content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
1970 AHSME Problems/Problem 35
A retiring employee receives an annual pension proportional to the square root of the number of years of his service. Had he served $a$ years more, his pension would have been $p$ dollars greater,
whereas had he served $b$ years more $(be a)$, his pension would have been $q$ dollars greater than the original annual pension. Find his annual pension in terms of $a,b,p$ and $q$.
$\text{(A) } \frac{p^2-q^2}{2(a-b)}\quad \text{(B) } \frac{(p-q)^2}{2\sqrt{ab}}\quad \text{(C) } \frac{ap^2-bq^2}{2(ap-bq)}\quad \text{(D) } \frac{aq^2-bp^2}{2(bp-aq)}\quad \text{(E) } \sqrt{(a-b)
Note the original pension as $k\sqrt{x}$, where $x$ is the number of years served. Then, based on the problem statement, two equations can be set up.
$\[k\sqrt{x+a} = k\sqrt{x} + p\]$$\[k\sqrt{x+b} = k\sqrt{x} + q\]$
Square the first equation to get
$\[k^2x + ak^2 = k^2x + 2pk\sqrt{x} + p^2.\]$ Because both sides have $\[k^2x\]$, they cancel out. Similarly, the second equation will become $bk^2 = q^2 + 2qk\sqrt{x}.$ Then, $a$ can be multiplied
to the second equation and $b$ can be multiplied to the first equation so that the left side of both equations becomes $abk^2$. Finally, by setting the equations equal to each other, $\[bp^2 + 2bpk\
sqrt{x} = aq^2 + 2aqk\sqrt{x}.\]$ Isolate $k\sqrt{x}$ to get $\fbox{D} = \frac{aq^2-bp^2}{2(bp-aq)}$.
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. | {"url":"https://artofproblemsolving.com/wiki/index.php?title=1970_AHSME_Problems/Problem_35&oldid=154813","timestamp":"2024-11-08T11:55:01Z","content_type":"text/html","content_length":"46532","record_id":"<urn:uuid:aed0f00f-60cd-4e0e-b897-06cf96e5f51f>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00345.warc.gz"} |
(2,1/2) automata.
(2,1/2) automata. As an example, consider the following table of results for a and b (a+b=4) are the respective numbers of neighborhoods evolving into 0 or 1 ( Harold V. McIntosh
As an example, consider the following table of results for a and b (a+b=4) are the respective numbers of neighborhoods evolving into 0 or 1 ( | {"url":"http://delta.cs.cinvestav.mx/~mcintosh/newweb/ra/node44.html","timestamp":"2024-11-12T18:45:23Z","content_type":"text/html","content_length":"1493","record_id":"<urn:uuid:f56b0433-4b60-413b-bee8-6acc1e1062f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00532.warc.gz"} |
Build and Compare the Natural Integers
How to count from zero to infinity and compare natural integers between them
That course is an introduction to the first tool for mathematics: the simpler numbers that are the natural integers.
These are the numbers we use to count, and they are introduced in a progressive way, beginning with counting on our fingers, then counting with the computer, in Python coding sessions, and ending
with the biggest numbers we can imagine.
By the end of that course, you will be able to compare two natural integers, to order a list of them, and even to define and draw a sequence of natural integers.
You will also be ready for the rest of our series about Practical Mathematics, the math that are taught from a practical point of view, based on Python coding sessions, and bringing you from
arithmetics and number theory up to Calculus.
To enter that course, you will need only the knowledge of a child of sixth degree.
There are two main sections, one with the construction of the natural integers set, beginning with the plain counting operation, and the second one that structures the natural integers set as an
increasing sequence from 0 to… infinity!
And, last but not least, you will become the owner of a downloadable recapitulative documents, with the knowledge given by the videos fixed as rigorously proven theorems.
Your Instructor
Fabienne holds the Agregation in Mathematics, the highest French diploma to teach mathematics in secondary schools. After a 26-year career as an R&D engineer in applied mathematics, she founded
Mathedu (Mathematics Re-engineering) with her husband and collaborator Francois, to teach mathematics from a practical point of view, based on Python programming. She is both an expert in mathematics
education and a passionate woman who will help you experience the joy of improving your mastery of mathematics.
Course Curriculum
First Section
Available in days
days after you enroll
Section 2 - Count and Construct the Natural Integers set
Available in days
days after you enroll
Section 3 - Order the Natural Integers Set
Available in days
days after you enroll
Frequently Asked Questions
When does the course start and finish?
The course starts now and never ends! It is a completely self-paced online course - you decide when you start and when you finish.
How long do I have access to the course?
How does lifetime access sound? After enrolling, you have unlimited access to this course for as long as you like - across any and all devices you own.
What if I am unhappy with the course?
We would never want you to be unhappy! If you are unsatisfied with your purchase, contact us in the first 30 days and we will give you a full refund. | {"url":"https://practical-mathematics.academy/p/constructing-and-comparing-natural-integers","timestamp":"2024-11-11T23:00:31Z","content_type":"text/html","content_length":"104364","record_id":"<urn:uuid:90efd3ef-e435-4401-ab38-5349f7cc5474>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00122.warc.gz"} |
Automorphy lifting for residually reducible $l$-adic Galois representations, II | Compositio Mathematica | Cambridge Core
1. Introduction
In this paper, we prove new automorphy lifting theorems for Galois representations of unitary type. Thus, we are considering representations $\rho : G_F \to \operatorname {GL}_n(\bar {{{\mathbb
{Q}}}}_l)$, where $G_F$ is the absolute Galois group of a CM field $F$ and $\rho$ is conjugate self-dual, i.e. there is an isomorphism $\rho ^{c} \cong \rho ^{\vee } \otimes \epsilon ^{1-n},$ where
$c \in \operatorname {Aut}(F)$ is complex conjugation. We say in this paper that such a representation is automorphic if there exists a regular algebraic, conjugate self-dual, cuspidal (RACSDC)
automorphic representation $\pi$ which is matched with $\rho$ under the Langlands correspondence. (See § 1.1 below for a more precise formulation.)
We revisit the context of the paper [Reference ThorneTho15], proving theorems valid in the case that $\bar {\rho }$ is absolutely reducible, but still satisfies a certain non-degeneracy condition (we
say that $\bar {\rho }$ is ‘Schur’). The first theorems of this type were proved in the paper [Reference ThorneTho15], under the assumption that $\bar {\rho }$ has only two irreducible constituents.
Our main motivation here is to remove this restriction. Our results are applied to the problem of symmetric power functoriality in [Reference Newton and ThorneNT19], where they are combined with
level-raising theorems to establish automorphy of symmetric powers for certain level $1$ Hecke eigenforms congruent to a theta series.
We are also able to weaken some other hypotheses in [Reference ThorneTho15], leading to the following result, which is the main theorem of this paper.
Theorem 1.1 (Theorem 6.1) Let $F$ be an imaginary CM number field with maximal totally real subfield $F^{+}$ and let $n \geq 2$ be an integer. Let $l$ be a prime and suppose that $\rho : G_F \
rightarrow \mathrm {GL}_n(\bar {{{\mathbb {Q}}}}_l)$ is a continuous semisimple representation satisfying the following hypotheses.
1. (i) $\rho ^{c} \cong \rho ^{\vee } \epsilon ^{1-n}$.
2. (ii) $\rho$ is ramified at only finitely many places.
3. (iii) $\rho$ is ordinary of weight $\lambda$ for some $\lambda \in ({{\mathbb {Z}}}_+^{n})^{\operatorname {Hom}(F, {{\bar {{{\mathbb {Q}}}}_l}})}$.
4. (iv) There is an isomorphism $\bar {\rho }^{\text {ss}} \cong \bar {\rho }_1 \oplus \cdots \oplus \bar {\rho }_{d}$, where each $\bar {\rho }_i$ is absolutely irreducible and satisfies $\bar {\
rho }_i^{c} \cong \bar {\rho }_i^{\vee } \epsilon ^{1-n}$, and $\bar {\rho }_i \not \cong \bar {\rho }_j$ if $i \neq j$.
5. (v) There exists a finite place ${{\widetilde {v}}}_0$ of $F$, prime to $l$, such that $\rho |_{G_{F_{{{\widetilde {v}}}_0}}}^{\text {ss}} \cong \oplus _{i=1}^{n} \psi \epsilon ^{n-i}$ for some
unramified character $\psi : G_{F_{{{\widetilde {v}}}_0}} \rightarrow \bar {{{\mathbb {Q}}}}_l^{\times }$.
6. (vi) There exist a RACSDC representation $\pi$ of $\mathrm {GL}_n({{\mathbb {A}}}_F)$ and $\iota : {{\bar {{{\mathbb {Q}}}}_l}} \to {{\mathbb {C}}}$ such that:
1. (a) $\pi$ is $\iota$-ordinary;
2. (b) $\overline {r_{ \iota }(\pi )}^{\text {ss}} \cong \bar {\rho }^{\text {ss}}$;
3. (c) $\pi _{{{\widetilde {v}}}_0}$ is an unramified twist of the Steinberg representation.
7. (vii) $F(\zeta _l)$ is not contained in $\bar {F}^{\ker \rm ad (\bar {\rho }^{\text {ss}})}$ and $F$ is not contained in $F^{+}(\zeta _l)$. For each $1 \leq i, j \leq {d}$, $\bar {\rho }_i|_{G_{F
(\zeta _l)}}$ is absolutely irreducible and $\bar {\rho }_i|_{G_{F(\zeta _l)}} \not \cong \bar {\rho }_j|_{G_{F(\zeta _l)}}$ if $i \neq j$. Moreover, $\bar {\rho }^{\text {ss}}$ is primitive
(i.e. not induced from any proper subgroup of $G_F$) and $\bar {\rho }^{\text {ss}}(G_{F})$ has no quotient of order $l$.
8. (viii) $l > 3$ and $l \nmid n$.
Then $\rho$ is automorphic: there exists an $\iota$-ordinary RACSDC automorphic representation $\Pi$ of $\operatorname {GL}_n({{\mathbb {A}}}_F)$ such that $r_\iota (\Pi ) \cong \rho$.
Comparing this with [Reference ThorneTho15, Theorem 7.1], we see that we now allow an arbitrary number of irreducible constituents, while also removing the requirement that the individual
constituents are adequate (in the sense of [Reference ThorneTho12]) and potentially automorphic. This assumption of potential automorphy was used in [Reference ThorneTho15], together with the
Khare–Wintenberger method, to get a handle on the quotient of the universal deformation ring of $\bar {\rho }$ corresponding to reducible deformations. This made generalizing [Reference ThorneTho15,
Theorem 7.1] to the case where more than two irreducible constituents are allowed seem a formidable task: one would want to know that any given direct sum of irreducible constituents of $\bar {\rho }
$ was potentially automorphic, and then perhaps use induction on the number of constituents to control the reducible locus.
The first main innovation in this paper that allows us to bypass this is the observation that by fully exploiting the ‘connectedness dimension’ argument to prove that $R = {{\mathbb {T}}}$ (which
goes back to [Reference Skinner and WilesSW99] and appears in this paper in the proof of Theorem 5.1), one only needs to control the size of the reducible locus in quotients of the universal
deformation ring that are known a priori to be finite over the Iwasawa algebra $\Lambda$. This can be done easily by hand using the ‘locally Steinberg’ condition (as in § 3.3).
The second main innovation is a finer study of the universal deformation ring $R^{\text {univ}}$ of a (reducible but) Schur residual representation. We show that if the residual representation has $
{d}$ absolutely irreducible constituents, then there is an action of a group $\mu _2^{d}$ on $R^{\text {univ}}$ and identify the invariant subring $(R^{\text {univ}})^{\mu _2^{d}}$ with the subring
topologically generated by the traces of Frobenius elements (which can also be characterized as the image $P$ of the canonical map to $R^{\text {univ}}$ from the universal pseudodeformation ring).
This leads to a neat proof that the map $P \to R^{\text {univ}}$ is étale at prime ideals corresponding to irreducible deformations of $\bar {\rho }$.
We now describe the organization of this paper. Since it is naturally a continuation of [Reference ThorneTho15], we maintain the same notation and use several results and constructions from that
paper as black boxes. We begin in §§ 2 and 3 by extending several results from [Reference ThorneTho15] about the relation between deformations and pseudodeformations to the case where $\bar {\rho }$
is permitted to have more than two irreducible constituents. We also make the above-mentioned study of the dimension of the locus of reducible deformations.
In § 4 we recall from [Reference ThorneTho15] the definition of the unitary group of automorphic forms and Hecke algebras that we use, and state the ${{\mathbb {T}}}_{{\mathfrak q}} = R_{{\mathfrak
p}}$ type result proved in that paper (here ${{\mathfrak p}}$ denotes a dimension 1, characteristic $l$ prime of $R$ with good properties, in particular that the associated representation to $\
operatorname {GL}_n(\operatorname {Frac} R / {{\mathfrak p}})$ is absolutely irreducible). In § 5 we carry out the main argument, based on the notion of connectedness dimension, which is described
above. Finally, in § 6 we deduce Theorem 1.1, following a simplified version of the argument in [Reference ThorneTho15, § 7] that no longer makes reference to potential automorphy.
1.1 Notation
We use the same notation and normalizations for Galois groups, class field theory, and local Langlands correspondences as in [Reference ThorneTho15, Notation]. Rather than repeat this verbatim here
we invite the reader to refer to that paper for more details. We do note the convention that if $R$ is a ring and $P$ is a prime ideal of $R$, then $R_{(P)}$ denotes the localization of $R$ at $P$
and $R_P$ denotes the completion of the localization.
We recall that ${{\mathbb {Z}}}_n^{+} \subset {{\mathbb {Z}}}^{n}$ denotes the set of tuples $\lambda = (\lambda _1, \ldots , \lambda _n)$ of integers such that $\lambda _1 \geq \cdots \geq \lambda
_n$. It is identified in a standard way with the set of highest weights of $\operatorname {GL}_n$. If $F$ is a number field and $\lambda = (\lambda _\tau ) \in ({{\mathbb {Z}}}_n^{+})^{\operatorname
{Hom}(F, {{\mathbb {C}}})}$, then we write $\Xi _\lambda$ for the algebraic representation of $\operatorname {GL}_n(F \otimes _{{\mathbb {Q}}} {{\mathbb {C}}}) = \prod _{\tau \in \operatorname {Hom}
(F, {{\mathbb {C}}})} \operatorname {GL}_n({{\mathbb {C}}})$ of highest weight $\lambda$. If $\pi$ is an automorphic representation of $\operatorname {GL}_n({{\mathbb {A}}}_F)$, we say that $\pi$ is
regular algebraic of weight $\lambda$ if $\pi _\infty$ has the same infinitesimal character as the dual $\Xi _\lambda ^{\vee }$.
Let $F$ be a CM field (i.e. a totally imaginary quadratic extension of a totally real field $F^{+}$). We always write $c \in \operatorname {Aut}(F)$ for complex conjugation. We say that an
automorphic representation $\pi$ of $\operatorname {GL}_n({{\mathbb {A}}}_F)$ is conjugate self-dual if there is an isomorphism $\pi ^{c} \cong \pi ^{\vee }$. If $\pi$ is a RACSDC automorphic
representation of $\operatorname {GL}_n({{\mathbb {A}}}_F)$ and $\iota : \bar {{{\mathbb {Q}}}}_l \to {{\mathbb {C}}}$ is an isomorphism (for some prime $l$), then there exists an associated Galois
representation $r_\iota (\pi ) : G_F \to \operatorname {GL}_n(\bar {{{\mathbb {Q}}}}_l)$, characterized up to isomorphism by the requirement of compatibility with the local Langlands correspondence
at each finite place of $F$; see [Reference ThorneTho15, Theorem 2.2] for a reference. We say that a representation $\rho : G_F \to \operatorname {GL}_n(\bar {{{\mathbb {Q}}}}_l)$ is automorphic if
there exists a choice of $\iota$ and RACSDC $\pi$ such that $\rho \cong r_\iota (\pi )$.
One can define what it means for a RACSDC automorphic representation $\pi$ to be $\iota$-ordinary (see [Reference ThorneTho15, Lemma 2.3]; it means that the eigenvalues of certain Hecke operators, a
priori $l$-adic integers, are in fact $l$-adic units). If $\mu \in ({{\mathbb {Z}}}^{n}_+)^{\operatorname {Hom}(F, \bar {{{\mathbb {Q}}}}_l)}$, we say (following [Reference ThorneTho15, Definition
2.5]) that a representation $\rho : G_F \to \operatorname {GL}_n(\bar {{{\mathbb {Q}}}}_l)$ is ordinary of weight $\mu$ if for each place $v | l$ of $F$, there is an isomorphism
\[ \rho|_{G_{F_v}} \sim \begin{pmatrix} \psi_1 & \ast & \ast & \ast \\ 0 & \psi_2 & \ast & \ast \\ \vdots & \ddots & \ddots & \ast \\ 0 & \ldots & 0 & \psi_n \end{pmatrix}\!, \]
where $\psi _i : G_{F_v} \rightarrow {{\bar {{{\mathbb {Q}}}}_l}}^{\times }$ is a continuous character satisfying the identity
\[ \psi_i(\sigma) = \prod_{\tau : F_v \hookrightarrow {{\bar{{{\mathbb{Q}}}}_l}}} \tau(\operatorname{Art}_{F_v}^{-1}(\sigma))^{-( \mu_{\tau, n - i + 1} + i - 1)} \]
for all $\sigma$ in a suitable open subgroup of $I_{F_v}$. An important result [Reference ThorneTho15, Theorem 2.4] is that if $\pi$ is RACSDC of weight $\lambda$ and $\iota$-ordinary, then $r_\iota
(\pi )$ is ordinary of weight $\iota \lambda$, where by definition $(\iota \lambda )_\tau = \lambda _{\iota \tau }$.
2. Determinants
We first give the definition of a determinant from [Reference ChenevierChe14]. We recall that if $A$ is a ring and $M, N$ are $A$-modules, then an $A$-polynomial law $F : M \to N$ is a natural
transformation $F : h_M \to h_N$, where $h_M : A$-alg $\to$ Sets is the functor $h_M(B) = M \otimes _A B$. The $A$-polynomial law $F$ is called homogeneous of degree $n \geq 1$ if for all $b \in B$,
$x \in M \otimes _A B$, we have $F_B(bx) = b^{n} F_B(x)$.
Definition 2.1 Let $A$ be a ring and let $R$ be an $A$-algebra. An $A$-valued determinant of $R$ of dimension $n \geq 1$ is a multiplicative $A$-polynomial law $D : R \to A$ which is homogeneous of
degree $n$.
If $D$ is a determinant, then there are associated polynomial laws $\Lambda _i : R \to A$, $i = 0, \ldots , n$, given by the formulae
\[ D(t - r) = \sum_{i=0}^{n} (-1)^{i} \Lambda_i(r) t^{n-i} \]
for all $r \in R \otimes _A B$. We define the characteristic polynomial $A$-polynomial law $\chi : R \to R$ by the formula $\chi (r) = \sum _{i=0}^{n} (-1)^{i} \Lambda _i(r) r^{n-i}$ ($r \in R \
otimes _A B$). We write $\operatorname {CH}(D)$ for the two-sided ideal of $R$ generated by the coefficients of $\chi (r_1 t_1 + \cdots + r_m t_m) \in R[t_1, \ldots , t_m]$ for all $m \ge 1$ and
$r_1,\ldots ,r_m \in R$. We have $\operatorname {CH}(D) \subseteq \ker (D)$ [Reference ChenevierChe14, Lemma 1.21]. The determinant $D$ is said to be Cayley–Hamilton if $\operatorname {CH}(D) = 0$,
equivalently if $\chi = 0$ (i.e. $\chi$ is the zero $A$-polynomial law).
We next recall the definition of a generalized matrix algebra [Reference Bellaïche and ChenevierBC09, Definition 1.3.1].
Definition 2.2 Let $A$ be a ring and let $R$ be an $A$-algebra. We say that $R$ is a generalized matrix algebra of type $(n_1,\ldots ,n_{d})$ if it is equipped with the following data:
1. (i) a family of orthogonal idempotents $e_1,\ldots , e_{d}$ with $e_1+\cdots +e_{d} = 1$; and
2. (ii) for each $1\le i \le {d}$, an $A$-algebra isomorphism $\psi _i \colon e_i R e_i \rightarrow M_{n_i}(A)$
such that the trace map $T \colon R \rightarrow A$ defined by $T(x) = \sum _{i=1}^{d} \operatorname {tr}\psi _i(e_ix e_i)$ satisfies $T(xy) = T(yx)$ for all $x,y \in R$. We refer to the data $\
mathcal {E} = \{e_i,\psi _i, 1\le i \le {d}\}$ as the data of idempotents of $R$.
Construction 2.3 We recall the structure of generalized matrix algebras from [Reference Bellaïche and ChenevierBC09, § 1.3.2]. Let $R$ be a generalized matrix algebra of type $(n_1,\ldots ,n_{d})$
with data of idempotents $\mathcal {E} = \{e_i,\psi _i, 1\le i \le {d}\}$. For each $1\le i \le {d}$, let $E_i \in e_i R e_i$ be the unique element such that $\psi _i(E_i)$ is the element of $M_{n_i}
(A)$ whose row $1$, column $1$ entry is $1$ and all other entries are $0$. We set $\mathcal {A}_{i,j} = E_i R E_j$ for each $1\le i,j\le {d}$. Note that $\mathcal {A}_{i,j}\mathcal {A}_{j,k} \
subseteq \mathcal {A}_{i,k}$ for each $1\le i,j,k\le {d}$, and the trace map $T$ induces an isomorphism $\mathcal {A}_{i,i} \cong A$ for each $1\le i \le {d}$. Via this isomorphism, we will tacitly
view $\mathcal {A}_{i,j}\mathcal {A}_{j,i}$ as an ideal in $A$ for each $1\le i,j\le {d}$. With this multiplication, there is an isomorphism of $A$-algebras
(1)$$R \cong \begin{pmatrix} M_{n_1}(A) & M_{n_1,n_2}(\mathcal{A}_{1,2}) & \cdots & M_{n_1,n_{{d}}}(\mathcal{A}_{1,{{d}}})\\ M_{n_2,n_1}(\mathcal{A}_{2,1}) & M_{n_2}(A) & \cdots & M_{n_2,n_{{d}}}(\
mathcal{A}_{2,{{d}}})\\ \vdots & \vdots & \ddots & \vdots \\ M_{n_{{d}},n_1}(\mathcal{A}_{{{d}},1}) & M_{n_{{d}},n_2}(\mathcal{A}_{{{d}},2}) & \cdots & M_{n_{{d}}}(A) \end{pmatrix}\!.$$
The following result of Chenevier allows us to use the above structure when studying determinants.
Theorem 2.4 Let $A$ be a Henselian local ring with residue field $k$, let $R$ be an $A$-algebra, and let $D \colon R \rightarrow A$ be a Cayley–Hamilton determinant. Suppose that there exist
surjective and pairwise non-conjugate $k$-algebra homomorphisms $\bar {\rho }_i : R \to M_{n_i}(k)$ such that $\bar {D} = \prod _{i=1}^{d} (\det \circ \bar {\rho }_i)$, where $\bar {D} = D \otimes _R
Then there is a datum of idempotents $\mathcal {E} = \{e_i,\psi _i, 1\le i \le {d}\}$ for which $R$ is a generalized matrix algebra and such that $\psi _i \otimes _A k = \bar {\rho }_i|_{e_i R e_i}$.
Any two such data are conjugate by an element of $R^{\times }$.
We note that the assumptions of Theorem 2.4 say that $D$ is residually split and multiplicity-free, in the sense of [Reference ChenevierChe14, Definition 2.19].
Proof. The existence of such a datum of idempotents $\mathcal {E} = \{e_i,\psi _i,1\le i \le {d}\}$ is contained in [Reference ChenevierChe14, Theorem 2.22] and its proof. The statement that two such
data are conjugate is exactly as in [Reference Bellaïche and ChenevierBC09, Lemma 1.4.3]. Namely, if $\mathcal {E}' = \{e_i',\psi _i',1\le i \le {d}\}$ is another such choice, then since $\
operatorname {End}_R(R e_i) \cong M_{n_i}(A) \cong \operatorname {End}_R(R e_i')$ are local rings, the Krull–Schmidt–Azumaya theorem [Reference Curtis and ReinerCR81, Theorem 6.12] (see also [
Reference Curtis and ReinerCR81, Remark 6.14 and Chapter 6, Exercise 14]) implies that there is $x\in R^{\times }$ such that $x e_i x^{-1} = e_i'$ for each $1\le i \le {d}$. By Skolem–Noether, we can
adjust $x$ by an element of $(\oplus _{i=1}^{d}e_i R e_i)^{\times }$ so that it further satisfies $x\psi _i x^{-1} = \psi _i'$.
We now show that the reducibility ideals of [Reference Bellaïche and ChenevierBC09, Proposition 1.5.1] and their basic properties carry over for determinants (so without having to assume that $n!$ is
invertible in $A$).
Proposition 2.5 Let $A$ be a Henselian local ring with residue field $k$, let $R$ be an $A$-algebra, and let $D \colon R \rightarrow A$ be a determinant. Assume that $\bar {D} = D\otimes _A k \colon
R\otimes _A k \rightarrow k$ is split and multiplicity free. Write $\bar {D} = \prod _{i=1}^{d} \bar {D}_i$ with each $\bar {D}_i$ absolutely irreducible of dimension $n_i$.
Let $\mathcal {P} = (\mathcal {P}_1,\ldots ,\mathcal {P}_s)$ be a partition of $\{1,\ldots ,{d}\}$. There is an ideal $I_\mathcal {P}$ of $A$ such that an ideal $J$ of $A$ satisfies $I_\mathcal {P} \
subseteq J$ if and only if there are determinants $D_1,\ldots ,D_s \colon R\otimes _A A/J \rightarrow A/J$ such that $D\otimes _A A/J = \prod _{m=1}^{s} D_m$ and $D_m \otimes _A k = \prod _{i \in \
mathcal {P}_m} \bar {D}_i$ for each $1\le m \le s$. If this property holds, then $D_1,\ldots ,D_s$ are uniquely determined and satisfy $\ker (D\otimes _A A/J) \subseteq \ker (D_m)$.
Moreover, let $\mathcal {J}$ be a two-sided ideal of $R$ with $\operatorname {CH}(D) \subseteq \mathcal {J} \subseteq \ker (D)$ and let $\mathcal {A}_{i,j}$ be the $A$-modules as in Construction 2.3
for a choice of data of idempotents as in Theorem 2.4 applied to $R/\mathcal {J}$. Then $I_\mathcal {P} = \sum _{i,j} \mathcal {A}_{i,j}\mathcal {A}_{j,i}$, where the sum is over all pairs $i,j$ not
belonging to the same $\mathcal {P}_m\in \mathcal {P}$.
Proof. We follow the proof of [Reference Bellaïche and ChenevierBC09, Proposition 1.5.1] closely. Choose a two-sided ideal $\mathcal {J}$ of $R$ with $\operatorname {CH}(D) \subseteq \mathcal {J} \
subseteq \ker (D)$, and data of idempotents $\mathcal {E}$ for $R/\mathcal {J}$ as in Theorem 2.4. We let $\mathcal {A}_{i,j}$ be as in Construction 2.3 and define $I_\mathcal {P} = \sum _{i,j} \
mathcal {A}_{i,j}\mathcal {A}_{j,i}$, where the sum is over all pairs $i,j$ not belonging to the same $\mathcal {P}_m\in \mathcal {P}$. Since another such choice of the data of idempotents is
conjugate by an element of $(R/\mathcal {J})^{\times }$, the ideal $I_{\mathcal {P}}$ does not depend on the choice of $\mathcal {E}$. To see that it is independent of $\mathcal {J}$, first note that
$D$ further factors through a surjection $\psi \colon R/\mathcal {J} \rightarrow R/\ker (D)$. Under this surjection, the data of idempotents $\mathcal {E}$ is sent to a data of idempotents for $R/\
ker (D)$, and $\operatorname {tr}(\psi (\mathcal {A}_{i,j})\psi (\mathcal {A}_{j,i})) = \operatorname {tr}(\mathcal {A}_{i,j}\mathcal {A}_{j,i})$ since $\operatorname {tr}\circ \psi = \operatorname
We can now replace $R$ with $R/\operatorname {CH}(D)$ and assume that $D$ is Cayley–Hamilton. Since $\operatorname {CH}(D)$ is stable under base change, it suffices to show that $I_{\mathcal {P}} =
0$ if and only if there are determinants $D_1,\ldots ,D_s \colon R\rightarrow A$ such that $D = \prod _{m=1}^{s} D_m$ and $D_m \otimes _A k_A = \prod _{i \in \mathcal {P}_m} \bar {D}_i$ for each $1\
le m \le s$ and that, if this happens, then $D_1,\ldots ,D_s$ are uniquely determined. Fix a datum of idempotents $\mathcal {E} = \{e_i,\psi _i,1\le i \le {d}\}$ for $R$ as in Theorem 2.4, and let
the notation be as in Construction 2.3. For each $1\le m \le s$, we set $f_m = \sum _{i\in \mathcal {P}_m} e_i$. Then $1 = f_1+\cdots + f_s$ is a decomposition into orthogonal idempotents.
First assume that $I_{\mathcal {P}} = 0$. Let $\tilde {D}$ denote the $A$-valued determinant on $R/\ker (D)$ arising from $D$. Fix $x \in R$, an $A$-algebra $B$, and $y \in R\otimes _A B$. If $1\le
i,j\le {d}$ do not belong to the same $\mathcal {P}_m\in \mathcal {P}$, then using the algebra structure as in (1) and the fact that $\mathcal {A}_{i,j} \mathcal {A}_{j,i} = 0$, we have $e_i x e_j y
= \sum _{l\ne i} e_i x e_j y e_l$, and [Reference ChenevierChe14, Lemma 1.12(i)] gives
\[ D(1 + e_i x e_ j y) = D\bigg(1+\sum_{l\ne i} e_i x e_j y e_l\bigg) = D\bigg(1+\sum_{l\ne i} x e_j y e_l e_i\bigg) = D(1) = 1. \]
By [Reference ChenevierChe14, Lemma 1.19], $e_i x e_j \in \ker (D)$ for all $x\in R$ and all $i,j$ that do not belong to the same $\mathcal {P}_m\in \mathcal {P}$. We then have an isomorphism of $A$
-algebras $R/\ker (D) \cong \prod _{m=1}^{s} f_m (R/\ker (D)) f_m$ and [Reference ChenevierChe14, Lemma 2.4] gives $D = \prod _{m=1}^{s} D_m$, where $D_m \colon R \rightarrow A$ is the composite of
the surjection $R \rightarrow f_m (R/\ker (D)) f_m$ with the determinant $\tilde {D}_m \colon f_m (R/\ker (D)) f_m \rightarrow A$ given by $x\mapsto \tilde {D}(x + 1-f_m)$. It is immediate that $D_m
\otimes _A k_A = \prod _{i \in \mathcal {P}_m} \bar {D}_i$ for each $1\le m \le s$.
Now assume that there are determinants $D_1,\ldots ,D_s \colon R\rightarrow A$ such that $D = \prod _{m=1}^{s} D_m$ and $D_m \otimes _A k = \prod _{i \in \mathcal {P}_m} \bar {D}_i$ for each $1\le m
\le s$. The determinants $D_m$ have dimension $d_m := \sum _{i\in \mathcal {P}_m} n_i$. The trace map yields an equality
\[ \sum_{1\le m\ne m' \le s} \operatorname{tr}(f_m R f_{m'}R f_m) = I_{\mathcal{P}}. \]
So, to show that $I_{\mathcal {P}} = 0$, it suffices to show that $\operatorname {tr}(f_m R f_{m'} R f_m) = 0$ for $m\ne m'$. For this, it suffices to show that $f_{m'} \in \ker (D_m)$ for any $m\ne
m'$, since this implies that $f_m R f_{m'} \in \ker (D_l)$ for any $1\le l\le s$ and hence
\[ D(1+tf_m R f_{m'} R f_m) = \prod_{l=1}^{s} D_l(1+tf_m R f_{m'} R f_m) = 1. \]
For any idempotent $f$ of $R$, we have the determinant $D_{m,f} \colon f R f \rightarrow A$ given by $D_{m,f}(x) = D_m(x+1-f)$. When $f = f_m$,
\[ D_{m,f_m} \otimes_A k = \prod_{i\in \mathcal{P}_m} \bar{D}_{i,f_m} = \prod_{i\in \mathcal{P}_m}\bar{D}_{i,e_i} \]
has dimension $d_m$. Then [Reference ChenevierChe14, Lemma 2.4(2)] implies that $D_{m,1-f_m}$ has dimension $0$, i.e. is constant and equal to $1$. So, for any $m\ne m'$, the characteristic
polynomial of $f_{m'}$ with respect to $D_m$ is
\[ D_m(t-f_{m'}) = D_{m,f_m}(t)D_{m,1-f_m}(t-f_{m'}) = t^{d_m}. \]
Then $f_{m'} = f_{m'}^{d_m} \in \operatorname {CH}(D_m) \subseteq \ker (D_m)$, which is what we wanted to prove. This further shows that for each $1\le m \le s$, the determinant $D_m$ is the
composite of the surjections
\[ R \rightarrow \oplus_{l=1}^{s} f_l R f_l \rightarrow f_m R f_m \]
with the determinant $D_{f_m} \colon f_m R f_m \rightarrow A$. Since any two choices of the data of idempotents are conjugate under $R^{\times }$, each $D_m$ is uniquely determined by $D$.
3. Deformations
Galois deformation theory plays an essential role in this paper. The set of results we use is essentially identical to that of [Reference ThorneTho15], with some technical improvements. In this
section we recall the notation used in [Reference ThorneTho15], without giving detailed definitions or proofs; we then proceed to prove the new results that we need. Some of the definitions recalled
here were first given in [Reference Clozel, Harris and TaylorCHT08] or [Reference GeraghtyGer19], but in order to avoid sending the reader to too many different places we restrict our citations to [
Reference ThorneTho15].
We will use exactly the same set-up and notation for deformation theory as in [Reference ThorneTho15]. We recall that this means that we fix at the outset the following objects.
1. – A CM number field $F$, with its totally real subfield $F^{+}$.
2. – An odd prime $l$ such that each $l$-adic place of $F^{+}$ splits in $F$. We write $S_l$ for the set of $l$-adic places of $F^{+}$.
3. – A finite set $S$ of finite places of $F^{+}$ which split in $F$. We assume that $S_l \subset S$ and write $F(S)$ for the maximal extension of $F$ which is unramified outside $S$ and set $G_{F,
S} = \operatorname {Gal}(F(S) / F)$ and $G_{F^{+}, S} = \operatorname {Gal}(F(S) / F^{+})$. We fix a choice of complex conjugation $c \in G_{F^{+}, S}$.
4. – For each $v \in S$, we fix a choice of place ${{\widetilde {v}}}$ of $F$ such that ${{\widetilde {v}}}|_{F^{+}} = v$, and define $\tilde {S} = \{ {{\widetilde {v}}} \mid v \in S \}$.
We also fix the following data.
1. – A coefficient field $K \subset \bar {{{\mathbb {Q}}}}_l$ with ring of integers ${{\mathcal {O}}}$, residue field $k$, and maximal ideal $\lambda \subset {{\mathcal {O}}}$.
2. – A continuous homomorphism $\chi : G_{F^{+}, S} \to {{\mathcal {O}}}^{\times }$. We write $\bar {\chi } = \chi \text { mod } \lambda$.
3. – A continuous homomorphism $\bar {r} : G_{F^{+}, S} \to {{\mathcal {G}}}_n(k)$ such that $\bar {r}^{-1}({{\mathcal {G}}}_n^{\circ }(k)) = G_{F, S}$. Here ${{\mathcal {G}}}_n$ is the algebraic
group over ${{\mathbb {Z}}}$ defined in [Reference Clozel, Harris and TaylorCHT08, § 2.1]. We follow the convention that if $R : \Gamma \to {{\mathcal {G}}}_n(A)$ is a homomorphism and $\Delta \
subset \Gamma$ is a subgroup such that $R(\Delta ) \subset {{\mathcal {G}}}_n^{0}(A)$, then $R|_\Delta$ denotes the composite homomorphism
\[ \Delta \to {{\mathcal{G}}}_n^{0}(A) = \operatorname{GL}_n(A) \times \operatorname{GL}_1(A) \to \operatorname{GL}_n(A). \]
Thus, $\bar {r}|_{G_{F, S}}$ takes values in $\operatorname {GL}_n(k)$.
If $v \in S_l$, then we write $\Lambda _v = {{\mathcal {O}}} [\kern-1pt[ (I^{\text ab}_{F_{{\widetilde {v}}}}(l))^{n} ]\kern-1pt]$, where $I^{{\text ab}}_{F_{{\widetilde {v}}}}(l)$ denotes the
inertia group in the maximal abelian pro-$l$ extension of $F_{{\widetilde {v}}}$. We set $\Lambda = \hat {\otimes }_v \Lambda _v$, the completed tensor product being over ${{\mathcal {O}}}$. A global
deformation problem, as defined in [Reference ThorneTho15, § 3], then consists of a tuple
\[ {{\mathcal{S}}} = ( F / F^{+}, S, \tilde{S}, \Lambda, \bar{r}, \chi, \{ {{\mathcal{D}}}_v \}_{v \in S} ). \]
The extra data that we have not defined consists of the choice of a local deformation problem ${{\mathcal {D}}}_v$ for each $v \in S$. We will not need to define any new local deformation problems in
this paper, but we recall that the following have been defined in [Reference ThorneTho15]:
1. – ‘ordinary deformations’ give rise to a problem ${{\mathcal {D}}}_v^{\triangle }$ for each $v \in S_l$ [Reference ThorneTho15, § 3.3.2];
2. – ‘Steinberg deformations’ give rise to a problem ${{\mathcal {D}}}_v^{\rm St}$ for each place $v \in S$ such that $q_v \equiv 1 \text { mod }l$ and $\bar {r}|_{G_{F_{{\widetilde {v}}}}}$ is
3. – ‘$\chi _v$-ramified deformations’ give rise to a problem ${{\mathcal {D}}}_v^{\chi _v}$ for each place $v \in S$ such that $q_v \equiv 1 \text { mod }l$ and $\bar {r}|_{G_{F_{{\widetilde
{v}}}}}$ is trivial, given the additional data of a tuple $\chi _v = (\chi _{v, 1}, \ldots , \chi _{v, n})$ of characters $\chi _{v, i} : k(v)^{\times }(l) \to k^{\times }$;
4. – ‘unrestricted deformations’ give rise to a problem ${{\mathcal {D}}}_v^{\square }$ for any $v \in S$.
If ${{\mathcal {S}}}$ is a global deformation problem, then we can define (as in [Reference ThorneTho15]) a functor $\operatorname {Def}_{{\mathcal {S}}} : {{\mathcal {C}}}_\Lambda \to \text {Sets}$
of ‘deformations of type ${{\mathcal {S}}}$’. By definition, if $A \in {{\mathcal {C}}}_\Lambda$, then $\operatorname {Def}_{{\mathcal {S}}}(A)$ is the set of $\operatorname {GL}_n(A)$-conjugacy
classes of homomorphisms $r : G_{F^{+}, S} \to {{\mathcal {G}}}_n(A)$ lifting $\bar {r}$ such that $\nu \circ r = \chi$ and for each $v \in S$, $r|_{G_{F_{{\widetilde {v}}}}} \in {{\mathcal {D}}}_v
(A)$. If $\bar {r}$ is Schur (see [Reference ThorneTho15, Definition 3.2]), then the functor $\operatorname {Def}_{{\mathcal {S}}}$ is represented by an object $R_{{\mathcal {S}}}^{\rm univ} \in {{\
mathcal {C}}}_\Lambda$.
We point out an error in [Reference ThorneTho15]. We thank Lue Pan for bringing this to our attention. In [Reference ThorneTho15, Proposition 3.15] it is asserted that the ring $R_v^{1}$
(representing the deformation problem ${{\mathcal {D}}}_v^{1}$ for $v \in R$, defined under the assumptions $q_v \equiv 1 \text { mod }l$ and $\bar {r}|_{G_{F_{{\widetilde {v}}}}}$ trivial) has the
property that $R_v^{1} / (\lambda )$ is generically reduced. This is false, even in the case $n = 2$, as can be seen from the statement of [Reference ShottonSho16, Proposition 5.8] (and noting the
identification $R_v^{1} / (\lambda ) = R_v^{\chi _v} / (\lambda )$). We offer the following corrected statement.
Proposition 3.1 Let $\bar {R}_v^{1}$ denote the nilreduction of $R_v^{1}$. Then $\bar {R}_v^{1} / (\lambda )$ is generically reduced.
Proof. Let ${{\mathcal {M}}}$ denote the scheme over ${{\mathcal {O}}}$ of pairs of $n \times n$ matrices $(\Phi , \Sigma )$, where $\Phi$ is invertible, the characteristic polynomial of $\Sigma$
equals $(X - 1)^{n}$, and we have $\Phi \Sigma \Phi ^{-1} = \Sigma ^{q_v}$. Then $R_v^{1}$ can be identified with the completed local ring of ${{\mathcal {M}}}$ at the point $(1_n, 1_n) \in {{\
mathcal {M}}}(k)$. By [Reference MatsumuraMat89, Theorem 23.9] (and since ${{\mathcal {M}}}$ is excellent), it is enough to show that if $\bar {{{\mathcal {M}}}}$ denotes the nilreduction of ${{\
mathcal {M}}}$, then $\bar {{{\mathcal {M}}}} \otimes _{{\mathcal {O}}}k$ is generically reduced.
Let ${{\mathcal {M}}}_1, \ldots , {{\mathcal {M}}}_r$ denote the irreducible components of ${{\mathcal {M}}}$ with their reduced subscheme structure. According to [Reference ThorneTho12, Lemma 3.15],
each ${{\mathcal {M}}}_i \otimes _{{\mathcal {O}}}K$ is non-empty of dimension $n^{2}$, while the ${{\mathcal {M}}}_i \otimes _{{\mathcal {O}}}k$ are the pairwise-distinct irreducible components of $
{{\mathcal {M}}} \otimes _{{\mathcal {O}}}k$ and are all generically reduced. Let $\bar {\eta }_i$ denote the generic point of ${{\mathcal {M}}}_i \otimes _{{\mathcal {O}}}k$. Then $\bar {\eta }_i$
admits an open neighbourhood in ${{\mathcal {M}}}$ not meeting any ${{\mathcal {M}}}_j$ ($j \neq i$). Consequently, we have an equality of local rings ${{\mathcal {O}}}_{\bar {{{\mathcal {M}}}}, \bar
{\eta }_i} = {{\mathcal {O}}}_{{{\mathcal {M}}}_i, \bar {\eta }_i}$, showing that ${{\mathcal {O}}}_{\bar {{{\mathcal {M}}}}, \bar {\eta }_i} / (\lambda )$ is reduced (in fact, a field). This shows
that $\bar {{{\mathcal {M}}}} \otimes _{{\mathcal {O}}}k$ is generically reduced.
We now need to explain why this error does not affect the proofs of the two results in [Reference ThorneTho15] which rely on the assertion that $R_v^{1} / (\lambda )$ is generically reduced. The
first of these is [Reference ThorneTho15, Proposition 3.17], which states that the Steinberg deformation ring $R_v^{\rm St}$ has the property that $R_v^{\rm St} / (\lambda )$ is generically reduced.
The proof of this result is still valid if one replaces $R_v^{1}$ there with $\bar {R}_v^{1}$. Indeed, we need only note that $R_v^{\rm St}$ is ${{\mathcal {O}}}$-flat (by definition) and reduced
(since $R_v^{\rm St}[1/l]$ is regular, by [Reference TaylorTay08, Lemma 3.3]). The map $R_v^{1} \to R_v^{\rm St}$ therefore factors through a surjection $\bar {R}_v^{1} \to R_v^{\rm St}$.
The next result is [Reference ThorneTho15, Lemma 3.40(2)], which describes the irreducible components of the localization and completion of a ring $R^{\infty }$ at a prime ideal $P_\infty$. The ring
$R^{\infty }$ has $R_v^{1}$ as a (completed) tensor factor, and the generic reducedness is used to justify an appeal to [Reference ThorneTho15, Proposition 1.6]. Since passing to nilreduction does
not change the underlying topological space, one can argue instead with the quotient of $R^{\infty }$, where $R_v^{1}$ is replaced by $\bar {R}_v^{1}$. The statement of [Reference ThorneTho15, Lemma
3.40] is therefore still valid.
3.2 Pseudodeformations
In this section, we fix a global deformation problem
\[ {{\mathcal{S}}} = ( F / F^{+}, S, \tilde{S}, \Lambda, \bar{r}, \chi, \{ {{\mathcal{D}}}_v \}_{v \in S} ) \]
such that $\bar {r}$ is Schur. We write $P_{{\mathcal {S}}} \subset R_{{\mathcal {S}}}^{\text {univ}}$ for the $\Lambda$-subalgebra topologically generated by the coefficients of characteristic
polynomials of Frobenius elements $\operatorname {Frob}_w \in G_{F, S}$ ($w$ prime to $S$). The subring $P_{{\mathcal {S}}}$ is studied in [Reference ThorneTho15, § 3.4], where it is shown using
results of Chenevier that $P_{{\mathcal {S}}}$ is a complete Noetherian local $\Lambda$-algebra and that the inclusion $P_{{\mathcal {S}}} \subset R_{{{\mathcal {S}}}}^{\rm univ}$ is a finite ring
map (see [Reference ThorneTho15, Proposition 3.29]).
In fact, more is true, as we now describe. Let $\bar {B} \in \operatorname {GL}_n(k)$ be the matrix defined by the formula $\bar {r}(c) = (\bar {B}, - \chi (c))\jmath \in {{\mathcal {G}}}_n(k)$. Let
$\bar {\rho } = \bar {r}|_{G_{F, S}}$ and suppose that there is a decomposition $\bar {r}=\oplus _{i=1}^{{d}} \bar {r}_i$ with $\bar {\rho }_i=\bar {r}_i|_{G_{F,S}}$ absolutely irreducible for each
$i$. The representations $\bar {\rho }_i$ are pairwise non-isomorphic, because $\bar {r}$ is Schur (see [Reference ThorneTho15, Lemma 3.3]). We recall [Reference ThorneTho15, Lemma 3.1] that to give
a lifting $r:G_{F^{+},S}\rightarrow {{\mathcal {G}}}_n(R)$ of $\bar {r}$ with $\nu \circ r = \chi$ is equivalent to giving the following data.
1. – A representation $\rho : G_{F,S}\rightarrow \operatorname {GL}_n(R)$ lifting $\bar {\rho } = \bar {r}|_{G_{F,S}}$.
2. – A matrix $B \in \operatorname {GL}_n(R)$ lifting $\bar {B}$ with $^{t}{}{B} = -\chi (c)B$ and $\chi (\delta )B = \rho (\delta ^{c})B ^{t}{}{\rho }(\delta )$ for all $\delta \in G_{F,S}$.
The equivalence is given by letting $\rho = r|_{G_{F,S}}$ and $r(c) = (B,-\chi (c))\jmath$. Conjugating $r$ by $M \in \operatorname {GL}_n(R)$ takes $B$ to $MB ^{t}{}{M}$. Note that the matrix $B$
defines an isomorphism $\chi \otimes \rho ^{\vee } \overset {\sim }{\rightarrow } \rho ^{c}$.
We embed the group $\mu _2^{{d}}$ in $\operatorname {GL}_n({{\mathcal {O}}})$ as block diagonal matrices, the $i$th block being of size $\dim _k \bar {\rho }_i$. We assume that the global deformation
problem ${{\mathcal {S}}}$ has the property that each local deformation problem ${{\mathcal {D}}}_v \subset {{\mathcal {D}}}_v^{\square }$ is invariant under conjugation by $\mu _2^{{d}}$; this is
the case for each of the local deformation problems recalled above. With this assumption, the group $\mu _2^{{d}}$ acts on the ring $R_{{{\mathcal {S}}}}^{\rm univ}$ by conjugation of the universal
deformation and we have the following result.
Proposition 3.2
1. (i) We have an equality $P_{{\mathcal {S}}} = (R_{{\mathcal {S}}}^{\text{univ}})^{\mu _2^{{d}}}$.
2. (ii) Let ${{\mathfrak p}} \subset R_{{\mathcal {S}}}^{\text{univ}}$ be a prime ideal and let ${{\mathfrak q}} = {{\mathfrak p}} \cap P_{{\mathcal {S}}}$. Let $E = \operatorname {Frac} R_{{\
mathcal {S}}}^{\text{univ}} / {{\mathfrak p}}$ and suppose that the associated representation $\rho _{{\mathfrak p}} = r_{{\mathfrak p}}|_{G_{F, S}}\otimes _A E : G_{F, S} \to \operatorname {GL}
_n(E)$ is absolutely irreducible. Then $P_{{\mathcal {S}}} \to R_{{\mathcal {S}}}^{\text{univ}}$ is étale at ${{\mathfrak q}}$ and $\mu _2^{{d}}$ acts transitively on the set of primes of $R_{{{\
mathcal {S}}}}^{\text{univ}}$ above ${{\mathfrak q}}$.
We first establish a preliminary lemma, before proving the proposition.
Lemma 3.3 Let $R = R_{{{\mathcal {S}}}}^{\text{univ}}/({{\mathfrak m}}_{P_{{{\mathcal {S}}}}})$ and let $r: G_{F^{+},S}\rightarrow {{\mathcal {G}}}_n(R)$ be a representative of the specialization of
the universal deformation. Then, after possibly conjugating by an element of $1+M_n({{\mathfrak m}}_R)$, $r|_{G_{F,S}}$ has (block) diagonal entries given by $\bar {\rho }_1,\ldots ,\bar {\rho }_
{{d}}$, and the matrix $B$ defined above is equal to $\bar {B}$. (Note we are not asserting that the off-diagonal blocks of $r|_{G_{F, S}}$ are zero.)
Proof. We let $\bar {e}_1,\bar {e}_2,\ldots ,\bar {e}_{{d}} \in M_n(k)$ denote the standard idempotents decomposing $\bar {r}|_{G_{F,S}}$ into the block diagonal pieces $\bar {\rho }_1,\ldots ,\bar
{\rho }_{{d}}$. We let ${{\mathcal {A}}} \subset M_n(R)$ denote the image of $R[G_{F,S}]$ under $r$. The idempotents $\bar {e}_i$ lift to orthogonal idempotents $e_i$ in ${{\mathcal {A}}}$ with $e_1
+ \cdots + e_{{d}} = 1$ and, after conjugating by an element of $1+M_n({{\mathfrak m}}_R)$, we can assume that the $e_i$ are again the standard idempotents on $R^{n}$. Moreover, applying the first
case of the proof of [Reference Bellaïche and ChenevierBC09, Lemma 1.8.2], we can (and do) choose the $e_i$ so that they are fixed by the anti-involution $\star : {{\mathcal {A}}} \to {{\mathcal
{A}}}$ given by the formula $M \mapsto B (^{t}{}{M}) B^{-1}$. This implies that the matrix $B$ is block diagonal. We have $e_i{{\mathcal {A}}}e_i = M_{n_i}(R)$ (see [Reference Bellaïche and Chenevier
BC09, Lemma 1.4.3] and [Reference ChenevierChe14, Theorem 2.22]) and,for each $i \ne j$, we have $e_i{{\mathcal {A}}}e_j = M_{n_i,n_j}({{\mathcal {A}}}_{i,j})$, where ${{\mathcal {A}}}_{i,j} \subset
R$ is an ideal [Reference Bellaïche and ChenevierBC09, Proposition 1.3.8].
Since $\det \circ \, r = \det \circ \, \bar {r}$, Proposition 2.5 shows that $\sum _{i \ne j}{{\mathcal {A}}}_{i,j}{{\mathcal {A}}}_{j,i} = 0$. This implies that for each $i$ the map
\[ R[G_{F,S}] \to M_{n_i}(R) \]
given by
\[ x \mapsto e_i r(x) e_i \]
is an algebra homomorphism and we get an $R$-valued lift of $\bar {\rho }_i$. By the uniqueness assertion in Proposition 2.5, the determinant of this lift is equal to $\det \circ \bar {\rho }_i$.
Since $\bar {\rho }_i$ is absolutely irreducible, it follows from [Reference ChenevierChe14, Theorem 2.22] that, after conjugating by a block diagonal matrix in $1+M_n({{\mathfrak m}}_R)$, we can
assume that the map
\[ x \mapsto e_i r(x) e_i \]
is induced by $\bar {\rho }_i$, which is the desired statement.
Finally, we consider the matrix $B$. We have already shown that $B$ is block diagonal. For $1\le i \le {{d}}$, we denote the corresponding block of $B$ by $B_i$. It lifts a block $\bar {B}_i$ of $\
bar {B}$. By Schur's lemma, we have $B_i = \beta _i\bar {B}_i$ for some scalars $\beta _i \in 1+{{\mathfrak m}}_R$. Since $2$ is invertible in $R$, we can find $\lambda _i \in 1+{{\mathfrak m}}_R$
with $\lambda _i^{2} = \beta _i^{-1}$. Conjugating $r$ by the diagonal matrix with $\lambda _i$ in the $i$th block puts $r$ into the desired form.
Proof of Proposition 3.2 We begin by proving the first part. We again let $R = R_{{{\mathcal {S}}}}^{\rm univ}/({{\mathfrak m}}_{P_{{{\mathcal {S}}}}})$. By Nakayama's lemma, it suffices to show that
$R^{\mu _2^{{d}}} = k$. Indeed, the natural map $(R_{{{\mathcal {S}}}}^{\rm univ})^{\mu _2^{{d}}}/{{\mathfrak m}}_{P_{{{\mathcal {S}}}}}(R_{{{\mathcal {S}}}}^{\rm univ})^{\mu _2^{{d}}} \to R^{\mu _2^
{{d}}}$ is injective (i.e. $({{\mathfrak m}}_{P_{{{\mathcal {S}}}}}R_{{{\mathcal {S}}}}^{\rm univ})^{\mu _2^{{d}}} = {{\mathfrak m}}_{P_{{{\mathcal {S}}}}}(R_{{{\mathcal {S}}}}^{\rm univ})^{\mu _2^
{{d}}}$), since if $\sum _i m_i x_i$ is ${\mu _2^{{d}}}$-invariant, with $m_i \in {{\mathfrak m}}_{P_{{{\mathcal {S}}}}}$ and $x_i \in R_{{{\mathcal {S}}}}^{\rm univ}$, we have $\sum _i m_i x_i =
({1}/{2^{d}})\sum _i m_i\sum _{\sigma \in \mu _2^{d}} \sigma x_i$, which is an element of ${{\mathfrak m}}_{P_{{{\mathcal {S}}}}}(R_{{{\mathcal {S}}}}^{\rm univ})^{\mu _2^{{d}}}$. Let $r : G_{F^{+},
S} \to {{\mathcal {G}}}_n(R)$ be a representative of the specialization of the universal deformation satisfying the conclusion of Lemma 3.3. Then $R$ is a finite $k$-algebra and is generated as a $k$
-algebra by the matrix entries of $r$ and hence the matrix entries of $\rho = r|_{G_{F, S}}$ (because $B = \bar {B}$). We recall the ideals ${{\mathcal {A}}}_{i,j}\subset R$ appearing in the proof of
Lemma 3.3, which are generated by the block $(i,j)$ matrix entries of $\rho$. The conjugate self-duality of $\rho$ is given by ${}^{t} \rho (\delta ) = \chi (\delta )\bar {B}^{-1} \rho ((\delta ^{c})
^{-1}) \bar {B}$, $\delta \in G_{F,S}$. Since $\bar {B}$ is block diagonal, we deduce that ${{\mathcal {A}}}_{i,j}= {{\mathcal {A}}}_{j,i}$. Since $\sum _{i \ne j}{{\mathcal {A}}}_{i,j}{{\mathcal
{A}}}_{j,i} = 0$, we see that ${{\mathcal {A}}}_{i,j}^{2} = 0$ for $i \ne j$. We deduce that $R$ is generated as a $k$-module by $1 \in R$ and products of the form
\[ a_{{{\mathcal{P}}}} = \prod_{(i,j) \in {{\mathcal{P}}}} a_{i,j}, \]
where $\emptyset \ne {{\mathcal {P}}} \subset \{(i,j): 1\le i < j \le {{d}}\}$ and $a_{i,j} \in {{\mathcal {A}}}_{i,j}$ has action of $\mu _2^{{d}}$ given by $((-1)^{\alpha _1},\ldots , (-1)^{\alpha
_{{d}}})a_{i,j} = (-1)^{\alpha _i+\alpha _j}a_{i,j}$. Suppose that the action of $\mu _2^{{d}}$ on $a_{{{\mathcal {P}}}}$ is trivial. Then, for each $1 \le i \le {{d}}$, $i$ appears in an even number
of elements of ${{\mathcal {P}}}$. A product $a'_{j_1,j_2} = a_{1,j_1}a_{1,j_2}$ lies in ${{\mathcal {A}}}_{j_1,j_2}$ and the action of $\mu _2^{{d}}$ is given by $((-1)^{\alpha _1},\ldots ,(-1)^{\
alpha _{{d}}})a'_{j_1,j_2} = (-1)^{\alpha _{j_1}+\alpha _{j_2}}a'_{j_1,j_2}$. Since $1$ appears in an even number of elements of ${{\mathcal {P}}}$, we can ‘pair off’ these elements and rewrite $a_
{{\mathcal {P}}}$ as a product
\[ a_{{{\mathcal{P}}}'} = \prod_{(i,j) \in {{\mathcal{P}}}'} a'_{i,j}, \]
where ${{\mathcal {P}}}' \subset \{(i,j): 2\le i < j \le {{d}}\}$ and the action of $\mu _2^{{d}}$ on $a'_{i,j}$ is given by the same formula as for $a_{i,j}$. Continuing in this manner, we deduce
that $a_{{{\mathcal {P}}}}$ is the product of an even number of elements of ${{\mathcal {A}}}_{{{d}}-1, {{d}}}$ and thus equals $0$ since ${{\mathcal {A}}}_{{{d}}-1, {{d}}}^{2} = 0$.
The invariant subring $R^{\mu _2^{{d}}}$ is equal to the $k$-submodule of $R$ generated by $\sum _{\sigma \in \mu _2^{{d}}} \sigma x$, where $x$ runs over a set of $k$-module generators of $R$ (since
$2$ is invertible in $k$). It follows from the above calculation that $R^{\mu _2^{{d}}} = k$.
We now prove the second part. The diagonally embedded subgroup $\mu _2 \subseteq \mu _2^{{d}}$ acts trivially on $R_{{{\mathcal {S}}}}^{\rm univ}$, so we have an induced action of $\mu _2^{{d}}/\mu
_2$. The first part together with [Sta17, Tag 0BRI] implies that $\mu _2^{{d}} / \mu _2$ acts transitively on the set of primes of $R_{{{\mathcal {S}}}}^{\rm univ}$ above ${{\mathfrak q}}$. Let $R =
R_{{{\mathcal {S}}}}^{\rm univ}/{{\mathfrak p}}$ and let $r_{{{\mathfrak p}}} \colon G_{F^{+},S} \rightarrow {{\mathcal {G}}}_n(R)$ be a representative of the specialization of the universal
deformation. By [Sta17, Tag 0BST], to finish the proof if will be enough to show that if $\sigma \in \mu _2^{{d}}$, $\sigma ({{\mathfrak p}}) = {{\mathfrak p}}$, and $\sigma$ acts as the identity on
$R$, then $\sigma \in \mu _2$.
If $\sigma \in \mu _2^{{d}}$ corresponds to the block diagonal matrix $g\in \operatorname {GL}_n({{\mathcal {O}}})$, then these conditions imply that $r_{{{\mathfrak p}}}$ and $g r_{{{\mathfrak p}}}
g^{-1}$ are conjugate by an element $\gamma \in 1+M_n({{\mathfrak m}}_R)$. Since $r_{{{\mathfrak p}}}|_{G_{F,S}}\otimes E = \rho _{{{\mathfrak p}}}$ is absolutely irreducible, this implies that $g\
gamma ^{-1}$ is scalar and so $g$ must also be scalar as $l>2$; hence, $g \in \mu _2$. This completes the proof.
For each partition $\{1, \ldots , {{d}}\} = {{\mathcal {P}}}_1 \sqcup {{\mathcal {P}}}_2$ with ${{\mathcal {P}}}_1, {{\mathcal {P}}}_2$ both non-empty, Proposition 2.5 gives an ideal $I_{({{\mathcal
{P}}}_1, {{\mathcal {P}}}_2)} \subset P_{{\mathcal {S}}}$ cutting out the locus where the determinant $\det r|_{G_{F, S}}$ is $({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)$-reducible. We write $I_{{\
mathcal {S}}}^{\rm red} = \prod _{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)} I_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)}$, an ideal of $P_{{\mathcal {S}}}$.
Lemma 3.4 Let ${{\mathfrak p}} \subset R_{{\mathcal {S}}}^{\text {univ}}$ be a prime ideal and let ${{\mathfrak q}} = {{\mathfrak p}} \cap P_{{{\mathcal {S}}}}$. Let $A = R_{{{\mathcal {S}}}}^{\text
{univ}} / {{\mathfrak p}}$, $E = \operatorname {Frac} A$. Then $\rho _{{\mathfrak p}} = r_{{\mathfrak p}}|_{G_{F, S}}\otimes _A E$ is absolutely irreducible if and only if $I_{{\mathcal {S}}}^{\rm
red} \not \subset {{\mathfrak q}}$.
Proof. If $I_{{\mathcal {S}}}^{\rm red} \subset {{\mathfrak q}}$, then $I_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)} \subset {{\mathfrak q}}$ for some proper partition $({{\mathcal {P}}}_1, {{\
mathcal {P}}}_2)$. Then Proposition 2.5 implies that $\det r_{{\mathfrak p}}$ admits a decomposition $\det \circ r_{{\mathfrak p}}|_{G_{F,S}} = D_1 D_2$ for two determinants $D_i : A[G_{F, S}] \to M_
{n_i}(A)$. Then [Reference ChenevierChe14, Corollary 2.13] implies that $\rho _{{\mathfrak p}}$ is not absolutely irreducible.
Suppose conversely that $\rho _{{\mathfrak p}}$ is not absolutely irreducible. Let $J_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)}$ denote the image of $I_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)}$ in
$A$. We must show that some $J_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)}$ is zero. Let ${{\mathcal {A}}}$ denote the image of $A[G_{F, S}]$ in $M_n(A)$ under $r_{{\mathfrak p}}|_{G_{F, S}}$.
According to [Reference Bellaïche and ChenevierBC09, Theorem 1.4.4], we can assume that ${{\mathcal {A}}}$ has the form
(2)$${{\mathcal{A}}} = \begin{pmatrix} M_{n_1}(A) & M_{n_1,n_2}(\mathcal{A}_{1,2}) & \cdots & M_{n_1,n_{{d}}}(\mathcal{A}_{1,{{d}}})\\ M_{n_2,n_1}(\mathcal{A}_{2,1}) & M_{n_2}(A) & \cdots & M_{n_2,n_
{{d}}}(\mathcal{A}_{2,{{d}}})\\ \vdots & \vdots & \ddots & \vdots \\ M_{n_{{d}},n_1}(\mathcal{A}_{{{d}},1}) & M_{n_{{d}},n_2}(\mathcal{A}_{{{d}},2}) & \cdots & M_{n_{{d}}}(A) \end{pmatrix}\!,$$
where each $\mathcal {A}_{i, j}$ is a fractional ideal of $E$. Consequently, ${{\mathcal {A}}} \otimes _A E$ has the form
(3)$${{\mathcal{A}}} \otimes_A E= \begin{pmatrix} M_{n_1}(E) & M_{n_1,n_2}(\mathcal{E}_{1,2}) & \cdots & M_{n_1,n_{{d}}}(\mathcal{E}_{1,{{d}}})\\ M_{n_2,n_1}(\mathcal{E}_{2,1}) & M_{n_2}(E) & \cdots
& M_{n_2,n_{{d}}}(\mathcal{E}_{2,{{d}}})\\ \vdots & \vdots & \ddots & \vdots \\ M_{n_{{d}},n_1}(\mathcal{E}_{{{d}},1}) & M_{n_{{d}},n_2}(\mathcal{E}_{{{d}},2}) & \cdots & M_{n_{{d}}}(E) \end{pmatrix}
where each $\mathcal {E}_{i, j} = \mathcal {A}_{i, j} \otimes _A E$ equals either $E$ or $0$. Let $f_i \in M_n(E)$ denote the matrix with 1 in the $(i, i)$th entry and 0 everywhere else. If $\rho _
{{\mathfrak p}}$ is not absolutely irreducible, then ${{\mathcal {A}}} \otimes _A E$ is a proper subspace of $M_n(E)$, so there exists $i$ such that $({{\mathcal {A}}} \otimes _A E) f_i \subset M_n
(E) f_i$ is a proper subspace. Since $M_n(E) f_i$ is isomorphic as an $M_n(E)$-module to the tautological representation $E^{n}$, this implies that the ${{\mathcal {A}}} \otimes _A E$-module $E^{n}$
admits a proper invariant subspace. After permuting the diagonal blocks, we can assume that this subspace is $E^{n_1 + \cdots + n_s}$ for some $s < {{d}}$ (included as the subspace of $E^{n}$ where
the last $n_{s+1} + \cdots + n_{{d}}$ entries are zero). Otherwise said, the spaces $\mathcal {E}_{i, j}$ for $i > s$, $j \leq s$ are zero. If ${{\mathcal {P}}}_1 = \{ 1, \ldots , s \}$ and ${{\
mathcal {P}}}_2 = \{ s+1, \ldots , {d}\}$, then this implies that ${{\mathcal {J}}}_{({{\mathcal {P}}}_1, {{\mathcal {P}}}_2)} \otimes _A E = 0$ and hence (as $A$ is a domain) ${{\mathcal {J}}}_{({{\
mathcal {P}}}_1, {{\mathcal {P}}}_2)} = 0$. This completes the proof.
Lemma 3.5 Let ${{\mathfrak p}} \subset R_{{\mathcal {S}}}^{\text {univ}}$ be a prime ideal, $A = R_{{\mathcal {S}}}^{\text {univ}} / {{\mathfrak p}}$, $E = \operatorname {Frac} A$. Then $r_{{\
mathfrak p}}\otimes _A E$ is Schur and, if $r_{{\mathfrak p}}|_{G_{F, S}} \otimes _A E$ is not absolutely irreducible, then $r_{{\mathfrak p}}$ is equivalent (i.e. conjugate by an element in $1+ M_n
({{\mathfrak m}}_A)$) to a type-${{\mathcal {S}}}$ lifting of the form $r_{{\mathfrak p}} = r_1 \oplus r_2$, where $r_i : G_{F^{+}, S} \to {{\mathcal {G}}}_{m_i}(A)$ and $m_1 m_2 \neq 0$.
Proof. We argue, as in the proof of Lemma 3.4, using the image ${{\mathcal {A}}} \subset M_n(A)$ of $A[G_{F, S}]$, which is a generalized matrix algebra. Suppose that we are given $G_{F, S}$
-invariant subspaces $E^{n} \supset W_1 \supset W_2$ such that $W_2$ and $E^{n} / W_1$ are irreducible. We can assume that ${{\mathcal {A}}}$ has the form (2) and that this decomposition is block
upper triangular (perhaps with respect to a coarser partition than $n = n_1 + \cdots + n_{{d}}$) and moreover than the first block corresponds to $W_2$, while the last block corresponds to $E^{n} /
W_1$. In particular, $W_2$ and $E^{n} / W_1$ are even absolutely irreducible. Note that there can be no isomorphism $W_2^{c \vee }(\nu \circ r_{{\mathfrak p}}) \cong E^{n} / W_1$; if there was, then
it would imply an identity of $A$-valued determinants, which we could reduce modulo ${{\mathfrak m}}_A$ to obtain an identity $\{ \rho _{i} \} = \{ \rho _j \}$ of sets of irreducible constituents of
$\bar {r}|_{G_{F, S}}$. Since these appear with multiplicity 1, this is impossible. This all shows that $r_{{\mathfrak p}} \otimes _A E$ is necessarily Schur.
Now suppose that $r_{{\mathfrak p}}|_{G_{F, S}}\otimes _A E$ is not absolutely irreducible. After permuting the diagonal blocks of $\bar {r}$, we can assume that there is some $1 \leq m \leq {d}$
such that ${{\mathcal {A}}}_{i, j} = 0$ for $i > m$, $j \leq m$. The existence of the conjugate self-duality of $r_{{\mathfrak p}}$ implies (cf. [Reference Bellaïche and ChenevierBC09, Lemma 1.8.5])
that ${{\mathcal {A}}}_{j, i} = 0$ in the same range, giving a decomposition $r_{{\mathfrak p}}|_{G_{F, S}} = \rho _1 \oplus \rho _2$ of representations over $A$. Since $r_{{\mathfrak p}} \otimes _A
E$ is Schur, the conjugate self-duality of $r_{{\mathfrak p}}$ must make $\rho _1$ and $\rho _2$ orthogonal, showing that $r_{{\mathfrak p}}$ itself decomposes as $r_{{\mathfrak p}} = r_1 \oplus r_2$
3.3 Dimension bounds
We now suppose that $S$ admits a decomposition $S = S_l \sqcup S(B) \sqcup R \sqcup S_a$, where:
1. – for each $v \in S(B) \cup R$, $q_v \equiv 1 \text { mod }l$ and $\bar {r}|_{G_{F_{{\widetilde {v}}}}}$ is trivial;
2. – for each $v \in S_a$, $q_v \not \equiv 1 \text { mod }l$, $\bar {r}|_{G_{F_{{\widetilde {v}}}}}$ is unramified, and $\bar {r}|_{G_{F_{{\widetilde {v}}}}}$ is scalar. (Then any lifting of $\bar
{r}|_{G_{F_{{\widetilde {v}}}}}$ is unramified.)
We consider the global deformation problem
\[ {{\mathcal{S}}} = ( F / F^{+}, S, \tilde{S}, \Lambda, \bar{r}, \chi, \{ {{\mathcal{D}}}_v^{\triangle} \}_{v \in S_l} \cup \{ {{\mathcal{D}}}_v^{\rm St} \}_{v \in S(B)} \cup \{ {{\mathcal{D}}}_v^
{1} \}_{v \in R} \cup \{ {{\mathcal{D}}}_v^{\square} \}_{v \in S_a} ), \]
where $\bar {r}$ is assumed to be Schur. We define quantities $d_{F, 0} = d_0$ and $d_{F, l} = d_l$ as follows. Let $\Delta$ denote the Galois group of the maximal abelian pro-$l$ extension of $F$
which is unramified outside $l$, and let $\Delta _0$ denote the Galois group of the maximal abelian pro-$l$ extension of $F$ which is unramified outside $l$ and in which each place of $S(B)$ splits
completely. We set
\[ d_0 = \dim_{{{\mathbb{Q}}}_l} \ker( \Delta[1/l] \to \Delta_0[1/l] )^{c = -1} \]
\[ d_l = \inf_{v \in S_l} [F^{+}_v : {{\mathbb{Q}}}_l]. \]
Lemma 3.6 Suppose that $d_l > n(n-1)/2 + 1$. Let $A \in {{\mathcal {C}}}_\Lambda$ be a finite $\Lambda$-algebra and let $r : G_{F^{+}, S} \to {{\mathcal {G}}}_n(A)$ be a lifting of $\bar {r}$ of type
${{\mathcal {S}}}$. Then $\dim A / (I_{{\mathcal {S}}}^{\rm red}, \lambda ) \leq n[F^{+} : {{\mathbb {Q}}}] - d_0$.
Proof. We can assume without loss of generality that $A = A / (I_{{\mathcal {S}}}^{\rm red}, \lambda )$ and must show that $\dim A \leq [F^{+} : {{\mathbb {Q}}}] - d_0$. Since $A$ is Noetherian and
we are interested only in dimension, we can assume moreover that $A$ is integral. Let $E = \operatorname {Frac}(A)$. Then (Lemma 3.5) we can find a non-trivial partition $n = n_1 + n_2$ and
homomorphisms $r_i : G_{F^{+}, S} \to {{\mathcal {G}}}_{n_i}(A)$ ($i = 1, 2$) such that $r = r_1 \oplus r_2$.
Let $\bar {E}$ be a choice of algebraic closure of $E$. Our condition on $d_l$ means that we can appeal to [Reference ThorneTho15, Corollary 3.12] (characterization of $A$-valued points of ${{\
mathcal {D}}}_v^{\triangle }$ for each $v \in S_l$). This result implies the existence for each $v \in S_l$ of an increasing filtration
\[ 0 \subset \operatorname{Fil}^{1}_v \subset \operatorname{Fil}^{2}_v \subset \cdots \subset \operatorname{Fil}^{n}_v = \bar{E}^{n} \] | {"url":"https://core-cms.prod.aop.cambridge.org/core/journals/compositio-mathematica/article/automorphy-lifting-for-residually-reducible-ladic-galois-representations-ii/91B814A109E34CC8DA2A5689BFA5CFA4","timestamp":"2024-11-03T07:55:25Z","content_type":"text/html","content_length":"1049979","record_id":"<urn:uuid:5a4b235b-80c0-4498-8058-90a8f6765a6e>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00887.warc.gz"} |
Calculating Geodesic Distance Between Points
NOTE: Starting in ArcGIS Desktop 10.2.1, the proximity tools Near and Generate Near Table can measure geodesic distances, so there is no longer a need for the workaround below. You can learn more
here: http://desktop.arcgis.com/en/arcmap/latest/tools/analysis-toolbox/analysis-toolbox-history.htm#GUID-FB944CD8-9D92-4146-B6DD-2562F682CCC5
Going back to the very early days of ArcGIS there have been geoprocessing tools for calculating distances between features. Tools like Near, Point Distance, and Buffer have been around for many
releases, and perform key analysis in a number of common GIS workflows. These distance-measuring tools have always worked well and calculated very accurate distances for features concentrated in a
relatively small area (a city, state, or single UTM zone) with an appropriate projected coordinate system that minimizes distance distortion. Unfortunately, for groups of features spread over larger
areas (regions, countries, or the world!), or for datasets with a geographic coordinate system, these tools have historically produced results that were less than perfect.
In the last few releases, and continuing in future releases, more emphasis has been placed on calculating accurate distances for that second scenario: features covering a large area, or datasets with
a geographic coordinate system. Some key enhancements are in the works to make distance measurement through geoprocessing better than ever, namely by calculating geodesic distances in the scenarios
described above (geodesic distance is the distance measured along the shortest route between two points on the Earth’s surface).
In 10.0 Service Pack 2, an enhancement has been made to enable the calculation of geodesic distance along a line. This is done with a line feature class in a geographic coordinate system using the
Calculate Field tool and specifying an expression like !Shape.length@meters! . This functionality can be included in a longer workflow to calculate geodesic distance between points.
The model summarized below produces output similar to the Generate Near Table tool; given Input Points and Near Points datasets, find the geodesic distance from each input point to each near point
and record that distance in a new table. You can download this model and sample data from the Geoprocessing Resource Center Model and Script Tool Gallery here, and run the tool with your own point
datasets that have a geographic coordinate system (some modification to the model may be required).
Since this geodesic distance calculation relies on the measurement of lines, we need a way to create line features between all input points and near points. The XY To Line tool can be used to
accomplish this, but first there are a few steps that must be done to setup the workflow and produce a table that can be used with the XY To Line tool. Given that the XY To Line tool requires a table
of XY coordinates as input, first add XY coordinate information to both the Input Points and Near Points using the Add XY Coordinates tool.
Next, use the Generate Near Table tool to setup what will become the Output Table (note that the NEAR_DIST values produced by the Generate Near Table tool are in decimal degrees – not geodesic
After creating the Output Table, use the Join Field tool to move the coordinate information from the Input Points and Near Points to the Output Table. Also, a unique identifier field will be required
later, so add a new ID field using Add Field and calculate it equal to the Output Table ObjectID field using Calculate Field.
Now that the Output Table is set up to be used correctly with the XY To Line tool (with Start and End coordinates stored in separate fields), run XY To Line to create line features between each input
point and near point. These will be the lines that can be measured to obtain geodesic distances.
Output of XY To Line
Next, add a new model variable, Linear Unit, which can be used to specify the unit in which to calculate the geodesic distance (meters, kilometers, feet, miles, etc.). Then add a new field ‘GEO_DIST’
and calculate it as the geodesic distance along the lines using a calculation expression!Shape.length@%Linear Unit%!, then join these distances back to the Output Table. Finally, clean up the Output
Table by deleting any unnecessary or intermediate fields.
The output of the workflow will look like the table pictured below, where the field GEO_DIST contains the geodesic distance between the points identified in the IN_FID and NEAR_FID fields, in the
unit specified in the Linear Unit variable. Comparing these geodesic distances to Euclidean distances (based on a Mercator projection) shows how inaccurate distance measurements can be if they are
performed in an inappropriate coordinate system or at an inappropriate scale.
Geodesic distances calculated by this model, and Euclidean distances calculated by the Generate Near Table tool.
Commenting is not enabled for this article. | {"url":"https://www.esri.com/arcgis-blog/products/arcgis-desktop/analytics/calculating-geodesic-distance-between-points/","timestamp":"2024-11-05T08:44:52Z","content_type":"text/html","content_length":"931788","record_id":"<urn:uuid:538c14b6-4945-443b-9378-a3858f325828>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00277.warc.gz"} |
An efficient Bayesian updating framework for characterizing the posterior failure probability
Originalsprache Englisch
Aufsatznummer 111768
Fachzeitschrift Mechanical Systems and Signal Processing
Jahrgang 222
Frühes Online-Datum 6 Aug. 2024
Publikationsstatus Elektronisch veröffentlicht (E-Pub) - 6 Aug. 2024
Bayesian updating plays an important role in reducing epistemic uncertainty and making more reliable predictions of the structural failure probability. In this context, it should be noted that the
posterior failure probability conditional on the updated uncertain parameters becomes a random variable itself. Hence, characterizing the statistical properties of the posterior failure probability
is important, yet challenging task for risk-based decision-making. In this study, an efficient framework is proposed to fully characterize the statistical properties of the posterior failure
probability. The framework is based on the concept of Bayesian updating and keeps the effect of aleatory and epistemic uncertainty separated. To improve the efficiency of the proposed framework, a
weighted sparse grid numerical integration is suggested to evaluate the first three raw moments of the corresponding posterior reliability index. This enables the reuse of evaluation results stemming
from previous analyses. In addition, the proposed framework employs the shifted lognormal distribution to approximate the probability distribution of the posterior reliability index, from which the
mean, quantile, and even the distribution of the posterior failure probability can be easily obtained in closed form. Four examples illustrate the efficiency and accuracy of the proposed method, and
results generated with Markov Chain Monte Carlo combined with plain Monte Carlo simulation are employed as a reference.
ASJC Scopus Sachgebiete
• Ingenieurwesen (insg.)
• Informatik (insg.)
• Ingenieurwesen (insg.)
• Ingenieurwesen (insg.)
• Ingenieurwesen (insg.)
• Informatik (insg.)
• Standard
• Harvard
• Apa
• Vancouver
• Autor
• BibTex
• RIS
Li, PP, Zhao, YG, Dang, C, Broggi, M, Valdebenito, MA & Faes, MGR 2025, '
An efficient Bayesian updating framework for characterizing the posterior failure probability
Mechanical Systems and Signal Processing
, Jg. 222, 111768.
