and graph the solutions. Component algebra 2 worksheets answer key guided practice suggestions 3 cont. 05 is owed according to the compound interest formula.
Then graph the solution set. Solving compound and absolute value inequafities write an absolute value inequality for each of the following. 5k 20 or 2k 8 10.
Solve the compound inequality. Algebra 2 compound interest worksheet answer key.
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Napierian Sentence Examples
• The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common
system, of which the base is io; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier /loge io, which is called
the modulus of the common system of logarithms.
• Napier's logarithms are not the logarithms now termed Napierian or hyperbolic, that is to say, logarithms to the base e where e= 2.7182818 ...; the relation between N (a sine) and L its
logarithm, as defined in the Canonis Descriptio, being N=10 7 e L/Ip7, so that (ignoring the factors re, the effect of which is to render sines and logarithms integral to 7 figures), the base is
• If 1 denotes the logarithm to base e (that is, the so-called "Napierian " or hyperbolic logarithm) and L denotes, as above, " Napier's " logarithm, the connexion between 1 and L is expressed by L
= r o 7 loge 10 7 - 10 7 / or e t = I 07e-L/Ia7 Napier's work (which will henceforth in this article be referred to as the Descriptio) immediately on its appearance in 1614 attracted the
attention of perhaps the two most eminent English mathematicians then living - Edward Wright and Henry Briggs.
• The logarithms are strictly Napierian, and the arrangement is identical with that in the canon of 1614.
• This is the largest Napierian canon that has ever been published.
• In the same year (1624) Kepler published at Marburg a table of Napierian logarithms of sines with certain additional columns to facilitate special calculations.
• In 1873 Charles Hermite proved that the base of the Napierian logarithms cannot be a root of a rational algebraical equation of any degree.3 To prove the same proposition regarding 7r is to prove
that a Euclidean construction for circle-quadrature is impossible.
• Similarly the continued fraction given by Euler as equivalent to 1(e - 1) (e being the base of Napierian logarithms), viz.
• The formula then becomes I = Ioe kt (2) where e is the base of Napierian logarithms, and k is a constant which is practically the same as j for bodies which do not absorb very rapidly.
• The logarithms introduced by Napier in the Descriptio are not the same as those now in common use, nor even the same as those now called Napierian or hyperbolic logarithms. The change from the
original logarithms to common or decimal logarithms was made by both Napier and Briggs, and the first tables of decimal logarithms were calculated by Briggs, who published a small table,
extending to 1000, in 1617, and a large work, Arithmetica Logarithmica, 1 containing logarithms of numbers to 30,000 and from 90,000 to Ioo,000, in 1624.
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ManPag.es -
sppequ.f −
subroutine SPPEQU (UPLO, N, AP, S, SCOND, AMAX, INFO)
Function/Subroutine Documentation
subroutine SPPEQU (characterUPLO, integerN, real, dimension( * )AP, real, dimension( * )S, realSCOND, realAMAX, integerINFO)
SPPEQU computes row and column scalings intended to equilibrate a
symmetric positive definite matrix A in packed storage and reduce
its condition number (with respect to the two-norm). S contains the
scale factors, S(i)=1/sqrt(A(i,i)), chosen so that the scaled matrix
B with elements B(i,j)=S(i)*A(i,j)*S(j) has ones on the diagonal.
This choice of S puts the condition number of B within a factor N of
the smallest possible condition number over all possible diagonal
UPLO is CHARACTER*1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.
N is INTEGER
The order of the matrix A. N >= 0.
AP is REAL array, dimension (N*(N+1)/2)
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.
S is REAL array, dimension (N)
If INFO = 0, S contains the scale factors for A.
SCOND is REAL
If INFO = 0, S contains the ratio of the smallest S(i) to
the largest S(i). If SCOND >= 0.1 and AMAX is neither too
large nor too small, it is not worth scaling by S.
AMAX is REAL
Absolute value of largest matrix element. If AMAX is very
close to overflow or very close to underflow, the matrix
should be scaled.
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, the i-th diagonal element is nonpositive.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 117 of file sppequ.f.
Generated automatically by Doxygen for LAPACK from the source code.
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Today in Science History - Quickie Quiz
Who said: “The path towards sustainable energy sources will be long and sometimes difficult. But America cannot resist this transition, we must lead it... That is how we will preserve our planet,
commanded to our care by God. That�s what will lend meaning to the creed our fathers once declared.”
Category Index for Science Quotations
Category Index T
> Category: Twice
Twice Quotes (20 quotes)
Here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!
Said by the fictional Red Queen character, in Through the Looking Glass and What Alice Found There (1872, 1896), 36.
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Fast (49)
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Want (504)
A metaphysician is one who, when you remark that twice two makes four, demands to know what you mean by twice, what by two, what by makes, and what by four. For asking such questions metaphysicians
are supported in oriental luxury in the universities, and respected as educated and intelligent men.
A previously unpublished epigram, added in A Mencken Chrestomathy (1949, 1956), 13-14.
Behold the mighty dinosaur,
Famous in prehistoric lore,
Not only for his power and strength
But for his intellectual length.
You will observe by these remains
The creature had two sets of brains—
One in his head (the usual place),
The other at his spinal base.
Thus he could reason 'A priori'
As well as 'A posteriori'.
No problem bothered him a bit
He made both head and tail of it.
So wise was he, so wise and solemn,
Each thought filled just a spinal column.
If one brain found the pressure strong
It passed a few ideas along.
If something slipped his forward mind
'Twas rescued by the one behind.
And if in error he was caught
He had a saving afterthought.
As he thought twice before he spoke
He had no judgment to revoke.
Thus he could think without congestion
Upon both sides of every question.
Oh, gaze upon this model beast
Defunct ten million years at least.
'The Dinosaur: A Poem' (1912). In E. H. Colbert (ed.), The Dinosaur Book (1951), 78.
Every man looks at his wood-pile with a kind of affection. … [T]hey warmed me twice, once while I was splitting them, and again when they were on the fire, so that no fuel could give out more heat.
In Walden: or, Life in the Woods (1854, 1899), 263.
Few people think more than two or three times a year. I have made an international reputation for myself by thinking once or twice a week.
As given in 'Quotable Quotes', Reader’s Digest (May 1933). It does not appear in a work written by Shaw. It may have been contributed to the magazine as a personal recollection, though that is not
specified in that source.
For a smart material to be able to send out a more complex signal it needs to be nonlinear. If you hit a tuning fork twice as hard it will ring twice as loud but still at the same frequency. That’s a
linear response. If you hit a person twice as hard they’re unlikely just to shout twice as loud. That property lets you learn more about the person than the tuning fork. - When Things Start to Think,
I just wish the world was twice as big and half of it was still unexplored.
Epigraph Bruce L. Smith, Stories from Afield: Adventures with Wild Things in Wild Places (2016), Chap. 16, citing the TV program Life on Earth.
If you see an antimatter version of yourself running towards you, think twice before embracing.
In general, mankind, since the improvement of cookery, eat about twice as much as nature requires.
Louis Klopsch, Many Thoughts of Many Minds (1896), 67.
Mathematics: A science that cannot explain what happens to a man if his wife is his better half and he marries twice.
In Esar’s Comic Dictionary (1943, 4th ed. 1983), 373.
Oddly enough, eccentrics are happier and healthier than conformists. A study of 1,000 people found that eccentrics visit a doctor an average of just once every eight years, while conformists go twice
a year. Eccentrics apparently enjoy better health because they feel less pressured to follow society’s rules, said the researcher who did the study at Royal Edinburgh Hospital in Scotland.
Eccentrics (1995).Study results in SELF magazine - 1992 National Enquirer.
Once when lecturing to a class he [Lord Kelvin] used the word “mathematician,” and then interrupting himself asked his class: “Do you know what a mathematician is?” Stepping to the blackboard he
wrote upon it:— [an integral expression equal to the square root of pi]
Then putting his finger on what he had written, he turned to his class and said: “A mathematician is one to whom that is as obvious as that twice two makes four is to you. Liouville was a
In Life of Lord Kelvin (1910), 1139.
Patience and tenacity of purpose are worth more than twice their weight in cleverness.
The difference between the long-term average of the graph and the ice age, 12,000 years ago, is just over 3°C. The IPCC 2001 report suggests that the line of the hockey stick graph might rise a
further 5°C during this century. This is about twice as much as the temperature change from the ice age to pre-industrial times.
In The Revenge of Gaia: Earth’s Climate Crisis & The Fate of Humanity (2006, 2007), 67.
The sea is not all that responds to the moon. Twice a day the solid earth bobs up and down, as much as a foot. That kind of force and that kind of distance are more than enough to break hard rock.
Wells will flow faster during lunar high tides.
Annals of the Former World
The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face a blackboard and to turn his back on
the class. He writes a, he says b, he means c, but it should be d. Some of his sayings are handed down from generation to generation:
“In order to solve this differential equation you look at it till a solution occurs to you.”
“This principle is so perfectly general that no particular application of it is possible.”
“Geometry is the science of correct reasoning on incorrect figures.”
“My method to overcome a difficulty is to go round it.”
“What is the difference between method and device? A method is a device which you used twice.”
In How to Solve It: A New Aspect of Mathematical Method (2004), 208.
We have an extraordinary opportunity that has arisen only twice before in the history of Western civilization—the opportunity to see everything afresh through a new cosmological lens. We are the
first humans privileged to see a face of the universe no earlier culture ever imagined.
As co-author with Nancy Ellen Abrams, in The View from the Center of the Universe: Discovering Our Extraordinary Place in the Cosmos (2006), 297.
We have little more personal stake in cosmic destiny than do sunflowers or butterflies. The transfiguration of the universe lies some 50 to 100 billion years in the future; snap your fingers twice
and you will have consumed a greater fraction of your life than all human history is to such a span. ... We owe our lives to universal processes ... and as invited guests we might do better to learn
about them than to complain about them. If the prospect of a dying universe causes us anguish, it does so only because we can forecast it, and we have as yet not the slightest idea why such forecasts
are possible for us. ... Why should nature, whether hostile or benign, be in any way intelligible to us? All the mysteries of science are but palace guards to that mystery.
What is right may well be said even twice.
Fragment 25, as translated by John Burnet in Oliver Joseph Thatcher (ed.) The Library of Original Sources (1907), Vol. 2, 162. Also translated as “What must be said, may well be said twice o’er”, in
William Ellery Leonard (trans.), The Fragments of Empedocles (1908), 27.
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Right (473)
Say (989)
With old inflation riding the headlines, I have read till I am bleary-eyed, and I can’t get head from tails of the whole thing. ... Now we are living in an age of explanations—and plenty of ’em,
too—but no two things that’s been done to us have been explained twice the same way, by even the same man. It’s and age of in one ear and out the other.
Newspaper column, for example in 'Complete Heads and Tails', St. Petersburgh Times (28 Jan 1934), 4. Collected in Will Rogers’ Weekly Articles: The Roosevelt Years (1933-1935) (1982), 91-92.
In science it often happens that scientists say, 'You know that's a really good argument; my position is mistaken,' and then they would actually change their minds and you never hear that old view
from them again. They really do it. It doesn't happen as often as it should, because scientists are human and change is sometimes painful. But it happens every day. I cannot recall the last time
something like that happened in politics or religion. (1987) --
Carl Sagan
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Financing Project Development and Pass-Throughs As.....
Problem 16-1
The Investor-developer would not be comfortable with a 7.8 percent return on cost because the margin for error is too risky. If
construction costs are higher or rents are lower than anticipated, the project may not be feasible. The asking price of the project is
$10,000,000 and the construction cost per unit is $81,600. The current rent to justify the land acqusition is $1.9 per square foot. The
weighted average is 900 square feet per unit. Average vacancy and Operating expenses are 5% and 35% of Gross Revenue
respectively. Use the following data to rework the calculations in Concept Box 16.2 in order to assess the feasibility of the project
a. Based on the fact that the project appears to have 9,360 square feet of surface area in excess of zoning requirements, the
developer could make an argument to the planning department for an additional 10 units, 250 units in total, or 25 units per acre. What
is the percentage return on total cost under the revised proposal? Is the revised proposal financially feasible?
b. Suppose the developer could build a 240-unit luxury apartment complex with a cost of $119,000 per unit. Given that NOI is 60% of
rents. What would such a project have to rent for (per square foot) to make an 8 percent return on total cost?
Complete this question by entering your answers in the tabs below.
Required A
Type here to search
Required B
Suppose the developer could build a 240-unit luxury apartment complex with a cost of $119000 per unit. Given that NOI is
60% of rents. What would such a project have to rent for (per square foot) to make an 8 percent return on total cost?
Note: Do not round intermediate calculations. Round your final answer to nearest whole dollar amount.
per month per unit
< Required A
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Penetrative Bénard-Marangoni Convection in a Micropolar Ferrofluid Layer via Internal Heating and Submitted to a Robin Thermal Boundary Conditions
Penetrative Bénard-Marangoni Convection in a Micropolar Ferrofluid Layer via Internal Heating and Submitted to a Robin Thermal Boundary Conditions ()
1. Introduction
Ferrofluids are colloidal suspensions of magnetic nanoparticles, as suggested by Rosensweig [1] in his monograph, it is pertinent to consider the effect of micro-rotation of the particles in the
study. Based on this fact, studies have been undertaken by treating ferrofluids as micropolar fluids and the theory of micropolar fluid proposed by Eringen [2] has been used in investigating the
problems. Micropolar fluids have been receiving a great deal of interest and research focus due to their applications like solidification of liquid crystals, the extrusion of polymer fluids, cooling
of a metallic plate in a bath colloidal suspension solutions and exotic lubricants. In the uniform magnetic field, the magnetization characteristic depends on particle spin but does not on fluid
velocity: Hence micropolar ferrofluid stability studies have become an important field of research these days. Although convective instability problems in a micropolar fluid layer subject to various
effects have been studied extensively, the works pertaining to micropolar ferrofluids are in much-to-be desired state. Many researchers (Lebon and Perez [3] , Payne and Straughan [4] , Siddheshwar
and Pranesh [5] , Idris et al. [6] , Mahmud et al. [7] , Sharma and Kumar [8] ) have been rigorously investigated the Rayleigh-Bénard situation in Eringen’s micropolar non-magnetic fluids. From all
these studies, they mainly found that stationary convection is the preferred mode for heating from below. Zahn and Greer [9] have considered interesting possibilities in a planar micropolar
ferromagnetic fluid flow with an AC magnetic field. Abraham [10] has investigated the problem of Rayleigh-Bénard convection in a micropolar ferromagnetic fluid layer permeated by a uniform magnetic
field for stress-free boundaries. Thermal instability problem in a rotating micropolar ferrofluid has also been considered by Sunil et al. [11] . Nanjundappa et al. [12] have investigated the onset
of ferromagnetic convection in a micropolar ferromagnetic fluid layer heated from below in the presence of a uniform applied vertical magnetic field.
The practical problems cited above require a mechanism to control thermomagnetic convection. One of the mechanisms to control (suppress or augment) convection is by maintaining a non-uniform
temperature gradient across the layer of ferrofluid. Such a temperature gradient may arise due to 1) uniform distribution of heat sources 2) transient heating or cooling at a boundary, 3) temperature
modulation at the boundaries and so on. Works have been carried out in this direction but it is still in much-to-be desired state. Rudraiah and Sekhar [13] have investigated convection in a
ferrofluid layer in the presence of uniform internal heat source. The effect of non-uniform basic temperature gradients on the onset of ferroconvection has been analyzed (Shivakumara et al. [14] ,
and Shivakumara and Nanjundappa [15] [16] ). Singh and Bajaj [17] have studied thermal convection of ferrofluids with boundary temperatures modulated sinusoidally about some reference value.
Nanjundappa et al. [18] have studied the effect of internal heat generation on the criterion for the onset of convection in a horizontal ferrofluid saturated porous layer Nanjundappa et al. [19] have
explored a model for penetrative ferroconvection via internal heat generation in a ferrofluid saturated porous layer. Nanjundappa et al. [20] have investigated the onset of penetrative
Bénard-Marangoni convection in a horizontal ferromagnetic fluid layer in the presence of a uniform vertical magnetic field via an internal heating model. Ram and Kumar [21] has carried out to examine
the effects of temperature dependent variable viscosity on the three dimensional steady axi-symmetric Ferrohydrodynamic (FHD) boundary layer flow of an incompressible electrically non conducting
magnetic fluid in the presence of a rotating disk. Ram and Kumar [22] have analyzed the analysis of three dimensional rotationally symmetric boundary layer flow of field dependent viscous ferrofluid
saturating porous medium. Ram et al. [23] have been made to describe the effects of geothermal viscosity with viscous dissipation on the three dimensional time dependent boundary layer flow of
magnetic nanofluids due to a stretchable rotating plate in the presence of a porous medium. Ram et al. [24] have investigated numerically on the convective heat transfer behaviour of time-dependent
three-dimensional boundary layer flow of nano-suspension over a radially stretchable surface. Kumar et al. [25] have studied the Bodewadt flow of a magnetic nanofluid in the presence of geothermal
viscosity. Very recently, Ram et al. [26] have studied the rheological effects due to oscillating field on time dependent boundary layer flow of magnetic nanofluid over a rotating disk.
The purpose of this paper is to study the penetrative Bénard-Marangoni convection in a micropolar ferromagnetic fluid layer via internal heat generation. Such a study helps in understanding control
of convection due to a non-uniform temperature gradient arising due to an internal heat source, which is important in the applications of ferrofluid technology. The linear stability problem is solved
numerically using the Galerkin method, and the results are presented graphically. Moreover, the stability of the system when heated from below and also in the absence of thermal buoyancy is discussed
in detail.
2. Mathematical Formulation
We consider an initially quiescent horizontal incompressible micropolar ferrofluid layer of characteristic thickness d in the presence of an applied uniform magnetic field H[0] in the vertical
direction with the angular momentum $\omega$ . Let ${T}_{0}\left(z=0\right)$ and ${T}_{1}<{T}_{0}\left(z=d\right)$ be the temperatures of the lower and upper rigid boundaries, respectively with $\
Delta T\left(={T}_{0}-{T}_{1}\right)$ being the temperature difference. A uniformly distributed overall internal heat source is present within the micropolar ferrofluid layer. The Cartesian
co-ordinate system $\left(x,y,z\right)$ is used with the origin at the bottom of the layer and z-axis is directed vertically upward. Gravity acts in the negative z-direction, $g=-g\stackrel{^}{k}$
where $\stackrel{^}{k}$ is the unit vector in the z-direction.
The upper free boundary is assumed to be flat and subjected to linearly temperature dependent surface tension σ is $\sigma ={\sigma }_{0}-{\sigma }_{T}\left(T-{T}_{0}\right)$ , ${\sigma }_{T}$ is the
rate of thermal surface tension.
The governing equations for the flow of an incompressible micropolar ferromagnetic fluid are:
$abla \cdot q=0$(1)
${\rho }_{0}\left[\frac{\partial q}{\partial t}+\left(q\cdot abla \right)q\right]=-abla p+\rho g+\left(B\cdot abla \right)H+\left(\eta +{\xi }_{r}\right){abla }^{2}q+2{\xi }_{r}\left(abla ×\omega \
${\rho }_{0}I\left[\frac{\partial \omega }{\partial t}+\left(q\cdot abla \right)\omega \right]={\mu }_{0}\left(M×H\right)+abla \left(abla \cdot \omega \right)+{\eta }^{\prime }\left({abla }^{2}\omega
\right)+2{\xi }_{r}\left[\left(abla ×q\right)-2\omega \right]$(3)
$\begin{array}{l}{k}_{1}{abla }^{2}T+\delta \left(abla ×\omega \right)\cdot abla T+{Q}^{″}\\ =+{\mu }_{0}T{\left(\frac{\partial M}{\partial T}\right)}_{V,H}\cdot \frac{DH}{Dt}+\left[{\rho }_{0}{C}_
{V,H}-{\mu }_{0}H\cdot {\left(\frac{\partial M}{\partial T}\right)}_{V,H}\right]\frac{DT}{Dt}\end{array}$(4)
$\rho ={\rho }_{0}\left[1-\alpha \left(T-{T}_{0}\right)\right]$(5)
$abla \cdot B=0$ , $abla ×H=0$ or $H=abla \varphi$(6)
$B={\mu }_{0}\left(M+H\right)$(7)
$M={M}_{0}+\chi \left(H-{H}_{0}\right)-K\left(\stackrel{¯}{T}-{T}_{0}\right)$(9)
The basic state is assumed to be quiescent and is given by
$\left[{q}_{b},{\omega }_{b},\rho ,T,H,M\right]=\left[0,0,{\rho }_{b}\left(z\right),{T}_{b}\left(z\right),{H}_{b}\left(z\right),{M}_{b}\left(z\right)\right]$(10)
Using Equation (10) in Equation (2) and (4) respectively yield
$\frac{\text{d}{p}_{b}}{\text{d}z}=-{\rho }_{0}\left[1-{\alpha }_{t}\left({T}_{b}-{T}_{0}\right)\right]g\stackrel{^}{k}+{\mu }_{0}\text{}{M}_{b}\frac{\text{d}{H}_{b}}{\text{d}z}$(11)
Solving Equation (12) subject to the boundary conditions ${T}_{b}={T}_{0}$ at $z=0$ and ${T}_{b}={T}_{0}-\Delta T$ at $z=d$ , we obtain
${T}_{b}\left(z\right)=-\frac{Q{z}^{2}}{2{k}_{1}}+\frac{Qdz}{2{k}_{1}}-\beta z+{T}_{0}$(13)
Substituting Equation (6) after using Equations (9) and (13), the basic state magnetic field intensity ${H}_{b}\left(z\right)$ and magnetization ${M}_{b}\left(z\right)$ are found to be (see Finlayson
[4] )
${H}_{b}\left(z\right)=\left[{H}_{0}-\frac{K}{1+\chi }\left(\frac{Q{z}^{2}}{2{k}_{1}}-\frac{Qdz}{2{k}_{1}}+\beta z\right)\right]\stackrel{^}{k}$(14)
${M}_{b}\left(z\right)=\left[{M}_{0}+\frac{K}{1+\chi }\left(\frac{Q{z}^{2}}{2{k}_{1}}-\frac{Qdz}{2{k}_{1}}+\beta z\right)\right]\stackrel{^}{k}$(15)
where ${M}_{0}+{H}_{0}={H}_{0}^{ext}$ .
Using Equations (13) and (14) in Equation (11) and integrating, we obtain
$\begin{array}{c}{p}_{b}\left(z\right)={p}_{0}-{\rho }_{0}gz-{\rho }_{0}\alpha g\left[\frac{Q{z}^{3}}{6{k}_{1}}-\frac{Qd{z}^{2}}{4{k}_{1}}+\frac{\beta {z}^{2}}{2}\right]-\frac{{\mu }_{0}{M}_{0}K}{1+\
alpha }\left[\frac{Q{z}^{2}}{2{k}_{1}}-\frac{Qdz}{2{k}_{1}}+\beta z\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\frac{{\mu }_{0}{K}^{2}}{{\left(1+\alpha \right)}^{2}}\left[\frac{{Q}^{2}{z}^
{4}}{8{k}_{1}^{2}}+\frac{{z}^{3}}{2}\left(\frac{Q\beta }{{k}_{1}}-\frac{{Q}^{2}d}{2{k}_{1}^{2}}\right)+\frac{{z}^{2}}{2}\left({\beta }^{2}+\frac{{Q}^{2}{d}^{2}}{4{k}_{1}^{2}}-\frac{Q\beta d}{{k}_{1}}
The pressure distribution is of no consequence here as we are eliminating the same. It may be noted that ${T}_{b}\left(z\right)$ , ${H}_{b}\left(z\right)$ and ${M}_{b}\left(z\right)$ are distributed
parabolically with the porous layer height due to the presence of internal heat generation. However, when $Q=0$ (i.e., in the absence of internal heat generation), the basic state temperature
distribution is linear in z. Thus the presence of internal heat generation plays a significant role on the stability of the system.
To study the stability of the system, we perturb all the variables in the form
$\begin{array}{l}\left[q,\omega ,\rho ,p,T,H,M\right]\\ =\left[{q}^{\prime },{\omega }^{\prime },{\rho }_{b}\left(z\right)+{\rho }^{\prime },{p}_{b}\left(z\right)+{p}^{\prime },{T}_{b}\left(z\right)+
{T}^{\prime },{H}_{b}\left(z\right)+{H}^{\prime },{M}_{b}+{M}^{\prime }\right]\end{array}$(17)
where ${q}^{\prime },{\omega }^{\prime },{\rho }^{\prime },{p}^{\prime },{T}^{\prime },{H}^{\prime }$ and ${M}^{\prime }$ are the perturbed quantities and are assumed to be very small. Substituting
Equation (17) into Equation (6) and using Equations (8) and (9) and assuming $K\beta d\approx \left(1+\chi \right){H}_{0}$ and $KQ{d}^{2}\approx 2\kappa \left(1+\chi \right){H}_{0}$ as propounded by
Finlayson [4] , we obtain (after dropping primes)
$\begin{array}{l}{H}_{x}+{M}_{x}=\left(1+{M}_{0}/{H}_{0}\right){H}_{x},\\ {H}_{y}+{M}_{y}=\left(1+{M}_{0}/{H}_{0}\right){H}_{y},\\ {H}_{z}+{M}_{z}=\left(1+\chi \right){H}_{z}-KT\end{array}$(18)
where, $\left({H}_{x},{H}_{y},{H}_{z}\right)$ and $\left({M}_{x},{M}_{y},{M}_{z}\right)$ are the $\left(x,y,z\right)$ components of the magnetic field and magnetization respectively. Thus the
analysis is restricted to physical situation in which the magnetization induced by the variations in temperature gradient and internal heating is small compared that induced by external magnetic
Substituting Equation (17) into Equation (2), linearizing, eliminating the pressure term by operating curl twice and using Equations (18) the z-component of the resulting equation can be obtained as
(after dropping the primes)
$\begin{array}{l}\left[{\rho }_{0}\frac{\partial }{\partial t}-\left(\eta +{\xi }_{r}\right){abla }^{2}\right]{abla }^{2}w\\ =2{\xi }_{r}{abla }^{2}{\Omega }_{3}{\rho }_{0}\alpha g{abla }_{1}^{2}T\
left[{\mu }_{0}K{abla }_{1}^{2}\left(\frac{\partial \varphi }{\partial z}\right)-\frac{{\mu }_{0}{K}^{2}}{1+\chi }{abla }_{1}^{2}T\right]\left[\frac{Qz}{{k}_{1}}-\frac{Qd}{2{k}_{1}}+\beta \right]\end
Substituting Equation (17) into Equation (3) we obtain (after dropping primes)
${\rho }_{0}I\left(\frac{\partial {\Omega }_{3}}{\partial t}\right)=-2{\xi }_{r}\left[{abla }^{2}w+2{\Omega }_{3}\right]+{\eta }^{\prime }{abla }^{2}{\Omega }_{3}$(20)
As before, substituting Equation (17) into Equation (4) and linearizing, we obtain (after dropping primes)
$\begin{array}{c}\left[{\rho }_{0}{C}_{0}\frac{\partial }{\partial t}-{k}_{1}{abla }^{2}\right]T=\left[{\rho }_{0}{C}_{0}-\frac{{\mu }_{0}{T}_{0}{K}^{2}}{1+\chi }\right]\left[\frac{Qz}{{k}_{1}}-\frac
{Qd}{2{k}_{1}}+\beta \right]w\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\mu }_{0}{T}_{0}K\frac{\partial }{\partial t}\left(\frac{\partial \varphi }{\partial z}\right)-\left[\frac{Qz}{{k}_{1}}-\
frac{Qd}{2{k}_{1}}+\beta \right]\delta {\Omega }_{3}\end{array}$(21)
where ${\rho }_{0}{C}_{0}={\rho }_{0}{C}_{V,H}+{\mu }_{0}{H}_{0}K$ .
Finally Equation (6), after using Equation (17) and (18), yield (after dropping primes)
$\left(1+\chi \right)\frac{{\partial }^{2}\varphi }{\partial {z}^{2}}+\left(1+\frac{{M}_{0}}{{H}_{0}}\right){abla }_{h}^{2}\varphi -K\frac{\partial T}{\partial z}=0$(22)
Since the principle of exchange of stability is valid, the normal mode expansion of the dependent variables takes the form
$\left\{w,T,\varphi ,{\Omega }_{3}\right\}=\left\{W\left(z\right),\Theta \left(z\right),\Phi \left(z\right),{\Omega }_{3}\left(z\right)\right\}\mathrm{exp}\left[i\left(lx+my\right)\right]$(23)
On non-dimensionalizing the variables by setting
\frac{d}{u }W,\\ {\Theta }^{*}=\frac{\kappa }{\beta u d}\Theta ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Phi }^{*}=\frac{\left(1+\chi \right)\kappa }{K\beta u {d}^{2}}\Phi ,\\ {\Omega }_{3}^{\
ast }=\frac{{d}^{3}}{u }{\Omega }_{3},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{I}^{*}=\frac{1}{{d}^{2}}I.\end{array}\right\}$(24)
Equation (23) is substituted into Equations (19)-(22) and then Equation (24) is used to obtain the stability equations in the following form
$\begin{array}{l}\left(1+{N}_{1}\right){\left({D}^{2}-{a}^{2}\right)}^{2}W\\ ={a}^{2}{R}_{t}\Theta -2{N}_{1}\left({D}^{2}-{a}^{2}\right){\Omega }_{3}-{a}^{2}{R}_{m}\left[1+{N}_{s}\left(2z-1\right)\
right]\left(D\Phi -\Theta \right)\end{array}$(25)
$2{N}_{1}\left[\left({D}^{2}-{a}^{2}\right)W+2{\Omega }_{3}\right]-{N}_{3}\left({D}^{2}-{a}^{2}\right){\Omega }_{3}=0$(26)
$\left({D}^{2}-{a}^{2}\right)\Theta +\left[{N}_{s}\left(2z-1\right)+1\right]\left[\left(1-{M}_{2}\right)W-{N}_{5}{\Omega }_{3}\right]=0$(27)
${D}^{2}\Phi -{a}^{2}{M}_{3}\Phi -D\Theta =0$(28)
The typical value of M[2] for magnetic fluids with different carrier liquids turns out to be of the order of 10^−^6 and hence its effect is neglected when compared to unity.
The above equations are to be solved subject to the rigid-paramagnetic boundary conditions:
$\begin{array}{l}W=DW={\Omega }_{3}=\Theta =\Phi =0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace
{\hspace{0.17em}}z=0\\ W={D}^{2}W+{a}^{2}Ma\text{\hspace{0.17em}}\Theta =D{\Omega }_{3}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text
{at}\text{\hspace{0.17em}}\text{\hspace{0.17em}}z=1\\ D\Theta +Bi\Theta =D\Phi =0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\
3. Numerical Solution
Equations (25)-(28) together with boundary conditions (29) constitute an eigenvalue problem with thermal Rayleigh number R[t] being an eigenvalue. Accordingly, $W,\Theta ,\Phi$ and ${\Omega }_{3}$
are written as
$\begin{array}{l}W\left(z\right)=\underset{i=1}{\overset{N}{\sum }}{A}_{i}{W}_{i}\left(z\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Omega }_{3}=\underset{i=1}{\overset
{N}{\sum }}{B}_{i}{\Omega }_{3}{}_{i}\left(z\right)\\ \Theta \left(z\right)=\underset{i=1}{\overset{N}{\sum }}{C}_{i}{\Theta }_{i}\left(z\right)\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\
hspace{0.17em}}\Phi \left(z\right)=\underset{i=1}{\overset{N}{\sum }}{D}_{i}{\Phi }_{i}\left(z\right)\end{array}\right\}$(30)
where ${A}_{i},{B}_{i},{C}_{i}$ and ${D}_{i}$ are the unknown constants to be determined. The basis functions ${W}_{i}\left(z\right)$ , ${\Theta }_{i}\left(z\right)$ , ${\Phi }_{i}\left(z\right)$ and
${\Omega }_{3}{}_{i}\left(z\right)$ are generally chosen such that they satisfy the corresponding boundary conditions but not the differential equations. Substituting Equation (30) into Equations
(25)-(28) and multiplying the resulting momentum Equation (25) by ${W}_{j}\left(z\right)$ , angular momentum Equation (26) by ${\Omega }_{3}{}_{j}\left(z\right)$ , energy Equation (27) by ${\Theta }_
{j}\left(z\right)$ and magnetic potential Equation (28) by ${\Phi }_{j}\left(z\right)$ , performing integration by parts with respect to z between $z=0$ and $z=1$ and using the boundary conditions
(29) we obtain the following system of linear homogeneous algebraic equations:
where the co-efficient ${C}_{ji}$ - ${L}_{ji}$ involve the inner product of the basis functions and are given by
${D}_{ji}=-{a}^{2}{R}_{t}{M}_{1}〈\left[{N}_{s}\left(2z-1\right)+1\right]{W}_{j}{\Theta }_{i}〉-{a}^{2}{R}_{t}〈{W}_{j}{\Theta }_{i}〉+{a}^{2}MaD{W}_{j}\left(1\right){\Theta }_{i}\left(1\right)$
${E}_{ji}={a}^{2}{R}_{t}{M}_{1}〈\left[{N}_{s}\left(2z-1\right)+1\right]{W}_{j}D{\Phi }_{i}〉$
${F}_{ji}=-2{N}_{1}\left[〈D{W}_{j}D{\Omega }_{3i}〉+{a}^{2}〈{W}_{j}{\Omega }_{3i}〉\right]$
${G}_{ji}=2{N}_{1}\left[〈D{\Omega }_{3j}D{W}_{i}〉+{a}^{2}〈{\Omega }_{3j}{W}_{i}〉\right]$
${H}_{ji}=-\left[4{N}_{1}〈{\Omega }_{3j}{\Omega }_{3i}〉+{N}_{3}〈D{\Omega }_{3j}D{\Omega }_{3i}〉+{N}_{3}{a}^{2}〈{\Omega }_{3j}{\Omega }_{3i}〉\right]$
${I}_{ji}=\left(1-{M}_{2}\right)〈\left[{N}_{s}\left(2z-1\right)+1\right]{\Theta }_{j}{W}_{i}〉$
${J}_{ji}=-\left[〈D{\Theta }_{j}D{\Theta }_{i}〉+{a}^{2}〈{\Theta }_{j}{\Theta }_{i}〉+Bi/4\right]$
${T}_{ji}=-{N}_{5}〈\left[{N}_{s}\left(2z-1\right)+1\right]{\Theta }_{j}{\Omega }_{3i}〉$
${K}_{ji}=〈{\Phi }_{j}D{\Theta }_{i}〉$
$\begin{array}{c}{L}_{ji}=a\left[{\Phi }_{j}\left(1\right){\Phi }_{i}\left(1\right)+{\Phi }_{j}\left(0\right){\Phi }_{i}\left(0\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+〈D{\Phi }
_{j}D{\Phi }_{i}〉+{a}^{2}{M}_{3}〈{\Phi }_{j}{\Phi }_{i}〉\end{array}$
where the inner product is defined as $<....>$$〈...〉=\underset{0}{\overset{1}{\int }}\left(...\right)\text{d}z$ . The set of
homogeneous algebraic equations can have non-trivial solutions if and only if
$|\begin{array}{cccc}{C}_{ji}& {D}_{ji}& {E}_{ji}& {F}_{ji}\\ {G}_{ji}& 0& 0& {H}_{ji}\\ {I}_{ji}& {J}_{ji}& 0& {T}_{ji}\\ 0& {K}_{ji}& {L}_{ji}& 0\end{array}|=0$(35)
The eigenvalue has to be extracted from the above characteristic equation. In Galerkin method, we choose the weighting function as the trial functions, thus:
$\begin{array}{l}{W}_{i}={z}^{2}{\left(z-1\right)}^{2}{z}^{i-1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Theta }_{i}=z\left(1-z/2\right){z}^{i-1},\\ {\Omega }_{3i}=z\left(1-z/2\right){z}^{i-1},\
text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Phi }_{i}=z\left(1-z/2\right){z}^{i-1}\end{array}\right\}$(36)
The velocity ( ${W}_{i}$ ), temperature ( ${\Theta }_{i}$ ), vorticity ( ${\Omega }_{3i}$ ) and magnetic potential ( ${\Phi }_{i}$ ) trail functions satisfy all the boundary condition while the
temperature ( ${\Theta }_{i}$ ) does not satisfy the boundary condition $D\Theta +Bi\Theta =0$ at $z=1$ . Therefore, following, the boundary residual technique is used for these functions. The
velocity, vorticity and the magnetic equations are made orthogonal to each of the corresponding trail functions. For the temperature trial the boundary residuals are added and their combined inner
product is set to zero to obtain
$〈D{\Theta }_{j}D{\Theta }_{i}〉+{a}^{2}〈{\Theta }_{j}{\Theta }_{i}〉+Bi{\Theta }_{j}\left(1\right){\Theta }_{i}\left(1\right)$ . Besides, the residual from this condition is included as residual
from the differential Equation (36) leads to a relation involving in the form
$f\left({R}_{t},{R}_{m},Ma,{N}_{s},{M}_{1},{M}_{3},{N}_{1},{N}_{3},{N}_{5},a\right)=0$ .
The critical values of R[t] (i.e., R[c]) or R[m] (i.e., R[mc]) or Ma (i.e., Ma[c]) is determined numerically with respect to a for different values of N[s], M[1], M[3], N[1], N[3] and N[5].
4. Result and Discussion
The classical linear stability analysis has been carried out to investigate the effect of internal heat source strength on the onset of Bénard-Marangoni ferroconvection in a horizontal micropolar
ferrofluid layer heated from below in the presence of a transverse uniform vertical magnetic field. The both the boundaries is considered to be rigid-ferromagnetic. The critical thermal Rayleigh
number (R[tc]), critical magnetic Rayleigh number (R[mc]) and critical Marangoni number (Ma[c]) and the corresponding critical wave number (a[c]) are used to characterize the stability of the system.
The critical stability parameters computed numerically by Galerkin technique as explained above, are found to converge by considering nine terms in the Galerkin expansion.
To validate the solution computed numerically for various values of R[t] and Bi in the absence of micropolar effects and internal heat source strength (i.e. ${N}_{1}={N}_{3}={N}_{5}=Ns=0$ ) are
compared in Table 1 with the previously published results of Davis [27] . In addition, the present method are compared with the previously published results of Char and Chiang [28] when ${N}_{1}={N}_
{3}={N}_{5}=0$ and ${R}_{m}={R}_{t}{M}_{1}=0$ (classical Rayleigh-Bénard problem) for various values of Ns (see Table 2). From the Tables, it is observed that our results are identical
Table 1. Comparison of Ma[c] for diff values of R[t] and Bi in the absence of micropolar ferrofluid.
Table 2. Comparison of R[tc] for diff values of Ns and Bi in the absence of micropolar ferrofluid.
with those obtained by Davis [27] as well as Char and Chiang [28] using different approaches.
The presence of internal heating makes the basic temperature, magnetic field and magnetization distributions to deviate from linear to parabolic with respect to micropolar ferrofluid layer height
which in turn have significant influence on the stability of the system. To assess the impact of internal heat source strength Ns on the criterion for the onset of ferroconvection, the distributions
of dimensionless basic temperature, ${T}_{b}\left(z\right)$ , magnetic field intensity, ${H}_{b}\left(z\right)$ and magnetization, ${M}_{b}\left(z\right)$ are exhibited graphically in Figure 1 for
various values of Ns. From the figure it is observed that increase in Ns amounts to large deviations in these distributions which in turn enhance the disturbances in the horizontal
Figure 1. Basic state temperature, magnetic intensity and magnetization distributions for different Ns.
porous layer and thus reinforce instability on the system.
Figures 2-4 depict the critical Ma[c] at the onset of ferroconvection as the function of “a”. It is noted that, as “a” decreases the Marangoni number decreases, attains a minimum at some critical
wave number, and increases again. The curves reported in figures have the shape is upward concave to that of Bénard-Marangoni-ferroconvection. For increasing R[m], N[s], N[3], R[t], and decreasing N
[1] is shifted to the neutral curves are slanted towards the higher wave number region.
Figure 5 represents the variation of critical Marangoni number Ma[c] as a function of N[1] for different values of R[m] and N[5] for ${N}_{3}=2$ , ${M}_{3}=5$ and $Ns=2$ . It is seen that Ma[c]
decreases with an increase in R[m] and hence its effect is to hasten the onset of ferroconvection due to an increase in the destabilizing magnetic force and the curve for ${R}_{m}=0$ corresponds to
non-magnetic micropolar fluid case. In other words, heat is transported more efficiently in magnetic fluids as compared to ordinary micropolar fluids. Also observed that Ma[c] increases with
increasing N[1]. This is because, as N[1] increases the concentration of microelements also increases and as a result a greater part of the energy of the system is consumed by these elements in
developing gravitational velocities in the fluid which ultimately leads to delay in the onset of ferromagnetic convection. Moreover, the system is found to be more stable if the micropolar heat
conduction of the parameter with ${N}_{5}=0.5$ as compared to the case of ${N}_{5}=0$ .
In Figure 6 Ma[c] is plotted as a function of N[1] for different values of spin diffusion (couple stress) parameter N[3] and R[m] when ${M}_{3}=5$ , ${N}_{5}=0.5$ and $Ns=2$ . Here, it is observed
that Ma[c] curves for different N[3] coalesce when ${N}_{1}=0$ . The impact of N[3] on the stability characteristics of the system is noticeable clearly with increasing N[1] and then it is seen that
the critical Marangoni
Figure 2. Neutral curves for different values of ${R}_{m}$ and ${N}_{5}$ with ${N}_{1}=0.5,{R}_{t}=50,{M}_{3}=5,\text{\hspace{0.17em}}Ns=2$ .
Figure 3. Neutral curves for different values of ${R}_{t}$ and ${N}_{1}$ for ${N}_{3}=2,\text{\hspace{0.17em}}{N}_{5}=0.5,{R}_{m}=50,\text{\hspace{0.17em}}{M}_{3}=5$ and $Ns=2$ .
number decreases with increasing N[3] indicating the spin diffusion (couple stress) parameter N[3] has a destabilizing effect on the system. This may be attributed to the fact that as N[3] increases,
the couple stress of the fluid increases, which leads to a decrease in micro-rotation and hence the system becomes more unstable.
Figure 7 shows the variation of critical Marangoni number Ma[c] and as a function of N[1] for various values of dimensionless internal heat source strength Ns when ${M}_{3}=5$ , ${N}_{3}=2$ and ${N}_
{5}=0.5$ . Figure 7 clearly indicates that Ma[c] decreases monotonically with Ns indicating the influence of increasing internal heating is to decrease the value of Ma[c] and thus destabilize the
system. This is because increasing Ns amounts to increase in energy supply to the system.
Figure 4. Neutral curves for different values of ${N}_{3}$ and $Ns$ with ${N}_{1}=0.2,\text{\hspace{0.17em}}{N}_{5}=0.5,{R}_{m}=50,\text{\hspace{0.17em}}{R}_{t}=50$ and ${M}_{3}=5$ .
Figure 5. Variation of $M{a}_{c}$ verses ${N}_{1}$ for different ${R}_{m}$ for $Ns=2,\text{\hspace{0.17em}}{M}_{3}=5,\text{\hspace{0.17em}}{N}_{3}=2$ .
The complementary effects of both buoyancy and magnetic forces are made clear in Figure 8 by displaying the locus of Ma[c] and magnetic Rayleigh number R[mc] for various values of Bi and N[5] when $
{N}_{1}=0.2$ , [c] is inversely proportional to R[mc] due to the destabilizing magnetic force. From the figure it is evident that, increasing in Bi is to increase Ma[c] and R[mc] and thus its effect
is to delay the onset of magnetic Bénard-Marangoni ferroconvection. This may be attributes to fact that with increasing Bi, the thermal disturbances can be easily dissipate in to the ambient
surrounding due to a better convective heat transfer co-efficient at the top surface and hence higher
heating is required at make the system unstable. It is also evident that micropolar ferrofluid saturated porous layer in the presence of vertical magnetic field becomes more stable with increasing in
The measure of non-linearity of fluid magnetization M[3], on the onset of ferroconvection is depicted in Figure 9. The curves of Ma[c] versus R[mc] shown in Figure 9 for various values of M[3] when
[3] has a destabilizing effect on the system. Nevertheless, the destabilization due to increase in M[3] is only marginal. This may be
attributed to the fact that the application of magnetic field makes the ferrofluid to acquire larger magnetization which in turn interacts with the imposed magnetic field and releases more energy to
drive the flow faster. Hence, the system becomes unstable with a smaller temperature gradient as the value of M[3] increases. Alternatively, a higher value of M[3] would arise either due to a larger
pyromagnetic coefficient or larger temperature gradient. Both these factors are conducive for generating a larger gradient in the Kelvin body force field, possibly promoting the instability.
5. Conclusions
The effect of internal heating and heat transfer coefficient on the onset of Bénard-Maranagoni-convection in a micropolar ferrofluid layer has been made theoretically. The solution of this problem is
obtained numerically using Galerkin-type of weighted residual technique by developing computer codes for MATTHEMAICA-11 software. Tabular and graphical method of appearance of the computed results
illustrates the details in this paper and their dependence on the physical parameters involved in the problem. The significant findings of this analysis are:
1) The system becomes more unstable with an increase in magnetic Rayleigh number R[m], nonlinearity of fluid magnetization parameter M[3], internal heat source strength Ns and spin diffusion (couple
stress) parameter N[3].
2) The effect of increasing the value of coupling parameter N[1], micropolar heat conduction parameter N[5], Biot number Bi and is to delay the onset of ferromagnetic convection.
3) The effect of increasing R[m] and Ns as well as decrease in N[1], M[3], N[3] and N[5] is to increase the critical wave number a[c] and hence there is to reduce the convection cells.
4) The magnetic and buoyancy forces are complementary with each other and the system is more stabilizing when the magnetic forces alone are present.
The authors gratefully acknowledged the financial support received in the form of a “Research Fund for Talented Teacher” scheme from Vision Group of Science & Technology, Government of Karnataka,
Bengaluru (No. KSTEPS/ VGST/06/2015-16).
Greek Symbols
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Installing NumPy, SciPy, matplotlib, and IPython
This article written by Ivan Idris, author of the book, Python Data Analysis, will guide you to install NumPy, SciPy, matplotlib, and IPython.
We can find a mind map describing software that can be used for data analysis at https://www.xmind.net/m/WvfC/. Obviously, we can't install all of this software in this article. We will install NumPy
, SciPy, matplotlib, and IPython on different operating systems.
[box type="info" align="" class="" width=""]Packt has the following books that are focused on NumPy:
• NumPy Beginner's Guide Second Edition, Ivan Idris
• NumPy Cookbook, Ivan Idris
• Learning NumPy Array, Ivan Idris
SciPy is a scientific Python library, which supplements and slightly overlaps NumPy. NumPy and SciPy, historically shared their codebase but were later separated.
matplotlib is a plotting library based on NumPy.
IPython provides an architecture for interactive computing. The most notable part of this project is the IPython shell.
Software used
The software used in this article is based on Python, so it is required to have Python installed. On some operating systems, Python is already installed. You, however, need to check whether the
Python version is compatible with the software version you want to install. There are many implementations of Python, including commercial implementations and distributions.
