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6600 | https://link.springer.com/article/10.1007/s10957-020-01711-z | Parameter-Dependent Stochastic Optimal Control in Finite Discrete Time | Journal of Optimization Theory and Applications
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Journal of Optimization Theory and Applications
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Parameter-Dependent Stochastic Optimal Control in Finite Discrete Time
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Published: 17 July 2020
Volume 186,pages 644–666, (2020)
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Journal of Optimization Theory and ApplicationsAims and scopeSubmit manuscript
Parameter-Dependent Stochastic Optimal Control in Finite Discrete Time
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Asgar Jamneshan1,
Michael Kupper2&
José Miguel Zapata-García2
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4 Citations
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Abstract
We prove a general existence result in stochastic optimal control in discrete time, where controls, taking values in conditional metric spaces, depend on the current information and past decisions. The general form of the problem lies beyond the scope of standard techniques in stochastic control theory, the main novelty is a formalization in conditional metric space and the use of conditional analysis. We illustrate the existence result by several examples such as wealth-dependent utility maximization under risk constraints and utility maximization with a conditional dimension. We also provide a discussion as to how our methods compare to techniques based on random sets.
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1 Introduction
The present work investigates parameter-dependent stochastic optimization in finite discrete time with the tools of conditional analysis. We consider a forward process ((x_t){t=0}^\mathrm{T}), for which (x{t+1}=v_{t}(x_t,z_t)) depends on (x_t) as a function of earlier decisions and an immediate decision (z_t) chosen recursively in a state-dependent control set (\varTheta _t(x_t)).
Given a filtered probability space ((\varOmega ,{\mathcal {F}},({\mathcal {F}}t){t=0}^\mathrm{T},{\mathbb {P}})), we assume that the forward process (x_t) and the control process (z_t) assume values in ({\mathcal {F}}t)-conditional metric spaces (X_t) and (Z_t), respectively. An ({\mathcal {F}}_t)-conditional metric space is a non-empty set _X endowed with a vector-valued metric (d\,{:}\,X\times X\rightarrow L^0_+(\varOmega ,{\mathcal {F}}t,{\mathbb {P}})), satisfying a concatenation property, which encodes information at time _t. An example is the space of strongly ({\mathcal {F}}t)-measurableFootnote 1 functions with values in a metric space with almost everywhere evaluation of the metric. Intuitively, one may imagine a conditional metric space to be a collection of classical metric spaces (X(\omega )) parameterized by the points (\omega \in \varOmega ) of a probability space and glued together in a measurable fashion. When the underlying probability space is not a standard Borel space, or satisfies a similar regularity assumption, and the “fibers” (X(\omega )) are not separable metric spaces, then this _pointwise perspective runs into unsolvable measurability problems, usually related to the lack of Borel (or analytic) measurable selectors, or the problem of handling an uncountable collection of null sets. Therefore, instead of trying to model the problem fiberwise, we directly work in the conditional metric space X and build instead on arguments in conditional analysis. Important is that conditional analysis works when we quotient out consequently all null sets. Then, we can rely on the Dedekind completeness of the space (L^0({\mathcal {F}})) and the existence of concatenations of sequences along countable measurable partitions of (\varOmega ), which allow for the conditional argumentation. For a methodological discussion, we refer the interested reader to Sect.5.
We focus on stochastic control problems, which by the Bellman principle can be reduced to a finite number of one-period conditional optimization problems. Our main result shows that the global maximizer is attained. By backward induction, we show that the optimal value function is upper semi-continuous on the conditional metric space (X_t). For this, we assume that the control sets (\varTheta _t(x_t)) are conditionally sequentially compact; for a discussion of the notion of conditional compactness, we refer to [1"), Sections 3 and 4] and [2, Sections 3.4 and 4]. Then, the existence of the one-period maximizers follows from a conditional version of the fact that a semi-continuous function on a compact space attains its extrema. Moreover, under a regularity condition on the control set—a conditional version of outer semi-continuity in set convergence (see, e.g., [3, Chapter 5, Section B] for the classical definition)—the value function is upper semi-continuous. Under stronger assumptions on the generators, the assumption of conditional compactness on the control set is relaxed in Proposition4.1 by modifying arguments in .
In Sect.3, we provide sufficient conditions for conditional compactness and conditional outer semi-continuity of the control set. We focus on conditionally finite dimensional control sets. The results are illustrated with applications in mathematical finance. In Example3.2, we study an optimal consumption problem with local risk constraints on the wealth process. Example3.3 indicates the importance of conditional Euclidean space with conditional dimension to model control processes with state-dependent dimension. As an application of Proposition4.1, we derive optimal portfolios w.r.t.dynamic risk measures, for which the risk aversion coefficient is influenced by the current wealth.
Normal integrands, random sets and measurable selection techniques are common tools in the study of parameterized stochastic optimization; see, e.g., 5,6,7,[8, 16–36 (1978)")]. In Sect.5, we discuss the connection of conditional analysis to random sets and normal integrands. In particular, we discuss a one-to-one correspondence between the set of measurable selections of Effros measurable and closed-valued mappings and stable and sequentially closed sets. This indicates that control problems formulated in the language of normal integrands and random sets can equally be formulated in the language of conditional analysis. For a formalization with normal integrands and random sets, measurable selection lemmas provide the main tool to secure measurability. The use of measurable selection arguments is enforced by a pointwise application of standard results in classical analysis, and relies on topological assumptions such as separability and standard Borel spaces. In this regard, conditional analysis provides a measure-theoretic alternative, which does not rely on any topological assumptions, and works as soon as a formalization within its language is reached, which is demonstrated in this article in discrete time stochastic control theory. Conditional analysis approaches measurable functions directly by working with a conditional version of results in classical analysis. The application of conditional versions of classical theorems preserves measurability, see for example the proofs below, in which a conditional version of the Bolzano–Weierstraß theorem, the maximum theorem and the Heine–Borel theorem are employed.
The remainder of this article is organized as follows. In Sect.2, we introduce the notion of conditional metric spaces and prove the main existence result. In Sects.3 and4, we discuss extensions of the main result and provide several examples. The link between conditional analysis and random set theory is established in Sect.5.
2 Main Result
Let ((\varOmega ,{\mathcal {F}},{\mathbb {P}})) be a probability space. Throughout, we identify two sets in ({\mathcal {F}}) whenever their symmetric difference is a null set, and identify two functions on (\varOmega ) if they coincide a.s.(almost surely). Let ({\mathcal {G}}) be a sub-(\sigma )-algebra of ({\mathcal {F}}). Denote by (\varPi {\mathcal {G}}) the set of partitions ((A_k)) of (\varOmega ) where (A_k\in {\mathcal {G}}) for all _k. Let (L^0_{\mathcal {G}}, L^0_{\mathcal {G}}({\mathbb {N}}), L^0_{{\mathcal {G}},+}, L^0_{{\mathcal {G}},++}, {{\underline{\hbox {L}}}}^0_{\mathcal {G}}), and ({\bar{L}}^0_{\mathcal {G}}) denote the spaces of ({\mathcal {G}})-measurable random variables with values in ({\mathbb {R}}, {\mathbb {N}}, [0,+\infty [, ]0,+\infty [, {\mathbb {R}}\cup {-\infty }), and ({\mathbb {R}}\cup {\pm \infty }), respectively. Recall that (L^0_{\mathcal {G}}) with the pointwise a.s.order is a Dedekind complete lattice-ordered ring. The essential supremum and the essential infimum are denoted by (\sup ) and (\inf ), respectively. Inequalities between random variables with values in an ordered set are always understood in the pointwise a.s.sense.
Definition 2.1
A ({\mathcal {G}})-conditional metric on a non-empty set X is a function (d\,{:}\,X\times X\rightarrow L^0_{{\mathcal {G}},+}), such that the following conditions hold:
(i)
(d(x,y)=0) if and only if (x=y),
(ii)
(d(x,y)=d(y,x),)
(iii)
(d(x,z)\le d(x,y)+d(y,z)),
(iv)
for every sequence ((x_k)) in X and ((A_k)\in \varPi {\mathcal {G}}), there exists exactly one element (x\in X) such that (1{A_k}d(x,x_k)=0) for all (k\in {\mathbb {N}}).
The pair (X,d) is called a ({\mathcal {G}})-conditional metric space.
In the following, we call the unique element in (iv) the concatenation of the sequence ((x_k)) along the partition ((A_k)), and denote it by (\sum \nolimits k 1{A_k}x_k). For a sequence ((x_n)) in a conditional metric space (X,d), we write (x_n\rightarrow x) whenever (d(x,x_n)\rightarrow 0) a.s.. Further, a conditional subsequence((x_{n_k})) of ((x_n)) is of the form (x_{n_k}:=\sum \nolimits {j\in {\mathbb {N}}} 1{{n_k=j}}x_j), where ((n_k)) is a sequence in (L^0_{{\mathcal {G}}}({\mathbb {N}})) such that (n_k<n_{k+1}) for all (k\in {\mathbb {N}}).
Definition 2.2
Let ((X,d_X)) and ((Z,d_Z)) be ({\mathcal {G}})-conditional metric spaces, and H and G subsets of X and Z, respectively.
The set H is called ({\mathcal {G}})-stable if (H \ne \emptyset ) and (\sum \nolimits k 1{A_k} x_k\in H) for all ((A_k)\in \varPi {\mathcal {G}}) and every sequence ((x_k)) in _H, and sequentially closed if H contains every (x\in X), such that there is a sequence ((x_k)) in H with (x_k\rightarrow x).
A function (f:H\rightarrow G) is said to be ({\mathcal {G}})-stable if (f\left( \sum k 1{A_k} x_k\right) =\sum \nolimits k 1{A_k} f(x_k)) for all ((A_k)\in \varPi {\mathcal {G}}) and every sequence ((x_k)) in _H, where H and G are assumed to be ({\mathcal {G}})-stable.
Remark 2.1
1.
If (X,d) is a ({\mathcal {G}})-conditional metric space, then the metric d is ({\mathcal {G}})-stable, i.e., (d\left( \sum \nolimits k 1{A_k}x_k, \sum \nolimits k 1{A_k}y_k\right) =\sum \nolimits k 1{A_k}d(x_k,y_k)) for all sequences ((x_k)) and ((y_k)) in X and ((A_k)\in \varPi {\mathcal {G}}). Indeed, denoting by (x=\sum \nolimits _k 1{A_k}x_k) and (y=\sum \nolimits k 1{A_k}y_k) the respective concatenations, it follows from the triangular inequality that
$$\begin{aligned} 1_{A_k}d(x,y)&\le 1_{A_k}d(x,x_k)+1_{A_k}d(x_k,y_k)+1_{A_k}d(y_k,y)=1_{A_k}d(x_k,y_k)\&\le 1_{A_k}d(x_k,x)+1_{A_k}d(x,y)+1_{A_k}d(y_k,y)=1_{A_k}d(x,y), \end{aligned}$$
which shows that (1_{A_k}d(x,y)=1_{A_k}d(\sum k 1{A_k}x_k,\sum \nolimits k 1{A_k}y_k)=1_{A_k}d(x_k,y_k)) for all (k\in {\mathbb {N}}). Summing up over all k yields the desired ({\mathcal {G}})-stability.
2.
Let ((X,d_X)) and ((Y,d_Y)) be two ({\mathcal {G}})-conditional metric spaces. Then, its product (X\times Y) endowed with the ({\mathcal {G}})-conditional metric
$$\begin{aligned}d_{X\times Y}\left( (x,y),(x^\prime ,y^\prime )\right) :=\max {d_{X}(x,x^\prime ),d_{Y}(y,y^\prime )}\end{aligned}$$
is a ({\mathcal {G}})-conditional metric space.
3.
Let ((X,d_X)) be a ({\mathcal {G}})-conditional metric space. Then, the set ({\mathbf {X}}) of all pairs ((x,A)\in X\times {\mathcal {G}}), where (x,A) and (y,B) are identified if (A=B) and (1_A d(x,y)=0), is a conditional set as introduced in ; in [2, Section 4] conditional metric spaces are defined.
We next introduce the parameter-dependent stochastic optimal control problem for conditional metric spaces. For a fixed finite time horizon (T\in {\mathbb {N}}), we consider a filtration ({\mathcal {F}}0\subset {\mathcal {F}}_1\subset \cdots \subset {\mathcal {F}}_T={\mathcal {F}}). For simplicity, we often abbreviate the index ({\mathcal {F}}_t) by _t, and write for instance (L^0_{t}) for (L^0_{{\mathcal {F}}t}). For each (t=0,\ldots ,T), let ((X_t,d{X_t})) and ((Z_t,d_{Y_t})) be ({\mathcal {F}}t)-conditional metric spaces. Our aim is to study control problems, for which the control set (\varTheta _t) depends on ({\mathcal {F}}_t), but also on a state parameter (x\in X_t). For every (t=0,\ldots ,T-1), we assume that the _state-dependent control set(\varTheta _t) satisfies
(c1)
(\emptyset \ne \varTheta _t(x)\subset Z_t) for all (x\in X_t),
(c2)
(\varTheta _t) is ({\mathcal {F}}_t)-stable, i.e.,
$$\begin{aligned} \varTheta t\left( \sum _k 1{A_k}x_k\right) =\sum k 1{A_k} \varTheta t(x_k):=\left{ \sum _k 1{A_k} z_k:z_k\in \varTheta _t(x_k) \text { for all }k\right} \end{aligned}$$
for all ((A_k)\in \varPi _t) and every sequence ((x_k)) in (X_t),
(c3)
for every (x\in X_t), the set (\varTheta t(x)) is _conditionally sequentially compact, i.e., for every sequence ((z_n)) in (\varTheta t(x)), there exists a conditional subsequence (n_1<n_2<\cdots ) with (n_k\in L^0_t({\mathbb {N}})) such that (z{n_k}\rightarrow z\in \varTheta _t(x)),
(c4)
for every sequence ((x_n)) in (X_t) such that (x_n\rightarrow x\in X_t) and every sequence ((z_n)) in (\varTheta t(x_n)), there exists a conditional subsequence (n_1<n_2<\cdots ) with (n_k\in L^0_t({\mathbb {N}})) and a sequence ((z_k^\prime )) in (\varTheta _t(x)) such that (d{Z_t}(z_{n_k},z_{k}^\prime )\rightarrow 0) a.s..
Note that ({\mathcal {F}}_t)-stability of (\varTheta _t) implies that (\varTheta _t(x)) is ({\mathcal {F}}_t)-stable for all (x\in X_t).
We consider forward generators
$$\begin{aligned} v_t:X_t\times Z_t\rightarrow X_{t+1},\quad t=0,\ldots ,T-1, \end{aligned}$$
which are
(v1)
({\mathcal {F}}t)-stable, i.e., (v_t\left( \sum _k 1{A_k}x_k, \sum \nolimits k 1{A_k}z_k\right) =\sum \nolimits k 1{A_k}v_t(x_k,z_k)) for every partition ((A_k)\in \varPi _t), and all sequences ((x_k)) in (X_t) and ((z_k)) in (Z_t),
(v2)
sequentially continuous, i.e., (v_t(x_n,z_n)\rightarrow v_t(x,z)) whenever (x_n\rightarrow x) in (X_t) and (z_n\rightarrow z) in (Z_t).
For every (x_t\in X_t), we consider the set
$$\begin{aligned} C_t(x_t)&:=\left{ ((x_s){s=t+1}^\mathrm{T},(z_s){s=t}^{T-1}) :x_{s+1}=v_s(x_s,z_s),z_s\in \varTheta _s(x_s)\right. \&\left. \qquad \text { for all }s=t,\ldots ,T-1\right} \end{aligned}$$
of all parameter processes ((x_s)_{s=t}^\mathrm{T}), which can be realized by the state-dependent controls (z_s\in \varTheta _t(x_s)) for (s=t,\ldots ,T-1).
As for the objective function, we consider backward generators
$$\begin{aligned} u_t:X_t\times {{\underline{\hbox {L}}}}^0_{t+1}\times Z_t \rightarrow {{\underline{\hbox {L}}}}^0_t,\quad t=0,1,\ldots ,T-1, \end{aligned}$$
which are
(u1)
({\mathcal {F}}t)-stable, i.e., (u_t\left( \sum _k 1{A_k}x_k,\sum \nolimits k 1{A_k} y_k,\sum \nolimits k 1{A_k} z_k\right) = \sum \nolimits k 1{A_k} u_t(x_k,y_k,z_k)) for all ((A_k)\in \varPi _t), and sequences ((x_k)) in (X_t, (y_k)) in (Y_t), and ((z_k)) in (Z_t),
(u2)
increasing in the second component, i.e., (u_t(x,y,z)\le u_t(x,y^\prime ,z)) whenever (y\le y^\prime ),
(u3)
sequentially upper semi-continuous, i.e.,
$$\begin{aligned}\limsup {n\rightarrow \infty } u_t(x_n,y_n,z_n)\le u_t(x,y,z),\end{aligned}$$
whenever (x_n\rightarrow x) in (X_t, y_n\rightarrow y) in ({\underline{\hbox {L}}}^0{t+1}), and (z_n\rightarrow z) in (Z_t).
We assume that (u_T\,{:}\,X_T\rightarrow {\underline{\hbox {L}}}^0_T) is ({\mathcal {F}}_{T})-stable and sequentially upper semi-continuous.Footnote 2
Given such a family ((u_t)_{t=0}^\mathrm{T}) of backward generators, our goal is to maximize
$$\begin{aligned} y_t(x_t):=\sup {((x_s){s=t+1}^\mathrm{T},(z_s){s=t}^{T-1})\in C_t(x_t) }u_t(x_t,\cdot ,z_t)\circ \cdots \circ u{T-1}(x_{T-1},\cdot ,z_{T-1})\circ u_T (x_T),\nonumber \ \end{aligned}$$
(1)
over all realizable state processes initialized at (x_t\in X_t). In (1) we consider the composition of the functions (u_T, u_{T-1}(x_{T-1},\cdot ,z_{T-1}),\ldots ,u_{t}(x_{t},\cdot ,z_{t})), where (u_s(x_s,\cdot ,z_s)) denotes the function ({\underline{\hbox {L}}}^0_{s+1}\rightarrow {\underline{\hbox {L}}}^0_{s}, y\mapsto u_s(x_s,y,z_s)).
Remark 2.2
The objective function in the stochastic control problem(1) is recursively defined. Its generators are functions between conditional metric spaces which are not necessarily (conditional) expected utilities. In case of (conditional) expected utility, the generators are closely related with dynamic and conditional risk measures; see 9,10,11,[12, 103–128 (2014)"),13,14]. The preferences which underly conditional expected utility functionals were studied in .
In decision theory, there is an extensive literature on recursive utilities starting with the seminal work [16, 17]. The preferences therein are defined on sets of temporal lotteries (probability trees), and follow a kind of Bellman recursive structure, which is similar (on a formal level) to the construction above; see [16, Theorem 1]. This was later extended in , where non-expected utilities were incorporated as well, and established under the name of Epstein–Zin utilities. See also for a survey on non-expected utility theory. With the techniques of conditional analysis and based on results in BSDE theory, solves a utility maximization problem in continuous time for Epstein–Zin utilities.
The following result shows that the global supremum in (1) is attained and can be reduced to local optimization problems by the following Bellman’s principle.
Theorem 2.1
Suppose that ((\hbox {c}1))–((\hbox {c}4)), ((\hbox {v}1))–((\hbox {v}2)), and ((\hbox {u}1))–((\hbox {u}3)) are fulfilled. Then, the functions (y_t:X_{t}\rightarrow {\underline{\hbox {L}}}^0_t) are ({\mathcal {F}}_t)-stable and sequentially upper semi-continuous for all (t=0,\ldots , T), and can be computed by backward recursion
$$\begin{aligned} y_T(x_T)&=u_T(x_T)\ y_t(x_t)&=\max {z_t\in \varTheta _t(x_t)} u_t(x_t,y{t+1}(v_t(x_t,z_t)),z_t), \quad t=0,\ldots ,T-1. \end{aligned}$$
Moreover, for every (x_t\in X_t) the process (((x^s){s=t}^\mathrm{T},(z^s){s=t}^{T-1})), given by (x_t^=x_t), and the forward recursion (x^_{s+1}=v_s(x^_s,z^_s)), where
$$\begin{aligned} z_s^\in \mathop {\hbox {argmax}}\limits _{z_s\in \varTheta _s(x_s^)} u_s\left( x_s^,y_{s+1}(v_t(x^_s,z_s)),z_s\right) ,\quad s=t,\ldots T-1, \end{aligned}$$
(2)
satisfies (((x^s){s=t+1}^\mathrm{T},(z^s){s=t}^{T-1})\in C_t(x_t)) and
$$\begin{aligned} y_t(x_t)= u_t(x_t,\cdot ,z^t)\circ \cdots \circ u{T-1}(x^_{T-1},\cdot ,z^_{T-1})\circ u_T (x^_T). \end{aligned}$$
Proof
The proof is by backward induction. For (t=T), it follows from (1) that (y_T=u_T), which by assumption is an ({\mathcal {F}}_t)-stable and sequentially upper semi-continuous function from (X_T) to ({\underline{\hbox {L}}}^0_T).
As for the induction step, assume that (y_{t+1}:X_{t+1}\rightarrow {\underline{\hbox {L}}}^0_{t+1}) is ({\mathcal {F}}{t+1})-stable and sequentially upper semi-continuous, and that for each (x{t+1}\in X_{t+1}) there exists (((x^s){s=t+2}^\mathrm{T},(z^s){s=t+1}^{T-1})\in C_{t+1}(x_{t+1})) such that
$$\begin{aligned} y_{t+1}(x_{t+1})=u_{t+1}(x_{t+1},\cdot ,z^_{t+1})\circ \cdots \circ u_T (x^_T). \end{aligned}$$
By (u1) and (v1), the function
$$\begin{aligned} X_t\times Z_t\ni (x,z)\mapsto u_t\left( x,y_{t+1}(v_t(x,z)),z\right) \end{aligned}$$
is ({\mathcal {F}}_t)-stable. Moreover, it is sequentially upper semi-continuous. Indeed, let ((x_k,z_k)) be a sequence in (X_t\times Z_t) such that (x_k\rightarrow x\in X_t) and (z_k\rightarrow z\in Z_t). Since (v(x_k,z_k)\rightarrow v(x,z)) by (v2), it follows from the induction hypothesis that
$$\begin{aligned} \limsup {k\rightarrow \infty } y{t+1}(v_t(x_k,z_k))\le y_{t+1}(v(x,z))<+\infty . \end{aligned}$$
Since
$$\begin{aligned} \left{ \sup {k\ge 1} y{t+1}(v_t(x_{k},z_{k}))=+\infty \right}&=\bigcap {k\ge 1}\left{ \sup {k^\prime \ge k} y_{t+1}(v_t(x_{k^\prime },z_{k^\prime }))=+\infty \right} \&=\left{ \limsup {k\rightarrow \infty } y{t+1}(v_t(x_k,z_k))=+\infty \right} , \end{aligned}$$
we have (\sup {k\ge 1} y{t+1}(v_t(x_{k},z_{k}))\in {\underline{\hbox {L}}}^0_{t+1}). Hence, by (u2), (u3) and (v2), we get
$$\begin{aligned} \limsup {k\rightarrow \infty } u_t\left( x_k,y{t+1}(v_t(x_k,z_k)),z_k\right)&\le \limsup {k\rightarrow \infty } u_t\left( x_k,\sup {k^\prime \ge k} y_{t+1}(v_t(x_{k^\prime },z_{k^\prime })),z_k\right) \nonumber \&\le u_t\left( x,\limsup {k\rightarrow \infty } y{t+1}(v_t(x_{k},z_{k})),z\right) \nonumber \&\le u_t\left( x, y_{t+1}(v_t(x,z)),z\right) , \end{aligned}$$
(3)
which shows the desired sequential upper semi-continuity. As a consequence, the supremum in
$$\begin{aligned} f_t(x_t):=\sup {z\in \varTheta _t(x_t)} u_t\left( x_t,y{t+1}(v_t(x_t,z)),z\right) \end{aligned}$$
(4)
is attained for each (x_t\in X_t). Indeed, since (z\mapsto u_t\left( x,y_{t+1}(v_t(x,z)),z\right) ) and (\varTheta _t(x_t)) are ({\mathcal {F}}_t)-stable, it follows from standard properties of the essential supremum that there exists a sequence (z_n\in \varTheta _t(x_t)) such that
$$\begin{aligned} u_t\left( x_t,y_{t+1}(v_t(x_t,z_n)),z_n\right) \rightarrow f_t(x_t). \end{aligned}$$
By (c3), there is a conditional subsequence (n_1<n_2<\cdots ) with (n_k\in L^0_t({\mathbb {N}})) such that (z_{n_k}\rightarrow z\in \varTheta t(x_t)) a.s.. Since (z\mapsto u_t\left( x,y{t+1}(v_t(x,z)),z\right) ) is sequentially upper semi-continuous and ({\mathcal {F}}_t)-stable, it follows that
$$\begin{aligned} u_t\left( x_t,y_{t+1}(v_t(x_t,z)),z\right) \ge \limsup {k\rightarrow \infty } u_t\left( x_t,y{t+1}(v_t(x_t,z_{n_k})),z_{n_k}\right) = f_t(x_t), \end{aligned}$$
which shows that the supremum in (4) is attained.
We next show that (f_t\,{:}\,X_t\rightarrow {\underline{\hbox {L}}}^0_{t+1}) is sequentially upper semi-continuous. By contradiction, suppose that ((x_k)) is a sequence in (X_t) such that (x_k\rightarrow x\in X_t) and (f_t(x)<\limsup {k\rightarrow \infty } f_t(x_k)) on some (A\in {\mathcal {F}}) with ({\mathbb {P}}(A)>0). Note that (f_t) is ({\mathcal {F}}_t)-stable. Thus, by possibly passing to a conditional subsequence , we can suppose that there exists (r\in L^0{t,++}) such that
$$\begin{aligned} f_t(x)+r< f_t(x_k)\text { on }A,\quad \text { for all }k\in {\mathbb {N}}. \end{aligned}$$
(5)
Denote by (z_k\in \varTheta t(x_k)) a respective maximizer of (f_t(x_k)). By (c4), there exists (z_k^\prime \in \varTheta _t(x)) such that (d{Z_t}(z_k,z^\prime k)\rightarrow 0) a.s.by possibly passing to a conditional subsequence . By (c3), there exists a conditional subsequence (k_1<k_2<\cdots ) with (k_l\in L^0_t({\mathbb {N}})) such that (z^\prime {k_l}\rightarrow z^\prime \in \varTheta t(x)). Since (d{Z_t}(z_{k_l},z^\prime {k_l})\rightarrow 0) a.s., by ({\mathcal {F}}_t)-stability of the conditional metric (d{Z_t}), it follows from the triangular inequality that (z_{k_l}\rightarrow z^\prime \in \varTheta t(x)). By the ({\mathcal {F}}_t)-stability of (f_t) and (c2), it follows that (z{k_l}) is in (\varTheta t(x{k_l})) and maximizes (f_t(x_{k_l})). Hence, it follows from (3) that
$$\begin{aligned} \limsup {l\rightarrow \infty } f_t(x{k_l})&=\limsup {l\rightarrow \infty } u_t\left( x{k_l},y_{t+1}(v_t(x_{k_l},z_{k_l})),z_{k_l}\right) \&\le u_t\left( x,y_{t+1}(v_t(x,z^\prime )),z^\prime \right) \&\le \sup {z\in \varTheta _t(x)} u_t\left( x,y{t+1}(v_t(x,z),z\right) =f_t(x). \end{aligned}$$
Notice that, due to the ({\mathcal {F}}t)-stability of (f_t), (5) is satisfied for any conditional subsequence of ((x_k)). Thus, we have that (f_t(x)+r\le \limsup {l\rightarrow \infty } f_t(x_{k_l})\le f_t(x)) on A, which is a contradiction. We conclude that (f_t) is sequentially upper semi-continuous.