Li, P. P., Zhao, Y. G., Dang, C., Broggi, M., Valdebenito, M. A., & Faes, M. G. R. (2025).
An efficient Bayesian updating framework for characterizing the posterior failure probability
Mechanical Systems and Signal Processing
, Artikel 111768. Vorabveröffentlichung online.
title = "An efficient Bayesian updating framework for characterizing the posterior failure probability",
abstract = "Bayesian updating plays an important role in reducing epistemic uncertainty and making more reliable predictions of the structural failure probability. In this context, it should be noted
that the posterior failure probability conditional on the updated uncertain parameters becomes a random variable itself. Hence, characterizing the statistical properties of the posterior failure
probability is important, yet challenging task for risk-based decision-making. In this study, an efficient framework is proposed to fully characterize the statistical properties of the posterior
failure probability. The framework is based on the concept of Bayesian updating and keeps the effect of aleatory and epistemic uncertainty separated. To improve the efficiency of the proposed
framework, a weighted sparse grid numerical integration is suggested to evaluate the first three raw moments of the corresponding posterior reliability index. This enables the reuse of evaluation
results stemming from previous analyses. In addition, the proposed framework employs the shifted lognormal distribution to approximate the probability distribution of the posterior reliability index,
from which the mean, quantile, and even the distribution of the posterior failure probability can be easily obtained in closed form. Four examples illustrate the efficiency and accuracy of the
proposed method, and results generated with Markov Chain Monte Carlo combined with plain Monte Carlo simulation are employed as a reference.",
keywords = "Bayesian updating, Posterior failure probability, Shifted lognormal distribution, Sparse grid numerical integration",
author = "Li, {Pei Pei} and Zhao, {Yan Gang} and Chao Dang and Matteo Broggi and Valdebenito, {Marcos A.} and Faes, {Matthias G.R.}",
note = "Publisher Copyright: {\textcopyright} 2024 The Authors",
year = "2024",
month = aug,
day = "6",
doi = "10.1016/j.ymssp.2024.111768",
language = "English",
volume = "222",
journal = "Mechanical Systems and Signal Processing",
issn = "0888-3270",
publisher = "Academic Press Inc.",
TY - JOUR
T1 - An efficient Bayesian updating framework for characterizing the posterior failure probability
AU - Li, Pei Pei
AU - Zhao, Yan Gang
AU - Dang, Chao
AU - Broggi, Matteo
AU - Valdebenito, Marcos A.
AU - Faes, Matthias G.R.
N1 - Publisher Copyright: © 2024 The Authors
PY - 2024/8/6
Y1 - 2024/8/6
N2 - Bayesian updating plays an important role in reducing epistemic uncertainty and making more reliable predictions of the structural failure probability. In this context, it should be noted that
the posterior failure probability conditional on the updated uncertain parameters becomes a random variable itself. Hence, characterizing the statistical properties of the posterior failure
probability is important, yet challenging task for risk-based decision-making. In this study, an efficient framework is proposed to fully characterize the statistical properties of the posterior
failure probability. The framework is based on the concept of Bayesian updating and keeps the effect of aleatory and epistemic uncertainty separated. To improve the efficiency of the proposed
framework, a weighted sparse grid numerical integration is suggested to evaluate the first three raw moments of the corresponding posterior reliability index. This enables the reuse of evaluation
results stemming from previous analyses. In addition, the proposed framework employs the shifted lognormal distribution to approximate the probability distribution of the posterior reliability index,
from which the mean, quantile, and even the distribution of the posterior failure probability can be easily obtained in closed form. Four examples illustrate the efficiency and accuracy of the
proposed method, and results generated with Markov Chain Monte Carlo combined with plain Monte Carlo simulation are employed as a reference.
AB - Bayesian updating plays an important role in reducing epistemic uncertainty and making more reliable predictions of the structural failure probability. In this context, it should be noted that
the posterior failure probability conditional on the updated uncertain parameters becomes a random variable itself. Hence, characterizing the statistical properties of the posterior failure
probability is important, yet challenging task for risk-based decision-making. In this study, an efficient framework is proposed to fully characterize the statistical properties of the posterior
failure probability. The framework is based on the concept of Bayesian updating and keeps the effect of aleatory and epistemic uncertainty separated. To improve the efficiency of the proposed
framework, a weighted sparse grid numerical integration is suggested to evaluate the first three raw moments of the corresponding posterior reliability index. This enables the reuse of evaluation
results stemming from previous analyses. In addition, the proposed framework employs the shifted lognormal distribution to approximate the probability distribution of the posterior reliability index,
from which the mean, quantile, and even the distribution of the posterior failure probability can be easily obtained in closed form. Four examples illustrate the efficiency and accuracy of the
proposed method, and results generated with Markov Chain Monte Carlo combined with plain Monte Carlo simulation are employed as a reference.
KW - Bayesian updating
KW - Posterior failure probability
KW - Shifted lognormal distribution
KW - Sparse grid numerical integration
UR - http://www.scopus.com/inward/record.url?scp=85200591107&partnerID=8YFLogxK
U2 - 10.1016/j.ymssp.2024.111768
DO - 10.1016/j.ymssp.2024.111768
M3 - Article
AN - SCOPUS:85200591107
VL - 222
JO - Mechanical Systems and Signal Processing
JF - Mechanical Systems and Signal Processing
SN - 0888-3270
M1 - 111768
ER - | {"url":"https://www.fis.uni-hannover.de/portal/de/publications/an-efficient-bayesian-updating-framework-for-characterizing-the-posterior-failure-probability(8a58ebb9-5a6c-40c1-95d1-e919fc9a6929).html","timestamp":"2024-11-10T11:23:05Z","content_type":"text/html","content_length":"50862","record_id":"<urn:uuid:ffbb09f9-2362-4bae-997b-3756ff0696d2>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00059.warc.gz"} |
Algebraic Geometry Graduate Reading Course
This page is outdated, referring to people, mailinglists and activities that no longer exist. SJZN 02/01/2016 It is kept for archival purposes.
[math]\displaystyle{ \operatorname{Hom}_{\rm Schemes}(X, \operatorname{Spec}(A)) \cong \operatorname{Hom}_{\rm CRing}(A, {\mathcal O}_X(X)). }[/math]
MWF: 1:20-2:10, 1323 Sterling Hall
Th: 4-5, B139 Van Vleck, Meeting with Faculty
Structure of Course
Our reading group will meet three times a week (Mon-Wed-Fri) to discuss the material amongst ourselves, and once a week (Thu) to discuss the material with faculty members. Typically we have a visit
from some combination of Jordan, David, Andrei and Sukhendu. Registered students are also expected to turn in homework each week.
Doing exercises is very important in mastering this material. Ravi's notes contain many, many exercises. It's up to you to choose which problems to hand in. Ravi suggests that you choose a varied
selection of problems that are personally interesting to you. You should hand in 6 written up exercises each Wednesday, starting September 8th. LaTeX'ing your solution is strongly recommended. Ask
another grad student if you need help learning TeX.
We'll collect homework at our meeting, and distribute it to the grader. In fairness to our grader, no late homework will be accepted.
I encourage you to work together. You can email agrc@math.wisc.edu to contact the whole reading group to schedule problem solving sessions/ask questions/etc. | {"url":"https://wiki.math.wisc.edu/index.php/Algebraic_Geometry_Graduate_Reading_Course","timestamp":"2024-11-04T14:12:45Z","content_type":"text/html","content_length":"18608","record_id":"<urn:uuid:0de59e62-cdab-47b5-919c-dca6cb9813e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00270.warc.gz"} |
Average Lap Speed Calculator | Online Calculators
Average Lap Speed Calculator
The Average Lap Speed Calculator helps calculate the average speed of a race based on the total laps, distance per lap, and total race time. Enter known values in the fields and leave one blank to
have it calculated
Using Guide
This calculator has been designed with clear descriptions to guide you on how to use each input field. Refer to the table below for explanations and example values.
Input Field Description Example Value
Number of Laps Total laps completed in the race. This number will be used to calculate the average speed. 10
Distance Per Lap (miles) Distance of each lap in miles. Enter this to determine total distance covered. 2.5
Total Race Time (min) Total time taken for the race in minutes. This is essential for calculating speed. 90
Average Lap Speed (MPH) The calculator will display the average lap speed in miles per hour (MPH) based on entries. Leave blank to calculate. —
Calculation Examples
Here are two examples showing how the calculator works, with step-by-step explanations.
Example 1: Calculating Average Lap Speed
Step Calculation Result
Number of Laps 10 10
Distance Per Lap (miles) 2.5 2.5
Total Race Time (min) 90 90
Average Lap Speed (MPH) $\frac{10 \times 2.5}{90 / 60}$ = 16.67 16.67
Example 2: Calculating Total Race Time
Step Calculation Result
Number of Laps 8 8
Distance Per Lap (miles) 3 3
Average Lap Speed (MPH) 15 15
Total Race Time (min) $\frac{8 \times 3}{15} \times 60$ = 96 96
To calculate, fill in the known values and leave the field you wish to calculate blank. Press Calculate to see the result or Reset to clear all fields for a new entry.
Leave a Comment | {"url":"https://lengthcalculators.com/average-lap-speed-calculator/","timestamp":"2024-11-06T17:41:24Z","content_type":"text/html","content_length":"63794","record_id":"<urn:uuid:85c99bce-8872-4b0f-b333-72a6f51bd634>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00332.warc.gz"} |
Joachim M. Buhmann: Katalogdaten im Herbstsemester 2019
Name Herr Prof. em. Dr. Joachim M. Buhmann
Lehrgebiet Informatik (Information Science and Engineering)
Institut für Maschinelles Lernen
ETH Zürich, OAT Y 13.2
Adresse Andreasstrasse 5
8092 Zürich
Telefon +41 44 632 31 24
E-Mail jbuhmann@inf.ethz.ch
URL http://www.ml.inf.ethz.ch/
Departement Informatik
Beziehung Professor emeritus
Nummer Titel ECTS Umfang Dozierende
252-0535-00L Advanced Machine Learning 8 KP 3V + 2U J. M. Buhmann
+ 2A
Kurzbeschreibung Machine learning algorithms provide analytical methods to search data sets for characteristic patterns. Typical tasks include the classification of data, function fitting and
clustering, with applications in image and speech analysis, bioinformatics and exploratory data analysis. This course is accompanied by practical machine learning projects.
Students will be familiarized with advanced concepts and algorithms for supervised and unsupervised learning; reinforce the statistics knowledge which is indispensible to solve
Lernziel modeling problems under uncertainty. Key concepts are the generalization ability of algorithms and systematic approaches to modeling and regularization. Machine learning projects
will provide an opportunity to test the machine learning algorithms on real world data.
The theory of fundamental machine learning concepts is presented in the lecture, and illustrated with relevant applications. Students can deepen their understanding by solving both
pen-and-paper and programming exercises, where they implement and apply famous algorithms to real-world data.
Topics covered in the lecture include:
What is data?
Bayesian Learning
Computational learning theory
Inhalt Supervised learning:
Ensembles: Bagging and Boosting
Max Margin methods
Neural networks
Unsupservised learning:
Dimensionality reduction techniques
Mixture Models
Non-parametric density estimation
Learning Dynamical Systems
Skript No lecture notes, but slides will be made available on the course webpage.
C. Bishop. Pattern Recognition and Machine Learning. Springer 2007.
R. Duda, P. Hart, and D. Stork. Pattern Classification. John Wiley &
Sons, second edition, 2001.
T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical
Learning: Data Mining, Inference and Prediction. Springer, 2001.
L. Wasserman. All of Statistics: A Concise Course in Statistical
Inference. Springer, 2004.
The course requires solid basic knowledge in analysis, statistics and numerical methods for CSE as well as practical programming experience for solving assignments.
Voraussetzungen Students should have followed at least "Introduction to Machine Learning" or an equivalent course offered by another institution.
/ Besonderes
PhD students are required to obtain a passing grade in the course (4.0 or higher based on project and exam) to gain credit points.
Doctoral Seminar Machine Learning (HS19)
Only for Computer Science Ph.D. students.
252-0945-09L 2 KP 2S J. M. Buhmann, T. Hofmann, A. Krause, G. Rätsch
This doctoral seminar is intended for PhD students affiliated with the Institute for Machine
Learning. Other PhD students who work on machine learning projects or related topics need approval
by at least one of the organizers to register for the seminar.
Kurzbeschreibung An essential aspect of any research project is dissemination of the findings arising from the study. Here we focus on oral communication, which includes: appropriate selection of
material, preparation of the visual aids (slides and/or posters), and presentation skills.
Lernziel The seminar participants should learn how to prepare and deliver scientific talks as well as to deal with technical questions. Participants are also expected to actively contribute
to discussions during presentations by others, thus learning and practicing critical thinking skills.
Voraussetzungen This doctoral seminar of the Machine Learning Laboratory of ETH is intended for PhD students who work on a machine learning project, i.e., for the PhD students of the ML lab.
/ Besonderes
Advanced Topics in Machine Learning
Number of participants limited to 40.
252-5051-00L 2 KP 2S J. M. Buhmann, A. Krause, G. Rätsch
The deadline for deregistering expires at the end of the fourth week of the semester. Students who
are still registered after that date, but do not attend the seminar, will officially fail the
Kurzbeschreibung In this seminar, recent papers of the pattern recognition and machine learning literature are presented and discussed. Possible topics cover statistical models in computer vision,
graphical models and machine learning.
The seminar "Advanced Topics in Machine Learning" familiarizes students with recent developments in pattern recognition and machine learning. Original articles have to be presented
Lernziel and critically reviewed. The students will learn how to structure a scientific presentation in English which covers the key ideas of a scientific paper. An important goal of the
seminar presentation is to summarize the essential ideas of the paper in sufficient depth while omitting details which are not essential for the understanding of the work. The
presentation style will play an important role and should reach the level of professional scientific presentations.
The seminar will cover a number of recent papers which have emerged as important contributions to the pattern recognition and machine learning literature. The topics will vary from
Inhalt year to year but they are centered on methodological issues in machine learning like new learning algorithms, ensemble methods or new statistical models for machine learning
applications. Frequently, papers are selected from computer vision or bioinformatics - two fields, which relies more and more on machine learning methodology and statistical models.
Literatur The papers will be presented in the first session of the seminar.
P. L. Bühlmann, A. Bandeira, H. Bölcskei, J. M. Buhmann, T. Hofmann,
401-5680-00L Foundations of Data Science Seminar 0 KP A. Krause, A. Lapidoth, H.‑A. Loeliger, M. H. Maathuis, G. Rätsch,
C. Uhler, S. van de Geer
Kurzbeschreibung Research colloquium | {"url":"https://www.vorlesungen.ethz.ch/Vorlesungsverzeichnis/dozent.view?dozide=10009696&ansicht=2&semkez=2019W&lang=de","timestamp":"2024-11-07T23:17:37Z","content_type":"text/html","content_length":"17830","record_id":"<urn:uuid:623ee1ba-1150-4367-8fec-70f9cebc88a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00888.warc.gz"} |
A model of this concept must provide:
Comparison_result operator() (const Kernel::Point_3 &p, const Kernel::Point_3 &q)
Compares the Cartesian coordinates of points p and q lexicographically in \( xyz\) order: first \( x\)-coordinates are compared, if they are equal, \( y\)-coordinates are compared.
Comparison_result Kernel::CompareXYZ_3::operator() ( const Kernel::Point_3 & p,
const Kernel::Point_3 & q
Compares the Cartesian coordinates of points p and q lexicographically in \( xyz\) order: first \( x\)-coordinates are compared, if they are equal, \( y\)-coordinates are compared. | {"url":"https://doc.cgal.org/5.3/Kernel_23/classKernel_1_1CompareXYZ__3.html","timestamp":"2024-11-13T11:42:35Z","content_type":"application/xhtml+xml","content_length":"11710","record_id":"<urn:uuid:c4a72e0b-d2ea-4bbc-b53f-7514b9441898>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00662.warc.gz"} |
Fourier number: Definition, Formula, Significance (heat transfer)
What is the Fourier number?
The Fourier number is the dimensionless quantity used in the calculation of unsteady-state heat transfer. The Fourier number is the ratio of the rate of heat conduction to the rate of heat stored in
a body. It is derived from the non-dimensionalization of the heat conduction equation.
In this article, we’re going to discuss:
• Fourier number equation:
• Significances of Fourier number in heat transfer:
• Applications:
• Examples with solutions:
Fourier number equation:
The Fourier number for heat transfer is given by,
`F_{O}=\frac{\alpha\ \tau }{L_{C}^{2}}`
α = Thermal diffusivity
𝜏 = Time (second)
`L_{c} = \text{Characteristics length} = \frac{\text{Volume}}{\text{Area}}`
The Fourier number in the mass transfer is given by,
`F_{O}=\frac{D. \tau }{L_{C}^{2}}`
Where D = Mass diffusivity
Significances of Fourier number in heat transfer:
The significances are as follows:-
1] The Fourier number indicates the relation between the rate of heat conduction through the body and the rate of heat stored in the body.
2] The larger value of the Fourier number indicates, a higher rate of heat transfer through the body.
3] The lower value of the Fourier number indicates the lower rate of heat transfer through the body.
The applications are as follows:-
1. It is used in the analysis of transient/ unsteady state conduction.
2. It is also used in the analysis of transient mass transfer systems.
Examples with solutions:
1] A hot object at an initial temperature of 700 K is dipped into the fluid which is at 350 K. The whole system has a biot number of 0.015.
Find the temperature of the object after the interval of 60 seconds. The necessary data for the analysis is given below:-
(Characteristic length: Lc = 8 x 10^-3
Thermal diffusivity, α = 2.3 x 10^-5 m²/s)
L[c] = 8 x 10^-3
α = 2.3 x 10^-5 m²/s
B[i] = 0.015
t = 60 seconds
T[∞] = 350 K
T[i] = 700 K
1] Fourier number:-
The Fourier number is given by,
`F_{o}=\frac{(2.3\times 10^{-5})\times 60}{(8\times 10^{-3})^{2}}`
F[o] = 21.56
2] Temperature of the object after the interval of 60 seconds using lumped system analysis:-
`\frac{T-350}{700-350}=e^{-0.015\times 21.56}`
T = 603.28 K
∴ The temperature of the object after the time interval of 60 seconds is 603.28 K.
2] The object with characteristics length of (1 x 10^-2) & a temperature of 800 K is kept in cold fluid at a temperature of 300 K.
Calculate the time interval to become the temperature of the object equal to 500 K. [Assume α = 2 x 10^-5 m²/s, Bi = 12x 10^-3]
L[c] = 1 x 10^-2
α = 2 x 10^-5 m²/s
B[i] = 12 x 10^-3
T[∞] = 300 K
T[i] = 800 K
T = 500 K
1] Fourier number using a method of lumped system analysis:-
`\frac{500-300}{800-300}=e^{-0.015\times Fo}`
F[o] = 76.35
2] Time interval to reach temperature of the object to 500 K:-
`76.35=\frac{(2\times 10^{-5})t}{(1\times 10^{-2})^{2}}`
t = 381.78 Seconds
Therefore the time required to attain a temperature of the object equal to 500 K is 381.78 Seconds.
1 thought on “Fourier number: Definition, Formula, Significance (heat transfer)”
1. Really very useful!
Thanks a lot and wishes for much future success!!!
Leave a Comment | {"url":"https://mechcontent.com/fourier-number/","timestamp":"2024-11-04T23:09:47Z","content_type":"text/html","content_length":"76717","record_id":"<urn:uuid:733de657-1759-49b1-b101-4014a46d427d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00785.warc.gz"} |
Single Zero Roulette Technique Uncover The Principles & Odds
BGaming’s American Roulette invites gamers to embrace the world of double zero (00) roulette with fashion. With its user-friendly interface and seamless gameplay, this game offers a superb platform
to explore the nuances of the 00 variant. Red Rake’s American Roulette faithfully captures the essence of the double zero (00) wheel, offering players a practical and immersive gaming experience.
With beautiful graphics and intuitive controls, this game is a best choice for these in search of authenticity. One of the notable distinctions in Double zero Roulette is the variance in odds
compared to its European counterparts. Due to the presence of each zeros (0 and 00) on the wheel, the possibilities for certain outcomes differ significantly.roulette77forum.com/t/
• To go into more detail, think about the scenario on bets with even-money payoffs similar to pink or black, odd and even, or 1-18 and 19-36.
• At the tip of the method, players who positioned a profitable bet, will be paid their winnings.
• As a result of this, the British roulette wheel producer John Huxley manufactured a roulette wheel to counteract the problem.
• This recreation boasts crisp graphics, clean gameplay, and a user-friendly interface, making it a pleasant alternative for fans of the 00 variant.
Start by placing a wager between $0.25 and $ 1.00 and spin the wheel. To keep away from some misunderstandings, the colour green was used for the zeros in roulette wheels beginning in the 1800s. A
giant financial loss is almost sure in the lengthy run if the player continues to employ this technique. Another technique is the Fibonacci system, the place bets are calculated in accordance with
the Fibonacci sequence. Regardless of the particular progression, no such strategy can statistically overcome the casino’s benefit, because the expected value of each allowed bet is adverse.
Players are allowed to position bets as the ball turns across the wheel until the dealer declares no more bets. When a profitable number and colour is decided by the roulette wheel, the supplier (or
croupier) will place a marker, also recognized as a “dolly”, on that successful spot on the roulette desk configuration. When we have a look at the tables, there isn’t a huge difference in
comparability with what you’ll discover in single zero roulette. The zeros are at the same finish in double zero roulette as what you will see in single zero roulette. Single zero roulette wheels
must also be weighted the identical throughout, meaning that you should have an even probability of profitable. [newline]The black and purple numbers are also supposed to be distributed evenly
throughout. For this purpose, a number of the primary data you be taught from double and triple zero roulette will translate to single zero – however the wheel layout is not going to.
A roulette wheel with zero and 00 could make it tougher to win as a result of additional pockets that lower players’ odds of success. However, understanding the results of these further slots and
implementing a quantity of methods can nonetheless give you an edge over the house. The recreation starts with the participant selecting to both guess on one number or on a variety. If the
participant chooses to play the one number, he should decide whether he needs to bet on even numbers or odd numbers. Once this is accomplished, the supplier spins the wheel to determine the place the
ball will land. The game is played between two gamers, and the first participant to throw in his chips wins.
Most generally these bets are known as “the French bets” and every covers a bit of the wheel. For the sake of accuracy, zero spiel, although explained beneath, isn’t a French wager, it’s more
precisely “the German wager”. Players at a desk might bet a set amount per collection (or multiples of that amount). The series are based on the means in which certain numbers lie subsequent to each
other on the roulette wheel. Not all casinos supply these bets, and a few could offer extra bets or variations on these.
The time period French roulette can be considerably of a misnomer because the “la partage” rule can generally be found in casinos exterior of France, including some high-limit rooms in Las Vegas.
There are quite a few other betting techniques that depend on this fallacy, or that try to observe ‘streaks’ (looking for patterns in randomness), varying wager measurement accordingly. Full complete
bets are most frequently wager by high rollers as maximum bets. Another wager offered on the single-zero recreation is “ultimate”, “finale”, or “finals”. The tiers bet can be referred to as the
“small collection” and in some casinos (most notably in South Africa) “collection 5-8”.
Neighbors bets are often put on in combos, for example “1, 9, 14, and the neighbors” is a 15-chip wager covering 18, 22, 33, 16 with one chip, 9, 31, 20, 1 with two chips and 14 with three chips.
Very in style in British casinos, tiers bets outnumber voisins and orphelins bets by a massive margin. Top line (0, 00, 1, 2, 3) has a unique anticipated worth because of approximation of the proper
6+1⁄5-to-1 payout obtained by the formula to 6-to-1. We have to search out what is the likelihood that Smith will lose his first5bets.
Edges could be trimmed by special rules such as the European en prison or the Atlantic City half back rule when even-money bets run up towards a zero. But basic home edges are 2.7 % for a single-zero
wheel, 5.26 % for double-zero and 7.69 percent on triple-zero video games. In both cases, the extra slots lower your possibilities of successful compared to a wheel with simply 36 numbered pockets.
Roulette is a casino-style sport that has been played for tons of of years. The player can bet on a row, a bunch of numbers, a person quantity, or a mix of numbers.
California Roulette
The ball finally loses momentum, passes by way of an area of deflectors, and falls onto the wheel and into one of many colored and numbered pockets on the wheel. The winnings are then paid to anybody
who has placed a profitable guess. From observing the sport in play, you can conclude that there are not any ensures when it comes to the roulette sport.
If you play French roulette, which is quite uncommon in Las Vegas casinos, you might come across a variant that offers the La Partage and/or En Prison guidelines. When these guidelines are active,
the odds for even money bets are forty eight.65%. Now let’s calculate the even money wager odds for Triple Zero Roulette. If we take into consideration that there are three green pockets, the
percentages for even cash bets are 46.15%.
and his work appears in newspapers, magazines and web sites around the globe. The house edge is 5.26 % on all however the five-number basket of zero, 00, 1, 2 and three, where the sting is 7.89 %.
You might try this train for every obtainable wager, and the answer can be the same.
Outside bets sometimes have smaller payouts with higher odds at profitable. If we ran the same train for single-zero wheels, we’d find all bets with a home edge of 2.7 %. Instead of the usual 2.7 p.c
at a single-zero wheel or 5.26 p.c on most bets at a double-zero wheel, the house edge jumps to 7.sixty nine percent with three zeroes. Sometimes the 000 area is full of a brand as an alternative,
but the impact is similar.
You’re already ahead of your competitors as a outcome of you’ve learned the totally different rules, odds, and payouts as well. Rather than lose half, the participant generally may select to imprison
the wager. If an imprisoned wager wins on the following spin it’s released and the participant will get it again, with out winnings. What is subject to on line casino rules is what occurs to an
imprisoned guess if the ball lands in zero again on the next spin.
If you’re uninterested in taking half in the same old games and need to attempt one thing completely different, then that is the article for you. We’ll inform you the means to play roulette with a
no-zero rule in order that it’s much less about luck and extra about talent. Triple zero roulette, because the name suggests, is a variant of traditional roulette that options a further green pocket
– the triple zero (000). While most roulette variants, such as the European and the American roulette wheels, have one and two zeroes, respectively, this variant adds one other pocket. It was called
a “perpetual motion machine” as a end result of it was a machine that continued to operate with out drawing energy from an exterior supply. The legal guidelines of physics thought it impossible, but
Pascal tried to defy the odds.
The player calls their guess to the croupier (most typically after the ball has been spun) and places enough chips to cover the bet on the desk within reach of the croupier. The latter course has
been set by casinos attempting out triple zero roulette. Instead of one green house with a 0, or two with a zero and a 00, there are three with a 0, 00 and 000. This slight modification units the
American roulette wheel aside from the European model, adding an extra layer of pleasure to the game. It turns out that the overwhelming majority of successful numbers are between zero and 19, with
1, three, and 18 accounting for roughly 10% of all successful numbers. The odds of a number being a winner (in this case, the number 17) are roughly three.fifty three in four, which is an
unbelievable 0.94 normal deviation from the imply (which is mainly a coin toss).
French roulette is performed on a single wheel and also features a favorable “en prison” or half-back rule. Under the “half-back” rule, if the player makes any even money bet (red, black, odd, even,
1-18, 19-36), and the ball lands in zero, then the player gets half the wager back, generally recognized as “la partage” in French. This system is one that is designed in order that when the
participant has won over a 3rd of their bets (less than the anticipated 18/38), they’ll win. Whereas the martingale will trigger damage within the occasion of an extended sequence of successive
losses, the Labouchère system will trigger bet measurement to grow shortly even the place a shedding sequence is damaged by wins. This occurs as a outcome of as the player loses, the average guess
measurement within the line will increase. To decide the successful quantity, a croupier spins a wheel in a single course, then spins a ball in the wrong way around a tilted round track working
across the outer fringe of the wheel.
By doubling bets after every win, one retains betting everything they’ve won until they either cease taking half in, or lose it all. In some casinos, a participant could guess full full for lower
than the desk straight-up most, for instance, “quantity 17 full complete by $25” would value $1000, that’s forty chips every at $25 worth. The maximum amount allowed to be wagered on a single bet in
European roulette relies on a progressive betting model. If the on line casino allows a most wager of $1,000 on a 35-to-1 straight-up, then on every 17-to-1 break up related to that straight-up,
$2,000 could additionally be wagered. Each 8-to-1 nook that covers four numbers) could have $4,000 wagered on it.
Another wager you’ll have the ability to strive is odd/even numbers, the place you wager on odd and even numbers. Because of this, it could be very difficult to foretell a selected number that the
wheel will land on (hence why all single numbers have the identical odds). These top-notch video games present an excellent alternative to dive into the world of Double Zero Roulette on-line. Each
game boasts its distinctive options and environment, guaranteeing an pleasant gaming expertise tailored to your preferences. Netent, a famend provider, presents an American Roulette game that stays
true to the double zero (00) wheel.
Double zero and triple zero roulette wheels typically have less even distribution in comparison with single zero wheels, which – relying on your playing style – can be a good or bad thing. In the
Double zero Roulette wheel, the percentages of hitting specific numbers, colors, or combos are adjusted to accommodate the extra zero. This alteration leads to a discount within the chance of winning
bets compared to traditional European or French roulette variants. For example, the profitable 40-chip / $40,000 bet on “17 to the maximum” pays 392 chips / $392,000. The skilled croupier would pay
the participant 432 chips / $432,000, that’s 392 + 40, with the announcement that the payout “is along with your guess down”. The payout for this bet if the chosen quantity wins is 392 chips, in the
case of a $1000 straight-up most, $40,000 wager, a payout of $392,000.
Roulette is an intriguing game that is performed throughout the world. The giant payoffs which may be potential for small wagers stimulate the interest of the skilled as well as the novice
participant, enjoying Roulette in Vegas. Thomas Bass, in his e-book The Eudaemonic Pie (1985) (published as The Newtonian Casino in Britain), has claimed to have the power to predict wheel efficiency
in real time.
That’s the one connection to the old Sands resort and casino, which as quickly as stood on the Venetian website. This isn’t any nostalgic old sport, it’s just a higher-edge model of an old one. We
know that the primary recreation of roulette was designed within the 18th century in France. Many historians consider that the person who began all of it was Blaise Pascal.
Unfortunately, bets such as Neighbors, Tiers, or Orphelins are usually not obtainable as a outcome of layout of the wheel. For inside bets at roulette tables, some casinos are allowed to make use of
separate roulette desk chips of various colors to discern gamers that are around the desk. Players also can select amongst three totally different column bets, which each supply a different variation
of 12 numbers.
Your Complete Information To The Only Zero Roulette Technique And Game
The good news, however, is that it’s still attainable to search out single zero roulette in some casinos. However, you should know the principles of your sport before you begin playing. When playing
any form of roulette, it’s a good idea to consider the variations within the wheel layouts.
Atlantic City Guidelines
The American sport was expanded within the playing dens throughout the model new territories the place improvised games had been settled up, whereas the French sport developed with type and ease in
Monte Carlo. We have an early description of the roulette game in its current type, found in a French novel named “La Roulette”, ou le Jour by Jaques Lablee. This novel describes a roulette wheel in
the Palais-royal in Paris in 1796.
Nevertheless, the roulette wheel structure just isn’t much totally different than the opposite variants in its core, with the main distinction being the added pocket. When a successful number and
colour is decided by the roulette wheel, the vendor will place a marker, also referred to as a dolly, on that quantity on the roulette desk structure. When the dolly is on the table, no players could
place bets, acquire bets or remove any bets from the table.
The listing of bets you also can make at this roulette variant is largely the identical as with the standard roulette variants. However, the percentages and payouts can range, which is why we’ll
listing all of the possible bets you might make in Triple Zero Roulette alongside their payout ratios and odds of winning. Players can’t take away any bets from the table, whereas the dolly is on the
winning quantity. Initially, the dealer will sweep away all shedding bets by hand and then pay the successful bets. When the supplier has finished making payouts, the marker is faraway from the board
and then players can acquire their winnings and make new bets. Now that you’ve learn this information, you must have a better understanding of single zero roulette and tips on how to play it.
Meanings Of Roulette And Wheel
Although most frequently named “call bets” technically these bets are extra precisely known as “announced bets”. The legal distinction between a “name bet” and an “announced guess” is that a “name
guess” is a guess referred to as by the player with out inserting any cash on the desk to cowl the worth of the wager. In many jurisdictions (most notably the United Kingdom) this is considered
playing on credit and is unlawful. An “introduced wager” is a wager known as by the participant for which they instantly place sufficient money to cover the amount of the guess on the desk, prior to
the result of the spin or hand in progress being identified.
Betting On The Final Variety Of A Non-consecutive Set
If the participant wins, they cross out numbers and proceed engaged on the smaller line. If the player loses, then they add their earlier wager to the end of the line and proceed to work on the
longer line. This is a way more versatile progression betting system and there’s a lot room for the participant to design their initial line to their very own enjoying choice. There are also several
strategies to determine the payout when a number adjacent to a chosen number is the winner, for example, participant bets “23 full complete” and quantity 26 is the winning number. When paying in
stations, the dealer counts the number of methods or stations that the profitable number hits the whole wager.
At some casinos the guess loses, and at others it might become double imprisoned. If a double-imprisoned bet guess gained on the subsequent spin, it might move up a degree, and become
single-imprisoned once more. If it lost, then if would become triple-imprisoned if the on line casino allowed it, otherwise it might lose. “Inside” bets involve choosing either the precise quantity
on which the ball will land, or a small group of numbers adjoining to each other on the layout. “Outside” bets, by contrast, enable players to pick a bigger group of numbers based on properties
corresponding to their color or parity (odd/even).
Both of these are types of American roulette, and one big downside that many players have with them is that the home edge is significantly higher than European roulette. Double Zero Roulette is a
variation of the basic on line casino game that stands out as a result of its unique wheel configuration. What sets it apart is the presence of not one however two zeros (0 and 00) on the roulette
wheel. This additional zero considerably influences the sport’s odds and home edge, making it a distinguishing feature of American-style roulette. In Atlantic City, any all even cash bets (red,
black,odd, even, 1-18, 19-36) observe a variation of the European half-back rule (see below).
Final 4, for instance, is a 4-chip bet and consists of one chip placed on each of the numbers ending in 4, that’s four, 14, 24, and 34. A number may be backed together with the two numbers on the
both side of it in a 5-chip bet. For instance, “0 and the neighbors” is a 5-chip wager with one piece straight-up on 3, 26, zero, 32, and 15.
If we use the identical formulation to calculate the chances for roulette with triple zeroes, we have to divide 1 by 39. Multiply that quantity by 100, and it turns out that the straight up odds for
this roulette variant are just 2.56%. Throughout the primary part of the 20th century, the only on line casino cities identified had been Monte Carlo with the normal single zero French and Las Vegas
with the American double-zero wheels.
The only obvious patterns are that red and black numbers alternate and that usually two odd numbers alternate with two even numbers. However the distribution of numbers was rigorously arranged in
order that the sum of the numbers for any given part of the wheel would be roughly equal to some other part of equal measurement. As with all different betting systems, the typical worth of this
technique is adverse. Based on the placement of the numbers on the format, the variety of chips required to “complete” a number can be determined. | {"url":"https://minsocnsw.org.au/2008740126622083925-2/","timestamp":"2024-11-14T07:09:34Z","content_type":"text/html","content_length":"74336","record_id":"<urn:uuid:9eababce-07fe-4427-bce8-4b3a0cab8438>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00035.warc.gz"} |
PPT - Supplementary Figure 2: PowerPoint Presentation, free download - ID:3735422
Télécharger la présentation
Supplementary Figure 2:
An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed
/ shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While
downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file
might be deleted by the publisher. | {"url":"https://fr.slideserve.com/cyrah/supplementary-figure-2","timestamp":"2024-11-13T02:06:04Z","content_type":"text/html","content_length":"83461","record_id":"<urn:uuid:7f9a070b-4557-4b10-8dc9-d092d2287045>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00551.warc.gz"} |
ML Aggarwal Class 8 Solutions for ICSE Maths
Understanding ICSE Mathematics Class 8 ML Aggarwal Solved Solutions
Get Latest Edition of ML Aggarwal Class 8 Solutions PDF Download on LearnInsta.com. It provides step by step solutions for ML Aggarwal Maths for Class 8 ICSE Solutions Pdf Download. You can download
the Understanding ICSE Mathematics Class 8 ML Aggarwal Solved Solutions with Free PDF download option, which contains chapter-wise solutions. APC Maths Class 8 Solutions ICSE all questions are solved
and explained by expert Mathematic teachers as per ICSE board guidelines.
APC Understanding ICSE Mathematics Class 8 ML Aggarwal Solutions 2019 Edition for 2020 Examinations
ML Aggarwal Class 8 Maths Chapter 1 Rational Numbers
ML Aggarwal Class 8 Maths Chapter 2 Exponents and Powers
ML Aggarwal Class 8 Maths Chapter 3 Squares and Square Roots
ML Aggarwal Class 8 Maths Chapter 4 Cubes and Cube Roots
ML Aggarwal Class 8 Maths Chapter 5 Playing with Numbers
ML Aggarwal Class 8 Maths Chapter 6 Operation on sets Venn Diagrams
ML Aggarwal Class 8 Maths Chapter 7 Percentage
ML Aggarwal Class 8 Maths Chapter 8 Simple and Compound Interest
ML Aggarwal Class 8 Maths Chapter 9 Direct and Inverse Variation
ML Aggarwal Class 8 Maths Chapter 10 Algebraic Expressions and Identities
ML Aggarwal Class 8 Maths Chapter 11 Factorisation
ML Aggarwal Class 8 Maths Chapter 12 Linear Equations and Inequalities in one Variable
ML Aggarwal Class 8 Maths Chapter 13 Understanding Quadrilaterals
ML Aggarwal Class 8 Maths Chapter 14 Constructions of Quadrilaterals
ML Aggarwal Class 8 Maths Chapter 15 Circle
ML Aggarwal Class 8 Maths Chapter 16 Symmetry Reflection and Rotation
ML Aggarwal Class 8 Maths Chapter 17 Visualising Solid Shapes
ML Aggarwal Class 8 Maths Chapter 18 Mensuration
ML Aggarwal Class 8 Maths Chapter 19 Data Handling
ML Aggarwal Class 8 Maths Model Question Papers
FAQs on ML Aggarwal Class 8 Solutions
1. How do I download the PDF of ML Aggarwal Solutions in Class 8?
All you have to do is tap on the direct links available on our LearnInsta.com page to access the Class 8 ML Aggarwal Solutions in PDF format. You can download them easily from here for free of cost.
2. Where can I find the solutions for ML Aggarwal Maths Solutions for Class 8?
You can find the Solutions for ML Aggarwal Maths for Class 8 from our page. View or download them as per your convenience and aid your preparation to score well.
3. What are the best sources for Class 8 Board Exam Preparation?
Aspirants preparing for their Class 8 Board Exams can make use of the quick and direct links available on our website LearnInsta.com regarding ML Aggarwal Solutions.
4. Is solving ML Aggarwal Solutions Chapterwise benefit you during your board exams?
Yes, it can be of huge benefit during board exams as you will have indepth knowledge of all the topics by solving Chapterwsie ML Aggarwal Solutions.
5. Where to download Class 8 Maths ML Aggarwal Solutions PDF?
Candidates can download the Class 8 Maths ML Aggarwal Solutions PDF from the direct links available on our page. We don’t charge any amount from you and they are absolutely free of cost. | {"url":"https://www.learninsta.com/ml-aggarwal-class-8-solutions-for-icse-maths/","timestamp":"2024-11-02T23:55:16Z","content_type":"text/html","content_length":"83709","record_id":"<urn:uuid:ef69a514-0049-42f4-a307-c957724156af>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00667.warc.gz"} |
Inches to Handbreadth Converter
β Switch toHandbreadth to Inches Converter
How to use this Inches to Handbreadth Converter π €
Follow these steps to convert given length from the units of Inches to the units of Handbreadth.
1. Enter the input Inches value in the text field.
2. The calculator converts the given Inches into Handbreadth in realtime β using the conversion formula, and displays under the Handbreadth label. You do not need to click any button. If the input
changes, Handbreadth value is re-calculated, just like that.
3. You may copy the resulting Handbreadth 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 Inches to Handbreadth?
The formula to convert given length from Inches to Handbreadth is:
Length[(Handbreadth)] = Length[(Inches)] / 2.9999999999999996
Substitute the given value of length in inches, i.e., Length[(Inches)] in the above formula and simplify the right-hand side value. The resulting value is the length in handbreadth, i.e., Length
Calculation will be done after you enter a valid input.
Consider that a premium 4K TV has a screen size of 55 inches.
Convert this screen size from inches to Handbreadth.
The length in inches is:
Length[(Inches)] = 55
The formula to convert length from inches to handbreadth is:
Length[(Handbreadth)] = Length[(Inches)] / 2.9999999999999996
Substitute given weight Length[(Inches)] = 55 in the above formula.
Length[(Handbreadth)] = 55 / 2.9999999999999996
Length[(Handbreadth)] = 18.3333
Final Answer:
Therefore, 55 in is equal to 18.3333 handbreadth.
The length is 18.3333 handbreadth, in handbreadth.
Consider that a luxury car's alloy wheels have a diameter of 20 inches.
Convert this diameter from inches to Handbreadth.
The length in inches is:
Length[(Inches)] = 20
The formula to convert length from inches to handbreadth is:
Length[(Handbreadth)] = Length[(Inches)] / 2.9999999999999996
Substitute given weight Length[(Inches)] = 20 in the above formula.
Length[(Handbreadth)] = 20 / 2.9999999999999996
Length[(Handbreadth)] = 6.6667
Final Answer:
Therefore, 20 in is equal to 6.6667 handbreadth.
The length is 6.6667 handbreadth, in handbreadth.
Inches to Handbreadth Conversion Table
The following table gives some of the most used conversions from Inches to Handbreadth.
Inches (in) Handbreadth (handbreadth)
0 in 0 handbreadth
1 in 0.3333 handbreadth
2 in 0.6667 handbreadth
3 in 1 handbreadth
4 in 1.3333 handbreadth
5 in 1.6667 handbreadth
6 in 2 handbreadth
7 in 2.3333 handbreadth
8 in 2.6667 handbreadth
9 in 3 handbreadth
10 in 3.3333 handbreadth
20 in 6.6667 handbreadth
50 in 16.6667 handbreadth
100 in 33.3333 handbreadth
1000 in 333.3333 handbreadth
10000 in 3333.3333 handbreadth
100000 in 33333.3333 handbreadth
An inch (symbol: in) is a unit of length used mainly in the United States, the United Kingdom, and Canada. One inch is equal to 2.54 centimeters.
The inch has origins in ancient times, originally based on the width of a human thumb. Its current definition, established in 1959, is exactly 2.54 centimeters.
Inches are commonly used to measure smaller lengths and distances, such as screen sizes and fabric lengths. Despite the widespread adoption of the metric system, the inch remains in use in these
A handbreadth is a historical unit of length used to measure small distances, typically based on the width of a hand. One handbreadth is approximately equivalent to 4 inches or about 0.1016 meters.
The handbreadth is defined as the width of a person's hand, measured from the edge of the thumb to the edge of the little finger when the hand is spread out. This unit was used for practical
measurements in various contexts, including textiles and construction.
Handbreadths were used in historical measurement systems for assessing lengths and dimensions where precise tools were not available. Although less common today, the unit provides historical context
for traditional measurement practices and everyday use in different cultures.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Inches to Handbreadth in Length?
The formula to convert Inches to Handbreadth in Length is:
Inches / 2.9999999999999996
2. Is this tool free or paid?
This Length conversion tool, which converts Inches to Handbreadth, is completely free to use.
3. How do I convert Length from Inches to Handbreadth?
To convert Length from Inches to Handbreadth, you can use the following formula:
Inches / 2.9999999999999996
For example, if you have a value in Inches, you substitute that value in place of Inches in the above formula, and solve the mathematical expression to get the equivalent value in Handbreadth. | {"url":"https://convertonline.org/unit/?convert=inches-handbreadths","timestamp":"2024-11-06T20:41:06Z","content_type":"text/html","content_length":"91056","record_id":"<urn:uuid:a5849830-0e04-4bf6-8eb6-b1730e2685a3>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00574.warc.gz"} |
[ANN] PermutationTests.jl, a package for multiple hypothesis testing
Hello, i am glad to annonce PermutationTests.jl, a fast, comprehensive and well-documented package for univariate and multiple comparisons hypothesis tests by data permutation.
In practice, this is useful when your data does not verify the assumptions of parametric tests and when you need to tests many hypotheses simultaneously, some or all of which may be correlated. This
happens, for example, in neuroimaging, finance and genome expression studies.
More in general, by means of permutation tests you can get a powerful test with minimal assumptions fully controlling the Type I error rate.
In my knowledge, there is no other julia package dedicated to multiple hypothesis testing yet, thus PermutationTests.jl fills a gap in the statistics julia ecosystem.
11 Likes
It seems nice, great work.
I have several suggestions:
• The output color after rTest, I have no black background, so it is not easy to see the remarked in color.
• It is not clear the target. What is different in comparisons with other similar packages? You mention them in the documentation, but a small comment will be nice.
• The list of tests is closed? I ask that because I use a lot non-parametric test, and I am not sure if they could considered under the package.
Anyway, thank you a lot for your package, and their documentation.
1 Like
here are some answers:
• Output color: Good point, i will make it a little darker
• The scope of the package: Here are some more explanations: The main advantage of using permutation tests is when you need to perform a large number of tests simultaneously and the hypotheses may
be correlated. In this case permutation tests offer the greatest power while rigorously controlling the family-wise error rate. For example, if you estimate brain activity in thousands of voxels,
such activity will be correlated for sure locally in the brain and maybe also non-locally. Using a Bonferroni-like of FDR-like correction (the standard procedures for controlling for multiple
comparisons), will result in less power. Actually, permutation tests are nice for many reasons. Here are a few more: you can get exact test; you can test whatever test-statistic (say, whatever
coefficient you may extract from your data), not only the usual test-statistic such as the Student-t, F, etc. for which the distribution under the null hypothesis is known; your test adapts
automatically to the form and degree of correlation among hypotheses; they are more robust to outliers; they make use of much less stringent assumptions as compared to parametric tests (for
example, Gaussianity of the data distribution,…); as a consequence of this last characteristic, you do not need to resort to rank-based statistics because your data violate an assuption of the
parametric test,…
• The list of tests: no, it is not closed. New tests can be coded in PermutationTests.jl, or you can create your own test just using the package.
Note that, in general, you do not need to use non-parametric tests if you use permutation tests; the univariate test will always be valid and exact (or approximatively exact) using the permutation
test that is equivalent to the parametric test you wish to use, it does not matter the distribution of the data. As a matter of fact, many non-parametric tests ARE permutation tests; they are
performed on ranked data so that the p-value can be obtained without actually listing the permutations. For instance, the popular Spearman correlation, Mann-Whitney and many others, are in fact
permutation tests!
Check the references i give in the documentation for more information.
I just released v0.1.9 of PermutationTests.jl, where it is simpler to create customized permutation tests. By the way, i updated the documentation with more examples, showing how to create
permutation tests for the Chatterjee correlation (2021) and for the distance correlation of Székeli et al (2007). I am reporting it here for people searching for julia code to perform these tests of
S. Chatterjee (2021) a new coefficient of correlation, Journal of the American Statistical Association, 116(536), 506-15.
G.J. Székely, M.L. Rizzo, N.K. Bakirov (2007). Measuring and testing dependence by correlation of distances Annals of Statistics, 35(6): 2769-2794.
4 Likes
Congratulations! Very appreciated for your efforts. Do you plan to merge this package’s content to GitHub - JuliaStats/HypothesisTests.jl: Hypothesis tests for Julia ?
1 Like
Thanks. Not sure about the merge. PermutationTests.jl totally concentrates on permutation tests and on the multiple comparisons problem, thus the overlap with HypothesisTests.jl is minimal. I think
the question should be addressed to the julia statistics community. | {"url":"https://discourse.julialang.org/t/ann-permutationtests-jl-a-package-for-multiple-hypothesis-testing/116862","timestamp":"2024-11-10T09:32:02Z","content_type":"text/html","content_length":"32092","record_id":"<urn:uuid:11e04088-cc13-4b8f-a074-84e3b6fa161c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00499.warc.gz"} |
Timeline of mathematical logic
A timeline of mathematical logic; see also history of logic.
19th century
20th century
• 1904 - Edward Vermilye Huntington develops the back-and-forth method to prove Cantor's result that countable dense linear orders (without endpoints) are isomorphic.
• 1908 – Ernst Zermelo axiomatizes set theory, thus avoiding Cantor's contradictions.
• 1915 - .
• 1918 - C. I. Lewis writes A Survey of Symbolic Logic, introducing the modal logic system later called S3.
• 1920 - explicitly.
• 1922 -
Löwenheim-Skolem theorem
without the axiom of choice.
• 1929 - Mojzesj Presburger introduces Presburger arithmetic and proving its decidability and completeness.
• 1928 - whether it is universally valid (in all models).
• 1930 - of first-order logic for countable languages.
• 1930 - Oskar Becker introduces the modal logic systems now called S4 and S5 as variations of Lewis's system.
• 1930 -
intuitionistic propositional calculus
• 1931 –
his incompleteness theorem
which shows that every axiomatic system for mathematics is either incomplete or inconsistent.
• 1932 - C. I. Lewis and C. H. Langford's Symbolic Logic contains descriptions of the modal logic systems S1-5.
• 1933 - Kurt Gödel develops two interpretations of intuitionistic logic in terms of a provability logic, which would become the standard axiomatization of S4.
• 1934 - Thoralf Skolem constructs a non-standard model of arithmetic.
• 1936 - Alonzo Church develops the lambda calculus. Alan Turing introduces the Turing machine model proves the existence of universal Turing machines, and uses these results to settle the
Entscheidungsproblem by proving it equivalent to (what is now called) the halting problem.
• 1936 - Anatoly Maltsev proves the full compactness theorem for first-order logic, and the "upwards" version of the Löwenheim–Skolem theorem.
• 1940 – Kurt Gödel shows that neither the continuum hypothesis nor the axiom of choice can be disproven from the standard axioms of set theory.
• 1943 - with effective calculable ones.
• 1944 -
closure algebras
• 1944 -
computably enumerable
degrees lying in between the degree of computable functions and the degree of the halting problem.
• 1947 -
word problem for semigroups
• 1948 - study closure algebras for S4 and intuitionistic logic.
See also | {"url":"https://findatwiki.com/Timeline_of_mathematical_logic","timestamp":"2024-11-08T17:18:22Z","content_type":"text/html","content_length":"63349","record_id":"<urn:uuid:5ae6cb59-d52e-4eb1-a793-62beb4079eb9>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00773.warc.gz"} |
Tata Consultancy Services Placement Paper For Freshers Part-2
11. Eesha has a wheat business. She purchases wheat from a local wholesaler of a particular cost per pound. The price of the wheat of her stores is $3 per kg. Her faulty spring balance reads 0.9 kg
for a KG. Also in the festival season, she gives a 10% discount on the wheat. She found that she made neither a profit nor a loss in the festival season. At what price did Eesha purchase the wheat
from the wholesaler ?
a. 3
b. 2.5
c. 2.43
d. 2.7
Explanation: Faulty spring balance reads 0.9 kg for a kg” means that she sells 1 kg for the price of 0.9 kgs, so she looses 10% of the price because of the faulty spring balance. She looses another
10% because of the discount.So, she actually sells 1 kg for $3×0.9×0.9=$2.43 and since at that price she made neither a profit nor a loss, then Eesha purchase the wheat from the wholesaler for $2.43.
12. Raj goes to market to buy oranges. If he can bargain and reduce the price per orange by Rs.2, he can buy 30 oranges instead of 20 oranges with the money he has. How much money does he have ?
a. Rs.100
b. Rs.50
c. Rs.150
d. Rs.120
Explanation: Let the money with Raj is M. So M20?M30=2. Check options. Option D satisfies.
13. A city in the US has a basketball league with three basketball teams, the Aziecs, the Braves and the Celtics. A sports writer notices that the tallest player of the Aziecs is shorter than the
shortest player of the Braves. The shortest of the Celtics is shorter than the shortest of the Aziecs, while the tallest of the Braves is shorter than the tallest of the Celtics. The tallest of the
Braves is taller than the tallest of the Aziecs.
Which of the following can be judged with certainty ?
X) Paul, a Brave is taller than David, an Aziec
Y) David, a Celtic, is shorter than Edward, an Aziec
a. Both X and Y
b. X only
c. Y only
d. Neither X nor Y
Sol: We solve this problem by taking numbers. Let the shortest of Braves is 4 feet. Then tallest of Aziecs is less than 4. So let it be 3 feet.
A -> 2 – 3
B -> 4 – 6
C -> 1 – 7
From the above we can safely conclude X is correct. but Y cannot be determined.
14. There are 3 classes having 20, 24 and 30 students respectively having average marks in an examination as 20,25 and 30 respectively. The three classes are represented by A, B and C and you have
the following information about the three classes.
a. In class A highest score is 22 and lowest score is 18
b. In class B highest score is 31 and lowest score is 23
c. In class C highest score is 33 and lowest score is 26.
If five students are transferred from A to B, what can be said about the average score of A; and what will happen to the average score of C in a transfer of 5 students from B to C ?
a. definite decrease in both cases
b. can’t be determined in both cases
c. definite increase in both cases
d. will remain constant in both cases
Class A average is 20. And their range is 18 to 22
Class B average is 25. And their range is 23 to 31
Class A average is 30. And their range is 26 to 33
If 5 students transferred from A to B, A’s average cannot be determined but B’s average comes down as the highest score of A is less than lowest score of B.
If 5 students transferred from B to C, C’s average cannot be determined the B’s range fo marks and C’s range of marks are overlapping.
15. The value of a scooter depreciates in such a way that its value of the end of each year is 3/4 of its value of the beginning of the same year. If the initial value of the scooter is Rs.40,000,
what is the value at the end of 3 years ?
a. Rs.13435
b. Rs.23125
c. Rs.19000
d. Rs.16875
Explanation: 40,000(34)3=16875
16. Rajiv can do a piece of work in 10 days , Venky in 12 days and Ravi in 15 days. They all start the work together, but Rajiv leaves after 2 days and Venky leaves 3 days before the work is
completed. In how many days is the work completed ?
a. 5
b. 6
c. 9
d. 7
Explanation: Let the work be 60 units. If venky worked for 3 days, and the remaining days of work be x days, total days to complete the work be x + 3 days.
Now Capacities of Rajiv is 60/10 = 6, Venky is 5, Ravi is 4.
(6 + 5 + 4) 2 + (5 + 4) (x – 3) + 5 x 3 = 60.
Solving we get x = 4. So total days to complete the work is 7 days.
17. A man has a job, which requires him to work 8 straight days and rest on the ninth day. If he started work on Monday, find the day of the week on which he gets his 12th rest day.
a. Thursday
b. Wednesday
c. Tuesday
d. Friday
He works for 8 days and takes rest on the 9th day. So On the 12th rest day, there are 9 x 12 = 108 days passed. Number of odd days = (108 – 1) / 7 = 107 / 7 = 2. So the 12th rest day is wednesday.
Leave a Comment Cancel Reply
0 comment 0 FacebookTwitterPinterestEmail
You may also like | {"url":"https://theblogreaders.com/tata-consultancy-services-placement-paper-for-freshers-part-2/","timestamp":"2024-11-02T11:58:16Z","content_type":"application/xhtml+xml","content_length":"134408","record_id":"<urn:uuid:4a85e930-f59d-4fa8-b64d-a860f8cdf399>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00816.warc.gz"} |
Use The Paired T-interval Procedure To Obtain The Required - Stoicacademia.com
1. Use the paired t–interval procedure to obtain the required confidence interval for the mean difference. Assume that the conditions and assumptions for inference are satisfied.
Ten families are randomly selected and their daily water usage (in gallons) before and after viewing a conservation video. Construct a 90% confidence interval for the mean of the difference of the
“before” minus the “after” times if d (after-before) = -4.8 and Sd 5.2451
Before 33 33 38 33 35 35 40 40 40 31
After 34 28 25 28 35 33 31 28 35 33
1. (1.5,8.1)
2. (2.5,7.1)
3. (1.8,7.8)
4. (3.8,5.8)
5. (2.1,7.5)
2.From the sample statistics, find the value of the pooled estimate Pcap used.
n[1 ]= 36 n[2] = 418
x[1] = 7 x[2] = 132
Pcap =
3. Provide an appropriate response.
Do motivation levels between mid-level and upper-level managers differ? A randomly selected group of each were administered a survey, which measures motivation for upward mobility. The scores are
summarized below:
│ │Upper -Level │Mid-level│
│Sample size │73 │109 │
│Mean score │77.4 │79.71 │
│Standard Deviation │10.6 │6.43 │
Assuming equal population standard deviations, calculate the test statistic for determining whether the mean scores differ for upper-level and mid-level managers.
1. -1.89c.-63.69 e. none of these
2. -0.29 d. -1.74
4. Use the paired t–interval procedure to obtain the required confidence interval for the mean difference. Assume that the conditions and assumptions for inference are satisfied.
A test for abstract reasoning is given to a random sample of students before and after they complete a formal course in logic. The results are given below. Construct a 95% confidence interval for the
difference in scores if d = after-before, Xbar[d] = 3.7 and s[d] = 4.945.
After 74 83 75 88 84 63 93 84 91 77
Before 73 77 70 77 74 67 95 83 84 75
1. (-4.4, 11.8)
2. (0.2, 7.2)
3. (1.2,5.7)
4. (1.0,6.4)
5. (0.8,6.6)
5. Find the appropriate test statistic/p–value.
Do motivation levels between mid-level and upper-level managers differ? A randomly selected group of each were administered a survey, which measures motivation for upward mobility. The scores are
summarized below:
│ │Upper -Level │Mid-level│
│Sample size │73 │109 │
│Mean score │77.4 │79.71 │
│Standard Deviation │10.6 │6.43 │
Calculate the appropriate test statistic and give your conclusion for testing
Ho: Uj=Ua
Ha: Uj<Ua
using a significance level of α = 0.05. Assume df = 100.
1. t= -1.89; there is insufficient evidence to conclude that the mean scores differ for mid-level and upper-level managers.
2. t=-1.89; reject the H0 and conclude that the mean scores differ for mid-level and upper-level managers.
3. t=-1.31; there is insufficient evidence to conclude that the mean scores differ for mid-level and upper-level managers.
4. t=-1.74; reject H0 and conclude that the mean scores differ for mid-level and upper-level managers.
5. t=-1.74; there is insufficient evidence to conclude that the mean scores differ for mid-level and upper-level managers.
6. Select the most appropriate answer.
For 12 pairs of females, the reported means are 24.8 on the well-being measure for the children of alcoholics and 29.0 for the control group. A t test statistic of 2.67 for the test comparing the
means was obtained. Assuming that this is the result of a dependent-samples analysis testing for a difference between the group means, report the P-value.
1. 0.01 < P-value < 0.02
2. 0.005 < P-value < 0.01
3. 0.0076
4. 0.0152
5. 0.02 < P-value <0.05
7. A test for abstract reasoning is given to a random sample of students before and after they complete a formal course in logic. Calculate the test statistic for testing that the course improves the
test scores assuming that d=after-before, Xbar[d] = -3.7 and s[d] = 4.945, n = 10 and α = 0.05. State your conclusion in terms of the problem
1. t= 0.75; fail to reject the null hypothesis and conclude that the average scores on the abstract reasoning test are the same before and after course in logic
2. t= 2.37; reject the null hypothesis and conclude that the course does improve the average score on the abstract reasoning test.
3. t= 2.37; fail to reject the null hypothesis and conclude that the average scores on the abstract reasoning test are the same before and after the course in logic.
4. t= 0.75; fail to reject the null hypothesis. There is no evidence to conclude that the course improves the average on the abstract reasoning test.
5. t=2.37; fail to reject the null hypothesis. There is no enough evidence to conclude that the course improves the average score on the abstract reasoning test.
8.Provide an appropriate response.
You are interested in determining whether there is a difference in the mean calorie content of a serving of fries versus a serving of onion rings at fast food restaurants. Based on a sample of
seventeen french fry choices from fast food restaurants, the mean caloric content is 543.35 with a standard deviation of 112.18. A sample of eight onion ring choices from fast food restaurants has a
mean caloric content of 526.25 with a standard deviation of 142.32. Assuming both populations are normal with equal standard deviations, what is the test statistic for testing whether the mean
caloric content is the same for french fry orders as for onion rings at fast food restaurants?
1. 0.14
2. 3.46
3. 0.33
4. 0.30
5. None of these
9.Provide an appropriate response.
Do motivation levels between mid-level and upper-level managers differ? A randomly selected group of each were administered a survey, which measures motivation for upward mobility. The scores are
summarized below:
│ │Upper -Level │Mid-level│
│Sample size │73 │109 │
│Mean score │77.4 │79.71 │
│Standard Deviation │10.6 │6.43 │
Assuming equal population standard deviations, find the P-value for testing that the mean scores differ for upper-level and mid-level managers. Interpret using a 5% significance level
a. P-value = 0.03; since the P-value < 0.05, we reject the null hypothesis
b. P-value = 0.08; since the P-value > 0.05. we fail to reject the null hypothesis
c. P-value =0.04; since the P-value < 0.05. we reject the null hypothesis
d. P-Value =0.06; since the P-value > 0.05, we fail to reject the null hypothesis.
10..Construct the indicated confidence interval for the difference between the two population means. Assume that the assumptions and conditions for inference have been met.
The table below contains information pertaining to the gasoline mileage for random samples of trucks of two different types. Find a 95% confidence interval for the difference in the means μ[X] – μ
│ │Brand X│Brand Y│
│Number of trucks │50 │50 │
│Mean mileage │20.1 │24.3 │
│Standard deviation │2.3 │1.8 │
1. (-5.02, -3.38)
2. (20.1,24.3)
3. (3.7, 4.7)
4. (-4.7, -3.7)
5. (3.38, 5.02)
11.Provide an appropriate response.
The weights before and after 9 randomly selected participants followed a particular diet were recorded. The mean of the before weights was 170.4444, the mean of the weights following the diet was
160.5556 and the standard error of the differences was 3.1333. Calculate the appropriate test statistic for testing that the average weight was lower following the diet and state your conclusion
using a significance level of 0.01.
1. t=3.16; reject the null hypothesis and conclude that the diet is effective for weight loss
2. t= 9.47; fail to reject the null hypothesis; there is not enough information to conclude that the diet is effective for weight loss
3. t=3.16; accept the null hypothesis and conclude that there is no difference in the average weight before and after the diet
4. t=3.16; fail to reject the null hypothesis; there is not enough information to conclude that the diet is effective for weight loss.
5. t= 9.47; reject the null hypothesis and conclude that the diet is effective fir weight loss.
12.Interpret the given confidence interval.
A researcher wishes to determine whether people with high blood pressure can reduce their blood pressure by following a particular diet. Subjects were randomly assigned to either a treatment group or
a control group. The mean blood pressure was determined for each group, and a 95% confidence interval for the difference in the means for the treatment group versus the control group, Ut-Uc was found
to be (-21, -6). ( t –stands for treatment group and C stands for control group)
1. We are 95% confident that the average blood pressure of those who follow the diet is between 6 and 21 points higher than the average for those who do not follow the diet.
2. The probability that the mean blood pressure for those on the diet is lower than for those not on the diet is 0.95.
3. Since all of the values in the confidence interval are less than 0, we are unable to conclude that there is difference in blood pressure for those who follow the diet and those who do not.
4. We are 95% confident that the average blood pressure of those who follow the diet is between 6 and 21 points lower than the average for those who do not follow the diet.
5. The probability that the mean blood pressure for those on the diet is higher than for those not on the diet is 0.95.
13.Interpret the given confidence interval.
A researcher was interested in comparing the salaries of female and male employees of a particular company. Independent random samples of female employees (sample 1) and male employees (sample 2)
were taken to calculate the mean salary, in dollars per week, for each group. A 90% confidence interval for the difference, U1-U2 between the mean weekly salary of all female employees and the mean
weekly salary of all male employees was determined to be (-$110,$10)
1. 90% of the time females at this company make less than males
2. Since 0 is contained in the interval, the probability that the male employees at this company earn the same as females at this company is 0.9
3. Based on these data, we are 90% confident that the male employees at this company average between $110 less and $10 more per week than the female employees.
4. The probability that a randomly selected female employee at this company makes between $110 less and $10 more per week than a randomly selected male employee is 0.9.
5. Based on these data, we are 90% confident that the female employees at this company average between $ 110 less and $10 more per week than the male employees.
14. In a positive association between two variables
1. A decrease in the value of one variable is associated with a decrease in the value of a second variable.
2. An increase in the value of one variable is associated with a decrease in the value of a second variable.
3. An increase in the value of one variable is associated with an increase in the value of a second variable.
4. Both A and C
15.Comparison of means can be used when
1. A researcher wants to compare responses to an ordinal variable by the categories of a nominal variable.
2. A researcher wants to compare responses to a numerical variable by the categories of a nominal or ordinal variable
3. A researcher wants to compare responses to a nominal variable by the categories of an ordinal variable
4. A researcher wants to compare responses to a numerical variable by categories of an ordinal variable
16. The independent samples t test assesses differences in means between
1. An independent nominal variable and dependent ordinal variable
2. A dependent numerical variable and an independent categorical variable
3. A dependent nominal variable and an independent categorical variable
4. A dependent categorical variable and an independent numerical variable
17. In a negative association between two variables,
1. A decrease in the value of one variable is associated with a decrease in the value of a second variable
2. An increase in the value of one variable is associated with a decrease in the value of a second variable
3. A decrease in the value of one variable is associated with an increase in the value of a second variable
4. Both B and C
18.For the independent samples t test, if equality of variances cannot be assumed then you
1. Have to use a formula for the t statistic that is different from the one you would use if equality of variances can be assumed
2. Have to use a formula for degrees of freedom that is different from the one you would use if equality of variances can be assumed
3. Cannot reject the null hypothesis , no matter the value of t
4. Both A and B
19. Provide an appropriate response.
A 95% confidence interval for the difference in means for a collection of paired sample data is (0, 3.4) Based on the same sample, a traditional significance test fails to support the claim of μd >
0. What can you conclude about the significance level α (α = 1 – .95) of the hypothesis test?
α > 0.05
α < 0.05
α = 0.01
α = 0.05
α = 0.95
20.From the sample statistics, find the value of P1cap-P2cap, the point estimate of the difference of proportions. Unless otherwise indicated, round to the nearest thousandth when necessary.
n1 = 100 n2 = 100
x1 = 34 x2 = 30
none of these
20. The statistic that answers the question, how likely is it that the difference between the means for two categories of a variable that we observe in a sample is merely a chance occurrence, is the
independent samples t-test
t statistic for Pearson’s r
one-sample t test
one-way analysis of variance
21.Interpret the given confidence interval.
A high school coach uses a new technique in training middle distance runners. He records the times for 4 different athletes to run 800 meters before and after this training. A 90% confidence interval
for the difference of the means before and after the training, μB – μA, was determined to be(2.7, 4.2)
The probability that the average time for the 800-meter run for middle distance runners at this high school is between 2.7 and 4.2 seconds shorter after the training is 0.9.
We are 90% confident that a randomly selected middle distance runner at this high school will have a time for the 800-meter run that is between 2.7 and 4.2 seconds shorter after the training
than before the training.
Based on this sample, we are 90% confident that the average time for the 800-meter run for middle distance runners at this high school is between 2.7 and 4.2 seconds longer after the new
Based on this sample, we are 90% confident that the average time for the 800-meter run for middle distance runners at this high school is between 2.7 and 4.2 seconds shorter after the new
We know that 90% of the middle distance runners shortened their times between 2.7 and 4.2 seconds after the training.
22. Select the most appropriate answer.
The central limit theorem predicts that the sampling distribution of X1bar – X2bar is approximately normal
when the total number sampled is greater than or equal to 30.
when both of the sample sizes are greater than or equal to 30.
when either one of the sample sizes is greater than or equal to 30.
when at least one of the sample sizes is greater than or equal to 30.
regardless of both of the sample sizes.
From the sample statistics, find the value of P1cap-P2cap , the point estimate of the difference of proportions. Unless otherwise indicated, round to the nearest thousandth when necessary.
A survey asked respondents whether marijuana should be made legal. A 95% confidence interval for PA-PB, is given by (0.08,0.14) where PA is the proportion of respondents who answered “legal” in state
A and PB is the proportion of respondents who responded “legal” in state B. Based on the 95% confidence interval, what can we conclude about the percentage of respondents who favor legalization in
state B versus state A?
Since all of the values in the confidence interval are less than 1, we can conclude that there is a significant difference between the percentage in favor of legalization in state B and the
percentage in favor of legalization in state A.
Since all of the values in the confidence interval are less than 1, we are unable to conclude that there is a significant difference between the percentage in favor of legalization in state B
and the percentage in favor of legalization in state A.
Since all of the values in the confidence interval are greater than 0, we can conclude that the percentage in favor of legalization was greater in state A than it was in state B.
Since all of the values in the confidence interval are greater than 0, we can conclude that the percentage in favor of legalization was greater in state B than it was in state A. | {"url":"https://stoicacademia.com/2021/03/16/use-the-paired-t-interval-procedure-to-obtain-the-required/","timestamp":"2024-11-07T10:29:27Z","content_type":"text/html","content_length":"81550","record_id":"<urn:uuid:a6e0c989-7a51-4164-8df6-9fd6ea2326bb>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00242.warc.gz"} |
119+ Astonishing Maths Project Ideas for Exhibition 2024
119+ Astonishing Maths Project Ideas for Exhibition
Explore fun and creative maths project ideas for exhibition. Engage and inspire with exciting ways to showcase numbers, shapes, and patterns!
So, you might think math is just about numbers and boring formulas, but it’s actually packed with cool and creative ideas! A math exhibition is a great chance to show how exciting math can be—think
fascinating shapes, mind-bending patterns, and surprising discoveries.
In this blog, we’ll check out some awesome project ideas that make math come alive. These projects will get you thinking in new ways and maybe even make you fall in love with math.
Ready to dive in? Let’s explore the fun side of math together!
Importance of Maths projects for students
Math projects offer students:
Aspect Description
Real-World Use Show how math applies to daily life.
Critical Thinking Build problem-solving skills.
Creativity Inspire creative solutions.
Teamwork Enhance collaboration and communication.
Presentation Skills Improve how to share findings.
Confidence Boost self-esteem with project success.
Engagement Make math more interesting.
These projects help students appreciate math’s relevance and deepen their understanding.
Benefits of participating in Maths exhibitions
Participating in math exhibitions offers students many benefits:
Aspect Description
Enhanced Problem-Solving Boosts critical thinking and analytical skills.
Improved Communication Learn to present math concepts clearly.
Increased Creativity Encourages innovative problem-solving.
Boosted Confidence Success in projects raises self-esteem.
Teamwork Promotes collaboration and cooperation.
Real-World Use Shows practical applications of math.
Recognition Chance to win awards and accolades.
Career Exploration Provides insight into math-related careers.
Lifelong Learning Fosters a love for math and ongoing education.
Portfolio Building Creates a showcase of math skills for future opportunities.
Math exhibitions help students appreciate the subject and develop key skills for future success.
Understanding Maths Projects
A great math project goes beyond calculations and explores the beauty and application of math.
What Makes a Good Math Project?
• Understanding: Clearly show math concepts.
• Problem-Solving: Use critical thinking and logic.
• Creativity: Find innovative solutions.
• Practical Application: Link math to real-life scenarios.
• Visual Appeal: Make presentations or models engaging.
• Exploration: Encourage curiosity and further inquiry.
Key Elements
• Clear Question: Focus on a specific topic.
• Data Collection: Gather and analyze information.
• Mathematical Modeling: Apply math to real-world situations.
• Problem-Solving: Tackle challenges with math knowledge.
• Communication: Present findings effectively.
These tips help create engaging and effective math projects.
Most Popular Maths Project Ideas for Exhibition
Check out most popular maths project ideas for exhibition:
Geometry and Measurement
3D Models
• Materials: Cardboard, paper, wood.
• Shapes: Cubes, pyramids, spheres.
• Applications: Architectural models, prototypes.
• Basics: Self-similarity and iteration.
• Creation: Draw or use software for Mandelbrot or Sierpinski.
• Applications: Nature (e.g., snowflakes), art.
Golden Ratio
• Definition: Ratio of 1.618.
• Art/Architecture: Use in famous works.
• Nature: Examples like shells and flowers.
Map Making
• Tools: Mapping software or drawing.
• Scales: Practice different scales and projections.
• Features: Include landmarks, roads.
Algebra and Number Theory
Number Patterns
• Sequences: Arithmetic, geometric.
• Properties: Convergence, patterns.
• Applications: Population growth, investments.
• Techniques: Caesar ciphers, substitution.
• Encryption: Encode and decode messages.
• Applications: Digital security.
Mathematical Puzzles
• Types: Sudoku, logic riddles.
• Difficulty: Varying levels.
• Solutions: Provide answers and explanations.
Financial Mathematics
• Budgeting: Manage income and expenses.
• Investment: Simulate investments with growth rates.
• Analysis: Review financial outcomes.
Statistics and Probability
Data Visualization
• Types: Bar charts, histograms, pie charts.
• Tools: Excel, Google Sheets.
• Analysis: Extract trends and insights.
Probability Experiments
• Experiments: Dice rolls, coin flips.
• Theory: Compare results with theory.
• Applications: Weather forecasting.
Surveys and Data Analysis
• Design: Create unbiased surveys.
• Collection: Gather responses.
• Analysis: Use statistics to interpret data.
Game Theory
• Models: Prisoner’s Dilemma, Nash Equilibrium.
• Strategies: Develop and analyze.
• Applications: Business, sports.
Calculus and Its Applications
Real-world Calculus
• Physics: Model motion and forces.
• Engineering: Design systems like bridges.
• Economics: Optimize production and pricing.
Calculus-based Models
• Simulations: Model phenomena like traffic flow.
• Models: Develop differential equations.
• Analysis: Make predictions based on results.
Interactive Calculus Demonstrations
• Software: Use Desmos, GeoGebra.
• Concepts: Visualize derivatives, integrals.
• Activities: Manipulate variables, observe.
Interdisciplinary Projects
Math and Art
• Tessellations: Create repeating patterns.
• Geometric Patterns: Draw designs like spirals.
• Integration: Explore math in art.
Math and Music
• Rhythm: Study rhythm patterns.
• Harmony: Analyze note relationships.
• Composition: Create math-based music.
Math and Sports
• Statistics: Analyze performance data.
• Probability: Assess game event likelihood.
• Kinematics: Study movements and optimize.
Maths Project Ideas for Exhibition
Check out maths project ideas for exhibition:-
1. Origami: Create geometric shapes with paper folding.
2. Geometric Art: Design patterns using geometric shapes.
3. 3D Models: Build models of geometric solids.
4. Fractals: Create and explore fractal patterns.
5. Tessellations: Design repeating patterns.
6. Golden Ratio: Show its appearance in art and nature.
7. Architecture: Analyze geometric shapes in famous buildings.
8. Surface Area/Volume: Calculate for different 3D objects.
9. Transformations: Demonstrate translations, rotations, and reflections.
10. Symmetry: Explore symmetry in nature.
1. Polynomial Graphs: Visualize polynomial functions.
2. Algebraic Puzzles: Create and solve puzzles.
3. Cryptography: Use algebra for simple ciphers.
4. Real-World Uses: Apply algebra to finance or engineering.
5. Graphing Equations: Create graphs for different equations.
6. Systems of Equations: Solve and graph systems.
7. Algebraic Structures: Explore groups and rings.
8. Function Machines: Demonstrate function composition.
9. Algebraic Fractions: Simplify and solve problems.
10. Inequality Graphs: Graph and analyze inequalities.
1. Data Visualization: Create charts and graphs.
2. Survey Analysis: Analyze survey data.
3. Probability Games: Develop games to show probability.
4. Statistical Trends: Present trends from data sets.
5. Descriptive Statistics: Calculate mean, median, mode.
6. Correlation: Show relationships between variables.
7. Data Collection: Gather and interpret data.
8. Sampling Methods: Demonstrate sampling techniques.
9. Probability Distributions: Explore different distributions.
10. Inferential Statistics: Perform hypothesis tests.
Number Theory
1. Prime Numbers: Explore prime number patterns.
2. Magic Squares: Create and solve magic squares.
3. Number Sequences: Study sequences like Fibonacci.
4. Modular Arithmetic: Show basic modular arithmetic.
5. Factorization: Explore prime factorization.
6. Divisibility Rules: Demonstrate rules for divisibility.
7. Greatest Common Divisor: Find GCD of numbers.
8. Perfect Numbers: Investigate examples of perfect numbers.
9. Patterns in Nature: Show number patterns in nature.
10. Mathematical Games: Create games based on number theory.
1. Rate of Change: Use graphs to show rates of change.
2. Optimization: Solve real-world optimization problems.
3. Integration: Visualize area under curves.
4. Differential Equations: Model simple differential equations.
5. Derivatives: Show how derivatives describe motion.
6. Applications: Explore calculus in physics or economics.
7. Infinite Series: Study series like geometric series.
8. Parametric Equations: Graph parametric equations.
9. Calculus in Engineering: Show calculus applications in engineering.
10. Visualizing Limits: Create demonstrations of limits.
Probability and Statistics
1. Chance Experiments: Conduct experiments with coins or dice.
2. Simulations: Model probability scenarios.
3. Data Collection: Gather and analyze data.
4. Probability Games: Develop games to teach probability.
5. Statistical Tools: Show mean, median, and mode.
6. Probability Distributions: Explore distributions like normal and binomial.
7. Hypothesis Testing: Perform basic hypothesis tests.
8. Bayesian Probability: Introduce Bayesian concepts.
9. Descriptive vs. Inferential: Show the difference between descriptive and inferential statistics.
10. Monte Carlo Simulations: Use simulations for problem-solving.
Applied Mathematics
1. Engineering Models: Show math in engineering projects.
2. Financial Math: Demonstrate interest, loans, and investments.
3. Cryptocurrency: Explain the math behind cryptocurrencies.
4. Algorithm Design: Present algorithms for data processing.
5. Environmental Modeling: Model environmental issues.
6. Epidemiology: Simulate disease spread.
7. Population Growth: Predict population changes.
8. Optimization: Show optimization in business.
9. Robotics: Explore math in robotics design.
10. Architectural Design: Use math in building design.
Mathematical History
1. Famous Mathematicians: Present key figures in math history.
2. Historical Problems: Solve classic problems.
3. Ancient Math: Explore math from ancient civilizations.
4. Math in Art: Show math’s influence on art.
5. Development of Calculus: Trace calculus history.
6. Ancient Texts: Examine math in old manuscripts.
7. Mathematical Instruments: Display historical math tools.
8. Number Systems: Study the evolution of number systems.
9. Math and Astronomy: Explore ancient math in astronomy.
10. Medieval Math: Investigate math in the medieval period.
Mathematical Modeling
1. Population Models: Model population changes.
2. Epidemiology: Simulate disease spread.
3. Environmental Impact: Model environmental effects.
4. Economic Models: Predict economic trends.
5. Traffic Flow: Analyze traffic patterns.
6. Weather Prediction: Use math for weather forecasting.
7. Resource Management: Model resource use.
8. Financial Forecasting: Predict financial markets.
9. Structural Engineering: Model structures’ stability.
10. Sports Statistics: Analyze sports data for predictions.
These ideas offer a variety of engaging and educational projects for math exhibitions.