[box type="note" align="" class="" width=""]You can download Python from https://www.python.org/download/. On this website, we can find installers for Windows and Mac OS X, as well as source archives
for Linux, Unix, and Mac OS X.[/box]
The software we will install has binary installers for Windows, various Linux distributions, and Mac OS X. There are also source distributions, if you prefer that. You need to have Python 2.4.x or
above installed on your system. Python 2.7.x is currently the best Python version to have because most Scientific Python libraries support it. Python 2.7 will be supported and maintained until 2020.
After that, we will have to switch to Python 3.
Installing software and setup on Windows
Installing on Windows is, fortunately, a straightforward task that we will cover in detail. You only need to download an installer, and a wizard will guide you through the installation steps. We will
give steps to install NumPy here. The steps to install the other libraries are similar. The actions we will take are as follows:
1. Download installers for Windows from the SourceForge website (refer to the following table). The latest release versions may change, so just choose the one that fits your setup best.
│ │ │Latest Version│
│Library │URL │ │
│ │ │ │
│ │ │1.8.1 │
│NumPy │http://sourceforge.net/projects/numpy/files/ │ │
│ │ │ │
│ │ │0.14.0 │
│SciPy │http://sourceforge.net/projects/scipy/files/ │ │
│ │ │ │
│ │ │1.3.1 │
│matplotlib│http://sourceforge.net/projects/matplotlib/files/ │ │
│ │ │ │
│ │ │2.0.0 │
│IPython │http://archive.ipython.org/release/ │ │
│ │ │ │
│ │ │ │
2. Choose the appropriate version. In this example, we chose numpy-1.8.1-win32-superpack-python2.7.exe.
3. Open the EXE installer by double-clicking on it.
4. Now, we can see a description of NumPy and its features. Click on the Next button.If you have Python installed, it should automatically be detected. If it is not detected, maybe your path
settings are wrong.
5. Click on the Next button if Python is found; otherwise, click on the Cancel button and install Python (NumPy cannot be installed without Python). Click on the Next button. This is the point of no
return. Well, kind of, but it is best to make sure that you are installing to the proper directory and so on and so forth. Now the real installation starts. This may take a while.
[box type="note" align="" class="" width=""]The situation around installers is rapidly evolving. Other alternatives exist in various stage of maturity (see https://www.scipy.org/install.html). It
might be necessary to put the msvcp71.dll file in your C:Windowssystem32 directory. You can get it from http://www.dll-files.com/dllindex/dll-files.shtml?msvcp71.[/box]
Installing software and setup on Linux
Installing the recommended software on Linux depends on the distribution you have. We will discuss how you would install NumPy from the command line, although, you could probably use graphical
installers; it depends on your distribution (distro). The commands to install matplotlib, SciPy, and IPython are the same – only the package names are different. Installing matplotlib, SciPy, and
IPython is recommended, but optional.
Most Linux distributions have NumPy packages. We will go through the necessary steps for some of the popular Linux distros:
• Run the following instructions from the command line for installing NumPy on Red Hat:
$ yum install python-numpy
• To install NumPy on Mandriva, run the following command-line instruction:
$ urpmi python-numpy
• To install NumPy on Gentoo run the following command-line instruction:
$ sudo emerge numpy
• To install NumPy on Debian or Ubuntu, we need to type the following:
$ sudo apt-get install python-numpy
The following table gives an overview of the Linux distributions and corresponding package names for NumPy, SciPy, matplotlib, and IPython.
│ │ │ │ │IPython│
│Linux distribution│NumPy │SciPy │matplotlib │ │
│ │ │ │ │ │
│ │ │ │ │Ipython│
│Arch Linux │python-numpy │python-scipy│python-matplotlib│ │
│ │ │ │ │ │
│ │ │ │ │Ipython│
│Debian │python-numpy │python-scipy│python-matplotlib│ │
│ │ │ │ │ │
│ │ │ │ │Ipython│
│Fedora │numpy │python-scipy│python-matplotlib│ │
│ │ │ │ │ │
│ │ │ │ │ipython│
│Gentoo │dev-python/numpy │scipy │matplotlib │ │
│ │ │ │ │ │
│ │ │ │ │ipython│
│OpenSUSE │python-numpy, python-numpy-devel │python-scipy│python-matplotlib│ │
│ │ │ │ │ │
│ │ │ │ │ipython│
│Slackware │numpy │scipy │matplotlib │ │
│ │ │ │ │ │
│ │ │ │ │ │
Installing software and setup on Mac OS X
You can install NumPy, matplotlib, and SciPy on the Mac with a graphical installer or from the command line with a port manager such as MacPorts, depending on your preference. Prerequisite is to
install XCode as it is not part of OS X releases. We will install NumPy with a GUI installer using the following steps:
1. We can get a NumPy installer from the SourceForge website http://sourceforge.net/projects/numpy/files/. Similar files exist for matplotlib and SciPy.
2. Just change numpy in the previous URL to scipy or matplotlib. IPython didn't have a GUI installer at the time of writing.
3. Download the appropriate DMG file usually the latest one is the best.Another alternative is the SciPy Superpack (https://github.com/fonnesbeck/ScipySuperpack).
Whichever option you choose it is important to make sure that updates which impact the system Python library don't negatively influence already installed software by not building against the Python
library provided by Apple.
1. Open the DMG file (in this example, numpy-1.8.1-py2.7-python.org-macosx10.6.dmg).
2. Double-click on the icon of the opened box, the one having a subscript that ends with .mpkg. We will be presented with the welcome screen of the installer.
3. Click on the Continue button to go to the Read Me screen, where we will be presented with a short description of NumPy.
4. Click on the Continue button to the License the screen.
5. Read the license, click on the Continue button and then on the Accept button, when prompted to accept the license. Continue through the next screens and click on the Finish button at the end.
Alternatively, we can install NumPy, SciPy, matplotlib, and IPython through the MacPorts route, with Fink or Homebrew. The following installation steps shown, installs all these packages.
[box type="info" align="" class="" width=""]For installing with MacPorts, type the following command:
sudo port install py-numpy py-scipy py-matplotlib py-
Installing with setuptools
If you have pip you can install NumPy, SciPy, matplotlib and IPython with the following commands.
pip install numpy
pip install scipy
pip install matplotlib
pip install ipython
It may be necessary to prepend sudo to these commands, if your current user doesn't have sufficient rights on your system.
In this article, we installed NumPy, SciPy, matplotlib and IPython on Windows, Mac OS X and Linux.
Resources for Article:
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What is the smallest value of angular displacement of the raft?
• Thread starter Lotto
• Start date
In summary, the smallest angular displacement of the raft from its original position during one cycle is the angle the raft makes with the vertical.
Homework Statement
There is a raft in the middle of the river. The mass of the raft is negligible, and it carries a crane on board. The crane moves boxes of building material of mass m from one river bank to
another. In one cycle, the crane loads material at one side of the river, rotates to the other river bank, unloads the material there, and rotates back. Calculate the smallest value of angular
displacement of the raft from its original position during one cycle. Approximate the crane by a homogenous cylinder of mass Mc and radius r, and a rotating jib in the shape of a slim rod of
length kr. Assume that the velocity of the river and the „friction“ between the raft and the water are negligible.
Relevant Equations
What is meant by "the smallest value of angular displacement of the raft from its original position during one cycle"? I understand that I am supposed to solve this problem using torques of the crane
and and of the boxes, but I am totally confused by that "smallest angular displacement". If it was the biggest angular displacement, I would understand it, but this is just strange. What does it
I suppose I should use an equation that torque1+torque2=0, then the crane is stable. Torque1= Mga (a = distance of the gravity force Mg from the rotation axis ). Torque2=-mgb (b= distance of the
gravity force mg from the rotation axis). But what am I to calculate anyway? I only need to know what that weird angle is.
Is there any diagram that is shown with this problem?
I can’t imagine what is holding the crane, so it can rotate 180°.
The massless raft could not do it, I believe.
Lnewqban said:
Is there any diagram that is shown with this problem?
I can’t imagine what is holding the crane, so it can rotate 180°.
The massless raft could not do it, I believe.
I am sorry but this is the whole problem, I do not have any other information or pictures. I am disappointed as well.
Lotto said:
I am sorry but this is the whole problem, I do not have any other information or pictures.
What is the source of the problem? Is it from a university class that you are taking (if so, ask the professor or TA for clarification)? Or did you get it from some online problem set that you want
to do on your own?
Lotto said:
I suppose I should use an equation that torque1+torque2=0, then the crane is stable. Torque1= Mga (a = distance of the gravity force Mg from the rotation axis ). Torque2=-mgb (b= distance of the
gravity force mg from the rotation axis). But what am I to calculate anyway? I only need to know what that weird angle is.
Make some assumption as to what is actually moving and in what direction as a starting point. Draw a picture to get your bearings.
bob012345 said:
as to what is actually moving
Yeah, I think that's got to be the key. It sounds like the crane "cylinder" is meant not to (counter-)rotate much, and only the thin "rod" is what rotates with the mass...
My interpretation is this...
I think ‘crane’ is a misnomer. It isn’t meant to do any vertical lifting or lowering. It can be thought of as a simple horizontal rod: one end is attached to the centre of the top of a cylinder (axis
of cylinder vertical); the rod is only able to rotate in a horizontal plane.
A mass is attached to the free end of the rod.
The rod is rotated 180º (in a horizontal plane) carrying the mass to the other bank.
The mass is then detached and the rod is rotated -180º (in a horizontal plane) to return it to its starting position.
The cylinder is fixed to the raft and they rotate together. As the rod rotates clockwise the cylinder must rotate anticlockwise so that the system's angular momentum remains zero.
Here's an interpretation that makes for a nice problem.
Assume that the length of the jib is variable via the parameter##k##. If the jib is longer than half the width of the river, then the jib will not need to rotate through 180 degrees to move the load
from one side of the river to the other side. I believe you can work out the amount of counter-rotation of the cylinder as a function of ##k## and the ratios ##m/M_c## and ##r/W##, where ##W## is the
width of the river. There will be a value of ##k## that minimizes the counter-rotation of the cylinder.
Of course, this interpretation might not be what was intended.
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Steve4Physics said:
The cylinder is fixed to the raft and they rotate together. As the rod rotates clockwise the cylinder must rotate anticlockwise so that the system's angular momentum remains zero.
Yes. When the mass is deposited on the opposite shore and the rod rotates to pick up the next mass, the raft does not return to its original orientation. Because the cylinder is fixed on the raft,
the two rotate together during the "go" trip but not during the return trip because there is no load on the massless rod. I think the problem is asking for the incremental angle by which the raft +
cylinder assembly rotates at the end of each "go" trip.
kuruman said:
the raft does not return to its original orientation
True, but if that becomes an issue it could counter that by occasionally going the long way around.
Unfortunately, neither will it return to its original location, and going the other way doesn't help. It will eventually reach the 'source' bank.
Anyway, I support
's reading in post #8, and suggest the easiest approach (ignoring the cross-river creep) is to find the equation relating the two angles (sweep of arm relative to river, ##\beta##, angular
displacement of raft, ##\alpha##) and set ##\frac{d\alpha}{d\beta}=0##.
I get ##\beta\sin(\beta)=1-\cos(\beta)##.
I think I understand the task now. When the jib rotates, the radius of a circle it circumscribes is ##kr## and the angel it circumscribes is ##\alpha_1=\pi ##. So the angular momentum ##L_1## of the
box dependent on time is ## L_1=\frac {m\alpha_1 k^2 r^2}{t} ##. The moment of inertia of the crane is ##\frac 12 Mr^2##, so its angular momentum is ##L_2=\frac{Mr^2\alpha_2}{2t}## (##\alpha_2## is
the angel we want to calculate). Because of the conservation of angular momentum, ##L_1=L_2##, so ##\alpha_2=\frac{2m\pi k^2}{M}##. This is the smallest angel because the angular acceleration is
zero. Is this solution right?
Lotto said:
I think I understand the task now. When the jib rotates, the radius of a circle it circumscribes is ##kr## and the angel it circumscribes is ##\alpha_1=\pi ##. So the angular momentum ##L_1## of
the box dependent on time is ## L_1=\frac {m\alpha_1 k^2 r^2}{t} ##. The moment of inertia of the crane is ##\frac 12 Mr^2##, so its angular momentum is ##L_2=\frac{Mr^2\alpha_2}{2t}## (##\
alpha_2## is the angel we want to calculate). Because of the conservation of angular momentum, ##L_1=L_2##, so ##\alpha_2=\frac{2m\pi k^2}{M}##. This is the smallest angel because the angular
acceleration is zero. Is this solution right?
Then, the relevant equations shown in the OP do not apply, but those for angular momentum.
, should not be a simultaneous translation of the raft as well?
Lnewqban said:
Then, the relevant equations shown in the OP do not apply, but those for angular momentum.
, should not be a simultaneous translation of the raft as well?
Yes, this problem seems not to be about torques, that I thought for the first time. Is my solution right?
Lotto said:
When the jib rotates, the radius of a circle it circumscribes is ##kr## and the angel it circumscribes is ##\alpha_1=\pi ##.
No, it would only have rotate that far relative to the line of the river if the jib only just reaches each bank. (Of course, it will have to rotate more relative to the raft since the raft is
rotating the other way.)
The point of the question is that increasing k reduces ##\alpha_1 ##, but it also increases ##\alpha_2 ## relative to ##\alpha_1 ##. So there can be an ideal ##\alpha_1 ## which minimises ##\alpha_2
Lotto said:
Yes, this problem seems not to be about torques, that I thought for the first time. Is my solution right?
It can be done using torques because the torque equations lead to conservation of angular momentum. But one could argue it is not about that either because time is irrelevant.
In linear motion, conservation of momentum leads to the fact that the mass centre ofban isolated system does not move. Something analogous applies here to angular displacement. The equation is
obtained by integrating the conservation of angular momentum equation wrt time.
Lnewqban said:
should not be a simultaneous translation of the raft as well?
Yes, as I noted in post #10, but I doubt the question is meant to be that hard and we don’t know the width of the river.
haruspex said:
(sweep of arm relative to river, ##\beta##, angular displacement of raft, ##\alpha##) and set ##\frac{d\alpha}{d\beta}=0##.
I get ##\beta\sin(\beta)=1-\cos(\beta)##.
I get the same condition, which can be written in other ways such as ##\beta = \tan (\frac{\beta}{2})##.
haruspex said:
Unfortunately, neither will it return to its original location, and going the other way doesn't help. It will eventually reach the 'source' bank.
Lnewqban said:
, should not be a simultaneous translation of the raft as well?
I completely overlooked this complication! I'm less certain than ever about the intended interpretation of the problem.
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Lnewqban said:
Then, the relevant equations shown in the OP do not apply, but those for angular momentum.
, should not be a simultaneous translation of the raft as well?
I have come to revise my thinking about this problem. When one talks about angular momentum, one must mention the reference point or axis about which this angular momentum is to be considered. Here,
I think the appropriate point is the CoM of the cylinder - load system. The two masses orbit their CoM. I don't think that there is spin angular momentum of the cylinder because I cannot see what
torque can change it. I am now considering that the angular displacement sought is that of the cylinder on the circle of its orbit around the CoM for a fixed length of the rod.
Lotto said:
Because of the conservation of angular momentum, ##L_1=L_2##
You mean ##L_1 = - L_2##. If (for example) the rod rotates clockwise then the cylinder must rotate anticlockwise. Or you could write ##|L_1|=|L_2|##.
Edit. Also, you may want to check the meanings of the two similar words 'angle' and 'angel'!
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kuruman said:
I have come to revise my thinking about this problem. When one talks about angular momentum, one must mention the reference point or axis about which this angular momentum is to be considered.
Here, I think the appropriate point is the CoM of the cylinder - load system. The two masses orbit their CoM. I don't think that there is spin angular momentum of the cylinder because I cannot
see what torque can change it. I am now considering that the angular displacement sought is that of the cylinder on the circle of its orbit around the CoM for a fixed length of the rod.
Interesting. For this interpretation, would it be necessary to state that the crane has the shape of a cylinder with a radius r?
Here's another option: assume that the raft is anchored in the middle of the river such that it can't translate through the water but it can freely rotate in the water.
Anyone's guess is at least as good as mine.
Now I understand that I want an ideal angle ##\alpha_1## so that the product of ##\alpha_1## and ##k^2## is as small as it can be. But how to calculate it? Is it possible to solve it without
calculus? Couldn't be the angle 90°?
kuruman said:
I don't think that there is spin angular momentum of the cylinder because I cannot see what torque can change it.
It's the rotation of the cylinder on its own axis that provides the torque forbtge rotation of the raft+boom+load system about its CoM. A motor on the raft will act to swing the boom around relative
to the raft. This will necessarily rotate the raft in the opposite direction relative to the banks. I think it is clear that this last is the angle to be minimised. In principle, the raft could
perform entire rotations for one sweep of the boom.
If the raft were mounted on an axle fixed to the bed of the river, that is all that would happen. As there is no such axle, yes, the raft+boom system will rotate about its CoM and this is what will
be exactly balanced by the raft's rotation on its own axis.
It's the rotation about the CoM which leads to the displacement of the raft. On the return, the CoM is the centre of the raft. Indeed, the raft does not move at all in that phase.
This is what led me to ignore the fact that the boom rotation is about the CoM, but I see now that there is a compromise. We can take that into account, but just overlook that it leaves the raft
displaced. I'll redo my analysis.
Edit: (I thought I added this yesterday but it has disappeared.)
Changing the pivot point to be the CoM produces the same result for ##\beta## as in posts #10 and #16. It does affect the answer for ##\alpha##; see post #33.
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Lotto said:
Now I understand that I want an ideal angle ##\alpha_1## so that the product of ##\alpha_1## and ##k^2## is as small as it can be. But how to calculate it? Is it possible to solve it without
calculus? Couldn't be the angle 90°?
To avoid subscripts I will call the angles α (the angle to be minimised) and β.
If the raft counter-rotates α on its own axis, what is the rotation angle β of the raft+boom+load system?
What width of river will be traversed by the load?
haruspex said:
It's the rotation of the cylinder on its own axis that provides the torque forbtge rotation of the raft+boom+load system about its CoM. A motor on the raft will act to swing the boom around
relative to the raft. This will necessarily rotate the raft in the opposite direction relative to the banks. I think it is clear that this last is the angle to be minimised. In principle, the
raft could perform entire rotations for one sweep of the boom.
If the raft were mounted on an axle fixed to the bed of the river, that is all that would happen. As there is no such axle, yes, the raft+boom system will rotate about its CoM and this is what
will be exactly balanced by the raft's rotation on its own axis.
It's the rotation about the CoM which leads to the displacement of the raft. On the return, the CoM is the centre of the raft. Indeed, the raft does not move at all in that phase.
This is what led me to ignore the fact that the boom rotation is about the CoM, but I see now that there is a compromise. We can take that into account, but just overlook that it leaves the raft
displaced. I'll redo my analysis.
Suppose the crane cylinder was just as wide as the river? In other words W=2r in the nice rendition by
in post #8?
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bob012345 said:
Suppose the crane cylinder was just as wide as the river? In other words W=2r in the nice rendition by
in post #8?
View attachment 316897
What are you suggesting? A way of preventing the linear displacement? But in that case, why not just anchor it to the bank so that it can't rotate either?
kuruman said:
I have come to revise my thinking about this problem. When one talks about angular momentum, one must mention the reference point or axis about which this angular momentum is to be considered.
Here, I think the appropriate point is the CoM of the cylinder - load system. The two masses orbit their CoM. I don't think that there is spin angular momentum of the cylinder because I cannot
see what torque can change it. I am now considering that the angular displacement sought is that of the cylinder on the circle of its orbit around the CoM for a fixed length of the rod.
There is also the problem of changing angular momentum with each load-unload cycle.
First loaded boom: Crane’s body-boom-load rotation happens about a common (loaded) COM; therefore, the crane’s body rotates mirroring the load, but describing a smaller circle.
First unloaded boom: Crane’s body-boom rotation happens about a new common (unloaded) COM’, this time located closer to the body, which describes a smaller circle in reverse.
Second loaded boom: As the second load is fixed respect to the ground, and next to where the previous load was located, but the crane’s body has relocated itself, the lenght of the boom must be
adjusted for the second cycle, thus modifying the angular momentum and subsequent circles.
Thank you,
Following this thread with interest and can’t stop myself sharing a few thoughts…
FWIW, I suspect that the question is simply a (very) poorly posed introductory-level problem with two key unstated assumptions:
1. Since the width of the river (or more correctly the distance from pick-up to drop-off points) is unspecified, we are meant to assume it is 2kR. The rod’s angles of rotation (relative to the
ground) can then be taken as 180º (transferring) and -180º (returning).
2. The linear displacement of the crane’s centre of mass during transfers can be ignored.
With these assumptions @Lotto’s answer in Post #11 would be the required one. (Though I fully appreciate the more rigorous analyses in the various posts.)
angle of rotation of the cylinder is achieved providing each mass (m) is released when the rod has completely stopped rotating – else the mass (m) will still have some angular momentum when released
– this would increase the necessary angle of rotation of the cylinder. That may answer the question originally asked by
about the use of the word 'minimum'.
Intuitively I don’t see how
's ##\beta\sin(\beta)=1-\cos(\beta)## can be correct because its solutions (##\beta = 0, 2.331rad, ...)## are discrete. Surely ##\beta## must be a (continuous) function of [some or all of] the masses
R and k? Or have I misunderstood something?
But, in the words of
, anyone's guess is at least as good as mine.
: If and when you get the official answer, I’m sure we would be interested to hear it. (And may the angles be with you!)
Edits: Minor changes.
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haruspex said:
What are you suggesting? A way of preventing the linear displacement? But in that case, why not just anchor it to the bank so that it can't rotate either?
Since the original wording asks to calculate the minimum
displacement I think it assumes no linear displacement. If we assume ##m<<M_c## we can ignore the CM translation and assume everything just rotates around the axis of the raft/crane. One way of
minimizing the angle of the jib is to shrink the river to ##2r## for a given k.
Lotto said:
Now I understand that I want an ideal angle ##\alpha_1## so that the product of ##\alpha_1## and ##k^2## is as small as it can be. But how to calculate it? Is it possible to solve it without
calculus? Couldn't be the angle 90°?
Does your course use Calculus? The ideal angle would not be 90° unfortunately. The question is whether the teacher/problem expects you to use ##m,k,M_c## as fixed constants or whether they (or at
least ##k##) can be optimized. If they are fixed constants and if only rotation is involved then I think you were on the right track earlier.
Steve4Physics said:
Following this thread with interest and can’t stop myself sharing a few thoughts…
FWIW, I suspect that the question is simply a (very) poorly posed introductory-level problem with two key unstated assumptions:
1. Since the width of the river (or more correctly the distance from pick-up to drop-off points) is unspecified, we are meant to assume it is 2kR. The rod’s angles of rotation (relative to the
ground) can then be taken as 180º (transferring) and -180º (returning).
2. The linear displacement of the crane’s centre of mass during transfers can be ignored.
With these assumptions @Lotto’s answer in Post #11 would be the required one. (Though I fully appreciate the more rigorous analyses in the various posts.)
Yes, I suspect this is the intended interpretation.
Steve4Physics said:
Since the width of the river (or more correctly the distance from pick-up to drop-off points) is unspecified, we are meant to assume it is 2kR.
I cannot agree.
We are asked to find the
angle, implying there is some variable which needs to be tuned to achieve that. Of the variables christened in post #1, only k seems like something the operator could tune.
It turns out that with this interpretation the width of the river, and r, M and m, are all irrelevant (Edit: for finding ##\beta## - thanks TSny),
[S:so the fact that we are not told the width of the river does not cast doubt on it:S]
. Moreover, it creates quite an interesting problem to solve.
Edit: Question is, did the author forget to specify a variable for the river width, or by mistake asked for the value of the angle minimised instead of the angle the boom sweeps for that minimum
(And, of course, if the width of the river is 2kr then the task becomes impossible. When the boom has swung 180° the raft will have shifted a bit the other way, leaving the load short of the target
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haruspex said:
We are asked to find the minimum angle, implying there is some variable which needs to be tuned to achieve that. Of the variables christened in post #1, only k seems like something the operator
could tune.
It turns out that with this interpretation the width of the river, and r, M and m, are all irrelevant
The value of ##\beta## that minimizes ##\alpha## doesn't depend on the width of the river, nor does it depend on r, M, m. But doesn't the minimum value of ##\alpha## itself depend on the width of the
river as well as r, M, and m? This is one thing that bothered me about the problem statement, which doesn't mention the width of the river as being relevant.
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Lotto said:
Relevant Equations:: torque1+torque2=0
What is meant by "the smallest value of angular displacement of the raft from its original position during one cycle"?
Since that is the
question asked, the planar movement around the COM of crane+load is not needed.
In the limit of a miniscle (0) load mass, the answer would be _____.
In the limit of an infinite load mass, the answer would be ______.
If the assumption of a massless boom is valid, I think that reduces to a function of the mass ratios, m and Mc, and the variable
TSny said:
The value of ##\beta## that minimizes ##\alpha## doesn't depend on the width of the river, nor does it depend on r, M, m. But doesn't the minimum value of ##\alpha## itself depend on the width of
the river as well as r, M, and m? This is one thing that bothered me about the problem statement, which doesn't mention the width of the river as being relevant.
Ah, good point.
But I still do not see any more reasonable interpretation.
Tom.G said:
Since that is the Only question asked, the planar movement around the COM of crane+load is not needed.
Except that the linear movement means the answer changes with each cycle, so it should ask for the first cycle, not "one" cycle,
Tom.G said:
In the limit of a miniscle (0) load mass, the answer would be
Tom.G said:
In the limit of an infinite load mass, the answer would be
How does that help?
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haruspex said:
By giving a hint of the form of the equation needed to answer the problem. (I hoped)
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Johnson-Kendall-Roberts (JKR) Contact Model
Cohesion or adhesion can be modeled by introducing an attractive force component to the contact model. The terms cohesion and adhesion are usually defined for attracting forces between similar and
dissimilar materials respectively. However, here the two terms are used interchangeably.
The Johnson-Kendall-Roberts (JKR) contact model is an extension of the well-known Hertz contact model proposed by [Johnson1971]. The model accounts for attraction forces due to van der Waals effects.
The model is, however, also used to model material where the adhesion is caused by capillary or liquid-bridge forces ([Hærvig2017], [Carr2016], [Morrisey2013b], [Xia2019]).
The implementation of this model in PFC was first proposed as a C++ Plugin for PFC version 6.0 by Prof. Corné Coetzee, whose contribution is gratefully acknowledged. The model for PFC 6.0 was made
publicly available in the UDM Library on the Itasca website.
This model is incorporated as a built-in model in PFC 7.0. It is referred to in commands and FISH by the name jkr.
Behavior Summary
The JKR contact model is an extension of the Hertz-Mindlin model, which allows for tensile forces to develop due to surface adhesion. This model also incorporates viscous damping and rolling
resistance mechanisms similar to those of the Rolling Resistance Linear Model.
Activity-Deletion Criteria
A contact with the JKR model becomes active when the surface gap becomes less or equal to the reference gap. Once activated, the contact remains active until the surface gap reaches a threshold
positive tear-off distance. Note that a simplified version of the model may be used where the tear-off distance is set equal to the contact reference gap.
Force-Displacement Law
The force-displacement law for the JKR model updates the contact force and moment as:
(1)\[\mathbf{F_{c}} =\mathbf{F^{JKR}} +\mathbf{F^{d}} ,\quad \mathbf{M_{c}} = \mathbf{M^r}\]
where \(\mathbf{F^{JKR}}\) is a non-linear force, \(\mathbf{F^{d}}\) is the dashpot force, and \(\mathbf{M^r}\) is the rolling resistance moment. The non-linear force (resolved into normal and shear
components \(\mathbf{F^{JKR}} = F_{n}^{JKR} {\kern 1pt} \hat{\mathbf{n}}_\mathbf{c} +\mathbf{F_{s}^{JKR}}\)), the dashpot force, and the rolling resistance moment are updated as detailed below.
Normal Force \(F_{n}^{JKR}\)
Following [Johnson1971] and [Chokshi1993], an attractive force component can be introduced to the classic Hertz model, which acts over the circular contact patch area with radius \(a\),
(2)\[F_{adh} = \pi a^2 \gamma^*\]
where \(\gamma^*\) is an effective surface energy per unit area. For two dissimilar surfaces the effective surface energy is given by
(3)\[\gamma^* = \gamma_1 + \gamma_2 - 2 \gamma_{12}\]
where \(\gamma_1\) and \(\gamma_2\) are the surface energy of each surface, and \(\gamma_{12}\) is the interface energy. When the two surface materials are identical, \(\gamma_1 = \gamma_2 = \gamma\)
, the interface energy is zero, \(\gamma_{12} = 0\), and the effective surface energy reduces to
(4)\[\gamma^* = 2 \gamma\]
and the attractive force becomes:
(5)\[F_{adh} = 2 \pi a^2 \gamma\]
Due to the contact stress distribution the total contact normal force is given by:
(6)\[F_{n}^{JKR} = \frac{4E^* a^3}{3 \bar{R}} - \sqrt{16 \pi \gamma E^* a^3}\]
where the first term is identical to the elastic Hertz force (with \(\bar{R}\) the effective contact radius), and the second term is the adhesion component. Due to the attractive force, the contact
area is larger than that predicted by the Hertz theory and the contact patch radius \(a\) is obtained by solving for the positive root of equation (6)
(7)\[a^3 = \frac{3 \bar{R}}{4E} \left[ F_{n}^{JKR} + 6 \pi \gamma \bar{R} + \sqrt{F_{n}^{JKR} 12 \pi \gamma \bar{R} + \left( 6 \pi \gamma \bar{R}\right)^2} \right]\]
The Hertz relation between the contact patch radius \(a\) and the normal deformation (\(a=\sqrt{\bar{R}\delta_n}\)) no longer holds. According to the JKR theory, the normal deformation is related to
the contact radius according to the following non-linear expression,
(8)\[\delta_n = \frac{a^2}{\bar{R}} - \sqrt{\frac{4 \pi \gamma a}{E^*}}\]
In PFC the contact overlap \(\delta_n\) is readily available, and from this the contact patch radius \(a\) should be calculated. Solving equation (8) for the contact patch radius is however not
trivial. [Parteli2014] proposed an analytical solution based on a fourth-order expansion of this equation and solving for the real root that is larger than the patch radius of the classic Hertz
model, which yields
(9)\[\begin{split}\begin{array}{lll} & a &= \frac{1}{2} \left(w + \sqrt{w^2 - 4(c_2+s+\lambda} \right) \\ \mbox{with} & c_0 &= \bar{R}^2 \delta_n^2 \\ & c_1 &= -8(1-\nu^2)\frac{\pi\gamma\bar{R}^2}{E^
*} \\ & c_2 &= -2\bar{R} \delta_n \\ & P &= -\frac{c_2^2}{12} - c_0 \\ & Q &= -\frac{c_2^3}{108} + \frac{c_0c_2}{3} - \frac{c_1^2}{8} \\ & U &= \left( - \frac{Q}{2} + \sqrt{\frac{Q^2}{4} + \frac{P^3}
{27}}\right)^{1/3} \\ & s &= \left\{ \begin{array}{ll} -\frac{5c_2}{6} + U - \frac{P}{3U} & \mbox{if $P \neq 0$} \\ -\frac{5c_2}{6} - Q^{1/3} & \mbox{if $P = 0$} \end{array} \right. \\ & w &= \sqrt
{c_2+2s} \\ & \lambda &= \frac{c_1}{2w} \end{array}\end{split}\]
This value of \(a\) is then used to compute \(F_{n}^{JKR}\) using equation (6).
The maximum tensile force is defined as the pull-off force \(F_{po}\) ([Chokshi1993]),
(10)\[F_{po} = 3 \pi \gamma \bar{R}\]
The pull-off force is independent on the elastic properties and only a function of particle size and surface adhesion energy. The contact force \(F_{n}^{JKR}\) can be normalized using \(F_{po}\) and
plotted as a function of the normalized overlap \(\delta_n / \delta_{to}\), where the tear-off distance \(\delta_{to}\) is the separation distance at which the contact breaks under tension:
(11)\[\delta_{to} = \frac{1}{2} \frac{1}{6^{1/3}} \frac{a_0^2} {\bar{R}}\]
where \(a_0\) is the contact patch radius where \(F_{n}^{JKR}=0\), i.e., at the point where the contact is in equilibrium when no other external forces are acting:
(12)\[a_0 = \left( \frac{9\pi \gamma \bar{R}^2}{E^*} \right)^{1/3}\]
Plotting the normalized force \(F_{n}^{JKR} / F_{po}\) an normalized patch radius \(a / a_0\) against the normalized overlap \(\delta_n / \delta_{to}\) results in Figure 1 below. When two particles
approach each other, the contact is formed when the two particles physically touch at \(\delta_n = 0\) (point 3 in Figure 1). At this point, the force suddenly jumps to a value \(F_{n}^{JKR} = - 8/9
F_{po}\), with the contact patch radius \(a = (2/3)^{2/3} a_0\). This attractive force will pull the particles closer until the equilibrium point is reached (assuming no other forces are acting)
where \(F_{n}^{JKR} =0\), \(a = a_0\), and \(\delta_n = (4/3)^{2/3} \delta_{to}\) (point 4 in Figure 1). Further loading due to external forces will result in an increased positive overlap \(\delta_n
\) and an increase in the compressive force as defined by equation (6).
Upon unloading, the same path is followed up to zero overlap \(\delta_n=0\), after which the force will further decrease down to a minimum value \(F_{n}^{JKR} = - F_{po}\) due to the necking effect
where \(a = (1/2)^{2/3} a_0\) (point 2 in Figure 1). With further unloading, the force increases to a value \(F_{n}^{JKR} = - 5/9 F_{po}\) where the contact suddenly breaks (tear-off) and the force
jumps to zero (point 1 in Figure 1). At this point the overlap is given by \(\delta_n = - \delta_{to}\) and the patch radius by \(a = (1/6)^{2/3} a_0\).
Note that the normal force is in tension when the overlap is positive \(\delta_n > 0\) and between points 3 and 4 in Figure 1. This is different from the Hertz model, where the normal force is always
compressive when the overlap is positive.
Also, in the Hertz model, the normal force becomes zero for \(\delta_n \leq 0\), while in the JKR model the contact remains active with a finite contact patch until the tear-off separation distance
is reached (point 1 in Figure 1).
A simplified version of the model can be used where the normal force becomes zero for \(\delta_n \leq 0\) by setting \(M_a = 0\).
Shear Force \(\mathbf{F_{s}^{JKR}}\)
In the shear direction, a trial shear force is first computed as:
(13)\[\mathbf{F_{s}^{JKR^*}} = \left(\mathbf{F_{s}^{JKR}}\right)_o + k_s^t \Delta \pmb{δ} _\mathbf{s}\]
where \(\left(\mathbf{F_{s}^{JKR}}\right)_o\) is the shear force at the beginning of the timestep, \(\Delta \pmb{δ} _\mathbf{s}\) is the relative shear-displacement increment, and the tangent shear
stiffness \(k_s^t\) is given by [Marshall2009b]:
(14)\[k_s^t = 8 G^* a\]
where \(G^*\) is the effective shear modulus. The tangent shear stiffness is a function of the patch radius \(a\) and not the normal overlap as in the Hertz model, because the patch radius can be
present for negative overlap until the contact breaks.
To account for cohesive effects in the tangential direction, the Coulomb friction limit is usually modified. This is a relatively simple approach that is easy to implement and is based on the work by
[Thornton1991] and [ThorntonAndYin1991], who showed that, with adhesion present, the contact area decreases with an increase in the tangential force. The tangential force reaches a critical value,
which defines the transition from a “peeling” action to a “sliding” action. Assuming the JKR theory ([Johnson1971a]), it is shown that at this critical point the contact area corresponds to a
Hertzian-like normal stress distribution under a load of \((F_n^{JKR} + 2 F_{po})\) ([Marshall2009b]):
(15)\[\begin{split}\mathbf{F_{s}^{JKR}} = \left\{ \begin{array}{l} \mathbf{F_{s}^{JKR^*}} , {\kern 5pt} \mbox{if $\left|\mathbf{F_{s}^{JKR^*}}\right| \leq \mu (F_n^{JKR} + 2 F_{po})$} \\ \mu (F_n^
{JKR} + 2 F_{po}) \left( \frac{\mathbf{F_{s}^{JKR^*}}}{\left|\mathbf{F_{s}^{JKR^*}}\right|} \right), {\kern 1pt} \mbox{otherwise} \end{array} \right.\end{split}\]
If the first condition is met, the boolean slip state is set to \(s=\mbox{false}\), and when the second condition is met it is set to \(s=\mbox{true}\). Whenever the slip states changes, the
slip_change callback event occurs.
Dashpot force
The dashpot force is resolved into normal and shear components:
(16)\[\mathbf{F^{d}} =-F_{n}^{d} {\kern 1pt} \hat{\mathbf{n}}_\mathbf{c} +\mathbf{F_{s}^{d}}\]
In the normal direction:
(17)\[F_{n}^{d} = 2 \beta_n \sqrt{m_c k_n^t} \dot{\delta}_n\]
where \(m_c = \frac{m_1 m_2}{m_1 + m_2}\) is the effective contact mass with \(m_1\) and \(m_2\) the mass of the two contacting particles (for a contact with a wall facet, \(m_c = m_p\) where \(m_p\)
is the mass of the particle), \(\dot{\delta}_n\) is the relative normal translational velocity, \(k_n^t\) is the tangent normal stiffness, and \(\beta_n\) is the normal critical damping ratio. This
ratio takes a value \(\beta_n=0\) when there is no damping, and \(\beta_n=1\) when the system is critically damped.
The tangent normal stiffness is given by:
(18)\[k_n^t = 2 E^* a\]
The dashpot force in the shear direction is given by:
(19)\[\begin{split}\mathbf{F_s^d} = \left\{ \begin{array}{ll} \left( - 2 \beta_s \sqrt{m_c k_s^t} \right) \dot{\pmb{δ}}_\mathbf{s}, & \mbox{$s =$ false or $M_d = 0$ (full shear)} \\ \mathbf{0}, & \
mbox{$s =$ true and $M_d = 1$ (slip-cut)} \end{array} \right.\end{split}\]
where \(M_d\) is the dashpot behavior mode, \(\dot{\pmb{δ}}_\mathbf{s}\) is the relative shear translational velocity, \(\beta_s\) is the shear critical damping ratio, and \(k_s^t\) is the tangential
shear stiffness given by equation (14).
Rolling Resistance
The rolling resistance moment is first incremented as:
(20)\[\mathbf{M^r} :=\mathbf{M^r} - k_r^t {\kern 1pt} \Delta \pmb{θ}_\mathbf{b}\]
where \(k_r^t\) is the tangent rolling resistance stiffness, and \(\Delta \pmb{θ}_\mathbf{b}\) is the relative bend-rotation increment of Equation (12) of the “Contact Resolution” section.
The tangent rolling resistance stiffness \(k_r^t\) is defined as:
(21)\[k_r^t = k_s^t {\kern 1pt} \bar{R}^{2}\]
with \(k_s^t\) the tangent shear stiffness given by equation (14) and \(\bar{R}\) the contact effective radius(equation :eq:’cmjkr_eqR’) CS: maybe this is cmeepa_eqR??? see line 262 (the cmeepa_Mr2
equation reference) for more badness).
The magnitude of the updated rolling resistance moment is then checked against a threshold limit:
(22)\[\begin{split}\mathbf{M^r} = \left\{ \begin{array}{l} \mathbf{M^r} , {\kern 5pt} \mbox{if $\| \mathbf{M^r} \| \le \mu_r \bar{R} (F_n^{JKR} + 2 F_{po})$} \\ \mu_r \bar{R} (F_n^{JKR} + 2 F_{po}) \
bigl( \frac{\mathbf{M^r}} {\| \mathbf{M^r} \|} \bigr), {\kern 1pt} \mbox{otherwise.} \end{array} \right.\end{split}\]
where the same arguments as used for the shear force are used to increase the limiting moment by the effects of the pull-off force \(F_{po}\). The normal force \(F_n^{JKR}\) is taken at the end of
the current step, and \(\mu_r\) is the rolling coefficient of friction. If the first condition in (22) is met, then the rolling slip state is set to \(s_r=\mbox{false}\), and when the second
condition is met it is set to \(s_r=\mbox{true}\)
Energy Partitions
The JKR model provides five energy partitions:
• strain energy, \(E_{k}\), stored in the non-linear springs;
• slip energy, \(E_{\mu}\), defined as the total energy dissipated by frictional slip;
• dashpot energy, \(E_{\beta}\), defined as the total energy dissipated by the dashpots;
• rolling strain energy, \(E_{k_r}\), stored in the rolling linear spring; and
• rolling slip energy, \(E_{\mu_r}\), defined as the total energy dissipated by rolling slip.
If energy tracking is activated, these energy partitions are updated as follows:
1. Incrementally update the strain energy:
(23)\[\begin{split}\begin{array}{ll} E_k := E_k & + \frac{1}{2} \left(\left(F_{n}^{JKR}\right)_o + F_{n}^{JKR}\right) \Delta \delta_n \\ & + \frac{1}{2} \left(\left(\mathbf{F_{s}^{JKR}}\right)_o
+ \mathbf{F_{s}^{JKR}}\right) \cdot \Delta \pmb{δ}_\mathbf{s}^\mathbf{k} \\ {\rm with} & \Delta \pmb{δ} _\mathbf{s}^{k } =\Delta \pmb{δ} _\mathbf{s} -\Delta \pmb{δ} _\mathbf{s}^{\mu} = \left(\
frac{\mathbf{F_{s}^{JKR}} -\left(\mathbf{F_{s}^{JKR}} \right)_{o} }{k_{s}^t } \right) \end{array}\end{split}\]
where \(\left(F_{n}^{JKR}\right)_o\) and \(\left(\mathbf{F_{s}^{JKR}}\right)_o\) are, respectively, the JKR normal and shear forces at the beginning of the step, and the relative
shear-displacement increment has been decomposed into an elastic \(\left(\Delta \pmb{δ} _\mathbf{s}^{k} \right)\) and a slip \(\left(\Delta \pmb{δ} _\mathbf{s}^{\mu } \right)\) component, with \
(k_s^t\) the tangent shear stiffness at the start of the current step (equation (14)).
2. Incrementally update the slip energy:
(24)\[E_{\mu } :=E_{\mu } - {\tfrac{1}{2}} \left(\left(\mathbf{F_{s}^{JKR}} \right)_{o} +\mathbf{F_{s}^{JKR}} \right)\cdot \Delta \pmb{δ} _\mathbf{s}^{\mu }\]
3. Incrementally update the dashpot energy:
(25)\[E_{\beta } :=E_{\beta } - \mathbf{F^{d}} \cdot \left(\dot{\pmb{δ} }{\kern 1pt} {\kern 1pt} \Delta t\right)\]
where \(\dot{\pmb{δ} }\) is the relative translational velocity of this equation of the “Contact Resolution” section.
4. Update the rolling strain energy:
(26)\[E_{k_r} = \frac{1}{2} \frac{\left\| \mathbf{M^{r}} \right\| ^{2} }{k_{r}}.\]
5. Incrementally update the rolling slip energy:
(27)\[\begin{split}\begin{array}{l} {E_{\mu_r} := E_{\mu_r} - {\tfrac{1}{2}} \biggl(\left(\mathbf{M^{r}} \right)_{o} +\mathbf{M^{r}} \biggr)\cdot \Delta \pmb{θ}_\mathbf{b}^{\mu_r} } \\ {{\rm
with}\qquad \Delta \pmb{θ}_\mathbf{b}^{\mu_r} = \Delta \pmb{θ}_\mathbf{b} -\Delta \pmb{θ}_\mathbf{b}^{k} =\Delta \pmb{θ}_\mathbf{b} -\left(\frac{\mathbf{M^{r}} -\left(\mathbf{M^{r}} \right)_{o} }
{k_{r}^t } \right)} \end{array}\end{split}\]
where \(\left(\mathbf{M^{r}} \right)_{o}\) is the rolling resistance moment at the beginning of the timestep, and the relative bend-rotation increment has been decomposed into an elastic \(\Delta
\pmb{θ}_\mathbf{b}^{k}\) and a slip \(\Delta \pmb{θ}_\mathbf{b}^{\mu_r}\) component, with \(k_{r}^t\) the tangent rolling stiffness. The dot product of the slip component and the rolling
resistance moment occurring during the timestep gives the increment of rolling slip energy.
Additional information, including the keywords by which these partitions are referred to in commands and FISH, is provided in the table below.
Keyword Symbol Description Range Accumulated
JKR Group:
energy‑strain‑jkr \(E_{k}\) strain energy \((-\infty,+\infty)\) YES
energy‑slip \(E_{\mu}\) total energy dissipated by slip \([0.0,+\infty)\) YES
Dashpot Group:
energy‑dashpot \(E_{\beta}\) total energy dissipated by dashpots \([0.0,+\infty)\) YES
Rolling-Resistance Group:
energy‑rrstrain \(E_{k_r}\) Rolling strain energy \([0.0,+\infty)\) NO
energy‑rrslip \(E_{\mu_r}\) Total energy dissipated by rolling slip \((-\infty,0.0]\) YES
The properties defined by the JKR contact model are listed in the table below for concise reference. See the “Contact Properties” section for a description of the information in the table columns.
The mapping from the surface inheritable properties to the contact model properties is also discussed below.
Keyword Symbol Description Type Range Default Modifiable Inheritable
jkr Model name
JKR Group:
jkr_shear \(G\) Shear modulus [stress] FLT \((0.0,+\infty) 0.0 YES YES
jkr_poiss \(\nu\) Poisson’s ratio [-] FLT \([0.0,0.5]\) 0.0 YES YES
ks_fac \(k_{sf}\) Shear stiffness scaling factor [-] FLT \((0.0,+\infty) 1.0 YES NO
fric \(\mu\) Sliding friction coefficient [-] FLT \([0.0,+\infty) 0.0 YES YES
surf_adh \(\gamma\) Surface adhesion energy [energy/area] FLT \([0.0,+\infty) 0.0 YES NO
a0 \(a_0\) Equilibrium contact patch radius [length] FLT \([0.0,+\infty) 0.0 NO N/A
pull_off_f \(F_{po}\) Pull-off force [force] FLT \((-\infty,0.0] 0.0 NO NO
tear_off_d \(\delta_{to}\) Tear-off distance [-] FLT \((0.0,1.0)\) 0.5 YES NO
active_mode \(M_a\) Active mode [-] INT {0;1} 0 YES NO
\(\;\;\;\;\;\;\begin{cases} \mbox{0: no negative overlap} \\ \mbox{1: allows negative overlap} \end{cases}\)
jkr_slip \(s\) Slip state [-] BOOL {false,true} false NO N/A
jkr_force \(\mathbf{F^{JKR}} JKR force (contact plane coord. system) VEC \(\mathbb{R}^3\) \(\mathbf{0} YES NO
\) \)
\(\left( -F_n^{JKR},F_{ss}^{JKR},F_{st}^{JKR} \right) \quad \left(\mbox{2D model: } F_{ss}^{JKR} \equiv 0 \
Rolling-Resistance Group:
rr_fric \(\mu_r\) Rolling friction coefficient [-] FLT \([0.0,+\infty\) 0.0 YES YES
rr_moment \(M^r\) Rolling Resistance moment (contact plane coord. system) VEC \(\mathbb{R}^3\) \(\mathbf{0} YES NO
\(\left( 0,M_{bs}^r,M_{bt}^r \right) \quad \left(\mbox{2D model: } M_{bt}^r \equiv 0 \right)\)
rr_slip \(s_r\) Rolling slip state [-] BOOL {false,true} false NO N/A
Dashpot Group:
dp_nratio \(\beta_n\) Normal critical damping ratio [-] FLT \([0.0,1.0]\) 0.0 YES NO
dp_sratio \(\beta_s\) Shear critical damping ratio [-] FLT \([0.0,1.0]\) 0.0 YES NO
dp_mode \(M_d\) Dashpot mode [-] INT {0;1} 0 YES NO
\(\;\;\;\;\;\;\begin{cases} \mbox{0: no cutt-off} \\ \mbox{1: cut-off in shear when sliding} \end{cases}\)
dp_force \(\mathbf{F^{d}}\) Dashpot force (contact plane coord. system) VEC \(\mathbb{R}^3\) \(\mathbf{0} NO N/A
\(\left( -F_n^d,F_{ss}^d,F_{st}^d \right) \quad \left(\mbox{2D model: } F_{ss}^d \equiv 0 \right)\)
Surface Property Inheritance
The shear modulus \(G\), Poisson’s ratio \(\nu\), friction coefficient \(\mu\), and rolling friction coefficient \(\mu_r\) may be inherited from the contacting pieces. Remember that for a property to
be inherited, the inheritance flag for that property must be set to true (default), and both contacting pieces must hold a property with the exact same name.