Finally, we show that (y_t=f_t). By induction hypothesis, for every (x_t\in X_t) and (z_t\in Z_t), there exists (((x^s){s=t+2}^\mathrm{T},(z^s){s=t+1}^{T-1} )\in C_{t+1}(v_t(x_t,z_t))) such that
$$\begin{aligned} y_{t+1}(v_t(x_t,z_t))=u_{t+1}(v_t(x_t,z_t),\cdot ,z^{t+1})\circ \cdots \circ u{T-1}(x^_{T-1},\cdot ,z^_{T-1})\circ u_T(z^_T). \end{aligned}$$
In particular, for (x_t\in X_t) and (z_t^\in Z_t) being a maximizer in (4), it holds
$$\begin{aligned} f_t(x_t)&=\sup {z\in \varTheta _t(x_t)} u_t\left( x_t,y{t+1}(v_t(x_t,z)),z\right) \&= u_t\left( x_t,y_{t+1}(v_t(x_t,z_t^)),z_t^\right) \&= u_t\left( x_t,\cdot ,z_t^\right) \circ u_{t+1}(v_t(x_t,z^t),\cdot ,z^{t+1})\circ \cdots \circ u{T-1}(x^{T-1},\cdot ,z^_{T-1})\circ u_T(z^T) \&= \sup {(x,z)\in C_{t+1}(v(x_t,z^_t)) } u_t\left( x_t,\cdot ,z_t^\right) \circ u_{t+1}(v_t(x_t,z_t),\cdot ,z_{t+1})\circ \cdots \circ \circ u_T(z_T) \&= \sup {z_t\in \varTheta _t(x_t)} \sup {(x,z)\in C_{t+1}(v(x_t,z_t)) } u_t\left( x_t,\cdot ,z_t\right) \circ u_{t+1}(v_t(x_t,z_t),\cdot ,z_{t+1})\circ \cdots \circ u_T(z_T) \&= \sup {(x,z)\in C{t}(x_t) } u_t\left( x_t,\cdot ,z_t\right) \circ u_{t+1}(v_t(x_t,z_t),\cdot ,z_{t+1})\circ \cdots \circ u_T(z_T) \&=y_t(x_t). \end{aligned}$$
This shows that (((x^s){s=t+1}^\mathrm{T},(z^s){s=t}^\mathrm{T})\in C_{t}(x_t)) is an optimizer of (1) whenever it satisfies the local optimality criterion
$$\begin{aligned} z^s\in {\mathop {\hbox {argmax}}\nolimits }{z\in \varTheta _t(x^_t)} u_s\left( x_s^,y_{s+1}(v_t(x^_s,z_s)),z_s\right) \quad \text{ and }\quad x^_{s+1}=v_s(x^_s,z^_s) \end{aligned}$$
for all (s=t,\ldots ,T), where (x_t^=x_t). In particular, every process which satisfies the forward recursion (2) is an optimizer for (1). (\square )
Example 2.1
As for the illustration, we provide examples of ({\mathcal {F}}_t)-conditional metric spaces, which are of interest for the control and parameter spaces.
1.
Given a nonempty metric space (X,d), denote by (L^0_t(X)) the set of all strongly ({\mathcal {F}}t)-measurable functions (x:\varOmega \rightarrow X). The metric _d extends from X to (L^0_t(X)) by defining
$$\begin{aligned} d_{L^0_t(X)}(x,{{\bar{x}}})(\omega ):=d(x(\omega ),{{\bar{x}}} (\omega ))\quad \text{ for } \text{ a.a. } \omega \in \varOmega \text{ and } \text{ all } x,{{\bar{x}}}\in L^0_t(X). \end{aligned}$$
Then, ((L^0_t(X),d_{L^0_t(X)})) is a ({\mathcal {F}}_t)-conditional metric space.
2.
The conditional Euclidean space with dimension (n=\sum k 1{A_k} n_k\in L^0_t({\mathbb {N}})) is defined as
$$\begin{aligned} L^0_t({\mathbb {R}})^n=\sum k 1{A_k} L^0_t({\mathbb {R}}^{n_k}):=\left{ \sum k 1{A_k} x_k :x_k\in L^0_t({\mathbb {R}}^{n_k}) \text { for all }k\right} . \end{aligned}$$
The ({\mathcal {F}}_t)-conditional metric on (L^0_t({\mathbb {R}})^n) is defined by
$$\begin{aligned} d_{L^0_t({\mathbb {R}})^n}(x,{{\bar{x}}}):=\sum k 1{A_k} d_{ L^0_t({\mathbb {R}}^{n_k}) }(x_k,{{\bar{x}}}k), \end{aligned}$$
where (x=\sum _k 1{A_k} x_k) and ({{\bar{x}}}=\sum k 1{A_k} {{\bar{x}}}k). Here, (d{L^0_t({\mathbb {R}}^{n_k})}) denotes the ({\mathcal {F}}t)-conditional metric on (L^0_t({\mathbb {R}}^{n_k})). Straightforward verification shows that ((L^0_t({\mathbb {R}})^n, d{L^0_t({\mathbb {R}})^n})) is a ({\mathcal {F}}_t)-conditional metric space.
3.
For (1\le p<\infty ), we define the conditional (L^p)-space
$$\begin{aligned} L^p_t:={x\in L^0_T :{\mathbb {E}}[|x|^p|{\mathcal {F}}t]<+\infty } \end{aligned}$$
with ({\mathcal {F}}_t)-conditional metric (d{L^p_t}(x,\bar{x}):={\mathbb {E}}[|x-{{\bar{x}}}|^p|{\mathcal {F}}t]^{1/p}). Then, ((L^p_t,d{L^p_t})) is a ({\mathcal {F}}_t)-conditional metric space.
3 Compactness Condition for the Control Set
3.1 The Finite Dimensional Case
Suppose that (Z_t=L^0_t({\mathbb {R}}^d)). As shown in Example 2.1, the Euclidean metric of ({\mathbb {R}}^d) extends to the ({\mathcal {F}}t)-conditional metric (d{L^0_t({\mathbb {R}}^d)}:L^0_t({\mathbb {R}}^d)\rightarrow L^0_{t,+}).
Proposition 3.1
Suppose that for each (t=0,\ldots ,T-1), the control set (\varTheta _t) satisfies (c1), (c2) and the following conditions:
(i)
(\left{ (x,z)\in X_t\times L^0_t({\mathbb {R}}^d):z\in \varTheta _t(x)\right} ) is sequentially closed,
(ii)
for every sequence ((x_n)) in (X_t) with (x_n\rightarrow x\in X_t) a.s.there exists (M\in L^0_{t,+}) such that (d_{L^0_t({\mathbb {R}}^d)}(z,0)\le M) for all (z\in \bigcup _n \varTheta _t(x_n)).
Then, the control set (\varTheta _t) satisfies (c1)–(c4).
Proof
Let ((x_n)) in (X_t) be a sequence such that (x_n\rightarrow x\in X_t) a.s., and ((z_n)) a sequence in (\varTheta t(x_n)). Since by assumption, (d{L^0_t({\mathbb {R}}^d)}(z_n,0)\le M) for some (M\in L^0_{t,+}), the conditional Bolzano–Weierstrass theorem [1"), Theorem 3.8] implies a conditional subsequence (n_1<n_2<\cdots ) with (n_k\in L^0_t({\mathbb {N}})) such that (d_{L^0_t({\mathbb {R}}^d)}(z_{n_k},z)\rightarrow 0) a.s. for some (z\in L^0_t({\mathbb {R}}^d)). Since (\varTheta t) satisfies (c2), it holds (z{n_k}\in \varTheta t(x{n_k})), and (i) implies (z\in \varTheta _t(x)). This shows (c4). Further, (c3) follows by considering the constant sequence (x_n=x) for all (n\in {\mathbb {N}}). (\square )
Example 3.1
Let ((S_t)_{t=0}^\mathrm{T}) be a (({\mathcal {F}}_t))-adapted price process with values in (]0,+\infty [^d). Given an initial investment (x_0>0), we consider the wealth process
$$\begin{aligned} x_{t+1}=v_t(x_t,z_t):=x_t+ \vartheta t\cdot \Delta S{t+1}, \end{aligned}$$
where the control is an investment strategy (\vartheta _t\in L^0_t({\mathbb {R}}^d)). We consider the (wealth-dependent) control set with short-selling constrains
$$\begin{aligned} \varTheta t(x_t):=\left{ \vartheta _t\in L^0_t({\mathbb {R}}^d):x_t=\vartheta {t}\cdot S_t,\, 0\le \vartheta ^i_t,\, i=1,\ldots ,d\right} . \end{aligned}$$
For each t, we consider a bounded measurable function (g_t:\varOmega \times {\mathbb {R}}^d\rightarrow {\mathbb {R}}) such that (g_t(\omega ,\cdot )) is upper semi-continuous for every (\omega \in \varOmega ). Next, we show that there exists an optimizer (((x_s^)_{s={t+1}}^\mathrm{T},(\vartheta _s^)_{s=t}^{T-1})\in C_t(x_t)) of the utility maximization problem
$$\begin{aligned} \underset{((x_s){s=t}^\mathrm{T},(\vartheta _s){s=t}^{T-1})\in C_t(x_t)}{\sup }{\mathbb {E}}\left[ \sum _{s=t}^\mathrm{T} g_s(\cdot ,x_s)\mid {\mathcal {F}}_t\right] . \end{aligned}$$
(6)
Inspection shows that (\varTheta t) satisfies (c1), (c2) and (i) of Proposition3.1. As for (ii), for every (\vartheta \in \varTheta _t(x)), it can be checked that (d{L^0_t({\mathbb {R}}^d)}(\vartheta ,0)\le M(x)) with (M(x):=\max {i=1,\ldots ,d}\frac{x}{S^i_t}). Let ((x_n)) be a sequence in (L^0_t) such that (x_n\rightarrow x\in L^0_t). Take ({\bar{x}}:=\sup _n x_n\in L^0+). Then, (d_{L^0({\mathbb {R}}^d)}(\vartheta ,0)\le M({\bar{x}})) for all (z\in \bigcup _n\varTheta _t(x_n)), which shows (ii). Finally, for fixed (\alpha >0), define (u_T(x):=g_T(\cdot ,x)) and for (t=0,1,\ldots ,T-1),
$$\begin{aligned} u_t(x,y,z):={\mathbb {E}}\left[ g_t(\cdot ,x)+\max \left{ -\alpha ,\min {\alpha ,y}\right} |{\mathcal {F}}_t\right] . \end{aligned}$$
Then, inspection shows that the backward process ((u_t)_{t=0}^\mathrm{T}) satisfies conditions (u1) and (u2), and due to Fatou’s lemma, it also satisfies (u3). Since (g_t) is bounded, for (\alpha >0) large enough, it can be checked by backward induction that
$$\begin{aligned} u_t(x_t,\cdot ,z_t)\circ \cdots \circ u_{T-1}(x_{T-1},\cdot ,z_{T-1})\circ u_T (x_T)= {\mathbb {E}}\left[ \sum _{s=t}^\mathrm{T} g_s(\cdot ,x_s)\mid {\mathcal {F}}_t\right] \end{aligned}$$
for every (((x_s){s=t+1}^\mathrm{T},(\vartheta _s){s=t}^{T-1})\in C_t(x_t)). By Proposition3.1 and Theorem 2.1 it follows that (6) has a global optimizer (((x^s){s=t+1}^\mathrm{T},(\vartheta ^s){s=t}^{T-1})\in C_t(x_t)).
Example 3.2
Let ((S_t){t=0}^\mathrm{T}) be a _d-dimensional (({\mathcal {F}}_t))-adapted price process. Given an initial investment (x_0>0), we consider the wealth process
$$\begin{aligned} x_{t+1}=v_t(x_t,z_t):=x_t+ \vartheta t\cdot \Delta S{t+1}-c_t, \end{aligned}$$
where the control (z_t=(\vartheta t,c_t)\in L^0_t({\mathbb {R}}^d)\times L^0+) consists of an investment strategy (\vartheta t\in L^0_t(\mathbb {R^d})) and a consumption (c_t\in L^0{t,+}). Further, the forward generator (v_t:L^0_t \times L^0_t({\mathbb {R}}^d\times {\mathbb {R}}+)\rightarrow L^0{t+1}) satisfies (v1) and (v2). We assume that the wealth process satisfies the regulatory restriction
$$\begin{aligned} \rho t(x{t+1})\le 0,\quad t=0,\ldots ,T-1. \end{aligned}$$
(7)
In other words, (x_{t+1}) is acceptable w.r.t.a ({\mathcal {F}}t)-conditional convex risk measure (\rho _t:L^0{t+1}\rightarrow {\bar{L}}^0_t) for all (t=0,\ldots ,T-1). Recall that a ({\mathcal {F}}_t)-conditional convex risk measure is
normalized,:
i.e., (\rho _t(0)=0),
monotone,:
i.e., (\rho t(x)\le \rho _t(y)) for all (x,y\in L^0{t+1}) with (x\ge y),
({\mathcal {F}}t)-translation invariant,:
i.e., (\rho _t(x+m)=\rho (x)-m) for all (x\in L^0{t+1}) and (m\in L^0_t),
({\mathcal {F}}t)-convex,:
i.e., (\rho _t(\lambda x+(1-\lambda )y)\le \lambda \rho _t(x)+(1-\lambda )\rho _t(y)) for all (x,y\in L^0{t+1}) and (\lambda \in L^0_t) with (0\le \lambda \le 1).
By ({\mathcal {F}}_t)-translation invariance, it follows that (7) is equivalent to
$$\begin{aligned}\rho t(\vartheta _t\cdot \Delta S{t+1})\le x_t-c_t.\end{aligned}$$
Moreover, (\rho _t) is ({\mathcal {F}}_t)-stable as it is ({\mathcal {F}}_t)-convex; see [1"), Lemma 4.3]. Hence, we consider the (wealth-dependent) ({\mathcal {F}}_t)-stable control set
$$\begin{aligned} \varTheta t(x_t)&:=\Big {z_t=(\vartheta _t,c_t)\in L^0_t({\mathbb {R}}^d\times {\mathbb {R}}+):\rho t(\vartheta _t\cdot \Delta S{t+1})\le x_t-c_t\&\qquad \text{ and } 0\le c_t\le x_t \Big }. \end{aligned}$$
Suppose that for every (\vartheta \in L^0_t({\mathbb {R}}^d)) it holds ({\mathbb {P}}(\vartheta \cdot \Delta S_{t+1}<0\mid {\mathcal {F}}t)>0) on ({\vartheta \ne 0}), and therefore ({\mathbb {P}}(\vartheta \cdot \Delta S{t+1}>0\mid {\mathcal {F}}t)>0) on ({\vartheta \ne 0}). Moreover, we assume that (\rho _t(\vartheta \cdot \Delta S{t+1})\in L^0_t) for all (\vartheta \in L^0_t({\mathbb {R}}^d)), and (\rho t) is ({\mathcal {F}}_t)-_sensitive to large losses, i.e., (\lim {m\rightarrow \infty }\rho _t(m y)=+\infty ) on ({{\mathbb {P}}(y<0\mid {\mathcal {F}}_t)>0}). Then, the control set (\varTheta _t) satisfies (i) and (ii) of Proposition 3.1. Indeed, consider the function (f_t:L^0_t({\mathbb {R}}^d\times {\mathbb {R}}+)\times L^0_t\rightarrow L^0_t) defined as (f_t(\vartheta ,c,x):=\rho t(\vartheta \cdot \Delta S{t+1})+c - x), which is ({\mathcal {F}}_t)-convex and therefore sequentially continuous by [1"), Theorem 7.2]. Hence, it follows that
$$\begin{aligned}&\left{ (x,\vartheta ,c)\in L^0_t\times L^0_t({\mathbb {R}}^d\times {\mathbb {R}}+):(\vartheta ,c)\in \varTheta _t(x)\right} \&\quad =\left{ (x,\vartheta ,c)\in L^0_t\times L^0_t({\mathbb {R}}^d\times {\mathbb {R}}+):f_t(\vartheta ,c,x)\le 0\right} \end{aligned}$$
is ({\mathcal {F}}_t)-convex and sequentially closed, which shows (i). As for (ii), let ((x_n)) be a sequence in (L^0_t) such that (x_n\rightarrow x\in L^0_t). For ({{\bar{x}}}:=\sup _n x_n\in L^0_t) one has
$$\begin{aligned} \varTheta _t(x_n)\subset \varTheta _t({{\bar{x}}}) \end{aligned}$$
for all (n\in {\mathbb {N}}). Hence, it remains to show that (\varTheta t({{\bar{x}}})) is ({\mathcal {F}}_t)-bounded, i.e., there is (M\in L^0{t,+}) such that (d_{L^0_t({\mathbb {R}}^d)}(\vartheta ,0)+c\le M) for all ((\vartheta ,c)\in \varTheta t({{\bar{x}}})). Since (\varTheta _t({{\bar{x}}})) contains ((0,0)\in L^0_t({\mathbb {R}}^d\times {\mathbb {R}}+)), by [1"), Theorem 3.13] it is enough to show that for each ((\vartheta ,c)\in \varTheta t({{\bar{x}}})) with ((\vartheta ,c)\ne (0,0)), there exists (k\in {\mathbb {N}}) such that (k(\vartheta ,c)\notin \varTheta _t({{\bar{x}}})). If (c\ne 0), this is obvious. Otherwise, it holds ({\mathbb {P}}(\vartheta \ne 0)>0), in which case (\lim {m\rightarrow \infty }\rho t(m\vartheta \cdot \Delta S{t+1})=+\infty ) on ({\vartheta \ne 0}).
By Proposition3.1 and Theorem 2.1, it follows that for each (x_0>0) and every recursive utility function with backward generators ((u_t){t=0}^\mathrm{T}), which satisfy (u1)–(u3), there exists a global optimizer (((x^s){s=1}^\mathrm{T},(\vartheta ^_s,c^_s){s=0}^{T-1})\in C_0(x_0)) of the utility maximization problem (1) such that the local criterion (2) holds.
3.2 Conditional Dimension
Suppose that (Z_t) is the conditional Euclidean space (L^0_t({\mathbb {R}})^{d_t}) with dimension (d_t=d_t(x)\in L^0_t({\mathbb {N}})), which depends on the parameter (x\in X_t); see Example2.1. Let (d_t:X_t\rightarrow L^0_t({\mathbb {N}})) be an ({\mathcal {F}}_t)-stable and sequentially continuous, where (L^0_t({\mathbb {N}})) is endowed with the ({\mathcal {F}}_t)-conditional metric which extends the discrete metric on ({\mathbb {N}}). The control set (\varTheta _t) is chosen such that the following conditions hold:
(c1)
(\emptyset \ne \varTheta _t(x)\subset L^0_t({\mathbb {R}})^{d_t(x)}) for all (x\in X_t),
(c2)
(\varTheta _t) is ({\mathcal {F}}_t)-stable, i.e.,
$$\begin{aligned} \varTheta t\left( \sum _k 1{A_k}x_k\right) =\sum k 1{A_k} \varTheta t(x_k)\subset L^0_t({\mathbb {R}})^{d_t(\sum _k 1{A_k} x_k)} \end{aligned}$$
for all ((A_k)\in \varPi _t) and every sequence ((x_k)) in (X_t).
Remark 3.1
Since (Z_t=L^0_t({\mathbb {R}})^{d_t(x)}) depends on the state (x\in X_t), we are in a more general setting as in Theorem 2.1. However, since (L^0_t({\mathbb {N}})) is endowed with the conditional discrete metric, for every sequence ((x_n)) in (X_t) such that (x_n\rightarrow x\in X_t), there exists (n_0\in L^0_t({\mathbb {N}})) such that (d_t(x_n)=d_t(x)) for all (n\ge n_0). In particular, (L^0_t({\mathbb {R}})^{d_t(x_n)}=L^0_t({\mathbb {R}})^{d_t(x)}) for all (n\ge n_0), and Theorem2.1 still holds true by exploring the arguments on (z_n\in \varTheta _t(x_n)) for sequences (x_n\rightarrow x) in the conditional space (L^0_t({\mathbb {R}})^{d_t(x)}).
A variant of Proposition3.1 for control sets with conditional dimension can be formulated as follows.
Proposition 3.2
Suppose that for each (t=0,\ldots ,T-1), the control set (\varTheta _t) satisfies (\text {(c1)}, \text {(c2)}) and the following conditions:
(i)
(\left{ (x,z)\in X_t\times L^0_t({\mathbb {R}})^{d_t(x)}:z\in \varTheta _t(x)\right} ) is sequentially closed.
(ii)
For every sequence ((x_n)) in (X_t) with (x_n\rightarrow x\in X_t), there is (M\in L^0_{t,+}) and a conditional subsequence (n_1<n_2<\cdots ) in (L^0_t({\mathbb {N}})), which satisfy (d_{L^0_t({\mathbb {R}}^{d_t(x)})}(z,0)\le M) for all (z\in \bigcup {k\ge k_0} \varTheta _t(x{n_k})) for some (k_0\in L^0_t({\mathbb {N}})), such that (\varTheta t(x{n_k})\subset Z_t(d_t(x))) for all (k\ge k_0).
Then, the control set (\varTheta _t) satisfies (\text {(c1)})–(\text {(c4)}).
Proof
Let ((x_n)) in (X_t) be a sequence such that (x_n\rightarrow x\in X_t), and (z_n\in \varTheta t(x_n)). By Remark3.1 there exists a conditional subsequence (n_1<n_2<\cdots ) with (n_k\in L^0_t({\mathbb {N}})) such that (z{n_k}\in \varTheta t(x{n_k})\subset L^0_t({\mathbb {R}})^{d_t(x)}) for all k. Hence, we can argue similar as in the proof of Proposition3.1. (\square )
Example 3.3
Consider a portfolio maximization problem, where the number of traded assets depends on past decision. More precisely, given a portfolio (x_t=z_{t-1}=(\vartheta {t-1},d{t-1})\in L^0_{t-1}({\mathbb {R}})^{d_{t-1}}\times L^0_{t-1}({\mathbb {N}})) chosen at time (t-1) (with initial value (x_{-1}=(\vartheta {-1},d{-1})\in {\mathbb {R}}^{d_{-1}}\times {\mathbb {N}})), the investor can rebalance the portfolio at time t to
$$\begin{aligned} x_{t+1}=z_t=(\vartheta t,d_t)\in \varTheta _t(x_t)\subset L^0_t({\mathbb {R}})^{d{t-1}}\times L^0_t({\mathbb {N}}). \end{aligned}$$
Here, the state spaces and the control spaces (X_{t+1}=Z_t=L^0_t({\mathbb {R}})^{d_{t-1}}\times L^0_t({\mathbb {N}})) both depend on the past decision (d_{t-1}). In line with Remark3.1, the convergence (x^n_t=(\vartheta ^n_{t-1},d^n_{t-1}) \rightarrow x_t=(\vartheta {t-1},d{t-1})) is understood as (\vartheta ^n_{t-1}\rightarrow \vartheta {t-1}) in the conditional metric space (L^0_t({\mathbb {R}})^{d{t-1}}), since (d^n_{t-1}=d_{t-1}) for all (n\ge n_0) for some (n_0\in L^0_{t-1}({\mathbb {N}})). Suppose that the control set (\varTheta t) satisfies (c1), (c2) as well as (i) and (ii) of Proposition3.2. Then, along the same argumentation as in Proposition3.2, it follows that (\varTheta _t) satisfies (c1)–(c4). Since (v_t(x_t,z_t):=z_t) satisfies (v1) and (v2), Theorem 2.1 is applicable whenever the backward generators ((u_t){t=0}^\mathrm{T}) satisfy (u1)–(u3).
The conditional dimension depending on past decisions allows for instance to add new assets at time t ((d_t>d_{t-1})) which are traded at (t+1). Notice that (\varTheta t(\vartheta {t-1},d_{t-1})) denotes the set of all attainable portfolios at time t. For instance, let (S_t\in L^0_{t,++}({\mathbb {R}}^d)) be a price process with fixed (d\in {\mathbb {N}}). Without frictions and short-selling constraints one has
$$\begin{aligned}\varTheta t(\vartheta {t-1}):={\vartheta t\in L^0{t,+}({\mathbb {R}}^d):\vartheta t\cdot S_t=\vartheta {t-1}\cdot S_t},\end{aligned}$$
which satisfies (c1)–(c4). Transaction costs can be included into the model by considering (\varTheta t(\vartheta {t-1}):={\vartheta t\in L^0{t,+}({\mathbb {R}}^d):\vartheta t-\vartheta {t-1}\in C_t}) for a solvency region (C_t\subset L^0_t({\mathbb {R}}^d)); see, e.g., for a discussion of different market models. Also, the solvency regions can be modeled state-dependently with conditional dimension (d_t\in L^0_t({\mathbb {N}})).
4 Unbounded Control Sets
In this section, we consider unbounded control sets (\varTheta _t\equiv L^0_t({\mathbb {R}}^d)) and do not assume constraints on the controls, but derive (c3) and (c4) for upper-level sets of (y_t) as a result of stronger assumptions on the forward and backward generators. In particular, we additionally need that the backward generators are ({\mathcal {F}}_t)-sensitive to large losses and increasing in the first argument. Suppose that the forward generators
$$\begin{aligned} v_t:L^0_t \times L^0_t({\mathbb {R}}^d) \rightarrow L^0_{t+1}, \quad t=0,1,\ldots ,T-1, \end{aligned}$$
satisfy (v1), (v2) and
(v3)
(v_t) is increasing in the first component,
(v4)
(v_t(x,\lambda z + (1-\lambda )z^\prime )\ge \lambda v_t(x,z) + (1-\lambda ) v_t(x,z^\prime )) for all (x\in L^0_t, z,z^\prime \in L^0_t({\mathbb {R}}^d)), and (\lambda \in L^0_t) with (0\le \lambda \le 1),
(v5)
({\mathbb {P}}(v_t(x,z)0) on ({z\ne 0}) for all (x\in L^0_t) and (z\in L^0_t({\mathbb {R}}^d)),
(v6)
(v_t(x,0)=x) for all (x\in L^0_t).
As for the backward generators, let (u_T:L^0_T\rightarrow L^0_T) be the identity mapping, and
$$\begin{aligned} u_t:L^0_t\times {\underline{\hbox {L}}}^0_{t+1}\times L^0_t({\mathbb {R}}^d)\rightarrow {\underline{\hbox {L}}}^0_t, \quad t=0,\ldots , T-1, \end{aligned}$$
satisfy (u1) and (u3) as well as
(u2’)
(u_t) is increasing in the first and second component,
(u4)
(u_t(x,\lambda y + (1-\lambda )y^\prime ,\lambda z + (1-\lambda )z^\prime )\ge \min \left{ u_t(x,y,z),u_t(x,y^\prime ,z^\prime )\right} ) for all (x\in L^0_t, y,y^\prime \in {\underline{\hbox {L}}}^0_{t+1}, z,z^\prime \in L^0_t({\mathbb {R}}^d)), and (\lambda \in L^0_t) with (0\le \lambda \le 1),
(u5)
(u_t(x,y+c,z)=u_t(x,y,z)+c) for all (x\in L^0_t, y\in {\underline{\hbox {L}}}^0_{t+1}, z\in L^0_t({\mathbb {R}}^d)) and (c\in L^0_t),
(u6)
(\lim {m\rightarrow \infty }u_t(x,m y,m z)=-\infty ) on ({{\mathbb {P}}(y<0\mid {\mathcal {F}}_t)>0}) for all (z\in L^0_t({\mathbb {R}}^d)) and (y\in {\underline{\hbox {L}}}^0{t+1}),
(u7)
(u_t(x,0,0)=0) for all (x\in L^0_t).