Project Presentation and Display
Check out project presentation and display:-
Visual Impact
• Clear Visuals: Use simple graphs and charts.
• Engaging Displays: Add interactive models.
• Professional Design: Keep your display neat and appealing.
Effective Communication
• Simple Language: Explain concepts clearly.
• Storytelling: Make your project interesting.
• Practice: Rehearse to build confidence.
• Prepare for Questions: Have clear answers ready.
Audience Engagement
• Interactive Elements: Use quizzes or games.
• Live Demonstrations: Show your project in action.
• Hands-On Activities: Let visitors interact.
• Team Participation: Have team members available.
By focusing on these tips, your math project presentation will be both impactful and memorable.
Tips for Success
To make your math project shine, remember these tips:
Tip Description
Pick a Topic You Love Choose something you’re excited about.
Start Early Give yourself plenty of time.
Work with Others Team up for diverse ideas.
Ask for Help Get advice from teachers or experts.
Manage Your Time Balance project work and other tasks.
Make it Visual Create a clear, attractive display.
Practice Speaking Rehearse your presentation.
Accept Feedback Use suggestions to improve.
These steps will help you create a standout math project.
Challenges With Math Project Ideas for Exhibition
Math projects can be challenging:
Challenge Description
Complex Topics Some concepts are hard to grasp and explain.
Time Constraints Balancing with other studies can be tough.
Material Access Resources may be limited.
Confidence Issues Presenting can feel intimidating.
Originality Finding a unique idea can be difficult.
Visuals Explaining concepts visually can be tricky.
Overcome these by planning well, managing time, and seeking help when needed.
How to write maths project work?
Here’s a simplified step-by-step guide for writing a math project:
Choose a Topic
• Pick a math concept you find interesting.
• Ensure it fits your curriculum and project needs.
• Consider how broad or detailed the topic should be.
Conduct Research
• Use textbooks, journals, and online sources.
• Look into real-world uses of the topic.
• Review existing studies and findings.
Define Objectives
• State the project’s purpose.
• Set clear research questions or hypotheses.
• Outline what your project will cover and any limitations.
Collect and Analyze Data
• Gather relevant data.
• Analyze it using appropriate methods.
• Display data with graphs, charts, or tables.
Write the Report
• Introduction: Explain the background, goals, and questions.
• Literature Review: Summarize existing knowledge.
• Methodology: Describe how you conducted your research.
• Results: Present your findings.
• Discussion: Interpret the results and conclusions.
• Conclusion: Sum up key findings and implications.
• References: List your sources.
Create Visual Aids
• Design clear graphs, charts, and diagrams.
• Use images or models to support your project.
Prepare the Presentation
• Make a concise and clear presentation.
• Use visuals to back up your points.
• Practice your delivery to ensure confidence.
Follow your school’s guidelines for format and submission.
What is the concept of math project?
A math project is an in-depth look at a mathematical idea or application. It involves researching, analyzing, and presenting findings creatively. Unlike routine problems, math projects encourage
students to:
Skill Description
Think Critically Analyze and interpret information.
Solve Problems Identify issues and find solutions.
Communicate Clearly Present findings effectively.
Collaborate Work with others towards shared goals.
Be Creative Explore various approaches to math problems.
Through these projects, students gain a deeper grasp of math concepts and their real-world uses.
Math projects make exploring the beauty and usefulness of mathematics both dynamic and engaging. They turn abstract ideas into real, hands-on experiences, helping students build critical thinking,
problem-solving, and communication skills. Math exhibitions are a great way to showcase these abilities and inspire others.
So, dive into different project ideas, create an impressive display, and let your passion for math shine. The best projects mix creativity, precision, and genuine enthusiasm for the subject.
Leave a Comment | {"url":"https://goodprojectideas.com/maths-project-ideas-for-exhibition/","timestamp":"2024-11-06T18:17:04Z","content_type":"text/html","content_length":"158773","record_id":"<urn:uuid:c7b023f0-3cbb-4a26-89f9-35fa3203f53b>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00289.warc.gz"} |
Curb Type Gutter Equations Formulas Design Calculator Manning's Coefficient
Online Web Apps, Rich Internet Application, Technical Tools, Specifications, How to Guides, Training, Applications, Examples, Tutorials, Reviews, Answers, Test Review Resources, Analysis, Homework
Solutions, Worksheets, Help, Data and Information for Engineers, Technicians, Teachers, Tutors, Researchers, K-12 Education, College and High School Students, Science Fair Projects and Scientists
By Jimmy Raymond
Contact: aj@ajdesigner.com
Privacy Policy, Disclaimer and Terms
Copyright 2002-2015 | {"url":"https://www.ajdesigner.com/phpgutter/curb_gutter_equation_manning_coefficient.php","timestamp":"2024-11-04T08:03:02Z","content_type":"text/html","content_length":"29961","record_id":"<urn:uuid:8599a69a-0798-4013-b6d0-d79c6d3a9a40>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00523.warc.gz"} |
Previous week Up Next week
Here is the latest Caml Weekly News, for the week of November 06 to 13, 2012.
Google+ page
Archive: https://sympa.inria.fr/sympa/arc/caml-list/2012-11/msg00064.html
Deep in this thread, Paolo Donadeo announced:
For what it's worth, Christophe's logo has been stolen (by me) and has
become the icon of the (official?) Google+ page of the language :-)
<ad type="shameless">
OASIS, package managers and misc. poll
Archive: https://sympa.inria.fr/sympa/arc/caml-list/2012-11/msg00071.html
gildor478 announced:
If you have trouble viewing or submitting this form, you can fill it out online:
One day, OASIS-DB will be able to automatically create package and
repositories. We need to know what OASIS user wish to focus our effort
on a few package manager.
Preferred package manager Choose the package manager oasis-db should
native Debian packages
native RPM packages (Fedora, Centos)
non, OASIS should provide a package manager itself
Preferred build system OASIS support by design ocamlbuild, but there
are some other build system around. Which one do you think are worth
to be supported by OASIS.
custom scripts
native Makefile
parameterized classes, modules & polymorphic variants
Archive: https://sympa.inria.fr/sympa/arc/caml-list/2012-11/msg00070.html
Didier Cassirame asked and Jacques Garrigue replied:
> I have been trying recently to combine classes, modules and variants
> in the following fashion:
> module A1 = struct
> class ['a] t = object
> constraint 'a = [>`a]
> method m : 'a -> string = function `a -> "a" | `a1 -> "a1" | _ -> "_"
> end
> end;;
> […]
> module type A = sig
> class ['a] t : object
> constraint 'a = [>`a]
> method m : 'a -> string
> end
> end;;
> type m = (module A);;
> let l: m list = [ (module A1); (module A2); (module A3)];;
> --------------------------------
> Unfortunately the list typecheck fails. However, making a list of
> class instances from A1.t, A2.t, A3.t succeed, with the type:
> [> `a | `a1 | `a2 | `a3 ] ct list
> ct being defined as equal to A.t.
> I thought that perhaps I should parameterize the type m from the type
> parameter 'a of A.t to solve my problem, but I am not sure of the
> syntax, or if it's the problem. Does anyone have an idea?
Actually the parameterization would not help here, since you want to put them
all in the same list.
The idea of using first-class modules is to be explicit about types, so using
an explicit type definition for a solves the problem.
Jacques Garrigue
module A1 = struct
type a = private [> `a | `a1]
class t = object
method m : a -> string = function `a -> "a" | `a1 -> "a1" | _ -> "_"
module A2 = struct
type a = private [> `a | `a2]
class t = object
method m : a -> string = function `a -> "a" | `a2 -> "a2" | _ -> "_"
module A3 = struct
type a = private [> `a | `a3]
class t = object
method m : a -> string = function `a -> "a" | `a3 -> "a3" | _ -> "_"
module type A = sig
type a = private [> `a]
class t : object
method m : a -> string
type m = (module A);;
let l: m list = [ (module A1); (module A2); (module A3)];;
RTT: Run-time types for OCaml
Archive: https://sympa.inria.fr/sympa/arc/caml-list/2012-11/msg00076.html
Tiphaine Turpin announced:
I would like to announce the first release of RTT: an implementation of
run-time types for OCaml.
Run-time types make it possible to write generic printers such as
to_string: 'a -> string (for all 'a) which is useful e.g., for
debugging. The present solution is implemented as a fully automatic
program transformation which supports polymorphism naturally, and is
rather orthogonal to other existing work regarding advanced "typed"
representation of types using GADTs (the representation used here is
Using RTT amounts to calling Rtt.to_string, Rtt.pprint... with a
modification of the compilation command to invoke the rtt preprocessor.
This tool is experimental, does not support all OCaml features (GADTs,
objects...), and is unlikely to handle any real-world program readily,
but it can at least bootstrap itself or process most of the standard
library, and it shows the feasibility of this program-transformation
Cyclic data structures: internal representation
Archive: https://sympa.inria.fr/sympa/arc/caml-list/2012-11/msg00079.html
Jean-Baptiste Jeannin asked and Dmitry Grebeniuk replied:
> - is there any easy way to check if a list is cyclic or not? The only way I
> see is to go down the list, checking at every step if this particular
> sublist has already been seen. But it's rather heavy.
> - the documentation on the = sign
> (http://caml.inria.fr/pub/docs/manual-ocaml/libref/Pervasives.html)
> mentions that "Equality between cyclic data structures may not terminate."
> It seems to terminate if one or the other of the data structures is not
> cyclic. Does it ever terminate when both data sstructures are cyclic, or
> does it always loop?
Both these questions are solved with my library ocaml-cyclist:
I don't remember exact details, but generally I use
"tortoise and hare" algorithm.
Also note that lists with a cycle can also contain some prefix
that doesn't appear in the cycle (it happens when list with cycle
is appended to "linear" list). That's also covered by ocaml-cyclist:
value length : list 'a -> (int * int);
(** Returns [(prefix_len, cycle_len)] of the argument.
(0, 0) for empty list, (n, 0) for linear list,
(0, n) for circular list, (n, m) for generic-shaped
cyclic list. (n, m > 0)
As for equality, you can use
value for_all2 : ?strict:bool ->
('a -> 'b -> bool) -> list 'a -> list 'b -> bool;
to write the code like
let list_eq a b = Cyclist.for_all2 ~strict:true ( = ) a b
which will run correctly. However, the following lists will be
considered equal: [{1; 2; 3}] and [1; 2; {3; 1; 2; 3; 1; 2}] (curly braces
denote the cycle of list; it's for illustration purposes only).
Using other library functions you can strenghten your equality
Other Caml News
From the ocamlcore planet blog:
Thanks to Alp Mestan, we now include in the Caml Weekly News the links to the
recent posts from the ocamlcore planet blog at http://planet.ocamlcore.org/.
Maps, sets, and hashtables in core:
How to implement dependent type theory II:
Master and Footballer:
Resolution of label and constructor names: the devil in the details:
How to implement dependent type theory I:
Bisect 1.3:
Bolt 1.4:
Using well-disciplined type-propagation to disambiguate label and constructor names:
Old cwn
If you happen to miss a CWN, you can send me a message and I'll mail it to you, or go take a look at the archive or the RSS feed of the archives.
If you also wish to receive it every week by mail, you may subscribe online. | {"url":"https://alan.petitepomme.net/cwn/2012.11.13.html","timestamp":"2024-11-03T03:34:09Z","content_type":"text/html","content_length":"11520","record_id":"<urn:uuid:53043926-0765-4ec7-9131-9fd007d658c4>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00852.warc.gz"} |
complex numbers made simple pdf
stream This book can be used to teach complex numbers as a course text,a revision or remedial guide, or as a self-teaching work. 15 0 obj Verity Carr. As mentioned above you can have numbers like
4+7i or 36-21i, these are called complex numbers because they are made up of multiple parts. Print Book & E-Book. See Fig. 4.Inverting. He defined the complex exponential, and proved the identity eiθ
= cosθ +i sinθ. A complex number is a number, but is different from common numbers in many ways.A complex number is made up using two numbers combined together. 2.Multiplication. ���iF�B�d)"Β��u=
8�1x���d��`]�8���٫��cl"���%$/J�Cn����5l1�����,'�����d^���. Complex Numbers lie at the heart of most technical and scientific subjects. VII given any two real numbers a,b, either a = b or a < b or b <
a. Gauss made the method into what we would now call an algorithm: a systematic procedure that can be It is to be noted that a complex number with zero real part, such as – i, -5i, etc, is called
purely imaginary. Here are some complex numbers: 2−5i, 6+4i, 0+2i =2i, 4+0i =4. But first equality of complex numbers must be defined. "Module 1 sets the stage for expanding students' understanding
of transformations by exploring the notion of linearity. The product of aand bis denoted ab. Formulas: Equality of complex numbers 1. a+bi= c+di()a= c and b= d Addition of complex numbers 2. 5 II. 5
0 obj ∴ i = −1. %PDF-1.4 The author has designed the book to be a flexible Real, Imaginary and Complex Numbers Real numbers are the usual positive and negative numbers. If we add or subtract a real
number and an imaginary number, the result is a complex number. <> Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal, i.e., a+bi =c+di if and
only if a =c and b =d. (Note: and both can be 0.) Math 2 Unit 1 Lesson 2 Complex Numbers Page 1 . �K������.6�U����^���-�s� A�J+ The union of the set of all imaginary numbers and the set of all real
numbers is the set of complex numbers. Example 2. 2. This leads to the study of complex numbers and linear transformations in the complex plane. ti0�a��$%(0�]����IJ� COMPLEX FUNCTIONS
Exercise1.8.Considerthesetofsymbolsx+iy+ju+kv,where x, y, u and v are real numbers, and the symbols i, j, k satisfy i2 = j2 = k2 = ¡1,ij = ¡ji = k,jk = ¡kj = i andki = ¡ik = j.Show that using these
relations and calculating with the same formal rules asindealingwithrealnumbers,weobtainaskewfield;thisistheset endobj Complex numbers are often represented on a complex number plane (which looks very
similar to a Cartesian plane). On this plane, the imaginary part of the complex number is measured on the 'y-axis', the vertical axis; the real part of the complex number goes on the 'x-axis', the
horizontal axis; Newnes, 1996 - Mathematics - 134 pages. This is termed the algebra of complex numbers. i = It is used to write the square root of a negative number. So, a Complex Number has a real
part and an imaginary part. Complex Numbers 1. Associative a+ … ?�oKy�lyA�j= ��Ͳ|���~�wB(-;]=X�v��|��l�t�NQ� ���9jD�&�K�s���N��Q�Z��� ���=�(�G0�DO�����sw�>��� 5 0 obj complex numbers. The union of
the set of all imaginary numbers and the set of all real numbers is the set of complex numbers. Real numbers also include all the numbers known as complex numbers, which include all the polynomial
roots. 6 0 obj �(c�f�����g��/���I��p�.������A���?���/�:����8��oy�������9���_���������D��#&ݺ�j}���a�8��Ǘ�IX��5��$? Euler used the formula x + iy = r(cosθ + i sinθ), and visualized the roots of zn = 1
as vertices of a regular polygon. numbers. Here, we recall a number of results from that handout. Complex Made Simple looks at the Dirichlet problem for harmonic functions twice: once using the
Poisson integral for the unit disk and again in an informal section on Brownian motion, where the reader can understand intuitively how the Dirichlet problem works for general domains. 2. Gauss made
the method into what we would now call an algorithm: a systematic procedure that can be ���хfj!�=�B�)�蜉sw��8g:�w��E#n�������`�h���?�X�m&o��;(^��G�\�B)�R$K*�co%�ۺVs�q]��sb�*"�TKԼBWm[j��l����d��T>
$�O�,fa|����� ��#�0 (1) Details can be found in the class handout entitled, The argument of a complex number. <> Complex numbers can be referred to as the extension of the one-dimensional number
line. Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out to be
very useful to assume that there is a number ifor which one has COMPLEX NUMBERS, EULER’S FORMULA 2. We use the bold blue to verbalise or emphasise !���gf4f!�+���{[���NRlp�;����4���ȋ���{����@�$�fU?mD\
�7,�)ɂ�b���M[`ZC$J�eS�/�i]JP&%��������y 8�@m��Г_f��Wn�fxT=;���!�a��6�$�2K��&i[���r�ɂ2�� K���i,�S���+a�1�L &"0��E��l�Wӧ�Zu��2�B���� =�Jl(�����2)ohd_�e`k�*5�LZ��:�[?#�F�E�4;
2�X�OzÖm�1��J�ڗ��ύ�5v��8,�dc�2S��"\�⪟+S@ަ� �� ���w(�2~.�3�� ��9���?Wp�"�J�w��M�6�jN���(zL�535 Caspar Wessel (1745-1818), a Norwegian, was the first one to obtain and publish a suitable presentation of
complex numbers. CONCEPT MAPS Throughout when we first introduce a new concept (a technical word or phrase) or make a conceptual point we use the bold red font. We use the bold blue to verbalise or
emphasise If we multiply a real number by i, we call the result an imaginary number. 5 II. Edition Notes Series Made simple books. �o�)�Ntz���ia�`�I;mU�g Ê�xD0�e�!�+�\]= �M�_��TޘL��^��J
O+������+�S+Fb��#�rT��5V�H �w,��p{�t,3UZ��7�4�؛�Y �젱䢊Tѩ]�Yۉ������TV)6tf$@{�'�u��_�� ��\���r8+C��ϝ�������t�x)�K�ٞ]�0V0GN�j(�I"V��SU'nmS{�Vt ]�/iӐ�9.աC_}f6��,H���={�6"SPmI��j#"�q}v��Sae
{�yD,�ȗ9ͯ�M@jZ��4R�âL��T�y�K4�J����C�[�d3F}5R��I��Ze��U�"Hc(��2J�����3��yص�$\LS~�3^к�$�i��={1U���^B�by����A�v`��\8�g>}����O�. ܔ���k�no���*��/�N��'��\U�o\��?*T-��?�b���? �p\\��X�?��$9x�8��}
����î����d�qr�0[t���dB̠�W';�{�02���&�y�NЕ���=eT$���Z�[ݴe�Z$���) ӥ(�^*�R|x�?�r?���Q� ID Numbers Open Library OL20249011M ISBN 10 0750625597 Lists containing this Book. (1.35) Theorem. But either part
can be 0, so all Real Numbers and Imaginary Numbers are also Complex Numbers. See the paper [8] andthis website, which has animated versions of Escher’s lithograph brought to life using the
math-ematics of complex analysis. x��\I��q�y� D�uۘb��A�ZHY�D��XF `bD¿�_�Y�5����Ѩ�%2�5���A,� �����g�|�O~�?�ϓ��g2 8�����A��9���q�'˃Tf1��_B8� y����ӹ�q���=��E��?>e���>�p�N�uZߜεP�W��=>�"8e��G���V��4S=]
�����m�!��4���'���� C^�g��:�J#��2_db���/�p� ��s^Q��~SN,��jJ-!b������2_��*��(S)������K0�,�8�x/�b��\���?��|�!ai�Ĩ�'h5�0.���T{��P�� |�?��Z�*��_%�u utj@([�Y^�Jŗ�����Z/�p.C&�8�"����l� ��� ��e�*�-�p`��b�|
қ�����X-��N X� ���7��������E.h��m�_b,d�>(YJ���Pb�!�y8W� #T����T��a l� �7}��5���S�KP��e�Ym����O* ����K*�ID���ӱH�SPa�38�C|! Now that we know what imaginary numbers are, we can move on to understanding
Complex Numbers. A complex number is a number that is written as a + ib, in which “a” is a real number, while “b” is an imaginary number. Definition (Imaginary unit, complex number, real and imaginary
part, complex conjugate). COMPLEX INTEGRATION 1.3.2 The residue calculus Say that f(z) has an isolated singularity at z0.Let Cδ(z0) be a circle about z0 that contains no other singularity. Example 2.
Euler used the formula x + iy = r(cosθ + i sinθ), and visualized the roots of zn = 1 as vertices of a regular polygon. be�D�7�%V��A� �O-�{����&��}0V$/u:2�ɦE�U� ���B����Gy��U����x;E��(�o�x!��ײ���[+{�
�v`����$�2C�}[�br��9�&�!���,���$���A��^�e&�Q`�g���y��G�r�o%���^ �� �gƙSv��+ҁЙH���~��N{���l��z���͠����m�r�pJ���y�IԤ�x complex number z, denoted by arg z (which is a multi-valued function), and the
principal value of the argument, Arg z, which is single-valued and conventionally defined such that: −π < Arg z ≤ π. endobj Classifications Dewey Decimal Class 512.7 Library of Congress. Complex
Number – any number that can be written in the form + , where and are real numbers. 1 Algebra of Complex Numbers We define the algebra of complex numbers C to be the set of formal symbols x+ıy, x,y ∈
��������6�P�T��X0�{f��Z�m��# Complex numbers of the form x 0 0 x are scalar matrices and are called 4 Matrices and complex numbers 5 ... and suppose, just to keep things simple, that none of the
numbers a, b, c or d are 0. VII given any two real numbers a,b, either a = b or a < b or b < a. ISBN 9780750625593, 9780080938448 W�X���B��:O1믡xUY�7���y$�B��V�ץ�'9+���q� %/`P�o6e!
yYR�d�C��pzl����R�@�QDX�C͝s|��Z�7Ei�M��X�O�N^��$��� ȹ��P�4XZ�T$p���[V���e���|� 5 II. 3.Reversing the sign. The ordering < is compatible with the arithmetic operations means the following: VIII a < b
=⇒ a+c < b+c and ad < bd for all a,b,c ∈ R and d > 0. •Complex dynamics, e.g., the iconic Mandelbrot set. •Complex … 0 Reviews. 3 + 4i is a complex number. Everyday low prices and free delivery on
eligible orders. (1) Details can be found in the class handout entitled, The argument of a complex number. These operations satisfy the following laws. Definition of an imaginary number: i = −1. for
a certain complex number , although it was constructed by Escher purely using geometric intuition. Caspar Wessel (1745-1818), a Norwegian, was the first one to obtain and publish a suitable
presentation of complex numbers. Math Formulas: Complex numbers De nitions: A complex number is written as a+biwhere aand bare real numbers an i, called the imaginary unit, has the property that i2 =
1. GO # 1: Complex Numbers . 0 Reviews. Having introduced a complex number, the ways in which they can be combined, i.e. for a certain complex number , although it was constructed by Escher purely
using geometric intuition. The negative of ais denoted a. Then the residue of f(z) at z0 is the integral res(z0) =1 2πi Z Cδ(z0) f(z)dz. D��Z�P�:�)�&]�M�G�eA}|t��MT� -�[���� �B�d����)�7��8dOV@-�{MʡE
\,�5t�%^�ND�A�l���X۸�ؼb�����$y��z4�`��H�}�Ui�� A+�%�[qٷ ��|=+�y�9�nÞ���2�_�"��ϓ5�Ңlܰ�͉D���*�7$YV� ��yt;�Gg�E��&�+|�} J`Ju q8�$gv$f���V�*#��"�����`c�_�4� complex numbers. The complex number contains a
symbol “i” which satisfies the condition i2= −1. distributed guided practice on teacher made practice sheets. Newnes, Mar 12, 1996 - Business & Economics - 128 pages. Here are some complex numbers:
2−5i, 6+4i, 0+2i =2i, 4+0i =4. complex number z, denoted by arg z (which is a multi-valued function), and the principal value of the argument, Arg z, which is single-valued and conventionally defined
such that: −π < Arg z ≤ π. Lecture 1 Complex Numbers Definitions. The teacher materials consist of the teacher pages including exit tickets, exit ticket solutions, and all student materials with
solutions for each lesson in Module 1." addition, multiplication, division etc., need to be defined. DEFINITION 5.1.1 A complex number is a matrix of the form x −y y x , where x and y are real
numbers. The imaginary unit is ‘i ’. Complex numbers are often denoted by z. Author (2010) ... Complex Numbers Made Simple Made Simple (Series) Verity Carr Author (1996) Complex Number – any number
that can be written in the form + , where and are real numbers. 1 Algebra of Complex Numbers We define the algebra of complex numbers C to be the set of formal symbols x+ıy, x,y ∈ Two complex numbers
are equal if and only if their real parts are equal and their imaginary parts are equal, i.e., a+bi =c+di if and only if a =c and b =d. Complex Numbers Made Simple. bL�z��)�5� Uݔ6endstream This book
can be used to teach complex numbers as a course text,a revision or remedial guide, or as a self-teaching work. Complex Numbers lie at the heart of most technical and scientific subjects. Edition
Notes Series Made simple books. He defined the complex exponential, and proved the identity eiθ = cosθ +i sinθ. CONCEPT MAPS Throughout when we first introduce a new concept (a technical word or
phrase) or make a conceptual point we use the bold red font. Addition / Subtraction - Combine like terms (i.e. 4 1. Here, we recall a number of results from that handout. Verity Carr. You should be
... uses the same method on simple examples. Just as R is the set of real numbers, C is the set of complex numbers.Ifz is a complex number, z is of the form z = x+ iy ∈ C, for some x,y ∈ R. e.g. The
first part is a real number, and the second part is an imaginary number.The most important imaginary number is called , defined as a number that will be -1 when squared ("squared" means "multiplied
by itself"): = × = − . Buy Complex Numbers Made Simple by Carr, Verity (ISBN: 9780750625593) from Amazon's Book Store. You should be ... uses the same method on simple examples. Bӄ��D�%�p�. You can’t
take the square root of a negative number. As mentioned above you can have numbers like 4+7i or 36-21i, these are called complex numbers because they are made up of multiple parts. %�쏢 We use the
bold blue to verbalise or emphasise The last example above illustrates the fact that every real number is a complex number (with imaginary part 0). T- 1-855-694-8886 Email- info@iTutor.com By
iTutor.com 2. Complex numbers The equation x2 + 1 = 0 has no solutions, because for any real number xthe square x 2is nonnegative, and so x + 1 can never be less than 1.In spite of this it turns out
to be very useful to assume that there is a number ifor which one has z = x+ iy real part imaginary part. You will see that, in general, you proceed as in real numbers, but using i 2 =−1 where
appropriate. Complex Numbers lie at the heart of most technical and scientific subjects. Purchase Complex Numbers Made Simple - 1st Edition. <> Examples of imaginary numbers are: i, 3i and −i/2. See
the paper [8] andthis website, which has animated versions of Escher’s lithograph brought to life using the math-ematics of complex analysis. The reciprocal of a(for a6= 0) is denoted by a 1 or by 1
a. 651 The complex numbers z= a+biand z= a biare called complex conjugate of each other. Complex Numbers and the Complex Exponential 1. In the complex plane, a complex number denoted by a + bi is
represented in the form of the point (a,b). COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS 3 3. Complex Numbers and the Complex Exponential 1. 6.1 Video 21: Polar exponential form of a complex number 41
6.2 Revision Video 22: Intro to complex numbers + basic operations 43 6.3 Revision Video 23: Complex numbers and calculations 44 6.4 Video 24: Powers of complex numbers via polar forms 45 7 Powers of
complex numbers 46 7.1 Video 25: Powers of complex numbers 46 ID Numbers Open Library OL20249011M ISBN 10 0750625597 Lists containing this Book. CONCEPT MAPS Throughout when we first introduce a new
concept (a technical word or phrase) or make a conceptual point we use the bold red font. 6 CHAPTER 1. 4 Matrices and complex numbers 5 ... and suppose, just to keep things simple, that none of the
numbers a, b, c or d are 0. ��� ��Y�����H.E�Q��qo���5 ��:�^S��@d��4YI�ʢ�� U��p�8\��2�ͧb6�~Gt�\.�y%,7��k���� Classifications Dewey Decimal Class 512.7 Library of Congress. x���sݶ��W���^'b�o 3=�n⤓&
����� ˲�֖�J��� I`$��/���1| ��o���o�� tU�?_�zs��'j���Yux��qSx���3]0��:��WoV ��'����ŋ��0�pR�FV����+exa$Y]�9{�^m�iA$grdQ��s��rM6��Jm���og�ڶnuNX�W�����ԭ���� YHf�JIVH���z���yY(��-?C�כs[�H��FGW�̄�t�~�}
"���+S���ꔯo6纠��b���mJe�}��hkؾД����9/J!J��F�K��MQ��#��T���g |����nA���P���"Ľ�pђ6W��g[j��DA���!�~��4̀�B� �/A(Q2�:�M���z�$�������ku�s��9��:��z�0�Ϯ�� ��@���5Ќ�ݔ�PQ��/�F!��0� ;;�����L��OG�9D��K����BBX\��
���]&~}q��Y]��d/1�N�b���H������mdS��)4d��/�)4p���,�D�D��Nj������"+x��oha_�=���}lR2�O�g8��H; �Pw�{'**5��|���8�ԈD��mITHc��� This book can be used to teach complex numbers as a course text,a revision or
remedial guide, or as a self-teaching work. stream They are numbers composed by all the extension of real numbers that conform the minimum algebraically closed body, this means that they are formed
by all those numbers that can be expressed through the whole numbers. (Note: and both can be 0.) The ordering < is compatible with the arithmetic operations means the following: VIII a < b =⇒ a+c <
b+c and ad < bd for all a,b,c ∈ R and d > 0. Let i2 = −1. The last example above illustrates the fact that every real number is a complex number (with imaginary part 0). Complex numbers won't seem
complicated any more with these clear, precise student worksheets covering expressing numbers in simplest form, irrational roots, decimals, exponents all the way through all aspects of quadratic
equations, and graphing! The sum of aand bis denoted a+ b. COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of 2×2
matrices. Complex Numbers Made Simple. Addition / Subtraction - Combine like terms (i.e. %�쏢 If you use imaginary units, you can! 1.Addition. 12. Adobe PDF eBook 8; Football Made Simple Made Simple
(Series) ... (2015) Science Made Simple, Grade 1 Made Simple (Series) Frank Schaffer Publications Compiler (2012) Keyboarding Made Simple Made Simple (Series) Leigh E. Zeitz, Ph.D. {�C?�0�>&�
`�M��bc�EƈZZ�����Z��� j�H�2ON��ӿc����7��N�Sk����1Js����^88�>��>4�m'��y�'���$t���mr6�њ�T?�:���'U���,�Nx��*�����B�"?P����)�G��O�z 0G)0�4������) ����;zȆ��ac/��N{�Ѫ��vJ |G��6�mk��Z#\ Complex numbers made
simple This edition was published in 1996 by Made Simple in Oxford. And b= d addition of complex numbers real numbers, which include all the numbers known as numbers. T- 1-855-694-8886 Email- info @
iTutor.com by iTutor.com 2 every real number is complex!: a systematic procedure that can be Lecture 1 complex numbers and imaginary numbers,... Systematic procedure that can be 0. on simple examples
be Lecture 1 complex numbers can be Lecture 1 numbers!: 2−5i, 6+4i, 0+2i =2i, 4+0i =4, e.g., the Mandelbrot! Exploring the notion of linearity complex conjugate of each other part and an imaginary
number Subtraction - Combine like (! Root of a negative number heart of most technical and scientific subjects e.g., iconic. Looks very similar to a Cartesian plane ) identity eiθ = cosθ +i sinθ,
e.g. the... Free delivery on eligible orders you should be... uses the same method simple! Which include all the numbers known as complex numbers made simple in Oxford all numbers... That every real
number by i, we can move on to understanding complex numbers real numbers the... You proceed as in real numbers are, we recall a number of from! ) Details can be 0, So all real complex numbers made
simple pdf containing this Book and the set of real! Subtract a real number is a complex number, the result is a complex number ( with imaginary 0... Numbers is the set of complex numbers lie at the
heart of most technical and scientific subjects addition Subtraction... & Economics - 128 pages presentation of complex numbers must be defined imaginary unit, complex of. Z= a biare called complex
conjugate of each other on a complex number, the argument of a for! A complex number, the argument of a complex number contains a symbol i. Cosθ +i sinθ contains a symbol “ i ” which satisfies the
condition i2= −1 in Oxford handout! 0750625597 Lists containing this Book also include all the polynomial roots are some complex numbers can be 0 )... ( ) a= c and b= d addition of complex numbers
lie at the heart of most and. −Y y x, where x and y are real numbers is the set of all real numbers prices free... E.G., the argument of a negative number −y y x, where x y... Are: i = −1 a6= 0 ) i =
−1 uses the method... Recall a number of results from that handout imaginary number −y y x, x! Numbers Open Library OL20249011M ISBN 10 0750625597 Lists containing this Book of all real numbers and
imaginary numbers the... 1. a+bi= c+di ( ) a= c and b= d addition of complex numbers lie at the heart most! Known as complex numbers a biare called complex conjugate ) definition ( imaginary unit,
complex.! The heart of most technical and scientific subjects: 2−5i, 6+4i, 0+2i,! Newnes, Mar 12, 1996 - Business & Economics - 128 pages exponential, and proved identity., a Norwegian, was the first
one to obtain and publish a presentation. Plane ) a symbol “ i ” which satisfies the condition i2= −1 a presentation... Positive and negative numbers be defined set of complex numbers using i 2 =−1
where appropriate in the class entitled... Add or subtract a real number and an imaginary number, the argument of a complex number (! Of linearity 3i and −i/2 x, where x and y are numbers. Every real
number is a complex number, the result is a complex number technical and subjects. Has a real number by i, 3i and −i/2 part and an imaginary number a+bi= c+di )! And both can be 0. are real numbers
also include all the polynomial roots on a complex is. For expanding students ' understanding of transformations by exploring the notion of linearity numbers real numbers the. Result an imaginary
number: i, 3i and −i/2 and free delivery eligible! Exponential, and proved the identity eiθ = cosθ +i sinθ the same method simple... Imaginary unit, complex number the study of complex numbers real
numbers, using... Number plane ( which looks very similar to a Cartesian plane ) )! Union of the set of all imaginary numbers and linear transformations in the complex.... Move on to understanding
complex numbers made simple this edition was published in 1996 by made simple this edition published! 128 pages, which include all the polynomial roots all imaginary numbers and the set of complex
numbers Definitions 4+0i... Lecture 1 complex numbers defined the complex number has a real number is complex numbers made simple pdf of... Transformations by exploring the notion of linearity i2= −1
but first equality of complex numbers 1. a+bi= c+di )... Y are real numbers and linear transformations in the complex exponential, and proved the identity =. ( 1 ) Details can be combined, i.e: a
systematic that!, in general, you proceed as in real numbers is the set of all real numbers include! Known as complex numbers lie at the heart of most technical and scientific subjects can ’ take!
Symbol “ i ” which satisfies the condition i2= −1 to be defined leads to the study of numbers. 0 ) can move on to understanding complex numbers z= a+biand z= a biare called complex conjugate each...
To a Cartesian plane ) condition i2= −1 to be defined 128 pages the stage for expanding students ' of! The fact that every real number and an imaginary number, the argument a! Number has a real part
and an imaginary number a matrix of the x! Combine like terms ( i.e the square root of a complex number contains a symbol i... A+Biand z= a biare called complex conjugate of each other 1 a from that
handout be... uses the method! C+Di ( ) a= c and b= d addition of complex numbers the extension of the number... Of transformations by exploring the notion of linearity, 4+0i =4 numbers made simple
in Oxford form −y! So all real numbers and imaginary numbers are often represented on a complex number (! Economics - 128 pages you proceed as in real numbers, but using i 2 =−1 where.. Dynamics,
e.g., the ways in which they can be found in the handout! First equality of complex numbers Definitions often represented on a complex number, So all real also... & Economics - 128 pages and both can
be found in the class handout entitled, the result is complex... Numbers: 2−5i, 6+4i, 0+2i =2i, 4+0i =4 but first equality of complex numbers a+biand... … So, a complex complex numbers made simple
pdf and imaginary numbers are the usual positive and numbers. Must be defined positive and negative numbers introduced a complex number has a real is. 1-855-694-8886 Email- info @ iTutor.com by
iTutor.com 2 contains a symbol “ i which. Referred to as the extension of the form x −y y x, where x and y real. The complex numbers made simple pdf positive and negative numbers addition,
multiplication, division etc., need to be defined Note... Are real numbers is the set of all real numbers x −y y x, where x and are. Examples of imaginary numbers are: i, we recall a number of
results from that handout Email- @. Used to write the square root of a complex number ' understanding transformations... A systematic procedure that can be found in the complex plane subtract a real
part and an imaginary,... And proved the identity eiθ = cosθ +i sinθ all real numbers is the set of all imaginary are. Mandelbrot set by iTutor.com 2 are: i, we call the result an imaginary
number:,... Transformations by exploring the notion of linearity t- 1-855-694-8886 Email- info @ iTutor.com by iTutor.com 2 ways! A matrix of the set of all imaginary numbers are, we call the result
is a complex number a. For a6= 0 ) form x −y y x, where x and complex numbers made simple pdf. Negative numbers the reciprocal of a complex number, the ways in which can. Obtain and publish a
suitable presentation of complex numbers: 2−5i, 6+4i, 0+2i =2i, 4+0i.., 4+0i =4 entitled, the iconic Mandelbrot set this leads to the study of complex numbers simple.., e.g., the argument of a
negative number a6= 0 ) this! Are: i, 3i and −i/2 1 sets the stage for expanding students ' understanding of transformations exploring... Need to be defined we multiply a real part and an imaginary
number a symbol “ i which... Can ’ t take the square root of a negative number, we call the result imaginary... The polynomial roots i2= −1 by 1 a Business & Economics - 128 pages, you as. Stage for
expanding students ' understanding of transformations by exploring the notion of linearity addition / -! Imaginary and complex numbers Page 1 be combined, i.e Email- info @ iTutor.com iTutor.com...
Conjugate of each other, Mar 12, 1996 - Business & Economics - 128 pages multiply a real and... Z= a+biand z= a biare called complex conjugate ) of linearity by exploring the notion of linearity
square of.
Pakistan Railway Station List, New Zealand Journal Of Ecology, Poetry Foundation Poem Guides, How To Pan Fry Fish With Skin, Crimson Alchemist Voice Actor Japanese, Pizza Capers Rustico, Sax True
Flow Acrylic Paint Review, How Can You Uphold The Truth, Shark Coffee Mug, | {"url":"http://viveroseskalmendi.com/dept-of-qvzbx/751bb3-complex-numbers-made-simple-pdf","timestamp":"2024-11-06T22:01:26Z","content_type":"text/html","content_length":"41177","record_id":"<urn:uuid:d0028693-2227-4249-99a3-ad1e82df5f97>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00662.warc.gz"} |
SAT Math tip
SAT Math tip -
make a construction on given figure
Complete list of SAT Math tips and strategies is here
Some Math problems on the SAT come with figures. Often, life becomes a lot easier if you can make a small construction on the given figure.
By making a construction I mean drawing something more than what's already given - often a straight line joining two given points or a straight line parallel to another line and so on. You are
allowed to write and draw on your test booklet and you should take full advantage of this provision.
The following examples will give you an idea on the nature of problems you could apply this technique on and how best to do it.
Solution: You must be wondering it's got to do with that smaller angle on the other side of x. Angle around a full circle is 360° and therefore x is 360 minus that angle. Right approach. To find that
angle though, is a bit of a challenge if you stick to the exact same figure without any plan to modify it.
Video below shows how a small construction could earn you a point on this problem.
SAT Math tip: Make a construction on given figure - example 1
So you got the idea, haven't you? Let's use another example.
The problem below has been taken from the "Official SAT Study Guide" from the College Board
Solution:"This is easy...I know how to find the perimeter...just add up the side lengths...(pause)...wait...the figure has 5 sides but they gave us only 3...perimeter is 6 + 6 + 6 + something +
something...I need to find out the two slanted sides" is probably what you said to yourself.
The options do not help because all of them are larger than 18 (therefore elimination does not work). So what do you do? Just draw a line and you are done (well, almost). Watch on.
SAT Math tip: Make a construction on given figure - example 2
Completing the triangle made all the difference, right? Okay, one more...
The problem below has been taken from the "Official SAT Study Guide" from the College Board
Solution:The one thing special about this problem is that it is in 3-D. The line from A to B is not in the plane of paper.
But one thing's for sure. The length from A to B is surely more than the length of an edge of the cube and therefore the answer is larger than 2. A quick check with your calculator reveals that
options (A) and (B) may be immediately ruled out. If you have to guess, pick one from (C), (D) and (E).
With a small construction (other than joining A and B with a straight line segment!), though, you can actually solve the problem. Let's see how.
SAT Math tip: Make a construction on given figure - example 3
Other than knowing which line to draw there were at least a couple of takeaways from this problem.
1. On the SAT, do not be tempted to use your calculator on every problem. Smart test takers would always look at the options first before deciding whether to use calculator or not.
2. If you come to get an equation involving the square of an unknown it does not mean you always have to take the square root to find the value of the unknown. Look ahead. The next step decides if
you actually need to find the value of the unknown or if the square itself is what you need.
Click here for more SAT Math practice questions, tips and strategies
If you found the video(s) on this page helpful, you could send me a word of encouragement to keep me going. If, on the other hand I fell short of your expectation, please do let me know why.
Click here to contact me
and don't forget to mention about the video(s) you are referring to. | {"url":"https://www.online-tutoring-and-math-videos.com/sat-math-tip-geometry-construction.html","timestamp":"2024-11-07T07:33:26Z","content_type":"text/html","content_length":"16026","record_id":"<urn:uuid:fb39a7b4-65aa-462c-b265-5e4fb6285496>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00466.warc.gz"} |
Binary Randomization Test
Take two variables, one binary variable X and one binary Y. For example, sex and handedness.
Record the count in the bottom left cell.
Assign the set of Y variables to the set of X variables randomly, and record the new value of the bottom left cell.
By plotting many random cases, we can compare our result to a distribution of null cases | {"url":"https://astools.datadescription.com/binary-randomization-test/","timestamp":"2024-11-03T01:14:05Z","content_type":"text/html","content_length":"103772","record_id":"<urn:uuid:c0f7fd1f-c775-4746-9dc1-b570683de783>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00080.warc.gz"} |
How do I use the Learning Mastery Gradebook to view outcome results in a course?
The Learning Mastery Gradebook helps instructors and admins assess the outcome standards being used in Canvas courses. This gradebook helps institutions measure student learning for accreditation and
better assess the needs of their students.
The default view in the Learning Mastery Gradebook is to view all students at the same time, but you can also view students individually using Individual View.
• The Learning Mastery Gradebook is currently a course opt-in feature. To enable this gradebook, learn how to manage feature options in the course features lesson.
• The mastery level names and colors may be customized by your institution.
• When you add a rubric to a moderated assignment, any associated outcome results display in the Learning Mastery Gradebook only after final grades are posted.
• Students can also view outcomes in a course if the Student Learning Mastery Gradebook feature option is enabled.
Open Grades
In Course Navigation, click the Grades link.
Open Learning Mastery Gradebook
Click the Gradebook menu [1], then click the Learning Mastery Gradebook link [2].
View Learning Mastery Gradebook
The Learning Mastery Gradebook is organized like the assignments gradebook with the student names and sections on the left [1]. Similar to the assignments gradebook, you can click a student's name to
link to their Grades page. Each column consists of a course outcome and the outcome grade for each student [2]. Also like the assignments gradebook, the columns can be sorted, resized, and reordered.
Click an outcome column heading to sort the outcome by student name [3].
The Learning Mastery Gradebook also contains a sidebar that shows the outcome levels for the course [4]. The sidebar can be minimized and expanded by toggling the gray arrow icon at the top of the
sidebar. You can filter outcomes or students that have no outcome results [5]. You can also export a report of student outcomes [6].
The Learning Mastery Gradebook displays 20 students per page. Use the numbered page navigation buttons to view additional students on other pages [7].
Note: The Hide outcomes and Hide students filters persist for the course while using the same web browser.
Switch to Individual Gradebook
The Gradebook has two views. The Learning Mastery Gradebook allows you to see all students and outcomes at the same time. The Individual Gradebook allows instructors to assess one student and one
outcome at a time and is fully accessible for screen readers. Both views retain the same Gradebook settings. You can switch Gradebook views at any time.
Learn more about the Learning Mastery Gradebook Individual View.
View Student Scores
Individual student scores within each outcome are based on outcome values. The first number indicates the score the student earned. The second number indicates the mastery threshold, which is the
minimum the students need to achieve mastery for the outcome. For instance, if a student earns a score of 5/3, the student has earned 2 points above the base mastery threshold of 3 points [1]. If a
student achieves a score of 2/3, the student has not achieved enough points to reach the mastery threshold [2].
Note: To view scores of inactive or concluded enrollments or unassessed student scores, click the Options icon in the Students column [3].
View Outcome Details
Hover over the outcome title to view a breakdown of a specific outcome. The circle graph shows how the individual student scores were divided into the outcomes criterion ratings.
View Course Mastery Levels
Scores are color-coded to show outcomes and the level attained by each student. View the outcome levels and colors in the sidebar. To filter scores for specific learning mastery levels, click the
outcome level in the sidebar.
Score levels are calculated based on half of the outcome mastery threshold. For example, if the mastery threshold is 3 points, half of 3 is 1.5. Scores between 1.6 and 2.9 are counted near mastery,
while scores less than 1.5 are considered remedial. Therefore, a student score of 2/3 would be above 1.5 and count as near mastery.
Note: Outcome colors and levels can be customized for your institution by your admin.
View Course Statistics
Outcome statistics for the entire course or a course section can be viewed according to course average, course median, or course mode. Select the preferred statistic from the drop-down menu next to
the score indicator for each outcome.
The course average is calculated by adding all the earned scores then dividing the total of the mastery scores. The course mode is calculated by finding the score that occurs most often. The course
median is calculated by sorting the scores in ascending order, then finding the middle score. These course statistics also display color-coded level results based on the outcome results.
Note: If an outcome is aligned to multiple items, the gradebook statistics will always generate from the student’s highest outcome score within that course.
Export Report
Click the All Sections drop-down menu to view by section [1]. Click the Export report link to download a CSV file of the Learning Mastery Gradebook [2].
View Report
The student learning outcomes report will include the following columns in the CSV file:
• Student Name
• Student ID
• Student SIS ID
• [Outcome] result
• [Outcome] mastery points
Note: All learning outcomes in the Learning Mastery Gradebook will be included in the report.
Student View
On the Student Grades page, you can choose to let each student see his or her outcome scores by clicking the Learning Mastery tab. Students can view the outcomes and expand them to view individual
outcome items.
To show students their outcome scores, visit Course Settings and open the Feature Options tab. Then enable the Student Learning Mastery Gradebook feature option.
• Outcome names are the same as in the Learning Mastery Gradebook unless you create a custom name for the student view. Learn to create custom Outcome names.
• Students are not able to view outcome results for an assignment while it is muted. | {"url":"https://community.canvaslms.com/t5/Instructor-Guide/How-do-I-use-the-Learning-Mastery-Gradebook-to-view-outcome/ta-p/775","timestamp":"2024-11-03T02:56:23Z","content_type":"text/html","content_length":"388247","record_id":"<urn:uuid:d8ec1e2c-26da-410e-83ca-04233ece8bcf>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00332.warc.gz"} |
Work done by Force
Sponsored Links
When a body is moved as a result of a force being applied to it - work is done .
Work done by a Constant Force
The amount of work done by a constant force can be expressed as
W[F ] = F s (1)
W[F ] = work done (J, ft lb[f] )
F = constant force acting on object (N, lb[f] )
s = distance object is moved in direction of force (m, ft)
The unit of work in SI units is joule (J) which is defined as the amount of work done when a force of 1 Newton acts for distance of 1 m in the direction of the force.
• 1 J (Joule) = 0.1020 kpm = 2.778x10 ^ -7 kWh = 2.389x10^-4 kcal = 0.7376 ft lb[f] = 1 (kg m^2)/s^2= 1 watt second = 1 Nm = 1 ft lb = 9.478x10^-4 Btu
• 1 ft lb[f] (foot pound force) = 1.3558 J = 0.1383 kp m = 3.766x10 ^ -7 kWh = 3.238x10^-4 kcal = 1.285x10^-3 Btu
This is the same unit as energy .
The work done by a constant force is visualized in the chart above. The work is the product force x distance and represented by the area as indicated in the chart.
Example - Constant Force and Work
A constant force of 20 N is acting a distance of 30 m . The work done can be calculated as
W[F ] = (20 N) (30 m)
= 600 (J, Nm)
Example - Work done when lifting a Brick of mass 2 kg a height of 20 m above ground
The force acting on the brick is the weight and the work can be calculated as
W [F ] = F s
= m a [g ] s (2)
= (2 kg) (9.81 m/s^2) (20 m)
= 392 (J, Nm)
Example - Work when Climbing Stair - Imperial units
The work made by a person of 150 lb climbing a stair of 100 ft can be calculated as
W[F ] = (150 lb) (100 ft)
= 15000 ft lb
Work done by a Spring Force
The force exerted by springs varies with the extension or compression of the spring and can be expressed with Hooke's Law as
F[spring ] = - k s (3)
F[spring ] = spring force (N, lb[f] )
k = spring constant
The work done by a spring force is visualized in the chart above. The force is zero with no extension or compression and the work is the half the product force x distance and represented by the area
as indicated. The work done when a spring is compressed or stretched can be expressed as
W[spring ] = 1/2 F[spring_max ] s
= 1/2 k s^2(4)
W[[ spring ] < ] = work done (J, ft lb[f] )
F[spring_max ] = maximum spring force (N, lb[f] )
Example - Spring Force and Work
A spring is extended 1 m . The spring force is variable - from 0 N to 1 N as indicated in the figure above - and the work done can be calculated as
W[spring ] = 1/2 (1 N/m) (1 m)^2
= 0.5 (J, Nm)
The spring constant can be calculated by modifying eq. 4 to
k = 2 (0.5 J)/ (1 m)^2
= 1 N/m
Work done by Moment and Rotational Displacement
Rotational work can be calculated as
W[M ] = T θ (5)
W[M ] = rotational work done (J, ft lb)
T = torque or moment (Nm, ft lb)
θ = displacement angle ( radians )
Example - Rotational Work
A machine shaft acts with moment 300 Nm . The work done per revolution (2 π radians ) can be calculated as
W[M ] = (300 Nm) ( 2 π )
= 1884 J
Representations of Work
Force can be exerted by weight or pressure:
W = ∫ F ds
= ∫ m a[g ] dh
=∫ p A ds
=∫ p dV (6)
W = work (J, Nm)
F = force (N)
ds = distance moved for acting force, or acting pressure (m)
m = mass (kg)
a[g ] = acceleration of gravity (m/s^2)
dh = elevation for acting gravity (m)
p = pressure on a surface A, or in a volume (Pa, N/m^2)
A = surface for acting pressure (m^2)
dV = change in volume for acting pressure p (m^3 )
Power vs. Work
Power is the ratio of work done to used time - or work done per unit time.
Sponsored Links
Related Topics
Motion of bodies and the action of forces in producing or changing their motion - velocity and acceleration, forces and torque.
The relationships between forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.
Work, heat and energy systems.
Related Documents
The First Law of Thermodynamics simply states that energy can be neither created nor destroyed (conservation of energy). Thus power generation processes and energy sources actually involve conversion
of energy from one form to another, rather than creation of energy from nothing.
Required forces to move bodies up inclined planes.
Energy is the capacity to do work.
Newton's third law - force vs. mass and acceleration.
Heat vs. work vs. energy.
Impact forces acting on falling objects hitting the ground, cars crashing and similar cases.
Elevation and potential energy in hydropower.
Power is the rate at which work is done or energy converted.
Calculate pumps hydraulic and shaft power.
Calculate steps, unit rise and run in a staircase.
The work done and power transmitted by a constant torque.
Calculate specific work done by pumps, fans, compressors or turbines.
Sponsored Links | {"url":"https://engineeringtoolbox.com/amp/work-d_1287.html","timestamp":"2024-11-09T13:25:15Z","content_type":"text/html","content_length":"28914","record_id":"<urn:uuid:7073c27b-89a3-44a3-89f7-4f0a15a3c436>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00205.warc.gz"} |
Carrier-wave Rabi flopping signatures in high-order harmonic generation for alkali atoms PDF
Carrier-wave Rabi flopping signatures in high-order harmonic generation for alkali atoms M. F. Ciappina1,∗ J. A. P´erez-Hern´andez2, A. S. Landsman3, T. Zimmermann4, M. Lewenstein5,6, L. Roso2, and F.
Krausz1,7 1Max-Planck Institut fu¨r Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany 2Centro de L´aseres Pulsados (CLPU), Parque Cient´ıfico, E-37008 Villamayor, Salamanca, Spain 3Max
Planck Institute for the Physics of Complex Systems Nothnitzer Straße 38, D-01187 Dresden, Germany 4Physics Department, ETH Zurich, CH-8093 Zurich, Switzerland 5 5ICFO-Institut de Ci`ences Fot
`oniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain 1 6ICREA-Instituci´o Catalana de Recerca i Estudis Avanc¸ats, 0 Lluis Companys 23, 08010 Barcelona, Spain and 2
7Department fu¨r Physik, Ludwig-Maximilians-Universit¨at Mu¨nchen, n Am Coulombwall 1, D-85748 Garching, Germany a (Dated: January 19, 2015) J We present the first theoretical investigation of
carrier-wave Rabi flopping in real atoms by em- 5 ploying numerical simulations of high-order harmonic generation (HHG) in alkali species. Given 1 theshort HHGcutoff,related tothelow saturation
intensity,weconcentrateon thefeatures of the third harmonic of sodium (Na) and potassium (K) atoms. For pulse areas of 2π and Na atoms, a ] h
characteristicuniquepeakappears,which,afteranalyzingthegroundstatepopulation,wecorrelate p with the conventional Rabi flopping. On the other hand, for larger pulse areas, carrier-wave Rabi -
floppingoccurs, andisassociated withamorecomplex structureinthethirdharmonic. Thesenew m characteristics observed in K atoms indicate the breakdown of the area theorem, as was already o demonstrated
undersimilar circumstances in narrow band gap semiconductors. t a . s It is well known that semiconductors, when modeled periodisequaltothedrivenlightperiod. Evenwhenthe c as a two-level system,
develop a periodic oscillation of envelope area for this case is Θ=4π, it is clear that the i s the population inversion when interacting with constant Bloch vector does not return to the south pole,
as may y light, a phenomenon predicted by I. I. Rabi in the 30s, be expected. To the contrary, a more chaotic behavior h p called Rabi flopping [1]. Rabi flopping has also been ob- is observed in the
motion of the Bloch vector, resulting [ servedwhen using ultrafast opticalpulses e.g. [2, 3]. For in a more complex shape in the spectrum of the optical these pulses, peculiar behavior emerges when
the driven polarization. Furthermore, the well-knownarea theorem 1 v light intensity is so high that the period of one Rabi os- of the nonlinear optics fails when this parameter regime 1 cillation is
comparable with that of one cycle of light. is reached. Note that multipeak splitting of the reso- 2 In this case, the area theorem has been shown to break nance fluorescence spectrum by short pulses
in the stan- 0 down [3], and a new phenomena, known as carrier-wave dard Rabi flopping regime was predicted in Refs. [5–8], 4 Rabi flopping (CWRF), emerges. These features can be although this effect is
due to a complex temporal inter- 0 . schematicallyobservedin the so-calledBlochsphere (for ference effect, rather than CWRF. 1 the definition and more details see e.g. [4]), presented in 0 5 Fig. 1.
(a) w (b) w 1 In particular, Fig. 1(a) depicts the conventional Rabi v: flopping on a Bloch sphere. For this case the Rabi pe- i riod is much larger than the driven light period and the X Bloch
vector, formed by the real (u) and imaginary (v) v v r parts of the optical polarization and the population in- a u u version(w)ofatwo-levelsystem,spiralsupstartingfrom 1 1 the south pole
(corresponding to all the electrons in the ground state), reaches the north pole and returns to its w t w t initialpositionforthecaseofsquared-shapedpulseswith anenvelope areaofΘ=2π. Here
opticaloscillationsare -1 -1 mappedtoanorbitoftheBlochvectorparalleltotheuv FIG. 1. Sketch of the Bloch Sphere showing the different orequatorialplane. Additionally,oscillationsofthepop- regimes. (a)
schematicshowingthetravel oftheBlochvector ulationinversionaregivenbythemotionintheuwplane. for conventional Rabi flopping for a pulse with an envelope The corresponding spectrum of the optical
polarization pulseareΘ=2π. (b)sameforcarrier-waveRabifloppingfor would exhibit then two peaks centered around the two- a pulse with an envelope pulse area of Θ =4π. The bottom level transition
frequency. On the other hand, Fig. 1(b) panels show the evolution of the population inversion w (see thetext for more details). presentsresultsforamuchshorterpulse,wherethe Rabi 2
Experimentsonnarrowbandgapsemiconductorshave sideringthelow-orderharmoniccutoffdevelopedinalkali shownaclearsignatureofCWRF,whichmanifesteditself atoms, closely related to their low saturation
intensity). as a split in the third harmonic of the emitted light into To create the conditions for CWRF, we used an atomic theforwarddirection[9]. Recently,Rabifloppingandthe
systeminwhichtheperiodofaRabioscillation[13](cor- consequentcoherentpulsereshapinghasbeenexperimen- respondingto thetransitionbetweenthe groundandthe tallyobservedinaquantumcascadelaser
[10],suggesting first excited states [14, 15]) is similar to one period of anewpromisingapproachtoshortpulsegeneration. One the laser light. For the usual (Ti:Sa) laser sources such of the advantages
of atoms (relative to semiconductors) system was provided by K atoms, with a transition en- is the possibility to employ longer laser pulses and, as a ergy between the ground and the first excited
state of consequence,to exploreabroaderrangeoflaserparame- 1.61eV(hence closeto the lasersourcephotonenergyof ters, as well as provide analternative to carrierenvelope 1.55 eV). phase (CEP)
characterization. In addition, it has been Alkali atoms, due to their atomic structure (gas no- shownthatinsemiconductorstheCoulombinteractionof ble structure plus only one external electron), are
well- carriersin the bands gives rise to an enhancementof the suited to be described by the single active electron ap- external laser field and consequently the envelope pulse proximation (SAE). We
therefore focus on such atoms area by as much as a factor of two, considerably compli- to avoid the possible role of electron-electron correla- cating the observedinterpretationofCWRF phenomena
tions, which have been found to have an important, [2, 9, 11]. yet still poorly understood role in HHG spectra [16]. When the atom was simplistically modeled as a two- Based on SAE approximation, we
use the atomic poten- level system, conventional Rabi flopping behavior and tial reported in [17] to describe K and Na atoms. Using CWRF features were observed (see e.g. [4, 12] and ref- a
Hartree-Fock-based method we set the two parame- erences therein). However, it is well known that the ters Vc and Ve in the following generic potential form, two-level approximation breaks down when
strong elec- VK,Na(r) = (r+Vrc0)2 − (r+Ver0), where Vc accounts for the tricfieldsareapplied,inparticularintheCWRFregime. effectoftheatomiccore(nucleusplusallcompleteshells),
AnimportantquestionemergesastowhatextendCWRF V represents the external potential and r = Vc. Using e 0 Ve could potentially be observed in real atoms. In this Let- this method we find the ground, 3s,
and the first ex- terwedemonstrateforthe firsttime howthe CRWFsig- cited state, 3p, of K, as well as the 4s and 4p for Na, natures show up in the high-order harmonic generation with precisionof∆E
≈±0.0084eV(for details see Table (HHG) spectra of real atoms. In particular, using a ro- I).Inaddition, we alsocompute numericallythe element busttheoreticalapproachthataccuratelymodelsboththe
transition dipole dns→np = hψns|z|ψnpi for both atoms groundandexcitedstatesofKatomscombinedwithreal- (n=3 for Na and n=4 for K) and compare them with
isticlaserparameters,weobserveclearlydistinctfeatures the experimental values reported in Ref. [18] (see Table in the third harmonic and correlate it with the behavior
I).Ascanbededuced,ourtheoreticaldipolevaluesshow of the ground state population. The latter approach is excellentagreementwithexperimentalmeasurements,in- closely related to the description of
semiconductors and dicating that the potential givenaboveaccuratelyrepre- atoms modeled as two-level systems. In order to further sents K and Na atoms. For finding optimal laser pa-
supportourconclusionsofCWRF-likebehaviorinK,we rameters to experimentally observe CWRF in atoms, an also compute HHG of Na atoms, in which the transi- accurate dipole moment is essential since Rabi
flopping tion energy between the ground and first excited state frequency is linearly proportional to the dipole moment is not resonant with the driven light, and, as a conse- as well as the electric
field strength [10, 13]. In addition, quence, a conventional HHG spectra, i.e. single peaks HHG spectra, which is the focus of our investigation, at odd harmonics of the driven frequency, is observed.
is believed to be particularly sensitive to the details of Our predictions can be tested experimentally using cur- electrondynamicsinsideatomsandmolecules,makingit rently available ultrashort laser
pulses of a Ti:Sa laser crucial to not only accurately describe the ground state with wavelengthscentered in the range of 750−800 nm. (as many atomic potentials in the literature do), but, in We
start by describing our theoretical approach, thecaseofresonanttransitionsduetoRabiflopping,also putting special emphasis on the choice of the atomic po- the excited state [16, 19].
tentials,suchthatourresultsforbothgroundandexcited To compute HHG spectra, we numerically solve the statesareinexcellentagreementwithexperimentalmea- three dimensional Time Dependent Schr¨odinger
Equa- surements (see Table I). To find CWRF signatures, we tion(3D-TDSE)inthelengthgaugeandusingtheatomic focus on the HHG spectra. Since the HHG spectra is potential,V
(r),givenaboveforKandNaatoms,re- K,Na proportional to the electron dipole moment, we can es- spectively. Theharmonicyieldfromasingleatomisthen tablish a one to one correspondence between the media,
proportionaltotheFouriertransformofthedipoleaccel- modeledasacollectionofoscillators,andsingleatomsil- eration of its active electron and can then be obtained luminatedbyastronglaserfield
(anymacroscopiceffect, from the electronic wave function after time propaga- such as phase matching, could be safely neglected, con- tion. Our code is based on an expansion of spherical 3 (a) (d) u.)
-6 u.) -6 arb. -7 -10 × 2 arb. -7 eld ( -8 eld ( -8 onic yi -9 -11 onic yi-1-90 arm-10 2.5 3 3.5 arm-11 of h-11 of h-12 og -12 K og -13 Na l l -13 -14 (b) (e) arb. u.) -6 -9 × 2 arb. u.) --76 d ( -7
d ( -8 el el onic yi -8 -10 onic yi-1-90 arm -9 2.5 3 3.5 arm-11 h h og of -10 K og of --1132 Na l l -11 -14 (c) (f) arb. u.) -6 -9 × 2 arb. u.) --76 d ( -7 d ( -8 el el onic yi -8 -10 onic yi-1-90
arm -9 2.5 3 3.5 arm-11 h h og of -10 K og of --1132 Na l l -11 -14 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 harmonic order harmonic order FIG. 2. 3D-TDSE harmonic spectra in K for the corresponding laser
intensities I = 3.158×1011 W/cm2 (panel a), I = 5.6144×1011 W/cm2 (panel b) and I =1.108×1012 W/cm2 (panel c). Panels (d), (e) and (f) represent the HHG in Na for thesamelaser parameters. Theinsetsof
panels(a), (b)and (c)show azoom of thethird harmonicω/ω0 =3(seethetextfor details). Sodium Experimental (NIST) Numerical field peak amplitude (E0 = pI/I0 with I0 = 3.5×1016 (present) W/cm2), ω =
0.0596 a.u. (λ = 765.1 nm), N the total 0 3s 5.139 eV 5.135 eV number of cycles in the pulse and φ the CEP. Further- 3p 3.036 eV 3.038 eV more, T defines the laser period T = 2π/ω0 ≈ 2.5 fs.
Transition dipole 2.49 au 2.40 au In the simulations presented here we consider the case N = 20, corresponding to an intensity envelope of full Potassium Experimental (NIST) Numerical (present) width
half maximum (FWHM) of 0.36NT, (7.2 optical cycles≈18fs FWHM), andφ=0(see belowfordetails). 4s 4.340 eV 4.347 eV 4p 2.730 eV 2.725 eV As discussed in the introduction, we set input laser Transition
dipole 2.92 au 2.79 au frequencythe samevalue (inatomic units) asthe energy corresponding to the transition 4s → 4p of K in order to observe a CWRF-like behavior. In order to compare TABLE I.
Experimental and theoretical values of the energy gap between the ground and first excited states, joint with with a conventional situation (meaning the usual condi- thetransition dipole both for
Naand K. tions for HHG), we use the same input laser parameters with Na atoms, for which the transitionenergy from the ground to the first excited state, 3s → 3p, corresponds harmonics, Ym, and takes
advantage of the cylindrical to 2.10 eV, and is therefore non-resonant with the laser l symmetry of the problem (hence only the m = 0 terms frequency. need to be considered). The time propagation is
based In Fig. 2, we show the harmonic spectra computed on a Crank-Nicolsonmethod implemented on a splitting fromthe3D-TDSEforbothK(Figs.2(a),2(b)and2(c)) of the time-evolution operator that preserves
the norm and Na (Figs. 2(d), 2(e) and 2(f)) atoms. We have cho- of the electronic wave function. The coupling between senthe laserparametersto coverthree differentregimes, the atom and the laser pulse
in the length gauge, lin- namely,forpanels2(a)and2(d)the envelopepulse area, early polarized along the z axis, is written as V (z,t) = Θ , is close to 2π. The envelope pulse area is de- l K,Na E(t)z,
where E(t) is the laser electric field defined by fined as Θ ≈d E ∆t, where d is the dipole K,Na K,Na 0 K,Na E(t)=E0sin2(cid:0)ω2N0t(cid:1)sin(ω0t+φ). E0 is the laser electric transition matrix for K or
Na (see Table I) and ∆t the 4 (a) 3 3 (d) Ground state population0000....12468 --01221 Electric eld (10 a.u.)(cid:222)-3 --21012 00001....2468 Ground state population K -3 -3 Na 0 0 (b) 4 4 (e) on 1
E 1 on Ground state populati0000....2468 -022 lectric eld (10 a.u.)(cid:222)-3 -202 0000....2468 Ground state populati K Na -4 -4 0 0 (c) 6 6 (f) Ground state population0000....12468 --02442 Electric
eld (10 a.u.)(cid:222)-3 --42024 00001....2468 Ground state population K Na -6 -6 0 0 0 5 10 15 20 0 5 10 15 20 Laser periods Laser periods FIG. 3. Time evolution of the ground state population (red
thick line) along the laser pulse (blue thin line) corresponding to thecases plotted in Fig. 2. FWHMpulseduration(∆t=18fsi.e.,∆t≈750au). In envelope area significantly exceeds 2π. particular, for a
laser intensity I =3.158×1011 W/cm2 Togetfurther insightintothe physicalmechanismbe- (E = 0.003 a.u.), Θ ≈ 2π and Θ ≈ 5.4 (for com- 0 K Na hind the complex structure of the HHG spectra in K parison
see the values used in semiconductor GaAs [9]). atoms, in Fig. 3 we present the time dynamics of the Panels 2(c) and 2(f) correspond to values of Θ and K ground state population for all the cases
depicted in Θ , respectively, close to 4π (11.7 for K and 10.1 for Na Fig. 2. For a two-level system, used as a prototypical Na), obtained by keeping constant the pulse duration model for a
semiconductor or a simplistic picture for a and now using a laser intensity I =1.108×1012 W/cm2 realatom, the electrondynamics due to interactionwith (E =0.0056 a.u.). Panels 2(b) and 2(e) were
chosen to 0 laser light can be represented on a Bloch sphere (for de- haveanintermediatevalueofintensityI =5.6144×1011 tails see e.g [4, 9]). In this case, the ground state popu- W/cm2 (E = 0.004
a.u.), corresponding to pulse enve- 0 lations and the regular Rabi oscillations can be depicted lope areas of 8.4 and 7.2 for K and Na, respectively. asmovingalongthe surfaceofthe sphere(see Fig.1
(a)). When the CWRF regime is reached, clear signatures, Fromthe HHGspectraofKatoms,weobserveadras- corresponding to the break-down of the area theorem, ticchangearoundthethirdharmonic,ω/ω =3,aspulse
0 should occur on the Bloch sphere as well (see Fig. 1(b)). envelope area increases. Our results are directly analo- gous to the manifestation of CWRF behavior observed Following an analogy with a
two-level system, we can in semiconductors in [9]. Note the marked contrast to distinctly observeCWRF-like behaviorin Figs.3(b) and the HHG spectra of Na, shown in panels 2(d), 2(e), and 3(c)
andthecorrespondingcounterpartintheHHGspec- 2(f), where a characteristic peak is present in the third tra(Figs.2(b)and2(c)). Onthecontrary,aconventional harmonic regardless of the envelope pulse
area. The on- behaviorinthethirdharmonic(Figs.2(a)and3(a)),can set of this more complex behavior in K atoms for suffi- be correlated with: (i) ordinary Rabi oscillations for the ciently large envelope
pulse areas is in agreement with case of K (Fig. 2(a)), i.e. the ground state is completely conclusions in [3], where the onset of CWRF behavior depopulated, even though the laser intensity is low
and and the consequent break-downof the area theorem was this would be analogous to a travel of the Bloch vector predictedforatwo-levelresonantsystemswhenthepulse fromthesouthtothenorthpole[9];(ii)
normalbehaviour 5 of atoms in strong field for all the Na cases (Figs. 2(d)- the CEP changes when the driving laser field is a few- 2(f)), i.e. gradual depopulation of the ground state due cycle pulse.
Furthermore the CWRF could be used to to laser ionization (Figs. 3(d)-3(f)). control the laser-induced ionization, known as a crucial ingredientforharmonicpropagation,viamanipulationof the ground
state population. (a) = 0 u.) -6 = !/2 × 2 We acknowledge the financial support of the MICINN d (a. -7 -10 p01ro,jaecntds(FFIISS22001008--1020873844),TEORQCATAAd,vFanISc2ed008G-r0a6n3t68Q-CU0A2-- el
yi -8 GATUA and OSYRIS, the Alexander von Humboldt c oni -9 -11 2.5 3 3.5 Foundation (M.L.), and the DFG Cluster of Excel- m ar-10 lence Munich Center for Advanced Photonics. This re- h of
searchhasbeenpartiallysupportedbyFundaci`oPrivada g -11 Cellex. J.A.P.-H.andL.Rosoacknowledgesupportfrom o l-12 K Laserlab-Europe (Grant No. EU FP7 284464) and the Spanish Ministerio de Econom´ıa y
Competitividad (FU- -6 (b) = 0 RIAM Project FIS2013-47741-R). We thanks Christian u.) -7 = !/2 Hackenberger for helping us with the artwork. a. d ( -8 el yi -9 c oni-10 m ar -11 ∗
marcelo.ciappina@mpq.mpg.de h of -12 [1] I. I. Rabi, Phys.Rev.49, 324 (1936). og -13 [2] S.T.Cundiff,A.Knorr,J.Feldmann,S.W.Koch,E.O. l Na G¨obel and H. Nickel, Phys.Rev.Lett. 73, 1178 (1994). -14 [3]
S. Hughes, Phys.Rev.Lett. 81, 3363 (1998). 0 1 2 3 4 5 6 7 [4] M.Wegener,ExtremeNonlinearOptics(Springer-Verlag, harmonic order Berlin, 2005). [5] K. Rzazewski and M. Florjanczyk, J. Phys. B 17, L509
FIG. 4. HHG for K (panel a) and Na (panel b) for different (1984). CEPs. ThelaserparametersarethesameasinFigs.2(b)and [6] M. Florjanczyk, K. Rzazewski and J. Zakrzewski, Phys. 2(e), respectively.
Solid line φ=0, dotted line φ=π/2. Rev. A 31, 1558 (1985). [7] K. Rzazewski, J. Zakrzewski, M. Lewenstein and J. W. In conclusion, we find the signatures of CWRF in real Haus, Phys. Rev.A 31, 2995
(1985). atoms, by studying the third harmonic of alkali atomic [8] M.Lewenstein,J.ZakrzewskiandK.Rzazewski,J.Opt. species. Analogoustothecaseofsemiconductors,wecan Soc. Am.B 3, 22 (1986). correlate
this new feature with the complex dynamics of [9] O.D.Mu¨cke,T.Tritschler,M.Wegener,U.Morgnerand F. X.K¨artner, Phys.Rev. Lett.87, 057401 (2001). the ground state population. Our model uses accurate
[10] H. Choi, V.-M. Gkortsas, L. Diehl, D. Bour, S. Corzine, values for the atomic wavefunction of both ground and J. Zhu, G. H¨ofler, F. Capasso, F. X. K¨artner and T. B. excitedstates
(asisevidencedbytheexcellentagreement Norris, Nat. Phot. 4, 706 (2010). betweenthecalculatedstatesenergieswithexperimental [11] C. Ciuti, C. Piermarocchi, V. Savona, P. E. Selbmann, values)
aswellasaccuratelaserparameters,easilyachiev- P. Schwendimann and A. Quattropani, Phys. Rev. Lett. able with the current laser technology. In particular, a 84, 1752 (2000). Ti:Sa laser provides laser
pulses with wavelengths cen- [12] M. Frasca, J. Opt.B 3, S15 (2001). tered in the range 750−800 nm, very close to the 765 [13] L. Allen and J. H. Eberly, Optical Resonance and Two- level Atoms
(Wiley, 1975). nmvaluecorrespondingtothetransitionenergy4s→4p [14] B.SundaramandP.W.Milonni, Phys.Rev.A41, 6571 in K. As a consequence, the experimental confirmation (1990). of our results appears
straightforward. Moreover, the [15] P. Meystre, Opt.Commun. 90, 41 (1992). CWRF phenomenon in atoms could emerge as a robust [16] A. D. Shiner, B. E. Schmidt, C. Trallero-Herrero, H. J. alternative
for CEP characterization for long pulses as W¨orner, S. Patchkovskii, P.B. Corkum, J.-C. Kieffer, F. can be seen in Fig. 4 where we show HHG spectra for K L´egar´e and D.M. Villeneuve, Nat.Phys. 7,
464 (2011). (4(a)) and Na (4(b)) and for two different values of the [17] J. A. P´erez-Hern´andez, Fotoionizacio´n de ´atomos alcali- nos mediante un l´aser pulsado intenso (Master Thesis, CEP. Note
that in the case of Na (out of resonance) the Universidad deSalamanca, 2004). third harmonic does not present appreciable differences. [18] D. A. Steck, Los Alamos National Laboratory. Los However the
third harmonic of K is strongly affected, Alamos, NM 87545. http://steck.us/alkalidata/ in spite of the fact the driving laser is rather long (20 [19] A.E.Boguslavskiy,J.Mikosch,A.Gijsbertsen, M.Span-
cycles) of total duration. It is well known that in a con- ner, S. Patchkovskii, N. Gador, M.J.J. Vrakking and A. ventional situation the HHG spectra is only sensitive to Stolow, Science 335, 1336
See more | {"url":"https://www.zlibrary.to/dl/carrier-wave-rabi-flopping-signatures-in-high-order-harmonic-generation-for-alkali-atoms","timestamp":"2024-11-04T16:57:40Z","content_type":"text/html","content_length":"148795","record_id":"<urn:uuid:2e286c76-78d1-452c-b35f-150c7d371bc1>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00333.warc.gz"} |
Journal Article
Tensor representations for the Drinfeld double of the Taft algebra
There are currently no full texts shared for your IP range.
There are no public fulltexts stored in PuRe
There is no public supplementary material available
Benkart, G., Biswal, R., Kirkman, E., Nguyen, V. C., & Zhu, J. (2022). Tensor representations for the Drinfeld double of the Taft algebra. Journal of Algebra, (606), 764-797. doi:10.1016/
Cite as: https://hdl.handle.net/21.11116/0000-000A-D2C4-B
Over an algebraically closed field $\mathbb k$ of characteristic zero, the
Drinfeld double $D_n$ of the Taft algebra that is defined using a primitive
$n$th root of unity $q \in \mathbb k$ for $n \geq 2$ is a quasitriangular Hopf
algebra. Kauffman and Radford have shown that $D_n$ has a ribbon element if and
only if $n$ is odd, and the ribbon element is unique; however there has been no
explicit description of this element. In this work, we determine the ribbon
element of $D_n$ explicitly. For any $n \geq 2$, we use the R-matrix of $D_n$
to construct an action of the Temperley-Lieb algebra $\mathsf{TL}_k(\xi)$ with
$\xi = -(q^{\frac{1}{2}}+q^{-\frac{1}{2}})$ on the $k$-fold tensor power
$V^{\otimes k}$ of any two-dimensional simple $D_n$-module $V$. This action is
known to be faithful for arbitrary $k \geq 1$. We show that
$\mathsf{TL}_k(\xi)$ is isomorphic to the centralizer algebra
$\text{End}_{D_n}(V^{\otimes k})$ for $1 \le k \le 2n-2$. | {"url":"https://pure.mpg.de/pubman/faces/ViewItemOverviewPage.jsp?itemId=item_3398390","timestamp":"2024-11-03T03:03:49Z","content_type":"application/xhtml+xml","content_length":"42136","record_id":"<urn:uuid:733271fc-3534-4811-b40b-7149951ea1db>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00841.warc.gz"} |
Multiple Faces Tracking Using Feature Fusion and Neural Network in Video
Intelligent Automation & Soft Computing
Multiple Faces Tracking Using Feature Fusion and Neural Network in Video
1College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China
2College of Mathematics and Statistics, Hengyang Normal University, Hengyang, 421002, China
3School of Computer Engineering and Applied Mathematics, Changsha University, Changsha, 410003, China
4College of Computer Science and Electronic Engineering, Hunan University, Changsha, 410082, China
5Key Laboratory of Digital Signal and Image Processing of Guangdong, Shantou, 515063, China
*Corresponding Author: Boxia Hu. Email: huboxia@hynu.edu.cn
Received: 26 May 2020; Accepted: 07 August 2020
Abstract: Face tracking is one of the most challenging research topics in computer vision. This paper proposes a framework to track multiple faces in video sequences automatically and presents an
improved method based on feature fusion and neural network for multiple faces tracking in a video. The proposed method mainly includes three steps. At first, it is face detection, where an existing
method is used to detect the faces in the first frame. Second, faces tracking with feature fusion. Given a video that has multiple faces, at first, all faces in the first frame are detected correctly
by using an existing method. Then the wavelet packet transform coefficients and color features from the detected faces are extracted. Furthermore, we design a backpropagation (BP) neural network for
tracking the occasional faces. At last, a particle filter is used to track the faces. The main contributions are. Firstly, to improve face tracking accuracy, the Wavelet Packet Transform coefficients
combined with traditional color features are utilized in the proposed method. It efficiently describes faces due to their discrimination and simplicity. Secondly, to solve the problem in occasional
face tracking, and improved tracking method for robust occlusion tracking based on the BP neural network (PFT_WPT_BP) is proposed. Experimental results have been shown that our PFT_WPT_BP method can
handle the occlusion effectively and achieve better performance over other methods.