The shear modulus and Poisson’s ratio are inherited as:
(28)\[\begin{split}\nu &= \frac{4 G^* - E^*}{2 G^* - E^*} \\ G &= 2 G^* (2 - \nu)\end{split}\]
(29)\[\begin{split}E^* &= \left( \frac{1-\nu^{(1)}}{2G^{(1)}} + \frac{1-\nu^{(2)}}{2G^{(2)}} \right)^{-1} \\ G^* &= \left( \frac{2-\nu^{(1)}}{G^{(1)}} + \frac{2-\nu^{(2)}}{G^{(2)}} \right)^{-1}\end
where (1) and (2) denote the properties of piece 1 and 2, respectively.
The friction and rolling friction coefficients are inherited using the minimum of the values set for the pieces:
(30)\[\begin{split}\mu &= \min \left( \mu^{(1)} ,\mu^{(2)} \right) \\ \mu_r &= \min \left( \mu_r^{(1)},\mu_r^{(2)}\right)\end{split}\]
No methods are defined by the JKR model.
Callback Events
Event Array Slot Value Type Range Description
contact_activated contact becomes active
1 C_PNT N/A contact pointer
slip_change Slip state has changed
1 C_PNT N/A contact pointer
2 INT {0;1} slip change mode
\(\;\;\;\;\;\;\begin{cases} \mbox{0: slip has initiated} \\ \mbox{1: slip has ended} \end{cases}\)
Model Summary
An alphabetical list of the JKR model properties is given here.
Carr, M. J., W. Chen, K. Williams, and A. Katterfeld. “Comparative investigation on modelling wet and sticky material behaviours with a simplified JKR cohesion model and liquid bridging cohesion
model in DEM,” ICBMH 2016 - 12th International Conference on Bulk Materials Storage, Handling and Transportation, Proceedings, pages 40–49, 2016.
[Chokshi1993] (1,2)
Chokshi, A.G., Tielens, G. M., and D. Hollenbach. “Dust Coagulation,” The Astrophysical Journal, 407:806-819, Apr. 1993.
Hærvig, J. , U. Kleinhans, C.Wieland, H. Spliethoff, A. L. Jensen, K. Sørensen, and T. J. Condra. “On the adhesive JKR contact and rolling models for reduced particle stiffness discrete element
simulations,” Powder Technology, 319:472–482, 2017. ISSN 1873328X. doi:10.1016/j.powtec.2017.07.006. URL http://dx.doi.org/10.1016/j.powtec.2017.07.006.
[Johnson1971] (1,2)
Johnson, K. L., K. Kendall, and A. D. Roberts. “Surface Energy and the Contact of Elastic Solids,” Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 324
(1558):301–313, Sept. 1971.
[Marshall2009b] (1,2)
Marshall, J. S., “Discrete-Element Modeling of Particulate Aerosol Flows,” Journal of Computational Physics, 228(5):1541-1561, Mar. 2009, doi:10.1016/j.jcp.2008.10.035.
Morrissey, J. P. “Discrete Element Modelling of Iron Ore Pellets to Include the Effects of Moisture and Fines,” PhD thesis, Edinburgh, Scotland: University of Edinburgh (2013).
Parteli, E. J., Schmidt, J., Blümel, C., Wirth, K. E., Peukert, W., and T. Pöshel. “Attractive particle interaction forces and packing density of fine glass powders,” Scientific Reports, 4:1-7,2014.
ISSN 20454322. doi:10.1038/srep06227.
Xia, R., B. Li, X.Wang, T. Li, and Z. Yang. “Measurement and calibration of the discrete element parameters of wet bulk coal,” Measurement: Journal of the International Measurement Confederation,
142:84–95, 2019. ISSN 02632241. doi: 10.1016/j.measurement.2019.04.069.
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How Systems of Equations Can Help You Become a Pro at Mathematics?
Mathematics is no doubt considered a very difficult & complex subject by many students around the globe. Blame it on the teachers that teach it or the crazy formulas you need to remember.
For all the children who don’t like Algebra & all the other concepts associated with it, we are here to help you with a very important topic of Algebra & that is systems of equations. Let’s look at
this concept in more detail!
What exactly systems of equations are?
When you solve two or more equations at the same time or we can say simultaneously, to get the common solution, it is called a system of equations. These equations can be made up of either two or
more variables.
To find the accurate solution for both the equations, you need to find a numerical value for each variable in the system, which will be capable of solving/satisfying all equations at the same time.
For example: (6,-1) is the right combination for the following equations:-
Because, when we put the value of x & y as 6 & – 1 in the above equations, the answer comes out to be equal.
2(6)+3(-1)=9 6+4(-1)=2
12-3=9 6-4=2
9=9 2=2
What are the various possible ways to solve a system of equations?
To solve any system of equations that are given to you, there are possibly 3 methods to solve them. They are:
1. Graphical method:
When you solve a system of equations, your solution might have only one set of correct answers or it may have 2 or more. The exact answer will depend on the equation that you are given.
For equations having one set of correct answers, there will only be a single point of intersection, when plotted at the graph.
But in case there are two sets of correct answers, there will be two points of intersection on the graph. This will depict that there are two solutions for the equations.
1. Substitution method:
One of the easiest methods you will ever come across. In this method, you need to solve your system of equations by expressing your equation in terms of one variable.
In this, you remove one variable from the equation & put it into another equation, which is hence called substitution.
• For example take two equations: 3x+2y=11& – x+y=3
• From equation 2, we will find the value of y i.e. y=x+3
• Now put the value of y in equation 1. It will be, 3x+2(x+3)=11
• On solving it, the solution will look like this:
5x=5, hence x=1
• Now we will put this value of x in equation 1. It will be, y=1+3 & therefore, the value of y=4
• The right set of answers for this system of equations is (1,4)
1. Elimination method:
In this method, a variable is removed from the system of equations, so that solving the remaining variables becomes easy. Once, the value of other variables is found, then the value of those
variables is substituted into the original equation, to find the remaining ones.
• Take two equations x+y=2 &x-y=14
• Now eliminate the y variable, by adding up the equations.
2x =16 hence x=8
• Now, put the value of X in equation 1, it will be 8+y=2
• Therefore, y=-6. The right pair for this system of equations is (8,-6)
Why study systems of equations online?
Long gone are the days when children used to wait for their teachers, to ask their doubts. This is the era of digitization & hence you can learn everything with the help of the internet. With the
help of Cuemath, you can learn online math to help clear your concepts of systems of equations. We provide you with the best subject matter relating to different topics of mathematics. Our team of
experts ensures to polish your aptitude skills by supplying you with the best knowledge without spending much.
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Change of Basis
Revision e0c8d4321c3d5fd522bfa51d231bf06622752661 (click the page title to view the current version)
Reading Ma 2004 Chapter 2.2-2.3
• Complex scene.
□ Lots and lots of points.
□ If an object moves, all its points changes.
□ This is unmanageable
• Modularise
□ Describe the object in local co-ordinates.
□ Describe the location and orientation of the object once in the global co-ordinate system.
□ When the object moves, its local co-ordinate system changes relative to the global one, but the internal description of the object does not.
Motion examples
1. Translation \[\vec{x}' = \vec{x}+\vec{t}\]
2. Rotation \[\vec{x}' = \vec{x}\cdot R\]
□ But note that \(R\) is not an arbitrary matrix.
□ We’ll return to the restrictions
Handdrawn illustration
Definition: Rigid Body Motion
1. 3D Object is a set of points in \(\mathbb{R}^3\)
2. If the object moves, the constituent points move
3. The points have to move so that they preserve the shape of the object
Let \(\vec{X}(t)\) and \(\vec{Y}(t)\) be the coordinates of points \(\vec{x}\) and \(\vec{x}\) at time \(t\).
1. Preserve distance between points
□ \(||\vec{X}(t)-\vec{Y}(t)||\) is constant
2. Preserve orientation
□ i.e. avoid mirroring
□ we have to preserve cross-products
□ If the right hand rule turns into a left hand rule, we have had mirroring.
Let \(u=\vec{X}-\vec{Y}\) be a vector, and \(g_*(u)=g(\vec{X})-g(\vec{Y})\) the corresponding vector after motion.
Preserving the cross-product means \[g_*(u)\times g_*(v) = g_*(u\times v), \forall u,v\in\mathbb{R}^3\]
Change of Basis
1. Basis aka. frame
□ Unit vectors: \(\vec{e}_1\), \(\vec{e}_2\), \(\vec{e}_3\)
2. The meaning of a tuple to denote a vector
□ \(\vec{x}=[x_1,x_2,x_3]= x_1\cdot\vec{e}_1+x_2\cdot\vec{e}_2+x_3\cdot\vec{e}_3\)
3. Orthonormal frame: orthogonal and unit length \[\vec{e}_i\vec{e}_j=\delta_{ij} = \begin{cases} 1 \quad\text{if } i=j\\ 0 \quad\text{if } i\neq j \end{cases} \]
Arbitrary Choice of Basis
• Any set of three linearly independent vectors \(\vec{u}_1\), \(\vec{u}_2\), \(\vec{u}_3\) can be used as a basis.
• Usually, we prefer an orthonormal basis, i.e.
□ the basis vectors are orthogonal
□ the basis vectors have unit length
\[ \langle\vec{u}_i, \vec{u}_j\rangle = \begin{cases} 1, \quad\text{when } i=j,\\ 0, \quad\text{when } i\neq j\end{cases}\]
• The basis is relative to a given Origin
Local and Global Basis
1. 3D Scenes are built hierarchically
2. Each object is described in a local basis
□ and then placed in the global basis.
3. Why?
□ Save computational work
□ Local changes affect only local co-ordinates
□ Component motion independent of system motion
Describing a Scene
• Each object described in its own basis
□ independently of its position and orientation in the scene
□ reusable objects
• Rotation and Deformation can be described locally
• Transformation from local to global co-ordinates
□ Rotation of the basis
□ Translation of the origin
• System of Systems
□ an object in the scene may itself be composed of multiple objects with different local frames
E.g. our co-ordinates: 62°28’19.3“N 6°14’02.6”E
Are these local or global co-ordinates?
Consider common origin first.
Working with Different Bases
Change of Basis
• Point \(\vec x\) represented in a basis \(\vec{e}_1\), \(\vec{e}_2\), \(\vec{e}_3\)
□ i.e. \(\vec x = x_1\vec{e}_1 + x_2\vec{e}_2 + x_3\vec{e}_3\)
• Translate to a representation in another basis \(\vec{u}_1\), \(\vec{u}_2\), \(\vec{u}_3\)
• Suppose we can write the old basis in terms of the new one
□ \(\vec{e_i} = e_{i,1}\vec{u}_1 + e_{i,2}\vec{u}_2 + e_{i,3}\vec{u}_3\)
\[\begin{split} p = & x_1(e_{1,1}\vec{u}_1 + e_{1,2}\vec{u}_2 + e_{1,3}\vec{u}_3) + \\ & x_2(e_{2,1}\vec{u}_1 + e_{2,2}\vec{u}_2 + e_{2,3}\vec{u}_3) + \\ & x_3(e_{3,1}\vec{u}_1 + e_{3,2}\vec{u}_2 +
e_{3,3}\vec{u}_3) \\ = &(x_1e_{1,1} + x_2e_{2,1} + x_3e_{2,1} )\vec{u}_1 + \\ &(x_1e_{1,2} + x_2e_{2,2} + x_3e_{2,2} )\vec{u}_2 + \\ &(x_1e_{1,3} + x_2e_{2,3} + x_3e_{2,3} )\vec{u}_3 \end{split}\]
Write \([x'_1, x'_2, x'_3]^\mathrm{T}\) for the coordinates in terms of the new basis.
\[ x'_i = x_1e_{1,i} + x_2e_{2,i} + x_3e_{2,i} \]
• In matrix form we can write \[p = [x_1',x_2',x_3'] = [ \vec{e}_1 | \vec{e}_2 | \vec{e}_3 ]\cdot\vec{x}\] where \(\vec{e}_i\) are written as column vectors.
Orthonormal matrix
\[R = [ \vec{e}_1 | \vec{e}_2 | \vec{e}_3 ]\]
• Orthonormal basis means that \(R\cdot R^T = R^T\cdot R=I\)
• Hence \(R^{-1}=R^T\)
• If \(\vec{x}'=\vec{x}\cdot R\) then \(\vec{x}=\vec{x}'\cdot R^T\) then
• If the columns of \(R\) make up the new basis,
□ then the rows make up the old basis
\[ \begin{split} R = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} & 0 \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} & 0 \\ 0 & 0 & 1 \end{bmatrix} \end{split} \]
Note this is orthogonal. Compare it to
Example in 2D
The principle is the same in 2D.
\[ R_1 = \begin{bmatrix} \frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \end{bmatrix} \quad R_2 = \begin{bmatrix} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ \
frac{\sqrt{2}}{2} & -\frac{\sqrt{2}}{2} \\ \end{bmatrix} \]
Note that
1. \(\frac{\sqrt2}{2}=\sin(\pi/4)=\cos(\pi/4)\).
2. \(R_1^TR_1=I\) and \(R_2R_2=I\)
3. The determinants \(|R_1|=1\) and \(|R_2|=-1\)
Consider a triangle formed by the points \((0,0)\), \((0,1)\), \((1,1)\), and consider each of them rotated by \(R_1\) and \(R_2\)
Rotation around an axis
A rotation by an angle \(\theta\) around the origin is given by \[ R_\theta = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{bmatrix} \] In 3D, we can rotate around a
given axis, by adding a column and row to the 2D matrix above. E.g. to rotate around the \(y\)-axis, we use \[ R_\theta = \begin{bmatrix} \cos\theta & 0 & -\sin\theta \\ 0 & 1 & 0 \\ \sin\theta & 0 &
\cos\theta \\ \end{bmatrix} \]
Operations on rotations
• If \(R_1\) and \(R_2\) are rotational matrices, then \(R=R_1\cdot R_2\) is a rotational matrix
• There is an identity rotation \(I\)
• If \(R_1\) is a rotational matrices, then there is an inverse rotation \(R_1^{-1}\)
• If \(R_1\), \(R_2\), and \(R_3\) are rotational matrices, then \(R_1(R_2R_3)= (R_1R_2)R_3\)
Rotation as Motion
Why is change of basis so important?
• It also describes rotation of rigid bodies
• To rotate a body, rotate its local basis
• Any orthonormal basis is a rotation of any other
□ The matrix \(R\) defines the rotation
• Any orthogonal matrix defines a rotation
Use Python Here
Moving the Origin
• A point is described relative to the origin
\[\mathbf x = \mathbf{0} + x_1\cdot\vec{e}_1 + x_2\cdot\vec{e}_2 + x_3\cdot\vec{e}_3\]
• Note that I write \(\mathbf{x}\) for a point and \(\vec{x}\) for a vector
• The origin is arbitrary
• The local co-ordinate system is defined by
1. the basis \(\vec{e}_1\), \(\vec{e}_2\), \(\vec{e}_3\)
2. the origin \(\mathbf{0}\)
• Move origin: \(\mathbf{0}'=\mathbf{0}+\vec{t}\)
□ for some translation vector \(\vec{t}\)
Arbitrary motion
• \(\mathbf{X}\mapsto \mathbf{X}\cdot R + \vec{t}\)
• or \(\mathbf{X}\mapsto (\mathbf{X}-\vec{t}_1)\cdot R + \vec{t}_2\)
□ remember, what is the centre of rotation?
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x root y on hp42s
Linear Mode
x root y on hp42s
« Next Oldest | Next Newest »
04-07-2012, 04:23 AM
I'm trying to turn the cube root program in the 42s's manual into an "x root y" program, but I'm having trouble understanding how to recall the Y register. Would please paste an nth-root program
04-07-2012, 04:39 AM
To recall X, Y, Z, T from the stack on th 42s, press
[RCL] and [.] and you will be presented with soft key choices. The same for STO.
Another way to access Y is to do X<>Y
04-07-2012, 11:51 PM
The following results match what I get with ^x|/y on the HP32 SII, except for the error messages. This can be optimized for size, of course.
00 { 41-Byte Prgm }
01>LBL "XROOT"
02 STO ST Z
03 X<>Y
04 X>=0?
05 GTO 00
06 X<>Y
08 MOD
09 X=0?
10 DOT
11 Rv
12 X<>Y
13 ENTER
14 FP
15 X!=0?
16 DOT
17 Rv
18 Rv
19>LBL 00
20 SIGN
21 LASTX
22 RCL ST Z
23 1/X
24 Y^X
25 ABS
26 ×
27 .END.
Keystrokes Display
27 +/- ENTER 3 XEQ XROOT -3
32 +/- ENTER 5 XEQ XROOT -2
27 +/- ENTER 3 +/- XEQ XROOT -3.33333333333E-1
32 +/- ENTER 5 +/- XEQ XROOT -4.99999999999E-1
32 ENTER 5 +/- XEQ XROOT 0.5
625 ENTER 4 XEQ XROOT 5
625 ENTER 4 +/- XEQ XROOT 0.2
625 +/- 4 XEQ XROOT Invalid Type
27.5 ENTER 3.5 +/- XEQ XROOT 3.87937790083E-1
27.5 +/- ENTER 3.5 XEQ XROOT Invalid Type
27.5 +/- ENTER 3.5 +/- XEQ XROOT Invalid Type
27.5 ENTER 3.5 XEQ XROOT 2.577732888
5 ENTER 0 XEQ XROOT Divide by 0
P.S.: Real arguments only.
Edited: 8 Apr 2012, 12:00 a.m.
04-08-2012, 06:17 AM
Hi Gerson,
Quote: The following results match what I get with x|/y on the HP32 SII ...
27 +/- ENTER 3 XEQ XROOT -3
Fine. And now try the cube root of 729. Does it still match ?-)
Quote: 32 +/- ENTER 5 +/- XEQ XROOT -4.99999999999E-1
My 35s here returns exactly -0,5. What does your 32sII say? I cannot imagine the result is any different.
BTW, it can be shown that evaluating roots this way may exhibit an error up to several units in the last significant digit. That's why I was so eager about CUBERT and XROOT in the 34s. ;-)
04-08-2012, 06:22 AM
The 34S will still exhibit the same error in the last place, but due to the increased internal precision, it will likely be hidden resulting in an effective +/- 1 ULP in the result.
- Pauli
04-08-2012, 12:06 PM
Yes, that's why there are guard digits. With its 39-digit precision and 16 resp. 32 digit output, I think the 34s may look at all this quite relaxed. ;-)
04-08-2012, 06:14 PM
Double precision carries 34 digits not 32 :-)
- Pauli
04-08-2012, 10:34 PM
... for that reason it's called 'double' ;-)
04-08-2012, 11:14 PM
It is called double because it occupies 128 bits of memory whereas single precision occupies 64 bits. There is logic here.
In reality, we're using double and quadruple precision. Single precision being 32 bits which allows seven digits.
The gain in digits is made possible by not doubling the size of the exponent field and not duplicating the sign bit with each doubling in length of the number.
- Pauli
04-08-2012, 11:37 AM
Hello Dieter,
Quote: And now try the cube root of 729. Does it still match ?-)
It practically matches: 8.99999999998.
Quote: 32 +/- ENTER 5 +/- XEQ XROOT -4.99999999999E-1
My 35s here returns exactly -0,5.
So does my HP-32SII. "Practically" is the word I missed. My fault, sorry!
Quote: That's why I was so eager about CUBERT and XROOT in the 34s. ;-)
It's a pity the latter is not available from the keyboard. IMHO, it should be a better companion to y^x than LOG[x], which of course should stay. Where to find room for both on the keyboard? I think
./, is superfluous as RDX. and RDX, are available under MODE. But I am aware time for keyboard layout suggestions is over...
04-08-2012, 12:17 PM
Quote: It practically matches: 8.99999999998.
Aaaaaha! ;-)
But it gets even worse. Try the 6th root of 531441. In cases like these the error may be as large at 5 ULP.
Quote: It's a pity the latter is not available from the keyboard. IMHO, it should be a better companion to y^x than LOGx, which of course should stay. Where to find room for both on the keyboard?
I think XROOT would fit nicely on the (green shifted) sqrt / x^2 key.
Oooops... guess what - that's exactly where it is placed on the 35s. :-)
Quote: But I am aware time for keyboard layout suggestions is over...
Walter always says that the 34s is a moving target. So nothing is fixed. ;-)
04-08-2012, 06:16 PM
Quote:Walter always says that the 34s is a moving target. So nothing is fixed. ;-)
The keyboard layout is a little less fluid than everything else.
- Pauli
04-08-2012, 12:21 PM
Quote: I think ./, is superfluous as RDX. and RDX, are available under MODE. But I am aware time for keyboard layout suggestions is over...
Sim, Gerson - you're right with your second sentence. And the label ./, was caused by the fact that at design time our North American friends kept asking for an easy way to change the radix mark
whenever they got a comma calculator ;-) And since we meanwhile put an alpha menu on said label, we can't change that easily (BTW I remember we had a discussion about that very label some (many?)
months ago, but I don't remember when).
04-08-2012, 12:39 PM
Quote: And the label ./, was caused by the fact that at design time our North American friends kept asking for an easy way to change the radix mark whenever they got a comma calculator ;-)
On the other hand, this also allows for an easy way to accidentally change the radix mark. I missed that discussion, however.
04-08-2012, 12:43 PM
On the other hand, this also allows for an easy way to accidentally change the radix mark. I missed that discussion, however.
You're right again, but you can correct that accident most easily :-)
I'm trying to turn the cube root program in the 42s's manual into an "x root y" program, but I'm having trouble understanding how to recall the Y register. Would please paste an nth-root program
To recall X, Y, Z, T from the stack on th 42s, press
[RCL] and [.] and you will be presented with soft key choices. The same for STO.
The following results match what I get with x|/y on the HP32 SII, except for the error messages. This can be optimized for size, of course.
00 { 41-Byte Prgm } 01>LBL "XROOT" 02 STO ST Z 03 X<>Y 04 X>=0? 05 GTO 00 06 X<>Y 07 2 08 MOD 09 X=0? 10 DOT 11 Rv 12 X<>Y 13 ENTER 14 FP 15 X!=0? 16 DOT 17 Rv 18 Rv 19>LBL 00 20 SIGN 21 LASTX 22 RCL
ST Z 23 1/X 24 Y^X 25 ABS 26 × 27 .END. Keystrokes Display 27 +/- ENTER 3 XEQ XROOT -3 32 +/- ENTER 5 XEQ XROOT -2 27 +/- ENTER 3 +/- XEQ XROOT -3.33333333333E-1 32 +/- ENTER 5 +/- XEQ XROOT
-4.99999999999E-1 32 ENTER 5 +/- XEQ XROOT 0.5 625 ENTER 4 XEQ XROOT 5 625 ENTER 4 +/- XEQ XROOT 0.2 625 +/- 4 XEQ XROOT Invalid Type 27.5 ENTER 3.5 +/- XEQ XROOT 3.87937790083E-1 27.5 +/- ENTER 3.5
XEQ XROOT Invalid Type 27.5 +/- ENTER 3.5 +/- XEQ XROOT Invalid Type 27.5 ENTER 3.5 XEQ XROOT 2.577732888 5 ENTER 0 XEQ XROOT Divide by 0
27 +/- ENTER 3 XEQ XROOT -3 32 +/- ENTER 5 XEQ XROOT -2 27 +/- ENTER 3 +/- XEQ XROOT -3.33333333333E-1 32 +/- ENTER 5 +/- XEQ XROOT -4.99999999999E-1 32 ENTER 5 +/- XEQ XROOT 0.5 625 ENTER 4 XEQ
XROOT 5 625 ENTER 4 +/- XEQ XROOT 0.2 625 +/- 4 XEQ XROOT Invalid Type 27.5 ENTER 3.5 +/- XEQ XROOT 3.87937790083E-1 27.5 +/- ENTER 3.5 XEQ XROOT Invalid Type 27.5 +/- ENTER 3.5 +/- XEQ XROOT Invalid
Type 27.5 ENTER 3.5 XEQ XROOT 2.577732888 5 ENTER 0 XEQ XROOT Divide by 0
Quote: The following results match what I get with x|/y on the HP32 SII ... 27 +/- ENTER 3 XEQ XROOT -3
BTW, it can be shown that evaluating roots this way may exhibit an error up to several units in the last significant digit. That's why I was so eager about CUBERT and XROOT in the 34s. ;-)
The 34S will still exhibit the same error in the last place, but due to the increased internal precision, it will likely be hidden resulting in an effective +/- 1 ULP in the result.
Yes, that's why there are guard digits. With its 39-digit precision and 16 resp. 32 digit output, I think the 34s may look at all this quite relaxed. ;-)
It is called double because it occupies 128 bits of memory whereas single precision occupies 64 bits. There is logic here.
In reality, we're using double and quadruple precision. Single precision being 32 bits which allows seven digits.
The gain in digits is made possible by not doubling the size of the exponent field and not duplicating the sign bit with each doubling in length of the number.
Quote: And now try the cube root of 729. Does it still match ?-)
Quote:Quote: 32 +/- ENTER 5 +/- XEQ XROOT -4.99999999999E-1 My 35s here returns exactly -0,5.
Quote: That's why I was so eager about CUBERT and XROOT in the 34s. ;-)
But it gets even worse. Try the 6th root of 531441. In cases like these the error may be as large at 5 ULP.
Quote: It's a pity the latter is not available from the keyboard. IMHO, it should be a better companion to y^x than LOGx, which of course should stay. Where to find room for both on the keyboard?
Quote: But I am aware time for keyboard layout suggestions is over...
Quote:Walter always says that the 34s is a moving target. So nothing is fixed. ;-)
The keyboard layout is a little less fluid than everything else.
Quote: I think ./, is superfluous as RDX. and RDX, are available under MODE. But I am aware time for keyboard layout suggestions is over...
Quote: And the label ./, was caused by the fact that at design time our North American friends kept asking for an easy way to change the radix mark whenever they got a comma calculator ;-)
On the other hand, this also allows for an easy way to accidentally change the radix mark. I missed that discussion, however.
Quote: On the other hand, this also allows for an easy way to accidentally change the radix mark. I missed that discussion, however.
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The Short Rook
The Short Rook is a piece that moves exactly like the FIDE Rook, except not as far.
Dynamics of the Short Rook
In a favorable position, a short Rook can do everything a normal Rook can do. The shortness of its move makes it more difficult to achieve such a favorable position.
If the Short Rook starts its life on b1, c1, f1, or g1, it gets in the way of the other pieces and impedes its whole army.
If the short Rook starts the game on a1 or h1, it is slow getting into the game; if it is an R5 or better, it is good that it is slow to develop because this helps to preserve it for the endgame.
The R4 (or shorter), however, is a minor piece, like a Bishop or a Knight, and it should get into the game as soon as possible in order to fight with the enemy's minor pieces. This tension between
the strategic role of a minor piece and the slow development suitable for a major piece helps to make the R4 (or shorter) an interesting piece to have.
Value of the Short Rook
Of course, a Short Rook is never quite as strong as a full Rook, and of course its value depends on how far it can move.
The Value of R1
R1, a Rook able to move just one square, is also known as the Wazir. As a piece by itself, it is too weak to be interesting, but in combination with other pieces it has roughly half the value of a
Knight or Bishop. (This is interesting because it suggests that one-third of a Rook's value comes from its ability to move to the square next to it.)
The Value of R2
R2, a short Rook able to move just two squares, is clearly worth less than a Knight. On an open board, it can barely get from e4 to e5 to e6 in one move, and if e5 is occupied it can't even get to
e6. (Of course, it can always get from e4 to e5; R2 includes the powers of R1.)
Arithmetical Calculation of Short-Rook Values
If we use the method described in A Better Way to Calculate Mobility, and assume that the probability of a square's being empty is 0.69, we find that the W (Wazir, R1 in this context) comes out to
have 0.335 times as much mobility as a full Rook (which of course is R7 in this context).
Similar calculations show the R2 has the mobility of 0.566 Rooks, whereas a Knight has a value of 0.666666 Rooks, or if you use the beginners' method of counting values it is worth 0.6 Rooks.
Comparing average mobility with piece values is dangerous; sometimes the mobility is a very good guide, and sometimes it is misleading. In this case it seems to be pretty good, so here is a list of
all the values:
RATIO OF PIECE TO
PIECE R7 R1 R(N-1) WIN
===== ===== ===== ===== ====
R1 0.335 1.000 ***** ....
R2 0.566 1.690 1.690 ....
R3 0.726 2.166 1.282 0.22
R4 0.836 2.495 1.152 0.33
R5 0.911 2.721 1.091 0.38
R6 0.964 2.878 1.057 0.45
R7 1.000 2.986 1.038 0.5
The R(N-1) column shows how much more mobility each piece has than the one before it; so for example R3 has 1.282 times the mobility of R2, and so on.
The WIN column shows the winning percentage when a weak computer program plays a few thousand games with White having short Rooks, Black having full Rooks, and of course the program believes thay are
of equal value. These numbers are very precise and repeatable, but I don't know exactly what they mean, or even if they do mean anything.
The calculated values correspond somewhat to the observed values of the short-Rooks R3, R4, and R5, but the calculations tend to greatly overestimate the values of the shorter pieces (R1, R2, and R3;
and, to a lesser extent, R4).
The Value of R3
Experience disagrees with calculation. Calculation shows that the R3 should be a bit stronger than a Bishop, but experience indicates that it may be a bit weaker; in both cases, it is fairly close.
The Value of R4
The R4 is worth more than a Bishop, but not so much that you should avoid trading them if there is anything at all to be gained by doing so; in other words, even the slightest positional
consideration is enough to make it worth while trading R4 for Bishop.
It is better to think of the R4 as a strong Bishop than as a weak Rook. In other words, you should treat the R4 as a "minor piece", in the same class as the N or B.
The Value of R5
The R5 can be treated as a weak Rook, and in fact it can very often be used to oppose the enemy Rooks and force a favorable trade.
It is interesting that the practical difference between R4 and R5 is so great, even though the mobility calculations show the R5 to be only "worth" 1.1 times as much. The main reason seems to be that
if you lift the R5 a mere two squares, it can reach all the way to the other side of the board.
For example, moving an R5 from e1 to e3 allows it to attack an enemy Rook on e8, but an R4 on e1 would have to advance at least to e4; it is much easier to have e3 defended by friends and not
attacked by enemy Pawns or by enemy minor pieces than it is to have e4 similarly available.
Another way of looking at this is that when a R4 is on one of the four center squares it can make all the moves that a Rook could make; a R5 can make all the R moves from 16 squares, and a R6 can do
it from 36.
The Value of R6
Most of the time, R6 is worth a Rook.
In a position with a White K on d2, a White R on e4, a Black K on d7, a Black Rook e8, and Black Pawns a7 and h7, White to play can draw simply by means of 1. Re4-a4 Re8-a8 2. Ra4-h4 and so on.
If White had a R6 on e4, the game would quite likely be lost.
The Value of R7
On the normal board of 64 squares, R7 is a Rook, plain and simple. On a bigger board, of course, all the values of all the pieces would be different.
Other Links In these Pages
This is a Mailme.
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A Self-Adaptive Multiple Exposure Image Fusion Method for Highly Reflective Surface Measurements
School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
Percipio Technology Limited, Shanghai 201203, China
School of Mechanical Engineering, Jinan University, Jinan 250022, China
School of Artificial Intelligence, OPtics and ElectroNics (iOPEN), Northwestern Polytechnical University, Xi’an 710072, China
Author to whom correspondence should be addressed.
Submission received: 22 September 2022 / Revised: 20 October 2022 / Accepted: 29 October 2022 / Published: 31 October 2022
Fringe projection profilometry (FPP) has been extensively applied in various fields for its superior fast speed, high accuracy and high data density. However, measuring objects with highly reflective
surfaces or high dynamic range surfaces remains challenging when using FPP. A number of multiple exposure image fusion methods have been proposed and successfully improved measurement performance for
these kinds of objects. Normally, these methods have a relatively fixed sequence of exposure settings determined by practical experiences or trial and error experiments, which may decrease the
efficiency of the entire measurement process and may have less robustness with regard to various environmental lighting conditions and object reflective properties. In this paper, a novel
self-adaptive multiple exposure image fusion method is proposed with two areas of improvement relating to adaptively optimizing the initial exposure and the exposure sequence. First, by introducing
the theory of information entropy, combined with an analysis of the characterization of fringe image entropy, an adaptive initial exposure searching method is proposed. Then, an exposure sequence
generation method based on dichotomy is further described. On the basis of these two improvements, a novel self-adaptive multiple exposure image fusion method for FPP as well as its detailed
procedures are provided. Experimental results validate the performance of the proposed self-adaptivity multiple exposure image fusion method via the measurement of objects with differences in surface
reflectivity under different ambient lighting conditions.
1. Introduction
Fringe projection profilometry (FPP) has been widely applied in three-dimensional shape measurement in many fields such as manufacturing [
], medicine [
], law enforcement [
] and entertainment [
], due to advantages such as its noncontact operation, full-field measurement, high accuracy and high efficiency. However, the measurement of highly reflective surfaces or high dynamic range surfaces
using FPP has always been a problem. Capturing surface details in dark areas requires higher camera exposure values, whereas recovering ground truth in bright regions needs lower exposure values. For
conventional cameras, this may lead to low contrast fringe regions or saturated regions existing in the captured images. These regions have a low signal–noise ratio, which may further result in
missing measurement data or decreasing accuracy. Therefore, finding effective measurement methods for highly reflective or high dynamic surfaces has always been an important research focus in the
context of FPP.
Various methods have been proposed to solve this problem. One category of the method, multiple exposure fusion methods, improves measurement performance by capturing fringe images with different
camera exposures. Zhang and Yau put forward a high dynamic range scanning method which takes advantage of the merits of pixel-by-pixel phase retrieval for the phase-shifting algorithm [
]. This technique captures a sequence of fringe images with different exposure times. The brightest but not saturated pixels are selected to construct the fused fringe images, which are used to
compute the final phase map. Ekstrand and Zhang proposed a method that could evaluate the effect of exposure according to feedback from the reflectivity of the measured object [
]. Defocusing binary projection is introduced to enlarge the camera exposure time selection range. Long et al. presented a method that uses the magnitude of a non-principal frequency component to
identify saturated pixels [
]. The magnitude function of the frequency component is presented by Fourier analysis. The magnitude of the non-principal frequency component is deduced and utilized as the saturation criteria.
Another category of the method involves measuring the shiny surfaces by adjusting the projected fringes accordingly. Waddington and Kofman presented a camera-independent method of avoiding image
saturation according to modifying projected fringes after complex calibration procedures [
]. Babaie et al. captured fringe images by recursively controlling the intensity of the projection pattern pixel-wisely [
]. The reflected images captured by the camera are used as references. Lin et al. proposed an adaptive method by adjusting the pixel-wise intensity of the projected fringe patterns based on the
saturated pixels in the captured images [
The third category of method is to extend the dynamic range by using hardware and can be categorized into two types: chip level extension and affiliated hardware extension. In terms of chip level
extension, Orly [
], Cheng [
] and Bub [
] proposed different methods to extend the dynamic response range of the pixels. When it comes to the affiliated hardware method, digital micromirror device (DMD) chips [
] and spatial light modulators [
] are usually implemented to modulate the intensity and direction of spatial light.
Among all these categories, the multiple exposure fusion methods have become one of the most important research directions thanks to their superior flexibility, high accuracy and hardware-independent
characteristics. There are two key factors in the sequence of exposure settings for multiple exposure fusion methods: the initial exposure and the exposure sequence. The initial exposure is
considered as the highest value in the exposure sequence, and it is critical for the quality of the fusion images and the overall processing efficiency. A higher initial exposure means more
unnecessary series of fringe images have to be captured, which contributes nothing to the quality of the fusion images but makes the overall process longer. A lower initial exposure may lead to poor
quality regarding the obtained fusion images. For the exposure sequence, a large sequence with small exposure steps improves the quality of the fusion images, but it requires a longer operating time.
A short sequence with large exposure steps may result in lower quality fusion images. There are no such criteria to balance the fusion image quality and the processing efficiency. Thus, the initial
exposure and the exposure sequence are very important with regard to the performance of the multiple exposure fusion method.
Current multiple exposure fusion methods always have a relatively fixed sequence of exposure settings which are normally determined by practical experience or multiple trial and error experiments.
However, when measuring new objects with differences in surface reflectivity or under different ambient lighting conditions, the initial exposure and the exposure sequence must be adjusted
accordingly to ensure multiple exposure fusion performance. In such cases, using trial and error experiments to adjust the initial exposure and the exposure sequence is time-consuming, and the
initial exposure and the exposure sequence cannot be precisely optimized based on manual adjustments based on practical experiences. This means that there is a critical demand for self-adaptive, fast
and automatic methods to optimize the sequence of exposure settings according to the measured object’s surfaces adaptively, thereby improving the overall processing efficiency as well as the
measurement quality.
In order to solve this problem, a novel self-adaptive multiple exposure image fusion method is proposed in this paper. By introducing the information entropy theory into the analysis of the fringe
images, which builds a theoretical foundation for the selection of the initial exposure, a self-adaptive initial exposure searching method is first presented. Additionally, an exposure sequence
generation method based on dichotomy is proposed. On the basis of the aforementioned two methods, a self-adaptive multiple exposure image fusion method for FPP as well as its detailed procedures are
presented. The remainder of this paper is organized as follows.
Section 2
presents the principle of the proposed method. Specifically,
Section 2.1
presents the initial exposure searching method based on information entropy,
Section 2.2
describes the exposure sequence generation method based on dichotomy and
Section 2.3
proposes the self-adaptive multiple exposure image fusion method for highly reflective surface measurements.
Section 3
presents the experiments and an analysis of the results.
Section 4
concludes this paper.
2. The Proposed Method
2.1. Initial Exposure Searching Based on Information Entropy
In the case of multiple exposure image fusion methods, the selection of the initial exposure value is one of the key aspects that determine the measurement performance. A high initial exposure may
increase pixel grayscale values in low-reflectivity areas, but it will also increase the number of overexposed pixels. Furthermore, multiple exposure fusion methods generally adopt a high to low
approach to vary the exposure values and analyze the fringe images. With a too high initial exposure, more exposure changes are required to obtain a fused image, reducing the overall measurement
efficiency. And if the initial exposure is set too low, the quality of the final fused image will be affected, leading to missing data and decreasing accuracy.
The present methods usually select an appropriate initial exposure through multiple experiments or practical experience. However, this kind of method is not very flexible if the illumination
environment changes or objects with differences in surface reflectivity are measured because the initial exposure has to be reset correspondingly. This reduces the adaptability of the current
multiple exposure methods. Therefore, there is an urgent need to develop a self-adaptive initial exposure searching method to enhance the adaptive capability of current multiple exposure image fusion
2.1.1. Selecting Criteria for Initial Exposure
An appropriate initial exposure value should not cause too many saturated or low gray value pixels in the captured images. As shown in
Figure 1
, in order to observe the fringe image intensity distribution in different exposure conditions, three simulated images are obtained by adjusting the modulated gray value in low, normal and high
exposure conditions. The upper row shows the simulated images, and the down row shows the intensity variation scatter diagram in one pixel row of the images. The scatters diagram in
Figure 1
a shows that most of the pixels in the image are concentrated in a narrow gray value interval in the case of low exposure. There is little difference in intensity between pixels, which makes the
simulated fringe image blurred.
Figure 1
b shows that the gray value of the fringe image pixels for normal exposure conditions are distributed in the entire interval between 0 and 255, and the stripes can be seen very clearly in the
corresponding image. In
Figure 1
c, it is illustrated that the intensity of most pixels is near 255 since the exposure value is too high. Therefore, the fringe is no longer sinusoidal. An appropriate initial exposure value should
maximize the variability between pixels in the acquired images and make the range of the pixels’ gray value as wide as possible. The opposite means that there are too many overflow pixels or too many
pixels with low gray values in the obtained initial exposure image. Whereas the former will lead to more adjustment times in terms of the exposure value and a longer measurement time, the latter will
easily lead to a reduction in quality in the fusion images, both of which should be avoided.
Based on the above analysis, the selection criterion for the initial exposure value for the multiple exposure image fusion method can be determined. That is, the selected initial exposure value
should ensure maximum variability in terms of the pixel grayscale values in the acquired sequence of images. The larger the variability in terms of the pixel gray values in the image, the more
obvious the sinusoidal fringe feature is. In FPP, fringe stripes are the carrier for encoding and decoding phase information, which itself is the information required for 3D reconstruction. In other
words, maximizing the variance in pixel gray values in an image means maximizing the amount of information in the image. Thus, the criterion for selecting the initial exposure value in the multiple
exposure image fusion method can be further clarified, as the initial exposure should be chosen in such a way as to maximize the amount of information in the acquired set of initial exposure value
2.1.2. Information Entropy Metric for Initial Exposure Selecting Criteria
As aforementioned, the criterion for selecting the initial exposure value in the multiple exposure image fusion method should be such that the amount of information in the corresponding image
sequence at this exposure is maximized. This implies the need to determine a quantitative index for measuring the amount of information in the images.
In physics, the concept of entropy is used to represent the uniformity of energy distribution in space. The more uniformly the energy is distributed, the higher the entropy of the system. Shannon,
the founder of information theory, introduced this concept into the field of informatics and proposed the use of information entropy to measure the uncertainty of an information source [
]. Suppose that the probability of occurrence of each event in a probabilistic event set is
$p 1$
$p 2$
, …
$p n$
, respectively, and suppose that there exists such a metric,
, that can describe the degree of uncertainty of the information, which can be given as follows [
], where n is the number of the event in the probabilistic event set.
$H = ∑ i = 1 n p i log 2 ( 1 p i )$
This formula is used to express the metric of information uncertainty in information theory, which is termed information entropy. The magnitude of information entropy can be expressed to express the
amount of information in the information source. Specifically, in the field of image processing, the pixel intensity is a probabilistic event in Shannon’s definition of information entropy. The
intensity of pixels in an image at different image coordinates is denoted by
$X i$
, where
$i = 1 , 2 , … , k$
is the number of pixel grayscale levels. Let
be the image information entropy, which can be expressed as follows.
$E = ∑ i = 1 k p i log 2 ( 1 p i ) = p 1 log 2 ( 1 p 1 ) + p 2 log 2 ( 1 p 2 ) + ⋯ p k log 2 ( 1 p k )$
$p i$
is the grayscale probability. For an 8-bit depth image, the maximal grayscale is 256 and the maximum image entropy equals 8 but only when the gray levels are evenly distributed in the image, which is
not a common case in real applications. For any given scene, the image information entropy will vary with variations in exposure. The amount of information will reach the maximum when the image
entropy becomes maximal.
Based on the above analysis, image information entropy can be used as the metric of the amount of information in an image. The greater the image entropy, the more information is contained in the
image. Thus, the selection criteria for the initial exposure in the multiple exposure image fusion method can be expressed as that which determines a suitable exposure such that the image information
entropy in the acquired image sequence at this exposure value reaches its maximum.
2.1.3. Variation Characteristics for Fringe Image Information Entropy
As one of the most common encoding and decoding methods for FPP, three-frequency three-step phase shifting is adopted in this study. When determining the initial exposure value, in order to improve
the algorithm efficiency, the fringe image at a certain frequency should be used as the reference image for the information entropy calculation. This requires the study of the information entropy
characteristics of the fringe images at different frequencies, the fringe images with different phase shift steps at the same frequency and the three-frequency three-step phase shift fringe images
captured when the exposure value changes.
Information entropy for multi-frequency fringe images
In the multi-frequency phase-shifting method, the frequency of fringe images is reflected in the stripe pitch used to produce the fringe image. To study the variation in fringe image information
entropy for different frequencies, a range of fringe images with varied stripe pitches are artificially produced. According to the Nyquist sampling theorem and further taking into account the effect
of noise [
], the minimum stipe pitch is set to 10. The maximum stripe pitch, meanwhile, is set to 1280, since this is the pixel column size for common DMD chips. A total of 1271 fringe images, with the strip
ranging from 10 to 1280, are used for further analysis. The image entropy of each fringe image was calculated, and the results are shown in
Figure 2
As can be seen from the figure, the information entropy of the striped image rises and fluctuates with the increase in the stripe pitch, showing an increasing trend. However, the curve plot has many
fluctuation intervals, in which the image information entropy fluctuates to some degree. As the fringe pitch increases, the period of the fluctuation interval also increases gradually. In terms of
the variation rate, it can be seen from the figure that in the interval $λ ∈ [ 10 , 100 ]$, the fringe image entropy increases rapidly with the increase in fringe pitch. When it comes to the interval
$λ ∈ [ 600 , 1280 ]$, the increasing speed of image information entropy reduces gradually. After that, the image entropy stabilizes between 7 and 8 and fluctuates in a small range.
Information entropy for three-step phase-shift fringe images
To study the information entropy variation for the phase shifting fringe images concerning the change in fringe pitch, three groups of fringe images with three-step phase shifts (
$δ 1 = 0$
$δ 2 = 2 π / 3$
$δ 3 = 4 π / 3$
) are produced. For each group of fringe images, its incremental step is set to 1, and a total of 3813 fringe images are produced. The information entropy for each phase shift concerning the
variation of fringe pitch is presented in
Figure 3
As shown in
Figure 3
, the information entropy of
$δ 1 = 2 π / 3$
$δ 2 = 4 π / 3$
is generally similar to that of
$δ 0 = 0$
, which shows the changing characteristics of the composite functions of logarithm and sine functions. There is also a certain degree of fluctuation, with its period becoming larger as the pitch
increases. In terms of the fluctuation in amplitude, in the range of
$λ ∈ [ 0 , 600 )$
, the curve of
$δ 0 = 0$
results in the smallest amplitude. The fluctuation of the other two curves is larger, and the fluctuation average is above that of
$δ = 0$
. This means that the phase-shifting images are more sensitive to fringe pitch changes. In most cases, the entropy of the fringe image with phase shift is larger than that of the fringe image without
phase shift. For the range of
$λ ∈ [ 600 , 1280 )$
, the fringe image entropy changes alternately among the three phase shifts.