Let (y_t:L^0_t\rightarrow {\underline{\hbox {L}}}^0_t) be given as in (1), where
$$\begin{aligned} C_t(x_t)&:=\left{ \left( (x_s){s=t+1}^\mathrm{T},(z_s){s=t}^{T-1}\right) :x_{s+1}=v_s(x_s,z_s),z_s\in L^0_t({\mathbb {R}}^d)\right. \&\left. \quad \text { for all }s=t,\ldots ,T-1\right} . \end{aligned}$$
Then, the following variant of Theorem2.1 holds.
Proposition 4.1
Suppose that (\text {(v1)})–(\text {(v6)}) and (\text {(u1)}), ((\hbox {u}2')), (\text {(u3)})–((\hbox {u}7)) are fulfilled, and there exists a constant (K>0) such that
$$\begin{aligned} \sup _{z\in L^0_t({\mathbb {R}}^d)} u_t(x,v_t(x,z),z) - x \le K \end{aligned}$$
for all (t=0,\ldots ,T-1), and (x\in L^0_t). Then, the functions (y_t:L^0_{t}\rightarrow {\underline{\hbox {L}}}^0_t) are ({\mathcal {F}}_t)-stable, increasing and sequentially upper semi-continuous for all (t=0,\ldots , T), and can be computed by backward recursion
$$\begin{aligned} y_T(x_T)&=u_T(x_T)=x_T\ y_t(x_t)&=\max {z_t\in L^0_t({\mathbb {R}}^d)} u_t(x_t,y{t+1}(v_t(x_t,z_t)),z_t), \quad t=0,\ldots ,T-1. \end{aligned}$$
Moreover, for every (x_t\in L^0_t) the process (((x^s){s=t}^\mathrm{T},(z^s){s=t}^{T-1})) given by (x_t^=x_t), and forward recursion (x^_{s+1}=v_s(x^_s,z^_s)), where
$$\begin{aligned} z_s^\in {\mathop {\hbox {argmax}}\limits }_{z_s\in L^0_t({\mathbb {R}}^d)} u_s\left( x_s^,y_{s+1}(v_t(x^_s,z_s)),z_s\right) ,\quad s=t,\ldots T-1, \end{aligned}$$
(8)
satisfies (((x^s){s=t+1}^\mathrm{T},(z^s){s=t}^{T-1})\in C_t(x_t)) and
$$\begin{aligned} y_t(x_t)= u_t(x_t,\cdot ,z^t)\circ \cdots \circ u{T-1}(x^_{T-1},\cdot ,z^_{T-1})\circ u_T (x^_T). \end{aligned}$$
Proof
The proof is similar to Theorem2.1. However, since the control set is not compact we have to argue differently to show the existence of (4), i.e., that the supremum in
$$\begin{aligned} y_t(x_t):=\sup {z\in L^0_t({\mathbb {R}}^d)} u_t\left( x_t,y{t+1}(v_t(x_t,z)),z\right) ,\quad x_t\in L^0_t, \end{aligned}$$
is attained. To do so, we first show that
$$\begin{aligned} 0\le y_t(x) - x \le K_t\quad \text{ for } \text{ all } x\in L^0_t, \end{aligned}$$
(9)
where (K_t:=(T-t)K) for all (t=0,1,\ldots ,T). For (t=T), one has (y_T(x)-x=0). By induction, suppose that (y_{t+1}(x)-x\le (T-t)K_{t+1}). Then, by (u2’) and (u5) for every (z\in L^0_t({\mathbb {R}}^d)), it holds
$$\begin{aligned}&u_t\left( x,y_{t+1}(v_t(x,z)\right) ,z)-x=u_t\left( x,y_{t+1}(v_t(x,z))-v_t(x,z)+v_t(x,z),z\right) -x \&\le u_t(x,v_t(x,z),z) - x + K_{t+1} \le K+K_{t+1}=K_t, \end{aligned}$$
so that (y_t(x)-x\le K_t). As for the lower bound, suppose by induction that (x\le y_{t+1}(x)). By (v6), (u2(^{\prime })), (u5) and (u7) it follows that
$$\begin{aligned} y_t(x)&\ge u_t\left( x,y_{t+1}(v_t(x,0)),0\right) \ge u_t\left( x,y_{t+1}(x),0\right) \ {}&\ge u_t\left( x,x,0\right) = u_t\left( x,0,0\right) +x= x. \end{aligned}$$
Fix (x\in L^0_t). For (z\in L^0_t({\mathbb {R}}^d)) with (u_t(x,y_{t+1}(v_t(x,0)),0)\le u_t(x,y_{t+1}(v_t(x,z)),z)), it follows from (9) (u5), (u7) and (v6) that
$$\begin{aligned} x&=u_t(x,x,0)\le u_t(x,y_{t+1}(x),0) \le u_t\left( x,y_{t+1}(v_t(x,0)),0\right) \&\le u_t\left( x,y_{t+1}(v_t(x,z)),z\right) \le u_t\left( x,v_t(x,z),z\right) +K_{t+1}. \end{aligned}$$
This shows that
$$\begin{aligned} y_t(x)=\sup {z\in \varTheta _t(x)} u_t\left( x,y{t+1}(v_t(x,z)),z\right) , \end{aligned}$$
for the ({\mathcal {F}}_t)-stable set
$$\begin{aligned} \varTheta t(x):=\left{ z\in L^0_t({\mathbb {R}}^d):u_t(x,v_t(x,z),z) \ge x - K{t+1}\right} . \end{aligned}$$
It remains to show that (\varTheta _t) satisfies (c1)–(c4). To that end, we verify (i) and (ii) of Proposition3.1. By (u3) and (v2), it follows that the set
$$\begin{aligned} \left{ (x,z)\in L^0_t\times L^0_t({\mathbb {R}}^d):z\in \varTheta _t(x)\right} \end{aligned}$$
is sequentially closed, which shows (i) of Proposition 3.1. As for (ii) of Proposition 3.1 let ((x_n)) be a sequence in (L^0_t) such that (x_n\rightarrow x\in L^0_t). Defining ({\underline{x}}:=\inf _n x_n\in L^0_t) as well as ({{\bar{x}}}:=\sup _n x_n\in L^0_t), it follows from (u2’) and (v3) that
$$\begin{aligned} \varTheta t(x_n)\subset \left{ z\in L^0_t({\mathbb {R}}^d):u_t(\bar{x},v_t({{\bar{x}}},z),z) \ge {\underline{x}} - K{t+1}\right} =:\varTheta _t({\underline{x}},{{\bar{x}}}) \end{aligned}$$
for all (n\in {\mathbb {N}}). Moreover, by (u4) and (v4), the set (\varTheta t({\underline{x}},{{\bar{x}}})) is ({\mathcal {F}}_t)-convex. It remains to show that there exists (M\in L^0_t) such that (d{L^0_t({\mathbb {R}}^d)}(z,0)\le M) for all (z\in \varTheta _t({\underline{x}},{{\bar{x}}})). This (L^0_t)-boundedness of (\varTheta _t({\underline{x}},{{\bar{x}}})) would follow from [1"), Theorem 3.13], if for all (z\in \varTheta _t({\underline{x}},{{\bar{x}}})) with (z\ne 0), there exists (A\in {\mathcal {F}}_t) with ({\mathbb {P}}(A)>0) such that
$$\begin{aligned} \lim _{m\rightarrow \infty } u_t({{\bar{x}}}, v_t({{\bar{x}}},m z), m z)=-\infty \quad \text { on }A. \end{aligned}$$
(10)
Indeed, since by (v5) one has ({\mathbb {P}}(v_t({{\bar{x}}},z)<\bar{x}\mid {\mathcal {F}}_t)>0) on ({z\ne 0}), there exists (l\in {\mathbb {N}}) such that (A:=\left{ {\mathbb {P}}\left( |{{\bar{x}}}| + l (v_t({{\bar{x}}},z)- {{\bar{x}}})<0 \mid {\mathcal {F}}_t\right) >0\right} \in {\mathcal {F}}_t) satisfies ({\mathbb {P}}(A)>0). By (v4), it follows that
$$\begin{aligned} v_t({{\bar{x}}},z)\ge \frac{1}{m}v_t({{\bar{x}}},mz)+\frac{m-1}{m}v_t(\bar{x},0), \end{aligned}$$
which by (v6) implies (m\left( v_t({{\bar{x}}},z)-{{\bar{x}}}\right) \ge v_t(\bar{x},mz)-{{\bar{x}}}) for all (m\in {\mathbb {N}}). This shows that
$$\begin{aligned} u_t({{\bar{x}}},v_t({{\bar{x}}},m z),m z)&\le u_t({{\bar{x}}},|{{\bar{x}}}| + v_t(\bar{x},m z)- {{\bar{x}}},m z)\&\le u_t\left( {{\bar{x}}},\frac{m}{l}\left( |\bar{x}| + l(v_t({{\bar{x}}},z)- {{\bar{x}}})\right) ,m z\right) \end{aligned}$$
for all (m\in {\mathbb {N}}) large enough. Hence, the condition (u6) implies (10). (\square )
Example 4.1
Let ((S_t)_{t=0}^\mathrm{T}) be a ({\mathbb {R}}^d)-valued adapted stochastic process modeling the discounted stock prices of a financial market model. Given a trading strategy (\vartheta _t\in L^0_t({\mathbb {R}}^d), t=0,\ldots ,T-1), and an initial investment (x_0\in L^0_0), we define recursively the wealth process
$$\begin{aligned} x_{t+1}=v_t(x_t,\vartheta t):=x_t + \vartheta _t\cdot \Delta S{t+1}, \quad t=0,\ldots ,T-1, \end{aligned}$$
where (\Delta S_{t+1}:=S_{t+1} - S_{t}) denotes the stock price increment. We assume the no-arbitrage condition:
$$\begin{aligned}\vartheta \cdot \Delta S_{t+1}\ge 0 \text{ for } \vartheta \in L^0_t({\mathbb {R}}^d)\quad \text{ implies }\quad \vartheta = 0\end{aligned}$$
for all (t=0,\ldots ,T-1). Then, the forward generator (v_t:L^0_t\times L^0_t({\mathbb {R}}^d)\rightarrow L^0_{t+1}) satisfies (v1)–(v6). As for the backward generators, let (u_T:L^0_T\rightarrow L^0_T) be the identity and
$$\begin{aligned} u_t:L_t^0\times {\underline{\hbox {L}}}^0_{t+1}\rightarrow {\underline{\hbox {L}}}^0_t, \quad u_t(x,y):=\frac{1}{\gamma _t(x)}g_t(\gamma _t(x)y), \quad t=0,\ldots ,T-1, \end{aligned}$$
where (g_t :{\underline{\hbox {L}}}^0_{t+1}\rightarrow {\underline{\hbox {L}}}^0_t) is increasing, ({\mathcal {F}}t)-concave, ({\mathcal {F}}_t)-translation invariant, sequentially upper semi-continuous, (g_t(0)=0) and (\lim {r\rightarrow \infty } g_t(ry)=-\infty ) on ({{\mathbb {P}}(y<0\mid {\mathcal {F}}t)>0}). The function (\gamma _t:L^0_t\rightarrow L^0{t,++}) is ({\mathcal {F}}t)-stable, decreasing and sequentially continuous and models the risk aversion depending on the wealth (x_t) at time _t. Then, (u_t) satisfies the conditions (u1), (u2(^{\prime })), (u3)–(u7). We only verify (u2(^{\prime })) and (u3). To prove (u2(^{\prime })) take (x_1\le x_2) and (y_1\le y_2). Let (\beta _i:=1/\gamma _t(x_i)), for (i=1,2). By using the monotonicity and ({\mathcal {F}}_t)-concavity of (g_t), we have
$$\begin{aligned} g_t\left( \frac{y_2}{\beta _2}\right) \ge g_t\left( \frac{y_1}{\beta _2}\right) \ge \frac{\beta _1}{\beta _2}g_t(\frac{y_1}{\beta _1}) + \frac{\beta _2-\beta _1}{\beta _2}g_t(0) = \frac{\beta _1}{\beta _2} g_t\left( \frac{y_1}{\beta _1}\right) . \end{aligned}$$
Multiplying by (\beta _2), we obtain (u_t(x_2,y_2)\ge u_t(x_1,y_1)). Due to the monotonicity of (u_t), it suffices to verify (u3) for decreasing sequences. Indeed, suppose that ({x}_k\searrow x) and ({y}_k\searrow y). Then, by the monotonicity of (u_t) we have
$$\begin{aligned} g_t(\gamma _t({x}_k){y}_k)\ge \frac{\gamma _t({x}_k)}{\gamma _t(x)}g_t(\gamma _t(x)y)\quad \text { for all }k. \end{aligned}$$
Thus, by using that (g_t) is sequentially upper semi-continuous and (\gamma ) is sequentially continuous we obtain
$$\begin{aligned} g_t(\gamma _t(x)y)&\ge \underset{k\rightarrow \infty }{\limsup }g_t(\gamma _t({x}_k){y}_k)\ge \underset{k\rightarrow \infty }{\liminf }g_t(\gamma _t({x}_k){y}_k) \ {}&\ge \underset{k\rightarrow \infty }{\lim }\frac{\gamma _t({x}_k)}{\gamma _t(x)}g_t(\gamma _t(x)y)=g_t(\gamma _t(x)y). \end{aligned}$$
This shows that (g_t(\gamma t({x}_k){y}_k)\rightarrow g_t(\gamma _t(x)y)), and therefore (u_t(x_k,y_k)\rightarrow u_t(x,y)). Given the wealth process ((x_t){t=0}^\mathrm{T}), define the backward process
$$\begin{aligned} y_t(x_t)=\underset{((x_s){s=t}^\mathrm{T},(\vartheta _s){s=t}^{T-1})\in C_t(x_t)}{\sup }u_t(x_t,\cdot )\circ \cdots \circ u_{T-1}(x_{T-1},\cdot )\circ u_T (x_T),\quad \end{aligned}$$
for (t=0,\ldots ,T-1), where (C_t(x_t)) consists of all (((x_s){s=t}^\mathrm{T},(\vartheta _s){s=t}^{T-1})) such that (x_{s+1}=x_s+\vartheta {s+1}\cdot \Delta S{s+1}) for all (s=t,\ldots ,T-1). By induction, one can verify that (y_t(x+c)=y_t(x)+c) for every (c\in L^0_{t-1}) with (t=1,\ldots ,T). Suppose there exists (K>0) such that
$$\begin{aligned} u_t\left( x,v_t(x,\vartheta ),\vartheta \right) -x\le \frac{1}{\gamma t(x)}g_t\left( \gamma _t(x)\vartheta \Delta S{t+1}\right) \le K \end{aligned}$$
(11)
for all (t=0,\ldots ,T-1, \vartheta \in L^0_t({\mathbb {R}}^d)), and (x\in L^0_t). Then, it follows from Proposition4.1 that
$$\begin{aligned} y_0(x_0)=\underset{((x_t){t=0}^\mathrm{T},(\vartheta _t){t=0}^{T-1})\in C_0(x_0)}{\sup }u_0(x_0,\cdot )\circ \ldots \circ u_{T-1}(x_{T-1},\cdot )\circ u_T (x_T) \end{aligned}$$
is attained for all (x_0\in L^0_0). For instance one could think of the dynamic entropic preference functional with generators (-\frac{1}{\gamma t}\log \left( {\mathbb {E}}[\exp (-\gamma _t y)\mid {\mathcal {F}}_t]\right) ), where the local risk aversion coefficient (\gamma _t=\gamma _t(x_t)) depends on the current wealth (x_t). Notice that (\lim {m\rightarrow \infty }-\log ({\mathbb {E}}[\exp (-my)\mid {\mathcal {F}}_t])=-\infty ) on ({{\mathbb {E}}[y<0\mid {\mathcal {F}}_t]>0}).
5 Connection to Random Set Theory
The aim of this section is to discuss methodological similarities and differences of conditional analysis and random set theory. Random sets are established on the basis of results in classical analysis. They seek a formalization of a randomized problem such that classical theorems can be applied in each state or point of the underlying probability space. Measurable selection lemmas then help to extract a measurable object. To guarantee that such selections exist usually topological countability assumptions such as separability on the involved spaces are necessary. On the other hand, conditional analysis relies on a conditional version of classical results which are directly applied to sets of measurable functions where we consequently identify functions which are equal almost surely. Therefore, the formalization mainly consists in describing those sets for which a conditional version of classical results can be proved . Measurability is then systematically preserved by the application of a conditional version of classical results by construction. A conditional version of classical theorems exists under measure-theoretic countability assumptions such as (\sigma )-finiteness and by working on the measure algebra by quotienting out the ideal of null sets from the underlying probability space, while the topological restrictions of random set techniques can be relaxed.
In the following, we show that conditional analysis extends measurable selections and random set theory. More precisely, we establish a correspondence between basic objects in random set theory and their analogues in conditional analysis under the hypothesis of separability. We fix a complete probability space ((\varOmega ,{\mathcal {F}},{\mathbb {P}})) and a Polish space E. Recall that a closed-valued map (S:\varOmega \rightrightarrows E) (i.e., (S(\omega )\subset E) is a closed set for all (\omega \in \varOmega )) is Effros measurableFootnote 3 whenever (S^{-1}(O):={\omega \in \varOmega :S(\omega )\cap O\ne \emptyset }\in {\mathcal {F}}) for all open sets O in E. Throughout, we assume that (S(\omega )\ne \emptyset ) for a.a.(\omega \in \varOmega ), and we identify (S_1) and (S_2) whenever (S_1(\omega )=S_2(\omega )) for a.a.(\omega \in \varOmega ). We say that (x\in L^0(E)) is an a.s.measurable selection of S if (x(\omega )\in S(\omega )) for a.a.(\omega \in \varOmega ). We denote by (X_S) the set of all a.s.measurable selections of S.
The following theorem is due to Kabanov and Safarian [22, Proposition 5.4.3] in the particular case (E={\mathbb {R}}^d); see also [23, Theorem 2.1.6] and [24"), Theorem 2.3]Footnote 4 and the Remark5.1 for further details on the relations of these results. Due to limitation of space, we drop our alternative proof of Theorem5.1 which uses conditional analysis techniques, which can be found in the arXiv-version.
Theorem 5.1
Let (S:\varOmega \rightrightarrows E) be a closed-valued and Effros measurable mapping, and let (X\subset L^0(E)) be decomposable and sequentially closed.Footnote 5 Then, there exist closed-valued and Effros measurable mappings (S_X:\varOmega \rightrightarrows E) and (S_{X_S}:\varOmega \rightrightarrows E) satisfying the reciprocality relations (S=S_{X_S}) and (X=X_{S_X}).
A frequently employed concept in stochastic optimal control is a normal integrand; see, e.g., [25, 26], that is a function (f:\varOmega \times E\rightarrow {\mathbb {R}}) whose epigraphical mapping (S_f:\varOmega \rightrightarrows E\times {\mathbb {R}}, S_f(\omega ):={(x,r)\in E\times {\mathbb {R}}:f(\omega ,x)\le r}), is closed-valued and Effros measurable. A consequence of normality of an integrand is that (f(\omega ,x(\omega ))) is measurable in (\omega ) whenever (x:\varOmega \rightarrow E) is a measurable function. Moreover, a normal integrand (f(\omega ,x)) is measurable in (\omega ) for fixed x and lower semi-continuous in x for fixed (\omega ); cf. [3, Proposition 14.28]. We obtain the following functional version of Theorem5.1, where two normal integrands (f:\varOmega \times E\rightarrow {\mathbb {R}}) and (g:\varOmega \times E\rightarrow {\mathbb {R}}) are identified if their epigraphical mappings coincide a.s..
Corollary 5.1
Let (u:L^0(E)\rightarrow L^0) be stable and sequentially lower semi-continuous and let (f:\varOmega \times E\rightarrow {\mathbb {R}}) be a normal integrand. Then, there exist a stable and sequentially lower semi-continuous function (u_f:L^0(E)\rightarrow L^0) and a normal integrand (f_u:\varOmega \times E\rightarrow {\mathbb {R}}) such that (u_{f_u}=u) and (f_{u_f}=f).
Proof
Due to normality, (u_f:L^0(E)\rightarrow L^0) given by (x\mapsto (\omega \mapsto f(\omega ,x(\omega )))) is well defined. Direct inspection shows that (u_f) is stable and sequentially lower semi-continuous. Conversely, put (X:={(x,r)\in L^0(E\times {\mathbb {R}}):u(x)\le r}). By assumption, X is a stable and sequentially closed subset of (L^0(E\times {\mathbb {R}})). By Theorem5.1, there exist an Effros measurable and closed-valued map (S_X:\varOmega \rightrightarrows E\times {\mathbb {R}}) corresponding to X. Thus (f_u:\varOmega \times E\rightarrow {\mathbb {R}}) defined by (f(\omega ,x):=\inf S(\omega )x) a.s.is a normal integrand where (S(\omega )_x) denotes the _x-section of (S(\omega )). It follows from the reciprocality relations in Proposition5.1 that (u_{f_u}=u) and (f_{u_f}=f). (\square )
We compare the assumptions which underly conditional analysis and random set theory. Conditional analysis is applicable under the following two purely measure-theoretic hypotheses:
A probability measure ({\mathbb {P}}) on ((\varOmega ,{\mathcal {F}})) needs to be fixed a priori in order to identifyFootnote 6 sets, functions, relations, etc. almost surely. By the almost sure identification, we work with equivalence classes of functions and sets, and thus basically change to a pointfree perspective.
One consequently works in the context of conditional sets . In particular, all involved sets must satisfy stability under countable concatenations; cf.,Definition2.2.
Conditional analysis does not rely on the following topological assumptions which are prevalent in random set theory:
standard Borel space,Footnote 7 measure completeness, closed-valued mappings and Polish spaces.
The connections provided by Theorem5.1 and Corollary 5.1 suggest that a stochastic control problem can equally be formalized in the language of conditional set theory. In, e.g., 6, 25,26,27,[28")] some form of integrability is always assumed, which leads to further technicalities in the proofs; see also [29, 30] and the references therein for basic studies on the relations of (conditional) expectations and integrands. The main results in Sect.2 are established for general utilities which are not necessarily in the form of expected utilities, and no integrability assumptions are required.
Remark 5.1
Let E be a separable Banach space, and let (L^p(E)) be the Bochner space of all p-integrable functions (x:\varOmega \rightarrow E) for (p\in [1,\infty ]). For a set-valued mapping (S:\varOmega \rightrightarrows E), denote by (X_S^p:=X_S\cap L^p(E)) the set of p-integrable selections of S. Let (X\subset L^p(E)) be norm-closed. The result of Kabanov and Safarian [22, Theorem 5.4.3] states that (X=X_S^p) for an Effros measurable closed-valued mapping (S:\varOmega \rightrightarrows E) if and only if X is finitely decomposable and (L^p)-closed, see also [23, Theorem 2.1.6], while [24"), Theorem 2.3] by Molchanov and Lépinette replace (L^p)-closed by closed with respect to convergence in probability.
6 Conclusions
Conditional analysis allows for a treatment of stochastic control problems, which offers a viable alternative to classical measurable selection arguments. In particular, as controls and states are modeled in general conditional metric spaces, the mathematical restrictions of random set techniques such as finite dimension, separability, and completeness can be relaxed. The existence result Theorem2.1 covers a wide variety of situations such as wealth-dependent utility maximization under risk constraints, or utility maximization, where the number of traded assets depends on past decisions. We applied the novel notion of conditional compactness , which works in finite and infinite dimensional settings thanks to a conditional version of the Heine–Borel theorem [2, Theorem 4.6]. Conditional compactness extends the notion of compact-valued and Effros measurable mappings; see [32")]. The control sets work in any conditional metric space. This involves many examples, which are out of reach of the existing technology, for example conditional (L^p)-spaces on general probability spaces, (L^0({\mathbb {R}})^n) with a conditional dimension, and (L^0(X)), where X is a non-separable metric space. Another example are conditional weak topologies, which are not included in this article for which conditional analysis offers extensive tools as well. We plan to explore this direction in future work.
Notes
That is a Borel measurable function with an essentially separable range.
(u_T: X_T\rightarrow {\underline{\hbox {L}}}^0_T) is sequentially upper semi-continuous if (\limsup _{n\rightarrow \infty } u_T(x_n)\le u_T(x)) whenever (x_n\rightarrow x) in (X_T).
There are other measurability concepts besides Effros measurability; see, e.g., [23, Section 1.2]. One of them is graph-measurability, i.e., ({(\omega ,x)\in \varOmega \times E:x\in S(\omega )}) is product-measurable. For closed-valued mappings, graph measurability is equivalent to Effros measurability whenever the underlying measurable space is complete; cf.,e.g., [23, Theorem 2.3].
We are indebted to an anonymous referee for these references.
It is easy to see that a decomposable and sequentially closed set is necessarily countably decomposable or stable under countable concatenations in our sense.
Actually, one needs to fix a (\sigma )-ideal ({\mathcal {I}}) of ({\mathcal {F}}) such that the quotient ({\mathcal {F}}/{\mathcal {I}}) is a complete Boolean algebra. The ideal of null sets of a (\sigma )-finite measure is one such example, see and the references therein for more examples.
For example, for purposes of a dynamical programming principle in finite discrete time stochastic optimal control, more precisely the existence of disintegration of measure, the underlying measure space is additionally assumed to be standard Borel in .
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Acknowledgements
Open Access funding provided by Projekt DEAL. The authors would like to thank two anonymous referees for helpful comments, and Ilya Molchanov for discussions on Sect.5. The first two authors gratefully acknowledge financial support from DFG-Project KU 2740/2-1. The third author was supported by the Grants MTM2014-57838-C2-1-P (MINECO) and Fundación Séneca 20903/PD/18.
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Department of Mathematics, UCLA, Los Angeles, CA, USA
Asgar Jamneshan
Department of Mathematics and Statistics, University of Konstanz, Constance, Germany
Michael Kupper&José Miguel Zapata-García
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Jamneshan, A., Kupper, M. & Zapata-García, J.M. Parameter-Dependent Stochastic Optimal Control in Finite Discrete Time. J Optim Theory Appl186, 644–666 (2020).