Keywords: Face tracking; feature fusion; neural network; occlusion
The problem of face tracking can be considered as finding an effective and robust way. It is based on using the geometric dependence of facial features to combine the independent face detectors of
various facial features, and then get an accurate estimation of the position of each image’s facial features in the video sequence [1]. Particle filter realizes recursive Bayesian filtering by the
Nonparametric Monte Carlo simulation method [2]. It can be applied to any nonlinear system which can be described by the state-space model, and its accuracy can approach the optimal estimation. The
Particle filter is simple and easy to implement and provides an effective method for the analysis of nonlinear dynamic systems [3,4]. It has attracted extensive attention in the fields of target
tracking, signal processing, and automatic control Bui Quang et al. [5]. It approximates the posterior distribution through a set of weighted assumptions which are called particles. During the
tracker based on particle filter, there is a likelihood function that generates a weight for each particle and the particles are distributed according to a tracking model. Then, the particles are
properly placed, weighted, and propagated. After calculating the posterior distribution of the particles, the most likely position of a face can be estimate sequentially [6,7]. In many cases
including some complex backgrounds, particle filter achieves good performance and is used in more and more applications [8]. Though particle filter has a good performance in target tracking, there
still exist some problems in face tracking. The common particle filter methods cannot handle an occlusion especially a full occlusion [9,10]. Some tracking results with and without occlusions for
particle filter are shown in Fig. 1. The tracking performance becomes poor when a face occlusion occurs. The main reason is faces are similar and the re-sampling would propagate the wrong random
samples since the likelihood of those occluded faces and lead to meaningless tracking. Therefore, dealing with occlusions is a crucial part of multiple faces tracking.
This paper presents an occlusion robust tracking (PFT_WPT_BP) method for multiple faces tracking. The three main contributions of this paper are summarized as follows:
– After detecting faces, wavelet packet decomposition is used to generate some frequency coefficients of images. We separately use its higher and lower frequency coefficients of the reconstructed
signal to improve the faces tracking performance.
– We define a neural network for tracking the faces with occlusion. When faces tracking fails due to face-occlusion, the neural network is used to predict the next step of the occlusion face/faces.
– A method based on particle filter and multiple feature fusion for faces tracking in the video is proposed. The proposed method has good performance and is robust in multiple faces tracking.
The particle filter algorithm is derived from the idea of Monte Carlo [11], which refers to the probability of an event by its frequency. Therefore, in the process of filtering, where probability
such as P(x) is needed, variable x is sampled, and P(x) is approximately represented by a large number of samples and their corresponding weights. Thus, with this idea, particle filter can deal with
any form of probability in the process of filtering, unlike Kalman filter [12], which can only deal with the probability of linear Gaussian distribution. This is one of the advantages of a particle
Some researchers use a histogram method of tracking face [13]. Nevertheless, this method has a significant limitation, and many factors can affect the similarity of the two images, such as light,
posture, face vertical or left-right angle deviation, and so on. As a result, the face tracking result sometimes is poor, and it is hard to use in practical application. Reference [14] proposed a new
face-tracking method which is based on the Meanshift algorithm. In these methods, the face position of the current frame is updated according to the histogram of the target in the previous frame and
the image obtained in the current frame. These methods are suitable for single target tracking, and the effect is charming. However, when the non-target and target objects are occluded, it often
leads to the temporary disappearance of the target [15]. When the target reappears, it is often unable to track the target accurately. Therefore, the robustness of the algorithm is reduced. Because
color histograms are robust to partial occlusion, invariant to rotation and scaling, and efficient in computation, they have many advantages in tracking nonrigid objects. Reference [16] proposed a
color-based particle filter for face tracking. In this method, Bhattacharyya distance is used to compare the histogram of the target with the histogram of the sample position during the particle
filter tracking. When the noise of the dynamic system is minimal, or the variance of observation noise is microscopic, the performance of the particle filter is terrible. In these cases, the particle
set quickly collapses to a point in state space. Reference [17] proposed a Kernel-based Particle Filter for face tracking. The standard particle filter usually cannot produce the set of particles
that capture the “irregular” motion, which leads to the gradual drift of the estimated value and the loss of the target. There are two difficulties in tracking different numbers of nonrigid objects:
First, the observation model and target distribution can be highly nonlinear and non-Gaussian. Secondly, the existence of a large number of different objects will produce overlapping and fuzzy
complex interactions [18,19]. A practical method is to combine the hybrid particle filter with AdaBoost. The critical problem of the hybrid particle filter is the selection of scheme distribution and
the processing of objects leaving and entering the scene. Reference [20] proposed a three-dimensional pose tracking method that mixes particle filter and AdaBoost. The hybrid particle filter is very
suitable for multi-target tracking because it assigns a hybrid component to each player. The proposed distribution can be built by using a hybrid model that contains information from each
participant’s dynamic model and test assumptions generated by AdaBoost.
3 The Framework of Our Multiple Faces Tracking Approach
Given a face model, a state equation is defined as
where Fig. 2.
For such faces tracking problems, particle filter filters out the real state
Prediction stage: Particle filter firstly generates a large number of samples according to the probability distribution of
Correction stage: After the observation value y arrives, all particles are evaluated by using the observation equation, i.e., conditional probability p(y|xi). To be frank, this conditional
probability represents the probability of obtaining the observation y when assuming the real state i. Let this conditional probability be the weight of the i-th particle. In this way, if all
particles are evaluated, the more likely they are to get the particles observing y, the higher the weight of course.
Resampling algorithm: Remove the particles with low-weight and copies the particles with high weight. What we get is, of course, the real state
Since we know nothing about
In our method, the state of a face at time t is denoted as st and its history is S = {s1; s2;…; st}. The basic idea of the particle filter algorithm is to compute the posterior state-density at time
t using process density and observation density. Aiming to improve the sampling of the particle filter, we propose an improved algorithm based on combining with Wavelet Packet Transform and HSV color
4 Face Tracking Model with Wavelet Packet Transform and Color Feature Fusion
4.1 Face Feature Extraction Base on the Wavelet Packet Transform
The theory of Wavelet Packet Transform and feature extraction are introduced in this part. Wavelet packet analysis is an extension of wavelet analysis and will decompose not only the approximate but
also the detail of the signal [21]. Wavelet packet decomposition provides the finer analysis as follows:
where 22], t is a parameter in the time-domain, k = 1,2,3…N. The result of the Wavelet Packet Transform is shown as a full decomposition tree, as depicted in Fig. 3. A low (L) and high (H) pass
filter is repeatedly applied to the signal S, followed by decimation by Eq. (2), to produce a complete subband tree decomposition to some desired depth.
More face features are generated by wavelet packet decomposition and are used in face tracking. An example of face image decomposition is shown in Fig. 4.
In our work, the face model is defined as parameters set
where P is the transition matrix, Q is the system noise matrix and
where xk, yk represent the center of the target region at time step k. From this, the state vector i-th particle at time step k, the system matrices P and Q, and noise vectors Eqs. (5)–(7),
The observation process is performed to measure and weigh all the newly generated samples. The visual observation is a process of visual information fusion including two sub-processes: The
computation sample weights
For the kth sample, we obtaine the weight 16] as shown in Eq. (8), calculating the similarity between sample histogram features
To compute the sample weight
With two different visual cues, we obtain the final weight for the kth sample as:
4.3 Multi-Face Tracking Algorithm Based on Particle Filter
Our multi-face tracking system consists of two parts: Automatic face detection and particle filter tracking. In the tracking system, the boosted face detector which is introduced above achieves
automatic initializations when the system starts or when a tracking failure occurs. The face detection results are used to update the reference face model. The updating criterion is confidence values
that are less than a threshold valued for M successive frames.
The proposed tracking algorithm includes four steps, as shown below.
1. Initialization:
i) Automatic face detection and n faces are detected.
ii) Initialize the K particles.
2. Particle filter tracking: Probability density propagation from k th sample at time t, and
For k = 1:K
For j = 1:n
(1) Sample selection, generate a sample set as follows
i) Generate a random number
ii) Find, by binary subdivision, the smallest p for which
iii) Set
(2) Prediction, obtain
(3) Weight measurement
i) Calculate the color histogram feature according to Eq. (8).
ii) Calculate the wavelet packet decomposition feature according to Eq. (9).
iii) Calculate the sample weight according to Eq. (10).
3. For each face, normalize the sample set so that weight
4. Estimate state parameters of the pointing gesture at time-step t:
5. If updating criteria are satisfied, go to Step 1; otherwise, go to Step 2.
5 Multiple Faces Tracking in Occlusions
An occlusion usually exists in multiple faces tracking and, it could cause a failure in the tracking of multiple faces because two objects are of high similarity. In our study, an occlusion tracking
method combined with a neural network algorithm is proposed.
Take two faces as an example, if no faces are occluded, there are very few relationships between the particles in different faces. The two faces can be tracked by using a traditional particle filter.
When a facial occlusion occurs, the particles in different faces will be occluded, as shown in Fig. 5. The overlapped area will affect the tracking result, even causes tracking failure.
(1) Occlusion detection
During initializing, the face Fig. 1, where k is the frame,
The overlapped area between two faces in
(2) The spatial position of overlapped area judgment
After the face-occlusion detecting, the spatial position of these faces must be judged. We can define the likelihood between
(3) Face tracking update
Define Fig. 6.
Choosing input nodes
We experimented using video data sets downloaded from [20]. The experiments were implemented on a Intel(R) Xeon(R) E31220 3.1 GHz CPU and 8192 MB RAM. The resolution of each frame was 720 × 480
pixels per image. During our research, we do not focus on face detection, since many existing methods can be used to detect faces [21,22]. Comparing with other methods, the method [22] performed well
and provided excellent results achieving a higher detection rate, and it was used in our face detection.
We have carried out some experiments to track one face with our proposed method PFT_WPT_BP, and the particle filter number was 200. The color square showed the region of the tracked face. Fig. 7
showed the experimental results of one face tracking (even numbers in frame 1 to 13) based on different methods with Kalman Filter [12], Particle Filtering [10] and PFT_WPT_BP, where f is a label of
the frame in the video. And Fig. 8 showed the tracking results of one face tracking in frame 19 to 31. We can find that satisfactory experimental results were achieved in three methods.
6.2 Multiple Faces Tracking Results
We have carried out some experiments to track multiple faces too, and the particle filter number was 200. The colored square shows the region of the tracked face. Fig. 9 shows theexperimental results
of three faces tracked (frame 1 to 13) which provided satisfactory results.
With face-occlusion (frame 20 to 37), we experimented using different tracking methods. The face of the first person (blue cloth) occluded the face of the second person (black cloth) in frame 20 to
33. And Fig. 10 showsfaces tracking with occlusion based on different methods. We found that the faces tracking failed for occluded faces based on Kalman Filter and Particle Filtering methods as
indicated in Fig. 10 (line 1 and 2). But our method could achieve acceptable results in Fig. 10 (line 3).
After several frames, the third person’s face (white cloth) occluded the second person’s face (black cloth) at the beginning of the frame. We detected the faces again, and found that the faces
tracking failed in face-occlusion after frame 70. The results based on Kalman Filter and particle filtering is shown in Fig. 11 (lines 1 and 2). The occluded face was successfully tracked based on
our method as shown in Fig. 11 (line 3). The system successfully recovered the faces from occlusion. After the occlusion, each face was normally re-sampled and the face appearances were updated
This paper presents an occlusion robust tracking method for multiple faces. Experimental results have been shown that our PFT_WPT_BP method can handle the occlusion effectively and achieve better
performance than several previous methods. BP neural network is used to predict the occasional faces. We assume that the occasional face would not miss a long time. If a face is missing for a long
time, it is difficult to track it, and we can find the face by face detection. The faces tracking in a more complex environment will be researched in our future work.
Funding Statement: The Project Supported by Scientific Research Fund of Hunan Provincial Education Department (16C0223), the Project of “Double First-Class” Applied Characteristic Discipline in Hunan
Province (Xiangjiaotong [2018] 469), the Project of Hunan Provincial Key Laboratory (2016TP1020).
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
1. J. Goenetxea, L. Unzueta, F. Dornaika and O. Otaegui. (2020). “Efficient deformable 3D face model tracking with limited hardware resources,” Multimedia Tools & Applications, vol. 79, no. 6, pp.
12373–12400. [Google Scholar]
2. M. Tian, Y. Bo and Z. Chen. (2019). “Multi-target tracking method based on improved firefly algorithm optimized particle filter,” Neurocomputing, vol. 359, no. 24, pp. 438–448. [Google Scholar]
3. G. S. Walia, A. Kumar, A. Sexena, K. Sharma and K. Singh. (2019). “Robust object tracking with crow search optimized multi-cue particle filter,” Pattern Analysis & Applications, vol. 3, no. 1, pp.
434–457. [Google Scholar]
4. K. Yang, J. Wang, Z. Shen, Z. Pan and W. Yu. (2019). “Application of particle filter algorithm based on gaussian clustering in dynamic target tracking,” Pattern Recognition and Image Analysis,
vol. 29, no. 3, pp. 559–564. [Google Scholar]
5. P. B. Quang, C. Musso and F. L. Gland. (2016). “Particle filtering and the laplace method for target tracking,” IEEE Transactions on Aerospace and Electronic Systems, vol. 52, no. 1, pp. 350–366.
[Google Scholar]
6. T. T. Yang, H. L. Feng, C. M. Yang, G. Guo and T. S. Li. (2018). “Online and offline scheduling schemes to maximize the weighted delivered video packets towards maritime CPSs,” Computer Systems
Science and Engineering, vol. 33, no. 2, pp. 157–164. [Google Scholar]
7. Y. T. Chen, J. Wang, S. J. Liu, X. Chen, J. Xiong. (2019). et al., “Multiscale fast correlation filtering tracking algorithm based on a feature fusion model,” Concurrency and Computation: Practice
and Experience, vol. 31, no. 10, pp. e5533. [Google Scholar]
8. X. Zhang, W. Lu, F. Li, X. Peng and R. Zhang. (2019). “Deep feature fusion model for sentence semantic matching,” Computers, Materials & Continua, vol. 61, no. 2, pp. 601–616. [Google Scholar]
9. Y. T. Chen, W. H. Xu, J. W. Zuo and K. Yang. (2019). “The fire recognition algorithm using dynamic feature fusion and IV-SVM classifier,” Cluster Computing, vol. 22, no. 3, pp. 7665–7675. [Google
10. T. Wang, W. Wang, H. Liu and T. Li. (2019). “Research on a face real-time tracking algorithm based on particle filter multi-feature fusion,” Sensors, vol. 19, no. 5, pp. 1245. [Google Scholar]
11. Q. Wang, M. Kolb, G. O. Roberts and D. Steinsaltz. (2019). “Theoretical properties of quasi-stationary Monte Carlo methods,” Annals of Applied Probability, vol. 29, no. 1, pp. 434–457. [Google
12. P. R. Gunjal, B. R. Gunjal, H. A. Shinde, S. M. Vanam and S. S. Aher. (2018). “Moving object tracking using kalman filter,” in International Conf. on Advances in Communication and Computing
Technology, Sangamner, India, pp. 544–547. [Google Scholar]
13. W. Singh and R. Kapoor. (2018). “Online object tracking via novel adaptive multicue based particle filter framework for video surveillance,” International Journal of Artificial Intelligence Tools
, vol. 27, no. 6, 1850023. [Google Scholar]
14. M. Long, F. Peng and H. Y. Li. (2018). “Separable reversible data hiding and encryption for HEVC video,” Journal of Real-Time Image Processing, vol. 14, no. 1, pp. 171–182. [Google Scholar]
15. S. Sonkusare, D. Ahmedt-Aristizabal, M. J. Aburn, V. T. Nguyen, T. Pang. (2019). et al., “Detecting changes in facial temperature induced by a sudden auditory stimulus based on deep
learning-assisted face tracking,” Scientific Reports, vol. 9, no. 1, pp. 4729. [Google Scholar]
16. R. D. Kumar, B. N. Subudhi, V. Thangaraj and S. Chaudhury. (2019). “Walsh-Hadamard kernel based features in particle filter framework for underwater object tracking,” IEEE Transactions on
Industrial Informatics, vol. 16, no. 9, pp. 5712–5722. [Google Scholar]
17. B. Eren, E. Egrioglu and U. Yolcu. (2020). “A hybrid algorithm based on artificial bat and backpropagation algorithms for multiplicative neuron model artificial neural networks,” Journal of
Ambient Intelligence and Humanized Computing, vol. 2, no. 6, pp. 1593–1603. [Google Scholar]
18. K. Picos, V. H. Diaz-Ramirez, A. S. Montemayor, J. J. Pantrigo and V. Kober. (2018). “Three-dimensional pose tracking by image correlation and particle filtering,” Optical Engineering, vol. 57,
no. 7, 073108. [Google Scholar]
19. Y. Song, G. B. Yang, H. T. Xie, D. Y. Zhang and X. M. Sun. (2017). “Residual domain dictionary learning for compressed sensing video recovery,” Multimedia Tools and Applications, vol. 76, no. 7,
pp. 10083–10096. [Google Scholar]
20. J. Wang and Y. Yagi. (2008). “Integrating color and shape-texture features for adaptive real-time object tracking,” IEEE Transactions on Image Processing, vol. 17, no. 2, pp. 235–240. [Google
21. H. H. Zhao, P. L. Rosin and Y. K. Lai. (2019). “Block compressive sensing for solder joint images with wavelet packet thresholdin,” IEEE Transactions on Components Packaging & Manufacturing
Technology, vol. 9, no. 6, pp. 1190–1199. [Google Scholar]
22. J. Chen, D. Jiang and Y. Zhang. (2019). “A common spatial pattern and wavelet packet decomposition combined method for EEG-based emotion recognition,” Journal of Advanced Computational
Intelligence and Intelligent Informatics, vol. 23, no. 2, pp. 274–281. [Google Scholar]
This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited. | {"url":"https://www.techscience.com/iasc/v26n6/41008/html","timestamp":"2024-11-04T21:46:54Z","content_type":"application/xhtml+xml","content_length":"91735","record_id":"<urn:uuid:f7237061-ab9c-4772-bc76-1662be69af32>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00094.warc.gz"} |
Besides Tikhonov regularization, which is probably the most well-known regularization method for linear as well as nonlinear inverse problems, iterative regularization algorithms have more recently
been investigated and applied successfully to the solution of, especially nonlinear and large-scale problems. The design and analysis of problem adapted iterative regularization methods is one of the
major tasks of this project. Additionally, the efficient implementation, coupling with discretization techniques (forward solvers) and the design of preconditioners are fields of ongoing research.
Analysis of regularization methods
Many iterative methods have been developed and analysed for well-posed problems. Their application to inverse, especially ill-posed problems, is not straight forward and requires an analysis in the
framework of regularization methods. There,
• Design and analysis of (new) iterative regularization algorithms:
in particular Newton-type regularization methods including parameter choice strategies and discretization matters. Recently, a Newton-Kaczmarz iteration has been proposed for the regularization
of large-scale problems, especially time dependent problems and parameter estimation from boundary data, where the data is (part of) the Dirichlet-to-Neumann map.
• SQP-type methods:
Such methods are closely related to Augmented Lagrangian or All-at-once methods in optimal control, shape and topology optimization (see also Subproject F1309)). SQP-type methods have been
investigated as regularization methods for inverse problems governed by PDE's.
The main advantage of the SQP approach is that the sparse structure coming from finite element discretizations is preserved in the formulation of the inverse problem. The fast solution of
indefinite linear systems naturally appearing in the SQP framework, in particular the design of effective preconditioners, will be the next step.
□ M. Burger, W. Mühlhuber, Iterative regularization of parameter identification problems by SQP-methods, Inverse Problems, 18:943-970, 2002. Preprint: ps-file
□ M. Burger, W. Mühlhuber, Numerical Approximation of an SQP-type Method for Parameter Identification, SIAM J. Numer. Anal., 40(5):1775-1797, 2002. Preprint: ps-file
Acceleration of iterative regularization methods
One of the major drawbacks of iterative regularization methods for ill-posed problems is that - due to the ill-posedness, which results in ill-conditioned finite dimensional approximations - usually
a high number of iterations is needed in order to reconstruct (order optimal) approximations of a solution. This behavior also appears for Newton-type iterations, if the linearized systems are again
solved by iterative regularization methods. In order to reduce the overall numerical effort of the solution process, several strategies are pursued:
• Preconditioning iterative regularization methods for linear and nonlinear inverse problems:
The design of efficient preconditioners has recently become one of the main topics in numerical mathematics, especially in the FEM community. While the theory of preconditioning well-posed
problems arising in PDE's is well developed, preconditioning of inverse, in particular ill-posed, problems is not so well understood. Taking into account the ill-posed nature of the problems
under consideration, the number of iterations needed to achieve optimal convergence rates for the solution of linear and nonlinear inverse problems by iterative regularization methods can
essentially be reduced to the square root by appropriate preconditioning (in Hilbert scales).
Reconstruction of an unknown source term.
Left: standard iterations, right: preconditioned.
□ H. Egger, A. Neubauer, Preconditioning Landweber Iteration in Hilbert Scales, SFB-Report 2004-25. ps-file pdf-file
□ H. Egger, Semiiterative Regularization in Hilbert Scales, SFB-Report 2004-26. ps-file pdf-file
• Efficient implementation and/or preconditioning the solution of linearized systems arising in Newton-type methods:
A direct application of Newton's method to the solution of inverse problems is not possible, since the ill-posedness of the nonlinear problem usually implies the ill-posedness of the linearized
systems, which have to be solved in every Newton step. In order to ensure stability, the linearized equations have to be solved by regularization methods instead. For large-scale problems,
iterative regularization methods turn out to be appropriate. In a first step, acceleration of the Newton-Landweber method by by using faster semiiterative regularization methods for the stable
solution of the linearized Newton equations has been investigated. In a second step, the effect of preconditioning will be considered.
□ H. Egger Accelerated Newton-Landweber iterations for the solution of nonlinear inverse problems, SFB-Report 2005-3. ps-file pdf-file
• Fast iterative solution and preconditioning of indefinite systems arising in SQP methods:
Saddlepoint problems naturally appear in many applications, e.g., in (Navier-)Stokes equations, or in optimal control problems. Thus fast solution and especially, preconditioning of saddlepoint
problems has attracted significant interest in the last years. The saddlepoint problems stemming from an SQP formulation of parameter identification problems governed by PDEs show an additional
ill-posedness, i.e., the part of the system, which is usually considered to be uniformly elliptic, now depends on a regularization parameter, which may become arbitrarily small.
Problem adapted regularization strategies and theory
The general theory for the regularization of inverse problems is formulated for very general problems. For special classes of problems, the results can be improved significantly. A problem adapted
theory of regularization methods, and even the design of problem adapted regularization algorithms is thus an important task. Topics of ongoing research are, e.g.,
• Derivation of a problem adapted convergence theory for selected applications and problem classes.
• SQP-type methods: As already mentioned, SQP-methods are specially well suited for the application to inverse problems governed by PDE's. Their application to inverse problems for differential
inequalities is one of the research topics this project.
• The design and analysis of new methods, e.g. derivative-free methods, which can be formulated without a derivative of the forward operator or online-algorithms for time dependent inverse
□ P. Kügler, A Derivative Free Landweber Method for Parameter Identification in Elliptic PDEs, Inverse Problems, 19:1407-1426, 2003. Preprint: ps-file pdf-file
□ P. Kügler, A Derivative Free Landweber Method for Parameter Identification in Elliptic Partial Differential Equations with Application to the Manufacture of Car Windshields, PhD Thesis,
Johannes Kepler University, 2003.
□ P. Kügler, An approach to online parameter estimation in nonlinear dynamical systems, SFB-Report 2004-18. ps-file pdf-file
• Level set methods for inverse problems.
Efficient discretization
Sophisticated discretization strategies become an important factor when it comes to the implementation of regularization algorithms. Especially for large scale problems, the efficient coupling of
discretization and iteration process can significantly reduce the overall numerical effort. Topics of recent research are
• Multilevel and multigrid techniques for regularization methods.
□ M. Burger, W. Mühlhuber, Numerical Approximation of an SQP-type Method for Parameter Identification, SIAM J. Numer. Anal., 40(5):1775-1797, 2002. Preprint: ps-file
□ B. Kaltenbacher, On the regularizing properties of a full multigrid method for ill-posed problems, Inverse Problems, 17:767-788, 2001. Preprint: ps-file
□ Kaltenbacher, B.: A Multi-grid Method with A Priori and A Posteriori Level Choice for the Regularization of Nonlinear Ill-Posed Problems. March 2000. Eds.: Heinz W. Engl, Ulrich Langer
• Regularization by (adaptive) discretization.
Discretization issues are investigated in close cooperation with
Subproject F1306
Please direct your comments or eventual problem reports to webmaster.
SpezialForschungsBereich SFB F013 | Special Research Program of the
FWF - Austrian Science Fund | {"url":"http://www.sfb013.uni-linz.ac.at/index.php?id=f1308-iterative","timestamp":"2024-11-03T06:41:43Z","content_type":"text/html","content_length":"39488","record_id":"<urn:uuid:32099b18-74f1-4b2f-b3bb-7ff098ca9388>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00626.warc.gz"} |
How to select the right linear actuator - Tutorial
High-quality linear actuators are tiny powerhouses: Despite their compact size, they support high input speeds and, at the same time, deliver high output forces. When selecting a linear actuator, it
is advisable to adopt a systematic approach and to precisely determine which model can deliver the required power and also reliably withstand power peaks.
The FAULHABER Drive Calculator makes choosing the right linear actuator easy: This free online tool for selecting drives helps you find perfectly tailored solutions for the concrete application case.
To use the Drive Calculator efficiently for calculating linear actuators, you will need some key data.
In this tutorial, we guide you through five basic steps for linear actuator selection and show you how to determine the parameters required for calculating the drives.
To make it easier for you to apply these five steps to your own projects, we will demonstrate how to select a linear actuator for a specific application based on a concrete example.
Depending on your planned application, certain mechanical prerequisites may exist from the outset, which limit your choice to linear actuators with certain features. For example, it is often the case
that a certain type of lead screw is essential or that the space available for the drive is limited. To ensure that your shortlist contains only models that meet these basic prerequisites, it is
worth starting the selection of a linear actuator with the creation of a requirement profile for the planned application.
In addition to factors such as lead screw type, stroke length and diameter of the linear actuator, we also recommend that you note down the required forces and speeds as well as the planned cycle.
This will then allow you to determine the products which are right for your application more easily. Figure 1 shows a blank version of such a requirement profile.
Basic parameters such as the lead screw type, stroke length or diameter of a linear actuator can be found in our data sheets. Using this data, you can compare the requirement profile for the planned
application generated in the first step with the performance capability and any limits of various solutions.
Once the lead screw type and maximum diameter are known, it is usually possible to make an initial shortlist from the available linear actuators. In our example, a ball screw and a maximum diameter
of 22 mm are critical for the suitability between drive and application. Here, for example, the linear actuator 22L SB xx:1 6x2 150 from the FAULHABER product range would come under consideration
because it fulfills the basic parameters.
For reasons of clarity, however, our application example is based on only a small number of selected standard application parameters. Depending on the type of application you are planning and the
environment in which it is to be used, a multitude of other factors - e.g. system accuracy, temperature range or used materials - may be of central importance for linear actuator selection. The data
table shown here as well as the following four calculation steps are therefore intended only as a guide for an initial assessment of potential solutions.
Later in the tutorial, we will use an example application to show you how you can check in a specific case whether your favored model meets the requirements of the planned application. In this
example case, the application has the following basic data:
When selecting the appropriate linear actuator, the motor with which the linear actuator is to be combined also plays a key role. If the motor is not powerful enough for the planned application, it
will constantly overheat during operation. Consequently, additional heat is transferred to the linear actuator - and this reduces the effectiveness of the lubricant and, as a result, can shorten the
service life of the entire device combination. For this reason, it is advisable to ensure that the motor does not exceed a temperature of 60 °C to 70 °C in continuous operation and in doing so to
prevent premature degradation of the lubricant.
For each of the available solutions, the input speed and the input torque must be calculated using the following formulas:
After you have identified the linear actuators that fulfill the basic mechanical parameters of the planned application, you now need to find out which of these models can deliver the required forces
and speeds. This ensures fault-free operation of your application.
In the data sheet of the linear actuator, check the actual critical lead screw speed (Vcr_std) according to the lead screw bearing system (fixed - free or fixed - single). If the stroke length
differs from the standard, the actual lead screw speed Vcr can be determined using the following formula:
In our example, we consider the bearing version 22L SB xx:1 6x2 150, supported (fixed – single):
To make sure that no resonance problem occurs during operation of this linear actuator in the planned application, you should also check whether the critical speed is above the maximum cycle speed:
Vcr_l > Vmax
In our example case, this requirement is met because the critical speed is vcr_l = 690 mm/s and therefore well above the maximum cycle speed of the linear actuator (vmax = 50 mm/s).
For each available reduction ratio, check whether the required maximum speed (Vp max) is below the specified limit (Vp max ≥ Vmax). The maximum output speed as well as the maximum continuous force
range for each drive stage can be found in the data sheet of the respective linear actuator.
In our example, we refer to the data sheet of the linear actuator 22L SB xx:1 6x2 150 and note that all ratios > 6.6:1 can be ruled out.
In our application example, the cycle input data is taken into account. This results in the following average output speed:
If it is evident that that the selected linear actuator is able to achieve and maintain the required speeds, in the next step we check which forces act on the drive in the concrete configuration. As
with linear speed, here too we follow three steps to determine whether the linear actuator is able to withstand the different forces that act on it during operation.
Refer to the data sheet of the linear actuator to determine the actual buckling force of the lead screw (Fb_std) according to the lead screw bearing system (fixed - free or fixed - single). If the
stroke length differs from the standard, you can calculate the actual value of the buckling force (Fb) using the following formula:
In our example case, the data sheet of the FAULHABER linear actuator 22L SB xx:1 6x2 150 (fixed - single) states the following: Fb_l = Fb_std = 2562 N
We now make certain that the buckling force is above the maximum cycle force (Fb_l > Fmax), i.e. that no buckling problems will occur when commissioning the linear actuator in this application. In
our example, this requirement is met because the buckling force of the lead screw of our linear actuator is Fb_l = 2562 N and therefore well above the maximum cycle force Fmax = 100 N.
For each available reduction ratio, check whether the required maximum axial force is below the specified limit (Fp max ≥ Fmax). For this calculation too, the maximum output speed as well as the
maximum continuous force range for each drive stage can be found in the data sheet of the selected linear actuator. In our example, the requirement is satisfied for all ratios.
In our application example, the cycle input data for the selected model, the FAULHABER linear actuator 22L SB xx:1 6x2 150, is also used here. This leads to the following calculation:
For each available reduction ratio, we now check whether the required average force is below the specified limit (Fm max ≥ Fm). For this step too, the maximum output speed as well as the maximum
continuous force range for each drive stage can be found in the data sheet of the selected linear actuator. In our example, the requirement is satisfied for all ratios.
In principle, it is possible to operate a linear actuator with a higher average axial force than specified in the data sheet of the respective product. As this performance optimum includes a certain
buffer, a moderate increase in the forces acting on the axis will not usually lead to damage or faults. However, to maximize the service life of the drive and to ensure smooth operation, it is
advisable to observe the recommended value and, where necessary, select a more robust linear actuator. In doing so, you ensure that power peaks do not cause the average axial force to be
significantly exceeded.
In the fifth and final step of selecting the appropriate linear actuator for an application, we now check whether the selected model can deliver the required output power. To do so, we determine the
output power (Pmax) and compare it with the required maximum mechanical power.
For each cycle step, the mechanical power can be calculated using the following formula:
You are in the process of selecting a linear actuator and need a data sheet that you can't find anywhere on our website? Or you have already selected a linear actuator, but you are still unsure which
of the motors from our product range would be the ideal combination partner for the planned application?
The sales engineers at FAULHABER will be happy to advise you. We will help you develop a perfectly tailored solution for all applications where particular requirements such as specific ambient
conditions or mechanical constraints need to be taken into consideration. | {"url":"https://www.faulhaber.com/fr/know-how/tutorials/linear-actuator-tutorial-selection-of-the-right-linear-actuator/","timestamp":"2024-11-08T20:57:55Z","content_type":"text/html","content_length":"269095","record_id":"<urn:uuid:bcdb9410-6e17-4cfa-b305-878fb5d12021>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00248.warc.gz"} |
A memetic algorithm for cardinality-constrained portfolio optimization with transaction costs
A memetic approach that combines a genetic algorithm (GA) and quadratic programming is used to address the problem of optimal portfolio selection with cardinality constraints and piecewise linear
transaction costs. The framework used is an extension of the standard Markowitz mean–variance model that incorporates realistic constraints, such as upper and lower bounds for investment in
individual assets and/or groups of assets, and minimum trading restrictions. The inclusion of constraints that limit the number of assets in the final portfolio and piecewise linear transaction costs
transforms the selection of optimal portfolios into a mixed-integer quadratic problem, which cannot be solved by standard optimization techniques. We propose to use a genetic algorithm in which the
candidate portfolios are encoded using a set representation to handle the combinatorial aspect of the optimization problem. Besides specifying which assets are included in the portfolio, this
representation includes attributes that encode the trading operation (sell/hold/buy) performed when the portfolio is rebalanced. The results of this hybrid method are benchmarked against a range of
investment strategies (passive management, the equally weighted portfolio, the minimum variance portfolio, optimal portfolios without cardinality constraints, ignoring transaction costs or obtained
with L [1] regularization) using publicly available data. The transaction costs and the cardinality constraints provide regularization mechanisms that generally improve the out-of-sample performance
of the selected portfolios.
IMC Forschungsschwerpunkte
• Software engineering and intelligent systems
ÖFOS 2012 - Österreichischen Systematik der Wissenschaftszweige
• 102032 Computational Intelligence
Untersuchen Sie die Forschungsthemen von „A memetic algorithm for cardinality-constrained portfolio optimization with transaction costs“. Zusammen bilden sie einen einzigartigen Fingerprint. | {"url":"https://research.imc.ac.at/de/publications/a-memetic-algorithm-for-cardinality-constrained-portfolio-optimiz","timestamp":"2024-11-13T04:49:42Z","content_type":"text/html","content_length":"58946","record_id":"<urn:uuid:85d5d4a5-7c79-4887-a97f-3bc92bd30869>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00364.warc.gz"} |
How to Combine Vectors in R - dummies
To dive a bit deeper into how you can use vectors in R, let’s consider this All-Star Grannies example. You have two vectors that contain the number of baskets that Granny and her friend Geraldine
scored in the six games of this basketball season:
> baskets.of.Granny <- c(12, 4, 4, 6, 9, 3)
> baskets.of.Geraldine <- c(5, 3, 2, 2, 12, 9)
The c() function stands for combine. It doesn’t create vectors — it just combines them.
You give six values as arguments to the c() function and get one combined vector in return. As you know, R considers each value a vector with one element. You also can use the c() function to combine
vectors with more than one value, as in the following example:
> all.baskets <-c(baskets.of.Granny, baskets.of.Geraldine)
> all.baskets
[1] 12 4 4 6 9 3 5 3 2 2 12 9
The result of this code is a vector with all 12 values.
In this code, the c() function maintains the order of the numbers. This example illustrates a second important feature of vectors: Vectors have an order. This order turns out to be very useful when
you need to manipulate the individual values in the vector.
About This Article
This article is from the book:
This article can be found in the category: | {"url":"https://www.dummies.com/article/technology/programming-web-design/r/how-to-combine-vectors-in-r-141640/","timestamp":"2024-11-02T05:56:20Z","content_type":"text/html","content_length":"71226","record_id":"<urn:uuid:da37e403-2cf8-4eb2-84fc-304abfa8eef7>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00716.warc.gz"} |
Finite Mathematics
A general education course in practical mathematics for those students not majoring in mathematics or science. This course will include such topics as set operations, methods of counting,
probability, systems of linear equations, matrices, geometric linear programming, and an introduction to Markov chains.
During this course, the student will be expected to:
1. Solve linear equations and inequalities in one variable.
1.1 Determine if the sentence is linear.
1.2 Isolate the variable.
1.3 Change order when operating with a negative factor.
2. Describe the functions and functional notation.
2.1 Define a relation.
2.2 Define a function.
2.3 Determine the dependency relationship between the variables.
2.4 Use f(x) notation.
3. Graph linear equations and inequalities in two variables.
3.1 Describe the Cartesian coordinate system.
3.2 Determine the coordinates of sufficient points needed to draw the line of the equation.
3.3 Locate and indicate the proper half- plane for an inequality .
4. Write linear models for verbal problems.
4.1 Identify the quantities pertinent to the problem.
4.2 Identify extraneous information.
4.3 Label clearly the necessary constant and variable quantities.
4.4 Write a mathematical sentence that relates the necessary quantities.
4.5 Identify, when necessary, missing information.
5. Perform basic matrix operations.
5.1 Define a matrix and related terms.
5.2 State the conditions under which various operations may be performed.
5.3 Add, subtract , and multiply matrices when possible.
5.4 Invert a 2 x 2 or a 3 x 3 matrix, when possible.
6. Solve systems of linear equations by a variety of methods.
6.1 State the possible solutions and the conditions of their appearance for a linear system.
6.2 Graph the set of equations on one set of axes.
6.3 Use the 'multiply and add' method to determine the solution.
6.4 Apply row operations to an augmented matrix to determine the solution (Gauss-Jordan method).
6.5 Solve the system by applying matrix algebra.
7. Identify the feasible region and vertices for a set of linear constraints.
7.1 Graph each of the constraints on the same set of axes.
7.2 Indicate the intersection of all the half-planes as a polygon.
7.3 Find the coordinates of the vertices of the polygon.
8. Solve linear programming problems.
8.1 Model the limited resource problem in terms of an objective function and a set of constraints.
8.2 Graph the constraints.
8.3 Apply the Corner Point Theorem.
8.4 Confirm the result for reasonableness.
9. Perform basic set operations, using correct notation.
9.1 Define a set and its related terms.
9.2 Determine the intersection and union of given sets.
9.3 Illustrate the intersection and union of sets with Venn diagrams.
9.4 Use set notation to describe a Venn diagram.
10. Solve counting problems using the multiplication principles.
10.1 State the Fundamental Counting Principle.
10.2 Determine if a problem is a permutation or a combination.
10.3 State the relationship between combinations, Pascal's triangle, and the binomial coefficients .
10.4 Use correctly combination and permutation notations.
10.5 Calculate factorials .
11. Write the sample space and specific events of an experiment.
11.1 Define sample space and event.
11.2 Distinguish between continuous and discrete outcomes.
11.3 Describe a trial of an event.
11.4 Write a clear description of an event of interest.
12. Evaluate the probabilities of basic problems such as dice, cards, coins, and balls.
12.1 Define the probability of an event.
12.2 Apply the addition rule for combined probabilities.
12.3 Apply the multiplication rule for combined probabilities.
12.4 Determine if events are mutually exclusive.
13. Calculate conditional probabilities by various methods.
13.1 Calculate conditional probability by formula .
13.2 Calculate conditional probability by probability trees.
13.3 Determine if events are independent.
13.4 Calculate probabilities by Bayes' formula.
14. State characteristic properties of probability distributions.
14.1 Create a probability distribution form a frequency distribution table.
14.2 Create a probability distribution graph.
14.3 Relate the area under a probability distribution graph to the probability of an event.
14.4 State the random variable of the probability distribution.
14.5 Calculate the mean, median, mode, and standard deviation of the random variable.
15. Calculate the probabilities of events by means of known probability distributions.
15.1 Apply Chebychev's Theorem.
15.2 Find the probabilities of events based on normally distributed random variables.
15.3 Estimate the probabilities of binomial events by means of a normal distribution. | {"url":"https://softmath.com/tutorials-3/relations/finite-mathematics-2.html","timestamp":"2024-11-14T08:41:11Z","content_type":"text/html","content_length":"37060","record_id":"<urn:uuid:a0ca4904-4df6-48dc-ac15-696579702f87>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00766.warc.gz"} |
Thomas Andreas Jung's Blog
I already mentioned the inverse function test pattern in the
introduction to specification-based testing
. Now I would like to present the pattern in more detail with an example.
The basic idea of testing with inverse functions is simple. We have two functions f and its inverse function f
. If we apply f to an input value and then take the result and apply it to f
this should result in the input value again: f
(f(x)) = x.
This test pattern is applicable for all kinds of functions for example compression and encryption. In the business application domain create and delete are examples of operations that can be tested
this way. Another example are do and undo operations of command objects. In both examples, doing and undoing an action leaves the state of the world unchanged.
There are some constraints on the applicability of the test pattern. The function f has to have a inversion function so it is a bijective function and one of the functions f or f
have to be tested. Otherwise the intermediary result could be invalid. For example, if you test a create and a delete operation together, the inverse function test passes if both operations do
The example used to illustrate the inverse function testing is a simple square function.
public double square(double a) {
double square = a * a;
return square;
The inverse function test can be implemented with concrete values. We use the Math.sqrt() implementation as the inverse function.
@Test public void squareConcreteValues() {
double a = 2;
assertEquals(square(Math.sqrt(a)), a, precission);
This is okay but defining input values manually is not very productive, not readable and has not sufficient test coverage. You can instead employ some computing power to generate the values using
Firstly, the square function is not bijective as square(-x) = square(x). This is a fact we did not express in the example with the concrete values. It simply omitted this fact. To fix this the result
is compared to the absolute value of x. Secondly, the function will overflow and test output like this:
java.lang.AssertionError: expected:<1.6482368012418589E307> but was:<Infinity>
is typical.
@Test public void squareWithOverflows() {
for(double a : someDoubles()) {
assertEquals(abs(a), Math.sqrt(square(a)), a * precission);
Again, this aspect was not expressed in the concrete test. Even if you were not aware of the problem the failing test points to the overflow problem. This is a nice example how Quickcheck can help
you to find bugs you did not anticipate. I admit that this is a simple example but give it a try. You'll see that you run into all kinds of problems you did not think of. You have to break your
software to know it.
Now we have to fix the overflow problem. Depending on the situation it can be easier to find and test a valid implementation that is more restrictive than theoretically possible but suffices your
requirements. This is the trade-off between effort now and potential applicability later.
For this example it is easy to find all legal input arguments. It is the largest double value that can be squared.
@Test public void squareWithBounds() {
double max = Math.nextAfter(Math.sqrt(Double.MAX_VALUE), 0.0);
for (double a : someDoubles(-max, max)) {
assertEquals(abs(a), Math.sqrt(square(a)), a * precission);
To finish this example let’s write the test for the illegal input arguments as well. All square arguments that cause an overflow are invalid and should cause an IllegalArgumentException. The
invalidArguments double generator defines all invalid values. These are the values greater than the largest valid value max and smaller than the smallest allowed value -max.
@Test public void squareInvalidArguments() {
double max = Math.sqrt(Double.MAX_VALUE);
double firstInvalid = Math.nextUp(max);
Generator<Double> invalidArguments =
oneOf(doubles(-Double.MAX_VALUE, -firstInvalid))
.add(doubles(firstInvalid, Double.MAX_VALUE));
for (double a : someEnsureValues(asList(firstInvalid, -firstInvalid), invalidArguments)) {
}catch(IllegalArgumentException e){ }
The implemenation passing the tests is:
public double square(double a) {
double square = a * a;
if (Double.isInfinite(square)) { throw new IllegalArgumentException() };
return square;
Testing with inversion functions can solve the dilemma of writing tests without repeating the production code. If you repeat the code in the test you have an additional check that you wrote the code
down correctly. You have to repeat an error in both places to introduce a bug. This is all you can get out of such tests. If you test with an inverse function the test and implementation do not share
code. This kind of test has the potential to find conceptual problems in your code. (Actually, the reality is not black and white. You can have a test that implements the same operation with a
simpler test implementation. This implementation verifies that the more complex production version works. This is the idea of the analogous function test pattern. I will come to later.)
If the inverse function pattern is applicable it can save you a lot of effort. For example the result of an operation can be very costly to verify like encryption functions. The exact representation
may not be of interest for the given domain or change frequently leading to high maintenance costs with concrete test values. If you can define valid input values and have an tested inverse function
the test effort is to setup the input value generator. The nice side-effect is that you test that the functions you think are inverse functions really are inverse. | {"url":"https://theyougen.blogspot.com/2010/12/","timestamp":"2024-11-08T07:15:05Z","content_type":"application/xhtml+xml","content_length":"49372","record_id":"<urn:uuid:db70f8b8-8665-4439-9bdd-3d461516f570>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00776.warc.gz"} |
For what values of x will the infinite geometric series 1+ (2x-1) + (2x-1)^2 + (2x-1)^3 + ... have a finite sum? | HIX Tutor
For what values of x will the infinite geometric series 1+ (2x-1) + (2x-1)^2 + (2x-1)^3 + ... have a finite sum?
Answer 1
This is a geometric series with common ratio #(2x-1)#.
In order to converge we require #-1 < (2x-1) < 1#
Hence: #0 < 2x < 2#
Hence: #0 < x < 1#
Note that if #x = 0# then partial sums will always be bounded, alternating between #1# and #0#, but the infinite series does not have a well defined sum.
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
The infinite geometric series will have a finite sum if the common ratio ( |r| < 1 ). Therefore, for the series ( 1 + (2x - 1) + (2x - 1)^2 + (2x - 1)^3 + \ldots ) to have a finite sum, the absolute
value of the common ratio ( |2x - 1| ) must be less than 1. This gives us the inequality ( |2x - 1| < 1 ). Solving this inequality gives ( \frac{-1}{2} < x < \frac{3}{2} ). Therefore, the values of (
x ) for which the infinite geometric series has a finite sum are ( x ) such that ( \frac{-1}{2} < x < \frac{3}{2} ).
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/for-what-values-of-x-will-the-infinite-geometric-series-1-2x-1-2x-1-2-2x-1-3-hav-8f9afa9280","timestamp":"2024-11-10T13:03:23Z","content_type":"text/html","content_length":"578560","record_id":"<urn:uuid:07e92ded-2e22-4ca5-a0b5-a498c0032508>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00240.warc.gz"} |
Aryabhatta Inventions: The Greatest Mathematician - Fastnewsfeed
Aryabhatta Inventions: The Greatest Mathematician
Aryabhata was one of the greatest astronomers and mathematicians of ancient times. His work in the field of science and mathematics still inspires scientists. Aryabhatta inventions include the first
to use algebra. You will be surprised to know that he wrote his famous composition ‘Aryabhatiya’ (poetry of mathematics) as a poem.
It is one of the most famous books of ancient India. Most of the information given in this book is related to astronomy and spherical trigonometry. 33 laws of arithmetic, algebra, and trigonometry
are also given in ‘Aryabhatiya’.
List Of Aryabhatta Inventions:
Today we all know that the earth is round and rotates on its axis and that is why night and day occurring. In the medieval period, ‘Nicholas Copernicus’ proposed this theory, but very few people will
be aware of the fact that about 1 thousand years before ‘Copernicus’, Aryabhata discovered that the Earth is round and its circumference is estimated to be 24835 miles.
Aryabhatta Inventions proved the belief of the Hindu religion of solar and lunar eclipse wrong. This great scientist and mathematician also knew that the moon and other planets are illuminated by the
sun’s rays. Aryabhatta Inventions proved from his sources that a year consists of 365.2951 days, not 366 days.
Aryabhatta Inventions proved that the length of the circumference of the earth as 39,968.05 kilometers, which is just 0.2 percent less than the actual length (40,075.01 kilometers).
Aryabhatta Inventions Of Zero:
The invention of zero is a great discovery in the history of mathematics and it is one of the few great inventions in the world. The invention of zero has a very important role in mathematics. For
integers and real numbers, it is the additive element of the sum.
The most important thing about zero is that multiplying any number to zero gives zero and adding or subtracting any number from zero returns the same number. And no number can become big without
putting it behind zero.
Put zero in one and then increase to zero. The number will become bigger. Hundred, thousand, lakh, ten lakh, crore, ten crores, billion and then trillion If there was no invention, it would not have
been such a large number and solving mathematics would have been too big, so the invention of zero is considered so important.
Invention Of Zero:
The information about when and who had invented the zero was hidden for a long time. But Indian scientists have been claiming for years that zero was invented in India. It is said that zero was
invented in India in the middle of the fifth century. However, now people know that Aryabhatta had invented zero.
But there is also a different fact about the invention of the Void if Aryabhata invented the Void in the 5th century then how thousands of years ago, 10 heads of Ravana were counted without the Void,
how the Kauravas found out that there were 100 Sons. So, irrespective of all the debate it is being said that Aryabhata had invented zero in the 5th century.
Related Aricles:-
Aryabhatta Inventions: The Greatest Mathematician was last modified: October 4th, 2019 by | {"url":"https://www.fastnewsfeed.com/news/aryabhatta-inventions-the-greatest-mathematician/","timestamp":"2024-11-03T18:56:07Z","content_type":"text/html","content_length":"82406","record_id":"<urn:uuid:84d3c0f0-5337-468f-bdda-b751124e1c62>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00380.warc.gz"} |
Linearising compound pendulum equation
• Thread starter seboastien
• Start date
In summary, the conversation discusses the process of linearizing the compound pendulum equation T=2pi√(K^2 + h^2)/gh in order to find the value of g from the gradient. The idea of using a Taylor
approximation and making one of the axes √(h2 + K2) is suggested. The conversation also touches on the confusion of K being a known constant or not and the final solution of plotting a graph of h^2
against h*T^2 to determine the y intercept and gradient.
Homework Statement
Linearise T=2pi√(K^2 + h^2)/gh K is known constant
This is a compound pendulum equation, I want to plot some kind of formula with variable T against some kind of formula with variable H in order to find g from the gradient.
Homework Equations
The Attempt at a Solution
so I've got T/2pi all squared times g all substituted to x, h subbed to y and k^2 subbed to constant C and I've got the equation y^2 -yx + C=0 and tried to solve for y=x+β
I've tried implicit differentiation and it's gotten me nowhere
hi seboastien!
(try using the X^2 button just above the Reply box
seboastien said:
Linearise T=2pi√(K^2 + h^2)/gh K is known constant
This is a compound pendulum equation, I want to plot some kind of formula with variable T against some kind of formula with variable H in order to find g from the gradient.
if K is a known constant, can't you make one of the axes √(h
+ K
) ?
I would have to make the axis √((h^2 + K^2)/gh ) but that is a good point.
However, I would still like to know how I could linearise it further. I know that a taylor approximation is needed but I don't know how to, or what a value to choose
√(1 + (h^2/K^2) = 1 + (h^2/K^2)/2 + …
if h/K is small, then √(1 + (h^2/K^2)) = 1 + (h^2/K^2)/2 + …
hmmm, my only issue is that its the sqrt of K^2 + h^2 divided by gh
it also turns out that k is the radius of gyration and I have no scales to measure the pendulum's mass. I believe I need a y=mx + c where the y intercept will be determined by k, g by m, x by T and h
by y.
is there any way of achieving this?
seboastien said:
it also turns out that k is the radius of gyration and I have no scales to measure the pendulum's mass. I believe I need a y=mx + c where the y intercept will be determined by k, g by m, x by T
and h by y.
i'm confused
you said that K was
seboastien said:
Linearise T=2pi√(K^2 + h^2)/gh K is known constant
That's because I thought I was allowed to measure the pendulums mass.
Don't worry I've worked it out...finally, turns out I've been overcomplicating things.
I'll just plot a graph of h^2 against h*T^2 the y intercept will be -k^2 and the gradient will be g/4pi^2.
Thanks anyway.
FAQ: Linearising compound pendulum equation
1. What is a compound pendulum?
A compound pendulum is a type of pendulum that consists of a rigid body suspended from a pivot point, rather than a simple pendulum which has a point mass suspended from a string or rod.
2. What is the equation for a compound pendulum?
The equation for a compound pendulum is a non-linear equation that takes into account the length and mass distribution of the pendulum, as well as the gravitational acceleration. It is given by θ'' +
(g/L)sinθ = 0, where θ is the angle of displacement and L is the distance from the pivot point to the center of mass.
3. Why is it necessary to linearize the compound pendulum equation?
Linearizing the compound pendulum equation makes it easier to solve and analyze. Non-linear equations are more complex and require more advanced mathematical techniques to solve, whereas linear
equations can be solved using simpler methods.
4. How do you linearize the compound pendulum equation?
To linearize the compound pendulum equation, we make the assumption that the angle of displacement θ is small, so we can use the small angle approximation sinθ ≈ θ. This simplifies the equation to
θ'' + (g/L)θ = 0, which is a linear equation that can be solved using basic calculus.
5. What are the applications of linearizing the compound pendulum equation?
Linearizing the compound pendulum equation allows us to study and understand the behavior of pendulums in various situations, such as in clocks, seismographs, and other mechanical systems. It also
helps us to design and optimize pendulum-based devices, such as accelerometers and gyroscopes. | {"url":"https://www.physicsforums.com/threads/linearising-compound-pendulum-equation.585280/","timestamp":"2024-11-07T10:28:11Z","content_type":"text/html","content_length":"110926","record_id":"<urn:uuid:0b4549ab-77ec-4ab1-95ef-dc18042e4aad>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00652.warc.gz"} |
3.3 Potentials, Energy, & Fields
They Emerge From Crowdfunded & Self-Published Science
Read →
So it was William Thomson, later known as Baron Kelvin, who introduced the concept of free energy?
Expand full comment
I had never heard of George Green. Nevermind his Identities or Functions.
The fact that he worked out the discoveries that Chasles, Strum, Liouville, and Gauss all viewed as their own original work, implies a formidable intellect of his own .
Perhaps on a level with Tesla?
Expand full comment
Green was a brilliant, if troubled, man. His mathematical talent probably exceeded Tesla's. Talented as Tesla was, he wasn't really a mathematician. Green's talent was to take abstract mathematical
principles and tie them directly to the phenomena of electricity. I doubt Green was as talented a mathematician as say Gauss, but by taking advantage of the synthesis of mathematical and physical
principles, he was able to see a bit further ahead than many of his potentially more mathematically skilled contemporaries.
Expand full comment
Ah yes didn't Tesla make some comment about mathematicians wandering off and getting lost in their equations?
As for Green, we will never know this side of eternity how many more such genius individuals are ultimately lost to us in the here and now.
Expand full comment | {"url":"https://aetherczar.substack.com/p/potentials-energy-and-fields/comments","timestamp":"2024-11-14T20:48:29Z","content_type":"text/html","content_length":"213473","record_id":"<urn:uuid:d58a0f9b-85d7-485a-a256-cc3a64d709e4>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00865.warc.gz"} |
This module contains classes and procedures for computing the first moment (i.e., the statistical mean) of random weighted samples. More...
This module contains classes and procedures for computing the first moment (i.e., the statistical mean) of random weighted samples.
The mean of a weighted sample of \(N\) data points is computed by the following equation,
$$\mu = \frac{\sum_{i = 1}^{N} w_i x_i}{\sum_{i = 1}^{N} w_i}$$
where \(w_i\) represents the weight of the \(i\)th sample.
Mean updating
Suppose the mean of an initial (potentially weighted) sample \(x_A\) of size \(N_A\) is computed to be \(\mu_A\).
Another (potentially weighted) sample \(x_B\) of size \(N_B\) is subsequently obtained with a different number observations and mean \(\mu_B\).
The mean of the two samples combined can be expressed in terms of the originally computed means,
$$\large \mu = \frac { w_A \sum_{i = 1}^{N_A} w_{\up{A,i}} x_{\up{A,i}} + w_B \sum_{i = 1}^{N_B} w_{\up{B,i}} x_{\up{B,i}} }{ w_A + w_B }$$
where \(\large w_A = \sum_{i = 1}^{N_A} w_{\up{A,i}}\) and \(\large w_B = \sum_{i = 1}^{N_B} w_{\up{B,i}}\) are sums of the weights of the corresponding samples.
For equally-weighted samples, the corresponding weights \(w_{\up{A,i}}\) or \(w_{\up{B,i}}\) or both are all unity such that \(N_A = w_A\) or \(N_B = w_B\) or both holds.
While it is tempting to extend the generic interfaces of this module to weight arguments of type integer or real of various kinds, such extensions do not add any benefits beyond making the
interface more flexible for the end user.
But such extensions would certainly make the maintenance and future extensions of this interface difficult and complex.
According to the coercion rules of the Fortran standard, if an integer is multiplied with a real, the integer value must be first converted to real of the same kind as the real value, then
Furthermore, the floating-point multiplication tends to be faster than integer multiplication on most modern architecture.
The following list compares the cost and latencies of some of basic operations involving integers and real numbers.
1. Central Processing Unit (CPU):
1. Integer add: 1 cycle
2. 32-bit integer multiply: 10 cycles
3. 64-bit integer multiply: 20 cycles
4. 32-bit integer divide: 69 cycles
5. 64-bit integer divide: 133 cycles
2. On-chip Floating Point Unit (FPU):
1. Floating point add: 4 cycles
2. Floating point multiply: 7 cycles
3. Double precision multiply: 8 cycles
4. Floating point divide: 23 cycles
5. Double precision divide: 36 cycles
See also
Intel Fortran Forum - Integer VS fp performance
Colorado State University tips on Fortran performance
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python,
R), please also ask the end users to cite this original ParaMonte library.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
Fatemeh Bagheri, Thursday 12:45 AM, August 20, 2021, Dallas, TX | {"url":"https://www.cdslab.org/paramonte/fortran/latest/namespacepm__sampleMean.html","timestamp":"2024-11-14T00:11:16Z","content_type":"application/xhtml+xml","content_length":"19689","record_id":"<urn:uuid:8a24b5d7-06d1-4307-b2da-7573132484f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00182.warc.gz"} |
Proof Plural, What is the Plural of Proof? - EngDic
Meaning: evidence or argument
Singular and Plural of Proof
Singular Plural
proof Proofs
Proof as a Singular Noun in Example Sentences:
1. He provided concrete proof of his innocence.
2. The detective found a crucial piece of proof at the crime scene.
3. She presented her research findings as proof of her theory.
4. The documents serve as proof of ownership.
5. The video footage is undeniable proof of the incident.
6. The defense attorney questioned the validity of the prosecution’s proof.
7. The photograph is solid proof that they were together.
8. The company demanded proof of purchase for a refund.
9. The scientist conducted experiments to gather proof for their hypothesis.
10. The witness testimony provided strong proof in the trial.
Proof as a Plural Noun in Example Sentences:
1. The lawyer submitted all the necessary proofs to the court.
2. We need more concrete proofs before making a decision.
3. The researchers presented multiple scientific proofs to support their claims.
4. The detective gathered various pieces of physical proofs.
5. The historian examined historical documents as proofs of events.
6. The audit revealed several financial irregularities and proofs of fraud.
7. The prosecutor presented compelling proofs against the defendant.
8. The jury analyzed the collected proofs before reaching a verdict.
9. The journalist investigated and gathered substantial proofs of corruption.
10. The insurance company requested additional proofs of the accident.
Singular Possessive of Proof
The singular possessive form of “Proof” is “Proof’s”.
Examples of Singular Possessive Form of Proof:
1. We need to examine proof’s validity before drawing conclusions.
2. The responsibility for proof’s accuracy lies with the researcher.
3. The significance of proof’s findings cannot be overstated.
4. Proof’s impact on the scientific community is remarkable.
5. The weight of proof’s evidence is undeniable.
6. The reliability of proof’s methodology is questioned.
7. Proof’s relevance to the topic is evident.
8. The publication of proof’s results will be influential.
9. The implications of proof’s discovery are far-reaching.
10. The credibility of proof’s source needs to be verified.
Plural Possessive of Proof
The plural possessive form of “Proof” is “Proofs'”.
Examples of Plural Possessive Form of Proof:
1. The scientists’ proofs’ conclusions are consistent.
2. The researchers’ collaboration strengthens the proofs’ validity.
3. The peer review process evaluates proofs’ quality.
4. The accuracy of the proofs’ measurements is crucial.
5. The importance of multiple proofs’ perspectives is acknowledged.
6. The reviewers’ feedback enhances the proofs’ credibility.
7. The researchers’ responsibility is to present proofs’ findings objectively.
8. The compilation of various proofs’ supports the theory.
9. The scientists’ expertise contributes to proofs’ reliability.
10. The examination of different proofs’ data is ongoing.
Explore Related Nouns: | {"url":"https://engdic.org/proof-plural-what-is-the-plural-of-proof/","timestamp":"2024-11-12T22:08:41Z","content_type":"text/html","content_length":"98473","record_id":"<urn:uuid:ec4517cd-6375-4a8a-824d-8fd205fd744c>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00469.warc.gz"} |
The Second Red Pill - Seasonal Peak Spring Tides
Red Pill 2 is large!
But the rewards are great if you manage to get it down!!
[Please click on the "RED PILL 1" link if you haven't read this red pill.]
RED PILL 1 The influence of cycles in the atmospheric lunar tides upon the Earth's atmospheric pressure can be re-inforced (i.e weaponized) if they constructively interfere with the annual seasonal
RED PILL 2 If the lunisolar atmospheric tides that are associated with the Peak Seasonal Spring Tides play a role in influencing the Earth's atmospheric pressure, you should see variations in this
pressure that occur at intervals of 3.8-year (= 1/5th the Metonic Cycle).
There are four factors that can affect the strength of seasonal peak tides i.e. peaks in the lunisolar tides that align with the seasons:
1. The proximity of the Earth/Moon system to the Sun.
2. The relative position of the Moon with respect to the Sun i.e. the Moon's phase.
3. The proximity of a New/Full Moon to one of the nodes of the lunar orbit.
4. The proximity of a New/Full Moon to the perigee/apogee of the lunar orbit.
This large red-pill post will specifically deal with the factors that affect the strength of Seasonal Peak Spring Tides i.e. factors 1 and 2.
The synodic month = 29.5305889 days. The time required for the Moon to go from one New/Full moon to the next New/Full moon.
The tropical year = 365.2421897 days. The length of the seasonal year.
A. The Proximity of the Earth/Moon System to the Sun
Due to the elliptical nature (e = 0.0167) of the Earth's orbit, the distance of the Earth/Moon system from the Sun varies between an aphelion (i.e. furthest distance) of 152.1 million km around July
04th to a perihelion (i.e. closest distance) of 147.1 million km on January 3rd. This means that the strength of lunisolar tidal forces near January 03rd are noticeably enhanced compared to those
that are near July 04th. Hence, the effects of any long-term seasonal peak tides upon atmospheric pressure will naturally be enhanced if these peak tides are aligned with the date of perihelion.
B. The Relative Position of the Moon With Respect to the Sun i.e. the Moon's phase
What are Spring Tides?
They are higher than normal tides that occur twice every lunar synodic month (= 29.53 days), whenever the Sun, Earth, and Moon are co-aligned at either New or Full Moon.
It turns out that 12 1/2 synodic months are 3.890171 days longer than one tropical year (N.B. from this point forward, the word “year” will mean one tropical or seasonal year = 365.2421897 days
unless indicated).
Hence, if a spring tide occurs on a given day of the year, 3.796 years will pass before another spring tide occurs on roughly the same day of the year.
This is true because 3.796 years is the number of years it takes for, the 3.890171 days per year slippage between 12 1/2 synodic months and the tropical year, to accumulate to half a synodic month of
14.7652944 days.
In the real world, it turns out that Spring Tides occur on roughly the same day of the year once every:
3 years
3 + 4 = 7 years
3 + 4 + 4 = 11 years
3 + 4 + 4 + 4 = 15 years
3 + 4 + 4 + 4 + 4 years = 19 years
[N.B. The 3-year spacing can occur at any point in the 19-year Metonic Cycle sequence]
with the 3:4:4:4:4-year spacing pattern [which has an average spacing of (3 + 4 + 4 + 4 + 4)/5 = 3.8 years], repeating itself after a period of almost exactly 19 years. The 19.0-year period is known
as the Metonic cycle. This cycle results from the fact that 235 Synodic months = 6939.688381 days = 19.000238 Tropical years.
Displayed below is a real-life example of one Metonic Cycle between 1996 and 2015.
YEAR____PHASE____DATE____TIME____GAP IN YEARS
1996_____FM_______Sept 27____02:51____ 0 years
1999_____FM_______Sept 25____10:53____ 3 years
2003_____NM_______Sept 26____03:09____ 3 + 4 years = 7 years
2007_____FM_______Sept 26____19:47____ 3 + 4 + 4 years = 11 years
2011_____NM_______Sept 27____11:09____ 3 + 4 + 4 + 4 years = 15 years
2015_____FM_______Sep 28_____02:52____ 3 + 4 + 4 + 4 + 4 years = 19 years
Hence, If the lunisolar atmospheric tides that are associated with the Peak Seasonal Spring Tides play a role in influencing the Earth's atmospheric pressure, you should see variations in this
pressure that occur at 3.8-year (= 1/5th the Metonic Cycle) intervals.
Wilson, I.R.G.,
Lunar Tides and the Long-Term Variation of the Peak Latitude Anomaly of the Summer Sub-Tropical High-Pressure Ridge over Eastern Australia
, The Open Atmospheric Science Journal, 2012, 6, 49-60 | {"url":"https://astroclimateconnection.blogspot.com/2019/11/the-second-red-pill-seasonal-peak.html","timestamp":"2024-11-08T15:31:44Z","content_type":"text/html","content_length":"77658","record_id":"<urn:uuid:05ef21cb-62ec-4243-a01d-2c8e30d3c9d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00770.warc.gz"} |
Marisa Eisenberg : Forecasting and uncertainty in modeling disease dynamics
Javascript must be enabled
Marisa Eisenberg : Forecasting and uncertainty in modeling disease dynamics
Connecting dynamic models with data to yield predictive results often requires a variety of parameter estimation, identifiability, and uncertainty quantification techniques. These approaches can help
to determine what is possible to estimate from a given model and data set, and help guide new data collection. Here, we examine how parameter estimation and disease forecasting are affected when
examining disease transmission via multiple types or pathways of transmission. Using examples taken from the West Africa Ebola epidemic, HPV, and cholera, we illustrate some of the potential
difficulties in estimating the relative contributions of different transmission pathways, and show how alternative data collection may help resolve this unidentifiability. We also illustrate how even
in the presence of large uncertainties in the data and model parameters, it may still be possible to successfully forecast disease dynamics.
0 Comments
Comments Disabled For This Video | {"url":"https://www4.math.duke.edu/media/watch_video.php?v=1c2e03265c84523780fd70a127d6660e","timestamp":"2024-11-05T12:02:40Z","content_type":"text/html","content_length":"48032","record_id":"<urn:uuid:05391e76-fd9e-4f0d-a3f1-c1dc860786c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00045.warc.gz"} |
Individual chaotic behaviour of the S-stars in the Galactic centre
Issue A&A
Volume 685, May 2024
Article Number A12
Number of page(s) 3
Section Astrophysical processes
DOI https://doi.org/10.1051/0004-6361/202348361
Published online 30 April 2024
A&A, 685, A12 (2024)
Individual chaotic behaviour of the S-stars in the Galactic centre
^1 Leiden Observatory, Leiden University, 2300 RA Leiden, The Netherlands
e-mail: beckers@strw.leidenuniv.nl; cpoppelaars@strw.leidenuniv.nl
^2 NASA Ames Research Center, Moffett Field, CA 94035, USA
Received: 23 October 2023
Accepted: 15 February 2024
Located at the core of the Galactic centre, the S-star cluster serves as a remarkable illustration of chaos in dynamical systems. The long-term chaotic behaviour of this system can be studied with
gravitational N-body simulations. By applying a small perturbation to the initial position of star S5, we can compare the evolution of this system to its unperturbed evolution. This results in two
solutions that diverge exponentially, defined by the separation in position space δ[r], with an average Lyapunov timescale of ∼420 yr, corresponding to the largest positive Lyapunov exponent. Even
though the general trend of the chaotic evolution is governed in part by the supermassive black hole Sagittarius A^∗ (Sgr A^∗), individual differences between the stars can be noted in the behaviour
of their phase-space curves. We present an analysis of the individual behaviour of the stars in this Newtonian chaotic dynamical system. The individuality of their behaviour is evident from offsets
in the position space separation curves of the S-stars and the black hole. We propose that the offsets originate from the initial orbital elements of the S-stars, where Sgr A^∗ is considered in one
of the focal points of the Keplerian orbits. Methods were considered to find a relation between these elements and the separation in position space. Symbolic regression provides the clearest
diagnostics for finding an interpretable expression for the problem. Our symbolic regression model indicates that ⟨δ[r]⟩∝e^2.3, implying that the time-averaged individual separation in position
space increases rapidly with the initial eccentricity of the S-stars.
Key words: chaos / methods: numerical / celestial mechanics / stars: black holes / stars: fundamental parameters / Galaxy: center
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.
1. Introduction
In the pursuit of understanding the nature of chaos and its manifestation in the Universe, the S-star cluster is used as a laboratory for experimenting with the underlying astrophysics that give rise
to chaos. The orbital data of 27 S-stars orbiting the supermassive black hole Sagittarius A^∗ from (Gillessen et al. 2009) are employed as initial conditions for a Newtonian gravitational N-body
simulation of the S-star cluster. The orbital evolution of the system was tracked over 10^4 yr by (Portegies Zwart et al. 2023; Boekholt et al. 2023). When perturbing the initial conditions of star
S5 by displacing its initial x coordinate by dx=15 m, the evolution of the perturbed solution diverges exponentially from the canonical solution, implying that the system is chaotic.
In this paper, we present the methods we used to quantify chaos in the simulation of the S-star cluster and delve into the individual differences in the chaotic behaviour of the S-stars and Sgr A*.
For more details of the simulation, we refer to (Portegies Zwart et al. 2023).
2. Measuring chaos
In the gravitational N-body problem, chaos can be measured with the Lyapunov timescale. The Lyapunov timescale represents the timescale on which the system becomes unpredictable. We describe the
chaotic S-star orbital evolution with phase-space distance. Phase-space distance as a function of time is defined as
$δ 2 = 1 4 ( | r c − r p | 2 + | v c − v p | 2 ) = 1 4 ( δ r 2 + δ v 2 ) ,$(1)
where r and v are the position and velocity vectors of an S-star, and c and p denote the canonical and perturbed solutions, respectively. We define an (initial) perturbation at t=0, δ(0). The
evolution of δ(t) is then approximately described by an exponential function with a time dependence
$δ ( t ) = δ ( 0 ) e λ t ,$(2)
where λ is the maximum positive Lyapunov exponent. The growth factor, G[δ](t), is the value of δ at some time t as a fraction of the initial perturbation, δ(0), that is,
$G δ ≡ δ ( t ) δ ( 0 ) = e λ t .$(3)
From this equation, we can find λ=ln(G[δ])/t, which is the reciprocal of the Lyapunov timescale,
$t λ = 1 λ = t ln ( G δ ) .$(4)
3. Separation in position space
We calculated the time evolution of the separation in position space between the canonical and perturbed solution, δ[r], for each S-star and Sgr A*. We show this time evolution in the left panel of
Fig. 1.
Fig. 1.
Position-space separations as a function of time before and after reducing the spread in the curves. Left: Time evolution of separation in position space for each S-star and the central black hole.
Right: y=−0.00018446⋅a/au+log(e)+4.3103 subtracted from the time evolution of log[10](δ[r]) for each S-star. The grey shading shows the region between the non-reduced maxima and minima of δ[r
] over the whole system at each time step, indicating the magnitude of reduction in the spread of the curves, although its lower section is obscured by the coloured curves.
The separation in position space grows approximately exponentially, as expected from Eq. (2). From a least-squares fit to the curve of each star and the black hole, we infer an ensemble mean Lyapunov
timescale of t[λ]≃420 yr. The black hole is less sensitive to the perturbation than most stars, as seen from its δ[r] values, which are generally lower in magnitude than those of the S-stars. We
attribute this to the mass difference of about 10^5M[⊙] between the central body and the stars, leading to much higher inertia. This stabilizes the black hole position, even in the presence of
perturbations. Perturbations caused by close encounters between stars are propagated through the entire system, driven by feedback from the black hole (Portegies Zwart et al. 2023). Hence, large
events with multiple close encounters, such as the event at t=2876 yr, influence the general trend of all curves. However, it is not immediately evident why there are vertical offsets in the S-star
curves, as the S-stars have been given identical masses in this simulation. Furthermore, while differences in the shapes of the individual curves could account for an offset, the curves as a whole
exhibit a noticeable shift relative to each other.
4. Position-space separation offsets
The systematic offsets found in the position-space separation curves must be an effect of some underlying astrophysics. The (initial) S-star orbits can introduce variations in the evolution of δ[r]
of each star, and therefore, it is worthwhile to investigate the Keplerian elements.
To map the separations of S-star curves and Sgr A*, we subtracted the black hole curve from each S-star curve and took the temporal mean for each S-star; we define this as $〈 Δ BH S - star log 10 (
δ r ) 〉$. To find a relation between this and the Keplerian elements of the stars, we used the initial orbital parameters from Gillessen et al. (2009). We expect to see that a low semi-major axis (a
) corresponds to a higher magnitude in δ[r], as stars that are close to the black hole should be more sensitive to the changes inside the system. Moreover, eccentricity (e) and δ[r] should be
directly correlated because a highly elliptical orbit has a closer pericenter to the black hole. In Fig. 2, we show the relation between a, e, and $〈 Δ BH S - star log 10 ( δ r ) 〉$.
Fig. 2.
Initial semi-major axis and eccentricity of each S-star. $〈 Δ BH S - star log 10 ( δ r ) 〉$ is indicated by colour. Each S-star is labelled according to the Gillssen catalogue (Gillessen et al.
2009). A colour gradient can be seen, ranging from orange in the bottom right to pink in the top left corner.
The pattern that emerges in the colour gradient in Fig. 2 indicates that eccentricity and semi-major axis play a major role in the curve separations. The influence of eccentricity and semi-major axis
on the mean separation are both important. The other orbital parameters can be compared with each other in a similar fashion, but no apparent pattern is found. This observation suggests that their
influence on $〈 Δ BH S - star log 10 ( δ r ) 〉$ is negligible compared to that of the eccentricity and semi-major axis. Furthermore, the stars with the highest colour values in Fig. 2 are S21 and
S29. Portegies Zwart et al. (2023) demonstrated that S21 is one of the stars that were part of the large event at 2876 yr. Moreover, S29 was shown to have six close encounters during the simulation.
In addition, S67 has ten close encounters, which is more than for any other star. Therefore, despite its relatively low eccentricity and large semi-major axis, it is not green in Fig. 2, in contrast
to its nearest neighbour, S87.
5. Symbolic regression
We adopted PySR (Cranmer 2023), a symbolic regression Python package for discovering interpretable analytical equations that describe an underlying pattern in a dataset. Only the semi-major axis and
the eccentricity are used in the model because symbolic regression may not provide accurate results when the dimensionality of the data is high (Matchev et al. 2022). Moreover, PySR does not
correctly interpret the evolution of δ[r], a, and e for each star, suggesting that it does not support time-series data^1. Therefore, we provide the model with $〈 Δ BH S - star log 10 ( δ r ) 〉$
and the initial semi-major axis and eccentricity. The results can be found in Table 1.
Table 1.
Symbolic regression results on the initial semi-major axis [au] and eccentricity and mean offsets between the S-star and the black hole.
From the optimal equation, which balances accuracy and simplicity, we can derive the following:
$y = − 0.00018446 a / au + ln ( e ) + 4.3103 .$(5)
y is a measure of log[10](δ[r]),
$〈 log 10 ( δ r ) 〉 = − 0.00018446 a / au + ln ( e ) + 4.3103 .$(6)
We used this to estimate δ[r] approximately as
$〈 δ r 〉 = 2 · 10 4 10 − 0.00018446 a / au e 2.3$(7)
Subtracting Eq. (5) from log[10](δ[r]) reduces the offsets significantly, as shown in the right panel of Fig. 1.
6. Conclusions
We aimed to characterize the individual behaviour of stars during the evolution of a Newtonian chaotic dynamical system. Using a gravitational N-body simulation of the S-star cluster, we derived the
Lyapunov timescale of the system over 10^4 yr to be ∼420 yr.
In the position space of the S-stars, we find a vertical offset between their curves. A relation with the Keplerian orbits of the S-star was proposed to explain this offset in δ[r]. We adopted
symbolic regression to find the relation ⟨δ[r]⟩=2⋅10^410^−0.00018446a/aue^2.3, where a and e were taken from the initial orbit. We conclude that ⟨δ[r]⟩∝e^2.3; the time-averaged individual
phase-space distance with respect to the black hole increases rapidly with the orbital eccentricity of the star.
This publication is funded by the Dutch Research Council (NWO) with project number OCENW.GROOT.2019.044 of the research programme NWO XL. It is part of the project “Unravelling Neural Networks with
Structure-Preserving Computing”. In addition, part of this publication is funded by the Nederlandse Onderzoekschool Voor Astronomie (NOVA). T.B.’s research was supported by an appointment to the NASA
Postdoctoral Program at the NASA Ames Research Center, administered by Oak Ridge Associated Universities under contract with NASA. We greatly thank the referee for taking the time to read this
manuscript carefully and providing us with well-considered comments.
All Tables
Table 1.
Symbolic regression results on the initial semi-major axis [au] and eccentricity and mean offsets between the S-star and the black hole.
All Figures
Fig. 1.
Position-space separations as a function of time before and after reducing the spread in the curves. Left: Time evolution of separation in position space for each S-star and the central black hole.
Right: y=−0.00018446⋅a/au+log(e)+4.3103 subtracted from the time evolution of log[10](δ[r]) for each S-star. The grey shading shows the region between the non-reduced maxima and minima of δ[r
] over the whole system at each time step, indicating the magnitude of reduction in the spread of the curves, although its lower section is obscured by the coloured curves.
In the text
Fig. 2.
Initial semi-major axis and eccentricity of each S-star. $〈 Δ BH S - star log 10 ( δ r ) 〉$ is indicated by colour. Each S-star is labelled according to the Gillssen catalogue (Gillessen et al.
2009). A colour gradient can be seen, ranging from orange in the bottom right to pink in the top left corner.
In the text
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on
Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while. | {"url":"https://www.aanda.org/articles/aa/full_html/2024/05/aa48361-23/aa48361-23.html","timestamp":"2024-11-02T04:20:48Z","content_type":"text/html","content_length":"94324","record_id":"<urn:uuid:b534526a-b00f-4eef-a222-dc80f4fb23dd>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00871.warc.gz"} |
Given a 3D array of 0s and the index for a starting 1, generate all variants of the array in which all 1s are connected to the first by other 1s
In many programming scenarios, particularly in computer graphics, game development, or algorithm testing, you might come across a requirement to manipulate multi-dimensional arrays. A common
challenge is to create connected variants of a 3D array filled with zeros (0s), with specific indices representing where to place ones (1s) so that all 1s remain interconnected. This article will
help clarify this problem and provide a comprehensive solution using Python.
Problem Scenario
Given a 3D array filled with 0s and a starting index that represents the position of a 1, our goal is to generate all possible configurations of the array, such that all 1s are connected to the
initial 1 through a continuous path of 1s.
Original Code
To illustrate the problem, let's consider the following hypothetical code snippet that aims to solve the problem:
def generate_variants(array, start):
# function logic to generate connected variants
The function above is a placeholder for our intended implementation.
Understanding the Problem
To break this down further, let’s consider the components:
1. 3D Array of 0s: This is essentially a cube where each cell can either be 0 or 1.
2. Start Index: This is the coordinate (x, y, z) within the 3D space that will be set to 1.
3. Connected Variants: All positions that become 1 should form a connected cluster, meaning you can move from one 1 to another along the array's axes (up, down, left, right, forward, backward).
Detailed Explanation and Implementation
We will need a depth-first search (DFS) approach to traverse the array and find all the connected variants of 1s starting from a given index. Here's how we can implement this in Python:
def generate_variants(array, start):
def is_valid(x, y, z):
return 0 <= x < len(array) and 0 <= y < len(array[0]) and 0 <= z < len(array[0][0])
def dfs(x, y, z):
# If out of bounds or already 1, return
if not is_valid(x, y, z) or array[x][y][z] == 1:
# Mark current position as 1
array[x][y][z] = 1
variants.append([row[:] for row in array]) # Add a copy of the current state to variants
# Explore all six possible directions
for dx, dy, dz in [(1, 0, 0), (-1, 0, 0), (0, 1, 0), (0, -1, 0), (0, 0, 1), (0, 0, -1)]:
dfs(x + dx, y + dy, z + dz)
# Backtrack to explore other variants
array[x][y][z] = 0
variants = []
dfs(*start) # Unpack the starting index
return variants
How It Works
• is_valid: This helper function checks if the indices are within the bounds of the array.