Information entropy for phase-shifting fringe images at different exposures
To further clarify the variation characteristics regarding the information entropy of three-frequency three-step phase-shifting fringe images when the exposure changes, simulation images and real
images are used to study the variation characteristics, respectively. Simulated images are used to study the variation characteristics without considering the surface optical properties of the
measured object. Then, real images are utilized to verify the applicability of the conclusions drawn from this ideal case.
The encoding scheme of the sinusoidal fringes can be expressed as
$I = I ′ + I ″ cos ( φ ( x ) + δ )$
is the fringe image intensity,
$I ′$
is the average intensity,
$I ″$
is the intensity modulation,
is the principle phase,
is the pixel position and
is the phase shift. The variation in intensity of the fringe images with the change in exposure can be simulated by modifying the intensity modulation. In the multiple exposure image fusion method,
the exposure values are usually adjusted in the order from the highest to the lowest values to obtain high-quality fused fringe images. Therefore, the light intensity modulation values are also
adjusted in the order from high to low. In the simulation, the fringe images are obtained at 500 light intensity modulation values, and the information entropy of a total number of 4500 fringe images
at different frequencies and phase shifts are calculated. The results are shown in
Figure 4
As can be seen from the figure, the information entropy of three-frequency three-step fringe images tends to increase and then decrease as the modulation value decreases. Specifically, the
information entropy of the fringe images first increases in a step-like manner. After reaching the maximum, it stabilizes in a small range of fluctuation as the modulation value decreases at the
beginning and descends rapidly when the modulation value is relatively small. In terms of the speed of reaching the maximum, the larger the fringe pitch is, the later the maximum is reached and the
smaller the corresponding modulation. For the same fringe pitch, the image entropy of the two steps with phase shifts reaches the maximum later than that of the step without phase shifts.
To verify the applicability of the above conclusions in a real scene, a white plane was measured using the multiple exposure image fusion method. The exposure was adjusted from a high value to a low
value, and three-frequency three-step phase-shifting fringe images at 207 different exposure were captured. The information entropy of the captured fringe images was calculated and the results are
shown in
Figure 5
. As can be seen, with the decrease in the exposure, the fringe image entropy also shows a trend of first increasing and then decreasing. And the larger the fringe pitch, the slower the stripe image
entropy reaches its maximum, which is the same as in the results obtained using the simulated fringes.
To enhance the measurement speed of the multiple exposure image fusion method for FPP, one phase-shifting image under a certain frequency should be selected and analyzed as the reference fringe image
for the initial exposure calculation. When the information entropy of this reference fringe image reaches the maximum, it should be ensured that the image entropy of all other fringe images is at or
adjacent to the maximum. Additionally, in order to obtain the highest possible quality fused fringe images, an as large as possible exposure value is preferred. Based on the above criteria, combined
with the analysis results for the variation characteristics of the fringe images’ entropy, the third phase-shifting fringe image with the largest fringe pitch is selected as the reference image for
initial exposure calculation.
2.1.4. The Initial Exposure Searching Algorithm Based on Information Entropy
Based on the above discussion, the variation characteristics of the fringe image information entropy concerning the change in pitch and exposure are clarified and the reference image used to search
the initial exposure is determined. Thus, an adaptive initial exposure searching algorithm based on information entropy for multiple exposure image fusion is proposed in this paper, as shown in
Figure 6
. The flow of the algorithm as well as its procedures can be detailly presented as follows.
Manually or automatically set an initial value for exposure searching according to the adjustment range of the camera. This initial value is denoted as $E x p o _ s e a r c h _ i n i t i a l$.
Acquire the reference fringe image of the third phase shift with the largest fringe pitch at the initial exposure. Calculate the image information entropy $E n _ i n i t i a l$.
Search along the direction the exposure reduces. Taking $S t e p$ as the exposure searching step, let $E x p o = E x p o _ s e a r c h _ i n i t i a l − S t e p$. Capture the images under this
exposure and calculate the entropy, $E n _ m i n o r _ 1$, for the reference fringe image.
Determine the searching direction according to the relationship between $E n _ m i n o r _ 1$ and $E n _ i n i t i a l$. If $E n _ m i n u s _ 1 < E n _ i n i t i a l$, according to the variation
characteristics of the fringe image entropy with the change in exposure, it will never reach the maximum in this direction. Therefore, searching in this direction stops and the procedure goes to
8. If $E n _ m i n u s _ 1 ≥ E n _ i n i t i a l$, this means that the entropy may reach the maximum in this direction. Continue to search in this direction and proceed to 5).
Continue to search in the direction of exposure reduction. Reduce the exposure by an interval of $S t e p$ and calculate the fringe image entropy. Then, compare the two values acquired before and
after the adjustment of exposure. If $E n _ m i n u s _ n + 1 ≥ E n _ m i n u s _ n$, the exposure is further reduced by $S t e p$ and the loop continues. If the two values are equal, the image
entropy and corresponding exposure will be saved to the data set $E q u a l _ E n$ in a form of key–value pair. If $E n _ m i n u s _ n + 1 < E n _ m i n u s _ n$, proceed to 6.
If $E n _ m i n u s _ n > E q u a l _ E n$, proceed to 7. Otherwise, the stop searching in the current direction and switch to the opposite direction to that in which the exposure increases.
Taking $E x p o _ s e a r c h _ i n i t i a l$ as the initial exposure searching value, proceed to 8.
If $E n _ m i n u s _ n$ belongs to the data set $E q u a l _ E n$, set $E n _ m a x$ to $E q u a l _ m i n u s _ n$. Select the exposure value of the first entropy–exposure pair in the last
equivalent entropy–exposure pair of $E q u a l _ E n$ as the desired exposure for the maximum entropy. The procedure then ends here. Otherwise, set $E n _ m a x$ equal to $E q u a l _ m i n u s _
n$ and output the corresponding exposure value. Then the procedure ends.
Search in the direction the exposure value increases. Let $E x p o = E x p o _ s e a r c h _ i n i t i a l + S t e p$, capture corresponding fringe images and calculate the image entropy. If $E n
_ p l u s _ 1 ≥ E n _ i n i t i a l$, then continue to search in this direction and proceed to 9. Otherwise, let $E n _ m a x = E n _ i n i t i a l$ and output the corresponding exposure, $E x p
o _ s e a r c h _ i n i t i a l$. This means that the initial exposure set at the beginning of the searching algorithm makes the image entropy reach the maximum. The procedure then ends here.
Continue the searching loop in the direction of rising exposure. Increase the exposure and calculate the image entropy. If $E n _ p l u s _ n + 1 ≥ E n _ p l u s _ n$, increase the exposure and
the loop continues. Otherwise, set $E n _ m a x$ to be equal to $E n _ p l u s _ n$. The corresponding exposure, $E x p o [ n ]$, will be outputted and the procedure ends here.
2.2. Exposure Sequence Generation Based on Dichotomy
In the case of multiple exposure fusion methods for highly reflective surface measurements, the selection of a proper exposure sequence is another key factor that affects the measurement effect. In
theory, the higher number of exposure value adjustments and the wider the exposure coverage range, the higher the quality of the fusion fringe image. However, from the perspective of improving the
measurement efficiency, it is not possible to increase the number of exposure value adjustments indefinitely. When the exposure adjustment step is too small, the change in the number of saturated
pixels will become insignificant. A balance between measurement efficiency and fusion image quality is generally achieved through experiments, which seriously reduces the algorithm efficiency. To
improve the automation of the multiple exposure method, an exposure sequence generation method based on dichotomy is proposed in this study.
In order to study the variation rule for the saturated pixel number concerning the exposure values for highly reflective metal surfaces, a reflective metal surface was selected for measurements at a
range of exposure values. The exposure value was adjusted from high to low in steps of 10 to obtain a raster stripe image at 1481 exposures, and the saturated pixels in the images were counted. As a
comparison, the same exposure values parameters were used to measure a white, uniformly scattering plane, and the number of saturated pixels was counted. The results are shown in
Figure 7
Compared to the saturated pixels number variation curve of the metallic highly reflective surface, the curve of the uniform scattering plane has an obvious slope. The reason for this difference is
that the uniformly scattering surface has a similar reflectance at all points, and, as the exposure value decreases, the grayscale values of the image pixels also tend to change uniformly. The number
of saturated pixels declines rapidly when the exposure value decreases to a certain level. When measuring the metallic highly reflective surface, the existence of a specular reflection lobe and
specular reflection crest in the direction of the camera means that there always exists saturated pixels in the images. This leads to a slight variation in the saturated pixel number curve, which
suits most of machined metallic surfaces. This makes the procedure suitable for constructing the exposure sequence based on the dichotomy method.
Let the upper limit and lower limit of the exposure be denoted as $E x p o _ b e g i n$ and $E x p o _ e n d$, respectively, and suppose that the exposure value in the exposure sequence is more than
three. Then, the exposure sequence generation procedure based on dichotomy is proposed as follows.
Set the desired total number of exposure N and determine $E x p o _ b e g i n$ and $E x p o _ e n d$;
Build the initial exposure sequence ${ E x p o _ b e g i n , E x p o _ e n d }$. The exposure number in the current exposure sequence is denoted as $N u m _ c o u n t$;
Update the exposure sequence. Compute the average of the upper and lower limit of exposure and insert it into the sequence. In this way, the updated exposure sequence S1 can be generated.
$S 1 = { E x p o _ b e g i n , E x p o _ 1 _ 1 , E x p o _ e n d }$
$E x p o _ 1 _ 1 = f l o o r [ ( E x p o _ b e g i n + E x p o _ e n d ) / 2 ]$
Continue to update the exposure sequence.
$E x p o _ 2 _ 1 = f l o o r [ ( E x p o _ b e g i n + E x p o _ 1 _ 1 ) / 2 ]$
$E x p o _ 2 _ 2 = f l o o r [ ( E x p o _ 1 _ 1 + E x p o _ e n d ) / 2 ]$
Then, the updated exposure sequence can be expressed as
$S 2 = { E x p o _ b e g i n , E x p o _ 2 _ 1 , E x p o _ 1 _ 1 , E x p o _ 2 _ 2 , E x p o _ e n d }$
If $N u m _ c o u n t > N$, then the procedure ends and outputs the exposure sequence. Otherwise, continue to execute the dichotomy method until the aforementioned inequation works. The final
generated exposure sequence can be denoted as $S _ f i n a l$.
2.3. Self-Adaptive Multiple Exposure Image Fusion Algorithm for FPP
Based on the previous initial exposure searching procedures based on information entropy and the exposure sequence generation procedures based on dichotomy, a self-adaptive multiple exposure image
fusion algorithm for FPP is finally proposed in this section, which is described in detail as follows.
Determine the beginning and ending exposure values. Determine the initial exposure,
$E x p o _ i n i t i a l$
, using the method proposed in
Section 2.1
. Then, the beginning exposure can be given by:
$E x p o _ b e g i n = k _ i n i t i a l ∗ E x p o _ i n i t i a l$
$k _ i n i t i a l$
is a weighting factor, which is normally greater than one. By using this item, enhance the intensity level of the pixels which are outside the saturated area of the image captured under the beginning
The procedure to determine the ending exposure,
$E x p o _ e n d$
, involves analyzing the number of saturated pixels, as shown in
Figure 8
. Let the current exposure be denoted by
$E x p o$
. Commence the searching procedure by capturing the images at the lower exposure limit of the camera,
$E x p o _ c a m _ l o w e r l i m i t$
, and calculating the number of saturated pixels,
$N _ s a t u r a t e d$
. If
$N _ s a t u r a t e d$
is greater than zero, and if
$E x p o = E x p o _ c a m _ l o w e r l i m i t$
, then set the ending exposure
$E x p o _ e n d$
to be
$E x p o$
. Otherwise,
$E x p o _ e n d = E x p o − M$
is the exposure adjustment step. If the number of saturated pixels is still equal to zero, then
$E x p o = E x p o + M$
. Then, repeat the aforementioned operations will be until the ending exposure is determined. The procedure then ends.
Thus, a complete set of the self-adaptive multiple exposure fusion methods for FPP for measuring highly reflective metal surfaces has been established. The self-adaptivity of this method is assured
by the initial exposure searching algorithm based on image information entropy and the exposure sequence generation algorithm based on dichotomy. This makes the method more flexible concerning the
measurement of surfaces with differences in reflectivity and usable in different lighting environments.
The developed algorithm needs to work in conjunction with FPP systems. It needs the fringe images captured by FPP systems as the inputs for analysis and generates the exposure values as the output to
instruct FPP systems to capture the images. This process is repeated during the initial exposure searching and exposure sequence generation. The algorithm can be developed using any programming
language depending on how easily it can be integrated into a specific FPP system and make the entire process automatic.
The total cost time of the entire process to determine the optimized initial exposure and exposure sequence mainly depends on the FPP hardware system because the process includes interacting with the
FPP hardware to project patterns and capture images. The computational time in terms of analyzing the fringe images is very little. For our system, the entire process is very fast and can output the
results within seconds. Once the optimized initial exposure and exposure sequence are obtained, the FPP system can start its measurement according to the exposure values.
3. Experiments and Discussions
To verify the performance of the proposed methods, two experiments were designed and implemented. The first experiment was designed for the verification of the initial exposure searching algorithm
and the second was for the verification of the proposed self-adaptive multiple exposure image fusion method for FPP by evaluating the measurement performance. The FPP system had two Basler aca1300-30
gm cameras with resolutions of 1280 × 960.
3.1. Adaptivity Verification of the Proposed Initial Exposure Searching Algorithm
To verify the adaptivity and accuracy of the proposed initial exposure searching algorithm based on information entropy, six metal workpieces with different sizes, shapes and surface reflectivity
were used as the experimental objects, as shown in
Figure 10
. The surfaces of workpieces no.1–no.4 are the original machined surfaces, workpiece no.5 is a cylinder after oxidation treatment and workpiece no.6 is an automobile body structural part treated by
the oxidation process. Workpieces no.1 to no.4 were utilized to verify algorithm performance in different lighting conditions. The suitability of the algorithm for surfaces with differences in
reflectivity was verified by using workpieces no.1 to no.6. Taking these workpieces as the test objects, the algorithm’s adaptivity in different lighting conditions and with differences in surface
reflectivity was verified as follows.
3.1.1. Algorithm Adaptivity Verification in Different Ambient Lighting Conditions
To verify the algorithm’s adaptivity in different ambient lighting conditions, workpieces no.1 to no.4 were selected as the experimental objects. The maximum information entropy of the fringe images
was searched in ambient lighting and normal ambient lighting, respectively. The ambient light scene is shown in
Figure 11
The fringe image entropy of each workpiece was obtained at a total of 229 exposures to verify the accuracy of the maximum entropy searching algorithm. This exposure sequence decreased from a maximum
of 15,000 to a minimum of 20, with a 100-step decrease when the exposure was greater than 1000 and a 10-step decrease when the exposure was less than 1000. Additionally, to observe the relationship
between the maximum in terms of the point cloud number and the initial exposure value obtained by the proposed algorithm, the point numbers of the point clouds obtained at these 229 exposures were
also counted. The searching algorithm was executed with 5000 as the initial exposure value and 50 as the searching step. The results are shown in
Figure 12
. The reason why a total of 229 exposures was used is because a complete distribution of image information entropy with respect to the exposure value needed to be obtained. It is considered as a
standard to evaluate the accuracy of results concerning the maximum entropy searching algorithm, and thus to further evaluate the proposed initial exposure searching algorithm. Regarding actual
measurements, these do not require as many exposures to optimize the initial exposure. Normally, twenty exposures are sufficient to accurately calculate the initial exposures in experiments.
As can be seen from the figure, the maximum value in terms of the fringe image entropy was accurately obtained for each of the workpieces and the corresponding initial exposure value is provided. For
workpieces no.1 and no.3, the fringe image entropy variation curves under normal ambient lighting are very close to that under low ambient lighting conditions, and thus the initial exposure values
obtained by the searching algorithm in different lighting conditions are equal. The image entropy change curves of workpiece no. 2 show a certain degree of difference between the different lighting
conditions, and, thus, the final obtained initial exposure values are also different. And as for workpiece 4, its fringe image entropy change curve obtained in normal ambient lighting is different
from that in low ambient illumination. Compared with the image entropy curve in low ambient illumination conditions, the curves in normal ambient illumination arrive at their peak later. The curves
in normal ambient illumination descend faster than those in low ambient illumination.
As for the variation characteristics of the point cloud number, it also rises at first and decreases after arriving at the peak. The curve of the point cloud number arrives at the peaks later than
the curve of the image entropy. It means that after the fringe image entropy reaches the maximum value, the point cloud number will continue to maintain the trend of increasing and reach the maximum
after the exposure value drops to a certain value. This illustrates that the three-frequency three-step phase shift method used in this study is adaptable to lighting conditions and can guarantee
decoding quality in a weaker lighting condition. Furthermore, it means that the exposure value corresponding to the maximum point number of the point cloud, denoted by $E x p o _ m a x _ p o i n t n
u m$, would be less than the initial exposure obtained from the proposed algorithm. This ensures that the exposure, $E x p o _ m a x _ p o i n t n u m$, is always within the upper limit and lower
limit of the exposure sequence, which helps the fusion algorithm make full use of the fringe images captured at this exposure value. Additionally, it ensures that more pixels with high intensity
could be selected for the fused image, which further improves the quality of the full-field phase map and point cloud.
3.1.2. Algorithm Verification for Surfaces with Differences in Reflectivity
In 3.1.1, the performance of the proposed algorithm was verified in different ambient lighting conditions using four workpieces with the original machining surfaces, which were highly reflective. To
verify the algorithm’s adaptivity concerning surfaces with differences in reflectivity, experiments were performed with workpieces no. 5 and no. 6, those with oxidized surfaces, under normal lighting
conditions. The relevant fringe image entropy curves were produced with the same parameters as those used in 3.1.1. The results are shown in
Figure 13
As can be seen from the figure, the initial exposure value search algorithm found the maximum entropy of the streak image and obtained the corresponding exposure value for both workpieces 5 and 6
with surface oxidation. This, combined with the previous results for the machined original surfaces of workpieces 1–4, demonstrates that the proposed algorithm can successfully find the maximum
entropy of the fringe images for measured objects with differences in surface reflectance and can obtain the corresponding exposure value as the initial exposure value. Thus, the proposed initial
exposure searching algorithm based on fringe image entropy can adapt to both different ambient lighting conditions and differences in surface reflectivity, which ensures the adaptiveness, robustness
and accuracy of the obtained results for the initial exposure value.
3.2. Measurement Verification of the Adaptive Multiple Exposure Fringe Fusion Method
To verify the measurement performance of the proposed adaptive multiple exposure fringe fusion method for FPP, workpieces no.1–no.4 were measured under normal ambient lighting conditions. To
demonstrate the algorithm’s performance sufficiently, the original fringe images with maximal entropy obtained in 3.1 were selected to be compared with the fused fringe images. The saturation
threshold was set to 250, and the overexposure pixels in the original images are represented in red. The area which failed to secure rational complemented pixels in the fused image is marked by a
green color. The image entropy before and after the fusion was calculated, and the results are shown in
Figure 14
As can be seen from the figure, there are overexposure areas, presented in red, in the original fringe images for all four workpieces. This may greatly reduce the sinusoidal intensity distribution in
these areas, which leads to significant phase errors during image decoding. In contrast, almost all of these areas are well restored in the fused fringe images, except for a small part of workpiece
1, presented as the green area, which could not be restored because of the extremely high surface reflection. Additionally, there are a large number of areas where the fringe contrast is very low due
to the darkness of the image, as shown in the right portion of the images of the four workpieces. The fringe contrast in these areas was significantly improved in the fused images. To evaluate this
effect quantitatively, the image entropy, E, was calculated before and after image fusion. It can be seen from the results that the entropy of each fused image is significantly improved compared to
the original image, with the improvement being 11.5%, 11.7%, 12.2% and 11.9%, respectively, which means that the fused images can provide more fringe information for later phase decoding, thus
improving the quality of the obtained phase map.
Phase unwrapping was performed using these fringe images before and after fringe fusion. The cross sections of the unwrapped phase maps at the same positions were obtained for evaluation, as shown in
Figure 15
. The quality of the full-field phase maps obtained using fused stripe images was significantly improved compared to the unwrapped results obtained using the original fringe images. Taking workpiece
no.1 as an example, in the upper left region of the phase map unwrapped from the original fringe images, there exists a distinct visual fold area. In this area, there is a large region where there is
a failure in phase unwrapping because of the strong surface reflection. This folded and black region in the phase map corresponds to the red region of workpiece 1 in
Figure 14
. It can be seen that there is a large error in this area due to the presence of high reflectivity. On the contrary, the fold areas in the original fringe phase map are effectively eliminated in the
fused phase map. As can be seen, all the regions of workpiece 1 have good unwrapping results, except for the extremely highly reflective region marked in green in
Figure 14
. The quality of the phase maps were also greatly improved for workpieces 2 to 4.
To compare the phase unwrapping error of the original fringe images and the fused fringe images more intuitively, the cross-section data was plotted in
Figure 15
. The selected section for the phase map before and after the fusion algorithm is marked with the red dotted line and green dotted line, respectively. As can be seen from the phase variation curve,
there exists a distinct phase disturbance in the cross-section phase curve of the original phase map. This means that there is an obvious phase error in this area, which would lead to an error in 3D
reconstruction. Comparatively, the section phase curve of the fused phase map is smooth, and the quality in terms of phase unwrapping was greatly improved.
As shown in
Figure 16
, the point cloud was reconstructed based on the original fringe images and the fused fringe images. As can be seen, many points are missing in the reconstructed point cloud reconstructed from the
original fringe images. This stems from the phase error which occurred because of the existence of a highly reflective area, which resulted in incorrect matching and matching failure in the stereo
match step and further lead to missing point cloud data. In contrast, the completeness of the point cloud reconstructed from the fused fringe images shows a dramatic improvement. The point numbers of
the original point cloud and the fused point cloud are given in
Table 1
To analyze the 3D reconstruction results more accurately, the aforementioned point clouds were fitted to obtain the fitting surface as the ground truth. Then, the errors in terms of the point cloud
before and after the fusion algorithm was calculated and the color range maps were constructed, as shown in
Figure 17
. Simultaneously, to quantitatively evaluate the error, the mean error and root mean square error (RMSE) were calculated, as listed in
Table 2
Compared to the point clouds of the original fringe images, the mean errors and RMSEs of the fused point clouds are greatly reduced. For example, the mean errors of the four workpieces are roughly
0.05 mm to 0.06 mm. Whereas the mean errors of the fused point clouds are reduced to approximately 0.03 mm. The RMSE of the original point cloud is roughly 0.09 mm, whereas the RMSE of the fused
point cloud is roughly 0.03 mm. This means that the accuracy of the fused point cloud is also improved compared with the original point cloud. This is mainly because the fused fringe images are of,
better quality especially in the reflective areas, and, as a result, the point noise can be greatly suppressed.
4. Conclusions
In this paper, a self-adaptive multiple exposure image fusion method has been proposed to solve the challenges of measuring highly reflective surfaces or high dynamic range surfaces for FPP. An
adaptive initial exposure searching method has been proposed by introducing the theory of information entropy to the fringe image combined with an analysis of the characterization of fringe image
entropy. To generate the proper exposure sequence automatically, an exposure sequence generation algorithm has been proposed, which needs only a small number of parameters for presetting. Based on
these two algorithms, a novel self-adaptive multiple exposure image fusion method for FPP as well as its detailed procedures have been provided. To verify the performance of the proposed method,
experiments were designed by measuring metal workpieces of different shapes and with differences in surface reflectivity. Compared with current multiple exposure image fusion methods, the
experimental results verify the self-adaptivity, efficiency and robustness of the proposed method when measuring for differences in reflectivity and in different ambient lighting conditions.
When measuring different objects with differences in surface reflectivity or in different ambient light conditions, the initial exposure and the exposure sequence must be adjusted accordingly. Most
existing multiple exposure image fusion methods typically have a relatively fixed sequence of exposure settings that are determined by practical experience or trial and error experiments. Trial and
error experiments used to adjust the sequence of exposure settings are time-consuming, and the initial exposure and the exposure sequence cannot be precisely optimized based on practical experience.
Compared with the existing methods, the core novelty and main contribution of the method proposed in this manuscript is that it can optimize the initial exposure and the exposure sequence
self-adaptively. This advantage makes the proposed method superior to others in terms of self-adaptivity for variations in surface reflectivity and ambient lighting conditions. It also has the
ability to optimize the exposure settings quickly and automatically rather than relying on manually adjustments based on practical experience or performing trial and error experiments.
Although the proposed method works well, a number of issues require further study, for example, the effect of the geometry complexity on measurements has not been studied in detail. In future works,
the possibility of applying the proposed solution in structured light measurement systems projecting other kinds of patterns needs to be studied. The influence of the object’s geometric complexity
and the intensity of the light on the proposed method need to be studied. The criteria to use when deciding whether to modify the exposure setting or continue using the old one when the relative
angle and position between the measured object and the light source changes is another area worthy of attention. Deep learning could also be utilized in the recovery of the saturation areas in the
captured images. Methods based on hardware could be studied. For example, taking the super high dynamic range characteristic of the chips into consideration, event-based cameras have great potential
when it comes to solving the problem of highly reflective surface measurements.
Author Contributions
Conceptualization, X.C., and J.X.; methodology, X.C., H.D. and J.Z.; software, X.C., H.D., J.Z. and X.Y.; validation, H.D. and X.Y.; data curation, H.D.; writing—original draft preparation, H.D.;
writing—review and editing, X.C. and H.D. All authors have read and agreed to the published version of the manuscript.
This research was funded by the National Natural Science Foundation of China (52175478), the Defense Industrial Technology Development Program (JCKY2020203B039), the Science and Technology Commission
of Shanghai Municipality (21511102602), the Doctoral Foundation of the University of Jinan (XBS1641) and the Joint Special Project of Shandong Province (ZR2016EL14).
Data Availability Statement
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
The authors thank Honghui Zhang for his support during the experiments.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 1. Simulated fringe images in different exposure conditions: (a) Low exposure fringes, (b) normal exposure fringes and (c) high exposure fringes.
Figure 4. Entropy variation diagram of three-frequency three-step phase-shifting simulated fringe images with the change in modulation.
Figure 5. Entropy variation of three-frequency three-step phase-shifting real captured fringe images with the change in exposure.
Figure 9. Steps (3)–(6) of the highly reflective surface measurement method based on self-adaptive multiple exposure image fusion: (a) Captured fringe images according to the exposure sequence; (b)
The results of the proposed multiple exposure image fusion method; (c) Left and right unwrapped full-field phase maps; (d) Obtained disparity map; (e) Reconstructed point cloud.
Figure 10. Metal test workpieces: workpieces nos. 1–4 possessing the original machining surface and nos. 5–6 possessing the oxidation treatment surface.
Figure 11. Experiment scenes with different ambient lighting conditions: (a) low lighting conditions and (b) normal lighting conditions.
Figure 12. Evaluation of the initial exposure searching algorithm in different lighting conditions for workpieces nos. 1-4.
Figure 13. Evaluation of the initial exposure searching algorithm for different surface reflectivity.
Figure 14. Comparison of the original fringe images with maximal entropy with the fused fringe images.
Figure 15. Unwrapped phase maps obtained from the initial fringe images versus those from the fused fringe images.
Figure 16. Point clouds reconstructed based on the original fringe images and the fused fringe images.
Figure 17. Errors of the point clouds reconstructed from the original fringe images and fused fringe images.
Workpiece No.1 Workpiece No.2 Workpiece No.3 Workpiece No.4
Initial point cloud 192535 23534 50454 4782
Fused point cloud 418859 129824 98169 33894
Error Item Workpiece No.1 Workpiece No.2 Workpiece No.3 Workpiece No.4
Mean 0.0639 0.0478 0.0537 0.0496
Initial point cloud Error/mm
RMSE/mm 0.1125 0.0942 0.0937 0.0786
Mean 0.0332 0.0308 0.0281 0.0313
Fused point cloud error/mm
RMSE 0.0358 0.0362 0.0297 0.0344
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© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:
Share and Cite
MDPI and ACS Style
Chen, X.; Du, H.; Zhang, J.; Yang, X.; Xi, J. A Self-Adaptive Multiple Exposure Image Fusion Method for Highly Reflective Surface Measurements. Machines 2022, 10, 1004. https://doi.org/10.3390/
AMA Style
Chen X, Du H, Zhang J, Yang X, Xi J. A Self-Adaptive Multiple Exposure Image Fusion Method for Highly Reflective Surface Measurements. Machines. 2022; 10(11):1004. https://doi.org/10.3390/
Chicago/Turabian Style
Chen, Xiaobo, Hui Du, Jinkai Zhang, Xiao Yang, and Juntong Xi. 2022. "A Self-Adaptive Multiple Exposure Image Fusion Method for Highly Reflective Surface Measurements" Machines 10, no. 11: 1004.
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
Article Metrics
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Question about meld
General questions (367)
I'm sorry for my carelessness. The right code is
ex:=A tr{u^{\nu} u^{\mu} u^{\mu} u^{\nu}}+B tr{ u^{\mu} u^{\mu} u^{\nu} u^{\nu}};
I want to use meld & factor_in to get the following result: $$ (A+B) tr(u^\mu u^\mu u^\nu u^\nu) $$ but I failed, is this a bug? I find the algorithm meld seems to be unsound, the following code is
another example:
ex:=a tr{A^{\mu\nu} B^{\mu\rho} B^{\nu\rho}}+b tr{C^{\mu\nu} A^{\mu\rho} B^{\nu\rho}}+c tr{C^{\mu\nu} B^{\mu\rho} A^{\nu\rho}};
The result is $$ \left(a+b+c\right) tr\left(A^{\mu \nu} B^{\mu \rho} B^{\nu \rho}\right). $$ This is very strange.
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measurements - MyStudyGeek.com
Identify each of the following as examples of nominal, ordinal, interval, or ratio scales of measurement. (4 points each)
1. A poll of registered voters in Florida asking which candidate they support
2. The length of time required for a wound to heal when using a new medicine
3. The number of telephone calls arriving at a switchboard per five-minute period
4. The distance first-year college football players can kick a ball
5. Mental health diagnoses present in an elderly population
6. The rankings of employees on their job performance
Question 2. 2.
Two hundred raffle tickets are sold. Your friend has five people in her family who each bought two raffle tickets. What is the probability that someone from her family will win the raffle?
Question 3. 3. Jolie has 45 minutes to do her statistics homework. If the mean is 38 minutes and the standard deviation is 3, calculate Jolie’s z score. Once calculated, interpret your findings in
terms of Jolie’s performance.
(HINT: use the normal distribution and the probability that other students performed better or worse.) (Points : 8)
Question 4. 4. A psychologist measures units of change for a memory test after students are given an opportunity to sleep only four hours. The following change units were obtained: 7, -12, 4, -7, 3,
-10. Find the a) mean, b) median, c) mode, d) standard deviation, e) range, and f) variance. (Points : 24)
Question 5. 5. A student scored 81 on a chemistry test and 75 on a history test. For the chemistry test, the mean was 70 and the standard deviation was 20. For the history test, the mean was 65 and
the standard deviation was 8. Did the student do better on the chemistry test or the history test? Explain your answer. (Points : 12)
Question 6. 6. Suppose you want to figure out what to do with your degree in psychology. You ask some fellow students from your psychology program who recently graduated to find out what they are
doing with their degree and how much it pays. What type of sampling is this? What are the limitations of this sampling approach? (Points : 8)
Question 7. 7. Variables in which the values are categories are known as (Points : 4)
Interval variables
Nominal variables
Ordinal variables
Ratio variables
Question 8. 8. Before the researcher can conduct a statistical test, the research question must be translated into (Points : 4)
A testable hypothesis
Additional observations
Mathematical symbols
Question 9. 9. The hypothesis stating that there are no differences, effects, or relationships is (Points : 4)
The alternative hypothesis
The baseline hypothesis
The null hypothesis
The reasonable hypothesis
Question 10. 10. A group of students made the following scores on a 10-item quiz in psychological statistics: {5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10} What is the mean score? (Points : 4)
Question 11. 11. A group of students made the following scores on a 10-item quiz in psychological statistics: {5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10} What is the median score? (Points : 4)
Question 12. 12. A group of students made the following scores on a 10-item quiz in psychological statistics: {5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10} What is the mode? (Points : 4)
Question 13. 13. A group of students made the following scores on a 10-item quiz in psychological statistics: {5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10} What is the range of scores? (Points : 4)
Question 14. 14. A group of students made the following scores on a 10-item quiz in psychological statistics: {5, 6, 7, 7, 7, 8, 8, 9, 9, 10, 10} What is the variance, treating these scores as a
sample? (Points : 4)
Question 15. 15. The standard normal distribution has all the following properties EXCEPT: (Points : 4)
The mean, mode, and median are all equal
The total area under the curve equals 1
The curve is specified by two parameters, the mean and the standard deviation
The curve extends to + and – 3 standard deviations from the mean
Question 16. 16. According to the Empirical Rule, approximately _______% of the data in a normal distribution will fall within ±1 standard deviation of the mean. (Points : 4)
Question 17. 17. In statistical computations, the number of values that are free to vary is known as (Points : 4)
Degrees of freedom
Freedom factor
Variability index
Variation quotient
Question 18. 18. Which of the following reflects a Type I error? (Points : 4)
Rejecting the null hypothesis when in reality the null hypothesis is true
Rejecting the null hypothesis when in reality the null hypothesis is false
Accepting the null hypothesis when in reality the null hypothesis is true
Accepting the null hypothesis when in reality the null hypothesis is false
Question 19. 19. Which type of sampling is used when the experimenter asks 5 area doctors to refer pregnant women to his study and accepts all women who offer to be in his study? (Points : 4)
purposive sampling
convenience sampling
cluster sampling
stratified sampling
Question 20. 20. In our statistics equations, n refers to: (Points : 4)
standard deviation
normal distribution
number of subjects
Question 21. 21. Which of the following is true regarding alpha? (Points : 4)
it is also known as the level of significance
value is set by the researcher
value is equal to the probability of a type I error
all of the above are true
Question 22. 22. Macy proposes that boys who play sports are viewed as more attractive than boys who do not play sports. What is her null hypothesis? (Points : 4)
Boys who play sports are not viewed as more attractive than boys who do not play sports
Playing sports will influence how attractively boys are viewed
Boys who play sports are more attractive than girls who play sports
There can be no null hypothesis
Question 23. 23. You calculate a t of 2.38 and note that the tabled value for .01 is 3.22 and for .05 is 2.19. You would conclude that the null hypothesis can be: (Points : 4)
Accepted at the .05 level
Rejected at the .01 level
Rejected at the .05 level
None of the above
Question 24. 24. A researcher is studying political conservatism among 11 engineering students and 11 humanities students. The number of degrees of freedom for a t test is: (Points : 4)
Question 25. 25. A t test for dependent groups should be used instead of a t test for independent samples: (Points : 4)
If each participant is measured twice
Whenever there are equal numbers of subjects in each group
Whenever there are only two groups
All of the above
Question 26. 26. In a normal distribution, what percent of the population falls within one and two standard deviations of the mean? (Points : 4)
cannot tell from the information given
Question 27. 27. Which of the following is more affected by extreme scores? (Points : 4)
None of the above are affected
Question 28. 28. On a histogram, what does the vertical (y) axis refer to? (Points : 4)
Individual scores
Deviation scores
Question 29. 29. Which statistic refers to the average amount by which the scores in the sample deviate from the mean? (Points : 4)
Standard deviation
Question 30. 30. Assume a normal distribution for N = 300. How many cases would one expect to find between +1 and -1 standard deviations around the mean? (Points : 4)
Question 31. 31. A z score of zero tells us that the score is at the________of the distribution. (Points : 4)
Very top
Very bottom
None of the above since z cannot be zero
Question 32. 32. In a unit normal curve, what goes on the x axis? (Points : 4)
Observed scores
z scores
33. Question : Which of the following is a measure of variability?
Student Answer: Mean
All of the above
Points Received: 4 of 4
34. Question : The only measure of central tendency that can be found for nominal data is the
Student Answer: Mean
Points Received: 4 of 4
35. Question : If the probability of event A is 0.45 and the probability of event B is 0.35 and the probability of A and B occurring together is 0.25, then the probability of A OR B is:
Student Answer: 0.8
Points Received: 4 of 4
36. Question A researcher knows that the average distance commuting students live from campus was previously 8.2 miles. Because of the rising prices of gasoline, the research wants to test the claim
: that commuting students now live closer to campus. What is the correct alternative hypothesis?
Student Answer: The new mean distance is 8.2 miles.
The new mean distance is less than or equal to 8.2 miles.
The new mean distance is less than 8.2 miles.
The new mean distance is greater than or equal to 8.2 miles.
Points Received: 4 of 4
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Talk:Munchausen numbers - Rosetta CodeTalk:Munchausen numbers
0 to the power 0 is considered as 0 for Munchausen numbers
...according to the Wikipedia page - might be worth mentioning in the task as it is normally 1 according to mathematicians. --Tigerofdarkness (talk) 18:25, 23 September 2016 (UTC)
This changes the output only for the number 0 which is not between 1 and 5000 (the task's requirement) --Walterpachl (talk) 19:16, 23 September 2016 (UTC)
True, it would only be an issue if anyone tried to extend this to the next Munchausen number (438 579 088 - also the last known one it seems). --Tigerofdarkness (talk) 21:12, 23 September
2016 (UTC)
Also, if you are using an algorithm where the digits are already split up (so not derived by using mod and division), you don't have to worry about leading zeros e.g. 0030 is "Munchausen" if
0^0 = 1. The non-standard 0^0 = 0 simplifies things. --Tigerofdarkness (talk) 21:55, 23 September 2016 (UTC)
Where would those leading zeros come from? --Rdm (talk) 00:33, 24 September 2016 (UTC)
See the current ALGOL 68 sample which holds the 4 digits as separate numbers and indexes a table of powers - the non-standard 0^0 allows indexing the table of powers without having to
worry whether the digit is a leading zero or not. The sample does this to avoids any multiplication, division or modulo operations. --Tigerofdarkness (talk) 19:51, 24 September 2016
Ok, so it's a hack that lets the programmer use a fixed width array of digits without having to bother with the relevant array length. --Rdm (talk) 20:22, 24 September 2016 (UTC)
Or it's a feature of the definition that can be exploited :) --Tigerofdarkness (talk) 09:40, 25 September 2016 (UTC)
How is that any different? Rdm (talk) 09:48, 25 September 2016 (UTC)
I would like to point out that +438579088, in the Algol60 output is not correct...
+/ x * x ← 4 3 8 5 7 9 0 8 8
438579089 Note that is APL so * is exponentiation not multiplication.
APL I'm sure, will be using the mathematical definition of exponentiation where 0^0 is defined to be 1. As the title of this section of this discussion page indicates, Munchausen numbers use an
alternative definition of 0^0 = 0 ( being vaguely from a mathematical background, I thought it was worth highlighting it ).
If you use 0^0 = 1, there are only two Munchausen numbers, 1 and 3435. If you use 0^0 = 0, there are four: 0, 1, 3435 and 438579088.
I see the Wikipedia page (Perfect digit-to-digit invariant) now discusses Munchausen numbers using either convention, as does the Mathworld page (which spells Munchhausen with two h letters).
There are a number of other language samples for this task that show the four Munchausen numbers, including Java, Kotlin and Pascal.
Incidentally, there currently isn't an Algol 60 sample, the language you are singling out for criticism is Algol 68 :)
BTW, please sign your posts.
--Tigerofdarkness (talk) 20:00, 17 August 2023 (UTC)
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Percentage Change
Practise calculating percentage increase and percentage decrease in this set of exercises.
This is level 1: find the new amount after a percentage increase. You can earn a trophy if you get at least 7 questions correct. You can use a calculator.
1. If 20 is increased by 20%, what is the result?
2. Shona ate 100 grapes yesterday. Today she eats 15% more than yesterday. How many grapes did she eat today?
3. John took 180 photographs last month. This month he took 50% more. How many photographs did he take this month?
4. What number is 95% more than 80?
5. Increase 150 by 14%.
6. Last time Orange Floyd released an album, 550 copies were downloaded in the first 24 hours. By the end of the first week fans had downloaded 8% more copies. How many albums had been downloaded
in total at the end of the first week?
7. Jamie decided to order 20% more flour for his restaurants than last year. If he ordered 400kg last year what weight of flour should he order this year? kg
8. Judith swam 22300 metres last month. Her target for this month is to swim 30% more metres than last month. How many metres must she swim this month to achieve her target? m
9. Find the new amount if 28400 is increased by 58%.
10. Find the new amount if 18300 is increased by 30%.
© Transum Mathematics 1997-2024
Scan the QR code below to visit the online version of this activity.
Description of Levels
Simple Percentages - A good place to start.
Level 1 - Find the new amount after a percentage increase.
Level 2 - Find the new amount after a percentage decrease.
Level 3 - Find the percentage increase given the original and final values.
Level 4 - Find the percentage decrease given the original and final values.
Level 5 - Find the original amount after a percentage increase or decrease given the final value.
Level 6 - Mixed questions.
Level 7 - Challenging questions.
Switch - Practise percentage increase and decrease calculations by completing this table.
Two-Step Percentages - Now things start to get a little more challenging!
Exam Style questions are in the style of GCSE or IB/A-level exam paper questions and worked solutions are available for Transum subscribers.
Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a
teacher, tutor or parent.
Curriculum Reference
You may want to use a calculator for some of the questions. See Calculator Workout skill 3.
Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly.
Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a
teacher, tutor or parent.
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A practical statistical polychromatic image reconstruction for computed tomography using spectrum binning
Wu M, Yang Q, Maier A, Fahrig R (2014)
Publication Type: Conference contribution
Publication year: 2014
Edited Volumes: Progress in Biomedical Optics and Imaging - Proceedings of SPIE
Pages Range: 9033-26
Conference Proceedings Title: Proc. SPIE Medical Imaging 2014
Event location: San Diego, California, United States
URI: http://www5.informatik.uni-erlangen.de/Forschung/Publikationen/2014/Wu14-APS.pdf
DOI: 10.1117/12.2043370
Polychromatic statistical reconstruction algorithms have very high computational demands due To The difficulty of The optimization problems and The large number of spectrum bins. We want To develop a
more practical algorithm That has a simpler optimization problem, a faster numerical solver, and requires only a small amount of prior knowledge. In This paper, a modified optimization problem for
polychromatic statistical reconstruction algorithms is proposed. The modified optimization problem utilizes The idea of determining scanned materials based on a first pass FBP reconstruction To fix
The ratios between photoelectric and Compton scattering components of all image pixels. The reconstruction of a density image is easy To solve by a separable quadratic surrogate algorithm That is
also applicable To The multi-material case. In addition, a spectrum binning method is introduced so That The full spectrum information is not required. The energy bins sizes and attenuations are
optimized based on The True spectrum and object. With These approximations, The expected line integral values using only a few energy bins are very closed To The True polychromatic values. Thus both
The problem size and computational demand caused by The large number of energy bins That are Typically used To model a full spectrum are reduced. Simulation showed That Three energy bins using The
generalized spectrum binning method could provide an accurate approximation of The polychromatic X-ray signals. The average absolute error of The logarithmic detector signal is less Than 0.003 for a
120 kVp spectrum. The proposed modified optimization problem and spectrum binning approach can effectively suppress beam hardening artifacts while providing low noise images. © 2014 SPIE.
Authors with CRIS profile
Involved external institutions
How to cite
Wu, M., Yang, Q., Maier, A., & Fahrig, R. (2014). A practical statistical polychromatic image reconstruction for computed tomography using spectrum binning. In Proc. SPIE Medical Imaging 2014 (pp.
9033-26). San Diego, California, United States.
Wu, Meng, et al. "A practical statistical polychromatic image reconstruction for computed tomography using spectrum binning." Proceedings of the SPIE Medical Imaging 2014, San Diego, California,
United States 2014. 9033-26.
BibTeX: Download
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Copyright (c) Colin Runciman et al.
License BSD3
Maintainer Roman Cheplyaka <roma@ro-che.info>
Safe Haskell Trustworthy
Language Haskell98
You need this module if you want to generate test values of your own types.
You'll typically need the following extensions:
{-# LANGUAGE FlexibleInstances, MultiParamTypeClasses #-}
SmallCheck itself defines data generators for all the data types used by the Prelude.
In order to generate values and functions of your own types, you need to make them instances of Serial (for values) and CoSerial (for functions). There are two main ways to do so: using Generics or
writing the instances by hand.
Generic instances
The easiest way to create the necessary instances is to use GHC generics (available starting with GHC 7.2.1).
Here's a complete example:
{-# LANGUAGE FlexibleInstances, MultiParamTypeClasses #-}
{-# LANGUAGE DeriveGeneric #-}
import Test.SmallCheck.Series
import GHC.Generics
data Tree a = Null | Fork (Tree a) a (Tree a)
deriving Generic
instance Serial m a => Serial m (Tree a)
Here we enable the DeriveGeneric extension which allows to derive Generic instance for our data type. Then we declare that Tree a is an instance of Serial, but do not provide any definitions. This
causes GHC to use the default definitions that use the Generic instance.
One minor limitation of generic instances is that there's currently no way to distinguish newtypes and datatypes. Thus, newtype constructors will also count as one level of depth.
Data Generators
Writing Serial instances for application-specific types is straightforward. You need to define a series generator, typically using consN family of generic combinators where N is constructor arity.
For example:
data Tree a = Null | Fork (Tree a) a (Tree a)
instance Serial m a => Serial m (Tree a) where
series = cons0 Null \/ cons3 Fork
For newtypes use newtypeCons instead of cons1. The difference is that cons1 is counts as one level of depth, while newtypeCons doesn't affect the depth.
newtype Light a = Light a
instance Serial m a => Serial m (Light a) where
series = newtypeCons Light
For data types with more than 4 fields define consN as
consN f = decDepth $
f <$> series
<~> series
<~> series
<~> ... {- series repeated N times in total -}
What does consN do, exactly?
consN has type (Serial t_1, ..., Serial t_N) => (t_1 -> ... -> t_N -> t) -> Series t.
consN f is a series which, for a given depth d > 0, produces values of the form
f x_1 ... x_N
where x_i ranges over all values of type t_i of depth up to d-1 (as defined by the series functions for t_i).
consN functions also ensure that x_i are enumerated in the breadth-first order. Thus, combinations of smaller depth come first (assuming the same is true for t_i).
If d <= 0, no values are produced.
Function Generators
To generate functions of an application-specific argument type, make the type an instance of CoSerial.