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Keywords
Conditional analysis
Stochastic optimal control
Conditional metric spaces
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1 Introduction
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Cheridito, P., Kupper, M., Vogelpoth, N.: Conditional analysis on ({\mathbb{R}}^d). Set Optim. Appl. Proc. Math. Stat. 151, 179–211 (2015)
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MATHGoogle Scholar
Rockafellar, R.T., Wets, R.J.B.: The optimal recourse problem in discrete time: (L^1)-multipliers for inequality constraints. SIAM J. Control Optim. 16(1), 16–36 (1978)
MathSciNetMATHGoogle Scholar
Bielecki, T., Cialenco, I., Drapeau, S., Karliczek, M.: Dynamic assessement indices. Stochastics 88(1), 1–44 (2016)
MathSciNetMATHGoogle Scholar
Filipović, D., Kupper, M., Vogelpoth, N.: Approaches to conditional risk. SIAM J. Financ. Math. 3(1), 402–432 (2012)
MathSciNetMATHGoogle Scholar
Frittelli, M., Maggis, M.: Conditional certainty equivalent. Int. J. Theor. Appl. Finance 14(1), 41–59 (2011)
MathSciNetMATHGoogle Scholar
Frittelli, M., Maggis, M.: Complete duality for quasiconvex dynamic risk measures on modules of the (L^{p})-type. Stat. Risk Model. 31(1), 103–128 (2014)
MathSciNetMATHGoogle Scholar
Acciaio, B., Penner, I.: Dynamic risk measures. In: Nunno, G.D., Öksendal, B. (eds.) Advanced Mathematical Methods for Finance, pp. 1–34. Springer, Berlin (2011)
Google Scholar
Detlefsen, K., Scandolo, G.: Conditional and dynamic convex risk measure. Finance Stoch. 9, 539–561 (2005)
MathSciNetMATHGoogle Scholar
Drapeau, S., Jamneshan, A.: Conditional preferences and their numerical representations. J. Math. Econom. 63, 106–118 (2016)
MathSciNetMATHGoogle Scholar
Kreps, D.M., Porteus, E.L.: Temporal resolution of uncertainty and dynamic choice theory. Econometrica 46(1), 185–200 (1978)
MathSciNetMATHGoogle Scholar
Kreps, D.M., Porteus, E.L.: Temporal von Neumann–Morgenstern and induced preferences. J. Econom. Theory 20(1), 81–109 (1979)
MathSciNetMATHGoogle Scholar
Epstein, L.G., Zin, S.E.: Substitution, risk aversion, and the temporal behavior of consumption and asset returns: a theoretical framework. Econometrica 57(4), 937–69 (1989)
MathSciNetMATHGoogle Scholar
Machina, M.J.: Non-expected utility theory. In: Durlauf, S.N., Blume, L.E. (eds.) The New Palgrave Dictionary of Economics. Macmillan, Basingstoke (2008)
Google Scholar
Cheridito, P., Hu, Y.: Optimal consumption and investment in incomplete markets with general constraints. Stoch. Dyn. 11(2), 283–299 (2011)
MathSciNetMATHGoogle Scholar
Pennanen, T., Penner, I.: Hedging of claims with physical delivery under convex transaction costs. SIAM J. Financ. Math. 1(1), 158–178 (2010)
MathSciNetMATHGoogle Scholar
Kabanov, Y., Safarian, M.: Markets with Transaction Costs: Mathematical Theory. Springer, Berlin (2009)
MATHGoogle Scholar
Molchanov, I.: Theory of Random Sets, 2nd edn. Springer, London (2017)
MATHGoogle Scholar
Lépinette, E., Molchanov, I.: Risk arbitrage and hedging to acceptability. arXiv:1605.07884 (2016)
Pennanen, T., Perkkiö, A.P.: Stochastic programs without duality gaps. Math. Program. 136(1), 91–110 (2012)
MathSciNetMATHGoogle Scholar
Rockafellar, R.T., Wets, R.J.B.: Deterministic and stochastic optimization problems of Bolza type in discrete time. Stochastics 10(3–4), 273–312 (1983)
MathSciNetMATHGoogle Scholar
Pennanen, T., Perkkiö, A.P.: Shadow price of information in discrete time stochastic optimization. Math. Program. 168(1–2), 347–367 (2018)
MathSciNetMATHGoogle Scholar
Rockafellar, R.T., Wets, R.J.B.: Continuous versus measurable recourse in (N)-stage stochastic programming. J. Math. Anal. Appl. 48, 836–859 (1974)
MathSciNetMATHGoogle Scholar
Hiai, F., Umegaki, H.: Integrals, conditional expectations, and martingales of multvalued functions. J. Multivar. Anal. 7, 149–182 (1977)
MATHGoogle Scholar
Lépinette, E., Molchanov, I.: Conditional cores and conditional convex hulls of random sets. J. Math. Anal. Appl. 478(2), 368–392 (2019)
MathSciNetMATHGoogle Scholar
Bertsekas, D., Shreve, S.: Stochastic Optimal Control: The Discrete Time Case. Mathematics in Science and Engineering. Academic Press, London (1978)
MATHGoogle Scholar
Jamneshan, A., Zapata, J.M.: On Compactness in (L^0)-Modules. arXiv:1711.09785 (2017)
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Table 1.
| |
| Polycystic ovary syndrome Idiopathic Late-onset or non-classic congenital adrenal hyperplasia Androgen-secreting tumours Ovarian tumours Adrenal tumours Cushing's syndrome Hyperprolactinaemia Acromegaly Thyroid dysfunction Insulin resistance syndromes Iatrogenic Androgen therapy (testosterone) Danazol Androgenic progestins (levonorgestrel, norethindrone and norgestrel) Glucocorticoids |
Causes of Hirsutism
Table 2.
| |
| History |
| Medication history Onset and progression of the hirsutism Anovulation (such as menstrual irregularity) and reproductive history Family history Galactorrhoea Symptoms of virilisation Symptoms of thyroid dysfunction Weight gain |
| Physical Examination |
| Assessment of the amount, distribution and severity of hirsutism Signs of virilisation Skin changes (i.e. acne, seborrhoea, temporal balding, striae, acanthosis nigricans) Signs of Cushing's syndrome(i.e. moon face, muscle atrophy, truncal obesity) Breast discharge (spontaneous or expressible) Height, weight and a calculation of body mass index Abdominal and pelvic examinations |
History and Physical Examination in the Evaluation of Hirsutism
Table 3.
| Class of drug | Drug | Indication | Dosage | Side effects | Comments |
--- --- --- |
| Oral contraceptives | Ethinyl oestradiol with norgestimate, desogestrel, norethindrone, ethynodiol diacetate | Generalised hirsutism | One tablet per day for 21 days, followed by 7-day pill-free interval | GI distress, breast tenderness, headache, weight gain, emotional lability, intolerance to contact lenses | Pregnancy category X. Least androgenic progestin component preferred |
| Ethinyl oestradiol with drospirenone | | | Hyperkalaemia may occur. Contraindicated with hepatic dysfunction, renal insufficiency, adrenal disease | Monitor serum potassium during first cycle with concurrent use of NSAIDs, ACE inhibitors, angiotensin-II receptor blockers, heparin, potassium supplements, potassium sparing diuretics |
| Anti-androgens | Spironolactone (Aldactone) | Moderate or severe hirsutism | 50–200 mg/day | Hyperkalaemia (rare), male pseudohermaphroditism in fetus, gynaecomastia, decreased libido, gastrointestinal discomfort, irregular menstrual bleeding, hypotension, liver dysfunction | Pregnancy category D, monitor electrolytes |
| Cyproterone acetate | Moderate or severe hirsutism | Induction: 50–100 mg by mouth at bedtime, days: 5–15. Maintenance: 5 mg by mouth at bedtime, days: 5–15 | Male pseudohermaphroditism in fetus, irregular menstrual bleeding, decreased libido, nausea, depression, fatigue, mood changes and weight gain | Contraception is mandatory when taking cyproterone acetate and is recommended for at least 3 months after stopping treatment. Liver function should be checked regularly during long-term use |
| Flutamide (Eulexin) | Severe hirsutism | 125–250 mg, two to three times daily | Male pseudohermaphroditism in fetus, hepatotoxicity | Combine with other method of contraception. Pregnancy category D. Monitor liver function |
| Finasteride (Proscar) | Hirsutism | 5 mg daily | Minimal gastrointestinal disturbances, headaches, dry skin and decreased libido | Pregnancy category X. Monitor liver function |
| Glucocorticoids | Dexamethasone prednisone | Congenital adrenal hyperplasia, may be combined with oral contraceptives or Gn-RH agonists for severe hirsutism | Dexamethasone 0.5 mg nightly, prednisone 5–7.5 mg by mouth at bedtime | Weight gain, hypokalaemia, impaired glucose tolerance, adrenal suppression, decreased bone density, immune suppression, changes typical of Cushing's syndrome | Pregnancy category C |
| Gn-RH agonist | Leuprolide acetate (Lupron) | Alternative to oral contraceptive | 7/5 mg monthly intramuscularly, with 25–50 ?g transdermal oestradiol | Hot flushes, decreased bone mineral density, atrophic vaginitis | Pregnancy category X. Use with caution for short periods because of hypoestrogenic effect |
| Insulin-sensitising agents | Metformin (Glucophage) | Hirsutism, polycystic ovary syndrome | 850 mg twice a day or 500 mg three times daily (maximal dosage 2.0–2.5 g/day) | GI distress, lactic acidosis (rare with mortality nearly 50%), numerous drug interactions | Pregnancy category B. Resumption of ovulation may occur. Monitor liver function, confirm normal renal function before starting, and monitor |
Medications Used in the Treatment of Hirsutism
Adapted from Refs[2,4];)
GI = gastrointestinal; NSAIDs = non-steroidal anti-inflammatory drugs; ACE = angiotensin-converting enzyme; Gn-RH = gonadotropin-releasing
hormone.
Table 4.
| Technique | Body area | Advantages | Disadvantages |
--- --- |
| Bleaching | Face Arms | The hydrogen peroxide will soften the hair Inexpensive | Skin irritation Hair discolouration Lack of effectiveness |
| Shaving | All areas | Easy Inexpensive Effective | Quick regrowth Risk of folliculitis Time consuming Beard stubble |
| Depilatories | Extremities Groin Face | Quick Inexpensive Effective | Can be irritating Regrowth in days |
| Plucking | Face Eyebrows Nipples Bikini area | Inexpensive Regrowth can take weeks | Painful Slow Skin irritation Postinflammatory pigmentation Folliculitis Ingrown hairs Scarring |
| Waxing | Face Eyebrows Groin Trunk Extremities | Regrowth can take weeks | Painful Slow Expense Irritation Risk of folliculitis Possible thermal burn from applying the wax when too hot |
| Electrolysis | All areas, but usually the face | May give permanent removal | Painful Repeat treatments needed Very time consuming Expensive Risk of scarring and skin pigment changes |
| Laser and intense pulsed light | All areas | May give permanent hair reduction Efficient | Painful Repeat treatments needed Dark hair required Expensive Risk of scarring and skin pigment changes Rare reports of paradoxical hypertrichosis |
| Eflornithine 13.9% cream | Face Neck | On stopping, regrowth can take 2 months Minimal adverse effects Can be used in conjunction with other treatments such as lasers and intense pulsed light | Must be continued indefinitely to prevent regrowth Skin-related side effects such as stinging, burning and tingling are seen occasionally |
Advantages and Disadvantages of Hair Removal Techniques
Adapted from Ref;)
Table 5.
| History and physical findings | Diagnosis | Blood tests |
---
| Hirsutism, regular menses | Idiopathic hirsutism | No tests are indicated |
| Hirsutism, anovulatory dysfunction (such as menstrual irregularity), acne, seborrhoea, temporal balding, infertility, obesity, insulin resistance (such as acanthosis nigricans), metabolic complications (type 2 diabetes mellitus, metabolic syndrome, sleep apnoea) | Polycystic ovary syndrome | Total testosterone, luteinising hormone, follicle-stimulating hormone |
| Hirsutism, menstrual irregularity or primary amenorrhoea | Late-onset or non-classic congenital adrenal hyperplasia | 17?-hydroxyprogesterone, ACTH stimulation may be necessary to make diagnosis |
| Abrupt onset, rapid pace of development or progression of hirsutism, evidence of virilisation, mass lesions in abdominal and pelvic examinations | Androgen-secreting tumours | Total testosterone, dehydroepiandrosterone sulphate |
| Hirsutism, moon face, muscle atrophy, striae, thin skin, bruising, truncal obesity, buffalo hump and supraclavicular fat | Cushing's syndrome | Urine analysis for free cortisol or an overnight dexamethasone-suppression test |
| Hirsutism, history of galactorrhoea, presence of any breast discharge (spontaneous or expressible) | Hyperprolactinaemia | Prolactin |
| Hirsutism, coarsening of facial features or hand enlargement | Acromegaly | IGF-I level |
Recommended Blood Tests Based on History and Physical Examination
ACTH = adrenocorticotropic hormone; IGF = insulin-like growth factor.
References
Faculty and Disclosures
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Author(s)
Alireza Mofid, MD
Department of Internal Medicine, Tehran University of Medical Sciences, Sina Hospital, Tehran, Iran
Department of Internal Medicine, Tehran University of Medical Sciences, Sina Hospital, Tehran, Iran
Disclosure: Alireza Mofid, MD, has disclosed no relevant financial relationships.
Seyyed Ahmad Seyyed Alinaghi, MD
Iranian Research Center for HIV/AIDS (IRCHA), Imam Khomeini Hospital, Tehran, Iran
Iranian Research Center for HIV/AIDS (IRCHA), Imam Khomeini Hospital, Tehran, Iran
Disclosure: Seyyed Ahmad Seyyed Alinaghi, MD, has disclosed no relevant financial relationships.
Saeed Zandieh, MD
Private Clinic, Tehran, Iran
Private Clinic, Tehran, Iran
Disclosure: Saeed Zandieh, MD, has disclosed no relevant financial relationships.
Tahereh Yazdani, MD
Obesity Research Center, Sina Hospital, Tehran, Iran
Obesity Research Center, Sina Hospital, Tehran, Iran
Disclosure: Tahereh Yazdani, MD, has disclosed no relevant financial relationships.
Hirsutism
processing....
Summary
Hirsutism is defined as the excessive growth of terminal hair on the face and body of a female in a typical male pattern distribution. Hirsutism is a common clinical problem in women and the treatment depends on the cause of hirsutism. Untreated hirsutism can be associated with considerable loss of self-esteem and psychological morbidity. Hyperandrogenemia is the key trigger for excess hair growth. Polycystic ovary syndrome and idiopathic hirsutism are the most common cause of hirsutism. As with all medical problems, investigation begins with a careful history, examination and then investigation directed at the possible cause. A raised serum testosterone level of > 150 ng/dl (5.2 nmol/l) should prompt further investigations to exclude an underlying androgen-secreting tumour. The treatment of hirsutism is most effective using combination therapy, including lifestyle therapies, androgen suppression, peripheral androgen blockage and cosmetic treatments. Women should be warned not to expect improvement or at least 3-6 months after therapy is begun and lifelong therapy may be needed to prevent recurrence. The current review discusses definition, pathogenesis, differential diagnosis, diagnostic strategies, management, guidelines and the authors’ recommendations about hirsutism.
Review Criteria
This article was built on a systematic Medline search using the following search terms: hirsutism, definition, pathogenesis,
hyperandrogenism, aetiology, polycystic ovary syndrome, idiopathic hirsutism, differential diagnosis, diagnosis, evaluation,
management, cosmetic therapy, hormonal therapy and guidelines. Relevant articles were ordered and referenced in this manuscript.
Websites were also searched.
Message for the Clinic
Although hirsutism is often resulting from conditions that are not life-threatening but other aetiologies such as tumours
also must be considered. So, clinical evaluation should differentiate benign causes from tumours or other conditions that
require specific treatment.
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MERCK MANUALConsumer Version
Shingles
(Herpes Zoster)
ByKenneth M. Kaye, MD, Harvard Medical School
Reviewed/Revised Modified Dec 2023
v788834
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GET THE QUICK FACTS
Shingles is a painful skin rash caused by a viral infection that results from reactivation of the varicella-zoster virus, the virus that causes chickenpox.
Symptoms and Complications|
Diagnosis|
Treatment|
Prevention|
What causes the virus to reactivate is usually unknown, but sometimes reactivation occurs when a disorder or medication weakens the immune system.
Shingles causes a painful rash of fluid-filled blisters and sometimes results in chronic pain in the affected area.
Doctors diagnose shingles when typical blisters appear on a strip of skin.
Antiviral medications, if started before blisters appear, can help relieve symptoms and help them resolve sooner, but pain relievers, including opioids, are often needed.
For people aged 50 and over, the shingles vaccine can help prevent shingles.
Chickenpox and shingles are caused by the varicella-zoster virus:
Chickenpox is the initial infection.
Shingles is a reactivation of the virus, usually years later.
The varicella-zoster virus is a member of the herpesvirus family (herpesvirus type 3). So shingles is sometimes called herpes zoster.
During chickenpox, the virus infects collections of nerve cells (ganglia) of the spinal or cranial nerves. The virus remains in the ganglia in an inactive (dormant or latent) state. The virus may never cause symptoms again, or it may reactivate many years later. When it reactivates, the virus travels down the nerve fibers to the skin, where it creates painful sores resembling those of chickenpox. This outbreak of sores (shingles) almost always appears on a strip of the skin over the infected nerve fibers and only on one side of the body. This strip of skin, the area supplied by nerve fibers from a single spinal nerve region, is called a dermatome. Sores may also appear on the dermatomes next to the affected dermatome.
Unlike herpes simplex virus infections, which can recur many times, there is usually only one outbreak of shingles in a person's lifetime. Fewer than 6% of people have more than one outbreak.
Did You Know...
| |
| Fewer than 6% of people have more than one outbreak of shingles. |
Shingles may develop at any age but is most common after age 50. The chance of developing shingles increases as people age.
Most often, the reason for reactivation is unknown. However, reactivation sometimes occurs when the immune system is weakened by another disorder, such as AIDS, or by use of medications that suppress the immune system (for example to prevent rejection of a transplanted organ). The occurrence of shingles does not usually mean that the person has another serious disease.
Symptoms and Complications of Shingles
During the 2 or 3 days before shingles develops, most people have pain, a tingling sensation, or itching in a strip of skin (a dermatome) on one side of the body. Clusters of small, fluid-filled blisters surrounded by a small red area then develop on the strip of skin. Typically, the blisters occur only on the limited area of skin supplied by the infected nerve fibers. Most often, blisters appear on the trunk, usually on only one side. However, a few blisters may also appear elsewhere on the body. Typically, blisters continue to form for about 3 to 5 days. The affected area is usually sensitive to any stimulus, including light touch, and may be very painful.
Shingles Rash on the Chest
Image
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Shingles symptoms are usually less severe in children than in adults.
The blisters begin to dry and form a scab about 5 days after they appear. Until scabs appear, the blisters are contagious and contain varicella-zoster virus, which, if spread to susceptible people, can cause chickenpox. Having many blisters outside the affected dermatome or having blisters that persist for more than 2 weeks usually indicates that the immune system is not functioning normally.
Rarely, the affected skin becomes infected by bacteria. Scratching the blisters increases this risk. Bacterial infections increase the risk of scarring.
If shingles affects the nerve that leads to the eye, the eye may be infected. Eye infection is more common when sores occur on the forehead, near the eye, and especially on the tip of the nose. This infection (called herpes zoster ophthalmicus) can be serious. Even with treatment, vision may be affected.
The nerve that leads to the ear may also be affected. This infection (called herpes zoster oticus, or Ramsay Hunt syndrome) can cause blisters in the ear canal, pain, partial paralysis of the face, hearing loss, ringing in the ears (tinnitus), and sometimes vertigo.
Scarring or hyperpigmentation of the skin may occur, but most people recover without lasting effects. A few people, more commonly older adults, continue to have chronic pain in the area (postherpetic neuralgia).
Postherpetic neuralgia develops in about 10% of people who have had shingles. It is more common among older adults. In postherpetic neuralgia, people who have had shingles continue to have pain long after the rash has gone away. The pain occurs in areas of skin supplied by nerves infected with herpes zoster. Postherpetic neuralgia can be very severe and even incapacitating.
Diagnosis of Shingles
A doctor's evaluation
Rarely analysis or biopsy of a sample taken from the blisters
People who suspect they have shingles should see a doctor right away because to be effective, treatment must be started early. Doctors ask them to precisely describe the location of the pain. Pain in a vague band on one side of the body suggests shingles. If characteristic blisters appear in the typical pattern (on a strip of skin representing a dermatome), the diagnosis is clear.
Rarely, doctors take a sample from the blisters to be analyzed or do a skin biopsy to confirm the diagnosis.
Treatment of Shingles
Antiviral medications
Pain relievers
Shingles treatments include several antiviral medications. Antivirals such as famciclovir or valacyclovir taken by mouth are often given, particularly to older adults and to people with a weakened immune system (see table . Antivirals such as famciclovir or valacyclovir taken by mouth are often given, particularly to older adults and to people with a weakened immune system (see tableSome Antiviral Medications for Herpesvirus Infections). For people who are severely immunocompromised, intravenous acyclovir is recommended. These antivirals are taken by mouth. ). For people who are severely immunocompromised, intravenous acyclovir is recommended. These antivirals are taken by mouth.
The medications should be started as soon as shingles is suspected, before blisters appear if possible. Antivirals are likely not to be effective if started more than 3 days after blisters appear. These medications do not cure the disease, but they can help relieve shingles symptoms and shorten their duration.
If an eye or ear is involved, the appropriate specialist (ophthalmologist or otolaryngologist) should be consulted.
Wet compresses are soothing, but pain-relieving medications are often required. Nonsteroidal anti-inflammatory drugs (NSAIDs) or acetaminophen may be tried, but opioid pain relievers, taken by mouth, are sometimes necessary.are often required. Nonsteroidal anti-inflammatory drugs (NSAIDs) or acetaminophen may be tried, but opioid pain relievers, taken by mouth, are sometimes necessary.
To prevent bacterial infections from developing, people with shingles should keep the affected skin clean and dry and should not scratch the blisters.
Prevention of Shingles
Preventing chickenpox by vaccinating children and adults who do not have immunity with the varicella vaccine is recommended.
There are two shingles vaccines: a newer recombinant vaccine and an older, weakened live-virus vaccine. The older, live-virus vaccine is no longer available in the United States, but it remains available in many other nations. The newer recombinant vaccine contains only pieces of a virus. The recombinant vaccine is preferred and is recommended for healthy people aged 50 or over, regardless of whether they recall having had chickenpox or shingles and regardless of whether they have been vaccinated with the older shingles vaccine. The recombinant vaccine is also recommended for adults 19 years and older who are or will be immunodeficient or immunosuppressed because of disease or therapy.
The recombinant herpes zoster vaccine is given in two doses, injected into a muscle. The doses are given 2 to 6 months apart and at least 2 months after the older (weakened live-virus) vaccine for people who have had that vaccine.
The recombinant vaccine significantly reduces the chance of getting shingles and postherpetic neuralgia.
When shingles occurs, taking antiviral medications may reduce the risk of developing postherpetic neuralgia.
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6603 | https://virtualnerd.com/common-core/hsn-number-quantity/HSN-Q-quantities/A/1/dimensional-analysis-convert-units-both-parts-rate | How Do You Use Dimensional Analysis to Convert Units on Both Parts of a Rate? | Virtual Nerd
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How Do You Use Dimensional Analysis to Convert Units on Both Parts of a Rate?
How Do You Use Dimensional Analysis to Convert Units on Both Parts of a Rate?
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Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship.
Image 2: What are Rates and Unit Rates?Image 3: What are Rates and Unit Rates?
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Can you do 100 sit-ups in 2 minutes? That's a rate! Driving a car going 40 miles per hour? That's a unit rate! Watch this tutorial to learn about rate and unit rate (and the difference!).
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Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Image 4: How Do You Use Dimensional Analysis to Convert Units on One Part of a Rate?Image 5: How Do You Use Dimensional Analysis to Convert Units on One Part of a Rate?
How Do You Use Dimensional Analysis to Convert Units on One Part of a Rate?
Word problems are a great way to see math in action! In this tutorial, learn how to use the information given in a word problem to create a rate. Then, find and use a conversion factor to convert a unit in the rate. Take a look!
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Contents
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1
Concept
Toggle Concept subsection
- 1.1
2-dimensional analogy
- 2
FLRW models
- 3
General metric
Toggle General metric subsection
- 3.1
Reduced-circumference polar coordinates
- 3.2
Hyperspherical coordinates
- 3.3
Cartesian coordinates
- 4
Curvature
Toggle Curvature subsection
- 4.1
Cartesian coordinates
- 4.2
Spherical coordinates
- 5
Name and history
- 6
Current status
- 7
References
- 8
Further reading
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Friedmann–Lemaître–Robertson–Walker metric
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From Wikipedia, the free encyclopedia
Metric based on the exact solution of Einstein's field equations of general relativity
| |
| Part of a series on |
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| hide Expansion · Future - Hubble's law · Redshift - Expansion of the universe - FLRW metric · Friedmann equations - Lambda-CDM model - Future of an expanding universe - Ultimate fate of the universe |
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The Friedmann–Lemaître–Robertson–Walker metric (FLRW; /ˈfriːdmən ləˈmɛtrə ... /) is a metric that describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected. The general form of the metric follows from the geometric properties of homogeneity and isotropy. Depending on geographical or historical preferences, the set of the four scientists – Alexander Friedmann, Georges Lemaître, Howard P. Robertson, and Arthur Geoffrey Walker – is variously grouped as Friedmann, Friedmann–Robertson–Walker (FRW), Robertson–Walker (RW), or Friedmann–Lemaître (FL). When combined with Einstein's field equations, the metric gives the Friedmann equation, which has been developed into the Standard Model of modern cosmology and further developed into the Lambda-CDM model.
Concept
[edit]
The metric is a consequence of assuming that the mass in the universe has constant density – homogeneity – and is the same in all directions – isotropy. Assuming isotropy alone is sufficient to reduce the possible motions of mass in the universe to radial velocity variations. The Copernican principle, that our observation point in the universe is equivalent to every other point, combined with isotropy, ensures homogeneity. Without the principle, a metric would need to be extracted from astronomical data, which may not be possible.: 408 Direct observation of stars has shown their velocities to be dominated by radial recession, validating these assumptions for cosmological models.: 65
To measure distances in this space, that is to define a metric, we can compare the positions of two points in space, moving along with their local radial velocity of mass. Such points can be thought of as ideal galaxies. Each galaxy can be given a clock to track local time, with the clocks synchronized by imagining the radial velocities run backwards until the clocks coincide in space. The equivalence principle applied to each galaxy means distance measurements can be made using special relativity locally. So a distance
d
τ
{\displaystyle d\tau }
can be related to the local time t and the coordinates:
c
2
d
τ
2
c
2
d
t
2
−
d
x
2
−
d
y
2
−
d
z
2
{\displaystyle c^{2}d\tau ^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}}
An isotropic, homogeneous mass distribution is highly symmetric. Rewriting the metric in spherical coordinates reduces four coordinates to three coordinates. The radial coordinate is written as a product of a comoving coordinate, r, and a time-dependent scale factor R(t). The resulting metric can be written in several forms. Two common ones are:
c
2
d
τ
2
c
2
d
t
2
−
R
2
(
t
)
(
d
r
2
+
S
k
2
(
r
)
d
ψ
2
)
{\displaystyle c^{2}d\tau ^{2}=c^{2}dt^{2}-R^{2}(t)\left(dr^{2}+S_{k}^{2}(r)d\psi ^{2}\right)}
or
c
2
d
τ
2
c
2
d
t
2
−
R
2
(
t
)
(
d
r
2
1
−
k
r
2
+
r
2
d
ψ
2
)
{\displaystyle c^{2}d\tau ^{2}=c^{2}dt^{2}-R^{2}(t)\left({\frac {dr^{2}}{1-kr^{2}}}+r^{2}d\psi ^{2}\right)}
where
ψ
{\displaystyle \psi }
is the angle between the two locations and
S
−
1
(
r
)
=
sinh
(
r
)
,
S
0
1
,
S
1
sin
(
r
)
.