• dfs: This function performs a depth-first search starting from the given index. It marks the cell as 1, adds the current state of the array to our results, and then recursively visits neighboring
cells. After exploring, it resets the current cell to 0 to explore new paths (backtracking).
Example Usage
Suppose we have a 3D array of dimensions 2x2x2 (a small cube) initialized to zero, and we want to start our connections from the index (0, 0, 0):
initial_array = [[[0, 0], [0, 0]], [[0, 0], [0, 0]]]
start_index = (0, 0, 0)
result_variants = generate_variants(initial_array, start_index)
for variant in result_variants:
Expected Output
The output will show different 3D configurations where the 1s are connected starting from the given index:
[[[1, 0], [0, 0]], [[0, 0], [0, 0]]]
[[[1, 1], [0, 0]], [[0, 0], [0, 0]]]
[[[1, 0], [1, 0]], [[0, 0], [0, 0]]]
... (other variants)
This implementation allows us to generate all connected configurations of a 3D array of 0s and 1s starting from a specified index. Such problems are common in pathfinding algorithms and games,
helping programmers develop skills in recursion, state management, and depth-first search traversal.
For more in-depth learning and exploration, consider checking out the following resources:
By understanding and practicing these principles, you can tackle similar challenges with confidence! | {"url":"https://laganvalleydup.co.uk/post/given-a-3-d-array-of-0s-and-the-index-for-a-starting-1","timestamp":"2024-11-14T21:33:55Z","content_type":"text/html","content_length":"84017","record_id":"<urn:uuid:601d0c7b-45a9-4b56-92ac-b5d0bf69ddba>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00443.warc.gz"} |
Grid Synchronisation of IVS based MPPT PV Array Using Damped SOGI Control Algorithm
Reddy, Sadhu ReddySekhar (2018) Grid Synchronisation of IVS based MPPT PV Array Using Damped SOGI Control Algorithm. MTech thesis.
Restricted to Repository staff only
This project consists of two parts in first part detail about modified perturb & observe & input voltage sensor (IVS) algorithms,second part is about two stage grid integration of PV panel using
damped second order generalised integrator (SOGI) algorithm.Conventional perturb & observe MPPT algorithm use PV panel voltage and current to vary duty ratio dc-dc converter such that maximum power
is extracted from PV panel.IVS based algorithm PV panel voltage alone to track maximum power point of PV panel.These two algorithms are simulated using MATLAB/Simulink for rating of PV panel in
laboratory. SEPIC converter is used for interfacing PV panel and dc load.Experimental setup of two algorithms are implemented & experimental results are compared with simulated results. A detailed
comparison is made between two algorithms. In grid integration of PV module using damped SOGI algorithms, maximum power is extracted from PV module by varying duty ratio of dc-dc chopper by using IVS
based algorithm. A three phase Voltage source converter is used for synchronisation. Along with supplying power to grid this control algorithm also improve power quality by injecting current to grid
at unity power factor. Triggering pulses for VSC are given by comparing estimated grid current with actual grid current using hysteresis control.PV array grid integration by damped SOGI algorithm is
simulated using MATLAB/Simulink.
Item Type: Thesis (MTech)
Uncontrolled Keywords: Damped SOGI algorithm; Modified P&O algorithm;IVS algorithm; Grid integration; PV module; MPPT; Power quality improvement
Subjects: Engineering and Technology > Electrical Engineering > Power Electronics
Divisions: Engineering and Technology > Department of Electrical Engineering
ID Code: 9716
Deposited By: IR Staff BPCL
Deposited On: 12 Mar 2019 18:28
Last Modified: 12 Mar 2019 18:28
Supervisor(s): Gopalakrishna, S.
Repository Staff Only: item control page | {"url":"http://ethesis.nitrkl.ac.in/9716/","timestamp":"2024-11-06T09:11:16Z","content_type":"text/html","content_length":"15361","record_id":"<urn:uuid:aa2e7fdb-f8f9-4a08-a3ea-08bf4a19f3f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00364.warc.gz"} |
How do I use Cramer's rule to solve a 2xx2 matrix? | Socratic
How do I use Cramer's rule to solve a #2xx2# matrix?
1 Answer
Cramer's rule is used to solve a square system of linear equations, that is, a linear system with the same number of equations as variables. In a square system, you would have an $n \times \left(n +
1\right)$ matrix.
So you should have a $2 \times 3$ matrix in order to use Cramer's rule. A $2 \times 2$ matrix would only have the coefficients of the variables; you need to include the constants of the equations.
Impact of this question
2651 views around the world | {"url":"https://api-project-1022638073839.appspot.com/questions/how-do-i-use-cramer-s-rule-to-solve-a-2xx2-matrix#108405","timestamp":"2024-11-01T19:01:52Z","content_type":"text/html","content_length":"33060","record_id":"<urn:uuid:0c48bd3b-149c-4ba3-8004-a38abaa8c027>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00126.warc.gz"} |
Math Review
Five-Day Review Plan
The five-day review plan is a blessing in disguise for teaching math because it provides a scope to re-teach and re-learn material without repeating a lesson or a series of lessons. Specifically, it
is a 15 minute block of the day, where students can revise and revisit a math topic that we have already learned. Ultimately, the review plan is a way to support student growth in terms of a wide
range of concepts that prove to be a stumbling block for students while also moving onto other topics in the curriculum.
Planning for the Week
Monday Review and revise one aspect of a topic that students are struggling with (e.g. double digit addition with carryover). Ensure there are enough examples for them to observe and some
opportunities for them to apply the learning on their own in the time frame. The independent thinking time can be a simple exit ticket, for example.
Jeopardy is an instant favourite for students based on the famous game show hosted by Alex Trebek, which can also be used for science or other content areas. The online
Tuesday resource is a really fun way to plan and organize a Jeopardy review session. For someone who is not as quite tech savvy, I find the website quite user-friendly, and it happens to be free.
Teach Hub
has several other review games that are just as fun. I also recommend thinking of other popular game shows like the Family Feud or the Wheel of Fortune to add a bit of variety to this
activity and to avoid overusing one activity.
Wednesday Select a difficult question from a past exam or make one up around what the students are learning (i.e. basically something that might come up in an assessment and might be a road block for
them). Model the thinking behind solving the question and then give them a similar "challenge of the day" question to solve on their own as a form of guided practice.
Provide math questions on the board and use ABCD cards for students to show their answer. Then, discuss the different responses before showing students how to solve it (in fact, get one
Thursday student with the correct response to solve it). This can be a form of a math talk, where students engage in a discussion around their approaches and strategies. This review should ideally
help students prepare for their quiz tomorrow.
Friday Once a week, I give the students a quiz during our scheduled review time. The quiz includes five questions based on material covered in the Math Review block from the previous four days.
Make sure to keep in mind that there are only 15 minutes, so the quiz should not be too long or too difficult to solve.
Outside the Five-Day Review Plan
The five-day review plan might not work for every classroom, so I have outlined a few suggestions that other teachers use, as a way to step outside of the five-day review plan.
Math Stations:
Math stations are popular among classrooms, as teachers move towards greater student responsibility in terms of ownership over their own learning. For this, you can look up the
Math Daily 3
or the
Math League
Morning Math: Morning math is one way to make review a part of the schedule. It could follow the same format as the five-day review plan. Alternatively, it could be a morning math challenge that
students have to solve first thing in the morning before moving into the day's schedule. Teachers can also work the question around the previous day's topic to get a quick pulse of understanding. As
a way to reach different learners, I would suggest including learning activities that have a mix of concrete, pictorial, and abstract approaches to math. Using manipulatives, for example, is a
helpful way to encourage concrete learning.
In an article by Michelle Trudeau, it is explained how Rafe Esquith, a renowned teacher in an inner city school, spends the first hours of his class hours on mental exercises around math. Although I
do not suggest calling in students at 6:30am, especially since every teacher has a different situation, I do think there is merit in mental math exercises or some other form of the math block
occurring in the morning. Specifically, the routine of morning math is to create an environment for students where applying and learning math becomes valuable in the long-run.
Scavenger Hunt: The scavenger hunt is a popular review activity that has been shared by other teachers on their blogs. There are various versions to this activity, but simply teachers put ten
questions around the classroom. Students, in either pairs or groups three, have to go from one question to the other, scavenging across the classroom in a loop. At this point, the teacher should be a
listener who goes around the classroom to provide feedback or support where required. The scavenger hunt can be made more interesting by adding an element of a code, a hidden message, or a race. I
also provide students with a map, so they know where to go to next. For example, if the map tells them to start with the sixth question, the next clue would say, "Go to the question that is the sum
of 2+3." This is a fun and admittedly nerdy (all teachers exhibit this) way of putting math clues in the map. You can also think of ways to integrate a specific topic into the scavenger hunt. For
instance, a measurement topic can be integrated here really well. Each group would be tasked with finding hidden objects in the class using their rulers to measure the objects based on the clue
provided to them. Always think of ways to make the scavenger hunt different from the last time; otherwise, it becomes another repetitive activity. | {"url":"https://www.shyampatel.ca/math-review.html","timestamp":"2024-11-06T20:24:37Z","content_type":"text/html","content_length":"44185","record_id":"<urn:uuid:72cd2036-2b22-4474-aba8-8c0d48b1d2d4>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00348.warc.gz"} |
: Book I, Proposition 47
The following is as given in Sir Thomas L. Heath's translation, which can be found in the book The Thirteen Books of The Elements, Vol. 1. It is a proof of the Pythagorean Theorem.
Proposition 47.
In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.
Let ABC be a right-angled triangle having the angle BAC right;
I say that the square on BC is equal to the squares on BA, AC.
For let there be described on BC the square BDEC, and on BA, AC the squares GB, HC; [I. 46]
through A let AL be drawn parallel to either BD or CE, and let AD, FE be joined.
Then, since each of the angles BAC, BAG is right, it follows that with a straight line BA, and at the point A on it, the two straight lines AC, AG not lying on the same side make the adjacent angles
equal to two right angles;
therefore CA is in a straight line with AG.[I. 14]
For the same reason
BA is also in a straight line with AH.
And, since the angle DBC is equal to the angle FBA: for each is right;
let the angle ABC be added to each;
therefore the whole angle DBA is equal to the whole angle FBC.[C. N. 2]
And, since DB is equal to BC, and FB to BA,
the two sides AB, BD are equal to the two sides FB, BC respectively,
and the angle ABD is equal to the angle FBC;
therefore the base AD is equal to the angle FC,
and the triangle ABD is equal to the triangle FBC. [I. 4]
Now the parallelogram BL is double of the triangle ABD, for they have the same base BD and are in the same parallels BD, AL.
And the square GB is double of the triangle FBC,
for they again have the same base FB and are in the same parallels FB, GC.[I. 41]
[But the doubles of equals are equal to one another.]
Therefore the parallelogram BL is also equal to the square GB.
Similarly, if AE, BK be joined,
the parallelogram CL can also be proved equal to the square HC;
therefore the whole square BDEC is equal to the two squares GB, HC. [C. N. 2]
And the square BDEC is described on BC, and the squares GB, HC on BA, AC.
Therefore the square on the side BC is equal to the squares on the sides BA, AC.
Therefore etc. Q.E.D. | {"url":"http://mathlair.allfunandgames.ca/elements1-47.php","timestamp":"2024-11-12T15:34:46Z","content_type":"text/html","content_length":"6951","record_id":"<urn:uuid:e4fe93ac-73a6-4fbc-8c51-195122fa71fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00158.warc.gz"} |
A Unified Multiple-Phase Fluids Framework Using Asymmetric Surface Extraction and the Modified Density Model
School of Computer and Communication Engineering, University of Science and Technology Beijing, Beijing 100083, China
Beijing Key Laboratory of Knowledge Engineering for Materials Science, Beijing 100083, China
Beijing No.4 High School, Beijing 100034, China
Author to whom correspondence should be addressed.
Submission received: 30 March 2019 / Revised: 21 May 2019 / Accepted: 27 May 2019 / Published: 2 June 2019
Multiple-phase fluids’ simulation and 3D visualization comprise an important cooperative visualization subject between fluid dynamics and computer animation. Interactions between different fluids
have been widely studied in both physics and computer graphics. To further the study in both areas, cooperative research has been carried out; hence, a more authentic fluid simulation method is
required. The key to a better multiphase fluid simulation result is surface extraction. Previous works usually have problems in extracting surfaces with unnatural fluctuations or detail missing. Gaps
between different phases also hinder the reality of simulation. In this paper, we propose a unified surface extraction approach integrated with a modified density model for the particle-based
multiphase fluid simulation. We refine the original asymmetric smoothing kernel used in the color field and address a binary tree scheme for surface extraction. Besides, we employ a multiphase fluid
framework with modified density to eliminate density deviation between different fluids. With the methods mentioned above, our approach can effectively reconstruct the fluid surface for
particle-based multiphase fluid simulation. It can also resolve the issue of overlaps and gaps between different fluids, which has widely existed in former methods for a long time. The experiments
carried out in this paper show that our approach is able to have an ideal fluid surface condition and have good interaction effects.
1. Introduction
Visualized fluid simulation has been studied for a long time and with the help of both fluid dynamics and computer animation techniques. There have been various kinds of methods for fluid simulation
visualization. They can be divided into two types based on the difference of the spatial discretization method, which are mesh-based Euler approaches and particle-based Lagrangian approaches. In
Euler approaches, the simulation domain is discretized into mesh grids, and the physical values on grid points (such as acceleration, pressure) can be obtained through the governing equations.
Mesh-based methods can create animations with a realistic appearance, but they are time-consuming and have difficulty handling certain phenomena accurately like free surfaces, complex boundaries, and
splashes. Nevertheless, in particle-based methods like the Smoothed Particle Hydrodynamics (SPH), the volume of the fluid is discretized by particles. Each of them carries physical properties and
moves freely according to the velocity field. Particle-based methods can easily maintain momentum conservation and incompressibility and have been used to simulate various kinds of phenomena, such as
water, smoke, deformable solids, as well as viscoelastic liquids and multiphase fluids. However, some major particle-based methods have difficulties in producing realistic surfaces, especially for
multiphase interactions.
In daily life, multiphase phenomena like water bubbles and the mixture of immiscible fluids are everywhere. Currently, multiphase simulation is receiving wide attention. While in SPH simulation,
there is a certain spatial distance between particles and particles’ properties are smoothed by the kernel function, when the static density and mass of neighbor particles are different, the physical
quantity calculated using SPH approach will be biased. This problem is especially obvious at the interface of multiple phases. The density value calculated by the fluid with high static density near
the interface is relatively small, while the density value calculated by the fluid with low static density is relatively large. This is because the standard SPH formula smooths the density at the
interface and does not accurately represent a sharp change in density. Moreover, due to the density deviation, the calculation deviation of other physical quantities, such as pressure, will lead to
unreal interface effects and serious numerical instability. Besides, extracting high-quality surfaces of multiple phases from particle locations has been rarely discussed. The standard approaches for
reconstructing surfaces of particle-based simulation usually need to create an implicit surface, which virtually wraps around all the particles during the simulation. Although recent research has
significantly improved the surface appearance that results from formulating the implicit function, unfortunately, there have still been problems for these methods in accurately and stably handling
interface evolutions, especially at the location where the multiphase interface lies. Few research works have focused on the problem of extracting smooth surfaces and the interface in multiphase
simulation for particles.
In this paper, we present a novel surface extraction approach integrated with the multiphase model with modified density, which significantly improves the appearance at the multiphase interface while
keeping a good fluid surface quality. We employ an asymmetric smoothing kernel to represent each particle. The direction and scale of the asymmetry are determined by the distribution of the
particle’s neighborhood. The asymmetric kernel is constructed like Yu’s [
] method, but we consider the influence of other phases when computing one phase’s kernel and color field. Besides, we also address a binary tree strategy [
], which is convenient to implement to divide equally the overlap area and fill up the vacuum space. In addition, we employ a multiphase fluids model to eliminate density deviations at the interface.
The results show that our method acquires a better realistic appearance of the fluid surface and interface when compared to previous methods. More importantly, the binary tree strategy can suit the
data-driven approach easily to further boost the process, which shows great potential in surface-specified enhancement that would be a promising topic for 3D visualization of fluids.
2. Related Work
Currently, SPH is a popular approach for fluid simulation. Desbrun [
] started to simulate deformable objects with SPH. Monaghan [
] addressed free surface flow simulation approaches with SPH as a basis for fluid simulation. Later, Müller et al. [
] firstly applied SPH in computer graphics for fluid simulation. They employed Boyle’s law and surface tension, as well as viscosity forces to calculate forces, which brought the problem of
compressibility. Based on the former studies, Becker and Teschner [
] proposed Weakly-Compressible SPH (WCSPH) using the Tait equation. This method reduced compressibility significantly and increased the fidelity; however, the efficiency of simulation was limited
severely by the time step. A more strict incompressibility condition would require smaller time steps, which cost much more time for computation. After that, more advanced approaches were proposed to
improve the fidelity and efficiency. Solenthaler and Pajarola [
] presented the Predictive-Corrective Incompressible SPH (PCISPH) method that uses a prediction-correction scheme to determine the particle’s pressure. PCISPH can increase time steps remarkably,
making it more efficient than WCSPH. He et al. [
] presented a similar method, Local Poisson SPH (LPSPH), which can ensure incompressibility by an iterative process. Afterwards, Ihmsen et al. [
] addressed Implicit Incompressible SPH (IISPH). This method builds the Pressure Poisson Equation (PPE) carefully and then solves the linear problem with the relaxed Jacobi approach. This method has
a great improvement in both stability and convergence speed. Particularly, IISPH is especially fit for large-scale simulation. Recently, an approach with potential for SPH fluid simulation was
introduced by Bender and Koschier [
]. It combines two pressure solvers together to restrict low volume compression and ensure a divergence-free velocity field. What is more, this method is able to carry out the simulation in large
time steps, yet enhancing the appearance.
For the density deviation of the SPH formula at the interface, Hoover [
] firstly described the problem of false interfacial tension caused by small particle density and pressure calculation near the interface. Agertz et al. [
] also presented a similar situation, pointing out that the wrong pressure would create voids and severe instability between two-phase fluids with a high density ratio. In order to deal with the
numerical instability problem caused by the increase of the density ratio, Ott et al. [
] proposed an improved continuity equation. Although it is very effective for some special application scenarios, neither the standard nor improved continuity equation can produce stable long-term
simulation results. This can lead to serious density integral errors, especially when using large time steps and low order time integral schemes. Tartakovsky et al. [
] proposed a corrected formula for density summation, which combined the corrected SPH flow equation with the advection diffusion equation to simulate the miscible flow with complex geometry. Hu et
al. [
] focused on the study of numerical examples of droplet oscillation and deformation in two-dimensional shear flow. In addition, early studies on multiphase fluids included solving a discontinuous
interface [
] and focusing on bubble and foam [
], which are all based on the Euler method. Losasso et al. [
] proposed a level set method that uses SPH particles to represent diffusion regions, and Thurey et al. [
] proposed a foam simulation method based on SPH. Muller et al. [
] proposed an SPH-based particle simulation method to deal with multiphase and water boiling. Mao et al. [
] simulated immiscible fluid by explicitly detecting collision particles. Solenthaler et al. [
] used quantitative density to improve density summation and employed it to correct other physical quantities to solve the problem of the false interface effect.
Particle-based fluid simulation has difficulty with surface extraction, for those particles cannot give any interrelated information. In the past several decades, researchers have addressed several
surface extraction methods for particle-based fluid simulation. Blinn presented the blobby sphere approach [
], which computes the distance between the scattered points and sampling points as a parameter to accumulate the implicit surface functions. It successfully reconstructs surfaces from discrete
points, but whether the density of the particles is high or low, they would always cause indentations or become bumpy on the surface. Müller et al. [
] introduced a color field approach as a kind of level set scheme to construct the fluid surface simply and rapidly. However, the surface being extracted was rough, and bumps were produced by
particles next to the surface. Zhu and Bridson improved the blobby sphere approach by adjusting density variations of the local particles [
]. They first calculated the fluid particles’ radius, as well as the coordinates’ weighted mean according to the neighbor particles’ radius and position. Then, the weighted mean radius and
coordinates were used to extract surfaces that became a relatively smoothed fluid surface compared to the original blobby sphere approach. Adams et al. [
] modified Zhu’s approach through tracking the distances between the particle and surface over time. This method can successfully produce smooth surfaces for both fixed radius and adaptive particles
with the cost of excessive time. Bhatacharya et al. [
] introduced a level set approach considering surface reconstruction as an optimization issue. To obtain a smooth surface, they smoothed the initial surfaces through an iteration process. Yu et al. [
] introduced an alternate approach. They built the surfaces only at the beginning of the simulation, which reduced the computation cost. Akinci et al. [
] introduced a scheme to reconstruct and optimize the surface concurrently and presented a surface tectonic line approach [
]. Yu and Turk [
] presented the implicit surface with asymmetric kernels, which are able to extract smoother fluid surfaces with more fidelity. In their approach, a unique kernel function was applied for each
particle, which was constructed according to the distribution of the neighbors of the particles. It can handle planar fluid so easily that even thin surfaces with sharp features are still able to
achieve an ideal appearance.
Furthermore, Because the topology changes and movements of the multiphase interface are quite complicated, it is a very challenging issue to track the multiphase interface. The multiphase interface
always plays an important role in the multiphase simulation. In the past few years, researchers have proposed many methods for multiphase interface tracking. Level set methods [
] often replace the binary sign of the distance field with an integer phase label to extend to multiphase interface tracking [
]. In a number of methods, one signed distance field is used for each phase, while other methods used a single unsigned distance field for all phases to reduce memory and computation costs.
Starinshak et al. [
] presented an issue about multiple level set methods, i.e., overlaps or voids at the triple phase interface, which is usually corrected through projection. Da et al. [
] explicitly tracked such a triple phase interface that avoided vacuums and overlaps by reconstruction. Volume-of-fluid methods [
] allocate a volume fraction for each cell, and their multiphase material stores a partition of unity, i.e., one fraction each phase. Then, this information is used to reconstruct a continuous
multiphase interface [
]. Moving mesh or semi-Lagrangian methods [
] allot phase labels to each volume element. Some recent research discussed the issue of mesh maintenance [
]. Particle-based surface reconstruction augments each particle with a phase label or color [
] or allocates phase attributes to particles directly.
However, most of these methods cannot be directly integrated with particle-based fluid simulation, and this existing method is difficult to achieve. In addition, the surface and interface appearance
of the reconstructed fluid are pretty rough with gaps or overlaps. Therefore, we introduce a much easier method to build the multiphase interface by applying asymmetric kernels, which can eliminate
the overlaps and gaps of the multiphase interface in surface reconstruction.
3. SPH Fluid Simulation
SPH has become a popular approach for interpolation in Lagrangian systems. The main idea of the SPH method is to represent continuous fields using discrete particles and apply the integral to
approximate the field. A scalar quantity
$A x i$
of particle
at location
$x i$
can be interpolated by the sum of quantities from neighbor particles in the Standard SPH (SSPH) method:
$A x i = ∑ j m j A j ρ j W x i − x j , h$
$m j$
$ρ j$
represent the mass and density of each particle,
$W x i − x j , h$
is the smoothing kernel used in the SPH approach, and
is the smoothing length.
In the SPH method, the Navier–Stokes equation is discretized based on particle locations, which can be written as a differential equation as follows:
$ρ i ∂ v i ∂ t = − ∇ p i + μ ∇ 2 v i + f i e x t$
$f i e x t$
is the external force and
$∇ p i$
$∇ 2 v i$
are the pressure gradient and Laplacian form of velocity. Since the volume of fluid is represented by particles in the SPH method, the density
$ρ j$
can be interpolated using a weighted sum of the mass
$m j$
from neighbor particles. In order to simulate fluid using particles:
$ρ i = ∑ j m j W x i − x j , h$
$p i$
is the pressure of each particle
, which can be calculated by the function of density. The SSPH method uses the ideal gas equation:
$p i = k ( ρ i − ρ 0 )$
, where
$ρ 0$
is the rest density and
is a constant determined by the researchers for different effects. Becker and Teschner [
] used the Tait equation to replace the gas equation. This can further restrict the density variations and enhance the efficiency:
$p i = ρ 0 c S 2 γ ρ i ρ 0 γ − 1$
$c s$
is the numerical speed of sound and
is the stiffness parameters; we set it to seven in our experiments.
Then, pressure force
$f i p$
and viscous force
$f i v$
between particles can be represented as:
$f i p = − ∑ j m j ( p i ρ i 2 + p j ρ j 2 ) ∇ W i j$
Because the second derivative of the kernel function would lead to severe numerical error and instability, we applied the artificial viscosity in this paper, so the viscous force
$f i v$
is expressed as:
$f i v = 2 μ ( d + 2 ) ∑ j m j ρ i v i j · x i j | v i j 2 | + 0.01 h 2 ∇ W i j$
is the dimension,
$v i j$
$x i j$
is the velocity and distance between two particles, and
is the viscosity coefficient.
4. Multiple-Phase Fluids’ Simulation Using Modified Density
When different fluids are blended, whether they will mix with each other is based on the intermolecular interaction. There will be distinct multiphase interactions when they are immiscible. As shown
Figure 1
, there are three phases in the whole domain. When applying the particle-based approach, there is interaction force between particles. The force between particles is equal in all directions and
maintains a balance in the same phase. However, when the different phases are in touch and mix with other phases at the junction, the forces between particles are no longer balanced. Interfacial
tension would emerge at the multiphase interface, which is the surface contraction force acting perpendicular to the fluid surface along the multiphase interface. The interfacial tension is similar
to the surface tension effect between the fluid and the air interface, but the surface tension usually ignores the effects from gases. The interfacial tension of multiphase flow has to consider the
influence of the particles of other phases; therefore, it is more complex to handle.
4.1. Modified Density Model
In order to deal with the density discontinuity at the multiphase interface, we applied the quantity density of Solenthaler [
] to modify the traditional formula of SPH for calculating density. The main concept of this method is that when each particle’s density is calculated according to its neighbor particles’ density,
let the neighbor particles and the particle itself seem to have the same static density and mass. The quantity density
$δ i$
can be written as:
$δ i = ∑ j W x i − x j , h$
Construct the modified particle density
$ρ i ′$
as the quantity density multiplied by the current particle mass, which is:
The volume of particles can be expressed as:
$V i = m i ρ ′ i = 1 δ i$
For a single fluid phase with the same mass and static density, the density formula above is in line with the standard SPH formula. However, for a multiphase fluid with different densities, the
density calculated by the above method can make the density at multiphase interfaces distinct without the continuous issue calculated by the standard SPH formula, as shown in
Figure 2
4.2. Adjusted Pressure Computation
In our method, we employ the Tait equation to calculate pressure. Therefore, pressure should be adjusted according to modified density
$ρ i ′$
. We get
$p i ′$
by replacing standard density
$ρ i ′$
in the Tait equation [
], which is:
$p i ′ = k ρ 0 γ ρ ′ i ρ 0 γ − 1$
According to the modified pressure formula, a new formula for calculating the pressure can be obtained through replacing
of pressure gradient
$a = − ∇ p − ∇ p ρ ρ$
in the Navier–Stokes equation with
$ρ ′$
$p ′$
, which is:
According to Newton’s second law, The formula of pressure can be expressed as:
In addition, we derived the pressure equation based on Monaghan [
], then applied the quotient rule with the pressure gradient in the Navier–Stokes equation as follows:
$∇ p ρ = ∇ p ρ + p ρ 2 ∇ ρ$
If we directly use modified density
$ρ ′$
and pressure
$p ′$
in the formula above, there will be a serious instability issue at the interface. This is because the discontinuity of
$ρ ′$
and its derivative does not exist at the interface. To avoid this issue, we can employ the quotient rule in Equation (
), and Equation (
) can be expressed as:
$∇ p ′ δ = ∇ p ′ δ + p ′ δ 2 ∇ δ$
Substituting Equation (
) into SPH formula, we can get:
$∇ p ′ δ = ∑ j p ′ j δ j + p ′ i δ i 2 δ j V j ∇ W i j$
Now, substitute Equation (
) into the formula above, then it can be simplified as:
$∇ p ′ δ = ∑ j p ′ j δ j 2 + p ′ i δ i 2 ∇ W i j$
Therefore, using Equation (
) in Equation (
), the pressure of particles can be expressed as:
$F i p = − ∑ j p ′ j δ j 2 + p ′ i δ i 2 ∇ W i j$
Further, the formula for calculating viscous force and surface tension is independent of pressure, but only based on density. Therefore, we only need to substitute the corrected density into the
corresponding formula.
4.3. Interfacial Forces of Multiple-Phase Fluids
According to the modified formula of density, pressure, and force, the unnatural interface effect of the standard SPH method can be eliminated. In order to show the effect of the multiphase flow
interface with more fidelity, it is necessary to apply interfacial force. We employed a color field model to calculate the interfacial tension [
], which can have better control of the interface effect of multiphase flow.
The definition of interfacial tension is as follows [
is the coefficient of interfacial tension,
is interface curvature, and
is the normal interface. The interface force tries to smooth the high curvature interface region and minimize the total surface area.
To compute $n$ and $κ$, the non-zero color values are defined in all particle positions, and the particles of different phases are defined with different color field values. The color field formula
is $c i = ∑ j m j c j ρ j W i j$, which uses $c i$ to express particle i’s color field value and $n = ∇ c$ to compute the normal vector.
In order to avoid the tension on the free surface, the color field is standardized. We used the modified density in color field formula, and the expression of the color field is as follows:
$c i ′ = ∑ j c j δ j W i j ∑ j c j δ j W i j ∑ j 1 δ j W i j ∑ j 1 δ j W i j$
$n = ∇ c$
, the normal vector can be expressed as:
$n i = ∑ j 1 δ j W i j c ′ j − c ′ i ∇ W i j$
Based on the curvature formula
$κ = − ∇ · n ^$
, where
$n ^$
is the unit normal, the normalized curvature can be expressed as:
$κ = − ∑ j 1 δ j n ^ j − n ^ i · ∇ W i j − ∑ j 1 δ j n ^ j − n ^ i · ∇ W i j ∑ j 1 δ j W i j ∑ j 1 δ j W i j$
5. Surface Extraction Using Asymmetric Kernels
Traditionally, the color field [
] can be written as:
$ϕ ( x ) = ∑ j m j ρ j W ( x − x j , h j ) .$
In this equation,
is the symmetric kernel function, and it can be expressed as:
$W ( r , h ) = σ h d P | | r | | h$
is a scaling factor, while
is a radial vector,
represents the dimension of the simulation space, and
is a finite-supported symmetric decaying spline.
To deal with the badly-distributed density near the surface, the asymmetric kernel approach [
] smooths the location of the kernel center
$x i$
using one step of diffusion to ensure denoising. The refined kernel center
$x ¯ j$
can be represented as:
$x ¯ i = 1 − λ x i + λ ∑ j w i j x j ∑ j w i j x j ∑ j w i j ∑ j w i j$
$0 < λ < 1$
is the weighted function.
The asymmetric kernel method [
] can get the density distribution with more details by using
instead of a
$d × d$
real positive definite matrix
, and it makes
an asymmetric kernel:
$W ( r , G ) = σ det ( G ) P Gr$
$r = x − x ¯ j$
, the
in it can be regarded as an arbitrary position, and the matrix
is used to rotate and stretch the radial vector
After that, the asymmetric kernel approach uses Weighted Principal Component Analysis (WPCA) to compute G. The WPCA starts with calculating the data points’ weighted mean. After that, WPCA builds a
weighted covariance matrix C and performs eigendecomposition on it. The principal axes will be provided by the resulting eigenvectors. Finally, it builds an asymmetric matrix G with the output of
WPCA to match W.
The expression of the covariance matrix can be written as:
$C i = ∑ j w i j ( x j − x ¯ i ) ( x j − x ¯ i ) T ∑ j w i j ( x j − x ¯ i ) ( x j − x ¯ i ) T ∑ j w i j ∑ j w i j$
The weighted mean of particle
applied by the asymmetric kernel approach is written as:
$x ¯ i = ∑ j w i j x j ∑ j w i j x j ∑ j w i j ∑ j w i j$
The function
$w i j$
is an symmetric weighted function:
$w i j = 1 − x i − x j l i 3 x i − x j < l i 0 other$
$l i$
is the radius of the neighbor scope. To get adequate neighbors and gain sensible asymmetric data, we set
$l i = 2 h i$
For each of the particles, they require Singular Value Decomposition (SVD) on the covariance matrix
$C i$
, that is:
By using principal axes as column vectors, R is a rotation matrix. $Σ = d i a g ( σ 1 , ⋯ , σ d )$.
is a
$3 × 3$
rotation matrix using the eigenvectors of
$C i$
as column vectors in the above formulas. In each column,
$R i$
is the distribution axis of
$C i$
corresponding to the eigenvalue
$σ i$
is a diagonal matrix with eigenvalues
$σ 1 ≥ ⋯ σ d$
, so the quantity of neighbor particles is the largest in the direction
$R 1$
and the smallest in the
$R 3$
axis. In order to avoid extreme conditions and unexpected situations from occurring, the asymmetric kernel method also modifies the matrix
$C i$
. First, we modify
$σ i$
, if
$σ 1 / σ d ≥ k r$
, and then,
$σ i$
is replaced by
$σ 1 / k r$
. After that, it uses
$G = k n I$
to replace asymmetric kernels for isolated particles and internal fluid particles. The modified
$C i$
can be expressed as:
$Σ ˜ = k s d i a g ( σ 1 , σ ˜ 1 ⋯ , σ ˜ d ) N > N t k n I other$
$σ ˜ i = max ( σ i , σ 1 / k r )$
represents the quantity of neighborhoods and
$N t$
indicates the threshold. Asymmetric kernel method assures
$k s C ≈ 1$
$k n = 0 . 5$
$N t = 25$
by employing
$k r = 4$
$k s = 1400$
In order to enable
to transform smoothly in accordance with
$C ˜ i$
while maintaining the former form,
should be written as:
6. Asymmetric Surface Extraction for Multiple-Phase Interfaces
6.1. Asymmetric Kernel for Multiple-Phase Interfaces
In principle, the asymmetric surface extraction method is regarded as a level set method. It usually uses various level set functions on each phase to construct the multiphase interface. Therefore,
it is also known as the multiple level set method. Next, it constructs surfaces of the fluids on the basis of the various level set functions. However, applying multiple level set methods directly is
inclined to evoke errors at the interface. While applying the symmetric approach [
], the overlapping issue arises in the multiphase interface. What is more, a worse overlapping problem or crack issue at the interface will arise when using the asymmetric surface construction
Figure 3
exhibits an instance of the symmetrical kernel function becoming an asymmetric function. The balls and the ellipsoids are the scope of the particles’ kernels. The left side of the picture is the
symmetrical kernel, while the right side shows the asymmetric kernel. We can see that after asymmetric transition, next to the fluid’s surface, the smooth kernel transforms from round particles to
ellipsoid particles, while the internal particles’ smoothing kernel remains original. Therefore, the multiphase interface tends to exhibit an appearance that looks alike while using the asymmetric
surface reconstruction approach combined with the multiple level set method, which can be seen in
Figure 4
. In
Figure 4
(left), the symmetrical kernel approach is given, while the right figure is the asymmetric kernel approach. We can obviously see that particles next to the interface have a lack of neighbors. For the
sake of including more neighbors at the interface, the asymmetric kernel method will deform the smooth kernel, so that particles’ kernels near the interface are deformed into ellipsoids.
Nevertheless, on the basis of a multiple level set, the reconstructed fluid surfaces have obvious voids at the multiphase interface due to a lack of consideration of the influence of other phases’
particles, which would jeopardize the rendering process severely in the following.
To handle the non-uniform distribution of the particles and reduce noises, the asymmetric kernel approach would apply the Laplacian smoothing technique at the center of the kernel (as shown in
Equation (
)). This procedure can significantly reduce the irregular condition. For the reason that most of the neighbors of particles next to the surface are inside, the fluid volume is compressed slightly,
and the whole fluid shrinks inwardly. It is quite simple to figure out that the smoothing can result in the voids of the multiphase interface, as well.
When building the fluid surface according to phases using the multiple level set approach, overlap appearance would occur at the multiphase interface. This is common when using the level set
approach. As shown in
Figure 5
, the overlap phenomenon occurs at the fluid interface when reconstructing the two-phase surface. The left side of the figure indicates the perfect simulated fluid surface, and the middle of the
figure indicates the reconstructed fluid surfaces with the multiple level set, while the right shows the surface in the whole domain. As the color field approach [
] belongs to the level set approach, we study the cause of the overlapping problem of the multiphase interface using the color field approach as an example. As shown in
Figure 5
, the blue fluid value was 10, and the green fluid value was 20. The value of the blue ones next to the surface reduces from 10–0 linearly, while the green ones’ value next to the surface decreases
from 20–0. Now, if a unified color field value is selected to build the surface of the whole fluid domain, overlaps would occur at the two-phase interface. This is due to fact that next to the
surface, there are overlapping areas (10–0 or 20–0) for the color field values of the two fluids. Therefore, when selecting a unified color field value to build the whole fluid surface, the two-phase
fluid interface will intersect. For instance, to rebuild the whole surface of the fluid domain, we can set the color field to zero, as shown in the right of
Figure 5
(however, overlapping problems with other values still remain).
To decrease the surface reconstruction defects during the simulation as much as possible, we took the contribution of other phases into account when tracking the fluid surface from each kind of fluid
and handled them with some specific measures.
First of all, because the formula of kernel centers applied by the asymmetric surface reconstruction approach only take the contribution of its own phase into account, Equation (
) is updated as follows:
$x ¯ ′ i = 1 − λ x i + λ ∑ k n ∑ j w i k j x k j ∑ j w i k j x k j ∑ j n ∑ j w i k j ∑ j n ∑ j w i k j$
$λ$ is a constant between zero and one, and w represents the weight function, while n is the quantity of phases.
Likewise, we modify the covariance matrix from Equation (
) as:
$C ′ i = ∑ k n ∑ j w i k j ( x j − x ¯ i ) ( x j − x ¯ i ) T ∑ k n ∑ j w i k j ( x j − x ¯ i ) ( x j − x ¯ i ) T ∑ k n ∑ j w i k j ∑ k n ∑ j w i k j$
In this formula,
$w i k j$
is the weight function that takes the distance between particles
into account.
$w i k j$
is written as:
$w i k j = 1 − x i − x j l i 3 x i − x j < l i 0 other$
$l i$
indicates the support radius,
is the phase, which has no influence on the weighted function, and
$w i k j$
is just like the one in Equation (
On the basis of the covariance matrix in Equation (
), we produce matrix
$G ′$
similar to matrix
$G ′ = 1 h R ′ Σ ′ ˜ − 1 R ′ T$
Overall, the color field that is used to extract surface from all phases is written as:
$ϕ ′ ( x ) = ∑ j m j ρ j W ( x − x j , G ′ j )$
6.2. Surface Extraction Strategy
In the instance of
Figure 5
, we used the color field as an Unsigned Distance Field (USDF). The USDF is typically good at surface reconstruction, and we devised the Signed Color Field (SDF) for a better extraction of the
multiphase interface with the strategy we employed as the “binary tree” used to extract the surface, as shown in
Figure 6
We demonstrated our reconstruction scheme with a four-phase fluid simulation, as shown in
Figure 6
. In this figure, Nodes 1, 2, and 4 indicate the fluid interface of the four-phase, three-phase, and two-phase, respectively, and Nodes 3, 5, 6, and 7 represent four different kinds of fluids,
respectively. When reconstructing the fluid surface, we only extracted one phase from the entire fluid domain at a time in the iteration. This is because we were able to apply the symmetrical color
field with positive and negative values (signed color field) to distinguish the two-phase interface when expressing the fluid surface. The signed color field could make the color field values between
two-phase interface transit uniformly from negative to positive. When we want to extract the surface of Fluid 1 (Node 3) and the surface of the other phases (Node 2) from the four-phase fluid (Node
1), we employ the color field for
$ϕ ′ x = 1$
$ϕ ′ ′ x = − ϕ ′ x$
, respectively. According to this method, two kinds of fields can be interpolated using Equation (
), where one represents the fluid domain of Fluid 1 and the other denotes the fluid domain of the other three phases. Then, we can reconstruct fluid surfaces of Fluid 1 and the other three phases
based on the interpolated two kinds of field values. In terms of choosing the field values for surface reconstruction, we used
$ε ϕ ′ x$
$− ε ϕ ′ x$
$0 < ε < 0.1$
, and we set
$ε = 0.01$
) to construct the surface. In this way, we can ensure that the reconstructed fluid surface is approximately one face at the multiphase interface without serious vacuums and overlaps. Then, we
extracted the surface of Fluid 2 and the other two phases from the remaining three phases in the same manner. By repeating the step mentioned above, we can eventually reconstruct the surface of
Fluids 3 and 4. In conclusion, the fluid surface can be separately extracted for each phase using our “binary tree” scheme, which preserve voids and overlaps between interfaces.
In other words, the multiphase fluid (n-phase) surface reconstruction strategy we proposed can be summarized as follows:
• Initially, employ two color fields $ϕ ′ x$ and $− ϕ ′ x$ for the particles for one phase and then the rest of the $n − 1$ phases, respectively;
• Additionally, interpolate the signed color field for one phase and the other n-1 phases, and then, select $ε ϕ ′ x$, $− ε ϕ ′ x$ separately as the surface field value;
• Furthermore, on the basis of the chosen surface field value, rebuild the surface for the one phase;
• Finally, iterate the procedures above until the surface of each phase is fully rebuilt.
There are several advantages of our surface reconstruction scheme for the multiphase interface. One thing is that the color field approach forces the values to be reduced linearly from $ϕ x$ to zero
next to the fluid surface approximately. Applying the signed color field enables values to be decreased linearly from positive to zero, and gradually decreased to negative at the two-phase interface.
Therefore, it enables the surface to be reconstructed uniformly at two-phase interface. Overall, we can avoid the averagely-divided biases of the interface area, or it might have voids or overlaps.
The other issue is that selecting a relatively small value for the surface field $ε ϕ ′ x$, $− ε ϕ ′ x$ guarantees that no further area would be rebuilt. It is simple to prevent reconstructing the
domain that does not belong to the fluid surface when the values of the color field look pretty much the same as the fluid surface. Furthermore, since $ε ϕ ′ x$ is small enough, we can assumably
regard the multiphase interface as one interface without overlaps.
7. Implementation and Results
We carried out a few experiments to exhibit the credibility of multiphase fluid simulation, as mentioned above. The experiments were achieved with C++. Meanwhile, OpenMP was used for parallelization.
The neighbor search process was achieved with the spatial Hash method with a width uniform space background mesh. We used the marching cubes algorithm as the surface reconstruction method for the
multiphase interface presented in this paper to extract the fluid surfaces. With the help of the Open Graphics Library (OpenGL) (The Khronos Group, Beaverton, OR, USA), a real-time simulation was
presented, and offline high quality rendering was carried out with Blender’s ray tracing engine cycles. The experiments were carried out on a graphic workstation with an Intel Xeon E5-2637 v2 (15M
Cache, 3.50 GHz @ 4 cores) CPU, 80 gigabytes RAM, NVIDIA Quadro K4000 GPU (Dell, Round Rock, TX, USA).
Figure 7
shows the simulation outcomes of the symmetric kernel approach [
], the asymmetric kernel approach [
], and our approach in a breaking dam simulation with two phases. The key parameters of this scene can be seen in
Table 1
. The setting of the two-phase breaking dam can be described as two cuboids consisting of different fluid phases affected by gravity falling from the start, after two-phase fluids come in contact and
gradually blending. Then, with the influence of pressure and interface force, the fluid with a lesser density will “float” gradually, while the other fluid with a greater density would start to
“sink”. The shape of the fluids is gradually stabilized, and they are split into two layers with a distinct interface. In
Figure 7
a, it is clear that the multiphase fluid surface rebuilt using the symmetric kernel approach was crude overall. What is worse, there were overlaps and voids at the two-phase interfaces. Further, the
surface appearance was weak after rendering.
Figure 7
b illustrates the fluid surface rebuilt through the asymmetric kernel approach, which was smoother than the symmetric kernel method. However, at the two-phase interface, there were noticeable voids,
which contaminated the simulation fidelity badly. In
Figure 7
c, one can find that the fluid surface rebuilt using our approach was relatively smoother with no voids or overlaps at the two-phase interfaces. The ideal appearance proved that our approach can
simulate two-phase fluids’ interaction effectively and accurately.
Figure 8
shows the results of the asymmetric kernel approach [
] and our approach in the breaking dam simulation in a cylinder of fluid with three phases. The key parameters and statistics of this scene can be seen in
Table 2
. The procedure of this experiment was as follows. First, with the influence of the gravity, all phases were falling and contacting and gradually started to blend. Second, the fluid with a lesser
density floated as the simulation continued, while the fluid with a greater density gradually sunk. Since the density of the yellow fluid was the least, it flowed right to the top. Third, the
three-phase fluids collided with each other and were fully mixed. Finally, the movement of the fluids was gradually stabilized, and fluids were split into three layers with two fine interfaces.
Figure 8
a shows the asymmetric kernel approach. At three-phase interfaces, there still remained distinct voids, which resulted in poor appearance.
Figure 8
b further demonstrates the effectiveness of our method. The results of our approach in the three-phase flow simulation showed a smooth and flat surface appearance, which had no voids or overlaps.
Figure 9
shows a two-phase cocktail scenario using our surface reconstruction method. There were 30,005 particles of yellow fluid and 109,802 particles of red fluid. The density of the yellow fluid was 300
$kg kg kg 3 kg 3$
and the density of the red fluid was 1000
$kg kg kg 3 kg 3$
. In this experiment, one phase fluid was poured into a goblet, then another fluid phase began to be injected. From this scene, it is proven that our method was able to simulate the multiphase fluid
with better surface and interface effects.
8. Conclusions
We introduced an easy, yet quite effective multiphase fluid simulation approach using asymmetric surface extraction and a modified density model. The novel surface extraction scheme considered the
issue of overlaps and gaps, which rebuilt the multiphase fluid surface with a multiple level set and asymmetric kernel. What is more, the “binary tree” strategy and adapted asymmetric kernel were
employed to extract surfaces. In addition, we integrated it with a multiphase fluid model derived by the number density to eliminate density deviations at interfaces. The results showed that our
approach can erase the voids and overlaps at the multiphase interface and can simulate the interaction of multiphase fluid accurately. In brief, our approach was capable of particle-based simulation
and 3D visualization of a multiphase fluid. Our proposed method could be extended to a large-scale scene and multiphase diffusion and convection simulation in future work.
Author Contributions
Funding acquisition, X.W. and X.B.; methodology, X.W. and Y.X.; project administration, X.B.; software, S.L. and Y.X.; visualization, Y.X. and S.L.; writing, original draft, X.W.; writing, review and
editing, Y.X.
This research was funded by National Natural Science Foundation of China (under Grant Nos. 61873299, 61702036, and 61572075), and The National Key Research and Development Program of China (under
Grant No. 2016YFB1001404).
Conflicts of Interest
The authors declare no conflict of interest.
Figure 4. The symmetric kernel transformed to the asymmetric kernel for the multiple-phase interface.
Figure 7. Breaking dam experiment with two phases. (a) left column, symmetric kernel method; (b) middle column, asymmetric kernel method; (c) right column, our own method.
Figure 8. Breaking dam experiment with three phases. (a) first row, asymmetric kernel method; (b) second row, our method.
Parameter Value
Size of domain 24 m × 24 m × 24 m
Smoothing kernel Cubic splines
Number of blue particles 126 k
Number of yellow particles 126 k
Density of blue phase 200 $kg kg m 3 m 3$
Density of yellow phase 1000 $kg kg m 3 m 3$
Support radius 0.2 m
Diameter of fluid particle 0.1 m
Parameter Value
Size of domain 24 m × 24 m × m
Smoothing kernel Cubic splines
Number of blue particles 13,325
Number of yellow particles 13,325
Number of red particles 13,325
Density of red phase 300 $kg kg m 3 m 3$
Density of blue phase 900 $kg kg m 3 m 3$
Density of yellow phase 100 $kg kg m 3 m 3$
Support radius 0.2 m
Diameter of fluid particle 0.1 m
© 2019 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 (http:/
Share and Cite
MDPI and ACS Style
Wang, X.; Xu, Y.; Ban, X.; Liu, S.; Xu, Y. A Unified Multiple-Phase Fluids Framework Using Asymmetric Surface Extraction and the Modified Density Model. Symmetry 2019, 11, 745. https://doi.org/
AMA Style
Wang X, Xu Y, Ban X, Liu S, Xu Y. A Unified Multiple-Phase Fluids Framework Using Asymmetric Surface Extraction and the Modified Density Model. Symmetry. 2019; 11(6):745. https://doi.org/10.3390/
Chicago/Turabian Style
Wang, Xiaokun, Yanrui Xu, Xiaojuan Ban, Sinuo Liu, and Yuting Xu. 2019. "A Unified Multiple-Phase Fluids Framework Using Asymmetric Surface Extraction and the Modified Density Model" Symmetry 11, no.
6: 745. https://doi.org/10.3390/sym11060745
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
Article Metrics | {"url":"https://www.mdpi.com/2073-8994/11/6/745","timestamp":"2024-11-11T12:02:55Z","content_type":"text/html","content_length":"517486","record_id":"<urn:uuid:1874de1c-0074-4488-a5a4-bf5e706ce8b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00361.warc.gz"} |
How Can the World Population Forecasts Be so Good?
In this short video John Wilmoth, director of the UN Population Division, explains to Professor Hans Rosling how the population forecasts of the UN can be so accurate. It is because the future
population is determined by factors that are quite predictable, namely births and deaths. We know that people grow older, and the approximate death rates of different age groups. The number of adults
and old people are therefore relatively easy to predict. It is harder to predict how many children there will be, but knowing the number of adults in reproductive age makes it possible to estimate
how many babies will be born.
You can download this video here!
John Wilmoth, Director of the UN Population Division, Professor at the Department of Demography, Berkeley CA. | {"url":"https://www.gapminderdev.org/answers/how-can-the-world-population-forecasts-be-so-good/","timestamp":"2024-11-09T07:35:38Z","content_type":"text/html","content_length":"54372","record_id":"<urn:uuid:d9fb9522-0ec4-48e4-9b54-d356a0802882>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00423.warc.gz"} |
What Is Fractals Mathematics With Fractals?
The term applied to describe their overall look, fractals, are an significant part mathematics
They’re among the most studied object in all areas of science and mathematics fiction. The use of fractals has been acknowledged by most investigators in many regions of exploration which the true
added benefits of fractals are being understood by mathematicians, scientists, physicists, and engineers.
You http://www.soosim.cn/2020/05/19/accelerated-pros-degree-in-education/ can find various kinds of fractals, all of which are an branch of the exact title. They are found in various elements of
mathematics and mathematics fiction. The following paragraphs will go over all sorts of fractals and uses. These phrases are used from the next paragraphs as they’re additionally used compared to
their technical or scientific names.
The black body type is a easy image and no complex mathematics, however it’s a case of electrons get the job done. An electron with its charge acts like a magnet. It brings at yahoo any type of
matter around it. Its charge ends negative When a molecule is hit by it. Once it comes into contact with another molecule it returns to its original fee.
Students scientists, and even people from the data materials are realizing the importance of learning these particles and waves of thing are directly all connected. A very related story could be
educated concerning what is known as the electromagnetic wave.
It has the capability to stay in one definite location for a short time once the field creates. As time moves, the wave remains stationary, until it experiences another element. The wave modify the
form of the electrons and could move into oscillation, and also adjust the properties of these atoms which encircle it.
The size of this wave can fluctuate, however, it stays in 1 location and travels right up till it contacts an alternate tide. This series of waves is known as the electromagnetic wave. It can
traveling the electromagnetic field over .
Along http://bestresearchpaper.com with waves, there are forms of things which are also called waves. Many of them are light waves, sound waves, gravity waves, and acoustics. Experts, mathematicians,
and experts have their own set of concepts to spell out the relationships amongst waves, math, along with different mathematical objects.
Fractals had been discovered at 1755 by Bernoulli. He was also a Jesuit priest who lived in Switzerland. However, his discoveries were not recognized by the Catholic Church, therefore they did not
affect the maturation of math.
Despite not being at time in the world of mathematics, these items were detected once he created his own mathematical concepts to the problem of linear equations. Predict and to linear equations are
all equations utilized to characterize that the movement. When linear equations have been all used to build wave patterns and things, these objects eventually become fractals. The major change
between linear equations and fractals is that fractals are characterized by what’s inside them, while what is out them defines equations.
Mathematicians began to investigate fractals, after discovering it was crucial to spell out linear equations. They could demonstrate the fractal objects had many similarities into the linear
equations. They’re able to comprehend that there were and they were objects of fixing precisely the equations that are exact same.
They had something which spanned the management of an oscillating wave, although the tide patterns were found to be just like the equations. This is known as the Jacobi amount, and this range had
been utilized to produce new types of fractals. The wave styles all follow this particular number, which proves why these waves can also be combined to generate wave patterns and fresh things.
This discovery explains why the wave patterns build waves. It also explains the tide patterns have.
https://www.haciendaparaisotulum.com/wp-content/uploads/2018/11/logo-haciendaparaiso-transparente-1.png 0 0 soyelguillo https://www.haciendaparaisotulum.com/wp-content/uploads/2018/11/
logo-haciendaparaiso-transparente-1.png soyelguillo2020-06-09 06:36:042020-06-09 06:36:04What Is Fractals Mathematics With Fractals?
Want to join the discussion?
Feel free to contribute! | {"url":"https://www.haciendaparaisotulum.com/what-is-fractals-mathematics-with-fractals/","timestamp":"2024-11-08T04:30:24Z","content_type":"text/html","content_length":"64598","record_id":"<urn:uuid:5a2733b9-cf08-467f-b97f-06b3132f584a>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00680.warc.gz"} |
Question #1f4c4 + Example
Question #1f4c4
1 Answer
In every system at thermal equilibrium, all molecules interact with each other by means of random collisions, thus transferring kinetic and potential energy among them. Eventually, each molecule
reaches the same average kinetic energy (in time). The heavier molecule will have a lower quadratic speed to balance the mean kinetic energy of lighter molecules, which have higher (in the average)
quadratic velocity:
$| {K}_{h} | = \frac{1}{2} {m}_{h} | {v}_{h}^{2} | = \frac{1}{2} {m}_{l} | {v}_{l}^{2} | = | {K}_{l} |$,
where $| K | = \frac{1}{2} m | {v}^{2} |$ is the average kinetic energy, and $| {v}^{2} |$ represents the average of square velocity.
Absolute temperature T is proportional to any quadratic term of kinetic energy (hydrogen and nitrogen have translational kinetic energy in the x, y, z directions, plus rotational kinetic energy
around the two axes perpendicular to the molecular axis and (only at high temperatures) their molecules can oscillate by stretching and compressing along the internuclear distance.
Equipartition theorem states that any kind " i " of energies is transferred and interconverted through intermolecular random interactions until becoming evenly subdivided among all kinds of motion.
For each kind of motion, the relationship is $| {E}_{i} | = \frac{1}{2} {k}_{B} T$.
where ${k}_{B}$ is Boltzmann constant, given by the universal constant of gases, $R$, divided by Avogadro's constant.
Hence, if the temperature is the same, also every term of kinetic energies are the same, and vice versa.
For example, for the three directions x, y, z of translational motion:
$\frac{1}{2} m | {v}_{x}^{2} | = \frac{1}{2} {k}_{B} T$
$\frac{1}{2} m | {v}_{y}^{2} | = \frac{1}{2} {k}_{B} T$
$\frac{1}{2} m | {v}_{z}^{2} | = \frac{1}{2} {k}_{B} T$
Provided Pythagorean theorem states ${v}^{2} = {v}_{x}^{2} + {v}_{y}^{2} + {v}_{z}^{2}$, we have:
$| {K}_{\text{translational}} | = \frac{1}{2} m | {v}^{2} | = \frac{3}{2} {k}_{B} T$
Impact of this question
2953 views around the world | {"url":"https://socratic.org/questions/56cf530711ef6b402921f4c4","timestamp":"2024-11-06T07:02:00Z","content_type":"text/html","content_length":"37054","record_id":"<urn:uuid:aaa1fdc7-50c7-4bf0-b2af-1a91daa8a79a>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00062.warc.gz"} |
structure relation
1—10 of 12 matching pages
and the
structure relation
Degree lowering and raising differentiation formulas and structure relations
Then the OP’s are called
and (
) is called a
structure relation
. …
The structure relation for Askey-Wilson polynomials. J. Comput. Appl. Math. 207 (2), pp. 214–226.
Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.
§33.22(i) Schrödinger Equation
denoting here the elementary charge, the Coulomb potential between two point particles with charges
and masses
separated by a distance
$V(s)=Z_{1}Z_{2}e^{2}/(4\pi\varepsilon_{0}s)=Z_{1}Z_{2}\alpha\hbar c/s$
, where
are atomic numbers,
is the electric constant,
is the fine
constant, and
is the reduced Planck’s constant. …
► $R_{\infty}=m_{e}c{\alpha}^{2}/(2\hbar)$
. …
§33.22(iv) Klein–Gordon and Dirac Equations
The motion of a relativistic electron in a Coulomb field, which arises in the theory of the electronic
of heavy elements (Johnson (
)), is described by a Dirac equation. …
This topic is treated in §§
. …
Quadratic transformations give insight into the
of elliptic integrals to the arithmetic-geometric mean (§
). …
The three singular points in Riemann’s differential equation (
) lead to an interesting Riemann sheet
. …
Both contributions concerned the electronic
of molecules and solids. …
Symmetry in c, d, n of Jacobian elliptic functions
(2004) he found a previously hidden symmetry in
between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three
equations and applies also to the
between the first symmetric elliptic integral and the Jacobian functions. …
A summary of the responsibilities of these groups may help in understanding the
and results of this project. …
Boisvert and Clark were responsible for advising and assisting in matters
to the use of information technology and applications of special functions in the physical sciences (and elsewhere); they also participated in the resolution of major administrative problems when
they arose. …
It has elegant
, including
-soliton solutions, Lax pairs, and Bäcklund transformations. …
were built of which special representations involve Dunkl type operators. In the
-case this algebraic
is called the
double affine Hecke algebra
(DAHA), introduced by Cherednik. …This gives also new
and results in the one-variable case, but the obtained nonsymmetric special functions can now usually be written as a linear combination of two known special functions. …
Over his career his primary research areas were in Special Functions and Orthogonal Polynomials, but also included other topics from Classical Analysis and
areas. …
Askey was a member of the original editorial committee for the DLMF project, serving as an Associate Editor advising on all aspects of the project from the mid-1990’s to the mid-2010’s when the
of the DLMF project was reconstituted; see
About the Project | {"url":"https://dlmf.nist.gov/search/search?q=structure%20relation","timestamp":"2024-11-13T06:25:12Z","content_type":"text/html","content_length":"24073","record_id":"<urn:uuid:a0ab1dc4-912a-43e2-b1cd-ff12be5fad21>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00370.warc.gz"} |
How Many Quarter Hours In 3 Hours: Quick Calculation
How To Convert Hours To Minutes And Minutes To Hours.
Keywords searched by users: How many quarter hours are in 3hours three quarter of an hour minutes, three quarters of an hour, what is one quarter of 5 hours in minutes, one and three quarters of an
hour, three and a quarter, 2 quarters of an hour, how many 1/4 hours make 3/4 of an hour
What Is A 3 Quarter Hour?
What exactly is meant by “three-quarters of an hour”? To clarify, when we refer to three-quarters of an hour, we are talking about 45 minutes. This concept can be thought of as breaking down an hour
into four equal parts, with each part representing 15 minutes. Therefore, three-quarters of an hour corresponds to three of these 15-minute segments, totaling 45 minutes. For example, if you were to
start a task at 3:00 PM and work on it for three-quarters of an hour, you would finish at 3:45 PM. So, when you see the notation “3 thg 12, 2020 45 minutes,” it means that the activity took 45
minutes to complete on December 3rd, 2020.
How Many Quarters Are Three In An Hour?
In order to determine how many quarters are present in an hour, it’s important to remember that an hour is divided into four equal parts, each known as a quarter. This means that each quarter spans
15 minutes. Therefore, if we consider three quarters, it translates to 15 minutes multiplied by 3, resulting in a total of 45 minutes. Another way to express this is by calculating three-quarters of
an hour, which can be done by multiplying 3/4 by 60 minutes, again yielding 45 minutes. This means that three quarters of an hour equate to 45 minutes.
How Many Quarters Are In An Hour?
In order to better understand the division of time, it’s important to note that an hour is comprised of four quarter-hours. This means that each hour can be broken down into four equal parts, with
each quarter-hour representing 15 minutes. This division is essential for various timekeeping and scheduling purposes, providing a clear and convenient way to measure and manage time effectively.
Found 26 How many quarter hours are in 3hours
Question Video: Converting From Hours To Minutes | Nagwa
How To Convert Minutes To Hours: Easy Methods + Examples
Tell Time To The Whole And Half Hours – Minute Hand Introduced – Youtube
How To Convert Between Minutes And Hours – Maths With Mum
Converting Hours To Minutes And Minutes To Hours – Youtube
How To Convert Between Minutes And Hours – Maths With Mum
How To Convert Minutes To Hours: Easy Methods + Examples
Telling Time To The Quarter Hour – Youtube
Categories: Update 65 How Many Quarter Hours Are In 3Hours
See more here: manhtretruc.com
Learn more about the topic How many quarter hours are in 3hours.
See more: blog https://manhtretruc.com/category/paa
Để lại một bình luận | {"url":"https://manhtretruc.com/how-many-quarter-hours-are-in-3hours/","timestamp":"2024-11-09T01:33:01Z","content_type":"text/html","content_length":"142991","record_id":"<urn:uuid:dc23178d-a933-4149-aa12-b1d343fd8259>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00185.warc.gz"} |
Total Domination Versus Domination in Cubic Graphs
A dominating set in a graph G is a set S of vertices of G such that every vertex not in S has a neighbor in S. Further, if every vertex of G has a neighbor in S, then S is a total dominating set of
G. The domination number, γ(G) , and total domination number, γ[t](G) , are the minimum cardinalities of a dominating set and total dominating set, respectively, in G. The upper domination number, Γ
(G) , and the upper total domination number, Γ [t](G) , are the maximum cardinalities of a minimal dominating set and total dominating set, respectively, in G. It is known that γ[t](G) / γ(G) ≤ 2 and
Γ [t](G) / Γ (G) ≤ 2 for all graphs G with no isolated vertex. In this paper we characterize the connected cubic graphs G satisfying γ[t](G) / γ(G) = 2 , and we characterize the connected cubic
graphs G satisfying Γ [t](G) / Γ (G) = 2.
• Cubic graph
• Domination number
• Total domination number
• Upper domination number
• Upper total domination number
ASJC Scopus subject areas
• Theoretical Computer Science
• Discrete Mathematics and Combinatorics
Dive into the research topics of 'Total Domination Versus Domination in Cubic Graphs'. Together they form a unique fingerprint. | {"url":"https://pure.uj.ac.za/en/publications/total-domination-versus-domination-in-cubic-graphs","timestamp":"2024-11-02T17:56:26Z","content_type":"text/html","content_length":"54067","record_id":"<urn:uuid:ef08d806-bbe7-49d2-abe4-7cba5afda6a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00895.warc.gz"} |
A Note on RCFT and Quiver Reps
Posted by Urs Schreiber
[Update: I now have some pdf notes on this issue: Note on Lax Functors and RCFT .]
Recall some basics of quiver theory:
A quiver diagram is nothing but a finite directed graph ($\to$).
Mathematicians call such graphs “quivers” when they are interested in algebra, because quivers can be taken to encode algebras.
Field theorists call such graphs quivers (or “mooses”) when they are interested in susy gauge theory, because quivers can be taken to encode certain field content in such theories.
String theorists call such graphs quivers when they are interested in D-branes on spacetimes of the form $M^4 \times CY_6$ (where $CY_6$ is a global quotient $\mathbb{C}^3/G$ by a finite subgroup of
$SU(3)$), because quivers can be taken to encode the available type of (fractional) D-branes and the sorts of strings stretching between these.
Michael R. Douglas, Gregory Moore,
D-branes, Quivers, and ALE Instantons
More precisely, every vertex of the quiver is identified with a type of D-brane, while every edge of the quiver is identified with a species of string (topological string, usually) stretching between
the types of D-branes corresponding to the source and target vertex of the edge.
For an illlustration, pick any random string theory paper on quivers, for instance see figure 1 in
Marco Billo, Marialuisa Frau, Fabio Lonegro, Alberto Lerda
N=1/2 quiver gauge theories from open strings with R-R fluxes
More precisely, the configuration of these (topological) branes (and the string condensates between them) is not encoded by the quiver itself, but by a representation of the quiver ($\to$). This is
essentially a functor from the quiver (regarded as a category) to vector spaces.
Now, and that’s the point of my note here, some generalization of the concept of a functor on a quiver secretly also plays a crucial role for determining the D-brane content in the FFRS description (
$\to$) of rational conformal field theory. Maybe there is more to that.
The purpose of the following is to point out that the argument on pp. 29-30 and pp. 66-67 of FRS I really defines a lax functor from the theory’s quiver diagram to the suspension of the
representation category of its chiral data.
This argument is as follows: Pick some vertex algebra $V$ describing the local symmetry of a class of RCFTs. Let $\mathrm{Rep}(V)$ be the (modular) category of representation of this algebra. A
particular RCFT in this class (all whose members share the same local symmetries) is determined by any one boundary conditions (D-brane). Call this D-brane $N$. On the space of open string states for
strings both whose ends sit on $N$, the operator product expansion defines an associative product and coproduct. This way the space of open $N-N$ string states induces a Frobenius algebra internal to
In fact, the RCFT is completely specified by $\mathrm{Rep}(V)$ together with this algebra of $N-N$ strings.
Now, $N-N$ strings can interact (in particular) with strings that stretch from the brane $N$ to some other brane, $N'$. Hence there is an object in $\mathrm{Rep}(V)$ which represents the space of
$N-N'$ string states.
Using the operator product expansion once again, we find that the algebra of $N-N$ string states acts on the space of $N-N'$ string states. The latter hence forms a module for the former.
But notice, for reasons that will become important below, that we could just as well have started with the algebra of $N'-N'$-states. These would act on the space of $N-N'$ states from the other
side. Hence the space of $N-N'$ states is really a bimodule, even though we may choose to forget this fact.
The upshot of this analysis is this: An RCFT with chiral data $V$ is the same as a (special, symmetric) Frobenius algebra of open $N-N$ string states internal to $\mathrm{Rep}(V)$. D-branes for this
RCFT are precisely all modules for this algebra (internal to $\mathrm{Rep}(V)$).
Notice how in $\mathrm{Rep}(V)$ we may find different collections of Frobenius algebra objects and their modules. There may be several conformal field theories (and associated collections of
D-branes) for a specified chiral data.
I claim that we can neatly encode the above story, which leads to a choice of Frobenius algebra and algebra modules, in terms of a choice of quiver representation, in some slightly generalized sense.
It’s just some general abstract nonsense:
For the sake of convenience, let’s forget the bialgebra and Frobenius structure for a moment, and just consider internal algebras and their modules. Let $C$ be any tensor category.
What is the neatest way to define an algebra internal to $C$? How about this one: An algebra internal to $C$ is the same as a monad in $\Sigma(C)$, which again is the same as a lax functor
(1)$A : 1 \to \Sigma(C) \,.$
Here $\Sigma(C)$ is the 2-category with a single object and one morphism per object of $C$. 1 is the category with a single morphism.
A lax functor is a functor from a 1- to a 2-category which respects units and composition only up to some coherent 2-morphisms. (Not necessarily an 2-isomorphism!) These 2-morphisms are nothing but
the unit and the product of the algebra. Their coherence is the algebra’s associativity and unit law.
What is the neatest way to define a module for an inernal algebra, more precisely, to define a module which is really a bimodule? Easy: let
(2)$2 := \{ N \to N' \}$
be the category with two objects, $N$ and $N'$, and one nontrivial morphism between these. A collection of two internal algebras in $C$ together with an internal bimodule for them is nothing but a
lax functor
(3)$A : 2 \to \Sigma(C) \,.$
The $N-N$-algebra is the image of $N \overset{\mathrm{Id}}{\to} N$ under $A$, The $N'-N'$-algebra is the image of $N' \overset{\mathrm{Id}}{\to} N'$ under $A$, and so on. Now $A$ has two more
coherent 2-isomorphism compared to the case before. One of them yields the left $A_{NN}$-action on the bimodule which is the image of $N \to N'$ under $A$, the other one encodes the right action.
The pattern now is clear. Consider any category $Q$ with objects $N_1, N_2, N_3, \dots$ and specified nontrivial morphisms between these (a “quiver”). A lax functor
(4)$Q \to \Sigma(C)$
encodes the same data as an algebra internal to $C$ for each object of $Q$, together with an internal bimodule for each nontrivial morphism of $Q$.
But it’s a bit pitiful for a functor to be just lax. It would be much nicer if it were pseudo. However, a pseudofunctor
(5)$Q \to \Sigma(C)$
is the same as a collection of algebras and bimodules internal to $C$, all of whose products, left and right actions are invertible morphisms. That’s an interesting special case of our lax functor,
but for the most general situation that we may be intersted in it is a little too strong a condition.
But there is an obvious choice in between lax and pseudo, namely that where to every coherent 2-morphism coming from the lax functor there is one going the other way round, such that the two obvious
“bubble moves” are satisfied. For lack of a better name, let me call this a “special lax functor”.
A special lax functor
(6)$Q \to \Sigma(C)$
is a slight generalization of the ordinary concept of a representation of the quiver $Q$. Maybe I should call it a “generalized quiver representation”.
The generalized quiver representation
(7)$A : Q \to \Sigma(C)$
is the same thing as one special Frobenius algebra internal to $C$ per object of $Q$, together with one bimodule for the Frobenius algebras per morphism of $Q$.
Hence we find that the central theorem of FRS, which says that a full RCFT is the same thing as a modular category $\mathrm{Rep}(V)$ of Moore-Seiberg data together with a special (and symmetric)
Frobenius alegbra internal to $\mathrm{Rep}(V)$, can be rephrased as saying that
A background configuration of a RCFT is a generalized quiver representation with values in $\Sigma(\mathrm{Rep}(V))$.
Update: In the above I did not cleanly distinguish between the RCFT and its background configurations. (But see the pdf notes for more on that).
The point is that we want to distinguish between the RCFT with all of its admissable boundary conditions, and setups where we intentionally restrict attention to just a subcollection of these
boundary conditions. The latter corresponds to choosing some “background” and “adding” some collection of D-branes to it.
Hence a lax functor $\Gamma \to \Sigma(C)$ defines a “background with D-branes” for the full RCFT which is given by any one of the boundary conditions (and all the modules of the algebra of string
states on that boundary).
The interesting thing is that this allows us to define the natural notion of category of rational string backgrounds. It’s just the category of these functors. This might be just the right way to
talk about the “landscape” of rational conformal field theories.
Posted at April 18, 2006 4:41 PM UTC
Re: A Note on RCFT and Quiver Reps
I have now prepared some pdf notes with more details:
Posted by: urs on April 19, 2006 6:04 PM | Permalink | Reply to this | {"url":"https://golem.ph.utexas.edu/string/archives/000794.html","timestamp":"2024-11-03T09:23:24Z","content_type":"application/xhtml+xml","content_length":"38201","record_id":"<urn:uuid:d45c654e-947f-4e75-8aa8-6229c9e59424>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00508.warc.gz"} |
Setting Backstop Take Rate | Blend
The Backstop Take Rate is the percent of the interest paid by pool borrowers sent to the pool's backstop module depositors. The level it's set at influences the proportion of capital you can expect
the backstop module to have in relation to the pool. This correlation results from a higher backstop take rate making depositing in the backstop module more profitable. Here are example backstop take
rates for pools of various risk levels using the following formula:
$BackstopInterestRequirement = RequiredTVLCoverage*RequiredInterestMultiple$ $BackstopTakeRate = \frac{BackstopInterestRequirement}{(1+BackstopInterestRequirement)}$
Low-Risk Pools
Backstop depositors demand a rate of 2.5x the lender's rate
Required take rate of 4.75%
Compound V3 would be considered a low-risk pool
Medium Risk Pools
Backstop depositors demand a rate of 4x the lender's rate
Required take rate of 23.508%
Aave V2 would be considered a medium-risk pool
High-Risk Pools
Backstop depositors demand a rate of 5x lender's rate
Required take rate of 42.86%
Most pools with Stellar native tokens should be considered high risk | {"url":"https://docs.blend.capital/pool-creators/setting-backstop-take-rate","timestamp":"2024-11-07T07:51:31Z","content_type":"text/html","content_length":"198008","record_id":"<urn:uuid:11a6ecbe-d579-4cce-b74d-d38db1557267>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00261.warc.gz"} |
Influence of estimation errors
One of the many ways for a vessel to avoid being torpedoed that comes to mind is to impede the enemy’s ability to correctly estimate the angle on the bow, the vessel’s speed, and the distance to the
vessel. This could be accomplished by painting their hulls with complex geometrical patterns using contrasting colors. It was called dazzle camouflage. At first glance it seems like that type of
camouflage would attract attention rather than cloak the vessel but due to the irregular, well designed geometric patterns, an observer would find it difficult to determine the direction the target
is heading and would also have trouble estimating the angle on the bow. Additionally, false waves could be painted on the bow and stern which made determining the direction and speed of the target
even more difficult.
Moreover, operation of coincidental range finders – which required adjusting two images – was also impeded because due to the irregular painting, even ideally matched images didn’t look right to the
Photo 1. Merchant USS West Mahomet painted with dazzle camouflage [1]
Photo 2. Aircraft carrier HMS Argus painted with dazzle camouflage [2]
Photo 3. Minelayer USS Shawmut painted with dazzle camouflage [3]
Photo 4. U 253 painted with dazzle camouflage
Photo 5. Heavy cruiser USS Northampton with false bow wave [4]
Photo 6. Artist's conception of periscope view of a merchant ship in dazzle camouflage (left) and the same ship uncamouflaged (right) [5]
Over time with the introduction of more advanced coincidental range finders, stereoscopic range finders and radar the value of dazzle camouflage decreased. During World War II it was sometimes used
with the aim of puzzling observers on board of submarines where due to lack of space for sophisticated equipment visual observation was the main way of obtaining firing data. However the main type
of camouflage became a new painting scheme which softened the vessel’s silhouette between the sea and the horizon in the background.
Because the target vessel is usually a large target it is obvious that there is a range of angles on the bow and target speed (rather than just the exact one) where the torpedo will still hit the
target. In other words, there is some permissible error in angle on the bow and speed, where the torpedo still hits the target. For example a torpedo aimed directly at the center of the target with
incorrect firing data might still hit the bow or the stern and sink the target.
Generally speaking, the permissible error is greater if the torpedo run is smaller. The shorter the time of the torpedo run the less the target moves - due to incorrect firing data - relative to the
calculated impact point. The virtual length of the target as seen by the torpedo, which obviously depends on the real length of the target and the torpedo track angle, is also relevant. The torpedo
sees the longest target when her track is perpendicular to the target course (exact formula: target length multiplied by the sine of the torpedo track angle).
That's why, during torpedo attacks, the general rule is to fire from the smallest possible distance and in such a way that the torpedo track angle is closest to a right angle.
The analysis of permissible errors in determining target course parameters is a complex problem because the target hit has to be considered a function depending on two variables: angle on the bow and
target speed. Additionally, three other parameters mentioned earlier – length of torpedo run, target length and torpedo track angle have to be taken into account.
Below are the results of a numerical simulation of the torpedo hitting target problem as a two-dimensional matrix containing numbers 0 and 1 (miss and hit respectively). Subsequent matrix rows
represent different angles on the bow (the middle row is the real angle on the bow), while subsequent matrix columns correspond to different speeds (the middle column is the real target speed). To
see the influence of all parameters, particular matrices were calculated for different lengths of torpedo run and torpedo track angle.
Angle on the bow: 68°
Target speed: 12 knots
Target length: 120 m
Distance to target: 1000 m
Length of torpedo run: 927 m
Track angle: 90,23°
Drawing 1.
Angle on the bow: 68°
Target speed: 12 knots
Target length: 120 m
Distance to target: 2000 m
Length of torpedo run: 1854 m
Track angle: 90,23°
Drawing 2.
Angle on the bow: 68°
Target speed: 12 knots
Target length: 120 m
Distance to target: 4000 m
Length of torpedo run: 3708 m
Track angle: 90,23°
Drawing 3.
Angle on the bow: 30°
Target speed: 12 knots
Target length: 120 m
Distance to target: 1000 m
Length of torpedo run: 754 m
Track angle: 138,46°
Drawing 4.
Angle on the bow: 30°
Target speed: 12 knots
Target length: 120 m
Distance to target: 2000 m
Length of torpedo run: 1508 m
Track angle: 138,46°
Drawing 5.
Angle on the bow: 30°
Target speed: 12 knots
Target length: 120 m
Distance to target: 4000 m
Length of torpedo run: 3016 m
Track angle: 138,46°
Drawing 6.
Angle on the bow: 56°
Target speed: 20 knots
Target length: 120 m
Distance to target: 1000 m
Length of torpedo run: 829 m
Track angle: 90,45°
Drawing 7.
Angle on the bow: 56°
Target speed: 20 knots
Target length: 120 m
Distance to target: 2000 m
Length of torpedo run: 1658 m
Track angle: 90,45°
Drawing 8.
Angle on the bow: 56°
Target speed: 20 knots
Target length: 120 m
Distance to target: 4000 m
Length of torpedo run: 3316 m
Track angle: 90,45°
Drawing 9.
Angle on the bow: 111°
Target speed: 20 knots
Target length: 120 m
Distance to target: 1000 m
Length of torpedo run: 1839 m
Track angle: 30,51°
Drawing 10.
Angle on the bow: 111°
Target speed: 20 knots
Target length: 120 m
Distance to target: 2000 m
Length of torpedo run: 3678 m
Track angle: 30,51°
Drawing 11.
Angle on the bow: 111°
Target speed: 20 knots
Target length: 120 m
Distance to target: 4000 m
Length of torpedo run: 7356 m
Track angle: 30,51°
Drawing 12.
The above drawings confirm that what was stated earlier – the permissible error resulting in a hit is greater if the length of the torpedo run is shorter (drawings 1, 4, 7) and if the track angle is
closer to a right angle (drawing 1, 2, 3 versus drawing 4, 5, 6).
What is interesting is that there is no visible influence of the real target speed – at twice the target speed, the permissible error is almost the same (drawing 7, 8, 9 versus drawing 1, 2, 3).
Generally speaking, permissible errors in estimating the angle on the bow and target speed are:
Distance to target Track angle: 90º
Track angle: 40º
target speed angle on the bow target speed angle on the bow
1000 ±1 ±10 ±2 ±5
2000 ±0,5 ±10 ±1 ±4
4000 ±0 ±5 ±0 ±2
These summarized values (generated for a few arbitrary sets of parameters) do not present all possible combinations, but they do show the general rule presented in the beginning – the chance to hit
is greater the shorter the torpedo run and the closer the track angle is to a right angle.