Again there is a standard pattern, this time using the altsN combinators where again N is constructor arity. Here are Tree and Light instances:
instance CoSerial m a => CoSerial m (Tree a) where
coseries rs =
alts0 rs >>- \z ->
alts3 rs >>- \f ->
return $ \t ->
case t of
Null -> z
Fork t1 x t2 -> f t1 x t2
instance CoSerial m a => CoSerial m (Light a) where
coseries rs =
newtypeAlts rs >>- \f ->
return $ \l ->
case l of
Light x -> f x
For data types with more than 4 fields define altsN as
altsN rs = do
rs <- fixDepth rs
(constM $ constM $ ... $ constM rs)
(coseries $ coseries $ ... $ coseries rs)
{- constM and coseries are repeated N times each -}
What does altsN do, exactly?
altsN has type (Serial t_1, ..., Serial t_N) => Series t -> Series (t_1 -> ... -> t_N -> t).
altsN s is a series which, for a given depth d, produces functions of type
t_1 -> ... -> t_N -> t
If d <= 0, these are constant functions, one for each value produced by s.
If d > 0, these functions inspect each of their arguments up to the depth d-1 (as defined by the coseries functions for the corresponding types) and return values produced by s. The depth to which
the values are enumerated does not depend on the depth of inspection.
Basic definitions
type Depth = Int Source #
Maximum depth of generated test values.
For data values, it is the depth of nested constructor applications.
For functional values, it is both the depth of nested case analysis and the depth of results.
data Series m a Source #
Series is a MonadLogic action that enumerates values of a certain type, up to some depth.
The depth bound is tracked in the SC monad and can be extracted using getDepth and changed using localDepth.
To manipulate series at the lowest level you can use its Monad, MonadPlus and MonadLogic instances. This module provides some higher-level combinators which simplify creating series.
A proper Series should be monotonic with respect to the depth — i.e. localDepth (+1) s should emit all the values that s emits (and possibly some more).
It is also desirable that values of smaller depth come before the values of greater depth.
MonadTrans Series Source #
Defined in Test.SmallCheck.SeriesMonad
Monad (Series m) Source #
Defined in Test.SmallCheck.SeriesMonad
Functor (Series m) Source #
Defined in Test.SmallCheck.SeriesMonad
Applicative (Series m) Source #
Defined in Test.SmallCheck.SeriesMonad
Alternative (Series m) Source #
Defined in Test.SmallCheck.SeriesMonad
MonadPlus (Series m) Source #
Defined in Test.SmallCheck.SeriesMonad
Monad m => MonadLogic (Series m) Source #
Defined in Test.SmallCheck.SeriesMonad
class Monad m => Serial m a where Source #
Monad m => Serial m Bool Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Char Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Double Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Float Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Word64 Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Int64 Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Word32 Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Int32 Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Word16 Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Int16 Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Word8 Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Int8 Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Word Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Int Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Natural Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m Integer Source #
Defined in Test.SmallCheck.Series
Monad m => Serial m () Source #
Defined in Test.SmallCheck.Series
Serial m a => Serial m (NonEmpty a) Source #
Defined in Test.SmallCheck.Series
(Num a, Ord a, Serial m a) => Serial m (NonNegative a) Source #
Defined in Test.SmallCheck.Series
(Num a, Ord a, Serial m a) => Serial m (Positive a) Source #
Defined in Test.SmallCheck.Series
Serial m a => Serial m [a] Source #
Defined in Test.SmallCheck.Series
Serial m a => Serial m (Maybe a) Source #
Defined in Test.SmallCheck.Series
(Integral i, Serial m i) => Serial m (Ratio i) Source #
Defined in Test.SmallCheck.Series
(CoSerial m a, Serial m b) => Serial m (a -> b) Source #
Defined in Test.SmallCheck.Series
(Serial m a, Serial m b) => Serial m (Either a b) Source #
Defined in Test.SmallCheck.Series
(Serial m a, Serial m b) => Serial m (a, b) Source #
Defined in Test.SmallCheck.Series
(Serial m a, Serial m b, Serial m c) => Serial m (a, b, c) Source #
Defined in Test.SmallCheck.Series
(Serial m a, Serial m b, Serial m c, Serial m d) => Serial m (a, b, c, d) Source #
Defined in Test.SmallCheck.Series
class Monad m => CoSerial m a where Source #
coseries :: Series m b -> Series m (a -> b) Source #
A proper coseries implementation should pass the depth unchanged to its first argument. Doing otherwise will make enumeration of curried functions non-uniform in their arguments.
coseries :: (Generic a, GCoSerial m (Rep a)) => Series m b -> Series m (a -> b) Source #
A proper coseries implementation should pass the depth unchanged to its first argument. Doing otherwise will make enumeration of curried functions non-uniform in their arguments.
Monad m => CoSerial m Bool Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Char Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Double Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Float Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Word64 Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Int64 Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Word32 Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Int32 Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Word16 Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Int16 Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Word8 Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Int8 Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Word Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Int Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Natural Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m Integer Source #
Defined in Test.SmallCheck.Series
Monad m => CoSerial m () Source #
Defined in Test.SmallCheck.Series
CoSerial m a => CoSerial m [a] Source #
Defined in Test.SmallCheck.Series
CoSerial m a => CoSerial m (Maybe a) Source #
Defined in Test.SmallCheck.Series
(Integral i, CoSerial m i) => CoSerial m (Ratio i) Source #
Defined in Test.SmallCheck.Series
(Serial m a, CoSerial m a, Serial m b, CoSerial m b) => CoSerial m (a -> b) Source #
Defined in Test.SmallCheck.Series
(CoSerial m a, CoSerial m b) => CoSerial m (Either a b) Source #
Defined in Test.SmallCheck.Series
(CoSerial m a, CoSerial m b) => CoSerial m (a, b) Source #
Defined in Test.SmallCheck.Series
(CoSerial m a, CoSerial m b, CoSerial m c) => CoSerial m (a, b, c) Source #
Defined in Test.SmallCheck.Series
(CoSerial m a, CoSerial m b, CoSerial m c, CoSerial m d) => CoSerial m (a, b, c, d) Source #
Defined in Test.SmallCheck.Series
Generic implementations
Convenient wrappers
newtype Positive a Source #
Positive x: guarantees that x > 0.
• getPositive :: a
(Num a, Ord a, Serial m a) => Serial m (Positive a) Source #
Defined in Test.SmallCheck.Series
Enum a => Enum (Positive a) Source #
Defined in Test.SmallCheck.Series
Eq a => Eq (Positive a) Source #
Defined in Test.SmallCheck.Series
Integral a => Integral (Positive a) Source #
Defined in Test.SmallCheck.Series
Num a => Num (Positive a) Source #
Defined in Test.SmallCheck.Series
Ord a => Ord (Positive a) Source #
Defined in Test.SmallCheck.Series
Real a => Real (Positive a) Source #
Defined in Test.SmallCheck.Series
Show a => Show (Positive a) Source #
Defined in Test.SmallCheck.Series
newtype NonNegative a Source #
NonNegative x: guarantees that x >= 0.
• getNonNegative :: a
(Num a, Ord a, Serial m a) => Serial m (NonNegative a) Source #
Defined in Test.SmallCheck.Series
Enum a => Enum (NonNegative a) Source #
Defined in Test.SmallCheck.Series
Eq a => Eq (NonNegative a) Source #
Defined in Test.SmallCheck.Series
Integral a => Integral (NonNegative a) Source #
Defined in Test.SmallCheck.Series
Num a => Num (NonNegative a) Source #
Defined in Test.SmallCheck.Series
Ord a => Ord (NonNegative a) Source #
Defined in Test.SmallCheck.Series
Real a => Real (NonNegative a) Source #
Defined in Test.SmallCheck.Series
Show a => Show (NonNegative a) Source #
Defined in Test.SmallCheck.Series
newtype NonEmpty a Source #
NonEmpty xs: guarantees that xs is not null
• getNonEmpty :: [a]
Serial m a => Serial m (NonEmpty a) Source #
Defined in Test.SmallCheck.Series
Show a => Show (NonEmpty a) Source #
Defined in Test.SmallCheck.Series
Other useful definitions
(>>-) :: MonadLogic m => m a -> (a -> m b) -> m b infixl 1#
Fair conjunction. Similarly to the previous function, consider the distributivity law for MonadPlus:
(mplus a b) >>= k = (a >>= k) `mplus` (b >>= k)
If 'a >>= k' can backtrack arbitrarily many tmes, (b >>= k) may never be considered. (>>-) takes similar care to consider both branches of a disjunctive computation.
decDepth :: Series m a -> Series m a Source #
Run a Series with the depth decreased by 1.
If the current depth is less or equal to 0, the result is mzero.
generate :: (Depth -> [a]) -> Series m a Source #
A simple series specified by a function from depth to the list of values up to that depth.
listSeries :: Serial Identity a => Depth -> [a] Source #
Given a depth, return the list of values generated by a Serial instance.
Example, list all integers up to depth 1:
• listSeries 1 :: [Int] -- returns [0,1,-1]
list :: Depth -> Series Identity a -> [a] Source #
Return the list of values generated by a Series. Useful for debugging Serial instances.
• list 3 series :: [Int] -- returns [0,1,-1,2,-2,3,-3]
• list 3 (series :: Series Identity Int) -- returns [0,1,-1,2,-2,3,-3]
• list 2 series :: [[Bool]] -- returns [[],[True],[False]]
The first two are equivalent. The second has a more explicit type binding.
fixDepth :: Series m a -> Series m (Series m a) Source #
Fix the depth of a series at the current level. The resulting series will no longer depend on the "ambient" depth.
decDepthChecked :: Series m a -> Series m a -> Series m a Source #
If the current depth is 0, evaluate the first argument. Otherwise, evaluate the second argument with decremented depth.
Orphan instances
(Serial Identity a, Show a, Show b) => Show (a -> b) Source #
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FIR Halfband Decimator
FIR Halfband Decimator
Decimate signal using polyphase FIR halfband filter
DSP System Toolbox / Filtering / Multirate Filters
The FIR Halfband Decimator block performs polyphase decimation of the input signal by a factor of 2. The block uses an FIR equiripple design or a Kaiser window design to construct the halfband
filters. The implementation takes advantage of the zero-valued coefficients of the FIR halfband filter, making one of the polyphase branches a delay. You can use the block to implement the analysis
portion of a two-band filter bank to separate a signal into lowpass and highpass subbands. For more information, see Algorithms.
The block supports fixed-point operations and ARM^® Cortex^® code generation. For more information on ARM Cortex code generation, see Code Generation for ARM Cortex-M and ARM Cortex-A Processors.
Design and Implement FIR Halfband Decimator in Simulink
Since R2023b
Design and implement an FIR halfband decimator using the FIR Halfband Decimator block. Pass a noisy input through the decimator. Plot the spectrum of the input and the decimated subband outputs in
the spectrum analyzer.
This example also shows how to use the Allow arbitrary frame length for fixed-size input signals parameter.
Open and inspect the DesignAndImplementFIRHalfbandDecimator model. The input signal in the model is a noisy sinusoidal signal with 513 samples per frame and contains two frequencies, one at 1 kHz and
the other at 15 kHz. The Random Source block adds white Gaussian noise with a mean of 0 and a variance of 0.05 to this signal. The input signal is a fixed-size signal, that is, the frame length of
the signal (513 samples per frame) does not vary during simulation.
To allow an arbitrary frame length for a fixed-size signal, that is, to allow a frame length that is not a multiple of the decimation factor 2, select the Allow arbitrary frame length for fixed-size
input signals parameter in FIR Halfband Decimator block dialog box.
The FIR halfband decimator has a transition width of 4.1 kHz and a stopband attenuation of 80 dB. The decimator outputs the highpass subband signal.
Visualize the magnitude response of the filter by clicking the View Filter Response button in the block dialog box.
Pass the noisy sinusoidal signal through the decimator. Plot the spectrum of the input and the lowpass and highpass subband outputs in two separate spectrum analyzer windows.
The block allows arbitrary frame lengths for variable-size signals irrespective of the setting of the Allow arbitrary frame length for fixed-size input signals parameter. When the input is a
variable-size signal, the frame length of the signal can vary during simulation.
Open the DesignAndImplementFIRHalfbandDecimator_Varsize model. The input is a variable-size signal. The input frame length can vary between 257 and 514 samples per frame.
Extract Low Frequency Subband From Speech in Simulink
Since R2023b
Use the FIR Halfband Decimator and FIR Halfband Interpolator blocks to extract and reconstruct the low-frequency subband from a speech signal.
Open and inspect the ExtractLowFrequencySubbandFromSpeechFIR.slx model. The input audio data is a single-channel speech signal with the sample rate of 22050 Hz.
Specify the Sample rate mode parameter of the FIR Halfband Decimator and FIR Halfband Interpolator blocks to Use normalized frequency (0 to 1). This option enables you to specify the transition width
of the decimation and interpolation filters in normalized frequency units. Set the transition width to 0.093 in normalized frequency units and the stopband attenuation to 80 dB. The design method is
set to Auto by default. In the Auto mode, the block selects the equiripple or Kaiser window design method based on the design parameters of the filter.
Read the speech signal from the audio file in frames of 1024 samples. The FIR Halfband Decimator block extracts and outputs the lowpass subband of the speech signal. The FIR Halfband Interpolator
block reconstructs the lowpass approximation of the speech signal by interpolating the lowpass subband.
The Audio Device Writer block plays the filtered output.
Design and Implement Two-Channel FIR Filter Bank in Simulink
Use the FIR Halfband Decimator and FIR Halfband Interpolator blocks to implement a two-channel filter bank. This example uses an audio file input and shows that the power spectrum of the filter bank
output does not differ significantly from the input. Play the output of the filter bank using the Audio Device Writer block.
Open and inspect the TwoChannelFIRFilterBank.slx model. The input audio data is a single-channel speech signal with a sample rate of 22050 Hz.
The FIR Halfband Decimator block acts as an FIR halfband analysis bank as the Output highpass subband parameter is selected in the block dialog box. The FIR Halfband Interpolator block acts as an FIR
halfband synthesis bank as the Input highpass subband parameter is selected in the block dialog box.
Set the Sample rate mode parameter in the FIR Halfband Decimator and FIR Halfband Interpolator blocks to Inherit from input port so that the blocks inherit the sample rate from the respective input
ports. Set the transition width to 4.1 kHz and the stopband attenuation to 80 dB. The design method is set to Auto by default. In the Auto mode, the block selects the equiripple or Kaiser window
design methods based on the design parameters of the filter.
Read the speech signal from the audio file in frames of 1024 samples. The FIR halfband analysis bank extracts the lowpass and highpass subbands of the speech signal. The FIR halfband synthesis filter
bank synthesizes the speech signal from the lowpass and highpass subbands.
Display the power spectrum of the audio input and the output of the synthesis filter bank in the spectrum analyzer. Play the synthesized speech signal using the Audio Device Writer block.
Input — Input signal
column vector | matrix
Specify the input signal as a column vector or a matrix of size P-by-Q.
The block treats each column of the input signal as a separate channel. If the input is a two-dimensional signal, the first dimension represents the channel length (or frame size) and the second
dimension represents the number of channels. If the input is a one-dimensional signal, then the block interprets it as a single-channel signal.
This block supports variable-size input signals (frame length changes during simulation).
When you input a variable-size signal, the frame length of the signal can be arbitrary, that is, the input frame length does not have to be a multiple of the decimation factor 2. When you input a
fixed-size signal (frame length does not change during simulation), the frame length can be arbitrary only when you select the Allow arbitrary frame length for fixed-size input signals parameter.
(since R2023b)
This block supports variable-size input signals.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | fixed point
Complex Number Support: Yes
LP — Lowpass subband of decimator output
column vector | matrix
Lowpass subband of the decimator output, returned as a column vector or a matrix. As the filter is a halfband filter, the downsampling factor is always 2.
The dimensions of the output signal depend on the dimensions of the input signal and on whether you select the Allow arbitrary frame length for fixed-size input signals parameter. (since R2023b)
This table provides more details on the dimensions of the lowpass subband output signal when you input a fixed-size input signal.
Fixed-Size Input Signal
Input Signal Output Signal
P-by-Q, where P is a multiple of the decimation factor 2 Fixed-size signal of size (P/2)-by-Q
Variable-size signal with an upper bound of size ceil(P/2)-by-Q when you select Allow arbitrary frame length for fixed-size input
P-by-Q, where P is not a multiple of the decimation factor 2 signals (since R2023b)
(since R2023b)
If you do not select Allow arbitrary frame length for fixed-size input signals, the block errors. (since R2023b)
This table gives more details on the dimensions of the lowpass subband output signal when you input a variable-size input signal. When you input a variable-size signal (frame length changes during
simulation), the Allow arbitrary frame length for fixed-size input signals parameter is visible in the block dialog box but does not have any impact on the input frame length. You can input a
variable-size signal of any frame length even if you do not select the Allow arbitrary frame length for fixed-size input signals parameter. (since R2023b)
Variable-Size Input Signal
Input Signal Output Signal
P-by-Q Variable-size signal with an upper bound of size ceil(P/2)-by-Q (since R2023b)
When the output is of fixed-point data type, it is signed only.
This port is unnamed until you select the Output highpass subband parameter.
Data Types: single | double | int8 | int16 | int32 | int64 | fixed point
Complex Number Support: Yes
HP — Highpass subband of decimator output
column vector | matrix
Highpass subband of the decimator output, returned as a column vector or a matrix. As the filter is a halfband filter, the downsampling factor is always 2.
The dimensions of the output signal depend on the dimensions of the input signal and on whether you select the Allow arbitrary frame length for fixed-size input signals parameter. (since R2023b)
This table provides more details on the dimensions of the highpass subband output signal when you input a fixed-size input signal.
Fixed-Size Input Signal
Input Signal Output Signal
P-by-Q, where P is a multiple of the decimation factor 2 Fixed-size signal of size (P/2)-by-Q
Variable-size signal with an upper bound of size ceil(P/2)-by-Q when you select Allow arbitrary frame length for fixed-size input
P-by-Q, where P is not a multiple of the decimation factor 2 signals (since R2023b)
(since R2023b)
If you do not select Allow arbitrary frame length for fixed-size input signals, the block errors. (since R2023b)
This table gives more details on the dimensions of the highpass subband output signal when you input a variable-size input signal. When you input a variable-size signal (frame length changes during
simulation), the Allow arbitrary frame length for fixed-size input signals parameter is visible in the block dialog box but does not have any impact on the input frame length. You can input a
variable-size signal of any frame length even if you do not select the Allow arbitrary frame length for fixed-size input signals parameter. (since R2023b)
Variable-Size Input Signal
Input Signal Output Signal
P-by-Q Variable-size signal with an upper bound of size ceil(P/2)-by-Q (since R2023b)
When the output is of fixed-point data type, it is signed only.
To enable this port, select the Output highpass subband parameter.
Data Types: single | double | int8 | int16 | int32 | int64 | fixed point
Complex Number Support: Yes
Main Tab
Filter specification — Filter design parameters
Transition width and stopband attenuation (default) | Filter order and transition width | Filter order and stopband attenuation | Coefficients
Select the parameters that the block uses to design the FIR halfband filter.
• Transition width and stopband attenuation (default) — Design the filter using Transition width (Hz) and Stopband attenuation (dB). This design is the minimum-order design.
• Filter order and transition width — Design the filter using Filter order and Transition width (Hz).
• Filter order and stopband attenuation — Design the filter using Filter order and Stopband attenuation (dB).
• Coefficients — Specify the filter coefficients directly through the Numerator parameter.
Filter order — Filter order
52 (default) | even positive integer
Specify the filter order as an even positive integer.
To enable this parameter, set Filter specification to Filter order and transition width or Filter order and stopband attenuation.
Transition width (Hz) — Transition width
4.1e3 (default) | positive scalar
Specify the transition width as a positive scalar in Hz or in normalized frequency units (since R2023b).
If you set the Sample rate mode parameter to:
• Specify on dialog or Inherit from input port –– The value of the transition width is in Hz and must be less than half the value of the input sample rate.
• Use normalized frequency (0 to 1) –– The value of the transition width is in normalized frequency units. The value must be a positive scalar less than 1.0.
(since R2023b)
To enable this parameter, set Filter specification to Filter order and transition width or Transition width and stopband attenuation.
Stopband attenuation (dB) — Stopband attenuation
80 (default) | positive real scalar
Specify the stopband attenuation as a real positive scalar in dB.
To enable this parameter, set Filter specification to Filter order and stopband attenuation or Transition width and stopband attenuation.
Numerator — FIR halfband filter coefficients
designHalfbandFIR('TransitionWidth',0.186, 'StopbandAttenuation',80, 'Structure','decim') (default) | row vector
Specify the FIR halfband filter coefficients directly as a row vector. The coefficients must comply with the FIR halfband impulse response format. If (length(Numerator) − 1)/2 is even, where (length
(Numerator) − 1) is the filter order, every other coefficient starting with the first coefficient must be 0 except the center coefficient which must be 0.5. If (length(Numerator) − 1)/2 is odd, the
sequence of alternating zeros with 0.5 at the center starts at the second coefficient.
To enable this parameter, set Filter specification to Coefficients.
Design method — Filter design method
Auto (default) | Equiripple | Kaiser
Specify the filter design method as one of the following:
• Auto –– The algorithm automatically chooses the filter design method depending on the filter design parameters. The algorithm uses the equiripple or the Kaiser window method to design the filter.
If the design constraints are very tight, such as very high stopband attenuation or very narrow transition width, then the algorithm automatically chooses the Kaiser method, as this method is
optimal for designing filters with very tight specifications. However, if the design constraints are not tight, then the algorithm chooses the equiripple method.
When you set the Design method parameter to Auto, you can determine the method used by the algorithm by examining the passband and stopband ripple characteristics of the designed filter. If the
block used the equiripple method, the passband and stopband ripples of the designed filter have a constant amplitude in the frequency response. If the filter design method the block chooses in
the Auto mode is not suitable for your application, manually specify the Design method as Equiripple or Kaiser.
• Equiripple –– The algorithm uses the equiripple method.
• Kaiser –– The algorithm uses the Kaiser window method.
To enable this parameter, set Filter specification to Filter order and stopband attenuation, Filter order and transition width, or Transition width and stopband attenuation.
Output highpass subband — Output highpass subband
off (default) | on
When you select this check box, the block acts as an analysis filter bank and analyzes the input signal into highpass and lowpass subbands. When you clear this check box, the block acts as an FIR
halfband decimator.
Sample rate mode — Mode to specify the input sample rate
Use normalized frequency (0 to 1) (default) | Specify on dialog | Inherit from input port
Since R2023b
Specify the input sample rate using one of these options:
• Use normalized frequency (0 to 1) –– Specify the transition width in normalized frequency units (0 to 1).
• Specify on dialog –– Specify the input sample rate in the block dialog box using the Input sample rate (Hz) parameter.
• Inherit from input port –– The block inherits the sample rate from the input signal.
To enable this parameter, set Filter specification to any value other than Coefficients.
Input sample rate (Hz) — Sample rate of input signal
44100 (default) | positive real scalar
Specify the sample rate of the input signal as a positive scalar in Hz.
To enable this parameter, set:
• Filter specification to any value other than Coefficients.
• Sample rate mode to Specify on dialog.
(since R2023b)
Allow arbitrary frame length for fixed-size input signals — Allow arbitrary frame length for fixed-size input signals
off (default) | on
Since R2023b
Specify whether a fixed-size input signal (whose size does not change during simulation) can have an arbitrary frame length, where the frame length does not have to be a multiple of the decimation
factor 2. The block uses this parameter only for fixed-size input signals and ignores it if the input data varies in size during simulation.
When the input signal is a variable-size signal, the signal can have an arbitrary frame length, that is, the frame length does not have to be a multiple of the decimation factor 2.
For fixed-size input signals, if you:
• Select the Allow arbitrary frame length for fixed-size input signals parameter, the frame length of the signal does not have to be a multiple of the decimation factor 2. If the input is not a
multiple of the decimation factor, then the output is generally a variable-size signal. Therefore, to support arbitrary input size, the block must also support variable-size operations, which you
can enable by selecting the Allow arbitrary frame length for fixed-size input signals parameter.
• Clear the Allow arbitrary frame length for fixed-size input signals parameter, the input frame length must be a multiple of the decimation factor 2.
View Filter Response — View Filter Response
Click this button to open the Filter Visualization Tool (FVTool) and display the magnitude and phase response of the FIR Halfband Decimator. The response is based on the values you specify in the
block parameters dialog box. Changes made to these parameters update FVTool.
To update the magnitude response while FVTool is running, modify the dialog box parameters and click Apply.
To view the magnitude response and phase response simultaneously, click the Magnitude and Phase responses button on the toolbar.
To visualize the filter response, set Sample rate mode to Specify on dialog or Use normalized frequency (0 to 1).
Simulate using — Type of simulation to run
Interpreted execution (default) | Code generation
Specify the type of simulation to run. You can set this parameter to:
• Interpreted execution –– Simulate model using the MATLAB^® interpreter. This option shortens startup time.
• Code generation –– Simulate model using generated C code. The first time you run a simulation, Simulink^® generates C code for the block. The C code is reused for subsequent simulations as long
as the model does not change. This option requires additional startup time but provides faster subsequent simulations.
Data Types Tab
Rounding mode — Rounding mode for output fixed-point operations
Floor (default) | Ceiling | Nearest | Round | Simplest | Zero
Select the rounding mode for output fixed-point operations. The default is Floor.
Coefficients — Word and fraction lengths of coefficients
fixdt(1,16) (default) | fixdt(1,16,0)
Specify the fixed-point data type of the coefficients as one of the following:
• fixdt(1,16) (default) — Signed fixed-point data type of word length 16 with binary point scaling. The block determines the fraction length automatically from the coefficient values in such a way
that the coefficients occupy maximum representable range without overflowing.
• fixdt(1,16,0) — Signed fixed-point data type of word length 16 and fraction length 0. You can change the fraction length to any other integer value.
• <data type expression> — Specify the coefficients data type by using an expression that evaluates to a data type object. For example, numerictype(fixdt([ ],18, 15)). Specify the sign mode of this
data type as [ ] or true.
• Refresh Data Type — Refresh to the default data type.
Click the Show data type assistant button to display the data type assistant, which helps you set the coefficients data type.
Block Characteristics
Data Types double | fixed point | integer | single
Direct Feedthrough no
Multidimensional Signals no
Variable-Size Signals yes
Zero-Crossing Detection no
More About
Halfband Filters
An ideal lowpass halfband filter is given by
$h\left(n\right)=\frac{1}{2\pi }{\int }_{-\pi /2}^{\pi /2}{e}^{j\omega n}d\omega =\frac{\mathrm{sin}\left(\frac{\pi }{2}n\right)}{\pi n}.$
An ideal filter is not realizable because the impulse response is noncausal and not absolutely summable. However, the impulse response of an ideal lowpass filter possesses some important properties
that are required in a realizable approximation. The impulse response of an ideal lowpass halfband filter is:
• Equal to 0 for all even-indexed samples.
• Equal to 1/2 at n=0 as shown by L'Hôpital's rule on the continuous-valued equivalent of the discrete-time impulse response
The ideal highpass halfband filter is given by
$g\left(n\right)=\frac{1}{2\pi }{\int }_{-\pi }^{-\pi /2}{e}^{j\omega n}d\omega +\frac{1}{2\pi }{\int }_{\pi /2}^{\pi }{e}^{j\omega n}d\omega .$
Evaluating the preceding integral gives the following impulse response
$g\left(n\right)=\frac{\mathrm{sin}\left(\pi n\right)}{\pi n}-\frac{\mathrm{sin}\left(\frac{\pi }{2}n\right)}{\pi n}.$
The impulse response of an ideal highpass halfband filter is:
• Equal to 0 for all even-indexed samples
• Equal to 1/2 at n=0
The FIR halfband decimator uses a causal FIR approximation to the ideal halfband response, which is based on minimizing the ${\ell }^{\infty }$ norm of the error (minimax). See Algorithms for more
Kaiser Window
The coefficients of a Kaiser window are computed from this equation:
$w\left(n\right)=\frac{{I}_{0}\left(\beta \sqrt{1-{\left(\frac{n-N/2}{N/2}\right)}^{2}}\right)}{{I}_{0}\left(\beta \right)},\text{ }\text{ }0\le n\le N,$
where I[0] is the zeroth-order modified Bessel function of the first kind.
To obtain a Kaiser window that represents an FIR filter with stopband attenuation of α dB, use this β.
$\beta =\left\{\begin{array}{ll}0.1102\left(\alpha -8.7\right),\hfill & \alpha >50\hfill \\ 0.5842{\left(\alpha -21\right)}^{0.4}+0.07886\left(\alpha -21\right),\hfill & 50\ge \alpha \ge 21\hfill \\
0,\hfill & \alpha <21\hfill \end{array}$
The filter order n is given by:
$n=\frac{\alpha -7.95}{2.285\left(\Delta \omega \right)},$
where Δω is the transition width.
Filter Design Method
The FIR halfband decimator algorithm uses the equiripple or the Kaiser window method to design the FIR halfband filter. When the design constraints are tight, such as very high stopband attenuation
or very narrow transition width, use the Kaiser window method. When the design constraints are not tight, use the equiripple method. If you are not sure of which method to use, set the design method
to Auto. In this mode, the algorithm automatically chooses a design method that optimally meets the specified filter constraints.
Halfband Equiripple Design
In the equiripple method, the algorithm uses a minimax (minimize the maximum error) FIR design to design a fullband linear phase filter with the desired specifications. The algorithm upsamples a
fullband filter to replace the even-indexed samples of the filter with zeros and creates a halfband filter. It then sets the filter tap corresponding to the group delay of the filter in samples to 1/
2. This yields a causal linear-phase FIR filter approximation to the ideal halfband filter defined in Halfband Filters. See [1] for a description of this filter design method using the Remez exchange
algorithm. Since you can design a filter using this approximation method with a constant ripple both in the passband and stopband, the filter is also known as the equiripple filter.
Kaiser Window Design
In the Kaiser window method, the algorithm first truncates the ideal halfband filter defined in Halfband Filters, then it applies a Kaiser window defined in Kaiser Window. This yields a causal
linear-phase FIR filter approximation to the ideal halfband filter.
For more information on designing FIR halfband filters, see FIR Halfband Filter Design.
Polyphase Implementation with Halfband Filters
The FIR halfband decimator uses an efficient polyphase implementation for halfband filters when you filter the input signal. The chief advantage of the polyphase implementation is that you can
downsample the signal prior to filtering. This allows you to filter at the lower sampling rate.
Splitting a filter’s impulse response h(n) into two polyphase components results in an even polyphase component with z-transform of
${H}_{0}\left(z\right)=\sum _{n}h\left(2n\right){z}^{-n},$
and an odd polyphase component with z-transform of
${H}_{1}\left(z\right)=\sum _{n}h\left(2n+1\right){z}^{-n}.$
The z-transform of the filter can be written in terms of the even and odd polyphase components as
You can represent filtering the input signal and then downsampling it by 2 using this figure.
Using the multirate noble identity for downsampling, you can move the downsampling operation before the filtering. This allows you to filter at the lower rate.
For a halfband filter, the only nonzero coefficient in the even polyphase component is the coefficient corresponding to z^0. Implementing the halfband filter as a causal FIR filter shifts the nonzero
coefficient to approximately z^-N/4 where N is the number of filter taps. This process is illustrated in the following figure. The top plot shows a halfband filter of order 52. The bottom plot shows
the even polyphase component. Both of these filters are noncausal. Delaying the even polyphase component by 13 samples creates a causal FIR filter.
To efficiently implement the halfband decimator, the algorithm replaces the delay block and downsampling operator with a commutator switch. When the first input sample is delivered, the commutator
switch feeds this input to the even branch and the halfband decimator computes the first output value. As more input samples come in, the switch delivers one sample at a time to each branch
alternatively. The decimator generates output every time the even branch generates an output. This halves the sampling rate of the input signal.
Which polyphase component reduces to a simple delay depends on whether the half order of the filter is even or odd. Here is the implementation when the filter half order is even. In this diagram, H
[0](z) becomes the gain followed by the delay.
When the filter half order is odd, H[1](z) becomes the gain followed by the delay. This is because the delay required to make the even polyphase component causal can be odd or even depending on the
filter half order.
To confirm this behavior, run the following code in the MATLAB command prompt and inspect the polyphase components of the following filters.
filterspec = "Filter order and stopband attenuation";
halfOrderEven = dsp.FIRHalfbandDecimator(Specification=filterspec,...
halfOrderOdd = dsp.FIRHalfbandDecimator(Specification=filterspec,...
Analysis Filter Bank
The FIR halfband decimator acts as an analysis filter bank and generates two power-complementary output signals by adding and subtracting the two polyphase branch outputs respectively.
For more information on filter banks, see Overview of Filter Banks.
To summarize, the FIR halfband decimator:
• Decimates the input prior to filtering and filters the even and odd polyphase components of the input separately with the even and odd polyphase components of the filter.
• Exploits the fact that one filter polyphase component is a simple delay for a halfband filter.
• Acts as an analysis filter bank.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
Version History
Introduced in R2015b
R2024a: Change in the default value of Simulate using parameter
The default value of the Simulate using parameter is now Interpreted execution. With this change, the block uses the MATLAB interpreter for simulation by default.
R2023b: Support for normalized frequencies
When you set the Sample rate mode parameter to Use normalized frequency (0 to 1), you can specify the transition width in normalized frequency units (0 to 1).
R2023b: Support for arbitrary input frame length
This block supports an input signal with an arbitrary frame length when the:
• Input signal is a fixed-size signal (frame length does not change during simulation) and you select the Allow arbitrary frame length for fixed-size input signals parameter.
• Input signal is a variable-size signal (frame length changes during simulation).
When this block supports an input signal with an arbitrary frame length, the input frame length does not have to be a multiple of the decimation factor 2 and the output signal is a variable-size
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[Solved] Two discs of moments of inertia I1 and I2 about thei... | Filo
Two discs of moments of inertia and about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds and are brought into contact face to face with
their axes of rotation coincident. (a) What is the angular speed of the two-disc system? (b) Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies
of the two discs. How do you account for this loss in energy? Take .
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(a) Total initial angular momentum=
When the discs are joined together, total moment of inertia about the axis becomes=
Let the angular speed of the two-disc system be .
Then from conservation of angular momentum=
(b) Total initial kinetic energy of the system=
Final kinetic energy of the system=
The loss of KE can be attributed to the frictional force that comes into play when the two discs come in contact with each other.
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Two discs of moments of inertia and about their respective axes (normal to the disc and passing through the centre), and rotating with angular speeds and are brought into contact face
Question to face with their axes of rotation coincident. (a) What is the angular speed of the two-disc system? (b) Show that the kinetic energy of the combined system is less than the sum of the
Text initial kinetic energies of the two discs. How do you account for this loss in energy? Take .
Updated On Jun 23, 2023
Topic System of Particles and Rotational Motion
Subject Physics
Class Class 11
Answer Text solution:1 Video solution: 2
Upvotes 267
Avg. Video 3 min
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Explain BPMN collaboration boundary completion rules. | BPMN Assignment Help
Explain BPMN collaboration boundary completion rules. Theorem 1.2 states that, for a configuration $Y$ of qubit one might have to solve the CME constraint $$\langle [p,m]\mid Y\mid Y_L\rangle_{c,\
vert B_\text{top}}=0. \label{eq:cmeeq1.2}$$ [Proof of Theorem 1.2 by Thm. 2.]{} #### Author. John Appling ##### Acknowledgements We are grateful to the reviewers of the original paper with whom we
have worked on the CMEB model: Alan Mironi, Alan Loomis, Matt Healy, Alan Lewis, Bill Moore, Andrew Shire, Ian Stoddard, Andrew Tyabrooper, Hugh Ryan, Nicholas Vaughan-Smith, Alan Sharman, Scott
Stevenson, Andrew Thomas, Mark White, Andreas Schneider, David Stone, Jonathan Schmidt, Paul Winterstein, Steve Verhaar, Thomas Thorntag, Steven Weinstein, and David read This work was done as part
of the project ‘Isabel’, a kind of collaboration between CIMINS, INRIA and Quantum Information Research Institute at LSI. Comments on the original paper and related contributions were greatly
appreciated. Further, David Soper, Alan Lewis, Jack Coddington, Chris Daff and Andrew Tully-Grigoriello were great constant and supportive figures in the proof of the CMLT equations. The authors
would additionally like to extend sincere thanks to David Bost of the Theano Centre Paul Scherrer Institute and Ben Altspace for their help in the proof of the CMLT equations. [^1]: Note that it is
by now assumed that $$\frac{l}{r}(\mathbf{T}\cdot\mathbf{n}) \to 1.\label{eq:1.6}$$ [^2]: The operator representing time evolution can also be represented by a dilation in the way we left it one, the
opposite one from $T=1$ in section \[sec:dilation\], as an example. Explain BPMN collaboration boundary completion rules. In VLC, the use of the WLAN, WiFi, EWP+EVP, and SIP prefixes are explicitly
described. In the future, it is possible to modify this proposal to include the global connection to EWP+EVP technology as well. New subnet-underlying connections do not require modification.
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Instead, these subnets are specified independently by the connections, or rather, at each subnet the topology is chosen as follows. Each subnet requires one large number of connections. my site the
WLAN, the three subnetologies are – *100xe2x88x9221, – *1186, and – *1023. The global connection is given in each of these four interfaces: *WLAN-10E11-,…, and *WLAN-10E12-. Extending the WLAN and
corresponding international interface through the *10xe2x80x2/WLAN-10E11-*: We have recently introduced a new protocol, in *WLAN-1-21, the *WLAN-1-21-1, *W/e * 1-100xe2x88x921-100xe2x88x9229. The
basic idea of this protocol is to use WLAN-1 and WLAN-20×0/WLAN-20-5 for WLAN-C1, WLAN-C2, WLAN-C3, and WLAN-C4 and to define some physical relations between the two-way interfaces the WLAN and (in
future) WLAN-C1 and WLAN-C2. Since it is possible to access multiple WAN networks by the visit site (20px/10×10) standard, it was proposed that the additional WLAN-C1 and WLAN-C2/C3 protocol together
with the *ZIP-20×0-20-10×10* (101xe2x88x921, or the combination of WLAN-10/WS0-1, WLAN-10/WS1-2, WLAN-10/WS2-3/W1/2, WLAN-10/WS3-5-, WLAN-10/WS4-1-, WLAN-10/WS4-2/) satisfy this requirement. Note in
this case the two-way L1 and the following L2-L3-L4 link would have no additional link to the WLAN-01 domain (WLAN-11, or WLAN-11-10), while the other link would have as little as the 0-5 level,
which is always assigned. The additional WLAN-10/WS12-10 link would have to be added (to WLAN-01) or removed (to WLAN-01-10×10). Alternatively, the network and users could have their own WLAN and
WLAN-C2 and WLAN-C3, or their own WLAN and WLAN-C2/C3, or their own WLAN-10/WS12/L1/14-10 link. EPSIP is now also implemented in *WLAN-E01x-21*, while EIMED is implemented in *WLAN-10-14x-11×2344.
The L1 and L2-L4 are described in more detail in Section \[sec:algo\_net\]. Therefore, the new L1-L2-L4 link is a *reactive* link, just as the two-way L2-L3-L4 link is one-way L3-L4 link. This allows
a more aggressive configuration of WLAN connections. This new network-underlying connection then provides aExplain BPMN collaboration boundary completion rules. Abstract: The experiment of combining
two different versions of the same atom into a single atom has been recently explored. Simulation has shown that a full atom is worth half of the total $\mathrm{nm}$ of $\text{Be}\nu/\text{Me}\nu$.
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Furthermore, simulations have shown that a full atom is worth all $\mathrm{nm}$ of $\text{Be}\nu/\text{Me}\nu$ when the dielectric length is the same as the length of the particle. However, the time
evolution of the atom is much slower than that of the particle. It was therefore suggested that there should be a significant time-dependent reduction in the number of layers of the atom or atoms
that is needed for calculating a full atom. From the results of a theoretical study by Pappano and coworkers [@PappanoPRB01], it became clear that the time-independent reduction of $0.05$, for a
total atom of about $25$ atoms: $$\frac{h^4}{m}\propto\frac{1}{\mathrm{Thc}}\left[\frac{\pi \text{nm}}{k_Bm}\right].\label{eqn:maxN}$$ Figure \[fig:Ais\] shows some comparison of the different
experimental results and theories. The most-expressed state was initially in a 1-D Ising model. The state was then subjected to single-particle dynamics as discussed in Sec. \[subsec:Complexons\],
using force-field methods on the CGO of BECs. Clearly, we are well informed about the physics underlying the non-trivial inversion of the BEC’s CGO by the fact that the exact initial state was not
recovered with the subsequent evolution of the Wannier functions. So, we conclude that very little is known beyond the relatively simple analysis which starts from Ref. . So, the study in Sec. \
[subsec:HexAlatomy\] comes with some basic limitations. This means that at the time in which the dynamics of the CGO is known, several models may or may see this here be valid and other steps are
needed to model it. In addition, several authors have attempted to model directly the same atom in a non-relativistic description for the atom-by-atom CGO interaction which relies upon the model and
the structure of the CGO. To find the potential energy of such model would require the understanding of the CGO’s coupling with the gas. So it is important to add some assumptions to make the CGOs
couplings so close to the equilibrium conditions. Although the CGO spectra were reported in Ref. , it was discovered that it involves the CGO solvation potential, which is related to the scattering
potential of the system, and
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Robodebt not only broke the laws of the land
Robodebt not only broke the laws of the land – it also broke laws of mathematics
published : 24 February 2024
Friday marked the end of the public hearings for the Royal Commission into the Robodebt Scheme. They painted a picture of a catastrophic program that was legally and ethically indefensible – an
example of how technological overreach, coupled with dereliction of duty can amount to immense suffering for ordinary people.
The artificial intelligence (AI) algorithm behind Robodebt has been called “flawed”. But it was worse than that; it broke laws of mathematics. A mathematical law called Jensen’s inequality shows the
Robodebt algorithm should have generated not only debts, but also credits.
What was Robodebt?
The Australian government’s Robodebt program was designed to catch people exploiting the Centrelink welfare system.
The system compared welfare recipients’ Centrelink-reported fortnightly income with their ATO-reported yearly income, the latter of which was averaged to provide fortnightly figures that could be
lined up with Centrelink’s system.
If the difference showed an overpayment by Centrelink, a red flag was raised. The AI system then issued a debt notice and put the onus on the recipient to prove they weren’t exploiting the welfare
A Robodebt victim
To understand the extent of the failure, let’s consider a hypothetical case study. Will Gossett was a university student from 2017-2019. He was single, older than 18, and living at home with his
Will received Centrelink payments according to his fortnightly income from a couple of casual jobs with highly variable work hours. In his first year at university his jobs didn’t pay much, so he
received more Centrelink payments in the 2018 financial year than the year following.
The Robodebt algorithm took Will’s ATO yearly income records for both the 2018 and 2019 financial years and, for each year, averaged them into a series of fortnightly “robo” incomes.
Inside Robodebt’s AI world, his fortnightly incomes were then the same throughout the 2018 financial year, and the same throughout the 2019 financial year.
Will was honest with his claims, but was stunned to receive a debt notice for Centrelink overpayments made in the 2019 financial year – the year in which he actually received lower welfare payments.
The income-averaging algorithm gave Will an average fortnightly income for 2019 that was above the threshold that made him eligible for Centrelink payments. As far as the Robodebt system was
concerned, Will shouldn’t have received any welfare payments that year.
Jensen’s inequality
The laws of mathematics tell us when two things are equal, but they can also tell us when one thing is bigger than another. This type of law is called an “inequality”.
To understand why and when Robodebt failed for Will, we need to understand a concept called Jensen’s inequality, credited to Danish mathematician Johan Jensen (1859-1925).
Jensen’s inequality explains how making a decision based on the averaging of numbers leads to either a negative bias or a positive bias under a “convexity condition”, which I’ll explain soon.
You’ll recall Will is a single university student, above 18, and living with his parents. Based on these factors, Centrelink has a fortnightly payment table for him, illustrated with the curve in the
figure below.
The figure shows the more income Will earns from his jobs, the less welfare payment he receives, until a specific income, after which he receives none. This graph, created from tables provided by
Centrelink, shows how certain factors determine the amount of welfare payments Will can receive depending on his income.
The parts of the curve where Jensen’s inequality is relevant are highlighted by two red squares. In the square on the left, the curve bends downwards (concave), and in the square on the right it
bends upwards (convex).
Because Will’s income was higher in 2019 and spread across the part where the payment curve is convex, Jensen’s inequality guarantees he would receive a Robodebt notice, even though there was no
In 2018, however, Will’s income distribution was spread around smaller amounts where the curve is concave. So if Jensen’s inequality was adhered to, the AI algorithm should have issued him a
“Robocredit” – but it didn’t.
It could be the algorithm contained a line of code that nullified Jensen’s inequality by instructing any credits be ignored.
Big data and a bad algorithm
The people responsible for the Robodebt system should have had a strong interest in keeping error rates low. Data scientists have a big red “stop” button when error rates of automated systems go
beyond a few percent.
It’s straightforward to estimate error rates for an AI scheme. Experts do this by running simulations inside a virtual model called a “digital twin”. These can be used to carry out statistical
evaluations, and expose conscious and unconscious biases in bad algorithms.
In Robodebt’s case, a digital twin could have been used to figure out error rates. This would have required running the Robodebt algorithm through representative incomes simulated under two different
Under the first scenario, incomes are simulated assuming no debt is owed by anyone. Every time a result is returned saying a debt is owed, a Type 1 (or false-positive) error is recorded. Under the
second scenario, incomes are simulated assuming everyone owes a debt (to varying degrees). If a no-debt result is returned, a Type 2 (false-negative) error rate is recorded.
Then an error rate is estimated by dividing the number of errors by the number of simulations, within each scenario.
Eye-watering inaccuracies
Although no consistently reliable error rates have been published for Robodebt, a figure of at least 27% was quoted in Parliament Question Time on February 7.
The reality was probably much worse. During the scheme, on the order of one million income reviews were performed, of which 81% led to a debt being raised.
Of these, about 70% (roughly 567,000 debts) were raised through the use of income averaging in the Robodebt algorithm.
In 2020, the government conceded about 470,000 debts had been falsely raised, out of a total of about 567,000.
Back-of-the-envelope calculations give a Type 1 (false-positive) error rate on the order of 80% (470,000/567,000). Compared to the usual target of a few percent, this is an eye-wateringly large error
If simulations had been run, or human intelligence used to check real cases, the “stop” button would have been hit almost immediately.
Jensen’s inequality establishes why and when income averaging will fail, yet income matching hasn’t gone away. It can be found in AI software used for official statistics, welfare programs, bank
loans and so forth.
Deeper statistical theory for this “change of support” problem — for example, going from data on yearly support to fortnightly support — will be needed as AI becomes increasingly pervasive in
essential parts of society.
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Linear Regression illustration
This GeoGebra worksheet allows you to control the line of best fit by means of two sliders giving the intercept (c) and the slope (m). The idea is to find a line that gives a "best" fit to the data.
The "residuals" are defined as the distance in the y direction between the data and this fitted line. The residuals are denoted by green vectors. The aim is to find a line fit that has the smallest
possible sum of squares of these residuals. In order to help you, the squares of the residuals are drawn on the page, and the sum of squares is given in a box. This best fit is defined as the
smallest possible sum of the squares of the residuals.