{\displaystyle S_{-1}(r)=\sinh(r),S_{0}=1,S_{1}=\sin(r).}
(The meaning of r in these equations is not the same). Other common variations use a dimensionless scale factor
a
(
t
)
=
R
(
t
)
R
0
{\displaystyle a(t)={\frac {R(t)}{R_{0}}}}
where time zero is now.: 70
2-dimensional analogy
[edit]
The time-dependent scale factor
R
(
t
)
{\displaystyle R(t)}
, which plays a critical role in cosmology, has an analog in the radius of a sphere. A sphere is a 2-dimensional surface embedded in a 3-dimensional space. The radius of a sphere lives in the third dimension: it is not part of the 2-dimensional surface. However, the value of this radius affects distances measured on the two-dimensional surface. Similarly, the cosmological scale factor is not a distance in our 3-dimensional space, but its value affects the measurement of distances.: 147
FLRW models
[edit]
Main article: Friedmann equations
Applying the metric to cosmology and predicting its time evolution requires Einstein's field equations and a way of calculating the density,
ρ
(
t
)
,
{\displaystyle \rho (t),}
such as a cosmological equation of state.
This process allows an approximate analytic solution of Einstein's field equations
G
μ
ν
+
Λ
g
μ
ν
=
κ
T
μ
ν
{\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }}
giving the Friedmann equations when the energy–momentum tensor is similarly assumed to be isotropic and homogeneous.
Models based on the FLRW metric and obeying the Friedmann equations are called FRW models.: 73
Direct observation of stars has shown their velocities to be dominated by radial recession, validating these assumptions for cosmological models.: 65
These models are the basis of the standard Big Bang cosmological model, including the current ΛCDM model.: 25.1.3
General metric
[edit]
The FLRW metric assumes homogeneity and isotropy of space.: 404 It also assumes that the spatial component of the metric can be time-dependent. The generic metric that meets these conditions is
−
c
2
d
τ
2
−
c
2
d
t
2
+
a
(
t
)
2
d
Σ
2
,
{\displaystyle -c^{2}\mathrm {d} \tau ^{2}=-c^{2}\mathrm {d} t^{2}+{a(t)}^{2}\mathrm {d} \mathbf {\Sigma } ^{2},}
where
Σ
{\displaystyle \mathbf {\Sigma } }
ranges over a 3-dimensional space of uniform curvature, that is, elliptical space, Euclidean space, or hyperbolic space. It is normally written as a function of three spatial coordinates, but there are several conventions for doing so, detailed below.
d
Σ
{\displaystyle \mathrm {d} \mathbf {\Sigma } }
does not depend on
t
{\displaystyle t}
– all of the time dependence is in the function
a
(
t
)
{\displaystyle a(t)}
, known as the "scale factor".
Reduced-circumference polar coordinates
[edit]
In reduced-circumference polar coordinates, the spatial metric has the form
d
Σ
2
d
r
2
1
−
k
r
2
+
r
2
d
Ω
2
,
where
d
Ω
2
d
θ
2
+
sin
2
θ
d
ϕ
2
.
{\displaystyle \mathrm {d} \mathbf {\Sigma } ^{2}={\frac {\mathrm {d} r^{2}}{1-kr^{2}}}+r^{2}\mathrm {d} \mathbf {\Omega } ^{2},\quad {\text{ where }}\mathrm {d} \mathbf {\Omega } ^{2}=\mathrm {d} \theta ^{2}+\sin ^{2}\theta \,\mathrm {d} \phi ^{2}.}
k
{\displaystyle k}
is a constant representing the curvature of the space. There are two common unit conventions:
k
{\displaystyle k}
may be taken to have units of length−2, in which case
r
{\displaystyle r}
has units of length and
a
(
t
)
{\displaystyle a(t)}
is unitless.
k
{\displaystyle k}
is then the Gaussian curvature of the space at the time when
a
(
t
)
=
1
{\displaystyle a(t)=1}
.
r
{\displaystyle r}
is sometimes called the reduced circumference because it is equal to the measured circumference of a circle (at that value of
r
{\displaystyle r}
), centered at the origin, divided by
2
π
{\displaystyle 2\pi }
(like the
r
{\displaystyle r}
of Schwarzschild coordinates). Where appropriate,
a
(
t
)
{\displaystyle a(t)}
is often chosen to equal 1 in the present cosmological era, so that
d
Σ
{\displaystyle \mathrm {d} \mathbf {\Sigma } }
measures comoving distance.
- Alternatively,
k
{\displaystyle k}
may be taken to belong to the set {−1, 0, +1} (for negative, zero, and positive curvature, respectively). Then
r
{\displaystyle r}
is unitless and
a
(
t
)
{\displaystyle a(t)}
has units of length. When
k
=
±
1
{\displaystyle k=\pm 1}
,
a
(
t
)
{\displaystyle a(t)}
is the radius of curvature of the space and may also be written
R
(
t
)
{\displaystyle R(t)}
.
A disadvantage of reduced circumference coordinates is that they cover only half of the 3-sphere in the case of positive curvature—circumferences beyond that point begin to decrease, leading to degeneracy. (This is not a problem if space is elliptical, i.e., a 3-sphere with opposite points identified.)
Hyperspherical coordinates
[edit]
In hyperspherical or curvature-normalized coordinates, the coordinate
r
{\displaystyle r}
is proportional to radial distance; this gives
d
Σ
2
d
r
2
+
S
k
(
r
)
2
d
Ω
2
{\displaystyle \mathrm {d} \mathbf {\Sigma } ^{2}=\mathrm {d} r^{2}+S_{k}(r)^{2}\,\mathrm {d} \mathbf {\Omega } ^{2}}
where
d
Ω
{\displaystyle \mathrm {d} \mathbf {\Omega } }
is as before and
S
k
(
r
)
=
{
k
−
1
sin
(
r
k
)
,
k
0
r
,
k
0
|
k
|
−
1
sinh
(
r
|
k
|
)
,
k
<
0.
{\displaystyle S_{k}(r)={\begin{cases}{\sqrt {k}}^{\,-1}\sin(r{\sqrt {k}}),&k>0\r,&k=0\{\sqrt {|k|}}^{\,-1}\sinh(r{\sqrt {|k|}}),&k<0.\end{cases}}}
As before, there are two common unit conventions:
k
{\displaystyle k}
may be taken to have units of length−2, in which case
r
{\displaystyle r}
has units of length and
a
(
t
)
{\displaystyle a(t)}
is unitless.
k
{\displaystyle k}
is then the Gaussian curvature of the space at the time when
a
(
t
)
=
1
{\displaystyle a(t)=1}
. Where appropriate,
a
(
t
)
{\displaystyle a(t)}
is often chosen to equal 1 in the present cosmological era, so that
d
Σ
{\displaystyle \mathrm {d} \mathbf {\Sigma } }
measures comoving distance.
- Alternatively, as before,
k
{\displaystyle k}
may be taken to belong to the set {−1 ,0, +1} (for negative, zero, and positive curvature respectively). Then
r
{\displaystyle r}
is unitless and
a
(
t
)
{\displaystyle a(t)}
has units of length. When
k
=
±
1
{\displaystyle k=\pm 1}
,
a
(
t
)
{\displaystyle a(t)}
is the radius of curvature of the space and may also be written
R
(
t
)
{\displaystyle R(t)}
. Note that when
k
=
+
1
{\displaystyle k=+1}
,
r
{\displaystyle r}
is essentially a third angle along with
θ
{\displaystyle \theta }
and
ϕ
{\displaystyle \phi }
. The letter
χ
{\displaystyle \chi }
may be used instead of
r
{\displaystyle r}
.
Though it is usually defined piecewise as above,
S
{\displaystyle S}
is an analytic function of both
k
{\displaystyle k}
and
r
{\displaystyle r}
. It can also be written as a power series
S
k
(
r
)
=
∑
n
0
∞
(
−
1
)
n
k
n
r
2
n
+
1
(
2
n
+
1
)
!
=
r
−
k
r
3
6
+
k
2
r
5
120
−
⋯
{\displaystyle S_{k}(r)=\sum _{n=0}^{\infty }{\frac {{\left(-1\right)}^{n}k^{n}r^{2n+1}}{(2n+1)!}}=r-{\frac {kr^{3}}{6}}+{\frac {k^{2}r^{5}}{120}}-\cdots }
or as
S
k
(
r
)
=
r
s
i
n
c
(
r
k
)
,
{\displaystyle S_{k}(r)=r\;\mathrm {sinc} \,(r{\sqrt {k}}),}
where
s
i
n
c
{\displaystyle \mathrm {sinc} }
is the unnormalized sinc function and
k
{\displaystyle {\sqrt {k}}}
is one of the imaginary, zero, or real square roots of
k
{\displaystyle k}
. These definitions are valid for all
k
{\displaystyle k}
.
Cartesian coordinates
[edit]
When
k
0
{\displaystyle k=0}
one may write simply
d
Σ
2
d
x
2
+
d
y
2
+
d
z
2
.
{\displaystyle \mathrm {d} \mathbf {\Sigma } ^{2}=\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}.}
This can be extended to
k
≠
0
{\displaystyle k\neq 0}
by defining
x
=
r
cos
θ
,
y
=
r
sin
θ
cos
ϕ
,
z
=
r
sin
θ
sin
ϕ
,
{\displaystyle {\begin{aligned}x&=r\cos \theta \,,\y&=r\sin \theta \cos \phi \,,\z&=r\sin \theta \sin \phi \,,\end{aligned}}}
where
r
{\displaystyle r}
is one of the radial coordinates defined above, but this is rare.
Curvature
[edit]
Cartesian coordinates
[edit]
In flat
(
k
=
0
)
{\displaystyle (k=0)}
FLRW space using Cartesian coordinates, the surviving components of the Ricci tensor are
R
t
t
=
−
3
a
¨
a
,
R
x
x
=
R
y
y
=
R
z
z
=
c
−
2
(
a
a
¨
+
2
a
˙
2
)
{\displaystyle R_{tt}=-3{\frac {\ddot {a}}{a}},\quad R_{xx}=R_{yy}=R_{zz}=c^{-2}\left(a{\ddot {a}}+2{\dot {a}}^{2}\right)}
and the Ricci scalar is
R
6
c
−
2
(
a
¨
(
t
)
a
(
t
)
+
a
˙
2
(
t
)
a
2
(
t
)
)
.
{\displaystyle R=6c^{-2}\left({\frac {{\ddot {a}}(t)}{a(t)}}+{\frac {{\dot {a}}^{2}(t)}{a^{2}(t)}}\right).}
Spherical coordinates
[edit]
In more general FLRW space using spherical coordinates (called "reduced-circumference polar coordinates" above), the surviving components of the Ricci tensor are[failed verification]
R
t
t
=
−
3
a
¨
a
,
R
r
r
=
c
−
2
(
a
a
¨
+
2
a
˙
2
)
+
2
k
1
−
k
r
2
R
θ
θ
=
r
2
[
c
−
2
(
a
a
¨
+
2
a
˙
2
)
+
2
k
]
R
ϕ
ϕ
=
r
2
sin
2
(
θ
)
[
c
−
2
(
a
a
¨
+
2
a
˙
2
)
+
2
k
]
{\displaystyle {\begin{aligned}R_{tt}&=-3{\frac {\ddot {a}}{a}},\[1ex]R_{rr}&={\frac {c^{-2}\left(a{\ddot {a}}+2{\dot {a}}^{2}\right)+2k}{1-kr^{2}}}\[1ex]R_{\theta \theta }&=r^{2}\left[c^{-2}\left(a{\ddot {a}}+2{\dot {a}}^{2}\right)+2k\right]\[1ex]R_{\phi \phi }&=r^{2}\sin ^{2}(\theta )\left[c^{-2}\left(a{\ddot {a}}+2{\dot {a}}^{2}\right)+2k\right]\end{aligned}}}
and the Ricci scalar is
R
6
c
2
(
a
¨
(
t
)
a
(
t
)
+
a
˙
2
(
t
)
a
2
(
t
)
+
c
2
k
a
2
(
t
)
)
.
{\displaystyle R={\frac {6}{c^{2}}}\left({\frac {{\ddot {a}}(t)}{a(t)}}+{\frac {{\dot {a}}^{2}(t)}{a^{2}(t)}}+{\frac {c^{2}k}{a^{2}(t)}}\right).}
Name and history
[edit]
See also: Friedmann equations § History
| | |
--- |
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In 1922 and 1924, the Soviet mathematician Alexander Friedmann and in 1927, Georges Lemaître, a Belgian priest, astronomer, and periodic professor of physics at the Catholic University of Leuven, arrived independently at results that relied on the metric.
Howard P. Robertson from the US and Arthur Geoffrey Walker from the UK explored the problem further during the 1930s. In 1935, Robertson and Walker rigorously proved that the FLRW metric is the only one on a spacetime that is spatially homogeneous and isotropic (as noted above, this is a geometric result and is not tied specifically to the equations of general relativity, which Friedmann and Lemaître always assumed).
This solution, often called the Robertson–Walker metric since they proved its generic properties, is different from the dynamical "Friedmann–Lemaître" models. These models are specific solutions for a(t) that assume that the only contributions to stress-energy are cold matter ("dust"), radiation, and a cosmological constant.
Current status
[edit]
See also: Shape of the universe
Unsolved problem in physics
Is the universe homogeneous and isotropic at large enough scales, as claimed by the cosmological principle? Is the CMB dipole purely kinematic, or does it signal a possible breakdown of the FLRW metric? Is the Friedmann–Lemaître–Robertson–Walker metric valid in the late universe?
More unsolved problems in physics
The current standard model of cosmology, the Lambda-CDM model, uses the FLRW metric. By combining the observation data from some experiments, such as WMAP and Planck, with theoretical results of the Ehlers–Geren–Sachs theorem and its generalization, astrophysicists now agree that the early universe is almost homogeneous and isotropic (when averaged over a very large scale) and thus nearly a FLRW spacetime. That being said, attempts to confirm the purely kinematic interpretation of the Cosmic Microwave Background (CMB) dipole through studies of radio galaxies and quasars show disagreement in the magnitude. Taken at face value, these observations are at odds with the Universe being described by the FLRW metric. Moreover, one can argue that there is a maximum value to the Hubble constant within an FLRW cosmology tolerated by current observations,
H
0
{\displaystyle H_{0}}
= 71±1 km/s/Mpc, and depending on how local determinations converge, this may point to a breakdown of the FLRW metric in the late universe, necessitating an explanation beyond the FLRW metric.
References
[edit]
^ For an early reference, see Robertson (1935); Robertson assumes multiple connectedness in the positive curvature case and says that "we are still free to restore" simple connectedness.
^ Lachieze-Rey, M.; Luminet, J.-P. (1995). "Cosmic Topology". Physics Reports. 254 (3): 135–214. arXiv:gr-qc/9605010. Bibcode:1995PhR...254..135L. doi:10.1016/0370-1573(94)00085-H. S2CID 119500217.
^ Ellis, G. F. R.; van Elst, H. (1999). "Cosmological models (Cargèse lectures 1998)". In Marc Lachièze-Rey (ed.). Theoretical and Observational Cosmology. NATO Science Series C. Vol. 541. pp. 1–116. arXiv:gr-qc/9812046. Bibcode:1999ASIC..541....1E. ISBN 978-0792359463.
^ Bergström, Lars; Goobar, Ariel (2008). Cosmology and particle astrophysics. Springer Praxis books in astronomy and planetary science (2. ed., reprinted ed.). Chichester, UK: Praxis Publ. p. 61. ISBN 978-3-540-32924-4.
^ Steven Weinberg (1972). Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. John Wiley & Sons, Inc. ISBN 978-0-471-92567-5.
^ Jump up to: a b c d Peacock, J. A. (1998-12-28). Cosmological Physics (1 ed.). Cambridge University Press. doi:10.1017/cbo9780511804533. ISBN 978-0-521-41072-4.
^ Tropp, Eduard A.; Frenkel, Viktor Ya.; Chernin, Artur D. (1993-06-03). Alexander A Friedmann: The Man who Made the Universe Expand. Translated by Dron, Alexander; Burov, Michael (1 ed.). Cambridge University Press. doi:10.1017/cbo9780511608131. ISBN 978-0-521-38470-4.
^ Navas, S.; Amsler, C.; Gutsche, T.; Hanhart, C.; Hernández-Rey, J. J.; Lourenço, C.; Masoni, A.; Mikhasenko, M.; Mitchell, R. E.; Patrignani, C.; Schwanda, C.; Spanier, S.; Venanzoni, G.; Yuan, C. Z.; Agashe, K. (2024-08-01). "Review of Particle Physics". Physical Review D. 110 (3): 030001. doi:10.1103/PhysRevD.110.030001. hdl:20.500.11850/695340. ISSN 2470-0010.{{cite journal}}: CS1 maint: article number as page number (link)
^ Weinberg, Steven (1972). Gravitation and cosmology: principles and applications of the general theory of relativity. New York: Wiley. ISBN 978-0-471-92567-5.
^ Wald, Robert M. (1984). General relativity. Chicago: University of Chicago Press. p. 116. ISBN 978-0-226-87032-8.
^ Carroll, Sean M. (2019). Spacetime and geometry: an introduction to general relativity. New York: Cambridge University Press. pp. 329–333. ISBN 978-1-108-48839-6.
^ Wald, Robert M. (1984). General relativity. Chicago: University of Chicago Press. p. 97. ISBN 978-0-226-87032-8.
^ "Cosmology" (PDF). p. 23. Archived from the original (PDF) on Jan 11, 2020.
^ Friedmann, Alexander (1922). "Über die Krümmung des Raumes". Zeitschrift für Physik A. 10 (1): 377–386. Bibcode:1922ZPhy...10..377F. doi:10.1007/BF01332580. S2CID 125190902.
^ Friedmann, Alexander (1924). "Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes". Zeitschrift für Physik A (in German). 21 (1): 326–332. Bibcode:1924ZPhy...21..326F. doi:10.1007/BF01328280. S2CID 120551579. English trans. in 'General Relativity and Gravitation' 1999 vol.31, 31–
^ Lemaître, Georges (1931), "Expansion of the universe, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulæ", Monthly Notices of the Royal Astronomical Society, 91 (5): 483–490, Bibcode:1931MNRAS..91..483L, doi:10.1093/mnras/91.5.483 translated from Lemaître, Georges (1927), "Un univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques", Annales de la Société Scientifique de Bruxelles, A47: 49–56, Bibcode:1927ASSB...47...49L
^ Lemaître, Georges (1933), "l'Univers en expansion", Annales de la Société Scientifique de Bruxelles, A53: 51–85, Bibcode:1933ASSB...53...51L
^ Robertson, H. P. (1935), "Kinematics and world structure", Astrophysical Journal, 82: 284–301, Bibcode:1935ApJ....82..284R, doi:10.1086/143681
^ Robertson, H. P. (1936), "Kinematics and world structure II", Astrophysical Journal, 83: 187–201, Bibcode:1936ApJ....83..187R, doi:10.1086/143716
^ Robertson, H. P. (1936), "Kinematics and world structure III", Astrophysical Journal, 83: 257–271, Bibcode:1936ApJ....83..257R, doi:10.1086/143726
^ Walker, A. G. (1937), "On Milne's theory of world-structure", Proceedings of the London Mathematical Society, Series 2, 42 (1): 90–127, Bibcode:1937PLMS...42...90W, doi:10.1112/plms/s2-42.1.90
^ Jump up to: a b c d Abdalla, Elcio; et al. (June 2022). "Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies". Journal of High Energy Astrophysics. 34: 49–211. arXiv:2203.06142v1. Bibcode:2022JHEAp..34...49A. doi:10.1016/j.jheap.2022.04.002. S2CID 247411131.
^ Billings, Lee (April 15, 2020). "Do We Live in a Lopsided Universe?". Scientific American. Retrieved March 24, 2022.
^ Migkas, K.; Schellenberger, G.; Reiprich, T. H.; Pacaud, F.; Ramos-Ceja, M. E.; Lovisari, L. (April 2020). "Probing cosmic isotropy with a new X-ray galaxy cluster sample through the L X – T scaling relation". Astronomy & Astrophysics. 636 (April 2020): A15. arXiv:2004.03305. Bibcode:2020A&A...636A..15M. doi:10.1051/0004-6361/201936602. ISSN 0004-6361. S2CID 215238834. Retrieved 24 March 2022.
^ Krishnan, Chethan; Mohayaee, Roya; Colgáin, Eoin Ó; Sheikh-Jabbari, M. M.; Yin, Lu (16 September 2021). "Does Hubble Tension Signal a Breakdown in FLRW Cosmology?". Classical and Quantum Gravity. 38 (18): 184001. arXiv:2105.09790. Bibcode:2021CQGra..38r4001K. doi:10.1088/1361-6382/ac1a81. ISSN 0264-9381. S2CID 234790314.
^ See pp. 351ff. in Hawking, Stephen W.; Ellis, George F. R. (1973), The large scale structure of space-time, Cambridge University Press, ISBN 978-0-521-09906-6. The original work is Ehlers, J., Geren, P., Sachs, R.K.: Isotropic solutions of Einstein-Liouville equations. J. Math. Phys. 9, 1344 (1968). For the generalization, see Stoeger, W. R.; Maartens, R; Ellis, George (2007), "Proving Almost-Homogeneity of the Universe: An Almost Ehlers-Geren-Sachs Theorem", Astrophys. J., 39: 1–5, Bibcode:1995ApJ...443....1S, doi:10.1086/175496.
^ See Siewert et al. for a recent summary of results Siewert, Thilo M.; Schmidt-Rubart, Matthias; Schwarz, Dominik J. (2021). "Cosmic radio dipole: Estimators and frequency dependence". Astronomy & Astrophysics. 653: A9. arXiv:2010.08366. Bibcode:2021A&A...653A...9S. doi:10.1051/0004-6361/202039840. S2CID 223953708.
^ Secrest, Nathan J.; Hausegger, Sebastian von; Rameez, Mohamed; Mohayaee, Roya; Sarkar, Subir; Colin, Jacques (2021-02-25). "A Test of the Cosmological Principle with Quasars". The Astrophysical Journal. 908 (2): L51. arXiv:2009.14826. Bibcode:2021ApJ...908L..51S. doi:10.3847/2041-8213/abdd40. S2CID 222066749.
^ Krishnan, Chethan; Mohayaee, Roya; Ó Colgáin, Eoin; Sheikh-Jabbari, M. M.; Yin, Lu (2021-05-25). "Does Hubble tension signal a breakdown in FLRW cosmology?". Classical and Quantum Gravity. 38 (18): 184001. arXiv:2105.09790. Bibcode:2021CQGra..38r4001K. doi:10.1088/1361-6382/ac1a81. S2CID 234790314.
Further reading
[edit]
North, John David (1990). The measure of the universe: a history of modern cosmology. New York: Dover Publications. ISBN 978-0-486-66517-7.
Harrison, E. R. (1967). "Classification of Uniform Cosmological Models". Monthly Notices of the Royal Astronomical Society. 137 (1): 69–79. Bibcode:1967MNRAS.137...69H. doi:10.1093/mnras/137.1.69. ISSN 0035-8711.
D'Inverno, Ray (1992). Introducing Einstein's relativity (Repr ed.). Oxford [England] : New York: Clarendon Press ; Oxford University Press. ISBN 978-0-19-859686-8.. (See Chapter 23 for a particularly clear and concise introduction to the FLRW models.)
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Sensor Types, edited by Kazunori Ikebukuro
Cloud chambers, bubble chambers, and superheated drop detectors
Charles Wilson invented the cloud chamber in 1911, a device used to detect ionizing charged particles (Wilson, 1911, 1912). A cloud chamber is a sealed transparent container holding a supercooled, supersaturated vapor of water, alcohol, or aromatic hydrocarbon. Alpha particles or beta particles ionize the saturated vapor as they pass through it. The ionized vapor molecules form nucleation centers for condensation and a visible mist forms along the ionization track. Wilson shared the 1927 Nobel Prize in Physics for inventing the cloud chamber.
Donald Glaser invented the bubble chamber in 1952, a type of detector closely related to the cloud chamber (Glaser, 1952). The bubble chamber is a vessel filled with a superheated transparent liquid used to detect charged particles as they move through the liquid. Charged particles deposit sufficient energy in the liquid so that it begins to boil along the ionization track, thereby forming a visible string of bubbles. Glaser received the 1960 Nobel Prize in Physics for inventing the bubble chamber.
In 1979, Robert Apfel invented the superheated drop detector, which is similar to the cloud chamber and the bubble chamber (Apfel, 1979). A fluid can be superheated to temperatures and pressures corresponding to the vapor region in the phase diagram. This metastable state is fragile and typically short-lived due to the high number of microscopic particles or gas pockets normally present at the container surfaces. However, a liquid may be kept in steady-state superheated conditions by fractionating it into droplets and dispersing them in an immiscible host fluid, for example, suspending freon droplets in a gel solution. This procedure creates perfectly smooth spherical interfaces, free of nucleating impurities or irregularities. Radiation particles, including neutrons, interacting in the fluid can disrupt this balance between the liquid and vapor phase of a superheated droplet, causing it to burst into a sizable bubble of vapor. Bubble formation is measured from the volume of vapor expelled or by detecting individual vaporizations acoustically, i.e., “listening” to the radiation interactions.
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Reference work2023, Encyclopedia of Sensors and BiosensorsDouglas S. McGregor, J. Kenneth Shultis
Chapter
Detection and Measurement Techniques
2002, Radiochemistry and Nuclear Chemistry (Third Edition)GREGORY R. CHOPPIN, ... JAN RYDBERG
8.1.1 Cloud and bubble chambers
The principle of a cloud chamber is shown in Figure 8.1. A volume of saturated vapor contained in a vessel is made supersaturated through a sudden adiabatic expansion. When ionizing radiation passes through such a supersaturated vapor the ionization produced in the vapor serves as condensation nuclei. As a result small droplets of liquid can be observed along the path of the radiation. These condensation tracks have a lifetime of less than a second and can be photographed through the chamber window. The density of the condensation depends on the ionization power of the projectile as well as on the nature of the vapor, which is often an alcohol or water. Cloud chamber photographs are shown in Figure 6.5.
In a similar fashion bubble chambers operate with superheated liquids in which gas bubbles are produced upon the passage of ionizing radiation. The most commonly used liquid in bubble chambers is hydrogen, and, as a consequence, the chamber must be operated at low temperatures (23 K for H2). Since the liquid medium in a bubble chamber is much denser than the vapor medium in a cloud chamber, the former are more suitable for studies of reactions of more energetic projectiles. The high energy p-p reaction shown in Figure 10.4 has been recorded in an 0.8 m diameter bubble chamber at Saclay, France; see also Figure 6.18.
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Book2002, Radiochemistry and Nuclear Chemistry (Third Edition)GREGORY R. CHOPPIN, ... JAN RYDBERG
Chapter
Nuclear Medicine and Molecular Imaging
2014, Comprehensive Biomedical PhysicsT.F. Budinger, T. Jones
1.01.3.4 Wilson Cloud Chamber
The first position-sensitive device for particle track visualization was the Wilson cloud chamber built in 1912 (Wilson, 1951). A cloud chamber is an enclosure containing a supersaturated vapor of water or alcohol. Radiation entering the chamber causes ionization, and these ions act as condensation loci around which tiny clouds are formed because the vapors are near a point of condensation. These nuclei leave tracks of the ionization. Light reflections from the liquid droplets provide visual evidence of the quantity and to some extent type of particle causing the ionization. The Wilson cloud chamber was the first detector used in human radiotracer studies (Blumgart and Yens, 1927). It allowed measurement of the transit time of blood from one arm to the opposite arm in a human subject injected with radioactivity (Appendix A-2.5). This experiment was the first physiological quantification of human blood flow speed. Cloud chambers were used extensively in particle physics until the 1950s, when the bubble chamber was introduced. The cloud chamber enabled the discoveries of the positron in 1932 and the muon in 1936, both by Carl Anderson (Anderson, 1933).