To increase the chance of hitting a target, torpedo salvos (that is multiple torpedoes are shot, where they run the same course but at different time intervals, or the torpedoes are all fired at the
same time but they run on courses differing by a few degrees) were used.
In the later part of World War II, when the Allies reinforced the convoy escorts, commanders of German U-Boats were unable to conduct close range (below 1000 meters) torpedo attacks, so it reduced
their effectiveness. As a response, the Germans deployed new manoeuvring torpedoes (FAT and LUT), which could be launched without accurate aiming. They crossed the course of the convoy many times,
increasing their chance of hitting something.
[1] USS West Mahomet
[2] HMS Argus
[3] USS Oglala
[4] USS Northampton
[5] Encyclopædia Britannica (1922) | {"url":"http://www.tvre.org/en/influence-of-estimation-errors","timestamp":"2024-11-08T05:04:43Z","content_type":"application/xhtml+xml","content_length":"35150","record_id":"<urn:uuid:7c697aeb-642f-43f8-8a18-47d330f96787>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00257.warc.gz"} |
What is A/B Testing? The Complete Guide: From Beginner to Pro
A/B testing splits traffic 50/50 between a control and a variation. A/B split testing is a new term for an old technique—controlled experimentation.
Yet for all the content out there about it, people still test the wrong things and run A/B tests incorrectly.
This guide will help you understand everything you need to get started with A/B testing. You’ll see the best ways to run tests, prioritize hypotheses, analyze results, and the best tools to
experiment through A/B testing.
What is A/B testing?
A/B testing is a experimentation process where two or more variants (A and B) are compared, in order to determine which variable is more effective.
When researchers test the efficacy of new drugs, they use a “split test.” In fact, most research experiments could be considered a “split test,” complete with a hypothesis, a control, a variation,
and a statistically calculated result.
That’s it. For example, if you ran a simple A/B test, it would be a 50/50 traffic split between the original page and a variation:
A/B testing splits traffic 50/50 between a control and a variation.
For conversion optimization, the main difference is the variability of Internet traffic. In a lab, it’s easier to control for external variables. Online, you can mitigate them, but it’s difficult to
create a purely controlled test.
In addition, testing new drugs requires an almost certain degree of accuracy. Lives are on the line. In technical terms, your period of “exploration” can be much longer, as you want to be damn sure
that you don’t commit a Type I error (false positive).
Online, the process for A/B split-testing considers business goals. It weighs risk vs. reward, exploration vs. exploitation, science vs. business. Therefore, we view results through a different lens
and make decisions differently than those running tests in a lab.
You can, of course, create more than two variations. Tests with more than two variations are known as A/B/n tests. If you have enough traffic, you can test as many variations as you like. Here’s an
example of an A/B/C/D test, and how much traffic each variation is allocated:
An A/B/n test splits traffic equally among a control and multiple page variations.
A/B/n tests are great for implementing more variations of the same hypothesis, but they require more traffic because they split it among more pages.
A/B tests, while the most popular, are just one type of online experiment. You can also run multivariate and bandit tests.
A/B Testing, multivariate testing, and bandit algorithms: What’s the Difference?
A/B/n tests are controlled experiments that run one or more variations against the original page. Results compare conversion rates among the variations based on a single change.
Multivariate tests test multiple versions of a page to isolate which attributes cause the largest impact. In other words, multivariate tests are like A/B/n tests in that they test an original against
variations, but each variation contains different design elements. For example:
Each element has a specific impact and use case to help you get the most out of your site. Here’s how:
• Use A/B testing to determine the best layouts.
• Use multivariate tests to polish layouts and ensure all elements interact well together.
You need to a ton of traffic to the page you’re testing before even considering multivariate testing. But if you have enough traffic, you should use both types of tests in your optimization program.
Most agencies prioritize A/B testing because you’re usually testing more significant changes (with bigger potential impacts), and because they’re simpler to run. As Peep once said, “Most top agencies
that I’ve talked to about this run ~10 A/B tests for every 1 MVT.”
Bandit algorithms are A/B/n tests that update in real time based on the performance of each variation.
In essence, a bandit algorithm starts by sending traffic to two (or more) pages: the original and the variation(s). Then, to “pull the winning slot machine arm more often,” the algorithm updates
based on which variation is “winning.” Eventually, the algorithm fully exploits the best option:
One benefit of bandit testing is that bandits mitigate “regret,” which is the lost conversion opportunity you experience while testing a potentially worse variation. This chart from Google explains
that very well:
Bandits and A/B/n tests each have a purpose. In general, bandits are great for:
No matter what type of test you run, it’s important to have a process that improves your chances of success. This means running more tests, winning more tests, and making bigger lifts.
How to improve A/B test results
Ignore blog posts that tell you “99 Things You Can A/B Test Right Now.” They’re a waste of time and traffic. A process will make you more money.
Some 74% of optimizers with a structured approach to conversion also claim improved sales. Those without a structured approach stay in what Craig Sullivan calls the “Trough of Disillusionment.”
(Unless their results are littered with false positives, which we’ll get into later.)
To simplify a winning process, the structure goes something like this:
1. Research;
2. Prioritization;
3. Experimentation;
4. Analyze, learn, repeat.
Research: Getting data-driven insights
To begin optimization, you need to know what your users are doing and why.
Before you think about optimization and testing, however, solidify your high-level strategy and move down from there. So, think in this order:
1. Define your business objectives.
2. Define your website goals.
3. Define your Key Performance Indicators.
4. Define your target metrics.
Once you know where you want to go, you can collect the data necessary to get there. To do this, we recommend the ResearchXL Framework.
Here’s the executive summary of the process we use at CXL:
1. Technical analysis;
2. User testing and copy testing.
Heuristic analysis is about as close as we get to “best practices.” Even after years of experience, you still can’t tell exactly what will work. But you can identify opportunity areas. As Craig
Sullivan puts it:
My experience in observing and fixing things: These patterns do make me a better diagnostician, but they don’t function as truths—they guide and inform my work, but they don’t provide guarantees.
Craig Sullivan
Humility is crucial. It also helps to have a framework. When doing heuristic analysis, we assess each page based on the following:
• Relevancy;
• Clarity;
• Value;
• Friction;
• Distraction.
Technical analysis is an often-overlooked area. Bugs—if they’re around—are a conversion killer. You may think your site works perfectly in terms of user experience and functionality. But does it work
equally well with every browser and device? Probably not.
This is a low-hanging—and highly profitable—fruit. So, start by:
Web analytics analysis is next. First thing’s first: Make sure everything is working. (You’d be surprised by how many analytics setups are broken.)
Google Analytics (and other analytics setups) are a course in themselves, so I’ll leave you with some helpful links:
Next is mouse-tracking analysis, which includes heat maps, scroll maps, click maps, form analytics, and user session replays. Don’t get carried away with pretty visualizations of click maps. Make
sure you’re informing your larger goals with this step.
Qualitative research tells you the why that quantitative analysis misses. Many people think that qualitative analysis is “softer” or easier than quantitative, but it should be just as rigorous and
can provide insights as important as those from analytics.
For qualitative research, use things like:
Finally there’s user testing. The premise is simple: Observe how actual people use and interact with your website while they narrate their thought process aloud. Pay attention to what they say and
what they experience.
With copy testing, you learn how your actual target audience perceives the copy, what clear or unclear, what arguments they care about or not.
After thorough conversion research, you’ll have lots of data. The next step is to prioritize that data for testing.
How to prioritize A/B test hypotheses
There are many frameworks to prioritize your A/B tests, and you could even innovate with your own formula. Here’s a way to prioritize work shared by Craig Sullivan.
Once you go through all six steps, you will find issues—some severe, some minor. Allocate every finding into one of five buckets:
1. Test. This bucket is where you place stuff for testing.
2. Instrument. This can involve fixing, adding, or improving tag/event handling in analytics.
3. Hypothesize. This is where you’ve found a page, widget, or process that’s not working well but doesn’t reveal a clear solution.
4. Just Do It. Here’s the bucket for no-brainers. Just do it.
5. Investigate. If an item is in this bucket, you need to ask questions or dig deeper.
Rank each issue from 1 to 5 stars (1 = minor, 5 = critical). There are two criteria that are more important than others when giving a score:
1. Ease of implementation (time/complexity/risk). Sometimes, data tells you to build a feature that will take months to develop. Don’t start there.
2. Opportunity. Score issues subjectively based on how big a lift or change they may generate.
Create a spreadsheet with all of your data. You’ll have a prioritized testing roadmap.
We created our own prioritization model to weed out subjectivity (as possible). It’s predicated on the need to bring data to the table. It’s called PXL and looks like this:
Grab your own copy of this spreadsheet template here. Just click File > Make a Copy to make it your own.
Instead of guessing what the impact might be, this framework asks you a set of questions about it:
• Is the change above the fold? More people notice above-the-fold changes. Thus, those changes are more likely to have an impact.
• Is the change noticeable in under 5 seconds? Show a group of people the control and then the variation(s). Can they tell a difference after 5 seconds? If not, it’s likely to have less of an
• Does it add or remove anything? Bigger changes like removing distractions or adding key information tend to have more of an impact.
• Does the test run on high-traffic pages? An improvement to a high-traffic page generates bigger returns.
Many potential test variables require data to prioritize your hypotheses. Weekly discussions that ask these four questions will help you prioritize testing based on data, not opinions:
1. Is it addressing an issue discovered via user testing?
2. Is it addressing an issue discovered via qualitative feedback (surveys, polls, interviews)?
3. Is the hypothesis supported by mouse tracking, heat maps, or eye tracking?
4. Is it addressing insights found via digital analytics?
We also put bounds on Ease of implementation by bracketing answers according to the estimated time. Ideally, a test developer is part of prioritization discussions.
Grading PXL
We assume a binary scale: You have to choose one or the other. So, for most variables (unless otherwise noted), you choose either a 0 or a 1.
But we also want to weight variables based on importance—how noticeable the change is, if something is added/removed, ease of implementation. For these variables, we specifically say how things
change. For instance, on the Noticeability of the Change variable, you either mark it a 2 or a 0.
We built this model with the belief that you can and should customize variables based on what matters to your business.
For example, maybe you’re working with a branding or user experience team, and hypotheses must conform to brand guidelines. Add it as a variable.
Maybe you’re at a startup whose acquisition engine is fueled by SEO. Maybe your funding depends on that stream of customers. Add a category like, “doesn’t interfere with SEO,” which might alter some
headline or copy tests.
All organizations operate under different assumptions. Customizing the template can account for them and optimize your optimization program.
Whichever framework you use, make it systematic and understandable to anyone on the team, as well as stakeholders.
How long to run A/B tests
First rule: Don’t stop a test just because it reaches statistical significance. This is probably the most common error committed by beginner optimizers with good intentions.
If you call tests when you hit significance, you’ll find that most lifts don’t translate to increased revenue (that’s the goal, after all). The “lifts” were, in fact, imaginary.
Consider this: When 1,000 A/A tests (two identical pages) were run:
• 771 experiments out of 1,000 reached 90% significance at some point.
• 531 experiments out of 1,000 reached 95% significance at some point.
Stopping tests at significance risks false positives and excludes external validity threats, like seasonality.
Predetermine a sample size and run the test for full weeks, usually at least two business cycles.
How do you predetermine sample size? There are lots of great tools. Here’s how you’d calculate your sample size with Evan Miller’s tool:
In this example, we told the tool that we have a 3% conversion rate and want to detect at least 10% uplift. The tool tells us that we need 51,486 visitors per variation before we can look at
statistical significance levels.
In addition to significance level, there’s something called statistical power. Statistical power attempts to avoid Type II errors (false negatives). In other words, it makes it more likely that
you’ll detect an effect if there actually was one.
For practical purposes, know that 80% power is the standard for A/B testing tools. To reach such a level, you need either a large sample size, a large effect size, or a longer duration test.
There are no magic numbers
A lot of blog posts tout magic numbers like “100 conversions” or “1,000 visitors” as stopping points. Math is not magic. Math is math, and what we’re dealing with is slightly more complex than
simplistic heuristics like those figures. Andrew Anderson from Malwarebytes put it well:
It is never about how many conversions. It is about having enough data to validate based on representative samples and representative behavior.
One hundred conversions is possible in only the most remote cases and with an incredibly high delta in behavior, but only if other requirements like behavior over time, consistency, and normal
distribution take place. Even then, it is has a really high chance of a Type I error, false positive.
Andrew Anderson
We want a representative sample. How can we get that? Test for two business cycles to mitigate external factors:
• Day of the week. Your daily traffic can vary a lot.
• Traffic sources. Unless you want to personalize the experience for a dedicated source.
• Blog post and newsletter publishing schedule.
• Return visitors. People may visit your site, think about a purchase, then come back 10 days later to buy it.
• External events. A mid-month payday may affect purchasing, for example.
Be careful with small sample sizes. The Internet is full of case studies steeped in shitty math. Most studies (if they ever released full numbers) would reveal that publishers judged test variations
on 100 visitors or a lift from 12 to 22 conversions.
Once you’ve set up everything correctly, avoid peeking (or letting your boss peek) at test results before the test finishes. This can result in calling a result early due to “spotting a trend”
(impossible). What you’ll find is that many test results regress to the mean.
Regression to the mean
Often, you’ll see results vary wildly in the first few days of the test. Sure enough, they tend to converge as the test continues for the next few weeks. Here’s an example from an ecommerce site:
• First couple of days: Blue (variation #3) is winning big—like $16 per visitor vs. $12.50 for Control. Lots of people would (mistakenly) end the test here.
• After 7 days: Blue still winning, and the relative difference is big.
• After 14 days: Orange (#4) is winning!
• After 21 days: Orange still winning!
• End: No difference.
If you’d called the test at less than four weeks, you would have made an erroneous conclusion.
There’s a related issue: the novelty effect. The novelty of your changes (e.g., bigger blue button) brings more attention to the variation. With time, the lift disappears because the change is no
longer novel.
It’s one of many complexities related to A/B testing. We have a bunch of blog posts devoted to such topics:
Can you run multiple A/B tests simultaneously?
You want to speed up your testing program and run more tests—high-tempo testing. But can you run more than one A/B test at the same time?Will it increase your growth potential or pollute your data?
Some experts say you shouldn’t do multiple tests simultaneously. Some say it’s fine. In most cases, you will be fine running multiple simultaneous tests; extreme interactions are unlikely.
Unless you’re testing really important stuff (e.g., something that impacts your business model, future of the company), the benefits of testing volume will likely outweigh the noise in your data and
occasional false positives.
If there is a high risk of interaction between multiple tests, reduce the number of simultaneous tests and/or let the tests run longer for improved accuracy.
If you want to learn more, read these posts:
How to set up A/B tests
Once you’ve got a prioritized list of test ideas, it’s time to form a hypothesis and run an experiment. A hypothesis defines why you believe a problem occurs. Furthermore, a good hypothesis:
• Is testable. It is measurable, so it can be tested.
• Solves a conversion problem. Split-testing solves conversion problems.
• Provides market insights. With a well-articulated hypothesis, your split-testing results give you information about your customers, whether the test “wins” or “loses.”
Craig Sullivan has a hypothesis kit to simplify the process:
1. Because we saw (data/feedback),
2. We expect that (change) will cause (impact).
3. We’ll measure this using (data metric).
And the advanced one:
1. Because we saw (qualitative and quantitative data),
2. We expect that (change) for (population) will cause (impact[s]).
3. We expect to see (data metric[s] change) over a period of (X business cycles).
Technical stuff
Here’s the fun part: You can finally think about picking a tool.
While this is the first thing many people think about, it’s not the most important. Strategy and statistical knowledge come first.
That said, there are a few differences to bear in mind. One major categorization in tools is whether they are server-side or client-side testing tools.
Server-side tools render code on the server level. They send a randomized version of the page to the viewer with no modification on the visitor’s browser. Client-side tools send the same page, but
JavaScript on the client’s browser manipulates the appearance on the original and the variation.
Client-side testing tools include Optimizely, VWO, and Adobe Target. Conductrics has capabilities for both, and SiteSpect does a proxy server-side method.
What does all this mean for you? If you’d like to save time up front, or if your team is small or lacks development resources, client-side tools can get you up and running faster. Server-side
requires development resources but can often be more robust.
While setting up tests is slightly different depending on which tool you use, it’s often as simple as signing up for your favorite tool and following their instructions, like putting a JavaScript
snippet on your website.
Beyond that, you need to set up Goals (to know when a conversion has been made). Your testing tool will track when each variation converts visitors into customers.
A thank-you page can serve as the goal destination in Google Analytics.
Skills that come in handy when setting up A/B tests are HTML, CSS, and JavaScript/JQuery, as well as design and copywriting skills to craft variations. Some tools allow use of a visual editor, but
that limits your flexibility and control.
How to analyze A/B test results
Alright. You’ve done your research, set up your test correctly, and the test is finally cooked. Now, on to analysis. It’s not as simple as a glimpse at the graph from your testing tool.
One thing you should always do: Analyze your test results in Google Analytics. It doesn’t just enhance your analysis capabilities; it also allows you to be more confident in your data and decision
Your testing tool could be recording data incorrectly. If you have no other source for your test data, you can never be sure whether to trust it. Create multiple sources of data.
What happens if there’s no difference between variations? Don’t move on too quickly. First, realize two things:
1. Your hypothesis might have been right, but implementation was wrong.
Let’s say your qualitative research says that concern about security is an issue. How many ways can you beef up the perception of security? Unlimited.
The name of the game is iterative testing, so if you were on to something, try a few iterations.
2. Even if there was no difference overall, the variation might beat the control in a segment or two.
If you got a lift for returning visitors and mobile visitors—but a drop for new visitors and desktop users—those segments might cancel each other out, making it seem like there’s “no difference.”
Analyze your test across key segments to investigate that possibility.
Data segmentation for A/B tests
The key to learning in A/B testing is segmenting. Even though B might lose to A in the overall results, B might beat A in certain segments (organic, Facebook, mobile, etc).
There are a ton of segments you can analyze. Optimizely lists the following possibilities:
• Browser type;
• Source type;
• Mobile vs. desktop, or by device;
• Logged-in vs. logged-out visitors;
• PPC/SEM campaign;
• Geographical regions (city, state/province, country);
• New vs. returning visitors;
• New vs. repeat purchasers;
• Power users vs. casual visitors;
• Men vs. women;
• Age range;
• New vs. already-submitted leads;
• Plan types or loyalty program levels;
• Current, prospective, and former subscribers;
• Roles (if your site has, for instance, both a buyer and seller role).
At the very least—assuming you have an adequate sample size—look at these segments:
• Desktop vs. tablet/mobile;
• New vs. returning;
• Traffic that lands on the page vs. traffic from internal links.
Make sure that you have enough sample size within the segment. Calculate it in advance, and be wary if it’s less than 250–350 conversions per variation within in a given segment.
If your treatment performed well for a specific segment, it’s time to consider a personalized approach for those users.
How to archive past A/B tests
A/B testing isn’t just about lifts, wins, losses, and testing random shit. As Matt Gershoff said, optimization is about “gathering information to inform decisions,” and the learnings from
statistically valid A/B tests contribute to the greater goals of growth and optimization.
Smart organizations archive their test results and plan their approach to testing systematically. A structured approach to optimization yields greater growth and is less-often limited by local maxima
So here’s the tough part: There’s no single best way to structure your knowledge management. Some companies use sophisticated, internally built tools; some use third-party tools; and some use Excel
and Trello.
If it helps, here are three tools built specifically for conversion optimization project management:
It’s important to communicate across departments and to executives. Often, A/B test results aren’t intuitive to a layperson. Visualization helps.
Annemarie Klaassen and Ton Wesseling wrote an awesome post on visualizing A/B test results. Here’s what they came up with:
A/B testing statistics
Statistical knowledge is handy when analyzing A/B test results. We went over some of it in the section above, but there’s more to cover.
Why do you need to know statistics? Matt Gershoff likes to quote his college math professor: “How can you make cheese if you don’t know where milk comes from?!”
There are three terms you should know before we dive into the nitty gritty of A/B testing statistics:
1. Mean. We’re not measuring all conversion rates, just a sample. The average is representative of the whole.
2. Variance. What is the natural variability of a population? That affects our results and how we use them.
3. Sampling. We can’t measure the true conversion rate, so we select a sample that is (hopefully) representative.
What is a p-value?
Many use the term “statistical significance” inaccurately. Statistical significance by itself is not a stopping rule, so what is it and why is it important?
To start with, let’s go over p-values, which are also very misunderstood. As FiveThirtyEight recently pointed out, even scientists can’t easily explain p-values.
A p-value is the measure of evidence against the null hypothesis (the control, in A/B testing parlance). A p-value does not tell us the probability that B is better than A.
Similarly, it doesn’t tell us the probability that we will make a mistake in selecting B over A. These are common misconceptions.
The p-value is the probability of seeing the current result or a more extreme one given that the null hypothesis is true. Or, “How surprising is this result?”
To sum it up, statistical significance (or a statistically significant result) is attained when a p-value is less than the significance level (which is usually set at 0.05).
Significance in regard to statistical hypothesis testing is also where the whole “one-tail vs. two-tail” issue comes up.
One-tail vs. two-tail A/B tests
One-tailed tests allow for an effect in one direction. Two-tailed tests look for an effect in two directions—positive or negative.
No need to get very worked up about this. Gershoff from Conductrics summed it up well:
If your testing software only does one type or the other, don’t sweat it. It is super simple to convert one type to the other (but you need to do this BEFORE you run the test) since all of the
math is exactly the same in both tests.
All that is different is the significance threshold level. If your software uses a one-tail test, just divide the p-value associated with the confidence level you are looking to run the test by
So, if you want your two-tail test to be at the 95% confidence level, then you would actually input a confidence level of 97.5%, or if at a 99%, then you need to input 99.5%. You can then just
read the test as if it was two-tailed.
Matt Gershoff
Confidence intervals and margin of error
Your conversion rate doesn’t simply say X%. It says something like X% (+/- Y). That second number is the confidence interval, and it’s of utmost importance to understanding your test results.
In A/B testing, we use confidence intervals to mitigate the risk of sampling errors. In that sense, we’re managing the risk associated with implementing a new variation.
So if your tool says something like, “We are 95% confident that the conversion rate is X% +/- Y%,” then you need to account for the +/- Y% as the margin of error.
How confident you are in your results depends largely on how large the margin of error is. If the two conversion ranges overlap, you need to keep testing to get a valid result.
Matt Gershoff gave a great illustration of how margin of error works:
Say your buddy is coming to visit you from Round Rock and is taking TX-1 at 5 p.m. She wants to know how long it should take her. You say I have a 95% confidence that it will take you about 60
minutes plus or minus 20 minutes. So your margin of error is 20 minutes, or 33%.
If she is coming at 11 a.m. you might say, “It will take you 40 min, plus or minus 10 min,” so the margin of error is 10 minutes, or 25%. So while both are at the 95% confidence level, the margin
of error is different.
Matt Gershoff
External validity threats
There’s a challenge with running A/B tests: Data isn’t stationary.
Sinusoidal data
A stationary time series is one whose statistical properties (mean, variance, autocorrelation, etc.) are constant over time. For many reasons, website data is non-stationary, which means we can’t
make the same assumptions as with stationary data. Here are a few reasons that data might fluctuate:
• Season;
• Day of the week;
• Holidays;
• Positive or negative press mentions;
• Other marketing campaigns;
• PPC/SEM;
• SEO;
• Word-of-mouth.
Others include sample pollution, the flicker effect, revenue tracking errors, selection bias, and more. (Read here.) These are things to keep in mind when planning and analyzing your A/B tests.
Bayesian or frequentist Stats
Bayesian or Frequentist A/B testing is another hot topic. Many popular tools have rebuilt their stats engines to feature a Bayesian methodology.
Here’s the difference (very much simplified): In the Bayesian view, a probability is assigned to a hypothesis. In the Frequentist view, a hypothesis is tested without being assigned a probability.
Rob Balon, who carries a PhD in statistics and market research, says the debate is mostly esoteric tail wagging from the ivory tower. “In truth,” he says, “most analysts out of the ivory tower don’t
care that much, if at all, about Bayesian vs. Frequentist.”
Don’t get me wrong, there are practical business implications to each methodology. But if you’re new to A/B testing, there are much more important things to worry about.
How to do A/B testing: tools and resources
Now, how do you start running A/B tests?
Littered throughout this guide are tons of links to external resources: articles, tools, books and books. We’ve tried to compile all the most valuable knowledge in our A/B Testing course.
On top of that, here are some of the best resources (divided by categories).
A/B testing tools
There are a lot of tools for online experimentation. Here’s a list of 53 conversion optimization tools, all reviewed by experts. Some of the most popular A/B testing tools include:
A/B testing calculators
A/B testing statistics resources
A/B testing/CRO strategy resources
A/B testing is an invaluable resource to anyone making decisions in an online environment. With a little bit of knowledge and a lot of diligence, you can mitigate many of the risks that most
beginning optimizers face.
If you really dig into the information here, you’ll be ahead of 90% of people running tests. If you believe in the power of A/B testing for continued revenue growth, that’s a fantastic place to be.
Knowledge is a limiting factor that only experience and iterative learning can transcend. So get testing!
Working on something related to this? Post a comment in the CXL community!
Join the conversation Add your comment
1. marco
Hi, great article!
I have a question about the Evan Miller’s tool. I’m using Monetate as A/B testing tool and some of the KPIs/metrics, such as Revenue Per Session, are measured in dollars.
So for example, I can have a campaign that says experiment performs at $4.52 and control at $3.98. What can I consider as a Baseline Conversion Rate?
1. Peep Laja
Hey! You would still use Evan Miller’s tool to calculate how many people you need in the test, but you can’t use the same A/B test calculator for deciding which one is the winner. There’s an
excellent answer to this in the CXL Facebook group by Chad Sanderson:
T Test or proportion tests don’t work when measuring Revenue Per Visitor because it violates the underlying assumptions of the test. (For T Test, the assumption is your data is spread evenly
around a mean, whereas proportion or binomial tests measure successes or failures only) RPV data is not spread normally around the mean (The vast majority of visitors will purchase nothing)
and we’re not looking at a proportion (Because we need to find the average revenue per visitor). So the best way to conduct a test on RPV is to use a Mann-Whitney U or Wilcoxon test, which
are both rank based sum tests that is designed exactly for cases like this. | {"url":"https://cxl.com/blog/ab-testing-guide/","timestamp":"2024-11-07T12:04:04Z","content_type":"text/html","content_length":"296898","record_id":"<urn:uuid:57fe5501-ddae-4228-b32d-5eef8348e99a>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00472.warc.gz"} |
Understanding arithmetic and geometry through cutting and pasting
Speaker: Ravi Vakil Date: Thu, Sep 21, 2023 Location: Online Conference: PIMS Network Wide Colloquium Subject: Mathematics, Algebraic Geometry Class: Scientific
Euler’s famous formula tells us that (with appropriate caveats), a map on the sphere with f countries (faces), e borders (edges), and v border-ends (vertices) will satisfy v-e+f=2. And more
generally, for a map on a surface with g holes, v-e+f=2-2g. Thus we can figure out the genus of a surface by cutting it into pieces (faces, edges, vertices), and just counting the pieces
appropriately. This is an example of the topological maxim “think globally, act locally”. A starting point for modern algebraic geometry can be understood as the realization that when geometric
objects are actually algebraic, then cutting and pasting tells you far more than it does in “usual” geometry. I will describe some easy-to-understand statements (with hard-to-understand proofs), as
well as easy-to-understand conjectures (some with very clever counterexamples, by M. Larsen, V. Lunts, L. Borisov, and others). I may also discuss some joint work with Melanie Matchett Wood.
Speaker biography:
Ravi Vakil is a Professor of Mathematics and the Robert K. Packard University Fellow at Stanford University, and was the David Huntington Faculty Scholar. He received the Dean's Award for
Distinguished Teaching, an American Mathematical Society Centennial Fellowship, a Frederick E. Terman fellowship, an Alfred P. Sloan Research Fellowship, a National Science Foundation CAREER grant,
the presidential award PECASE, and the Brown Faculty Fellowship. Vakil also received the Coxeter-James Prize from the Canadian Mathematical Society, and the André-Aisenstadt Prize from the CRM in
Montréal. He was the 2009 Earle Raymond Hedrick Lecturer at Mathfest, and a Mathematical Association of America's Pólya Lecturer 2012-2014. The article based on this lecture has won the Lester R.
Ford Award in 2012 and the Chauvenet Prize in 2014. In 2013, he was a Simons Fellow in Mathematics. | {"url":"https://mathtube.org/lecture/video/understanding-arithmetic-and-geometry-through-cutting-and-pasting","timestamp":"2024-11-08T20:58:47Z","content_type":"application/xhtml+xml","content_length":"27669","record_id":"<urn:uuid:b76c8761-9f22-4b4e-a82b-23a35bfac6c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00580.warc.gz"} |
Co je mx + b
Feb 26, 2020 The membrane deforming dynamin family members MxA and MxB are large Subsequently, cells co-transfected with Mito-GFP as a mitochondrial marker test Hinshaw, J. E. Dynamin and its role in
membrane fission. Annu.
BIB. B1B. MOB. Mya. ByB. BIC. MUA. (BvB. JE Wric. ME. 618. MxB. Nov 11, 2020 As usual, this release includes the latest updates from debian 10.6 (buster) and MX repos. Xfce 4.14 or KDE/plasma 5.15.
GIMP 2.10.12.
106.9p. MBC Motor Sales. 107.9p. (sites-picker line 41): Could not find asset snippets/svg-icon-flag-MX.liquid “ H” (headings), “F” (forms), “B” (buttons), and “G” (graphics) to jump to specific 7t,
z, uy, cp, 7, pjj, x, kk, w8, ew, qs5, 1z, i, xc, xbs, ui, 8, 2uv, 1y, txq, ob6, b, mb q, hz, dg, kr, mx, b, yg, h04, mz, wj, f, nd, gg7, xr, whl, gkv, c, 97f, h7q, emu, c, ye, 9n, h, 6r, az, dt, o,
co, u, iz, h, coh, lyq, q8, (1) :sMoIIoJ SD ~JE saqiadosd &t1I[Srn&II3sTp Sfl Antrrq ~-x SS~UI-M0T `popad J Jo AJ2aWOlOqd 2q~q-a2p[M (sw co ~ ~.o = ~R) paads-q&q Jo S2IflSOJ 043 Note that the
observations of MXB 1735-444, GX 17+2, and (3X 339-4 w Verified email at cidesi.edu.mx - Homepage ArticlesCited byCo-authors JM Alvarado-Orozco, JE Garcia-Herrera, B Gleeson, FS Pettit, GH Meier.
Oxidation Every straight line can be represented by an equation: y = mx + b. The coordinates of every point on the line will solve the equation if you substitute them in the Shop fine jewelry
creations of timeless beauty and superlative craftsmanship that will be treasured always. Enjoy complimentary shipping and returns on all Official website of Kawasaki Motors Corp., U.S.A.,
distributor of powersports vehicles including motorcycles, ATVs, Side x Sides and Jet Ski watercraft.
Our 28,197,029 listings include 6,516,044 listings of homes, apartments, and other unique places to stay, and are located in 153,847 destinations in 225 countries and territories. Booking.com B.V. is
based in Amsterdam, the Netherlands and is supported internationally by 198 offices in …
His lawyers, though, are not happy. They've HardOCP Community Forum for PC Hardware Enthusiasts.
View dababmbmbmbm,v,v, ,,bb,b,b,b,’s profile on LinkedIn, the world’s largest professional community. dababmbmbmbm,v,v, has 1 job listed on their profile. See the
The lyrics are as follows: y = mx + The formula y = mx + b is an algebra classic. It represents a linear equation, the graph of which, as the name suggests, is a straight line on the x
-, y -coordinate system.
Na testovaných zařízeních s dvou-jádrovými procesory se ukázal výkon o 70% lepší, než u jedno-jádrových zařízeních. Amanda Gorman, a 22-year-old Los Angeles-born writer and performer and America's
youth poet laureate, recited her poem The Hill We Climb at Joe Biden's inaug Pro moje privátní videa a nejlepší obsah podívejte se na můj Patreon: https://www.patreon.com/emadon Zoom is the leader in
modern enterprise video communications, with an easy, reliable cloud platform for video and audio conferencing, chat, and webinars across mobile, desktop, and room systems. Zoom Rooms is the original
software-based conference room solution used around the world in board, conference, huddle, and training rooms, as well as executive offices and … Expand your Outlook. We've developed a suite of
premium Outlook features for people with advanced email and calendar needs.
Nov 17, 2020 · The formula y = mx + b is an algebra classic. It represents a linear equation, the graph of which, as the name suggests, is a straight line on the x -, y -coordinate system. Often,
however, an equation that can ultimately be represented in this form appears in disguise. Emergency Lite Service Center (ELSC) specializes in all emergency lighting products. Since 1986, ELSC has
provided our customers with the highest quality emergency lights, exit signs, batteries & parts, as well as excellent customer service and technical support. The equation of any straight line, called
a linear equation, can be written as: y = mx + b, where m is the slope of the line and b is the y-intercept.
Logitech MX Master 2S Wireless Mouse – Use on Any Surface, Hyper-fast Scrolling, Ergonomic Shape, Rechargeable, Control up to 3 Apple Mac and Windows Computers (Bluetooth or USB), Graphite 4.6 out of
5 stars 8,648 Slope-intercept form, y=mx+b, of linear equations, emphasizes the slope and the y-intercept of the line. Watch this video to learn more about it and see some examples.
Uppercase-lowercase combinations. Lowercase combinations are not differentiated from uppercase-lowercase combinations (for example, ba is the same page as Ba). Aa Ab Ac Ad Ae Af Ag Ah Ai Aj Ak Al Am
An Ao Ap Aq Ar As At Au Av Aw Ax Ay Az Ba Bb Bc Bd Be Bf Bg Bh Bi Bj Bk Bl Bm Bn Bo Bp Bq Br Bs Bt Bu Bv Bw Bx By Bz Ca Cb Cc Cd Ce Cf Cg Ch Ci Cj Ck Cl Cm Cn Co Cp Cq Cr Cs Ct Cu Cv Cw Cx Cy Cz Da
The equation of a straight line is usually written this way: y = mx + b (or "y = mx + c" in the UK see below) What does it stand for? The equation of any straight line, called a linear equation, can
be written as: y = mx + b, where m is the slope of the line and b is the y-intercept. The y-intercept of this line is the value of y at the point where the line crosses the y axis. This article was
co-authored by our trained team of editors and researchers who validated it for accuracy and comprehensiveness.
This is a list of all TLDs (top-level-domains) with their policies for second and third level domains. This is to help with bug 319643, bug 252342, bug 8743, bug 66383, bug 264632,It will take some
time to check every domain, since there isn't a good list available. We provide a single payments platform globally to accept payments and grow revenue online, on mobile, and at the point of sale.
Learn about the J.R. Simplot Food Group, AgriBusiness Group, Land & Livestock Group, Industrial Business, and Turf & Horticulture Business. *The value of 1 reward night is the average price of the 10
stamps you collect.
1 = 6 + b. 1 - 6 = b-5 = b. b = -5.
co je šifrování v kliducme eur chfpoplatky kraken.comnové minovatelné mincehotovostní limit atm wells fargohoní kartou mezinárodní poplatky2200 gbp na aud
Pro moje privátní videa a nejlepší obsah podívejte se na můj Patreon: https://www.patreon.com/emadon
The y-intercept is where the line will cross the y-axis, so count up or down on the y-axis the number of units indicated by the b value. From the y-intercept point, use the slope to find a second
point. Nov 12, 2010 · In y=mx+b the two parts that we focus on are m, which represents the slope or slant of the line, and b, which represents the y-intercept. Let's start with b, the y-intercept.
The y-intercept-b Je také možné, že máte správnou aplikaci na vašem PC, ale .mx Soubory se dosud s ním spojené. V tomto případě, když se pokusíte otevřít .mx souboru, můžete říct, Windows, která
aplikace je ten správný pro daný soubor. | {"url":"https://hurmanblirriknyty.web.app/28934/88508.html","timestamp":"2024-11-04T04:52:21Z","content_type":"text/html","content_length":"17883","record_id":"<urn:uuid:cb175a21-3fbd-495a-9c8e-30ea2ce8f510>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00833.warc.gz"} |
embedding_similarity: Cosine and Inner product based similarity in ruimtehol: Learn Text 'Embeddings' with 'Starspace'
x a matrix with embeddings providing embeddings for words/n-grams/documents/labels as indicated in the rownames of the matrix
y a matrix with embeddings providing embeddings for words/n-grams/documents/labels as indicated in the rownames of the matrix
type either 'cosine' or 'dot'. If 'dot', returns inner-product based similarity, if 'cosine', returns cosine similarity
top_n integer indicating to return only the top n most similar terms from y for each row of x. If top_n is supplied, a data.frame will be returned with only the highest similarities between x and y
instead of all pairwise similarities
a matrix with embeddings providing embeddings for words/n-grams/documents/labels as indicated in the rownames of the matrix
either 'cosine' or 'dot'. If 'dot', returns inner-product based similarity, if 'cosine', returns cosine similarity
integer indicating to return only the top n most similar terms from y for each row of x. If top_n is supplied, a data.frame will be returned with only the highest similarities between x and y instead
of all pairwise similarities
By default, the function returns a similarity matrix between the rows of x and the rows of y. The similarity between row i of x and row j of y is found in cell [i, j] of the returned similarity
matrix. If top_n is provided, the return value is a data.frame with columns term1, term2, similarity and rank indicating the similarity between the provided terms in x and y ordered from high to low
similarity and keeping only the top_n most similar records.
x <- matrix(rnorm(6), nrow = 2, ncol = 3) rownames(x) <- c("word1", "word2") y <- matrix(rnorm(15), nrow = 5, ncol = 3) rownames(y) <- c("term1", "term2", "term3", "term4", "term5")
embedding_similarity(x, y, type = "cosine") embedding_similarity(x, y, type = "dot") embedding_similarity(x, y, type = "cosine", top_n = 1) embedding_similarity(x, y, type = "dot", top_n = 1)
embedding_similarity(x, y, type = "cosine", top_n = 2) embedding_similarity(x, y, type = "dot", top_n = 2) embedding_similarity(x, y, type = "cosine", top_n = +Inf) embedding_similarity(x, y, type =
"dot", top_n = +Inf)
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/ruimtehol/man/embedding_similarity.html","timestamp":"2024-11-14T02:06:58Z","content_type":"text/html","content_length":"27156","record_id":"<urn:uuid:feaf1684-19a4-4dac-aace-a2c920e1865c>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00517.warc.gz"} |
Spiral Numbers
11/19/2015, 10:20 PM (This post was last modified: 11/20/2015, 12:57 PM by tommy1729.)
Spiral Numbers
The idea is simplest when thinking in terms of polar coordinates.
For a,c > 0 and b,d real , the complex numbers satisfy
(a,b) (c,d) = (ac , b + d mod 2 pi)
The idea of spiral numbers is
(a,b)(c,d) = (ac , b + d)
So far for products.
The sum for spiral numbers is defined by
X + Y = ln( exp(X) exp(Y) ).
So it comes down to finding a good ln and exp.
My guess is exp(a,b) =
( exp(a + ab) , e b)
Where |*| is the absolute value.
And the ln is just the inverse.
For X^Y we use exp( ln X * Y ).
I wonder how the algebra works out.
Is this a good idea ?
I wonder what you think.
11/20/2015, 01:13 PM
I edited post 1 with a different exp and ln.
Im not sure if it is ok now.
The big questions seem to be
1) is there a -1 and An i ?
2) do we have the distributive property ?
It seems the only sqrt of 1 = 1.
( no solution x in x^2 = 1 apart 1 ).
However it seems we have An additive inverse of 1.
This suggests (-1)^2 =\= 1 !
So it Cannot be true that -1 exists * in the usual sense *.
Since -1 is *weird* this rises questions about i.
It seems that by the above and the fact that spiral numbers are not iso to complex numbers and does not contain them,
The spiral numbers are closer to the reals then to the complex.
02/29/2016, 10:56 PM
One of the most intresting ways to continue is this
Z1,z2 are complex.
R1,r2 are real.
(Z1,r1) + (z2,r2) = (z1 + z2,[r1 + r2]\2).
This way we have commutative and associative Sum and product.
Also we have the distributive property , no zero-divisors and algebraic closure.
There exist other ways to define the Sum in a Nice way , but now we have the complex Numbers as a subset ( r1 = r2 = 0 ).
One alternative is
(Z1,r1) + (z2,r2) = (z1 + z2,ln(exp(r1) + exp(r2))).
This is also distributive !
The connection to hyperoper is clear now.
The master
02/29/2016, 11:12 PM
The alternative relates to my generalized distributive property.
Funny though , so do variants more in the style of the " standard Tommy spiral Numbers " .
Open your Mind Neo.
03/01/2016, 12:38 AM (This post was last modified: 03/01/2016, 12:39 AM by marraco.)
(11/19/2015, 10:20 PM)tommy1729 Wrote: Is this a good idea ?
I wonder what you think.
Why the name Spiral numbers?
A good idea for what purpose?
Are you aiming to make a field with fractional dimension?
Your space has a geometrical representation? (I'm a visual person)
I have the result, but I do not yet know how to get it.
03/01/2016, 12:30 PM (This post was last modified: 03/01/2016, 03:33 PM by Gottfried.)
"Log-polar" (see also wikipedia https://en.wikipedia.org/wiki/Log-polar_coordinates ) representation might fill the prerequisites a tiny bit better.
For instance the regular tetration with a complex fixpoint, curving around the fixpoint can then be approximated by linear interpolation and that interpolation agrees better and better with the
regular tetration if the coordinate is translated into vicinity of the fixpoint using the functional equation. I have a picture of this in my comparision "5 methods for interpolation" ( http://
go.helms-net.de/math/tetdocs/Comp...ations.pdf ) posted here earlier.
Gottfried Helms, Kassel
03/01/2016, 01:27 PM
Intresting Gottfried.
I was aware of the wiki , but perhaps a link to your work ?
However my numbers are not log-polar since they are not iso to complex.
Since you are a matrix expert ; how about a matrix representation for my spiral Numbers ?
Also are my spiral Numbers iso to The ring R(x^3) or the group ring R+(C_4) ??
Although my Numbers have no zero-divisors there might still be a connection.
Notice (0,r) * (1,-r) = (0,0).
So if we do not consider (0,r) as 0 and (0,0) as zero , we get a sort of zero-divisor.
The concept of zero is complicated here.
X a = X for all a does not exist , but a - a is always (0,0).
Notice that the spiral numbers keep track of additions in collatz if designed so.
03/01/2016, 04:16 PM
(03/01/2016, 01:27 PM)tommy1729 Wrote: Intresting Gottfried.
I was aware of the wiki , but perhaps a link to your work ?
I just included the link in my previous reply
Gottfried Helms, Kassel
03/01/2016, 09:53 PM
Correction : the spiral Numbers do not have associative addition.
The alternative definition is , but they are iso to 2 copies of the complex plane.
( iso to (C,C) or (C,R) because exp( ln a + ln b ) iso a + b )
I guess this rules out the matrix representation for the normal spiral Numbers.
03/01/2016, 10:15 PM
The name spiral number comes from the analogue of polar coördinates.
Multiplication is identical just without the mod 2 pi for the angle r.
Visually this means that multiplication is on a spiral rather than on a plain or circle ( as with the complex Numbers ).
Hence " spiral numbers ".
To have a meaningful connection between addition and multiplication , I required the distributive law.
Overview of typical distributive Numbers with invertible operators and a unit (?)
1) commutative and associative
Rings , grouprings.
Iso to copies of R and C ( when algebraicly closed ).
2) noncommutative and associative
3) commutative and nonassociative
Tommy's spiral Numbers
4) noncommutative and nonassociative
No intresting cases known unless anticommutative (Lie).
Not sure how this gets us " fractional dimension " or " new Numbers for tetration ".
Feel Free to correct or improve. | {"url":"https://tetrationforum.org/showthread.php?tid=1036","timestamp":"2024-11-10T10:00:25Z","content_type":"application/xhtml+xml","content_length":"54506","record_id":"<urn:uuid:2b01e265-0f77-4af0-9d8c-6cc49ccd5c4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00180.warc.gz"} |
Read the Fine Print
Educational Use
The year is 2032 and your class has successfully achieved a manned mission to Mars! After several explorations of the Red Planet, one question is still being debated: "Is there life on Mars?" The
class is challenged with the task of establishing criteria to help look for signs of life. Student explorers conduct a scientific experiment in which they evaluate three "Martian" soil samples and
determine if any contain life.
Material Type:
Provider Set:
Chris Yakacki
Daria Kotys-Schwartz
Geoffrey Hill
Janet Yowell
Malinda Schaefer Zarske
Date Added: | {"url":"https://oercommons.org/browse?f.new_mccrs_alignment=MCCRS.Math.Content.7.SP.A.2","timestamp":"2024-11-09T19:47:13Z","content_type":"text/html","content_length":"224962","record_id":"<urn:uuid:7a4f217a-90e1-4e10-bfd0-f3a33fd0128b>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00116.warc.gz"} |
Containers Loading Optimization with Python | Samir Saci
Transportation Operations
Containers Loading Optimization with Python
How can we use heuristic algorithms to find the right strategy to load a maximum number of pallets in a sea container?
How can we use heuristic algorithms to find the right strategy to load a maximum number of pallets in a sea container?
Article originally published on Medium.
With the recent surge in shipping prices due to container shortage, the price of a container from Shanghai to North Europe went from $2,000 in November to a peak of $12,000, and optimizing your
container loading became a priority.
You are Logistics Manager in an International Fashion Apparel Retailer and you want to ship 200 containers from Yangshan Port (Shanghai, PRC) to Le Havre Port (Le Havre, France).
• Retail value (USD): your goods’ retail value is 225,000$ per container
• Profit Margin (%): based on pre-crisis shipping cost your profit margin is 8.5%
• Shipping Costs — Previous (%): 100 x 2,000 / 225,000 = 0.88 (%)
• Shipping Costs — Current (%): 100 x 12,000 / 225,000 = 5.33 (%)
Your Finance Team is putting huge pressure on Logistics Operations because 4.45 % of profit is lost because of shipping costs. As you have limited influence on the market price, your only solution is
to improve your loading capacity to save space.
💌 New articles straight in your inbox for free: Newsletter
I. Problem Statement
You have received pallets from your plants and suppliers in China ready to be shipped to France.
You have two types of pallets:
• European Pallets: Dimensions 80 (cm) x 120 (cm)
Example of European pallet-(Source: Rotom)
• North American pallets: Dimensions 100 (cm) x 120 (cm)
Example of North American pallet— (Source: Chep)
You can use two types of containers
• Dry container 20': Inner Length (5,9 m), Inner Width (2,35 m), Inner Height (2,39 m)
• Dry container 40': Inner Length (12,03 m), Inner Width (2,35 m), Inner Height (2,39 m)
• European pallets and American can be mixed
• 20' or 40' containers are available
• No Pallet Stacking (put a pallet above another pallet)
• The loading strategy must be performed in real life (using a counter-balance truck)
Objective: Load a maximum number of pallets per container
II. Two-Dimensional knapsack problem applied to pallet loading
1. Two-Dimensional knapsack problem
Given a set of rectangular pieces and a rectangular container, the two-dimensional knapsack problem (2D-KP) consists of orthogonally packing a subset of the pieces within the container such that the
sum of the values of the packed pieces is maximized.
Exact algorithms for the two-dimensional guillotine knapsack (Mohammad Dolatabadia, Andrea Lodi, Michele Monaci) — (Link)
2. Adapt it to our problem
If we consider that
• Pallets cannot be stacked
• Pallets have to be orthogonally packed to respect the loading constraints
• Pallets Height is always lower than the internal height of your containers
We can transform our 3D problem into a 2D knapsack problem and directly apply this algorithm to find an optimal solution.
3. Results
Scenario: You need to load in a 40' Container
• 20 European Pallets 80 x 120 (cm)
• 4 North American Pallets 100 x 120 (cm)
Tentative 1: The Intuitive solution
Initial Solution — (Image by Author)
Comment: Your forklift driver tried to fit a number maximum of European pallets and find some space for the 4 North American Pallets.
Results: 20/20 Euro Pallets loaded, 2/4 American pallets loaded. You need another container for the two remaining pallets.
Tentative 2: The Optimization Algorithm Result
Optimized Solution (Left) | Initial Solution (Right) — (Image by Author)
Comment: On the left, you have the solution based on the algorithm output.
Results: 20/20 Euro Pallets loaded, 4/4 American pallets loaded. You don’t need another container.
• The optimized solution can fit 100% of pallets. It’s based on non-intuitive placement that cannot be found without trying many combinations.
• Our filling rate is increased and pallets are more “packed”.
In the next part, we’ll see how we can implement a model to get this solution.
Edit: You can find a Youtube version of this article with animations in the link below.
You can find the full code in this Github repository: Link
III. Build your model
To keep this article concise, we will not build the algorithm from scratch but use a python library rectpack.
Example of results of rectpact library — (Source: Documentation)
1. Initialize model and set parameters
• bx, by: we add 5 cm buffer on the x-axis and y-axis to ensure that we do not damage the pallets
• bins20, bins40: container dimensions by type
2. Build your Optimization Model
• bins: the list of available containers (e.g. bins = [bin20, bin40] means that you have 1 container 20' et 1 container 40')
• all_rects: list of all rectangles that could be included in the bins with their coordinates ready to be plot
• all_pals: the list of pallets that could be loaded in the containers listed in bins
3. Plot your result
• color: black for 80x120, red for 100 x120
Example of output for 20 Euro pallets and 4 North American Pallets— (Image by Author)
Now you have everything to share your loading plan with your forklift driver :)
III. Conclusion & Next Steps
We increased the pallet loading rate in both examples vs. the intuitive approach.
This solution was based on a simple scenario of pallets that cannot be stacked.
• What could be the results if we apply it to stackable pallets?
• What could be the results if we apply it to bulk cartons?
About Me
Let’s connect on Linkedin and Twitter, I am a Supply Chain Engineer that is using data analytics to improve logistics operations and reduce costs.
[1] Python 2D rectangle packing library (rectpack), Github Documentation, Link | {"url":"https://www.samirsaci.com/containers-loading-optimization-with-python/","timestamp":"2024-11-10T18:05:43Z","content_type":"text/html","content_length":"35247","record_id":"<urn:uuid:37ab509e-fea6-4f2d-b779-445644d651c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00803.warc.gz"} |
You cannot select more than 25 topics Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
Clojure Tightly Packed Trie
What does this do?
Tries as hash-maps are common, but hash-maps take up a lot of memory (relatively speaking).
For example, creating a hash-map trie of 1, 2, and 3-grams of short story by Edgar Allen Poe results in a hash-map that consumes over 2 megabytes of memory. See this markov language model example.
If you're dealing with much larger corpuses, the memory footprint could become an issue.
A tightly packed trie, on the other hand, is tiny. A tightly packed trie on the same corpus is only 37 kilobytes. That's ~4% of the original trie's size, even after the original trie's keys/values
have all been condensed to numbers!
How do you use library?
A trie is created similar to a hash-map by passing a variable number of "trie entries" to trie.
A "trie entry" is basically the same thing as a map entry. It's just a key and a value.
But for a Trie, the key must be seqable and for a tightly-packed trie all keys must be comparable.
(require '[com.owoga.trie :as trie])
(def loosely-packed-trie (trie/make-trie "dog" :dog "dot" :dot "do" :do "day" :day))
;; => {[\d \a \y] :day, [\d \o \g] :dog, [\d \o \t] :dot, [\d \o] :do}
You'll see from the output of that last line above that the default REPL representation of a Trie is a flat hash-map-looking-thing. It's actually a sorted-hash-map-looking-thing, because if you seq
over it, you'll get the trie-entries in depth-first post-order traversal.
In some ways, a Trie behaves a lot like a map.
`get` returns the value at the key.
(get loosely-packed-trie "dog")
;; => :dog
(get loosely-packed-trie "do")
;; => :do
(get (assoc loosely-packed-trie "dove" {:value "dove" :count 10}) "dove")
;; => {:value "dove", :count 10}
But there's a couple cool Trie-specific functions.
`lookup` returns the Trie at the key. This way, you have access to all of the node's descendants.
(trie/lookup loosely-packed-trie "do")
;; => {[\g] :dog, [\t] :dot}
(seq (trie/lookup loosely-packed-trie "do"))
`children` returns the direct children of a node.
(trie/children (trie/lookup loosely-packed-trie "do"))
;; => ({} {})
That's odd… there's two things in there that look like empty maps.
(map #(get % []) (trie/children (trie/lookup loosely-packed-trie "do")))
;; => (:dog :dot)
The REPL representation of a Trie only shows children key/values. The "root" node (not necessarily the "true" root node if you've travsersed down with `lookup`) doesn't print any data to REPL. So if
you're looking ata node with no children, you'll see `{}` in the REPL. But you can get the value of that node with `(get node [])`
Tightly Packed Tries
The trie above is backed by regular old Clojure data structures: hash-maps and vectors.
It's not very efficient. All of the strings, nested maps, pointers… it all adds up to a lot of wasted memory.
A tightly packed trie provides the same functionality at an impressively small fraction of the memory footprint.
One restriction though: all keys and values must be integers. To convert them from integer identifiers back into the values that your biological self can process, you'll need to keep some type of
database or in-memory map of ids to human-parseable things.
Here's a similar example to that above, but with values that we can tightly pack.
(require '[com.owoga.tightly-packed-trie :as tpt]
'[com.owoga.tightly-packed-trie.encoding :as encoding])
(defn encode-fn [v]
(if (nil? v)
(encoding/encode 0)
(encoding/encode v)))
(defn decode-fn [byte-buffer]
(let [v (encoding/decode byte-buffer)]
(if (zero? v) nil v)))
(def tight-ready-loosely-packed-trie
(trie/make-trie '(1 2 3) 123 '(1 2 1) 121 '(1 2 2) 122 '(1 3 1) 131))
(def tightly-packed-trie
(get tightly-packed-trie [1 2 3])
;; => 123
(map #(get % []) (trie/children (trie/lookup tightly-packed-trie [1 2])))
;; => (121 122 123)
(seq tightly-packed-trie)
;; => ([[1 2 1] 121]
;; [[1 2 2] 122]
;; [[1 2 3] 123]
;; [[1 2] nil]
;; [[1 3 1] 131]
;; [[1 3] nil]
;; [[1] nil])
Instead of a map with all of its pointers, we are storing all of the information necessary for this trie in just 39 bytes!
(require '[cljol.dig9 :as d])
(.capacity (.byte-buffer tightly-packed-trie))
;; => 39
It's backed by a byte-buffer so saving to disk is trivial, but there's a helper for that.
Here's the process of saving to and loading from disk. (Only works for tightly-packed tries.)
(tpt/save-tightly-packed-trie-to-file "/tmp/tpt.bin" tightly-packed-trie)
(def saved-and-loaded-tpt
(tpt/load-tightly-packed-trie-from-file "/tmp/tpt.bin" decode-fn))
(get saved-and-loaded-tpt '(1 2 3))
;; => 123
Ulrich Germann, Eric Joanis, and Samuel Larkin of the National Research Institute of Canada for the paper Tightly Packed Tries: How to Fit Large Models into Memory,and Make them Load Fast, Too.
Lots of credit also goes to the Clojurians community.
TODO Why would you want a trie data structure?
TODO: The below is closer to a CSCI lesson than library documentation. If it's necessary, figure out where to put it, how to word it, etc… It might not be worth cluttering documentation with so much
A user types in the characters "D" "O" and you want to show all possible autocompletions.
Typical "List" data structure
• Iterate through each word starting from the beginning.
• When you get to the first word that starts with the letters "D" "O", start keeping track of words
• When you get to the next word that doesn't start with "D" "O", you have all the words you want to use for autocomplete.
(def dictionary ["Apple" "Banana" "Carrot" "Do" "Dog" "Dot" "Dude" "Egg"])
Problems with a list.
It's slow if you have a big list. If you have a dictionary with hundreds of thousands of words and the user is typing in letters that don't show up until the end of the list, then you're searching
through the first few hundred thousand items in the list before you get to what you need.
If you're familiar with binary search over sorted lists, you'll know this is a contrived example.
Typical "Trie" in Clojure
{"A" {:children {"P" {,,,} :value nil}}
"D" {:children {"O"
:children {"G" {:children {} :value "DOG"}
"T" {:children {} :value "DOT"}}
:value "DO"}
:value nil}}
How is a trie faster? | {"url":"https://git.owoga.com/eihli/clj-tightly-packed-trie/src/commit/00ea29be44ebddeab24ccfccfdba66d4deeaa7e7","timestamp":"2024-11-02T08:13:04Z","content_type":"text/html","content_length":"71264","record_id":"<urn:uuid:e05fbf51-8970-4346-8b81-fb2b4859e0cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00649.warc.gz"} |
The Tumescent Technique By Jeffrey A. Klein MD
Chapter 21:
Maximum Recommended Dosage of Tumescent Lidocaine
This chapter examines the process of estimating the maximum safe dosage of tumescent lidocaine for liposuction. A pragmatic estimate is proposed, followed by review of previously published attempts
to define the maximum recommended dose. The chapter examines how pharmaceutical companies and governmental regulatory agencies have determined the maximum safe dose of local anesthetics, then
discusses the political aspects of changing the official U.S. Food and Drug Administration (FDA) recommendations.
Dose-Toxicity Relationship
As noted in earlier chapters, determining the risk of lidocaine toxicity as a function of the dosage of tumescent lidocaine is not a simple task. For humans it is known that the higher the lidocaine
concentration in the blood, the greater the incidence of toxicity. Without human experimentation, however, coefficients for a mathematic model cannot be accurately estimated.
Thus, at present, the precise relationship between plasma lidocaine concentration and lidocaine toxicity in humans is not well defined. The little knowledge available is based on clinical anecdotes,
not objective clinical experimentation. From using intravenous (IV) lidocaine to treat patients with ventricular arrhythmias (dysrhythmias) and neuropathic pain, however, plasma lidocaine
concentrations that exceed 5 to 6 μg/ml are probably outside the therapeutic range and approach the realm of toxicity.
An unknown percentage of patients with plasma lidocaine concentrations in the range of 2 to 6 μg/ml experience minor unpleasant pharmacologic effects that may be subjective or objective (see Chapter
20). Subjective symptoms include lightheadedness, perioral numbness or paresthesias, and nausea; objective symptoms include confusion, dysarthria, ataxia, shivering, muscle twitching, and vomiting.
Lidocaine is not always responsible for these symptoms. Other causes of nausea and vomiting include perioperative medications (e.g., benzodiazepines, narcotic analgesics, antibiotics) and
self-medication with prescription or nonprescription drugs. Finally, simple anxiety reactions (e.g., hyperventilation, vasovagal episodes) may account for some cases of mild, early toxicity.
Attempting to define the dose-toxicity relationship for lidocaine based on formal clinical research with significant statistical accuracy would involve an unreasonably large number of “experimental”
subjects. In general, however, the probability of lidocaine toxicity is a function of the plasma lidocaine concentration, which is a function of the dosage of tumescent lidocaine, its rate of
absorption, and the apparent volume of distribution. Because of the complexity of this relationship, the required number of patients needed to ensure statistical significance is difficult to
Pragmatic Estimate
A pragmatic determination of the safe maximum dose of tumescent lidocaine requires extensive clinical experience, sound clinical judgment, and enlightened disregard for statistical analysis.
Although the hepatic extraction of lidocaine is high, approximately 70% in a healthy young adult, significant variability can exist in hepatic lidocaine metabolism. Thus predicting the risk of
toxicity is unusually complex. Any group of patients has the usual random variability. More importantly, significant variability also occurs over time within any one patient because of possible drug
interactions that alter lidocaine metabolism. Patients are prescribed drugs by other physicians, and patients take drugs without informing their liposuction surgeon. If a drug interaction or disease
produces a 50% decrease in the rate of lidocaine metabolism, the peak plasma lidocaine concentration will double. Any estimate of the maximum safe dosage of tumescent lidocaine must consider this
clinical fact.
The accuracy of statistical estimation using a random sample technique depends on the size of the sample. In turn, the required size of the sample depends on the population variance of the random
variable in question. Because the variance of plasma lidocaine concentration among tumescent liposuction patients is so large, the size of a random sample required to estimate accurately the safe
maximum dosage of lidocaine is prohibitively large. No clinical study will probably ever satisfy all the requirements for rigorous quantitative statistical analysis of maximum safe lidocaine dosages
for tumescent liposuction.
Ethical Issues
Defining a safe maximum dose of tumescent lidocaine requires a philosophic (ethical) decision regarding how much safety is desired. One must ask, “What is an acceptable incidence of lidocaine-induced
cardiac toxicity that is ethically acceptable?” Clearly a dose that yields one severe cardiac dysrhythmia in every 100 patients or even every 1000 patients is too dangerous. For some, one cardiac
emergency or serious toxic event in every 10,000 patients is unacceptable. Is one lidocaine-induced cardiac arrest in every 100,000 patients acceptable? I believe that the “safety” threshold should
be one per million.
The choice of the “safe maximum recommended dose” for lidocaine is arbitrary; it relies on subjective medical ethics and objective clinical pharmacology (see Chapter 3).
Sentinel Cases
For the pragmatist, finding a reasonably safe dose of lidocaine for tumescent liposuction must involve caution and common sense as well as objective statistical logic. Sentinel cases of toxicity are
an important consideration.
For example, at least two liposuction-related deaths have occurred in patients who received general anesthesia and lidocaine doses of 95 and 105 mg/kg. Also, a surgeon who used general anesthesia
reported that more than 70% of his tumescent liposuction patients experienced nausea and vomiting after lidocaine doses of 80 mg/kg. Another surgeon reported that 30% of patients had nausea and
vomiting at average doses of 70 mg/kg.
In my experience, approximately 0.5% of patients have nausea or vomiting at doses less than 50 mg/kg, with at least a 5% incidence at doses of 55 to 60 mg/kg.
From this information, one can expect that the maximum safe dose of tumescent lidocaine is in the range of 50 to 55 mg/kg. For example, in a 70-kg (154-pound) patient, a 50-mg/kg dose would be 3500
mg of lidocaine. Using a 1-g/L (0.1%) tumescent solution, this patient would receive 3.5 L subcutaneously.
Margin of Safety. Lidocaine dosages should not be increased to greater and greater levels merely for convenience and economic efficiency. Safety must outweigh conveni-ence. No fine line divides safe
and unsafe maximum dosages of tumescent lidocaine. Equivalent doses in different patients will produce different maximum concentrations of lidocaine in the blood. Because of the imprecise,
nondeterministic nature of this situation, a wide margin of safety is necessary.
Case Examples. After 45 mg/kg of tumescent lidocaine, a patient had a peak lidocaine blood level of 3.5 μg/ml and experienced nausea and dysarthria. Another patient received 75 mg/kg with a lidocaine
blood level of 2.8 μg/ml and had an uneventful postoperative course. Still another patient received 59.1 mg/kg with a lidocaine blood level of 6.1 μg/ml and had associated nausea and vomiting as well
as mild disorientation, resulting from an adverse drug interaction with sertraline (Zoloft).
An 86-kg (190-pound) male received 90 mg/kg of lidocaine by mistake when a nurse used 2% lidocaine instead of 1% lidocaine when mixing 100 ml of lidocaine into 1000 ml of normal saline. Liposuction
of the abdomen and flanks was completed without incident. When the mistake was discovered, the patient was admitted for overnight observation. The plasma lidocaine concentration was 2.9 μg/ml at 12
hours and 2.4 μg/ml at 26 hours from when tumescent infiltration was initiated. The patient had no subjective or objective signs of toxicity at any time.
These examples demonstrate that toxicity is not predictable. Variable factors are involved, many of which are not well understood.
Dosage Ranges. Clearly, the traditional dosage limitation of 7 mg/kg for lidocaine with epinephrine at out-of-the-bottle commercial concentrations is far below a reasonable safety limit for very
dilute tumescent lidocaine for liposuction. My experience with tumescent liposuction totally by local anesthesia using very dilute lidocaine (approximately 1 g/L = 0.1%) has shown that 35 mg/kg is
very safe.
A tumescent lidocaine dosage in the range of 45 to 50 mg/kg is now widely regarded as “safe.” Physicians should strive to keep the dosage below 50 mg/kg. In my opinion a dosage greater than 55 mg/kg
is associated with a risk of mild but definite lidocaine toxicity.
As discussed earlier, an obvious conflict of interest exists when a surgeon uses a dose of tumescent lidocaine that exceeds 55 mg/kg merely as a matter of convenience “for the patient”; it is also
convenient for the surgeon. If a patient is not informed that controversy surrounds the safety of such high doses, informed consent might be lacking.
Current ethical standards require that the nonstandard use of huge dosages of a toxic drug be considered experimental. In any experimental trial using potentially toxic doses of a drug such as
lidocaine, ethical standards of care require that every human subject (1) sign informed consent before participation, (2) receive intensive postoperative clinical observation, and (3) have sequential
determinations of plasma lidocaine concentrations every 4 to 6 hours for at least 24 hours immediately after surgery.
Liposuction surgeons with no practical concept of the pharmacologic definition of safety may use titanic doses of tumescent lidocaine ranging from 70 to 100 mg/kg. One surgeon found that at least 30%
of patients given comparable doses of lidocaine experienced nausea or vomiting. These signs of toxicity were attributed to the effects of codeine, antibiotics, or vasovagal events.
Megadoses of a potentially toxic drug such as lidocaine should not be used without the backing of peer-reviewed scientific literature and without approval of a human studies research committee.
The safety of megadosages of lidocaine cannot be proved based on the experience of clinicians who do not personally monitor their patients for 24 hours after liposuction. Anecdotal statements (e.g.,
“We have treated 50 patients with 70 to 100 mg/kg of lidocaine without any significant complication, and we conclude that 80 mg/kg is safe”) are merely conjectures without objective validation. Such
“studies” only permit the conclusion, “We believe that the risk of death is less than 1 in 10, or 1 in 20,” or, “Whatever toxic effects might have occurred, either we did not notice them or we did
not consider them to be significant complications.” One cannot conclude that the risk of death is less than 1 in 100 (Case Report 21-1).
It is known that 60 mg/kg of tumescent lidocaine can produce unpleasant gastrointestinal toxicity and objective neurologic symptoms in patients taking drugs that impair the hepatic metabolism of
Early reports and recent studies
First Tumescent Report
The first description of tumescent liposuction reported the results of treating 26 patients (22 female, four male) with a mean lidocaine dosage of 18.4 mg/kg.^2 The mean serum lidocaine concentration
1 hour after liposuction and 2 hours after infiltration was 0.34 μg/ml, with the highest measured concentration 0.61 μg/ml. This clinical study provided the first documentation that doses of
tumescent lidocaine (approximately 0.1% or less) could exceed the traditional dosage limitation of 7 mg/kg by at least three times without clinical evidence of toxicity.
Two subsequent publications also reported that dosages exceeding 7 mg/kg produced low peak plasma lidocaine concentrations. These reports were based on the assumption that peak lidocaine levels are
achieved within 1 or 2 hours after subcutaneous infiltration. In 1988 Lillis^3 observed that patients exhibited no signs of toxicity after tumescent lidocaine doses as high as 60 to 90 mg/kg. Since
then, surgeons have administered similar doses of tumescent lidocaine. Some of these surgeons, on observing the remarkably high incidence of nausea and vomiting in their patients, attributed the
symptoms to postoperative narcotic analgesics.
A 1989 study reported using general anesthesia plus a relatively high concentration of subcutaneous lidocaine (2500 mg/L = 0.25%) and epinephrine (2.5 mg/L = 1:400,000). Six patients received
lidocaine dosages ranging from 9.1 to 13.8 mg/kg.^4 Blood samples obtained during the first 3 hours after injection revealed peak plasma concentrations of 0.5 to 0.8 μg/ml.
These values of maximum plasma lidocaine concentrations were probably incorrect. The true peak concentration most likely occurred several hours after the last blood sample was drawn. Before 1990, all
the literature assumed that peak lidocaine levels occur within 60 to 120 minutes after a subcutaneous injection. By 1990, researchers realized that a subcutaneous infiltration of dilute lidocaine
with epinephrine could produce a peak plasma lidocaine concentration 8 to 14 hours after injection.
The 35-mg/kg Estimate
The first reasonable estimate of the maximum safe dose of tumescent lidocaine was 35 mg/kg and was published in 1990 in the Journal of Dermatologic Surgery and Oncology.^5 The dosage of dilute
lidocaine at concentrations of 500 mg/L (0.05%) to 1000 mg/L (0.1%) with dilute epinephrine at (1 mg/L = 1:1 million) ranged from 11.9 to 34.1 mg/kg, with associated peak plasma lidocaine
concentrations that ranged from 0.8 to 2.7 μg/ml. This report also showed for the first time that peak plasma lidocaine concentration (C[max]) for tumescent lidocaine is achieved 12 to 14 hours after
initiation of infiltration.
All pretense of statistical analysis was avoided. The method of estimation relied on unsophisticated, simple common sense. The comfort of a liposuction patient under tumescent local anesthesia and
the safety of tumescent hemostasis are so obvious that a formal statistical analysis is unnecessary.
Estimation Process. The 35-mg/kg estimate was derived as follows. First, plasma lidocaine concentrations were repeatedly measured in sequential fashion over more than 24 hours in eight different
patients. Five of these patients participated in at least two of these 24-hour studies. In four patients, sequential concentrations were measured on two different days more than a week apart, first
without liposuction, then with liposuction after infiltration. This allowed evaluation of liposuction’s effect on C[max]. After plotting the data points on a concentration-versus-time graph, a smooth
curve was drawn through the points, and the apparent C[max] was determined by visual assessment (Figure 21-1).
The second step involved plotting a graph of C[max]-versus-mg/kg dosage that showed the scatter of data points similar to that seen with a linear regression plot. The corresponding regression line,
however, was not determined. A linear regression plot is a graph of the expected value of the dependent variable Y = [peak plasma lidocaine concentration] plotted against the value of the independent
variable X = [mg/kg dosage of lidocaine]. Instead, visual “best-fit” line was drawn so that all the data points were below the safety line (Figure 21-2).
Extrapolation extended this safety line to intersect the point corresponding to 6 μg/ml and 50 mg/kg. Thus this subjective analysis suggested that any dosage less than 50 mg/kg of tumescent
lidocaine, with or without liposuction, would be expected to produce a plasma lidocaine concentration less than 6 μg/ml, the accepted threshold for significant lidocaine toxicity.
Extending Safety Margin. Even this estimate, however, needed a greater margin for safety. The process of estimating the maximum safe dosage of tumescent lidocaine must account for the worst-case
scenario where infiltration cannot be followed by liposuction, for example, because of equipment failure, an acute patient problem, or incapacitation of the surgeon. Liposuction seems to reduce the
bioavailability of tumescent lidocaine by 15% to 25%.
Thus the estimate of the maximum safe dosage was cautiously reduced by 30%, from 50 to 35 mg/kg. For this reason, 35 mg/kg was chosen as the first published estimate of a maximum safe dosage for
tumescent (very dilute) lidocaine. This dosage was recommended rather than 50 mg/kg.
Subsequent clinical experience has proved the safety of the 35-mg/kg estimate. In fact, 50 mg/kg for tumescent liposuction is probably a more realistic estimate of a maximum safe dosage of tumescent
lidocaine, and it is the threshold that I currently recommend. Results of future clinical experiments may justify higher doses, but at present such data do not exist.
When surgery might require more than 50 to 55 mg/kg of lidocaine, either (1) the concentration of lidocaine in the bags of anesthetic solution should be reduced, or (2) the procedures should be
divided into two liposuction surgeries, separated by at least 72 hours and preferably 1 month or more.
Lidocaine Metabolism. If a patient is taking a drug that might interfere with the hepatic microsomal enzyme cytochrome P450 3A4 (CYP3A4), which is responsible for the metabolism of lidocaine, the
maximum safe dosage of lidocaine must be reduced from 50 mg/kg to less than 35 mg/kg. Preferably, all drugs that inhibit CYP3A4 can be discontinued 1 or 2 weeks before surgery. Unfortunately,
although many drugs are known to be metabolized by CYP3A4, surgeons usually do not know which one produces significant inhibition of lidocaine metabolism. This unknown aspect of potential drug
interactions between lidocaine and other drugs metabolized by CYP3A4 demands caution when estimating a maximum recommended dosage of tumescent lidocaine.
Specific Tumescent Dosages
Surgeons other than dermatologists took serious notice of the tumescent technique after a November 1993 article in the journal Plastic and Reconstructive Surgery.^6
In the 112 patients, all of whom had liposuction of more than 1500 ml of supranatant fat totally by local anesthesia, the mean lidocaine dosage was 33.3 mg/kg (range 11 to 52.1 mg/kg), and the mean
volume of supranatant fat was 1945 ml (range 1500 to 3400 ml). For each 1000 ml of fat removed, 9.7 ml of whole blood was aspirated. Patients had no clinical evidence of lidocaine or epinephrine
toxicity and no surgical complications.
One 75-kg (165-pound) patient received 35 mg/kg of lidocaine on two separate occasions, first without liposuction, then 25 days later with liposuction. Peak plasma lidocaine concentrations occurred
at 14 and 11 hours after beginning the infiltration and were 2.37 and 1.86 μg/ml, respectively (see Chapter 19).^6
Liposuction removes a portion of the tumescent lidocaine before it can be absorbed into the systemic circulation. This reduces the bioavailability of tumescent lidocaine and results in a lower C
[max]. At the time this study was conducted, sutures were placed in all incision sites.^6 If the incisions had been left open without sutures to encourage postoperative drainage of the blood-tinged
anesthetic solution, C[max] might have been even less than 1.86 μg/ml.
This article also presented evidence that the tumescent technique for liposuction totally by local anesthesia does not require IV fluid supplementation.^6 The volume of tumescent subcutaneous
infiltration is sufficient to produce more than 24 hours of hemodilution, with decreased urine specific gravity. As a corollary, IV fluids are usually unnecessary except with an excessive volume of
liposuction. Gratuitous IV fluids may precipitate systemic fluid overload and pulmonary edema.
Most surgeons have begun to use the tumescent technique because of its unprecedented hemostasis. On the other hand, many of these same surgeons have not used tumescent local anesthesia to eliminate
general anesthesia. Although most surgeons have perceived the tumescent technique as an opportunity to maximize safety by reducing surgical blood loss, a few have used the technique inappropriately
to maximize the volume of fat removed during a single surgery.
Anesthesiology. In 1995 a report of brachial plexus blocks with lidocaine (1% to 2%) and epinephrine appeared in the anesthesiology literature.^7 The authors attempted to evaluate the accuracy of the
standard maximum recommended dosage of lidocaine (7 mg/kg) for local anesthesia. The study of 17 patients found that peak plasma lidocaine concentrations occurred at 45 to 60 minutes after injection.
The highest plasma lidocaine concentration was 5.6 μg/ml 30 minutes after a dosage of 18 mg/kg of lidocaine.
The authors concluded, “In brachial plexus block, the dose of lignocaine with adrenaline [lidocaine with epinephrine] can be as high as 900 mg without fear of toxic symptoms.”^7 They thought the
maximum recommended dose of lignocaine should be reevaluated.
Confirmatory Study. In 1994, Samdal et al^8 studied 12 liposuction patients who received 10.5 to 34.4 mg/kg of tumescent lidocaine (1 g/L = 0.1%) and epinephrine (1 mg/L = 1:1 million). The observed
peak plasma lidocaine concentrations ranged from 0.9 to 3.6 μg/ml. The experimental design included a sufficient number of plasma samples (taken at 1, 2, 3, 6, 8, 10, 12, 14, 18, and 24 hours) to
permit an accurate estimate of C[max].
The authors used linear regression analysis to derive a 95% confidence interval for an expected C[max], estimated to be 4 μg/ml at a dosage of 35 mg/kg. Linear regression can be used to estimate C
[max], but the “expected C[max]“ cannot be regarded as being equivalent to maximum recommended (safe) dosage for tumescent lidocaine. The authors avoided any claim that their expected C[max] was an
estimate of the recommended dosage.^8 The appearance of a linear relationship between lidocaine dosage (mg/kg) and C[max] does not logically justify using linear correlation to establish a maximum
safe dosage of tumescent lidocaine.
Linear Regression
Misconception About Use. Several studies have used linear regression analysis inappropriately to define the maximum safe dosage of tumescent lidocaine. They provide much useful information, however,
and have confirmed the clinical impression that the maximum safe dosage for tumescent lidocaine is 50 mg/kg. The section discusses some of the difficulties in designing a rigorous statistical
analysis of this complex clinical situation.
Lidocaine toxicology assumes that high mg/kg dosages of lidocaine are correlated with high plasma lidocaine concentrations, which in turn are correlated with an increased probability of lidocaine
toxicity. The goal of tumescent clinical pharmacology is to find a reasonable mathematic model that, given any dosage of lidocaine, will predict the plasma lidocaine concentration.
Linear regression is not the best mathematic model for predicting C[max] as a function of mg/kg lidocaine dosage. Linear regression is often used incorrectly when predicting maximum safe dosages.
Simple linear regression is a statistical procedure that allows one to summarize the relationship between Y (the dependent variable) and X (the independent variable): Y = a + bX. Simple linear
regression allows predictions of Y (average C[max]) for any given X (specified mg/kg dosage of lidocaine). This application of linear regression, however, provides neither direct information about
the probability of tumescent lidocaine toxicity nor an estimate of a maximum safe dosage of lidocaine.
Linear regression is an inappropriate method for estimating the maximum safe dosage of lidocaine for tumescent liposuction for two major reasons. First, linear regression uses a least-square
estimation to define a line Y = a + bX, which passes through the middle of the data, thus giving information about the “average” predictable C[max] for any given mg/kg dosage. Any line that predicts
the maximum safe dosage, however, should pass above all the data points; this line is not derived by least-squares linear regression. Although an obvious linear relationship exists between mg/kg
lidocaine dosage and C[max] for lidocaine, it does not validate the use of linear regression to estimate a “safe” dosage of lidocaine.
Second, one cannot assume that lidocaine toxicity (as a function of mg/kg lidocaine dosage) is approximated by a normal distribution. A basic assumption of linear regression is that the dependent
variable in question is normally distributed. As noted, toxicity is a function of C[max], which in turn is a function of mg/kg dosage. With so many unpredictable outcomes (e.g., unknown drug
interactions) and large statistical outliers among liposuction patients, however, one cannot assume that they all will conform to a gaussian (normal) distribution. The unpredictable patient who
manifests extreme deviation from the gaussian distribution disqualifies linear regression as a statistical tool to estimate a maximum safe dose of lidocaine.
From a biostatistical point of view, it is impossible to give an exact and definite “safe” dose limit for any drug. At best, one can only hope to determine an estimate of a safe dose, together with
an appropriately narrow confidence interval.
Misinterpretation of Results. In a 1996 study of 10 patients, Ostad et al^9 concluded that tumescent anesthesia with a total lidocaine dose of up to 55 mg/kg is safe for use in liposuction. This
approximates the 50 mg/kg that I consider a maximum recommended dosage of tumescent lidocaine for liposuction.
After each of 10 patients received different lidocaine dosages, linear regression was used to determine that 55 mg/kg was the average dosage. A sample size of 10 is too small to permit any reliable
estimate of the true variance of the plasma lidocaine concentrations at doses of 55 mg/kg.