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Symposium on `Quarks to Universe in Computational Science (QUCS 2015)’
Deadline extended of abstract submission: September 25th, 2015
Dates: November 4-8, 2015
Location: Nara Prefectural New Public Hall, Nara, Japan
Web: https://www.jicfus.jp/en/qucs2015/
The aim of the symposium is to summarize the HPCI Strategic Program “The Origin of Matter and the Universe” (2011-2015).
This is a joint program of computational particle/nuclear/astro physics. By using 10 PFlops K-computer, and will be continued further in a different scheme even beyond 2015. At this symposium, we
will discuss the current topics in particle physics, nuclear physics and astrophysics from the point of view of large scale numerical simulations.
The major topics will include
• hadron physics from lattice QCD
• nuclear physics from lattice QCD
• high precision quantum few-body calculations
• large scale simulations of nuclear many-body problems
• supernova simulations
• compact star mergers
• formation of black holes and gravitational waves
• algorithms for computational physics
• etc.
Now, the registion and submission of the abstract is open.
The important dates:
Deadline of abstract submission: September 15th, 2015; Deadline extended : September 25th
Deadline of Registration fee: October 15th , 2015
We look forward to seeing you in Nara.
Shinya Aoki (chair, YITP), Tetsuo Hatsuda (vice-chair, RIKEN), Masayuki Umemura (vice-chair, Tsukuba)
Scientific secretaries:
Emiko Hiyama(RIKEN), Hideo Matsufuru(KEK)
contacting e-mail address: qucs2015 ’at’ ml.kek.jp
Wednesday 04 November 2015
Registration – Registration desk (08:40-09:15)
Plenary session 1 – Noh theatre (09:15-10:50)
– Conveners: Umemura, Masayuki
09:15 Opening address AOKI, Sinya
09:20 Summary of HPCI Strategy Field 5 AOKI, Sinya
09:25 Report on HPCI (Lattice QCD) HATSUDA, Tetsuo
09:40 Report on HPCI (Quantum Many-Body) OTSUKA, Takaharu
09:55 Report on HPCI (First Generation Objects) MAKINO, Junichiro
10:10 Report on HPCI (Supernova Explosion and Blackhole) SHIBATA, Masaru
10:25 Report on HPCI (Computational Science) HASHIMOTO, Shoji
Plenary session 2 – Noh theatre (11:10-12:40)
– Conveners: Aoki, Sinya
11:10 Ab initio calculation of the neutron proton mass difference FODOR, Zoltan
12:10 Recent results of Lattice QCD using chiral quarks IZUBUCHI, Taku
Plenary session 3 – Noh theatre (14:00-16:00)
– Conveners: Hatsuda, Tetsuo
14:00 Overview of Strangeness Nuclear Physics NAGAE, Tomofumi
14:30 Radiative Transfer Simulations in Compact Star Mergers TANAKA, Masaomi
15:00 RECENT RESULTS FROM SUPER-KAMIOKANDE NAKAHATA, Masayuki
15:30 Nuclear structure and excitations clarified by Monte Carlo Shell Model SHIMIZU, Noritaka
calculation on K computer
Plenary session 4 – Noh theatre (16:20-17:50)
– Conveners: Shibata, Masaru
16:20 Planet Formation and Its Simulations IDA, Shigeru
16:50 Simulations of planetesimal formation JOHANSEN, Anders
17:20 Simulations of Star formation MACHIDA, Masahiro
Thursday 05 November 2015
Registration – Registration desk (08:40-09:00)
Plenary session 5 – Noh theatre (09:00-11:00)
– Conveners: Otsuka, Takaharu
09:00 Computational nuclear structure in the eve of exascale NAZAREWICZ, Witold
10:00 Topology in lattice QCD FUKAYA, Hidenori
10:30 A glimpse into the cold atom world: Results from small and large(r) scale computations BLUME, Doerte
Plenary session 6 – Noh theatre (11:20-12:20)
– Conveners: Ishikawa, Kenichi
11:20 Cluster model approaches to nuclear many-body dynamics FUNAKI, Yasuro
11:50 Observations of High-z Galaxies OUCHI, Masami
Parallel session 1A – Noh theatre (13:50-15:50)
– Conveners: Ukita, Naoya
13:50 Chemical enrichment of passive galaxies in cosmological simulatins OKAMOTO, Takashi
14:10 Radiation hydrodynamic simulations on the possibility of radiation-supported AGN tori NAMEKATA, Daisuke
14:30 Time evolution of the Sgr A* accretion flow interacting with the G2 cloud KAWASHIMA, Tomohisa
14:50 Relativistic Radiation Magnetohydrodynamic Simulations of the Black Hole Accretion Disks and Outflows TAKAHASHI, Hiroyuki
15:10 Radiation hydrodynamic simulations of line-driven disk winds around super massive black holes NOMURA, Mariko
15:30 General relativistic magnetohydrodynamics simulations of compact binary mergers on K KIUCHI, Kenta
Parallel session 1B – Conference room 1 (13:50-15:50)
– Conveners: Doi, Takumi
13:50 Lattice QCD studies of Omega-Omega and Delta-Delta interactions including results at physical point GONGYO, Shinya
14:10 A study of LcN 2-body system on the lattice MIYAMOTO, Takaya
14:30 Light nuclei and nucleon form factors from lattice QCD YAMAZAKI, Takeshi
14:50 Lattice QCD studies of baryon-baryon interactions: the potential method and the direct method ISHII, Noriyoshi
15:10 Non-locality of a wave-function-equivalent potential using the derivative expansion SUGIURA, Takuya
15:30 Algorithm, benchmarks, and hyperon potentials with strangeness S=-1 at almost physical point NEMURA, Hidekatsu
Parallel session 2A – Noh theatre (16:10-18:10)
– Conveners: TBA
16:10 Dynamical mass ejection from black hole-neutron star binaries KYUTOKU, Koutarou
16:30 3D GRMHD simulations of jets from black hole and accretion disk MIZUTA, Akira
16:50 A new class of rotational explosion in core-collpase supernovae TAKIWAKI, Tomoya
17:10 Systematic Features and Progenitor Dependences of Core-collapse Supernovae NAKAMURA, Ko
17:30 Equation of state including full nuclear ensemble in supernova simulations FURUSAWA, Shun
17:50 Microscopic equation of state for supernova matter with realistic nuclear forces TOGASHI, Hajime
Parallel session 2B – Conference room1 (16:10-18:10)
– Conveners: Ishii, Noriyoshi
16:10 Physical point lattice QCD simulation on the S = -2 baryon-baryon interactions SASAKI, Kenji
16:30 Structure of Zc(3900) from coupled-channel scattering on the lattice IKEDA, Yoichi
16:50 The nuclear matrix element of double beta decay IWATA, Yoritaka
17:10 Effects of nuclear many-body correlations on neutrinoless double-beta decay TERASAKI, Jun
in quasiparticle random-phase approximaion
17:30 Monte Carlo shell model for electric dipole strength distribution in medium-heavy nuclei TOGASHI, Tomoaki
17:50 Collaborative code development, through the development of the lattice common code “Bridge++” UEDA, Satoru
Poster session – Conference room 2 (18:20-19:30)
P01: Accretion versus merger in the early growth of massive black holes TAGAWA, Hiromichi
P02: CORE-K Simulation: COsmic REionization simulation with K-Computer HASEGAWA, Kenji
P03: Comparative study of topological charge in lattice QCD NAMEKAWA, Yusuke
P04: Supercomputing for exploring electron accelerations in strong shock waves MATSUMOTO, Yosuke
P05: The Strategy for 6D Simulations of Core-Collapse Supernovae with Boltzmann-Hydro Code IWAKAMI, Wakana
P06: General relativistic radiative transfer simulations around a Kerr black hole TAKAHASHI, Rohta
P07: Density Independent Formulation of SPH SAITOH, Takayuki
P08: Alpha-cluster structure for Be isotopes appeared in the wave function of Monte Carlo shell model YOSHIDA, Tooru
P09: Geometrical structure of helium triatomic systems: comparison with the neon trimer SUNO, Hiroya
P10: Signature of (1405) in d(K-; n) reaction OHNISHI, Shota
P11: Center for Computational Astrophysics at National Astronomical Observatory of Japan KOKUBO, Eiichiro
P12: Effective restoration of axial symmetry at finite temperature COSSU, Guido
P13: Japan Lattice Data Grid MATSUFURU, Hideo
Friday 06 November 2015
Plenary session 7 – Noh theatre (09:00-11:00)
– Conveners: Makino, Junichiro
09:00 Core-Collapse Supernova Theory: the current status and future prospects YAMADA, Shoichi
10:00 Nuclear Physics from Lattice QCD DOI, Takumi
10:30 Connecting the quarks to the cosmos and beyond WALKER-LOUD, Andre
Plenary session 8 – Noh theatre (11:20-12:50)
– Conveners: Hashimoto, Shoji
11:20 Novel applications of gradient flow KAPLAN, David, B.
11:50 Excited state energies and scattering phase shifts from lattice QCD with the stochastic LapH method MORNINGSTAR, Colin
12:20 Binary neutron star mergers and r-process nucleosynthesis SEKIGUCHI, Yuichiro
lunch – (12:50-14:20)
Parallel session 3A – Noh theatre (14:20-16:00)
– Conveners: Yamazaki, Takeshi
14:20 Light and Heavy decay constants from Lattice QCD using Domain-Wall fermions FAHY, Brendan
14:40 Improved lattice fermion action for heavy quarks CHO, Yong-Gwi
15:00 Charmonium current-current correlators with Mobius domain wall fermion NAKAYAMA, Katsumasa
15:20 Analysis of short distance current correlators using OPE TOMII, Masaaki
15:40 The perturbation analysis of the Moebius Domain Wall Fermion with the Schroeginger functional scheme MURAKAMI, Yuko
Parallel session 3B – Conference room 1 (14:20-16:00)
– Conveners: Ohnishi, Akira
14:20 Galactic Evolution of Supernova and Merger R-process KAJINO, Taka
14:40 Solar global convection and dynamo with the Reduced Speed of Sound Technique HOTTA, Hideyuki
15:00 Athena++: a New RMHD Simulation Code with Adaptive Mesh Refinement TOMIDA, Kengo
15:20 High-Resolution Global N-body Simulation of Planetary Formation: Outward Migration of a Protoplanet KOMINAMI, Junko
15:40 –
Parallel session 4A – Noh theatre (16:20-18:40)
– Conveners: Namekawa, Yusuke
16:20 Determination of the ratio between the $\Lambda$-parameter ratio associated with the Schr\”{o}dinger functional and the twisted gradient flow in the pure SU(3) gauge theory UENO, Ryoichiro
16:40 Thermodynamics of SU(3) gauge theory using gradient flow ITOU, Etsuko
17:00 Strong-Coupling Lattice QCD with fluctuation and plaquette effects OHNISHI, Akira
17:20 Study of high density lattice QCD with canonical approach TANIGUCHI, Yusuke
17:40 Universality test of Complex Langevin approach for chiral random matrix theories ICHIHARA, Terukazu
18:00 The application of the complex Langevin method to a matrix model with spontaneous rotational symmetry breaking ITO, Yuta
18:20 Precision test of gauge/gravity duality by lattice simulation SHIMASAKI, Shinji
Parallel session 4B – Conference room1 (16:20-18:40)
– Conveners: Shimizu, Noritaka
16:20 Monte Carlo shell model calculations for structure of nuclei around Z=28 TSUNODA, Yusuke
16:40 Calculations for medium-mass nuclei with the chiral EFT interactions in the unitary-model-operator approach MIYAGI, Takayuki
17:00 Recent development of finite-amplitude method for nuclear collective excitation HINOHARA, Nobuo
17:20 Impurity effects in deformed/clustering hypernuclei with antisymmetric molecular dynamics ISAKA, Masahiro
17:40 Weak decay of Lambda_c for the study of Lambda resonances MIYAHARA, Kenta
18:00 Compositeness of near-threshold quasi-bound states KAMIYA, Yuki
18:20 Infinite basis-space extrapolation of ground-state energies in no-core Monte Carlo shell model ABE, Takashi
Saturday 07 November 2015
Plenary session 9 – Noh theatre (09:00-11:00)
– Conveners: Nakatsukasa, Takashi
09:00 Radiation Magnetohydrodynamic (MHD) Simulations of Astrophysical Accretion STONE, James M.
10:00 The r-process in the ejecta of neutron star mergers WANAJO, Shinya
10:30 Nuclear physics from chiral effective field theory SCHWENK, Achim
Plenary session 10 – Noh theatre (11:20-12:50)
– Conveners: Ishizuka, Naruhito
11:20 Simulation Study of Solar-Terrestrial Environment KUSANO, Kanya
11:50 2+1 avor QCD simulation near the physical point on a 96^4 lattice UKITA, Naoya
12:20 From nuclear force to neutron-rich nuclei TSUNODA, Naofumi
Public lecture – (15:00-17:00)
HASHIMOTO, Koji
15:00 Public lecture (Japanese) HATSUDA, Tetsuo
YOSHIDA, Naoki
Banquet – (18:00-20:00)
Banquet is held at Hotel Nikko Nara.
Sunday 08 November 2015
Plenary session 11 – Noh theatre (09:00-11:00)
– Conveners: Tomisaka, Kohji
09:00 Skyrme energy-density-functional method for large-scale linear-response calculations YOSHIDA, Kenichi
09:30 Finite density lattice QCD simulations towards the understanding of QGP, heavy ion collisions and nuclear matter NAGATA, Keitaro
10:00 Observational constraints on r-process AOKI, Wako
10:30 Full Boltzmann-Hydrodynamic Simulations for Core Collapse Supernovae on K computer NAGAKURA, Hiroki
Plenary session 12 – Noh theatre (11:20-13:00)
– Conveners: Sakurai, Tetsuya
11:20 Numerical test of gauge/gravity duality – from lattice gauge theory to black hole physics – KADOH, Daisuke
11:50 Radiation magnetohydrodynamics simulations of accretion ows and out ows around black holes OHSUGA, Ken
12:20 Formation of the First Stars in the Universe YOSHIDA, Naoki
12:50 Closing
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Understanding Mathematics: The Fundamental Differences between Axioms and Theorems Explained
Ever found yourself tangled in the intricate web of mathematical terminologies? You’re not alone. Understanding these terms, such as ‘axiom’ and ‘theorem’, can sometimes feel like learning a new
language altogether! But don’t worry – we’ve got your back.
In the world of mathematics, axioms and theorems are fundamental pillars that hold up complex theories. They may seem similar at first glance but they play distinct roles in building logical
structures. So what’s exactly is an axiom? And how does it differ from a theorem?
Understanding Axioms and Theorems
What Is an Axiom?
An axiom, often regarded as a self-evident truth in mathematics, forms the base for logical reasoning. This principle doesn’t require proof due to its universally accepted nature. For instance,
consider the zero property of multiplication stating that any number multiplied by zero equals zero – it’s an axiom because you accept this without demanding evidence.
A mathematical system typically starts with axioms; they act like building blocks laying out fundamental concepts or assumptions. To further clarify: Euclidean Geometry begins with five key axioms
known as “Euclid’s Postulates,” such as ‘a straight line segment can be drawn joining any two points.’
Without these foundational statements (axioms), developing complex theories becomes virtually impossible.
What Is a Theorem?
In contrast to an axiom, a theorem isn’t inherently true but rather requires verification through rigorous proof before acceptance within the mathematical community. Essentially derived from
previously established truths or axioms using deductive logic processes are these theoretical propositions called theorems.
Take Pythagoras’ theorem for example—it asserts that in right-angled triangles, square on hypotenuse equates sum squares of other two sides—this didn’t simply emerge out-of-the-blue nor is it taken
at face value! It demanded meticulous demonstration based upon existing geometrical facts before becoming recognized universally.
The Origin of Axioms and Theorems
The birthplace of axioms and theorems harks back to a period when humans began trying to understand, quantify, and rationalize their surroundings. Let’s investigate deeper into these origins.
Historical Development of Axioms
A peek into antiquity reveals that axioms’ origin traces back to ancient Greek mathematicians like Euclid (circa 300 BC). His seminal work “Elements,” comprised thirteen books filled with geometric
truths he believed were self-evident – essentially our first recorded set of mathematical axioms. These foundational principles guided thinkers for centuries, forming the backbone for logical
deductions in mathematics even today.
Yet it wasn’t all smooth sailing! By introducing non-Euclidean geometries during the nineteenth century – ones where Euclid’s fifth postulate didn’t hold true – mathematicians such as Lobachevsky
stirred quite a controversy. This shift demonstrated how some truths taken as given might not be universal after all!
As you traverse through time, there is an unmistakable evolution witnessed in defining what qualifies as an axiom. While initially seen purely from intuitive or empirical perspectives within specific
systems (Euclidean geometry), modern viewpoints view them more abstractly — applicable across various structures without necessarily being ‘self-evidently true’.
Historical Context of Theorems
While on this historical journey let us also shed light on theorem development over ages.
It was again under Greece’s azure skies that Thales kickstarted formal reasoning processes using deductive logic around 600 BC—considered by many historians his eponymous theorem’s establishment
marks humanity’s initial leap towards rigorous proof building.
This trend continued unabated till Pythagoras formulated his famous theorem about right-angled triangles; offering perhaps history’s most recognized example demonstrating deduction use from existing
facts or propositions i.e., application derived from previously established premises—the cornerstone upon which any robust mathematical argument stands.
In later centuries, theorems grew increasingly complex and abstract with developments in mathematical branches like calculus or number theory. The path wasn’t always straightforward though; remember
Fermat’s Last theorem? Simple to state but its proof evaded mathematicians for more than three centuries till Andrew Wiles finally solved it using modern techniques!
Through these historical journeys of axioms and theorems, we appreciate their significant roles within mathematics’ evolution – from creating fundamental concepts defining structures to offering a
rigorous framework enabling logical deductions.
Key Differences Between Axioms and Theorems
Basis of Truth
The distinction between axioms and theorems originates from their respective bases of truth. While an axiom presents itself as a self-evident fact, it’s not subject to proof or disproof due its
acceptance as a fundamental premise in mathematical reasoning.
Consider Euclid’s postulate: “A straight line segment can be drawn joining any two points,” recognized universally without demanding evidence. But, each theorem stands on solid grounds only after
undergoing rigorous scrutiny via logical deduction based on previously established statements – which could include both axioms and other proven theorems.
Take Pythagoras’ theorem for instance: “In a right-angled triangle, square off the hypotenuse equals summing up squares off other two sides.” This statement doesn’t claim instant acceptance but
requires demonstrative proof starting from accepted facts – those very axioms we spoke about earlier!
Role in Mathematical Reasoning
Diving deeper into roles they play in mathematics reveals more differences. In essence, you’d find that while every set theory uses some form of axiom(s) to lay down foundational principles (like
Zermelo-Fraenkel Set Theory), there isn’t really such thing called ‘theorem theory’. That’s because by nature, all theories employ certain ‘given truths’ before building complex structures atop them
using deductive logic.
This is where function begins differing markedly for these entities! An axiom helps provide framework inside which mathematical discourse happens whereas every new theorem brings forth enriched
understanding within this predefined structure – expanding our knowledge frontiers incrementally yet significantly.
For example – Fermat’s Last Theorem (originally conjecture turned theorem!) might have been rooted originally upon base givens like definitions of whole numbers or addition operation; but over
centuries its eventual confirmation has propelled mathematicians towards uncharted territories involving advanced concepts such as elliptic curves and modular forms.
And so, while axioms and theorems both hold pivotal roles within mathematical world, they indeed differ in how their truths are established as well as ways they contribute towards evolution of this
abstract universe.
Examples in Mathematics
Building on the foundational understanding of axioms and theorems, let’s investigate into some specific examples to further clarify these mathematical concepts.
Examples of Axioms
A popular example comes from Euclid’s five postulates. The fifth one—known as the Parallel Postulate—states that if a straight line crossing two other lines makes interior angles on the same side
less than 180 degrees, then those two lines will meet on that side if extended far enough. This axiom is universally accepted without proof; it forms an integral part of geometric reasoning.
Another classic example pertains to arithmetic—the Commutative Property of Addition. It simply states: for any numbers ‘a’ and ‘b’, a + b equals b + a. Although this may seem like common sense or
instinctive knowledge, it’s technically an axiom since we accept it without requiring a formal proof!
Examples of Theorems
In contrast with axioms, Pythagoras’ theorem stands out among others—it needs demonstrable proof before acceptance! As you might recall from your geometry lessons at school, this theorem declares: In
right-angled triangles (triangle ABC), square AC is equal to squares AB plus BC (AC² = AB² + BC²).
Let us look at another theorem – Fermat’s Last Theorem—a notoriously challenging statement proposed by Pierre de Fermat back in 1637 but only proven centuries later by Andrew Wiles in 1994!
Essentially stating there are no three positive integers (a,b,c) satisfying an+bn=cn when n>2; demonstrating its truth required advanced techniques beyond traditional mathematics available during
Fermat’s time.
Through each unique instance mentioned above—from Euclidean geometry rules being declared self-evident truths (axioms), through classical triangle properties needing rigorous validation(theorems)—you
can better grasp how both play distinct yet interrelated roles in mathematical discourse.
You’ve journeyed through the world of axioms and theorems, understanding their fundamental differences. You’ve seen how axioms stand as self-evident truths requiring no proof – like Euclid’s
postulates or the Commutative Property of Addition. On the other hand, you learned that a theorem is something entirely different; it needs validation such as Pythagoras’ theorem or Fermat’s Last
Theorem. These examples show us not only what sets them apart but also their interconnected roles in shaping mathematical discourse throughout history.
So remember: while they may seem similar at first glance, there’s a whole universe between an axiom and a theorem waiting for your exploration! Embrace these concepts to deepen your grasp on
mathematics and further unlock its vast potential.
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guide Archives
The scientific method is a proven procedure for expanding knowledge through experimentation and analysis. It is a process that uses careful planning, rigorous methodology, and thorough assessment.
Statistical analysis plays an essential role in this process.
In an experiment that includes statistical analysis, the analysis is at the end of a long series of events. To obtain valid results, it’s crucial that you carefully plan and conduct a scientific
study for all steps up to and including the analysis. In this blog post, I map out five steps for scientific studies that include statistical analyses. [Read more…] about 5 Steps for Conducting
Scientific Studies with Statistical Analyses
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A quadrature that implements the Duffy transformation from a square to a triangle to integrate singularities in the origin of the reference simplex.
The Duffy transformation is defined as
\[ \begin{pmatrix} x\\ y \end{pmatrix} = \begin{pmatrix} \hat x^\beta (1-\hat y)\\ \hat x^\beta \hat y \end{pmatrix} \]
with determinant of the Jacobian equal to \(J= \beta \hat x^{2\beta-1}\). Such transformation maps the reference square \([0,1]\times[0,1]\) to the reference simplex, by collapsing the left side of
the square and squeezing quadrature points towards the origin, and then shearing the resulting triangle to the reference one. This transformation shows good convergence properties when \(\beta = 1\)
with singularities of order \(1/R\) in the origin, but different \(\beta\) values can be selected to increase convergence and/or accuracy when higher order Gauss rules are used (see "Generalized
Duffy transformation for integrating vertex singularities", S. E. Mousavi, N. Sukumar, Computational Mechanics 2009).
When \(\beta = 1\), this transformation is also known as the Lachat-Watson transformation.
Definition at line 802 of file quadrature_lib.h.
Quadrature< spacedim > QSimplex< dim >::compute_affine_transformation ( const std::array< Point< spacedim >, dim+1 > & vertices ) const inherited
Return an affine transformation of this quadrature, that can be used to integrate on the simplex identified by vertices.
Both the quadrature point locations and the weights are transformed, so that you can effectively use the resulting quadrature to integrate on the simplex.
The transformation is defined as
\[ x = v_0 + B \hat x \]
where the matrix \(B\) is given by \(B_{ij} = v[j][i]-v[0][i]\).
The weights are scaled with the absolute value of the determinant of \(B\), that is \(J \dealcoloneq |\text{det}(B)|\). If \(J\) is zero, an empty quadrature is returned. This may happen, in two
dimensions, if the three vertices are aligned, or in three dimensions if the four vertices are on the same plane. The present function works also in the codimension one and codimension two case. For
instance, when dim=2 and spacedim=3, we can map the quadrature points so that they live on the physical triangle embedded in the three dimensional space. In such a case, the matrix \(B\) is not
square anymore.
[in] vertices The vertices of the simplex you wish to integrate on
A quadrature object that can be used to integrate on the simplex
Definition at line 709 of file quadrature_lib.cc.
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Introductory Econometrics Examples
Example 4.1 Hourly Wage Equation
Using the same model estimated in example: 3.2, examine and compare the standard errors associated with each coefficient. Like the textbook, these are contained in parenthesis next to each associated
Dependent variable:
educ 0.09203^*** (0.00733)
exper 0.00412^** (0.00172)
tenure 0.02207^*** (0.00309)
Constant 0.28436^*** (0.10419)
Observations 526
R^2 0.31601
Adjusted R^2 0.31208
Residual Std. Error 0.44086 (df = 522)
F Statistic 80.39092^*** (df = 3; 522)
Note: ^p<0.1; ^p<0.05; ^p<0.01
For the years of experience variable, or exper, use coefficient and Standard Error to compute the \(t\) statistic:
\[t_{exper} = \frac{0.004121}{0.001723} = 2.391\]
Fortunately, R includes \(t\) statistics in the summary of model diagnostics.
(Intercept) 0.28436 0.10419 2.72923 0.00656
educ 0.09203 0.00733 12.55525 0.00000
exper 0.00412 0.00172 2.39144 0.01714
tenure 0.02207 0.00309 7.13307 0.00000
Plot the \(t\) statistics for a visual comparison:
Example 4.7 Effect of Job Training on Firm Scrap Rates
Load the jtrain data set.
From H. Holzer, R. Block, M. Cheatham, and J. Knott (1993), Are Training Subsidies Effective? The Michigan Experience, Industrial and Labor Relations Review 46, 625-636. The authors kindly provided
the data.
\(year:\) 1987, 1988, or 1989
\(union:\) =1 if unionized
\(lscrap:\) Log(scrap rate per 100 items)
\(hrsemp:\) (total hours training) / (total employees trained)
\(lsales:\) Log(annual sales, $)
\(lemploy:\) Log(umber of employees at plant)
First, use the subset function and it’s argument by the same name to return observations which occurred in 1987 and are not union. At the same time, use the select argument to return only the
variables of interest for this problem.
jtrain_subset <- subset(jtrain, subset = (year == 1987 & union == 0),
select = c(year, union, lscrap, hrsemp, lsales, lemploy))
Next, test for missing values. One can “eyeball” these with R Studio’s View function, but a more precise approach combines the sum and is.na functions to return the total number of observations equal
to NA.
## [1] 156
While R’s lm function will automatically remove missing NA values, eliminating these manually will produce more clearly proportioned graphs for exploratory analysis. Call the na.omit function to
remove all missing values and assign the new data.frame object the name jtrain_clean.
Use jtrain_clean to plot the variables of interest against lscrap. Visually observe the respective distributions for each variable, and compare the slope (\(\beta\)) of the simple regression lines.
Now create the linear model regressing hrsemp(total hours training/total employees trained), lsales(log of annual sales), and lemploy(the log of the number of the employees), against lscrap(the log
of the scrape rate).
\[lscrap = \alpha + \beta_1 hrsemp + \beta_2 lsales + \beta_3 lemploy\]
Finally, print the complete summary diagnostics of the model.
Dependent variable:
hrsemp -0.02927 (0.02280)
lsales -0.96203^** (0.45252)
lemploy 0.76147^* (0.40743)
Constant 12.45837^** (5.68677)
Observations 29
R^2 0.26243
Adjusted R^2 0.17392
Residual Std. Error 1.37604 (df = 25)
F Statistic 2.96504^* (df = 3; 25)
Note: ^p<0.1; ^p<0.05; ^p<0.01
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Vedic Maths DVD with Tips and Tricks | Free Phone Support
You will be learning how to speed up Multiplication, Division, Addition, Subtraction, How to find squares in less than 5 seconds, Cubes, find Cube Roots in 3 seconds flat. Find Square Roots, sharpen
your skills of Algebraic Equations and do it mentally. Learn Divisibility rules for every number in the number system. Speed up concepts of Geometry, Find out Divisibility rules for every number in
the number system, Calendars and much much more. This system will change your paradigm about mathematics and you will be able to enjoy it more than you ever did.
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Bubbles and stacks - visualising recursive functions
Recursive functions can be a bit mindbending when you first encounter them. As part of my process to understand them better I looked at some simple examples and tried to step through each function
stage by stage to get an idea of how the seemingly paradoxical process of a function calling itself can actually return a result. My intial way into this was through the “bubbling up” analogy, which
really intuitive and easy to grasp. Once that makes sense, there’s a refinement to the analogy which looks at the way the call stack manages function calls when a script is being run in browser.
Check out this example of a really simple recursive function:
function factorial(n){
if (n === 1){
return 1;
else {
return n * factorial (n-1);
The function factorial(n) simply calculates n!, factorial of the integer n, which can be found by evaluating n * (n-1) * (n-2) * (n-3) ... * 1. There are ways to evaluate the factorials of 0 and of
minus numbers, but for simplicity this function will only work as long as n>=1.
Clearly theres a lot of repetition required to evaluate a factorial, and that’s why it makes sense to write it as a recursive function. Just like all recursive functions this one includes a recursive
condition (the condition under which it will call itself again, creating a loop) and a break condition (the condition where the function eventually terminates, breaking out of the loop). Here, the
break condition is met when n === 1 since there are no more integers left to multiply after this.
Plugging in a value and stepping through the function in your head might go something like this:
//5 != 0 so ignore the break condition
return 5 * factorial(4)
// cannot evaluate!
At this point the universe should crack open because surely there’s no way to evaluate a function whose value is another function call! The solution is to keep going deeper: if factorial(5) can only
be evaluated when we know the value of factorial(4) then we need to evaluate that too. If factorial(4) can only be evaluated when we know the value of factorial(3) … well you get it. We need to keep
descending further and further until we hit something we can actually know the value of … which is where the break condition comes in.
When we hit factorial(1) we break out of the loop and finally return the value of 1. Now we can start the long climb back up to the top of this tree of nested function calls. Using our value for
factorial(1) we can evaluate factorial(2) (2 * 1 = 2). This lets us evaluate factorial(3) (3 * 2 * 1 = 6). This then gives factorial(4), which then gives factorial(5).
factorial(5) = 5*---------------120 // finally!
↓ ↑
factorial(4) = 4*-------------24
↓ ↑
factorial(3) = 3*-----------6
↓ ↑
factorial(2) = 2*---------2
↓ ↑
factorial(1) = -----→---1
The analogy of “bubbling up” which I first came across in This video was a huge help in understanding how this kind of function works. When we have a whole tree of nested function calls waiting to be
resolved, the bottom of the tree is where the chain of evaluations starts. As we successfully evaluate and close function calls, the result “bubbles” up to the top of the tree, eventually returning a
result for our original function call.
This seems neat and tidy, but what happens when we move onto more complex recursive functions? Have a look at the range() function below:
function range(a,b){
if (b-a === 1){
return [];
else {
let list = range(a,b-1);
return list;
This function returns the list of integers that fall between the values a and b, where a and b are both integers, and a < b. The expected result for range(2,9) would be [3,4,5,6,7,8].
The break condition for this function is b-a === 1 , since at this point there are no integers left between a and b so there’s nothing else for us to log as part of our result. Later, we’ll see why
it’s essential here that we return an empty array rather than “0” or an empty string.
The recursive part of the function has three steps:
1. Generate the variable list
2. push a value onto list
3. return list
The tricky part here is in step 1: list needs to be assigned as range(a,b-1), which we haven’t yet evaluated. Like before, we are going to have to build a tree of nested functions, going deeper until
we hit something we can evaluate; only at this point can we complete the original let list... instruction and assign the variable list. Using range(2,6) as an example we would expect to get a tree
range (2,6) → let list= -----[3,4,5] <<list.push(b-1)
↓ ↑
range (2,5)----------------[3,4] <<list.push(b-1)
↓ ↑
range (2,4)--------------[3] <<list.push(b-1)
↓ ↑
range (2,3)------→-----[] <<list gets assigned here!
Again, although we attempt to create list right at the top of the tree, we only complete that assignment right at the bottom. Only now can we move onto step 2, and start pushing values into list. Now
it’s clear why we needed to return an empty array as part of the break condition: there wouldn’t be an array to push variables into if we didn’t define it here. Now we have our array, we can carry
out step 2 for each of our nested function calls. Once this is done, we can finally move onto step 3, and return our completed list.
This order of progress is not entirely intuitive, so adding a couple of console.logs into the function body can help to clarify things:
function range(a,b){
if (b-a === 1){
return [];
else {
let list = range(a,b-1);
return list;
//expected result
The “bubbling up” analogy from earlier really makes sense in this function. Intuitively we might expect the result [5,4,3] when we call range(2,6) since b = 6 is the first parameter we attempt to
evaluate. Looking at the entire function tree we can see that we only start to push values into the array right at the bottom of the tree, filling the array in reverse order, from bottom to top.
“Bubbling up” is a really useful way to conceptualise recursive functions, but there’s an alternative way to look think about this with reference to the data structures at work under the hood while
our function is running. When I run a script in my browser, it uses a call stack to keep track of which functions need to be called, and manage which function is being run at any particular time. The
stack is a bit like a pile of dishes - items are constantly being taken off the top of the stack to tidy them away, and at the same time new items are added back on to the top. If a script is run
with multiple function calls inside it, each of these will be evaluated and tidied away as necessary before progressing through the script. If a function call cannot be evaluated immediately, it
stays on the bottom of the stack until we have the information we need to get a result. Have a look at this script:
function total(){
function first(){
return 5;
function second(){
return 6;
First we call total(), but this cannot be evaluated until we’ve called first() and second(). So total() goes onto the bottom of the stack, while first() and second() are added above it. When first()
and second() are evaluated we can clear them off the stack and pass their values into evaluate.
+---------+ +---------+
|second() | | = 6 |
+---------+ +---------+ +---------+
| first() | |first() | | = 5 |
+---------+ +---------+ +---------+ +---------+ +---------+
| total() | → | total() | → | total() | → | total() | → | = 11 |
+---------+ +---------+ +---------+ +---------+ +---------+
How does this work for a recursive function? There’s no difference really. Here’s an animation to demonstrate how the call stack manages the factorial() function from above:
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Covering a Chessboard with a Hole with L-Trominoes
The applet below is an interactive illustration for the problem #678 proposed by Charles W. Trigg, San Diego, California (Mathematics Magazine, Vol. 41, No. 1 (Jan., 1968), p. 42):
From an 8 by 8 checkerboard, the four central squares are removed.
a. Show how to cover the remainder of the board with right trominoes so as to have no fault line, or exactly two fault lines, or three fault lines.
b. Show that no covering with right trominoes can have four fault lines.
A right tromino is a nonrectangular assemblage of three adjoining squares mostly referred to as L-tromino at this site. A fault line has its extremities on the perimeter so that a portion of the
configuration may be slid along it in either direction without otherwise disturbing the relative position of its parts.
(Two draw a tromino, click on one of the squares, move (not drag) the cursor to the next one and click again, move to a third (suitable) square and - to place a tromino - click the third time.)>
Golomb's theory assures us that the 8×8 board with a 2×2 hole - whenever the latter is cut off - can be covered with L-trominoes and that in multiple ways. Indeed, place one tromino into the cut-off
square will leave a 1×1 hole. Now Golomb's theorem shows that an 8×8 board with a 1×1 hole can be covered with L-trominoes. So the question is clearly about counting the fault lines.
Related material
Read more...
Covering A Chessboard With Domino
Dominoes on a Chessboard
Tiling a Chessboard with Dominoes
Vertical and Horizontal Dominoes on a Chessboard
Straight Tromino on a Chessboard
Golomb's inductive proof of a tromino theorem
Tromino Puzzle: Interactive Illustration of Golomb's Theorem
Tromino as a Rep-tile
Tiling Rectangles with L-Trominoes
Squares and Straight Tetrominoes
Tromino Puzzle: Deficient Squares
Tiling a Square with Tetrominoes Fault-Free
Tiling a Square with T-, L-, and a Square Tetrominoes
Tiling a Rectangle with L-tetrominoes
Tiling a 12x12 Square with Straight Trominoes
Bicubal Domino
|Contact| |Front page| |Contents| |Games|
Copyright © 1996-2018
Alexander Bogomolny
Solution below is by Benjamin L. Schwartz, The MITRE Corporation, Virginia.
The problem as stated leaves unsettled the question of the existence of exactly one fault line. In the discussion below, we shall also resolve this question affirmatively with an example.
Notation. Denote the 8 horizontal rows of squares by letters A through H. Denote the vertical files by numbers 1 through 8. Denote a line between rows by the names of the two bordering rows (e.g.,
line BC).
a. The diagrams (1), (2), (3) and (4) show examples with zero, one, two and three fault lines, respectively. In (2), the fault line is 34. In (3), the fault lines are DE and 45; and, in (4), they
are CD, EF and 45. As shown later, these arrangements of fault lines are essentially unique.
b. To prove other cases impossible, we introduce a few easily proved lemmas about coverings with trominoes. Proofs are omitted.
I. A 2×n area can be covered iff 31 n.
II. Two adjacent lines (e.g., CD and DE) cannot both be fault lines.
III. A 3×3 area cannot be covered exactly.
IV. A fault line cannot occur adjacent to an edge.
Lemma IV eliminates A F, GH, 12 and 78 as candidates. But Lemma I also eliminates BC, FG, 23 and 67. Hence the only horizontal candidates are CD, DE and EF; and verticals 34, 45 and 56. Furthermore,
by Lemma II, if DE is a fault line, it is the only horizontal one. Similarly, if 45 is a vertical fault line, it is the only one.
Thus, the only prospect for a 4-fault-line configuration requires that these lines be CD, EF, 34 and 56. But this means the 3×3 areas in the four corners must be covered exactly, which violates Lemma
III. An additional question is whether the arrangement of fault lines in each of the various cases is essentially unique. The affirmative answer to this follows from the following theorem:
If CD is a fault line, so is EF. (Similarly 34 and 56.)
To see this, suppose there is a covering with CD as a fault line. Then consider how square Dl could be covered. Place a tromino to cover it with each of the three possible orientations of the other
two squares. It follows that either the other squares of files 1 and 2 in rows D and E cannot be covered, or can only be covered in such a way as to fill exactly the 2×3 rectangle (123)×(DE). The
same argument applies to (678)×(DE). Hence EF must be a fault line.
A more laborious and detailed analysis concerns the actual arrangement of the trominoes in the coverings. It has shown that except for "flipover" of the coverings of 2×3 rectangles, and rotations of
the entire board, the solutions of (2), (3) and (4) are unique. A similar statement is believed to hold for (1), but has not yet been proved.
|Contact| |Front page| |Contents| |Games|
Copyright © 1996-2018
Alexander Bogomolny
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Book Club: Mathematical Go, Chilling Gets the last point by Elwyn Berlekamp and David Wolfe
I tried writing down a few reponses and ideas, none of which really made sense in the end. Probably to get to the bottom of it, it makes sense to just take final positions of a game, where one knows
what the winner should be, and then try to extrapolate from there.
Let’s say this was a game, where both players have played 20 moves each (which they have), only black played in white’s area a lot, before playing twice in their own, and white removed all the
prisoners. There have been no passes. White has 10 more prisoners, and no komi.
Territory scoring would say 19 points to white, and 12 to black. If they play it out with stone scoring and prisoner return, then white does win by 7.
We expect the score with area scoring to be the same, since both played the same number of stones, 28 to 21. I think then to make it combinatorial, you need to some how turn that score of 1 point per
stone, and 1 point per area, into moves (ignoring prisoners).
Maybe when it comes to filling in territory you can think of it like each player takes turns placing a neutral (red/grey) stone into each point of their territory, and when they’ve finished doing
that, they take turns removing their stones from the board.
The reason is to come up with rules of a game, which is effectively just the counting process.
That way the numbers kind of make sense, if we modify them to the above.
On the left, Black and white would take turns removing stones, and white would have two extra left over.
On the right, White would first place a netural coloured stone (red or grey or something) into the open area, while black removes a stone. Then both continue to remove stones and White has 6 extra
At least the result of the game then is the actual area score?
I think of it as a directed graph, with nodes being the possible game states, and edges being all possible game state transitions (moves). Some nodes are terminal, no outgoing edges, i.e. the game
ends. Those nodes carry a result of the game.
I think thats all the assumptions you need.
I suppose that is technically all you need, but with the terminal positions (for one or both players), you do have to define why the value is what it is. I suppose it can be arbitrary, but not when
you’re trying to match it to Go specifically.
To make it look like a combinatorial game, more so than just the game tree means interpreting the final scores, where neither side no longer wishes to play. Equally, there’s no passing in
combinatorial games so you have to deal with that.
So that’s what’s prompting the whole discussion of Japanese vs Chinese rules, their mathematical versions where you use prisoner return, no passing, and a special way to fill in eyes in order to turn
scoring into part of the game.
Then when you say B+3 in the “terminal” position, it translates to Black have three more moves left than white, as well as being interpreted in Go as having three more points.
1 Like
Ok, I believe we are working with different definitions for “combinatorial game”. Sorry for interrupting.
But I am curious as to how your definition works. In my mind, “pass” is just another possible move option.
I wouldn’t say you’re interrupting
The problem with pass, is that the game immediatley becomes loopy, which makes it harder to work with. The game is no longer short then either, because there’s no rule generally that says “two
passes” end a game, so you don’t have a finite number of positions in the game.
So I would think, at least with trying to apply it to Go, and not like infinite chess or something, one would use:
G is finite : it has just finitely many subpositions.
G is loop free: it admits no infinite run → there is no sequence of moves proceding from G that repeats a position.
It becomes a bit strange mathematically to have G as an element of its left or right options, which then again must have G as an element of its left or right options, … etc.
Just for clarity, at least in the Berlekamp book, the work with G being finite and loopfree page 46, section 3.5.1, with G={G^L | G^R}, the game G is a pair of sets, the left set being left’s moves
from G, and the right set being right’s moves. Typically L is assigned the colours bLack, bLue etc in game, and right gets assinged white Red etc.
I mean they make it work somehow in CGT, I just haven’t started reading those sections yet. (I’ll have a glance now).
Right I see what you’re saying. You’re already thinking of it more generally, allowing for possibilities of loops and other things, not just as a game tree like you might do in a finite game.
1 Like
One can enforce that the graph is a tree by defining it such that each game state is uniquely defined by a path starting from a start state (think of the game state as including all the game history,
e.g all previous moves), and fixing one start state.
For a finite game we’d need to require the graph to be finite.
As far as I know, with japanese rules go is not a finite game, since board-state-loops are allowed.
I think a major problem with passes in combinatorial game theory is that they would eliminate the possibility of a “zero game”, in which the first player loses.
Without a zero game, the set of games no longer forms a group, so you can’t do stuff like comparing games by analyzing G - H, right?
2 Likes
If by pass you mean a move that transitions a game state S into S, i.e. a single-edge loop, then yes.
However, in Go a pass is not like that. Note that the board state alone is not sufficient to represent the game state in Go.
Certainly a tree is a kind of graph, even a directed graph. The graph approach sounds more general, it’s just maybe not necessarily needed, unless one wants to treat ko properly I would think? Since
then as you said, that involves potential loops. Ko, send two return one without a superko etc.
But in game theory it is, which is the problem. Games don’t seem to encode whose move it is, so when you pass you just get the exact same game state.
You get the same board state, but board state is not equivalent to game state.
Except it is in the examples of combinatorial game theory we’ve been working with. I’m including things like prisoners in that. So if you passed like in AGA rules, where you gift a prisoner, then
sure, the game state changes, but like in Chinese and Japanese rules, the game state looks identical.
Players can take two moves in a row in the mathematical formulation without the other player passing, also something that isn’t legal in Go
1 Like
Ah interesting, I think I’m starting to understand where this is coming from. We split the board position into multiple local positions (e.g. in the endgame), and view each position as a single game.
2 Likes
I think this is indeed supposed to be the utility of this formulation, particularly for Go endgame, and other games (Amazons endgame, certain chess positions involving King and pawns and zugzwang
etc), that the game splits up into several smaller games that you can calculate separately. Then you sum them back up to figure out who wins in the overall game.
1 Like
I don’t see any general solution for non-trivial loops
- since for example, these could combine into triple-ko -
but when passes are the only loops, it seems to me that one still gets a zero game:
Each game consists of
a finite set of positions
an element of that set (the starting position)
two sets of directed edges, neither of which has an edge whose start and end vertices are equal
(Black’s non-pass moves and White’s non-pass moves)
three functions to scores, each from a different one of
the set of positions , the set of black non-pass moves , the set of white non-pass moves
Play ends when both players pass consecutively. When that happens, the overall score is
the sum of scores of the ending positions (in the component games)
the sum of the scores for the moves made (this is how one can account for captures)
The zero game is the one such that
its set of positions is {{}}
the score of its position is zero
1 Like
Well in any case, I think that passing doesn’t add much except in some unusual situations that also involve ko.
Usually the passes won’t happen until the end of a game, and then the final pass-pass can just be reduced to a zero. So why bother with them at all?
If you want to use combinatorial game theory, you also need to do away with the concept of a score that is separate from the game tree, but it works fine to just put the scores at the leaves like I
was saying, and I think that corresponds to how it works in the book. Maybe you need to think a little about how half-point komi or a tie gets represented, but it’s not a big deal.
By the way, for my diagrams, I wasn’t intending to imply that the stones shown were the entire game in any sense. There could be any number of live stones connected to them, and I was just drawing a
minimal number of alive stones around the edges so that you can know what’s going on with the contested region in the middle.
To analyze an whole-board position you’ll always need to add in some extra points to account for all the other stones/territory/captures/komi that are already resolved and not part of the regions
under analysis. So it’s simpler to leave out as much as possible and keep the scores for those regions close to zero.
I think that’s how they do it in the book, but Japanese rules makes it a little more straightforward since you only have to worry about the territory part and all those stones around the border don’t
2 Likes
Are your two functions to score in edges (the “moves”) in order to define move values? They don’t seem necessary otherwise. Surely you only need to score terminal positions?
What kinds of scores do the functions map unsettled positions to and what’s their purpose otherwise? Do you want to score unfinished Go game positions?
I think essentially, I’m sure it’s possible. The idea of treating a game as a directed graph seems more general, and it’ll work for some loopy games, but the point is it’s not needed for Go without
It might be simpler, but then is it actually accurate to the idea of what the “game” is though?
Playing the game should reflect the score in some way. And playing the combinatorial game shouldn’t disagree with playing the Go game.
At that’s why I think one should probably include the stones as points if one really wants to do area scoring and not just territory scoring.
Well we might be done already with analyzing Chinese rules games so [S:there’s not much point in debating further:S], but yes, as long as you have a clear way of combining the regions to form a
whole-board game it should be okay to shift the scores for the regions any amount in any direction.
Any offset also doesn’t affect optimal play either, so worst case you can always just play out the optimal solution and then score the whole board normally.
EDIT: sorry, I must have forgotten briefly that we’re on OGF debating a small piece of an obscure topic, we should keep it up!
1 Like
I had not realized that my no non-trivial loops condition made those two unnecessary.
However, for Japanese rules, they can let game-descriptions be
overall significantly smaller. (via a reduced number of positions)
(Also, they would be necessary for more complicated loops,
and even for basic kos if one doesn’t make those a special case.)