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Reference work2014, Comprehensive Biomedical PhysicsT.F. Budinger, T. Jones
Chapter
Cloud Modeling
2024, Reference Module in Earth Systems and Environmental SciencesWojciech W. Grabowski, Piotr K. Smolarkiewicz
Modeling of Clouds in the Laboratory
Laboratory studies of cloud processes are arguably the only way to directly observe formation and growth of cloud particles. Those studies are conducted applying devices referred to as “cloud chambers”. They range from large multi-chamber facilities like the AIDA presented below, through cloud chambers with volumes between fraction of a cubic meter to a few cubic meters, to small laboratory devices designed to study growth of individual cloud particles, both droplet and ice crystals, suspended in the vertical air flow, attached to a small rod or a thread, or kept motionless by electrostatic levitation. With changes of temperature and humidity conditions, these techniques allow studies of particle formation and growth in environments mimicking those of natural clouds.
AIDA (Aerosol Interaction and Dynamics in the Atmosphere) is a large facility at the Institute of Meteorology and Climate Research, Karlsruhe Institute of Technology (KIT), Germany, see with a rich history of cloud physics and aerosol research going back to the 1970s. There are currently two main cloud chambers at the AIDA facility. AIDAc, in operation since late 1990s, is a large-volume (several tens of cubic meters) classical cloud chamber that can simulate conditions within a rising air parcel by reducing chamber pressure and thus lowering the air temperature. AIDAc can be operated in a wide range of atmospheric pressures (from 1 to 1000 hPa), temperatures from −90 to 60°C, and relative humidity from close to 0 to 100%. AIDAd, a smaller device in operation since 2020, adds temperature-controlled walls that eliminate some problems with the AIDAc operations. The two chambers provide a unique facility to study formation of water droplets and ice crystals in the atmosphere, and to investigate physical and chemical processes related to atmospheric trace gasses and aerosols. See an extensive and continuously growing list of research publications at
Examples of smaller cloud chamber facilities that address specific cloud-physics problems are in operation at the University of Warsaw (Poland; e.g., Malinowski et al., 2008) and at the Michigan Technological University, Houghton (USA; e.g., Chang et al., 2016), see Fig. 5. The chamber at the University of Warsaw allows studying mixing between a stream of cloudy air produced in the upper part of the chamber with cloud-free subsaturated air in the main chamber volume. As illustrated in the previous section, mixing of the cloudy and cloud-free air, because of cloud entrainment, is an important factor in cloud dynamics and microphysics. Due to its small-scale nature, impact of mixing on cloud microscale is virtually impossible to observe in natural clouds. Turbulence generated by the descending cloudy plume in the chamber and microphysical transformations resulting from the mixing between the plume and cloud-free air within the chamber represent laboratory analogs of natural cloud processes. One of the key questions such experiments address is the so-called homogeneity of the cloud-clear air mixing. Since the entrainment and mixing usually leads to some evaporation of the cloud water, mixing homogeneity refers to specific changes of the cloud droplet size spectrum. One can consider two limiting situations, with natural processes arguably taking place between those two limits. In the first limit, all cloud droplets similarly reduce their size to provide required reduction of the cloud water. This is a simple model of the homogeneous mixing in which droplet concentration changes insignificantly (i.e., only due the dilution), and the mean droplet size is reduced. The opposite limit involves complete evaporation of some droplets, with the rest not reducing their sizes at all. This is the extremely inhomogeneous mixing in which droplet concentration is reduced and the mean droplet radius does not change. On theoretical grounds, homogeneity of mixing depends on the relative magnitudes of the time scale of the turbulent mixing and the time scale of the droplet evaporation. If turbulent mixing is fast compared to droplet evaporation, droplets adjust their size after the air temperature and humidity become uniform (i.e., when the turbulent mixing homogenizes the volume). If droplet evaporation is fast compared to the turbulent mixing, only droplets near the cloud-cloud free interface evaporate. Details of these processes have been investigated using cloud chamber observations and numerical simulations. Another aspect investigated in the University of Warsaw cloud chamber concerns the impact of turbulence on local droplet concentration fluctuations that result from more involved motion of cloud droplets in the turbulent environment.
Cloud chamber at the Michigan Technological University is referred to as the Pi chamber because its volume is Pi cubic meters in one of possible configurations, see The chamber simulates turbulent moist (i.e., with water phase changes) Rayleigh-Bénard convection, that is, turbulent fluid motions resulting from heating the fluid from below and cooling it from the top. In contrast to the classical (i.e., without water phase changes) Rayleigh-Bénard convection, bottom and top horizontal boundaries serve as source of the water vapor as well as the temperature. The operating principle of the Pi chamber is cloud formation resulting from mixing between warm and humid air rising from the lower boundary, and cold and less humid air descending from the upper boundary. The chamber can operate in diverse thermodynamic conditions, with surface, top, and vertical boundaries capable of maintaining a steady temperature in the range between −50 and 55 °C, and chamber pressure in the range corresponding to atmospheric pressures from the lower to the upper troposphere. These conditions span ranges typical for clouds forming in most of the Earth troposphere. Pi chamber investigations apply instrumentation for measurements of thermodynamic, turbulence, aerosol, chemical, and cloud properties. The chamber has been used for a wide range of scientific studies, including aerosol-cloud interactions, influence of turbulence on cloud droplet formation and diffusional growth, light propagation through clouds, cloud chemistry, and ice initiation. Pi chamber investigations together with accompanying theoretical and numerical modeling studies document the key impact of turbulent temperature and humidity fluctuations on cloud droplet and ice crystal formation in the atmosphere. For cloud droplet formation, the impact of cloud condensation nuclei (CCN3) characteristics (i.e., size, chemical composition, etc.) is particularly relevant. One of specific observations in the Pi chamber is that having the air completely void of CCN (by an efficient air filtration) allows obtaining high supersaturations without cloud droplet formation. This documents the key role of CCN in the formation of cloud droplets. By allowing environmental air with controlled thermodynamic properties to enter the chamber, studies related to cloud entrainment and mixing can also be conducted. A list of selected publications resulting from studies conducted within the Pi chamber is available at
Over the last few decades, several laboratory facilities have been in operation to study formation and growth of individual cloud particles. This especially applies to ice crystals as formation and growth of cloud droplets is relatively well understood. That said, laboratory studies of droplet collisions did provide information required for numerical models of precipitation formation described in the next section. Laboratory studies of individual ice particles contribute to the fundamental understanding of complex ice forms and shapes, referred to as habits, and their growth dependence on environmental conditions (e.g., temperature, humidity). These studies applied laboratory instruments that allow formation and growth of individual ice particles in controlled environments. Fig. 6 reproduced from Bailey and Hallett (2002) shows a schematic of one of such instruments that was in operation at the Division of Atmospheric Sciences of the Desert Research Institute (DRI) in Reno (Nevada). The upper panel shows details of the instrument design, and the lower panel documents the principle of its operation. The instrument, referred to as the static diffusion chamber, allows growing individual ice crystals between two plates with controlled pressure, temperature, and humidity conditions. The plates are covered with frozen solid ice that ensures ice-saturated conditions at both plates. Ice crystals are suspended on a glass filament and their growth is observed using a microscope and recorded using a video camera. The principle of the chamber operation is the nonlinear temperature-dependence of the saturated water vapor pressure. Molecular diffusion of the temperature and water vapor between the two plates results in linear profiles of both the temperature and water vapor mixing ratio (as illustrated by the measured temperature profile in panel c). Because of the nonlinearity, the supersaturation field has a peak between the two plates (of about 0.3% for the selected plates temperatures shown in the figure) as illustrated in the panel b. Observations of ice crystal growth at DRI's diffusion chamber provide additional details of the ice crystal diffusional growth, expanding the classical ice crystal classification developed in the middle of the 20th century by Ukichiro Nakaya, and followed by Choji Magono and Chung Woo Lee, Japanese researchers from the Hokkaido University.
More advanced diffusion chambers to study growth individual ice crystals are in operation at the cloud physics laboratory in the Department of Meteorology and Atmospheric Science at the Pennsylvania State University. One of them is the static button electrode (SBE) diffusion chamber (Harrison et al., 2016) that allows holding the particle without mechanical supports (i.e., the glass filament in Fig. 6A) through electrodynamic levitation using button electrodes. Studies using SBE diffusion chamber quantify details of the ice crystal characteristics at temperatures corresponding to the upper troposphere in the Earth atmosphere (−30 °C and colder) and provide information about parameters needed to describe ice crystals growth rate in numerical models.
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Reference work2024, Reference Module in Earth Systems and Environmental SciencesWojciech W. Grabowski, Piotr K. Smolarkiewicz
Chapter
Cloud Modeling
2024, Reference Module in Earth Systems and Environmental SciencesWojciech W. Grabowski, Piotr K. Smolarkiewicz
Conclusions
Cloud modeling is a multifaceted problem with multiple foci of interests, mathematical representations and tools of research—all dependent on the scale of addressed phenomena and targeted applications. Here it has been discussed in the context of theory, laboratory experiments, and numerical simulation. For the theory, marine shallow convective clouds in the Earth tropics were discussed first, applying a concept of an air parcel rising from the vicinity of the ocean surface. Condensation of the water vapor within the air parcel and the concomitant formation of cloud droplets provide the latent heating and positive parcel buoyancy driving the cloud updraft. Secondly, a theory explaining the organization and longevity of mesoscale convective systems (e.g., squall lines) has been presented. The RKW theory focuses on the interaction between the lower-tropospheric shear of the horizontal wind and a convective cold pool. Fig. 9 (Grabowski et al., 1998) illustrates a numerical simulation of both organized deep convection and scattered shallow convection in the tropics. Laboratory experiments allow modeling of relevant cloud processes in detail impossible for natural clouds. The experiments concern mixing between cloudy and cloud-free air, formation and growth of either individual cloud particles (in small laboratory devices) or particle populations in larger cloud chambers. Those devices allow studies of physical and chemical cloud processes with unprecedented fidelity. Finally, numerical modeling brings together theory and laboratory studies into simulation of cloud and precipitation processes that provide understanding of clouds as part of the Earth weather and climate. Rapidly growing cloud modeling efforts at global scales are especially important in the era of anthropogenic climate change, with many weather-related phenomena providing threats to human activities.
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Reference work2024, Reference Module in Earth Systems and Environmental SciencesWojciech W. Grabowski, Piotr K. Smolarkiewicz
Chapter
Cloud Modeling
2024, Reference Module in Earth Systems and Environmental SciencesWojciech W. Grabowski, Piotr K. Smolarkiewicz
Introduction
Clouds are collections of droplets (water clouds) or ice crystals (ice clouds), or mixtures of water droplets and ice crystals (so-called mixed-phase clouds). Clouds form when the air carrying water vapor is brought to saturation. This can happen because of a cooling of the air or because of mixing between warm and cold unsaturated air volumes (or parcels). The cooling may be associated with radiative processes (e.g., nighttime cooling of the air near the earth surface leading to fog formation) or it can happen because of the rising air motion in the atmosphere. Rising motion in a stably-stratified atmosphere is accompanied by the air expansion that lowers its temperature. Rising motion may be forced by environmental conditions (e.g., flow over topography or a frontal circulation) or may come from atmospheric dynamics as in the case of moist convection. Cloud formation resulting from mixing between two air parcels is the effect of the nonlinear temperature dependence of the saturated water vapor pressure. Examples of clouds that form due to mixing include upper-tropospheric contrails resulting from mixing of the hot and humid jet engine exhaust and environmental cold air, or steam that forms when water is boiled in a kettle or when a person breathes in cold weather.
Phase changes of the water substance are a unique feature of the Earth atmosphere. Without them, Earth climate would be relatively straightforward to understand and predict. This is because the key aspect of every planet's climate is the radiative balance between the solar (shortwave) radiation reaching the planet from the Sun, and the planet's thermal (longwave) radiation that the planet emits back to space. For some planets, the thermal energy transported from their interior may be important as well. Geothermal energy, important for some regions of the Earth like Iceland, contributes insignificantly to the Earth mean energy budget. For a planet with its axis of rotation approximately perpendicular to the planet orbit around the Sun (like planets in our solar system), most of the solar radiation arrives at low latitudes, that is, not far from the equator. In contrast, planet's thermal radiation is emitted in a similar way at all latitudes. It follows that the planet's long-term mean energy balance at any latitude can only be closed by the energy transport from low latitudes, where the incoming solar radiation is at its peak, toward the poles, where the solar radiation reaching the planet is close to zero. At any given location, the vertical energy transport, either through radiative transfer or by the air flow, determines specific aspects of the atmosphere's vertical structure. On Earth, these horizontal and vertical transports take place mostly in the troposphere, the lowest part of the Earth's atmosphere (about 10 km deep), and they are significantly affected by phase changes of the water substance that are responsible for cloud formation. Clouds significantly impact shortwave and longwave radiative transfer through the Earth atmosphere, but they also are source of precipitation and thus they play the key role in the water cycle on Earth. It follows that understanding and predicting the Earth climate, as well as day-to-day weather, requires appropriate knowledge of cloud processes. This is difficult because clouds are small-scale features compared to the size of the planet, they consist of particles that are microscopic compared to the size of a single cloud, and cloud particle formation and growth involve processes at sub-centimeter spatial scales. Understanding of cloud processes comes from theoretical, laboratory, and modeling studies. Examples of those are discussed in this chapter.
Theoretical studies typically involve applications of simple concepts that describe conservation of energy and water substance during cloud formation. The next section provides an example of such concepts applying a simple model of convective cloud formation. A simple idea is also presented that explains a unique feature of precipitating moist convection in the Earth atmosphere, that is, convection organization into long-lived mesoscale convective systems. Subsequently, modeling of clouds in laboratory devices is discussed with specific examples representing just a small fraction of laboratory instruments available in various scientific institutions around the world. Cloud chambers and other laboratory devices allow studying cloud dynamics and microphysics in a controlled environment. The final section provides a discussion of three-dimensional numerical modeling of clouds. Numerical modeling concerns solving fluid flow equations combined with the representation of thermodynamics of phase changes (condensation and evaporation, freezing and melting, sublimation and resublimation) together with precipitation formation and fallout.
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Reference work2024, Reference Module in Earth Systems and Environmental SciencesWojciech W. Grabowski, Piotr K. Smolarkiewicz
Chapter
Interaction of Radiation with Matter
2012, Physics in Nuclear Medicine (Fourth Edition)Simon R. Cherry PhD, ... Michael E. Phelps PhD
3 Charged-Particle Tracks
A charged particle passing through matter leaves a track of secondary electrons and ionized atoms in its path. In soft tissue and materials of similar density, the tracks are typically approximately 100 µm wide, with occasionally longer side tracks generated by energetic δ rays. The tracks are studied in nuclear physics using film emulsions, cloud chambers, and other devices.
When a heavy particle, such as an α particle, collides with an orbital electron, its direction is virtually unchanged and it loses only a small fraction of its energy (rather like a bowling ball colliding with a small lead shot). The maximum fractional energy loss by a heavy particle of mass M colliding with a light particle of mass m is approximately 4 m/M. For an α particle colliding with an electron, this amounts to only approximately 0.05% [4 × (1/1840)/4 ≈ (1/2000)]. Heavy particles also undergo relatively few bremsstrahlung-producing collisions with nuclei. As a result, their tracks tend to be straight lines, and they experience an almost continuous slowing down in which they lose small amounts of energy in a large number of individual collisions.
By contrast, electrons can undergo large-angle deflections in collisions with orbital electrons and can lose a large fraction of their energy in these collisions. These events are more like collisions between billiard balls of equal mass. Electrons also undergo occasional collisions with nuclei in which they are deflected through large angles and bremsstrahlung photons are emitted. For these reasons, electron tracks are tortuous, and their exact shape and length are unpredictable.
An additional difference between electrons and heavy particles is that for a given amount of kinetic energy, an electron travels at a much faster speed. For example, a 4-MeV α particle travels at approximately 10% of the speed of light, whereas a 1-MeV electron travels at 90% of the speed of light. As a result, an electron spends a much briefer time in the vicinity of an atom than does an α particle of similar energy and is therefore less likely to interact with the atom. Also, an electron carries only one unit of electrical charge, versus two for α particles, and thus exerts weaker forces on orbital electrons. For these reasons, electrons experience less frequent interactions and lose their energy more slowly than α particles; they are much less densely ionizing, and they travel farther before they are stopped than α particles or other heavy charged particles of similar energy.
To illustrate these differences, Figure 6-4 shows (in greatly enlarged detail) some possible tracks for β particles and for α particles in water. The actual track lengths are on the order of microns for α particles and fractions of a centimeter for β particles. This is discussed further in Section B.
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Book2012, Physics in Nuclear Medicine (Fourth Edition)Simon R. Cherry PhD, ... Michael E. Phelps PhD
Chapter
Detection and Measurement Techniques
2002, Radiochemistry and Nuclear Chemistry (Third Edition)GREGORY R. CHOPPIN, ... JAN RYDBERG
8.1 Track measurements
The most striking evidence for the existence of atoms comes from the observation of tracks formed by nuclear particles in cloud chambers, in solids and in photographic emulsions. The tracks reveal individual nuclear reactions and radioactive decay processes. From a detailed study of such tracks, the mass, charge and energy of the particle can be determined.
The tracks formed can be directly observed by the naked eye in cloud and bubble chambers, but the tracks remain only for a short time before they fade. For a permanent record we must use photography. On the other hand, in solid state nuclear track detectors (SSNTD), of which the photographic emulsion is the most common variant, the tracks have a much longer lifetime during which they can be made permanent and visible by a suitable chemical treatment. Because of the much higher density of the absorber, the tracks are also much shorter and often therefore not visible for the naked eye. Thus the microscope is an essential tool for studying tracks in solids.
8.1.1 Cloud and bubble chambers
The principle of a cloud chamber is shown in Figure 8.1. A volume of saturated vapor contained in a vessel is made supersaturated through a sudden adiabatic expansion. When ionizing radiation passes through such a supersaturated vapor the ionization produced in the vapor serves as condensation nuclei. As a result small droplets of liquid can be observed along the path of the radiation. These condensation tracks have a lifetime of less than a second and can be photographed through the chamber window. The density of the condensation depends on the ionization power of the projectile as well as on the nature of the vapor, which is often an alcohol or water. Cloud chamber photographs are shown in Figure 6.5.
In a similar fashion bubble chambers operate with superheated liquids in which gas bubbles are produced upon the passage of ionizing radiation. The most commonly used liquid in bubble chambers is hydrogen, and, as a consequence, the chamber must be operated at low temperatures (23 K for H2). Since the liquid medium in a bubble chamber is much denser than the vapor medium in a cloud chamber, the former are more suitable for studies of reactions of more energetic projectiles. The high energy p-p reaction shown in Figure 10.4 has been recorded in an 0.8 m diameter bubble chamber at Saclay, France; see also Figure 6.18.
8.1.2 Solid state nuclear track detectors (SSNTD)
The main types of SSNTD (or DTD, for dielectric track detector) are photographic emulsions, crystals, glasses, and plastics. Because the density of these materials is much higher than for the previous group (§8.1.1), nuclear particles can spend all their kinetic energy in these detectors, allowing identification of the particle. Since the SSNTD retains the particle path, they can be used to record reactions over a long time period. These advantages have made SSNTD especially valuable in the fields of cosmic ray physics, radiochemistry, and earth sciences.
Nuclear emulsions are similar to optical photographic emulsions. They contain AgBr crystals embedded in gelatin to which small amounts of sensitizing agents have been added. The AgBr content is as much as four times (i.e. 80% AgBr) greater than in optical film. Also the crystals are much smaller (developed grain 0.1 – 0.6 µm) and well separated. The emulsions come in thicknesses from a few µm up to 1 mm. Nuclear radiation passing through the emulsion causes ionization and excitation which activates the AgBr crystals, producing a latent image of the particle path. Upon development the activated crystals serve as centers for further reduction of silver, leading to visible grains. It is assumed that at least 3 silver atoms must be activated to produce a visible grain, while about 30 atoms are needed for normal blackening. Each activated grain seems to require about 2.5 eV to be absorbed on the average.
While the memory effect of the developed film is almost infinite, this is not the case for the latent image which slowly fades, depending on the number of originally activated silver atoms, the film type and external conditions like temperature, humidity, etc. When stored under ambient conditions, about 80% of the latent image disappears in half a year.
The developed grains form an interrupted track along the original path of the energetic particle (Fig. 8.2(a)). The specific energy loss of the particle, dE/dx (i.e. the stopping power of the absorber), depends on the mass, charge, and velocity of the particle, and on the composition of the absorber. From the track length, grain density (i.e. grains per track length), and gap length between the grains, the particle and its energy can be determined (cf. §§6.3 and 7.1). For a given particle, the range is proportional to the energy as shown in Table 8.1. The range decreases with increasing mass of the particle and density of the absorber. The grain density depends on the specific ionization of the particle which does not vary linearly with the particle energy (or velocity), as seen from Figure 6.7; thus the grain density changes along the track.
Table 8.1. Range of energetic high-ionizing particles in various solids
| Particle | Energy (MeV) | Absorber (density) | Range (µm) |
--- --- |
| 1H | 10 | Ilford C2 (3.8)(a) | 540 |
| 3H | 10 | Ilford C2 (3.8)(a) | 230 |
| 4He | 10 | Ilford C2 (3.8)(a) | 57 |
| 4He (214Bi) | 7.7 | Eastman NTA (3.6) | 38 |
| | | Mica (3.1) | 36 |
| | | Glass (2.5) | 41 |
| | | Water (1.0) | 60 |
| 4He (238U) | 4.2 | Mica (3.1) | 13 |
| 4He (U-series) | | Pitchblende (7.0) | 23(b) |
| | | Carnotite (4.1) | 32(b)) |
| | -150 | Eastman NTC (∼3.4) | |
| | Eastman NTC (∼3.4) |
| 238U. Both fiss. fragm. | -160 | Leopoldite (∼4) | -20 |
(a)
: Density of AgBr 6.47; of gelatin 1.31.
(b)
: Range of the predominating α-particles.
Other solid material may be used as SSNTD instead of AgBr emulsions: plastics (cellulose nitrate and polycarbonate films), glass, crystals, etc. In order to make the tracks visible in the microscope the surface of the SSNTD must be polished and etched, usually with alkali.
Because of the natural radiation background, every SSNTD has a memory of past nuclear events, which must be erased as far as possible before a new exposure. In nuclear emulsions an α-radiation background of 20 – 60 tracks cm−2 per day is normal. The technique of background eradication prior to exposure may consist of treating an emulsion with chromic acid, H2O2-vapor or heating (annealing) a glass plate. Because this technique more easily removes weak images, it may also be used after exposure, e.g. to remove fainter α-tracks from heavier fission tracks.
Let us consider some examples of uses of SSNTD. Tracks obtained under various conditions are shown in Figure 8.2.
(i).
: As mentioned SSNTD has been used in cosmic ray experiments at high altitudes and in space journeys where memory effect and simple construction make them especially suitable. Many elementary particles have been discovered by this technique, notably the π-and µ-mesons. Figure 10.2 shows tracks of high energy cosmic ray particles, probably iron atoms, which have been stopped in Apollo astronaut helmets.
(ii).
: Nuclear reactions can be studied by SSNTD. The target material may either be regular atoms of the detector (H, O, Ag, Si, etc.)or material introduced into the matrix, e.g. thin threads of target metals or uranium atoms. The former have been used in high energy physics for hadron-induced reactions, and the latter for studying fission processes. From experiments with uranium the frequency of spontaneous fission of 238U has been determined, and also the rate of ternary fission and emission of long range α's.
(iii).
: When emulsions are dipped into solution, some of the dissolved material is soaked up or absorbed in the emulsion. For example, if the solution contained samarium, some α-tracks of its spontaneous decay (decay rate 127 Bq g−1) appear in the emulsion (cf. Fig. 8.2(a)). Since 147Sm has an isotopic abundance of 15%, its half-life is calculated to be 1.1 × 1011 y. The lower limit of detection is about 500 tracks cm2 d−1, so quite low decay rates can be accurately measured, making this a valuable technique for determination of long half-lives.
(iv).
: 222Rn is released through the earth's surface from uranium minerals. The amount released varies not only with the uranium content and mineral type, but also with the time of the day; variation from 1 - 75 Bq 1−1 has been registered during a 24 h period. To avoid this variation, cups containing a piece of plastic TD are placed upside down in shallow (0.5 – 1.0 m) holes for about 3 weeks, after which the SSNTD are etched and α-tracks from radon counted. Mineral bodies hundreds of meters underground can be mapped with this technique in great detail in a reasonably short time. The US Geological Survey uses the same technique to predict earthquakes; it has been observed that just before earthquakes the radon concentration first increases, then suddenly decreases, the minimum being observed about one week before the earthquake appears.
(v).
: The average radon concentration in houses can be measured by hanging a plastic film inside the house over a time period of some weeks. The film is returned to a laboratory where it is etched and the number of α-tracks per unit area counted.
(vi).
: Fission fragments make dense tracks in all solid material. The tracks are short and thick: in a crystal material like zircon (a common mineral of composition ZrSiO4) they may not be more than 10−2 µm (10 nm) in diameter, and 10 – 20 µm in length. They are therefore not visible even in the best optical microscopes. Using scanning electron microscopy, it has been found that the hole formed retains the crystal structure or regains it (Fig. 8.2(d)). On the other hand, if the track is formed in glass, a gas bubble appears instead of a track (Fig. 8.2(e)); these slightly elongated bubbles can be distinguished from other completely spherical bubbles formed by other processes. To make the tracks visible, the specimen is embedded in a resin, then one surface is ground and carefully polished after which it is dipped in an acid, e.g. HF. Because of defects in the crystal structure along the fission track, the track and its close surroundings are attacked by the acid, and the diameter of the track increases a hundredfold to a micron or so. The tracks are then visible under a microscope with a magnification of 500 – 1000 x.
This procedure has been used as an analytical tool for determination of uranium and plutonium in geological and environmental samples. In this technique, the sample (either a ground and polished surface of a mineral, or a dust sample on tape) is firmly pressed against a photographic film, and the package is irradiated by slow neutrons. From the fission track count of the developed film the uranium or plutonium content can be calculated. Thus a Swedish shale was found to contain 4 ± 1 ppm U, and a bottom sediment in a Nagasaki water reservoir 0.44 ± 0.04 Bq 239Pu per kg sediment. In the latter case, the ratio between the number of fission tracks Nft and α-tracks Nαt from 239Pu is
(8.1)
This technique is very useful for routine measurements of fissionable material in very low concentrations. Figure 8.2(c) shows fission tracks in uranium-containing mineral which has been exposed to neutrons.
Fission track counting is also important for dating of geological samples (Ch. 5) and for estimation of the maximum temperature experienced by sedimentary rocks. The latter is important in oil prospecting operations as the maximum temperature seems to be a useful indicator on whether to expect oil, natural gas or nothing. For too low temperatures neither oil nor gas is expected, for intermediate temperatures oil may be present and for high temperatures only natural gas. Temperature history information can obtained from a combination of age (estimated by other radioactive methods, cf. Ch. 5), uranium content of crystals of several minerals in the rock, their fission track count and track-length distribution caused by thermal annealing. Zircon, titanite and apatite are examples of three such minerals in order of increasing annealing temperature.