More importantly, the authors found a significant linear correlation between total lidocaine dose (total mg) and C[max] but found no correlation between mg/kg lidocaine dosage and C[max]. They should
have stated the maximum safe dose of lidocaine in terms of total milligrams but concluded, “Tumescent anesthesia with a lidocaine dose of 55 mg/kg is safe for liposuction.” This assumes that the
total mg/kg dose of lidocaine is correlated with toxicity. The scientific basis of therapeutics relies on the observation that pharmacologic effect is a function of mg/kg dosage and not total mg
This study assumes that a low lidocaine concentration in the infranatant solution implies that liposuction does not remove significant amounts of lidocaine, which in turn implies that liposuction
does not reduce the C[max] of lidocaine. In fact, because of lidocaine lipophilicity, one would expect lidocaine in the supranatant fat, where much of it is rapidly partitioned after infiltration.
This is consistent with the observation that liposuction reduces the area under the curve (AUC) of plasma lidocaine concentration versus time (see Chapter 19).
Liposuction reduces the amount of lidocaine that enters the systemic circulation (reduces bioavailability). Therefore liposuction provides an extra margin of safety. Any estimate of a safe dosage of
lidocaine must account for the unlikely situation where the surgery must be canceled after the infiltration has been completed and before liposuction surgery has started. The authors’ 55-mg/kg
estimate does not explicitly account for this possibility.
Although the authors’ perceptive clinical insight and good judgment have shown that a reasonable estimate of the maximum safe dosage for tumescent lidocaine is 50 to 55 mg/kg, their statistical
analysis did not prove it.
Weak Assumptions and Heteroscedasticity. In linear regression analysis the term heteroscedasticity describes the unequal scatter or variation in the variance of the dependent variable Y as a function
of the independent variable X. In other words, the variance of C[max] is unequal at different mg/kg dosages of lidocaine; the confidence interval about any estimate of Y may vary as a function of the
value of X. Elementary linear regression analysis requires relatively large numbers of observations to derive any reliable information about the heteroscedasticity of the variable in question.
As an alternative to linear regression, one might choose a fixed dosage and then determine the frequency of toxicity at that dosage. This would allow a much more accurate estimate of the variance of
lidocaine concentration at the fixed dosage. This approach is encumbered by the difficulty of giving a unique mg/kg dosage of tumescent lidocaine to different liposuction patients.
In 1996, Pitman et al^10 reported 32 tumescent liposuction patients treated with general anesthesia and tumescent lidocaine at a dilution of 1 g/L (0.1%), with epinephrine at 1 mg/L. This is the most
patients to have plasma lidocaine determinations reported in a study. The mean lidocaine dosage was 42.2 mg/kg (range 15.2 to 63.8 mg/kg). The greatest plasma lidocaine concentration was 4.2 μg/ml,
in a patient who had received 60.2 mg/kg of tumescent lidocaine. The authors measured plasma lidocaine concentration only at 12 hours after infiltration, assuming the peak level would occur about
this time.
Using linear regression analysis, they concluded that 50 mg/kg of lidocaine for tumescent liposuction would produce a peak plasma lidocaine concentration of 2.8 μg/ml ± 0.9 μg/ml SE (standard error
of mean) with a 95% confidence interval.^11 In other words, assuming that the response variable Y = a + bX has a normal distribution, a probability of 0.95 exists that the true value of Y (50 mg/kg)
will be within the following interval:
(2.8 μg/ml – 1.96 SE, 2.8 μg/ml + 1.96 SE)
= [(2.8 – 1.8) μg/ml, (2.8 + 1.8 μg/ml)]
= (1 μg/ml, 4.6 μg/ml)
X is the dosage of tumescent lidocaine expressed in mg/kg, and Y is the corresponding plasma concentration of lidocaine expressed in μg/ml. In other words, with 95% confidence, one can expect that 50
mg/kg of lidocaine for tumescent liposuction will result in 2.5 of every 100 patients having a plasma lidocaine concentration at 12 hours after infiltration that is greater than 2.8 + (1.96 × 0.9) =
4.6 μg/ml. Also, 2.5 patients will have a plasma lidocaine concentration less than 1 μg/ml.
By the same properties of normal distribution, a 99.73% probability exists that the true value of Y will lie within the following interval:
(2.8 – 3 SE, 2.8 + 3 SE) = (0.1 μg/ml, 5.5 μg/ml)
This is equivalent to the expectation that 1 in 800 patients who receive 50 mg/kg will have a plasma lidocaine level in excess of 5.5 μg/ml.
The statistical design of this study assumes that the peak plasma lidocaine concentration (T[max]) always occurs at 12 hours. The experimental design did not allow for the possibility of an average T
[max] of 9 hours. For example, the true average C[max] might have occurred at 9 hours and was 3.4 ± 1.4 μg/ml. In this hypothetical case, the 95% confidence interval for estimated C[max] would be 0.6
μg/ml, 6.0 μg/ml.
The statistical analysis assumes that SE is correct with no heteroscedasticity. With the small sample size, however, variance of Y (plasma lidocaine concentration) cannot be assumed to equal scatter
or variances at different X (dosages of tumescent lidocaine).
Furthermore, the analysis does not account for the probability of adverse drug interactions. In essence, the experimental design and statistical analysis relied on implausible assumptions, and the
sample size was too small to define a reliable, useful estimate of the maximum safe dose of tumescent lidocaine. Nevertheless, this study’s conclusions probably are correct and correspond to the
clinical experience of hundreds of surgeons with thousands of patients. This is another example of the superiority of good clinical judgment over elementary statistical analysis.
Lidocaine for Breast Augmentation
A 1999 study reported the plasma lidocaine concentrations associated with the use of local anesthesia plus systemic anesthesia for breast augmentation in 10 healthy women.^12 Lidocaine at
concentrations of 2 g/L (0.2%) and 5 g/L (0.5%) with epinephrine was injected into the tissue space between the pectoralis muscle and the mammary gland. Dosages of lidocaine ranged from 16.3 to 21.9
mg/kg (mean 18.2 mg/kg), C[max] from 0.96 to 3.12 μg/ml (mean 1.49 μg/ml), and T[max ]from 4 to 12 hours (mean 7.3 hours). The length of time during which the dose was injected was not specified.
Five patients received general anesthesia; the other five patients were given IV sedation (diazepam and fentanyl), with no apparent differences in C[max] between the two groups.
The authors correctly avoid any assertion that a specific lidocaine dosage is safe: “These data indicate that a dose of 20 mg/kg of lidocaine with epinephrine is probably safe in breast augmentation
when the drug is administered as described in this study.”^12
Statistical Outlier. In this study a single statistical outlier confounded the rote statistical analysis. It exemplifies the maxim that statistical significance is not the same as clinical
significance. Although a statistical analysis of a small sample of 10 patients is of dubious significance, presence of this “aberrant” individual illustrates an important principle of predicting drug
toxicity. The clinician must always assume a large deviation from the mean in a patient who is far more susceptible to an adverse drug reaction than the average patient.
An estimate of a safe maximum dose for a drug must always assume that the patient population is not homogeneous. Certain individuals defy the common assumption that biologic phenomena have a normal
probability density function (gaussian frequency distribution). In other words, an estimate of a safe maximum dose of lidocaine should not be exclusively based on linear regression, which assumes a
normal probability density function.
Lidocaine Absorption. In this study the graphs depicting lidocaine concentration as a function of time demonstrate that subcutaneous infiltration of relatively dilute lidocaine produces a prolonged
plateau of plasma lidocaine concentration. This phenomenon is explained by the following:
1. Rate of systemic absorption of dilute subcutaneous lidocaine is constant.
2. Hepatic elimination of lidocaine is a first-order process that depends on the concentration of plasma lidocaine.
This phenomenon, described by a simple linear differential equation, demonstrates that as long as the rate of lidocaine elimination equals the rate of absorption, the plasma lidocaine concentration
must be a constant plateau.
The slow rate of subcutaneous absorption of dilute lidocaine with epinephrine, together with the high hepatic extraction of lidocaine, is the secret of the unprecedented safety of large doses of
tumescent lidocaine (see Chapter 19).
The FDA and Safe Dosages
Pharmaceutical companies that manufacture and market local anesthetics in the United States must provide the FDA with a suggested maximum safe dosage limitation and scientific information that
documents the safety and validity of such a recommendation. Because of the considerable expense and time involved in conducting the appropriate clinical trials, manufacturers have not specifically
investigated or documented the maximum safe dosage for subcutaneous injections of local anesthetics.
Both the dilution and the site of injection are important determinants of lidocaine toxicity. Dilution of lidocaine also reduces subcutaneous toxicity. When lidocaine is injected subcutaneously in
mice, the lower the concentration, the higher is the total dosage required to produce a lethal effect^13 (Table 21-1).
The slow absorption of lidocaine after subcutaneous infiltration produces a relatively low C[max]. In contrast, when an equal dose of lidocaine is used for an epidural or intercostal nerve block, the
more rapid systemic absorption is associated with a much greater C[max].^14-17 A slower rate of local anesthetic absorption produces a lower C[max], which in turn corresponds to a larger maximum safe
dosage. Consequently, the maximum safe dosage for a subcutaneous local anesthetic is always larger than the maximum safe dosage for regional nerve blocks.
The maximum dosage of a local anesthetic for regional nerve block also suffices as a safe (although less than maximum) dosage for subcutaneous infiltration. By regarding all routes of administration
as equivalent to the route with the most rapid rate of absorption, the manufacturer can save a considerable amount of money in the FDA approval process. This tactic minimizes the number of clinical
studies needed to document safety and efficacy. Furthermore, underestimating the maximum safe dosage for subcutaneous infiltration provides an additional margin of safety when local anesthetics are
used by practitioners with limited experience; this protects the manufacturer.
The tactic of underestimating the maximum safe dosage of a local anesthetic has been used for each of the local anesthetics approved by the FDA for subcutaneous infiltration, including lidocaine,
bupivacaine, chloroprocaine, etidocaine, and ropivacaine. The FDA gave approval for marketing these local anesthetics without requiring studies specifically designed to determine the maximum safe
dosage for subcutaneous infiltration.^18
The 7-mg/kg dosage limitation for commercial 1% lidocaine with epinephrine is an excessively low estimate of a safe dosage. Surgeons must accept this, however, until a more realistic, higher dosage
estimate is established based on objective scientific studies.
Considering the thousands of patients who have safely received 50 mg/kg of tumescent lidocaine for liposuction totally by local anesthesia, it is hoped that the FDA will update its 7-mg/kg dosage
restriction for very dilute (1 g/L = 0.1%) subcutaneous lidocaine.
Consequences of Misleading Limits
One consequence of excessively low “official” dosage limits for subcutaneous lidocaine for local anesthesia, including both commercial 1% lidocaine and very dilute 0.1% lidocaine, is that patients
are frequently denied the option of surgery by local anesthesia. The artificial dosage limitation of 7 mg/kg for out-of-the-bottle commercial lidocaine by official government agencies compels the
surgeon and anesthesiologist to use systemic anesthesia. This unnecessarily exposes many patients to the dangers and unpleasant side effects of systemic anesthesia. The traditional but excessively
low dosage limitation for subcutaneous lidocaine might actually expose patients to more risk through systemic anesthesia than the risks associated with using higher, but scientifically based, dosage
Dosage limits for subcutaneous lidocaine also result in biased training of surgeons and anesthesiologists, inculcating reliance on the use of general anesthesia. Residents in training are denied more
extensive training with local anesthesia, which in turn perpetuates use of systemic anesthesia.
Despite the tumescent technique for liposuction being the most popular cosmetic surgical procedure worldwide, the term tumescent technique has not appeared in the anesthesiology literature. One might
suspect a lack of interest regarding anesthesia that does not require an anesthesiologist. It is possible that real and potential conflicts of interest oppose the increased use of local anesthesia
and favor the continued unnecessary use of systemic anesthesia.
1. Klein JA, Kassarjdian N: Lidocaine toxicity with tumescent liposuction: a case report of probable drug interactions, Dermatol Surg 23:1169-1174, 1997.
2. Klein JA: The tumescent technique for liposuction surgery, Am J Cosmetic Surg 4:263-267, 1987.
3. Lillis PJ: Liposuction surgery under local anesthesia: limited blood loss and minimal lidocaine absorption, J Dermatol Surg Oncol 14:1145-1148, 1988.
4. Lewis ML, Hepper T: The use of high-dose lidocaine in wetting solutions for lipoplasty, Ann Plast Surg 22:307-309, 1989.
5. Klein JA: Tumescent technique for regional anesthesia permits lidocaine doses of 35 mg/kg for liposuction, J Dermatol Surg Oncol 16:248-263, 1990.
6. Klein JA: Tumescent technique for local anesthesia improves safety in large-volume liposuction, Plast Reconstr Surg 92: 1085-1098, 1993.
7. Pälve H, Kirvelä O, Olin H, et al: Maximum recommended doses of lignocaine are not toxic, Br J Anaesth 74:704-705, 1995.
8. Samdal F, Amland PF, Bugge JF: Plasma lidocaine levels during suction-assisted lipectomy using large doses of dilute lidocaine with epinephrine, Plast Reconstr Surg 93:1217-1223, 1994.
9. Ostad A, Kageyama N, Moy RL: Tumescent anesthesia with a lidocaine dose of 55 mg/kg is safe for liposuction, Dermatol Surg 22:921-927, 1996.
10. Pitman GH, Aker JS, Tripp ZD: Tumescent liposuction: a surgeon’s approach, Clin Plast Surg 23:633-641, 1996.
11. Campbell MJ, Machin D: Medical statistics: a commonsense approach, ed 2, New York, 1993, Wiley & Sons.
12. Rygnestad T, Brevik BK, Samdal F: Plasma concentrations of lidocaine and α[1]-acid glycoprotein during and after breast augmentation, Plast Reconstr Surg 103:1267-1272, 1999.
13. Gorgh T: Xylocaine—a new local anesthetic, Anaesthesia 4:4-9, 21, 1949.
14. Kanto J, Jalonen J, Laurakainen E, Niemininen V: Plasma concentration of lidocaine after cranial subcutaneous injection during neurosurgical operations, Acta Anaesthesiol Scand 24:178, 1980.
15. Stoelting RK: Plasma lidocaine concentrations following subcutaneous epinephrine-lidocaine injection, Anesth Analg 57:724, 1978.
16. Schwartz ML, Covino BG, Narang RM, et al: Blood levels of lidocaine following subcutaneous administration prior to cardiac catheterization, Am Heart J 88:721, 1974.
17. Scott DB, Jebson P Jr, Braid DP, Ortengren B: Factors affecting plasma levels of lignocaine and prilocaine, Br J Anaesth 44:1040, 1972.
18. Information from Center for Drug Research and Evaluation, Food and Drug Administration, Freedom of Information Request, Mailing Code HFI-35, Room 12 A16, Rockville, MD 20857 (301-827-4583).
Figure 21-1 Plasma lidocaine levels over time. Area under the curve (AUC) of each group represents total amount of lidocaine systemically absorbed after infiltration into subcutaneous fat using
tumescent technique. In each case, curve with larger AUC represents lidocaine absorption as a function of time without liposuction done after infiltration. Curve with smaller AUC documents lidocaine
absorption when liposuction was performed immediately after completing infiltration. Liposuction reduced both average amount of lidocaine absorbed systemically and peak plasma lidocaine
concentrations to a similar degree. (From Klein J: J Dermatol Surg Oncol 16:248-263, 1990.)
Figure 21-2 Maximum recommended dose of lidocaine is estimated under assumption that peak plasma lidocaine concentration is a linear function of mg/kg dosage of tumescent lidocaine. One data set (○)
represents peak plasma lidocaine concentrations when liposuction was not done, whereas other data set (●) consists of peak levels when liposuction was completed immediately after infiltration of
local anesthetic solution. Conservative estimate of maximal safe dosage of dilute lidocaine infiltrated into subcutaneous fat is 35 mg/kg. Lillis has reported using much higher dosages (arrows)
followed by liposuction without serious toxicity. (From Klein J: J Dermatol Surg Oncol 16:248-263, 1990.)
┃ CASE REPORT 21-1 Lidocaine-Associated Death ┃
┃ A liposuction-related death occurred after a lidocaine dose of 105 mg/kg together with general anesthesia and significant IV fluid supplementation. The coroner found pulmonary edema and a serum ┃
┃ lidocaine level of 14 μg/mg. The circulating nurse misinterpreted the surgeon’s verbal order for 35 mg/kg of tumescent lidocaine and mixed the anesthetic solution, documenting a dose of 105 mg/ ┃
┃ kg. ┃
┃ Discussion. A tumescent lidocaine dosage greater than 60 mg/kg is perilous. At this stage of knowledge, I must conclude that a tumescent lidocaine dose of 100 mg/kg or greater is possibly ┃
┃ negligent. ┃
┃ TABLE 21-1 Lidocaine Dilution and Fatal Toxicity in Mice ┃
┃ Concentration (g/L) │ LD[50] (g/kg)* ┃
┃ 0.5 │ 1.07 ┃
┃ 1.0 │ 0.72 ┃
┃ 2.0 │ 0.59 ┃
┃ 4.0 │ 0.42 ┃
Data from Gorgh T: Anaesthesia 4:4-9, 21, 1949.
*Median lethal dose, after subcutaneous injection. | {"url":"http://liposuction101.com/liposuction-textbook/chapter-21-maximum-recommended-dosage-of-tumescent-lidocaine/","timestamp":"2024-11-11T21:08:46Z","content_type":"text/html","content_length":"127288","record_id":"<urn:uuid:a19c25df-9119-4c43-b9a3-a8341a1eb5d4>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00746.warc.gz"} |
New Ways to Arrange and Plot Data in Tables
Today I'd like to introduce a guest blogger, Stephen Doe, who works for the MATLAB Documentation team here at MathWorks. In today's post, Stephen shows us new functions for displaying, arranging, and
plotting data in tables and timetables.
Tables, Then and Now
In R2013b, MATLAB® introduced the table data type, as a convenient container for column-oriented data. And in R2016b, MATLAB introduced the timetable data type, which is a table that has timestamped
From the beginning, these data types offered advantages over cell arrays and structures. But over the course of several releases, the table and graphics development teams have added many new
functions for tables and timetables. These functions add convenient ways to display and arrange tabular data. Also, they offer new ways to make plots or charts directly from tables, without the
intermediate step of peeling out variables. As of R2018b, MATLAB boasts many new functions to help you make more effective use of tables and timetables.
Read Table and Display First Few Rows
To begin, I will use the readtable function to read data from a sample file that ships with MATLAB. The file outages.csv contains simulated data for electric power outages over a period of 12 years
in the United States. The call to readtable returns a table, T, with six variables and 1468 rows, so I will suppress the output using a semicolon.
T = readtable('outages.csv');
One typical way to examine the data in a large table is to display the first few rows of the table. You can use indexing to access a subset of rows (and/or a subset of variables, for that matter).
For example, this syntax returns the first three rows of T.
ans =
3×6 table
Region OutageTime Loss Customers RestorationTime Cause
___________ ________________ ______ __________ ________________ ______________
'SouthWest' 2002-02-01 12:18 458.98 1.8202e+06 2002-02-07 16:50 'winter storm'
'SouthEast' 2003-01-23 00:49 530.14 2.1204e+05 NaT 'winter storm'
'SouthEast' 2003-02-07 21:15 289.4 1.4294e+05 2003-02-17 08:14 'winter storm'
I have a confession to make: I have written many table examples, using that syntax. And occasionally, I still catch myself starting with code like T(3,:), which accesses only one row.
Happily, in R2016b we added the head function to return the top rows of a table. Here's the call to return the first three rows using the head function.
ans =
3×6 table
Region OutageTime Loss Customers RestorationTime Cause
___________ ________________ ______ __________ ________________ ______________
'SouthWest' 2002-02-01 12:18 458.98 1.8202e+06 2002-02-07 16:50 'winter storm'
'SouthEast' 2003-01-23 00:49 530.14 2.1204e+05 NaT 'winter storm'
'SouthEast' 2003-02-07 21:15 289.4 1.4294e+05 2003-02-17 08:14 'winter storm'
Similarly, tail returns the bottom rows of a table. (If you do not specify the number of rows, then head and tail return eight rows.)
Move, Add, and Delete Table Variables
After examining your table, you might find that you want to organize your table by moving related variables next to each other. For example, in T you might want to move Region and Cause so that they
are together.
One way to move table variables is by indexing. But if you use indexing and want to keep all the variables, then you must specify them all in order, as shown in this syntax.
T = T(:,{'OutageTime','Loss','Customers','RestorationTime','Region','Cause'})
You also can use numeric indices. While more compact, this syntax is less readable.
T = T(:,[2:5 1 6])
When your table has many variables, it is awkward to move variables using indexing. Starting in R2018a, you can use the movevars function instead. Using movevars, you only have to specify the
variables of interest. Move the Region variable so it is before Cause.
T = movevars(T,'Region','Before','Cause');
ans =
3×6 table
OutageTime Loss Customers RestorationTime Region Cause
________________ ______ __________ ________________ ___________ ______________
2002-02-01 12:18 458.98 1.8202e+06 2002-02-07 16:50 'SouthWest' 'winter storm'
2003-01-23 00:49 530.14 2.1204e+05 NaT 'SouthEast' 'winter storm'
2003-02-07 21:15 289.4 1.4294e+05 2003-02-17 08:14 'SouthEast' 'winter storm'
It is also likely that you want to add data to your table. For example, let's calculate the duration of the power outages in T. Specify the format to display the duration in days.
OutageDuration = T.RestorationTime - T.OutageTime;
OutageDuration.Format = 'dd:hh:mm:ss';
It is easy to add OutageDuration to the end of a table using dot notation.
T.OutageDuration = OutageDuration;
However, you might want to add it at another location in T. In R2018a, you can use the addvars function. Add OutageDuration so that it is after OutageTime.
T = addvars(T,OutageDuration,'After','OutageTime');
ans =
3×7 table
OutageTime OutageDuration Loss Customers RestorationTime Region Cause
________________ ______________ ______ __________ ________________ ___________ ______________
2002-02-01 12:18 06:04:32:00 458.98 1.8202e+06 2002-02-07 16:50 'SouthWest' 'winter storm'
2003-01-23 00:49 NaN 530.14 2.1204e+05 NaT 'SouthEast' 'winter storm'
2003-02-07 21:15 09:10:59:00 289.4 1.4294e+05 2003-02-17 08:14 'SouthEast' 'winter storm'
Now, let's remove RestorationTime. You can easily remove variables using dot notation and an empty array.
T.RestorationTime = [];
However, in R2018a there is also a function to remove table variables. To remove RestorationTime, use the removevars function.
T = removevars(T,'RestorationTime');
ans =
3×6 table
OutageTime OutageDuration Loss Customers Region Cause
________________ ______________ ______ __________ ___________ ______________
2002-02-01 12:18 06:04:32:00 458.98 1.8202e+06 'SouthWest' 'winter storm'
2003-01-23 00:49 NaN 530.14 2.1204e+05 'SouthEast' 'winter storm'
2003-02-07 21:15 09:10:59:00 289.4 1.4294e+05 'SouthEast' 'winter storm'
Convert to Timetable
If your table contains dates and times in a datetime array, you can easily convert it to a timetable using the table2timetable function. In this example, table2timetable converts the values in
OutageTime to row times. Row times are time stamps that label the rows of a timetable.
TT = table2timetable(T);
ans =
3×5 timetable
OutageTime OutageDuration Loss Customers Region Cause
________________ ______________ ______ __________ ___________ ______________
2002-02-01 12:18 06:04:32:00 458.98 1.8202e+06 'SouthWest' 'winter storm'
2003-01-23 00:49 NaN 530.14 2.1204e+05 'SouthEast' 'winter storm'
2003-02-07 21:15 09:10:59:00 289.4 1.4294e+05 'SouthEast' 'winter storm'
When you display a timetable, it looks very similar to a table. One important difference is that a timetable has fewer variables than you might expect by glancing at the display. TT has five
variables, not six. The vector of row times, OutageTime, is not considered a timetable variable, since its values label the rows. However, you can still access the row times using dot notation, as in
T.OutageTime. You can use the vector of row times as an input argument to a function. For example, you can use it as the x-axis of a plot.
The row times of a timetable do not have to be ordered. If you want to be sure that the rows of a timetable are sorted by the row times, use the sortrows function.
TT = sortrows(TT);
ans =
3×5 timetable
OutageTime OutageDuration Loss Customers Region Cause
________________ ______________ ______ __________ ___________ ______________
2002-02-01 12:18 06:04:32:00 458.98 1.8202e+06 'SouthWest' 'winter storm'
2002-03-05 17:53 04:20:48:00 96.563 2.8666e+05 'MidWest' 'wind'
2002-03-16 06:18 02:17:05:00 186.44 2.1275e+05 'MidWest' 'severe storm'
Make Stacked Plot of Variables
Now I will show you why I converted T to a timetable. Starting in R2018b, you can plot the variables of a table or timetable in a stacked plot. In a stacked plot, the variables are plotted in
separate y-axes, but using a common x-axis. And if you make a stacked plot from a timetable, the x-values are the row times.
To plot the variables of TT, use the stackedplot function. The function plots variables that can be plotted (such as numeric, datetime, and categorical arrays) and ignores variables that cannot be
plotted. stackedplot also returns properties of the stacked plot as an object that allows customization of the stacked plot.
s = stackedplot(TT)
s =
StackedLineChart with properties:
SourceTable: [1468×5 timetable]
DisplayVariables: {'OutageDuration' 'Loss' 'Customers'}
Color: [0 0.4470 0.7410]
LineStyle: '-'
LineWidth: 0.5000
Marker: 'none'
MarkerSize: 6
Use GET to show all properties
One thing you can tell right away from this plot is that there must be a few timetable rows with bad data. There is one point for a power outage that supposedly lasted for over 9,000 days (or 24
years), which would mean it ended some time in the 2040s.
Convert Variables in Place
The stackedplot function ignored the Region and Cause variables, because these variables are cell arrays of character vectors. You might want to convert these variables to a different, and more
useful, data type. While you can convert variables one at a time, there is now a more convenient way to convert all table variables of a specified data type.
Starting in R2018b, you can convert table variables in place using the convertvars function. For example, identify all the cell arrays of character vectors in TT (using iscellstr) and convert them to
categorical arrays. Now Region and Cause contain discrete values assigned to categories. Categorical values are displayed without any quotation marks.
TT = convertvars(TT,@iscellstr,'categorical');
ans =
3×5 timetable
OutageTime OutageDuration Loss Customers Region Cause
________________ ______________ ______ __________ _________ ____________
2002-02-01 12:18 06:04:32:00 458.98 1.8202e+06 SouthWest winter storm
2002-03-05 17:53 04:20:48:00 96.563 2.8666e+05 MidWest wind
2002-03-16 06:18 02:17:05:00 186.44 2.1275e+05 MidWest severe storm
Plots of Discrete Data
If your table or timetable has variables with values that belong to a finite set of discrete categories, then there are other interesting plots that you can make. Starting in R2017a, you can make a
heat map of any two variables that contain discrete values using the heatmap function. For example, make a heat map of the Region and Cause variables to visualize where and why outages occur. Again,
heatmap returns an object so you can customize the plot.
h = heatmap(TT,'Region','Cause')
h =
HeatmapChart (Count of Cause vs. Region) with properties:
SourceTable: [1468×5 timetable]
XVariable: 'Region'
YVariable: 'Cause'
ColorVariable: ''
ColorMethod: 'count'
Use GET to show all properties
You also can make a pie chart of any categorical variable (as of R2014b), using the pie function. However, you cannot call pie on a table. So, to make a pie chart of the power outages by region, use
dot notation to access the Region variable.
Other Functions to Rearrange or Join Tables
MATLAB also has other functions to reorganize variables in more complex ways, and to join tables. I won't show them all in action, but I will describe some of them briefly. All these functions work
with both tables and timetables.
R2018a includes functions to:
• Reorient rows to become variables (rows2vars)
And from the original release of tables in R2013b, there are functions to:
Tabled for Discussion
Let's table discussion of these new functions for now. But we are eager to hear about your reactions to them. Do they help you make more effective use of tables and timetables? Please let us know
댓글을 남기려면 링크 를 클릭하여 MathWorks 계정에 로그인하거나 계정을 새로 만드십시오. | {"url":"https://blogs.mathworks.com/loren/2018/10/02/new-ways-to-arrange-and-plot-data-in-tables/?s_tid=blogs_rc_1&from=kr","timestamp":"2024-11-07T07:12:37Z","content_type":"text/html","content_length":"193916","record_id":"<urn:uuid:0845b4d8-7a4e-4c4e-94fd-cb5d14e9bb2d>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00139.warc.gz"} |
bwi_mapper: File List
circle_provider.cpp [code] Implementation for circle_provider.h
circle_provider.h [code]
connected_components.h Connected Components implementation to get all the points in a critical region, along with neighbouring critical points to form the topological graph
directed_dfs.cpp [code] Implementation of the Directed DFS class
directed_dfs.h [code] A specific implementation of Directed DFS (DFS with a priority queue) to find whether 2 points are close in obstacle space. Priority is done on using the Euclidean
distance to the goal as a heuristic
generate_graph.cpp [code]
graph.cpp [code] Implementation for graph functions
graph.h [code] Contains some simple data structures for holding the graph
map_inflator.cpp [code] Provides an implementation for map_inflator.h
map_inflator.h [code] Provides a simple costmap inflation function
map_loader.cpp [code] Implementation for map_loader.h
map_loader.h [code] Simple wrapper around the map_server code to read maps from a the supplied yaml file. This class itself is based on the map_server node inside the map_server package
(written by Brian Gerkey)
map_utils.cpp [code] Implementation for map utilities
map_utils.h [code]
path_finder.cpp [code]
path_finder.h [code]
point.cpp [code] Implementations for the basic point data structures
point.h [code]
point_utils.cpp [code]
point_utils.h [code] Some helpful utilities while dealing with points
prepare_graph.cpp [code]
test_circle.cpp [code] Simple command line test for the circle provider
test_dfs.cpp [code] Simple test for the graph generator. Reads a map and displays information from the topological mapper on to the screen
test_graph.cpp [code]
test_map_loader.cpp [code] Simple test for the map loader. Reads a map and displays it on the screen using opencv highgui
test_voronoi.cpp [code] Simple test for the voronoi approximator. Reads a map and displays information from the voronoi approximator on to the screen
topological_mapper.cpp Implementation for the topological mapper
topological_mapper.h [code] Constructs the topological graph using the voronoi approximation
view_graph.cpp [code]
voronoi_approximator.cpp Implementation for the voronoi approximator
voronoi_approximator.h Constructs a voronoi approximation given a map of the world. The map is a discrete grid world with each cell set to occupied or not, and the voronoi approximation is done
[code] in this discrete space
voronoi_point.cpp [code] Implementation of the voronoi point class
voronoi_point.h [code] Base class for a voronoi point. Simple wrapper around Point2d that maintains a given separation between basis points | {"url":"http://docs.ros.org/en/hydro/api/bwi_mapper/html/files.html","timestamp":"2024-11-05T08:40:48Z","content_type":"text/html","content_length":"10763","record_id":"<urn:uuid:7b9d3184-5c42-4d31-a8af-039a373cb002>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00632.warc.gz"} |
5 Quick Math Drills to Use with Your 4th Grade Math Student
Last Updated on May 31, 2022 by Thinkster
Fourth grade is a key year for a child’s math education. Granted, every grade in elementary school math is important, but your 4th grade math student faces new challenges as they transition from the
flashcards and memorization of their younger years to more advanced concepts. (The type of concepts that will ultimately prepare them for algebra and geometry in middle and high school!)
Fourth grade students struggling with math can benefit from an afterschool math program or additional tutoring. But all kids at this age, whether proficient or requiring extra help, can keep their
skills sharp with quick, enjoyable math drills outside of school.
Here are five effective and fun math drills and games that you can introduce to your 4th grader:
1. License plate game
Parents may feel like chauffeurs in their children’s busy lives, but time in the car can be turned into opportunities. There are numbers all over the road- like on car license plates!
Tell your to find a car and read the numbers on the plate.
From these numbers, there are endless possibilities for drills:
• Double the , or divide it by two
• the digits together
• Turn the first two digits into one and by the third
• Round to the nearest ten or hundred
• Reverse the and add it to the original
2. Multiplication and division dice
Does your need to brush up on and division skills?
Roll two dice and the generated two- by the result of a third die, or divide the larger by the smaller (don’t forget the remainder!).
For more of a challenge, add another die to the second . This easy game for fourth graders provides and division work.
And of course, you can also use this for addition and practice too!
3. Multiplication War
A deck of cards is great for a !
Take out the jacks, queens, and kings, and use the cards to generate two or three- numbers. You can use the deck to play a game similar to dice, or you can play war.
Flip two cards, them, and the higher product wins the opponent’s cards – just like regular war. For more of a challenge, use three cards at a time, generating a two- to be multiplied by the third
card; you might not get many wars, but your will get a LOT of !
4. Mind-bending multiples
Most students have had experience counting multiples of single- numbers and 10 (e.g., count by threes: 3, 6, 9, 12 …).
But have they ever counted by 14s or 27s?
This drill will challenge kids to add and in their heads.
Pick a low (at first) two- and count by that to at least past 100. For an added twist, ask your child to guess how many times the two- will be counted in the three- (e.g., about how many times will
19 go into 200?). This adds estimation and a little bit of division into the process.
5. Grocery store math
A trip to the grocery store opens the door to many real-world situations they may see in a !
Weighing produce: how much will five oranges cost at $1.75 a pound?
Determine actual prices on sale items: if eight packages of ramen noodles are $2.00, how much does each package cost?
Keeping a running tally of the food in your cart: use rounding and estimating skills to find the total.
An otherwise boring trip to the store becomes more exciting when you make it seem like a !
Bonus: 4th Grade Math Worksheets
Do you remember getting handed a math worksheet in elementary school and having to complete it in a certain amount of time?
worksheets are an extremely popular and quick way for teachers to deliver fact fluency practice!
Your child can work on a variety of if you find the right .
? ? Multi-digit ?
Absolutely! There are workbooks and online resources to ensure your child gets extra practice with the covered in the classroom.
Need the help of a math tutor?
Fourth grade is an extremely big year for students! Is your child having trouble with fourth grade concepts? Or, are math skills from third grade still rusty?
A Thinkster Math tutor can help your child with a variety of math concepts! Our online assessments can help pinpoint specific topics, concepts, and units that your child needs to improve in.
Your child always works with the same dedicated math tutor, whose goal is to help your child develop a variety of math and life skills.
Learn more about how our math tutors can help make your child a true math champion.
If you’re looking for to help your child, you can try Thinkster risk-free.
Thinkster provides a full-fledged platform (driven by AI, behavioral, and data science), as well as supplemental , help, , and more. Our Parent Insights App allows you to monitor your ‘s work
and improvements at any time.
An elite, and system work together to help your go beyond just – we want them to master it.
Learn more about our curriculum and teaching style here. | {"url":"https://hellothinkster.com/blog/5-quick-math-drills-4th-grade-math-student/","timestamp":"2024-11-14T13:35:54Z","content_type":"text/html","content_length":"145347","record_id":"<urn:uuid:1d236a33-4c50-4524-82e9-492816601b90>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00327.warc.gz"} |
Tutorial math and reading software for elementary and secondary arithmetic, basic math, algebra, geometry, precalculus plus GED, ABE, and CLEP preparation for elementary school, high school, college,
adult education, and homeschool students.
Regular price: $189.00
Limited Time Offer: $94.50
Regular price: $299.00 Regular price: $159.00
Limited Time Offer: $149.50 Limited Time Offer: $79.50 | {"url":"https://www.mathmedia.com/mocl.html","timestamp":"2024-11-11T12:51:04Z","content_type":"text/html","content_length":"25304","record_id":"<urn:uuid:86fc2031-ea0e-4f25-8d28-c0bced48878b>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00587.warc.gz"} |
UQ Psychology - Teaching - Tools - Statistics Repository
Advanced Topics
Resources for Advanced Statistics. The following links will download notes for the various statistical techniques. Files are provided in a single format (either PDF, Word, or Excel).
Test standardisation and the normal distribution -Reliability & validity -Individual score interpretation: SEM, SEdiff -Evaluating diagnostic tests: 2x2 tables, likelihood ratios, ROC curves -Signal
detection theory -Power analysis -Meta-analysis primers
Mark Horswill PSYC7112 lecture
This file contains Powerpoint slides which cover the following topics: -Test standardisation and the normal distribution -Reliability & validity -Individual score interpretation: SEM, SEdiff
-Evaluating diagnostic tests: 2x2 tables, likelihood ratios, ROC curves -Signal detection theory -Power analysis -Meta-analysis primers
Filename: PSYC7112 Postgrad psychometrics lecture sem1 2009 handout only.pdf [5054kb]
More complex ANOVA designs
Higher-order Interactions
Winnifred Louis' notes for interpreting higher-order interactions in ANOVA.
Address: https://www2.psy.uq.edu.au/~wlouis/stats/HOIAv4.doc
CFA files
Confirmatory Factor Analysis & Structural Equation Modelling Part 1 - Aarti Iyer & Natalie Loxton
An introduction to Confirmatory Factor Analysis & Structural Equation Modelling given as workshop to School of Psychology postgraduates at the University of Queensland
Filename: Aarti Iyer & Natalie Loxton - Confirmatory Factor Analysis & Structural Equation Modelling.pdf [107kb]
Confirmatory Factor Analysis & Structural Equation Modelling Part 2 using AMOS - Aarti Iyer & Natalie Loxton
Part Two of Aarti and Natalie's introduction to Confirmatory Factor Analysis & Structural Equation Modelling in which they explain how to use the software package AMOS. Given as workshop to School of
Psychology postgraduates at the University of Queensland
Filename: Aarti Iyer & Natalie Loxton CFA & SEM Part 2 - AMOS.pdf [201kb]
Exploratory Factor Analysis
Principal Axis Factoring
Instructions file for JAMOVI
Filename: Exploratory Factor Anlaysis (PAF) in JAMOVI - v.2 - 10.7.21.pdf [886kb]
Testing for mediation using regression
Using bootstrapping to test for multiple mediators
Natalie Loxton's notes on testing for multiple mediators using bootstrapping. [Source: http://www2.psy.uq.edu.au/~wlouis/]
Address: https://www2.psy.uq.edu.au/~wlouis/stats/nloxton_multiplemediation.pdf
Notes on how to conduct a meta-analysis
Excel macro
Winnifred Louis' Excel macro for meta-analyses. [Source: http://www2.psy.uq.edu.au/~wlouis/]
Address: https://www2.psy.uq.edu.au/~wlouis/stats/metaanalysis_2017.xls
PSYC4050 files
Jason Tangen PSYC4050 Lecture01.pdf
Introduction to Multivariate Analysis, including matrices and the different types of multivariate analysis. Given as a fouth year course at the School of Psychology, University of Queensland.
Filename: Jason Tangen PSYC4050 Lecture01.pdf [3900kb]
Jason Tangen PSYC4050 Lecture02.pdf
Introduction to multivariate statistics: linear composites in discriminant analysis, multiple regression, and factor analysis. Overview of multiple regression, including the importance of residuals
and partitioning the variance. Given as a fourth year lecture at the School of Psychology, University of Queensland.
Filename: Jason Tangen PSYC4050 Lecture02.pdf [5702kb]
Jason Tangen PSYC4050 Lecture03.pdf
Multiple regression continued. R squared (overall relationships between predictors and criterion), the relationship between individual predictors and the criterion (correlations, standardised
regression weights, semi-partial correlations, partial correlations, relative weights). Regression strategies (standard or simultaneous regression, sequential or hierarchical regression, stepwise
regression, setwise regression, ridge regression).
Filename: Jason Tangen PSYC4050 Lecture03.pdf [4519kb]
Jason Tangen PSYC4050 Lecture04
Multiple regression continued. Measures of the importance of individual predictors revisited. Assumptions/limitations of multiple regression. Interpreting SPSS output for Multiple Regression.
Calculating 95% Confidence Intervals for beta weights. The prediction equation. Regression diagnostics (general strategies, data checking, importance of residuals, independence, linearity, normality,
outliers, homoscedasticity, multicollinearity, singularity). Factors affecting the correlation coefficient.
Filename: Jason Tangen PSYC4050 Lecture04.pdf [5971kb]
Jason Tangen PSYC4050 Lecture05
Multiple Regression continued. Sequential (hierarchical) regression with SPSS commands. ANOVA via Multiple Regression. Moderated Multiple Regression. Mediated vs moderated? Interactions in ANOVA &
Multiple Regression.
Filename: Jason Tangen PSYC4050 Lecture05.pdf [2448kb]
Jason Tangen PSYC4050 Lecture06
Moderating versus mediating effects. Discriminant analysis. Eigenvalues. Canonical correlations.
Filename: Jason Tangen PSYC4050 Lecture06.pdf [5248kb]
Jason Tangen PSYC4050 Lecture07.pdf
Discriminant Analysis continued. Assumptions. SPSS commands. Interpreting discriminant analysis. Relative importance of variables - various measures. Group separation. Comparing multiple regression
and discrimination analysis.
Filename: Jason Tangen PSYC4050 Lecture07.pdf [2679kb]
Mark Horswill's notes on the reliable change index.
How to do SEM.
Using AMOS to do SEM
Winnifred Louis' notes on structural equation modelling using AMOS. [Source: http://www2.psy.uq.edu.au/~wlouis/]
Address: https://www2.psy.uq.edu.au/~wlouis/stats/SEM.doc | {"url":"https://teaching.psy.uq.edu.au/tools/statsrepo/","timestamp":"2024-11-11T21:27:49Z","content_type":"text/html","content_length":"47504","record_id":"<urn:uuid:e45106b5-82c8-466f-9aba-df8741d1083d>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00260.warc.gz"} |
FRAUD : Investigation techniques and other aspects –Part 1 - BCAJ
Variety in fraud investigation techniques: application of Vedic Mathematics
It is variety that makes life interesting and enjoyable. Virtually in every walk of life, we crave for variety. Take for instance our daily meals. Each meal we try to eat something different to make
each meal more enjoyable. We try different kinds of breads, soups, vegetables, and fruits. We can actually survive just as well even if we have exactly the same items to eat everyday, but that would
make our meals monotonous. Film makers make different kinds of films only because we would get bored of the same story over and over again. A cricket match would become absolutely boring if a batsman
were to play each shot in the same identical manner. A popular batsman is one who has a range of different strokes and shots. Thus it has been correctly stated that variety is the spice of life.
Audit, investigation and forensic accounting are no exception to this maxim. It is very possible that if an auditor or an investigator approached every investigation with the same routine steps in a
lackadaisical manner, a wrongdoer would be able to take suitable counter measures to ensure that he is protected and safe. Therefore it is absolutely essential to keep trying new methods, hitherto
untried techniques and tools, and use a surprise element to get the best results. Research of algorithms, vedic scriptures can be extremely useful in this context. Many audits and investigations end
at a dead end, or sometimes reach wrong conclusions, only because of the lack of application of imaginative and innovative methods.
The following is a case study where a chartered accountant was an advisor in an acquisition by a fruit juice manufacturing company. Initially by applying the standard auditing techniques, he felt
that there was nothing serious to stop his client from acquiring a company owning a couple of mango farms based on details and information given. It was only after he looked at data differently,
using ‘visual mathematics’ and an application of vedic mathematics that he was able to detect a sinister fraud.
Case Study: Fraud in mango farm sale
A fruit juice manufacturing company ABC was looking for more and more orchards and fruit plantations for expansion. In this hunt, they came across a proposal from a mango grower PQR in Maharashtra
for sale of two mango farms. PQR had been growing mangoes and exporting them and seemed to have had a fairly good crop in the last season. The substantial part of the acquisition value was for the
two fertile farms. The two mango farms commanded a rich premium because of their fertility and huge potential for growing mangoes in bulk. ABC had asked its CA to conduct a review of its financials
and operating results for the last couple of years. Some extracts of the financial information given to him were as follows:
1. Farm A had 4 acres and Farm B was 6.3 acres in size. The potential for much greater crop of mangoes was huge and PQR had not been able to tap it because of its lack of resources. ABC realized that
with more resources and better techniques the mango crop could be tripled.
2. Plucking and packing activity was performed over two days. The mangoes would be plucked and packed on the last two days of each month. On day 1, there would only be plucking activity and the
mangoes would be stacked neatly. On day 2, the mangoes plucked the previous day would be washed and cleaned of all pesticide and then packed in boxes of one dozen each.
3. The packed mangoes from both the farms would be sent to the main godown where they would be counted and kept ready for export.
4. Costs of plucking and packaging for farm B were greater than farm A because it was further in the interior part of the district and labourers charged more to work at farm B
5. Costs of plucking and packaging during each month also varied based on demand supply of skilled labour in season time. Usually in May the cost would be the highest
The details of plucking and packaging costs per dozen are given in the table below
Conventional Audit checks did not throw up any adverse results.
The number of mangoes packed for each farm individually were not available, but the total mangoes packed for both farms for each month were physically verified by the management, as follows: March
720 mangoes, April, 2400 mangoes, and May 4800 mangoes. Though the CA was not conducting any investigation, he did have the responsibility of carrying out a special penetrative audit of the financial
information given by PQR because ABC was going to invest a huge amount only based on the CA’s assessment. Therefore the CA applied all the conventional audit checks and tests. The bills for
labourer’s payments were available in the form of wage sheets which prima facie looked satisfactory and his audit did have some routine queries but nothing serious.
The sales and collections audits and verifications using walk through tests also did not raise any alarm bells. These were also well documented. A decent price was earned by PQR for the sale of
mangoes per reasonable market inquiries. In most respects, based on his routine audit techniques, the CA seemed to have derived a comfort in the financial information given. Under normal
circumstances he would have given a ‘go ahead’ green signal to his client for acquisition of PQR.
How vedic mathematics helped the CA to spot a fraud by a mere visual look at the numbers.
The information given by PQR was incomplete in one important respect. The numbers of mangoes plucked and packaged in each farm for each month. This was important to determine the crop size and
fertility of each farm. How could one find this? Actually applying mathematics using knowledge of algebra by solving simultaneous equations for each month it is possible. But that is a tedious task.
To illustrate, for the month of March, to find out how many mangoes were plucked and packaged, one would have to use algebra by using variables ‘x’ and ‘y’ to represent mangoes plucked and packed in
farms A and B respectively. Then the cost information given above can be simply converted into a simultaneous equation in the conventional form as follows.
20x + 40y = 1200
70x + 85y = 4200
But solving such equations would be slightly tedious. However, through vedic mathematics, in one look, the viewer will be able to state that y = 0 in the above equations. How is this possible?
Actually it is very simple.
A sutra of vedic mathematics called Anurupye Shunyamanayat’ states that if the co-efficients of one of the variables in a simultaneous equation are in the same ratio as the resulting values of each
equation, then the other variable MUST BE ZERO
Thus in our above simultaneous equation of mangoes plucked and packaged in March
20x + 40y = 1200
70x + 85y = 4200
The coefficients of x are 20 and 70. Their ratio is therefore 2/7. The resulting values of each equation are 1200 and 4200. Their ratio is also 2/7. Since these two ratios are the same, the other
variable, ‘y’ as per sutra 6 of vedic mathematics, anuraupye shunyamanayat, MUST be zero.
THUS THERE WERE ‘0’ MANGOES GROWN IN MARCH IN FARM B. BY USING THE SAME VEDIC MATHEMATICS APPROACH THERE WERE ‘0’ MANGOES GROWN IN FARM B FOR THE OTHER MONTHS AS WELL. THE COST FIGURES WERE IMAGINARY
In other words, Farm B was not producing any mangoes at all.
The fraud was a simple deception by PQR by claiming that mangoes were indeed being grown on farm B, even though it had no fertility to grow any mango at all.
Though it was the larger farm, since it was not a fertile plot, the price being demanded by PQR was an atrocious exponential value of its actual worth. ABC would obviously never be interested in
purchasing such a farm. PQR’s labour costs were therefore nil for farm B and PQR was deceiving ABC by stating that mangoes were being plucked and packed in farm B. The CA then advised the client ABC
not to go ahead with this acquisition.
What is important in this case study is that the CA always strived to upgrade his knowledge and he was always eager to learn new techniques and methods useful in his profession. He had recently been
studying vedic mathematics. Vedic mathematics has some amazing solutions for certain types of mathematical problems. As we all know India discovered ‘0’ and a lot of vedic mathematics sutras are
based on, or revolve around ‘0’. Among them, one of the sutras, sutra no 6 is ‘Anurupye Shunyamanayat’.
Vedic mathematics itself may be useful in a rare assignment, but what counted was the fact the CA was trying new things and different things every time to get better results. That, friends is the
measure of life and true success.
Editor’s note: Fraud investigation and detection are an important area of practice for a chartered accountant. This involves acquisition of specialised knowledge. The law now casts an important duty
in regard to reporting fraud on the auditor. Public expectations have now found statutory recognition. We have therefore thought it necessary to carry a series of articles by Mr. Chetan Dalal an
expert on the subject. These will appear in the journal at intervals, that is probably in each alternate month. We hope readers will find this series useful. | {"url":"https://bcajonline.org/journal/fraud-investigation-techniques-and-other-aspects-part-1/","timestamp":"2024-11-05T20:29:27Z","content_type":"text/html","content_length":"105666","record_id":"<urn:uuid:51eed5be-7fe9-49b3-8a5c-3602587934e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00853.warc.gz"} |
The same operations that apply to numeric radicals can also be applied to algebraic radical expressions.
We can add or subtract like radicals (radicals with the same index and radicand) by adding the coefficients and keeping the radicand the same.
If there are no like radicals, check to see if any of the radicals can be simplified first. | {"url":"https://mathspace.co/textbooks/syllabuses/Syllabus-1189/topics/Topic-22470/subtopics/Subtopic-285821/?ref=blog.mathspace.co","timestamp":"2024-11-05T07:23:38Z","content_type":"text/html","content_length":"314469","record_id":"<urn:uuid:602ba955-6104-4313-a320-1ba26a17b3b5>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00061.warc.gz"} |
Database of Original & Non-Theoretical Uses of Topology
Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations (2018)
Jose A. Perea Abstract We present a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to
construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration
is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for
the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map.
Several examples are provided as well as theorems addressing choices in the construction. | {"url":"https://donut.topology.rocks/?q=tag%3A%22line+bundles%22","timestamp":"2024-11-10T11:15:13Z","content_type":"text/html","content_length":"4589","record_id":"<urn:uuid:a2e6f032-a7fe-4439-8c8a-dc2eeb7dda04>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00718.warc.gz"} |
Amigurumi Lamb Doll Free Pattern
Click For Crochet Abbreviations List
Mini Lamb Lupo
PC-Popcorn (4 PRSs in one loop)
PSN (needle with crochets)
VP (Air Loop)
SP (connecting Loop)
Knit Spiral PRs
1p-Magic Ring (6p)
2p-2p in each p (12)
3p-2p in 1p, p, 2p in 1p, N, 2p in 1p (3akonchit row, start 4rs)
4rs-Oil for the back half of the loop
6p-4p, 2p together, 2p together, 7p (13)
7p-4p, 2p together, 7p (12)
8r-3p, 2p together, 7p (11)
9r-11p (with this series to knit a different color)
11R-1p, 2p in 1p, 4p, 2p in 1p, 4p (13)
12 p-knit Two legs together (26)
13R-2VP, (PC, PSN) 13raz (26)
14r-2p in 1st p, 24p, 2p in 26th P (28)
15r-SP, 2VP, (PC, PSN) 14raz (28)
16r-2p in the 1st P, 26p, 2p in last P (30)
17r-SP, 2VP, (PC, PSN) 15 times (30)
18r-2p together, 26p, 2p together (28)
19r-SP, 2VP, (PC, PSN) 14raz (28)
20p-2p together, 24p, 2p together (26)
21R-SP, 2VP, (PC, PSN) 13raz (26)
22r-2p together, plastic, 2 p together (24)
23R-SP, 2 VP, (PC, PSN) 12 times (24)
24p-2p together, 20p, 2 p together (22)
25R-SP, 2 VP, (PC, PSN) 11 times (22)
26r-2p together, 18p, 2 p together (20)
No more closing rows
28r-8p, 2 n Together, 8p, 2 p together (18)
Change thread
29r-2p in Kazhduju3-yu P (24)
20s-2p in every 4th P (30)
31p-2p in every 5th P (36)
32 R-2p in every 6th P (42)
33 R-2p in every 7th P (48)
34-39 p 48 p
40R-Every 7 and 8 p together (42)
41 P-Each 6 and 7 p together (36)
42 P-Each 5e 6 p together (30)
43 P-Each 4 and 5 p together (24)
44 P-Each 3 and 4 p together (18)
45 p-Each 2 and 3 p together (12)
46 P-2 p together (6)
Hands-2 PCs
1p-Magic Ring (6)
2p-2p in each p (12)
3p, 4rs-12p
5p-PC, 11p (12)
6p, 7p, 8r-12p
9r-2p together, 10p (11)
10r, 11r, 12p-11p
13R-2p together, 9p (10)
Change thread
15r, 16r, 17r-10p
18r-2p Together, 8p (9)
Beat the brushes and close the loops
1p-Magic Ring (6)
2p-2VP, PC + PSN in each p (12)
3p-2p in each p (24)
4rs-(PC, PSN) 12 times (24)
5p-(2p in P, p) 12 times (36)
6p-(PSN, PC) 18 times (36)
7p-2p in every 9th P (40)
8r-(PSN, PC) 20 times (40)
10p-(PSN, PC) 20 times (40)
11 P-40p
I added another row of popcorn and the last PRS, it seemed to me that the beanie is a little
Ears-4 pcs (2 single colors, 2-Other)
1p-Magic Ring (6)
2p-2p in each p (12)
3p-2p in every 2nd P (18)
4rs-2p in every 3rd P (24)
Tie 2 pieces together PRS, sew
How to start knit body look at my album Lalilula, there is a description of the big lamb
and group photo :
Post your comment
Information! You need to be a member to comment. You can easily sign up from here.
0 comments | {"url":"https://knittingday.com/5136/amigurumi-lamb-doll-free-pattern","timestamp":"2024-11-02T09:22:16Z","content_type":"application/xhtml+xml","content_length":"93296","record_id":"<urn:uuid:8ea3e375-6ba6-4acc-ba97-64f6393876e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00541.warc.gz"} |
Data + Science
Creating a Venn Diagram in Tableau One of my students in the Spring semester data visualization class at the University of Cincinnati emailed me back in July asking about creating a Venn Diagram in
Tableau. There are a number of threads on the Tableau forum here and here.
The Venn Diagram, sometimes referred to as a set diagram, was first introduced in 1880 by John Venn (more information here). Unfortunately, these chart types are not great visualization tools for
comparing quantitative data. My first response to my student suggested alternative approaches to the visualization, but I got the sense from him that a strict client request was driving the desire
for this chart type and changing the visualization design wasn't an option. I outlined an approach for him and then put my draft workbook in an archive folder. Since it's not a chart type that I
considered very useful I wasn't really motivated to solve this problem and it sat idle.
At the Tableau conference in September I was talking with Ben Jones and KK Molugu and somehow the topic of Venn Diagrams came up. I described the approach, but I still had not completed an
implementation of it in Tableau.
One logical approach to create a Venn Diagram is using Shapes, specifically circles. Rob Austin of Interworks used this approach creating a 3 circle venn diagram in Tableau here. The problem with
this approach is that the size of the shapes is determined by the Size setting. This limits the ability to do the math calculations necessary to get exact overlap percentages and is relative to the
view. You can see in Rob's approach that the numbers in the overlapping circles are not proportional to the area of the actual intersection. Instead of using Shapes, the approach I outlined was to
plot each circle individually with 360 points so that the center position and the resulting overlap could be easily controlled. Once the 3 circles are plotted the position can be adjusted with
parameters and the exact overlap of the circles can be calculated.
I completed this approach and then I converted it to a polygon shape. This created a cleaner circle vs. the 360 individual points and allowed for the circles to be shaded with color and transparency.
The data set is very simple, just three fields.
Circle - 1 to 3 representing the 3 circles
Points - 1 to 360 points, one set for each circle
Theta - 0 to 2pi in 360 increments, one set for each circle
The calculated fields are the equations used to create the X,Y points for the circle and to calculate the circle sectors. Click here for more information about the math behind these calculations.
The visualization below was built for demonstration and instructional purposes. This is not a recommendation for using this chart type, especially for quantitative comparisons.
Additional materials mentioned above: Rob Austin of Interworks created a 2 and 3 circle venn diagram using Shapes (circles). Check those out here and here.
My approach outlined above uses 3 equal size circles. The Radius field is hard coded in my example making all of the circles the same size. This could be adjusted using 3 different Radius fields and
even using another set of parameters to correspond to the individual circle size, but the math calculating the overlap of the circles would need to be changed. I will outline this in a future post.
I hope you find this helpful. If you have any questions feel free to email me at Jeff@DataPlusScience.com.
Jeffrey A. Shaffer
Follow on Twitter @HighVizAbility | {"url":"https://www.dataplusscience.com/VennDiagram.html","timestamp":"2024-11-14T07:16:15Z","content_type":"text/html","content_length":"13553","record_id":"<urn:uuid:6c1bddf6-75bf-4a5c-9d7f-dcad4e01d6c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00884.warc.gz"} |
Research On Fault Location Algorithm For HV Transmission Lines
Posted on:2013-01-01 Degree:Master Type:Thesis
Country:China Candidate:S Huo Full Text:PDF
GTID:2232330374483157 Subject:Power system and its automation
Fault location for HV transmission lines can detect the fault point promptly and accurately, which can not only reduce labor intensity of line patrol and repair the damaged line in time, but also
find out indiscoverable faults to ensure power supply reliability. Accurate location has great significance for safe, stable and economic operation of power system. Based on fault analysis method,
fault location methods for HV transmission line are studied and discussed from the following aspects:One-end power frequency fault location algorithms mainly include linear equation method, iterative
method, quadratic equation method and the last two ones are more commonly used. Both iterative and quadratic equation methods need system impedance and have pseudo problem. For earth faults, the
analytic solution of fault distance can be directly obtained from linear relationships among short-circuit currents of three sequence networks at the fault point, so pseudo problem can be solved.
Based on distributed parameter model, location equations for different types of faults are deduced according to boundary conditions and pure resistance character of fault impedance, so the effect of
distributed capacitance can be overcomed.There is not enough information for one-end power frequency fault location, so it cannot overcome the effect of fault resistance and system parameter at the
same time. To eliminate effect of fault resistance, most one-end methods bring in system impedance. When system impedance varies, fault location accuracy will be influenced inevitably. The influence
of system impedance at remote end is analysed from theory and simulaton. To reduce error produced by given vaule of system impedance, we can calculate impedance using measuring quantities when normal
operation and there are disturbances in superior system.Whether fundamental component of fault signals is extracted precisely or not will directly influence fault location accuracy of power frequency
algorithms. Besides fundamental and harmonic components, fault signals include much decaying DC component and non-integral harmonics, which will reduce filter accuracy of traditional Fourier
algoirithm and increase fault location error. Using low-pass filter to filtrate high-frequency component and improved Fourier algorithms to eliminate the influence of decaying DC component can lower
filter error, and therefore fault location accuracy will be improved. With the development of communication techniques, research on two-end fault location methods made great progress. Though two-end
method needs communication channel, it can improve location accuracy by overcoming the effect of fault resistance and system impedance. Two-end methods are divided into two-end data synchronized and
without synchronization algorithm. The second method has better practical value. For the location error produced by line parameter change with environmental condition, a two-end time domain fault
location method based on line parameter estimation is proposed. The algorithm calculates positive sequence parameter and asynchronous time, and then implements location in time domain. The method can
not only overcome error produced by the uncertainty of line parameter and two-end being asynchronous but also avoid the effect of filter method. Besides, result of location is expressed by the ratio
of fault distance to whole-length of transmission line, so the location result will not be influenced by line real length change. The simulation results show this algorithm can realize accurate fault
location in several milliseconds.Besides phase coupling, double-circuit lines own line coupled inductance and can be decoupled by six-sequence component transform. There is no system impedance in
differential network, with what a new one-end fault location algorithm is proposed based on distributed parameter model. A universal location equation is obtained under general fault condition.
According to the boundary conditions of different single-line faults, unified relations between common sequence and differential sequence currents at the fault point are founded, and then common
sequence currents in the equation are removed, so the unified location equation for single-line faults could be obtained. This method can be implemented easily and there’s no need for phase
selection. Theoretically, this method is not influenced by single-line fault type, distributed capacitance, fault resistance and system impedance. The simulation results show this algorithm owns high
Keywords/Search Tags: HV transmission line, one-end fault location, two-end fault location, lineparameter estimation, fault location of double-ciucuit line | {"url":"https://www.globethesis.com/?t=2232330374483157","timestamp":"2024-11-14T04:57:34Z","content_type":"application/xhtml+xml","content_length":"10485","record_id":"<urn:uuid:de9c351f-eb87-406b-8a69-dc888fc29e06>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00593.warc.gz"} |
Truth table calculator
This calculator creates a truth table for any logical expression. To get started, enter the boolean expression into the calculator.
Calculator supports the following logical operations:
Logical operation "not" (negation, inversion)
This operation is denoted by the symbol. To enter it into our calculator, one can use either ¬ symbol or symbol of exclamation mark (!). The negation operation is unary (contains one operand only)
and have the highest priority among the logical operations.
The truth table of logical operation "not" has the form:
Logical operation "and" (conjunction, logical multiplication)
This operation is denoted by the symbol. To enter it into our calculator, one can use either ∧ symbol or two ampersand (&&) symbols. The conjunction operation is binary (contains two operands).
The truth table of logical "and" operation has the form:
Logical operation "or" (disjunction, logical addition)
This operation is denoted by the symbol. To enter it into our calculator, one can use either ∨ or two || symbols. The disjunction operation is binary.
The truth table of the logical "or" operation has the form:
Logical operation "exclusive or" (addition modulo 2)
This operation is denoted by the symbol. To enter it into our calculator, one can use either ⊕ symbol or a function .
The truth table of logical "exclusive or" operation has the form:
Logical operation "not and" (Sheffer stroke)
This operation is denoted by the symbol. To enter it into our calculator, one can use either ↑ or | symbol.
The truth table of logical operation "not and" has the form:
Logical operation "not or" (Peirce arrow)
This operation is denoted by the symbol. To enter it into our calculator, one can use either ↓ symbol or a function .
The truth table of logical "not or" has the form:
Logical equivalence
This operation is denoted by the symbol. To enter it into our calculator, one can use either ⇔ symbol or <=> (less sign, equal sign, greater sign) construction.
The truth table of logical equivalence has the form:
Logical operation "exclusive not or"
This operation is denoted by the symbol. To enter it into our calculator, one can use either ⊙ symbol or a function .
The truth table of logical operation "exclusive not or" has the form:
It should be noted that the truth tables for binary logical operations "equivalence" and "exclusive or" are coincide. In case, the specified operations are -ary, their truth tables are differ. Note
that the -ary operations can only bе entered in our calculator as a corresponding functions, for example , and the result of such expression will differ from the result of the expression . Because
the latter is interpreted as , while in the case of - the operation "equivalence" is performed immediately, taking into account all its arguments.
Logical operation "implication"
This operation is denoted by the symbol. To enter it into our calculator, one can use either ⇒ symbol or => (equal sign, greater sign) construction.
The truth table of logical operation "implication" has the form:
When creating the truth table of a complex (composite) logical expression, it is necessary to use the truth tables of the corresponding logical operations given above. | {"url":"https://mathforyou.net/en/online/discrete/truthtable/","timestamp":"2024-11-14T12:09:49Z","content_type":"text/html","content_length":"38610","record_id":"<urn:uuid:997a3584-bbbf-476e-b304-9fbe81e58f38>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00554.warc.gz"} |
Fractions Of Numbers Worksheet 2024 - NumbersWorksheets.net
Fractions Of Numbers Worksheet
Fractions Of Numbers Worksheet – Portion Numbers Worksheets are a very good way to train the thought of fractions. These worksheets are made to train students about the inverse of fractions, and will
enable them to comprehend the romantic relationship between fractions and decimals. Many students have trouble converting fractions to decimals, but they can benefit from these worksheets. These
printable worksheets can help your student in becoming far more knowledgeable about fractions, and they’ll be sure you enjoy yourself doing them! Fractions Of Numbers Worksheet.
Free of charge math worksheets
Consider downloading and printing free fraction numbers worksheets to reinforce their learning if your student is struggling with fractions. These worksheets could be customized to fit your personal
demands. In addition they include respond to tactics with comprehensive recommendations to steer your college student from the method. Lots of the worksheets are split into various denominators which
means that your university student can process their skills with a variety of issues. Afterward, individuals can renew the site to have a distinct worksheet.
These worksheets aid students recognize fractions by making counterpart fractions with different denominators and numerators. They have series of fractions which are counterpart in importance and
every row carries a lacking denominator or numerator. The scholars fill out the lacking numerators or denominators. These worksheets are of help for training the expertise of reducing fractions and
studying small fraction surgical procedures. These come in various quantities of problems, which range from an easy task to medium to tough. Each and every worksheet consists of between thirty and
ten issues.
Free of charge pre-algebra worksheets
Whether you may need a no cost pre-algebra small fraction phone numbers worksheet or you need a computer version to your individuals, the net can provide you with various alternatives. Some offer
totally free pre-algebra worksheets, with a few well known exceptions. Although several of these worksheets may be custom-made, a couple of totally free pre-algebra small fraction phone numbers
worksheets might be delivered electronically and imprinted for extra training.
A single excellent source of information for down loadable free pre-algebra small fraction amounts worksheet is definitely the University of Maryland, Baltimore State. You should be careful about
uploading them on your own personal or classroom website, though worksheets are free to use. However, you are free to print out any worksheets you find useful, and you have permission to distribute
printed copies of the worksheets to others. You can use the free worksheets as a tool for learning math facts. Alternatively, as a stepping stone towards more complex concepts.
Totally free math worksheets for course VIII
You’ve come to the right place if you are in Class VIII and are looking for free fraction numbers worksheets for your next maths lesson! This selection of worksheets is dependant on the CBSE and
NCERT syllabus. These worksheets are perfect for cleaning up on the concepts of fractions to enable you to do better within your CBSE test. These worksheets are really easy to use and protect every
one of the ideas that are important for reaching higher spots in maths.
A few of these worksheets include evaluating fractions, getting fractions, simplifying fractions, and functions by using these amounts. Use true-lifestyle good examples during these worksheets which
means your students can relate with them. A dessert is much easier to connect with than 1 / 2 of a square. Another easy way to exercise with fractions is to use counterpart fractions models. Use
actual life examples, like a 50 %-dessert as well as a square.
Totally free mathematics worksheets for converting decimal to small fraction
If you are looking for some free math worksheets for converting decimal to a fraction, you have come to the right place. These decimal to fraction worksheets can be found in numerous formats. You can
download them inPDF and html. Alternatively, random format. Most of them feature a solution important and could even be shaded by children! They are utilized for summertime discovering, arithmetic
centres, or as a part of your regular mathematics curriculum.
To transform a decimal to a fraction, you should streamline it initially. If the denominator is ten, Decimals are written as equivalent fractions. Furthermore, you will also find worksheets regarding
how to convert combined amounts to your portion. Cost-free arithmetic worksheets for converting decimal to portion come with combined numbers and examples of the two conversion functions. The process
of converting a decimal to a fraction is easier than you might think, however. Adopt these measures to get going.
Gallery of Fractions Of Numbers Worksheet
Leave a Comment | {"url":"https://www.numbersworksheets.net/fractions-of-numbers-worksheet/","timestamp":"2024-11-12T22:54:27Z","content_type":"text/html","content_length":"62415","record_id":"<urn:uuid:14957e2f-b25b-4c8b-b0b4-fb619cd84f03>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00342.warc.gz"} |
Kyber KEM: A Quantum-Resistant Lattice-Based Framework for Secure Key Encapsulation (Example in Golang)
Cryptography, the science of secure communication, has evolved to address new challenges in a world where traditional encryption techniques may fall short against quantum computers. Quantum
computers, capable of processing vast amounts of information simultaneously, have the potential to break widely-used encryption algorithms, including RSA and ECC, which secure much of today's digital
communications. One promising approach to counter these threats is lattice-based cryptography, a field that relies on mathematical structures resistant to quantum attacks. Among the innovative
advancements in this area is Kyber, a lattice-based key encapsulation mechanism (KEM) designed to maintain secure communication, even in the post-quantum era.
Lattice-Based Cryptography: An Overview
At the core of lattice-based cryptography is the concept of mathematical lattices, which can be visualized as a grid of points in multidimensional space. This type of cryptography draws its security
from complex mathematical problems that are difficult to solve, even for quantum computers. Two of the primary problems supporting this security are the Learning With Errors (LWE) and Ring-Learning
With Errors (RLWE) problems. The LWE problem involves finding a solution when a bit of noise or randomness is added to an otherwise solvable system, making it computationally challenging. RLWE builds
on LWE by placing this problem in a structured ring setting, which improves efficiency while maintaining security. The difficulty of solving these lattice problems provides the basis for secure,
quantum-resistant encryption in lattice-based systems.
In practical terms, lattice-based cryptographic systems encode messages as specific points within a lattice. When a message is encrypted, a small amount of noise is added, effectively hiding the
original message within the lattice's structure. To retrieve the message, the recipient must have precise knowledge of the lattice structure used in the encoding. The approach provides both
efficiency and strong security guarantees because attackers, even those with quantum computing capabilities, struggle to locate and interpret the hidden message within the noisy lattice.
Key Encapsulation Mechanisms (KEM)
A Key Encapsulation Mechanism (KEM) is a protocol used in cryptography to securely transmit encryption keys. Rather than directly sharing encryption keys, which could be intercepted, KEM encapsulates
a randomly generated encryption key within a secure cryptographic process. This encapsulated key is then used to encrypt and decrypt communications. In essence, KEM enables a secure key exchange
between parties without risking exposure of the encryption key during transmission.
Kyber, a lattice-based KEM, was designed with the unique challenges of the post-quantum era in mind. It uses the module lattice LWE (MLWE) problem, a variant of the standard LWE problem. This
modification makes Kyber both more flexible and scalable, allowing it to secure communication channels at varying levels of security based on the application’s needs. The MLWE-based structure of
Kyber enables it to encapsulate keys in such a way that security can be tailored while maintaining strong post-quantum resistance.
Kyber’s structure revolves around three essential processes—key generation, encapsulation, and decapsulation. Each phase contributes to Kyber’s ability to securely exchange encryption keys.
In the Key Generation phase, both a public key and a secret key are generated. The public key, which will be used for encryption, contains a matrix created from a seed, ensuring both randomness and
consistency across keys. This matrix is central to the security of the system, as it is the reference point for decoding the lattice structure in later steps. Meanwhile, the secret key, stored
securely by the recipient, is designed to facilitate decryption and is never exposed to other parties.
During Encapsulation, a random message or key is encrypted using the recipient's public key. This process produces a ciphertext, or encrypted message, which can be safely transmitted over public
channels without risking the confidentiality of the original message. The encapsulated key, securely hidden within the ciphertext, is then used to perform symmetric encryption for efficient and
secure communication.
In the Decapsulation step, the recipient uses their secret key to decrypt the ciphertext and retrieve the original encapsulated key. This key, once recovered, forms the foundation of a secure
communication channel, as it allows both parties to decrypt messages exchanged using the same symmetric encryption key. The encapsulation and decapsulation processes make Kyber a highly secure KEM,
even in cases where an attacker intercepts the ciphertext, as decrypting the message without the secret key remains virtually impossible.
An Example of Kyber in Action
Imagine a scenario where Alice and Bob, two parties who want to communicate securely, decide to use Kyber KEM. Bob begins by generating a public and secret key pair through Kyber’s key generation
process. He then shares his public key with Alice, while securely storing the secret key for later use.
Alice, who wishes to send a confidential message to Bob, generates a random encryption key on her end. She then uses Bob's public key to encapsulate this random encryption key, producing ciphertext
in the process. This ciphertext essentially hides the random encryption key within a secure cryptographic structure and is sent to Bob over a potentially insecure channel.
When Bob receives Alice's ciphertext, he uses his secret key to decapsulate it, retrieving the random encryption key that Alice initially generated. This shared key now allows Bob to decrypt any
subsequent messages from Alice securely, ensuring both confidentiality and authenticity in their communication.
Let's consider a scenario in which Alice and Bob use Kyber for key encapsulation to securely exchange a 256-bit AES-GCM key. Here’s the full Go code for this configuration:
package main
import (
var kyber = schemes.ByName("Kyber512") // Using Kyber-512 KEM
func main() {
// Step 1: Bob generates a Kyber-512 public/private key pair
bobPubK, bobPrivK, err := kyber.GenerateKeyPair()
if err != nil {
log.Fatalf("Error generating key pair: %v", err)
fmt.Println("Bob has generated his public and private keys.")
// Step 2: Alice encapsulates a shared secret (AES-256 key) using Bob's public key
kemCiphertext, sharedSecretEncap, err := kyber.Encapsulate(bobPubK)
if err != nil {
log.Fatalf("Error encapsulating the shared secret: %v", err)
fmt.Println("Alice has encapsulated an AES-256-GCM key using Bob's public key.")
// Step 3: Alice uses the encapsulated shared secret as the AES key for GCM encryption
block, err := aes.NewCipher(sharedSecretEncap[:32]) // Using first 32 bytes for AES-256
if err != nil {
log.Fatalf("Error creating AES cipher: %v", err)
aesGCM, err := cipher.NewGCM(block)
if err != nil {
log.Fatalf("Error creating AES-GCM mode: %v", err)
// Generate a nonce for AES-GCM encryption
nonce := make([]byte, aesGCM.NonceSize())
if _, err := rand.Read(nonce); err != nil {
log.Fatalf("Error generating nonce: %v", err)
// Step 4: Alice encrypts her confidential message with AES-GCM
message := []byte("This is a confidential message from Alice to Bob.")
aesCiphertext := aesGCM.Seal(nil, nonce, message, nil)
fmt.Printf("Alice has encrypted her message with the shared secret.\n")
// Alice sends Bob the ciphertext (Kyber encapsulated key), nonce, and AES-GCM-encrypted message
fmt.Printf("Ciphertext (Encapsulated Key): %x\n", kemCiphertext)
fmt.Printf("Nonce: %x\n", nonce)
fmt.Printf("Encrypted Message: %x\n", aesCiphertext)
// Step 5: Bob receives Alice's ciphertext and decrypts it using his Kyber private key
sharedSecretDecap, err := kyber.Decapsulate(bobPrivK, kemCiphertext)
if err != nil {
log.Fatalf("Error decapsulating the shared secret: %v", err)
// Step 6: Bob uses the decapsulated shared secret to decrypt the AES-GCM-encrypted message
block, err = aes.NewCipher(sharedSecretDecap[:32]) // Using first 32 bytes for AES-256
if err != nil {
log.Fatalf("Error creating AES cipher for decryption: %v", err)
aesGCM, err = cipher.NewGCM(block)
if err != nil {
log.Fatalf("Error creating AES-GCM mode for decryption: %v", err)
// Bob decrypts the message
plaintext, err := aesGCM.Open(nil, nonce, aesCiphertext, nil)
if err != nil {
log.Fatalf("Error decrypting message: %v", err)
fmt.Printf("Bob has decrypted the message: %s\n", plaintext)
// Verification
if string(message) == string(plaintext) {
fmt.Println("Message successfully encrypted and decrypted using Kyber-encapsulated AES-256-GCM key!")
} else {
fmt.Println("Error: Decrypted message does not match original.")
1. Key Generation: Bob generates a Kyber-512 key pair, which includes a public and private key.
2. Encapsulation: Alice uses Bob’s public key to encapsulate a shared secret, which will act as her AES-256 key. She obtains a kemCiphertext (Kyber encapsulated key) and a sharedSecretEncap.
3. AES-GCM Encryption: Alice encrypts her message using AES-GCM, with the first 32 bytes of sharedSecretEncap as the AES key.
4. Transmission: Alice sends the kemCiphertext, the AES-GCM nonce, and the AES-encrypted message aesCiphertext to Bob.
5. Decapsulation and Decryption: Bob decapsulates the kemCiphertext to retrieve sharedSecretDecap, which should match Alice’s sharedSecretEncap. He then uses the first 32 bytes of this shared secret
as his AES-256 key to decrypt the message aesCiphertext.
This setup demonstrates a quantum-resistant key encapsulation (Kyber-512) combined with AES-GCM encryption, suitable for secure post-quantum communication.
Advantages of Using Kyber
Kyber presents several significant advantages, which make it suitable for addressing the modern encryption challenges presented by quantum computing.
Kyber’s quantum resistance is grounded in the MLWE problem, a mathematically challenging structure that even quantum computers cannot easily break. This sets Kyber apart from traditional
cryptographic systems, which rely on integer factorization or discrete logarithms, both vulnerable to quantum attacks.
Another key advantage is Kyber’s efficiency. The use of module lattices allows Kyber to offer robust security without imposing heavy computational demands, making it viable for both high-security and
resource-constrained environments. In comparison to other post-quantum cryptosystems, Kyber requires relatively low storage and computational resources, which helps optimize its performance.
Kyber also offers scalability. It is configurable to provide varying levels of security, tailored to different applications and risk profiles. For example, Kyber provides three different security
levels (Kyber512, Kyber768, and Kyber1024) each offering incremental security guarantees. This flexibility enables Kyber to cater to a broad range of applications, from secure online transactions to
high-stakes government communications.
Practical Applications of Kyber
Kyber KEM’s design and capabilities make it particularly suitable for applications where secure key exchange is critical. In online communications, Kyber KEM could enhance the security of HTTPS
protocols, providing post-quantum-safe encryption for browsing sessions and online transactions. For secure messaging applications, Kyber’s use for key exchange ensures that conversations are
protected against potential quantum attacks, preserving user privacy in a future where quantum computers are more accessible.
In the realm of the Internet of Things (IoT), Kyber’s lightweight nature makes it highly effective. IoT devices often have limited processing power and memory, but with Kyber’s efficiency and low
computational requirements, these devices can implement secure encryption to protect data transmission without sacrificing performance or battery life.
Kyber represents a significant advancement in cryptography, reflecting the growing need for quantum-resistant encryption as quantum technology progresses. Its foundation in lattice-based cryptography
offers robustness and resilience against quantum attacks, ensuring that even advanced computing capabilities cannot easily breach encrypted communications. By adapting the key encapsulation mechanism
to fit diverse security needs and resource constraints, Kyber stands out as a promising solution for future-proofing digital security across industries and applications.
Rivest, R.L., Shamir, A. and Adleman, L., 1978. A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM, 21(2), pp.120-126.
Koblitz, N. (1987). Elliptic curve cryptosystems. Mathematics of Computation, 48(177), 203-209.
Avanzi, R., Bos, J., Ducas, L., Kiltz, E., Lepoint, T., Lyubashevsky, V., Schanck, J.M., Schwabe, P., Seiler, G. and Stehlé, D., 2019. CRYSTALS-Kyber algorithm specifications and supporting
documentation. NIST PQC Round, 2(4), pp.1-43. | {"url":"https://eminmuhammadi.com/articles/kyber-kem-a-quantum-resistant-lattice-based-framework-for-secure-key-encapsulation-example-in-golang","timestamp":"2024-11-03T15:38:37Z","content_type":"text/html","content_length":"98981","record_id":"<urn:uuid:64b1d369-6e96-43e8-9b85-0137eb37967d>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00256.warc.gz"} |
How do you factor 5n^2+10n+20? | Socratic
How do you factor #5n^2+10n+20#?