A pass in a subgame may not be a global pass, but just a tenuki (playing in a different subgame). So considering sequences where a player gets several moves in a row in a particular subgame is not
illegal or even abnormal I think.
I’ll just note that the proper “summation” of subgames is not as straightforward as summation of regular numbers, as it may involve tree recursion (sort of like MiniMax) from surreal numbers.
2 Likes
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On a family of cheap symmetric one-step methods of order four
Kulikov, Gennady Yu; Shindin, S.K.
Lecture Notes in Computer Science , 3991 (2006), 781-785
In the paper we present a new family of one-step methods. These methods are of the Runge-Kutta type. However, they have only explicit internal stages that leads to a cheap practical implementation.
On the other hand, the new methods are of classical order 4 and stage order 2 or 3. They are A-stable and symmetric.
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kyceye's blog 金睛塔
The articles on this blog are excerpts and summaries of some of the academically original content published in the books 'Relativistic Universe and Forces'(ISBN 979-8865126171) or '상대론적 우주와 힘
'(ISBN 9791172187644).…
Cosmological Principle Explanation with the Special Relativity
Using special relativity, I succeeded in mathematically proving the cosmological principles regarding the uniformity and isotropy of the universe. The particle density distribution, centered on
itself in all primordial inertial systems in this universe, was calculated as $$\frac{1}{8} \left( \frac{r}{1 - r} \right)^2$$ in polar coordinates. ---- Originally in English
Relativistic Consistency of Electromagnetic Force
According to special relativity and electromagnetism, the electromagnetic force, also known as the Lorentz force \[ \vec{F} = q (\vec{E} + \vec{v} \times \vec{B})\], was presumed to take the same
form in every inertial frame. However, actual theoretical validation has not been conducted until now. Hence, I undertook this task independently, introducing a new form of the Heaviside-Feynman
formula. \[\vec{E} = \frac{q}{4 \pi \varepsilon_0 r^2 \left( 1 + \frac{\dot{r}}{c} \right)^3} \left( \left( 1 - \frac{v^2}{c^2} + \frac{\vec{a} \cdot \vec{r}}{c^2} \right) \left( \hat{r} - \frac{\vec
{v}}{c} \right) - \left( 1 + \frac{\dot{r}}{c} \right) \frac{r \vec{a}}{c^2} \right)\] ---- Originally in English
Mercury’s Perihelion Advancement Due to Special Relativistic Effects
I discovered that the cause of Mercury's perihelion advance could be explained by the sum of Maxwellian gravity and several special relativity factors. The simulation results, taking into account all
these factors, were demonstrated to align precisely with the outcomes predicted by the traditional Gerber-Einstein formula. ---- Originally in English
Answering Laplace’s Problem
I discovered a thorough explanation addressing the issue raised by Laplace, in the book "Celestial Mechanics" in 1805. He had contended that it was impossible to sustain a stable orbit for a
celestial body with a force transmitted at a finite speed. This part had been roughly guessed after Purcell's formula for electromagnetic fields appeared, but it was a part that could not be handled
accurately with Purcell's formula, which could not handle acceleration. Therefore, the essential aspects of the phenomenon were completely inaccessible. I solved this problem by using Maxwelian
gravity and converting Feynman's formula into a more practical form. $$\vec{E} = \frac{q}{4 \pi \varepsilon_0 r^2 \left( 1 + \frac{\dot{r}}{c} \right)^3} \left( \left( 1 - \frac{v^2}{c^2} + \frac{\
vec{a} \cdot \vec{r}} {c^2} \right) \left( \hat{r} - \frac{\vec{v}}{c} \right) - \left( 1 + \frac{\dot{r}}{c} \right) \frac{r \vec{a}}{c^2} \right)$$ Consequently, I discovered that the force
transmitted at a finite speed of light produces a subtle resistance component. The energy loss attributed to this resistance component consistently remains smaller than the loss incurred due to the
wave resulting from force-induced acceleration, thereby being overshadowed by the greater loss. Given that the energy loss from wave loss energy due to general relativity is significantly smaller
compared to the loss stemming from Maxwellian gravity, this serves as compelling evidence against the validity of general relativity : ---- Originally in English
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Transportation Conversion Tables and Calculations
Physical Weight
From To Multiply By
Pounds Kilos 0.4536
Kilos Pounds 2.2046
Metric Ton Kilos 1,000
Long Ton Pounds 2,240
Short Ton Pounds 2,000
From To Multiply By
Cubic Meter Cubic Feet 35.3147
Cubic Feet Cubic Inches 1,728
Cubic Meter Cubic Centimeters 1,000,000
From To Calculation
Fahrenheit (F) Celsius (C) (F-32) x 5/9
Celsius (c) Fahrenheit (F) (C x 9/5) + 32
Seafreight Trade Lane Conversions
Weight Or Measure Metric (W/M) 1,000 Kilos or 1 Cubic Meter
US Domestic 100 Pounds or 1 Cubic Foot
Caribbean 2,000 Pounds or 40 Cubic Feet
Linear Measure
From To Multiply By
Inches Centimeters 2.54
Centimeters Inches 393701
Meters Feet 3.281
Air to Sea Conversions
From To Multiply By If IATA
Volume Kilos Cubic Meters 0.006 6,000
Volume Kilos Cubic Meters 0.007 7,000
When you do not have enough cargo to fully use all the space or physical weight limitations of an entire ocean container you have what is called less-than-containerload (LCL) cargo. If your cargo is
too large to fit inside any type of ocean container, you have what is called break-bulk ocean cargo. Either way, the cost of the “space” your cargo will utilize inside a consolidated ocean container
or loose on a breakbulk ocean vessel compared to cost associated with the physical weight of your cargo is used in calculating ocean freight cost.
Most LCL freight cost is based on the higher of 1,000 kilos or 1 cubic meter and referred to as weight or measure (W/M) metric.
Nine pallets, each 150kgs and 122cm x 101.5cm x 127cm
(English Standard Measure, each 330.7lbs and 48in x 40in x 50in)
9 pallets x 122cm x 101.5cm x 127cm / 1,000,000 cubic centimeters = 14.15 cubic meters
9 pallets x 48in x 40in x 50in = cubic inches / 1,728 = cubic feet / 35.314 = 14.15 cubic meters
The physical weight of this shipment is 9 pallets x 150 kilos = 1,350 physical kilos. For the volume of this cargo not to exceed the physical weight, the physical weight would need to be at least
14,150 kilos. Since this is not the case, the ocean freight would be calculated based on 14.15 cubic meters.
The most commonly used calculation in the US domestic LCL markets of Hawaii, Alaska and Puerto Rico is the greater of 100 pounds or 1 cubic foot, and in the Caribbean LCL market is the greater of
2,000 pounds or 40 cubic feet. Metric Ton, Short Ton and Long Ton values are used as the basis of breakbulk ocean cargo freight calculations.
Air/Sea Freight Combination
Air Freight, Sea Freight and Air/Sea Combination services can be calculated and quickly compared using the above provided AIR/SEA FREIGHT CONVERSION table. Use 0.006 factor if comparing to airfreight
based on IATA standard of 6,000 cubic centimeters per one physical kilogram and use 0.007 factor if comparing to airfreight based on 7,000 cubic centimeters per one physical kilogram. In the Example,
2,359 volume kilos of airfreight (based on IATA standard) x 0.006 = 14.15 cubic meters sea freight.
Physical Weight
From To Multiply By
Pounds Kilos 0.4536
Kilos Pounds 2.2046
Metric Ton Kilos 1,000
Air Freight Dim Factors
Using Centimeters to Calculate Volume Kilos Using Inches to Calculate Volume Pounds Using Inches to Calculate Volume Kilos
6,000 166 366
7,000 194 428
Linear Measure
From To Multiply By
Inches Centimeters 2.54
Centimeters Inches 393701
Linear Measure
From To Calculation
Fahrenheit (F) Celsius (C) (F-32) x 5/9
Celsius (C) Fahrenheit (F) (C x 9/5) + 32
Dimensional weight, also called dim weight or volume weight, is used because the space a package takes on an aircraft may cost more than the physical weight of the package. For every shipment
dimensional weight is compared to the physical weight, and the greater of the two is used to determine the shipment cost. IATA standard dimensional weight is based on 6,000 cubic centimeters per one
physical kilogram and calculated as follows:
Length (cm) x width (cm) x height (cm) / 6,000 = volume kilos
International transportation rates are predominately expressed in metric measure. For countries in which English Standard measure is more commonly used, the same dimensional weight formula is used,
but with different factors or divisors. Using inches, the same volume weight can be expressed as either volume pounds by using a divisor of 166 or as volume kilos using a divisor of 366.
Nine pallets, each 150kgs and 122cm x 101.5cm x 127cm
(English Standard Measure, each 330.7lbs and 48in x 40in x 50in)
9 pallets x 122cm x 101.5cm x 127cm / 6,000 = 2,359 volume kilos
9 pallets x 150kgs = 1,350 physical kilos
9 pallets x 48in x 40 in x 50in / 366 = 2,359 volume kilos
9 pallets x 150kgs = 1,350 physical kilos
9 pallets x 48in x 40in x 50in / 166 = 5,205 volume pounds
9 pallets x 330.7lbs = 2,976 physical pound
The chargeable weight of the nine pallets is expressed as either 2,359 chargeable kilos or 5,205 chargeable pounds. To verify the accuracy of the calculations—2,359 volume kilos x 2.2046 = 5,205
volume pounds.
In some trades, particularly in the US domestic airfreight market, the more commonly used dimensional factor is based on 7,000 cubic centimeters per one physical kilogram. Volume weight is calculated
using the same formula, but with different factors of 7,000, 194 or 428 per the table provided above.
Typically, large airfreight cargos are expressed as tons referring to the higher of either physical metric tons or volume metric tons. One metric ton = 1,000 kilograms, therefore the example cargo
would be referred to as just under 2 1/2 tons.
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What is Windowing and when/why do we need it?
7 years ago
●8 replies●
latest reply 7 years ago
7829 views
This is the first question to make it into the new #FAQ section.
The hope is that your collective insights will make this page a great resource for the EE community to learn about the basics of Windowing in the DSP world: what it is, when and why we need it, when
we don't need it, Matlab examples, etc.
Please don't forget:
1. that you can add images to your post simply by drag and dropping them into the editor
2. that you can add great looking equations to your post through Mathjax (read here for details)
3. to click on the 'thumbs up' button for the contributions that you find the most insightful
In the hope to encourage the participation of many, I will start keeping track of the total number of thumbs-ups every user receives for FAQs. From time to time, I will draw a prize (a dsp book
probably) to be sent to the winner. The more thumbs-ups you accumulate, the more chances you have to win the prize.
Thanks a lot for your time!
[ - ]
Reply by ●November 15, 2017
Sometimes we have to teach new hires about windowing on a basic level. They've already seen it in school, but maybe don't "feel" the need, and just think of it on a list that includes lots of signal
processing concepts. We show them a waveform, say 10000 samples, and ask them how they would apply an FFT to it in Matlab. Of course they always know Matlab (or Octave), so they give a Matlab'ish
type answer and it's easy. Then we ask, now what if you have to do this to continuous data -- you can't see the end of it, and it may never stop ?
Most say something like "process it in chunks". Some add something like "make sure we keep up with the data". And that starts a conversation about how to determine the chunk size, how to deal with
the edges of the chunk ("truncation" has been mentioned already), how to sew the chunks together, and how to make processing each chunk efficient. What we want them to keep in mind is that windowing
is fundamental to handling continuous streams of data.
[ - ]
Reply by ●November 15, 2017
"Windowing" can mean a couple of things:
1) A limitation in temporal or spatial or spectral extent (etc.) - whether intentional or simply a function of the limitations of observation or measurement. The "limitation" may also be a single
period of a periodic function or record. [From now on, I'll use "time" with the understanding that it may be "frequency" or "space" or ....]
2) A finite function applied to (multiplied) over a time function or record in order to achieve a desired effect.
So, in (1), we may refer to an "observation window" meaning that there is an observation or measurement time that is either selected or imposed. This by itself has no weighting involved and is
called a "rectangular window" or "gate function", etc. Importantly, the Fourier transform of a rectangular window is of the form sinx/x or what's called a "sinc" function. This is a fundamental
characteristic. The shorter the rectangular window, the wider the sinc and vice versa. Thus the reference to the Heisenberg uncertainty principle.
In (2), we impose a weighting function over the rectangular window. Why would we do this? For example, the sharp edges of the rectangular window in time introduce broad spectral content. And, broad
spectral content is detrimental to spectral analysis.
Imagine a rectangular window that's multiplied by a bell-shaped curve - clearly this will remove the sharp transitions at the ends and reduce the broad spectral energy that the sharp edges contain.
A good deal of effort has gone into analyzing and designing such "windows" or "weighting functions" with the idea of helping one understand their characteristics and to help one implement them
efficiently. Some key references are:
"On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform" Fredric J. Harris, Proceedings IEEE, vol. 66, pp 51-83
Nuttall, A. H. (February 1981). "Some windows with very good sidelobe behavior". IEEE Transactions on Acoustics, Speech, Signal Processing. ASSP-29: 84–91.
G.C. Temes, V. Barcilon, and F.C. Marshall III. The optimization of bandlimited systems. Proc. IEEE, Vol 61, pages 196-234, 1973.
[ - ]
Reply by ●November 15, 2017
Here are two on-line references on how windows are used in DSP:
In spectral analysis:
In filter design:
[ - ]
Reply by ●November 15, 2017
Hi Steve. Nice to hear from you.
[ - ]
Reply by ●November 15, 2017
Windowing, to my best understanding, is
In Signal processing:
1.to trancate signals which are of your interest
2.to reduce memory requirement and accelerate computation process
In filter design: to make FIR filter
I think windowing is needed out of practical reasons. In the context of signal processing, almost all signals we are interested in are restrained to a certain period of time(For example, In a radar
system, we usually analysis the received signal within a duration of a few pulses), thus by windowing we get useful signals. In the context of filter design, there is a filter type of FIR(finite
impluse reponse), the design technique lies on windowing. There are plenty of prototypes out there, all you need to do is to apply windowing and make it implementable and computational effective.
From a university student in his forth year
[ - ]
Reply by ●November 15, 2017
It is deeply and fundamentally mathematical. It is actually related to Heisenberg's uncertainty and is called The Gabor Limit. The more precise you know things in time, the less precisely you can
know their frequency ... and ... vice-versa.
Any truncated time-domain sequence MUST have an infinite frequency-domain sequence.
Any truncated frequency-domain sequence MUST have an infinite time-domain sequence.
To demonstrate ... as you crank up the Alpha of a Keyser or Dolph filter, the time window gets narrower and making time more accurate (throwing away information, by the way) while the stopband gets
deeper and deeper with the main lobe widening, so the frequency information becomes LESS specific.
The best we can hope for is a tapering where some time or frequency region is significant and outside of that region the information has been "Windowed-out" to become insignificant. Window shapes in
the time or frequency domains are chosen to optimize some needed parameter. Sometimes that optimization is selecting single-frequency boomer out of the otherwise flat background noise, for this a
Dolph is optimal, and a Blackman-Harris is really fast and easy to calculate. Sometimes, we need the least distortion of the peak amplitude for some spectral analysis reason. At this point you
might get Remez going or use one of the Flat-Top windows. The rectangular window (also known as Dirichlet) is the best there is at providing an unbiased estimator when you DON'T KNOW whether you are
working on broadband or narrowband signals.
The point is that you have to look at the work you are doing to choose the right Window function.
The most egalitarian window is Gaussian. It gives the same tail shapes in time as it gives in frequency and they tail off really fast.
Hope this helps.
from a 40-year DSP veteran.
[ - ]
Reply by ●November 15, 2017
To keep my life simple I first avoid analogy with my bedroom window and second I view it as nothing more than a scale factor over a finite given samples. This scale factor usually goes up then down
under different names. It may be square and then this window is same as no-window, a bit silly terminology gymnastics. As far as I know it is useful when applied on truncated (well any) stream before
FFT to reduce the phase discontinuity at start-to-end of stream and therefore should not be applied if phase is already ok or that we are targetting equivalent operation between time and frequency
[ - ]
Reply by ●November 15, 2017
I used windowing for multiple signal processing purposes. the most frequent one is windowing prior to FFT. I was using FFT to determine the spectral strength of fundamental frequency. I applied
windowing prior to FFT as I was not in a position to ensure coherent sampling. If had my fundamental aligned with the sampling frequency i'll get frequency in a bin. for any other case, frequency
peak will reflect more than one bin. use of appropriate window will make the frequency more concentrated in less number of bins - practically 3 bin. with the amplitude value of these 3 bins we can
determine the spectral position and the amplitude more accurately using simple parabolic or Gaussian interpolation. there are more than 16 commonly used windows for this purpose or similar, I tried
some of them- Blackman Harris and Hann window are two cases I find it gave me accurate results for my case.
Another case where I used windowing is for overlap and add case. in that case, the window has to have an additional property such that the a^2+b^2 = 1. where a and b are the lower and upper half of
the window in the overlapping order. similar windowing is used in the time-to frequency transforms in most of the audio codecs.
There are other purposes for windowing, like synthesis and analysis windows for multirate processing. looks like they are called window just because they are time domain dot-product. some of the
window shapes are not that intuitive as the overlap and add case. you will understand the point if you just plot the polyphase synthesis C-window used in MP3 :).
Coming back to the common windowing used prior to FFT, the shape is decided based on the time-frequency resolution requirement. one thing is for sure, it is worth spending some time to decide your
window for your application based on the specific signal characteristics which you are expecting as input. common signal types are tones/harmonics, noise, voice, glitches/attacks, and mix of one or
more... It is not uncommon to spend your time on deciding the type of window depending on the type of signal. recollecting classical short window / long window switching in MPEG audio codecs are
example. I'm sure positioning the windowing is also worth considering for your specific needs !
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The annual maximum loan amount an undergraduate student may receive must be prorated (reduced) when the borrower is:
• Enrolled in a program that is shorter than a full academic year; or
• Enrolled in a program that is one academic year or more in length, but is in a remaining period of study (a period of study at the end of which a student will have completed all requirements of
the program) that is shorter than a full academic year.
With one exception (see "Proration of the annual loan limit for students who graduate early from clock-hour programs" later in this chapter), the annual loan limits for Direct Subsidized Loans and
Direct Unsubsidized Loans are prorated only in these two situations (. Loan limits are not prorated based on a student’s enrollment status, such as when a student is enrolled less than full time or
is enrolled for a period of less than a full academic year that is not a remaining period of study.
The annual loan limit for Direct Unsubsidized Loans is not prorated for students enrolled in graduate or professional level programs. Loan proration requirements also do not apply to students taking
preparatory coursework or coursework necessary for teacher certification. The annual loan limit must be prorated only when a student is enrolled in a program or remaining portion of a program that is
shorter than an academic year. For purposes of awarding Title IV aid, students taking preparatory coursework or coursework needed for teacher certification are not considered to be enrolled in a
It’s important to understand that loan limit proration determines the maximum loan amount that a student may borrow for a program or remaining balance of a program, not the loan amount that the
student actually receives. In some cases, the actual loan amount that a student is eligible to receive (based on costs, EFC, and other aid) may be less than the prorated loan limit.
Use of fractions vs. decimals when prorating loan limits
As we explain in more detail below, proration involves multiplying the annual loan limit by a fraction. It’s acceptable to convert the fraction to a decimal and then multiply the annual loan limit by
the decimal, but this conversion is not a requirement. However, you should be consistent in the method you use, since the fraction and decimal calculations sometimes result in slightly different
prorated loan limits, as shown in the examples later in the chapter.
Using school’s definition of academic year if longer than the Title IV minimum
As explained above, proration of the annual loan limit is required when an undergraduate student is enrolled in a program that is shorter than an academic year or is enrolled in a remaining period of
study that is shorter than an academic year. A school may choose to define its academic year as longer in weeks or hours than the minimum statutory requirements. If so, the school’s standard – not
the statutory minimum – determines whether a program or a final period of study is shorter than an academic year.
Separate calculations for combined subsidized/unsubsidized annual loan limit and maximum subsidized annual loan limit
As explained in Chapter 4 of this volume, for undergraduate students there is a maximum combined annual loan limit for Direct Subsidized Loans and Direct Unsubsidized Loans, and a maximum portion of
that combined annual loan limit that a student may receive in Direct Subsidized Loans. If the annual loan limit for an undergraduate student must be prorated, you must first determine the combined
Direct Subsidized Loan and Direct Unsubsidized Loan prorated annual loan limit, and then separately determine the Direct Subsidized Loan prorated annual loan limit. This is illustrated in the
proration examples below.
Prorating loan limits for programs of study shorter than an academic year
If an academic program is shorter than a full academic year in length, you must multiply the applicable loan limit(s) by the lesser of —
Semester, trimester, quarter, or clock hours enrolled in program
Semester, trimester, quarter, or clock hours in academic year
Weeks enrolled in program
Weeks in the academic year
The result is the prorated annual loan limit for that program.
Proration examples: programs shorter than an academic year
Examples 6 and 7 illustrate how the prorated annual loan limit is determined when a student is enrolled in a program that is shorter than an academic year.
A dependent student is enrolled in a 400 clock-hour, 12-week program (a “short-term program” as described in Volume 2, Chapter 2). The school defines the academic year for this program as 900 clock
hours and 26 weeks of instructional time.
To determine the maximum loan amount the student can borrow, convert the fractions based on weeks and hours to decimals:
12/26 = 0.46
400/900 =0 .44
Multiply the smaller decimal (0.44) by the combined Direct Subsidized Loan and Direct Unsubsidized Loan annual loan limit for a first-year dependent undergraduate ($5,500, not more than $3,500 of
which may be subsidized):
$5,500 x 0.44 = $2,420 combined subsidized/unsubsidized prorated annual loan limit
To determine the maximum portion of the $2,420 prorated annual loan limit that the student may receive in subsidized loan funds, multiply the maximum subsidized annual loan limit of $3,500 by the
smaller decimal (0.44):
$3,500 x 0.44 = $1,540 subsidized prorated annual loan limit
The maximum combined Direct Subsidized Loan and Direct Unsubsidized Loan amount the student can borrow for the program is $2,420, but no more than $1,540 of this amount may be in subsidized loans.
Note: In Example 6 above and in the other proration examples that follow, the fractions are converted to decimals. As an alternative you could choose to multiply the annual loan limit by the original
fraction, though you should be consistent in using one method or the other. Using the fraction 400/900 in Example 6 instead of the decimal 0.44 would result in a slightly higher prorated loan limit:
$5,500 x 400/900 = $2,444.
An independent student is enrolled in a 24 quarter-hour, 20-week program. The school defines the academic year for this program as 36 quarter hours and 30 weeks of instructional time.
To determine the maximum loan amount the student can borrow, convert the fractions based on weeks and quarter-hours to decimals:
20/30 = 0.67
24/36 = 0.67
Multiply the smaller decimal (in this case, both are 0.67) by the combined Direct Subsidized Loan and Direct Unsubsidized Loan annual loan limit for a first-year independent undergraduate ($9,500,
not more than $3,500 of which may be subsidized):
$9,500 x 0.67 = $6,365 combined subsidized/unsubsidized prorated annual loan limit
To determine the maximum portion of the $6,365 prorated annual loan limit the student may receive in subsidized loan funds, multiply the maximum subsidized annual loan limit of $3,500 by the same
decimal (0.67):
$3,500 x 0.67 = $2,345 subsidized prorated annual loan limit
The maximum combined Direct Subsidized Loan and Direct Unsubsidized Loan amount the student can borrow for the program is $6,365, not more than $2,345 of which may be in subsidized loans.
Note: Using the fraction 24/36 in Example 7 instead of the decimal 0.67 would result in a slightly lower prorated loan limit: $9,500 x 24/36 = $6,333.
Prorating loan limits for remaining periods of study shorter than an academic year
You must also prorate loan limits for students enrolled in remaining periods of study shorter than an academic year. This circumstance can occur when a student is enrolled in a program that is one
academic year or more in length, but the remaining period of study needed to complete the program (sometimes called a “final” period of study) will be shorter than an academic year.
Proration is required only when you know in advance that a student will be enrolled for a remaining period of study that is shorter than an academic year. If a student originally enrolls for a
remaining period of study that is a full academic year in length, but completes the program early in less than a full academic year, you're not required to retroactively prorate the annual loan limit
(but see the discussion under "Proration of the annual loan limit for students who graduate early from clock-hour programs" later in this chapter for a limited exception to this general rule).
In a standard-term program, or a credit-hour program using SE9W nonstandard terms, a remaining period of study is considered shorter than an academic year if the remaining period contains fewer terms
than the number of terms covered by the school’s Title IV academic year. For programs that are offered in a Scheduled Academic Year (SAY; see Chapter 6), the number of terms covered in the school’s
Title IV academic year usually does not include a summer “header” or “trailer” term.
Consider a student who is enrolled in a four-year program that is offered in an SAY consisting of three quarters plus a summer “trailer,” and who has completed four academic years of study. However,
the student needs to attend an additional quarter term to complete the program requirements. The final quarter term would fall in a new academic year, and thus the annual loan limit would have to be
prorated, because the remaining period of study (a single quarter) is less than a full academic year.
Similarly, if a student enrolled in a two-year program not offered in an SAY (where the Title IV academic year covers two 15-week semesters) has completed two academic years of study, but needs to
return for an additional semester to complete the program requirements, the loan limit would have to be prorated if the student receives a loan for the final semester.
Note that for standard-term programs or credit-hour programs with SE9W nonstandard terms, the length of the loan period does not determine whether a student is enrolled in a remaining period of study
that is shorter than an academic year. The determining factor is the length of the remaining period of study in which the student is enrolled, which may not be the same as the loan period. For
example, if an undergraduate student is enrolled for a full SAY consisting of fall and spring semesters, and will complete the program at the end of the spring term, but is enrolled less than half
time during the spring, the student is eligible to receive a Direct Loan only for the fall semester. Although the loan period (fall only) would be shorter than an academic year, the remaining period
of study (fall through spring) is a full academic year. Therefore, if the student receives a Direct Loan in the fall, proration of the annual loan limit is not required.
In a clock-hour program, non-term program, or a program with non-SE9W nonstandard terms, a remaining period of study is considered less than an academic year if it consists of fewer clock or credit
hours than the program’s defined Title IV academic year. In contrast to standard term and SE9W nonstandard term programs, if a student enrolled in a clock-hour, non-term, or non-SE9W nonstandard term
program is in a remaining period of study shorter than an academic year and receives a Direct Loan, the loan period and the remaining period of study will always be the same. This is because for
these programs the minimum loan period is the lesser of the length of the program (or remaining portion of a program) or the academic year.
For all types of programs, where there is a remaining period of study less than an academic year, the annual loan limit for the student’s grade level is multiplied by the following fraction to
determine the prorated loan limit:
Semester, trimester, quarter, or clock hours enrolled in program
Semester, trimester, quarter, or clock hours in academic year
Unlike proration for programs that are shorter than an academic year, there is no comparison of weeks and hours. Only the credit or clock hours that the student is scheduled to attend or is actually
attending at the time of origination are used in the calculation.
Proration examples: remaining periods of study shorter than an academic year
Examples 8 through 12 illustrate how the prorated annual loan limit is determined when a student is enrolled in a remaining period of study shorter than an academic year.
EXAMPLE 8: REMAINING PERIOD = ONE QUARTER
A dependent student is enrolled in a 2-year credit-hour program offered in standard terms (quarters). The school defines the academic year for the program as 36 quarter hours and 30 weeks of
instructional time (covering three quarters: fall, winter, and spring).
The student has attended the program for six quarters (two academic years), but to finish the program needs to complete an additional six hours (half time) in the fall quarter of the next academic
To determine the prorated Direct Loan limit for the student’s remaining period of study (one quarter), convert the fraction based on the hours that the student is expected to attend and the hours in
the academic year to a decimal:
6/36 = 0.17
Multiply this decimal by the combined Direct Subsidized Loan and Direct Unsubsidized Loan annual loan limit for a dependent second-year undergraduate ($6,500, not more than $4,500 of which may be
$6,500 x 0.17 = $1,105 combined subsidized/unsubsidized prorated annual loan limit
To determine the maximum portion of the $1,105 prorated annual loan limit that the student may receive in subsidized loan funds, multiply the maximum subsidized annual loan limit of $4,500 by the
same decimal (0.17):
$4,500 x 0.17 = $765 subsidized prorated annual loan limit
The maximum combined Direct Subsidized Loan and Direct Unsubsidized Loan amount the student can borrow for the remaining portion of the program is $1,105, not more than $765 of which may be
EXAMPLE 9: REMAINING PERIOD = TWO SEMESTERS, WITH LESS THAN HALF-TIME ENROLLMENT IN ONE TERM
The student from Example 8 transfers to a BA program at a different school. The academic year for the program contains two semesters, fall and spring. During the student’s second year of study in the
BA program, they will be enrolled full time in the fall and less than half time in the spring, and will graduate at the end of the spring term:
(less than half time)
Although the student is not eligible to receive a Direct Loan for the spring term, the remaining period of study (two semesters) is equal to a full academic year. Therefore, proration of the annual
loan limit is not required if the student receives a Direct Loan for the fall term.
A dependent fourth-year undergraduate is enrolled in a program with a defined academic year of 36 quarter hours and 30 weeks of instructional time, covering three quarters (fall, winter, and
spring).The student will be enrolling in the fall and winter quarters, but not the spring quarter, and will graduate at the end of the winter term.
The student will be enrolled for 12 quarter hours (full time) during the fall quarter, but will be enrolled for only three hours (less than half time) in the winter quarter:
Fall Winter
(full time: 12 hours) (less than half time: 3 hours)
The student’s final period of study (two terms) is shorter than an academic year, so the annual loan limit must be prorated. However, because the student will be enrolled less than half time during
the winter quarter (and therefore ineligible to receive Direct Loan funds for that term), the loan period will cover the fall quarter only, and only the 12 quarter hours for the fall term are used to
determine the prorated annual loan limit.
To determine the prorated loan limit for the final period of study, convert the fraction based on the hours that the student is expected to attend in the fall quarter and the hours in the academic
year to a decimal:
12/36 = 0.33
Multiply this decimal by the combined Direct Subsidized Loan and Direct Unsubsidized Loan annual loan limit for a dependent fourth-year undergraduate ($7,500, not more than $5,500 of which may be
$7,500 x 0.33 = $2,475 combined subsidized/unsubsidized prorated annual loan limit.
To determine the maximum portion of the $2,475 prorated annual loan limit that the student may receive in subsidized loan funds, multiply the maximum subsidized annual loan limit of $5,500 by the
same decimal (0.33):
$5,500 x 0.33 = $1,815 subsidized prorated annual loan limit
The total prorated annual loan limit for the fall quarter loan is $2,475, not more than $1,815 of which may be subsidized.
EXAMPLE 11: REMAINING PERIOD = TWO QUARTERS, SEPARATED BY A PERIOD OF NON-ENROLLMENT
A school has an academic year that covers three quarters: fall, winter, and spring. An independent fourth-year undergraduate will be enrolling full time in the fall and spring quarters, but will not
be enrolled in the winter quarter, and will graduate at the end of the spring quarter:
Fall Spring
(full time: 12 hours) (full time: 12 hours)
Because the fall quarter is in the same academic year as the student’s final quarter of attendance, it is part of the remaining period of study, even though there is a term between the fall and
spring quarters in which the student will not be enrolled. The school must award separate loans for fall and spring.
The remaining period of study (two terms) is shorter than an academic year, so the annual loan limit for each loan must be prorated based on the number of hours for which the student is enrolled in
each term.
The prorated loan limit is determined separately for each term by converting the fraction based on the number of hours in each term to a decimal:
12/36 = 0.33
Multiply this decimal by the combined Direct Subsidized Loan/Direct Unsubsidized Loan annual loan limit for an independent fourth-year undergraduate ($12,500, not more than $5,500 of which may be
$12,500 x 0.33 = $4,125 combined subsidized/unsubsidized prorated annual loan limit for a single term (fall or spring)
To determine the maximum portion of the $4,125 prorated annual loan limit that the student may receive in subsidized loan funds for a single term, multiply the maximum subsidized annual loan limit of
$5,500 by the same decimal (0.33):
$5,500 x 0.33 = $1,815 subsidized prorated annual loan limit for a single term (fall or spring)
The combined total prorated loan limit for the two single-term loans (fall-only and spring-only) in the remaining period of study is $4,125, not more than $1,815 of which may be subsidized. This
means that the maximum loan amount the student may receive for the two terms in the final period of study combined is $8,250, not more than $3,630 of which may be subsidized.
A school has an 1800 clock-hour program with a defined academic year of 900 clock hours and 26 weeks of instructional time. A dependent undergraduate student successfully completes the first 900
clock hours of the program in 22 weeks of instructional time. However, the student must complete an additional four weeks of instructional time before receiving a second loan (see Chapter 6 of this
After 26 weeks of instructional time have elapsed, the student has successfully completed 1040 clock hours and may then receive a second loan. However, the loan limit must be prorated based on the
number of clock hours remaining in her program at this point (760). To determine the prorated loan limit for the student’s second loan, convert the fraction based on the number of clock hours
remaining to a decimal:
760/900 = 0.84
Multiply this decimal by the combined Direct Subsidized Loan and Direct Unsubsidized Loan annual loan limit for a dependent second-year undergraduate ($6,500, not more than $4,500 of which may be
$6,500 x 0.84 = $5,460 combined subsidized/unsubsidized prorated annual loan limit
To determine the maximum portion of the $5,460 prorated annual loan limit that the student may receive in subsidized loan funds, multiply the maximum subsidized annual loan limit of $4,500 by the
same decimal (0.84):
$4,500 x 0.84 = $3,780 subsidized prorated annual loan limit
The total prorated loan limit for the remaining period of study is $5,460, not more than $3,780 of which may be subsidized.
Proration of the annual loan limit for students who graduate early from a clock-hour program
Under the regulations that govern the treatment of Title IV funds when a student withdraws, a student who completes all the requirements for graduation from a program before completing the days or
hours that they were scheduled to complete is not considered to have withdrawn, and no return of Title IV funds calculation is required (see Volume 5 for more detail). However, a school may be
required to return a portion of the Direct Loan funds that were disbursed to a student who successfully completes the requirements for graduation from a clock-hour program before completing the
number of clock hours that they were scheduled to complete.
A student's eligibility to receive Title IV aid for a clock-hour program is based, in part, on the total number of clock hours in the program. If a school allows a student to graduate from a
clock-hour program without completing all of the originally established hours for the program, the school has effectively shortened the program length and reduced a student's Title IV aid eligibility
for the program. In this circumstance, the school must prorate (or re-prorate) the annual loan limit for the student based on the number of hours the student actually completed, and after this
recalculation, the school must return to the Department any portion of the Direct Loan funds the student received that exceed the newly prorated (or re-prorated) annual loan limit. (For a student who
received a Pell Grant, the school must also recalculate the student's Pell Grant award in this situation. See Volume 7 for more information.)
This requirement applies only to clock-hour programs, and it applies regardless of the length of the program or remaining portion of a program. In some cases, this means that previously prorated
annual loan limit must be re-prorated.
Examples 13 and 14 illustrate the requirement described above.
A dependent student enrolls in a 900 clock-hour program, with the academic year defined as 900 clock hours and 26 weeks of instructional time. The school assumes that the student will complete 900
clock hours.
Based on EFC and COA, the student qualifies to receive the maximum annual combined Direct Subsidized Loan/Direct Unsubsidized Loan limit of $3,500 in the form of a Direct Subsidized Loan and the
maximum additional Direct Unsubsidized Loan amount of $2,000. Each loan is paid in two equal disbursements, as shown below.
Combined Subsidized/Unsubsidized Annual Loan Limit Direct Subsidized Loan First Disbursement Direct Subsidized Loan Second Disbursement
$3,500 $1,750 $1,750
Additional Unsubsidized Annual Loan Limit Direct Unsubsidized Loan First Disbursement Direct Unsubsidized Loan Second Disbursement
$2,000 $1,000 $1,000
The school considers the student to have met the requirements for graduation from the program after the student has completed only 750 of the originally scheduled 900 clock hours. As soon as
practicable after determining that the student will meet the graduation requirements after completing only 750 clock hours, the school must prorate the student's Direct Loan annual loan limit,
because the student is now treated as having been enrolled in a program shorter than an academic year in length (i.e. as though the student had originally enrolled in a 750 clock-hour program).
However, in this circumstance only the number of clock hours that the student completed are used to determine the prorated loan limit. There is no comparison of hours and weeks fractions, as is
normally required when prorating the Direct Loan annual loan limit for students who are enrolled in programs shorter than an academic year.
The school determines the prorated annual loan limit by multiplying the applicable annual loan limit by the number of clock hours the student actually completed, then dividing the result by the
number of clock hours in the program's academic year definition:
($3,500 x 750) ÷ 900 = $2,917 prorated combined subsidized/unsubsidized annual loan limit
($2,000 x 750) ÷ 900 = $1,667 prorated additional unsubsidized annual loan limit
(As noted earlier in this chapter, the prorated loan limit may also be determined by converting the fraction consisting of the number of clock hours the student completed in the program over the
number of clock hours in the program's academic year to a decimal, and then multiplying the decimal by the applicable annual loan limit. Whatever approach a school chooses should be applied
consistently, as the fraction method shown above and the decimal method may produce slightly different results.)
The school reduces each disbursement of the student’s Direct Subsidized Loan and Direct Unsubsidized Loan as shown below and returns the excess loan funds to the Department. Note that the school –
not the student – is responsible for returning the excess Direct Loan funds in this situation.
Prorated Combined Subsidized/Unsubsidized Annual Loan Direct Subsidized Loan Adjusted First Direct Subsidized Loan Adjusted Second
Limit Disbursement Disbursement Direct Subsidized Loan Funds Returned to
$2,917 $1,458 $1,459
($583 reduction from original amount disbursed) (original disbursement reduced by $292) (original disbursement reduced by $291)
Prorated Additional Unsubsidized Annual Loan Limit Direct Unsubsidized Loan Adjusted First Direct Unsubsidized Loan Adjusted First
Disbursement Disbursement Direct Unsubsidized Loan Funds Returned to
$1,667 Department
$833 $834
($333 reduction from original amount disbursed) $333
(original disbursement reduced by $167) (original disbursement reduced by $166)
A dependent student is enrolled in the remaining 500 clock hours of a 1500 clock-hour program that has a defined academic year of 900 clock hours and 26 weeks of instructional time. Because the
student is enrolled in a final period of study shorter than an academic year, the school prorates the annual loan limit based on the 500 hours that it expects the student to complete:
($4,500 x 500) ÷ 900 = $2,500 prorated combined subsidized/unsubsidized annual loan limit
($2,000 x 500) ÷ 900 = $1,111 prorated additional unsubsidized annual loan limit
Based on EFC and COA, the student qualifies to receive the maximum annual combined prorated Direct Subsidized Loan/Direct Unsubsidized Loan limit of $2,500 in the form of a Direct Subsidized Loan and
the maximum additional prorated Direct Unsubsidized Loan limit of $1,111. Each loan is paid in two equal disbursements, as shown below.
Prorated Combined Subsidized/Unsubsidized Annual Loan Limit Direct Subsidized Loan First Disbursement Direct Subsidized Loan Second Disbursement
$2,500 $1,250 $1,250
Prorated Additional Unsubsidized Annual Loan Limit Direct Unsubsidized Loan First Disbursement Direct Unsubsidized Loan Second Disbursement
$1,111 $556 $555
The student successfully meets the requirements for graduation from the program after completing only 400 clock hours. This means that the school must re-prorate the annual loan limit based on the
400 hours that the student actually completed:
($4,500 x 400) ÷ 900 = $2,000 re-prorated combined subsidized/unsubsidized annual loan limit
($2,000 x 400) ÷ 900 = $889 re-prorated additional unsubsidized annual loan limit
Since the student originally received Direct Loan amounts in excess of the re-prorated loan limit, the school must adjust the original disbursements and return the difference to the Department, as
shown below. The school – not the student – is responsible for returning the excess funds.
Re-Prorated Combined Subsidized/Unsubsidized Annual Direct Subsidized Loan Adjusted First Direct Subsidized Loan Adjusted Second
Loan Limit Disbursement Disbursement Direct Subsidized Loan Funds Returned to
$2,000 $1,000 $1,000
($500 reduction from original amount disbursed) (original disbursement reduced by $250) (original disbursement reduced by $250)
Re-prorated Additional Unsubsidized Annual Loan Limit Direct Unsubsidized Loan Adjusted First Direct Unsubsidized Loan Adjusted First
Disbursement Disbursement Direct Unsubsidized Loan Funds Returned to
$889 Department
$445 $444
($222 reduction from original amount disbursed) $222
(original disbursement reduced by $111) (original disbursement reduced by $111)
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The anisotropic Higgs oscillator on the two-dimensional sphere and the hyperbolic plane
The anisotropic Higgs oscillator on the two-dimensional sphere and the hyperbolic plane
An integrable generalization on the two-dimensional sphere S^2 and the hyperbolic plane H^2 of the Euclidean anisotropic oscillator Hamiltonian with "centrifugal" terms given by $H=1/2(p_1^2+p_2^2)+
\delta q_1^2+(\delta + \Omega)q_2^2 +\frac{\lambda_1}{q_1^2}+\frac{\lambda_2}{q_2^2}$ is presented. The resulting generalized Hamiltonian H_\kappa\ depends explicitly on the Gaussian curvature \kappa
\ of the underlying space, in such a way that all the results here presented hold simultaneously for S^2 (\kappa>0), H^2 (\kappa<0) and E^2 (\kappa=0). Moreover, H_\kappa\ is explicitly shown to be
integrable (albeit not superintegrable) for any values of the parameters \delta, \Omega, \lambda_1 and \lambda_2. Therefore, H_\kappa\ can also be interpreted as an anisotropic generalization of the
curved Higgs oscillator, that is recovered as the isotropic limit \Omega = 0 of H_\kappa. Furthermore, numerical integration of some of the trajectories for H_\kappa\ are worked out and the dynamical
features arising from the introduction of a curved background are highlighted. In particular we focus on the case \Omega=3\delta, whose Euclidean limit \kappa = 0 is the superintegrable 1:2
oscillator. In this case we illustrate how the Euclidean superintegrability of H is broken when the curvature \kappa\ is non-zero. Since for the specific \Omega=3\delta\ case another superintegrable
curved generalization is already known in the literature, the existence of an \Omega-dependent plurality of integrable curved systems whose Euclidean limit is the same anisotropic oscillator H can be
conjectured. Finally, the geometric interpretation of the curved "centrifugal" terms appearing in H_\kappa\ is also discussed in detail
The anisotropic Higgs oscillator on the two-dimensional sphere and the hyperbolic plane
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Parsing expression grammars
PEGs are an interesting new formalism for grammars. Unlike most older formalisms, which are based on Chomsky’s generative grammars, their starting point is recognizing strings that are in a language,
not generating them. As such they are a closer match to what we usually want a grammar for. The practical effect of this is that they naturally avoid common ambiguities without external rules, such
as C’s if/else ambiguity or the various rules about greediness imposed on regexes (e.g. perl’s matching rules versus POSIX’s longest-leftmost rule, discussed in Friedl’s book). Even though PEGs can
recognize some non-context-free languages (e.g. a^nb^nc^n) they can be matched in linear time using a packrat parser (which can be implemented very beautifully in Haskell).
Bryan Ford’s 2004 POPL paper establishes the formal foundations of PEGs and defines a concrete syntax for them, fairly similar to ABNF. The key differences are: the choice operator is ordered
(prefers to match its left-hand sub-expression); repetition operators are maximally greedy and don’t backtrack (so the second a in a*a can never match); and it includes positive and negative
lookahead assertions of the form &a and !a (like (?=a) and (?!a) in perl).
It occurs to me that there is a useful analogy hidden in here, that would be made more obvious with a little change in syntax. Instead of a / b write a || b, and instead of &a b write a && b. With ||
and && I am referring to C’s short-cutting “orelse” and “andalso” boolean operators - or rather the more liberal versions that can return more than just a boolean, since a PEG returns the amount of
input matched when it succeeds. This immediately suggests some new identities on grammars based on De Morgan’s laws, e.g. !a || !b === !(a && b). Note that !!a =/= a because the former never consumes
any input, so not all of De Morgan’s laws work with PEGs. This also suggests how to choose the operators to overload when writing a PEG parser combinator library for C++ (which has a much wider range
of possibilities than Lua).
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XGBoost | Exploratory
Create extreme gradient boosting model regression, binary classification and multiclass classification.
How to Access?
There are two ways to access. One is to access from 'Add' (Plus) button.
Another way is to access from a column header menu.
How to Use?
Column Selection
There are two ways to set what you want to predict by what variables.
If you are on "Select Columns" tab, you can set them by column selector.
If you are on "Custom" tab, you can type a formula directly.
Train Test Split
You can split the data into training and test to evaluate the performance of the model. You can set
Test Data Set Ratio - Ratio of test data in the whole data.
Random Seed to Split Training/Test - You can change random seed to try other training and test data combination.
Use Validation Data (Optional) - You can set data randomly selected to use as validation data set to watch the performance of the model against data that is not used for learning process. It
prevents overfitting.
How to treat NA? (Optional) - "na.action" parameter of glm. function. The default is "na.pass". This changes the behaviour of NA data. Can be one of the following.
Use Sparse Matrix (Optional) - If TRUE, it uses sparse matrix internally. This is memory efficient when the data becomes sparse, which means it has a lot of zero values. You can set this
implicitly but as default, sparse matrix is used when categorical values are used because the model matrix is often sparse in such case.
Type of Output (Optional) - The default is "linear". What distribution the target variable follows. This can be
Max Number of Iterations (Optional) - The default is 10. Max number of iterations for training.
Booster Type (Optional) - The default is "gbtree". Distribution that the target variable follows. This can be
Weight Column (Optional) - The default is NULL. A column with weight for each data.
Number of Early Stopping Rounds (Optional) - The default is NULL. The number of iterations to stop after the performance doesn't improve.
Minimum Child Weight (Optional)
Column Sample by Tree (Optional)
Evaluation Metrics (Optional)
How to Read Summary
Summary of Fit
Number of Iteration - Number of training iteration
Root Mean Square Error - Root mean square error to training data.
Mean Absolute Error - Mean absolute to training data.
Feature Importance
Feature - Name of the feature.
Importance - Improvement in accuracy for predicting the outcome by the feature.
Coverage - The ratio of the data covered by the feature.
Frequency - How many times each feature is used in all generated trees for training the model in a relative quantity scale.
Binary Classification
Use Validation Data (Optional) - You can set data randomly selected to use as validation data set to watch the performance of the model against data that is not used for learning process. It
prevents overfitting.
How to treat NA? (Optional) - "na.action" parameter of glm. function. The default is "na.pass". This changes the behaviour of NA data. Can be one of the following.
Use Sparse Matrix (Optional) - If TRUE, it uses sparse matrix internally. This is memory efficient when the data becomes sparse, which means it has a lot of zero values. You can set this
implicitly but as default, sparse matrix is used when categorical values are used because the model matrix is often sparse in such case.
Type of Output (Optional) - The default is "softprob". What distribution the target variable follows. This can be
Max Number of Iterations (Optional) - The default is 10. Max number of iterations for training.