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Book2002, Radiochemistry and Nuclear Chemistry (Third Edition)GREGORY R. CHOPPIN, ... JAN RYDBERG
Chapter
ATOMIC ABSORPTION SPECTROMETRY | Flame
2005, Encyclopedia of Analytical Science (Second Edition)S.J. Hill
Sample Introduction and Burner System
In order for the atom cell, i.e., the flame, to function effectively and produce an atomic vapor, the sample must be presented in the form of a fine aerosol. This is achieved using a nebulizer and spray chamber prior to the burner. A typical pneumatic nebulizer system for a premixed flame is shown in Figure 1. The sample solution is sucked up a plastic capillary tube by the reduced pressure created at the end of the tube by the oxidant or carrier gas flow, i.e., the Venturi effect. During this process the sample is shattered into tiny droplets as it exits the capillary tube to produce an aerosol. This process may be optimized with respect to sample uptake and drop size by adjusting the position of the nebulizer capillary. Additionally, an impact bead may be placed in the path of the initial aerosol to provide secondary fragmentation. Both the nebulizer and impact bead must be made of corrosion-resistant material such as a platinum–iridium alloy (90:10) or stainless steel coated with an inert plastic.
Once formed, the aerosol passes into the burner or spray chamber (sometimes also called the cloud chamber). The role of the spray chamber is to homogenize both the aerosol and gases that tend to dampen fluctuations in nebulizer efficiency, and to remove any large droplets before they reach the flame. Large droplets (diameter >10 μm) collect on the sides of the chamber and then drain to waste. Spoilers and baffles placed at the end of the spray chamber aid this process. Because the spray chamber will fill with flammable gas, modern instruments will also incorporate some form of antiflashback protection from the flame.
Although the nebulizer and spray chamber arrangement described allows only small droplets to reach the flame, the efficiency in terms of sample transport is only ∼10%. Ultrasonic nebulizers offering higher efficiency are available, but these are more complicated and have a tendency to suffer from memory effects, i.e., contamination from previous samples. Organic solvents also improve the efficiency, probably forming smaller droplets as a result of their lower surface tension. However, organic solvents may also affect nebulization because of their different density, viscosity, and saturated vapor pressure. In addition, they may also act as secondary fuels, thus affecting the flame.
The introduction of solid powder samples is possible using pneumatic sampling devices. Various graphite capsules and vibration tubes have also been used on this form of sample. In both cases the precision is very poor and the introduction of powders is thus seldom attempted these days.
The design of the burner employed in AAS depends on the oxidant–fuel gas mixture. Slot burners are now used almost exclusively, the length of the slot being 10 cm for use with air–acetylene and multishot burners, and 5 cm for nitrous oxide–acetylene burners, reflecting the higher burning velocity of the flame as shown in Table 1. The slot width and the conductivity of the metal used for the construction of the burner are also important in terms of stability of operation and prevention of clogging. The burner should also be constructed or coated in an inert material to avoid corrosion.
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Reference work2005, Encyclopedia of Analytical Science (Second Edition)S.J. Hill
Chapter
Radiation Effects on Matter
2002, Radiochemistry and Nuclear Chemistry (Third Edition)GREGORY R. CHOPPIN, ... JAN RYDBERG
7.2 Radiation tracks
The energy lost when a high energy charged particle is slowed in matter gives rise to a trail (or, more commonly, track) of ionized and excited molecules along the path of the particle; Figure 7.1 schematically depicts a track. Figure 6.5 shows cloud chamber photographs of such tracks. A photon imparts a large fraction of its energy to a single electron which subsequently ionizes and excites many other molecules along its path. The absorption of any type of ionizing radiation by matter thus leads to the formation of tracks of ionized and excited species. Whereas these species generally are the same in a particular sample of matter regardless of the type of radiation, the tracks may be sparsely or densely populated by these species. Expressions such as linear energy transfer (LET) and stopping power are based on the implicit assumption of a continuous slowing down process and thus gives quantitative information on the average energy loss but only qualitative information on the densities of reactive species.
Electrons are liberated in the ionization process with greatly varying kinetic energy. If the energy of these secondary electrons is relatively low (< 100 eV) their range in liquids and solids is short and the ionizations and excitations caused by these electrons take place close to the primary ionizations leading to the formation of small spurs containing ionized and excited species. Secondary electrons with high kinetic energy form tracks of their own branching from the primary track. Such electrons are called δ-rays. For high energy electrons spurs are formed at well separated intervals along the track whereas for densely ionizing radiation such as a particles, protons and recoil atomic ions the spurs overlap and form columns of excited and ionized species. Differences in chemical and biological effects caused by different radiations reflects the varying track structures.
The tracks of X-ray and γ-radiation results in tracks of fast electrons. The energy of these fast electron is consumed by the formation of spurs (6 – 100 eV), blobs (100 – 500 eV) and short tracks (500 – 5000 eV), see Figure 7.1. For a primary electron in the range 104 – 107 eV the distribution of energy of secondary electrons is approximately 40% < 3.4 eV, 20% 3.4 – 6.8 eV, 18% 6.8 – 13.5 eV and 12% 13.5 – 27.1 eV. The radius of the core at low electron velocities is of the order of 1 nm.
The energy transferred to the electrons by an energetic ion depends on the mass, the charge (Z) and velocity (v) of the ion, cf. (6.13): the probability of interaction is proportional to the ratio Z2/mv2. Heavy ions produce track structures similar to those of fast electrons. However, the spurs are quite close to each other and, for the case of α-particles, the original distance between them is of the order of fractions of a nanometer. Immediately at their formation they comprise a continuous cylindrical column. The column consists of a dense core surrounded by a more diffuse shell of tracks of high energy δ-electrons.
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Book2002, Radiochemistry and Nuclear Chemistry (Third Edition)GREGORY R. CHOPPIN, ... JAN RYDBERG
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6606 | https://en.wikipedia.org/wiki/Hausdorff_space | Jump to content
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Contents
(Top)
1 Definitions
2 Equivalences
3 Examples of Hausdorff and non-Hausdorff spaces
4 Properties
5 Preregularity versus regularity
6 Variants
7 Algebra of functions
8 Academic humour
9 See also
10 Notes
11 References
Hausdorff space
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From Wikipedia, the free encyclopedia
Type of topological space
| Separation axiomsin topological spaces |
| Kolmogorov classification |
| T0 | (Kolmogorov) |
| T1 | (Fréchet) |
| T2 | (Hausdorff) |
| T2½ | (Urysohn) |
| completely T2 | (completely Hausdorff) |
| T3 | (regular Hausdorff) |
| T3½ | (Tychonoff) |
| T4 | (normal Hausdorff) |
| T5 | (completely normal Hausdorff) |
| T6 | (perfectly normal Hausdorff) |
| History |
In topology and related branches of mathematics, a Hausdorff space (/ˈhaʊsdɔːrf/ HOWSS-dorf, /ˈhaʊzdɔːrf/ HOWZ-dorf), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" (T2) is the most frequently used and discussed. It implies the uniqueness of limits of sequences, nets, and filters.
Hausdorff spaces are named after Felix Hausdorff, one of the founders of topology. Hausdorff's original definition of a topological space (in 1914) included the Hausdorff condition as an axiom.
Definitions
[edit]
Points and in a topological space can be separated by neighbourhoods if there exists a neighbourhood of and a neighbourhood of such that and are disjoint . is a Hausdorff space if any two distinct points in are separated by neighbourhoods. This condition is the third separation axiom (after T0 and T1), which is why Hausdorff spaces are also called T2 spaces. The name separated space is also used.
A related, but weaker, notion is that of a preregular space. is a preregular space if any two topologically distinguishable points can be separated by disjoint neighbourhoods. A preregular space is also called an R1 space.
The relationship between these two conditions is as follows. A topological space is Hausdorff if and only if it is both preregular (i.e. topologically distinguishable points are separated by neighbourhoods) and Kolmogorov (i.e. distinct points are topologically distinguishable). A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.
Equivalences
[edit]
For a topological space , the following are equivalent:
is a Hausdorff space.
Limits of nets in are unique.
Limits of filters on are unique.
Any singleton set is equal to the intersection of all closed neighbourhoods of . (A closed neighbourhood of is a closed set that contains an open set containing .)
The diagonal is closed as a subset of the product space .
Any injection from the discrete space with two points to has the lifting property with respect to the map from the finite topological space with two open points and one closed point to a single point.
Examples of Hausdorff and non-Hausdorff spaces
[edit]
See also: Non-Hausdorff manifold
Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers (under the standard metric topology on real numbers) are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions.
A simple example of a topology that is T1 but is not Hausdorff is the cofinite topology defined on an infinite set, as is the cocountable topology defined on an uncountable set.
Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge spaces. Indeed, when analysts run across a non-Hausdorff space, it is still probably at least preregular, and then they simply replace it with its Kolmogorov quotient, which is Hausdorff.
In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry, in particular as the Zariski topology on an algebraic variety or the spectrum of a ring. They also arise in the model theory of intuitionistic logic: every complete Heyting algebra is the algebra of open sets of some topological space, but this space need not be preregular, much less Hausdorff, and in fact usually is neither. The related concept of Scott domain also consists of non-preregular spaces.
While the existence of unique limits for convergent nets and filters implies that a space is Hausdorff, there are non-Hausdorff T1 spaces in which every convergent sequence has a unique limit. Such spaces are called US spaces. For sequential spaces, this notion is equivalent to being weakly Hausdorff.
Properties
[edit]
Subspaces and products of Hausdorff spaces are Hausdorff, but quotient spaces of Hausdorff spaces need not be Hausdorff. In fact, every topological space can be realized as the quotient of some Hausdorff space.
Hausdorff spaces are T1, meaning that each singleton is a closed set. Similarly, preregular spaces are R0. Every Hausdorff space is a Sober space although the converse is in general not true.
Another property of Hausdorff spaces is that each compact set is a closed set. For non-Hausdorff spaces, it can be that each compact set is a closed set (for example, the cocountable topology on an uncountable set) or not (for example, the cofinite topology on an infinite set and the Sierpiński space).
The definition of a Hausdorff space says that points can be separated by neighborhoods. It turns out that this implies something which is seemingly stronger: in a Hausdorff space every pair of disjoint compact sets can also be separated by neighborhoods, in other words there is a neighborhood of one set and a neighborhood of the other, such that the two neighborhoods are disjoint. This is an example of the general rule that compact sets often behave like points.
Compactness conditions together with preregularity often imply stronger separation axioms. For example, any locally compact preregular space is completely regular. Compact preregular spaces are normal, meaning that they satisfy Urysohn's lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers. The Hausdorff versions of these statements are: every locally compact Hausdorff space is Tychonoff, and every compact Hausdorff space is normal Hausdorff.
The following results are some technical properties regarding maps (continuous and otherwise) to and from Hausdorff spaces.
Let be a continuous function and suppose is Hausdorff. Then the graph of , , is a closed subset of .
Let be a function and let be its kernel regarded as a subspace of .
If is continuous and is Hausdorff then is a closed set.
If is an open surjection and is a closed set then is Hausdorff.
If is a continuous, open surjection (i.e. an open quotient map) then is Hausdorff if and only if is a closed set.
If are continuous maps and is Hausdorff then the equalizer is a closed set in . It follows that if is Hausdorff and and agree on a dense subset of then . In other words, continuous functions into Hausdorff spaces are determined by their values on dense subsets.
Let be a closed surjection such that is compact for all . Then if is Hausdorff so is .
Let be a quotient map with a compact Hausdorff space. Then the following are equivalent:
is Hausdorff.
is a closed map.
is a closed set.
Preregularity versus regularity
[edit]
All regular spaces are preregular, as are all Hausdorff spaces. There are many results for topological spaces that hold for both regular and Hausdorff spaces. Most of the time, these results hold for all preregular spaces; they were listed for regular and Hausdorff spaces separately because the idea of preregular spaces came later. On the other hand, those results that are truly about regularity generally do not also apply to nonregular Hausdorff spaces.
There are many situations where another condition of topological spaces (such as paracompactness or local compactness) will imply regularity if preregularity is satisfied. Such conditions often come in two versions: a regular version and a Hausdorff version. Although Hausdorff spaces are not, in general, regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular. Thus from a certain point of view, it is really preregularity, rather than regularity, that matters in these situations. However, definitions are usually still phrased in terms of regularity, since this condition is better known than preregularity.
See History of the separation axioms for more on this issue.
Variants
[edit]
The terms "Hausdorff", "separated", and "preregular" can also be applied to such variants on topological spaces as uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that unites the concept in all of these examples is that limits of nets and filters (when they exist) are unique (for separated spaces) or unique up to topological indistinguishability (for preregular spaces).
As it turns out, uniform spaces, and more generally Cauchy spaces, are always preregular, so the Hausdorff condition in these cases reduces to the T0 condition. These are also the spaces in which completeness makes sense, and Hausdorffness is a natural companion to completeness in these cases. Specifically, a space is complete if and only if every Cauchy net has at least one limit, while a space is Hausdorff if and only if every Cauchy net has at most one limit (since only Cauchy nets can have limits in the first place).
Algebra of functions
[edit]
The algebra of continuous (real or complex) functions on a compact Hausdorff space is a commutative C-algebra, and conversely by the Banach–Stone theorem one can recover the topology of the space from the algebraic properties of its algebra of continuous functions. This leads to noncommutative geometry, where one considers noncommutative C-algebras as representing algebras of functions on a noncommutative space.
Academic humour
[edit]
Hausdorff condition is illustrated by the pun that in Hausdorff spaces any two points can be "housed off" from each other by open sets.
In the Mathematics Institute of the University of Bonn, in which Felix Hausdorff researched and lectured, there is a certain room designated the Hausdorff-Raum. This is a pun, as Raum means both room and space in German.
See also
[edit]
Fixed-point space – Space where all functions have fixed points, a Hausdorff space X such that every continuous function f : X → X has a fixed point.
Locally Hausdorff space – Space such that every point has a Hausdorff neighborhood
Non-Hausdorff manifold
Quasitopological space – Function in topology
Separation axiom – Axioms in topology defining notions of "separation"
Weak Hausdorff space
Notes
[edit]
^ "Hausdorff space Definition & Meaning". www.dictionary.com. Retrieved 15 June 2022.
^ a b "Separation axioms in nLab". ncatlab.org. Archived from the original on 2020-09-30. Retrieved 2019-10-16.
^ Hausdorff, Felix (1914). Grundzüge der Mengenlehre (in German). Leipzig: Veit & Comp. p. 213.
^ a b Willard 2004, pp. 86–87
^ Bourbaki 1966, p. 75
^ See for instance Lp space#Lp spaces and Lebesgue integrals, Banach–Mazur compactum etc.
^ van Douwen, Eric K. (1993). "An anti-Hausdorff Fréchet space in which convergent sequences have unique limits". Topology and Its Applications. 51 (2): 147–158. doi:10.1016/0166-8641(93)90147-6.
^ Wilansky, Albert (1967). "Between T1 and T2". The American Mathematical Monthly. 74 (3): 261–266. doi:10.2307/2316017. JSTOR 2316017.
^ Shimrat, M. (1956). "Decomposition spaces and separation properties". Quarterly Journal of Mathematics. 2: 128–129. doi:10.1093/qmath/7.1.128.
^ Willard 2004, pp. 124
^ Schechter 1996, 17.14(d), p. 460.
^ "Locally compact preregular spaces are completely regular". math.stackexchange.com.
^ Schechter 1996, 17.7(g), p. 457.
^ Adams, Colin; Franzosa, Robert (2008). Introduction to Topology: Pure and Applied. Pearson Prentice Hall. p. 42. ISBN 978-0-13-184869-6.
References
[edit]
Arkhangelskii, A.V.; Pontryagin, L.S. (1990). General Topology I. Springer. ISBN 3-540-18178-4.
Bourbaki (1966). Elements of Mathematics: General Topology. Addison-Wesley.
"Hausdorff space", Encyclopedia of Mathematics, EMS Press, 2001
Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365.
Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.
| v t e |
| Fields | General (point-set) Algebraic Combinatorial Continuum Differential Geometric + low-dimensional Homology + cohomology Set-theoretic Digital |
| Key concepts | Open set / Closed set Interior Continuity Space + compact + connected + Hausdorff + metric + uniform Homotopy + homotopy group + fundamental group Simplicial complex CW complex Polyhedral complex Manifold Bundle (mathematics) Second-countable space Cobordism |
| Metrics and properties | Euler characteristic Betti number Winding number Chern number Orientability |
| Key results | Banach fixed-point theorem De Rham cohomology Invariance of domain Poincaré conjecture Tychonoff's theorem Urysohn's lemma |
| Category Mathematics portal Wikibook Wikiversity Topics + general + algebraic + geometric Publications |
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4 key differences between liabilities and expenses
Capital One Business
June 17, 2025|7 min read
What’s the difference between a business expense and a business liability? When you’re running a business, chances are you’ll have to deal with both. By having a clear picture of your business’s liabilities and expenses, you’ll have a head start in managing your company’s finances effectively, including financial reporting and planning. There are a few key differences between liabilities and expenses, and knowing what they are can help ensure you’re making smart business decisions—now and in the future.
In this article, we’ll explore the different types of business liabilities and expenses and what sets these two financial concepts apart.
What you’ll learn:
Liabilities are financial obligations or debts a business owes to an outside party, such as accounts payable, short-term loans and other commitments that require future payment or services.
Expenses are the costs associated with running a business and generating revenue, like office supplies, rent and cost of goods sold (COGS).
Liabilities and expenses have several key differences, but both are essential for managing your company’s finances and planning for the future.
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What are liabilities?
Liabilities are debts or financial obligations that a business owes to outside parties, such as vendors or suppliers. These can include accounts payable, outstanding invoices and short- and long-term loans—any type of transaction that will require future payment or services.
Liabilities are listed on your company’s balance sheet and directly impact your business’s assets and equity. There are several types of liabilities, and understanding each one is essential for assessing your business’s financial health and managing future cash flow.
Types of liabilities
Common types of liabilities include current liabilities, noncurrent liabilities (also known as long-term liabilities) and contingent liabilities.
Current liabilities are often paid with current assets and are due within a year. These can include financial obligations such as:
Accounts payable or money owed to suppliers
Payroll expenses
Short-term loans
Taxes owed
Monthly utilities
Other examples of current liabilities include wages payable, dividends payable, interest payable and unearned revenues—money received in advance for services yet to be completed.
Noncurrent liabilities are obligations due after one year or longer. It’s common for businesses to incur long-term debt, also known as bonds payable, to finance their ongoing operations. Examples of long-term liabilities include:
Long-term loans
Leases
Pension obligations
Deferred tax liabilities
Contingent liabilities are possible obligations that could become actual liabilities. They aren’t guaranteed and depend on a specific event happening, such as a lawsuit or a customer invoking a warranty.
What are expenses?
Expenses are costs a business incurs to operate and generate revenue. Understanding your expenses is essential because they reduce your net income. Tracking expenses can help with budgeting and tax deductions and provide an overview of your finances. Properly managing expenses can also lead to higher profitability.
Businesses report expenses on their income statements. Some expenses may be tax-deductible, as long as they are considered “ordinary and necessary” for the business, according to the IRS.
Types of expenses
Expenses are typically recurring payments that are necessary to run a business. There are two main types of expenses.
Operating expenses: Operating expenses are the regular costs involved in running a business. These include rent or lease payments, employee salaries, utilities, marketing and advertising costs, administrative fees, office supplies, COGS and more.
Nonoperating expenses: Nonoperating expenses are costs not directly tied to the business’s daily operations, such as restructuring costs or interest charges on loans and other debt.
What is the difference between liabilities and expenses?
Think of expenses as the costs of running the business now and liabilities as financial commitments that need to be paid in the future. While both involve money the business has to pay, liabilities and expenses serve different purposes in accounting and financial analysis.
Here’s an example of how liabilities and expenses might impact a small business, such as a boutique clothing retailer. If the business owner purchases inventory and pays for it immediately, that’s an expense. The money is spent to operate the business now, and the cost will appear on the income statement.
However, if the same business owner chooses to buy inventory on credit, with the bill due in 60 days, this is considered a liability because no money has exchanged hands yet—it’s owed to the supplier. This liability will show up on the balance sheet as accounts payable. When the invoice is paid, the liability is removed and the cost is recorded as an expense.
Here are some key differences between liabilities and expenses that all business owners should understand.
Difference 1: Purpose and timing
A key difference between liabilities and expenses is their purpose and what they represent. Liabilities are amounts the company owes to others, such as lenders, creditors and suppliers. On the other hand, expenses are costs incurred to generate revenue and keep operations running. In this sense, you can consider timing to be an important distinction between liabilities and expenses.
Liabilities are recognized when an obligation is created that will require payment in the future. For example, if you receive a bill from the electric company on June 1, that’s considered a liability because it’s recorded in accounts payable. Once the bill is paid, it becomes an expense on the income statement and the liability is removed.
Expenses are recognized as soon as the resource or service is used, not necessarily when the payment is made. For example, rent is considered an expense during the month you occupy the space, even if the bill hasn’t been paid yet.
Difference 2: Reporting and financial statements
Liabilities are listed on the balance sheet and represent what the business owes. They help business owners understand the company’s ability to meet financial obligations and how much it relies on outside financing.
Expenses are reported on a company’s income statement and represent the costs of running the business during a specific accounting period. Some examples include marketing, rent and salaries. These costs are subtracted from the business’s revenue to calculate net income, which is a key indicator of the company’s performance. By accurately tracking expenses, business owners have a clearer picture of their profitability and efficiency.
It’s essential to properly report both liabilities and expenses. Inaccurately or incorrectly reporting them can impact your company’s financial well-being and decision-making.
Difference 3: How they’re classified
Liabilities are classified as current or noncurrent. Current liabilities are due within a year and include costs such as payroll, accounts payable and other short-term obligations. Noncurrent liabilities are long-term debts or obligations that are due beyond a 12-month period.
Expenses are classified as operating and nonoperating. Operating expenses are the costs incurred from daily business activities, while nonoperating expenses include things like interest charges and other costs not related to core operations.
Properly classifying liabilities and expenses is key to assessing your company’s short- and long-term financial health. They’re also essential to understanding its performance and ensuring accurate financial analysis.
Difference 4: Impact on profits and cash flow
Liabilities and expenses both influence your business’s financial health but in different ways. Since liabilities aren’t part of the income statement, they don’t directly impact profit—but they do affect equity and assets. On the other hand, expenses directly affect profits—especially when operating costs increase.
Changes in liabilities can also affect cash flow. For example, if the company delays paying a vendor, accounts payable increases, which could temporarily improve cash flow. However, when the invoice is paid, it becomes a cash outflow, reducing the company’s available funds.
It’s essential to understand these differences. Underreporting expenses will overstate profits. At the same time, ignoring liabilities until they’re due can lead to poor cash flow management and planning—both of which lead to inaccurate budgeting and forecasting.
Key takeaways
Liabilities and expenses both play a role in accounting for and managing your business’s finances, but they serve different purposes. They appear in different financial statements, have varying durations and impact your company’s bottom line in different ways. A liability is money owed in the future, while an expense is a cost incurred to help the business run smoothly and generate revenue.
If you’re considering a business credit card to help you manage your finances and track expenses, you can get pre-approved with Capital One to find a card that meets your company’s needs and supports business growth.
Capital One Business
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6608 | https://www.randomservices.org/random/prob/Independence.html | Independence
Random
1. Probability Spaces
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5. Independence
In this section, we will discuss independence, one of the fundamental concepts in probability theory. Independence is frequently invoked as a modeling assumption, and moreover, (classical) probability itself is based on the idea of independent replications of the experiment. Conditional probability is an important prerequisite for this section, but as usual, if you are a new student of probability, you may want to ignore the measure-theoretic details.
Basic Theory
As usual, our starting point is a random experiment modeled by a probability space(S,S,P). So to review, S is the set of outcomes, S the collection of events, and P the probability measure on the sample space (S,S). We will define independence for two events, then for collections of events, and then for collections of random variables. In each case, the basic idea is the same.
Independence of Two Events
Events A and B are independent if P(A∩B)=P(A)P(B).
If both of the events have positive probability, then independence is equivalent to the statement that the conditional probability of one event given the other is the same as the (unconditional) probability of the event. That is, P(A∣B)=P(A)⟺P(B∣A)=P(B)⟺P(A∩B)=P(A)P(B) This is how you should think of independence: knowledge that one event has occurred does not change the probability assigned to the other event. Independence of two event was studied in the last section in the context of correlation. In particular, for two events, independent and uncorrelated mean the same thing.
The terms independent and disjoint sound vaguely similar but they are actually very different. First, note that disjointness is a purely set-theory concept while independence is a probability (measure-theoretic) concept. Indeed, two events can be independent relative to one probability measure and dependent relative to another. But most importantly, two disjoint events can never be independent, except in the trivial case that one of the events is null.
Suppose that A and B are disjoint events, each with positive probability. Then A and B are dependent, and in fact are negatively correlated.
Details:
Note that P(A∩B)=P(∅)=0 but P(A)P(B)>0.
If A and B are independent events then intuitively it seems clear that any event that can be constructed from A should be independent of any event that can be constructed from B. This is the case, as the next result shows. Moreover, this basic idea is essential for the generalization of independence that we will consider shortly.
If A and B are independent events, then the following pairs of events are independent:
A c, B
B, A c
A c, B c
Details:
Suppose that A and B are independent. Then by the difference rule and the complement rule, P(A c∩B)=P(B)−P(A∩B)=P(B)−P(A)P(B)=P(B)[1−P(A)]=P(B)P(A c) Hence A c and B are equivalent. Parts (b) and (c) follow from (a).
An event that is essentially deterministic, that is, has probability 0 or 1, is independent of any other event, even itself.
Suppose that A and B are events.
If P(A)=0 or P(A)=1, then A and B are independent.
A is independent of itself if and only if P(A)=0 or P(A)=1.
Details:
1. Recall that if P(A)=0 then P(A∩B)=0, and if P(A)=1 then P(A∩B)=P(B). In either case we have P(A∩B)=P(A)P(B).
2. The independence of A with itself gives P(A)=[P(A)]2 and hence either P(A)=0 or P(A)=1.
General Independence of Events
To extend the definition of independence to more than two events, we might think that we could just require pairwise independence, the independence of each pair of events. However, this is not sufficient for the strong type of independence that we have in mind. For example, suppose that we have three events A, B, and C. Mutual independence of these events should not only mean that each pair is independent, but also that an event that can be constructed from A and B (for example A∪B c) should be independent of C. Pairwise independence does not achieve this. Exercise gives a counterexample with three events that are pairwise independent, but the intersection of two of the events is related to the third event in the strongest possible sense.
Another possible generalization would be to simply require the probability of the intersection of the events to be the product of the probabilities of the events. However, this condition does not even guarantee pairwise independence. Exericse gives a counterexample. However, the definition of independence for two events does generalize in a natural way to an arbitrary collection of events.
Suppose that A i is an event for each i in a nonempty index set I. Then the collection A={A i:i∈I} is independent if for every finite J⊆I, P(⋂j∈J A j)=∏j∈J P(A j)
Independence of a collection of events is much stronger than mere pairwise independence of the events in the collection. The basic inheritance property in below follows immediately from the definition.