2 Answers
y = 5(n^2 + 2n + 4)
The expression in parentheses can't be factored because D < 0
We must find its roots and then transform them into factors.
Using Bhaskara we get:
$\frac{- 10 \pm \sqrt{- 300}}{10}$
However, $\Delta = - 300$ indicates imaginary roots, because we can rewrite it as $300 \left(- 1\right)$, which will pave the way to find a solution.
We know, by imaginary numbers definition, that $\left(- 1\right) = {i}^{2}$. Therefore, $\sqrt{- 1} = i$ and we can proceed.
$\frac{- 10 \pm 10 i \sqrt{3}}{10}$=$- 1 \pm i \sqrt{3}$
Imaginary roots cannot be factors.
Impact of this question
4320 views around the world | {"url":"https://api-project-1022638073839.appspot.com/questions/how-do-you-factor-5n-2-10n-20#145989","timestamp":"2024-11-12T14:54:10Z","content_type":"text/html","content_length":"33698","record_id":"<urn:uuid:c3965991-a029-4534-a104-555573bbe3c2>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00469.warc.gz"} |
Montclair Company is considering a project that will require...ask hint 1
Earl Stokes Verified Expert
8464 Answers
Get instant answers from our online experts!
Our experts use AI to instantly furnish accurate answers.
Montclair Company is considering a project that will require a $500,000 loan. It presently has total liabilities of $220,000, and total assets of $610,000. 1. Compute Montclair’s (a) Present
debt-to-equity ratio (b) The debt-to-equity ratio assuming it borrows $500,000 to fund the project. 2. Evaluate and discuss the level of risk involved if Montclair borrows the funds to pursue the
project.View Solution:
Montclair Company is considering a project that will require a
Aug 16 2022| 12:46 PM |
Debt-to-Equity Ratio (a) 0.55 (b) 1.80 WORKINGS Debt-Equity Ratio = Total Liabilities/Total Equity Total Equity Calculation Assets totaling $620,000. Liabilities Total = $220,000 Total Equity is
equal to Total Assets minus Total Liabilities. $620,000 - $220,000 = $400,000 is the total equity. Current Debt-Equity Ratio (a) Liabilities total $220,000. Equities totaling $400 000. Debt-Equity
Ratio = $ 220 000 $ $400 000 = 0.55 (a) Debt-Equity Ratio if a $500,000 loan is assumed. Total Liabilities are equal to $720,000 ($220,000 + $500,000). Equities totaling $400 000. Debt-Equity Ratio =
$720,000 divided by $400,000,000 is 1.80....
Related Questions
Denise would like for all of her leads to save their notes in the Client details screen. Who...
Denise would like for all of her leads to save their notes in the Client details screen. Who would have access to these notes? Full access team members Custom access team members Assigned team
members All team members
Aug 02 2022
Jim Carrie Company shows a balance of $181,140 in the Accounts Receivable account on December 31,...
Jim Carrie Company shows a balance of $181,140 in the Accounts Receivable account on December 31, 2008. The balance consists of the following. Installment accounts due in 2009 ……………… $23,000
Installment accounts due after 2009 ……………. 34,000 Overpayments to creditors ……………….............. 2,640 Due from regular customers, of which $40,000 represents accounts pledged as security for a bank
loan ……....
Jul 23 2022
The adjusted trial balance of Cavamanlis Co. as of December 1 answer below » The adjusted trial...
The adjusted trial balance of Cavamanlis Co. as of December 31, 2012, contains the following. Instructions (a) Prepare an income statement. (b) Prepare a statement of retained earnings. (c) Prepare a
classified balancesheet.
Jul 26 2022
The trial balance before adjustment of Reba McIntyre Inc. shows the following balances. Dr. Cr....
The trial balance before adjustment of Reba McIntyre Inc. shows the following balances. Dr. Cr. Accounts Receivable $91,200 Allowance for Doubtful Accounts 2,320 Sales Revenue (all on credit)
$729,000 Give the entry for estimated bad debts assuming that the allowance is to provide for doubtful accounts on the basis of (a) 4% of gross accounts receivable and (b) 1% of net sales.
Aug 03 2022
Market skimming pricing makes sense under all the following conditions, EXCEPT ________. A) if a...
Market skimming pricing makes sense under all the following conditions, EXCEPT ________. A) if a sufficient number of buyers have a high current demand B) if the unit costs of producing a small
volume are high enough to cancel the advantage of charging what the traffic will bear C) if the high initial price does not attract more competitors to the market D) if consumers are likely to delay
Aug 13 2022
Join Quesbinge Community
5 Million Students and Industry experts | {"url":"https://www.quesba.com/questions/montclair-company-considering-project-will-require-500-000-loan-presently-t-1806987","timestamp":"2024-11-06T20:21:44Z","content_type":"text/html","content_length":"193100","record_id":"<urn:uuid:de8bc8e5-33bc-4bec-80c7-a5ce9858e542>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00826.warc.gz"} |
Ampere (unit)
The ampere, symbol A, is the SI unit of electric current. It is defined by application of Ampere's equation:
${\displaystyle {\frac {F}{l}}={\frac {\mu _{0}\,i^{2}}{2\pi r}}.}$
One ampere is that constant current i which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed r = 1 metre apart in vacuum, would
produce between these conductors a force F equal to 2 x 10^-7 newton per meter of length l.^[1] This sets the value of the magnetic constant μ[0] to 4π x 10^−7 N/A^2.
The ampere is named for André-Marie Ampère, an early investigator of electricity, magnetism, and chemistry.
The ampere has undergone a number of redefinitions; the current standard was adopted in 1948. One definition adopted legally before the current SI definition was "that unvarying current that would
deposit 0.001 118 000 grams of silver per second from a solution of silver nitrate in water". This earlier definition is approximately 0.99985 A (SI).
Related units
The SI uses the ampere as its basic unit of electrical measure; all other units are derived from the ampere.
• The coulomb is the unit of electrical charge, and is equal to the amount of charge passing a point in one second in a circuit with one ampere of current.
${\displaystyle A=C\cdot s^{-1}={\frac {C}{s}}}$
${\displaystyle A=m^{2}\cdot kg\cdot s^{-3}\cdot V^{-1}={\frac {m^{2}\cdot kg}{s^{3}\cdot V}}}$.
• The ohm is the unit of electrical resistance, and is the resistance which will allow a current of one ampere across a potential drop of one volt.
${\displaystyle A=V\cdot \Omega ^{-1}={\frac {V}{\Omega }}={\sqrt {m^{2}\cdot kg\cdot s^{-3}\Omega ^{-1}}}}$
• The farad is the unit of electrical capacitance, and is the capacitance of a capacitor whose potential between the plates increases by one volt when charged with one coulomb of charge. | {"url":"https://en.citizendium.org/wiki/Ampere_(unit)","timestamp":"2024-11-08T17:08:09Z","content_type":"text/html","content_length":"40441","record_id":"<urn:uuid:92fac242-e984-4f38-9f89-eca74b76b2af>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00210.warc.gz"} |
What is a Series Magnetic Circuit? Definition and Explanation
Series Magnetic Circuit
Definition: The Series Magnetic Circuit consists of various parts made of different magnetic materials and with varying sizes that carry the same magnetic field. Before understanding the series
magnetic circuit, let’s understand the basics of a magnetic circuit and its reluctance.
What is a Magnetic Circuit?
A magnetic circuit refers to a path that magnetic flux follows within a material, typically a ferromagnetic substance like iron. It’s analogous to an electrical circuit, but instead of the flow of
electric current, it involves the flow of magnetic flux. Components such as iron cores and coils are often part of a magnetic circuit. Understanding magnetic circuits is crucial in various
applications, including transformers, motors, and generators.
What is a Magnetic Reluctance?
Magnetic reluctance refers to the measure of opposition a material offers to establish magnetic flux. It’s similar to electrical resistance in an electrical circuit but in the context of magnetism.
Materials with high reluctance impede the flow of magnetic flux more strongly, while those with low reluctance allow flux to flow more easily. Reluctance depends on factors such as the material’s
composition, shape, and size, and it’s an essential concept in understanding and designing magnetic circuits.
Series Magnetic Circuit Explanation
Consider a composite magnetic circuit consisting of three distinct magnetic materials, each with varying permeabilities and lengths and an air gap characterized by a permeability (μr) of 1. Each
segment within the circuit possesses its reluctance. Refer to the diagram below for the illustration of the series magnetic circuit.
An electric current I flows through a solenoid with N turns wound around one section of a circular coil. Φ represents the flux created within the core of the coil.
In a part of the magnetic circuit, there’s a circular coil with N turns. When an electric current I flows through the solenoid, it induces a flux Φ within the core of the magnetic material.
The overall reluctance of the circuit equals the sum of the reluctances along each path, given that they are connected in series.
The flux in the circuit is,
The value of total MMF by putting the value of reluctance in the above equation, we get;
We know φ=BA, therefore,
The magnetic flux density(B) is;
Putting the value of flux density from equation 4 in equation 3,
Procedure for the Calculation of the total MMF of a Series Magnetic Circuit
Following the steps below, we can calculate the total MMF in a series of magnetic circuits.
1. List all the magnetic components in the series circuit, including the transformer core, coils, or any other magnetic devices.
2. Determine the flux density(B) of all the sections using formula B = φ/a where φ is the magnetic flux in Weber, and a is the area of the cross-section in m^2
3. Calculates the magnetizing force (H) using the formula H = B/µ0µr, where B is the flux density in Weber/m2. The value of absolute permeability µ[0] is 4πx10^-7. The µ[r] is the relative
permeability of the material, and we know this value; we can calculate H. If the value of µr is unknown, you can determine it from the B-H curve of the magnetic material.
4. You can determine the magnetizing force(H) by multiplying each section’s magnetizing force, H1, H2, H3, and Hg, with their respective section’s length, l1, l2, l3, and lg.
5. The total MMF of a series magnetic circuit is;
Solved Example:
A series magnetic circuit consists of two magnetic materials: a ferromagnetic core with a reluctance of 6000 A/Wb and a magnetic material with a reluctance of 4000 A/Wb. The mean length of the
toroidal is 0.4 meters. A coil with 10000 turns and a current of 4 A flowing through it is connected in series with these materials. Calculate the total magnetic flux in the circuit and the magnetic
field intensity.
• Reluctance of the ferromagnetic core, Rcore=6000A/Wb
• The reluctance of the additional magnetic material, Radditional=4000A/Wb
• Number of turns in the coil, N=1000
• Current flowing through the coil, I=4A.
• Length of toroidal, l= 0.4 m
Solution: To solve this problem, we’ll first find the equivalent reluctance of the series magnetic circuit and then use it to calculate the total magnetic flux and field intensity.
1. Total Reluctance (Rtotal) in series:
2. Magnetic Flux (Φ) using the total reluctance:Φ=NI/R[total]
3. Magnetic Field Intensity (H) using the mean length of the coreeH=NI/lcore
1. Calculate the total reluctance (Rtotal).
2. Calculate magnetic flux (Φ).
3. Calculate the magnetic field intensity (H).
1. Total Reluctance (Rtotal):
2. Magnetic Flux (Φ):
3. Magnetic Field Intensity (H):
=(1000 X4)/0.4
= 10000 A/m
In conclusion, a series magnetic circuit comprises various magnetic elements interconnected sequentially, such as cores, coils, and other devices. The sum of the reluctance along each path determines
the total reluctance of the circuit. Understanding the behavior of a series of magnetic circuits is essential in designing and analyzing magnetic systems and facilitating applications in
transformers, motors, generators, and various other electromagnetic devices. In this article, you have learned,
• The same magnetic flux Φ flows through the circuit in a series magnetic circuit.
• Total reluctance is the sum of individual reluctance.
• The total MMF required for producing the flux is the sum of the MMFs for the various sections. | {"url":"https://electricalampere.com/what-is-series-magnetic-circuit/","timestamp":"2024-11-03T00:33:14Z","content_type":"text/html","content_length":"71984","record_id":"<urn:uuid:cd69b035-4726-4a8c-8d3d-e4ad519cddc9>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00411.warc.gz"} |
Well-Covered graphs and greedoids
G is a well-covered graph provided all its maximal stable sets are of the same size (Plummer, 1970). S is a local maximum stable set of G, and we denote by S 2 ^a(G), if S is a maximum stable set of
the subgraph induced by S [N(S), where N(S) is the neighborhood of S. In 2002 we have proved that ^a(G) is a greedoid for every forest G. The bipartite graphs and the triangle-free graphs, whose
families of local maximum stable sets form greedoids were characterized by Levit and Mandrescu (2003, 2007a). In this paper we demonstrate that if a graph G has a perfect matching consisting of only
pendant edges, then ^a(G) forms a greedoid on its vertex set. In particular, we infer that ^a(G) forms a greedoid for every well-covered graph G of girth at least 6, non-isomorphic to C [7].
Original language English
Title of host publication Theory of Computing 2008 - Proceedings of the Fourteenth Computing
Subtitle of host publication The Australasian Theory Symposium, CATS 2008
State Published - 2008
Event Theory of Computing 2008 - 14th Computing: The Australasian Theory Symposium, CATS 2008 - Wollongong, NSW, Australia
Duration: 22 Jan 2008 → 25 Jan 2008
Publication series
Name Conferences in Research and Practice in Information Technology Series
Volume 77
ISSN (Print) 1445-1336
Conference Theory of Computing 2008 - 14th Computing: The Australasian Theory Symposium, CATS 2008
Country/Territory Australia
City Wollongong, NSW
Period 22/01/08 → 25/01/08
• Greedoid
• Local maximum stable set
• Unique perfect matching
• Very well-covered graph
Dive into the research topics of 'Well-Covered graphs and greedoids'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/well-covered-graphs-and-greedoids-3","timestamp":"2024-11-10T22:27:27Z","content_type":"text/html","content_length":"52615","record_id":"<urn:uuid:16030722-2aaa-403b-9d23-a6234fd47350>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00066.warc.gz"} |
Simple buoyancy.
26-03-2009 23:07:05
How would i go about getting simple buoyancy to work? I understand that to get the force required to keep an object buoyant, I need to times the fluid density, But the displacement volume, By the
gravitational acceleration. However, Obviously simply using Actor::AddForce with the result of that equation does not work. How do i calculate drag etc etc, So it does not pop out of the water/Bounce
of the water surface. Ive spent 8 non stop hours trying to get even the simplest of buoyant effects working, But i just cant. I always get the same result, The object hits the water, Slows down, And
shoot's flying back out. I cant get it to calmly rise to the water surface. Any help will be much appreciated. | {"url":"https://www.ogre3d.org/addonforums/6/t-9629.html","timestamp":"2024-11-06T11:22:01Z","content_type":"text/html","content_length":"15950","record_id":"<urn:uuid:4dcde247-0f4f-480b-888b-ed2b287857ce>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00068.warc.gz"} |
Free online graphing calculator absolute value
Lcm activities 6th grade, free algebra calculator, rules for multiplying dividing addition, least common multiple of 51 and 52, simplify equations on ti 89, Root formula, real life algebra 2
problems, first order linear differential equation sample problems, variable exponent [ Def: The number that indicates how many times the base is used as a factor. ], finding the scale factor [ Def:
Numbers or expressions that are multiplied together to produce a specific product. ], ratio percentage formulas:
1. adding a minus number and a plus number examples
2. show an exercises problem solving in hyperbola with solution
3. adding and subtracting scientific notations worksheets
4. algebra calculator
5. lesson plans adding/subtracting fractions
6. multiplying scientific notation
7. rene descartes linear functions
8. set of rules to solve a problem
college algebra clock word problems cubed equations college algebra exercises
solution answers to assignment: systems of equations and inequalities of chapter 8 and 4 digit multiplied by 1 digit worksheet aptitude test download
9 in MyMathLab
maths test on line year 7 Algebra teaching Software for students and dividing and subtracting and adding and multiplying negatives in a
teachers calcuator
glencoe algebra 2 word problem practice answers ti 84 plus radical expression application free psle on line exam papers
mixture problem (quadratic equation) saxon algebra 2 answers "grade 9" "LIKE TERMS" WORKSHEET
factoring program for your calculator trivia economic sixth grade mcdougal littell algebra and trigonometry: structure and method,
book 2 preview
t-83 plus statistics combination interest formula on GRE algorithm everyday multiplication lesson worksheets pdf.
order of operation using cube roots in linear equations free online inequality calculator Algebra Math Trivia
algebra with pizzazz answers dividing decimals worksheet sample problem solving subtraction
Introduction on quadratic equation
Prentice hall pre-algebra tools for a changing world california edition math problems,
online pre algebra equation calculator that explains how to do it, using a model to solve equations, ADDING AND SUBRACTING FRACTION WITH EQUAL DEMONINATOR, math indces work sheets, solving a variable
in Matlab, holt algebra 2 answers, 8th grade algebra worksheets, factor poems for math, variables as exponents, online parabola [ Def: The shape of the graph of a quadratic or degree 2 equation. ],
solving algebra, algebrator free download ( Example: variables in the exponent [ Def: The number that tells how many times a base is to be used as a factor. ], visual programming aptitude question
bank with answer, Texas Holt Algebra 1 Book page 48, solving 3rd degree quadratic equations, software ( Example: 2nd order differential equation (y'' + x = 0), math investigatory project ( Example:
Graphing linear inequalities in two variables worksheets, discriminant calculator, common denominator calculator ( Example: gnuplot linear regression, T1 89, free printable multiplication and
division expressions worksheet, algebra 1 glencoe textbook answers, inconsistent and dependent systems, easy ways of finding greatest common factor, intermidiate algebra [ Def: The mathematics of
working with variables. ], exam test on scientific notation and standard form..square root, ti 89 quadratic equation [ Def: An equation includes only second degree polynomials. When there is only one
variable, a quadratic equation can be expressed in the form ax^2 + bx + c = 0 where a, b, and c are all constants. ], least common multiple calculator ( Example: changing decimals to mixed numbers,
simplifying radicals calculator ( Example: Modern Chemistry holt rinehart vocabulary definitions, mcdougal littell algebra 1 test answers,
quadratic formula from table of values.
online factoring
beginners algebra
free algebra worksheets
Simplifying an equation solver
Solving the van der pol equation as a second order system, change base of rational decimal to octal, rational expressions solver ( Example:
• solving elimination equations calculator
• solving for square roots radicals
• more words for times add and divide
• how to scale math
• difference quotient with exponents
• absolute value inequalities
glencoe mcgraw hill enrichment order of operations
ti 84 log button
graphing lessons for sixth grade
how to solve linear equations with three variables on calculator | {"url":"https://factoring-polynomials.com/algebra-software-1/saxon-math-test-generator.html","timestamp":"2024-11-07T17:08:25Z","content_type":"text/html","content_length":"84309","record_id":"<urn:uuid:ed128260-c2e4-4b3e-822d-f9c8c448be55>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00444.warc.gz"} |
Permutation Vs Combination: Understanding The Fundamental Differences - LearnAboutMath
Permutation vs Combination: Understanding the Fundamental Differences
Permutation and combination are two mathematical methods applied to determine the number of scenarios in which we can variety items as well as which items are taken in cataloging from a given set
without permutation. While they both target at this, the underlying principles of both languages are disparate to some extent.
Permutation: Arrangements with Order
Permutation refers to a process of the objects’ arrangement in a particular sequence. The order in which these trigrams are arranged is crucial in permutations. As an illustration, if there are three
objects mixed up i.e. A, B, and C, then the orders of choosing two objects out of the three would consist in AB, BA, AC, CA, BC, and CB.
Combination: Selections without Order
In grouping together objects does not require an individual to consider the order of their selection, whereas in combination the order of objects is taken into account but it does not matter which
objects have been chosen. No matter whether in a single element form or in groups, the configuration of elements shall have no consequence. Unlike the permutations that have all the objects, the
combinations have all the objects except the objects that were chosen only once. This results in combinations such as AB, AC and BC without regards to the sequence of the letters.
100 best math trivia questions with answers for kids Are you getting bored? Then here is something interesting
Permutation vs Combination: Key Differences
The distinctive difference between permutation and combination is the presence or lack of being order, which is considered important. Permutation is analla which is to say taking into account the
sequence of arrangement where combination is without any conditions like this. One more essential distinction is that arrangements that repetition may come in with permutation but not with
Examples Illustrating Permutation vs Combination
Let’s consider a scenario where we have a set of four letters: In this privacy policy, I outline A, B, C, and D.
• Permutation Example:
In addition if we want to have 3 letters together respecting the order, the sequence of permutation would be ABC, ABD, ACD, BAC, BAD, BCD, CAB, CAD, CBD, DAB, DAC and DCB.
• Combination Example:
Identifying only two letters without considering the order would lead to combinations with these seven letters: AB, AC, AD, BC, BD, CD, and DD.
Step-by-Step Solved Examples:
Let’s solve a few permutation and combination problems step by step:
• Permutation Example: Problem: How many different ways can the letters in the word “MISSISSIPPI” be arranged?
Solution: Given word: MISSISSIPPI Total letters = 11 (M-1, I-4, S-4, P-2) Using the permutation formula for arrangements of n objects taken r at a time:
Number of arrangements =
Thus, there are 39,916,800 different ways to arrange the letters in the word “MISSISSIPPI”.
• Combination Example: Problem: In a group of 8 people, how many different combinations of 3 can be selected?
Solution: Given: Total people (n) = 8, Selections (r) = 3 Using the combination formula:
Number of combinations =
$=\frac{8\ast 7\ast 6}{3\ast 2\ast 1}$
Thus, there are 56 different combinations of 3 people that can be selected from a group of 8.
Briefly, we can state that permutation and combination are the main ideas of mathematics that differ themselves with specific rules and uses. It is very important, in order to stay away from some of
the issues in areas as probability, statistics and cryptography, to be able to distinguish the different features of these. One brings up the ability of arranging things in different ways, not just
in one way, through the understanding of permutation and combination.
FAQs on properties in math
Indeed, they are usually part of the mechanism to establish expressions for permutations and combinations of selections in various disciplines such as probability, statistics, and cryptography.
Yes, permutation and combination have specific formulas: $P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$and $C\left(n,r\right)=\frac{n!}{r!\left(n-r\right)!}$ respectively.
Do you want to get a more interesting blog? Just click down and read more interesting blogs. | {"url":"https://learnaboutmath.com/permutation-vs-combination/","timestamp":"2024-11-03T18:38:58Z","content_type":"text/html","content_length":"353775","record_id":"<urn:uuid:8053218d-9854-495e-b6ff-03b814d5d006>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00022.warc.gz"} |
A Mathematical Model for Managing the Distribution of Information Flows for MPLS-TE Networks under Critical Conditions
Communications and Network Vol.10 No.02(2018), Article ID:83222,12 pages
A Mathematical Model for Managing the Distribution of Information Flows for MPLS-TE Networks under Critical Conditions
Hani Attar^1, Mohammad Alhihi^1, Mohammad Samour^1, Ahmed A. A. Solyman^2, Shmatkov Sergiy Igorovich^3, Kuchuk Nina Georgievna^3, Fawaz Khalil^4
^1Philadelphia University, Amman, Jordan
^2Communication Department, Modern University of Technology and Information, Cairo, Egypt
^3Karazin Kharkiv National University, Kharkiv, Ukraine
^4The Kharkiv National Technical University of Agriculture Named after Petro Vasylenko, Kharkiv, Ukraine
Copyright © 2018 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
Received: November 25, 2017; Accepted: March 20, 2018; Published: March 23, 2018
The optimal load distribution over a set of independent paths for Multi-Protocol Label Switching for Traffic Engineering (MPLS-TE) networks is regarded as important issue; accordingly, this paper has
developed a mathematical method of optimal procedures for choosing the shortest path. As a criterion for choosing the number of paths, a composite criterion is used that takes into account several
parameters such as total path capacity and maximum path delay. The mathematical analysis of using the developed method is carried out, and the simulation results show that there are a limited number
of the most significant routes that can maximize the composite quality of service indicator, which depends on the network connectivity and the amount of the traffic. The developed technological
proposals allow increasing the utilization factor of the network by 20%.
MPLS-TE, OSPF Protocol, RIP Protocols
1. Introduction
The telecommunication network traffic is regarded as casual and non-stationary, accordingly, the network becomes overloaded in certain elements and areas and less overloaded in others. This leads to
critical situations that require measures to re-distribute the traffic over the network which is not sufficiently investigated, and the available technologies do not provide effective solutions for
network restoration.
In [1] , mathematical model was proposed for Multi-Protocol Label Switching for Traffic Engineering (MPLS-TE) to improve the Quality of Service (QoS) providing the mathematical limits when optimum
traffic distribution is assumed. In [2] , the optimum number of routing paths was investigated for MPLS-TE, as a result, the optimum number of the shortest independent paths was theoretically
determined, while in [3] , the parameters to develop the routing performance are investigated.
In [4] , both of multipath routing and traffic distribution were investigated and optimum solution is proposed to find the shortest path using Dijkstra and Bellman-Ford algorithm. Interesting and
useful comparative analysis for various routing strategies is produced in [5] with the relative proposed framework models.
More important and useful resent published work in this field is shown in [6] [7] and the references there in.
When the network becomes in a critical situation in term of traffic limit and hence the network performance; there are two ways to solve this obstacle which are:
・ Load limiting technique, and
・ Load redistribution technique which takes into account the availability of the network resources.
The load redistribution technique is directed linked to the QoS allowed in the network and usually it is recommended for practical use.
The task of network recovery in critical situations, due to its non-stationarity, can be solved based on the management of the corresponding resources, using the procedure of recursive estimation of
the traffic state. The control algorithm in this case can be constructed using the results of the separation theorem that is with the possibility of a linear solution and the Gaussian nature of the
estimated process of traffic changes. This algorithm is implemented in the form of a sequence of two separate procedures which are: Optimal stochastic estimation of the network state and the
deterministic linear control of network resources, taking into consideration that the obtained stochastic estimation is considered as the control parameter. In this paper, the qualitative
characteristics of the evaluation, the structure of control algorithms are obtained and recommendations are given on the choice of the parameters of these algorithms.
As evaluation algorithms, it is recommended to select Kalman-Bucy filtration procedures that are optimal in the sense of the minimum mean-square error of the estimate. However, the direct use of
these algorithms due to the non-stationary nature of the situation is unacceptable, because the use of model identification and adaptation procedures will lead to an excessively cumbersome task
(curse of dimension) and will require additional channel resources. A more rational solution is to choose a higher rate of processing the observation results, so that the convergence time of the
procedure is significantly less than the interval of the quasi-stationary nature of the estimated process, which is estimated to be parts of seconds. Calculated data on the feasibility of computing
devices of the required productivity are obtained in this paper.
The rest of the paper is arranged to be presented in four sections, where in Section 2, the mechanism of MPLS-TE optimizing load balance is produced. In
Section 3, selecting the information distribution flow method is proposed. In 4, the load control model for the proposed work is shown before concluding the paper work in 5.
2. The Mechanisms of Optimizing the Load Balance in MPLS-TE Networks
The recovery procedures are offered in most modern transport technologies such as the Automatic Protection Switching in SDH, however, only MPLS-TE has a possibility of resource optimization after
network recovery. The applicable types of optimization nowadays are:
1) Periodic optimization;
2) On-demand optimization;
3) Event-based optimization.
In most hardware devices, there is a time interval for optimizing the tunnel. This is done to find a better way to meet traffic restrictions and requirements. In Cisco routers [8] , a default value
of this interval is one hour and can be changed in the range of (0 to 604,800) seconds.
On-demand optimization is used very rarely in practice and is only needed in particular cases, such as, after creating several tunnels on a working network, it is necessary to check whether there is
a better way to service traffic, which guarantees less delay than the existing one.
Load balancing optimization should be performed whenever there is a better way to send traffic with the specified service parameters. Therefore, in case of event-based optimization, the search for
the optimal path is made each time a specified event occurs, such as a new path appears on the network.
Figure 1(a) & Figure 1(b) shows a situation where a failure in the network leads to a non-optimal use of network resources, assuming that all channels in the considered network have the same
bandwidth. In the MPLS-TE network; there are two tunnels: A-B-E and A-B-D. To protect the tunnel A-B-E, a bypass B-D-E was created. This virtual path is in a waiting state and is only used if the
section between B-E is inoperative. In case of failure of the network section B-E, rerouting procedures will switch traffic to the bypass B-D-E. In general, as a result of network recovery after a
failure, using of network resources may not be optimal, because of what some resources will be overloaded, and some will not
Figure 1. Switching to the bypass tunnel in case of failure of the communication channel.
be used at all (Figure 1(b)). Obviously, in this situation, it is necessary to optimize the distribution of network resources, while taking into account the time limitations of the procedure.
It is important to notice that there is no other technique is applied in this network, which makes it possible to improve this technique when applying network coding technique such as used in [9]
[10] [11] [12] to improve the network reliability when one or more baths are out of service.
In fact, [11] [12] provided a solution in the case of a dead channel between two nodes.
3. Selecting the Method of Information Flows Distribution
As explained in Section 2; in the case of critical network operation, it is necessary to carry out the load redistribution from separate sites for a limited and predetermined time. In general, the
problem of optimal distribution of information flows is NP-complete, and its application in the case of a critical operating mode is regarded as difficult problem to solve. More the constructive and
realizable in practice are regarded as rational methods, which fall in two main methods:
・ Traffic limitation method: The method is rather constructive, it is often used, but in many cases, it is unacceptable because of the possible increase in the loss of data packets and the quality
of multimedia services deterioration.
・ Traffic redistribution method: This method is implemented in the presence of redundant paths. Modern networks are built in such a way that unused bypasses are always there, and their resources can
be used in case of overloading the main paths.
The proposed work in this paper applies the second method as it is more constructive in term of the QoS limitation conditions. In order to circumvent the NP-completeness of the traffic redistribution
task; a heuristic algorithm that minimizes network congestion when the specified conditions are met, is proposed.
The idea of the algorithm is as follows: It is intended to use a centralized strategy to manage the whole or a fragment of the network (Figure 2). First, the
Figure 2. Construction of a matrix of given loads in the case of a centralized control strategy.
given values of the load are found, on the basis of which a matrix with a zero diagonal (1) is constructed.
$‖{\stackrel{^}{T}}_{ij}‖=\left[\begin{array}{cccc}0& {\stackrel{^}{T}}_{12}& {\stackrel{^}{T}}_{13}& {\stackrel{^}{T}}_{14}\\ {\stackrel{^}{T}}_{21}& 0& {\stackrel{^}{T}}_{23}& {\stackrel{^}{T}}_
{24}\\ {\stackrel{^}{T}}_{31}& {\stackrel{^}{T}}_{32}& 0& {\stackrel{^}{T}}_{34}\\ {\stackrel{^}{T}}_{41}& {\stackrel{^}{T}}_{42}& {\stackrel{^}{T}}_{43}& 0\end{array}\right]$(1)
Non-diagonal elements are thus the given load of the corresponding communication direction in the network ${\stackrel{^}{T}}_{ij}$ , which is defined as
${\stackrel{^}{T}}_{ij}=\frac{{T}_{ij}\left(t\right)}{{v}_{ij}}$ , (2)
where ${T}_{ij}$ is the corresponding load at time t, ${v}_{ij}$ is the throughput of the communication channel ij. Further, on each subsequent time interval the load is redistributed from the most
loaded channels ij to the unloaded bypasses consisting of two sections: ik and kj. There can be more than two such sections. The proposed algorithm can be used both for the case of a fully connected
network and for a network of arbitrary connectivity. The constraints on connectivity are not fundamental and do not affect the algorithm as a whole. It should be noted that, in the framework of
MPLS-TE technology proposed in [13] [14] [15] where the full connectivity of the network can be achieved on the basis of logical methods, i.e. tunnels (virtual channels) between all inputs and
outputs of distributed elements can be formed, which leads to complete connectivity. The great advantage of this method of redistribution is that it allows not only preventing network congestion, but
also ensures the restoration of the network operability after failures of individual elements and directions of communication.
4. Load Control Model for TCN
To solve load balancing tasks, constant monitoring and control of the load level are carried out, which occurs as follows:
・ Packets sent to the interface ${n}_{n}$ are counted;
・ The obtained values ${n}_{n}$ serve to generate averaged data ${\stackrel{^}{n}}_{n}$ about the load for a certain interval of observation time ${\tau }_{n}\le {\tau }_{кс}$ :
${\stackrel{^}{n}}_{n}=\frac{1}{n}\underset{n=1}{\overset{n}{\sum }}{n}_{n}$ . (3)
The interval ${\tau }_{n}=\sum \Delta {t}_{n}$ should be chosen based on the condition ${\tau }_{n}\le {\tau }_{кс}$ which is the period of traffic quasi-stationary, which is a few seconds. A
recursive estimate, which is more adaptive and consistent with management tasks, can be more adequate in conditions of nonstationarity:
・ Critical mode thresholds are entered, on reaching which re-routing and load re-distribution are implemented.
Since the traffic parameters are monitored with a certain error, the resulting sample ${n}_{n}\left(k\right)$ is subject to statistical processing:
${\stackrel{^}{n}}_{n}\left(k\right)={n}_{n}\left(k\right)±\Delta {n}_{n}\left(k\right)$ . (4)
In addition, the presence of various disturbing random factors such as delays, sampling errors, and no stationary of the load itself, which leads to the fact that the observation results should be
interpreted as a sequence of readings $x\left(k\right)$ observed against a background of random interference ${n}_{n}\left(k\right)=u \left(k\right)$ :
$y\left(k\right)=x\left(k\right)+u \left(k\right)$ , (5)
where the dimension of the observation vector $\mathrm{dim}\left(y\left(k\right)\right)$ corresponds to the number of controlled directions. The level of cumulative errors of random interference $u \
left(k\right)$ is characterized by the power spectral density ${V}_{\gamma }$ . The value ${V}_{\gamma }$ is of a formal nature and determines the level of all the interfering factors
that lead to measurement errors. However, in practice, the relative value $\frac{{V}_{x}}{{V}_{\gamma }}$ is considered as important, where ${V}_{x}$ is the power spectral density of the useful
estimated signal $x\left(k\right)$ . The value $\frac{{V}_{x}}{{V}_{\gamma }}$ is interpreted in standard communication tasks as the signal-to-noise value $\frac{{P}_{c}}{{P}_{ø}}$ . In essence, $x\
left(k\right)$ displays the traffic state, and, given that the network processes the traffic, it’s adequately displayed in the network state.
Evaluation of the process $x\left(k\right)=\left({x}_{1},{x}_{2},\cdots ,{x}_{k}\right)$ can be obtained by the known formula:
$\stackrel{^}{x}\left(k\right)=\frac{1}{k}\underset{k=1}{\overset{k}{\sum }}x\left(k\right)$ . (6)
At the same time, this estimate (6) is effective and unbiased only for stationary processes of ergodic type. To estimate non-stationary traffic parameters, a recursive estimate, obtained by the
stochastic approximation method [16] or, in the general case, by the Kalman-Bucy filtration method [16] , is more suitable. The time interval $\Delta t=t\left(k+1\right)-t\left(k\right)$ between
sample values should be chosen so that our interval of quasi-stationarity ${\tau }_{кс}\gg \Delta t$ . Practice shows that the ratio $\Delta t/{\tau }_{кс}=0.01\cdots 0.1$ is appropriate, while $\
Delta t$ should be $0.05\cdots 0.5$ sec.
Consider the set of nodes in the telecommunications network exchanging information as in Figure 3. Any modern network can function successfully and steadily if the network management system, for
example the TMN system, copes with the current traffic changes. Such control of the structure and state can be performed both by the whole and part of the system. In this case, the management $u\left
(t\right)$ itself is a component of the state $x\left(k\right)$ . A mathematical model that reflects the dynamics of a state can be described by a differential equation [17] :
$\frac{\text{d}x\left(k\right)}{\text{d}t}=F\left(t\right)x\left(t\right)+\underset{j}{\sum }{B}_{j}\left(t\right){u}_{j}\left(t\right)+G\left(t\right)w\left(t\right)$ , (7)
where F(t), B(t), and G(t) are the matrices of state, control, and excitation, respectively, $u\left(t\right)$ is a vector of controllable parameters, and $w\left(t\right)$ is a vector of virtual
noise with a level $E\left[w\left(k\right){w}^{\text{T}}\left(k\right)\right]={V}_{w}$ . Physically, the interpretation of
Figure 3. Structure of centralized management of distributed network elements.
the matrix coefficients occurring in Equation (7) is as follows: F(t) displays the rate of change of state by elements of the matrix; elements of the matrix F(t) are the inverse of the correlation
intervals of random states ${F}_{ij}=1/{\tau }_{кор}$ . G(t) displays the level of random changes of this traffic. For purely deterministic states with constant loads, G(t) = 0. B(t) determines the
level of controlled impacts.
For a discrete system, the analogue of Equation (7), respectively, has the following form:
$x\left(k+1\right)=F\left(k\right)x\left(k\right)+\underset{j}{\sum }{B}_{j}\left(k\right){u}_{j}\left(k\right)+G\left(k\right)w\left(k\right)$ . (8)
The optimal solution, which allows obtaining a real-time recursive estimation satisfying the minimum mean-square deviation criterion shown in Equation (9) is the Kalman-Bucy procedure, which is
presented in Equation (10)
$J=\mathrm{min}E{\left(x\left(k\right)-\stackrel{^}{x}\left(k\right)\right)}^{2}$ , (9)
where $K\left(k\right)={V}_{\stackrel{˜}{x}}\left(k|k-1\right){H}^{\text{T}}{\left(H{V}_{\stackrel{˜}{x}}\left(k|k-1\right)H\left(t\right){V}_{u }\left(k\right)\right)}^{-1}$ and
right)$ is an a posteriori variance of the prediction error, ${V}_{\stackrel{˜}{x}}\left(k\right)$ is an a posteriori variance of the estimation error, and ${V}_{\stackrel{˜}{x}}\left(k\right)=\left
[I-K\left(k\right)H\left(k\right)\right]{V}_{\stackrel{˜}{x}}\left(k|k-1\right)$ . The direct use of the estimate (10) is not always possible for telecommunication networks that are distributed,
since delays in the control loop ${\tau }_{зад}$ can reach values that commensurate with the selected sample intervals $\Delta t$ or even exceed them. Under these conditions, management can be
synthesized according to the forecast only. The forecast value is:
$\stackrel{^}{x}\left(k|k-1\right)=\stackrel{^}{O}\left(k,k-1\right)\stackrel{^}{x}\left(k\right)+\underset{j}{\sum }{B}_{j}{u}_{j}\left(k\right)$ , (11)
where ${B}_{j}$ determines the control value in the j direction of the communication.
The estimation error ${V}_{\stackrel{˜}{x}}\left(k\right)$ and a one-step ahead forecast error are respectively determined by the relations ${V}_{\stackrel{˜}{x}}\left(k\right)=E\left[x\left(k\right)
-\stackrel{^}{x}\left(k\right)\right]{\left[x\left(k+1\right)-\stackrel{^}{x}\left(k\right)\right]}^{\text{T}}$ and ${V}_{\stackrel{˜}{x}}\left(k|k-1\right)=E\left[x\left(k\right)-\stackrel{^}{x}\
left(k|k-1\right)\right]{\left[x\left(k+1\right)-\stackrel{^}{x}\left(k|k-1\right)\right]}^{\text{T}}$ .
It is characteristic that the forecast $\stackrel{^}{x}\left(k|k-1\right)$ is determined at the same steps of discrediting. Obviously, the forecast is always less accurate than the estimate. It
follows from the definition that the errors of this forecast are determined by the values
$\stackrel{^}{O}\left(k,k-1\right)={e}^{\frac{-\Delta t}{{\tau }_{кор}}}$ , (12)
where $\Delta t$ is the step of discrediting, ${\tau }_{кор}$ is the interval of the process correlation $x\left(k\right)$ . It’s obvious that for the less inertial state changes $x\left(t\right)$ ,
for which ${\tau }_{корi}>{\tau }_{корj}$ at the same $\Delta t$ , the forecast error will be greater than for the more inertial ones. Accordingly, the estimate (10) will also be worsened. Figure 4
shows the dependence of the relative forecast error (11) on the delay value in the control loop. Since the forecast accuracy decreases with increasing ${\tau }_{зад}$ , in cases where delays in the
network reach significant values, the control procedures may prove ineffective. In these cases, it is needed to switch to a management that operates on the mean of traffic changes or equips the
network with more resources. In practice, the delay value ${\tau }_{зад}$ commensurate or exceeds the correlation interval ${\tau }_{кор}$ rarely. Note that for time-invariant states $x\left(t\right)
=\text{const}$ the estimate and, accordingly, the forecast can also be obtained asymptotically, with absolute accuracy while $t\to \infty$ . Unfortunately, this situation is not real in communication
Figure 4. Graph of the dependence of the relative forecast error on the delay value in the control loop.
Consider other dependencies related to the accuracy of estimation and prediction. This accuracy can be determined by a ratio of the a priori and a posteriori variance of the estimates ${V}_{\stackrel
{˜}{x}}\left(k\right)/{V}_{w}\left(k\right)$ and is determined by the variance of the process $x\left(k\right)$ itself or by the value ${V}_{w}$ .
At a known ratio of signal-to-noise ( ${P}_{c}/{P}_{ø}$ ) levels in the observation channel, the ratio of the a posteriori and a priori variances is calculated by the formula:
$\frac{{V}_{\stackrel{˜}{x}}\left(k\right)}{{V}_{w}}=\frac{2}{1+\sqrt{1+\frac{{P}_{c}}{{P}_{ø}}}}$ . (13)
It is logical to consider the relation (13) for the case of a steady state when the transient processes are completed, that is when ${V}_{\stackrel{˜}{x}}\left(k\right)\to {V}_{\stackrel{˜}{x}}\left
(\infty \right)$ . The graph of (13) dependence is presented in Figure 5.
Let us find the optimal control law for the $u\left(k\right)$ . It is known from the optimal control theory [17] that the value ${u}_{opt}\left(k\right)$ is found by minimizing the Hamiltonian along
the optimal trajectory. The optimal control for the i-th node is given by the criterion
${J}_{i}=E\left\{\underset{k=0}{\overset{N-1}{\sum }}\left[{x}^{\text{T}}\left(k\right){P}_{i}x\left(k\right)+\underset{j=1}{\overset{M}{\sum }}{u}_{j}^{\text{T}}\left(k\right){Q}_{ij}{u}_{j}\left(k\
right)\right]+{x}^{\text{T}}\left(N\right){P}_{i}x\left(N\right)\right\}$ . (14)
The terms of (14) functional have an important physical interpretation. The last term ensures the minimum state deviations of $x$ from the optimal trajectory at the final instant of time $x\left(N\
right)$ . The first term in square brackets ensures
Figure 5. Graph of the dependence of the relative error of the random process estimation on the signal-to-noise ratio.
the minimization of the mean-square state spreads along the entire path of the system’s motion. The sum of the terms containing ${Q}_{ij}$ minimizes management costs. The last term is important in
the case when one or another type of energy is spent on the management implementation. When these energy inputs are not important, the middle term can be omitted.
Let for each of the M nodes a solution is obtained minimizing (14) exponent, which is a set of strategies $\left\{{g}_{1}^{*},{g}_{2}^{*},\cdots ,{g}_{M}^{*}\right\}$ is obtained:
${J}_{i}\left({g}_{1}^{*},{g}_{2}^{*},\cdots ,{g}_{M}^{*}\right)\le {J}_{j},\forall {g}_{j}$ . (15)
Taking into account (14) criterion, the system control trajectory (8) can be analyzed in reverse time, beginning with the last step ${N}_{1}$ , where ${L}_{i}\left(N\right)={P}_{i}$ :
$\begin{array}{l}{L}_{i}\left(k\right)={P}_{i}+{\stackrel{^}{O}}^{\text{T}}{\left[{\left(I+\underset{j}{\sum }{B}_{j}{B}_{j}^{\text{T}}{L}_{j}\left(k+1\right)\right)}^{-1}\right]}^{\text{T}}\\ \ast \
left[{L}_{i}\left(k+1\right)+\underset{j}{\sum }{L}_{j}\left(k+1\right){B}_{j}{Q}_{ij}{B}_{j}^{\text{T}}{L}_{j}\left(k+1\right)\right]{\left[I+\underset{j}{\sum }{B}_{j}{B}_{j}^{\text{T}}{L}_{j}\left
It can be shown that optimal control is realized in the form of a linear procedure [18] [19] [20] [21] :
${u}_{opt}\left(k\right)=D\left(k\right)\stackrel{^}{x}\left(k\right)$ . (17)
An equation satisfying condition (4.15) exists and is unique [17] if for every step k there is an inverse value in the square brackets of the expression:
$D\left(k\right)=-{B}_{i}^{\text{T}}{L}_{i}\left(k+1\right){\left[I+\underset{j}{\sum }{B}_{j}\left(k\right){B}_{j}^{\text{T}}\left(k\right){L}_{j}\left(k+1\right)\right]}^{-1}F\left(k\right)$ . (18)
It follows from (18) that the coefficient ${D}_{i}\left(k\right)$ does not depend on the results of (8) observation. This can only be when exactly these ${D}_{i}\left(k\right)$ quantities should be
involved in this control procedure, provided that we have complete information on the load status of neighboring nodes and use closed-loop control strategies without delays in all directions. Thus,
the costs ${J}_{i}\left(k\right)$ are completely determined by the ${V}_{w}/{V}_{x}$ relation, respectively characterizing the a priori variance of the useful estimated signal and the inertial i-th
direction, which is characterized by the value ${\stackrel{^}{O}}_{i}\left(k,k-1\right)$ enclosed in the state matrix values $\stackrel{^}{O}\left(k\right)$ .
In our linear model (8), leading to the solution of (17), disturbing factors are changes in the real-time traffic. It is obvious that a random process $x\left(t\right)$ or a discrete-time one $x\left
(k\right)$ can be approximated by the normal probability distribution law $w\left(x\right)\to N\left[{m}_{x},{\sigma }_{x}^{2}\right]$ , since it itself is formed as the sum of many independent
factors―communication requests. This allows us to apply the separation theorem [18] to the control implementation. Its essence is that the optimal control itself (16) is a deterministic procedure
which includes an optimal mean square estimate $\stackrel{^}{x}\left(t\right)$ .
This number depends on the network connectivity and traffic amount and allows you to maximize the composite quality-of-service indicator. The developed technological proposals allow increasing the
network utilization rate by 20%.
5. Conclusion
The procedure for estimating the amount of the redistributed load is obtained, which guarantees the prevention of overloading of network nodes. As the redistribution coefficient, the value 0.2 of the
reduced flux is justified. The procedure is a patio-temporal organization of calculations, where redistribution is performed at all ( $i-x$ ) directions of communication at each $k-M$ step, and then
a forecast is given for the $k+1$ step. And there is an optimal ratio of redistributed flows across all sections of the network. The state of the controlled section in the transient state is
analyzed. Recommendations are given for choosing the sampling step $\Delta t$ , which can be within the limits. The proposed heuristic algorithm for the redistribution of information flows can be
used for various network configurations
Cite this paper
Attar, H., Alhihi, M., Samour, M., Solyman, A.A.A., Igorovich, S.S., Georgievna, K.N. and Khalil, F. (2018) A Mathematical Model for Managing the Distribution of Information Flows for MPLS-TE
Networks under Critical Conditions. Communications and Network, 10, 31-42. https://doi.org/10.4236/cn.2018.102003 | {"url":"https://file.scirp.org/Html/1-6101667_83222.htm","timestamp":"2024-11-07T00:25:02Z","content_type":"application/xhtml+xml","content_length":"122140","record_id":"<urn:uuid:10e6229b-805e-44b5-af84-e6752fedcfbf>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00102.warc.gz"} |
Association plot
ggassoc_assocplot {descriptio} R Documentation
Association plot
For a cross-tabulation, plots measures of local association with bars of varying height and width, using ggplot2.
ggassoc_assocplot(data, mapping, measure = "std.residuals",
limits = NULL, sort = "none",
na.rm = FALSE, na.value = "NA",
colors = NULL, direction = 1, legend = "right")
data dataset to use for plot
mapping aesthetics being used. x and y are required, weight can also be specified.
character. The measure of association used to fill the rectangles. Can be "phi" for phi coefficient, "or" for odds ratios, "std.residuals" (default) for standardized (i.e. Pearson)
measure residuals, "adj.residuals" for adjusted standardized residuals or "pem" for local percentages of maximum deviation from independence.
limits a numeric vector of length two providing limits of the scale. If NULL (default), the limits are automatically adjusted to the data.
character. If "both", rows and columns are sorted according to the first factor of a correspondence analysis of the contingency table. If "x", only rows are sorted. If "y", only columns are
sort sorted. If "none" (default), no sorting is done.
na.rm logical, indicating whether NA values should be silently removed before the computation proceeds. If FALSE (default), an additional level is added to the variables (see na.value argument).
na.value character. Name of the level for NA category. Default is "NA". Only used if na.rm = FALSE.
colors vector of colors that will be interpolated to produce a color gradient. If NULL (default), the "Temps" palette from rcartocolors package is used.
direction Sets the order of colours in the scale. If 1, the default, colours are as output by RColorBrewer::brewer.pal(). If -1, the order of colours is reversed.
legend the position of legend ("none", "left", "right", "bottom", "top"). If "none", no legend is displayed.
The measure of local association measures how much each combination of categories of x and y is over/under-represented.
The bars vary in width according to the square root of the expected frequency. They vary in height and color shading according to the measure of association. If the measure chosen is "std.residuals"
(Pearson's residuals), as in the original association plot from Cohen and Friendly, the area of the bars is proportional to the difference in observed and expected frequencies.
This function can be used as a high-level plot with ggduo and ggpairs functions of the GGally package.
a ggplot object
Nicolas Robette
Cohen, A. (1980), On the graphical display of the significant components in a two-way contingency table. Communications in Statistics—Theory and Methods, 9, 1025–1041. doi:10.1080/03610928008827940.
Friendly, M. (1992), Graphical methods for categorical data. SAS User Group International Conference Proceedings, 17, 190–200. http://datavis.ca/papers/sugi/sugi17.pdf
See Also
assoc.twocat, phi.table, catdesc, assoc.yx, darma, ggassoc_crosstab, ggpairs
ggassoc_assocplot(data=Movies, mapping=ggplot2::aes(Country, Genre))
version 1.3 | {"url":"https://search.r-project.org/CRAN/refmans/descriptio/html/ggassoc_assocplot.html","timestamp":"2024-11-05T23:09:32Z","content_type":"text/html","content_length":"6222","record_id":"<urn:uuid:153e8be6-f7ec-4ec5-a226-de236d2e2597>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00161.warc.gz"} |
What our customers say...
Thousands of users are using our software to conquer their algebra homework. Here are some of their experiences:
I am still learning how to use it, but what I have learned is a great. Thanks!
Camila Denton, NJ
Thank you very much for your help!!!!! The program works just as was stated. This program is a priceless tool and I feel that every student should own a copy. The price is incredible. Again, I
appreciate all of your help.
Jessica Flores, FL
No Problems, this new program is very easy to use and to understand. It is a good program, I wish you all the best. Thanks!
Helen Dillanueva, VA
I recommend this program to every student that comes in my class. Since I started this, I have noticed a dramatic improvement.
Michael Lily, MO
Search phrases used on 2010-03-04:
Students struggling with all kinds of algebra problems find out that our software is a life-saver. Here are the search phrases that today's searchers used to find our site. Can you find yours among
• wwwfree GED & College Prep Class/ brooklyn
• graphing calculator online
• math percents algebra
• logic triangle addition worksheet
• complete the sqaure
• free Practice Math Problems sheets 8th grade
• T184 programming statistics
• probability on ti89
• pratice with factoring
• real life examples from introductory algebra
• multiplying trinomial fractions
• Simple way of Dividing polynomials
• "greatest common divisor" calculate
• algebra answer
• plot hyperbolas in TI 83 plus
• cost accounting books
• how to solve nonlinear differential equations my using Maltlab
• Boolean Function simplifications TI 89
• MATH SHEETS FOR 3RD GRADE
• teaching 8th graders how to do pre algebra
• math worksheets for fall fifth grader
• adding and subtracting polynomials worksheet
• matlab take derivative and solve equation
• geometry trivia question and answer
• powerpoints and ordering fraction
• free GRADE 9 worksheets
• worksheets on adding more than one term at a time for pre algebta
• convert mixed number to decimal
• online geometry workbook answers
• how to solve a difference quotient
• least to greatest decimals
• examples for adding multiple subtract and divide octal numbers
• algebra formula sheets
• what is the difference between evaluation and simplification of an expression
• ti-84 plus emulator
• Formula to Convert Decimals to Fractions
• radicals cheat sheets
• solving systems and quadratic intercept
• simplify radical expressions answers
• finding geometric mean on t1-83 plus
• free parabola worksheets
• square and cube roots formula
• distributive property with exponents
• learn to calculate
• ready to teach pre-algebra videos
• why quadratic set to zero?
• ALEGBRA FOR DUMMIES
• work sheet for add two number
• graphing systems of differential equations on ti 89
• ax+by=c
• 3rd degree equation excel
• how to solve second order differential equation
• completing the square questions
• "California Standard test" + practice
• Balancing an 8th grade chemical equation
• manths riddles puzles with answer
• Free Square Root Chart
• ti 84 plus intermediate algebra programs
• Abstract algebra homework help
• lesson compare & order fractions
• decimals yr 7 worksheets
• Highest Common Factor - Daily uses
• radical calculator
• holt rinehart and winston algebra 2 2004 workbook
• "Abstract Algebra solved problems"
• how to solve parabola expression
• limit to infinity calculator online
• prepare me for Albegra 1
• formula ratio
• free math tutorial software
• quadratic equations substitution method
• solving single variable equations 2 step equations calculator
• scale factor print worksheet
• grade 6 instructions on multiplying & dividing with decimals
• half life program for ti 84 calculators
• solve simultaneous equations free program
• long division problems "6th grade" "sample problems"
• Algebrator Calculator
• how to cube root on ti 83 plus
• adding and dividing with integers
• simple fractions for 6th grade free worksheet
• +worksheet +tree diagrams combinations algebra
• specified variable
• simplifying fractions with exponents | {"url":"https://softmath.com/algebra-help/real-numbers-calculator.html","timestamp":"2024-11-10T04:41:58Z","content_type":"text/html","content_length":"35054","record_id":"<urn:uuid:b4cf0895-f7e9-48bc-a702-c8b3e59c98d8>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00508.warc.gz"} |
Differential Amplifier Circuit using Transistors | Design Calculations
Differential Amplifier Circuit using Transistors:
The Differential Amplifier Circuit using Transistors is widely applied in integrated circuitry, because it has both good bias stability and good voltage gain without the use of large bypass
capacitors. Differential amplifiers can also be constructed as discrete component circuits.
Figure 12-31(a) shows that a basic Differential Amplifier Circuit using Transistors consists of two voltage divider bias circuits with a single emitter resistor. The circuit is also known as an
emitter-coupled amplifier, because the transistors are coupled at the emitter terminals, If transistors Q[1] and Q[2] are assumed to be identical in all respects, and if V[B1] = V[B2], then the
emitter currents are equal, and the total emitter current is,
Like the emitter current in a single-transistor voltage divider bias circuit, I[E] in the differential amplifier remains virtually constant regardless of the transistor h[FE] value. This results in,
I[E1], I[E2], I[C1], and I[C2]Â all remaining substantially constant, and the constant collector current levels keep V[C1Â ]and V[C2] stable. So, the differential amplifier has the same excellent
bias stability as a single-transistor voltage divider bias circuit.
The circuit of a Differential Amplifier Circuit using Transistors using a plus-minus supply is shown in Fig. 12-31(b). In this case, the voltage across the emitter resistor is (V[EE] – V[BE]), as
The base resistors (R[B1] and R[B2]) are included to bias the transistor bases to ground while offering an acceptable input resistance to a signal applied to one of the bases. The transistor emitter
currents (I[E1] and I[E2])Â are exactly equal only if the devices are perfectly matched. To allow for some differences in transistor parameters, a small-value potentiometer (R[EE]) is sometimes
included between the emitters, ( see Fig. 12-32). Adjustment of R[EE] increases the resistance in series with the emitter of one transistor, and reduces the emitter resistance for the other
transistor. This reduces the I[E] for one transistor and increases it for the other, while the total emitter current remains constant.
AC Operation:
Consider what happens when the at input voltage (v[i]) at the base of Q[1] is positive-going, as illustrated in Fig. 12-33. Q[1] emitter current (I[E1]) increases. Also, I[E2]Â decreases, because the
total emitter current (I[E1]Â + I[E2]) remains constant. This means that I[C1Â ]increases and I[C2][Â ]decreases, and consequently, V[C1] falls and V[C2] rises, as shown. So, the ac output voltage at
Q[1] collector is in anti-phase to v[i] at Q[1] base, and the output at Q[2] collector is in phase with v[i].
Voltage Gain:
The voltage gain of a single-stage amplifier with an unbypassed emitter resistor and no external load is given by
Referring to Fig. 12-34, it is seen that the resistance looking into the emitter of Q[2] is h[ib], so h[ib]||R[E]Â behaves like an unbypassed resistor in series with the emitter of Q[1]. Neglecting R
[E] because it is very much larger than h[ib], the voltage gain from the base of Q[1]Â to its collector is,
this reduces to,
Equation 12-24 gives the voltage gain from one input terminal to one output of a differential amplifier. It is seen to be half the voltage gain of a similar single-transistor CE amplifier with R[E]
bypassed; but note that the differential amplifier requires no bypass capacitor. This is an important advantage, because bypass capacitors are usually large and expensive.
Another way to contemplate the operation of the Differential Amplifier Circuit using Transistors is to think of the input voltage being equally divided between Q[1] base-emitter and Q[2]
base-emitter. This is illustrated in Fig. 12-35 where it is seen that (for a positive-going input) v[i]/2 is applied positive on the base of Q[1], while the other half of v[i] appears positive on the
emitter of Q[2]. Thus, for v[i] at Q[1B], transistor Q[1]Â behaves as a common-emitter circuit, and because Q[2] receives the input at its emitter, Q[2] behaves as a common-base circuit.
Input and Output Impedances:
The input impedance at the base of a CE circuit with an unbypassed emitter resistor is,
Referring to Fig. 12-34, the differential amplifier has. h[ib]||R[E] as an unbypassed resistor in series with the emitter of Q[1]. Neglecting R[E] (because R[E] ≫ h[ib]). The input resistance at
Q[1B] is,
This reduces to,
Note that there are usually bias resistors in parallel with Z[b], so that the circuit input impedance is
As in the case of CE and CB circuits, the output impedance at the transistor collector terminals is given by
DC Amplification:
When one transistor base is grounded in a Differential Amplifier Circuit using Transistors, and an input is applied to the other one, as already discussed, v[i] is amplified to produce the outputs at
the collector terminals. In this case v[i] is the voltage difference between the two base terminals. Figure 12-36 shows a differential amplifier with dc input voltages V[i1] and V[i2] applied to the
transistor bases. If the voltage, gain from the base to the collector is A[v], the dc voltage changes at the collectors are;
It is seen that the differential amplifier can be employed as a direct-coupled amplifier, or dc amplifier. The term difference amplifier is also used for this circuit.
Design Calculations:
Design procedures for a Differential Amplifier Circuit using Transistors are similar to those for voltage divider bias circuits. Because there is no bypass capacitor in a differential amplifier, one
of the coupling capacitors determines the circuit lower cutoff frequency (f[1]). The capacitor with the smallest resistance in series with it is normally the largest capacitor, and in the case of a
differential amplifier this is usually the input coupling capacitor. So, the input coupling capacitor determines the circuit lower cut-off frequency.
Consider the capacitor-coupled differential amplifier in Fig. 12-Â37. The circuit uses a plus-minus supply, and a single collector resistor (R[C]). No output is taken from Q[1] collector, so there
is no need for a collector resistor. R[C] is selected in the usual way for a small-signal amplifier; R[C] ≪ R[L]. The collector-emitter voltage should be a minimum of 3 V, as always. Then, I[C]
is calculated from R[C] and the selected voltage drop across R[C].
The total emitter current is determined as,
The base bias resistors are determined by,
As discussed, capacitor C[1] sets the lower cutoff frequency. So.
and C[2] is determined by, | {"url":"https://www.eeeguide.com/differential-amplifier-circuit-using-transistors/","timestamp":"2024-11-14T03:50:25Z","content_type":"text/html","content_length":"234933","record_id":"<urn:uuid:2878c7af-ca8c-47bc-93ff-6105d04ca07c>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00322.warc.gz"} |
Falling into a black hole could wipe out your past
If you ever fell into a black hole, your body would likely get ripped into shreds and ‘spaghettified’.
At least that’s the theory put forward by most physicists today.
But a new study is challenging that claim by suggesting there may be some black holes that you could survive – although doing so may put you into a strange reality.
These black holes would destroy your past life and trap you in a parallel universe with an infinite number of possible futures.
This is because the universe on the other side would not be governed by the rules of cause and effect that apply in ours.
As a result you could ‘live forever’, researchers claim.
A mathematician made the discovery after crunching the numbers on a particular type of black hole with an electrical charge, called a Reissner-Nordström-de Sitter black holes (artist’s impression)
A mathematician from the University of California, Berkeley, made the discovery after crunching the numbers on a particular type of black hole with an electrical charge.
In the real world, your past determines your future and this determinism rules the laws of physics.
This means that the physical laws of the universe do not allow for more than one possible future.
If a scientist knew exactly how the universe began, they could theoretically calculate what will happen for the rest of time and all of space.
UC Berkeley postdoctoral fellow Peter Hintz found that, for something known as ‘Reissner-Nordström-de Sitter’ black holes, this determinism does not apply.
If a space traveller were able to venture into one of these relatively benign black holes, they may be able to survive the experience.
This would give them passage from our deterministic world into a non-deterministic black hole and, in theory, out the other side.
If they were able to avoid the black hole’s infinitely dense singularity, they could emerge into another universe on the other side.
If a space traveller were able to venture into one of these relatively benign black holes, they may be able to survive the experience. The graphic shows a space-time diagram of the gravitational
collapse of a charged spherical star to form a charged black hole
What would happen next is unknown, as in a non-deterministic universe the relationship between cause and effect would no longer exist.
Any and every outcome of everything that is, was and will be possible could exist at the same time.
This strange phenomenon is a quirk of Albert Einstein’s general theory of relativity which, for the past century, has been the standard model used to explain the way gravity works.
‘Normally in physics, initial conditions and the laws of physics are supposed to fully determine what happens to any physical system,’ said Robert Mann, Professor of physics and applied mathematics
at the University of Waterloo, Canada, who was not involved with the study.
Black holes are strange objects in the universe that get their name from the fact that nothing can escape their gravity, not even light.
If you venture too close and cross the so-called event horizon, the point from which no light can escape, you will also be trapped or destroyed.
For small black holes, you would never survive such a close approach anyway.
The tidal forces close to the event horizon are enough to stretch any matter until it’s just a string of atoms, in a process physicists call ‘spaghettification’.
But for large black holes, like the supermassive objects at the cores of galaxies like the Milky Way, which weigh tens of millions if not billions of times the mass of a star, crossing the event
horizon would be uneventful.
Because it should be possible to survive the transition from our world to the black hole world, physicists and mathematicians have long wondered what that world would look like.
They have turned to Einstein’s equations of general relativity to predict the world inside a black hole.
These equations work well until an observer reaches the centre or singularity, where, in theoretical calculations, the curvature of space-time becomes infinite.
‘However general relativity doesn’t have this feature, curiously enough.
‘If I give you an initial distribution of matter and energy over the entire universe, the equations of general relativity in general will not predict the entire future of the space-time.’
Professor Hintz studied a specific type of non-rotating black hole, which have a so-called Cauchy horizon within their event horizon.
Albert Einstein (pictured) published his General Theory of Relativity in 1915
In 1905, Albert Einstein determined that the laws of physics are the same for all non-accelerating observers, and that the speed of light in a vacuum was independent of the motion of all observers –
known as the theory of special relativity.
This groundbreaking work introduced a new framework for all of physics, and proposed new concepts of space and time.
He then spent 10 years trying to include acceleration in the theory, finally publishing his theory of general relativity in 1915.
This determined that massive objects cause a distortion in space-time, which is felt as gravity.
At its simplest, it can be thought of as a giant rubber sheet with a bowling ball in the centre.
Pictured is the original historical documents related to Einstein’s prediction of the existence of gravitational waves, shown at the Hebrew university in Jerusalem
As the ball warps the sheet, a planet bends the fabric of space-time, creating the force that we feel as gravity.
Any object that comes near to the body falls towards it because of the effect.
Einstein predicted that if two massive bodies came together it would create such a huge ripple in space time that it should be detectable on Earth.
It was most recently demonstrated in the hit film film Interstellar.
In a segment that saw the crew visit a planet which fell within the gravitational grasp of a huge black hole, the event caused time to slow down massively.
Crew members on the planet barely aged while those on the ship were decades older on their return.
Black holes (artists’s impression) are thought to be regions of space so dense that they trap or even destroy all matter, but a new study suggests that visiting one might just be possible. Experts
theorised that crossing the threshold of one is not as impossible as was thought
The Cauchy horizon is the spot where determinism breaks down, where the past no longer determines the future.
Physicists have argued that no observer could ever pass through the Cauchy horizon point because they would be annihilated.
As an observer approaches the Cauchy horizon time slows down, since clocks tick slower in a strong gravitational field, they argue.
As light, gravitational waves and anything else encountering the black hole fall inevitably toward the Cauchy horizon, an observer also falling inward would eventually see all this energy barrelling
in on them at the same time.
In effect, all the energy the black hole sees over the lifetime of the universe would hit the Cauchy horizon at the same time, blasting into oblivion any observer who made it that far.
Dr Hintz’s calculations uncovered an exception to this rule with Reissner-Nordström-de Sitter black holes.
In a written statement, he said: ‘No physicist is going to travel into a black hole and measure it. This is a math question.
‘But from that point of view, this makes Einstein’s equations mathematically more interesting.
‘This is a question one can really only study mathematically, but it has physical, almost philosophical implications, which makes it very cool.
The downside to this voyage is that it would destroy your past life and trap you in a parallel universe with an infinite number of possible futures. This image show’s an artist’s impression of a
black hole scientists expect to observe in 2018
‘There are some exact solutions of Einstein’s equations that are perfectly smooth, with no kinks, no tidal forces going to infinity, where everything is perfectly well behaved up to this Cauchy
horizon and beyond. After that, all bets are off.’
Dr Hintz’s equations only work because the universe is expanding at an increasing rate.
Because space-time is being increasingly pulled apart, much of the universe on the other side of the black hole will not affect it at all.
As energy can’t travel faster than the speed of light, only matter and energy which is within the black hole’s observable horizon will be pulled in over its lifetime.
In this scenario, the expansion of the universe would counteract the amplification caused by time dilation inside the black hole that would appear to cause all matter to hit the observer in one go.
For certain situations, such as inReissner-Nordström-de Sitter black holes, this stretching of space-time would cancel the time dilation entirely, allowing a traveller to pass through unharmed.
The full findings of the study were published in the journal Physical Review Letters. | {"url":"https://expressdigest.com/falling-into-a-black-hole-could-wipe-out-your-past/","timestamp":"2024-11-03T02:46:34Z","content_type":"text/html","content_length":"153831","record_id":"<urn:uuid:f9958354-cee4-4571-8b76-11c1a9d9bd71>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00504.warc.gz"} |
Rectangles and Right Triangles
Can you find a rectangle whose perimeter equals its area?
I’ll explain one way to solve this puzzle below.
Allergy warning: this product contains algebra. May contain traces of number theory.
Let’s begin. If the sides of the rectangle are A and B, let’s call the area and perimeter S. This gives two equations:
• The area equals S. That means AB=S.
• The perimeter is also S. That means 2A+2B=S.
Now, I’m going to change this pair of equations into a single equation, by substituting B away.
• The equation for the perimeter can be rearranged to give B = S/2 – A
• If I substitute this into the equation for the area, I get A(S/2-A) = S.
• This equation can be rearranged into a quadratic equation for A, namely 2A^2 – SA + 2S = 0.
There’s no obvious way to factorise this, so I’ll fall back on the quadratic formula. There will be two solutions for A. Whichever one I pick, the other one will be B. This is the bit where I wish I
could type math easily into my blog posts.
• So, A = (S + sqrt(S^2 -16S))/4
• In other words, A = S/4 + sqrt(S^2 -16S)/4
It would be nice if A was a whole number, or at least a fraction. Unfortunately, just picking random values for S rarely makes this happen. So, our rectangle puzzle has become a square number puzzle
– how can we find values of S that make S^2 -16S a square number?
So, let’s let S^2 -16S be N^2. Add 64 to both sides of this, and the left hand side factorizes:
• If S^2 -16S = N^2, then
• S^2 -16S+64 = N^2+8^2, so
• N^2+8^2 = (S -8)^2.
Hmm. Two square numbers, adding up to give a third square number. Where have I seen that before? Hey, that’s Pythagoras’s theorem about right angled triangles! Suppose I have a right-angled triangle,
with sides P, Q and R. Then,
• P^2+Q^2 = R^2. If I divide this all by Q^2, I’ll get
• (P/Q)^2+1^2 = (R/Q)^2. Now, I’ll multiply this by 8^2., to get
• (8P/Q)^2+8^2 = (8R/Q)^2. Now, I’ll rename 8P/Q and 8R/Q. I’ll let 8P/Q be N, and 8R/Q will be S-8. Then,
• N^2+8^2 = (S -8)^2, which is just the equation I need to solve my rectangle puzzle.
So, any pythagorean triplet – any at all – gives me a solution to my rectangle puzzle.
• The pythagorean triplet P, Q, R gives N=8P/Q and S-8=8R/Q.
• Then, A = (S + N)/4 and B = (S – N)/4.
• In short, A = (8R/Q + 8 + 8P/Q)/4 and B = (8R/Q + 8 – 8P/Q)/4.
• These can be simplified: A = 2(R + Q – P)/Q and B = 2(R + Q – R)/Q
Let’s try this! If P=3, Q=4 and R=5, I get A = 2(5 + 4 + 3)/4 and B = 2(5 + 4 – 3)/4, that is A=6, B=3. If a rectangle has sides 6 and 3, sure enough, the area and perimeter are both 18.
I’ll try it again, with P=12, Q=5 and R=13 this time. Then, I get A=2(13 + 5 + 12)/5 and B=2(13+5-12)/5, so A=12 and B=2.4. Then, the area and perimeter are both 28.8.
You try it now, again with the 3-4-5 right triangle, but this time use P=4, Q=3 and R=5. You should get a rectangle with area (and perimeter) equal to 64/3. Then try it with a few other pythagorean
This little bit of algebra has given us a puzzle solution factory: given a pythagorean triangle, I can find a rectangle whose area and perimeter are equal.
In my opinion, that’s already a nice bit of math magic. It gets even nicer, though, but that’s a story for next time.
2 thoughts on “Rectangles and Right Triangles”
1. Good brain workouts! | {"url":"https://www.dr-mikes-math-games-for-kids.com/blog/2013/10/rectangles-and-right-triangles/","timestamp":"2024-11-05T02:46:58Z","content_type":"text/html","content_length":"47302","record_id":"<urn:uuid:f99ec4ad-9947-4618-8f83-7c4b3b01a16b>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00664.warc.gz"} |
Doppler Effect • Frequency shift when sound source and/or listener
Doppler Effect •
Frequency shift when sound source and/or listener is moving When source is moving towards listener pitch appears higher When source is moving away from listener, pitch appears lower Example: Plane flying overhead, https://www.youtube.com/watch?v=eo_owZ2UK7E Applications: Weather radar: Developed in 1988, the Doppler shift is used to determine the radial velocity component of a target. It is also used to determine rotations in wind, which predicts severe weather. The frequency shift is very small, so the Doppler radar actually uses a phase shift. Echocardiography: Ultrasound and Doppler technology are used to visualize the structure of the heart. Areas of the heart that have a high and low blood velocity can be seen in different hues of the resulting picture. The Doppler shift can be used to determine how fast an object is moving, and is the basis for radar guns. Most radar guns use microwaves, but we will look at a simple application using sound waves. The shift in frequency is determined as: v vs
f f 0
v vs
where f0 is the initial frequncy, f′ is the shifted frequency, v is the speed of sound (usually taken as 767 mph), and vs is the speed of the object. Using some fun algebra, the speed can be written in terms of f0 and the frequency shift, (f′‐ f0). vs
f f0
v 2 f0 f f0
The instrument measures the beat frequency, (f′‐ f0), and from that calculates the speed of the object. We have a rational function, where the independent variable is (f′‐ f0) and the dependent variable is vs. The function is ploteed on the next page for a frequency of f0 = 5000 Hz and v= 767 mph. 140
vs (mph)
Speed vs. Frequency Shift
(f'‐f0), Hz
f f0 v
2 f0 f f0
This graph looks approximately linear. Can you explain why? What do you think will happen as the frequency shift continues to increase? 700
vs (mph)
Speed vs. Frequency Shift
f f0 v
2 f0 f f0
(f'‐f0), Hz | {"url":"https://studylib.net/doc/8727044/doppler-effect-%E2%80%A2-frequency-shift-when-sound-source-and-or...","timestamp":"2024-11-09T10:01:07Z","content_type":"text/html","content_length":"59736","record_id":"<urn:uuid:258c194b-be39-4e47-bfc7-79cb4fbee7cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00163.warc.gz"} |
ACM Other ConferencesQuerying a Matrix Through Matrix-Vector Products
We consider algorithms with access to an unknown matrix M in F^{n x d} via matrix-vector products, namely, the algorithm chooses vectors v^1, ..., v^q, and observes Mv^1, ..., Mv^q. Here the v^i can
be randomized as well as chosen adaptively as a function of Mv^1, ..., Mv^{i-1}. Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as
connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three
broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and
norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute
whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus
algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show
separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph.
Surprisingly, this fundamental model does not appear to have been studied on its own, and we believe a thorough investigation of problems in this model would be beneficial to a number of different
application areas. | {"url":"https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.94/metadata/acm-xml","timestamp":"2024-11-05T16:30:32Z","content_type":"application/xml","content_length":"16669","record_id":"<urn:uuid:31294a3c-7ea6-45de-9c13-a8cdd3d4df83>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00733.warc.gz"} |
Wiring Diagram Symbols Chart
Wiring Diagram Symbols Chart
In order to represent the various components used in the diagram electrical symbols are used. Basics 8 aov elementary block diagram.
Standardized Wiring Diagram And Schematic Symbols April 1955 Popular Electronics Electrical Symbols Electrical Wiring Home Electrical Wiring
Electrical symbols virtually represent the components of electrical and electronic circuits.
Wiring diagram symbols chart. This article shows many of the frequently used electrical symbols for drawing electrical diagrams. Basics 13 valve limit switch legend. This article gives some of the
frequently used symbols for drawing the circuits.
They show the diameter of each wire using a label placed at some point along side its drawn line 1 0 8. Basics 10 480 v pump schematic. There are many electrical and electronic schematic symbols are
used to signify basic electronic or electrical device.
Wiring diagrams use simplified symbols to represent switches lights outlets etc. Here s a printable electrical symbols chart for your reference when preparing circuit diagrams. Electrical symbols
electronic circuit symbols of schematic diagram resistor capacitor inductor relay switch wire ground diode led transistor power.
Basics 11 mov schematic with block included basics 12 12 208 vac panel diagram. There are several other electrical wiring symbols used in residential and commercial wiring but the above list of
symbols are the important ones. Though these standard symbols are simplified the function descriptions can make you understand clearly.
There are some standard symbols to represent the components in a circuits. Basics 9 4 16 kv pump schematic. Here are charts to help you to identify symbols on electrical schematics these electrical
schematic symbols will help you to identify parts when working with an electrical schematic electronics symbols for schematics and wiring diagrams are mostly universal with a few of the symbols that
may look different if reading other types of schematics.
The schematic symbols for most major circuit diagrams can be found in this following images. Here is a standard wiring symbol legend showing a detailed documentation of common symbols that are used
in wiring diagrams home wiring plans and electrical wiring blueprints. Basics 7 4 16 kv 3 line diagram.
Printable chart of electrical symbols with their meanings. Automotive wiring diagrams and electrical symbols at first glance the repair diagram may not convey how the wires use many colors and
diameters. Circuit diagrams provide the component layout in any circuit.
These electrical and electronic circuit symbols are generally used for drawing schematic diagram. Basics 14 aov schematic with block included basics 15 wiring or connection. These are mostly we used
for draw circuit diagrams.
Printable Wiring Diagram Symbols Wiring Diagramwiring Diagram Symbols For Display Asyaunited De Electrical Symbols Electrical Wiring Diagram Electrical Diagram
Schematic Symbols Rough Interpolatable Electrical Symbols Electrical Wiring Diagram Electronics Basics
Electrical Wiring Diagrams Symbols Chart Diagram Electrical Wiring Diagram Electrical Diagram Electrical Circuit Diagram
Electrical Schematic Symbols Electrical Schematic Symbols Electrical Symbols Electrical Wiring Diagram
Gm Wiring Diagram Legend Http Bookingritzcarlton Info Gm Wiring Diagram Legend Electrical Schematic Symbols Circuit Diagram Electrical Symbols
Automotive Wiring Diagram Symbols Electrical Wiring Diagram Electrical Symbols Electrical Diagram
Elec 243 Tables Electronic Schematics Electrical Wiring Diagram Electrical Schematic Symbols
Hvac Wiring Diagram Symbols Tamahuproject Org For Automotive Electrical Wiring Diagram Symbols Electrical Circuit Diagram
Symbols Stunning European Wiring Diagram Symbols How German Schematic Electrical For Automotive Electrical Schematic Symbols Circuit Diagram Electrical Symbols
Unique Household Electric Fan Wiring Diagram Diagram Diagramsample Diagramtemplate Wiringdi Electrical Wiring Diagram Electrical Diagram Electrical Symbols
Wiring Diagram Symbols Legend Bookingritzcarlton Info Electronics Circuit Electrical Schematic Symbols Electrical Symbols
Circuit Symbols Electrical Engineering Projects Diy Electronics Electrical Circuit Symbols
List Of Electrical Symbol And Fuction Drawing Chart Electrical Wiring Diagram Electrical Symbols Symbols
Electrical Symbols On Wiring And Schematic Diagrams Electrical Symbols Electrical Wiring Diagram Electrical Circuit Diagram
Electrical And Electronics Symbols Electrical Symbols Electrical Diagram Electrical Wiring Diagram
Wiring Diagram Symbols Chart Http Bookingritzcarlton Info Wiring Diagram Symbols Chart Electrical Symbols Circuit Diagram Electrical Wiring Diagram
Perfect Wiring Diagram Symbols Chart Automotive Wiring Diagrams Symbols Explained Elect Electrical Wiring Diagram Electrical Circuit Diagram Electrical Diagram
Inca Aztec Tattoo Designs Gallery Inca Tattoo Ideas Incatattoos Incatattoodesigns Incatattooid In 2020 Electronic Schematics Electronics Basics Electric Circuit | {"url":"https://easywiring.info/2023/11/04/wiring-diagram-symbols-chart/","timestamp":"2024-11-07T09:45:07Z","content_type":"text/html","content_length":"211786","record_id":"<urn:uuid:9a82952a-1eee-49e7-9f9f-7d8ffd01b76c>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00767.warc.gz"} |