Booster Type (Optional) - The default is "gbtree". Distribution that the target variable follows. This can be
Weight Column (Optional) - The default is NULL. A column with weight for each data.
Number of Early Stopping Rounds (Optional) - The default is NULL. The number of iterations to stop after the performance doesn't improve.
Minimum Child Weight (Optional)
Column Sample by Tree (Optional)
Evaluation Metrics (Optional)
How to Read Summary
Summary of Fit
Number of Iteration - Number of training iteration
AUC - Area under curve score to training data.
Misclassification Rate - Ratio of wrong classification to training data.
Negative Log Likelihood - Negative log likelihood score to training data.
Feature Importance
Feature - Name of the feature.
Importance - Improvement in accuracy for predicting the outcome by the feature.
Coverage - The ratio of the data covered by the feature.
Frequency - How many times each feature is used in all generated trees for training the model in a relative quantity scale.
Multiclass Classification
Use Validation Data (Optional) - You can set data randomly selected to use as validation data set to watch the performance of the model against data that is not used for learning process. It
prevents overfitting.
How to treat NA? (Optional) - "na.action" parameter of glm. function. The default is "na.pass". This changes the behaviour of NA data. Can be one of the following.
Use Sparse Matrix (Optional) - If TRUE, it uses sparse matrix internally. This is memory efficient when the data becomes sparse, which means it has a lot of zero values. You can set this
implicitly but as default, sparse matrix is used when categorical values are used because the model matrix is often sparse in such case.
Type of Output (Optional) - The default is "linear". What distribution the target variable follows. This can be
Max Number of Iterations (Optional) - The default is 10. Max number of iterations for training.
Booster Type (Optional) - The default is "gbtree". Distribution that the target variable follows. This can be
Weight Column (Optional) - The default is NULL. A column with weight for each data.
Number of Early Stopping Rounds (Optional) - The default is NULL. The number of iterations to stop after the performance doesn't improve.
Minimum Child Weight (Optional)
Column Sample by Tree (Optional)
Evaluation Metrics (Optional)
How to Read Summary
Summary of Fit
Number of Iteration - Number of training iteration.
Misclassification Rate - Ratio of wrong classification to training data.
Multiclass Logloss - Negative log likelihood score to training data.
Feature Importance
Feature - Name of the feature.
Importance - Improvement in accuracy for predicting the outcome by the feature.
Coverage - The ratio of the data covered by the feature.
Frequency - How many times each feature is used in all generated trees for training the model in a relative quantity scale.
Here's a step-by-step tutorial guide on how you can build, predict and evaluate logistic regression model.
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Data Science Learning Roadmap for 2021 - KDnuggets
Venturing into the world of Data Science is an exciting, interesting, and rewarding path to consider. There is a great deal to master, and this self-learning recommendation plan will guide you toward
establishing a solid understanding of all that is foundational to data science as well as a solid portfolio to showcase your developed expertise.
Although nothing really changes but the date, a new year fills everyone with the hope of starting things afresh. If you add in a bit of planning, some well-envisioned goals, and a learning roadmap,
you'll have a great recipe for a year full of growth.
This post intends to strengthen your plan by providing you with a learning framework, resources, and project ideas to help you build a solid portfolio of work showcasing expertise in data science.
Just a note: I've prepared this roadmap based on my personal experience in data science. This is not the be-all and end-all learning plan. You can adapt this roadmap to better suit any specific
domain or field of study that interests you. Also, this was created with Python in mind, as I personally prefer it.
What is a learning roadmap?
A learning roadmap is an extension of a curriculum. It charts out a multi-level skills map with details about what skills you want to hone, how you will measure the outcome at each level, and
techniques to further master each skill.
My roadmap assigns weights to each level based on the complexity and commonality of its application in the real-world. I have also added an estimated time for a beginner to complete each level with
exercises and projects.
Here is a pyramid that depicts the high-level skills in order of their complexity and application in the industry.
Data science tasks in the order of complexity.
This will mark the base of our framework. We’ll now have to deep dive into each of these strata to complete our framework with more specific, measurable details.
Specificity comes from examining the critical topics in each layer and the resources needed to master those topics.
We’d be able to measure the knowledge gained by applying the learned topics to a number of real-world projects. I’ve added a few project ideas, portals, and platforms that you can use to measure your
Important NOTE: Take it one day at a time, one video/blog/chapter a day. It is a wide spectrum to cover. Don’t overwhelm yourself!
Let’s deep dive into each of these strata, starting from the bottom.
1. How to Learn About Programming or Software Engineering
(Estimated time: 2-3 months)
First, make sure you have sound programming skills. Every data science job description will ask for programming expertise in at least one language.
Specific programming topics to know include:
• Common data structures (data types, lists, dictionaries, sets, tuples), writing functions, logic, control flow, searching and sorting algorithms, object-oriented programming, and working with
external libraries.
• SQL scripting: Querying databases using joins, aggregations, and subqueries
• Comfort using the Terminal, version control in Git, and using GitHub
Resources to learn Python:
Resources for learning Git and GitHub
• Guide for Git and GitHub [free]: complete these tutorials and labs to develop a firm grip over version control. It will help you further in contributing to open-source projects.
• Here's a Git and GitHub crash course on the freeCodeCamp YouTube channel
Resources for learning SQL
Measure your expertise by solving a lot of problems and building at least 2 projects:
• Solve a lot of problems here: HackerRank (beginner-friendly) and LeetCode (solve easy or medium-level questions)
• Data Extraction from a website/API endpoints—try to write Python scripts from extracting data from webpages that allow scraping like soundcloud.com. Store the extracted data into a CSV file or
a SQL database.
• Games like rock-paper-scissor, spin a yarn, hangman, dice rolling simulator, tic-tac-toe, and so on.
• Simple web apps like a YouTube video downloader, website blocker, music player, plagiarism checker, and so on.
Deploy these projects on GitHub pages or simply host the code on GitHub so that you learn to use Git.
2. How to Learn About Data Collection and Wrangling (Cleaning)
(Estimated time: 2 months)
A significant part of data science work is centered around finding apt data that can help you solve your problem. You can collect data from different legitimate sources—scraping (if the website
allows), APIs, Databases, and publicly available repositories.
Once you have data in hand, an analyst will often find themself cleaning dataframes, working with multi-dimensional arrays, using descriptive/scientific computations, and manipulating dataframes to
aggregate data.
Data are rarely clean and formatted for use in the “real world”. Pandas and NumPy are the two libraries that are at your disposal to go from dirty data to ready-to-analyze data.
As you start feeling comfortable writing Python programs, feel free to start taking lessons on using libraries like pandas and numpy.
Resources to learn about data collection and cleaning:
Data collection project Ideas:
• Collect data from a website/API (open for public consumption) of your choice, and transform the data to store it from different sources into an aggregated file or table (DB). Example APIs
include TMDB, quandl, Twitter API, and so on.
• Pick any publicly available dataset and define a set of questions that you’d want to pursue after looking at the dataset and the domain. Wrangle the data to find out answers to those questions
using Pandas and NumPy.
3. How to Learn About Exploratory Data Analysis, Business Acumen, and Storytelling
(Estimated time: 2–3 months)
The next stratum to master is data analysis and storytelling. Drawing insights from the data and then communicating the same to management in simple terms and visualizations is the core
responsibility of a Data Analyst.
The storytelling part requires you to be proficient with data visualization along with excellent communication skills.
Specific exploratory data analysis and storytelling topics to learn include:
• Exploratory data analysis—defining questions, handling missing values, outliers, formatting, filtering, univariate and multivariate analysis.
• Data visualization—plotting data using libraries like matplotlib, seaborn, and plotly. Know how to choose the right chart to communicate the findings from the data.
• Developing dashboards—a good percent of analysts only use Excel or a specialized tool like Power BI and Tableau to build dashboards that summarise/aggregate data to help management make
• Business acumen: Work on asking the right questions to answer, ones that actually target the business metrics. Practice writing clear and concise reports, blogs, and presentations.
Resources to learn more about data analysis:
Data analysis project ideas
• Exploratory analysis on movies dataset to find the formula to create profitable movies (use it as inspiration), use datasets from healthcare, finance, WHO, past census, e-commerce, and so on.
• Build dashboards (jupyter notebooks, excel, tableau) using the resources provided above.
4. How to Learn About Data Engineering
(Estimated time: 4–5 months)
Data engineering underpins the R&D teams by making clean data accessible to research engineers and scientists at big data-driven firms. It is a field in itself, and you may decide to skip this part
if you want to focus on just the statistical algorithm side of the problems.
Responsibilities of a data engineer comprise building an efficient data architecture, streamlining data processing, and maintaining large-scale data systems.
Engineers use Shell (CLI), SQL, and Python/Scala to create ETL pipelines, automate file system tasks, and optimize the database operations to make them high-performance.
Another crucial skill is implementing these data architectures, which demand proficiency in cloud service providers like AWS, Google Cloud Platform, Microsoft Azure, and others.
Resources to learn Data Engineering:
• Data Engineering Nanodegree by Udacity—as far as a compiled list of resources is concerned, I have not come across a better-structured course on data engineering that covers all the major
concepts from scratch.
• Data Engineering, Big Data, and Machine Learning on GCP Specialization—You can complete this specialization offered by Google on Coursera that walks you through all the major APIs and services
offered by GCP to build a complete data solution.
Data Engineering project ideas/certifications to prepare for:
• AWS Certified Machine Learning (300 USD)—A proctored exam offered by AWS that adds some weight to your profile (doesn’t guarantee anything, though), requires a decent understanding of AWS
services and ML.
• Professional Data Engineer—Certification offered by GCP. This is also a proctored exam and assesses your abilities to design data processing systems, deploying machine learning models in a
production environment, and ensure solutions quality and automation.
5. How to Learn About Applied Statistics and Mathematics
(Estimated time: 4–5 months)
Statistical methods are a central part of data science. Almost all data science interviews predominantly focus on descriptive and inferential statistics.
People often start coding machine learning algorithms without a clear understanding of underlying statistical and mathematical methods that explain the working of those algorithms. This, of course,
isn't the best way to go about it.
Topics you should focus on in Applied Statistics and math:
• Descriptive Statistics—to be able to summarise the data is powerful, but not always. Learn about estimates of location (mean, median, mode, weighted statistics, trimmed statistics), and
variability to describe the data.
• Inferential statistics—designing hypothesis tests, A/B tests, defining business metrics, analyzing the collected data and experiment results using confidence interval, p-value, and alpha
• Linear Algebra, Single and multi-variate calculus to understand loss functions, gradient, and optimizers in machine learning.
Resources to learn about Statistics and math:
Statistics project ideas:
• Solve the exercises provided in the courses above, and then try to go through a number of public datasets where you can apply these statistical concepts. Ask questions like “Is there sufficient
evidence to conclude that the mean age of mothers giving birth in Boston is over 25 years of age at the 0.05 level of significance”?
• Try to design and run small experiments with your peers/groups/classes by asking them to interact with an app or answer a question. Run statistical methods on the collected data once you have a
good amount of data after a period of time. This might be very hard to pull off but should be very interesting.
• Analyze stock prices, cryptocurrencies, and design hypothesis around the average return or any other metric. Determine if you can reject the null hypothesis or fail to do so using critical
6. How to Learn About Machine Learning and AI
(Estimated time: 4–5 months)
After grilling yourself and going through all the aforementioned major concepts, you should now be ready to get started with the fancy ML algorithms.
There are three major types of learning:
1. Supervised Learning—includes regression and classification problems. Study simple linear regression, multiple regression, polynomial regression, naive Bayes, logistic regression, KNNs, tree
models, ensemble models. Learn about evaluation metrics.
2. Unsupervised Learning—Clustering and dimensionality reduction are the two widely used applications of unsupervised learning. Dive deep into PCA, K-means clustering, hierarchical clustering, and
gaussian mixtures.
3. Reinforcement learning (can skip*)—helps you build self-rewarding systems. Learn to optimize rewards, use the TF-Agents library, create Deep Q-networks, and so on.
The majority of the ML projects need you to master a number of tasks that I’ve explained in this blog.
Resources to learn about Machine Learning:
Deep Learning Specialization by deeplearning.ai
For those of you who are interested in further diving into deep learning, you can start off by completing this specialization offered by deeplearning.ai and the Hands-ON book. This is not as
important from a data science perspective unless you are planning to solve a computer vision or NLP problem.
Deep learning deserves a dedicated roadmap of its own. I’ll create that with all the fundamental concepts soon.
Track your learning progress
I’ve also created a learning tracker for you on Notion. You can customize it to your needs and use it to track your progress, have easy access to all the resources and your projects.
Find the learning tracker here.
Also, here's the video version of this blog:
This is just a high-level overview of the wide spectrum of data science. You might want to deep dive into each of these topics and create a low-level concept-based plan for each of the categories.
Original. Reposted with permission.
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Re: Derivative of a Conjugate
• To: mathgroup at smc.vnet.net
• Subject: [mg87256] Re: [mg87208] Derivative of a Conjugate
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sat, 5 Apr 2008 04:25:40 -0500 (EST)
• References: <200804040757.CAA03892@smc.vnet.net>
On 4 Apr 2008, at 09:57, David Forehand wrote:
> Hi All,
> My first posting here, so please forgive me if I am being a bit
> stupid.
> I'm entering the following input:
> D[f[t0, t1], t0, t1] /. {f -> ((#1^2)*Conjugate[a[#2]] &)}
> and Mathematica gives the following output:
> 2 t0 a'[t1] Conjugate'[a[t1]]
> I would have expected:
> 2 t0 Conjugate[a'[t1]]
> i.e. the derivative of a conjugate is the conjugate of the derivative.
> Any idea how a force Mathematica to give the result I am expecting?
> In the above, I am assuming the variables "t0" and "t1" are real and
> the variable "a" is complex, although I have not explicitly told
> Mathematica this.
> Thanks Very Much in advance,
> David
Mathematica always assumes that the derivative is computed in the
complex plane and it automatically applies the chain rule. It
certianly would be wrong to assume that the derivative commutes with
conjugation, as that is not true in general in the complex plane, so
if Mathematica did not automatically apply the chain rule it would
have to return your input unevaluated. One way to avoing your problem
is to write your function a[x] explicitly as a sum of its real and
imaginary parts, say c[x] and d[x]. First make a list of replacement
rules for getting the output back in terms of a[x]:
rules = {c[x_] :> Re[a[x]], d[x_] :> Im[a[x]],
Derivative[n_][c][x_] :> Re[Derivative[n][a][x]],
Derivative[n_][d][x_] :> Im[Derivative[n][a][x]]};
Now, we differentiate usign c[x]+I d[x] in place of a[x], apply rules
and FullSimplify:
FullSimplify[D[f[t0, t1], t0, t1] /.
{f -> (#1^2*(c[#2] - I*d[#2]) & )} /. rules]
Andrzej Kozlowski
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Answer in Mechanical Engineering for Uday sharma #150540
Answer to Question #150540 in Mechanical Engineering for Uday sharma
A rope pulls a 200 kg sleigh up a slope at incline angle θ=30 deg, through distance
d=20m. The sleigh and its contents have a total mass of 200kg. The snowy slope is so
slippery that we take it to be frictionless. How much work is done by each force
(Normal, gravitational, rope’s) acting on the sleigh?
Total mass of content is 200 Kg, angle of inclination is 30^0,distance of travelled = 20 m
Now we can use the below diagram for it
Now tension force= T= mgsin"\\theta"
Work done by tension force="mg sin\\theta \\times s= 200\\times 9.8\\times sin 30^0\\times 20 = 1960\\times 0.5\\times 20=19600 J"
Gravitation work="mgh= 200\\times 9.8\\times 20 \\times sin 30^0= 19600" J
Normal reaction force = mgcos"\\theta= 200\\times 9.8\\times 0.866=1697.36 N"
Learn more about our help with Assignments:
Mechanical Engineering
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Electromotive force for an anisotropic turbulence: Intermediate nonlinearity
A nonlinear electromotive force for an anisotropic turbulence in the case of intermediate nonlinearity is derived. The intermediate nonlinearity implies that the mean magnetic field is not strong
enough to affect the correlation time of a turbulent velocity field. The nonlinear mean-field dependencies of the hydrodynamic and magnetic parts of the [Formula Presented] effect, turbulent
diffusion, and turbulent diamagnetic and paramagnetic velocities for an anisotropic turbulence are found. It is shown that the nonlinear turbulent diamagnetic and paramagnetic velocities are
determined by both an inhomogeneity of the turbulence and an inhomogeneity of the mean magnetic field [Formula Presented] The latter implies that there are additional terms in the turbulent
diamagnetic and paramagnetic velocities [Formula Presented] and [Formula Presented] These effects are caused by a tangling of a nonuniform mean magnetic field by hydrodynamic fluctuations. This
increases the inhomogeneity of the mean magnetic field. It is also shown that in an isotropic turbulence the mean magnetic field causes an anisotropy of the nonlinear turbulent diffusion. Two types
of nonlinearities in magnetic dynamo determined by algebraic and differential equations are discussed. Nonlinear systems of equations for axisymmetric [Formula Presented] dynamos in both spherical
and cylindrical coordinates are derived.
ASJC Scopus subject areas
• Statistical and Nonlinear Physics
• Statistics and Probability
• Condensed Matter Physics
Dive into the research topics of 'Electromotive force for an anisotropic turbulence: Intermediate nonlinearity'. Together they form a unique fingerprint.
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18.1: What’ll It Be? (10 minutes)
This warm-up helps students get oriented to the situation they will be working with throughout this lesson. Students also make initial predictions based on very limited information.
Student Facing
The points on the graph represent the average resale price of a toy in dollars as a function of time.
1. Use the information to predict the average resale price of the toy on day 12. Explain your reasoning.
2. How confident are you in your predictions? Explain your reasoning.
Activity Synthesis
The purpose of the discussion is to recognize that it can be difficult to make predictions based on very limited data and information.
Poll the class on their predictions. Ask selected students to explain why they made that prediction. Record and display the responses for all to see. If possible, display and reference the graph as
students explain their reasoning.
Ask students, “What other information would help improve your predictions?” (More data or a more detailed explanation of what the toy is and what is happening would help me make a better prediction.)
18.2: Collectable Toy Price (15 minutes)
In this activity students use some data to find an average rate of change and write a linear function to model data for the price of a collectable toy over several days. In the associated Algebra 1
lesson students examine battery life on a phone by modeling a graph with a function. Students are supported by this activity by being given some additional steps to think through while modeling a
If possible, do not allow students to look at the next activity while they work on this activity. The next activity gives students additional information about the price that may affect their
thinking for the questions here.
Student Facing
The graph shows the average resale price for a toy in dollars as a function of time in days.
1. Estimate the average rate of change for the first 10 days.
2. Estimate the rate of change between days 9 and 10.
3. Write a linear function, \(f\), that models the data.
4. Predict the price of the toy after 12 days.
Activity Synthesis
The purpose of the discussion is to provide insight into how to model data with functions. Select students to share their predictions, estimates, and model functions. After each response is shared,
ask if there are other possible solutions from other students. Ask students,
• “How did you come up with your model function?” (I drew a line that went through the middle of the data and it was close to going through the points \((0,5)\) and \((10,25)\), so I wrote the
equation of the line going through those points.)
• “On day 0 the price of the toy was $5, does that mean your linear function should also have 5 as the vertical intercept?” (Not necessarily. It did work out in this case, but there could be
another vertical intercept, especially if the point is very different from the rest of the data near zero.)
• “How did you use the average rate of change in your model function?” (I used the average rate of change for the 10 days as the slope of my function because it seemed to be a good fit.)
• “How confident are you in your prediction for the price of the toy after 12 days?” (I’m not very confident. While there does seem to be a nice upward trend from the data we have, at any point the
price could drop significantly.)
18.3: More Information (15 minutes)
In this activity students get additional information about the average price of the toy which does not follow the same trend as in the previous activity. The additional information shows students
that trends can change and that knowing additional data is helpful, but also knowing the situation can help make better predictions.
After a few initial questions, students are asked to pause to get additional information from the teacher. During the pause, tell students that on day 13 the company that makes the toy released
another shipment of the toy.
Student Facing
After a few more days, a graph of the average price of the toy looks like this.
1. Draw a function (it does not need to be linear) that could model the data.
2. Use your graph to predict the average price of the toy after 12 days. How confident are you in this answer?
3. Pause here to get additional information from your teacher about the price of the toy. Based on the new information, do you have a new prediction for what happens to the average price of the toy
after 12 days? Explain your reasoning.
Activity Synthesis
The purpose of the discussion is to recognize that additional data can provide a better understanding of a situation, but understanding what is actually happening in the situation can help a lot as
well. Select students to share their responses. After each response, ask if there are additional solutions. Ask students,
• “How does knowing additional numerical data about the average price of the toy help your prediction?” (It shows me that the trend I saw at the beginning was not expected to continue.)
• “How does knowing the actual situation about the additional shipment help your prediction?” (It let me know that the trend for the price increase over the first 10 days is likely to continue
until the new shipment arrived, when the average price dropped down and began rising again.)
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Analyzing value distributions with box plots
Greetings, fellow data analysts!
Statistical distribution wasn’t exactly a hot conversation topic a few years back. Yet in Germany, it was almost omnipresent – at least as soon as we opened our wallets. A portrait of Carl Friedrich
Gauß, the inventor of the normal distribution curve, graced the front side of the fourth (and last series) of the ten D-mark bill.
Although this bill is now history, delving into statistical distributions of measures is still well worth the effort. Sometimes, we want to analyze distributions and statistical spreads beyond
standard reports with dense measures: Are our processing times stable or do they vary in larger intervals? Can we minimize the outliers in our error rates? Do our delivery times vary greatly – or are
they only satisfactory on average? Are our sales markets homogeneous?
To answer these and other complex questions, we need more than just a simple average. Statisticians often rely on box plots to describe and explain distributions. In this edition of clicks!, you will
see how quick and easy it is to create them with DeltaMaster. Maybe, you’ll even add them to your standard reports and distribute them just as you normally would, for example, with ReportServer.
Best regards,
Your Bissantz & Company Team
Controlling and statistics sometimes have more and sometimes less common ground. The parallels are greater when we deviate from standardized list reporting and move more into analysis. A sorted list
of customer revenues, for example, certainly has its value – but it can’t easily provide insight on how sales are distributed or if there have been movements in that distribution over time. In the
next few pages, we would like to present the box plot, a type of visualization which many DeltaMaster users have long since added to their standard reports and cockpits. Box plots help you examine a
large amount of values with regards to their distribution or statistical spread. Most likely, you have already seen these types of charts before.
Box plots visualize the position and spread of values in a distribution (a random sample). Unlike many other types of charts, they do not show individual objects such as customers, products,
production orders, shipments, or service cases. Instead, the presentation is based on five statistical measures that characterize the distribution as a whole: the minimum, the lower quartile, the
median, the upper quartile, and the maximum. (We’ll explain the statistical background in more detail below.)
The visualization above shows the margin rates of products. You can either analyze each plot alone or observe the development of the distributions over time. As you have probably noticed, the range
has barely changed but the median has increased from August to November. This means that more products were sold at a higher margin. In January, however, the margin of some products sank again. This
should warn a product manager that costs might be getting out of control or that the price elasticity has changed.
To explain how to use box plots, let’s once again use our ‘Chair’ reference model, which contains financial and sales measures. You can, however, use box plots to analyze many other aspects of your
□ In production analysis, you can evaluate processing times, error rates, maintenance intervals, or buffer stock. You could also see if changes in the process parameters have lead to more stable
processes, for example, if the respective production measures weren’t as scattered or the outliers shifted closer to the box or median. How did the median move? Can you see improvements or
declines ‘in the middle’ (see definition below)? How strongly are the measurements scattered around the middle?
□ Many similar questions often arise in logistics analysis. erHHere, for example, you might want to know more about delivery times or availability. Major costs caused by inadequate or poor capacity
utilization are often hidden in these processes.
The example above, taken from an application for transport analysis, is used to examine the measure ‘Service costs per container by customers’. Here, you can see that the cost distribution was
relatively constant over a period of several months. The median did not move much and the box with the mid 50% of the values only showed minimal changes in size or position. In other words, it was
business as usual. In June, however, the costs per means of transportation for the customers were higher and, most of all, the range was much wider – a trend which started back in May. These
developments are grounds for further investigation.
□ Going back to our example in sales, you might ask yourself if different market segments react similarly. The reasons might be caused by the nature of the respective segments themselves or can be
an indication of a very (or not so) effective segment management, which would be indicated by higher uniformity and a smaller spread.
Creating box plots
Box plots are also known as box-whisker plots, with the box being in the middle and the whiskers pointing up and down.
You can easily identify the five statistical measures in the screenshot on your right. The upper/lower corners of the box represent the upper/lower quartiles. The line in the middle of the box shows
the position of the median. The bars on the ends of both lines mark the maximum and minimum values. The gaps between each of the five markings are important in interpreting the visualization; the
width of the box, however, is irrelevant. You will also note that this chart displays percentage values on the Y axis. This has nothing to do with the method itself; the measure you are analyzing in
this case (i.e. the margin rate) is a percentage. In other scenarios, the box plot would display this data using the measure’s units, for example, in Euros, pieces, minutes, etc.
Medians and quartiles – what were they again?
Although we don’t want to digress in the depths of descriptive statistics, a small recap of the basics can’t hurt either. After all, even if you can create box plots easily in DeltaMaster, you still
need a bit more background knowledge to interpret them or respond to inquiries than with simple columns or bars in charts and graphic tables.
□ The median lies in the middle of a sorted series of values. Half of the values are larger (or equally large) than it and the other half are smaller (or equally large). If you take the values 10,
20, 30, 40 and 1,000, the median is 30. Unlike an arithmetic mean, you cannot determine the median through addition and division; instead you simply ‘count down’ from the smallest to the largest
values until you find the place where you can divide the series into two sections of equal size. In many cases, the median provides more information than the arithmetic mean because it is less
sensitive to outliers. This is also the case in the series described above; a mean of 220 does a poor job of describing the four small values as well as the extreme outlier of 1,000. The one
outlier raised the arithmetic mean to a value that has nothing to do with any of the other measurements. The median, in contrast, represents the surrounding values quite well and displays the
outliers as such.
□ Quartiles follow the same principle. They, too, are values that divide a sorted series – not just in the middle but also into upper and lower quarters. The lower quartile (25-percent quartile) is
nevertheless a value that is larger than a quarter of the values and smaller than the remaining three quarters. The upper quartile (75-percent quartile) is a value that is larger than three
quarters of the values and smaller than the remaining quarter. As a result, you can also describe the median as the mid quartile or the 50-percent quartile.
Based on these pragmatic explanations, you can determine that the mid 50 percent of the values lie between the upper and the lower quartiles (see diagram on the next page). This area is drawn as a
‘box’ in your box plot. A line within this box marks the median. Since the median is not the average, it does not have to lie in the middle of the box. Instead, the box and the median marking show
how the mid 50 percent of the values are distributed around the median. The 25 percent of the smallest values lie between the minimum and the lower quartile. This is equivalent to the area between
the end of the lower ‘whisker’ and the lower end of the box. 25 percent of the largest values lie between the upper quartile and the maximum. This is equivalent to the area between the upper end of
the box and the end of the upper ‘whisker’.
How the values are calculated mathematically is a science of its own. It varies, for example, if the number of values is even, uneven, or divisible by four. The programming language ‘R’ uses nine
different ways to calculate quartiles. For your reporting purposes, however, you don’t need to go to extremes. You can probably live with slight inaccuracies in decimal places and different nuances
in the definitions if your series of values is large enough. In most cases, that already applies with 30 values or more – a relatively small amount considering the volumes of data that you use for
analysis. Identifying distribution measures down to the exact decimal value is not relevant for management information. Instead, you simply want to visualize, assess, and compare the distribution
with other ones.
How to create box plot Charts
You can create a box plot chart with DeltaMaster in three easy steps:
1. You need to create five statistical measures as individual measures in the analysis model. You can do this quickly using the built-in wizards in DeltaMaster.
2. You must transfer the measures into a pivot table. This, too, is simple. You simply select the measures as you normally would in the Axis definition.
3. You need to change a few of the format settings of the graphical visualization of the pivot table (i.e. pivot graphic) to create the typical box plot chart. This, too, only involves a few mouse
clicks which, at most, might be somewhat unaccustomed to you.
But let’s not get ahead of ourselves. The box plot is a pivot graphic and, like all pivot graphics, is based on a pivot table. As a result, you can already create box plot charts as well as the
necessary measures starting on the Pivotizer level.
Creating measures
Before you can start, the five measures must be defined in your data model. If you don’t already have them, you can easily create these as univariate statistical measures in DeltaMaster (Model menu,
Create new measure).
The respective wizard generates all of the desired measures at once. For the Dimension, simply select the one in which the members that you are examining are distributed, for example, products,
customers, offices, or orders. For more information on working with univariate statistical measures, please refer to DeltaMaster clicks! 7/2009.
Creating a pivot table
The box plot requires a standardized table construction with these five rows:
□ Row 1: the minimum
□ Row 2: the lower quartile
□ Row 3: the median
□ Row 4: the upper quartile
□ Row 5: the maximum
The values were offered in the same order in the New measure wizard and are generally shown that way in the Measure browser as well. This makes it easier to select them.
The column axis can remain empty. If you use it, DeltaMaster will create a separate box plot for each member and place them next to each other in the same chart. This makes it easier to compare
differences in the distribution across various countries, offices, product lines, order types, or other report components. Now, simply place the time dimension or the time utility dimension in the
column axis so that you can observe how the distribution has changed over a stretch of time.
Formatting data series
From your pivot table, go the View menu, switch your view to Chart, and open the menu bar (context menu, I want to… menu). On the bottom-right corner, you can select the box plot
from the different types of charts that are available. The visualization that you will see at first, however, will not look like a typical box plot chart.
To create the typical outline form, you will now need to edit the Settings (context menu) of the data series. Let’s start with the red series. In the default setting, this stands for the median due
to the standard table construction.
Under Settings on the Series tab, you can now set the Fill to the Color white. To create a Frame, you simply tick on the box and select No effect. Now, set the Color of the Frame to either gray or
black. If you want the box plot to be very aesthetic, change the Width and select the second thinnest line size which best resembles the ‘whisker’ look.
Now, repeat the same steps for the lower quartile that was colored green in the default view: white Fill, Frame with No effect in the Color black or gray and a somewhat larger Width.
Formatting the whiskers is easier. Since you only have to change the fill and not the frame, you can open the context menu of the pivot graphic to apply a different color to the series. Here, you
should use the same black or gray shade that you used for the frame.
If desired, you can also display the Values of the individual sections (context menu of the graphic). To format the labeling, simply open the Graphic settings (context menu, I want to… menu or F4
key) and change the Point labels on the respective tab. Here, you can also suppress the label for individual measures. In many cases, for example, you may want to omit the upper and lower quartiles
but show the minimum, maximum, and median values. To do this, simply open your Chart properties, select the respective Series, deactivate the box to Show the point label, and Apply this change for
every series.
For advanced users
If you use box plots regularly, you may sometimes wish that you could show the arithmetic means in your chart as well. You can do this by adding a sixth row to your pivot table. In this case,
DeltaMaster will draw all other rows as lines in the box plot. Please note, however, that you should only offer this type of chart to more experienced readers. Some readers might be irritated by the
additional markings – especially when the arithmetic means lies outside of the box. Before using this option, you may want to inform your audience that this occurs occasionally (and rightly so due to
the statistical relationships).
Using box plots
You can certainly save box plots as reports and distribute them accordingly. This way, users on Reader and Viewer levels can assess the results as well. Viewer readers can even dynamically change
which box plots should be contained in the chart depending on the setup of the pivot table. In this case, there are two important options which you can define in the Axis definition of the column
axis from Pivotizer or a higher user level. If you define the axis by Level selection (General tab) and select the dynamic synchronization, Viewer users can select which members they want to display
on the axis – as well as which and how many plots they would like to see in the chart – all from the View window (see DeltaMaster clicks! 4/2009). If you now go to the Axis definition on the Options
tab, you can also Allow drill downs for Viewer mode so that your users can determine the contents of the chart themselves (see DeltaMaster clicks! 6/2009 for more information). Each user can then
switch the View (menu in the Report window) from Chart to Table, drill down as desired and then switch back to the Chart.
Just like all other pivot tables and charts, you can also integrate box plots into Combination cockpits. This is especially helpful when you want to create a visual comparison of multiple measures.
In addition, you can multiply this visualization as well using Small multiples as described in DeltaMaster clicks! 11/2010.
Questions? Comments?
Just contact your Bissantz team for more information.
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Math Contest Repository
Fermat 2024 Part A - Question 4, CEMC UWaterloo
(Fermat 2024, Part A, Question 4, CEMC - UWaterloo)
Shuxin begins with $10$ red candies, $7$ yellow candies, and $3$ blue candies. After eating some of the candies, there are equal numbers of red, yellow, and blue candies remaining. What is the
smallest possible number of candies that Shuxin ate?
$(A)$ $11$
$(B)$ $7$
$(C)$ $17$
$(D)$ $20$
$(E)$ $14$
Answer Submission Note(s)
Your answer should be a single capital letter, e.g. 'A'.
Please login or sign up to submit and check if your answer is correct.
flag Report Content
You should report content if:
• It may be offensive.
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Thanks for keeping the Math Contest Repository a clean and safe environment!
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Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. It is related to many other areas of mathematics, such as algebra, probability theory,
ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics. Aspects of combinatorics include "counting" the objects satisfying certain criteria (
enumerative combinatorics), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest",
"smallest", or "optimal" objects ( extremal combinatorics and combinatorial optimization), and finding algebraic structures these objects may have ( algebraic combinatorics).
Combinatorics is as much about problem solving as theory building, though it has developed powerful theoretical methods, especially since the later twentieth century (see the page List of
combinatorics topics for details of the more recent development of the subject). One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural
connections to other areas.
There are many combinatorial patterns and theorems related to the structure of combinatoric sets. These often focus on a partition or ordered partition of a set. See the List of partition topics for
an expanded list of related topics or the List of combinatorics topics for a more general listing. Some of the more notable results are highlighted below.
An example of a simple combinatorial question is the following: What is the number of possible orderings of a deck of 52 distinct playing cards? The answer is 52! (52 factorial), which is equal to
about 8.0658 × 10^67.
Another example of a more difficult problem: Given a certain number n of people, is it possible to assign them to sets so that each person is in at least one set, each pair of people is in exactly
one set together, every two sets have exactly one person in common, and no set contains everyone, all but one person, or exactly one person? The answer depends on n. See "Design theory" below.
Combinatorics is used frequently in computer science to obtain estimates on the number of elements of certain sets. A mathematician who studies combinatorics is often referred to as a
combinatorialist or combinatorist.
History of Combinatorics
Earliest uses
The earliest books about combinatorics are from India. A Jainist text, the Bhagabati Sutra, had the first mention of a combinatorics problem; it asked how many ways one could take six tastes one,
two, or three tastes at a time. The Bhagabati Sutra was written around 300 BC, and thus was the first book to mention the choice function . The next ideas of Combinatorics came from Pingala, who
was interested in prosody. Specifically, he wanted to know how many ways a six syllable meter could be made from short and long notes. He wrote this problem in the Chanda sutra (also
Chandahsutra) in the second century BC . In addition, he also found the number of meters that had n long notes and k short notes, which is equivalent to finding the binomial coefficients.
The ideas of the Bhagabati were generalized by the Indian mathematician Mahariva in 850 AD, and Pingala's work on prosody was expanded by Bhaskara and Hemacandra in 1100 AD. Bhaskara was the
first known person to find the generalized choice function, although Brahmagupta may have known earlier. Hemacandra asked how many meters existed of a certain length if a long note was considered
to be twice as long as a short note, which is equivalent to finding the Fibonacci numbers.
While India was the first nation to publish results on Combinatorics, there were discoveries by other nations on similar topics. The earliest known connection to Combinatorics comes from the
Rhind papyrus, problem 79, for the implementation of a geometric series. The next milestone is held by the I Ching. The book is about what different hexagrams mean, and to do this they needed to
know how many possible hexagrams there were. Since each hexagram is a permutation with repetitions of six lines, where each line can be one of two states, solid or dashed, combinatorics yields
the result that their are $2^6=64$ hexagrams. A monk also may have counted the number of configurations to a game similar to Go around 700 AD. Although China had relatively few advancements in
enumerative combinatorics, they solved a combinatorial design problem, the magic square, around 100 AD.
In Greece, Plutarch wrote that the Xenocrates discovered the number of different syllables possible in the Greek language. This, however, is unlikely because this is one of the few mentions of
Combinatorics in Greece. The number they found, $1.002 \cdot 10^{12}$ also seems too round to be more than a guess. .
Magic squares remained an interest of China, and they began to generalize their original 3×3 square between 900 and 1300 AD. China corresponded with the Middle East about this problem in the 13th
century. The Middle East also learned about binomial coefficients from Indian work, and found the connection to polynomial expansion.
Combinatorics in the West
Combinatorics came to Europe in the 13th century through two mathematicians, Leonardo Fibonacci and Jordanus de Nemore. Fibonacci's Liber Abaci introduced many of the Arabian and Indian ideas to
Europe, including that of the Fibonacci numbers. Jordanus was the first person to arrange the Binomial coefficient's in a triangle, as he did in proposition 70 of De Arithmetica. This was also
done in the Middle East in 1265, and China around 1300. Today, this triangle is known as Pascal's triangle.
Pascal's contribution to the triangle that bears his name comes from his work on formal proofs about it, in addition to his connection between it and probability. Together with Leibniz and his
ideas about partitions in the 17th century, they are considered the founders of modern combinatorics.
Both Pascal and Leibniz understood that algebra and combinatorics corresponded (aka, binomial expansion was equivalent to the choice function.) This was expanded by De Moivre, who found the
expansion of a multinomial. De Moivre also found the formula for derangements using the principle of inclusion-exclusion, a method different from Nikolaus Bernouli, who had found them previously.
He managed to approximate the binomial coefficients and factorial. Finally, he found a closed form for the Fibonacci numbers by inventing generating functions.
In the 18th century, Euler worked on problems of combinatorics. In addition to working on several problems of probability which link to combinatorics, he worked on the knights tour, Graeco-Latin
square, Eulerian numbers, and others. He also invented graph theory by solving the Seven Bridges of Königsberg problem, which also lead to the formation of topology. Finally, he broke ground with
partitions by the use of generating functions.
Enumerative combinatorics
Counting the number of ways that certain patterns can be formed is the central problem of enumerative combinatorics. Two examples of this type of problem are counting combinations and counting
permutations (as discussed in the previous section). More generally, given an infinite collection of finite sets {S[i]} indexed by the natural numbers, enumerative combinatorics seeks to describe a
counting function which counts the number of objects in S[n] for each n. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in
applications have a relatively simple combinatorial description.
The simplest such functions are closed formulas, which can be expressed as a composition of elementary functions such as factorials, powers, and so on. For instance, as shown below, the number of
different possible orderings of a deck of n cards is f(n) = n!. Often, no closed form is initially available. In these cases, we frequently first derive a recurrence relation, then solve the
recurrence to arrive at the desired closed form.
Finally, f(n) may be expressed by a formal power series, called its generating function, which is most commonly either the ordinary generating function
$\sum_{n\ge 1} f(n) x^n$
or the exponential generating function
$\sum_{n \ge 1} f(n) \frac{x^n}{n!}.$
Often, a complicated closed formula yields little insight into the behaviour of the counting function as the number of counted objects grows. In these cases, a simple asymptotic approximation may be
preferable. A function $g(n)$ is an asymptotic approximation to $f(n)$ if $f(n)/g(n)\rightarrow 1$ as $n\rightarrow$infinity. In this case, we write $f(n) \sim g(n)\,$.
Once determined, the generating function may allow one to extract all the information given by the previous approaches. In addition, the various natural operations on generating functions such as
addition, multiplication, differentiation, etc., have a combinatorial significance; this allows one to extend results from one combinatorial problem in order to solve others.
When the order matters, and an object can be chosen more than once, the number of permutations is
$n^r \,\!$
where n is the number of objects from which you can choose and r is the number to be chosen.
For example, if you have the letters A, B, C, and D and you wish to discover the number of ways to arrange them in three letter patterns ( trigrams)
1. order matters (e.g., A-B is different from B-A, both are included as possibilities)
2. an object can be chosen more than once (A-A possible)
you find that there are 4^3 or 64 ways. This is because for the first slot you can choose any of the four values, for the second slot you can choose any of the four, and for the final slot you can
choose any of the four letters. Multiplying them together gives the total.
Permutations without repetitions
When the order matters and each object can be chosen only once, then the number of permutations is
$(n)_{r} = \frac{n!}{(n-r)!}$ where n is the number of objects from which you can choose, r is the number to be chosen and "!" is the standard symbol meaning factorial.
For example, if you have five people and are going to choose three out of these, you will have 5!/(5 − 3)! = 60 permutations.
Note that if n = r (meaning the number of chosen elements is equal to the number of elements to choose from; five people and pick all five) then the formula becomes
$\frac{n!}{(n-n)!} = \frac{n!}{0!} = n!$
where 0! = 1.
For example, if you have the same five people and you want to find out how many ways you may arrange them, it would be 5! or 5 × 4 × 3 × 2 × 1 = 120 ways. The reason for this is that you can choose
from 5 for the initial slot, then you are left with only 4 to choose from for the second slot etc. Multiplying them together gives the total of 120.
Combinations without repetitions
When the order does not matter and each object can be chosen only once, the number of combinations is the binomial coefficient:
${n\choose k} = {{n!} \over {k!(n - k)!}}$
where n is the number of objects from which you can choose and k is the number to be chosen.
For example, if you have ten numbers and wish to choose 5 you would have 10!/(5!(10 − 5)!) = 252 ways to choose. The binomial coefficient is also used to calculate the number of permutations in a
Combinations with repetitions
When the order does not matter and an object can be chosen more than once, then the number of combinations is
${{(n + k - 1)!} \over {k!(n - 1)!}} = {{n + k - 1} \choose {k}} = {{n + k - 1} \choose {n - 1}}$
where n is the number of objects from which you can choose and k is the number to be chosen.
For example, if you have ten types of donuts (n) on a menu to choose from and you want three donuts (k) there are (10 + 3 − 1)! / 3!(10 − 1)! = 220 ways to choose (see also multiset).
Fibonacci numbers
Let f(n) be the number of distinct subsets of the set $S(n)=\{1,2,3, \ldots ,n \}$ that do not contain two consecutive integers. When n = 4, we have the sets {}, {1}, {2}, {3}, {4}, {1,3}, {1,4},
{2,4}, so f(4) = 8. We count the desired subsets of $S(n)$ by separately counting those subsets that contain element $n$ and those that do not. If a subset contains $n$, then it does not contain
element $n-1$. So there are exactly $f(n-2)$ of the desired subsets that contain element $n$. The number of subsets that do not contain $n$ is simply $f(n-1)$. Adding these numbers together, we get
the recurrence relation:
$f(n) = f(n-1) + f(n-2)\, ,$
where $f(1)=2$ and $f(2)=3$.
As early as 1202, Leonardo Fibonacci studied these numbers. They are now called Fibonacci numbers; in particular, $f(n)$ is known as the $n+2$nd Fibonacci number. Although the recurrence relation
allows us to compute every Fibonacci number, the computation is inefficient. However, by using standard techniques to solve recurrence relations, we can reach the closed form solution:
$f(n) = \frac{\phi^{n+2}-(1-\phi)^{n+2}}{\sqrt{5}}$
where $\phi = (1 + \sqrt 5) / 2$, the golden ratio.
In the above example, an asymptotic approximation to $f(n)$ is:
$f(n) \sim \frac{\phi^{n+2}}{\sqrt{5}}$
as n becomes large.
Structural combinatorics
Graph theory
Graphs are basic objects in combinatorics. The questions range from counting (e.g. the number of graphs on n vertices with k edges) to structural (e.g. which graphs contain Hamiltonian cycles).
Design theory
A simple result in the block design area of combinatorics is that the problem of forming sets, described in the introduction, has a solution only if n has the form q^2 + q + 1. It is less simple to
prove that a solution exists if q is a prime power. It is conjectured that these are the only solutions. It has been further shown that if a solution exists for q congruent to 1 or 2 mod 4, then q is
a sum of two square numbers. This last result, the Bruck-Ryser theorem, is proved by a combination of constructive methods based on finite fields and an application of quadratic forms.
When such a structure does exist, it is called a finite projective plane; thus showing how finite geometry and combinatorics intersect.
Matroid theory
Matroid theory abstracts part of geometry. It studies the properties of sets (usually, finite sets) of vectors in a vector space that do not depend on the particular coefficients in a linear
dependence relation. Not only the structure but also enumerative properties belong to matroid theory.
For instance, given a set of n vectors in Euclidean space, what is the largest number of planes they can generate? Answer: the binomial coefficient
Is there a set that generates exactly one less plane? (No, in almost all cases.) These are extremal questions in geometry, as discussed below.
Extremal and probabilistic combinatorics
Many extremal questions deal with set systems. A simple example is the following: what is the largest number of subsets of an n-element set one can have, if no two of the subsets are disjoint?
Answer: half the total number of subsets. Proof: Call the n-element set S. Between any subset T and its complement S − T, at most one can be chosen. This proves the maximum number of chosen subsets
is not greater than half the number of subsets. To show one can attain half the number, pick one element x of S and choose all the subsets that contain x.
A more difficult problem is to characterize the extremal solutions; in this case, to show that no other choice of subsets can attain the maximum number while satisfying the requirement.
Often it is too hard even to find the extremal answer f(n) exactly and one can only give an asymptotic estimate.
Ramsey theory
Ramsey theory is a celebrated part of extremal combinatorics. It states that any sufficiently large random configuration will contain some sort of order.
Frank P. Ramsey proved that for every integer k there is an integer n, such that every graph on n vertices either contains a clique or an independent set of size k. This is a special case of Ramsey's
theorem. For example, given any group of six people, it is always the case that one can find three people out of this group that either all know each other or all do not know each other. The key to
the proof in this case is the Pigeonhole Principle: either A knows three of the remaining people, or A does not know three of the remaining people.
Here is a simple proof: Take any one of the six people, call him A. Either A knows three of the remaining people, or A does not know three of the remaining people. Assume the former (the proof is
identical if we assume the latter). Let the three people that A knows be B, C, and D. Now either two people from {B,C,D} know each other (in which case we have a group of three people who know each
other - these two plus A) or none of B,C,D know each other (in which case we have a group of three people who do not know each other - B,C,D). QED.
Extremal combinatorics
The types of questions addressed in this case are about the largest possible graph which satisfies certain properties. For example, the largest triangle-free graph on 2n vertices is a complete
bipartite graph K[n,n].
Probabilistic combinatorics
Here the questions are of the following type: what is the probability of a certain graph property for a random graph (within a certain class) E.g. what is the average number of triangles in a random
Geometric combinatorics
Geometric combinatorics is related to convex and discrete geometry. It asks, e.g. how many faces of each dimension can a convex polytope have. Metric properties of polytopes play an important role as
well, e.g. the Cauchy theorem on rigidity of convex polytopes. Special polytopes are also considered, such as permutohedron, associahedron and Birkhoff polytope.
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