Suppose that A is a collection of events.
If A is independent, then B is independent for every B⊆A.
If B is independent for every finite B⊆A then A is independent.
For a finite collection of events, the number of conditions required for mutual independence grows exponentially with the number of events.
For n∈N+, there are 2 n−n−1 non-trivial conditions in the definition of the independence of n events.
Explicitly give the 4 conditions that must be satisfied for events A, B, and C to be independent.
Explicitly give the 11 conditions that must be satisfied for events A, B, C, and D to be independent.
Details:
There are 2 n subcollections of the n events. One is empty and n involve a single event. The remaining 2 n−n−1 subcollections involve two or more events and correspond to non-trivial conditions.
A, B, C are independent if and only if P(A∩B)=P(A)P(B)P(A∩C)=P(A)P(C)P(B∩C)=P(B)P(C)P(A∩B∩C)=P(A)P(B)P(C)
A, B, C, D are independent if and only if P(A∩B)=P(A)P(B)P(A∩C)=P(A)P(C)P(A∩D)=P(A)P(D)P(B∩C)=P(B)P(C)P(B∩D)=P(B)P(D)P(C∩D)=P(C)P(D)P(A∩B∩C)=P(A)P(B)P(C)P(A∩B∩D)=P(A)P(B)P(D)P(A∩C∩D)=P(A)P(C)P(D)P(B∩C∩D)=P(B)P(C)P(D)P(A∩B∩C∩D)=P(A)P(B)P(C)P(D)
If the events A 1,A 2,…,A n are independent, then it follows immediately from the definition that P(⋂i=1 n A i)=∏i=1 n P(A i) This is known as the multiplication rule for independent events. Compare this with the general multiplication rule for conditional probability.
The collection of essentially deterministic events D={A∈S:P(A)=0 or P(A)=1} is independent.
Details:
Suppose that n∈N+ and that {A 1,A 2,…,A n}⊆D. If P(A i)=0 for some i∈{1,2,…,n} then P(A 1∩A 2∩⋯∩A n)=0. If P(A i)=1 for every i∈{1,2,…,n} then P(A 1∩A 2∩⋯∩A n)=1. In either case, P(A 1∩A 2⋯∩A n)=P(A 1)P(A 2)⋯P(A n)
The next result generalizes on the complements of two independent events.
Suppose that A={A i:i∈I} and B={B i:i∈I} are two collections of events with the property that for each i∈I, either B i=A i or B i=A i c. Then A is independent if and only if B is independent.
Details:
The proof is actually very similar to the proof for two events in , except for more complicated notation. First, by the symmetry of the relation between A and B, it suffices to show A indpendent implies B independent. Next, by , it suffices to consider the case where the index set I is finite.
Fix k∈I and define B k=A k c and B i=A i for i∈I∖{k}. Suppose now that J⊆I. If k∉J then trivially, P(⋂j∈J B j)=∏j∈J P(B j). If k∈J, then using the difference rule, P(⋂j∈J B j)=P(⋂j∈J∖{k}A j)−P(⋂j∈J A j)=∏j∈J∖{k}P(A j)−∏j∈J P(A j)=[∏j∈J∖{k}P(A j)][1−P(A k)]=∏j∈J P(B j) Hence {B i:i∈I} is a collection of independent events.
Suppose now that B={B i:i∈I} is a general collection of events where B i=A i or B i=A i c for each i∈I. Then B can be obtained from A by a finite sequence of complement changes of the type in (a), each of which preserves independence.
Theorem in turn leads to the type of strong independence that we want. The following exercise gives examples.
If A, B, C, and D are independent events, then
A∪B, C c, D are independent.
A∪B c, C c∪D c are independent.
Details:
We will give proofs that use the complement theorem in , but to do so, some additional notation is helpful. If E is an event, let E 1=E and E 0=E c.
Note that A∪B=⋃(i,j)∈I A i∩B j where I={(1,0),(0,1),(1,1)} and note that the events in the union are disjoint. By the distributive property, (A∪B)∩C c=⋃(i,j)∈I A i∩B j∩C 0 and again the events in the union are disjoint. By additivity and , P[(A∪B)∩C c]=∑(i,j)∈I P(A i)P(B j)P(C 0)=(∑(i,j)∈I P(A i)P(B j))P(C 0)=P(A∪B)P(C c) By exactly the same type of argument, P[(A∪B)∩D]=P(A∪B)P(D) and P[(A∪B)∩C c∩D]=P(A∪B)P(C c)P(D). Directly from , P(C c∩D)=P(C c)P(D).
Note that A∪B c=⋃(i,j)∈I A i∩B j where I={(0,0),(1,0),(1,1)} and note that the events in the union are disjoint. Similarly C c∪D c=⋃(k,l)∈J C i∩D j where J={(0,0),(1,0),(0,1)}, and again the events in the union are disjoint. By the distributive rule for set operations, (A∪B c)∩(C c∪D c)=⋃(i,j,k,l)∈I×J A i∩B j∩C k∩D l and once again, the events in the union are disjoint. By additivity and , P[(A∪B c)∩(C c∪D c)]=∑(i,j,k,l)∈I×J P(A i)P(B j)P(C k)P(D l) But also by additivity, , and the distributive property of arithmetic, P(A∪B c)P(C c∪D c)=(∑(i,j)∈I P(A i)P(B j))(∑(k,l)∈J P(C k)P(D l))=∑(i,j,k,l)∈I×J P(A i)P(B j)P(C k)P(D l)
The complete generalization of these results is a bit complicated, but roughly means that if we start with a collection of indpendent events, and form new events from disjoint subcollections (using the set operations of union, intersection, and complment), then the new events are independent. The importance of lies in the fact that any event that can be defined in terms of a finite collection of events {A i:i∈I} can be written as a disjoint union of events of the form ⋂i∈I B i where B i=A i or B i=A i c for each i∈I.
Another consequence of is a formula for the probability of the union of a collection of independent events that is much nicer than the inclusion-exclusion formula.
If A 1,A 2,…,A n are independent events, then P(⋃i=1 n A i)=1−∏i=1 n[1−P(A i)]
Details:
From De Morgan's law and the independence of A 1 c,A 2 c,…,A n c we have P(⋃i=1 n A i)=1−P(⋂i=1 n A i c)=1−∏i=1 n P(A i c)=1−∏i=1 n[1−P(A i)]
Independence of Random Variables
Suppose now that (T,T) is another measurable space so that T is a set and T is a σ-algebra of subsets of T (the admissible subsets of T). Recall that a random variable with values in T is a measurable function from S to T, so that {X∈A}∈S for every A∈T. In words, a meaningful statment about X is an event. Intuitively, random variables are independent if information about some of the variables tells us nothing about the other variables. Mathematically, independence of random variables can be reduced to the independence of events. We start with two random variables, where the notation is simplest.
Suppose that (T,T) and (U,U) are measurable spaces and that X and Y are random variables with values in T and U, respectively. Then X and Y are independent if P(X∈A,Y∈B)=P(X∈A)P(Y∈B),A∈T,B∈U
That is, X and Y are independent if the events {X∈A} and {Y∈B} are independent for every A∈T and B∈U. The generalization to arbitrary collections of random variables is straightforward; only the notation is more complicated.
Suppose that (T i,T i) is a measurable space for each i in a nonempty index set I and that X i is a random variable with values in T i for each i∈I. The collection of random variables X={X i:i∈I} is independent if the collection of events {{X i∈A i}:i∈I} is independent for every choice of A i∈T i for i∈I. Equivalently then, X is independent if for every finite J⊆I, and for every choice of A j∈T j for j∈J we have P(⋂j∈J{X j∈A j})=∏j∈J P(X j∈A j)
Theorem is the inheritance property for independent random variables, analogous to for random variables:
Suppose that X is a collection of random variables.
If X is independent, then Y is independent for every Y⊆X
If Y is independent for every finite Y⊆X then X is independent.
It would seem almost obvious that if a collection of random variables is independent, and we transform each variable in deterministic way, then the new collection of random variables should still be independent.
In the setting of definition , suppose now that (U i,U i) is another measurable space for i∈I and that g i is a measurable function from T i into a U i for each i∈I. If {X i:i∈I} is independent, then {g i(X i):i∈I} is also independent.
Details:
Except for the abstract setting, the proof of independence is easy. Suppose that A i∈U i for each i∈I. Then {g i(X i)∈A i}={X i∈g i−1(A i)} for i∈I. Since the functions are measurable, g−1(A i)∈T i for i∈I and then by the independence of {X i:i∈I}, the collection of events {{X i∈g i−1(A i)}:i∈I} is independent.
Suppose that (T,T) is a measurable space and that X is a random variable for the experiment with values in T.
X is independent of itself if and only if P(X∈A)=0 or P(X∈A)=1 for every A∈T.
If X is independent of itself then X is independent of any other random variable Y for the experiment.
Proof;
The results follow immediately from the corresponding results for events in .
A constant random variable is independent of itself, and hence independent of any other random variable. To make this statement precies, suppose that (T,T) is a measurable spaces and that {x}∈T for every x∈T, a condition that is satisfied for the usual spaces that arise in applications.
Suppose that X is a random variables with values in T and that P(X=x)=1 for some x∈T so that X is constant with probability 1. Then X is independent of itself.
Details:
The events associated with X have probability 0 or 1: If A∈T then P(X∈A)=1(x∈A). So the result follows from the previous one.
Conversely, under certain conditions, a random variable that is independent of itself must be constant with probability 1. We will return to this point in the more advance section on probability spaces.
As with events, the (mutual) independence of random variables is a very strong property. If a collection of random variables is independent, then any subcollection is also independent. New random variables formed from disjoint subcollections are independent. Here is a simple example
Suppose that X, Y, and Z are independent real-valued random variables. Then
sin(X), cos(Y), and e Z are independent.
(X,Y) and Z are independent.
X 2+Y 2 and arctan(Z) are independent.
X and Z are independent.
Y and Z are independent.
In particular, note that statement (b) in the list above is much stronger than the conjunction of statements (d) and (e). Contrapositively, if X and Z are dependent, then (X,Y) and Z are also dependent. Independence of random variables subsumes independence of events.
A collection of events A is independent if and only if the corresponding collection of indicator variables {1 A:A∈A} is independent.
Details:
Let A={A i:i∈I} where I is a nonempty index set. For i∈I, the only non-trivial events that can be defined in terms of 1 A i are {1 A i=1}=A i and {1 A i=0}=A i c. So {1 A i:i∈I} is independent if and only if every collection of the form {B i:i∈I} is independent, where for each i∈I, either B i=A i or B i=A i c. But by , this is equivalent to the independence of {A i:i∈I}.
Many of the concepts that we have been using informally can now be made precise. A compound experiment that consists of independent stages is essentially just an experiment whose outcome is a sequence of independent random variables X=(X 1,X 2,…) where X i is the outcome of the i th stage.
In particular, suppose that we have a basic experiment with outcome variable X. By definition, the outcome of the experiment that consists of independent replications of the basic experiment is a sequence of independent random variables X=(X 1,X 2,…) each with the same probability distribution as X. This is fundamental to the very concept of probability, as expressed in the law of large numbers. From a statistical point of view, suppose that we have a population of objects and a vector of measurements X of interest for the objects in the sample. The sequence X above corresponds to sampling from the distribution of X; that is, X i is the vector of measurements for the i th object drawn from the sample. When we sample from a finite population, sampling with replacement generates independent random variables while sampling without replacement generates dependent random variables.
Conditional Independence and Conditional Probability
As noted at the beginning of our discussion, independence of events or random variables depends on the underlying probability measure. Thus, suppose that B is an event with positive probability. A collection of events or a collection of random variables is conditionally independent given B if the collection is independent relative to the conditional probability measure A↦P(A∣B). For example, a collection of events {A i:i∈I} is conditionally independent given B if for every finite J⊆I, P(⋂j∈J A j|B)=∏j∈J P(A j∣B) Note that the definitions and theorems of this section would still be true, but with all probabilities conditioned on B.
Conversely, conditional probability has a nice interpretation in terms of independent replications of the experiment. Thus, suppose that we start with a basic experiment with S as the set of outcomes. We let X denote the outcome random variable, so that mathematically X is simply the identity function on S. In particular, if A is an event then trivially, P(X∈A)=P(A). Suppose now that we replicate the experiment independently. This results in a new, compound experiment with a sequence of independent random variables (X 1,X 2,…), each with the same distribution as X. That is, X i is the outcome of the i th repetition of the experiment.
Suppose now that A and B are events in the basic experiment with P(B)>0. In the compound experiment, the event that when B occurs for the first time, A also occurs has probability P(A∩B)P(B)=P(A∣B)
Details:
In the compound experiment, if we record (X 1,X 2,…) then the new set of outcomes is S∞=S×S×⋯. The event that when B occurs for the first time, A also occurs is ⋃n=1∞{X 1∉B,X 2∉B,…,X n−1∉B,X n∈A∩B} The events in the union are disjoint. Also, since (X 1,X 2,…) is a sequence of independent variables, each with the distribution of X we have P(X 1∉B,X 2∉B,…,X n−1∉B,X n∈A∩B)=[P(B c)]n−1 P(A∩B)=[1−P(B)]n−1 P(A∩B) Hence, using geometric series, the probability of the union is ∑n=1∞[1−P(B)]n−1 P(A∩B)=P(A∩B)1−[1−P(B)]=P(A∩B)P(B)
Here is the heuristic argument: Suppose that we create a new experiment by repeating the basic experiment until B occurs for the first time, and then record the outcome of just the last repetition of the basic experiment. Now the set of outcomes is simply B and the appropriate probability measure on the new experiment is A↦P(A∣B).
Suppose that A and B are disjoint events in a basic experiment with P(A)>0 and P(B)>0. In the compound experiment obtained by replicating the basic experiment, the event that A occurs before B has probability P(A)P(A)+P(B)
Details:
Note that the event A occurs before B is the same as the event when A∪B occurs for the first time, A occurs.
Examples and Applications
Basic Rules
Suppose that A, B, and C are independent events in an experiment with P(A)=0.3, P(B)=0.4, and P(C)=0.8. Express each of the following events in set notation and find its probability:
All three events occur.
None of the three events occurs.
At least one of the three events occurs.
At least one of the three events does not occur.
Exactly one of the three events occurs.
Exactly two of the three events occurs.
Details:
1. P(A∩B∩C)=0.096
2. P(A c∩B c∩C c)=0.084
3. P(A∪B∪C)=0.916
4. P(A c∪B c∪C c)=0.904
5. P[(A∩B c∩C c)∪(A c∩B∩C c)∪(A c∩B c∩C)]=0.428
6. P[(A∩B∩C c)∪(A∩B c∩C)∪(A c∩B∩C)]=0.392
Suppose that A, B, and C are independent events for an experiment with P(A)=1 3, P(B)=1 4, and P(C)=1 5. Find the probability of each of the following events:
(A∩B)∪C
A∪B c∪C
(A c∩B c)∪C c
Details:
1. 4 15
2. 13 15
3. 9 10
Simple Populations
A small company has 100 employees; 40 are men and 60 are women. There are 6 male executives. How many female executives should there be if gender and rank are independent? The underlying experiment is to choose an employee at random.
Details:
9
Suppose that a farm has four orchards that produce peaches, and that peaches are classified by size as small, medium, and large. The table below gives total number of peaches in a recent harvest by orchard and by size. Fill in the body of the table with counts for the various intersections, so that orchard and size are independent variables. The underlying experiment is to select a peach at random from the farm.
| Orchard | Size | Small | Medium | Large | Total |
--- ---
| 1 | | | | 400 |
| 2 | | | | 600 |
| 3 | | | | 300 |
| 4 | | | | 700 |
| Total | 400 | 1000 | 600 | 2000 |
Details:
| Orchard | Size | Small | Medium | Large | Total |
--- ---
| 1 | 80 | 200 | 120 | 400 |
| 2 | 120 | 300 | 180 | 600 |
| 3 | 60 | 150 | 90 | 300 |
| 4 | 140 | 350 | 210 | 700 |
| Total | 400 | 1000 | 600 | 2000 |
Note from exercises and that you cannot see independence in a Venn diagram. Again, independence is a measure-theoretic concept, not a set-theoretic concept.
Bernoulli Trials
A Bernoulli trials sequence is a sequence X=(X 1,X 2,…) of independent, identically distributed indicator variables. Random variable X i is the outcome of trial i, where in the usual terminology of reliability theory, 1 denotes success and 0 denotes failure. The canonical example is the sequence of scores when a coin (not necessarily fair) is tossed repeatedly. Another basic example arises whenever we start with an basic experiment and an event A of interest, and then repeat the experiment. In this setting, X i is the indicator variable for event A on the i th run of the experiment. The Bernoulli trials process is named for Jacob Bernoulli), and has a single basic parameter p=P(X i=1).
For (x 1,x 2,…,x n)∈{0,1}n, P(X 1=x 1,X 2=x 2,…,X n=x n)=p x 1+x 2+⋯+x n(1−p)n−(x 1+x 2+⋯+x n)
Details:
If X is a generic Bernoulli trial, then by definition, P(X=1)=p and P(X=0)=1−p. Equivalently, P(X=x)=p x(1−p)1−x for x∈{0,1}. Thus the result follows by independence.
Note that the sequence of indicator random variables X is exchangeable. That is, if the sequence (x 1,x 2,…,x n) in is permuted, the probability does not change. On the other hand, there are exchangeable sequences of indicator random variables that are dependent, as Pólya's urn model so dramatically illustrates.
Let Y denote the number of successes in the first n trials. Then P(Y=y)=(n y)p y(1−p)n−y,y∈{0,1,…,n}
Details:
Note that Y=∑i=1 n X i, where X i is the outcome of trial i, as in the previous result. For y∈{0,1,…,n}, the event {Y=y} occurs if and only if exactly y of the n trials result in success (1). The number of ways to choose the y trials that result in success is (n y), and by , the probability of any particular sequence of y successes and n−y failures is p y(1−p)n−y. Thus the result follows by the additivity of probability.
The distribution of Y is called the binomial distribution with parameters n and p. More generally, a multinomial trials sequence is a sequence X=(X 1,X 2,…) of independent, identically distributed random variables, each with values in a finite set S. The canonical example is the sequence of scores when a k-sided die (not necessarily fair) is thrown repeatedly.
Cards
Consider the experiment that consists of dealing 2 cards at random from a standard deck and recording the sequence of cards dealt. For i∈{1,2}, let Q i be the event that card i is a queen and H i the event that card i is a heart. Compute the appropriate probabilities to verify the following results. Reflect on these results.
Q 1 and H 1 are independent.
Q 2 and H 2 are independent.
Q 1 and Q 2 are negatively correlated.
H 1 and H 2 are negatively correlated.
Q 1 and H 2 are independent.
H 1 and Q 2 are independent.
Details:
1. P(Q 1)=P(Q 1∣H 1)=1 13
2. P(Q 2)=P(Q 2∣H 2)=1 13
3. P(Q 1)=1 13, P(Q 1∣Q 2)=1 17
4. P(H 1)=1 4, P(H 1∣H 2)=4 17
5. P(Q 1)=P(Q 1∣H 2)=1 13
6. P(Q 2)=P(Q 2∣H 1)=1 13
In the card experiment), set n=2. Run the simulation 500 times. For each pair of events in exercise , compute the product of the empirical probabilities and the empirical probability of the intersection. Compare the results.
Dice
Exercise gives three events that are pairwise independent, but not (mutually) independent.
Consider the dice experiment that consists of rolling 2 standard, fair dice and recording the sequence of scores. Let A denote the event that first score is 3, B the event that the second score is 4, and C the event that the sum of the scores is 7. Then
A, B, C are pairwise independent.
A∩B implies (is a subset of) C and so these events are dependent in the strongest possible sense.
Details:
Note that A∩B=A∩C=B∩C={(3,4)}, and the probability of the common intersection is 1 36. On the other hand, P(A)=P(B)=P(C)=6 36=1 6.
In the dice experiment), set n=2. Run the experiment 500 times. For each pair of events in exercise , compute the product of the empirical probabilities and the empirical probability of the intersection. Compare the results.
Exercise gives an example of three events with the property that the probability of the intersection is the product of the probabilities, but the events are not pairwise independent.
Suppose that we throw a standard, fair die one time. Let A={1,2,3,4}, B=C={4,5,6}. Then
P(A∩B∩C)=P(A)P(B)P(C).
B and C are the same event, and hence are dependent in the strongest possbile sense.
Details:
Note that A∩B∩C={4}, so P(A∩B∩C)=1 6. On the other hand, P(A)=4 6 and P(B)=P(C)=3 6.
Suppose that a standard, fair die is thrown 4 times. Find the probability of the following events.
Six does not occur.
Six occurs at least once.
The sum of the first two scores is 5 and the sum of the last two scores is 7.
Details:
1. (5 6)4≈0.4823
2. 1−(5 6)4≈0.5177
3. 1 54
Suppose that a pair of standard, fair dice are thrown 8 times. Find the probability of each of the following events.
Double six does not occur.
Double six occurs at least once.
Double six does not occur on the first 4 throws but occurs at least once in the last 4 throws.
Details:
1. (35 36)8≈0.7982
2. 1−(35 36)8≈0.2018
3. (35 36)4[1−(35 36)4]≈0.0952
Let n,k∈N+ and consider the experiment that consists of rolling n dice, each k-sided and recording the sequence of scores X=(X 1,X 2,…,X n). The following conditions are equivalent (and correspond to the assumption that the dice are fair):
X is uniformly distributed on {1,2,…,k}n.
X is a sequence of independent variables, and X i is uniformly distributed on {1,2,…,k} for each i.
Details:
Let S={1,2,…,k} and note that S n has k n points. Suppose that X is uniformly distributed on S n. Then P(X=x)=1/k n for each x∈S n so P(X i=x)=k n−1/k n=1/k for each x∈S. Hence X i is uniformly distributed on S. Moreover, P(X=x)=P(X 1=x 1)P(X 2=x 2)⋯P(X n=x n),x=(x 1,x 2,…,x n)∈S n so X is an independent sequence. Conversely, if X is an independent sequence and X i is uniformly distributed on S for each i then P(X i=x)=1/k for each x∈S and hence P(X=x)=1/k n for each x∈S n. Thus X is uniformly distributed on S n.
A pair of standard, fair dice are thrown repeatedly. Find the probability of each of the following events.
A sum of 4 occurs before a sum of 7.
A sum of 5 occurs before a sum of 7.
A sum of 6 occurs before a sum of 7.
When a sum of 8 occurs the first time, it occurs the hard way as (4,4).
Details:
1. 3 9
2. 4 10
3. 5 11
4. 1 5
Problems of the type in exercise are important in the game of craps.
Coins
A biased coin with probability of heads 1 3 is tossed 5 times. Let X denote the outcome of the tosses (encoded as a bit string) and let Y denote the number of heads. Find each of the following:
P(X=x) for each x∈{0,1}5.
P(Y=y) for each y∈{0,1,2,3,4,5}.
P(1≤Y≤3)
Details:
1. 32 243 if x=00000, 16 243 if x has exactly one 1 (there are 5 of these), 8 243 if x has exactly two 1s (there are 10 of these), 4 243 if x has exactly three 1s (there are 10 of these), 2 243 if x has exactly four 1s (there are 5 of these), 1 243 if x=11111
2. 32 243 if y=0, 80 243 if y=1, 80 243 if y=2, 40 243 if y=3, 10 243 if y=4, 1 243 if y=5
3. 200 243
A box contains a fair coin and a two-headed coin. A coin is chosen at random from the box and tossed repeatedly. Let F denote the event that the fair coin is chosen, and let H i denote the event that toss i results in heads for i∈N+. Then
(H 1,H 2,…) are conditionally independent given F, with P(H i∣F)=1 2 for each i.
(H 1,H 2,…) are conditionally independent given F c, with P(H i∣F c)=1 for each i.
P(H i)=3 4 for each i.
P(H 1∩H 2∩⋯∩H n)=1 2 n+1+1 2.
(H 1,H 2,…) are dependent.
P(F∣H 1∩H 2∩⋯∩H n)=1 2 n+1.
P(F∣H 1∩H 2∩⋯∩H n)→0 as n→∞.
Details:
Parts (a) and (b) are essentially modeling assumptions, based on the design of the experiment. If we know what kind of coin we have, then the tosses are independent. Parts (c) and (d) follow by conditioning on the type of coin and using parts (a) and (b). Part (e) follows from (c) and (d). Note that the expression in (d) is not (3/4)n. Part (f) follows from part (d) and Bayes' theorem. Finally part (g) follows from part (f).
Consider again the box in exercise , but we change the experiment as follows: A coin is chosen at random from the box and tossed and the result recorded. The coin is returned to the box and the process is repeated. As before, let H i denote the event that toss i results in heads for i∈N+. Then
(H 1,H 2,…) are independent.
P(H i)=3 4 for each i.
P(H 1∩H 2∩⋯H n)=(3 4)n.
Details:
Again, part (a) is essentially a modeling assumption. Since we return the coin and draw a new coin at random each time, the results of the tosses should be independent. Part (b) follows by conditioning on the type of the i th coin. Part (c) follows from parts (a) and (b).
Think carefully about the results in exercises and , and the differences between the two models. Tossing a coin produces independent random variables if the probability of heads is fixed (that is, non-random even if unknown). Tossing a coin with a random probability of heads generally does not produce independent random variables; the result of a toss gives information about the probability of heads which in turn gives information about subsequent tosses.
Uniform Distributions
Recall that Buffon's coin experiment consists of tossing a coin with radius r≤1 2 randomly on a floor covered with square tiles of side length 1. The coordinates (X,Y) of the center of the coin are recorded relative to axes through the center of the square in which the coin lands. The following conditions are equivalent:
(X,Y) is uniformly distributed on [−1 2,1 2]2.
X and Y are independent and each is uniformly distributed on [−1 2,1 2].
Buffon's coin experiment
Details:
Let S=[−1 2,1 2] so that the square S 2 is the set of outcomes of Buffon's experiment. The standard (Borel) σ-algebra S of subsets of S is the one generated by intervals, and so the standard σ-algebra S 2 of subsets of S 2 is the product σ-algebra, generated by rectangles. The standard (Lebesgue) measure λ 1 on (S,S) generalizes length, and the standard measure λ 2 on (S 2,S 2) is the product measure that generalizes area. Note that λ 1(S)=λ 2(S 2)=1. With the technicalities out of the way, suppose that (X,Y) is uniformly distributed on S 2, so that P[(X,Y)∈C]=λ 2(C) for C∈S 2. For A∈S, P(X∈A)=P[(X,Y)∈A×S]=λ 2(A×S)=λ 1(A) Hence X is uniformly distributed on S. By a similar argument, Y is also uniformly distributed on S. Moreover, for A∈S and B∈S, P(X∈A,Y∈B)=P[(X,Y)∈A×B]=λ 2(A×B)=λ 1(A)λ 1(B)=P(X∈A)P(Y∈B) so X and Y are independent. Conversely, if X and Y are independent and each is uniformly distributed on S, then for A∈S and B∈S, P[(X,Y)∈A×B]=P(X∈A)P(Y∈B)=λ 1(A)λ 1(B)=λ 2(A×B) It then follows that P[(X,Y)∈C]=λ 2(C) for every C∈S 2. For more details about this last step, see the section on existence and uniqueness or positive measures.
Compare exercise with exercise on fair dice.
In Buffon's coin experiment), set r=0.3. Run the simulation 500 times. For the events {X>0} and {Y<0}, compute the product of the empirical probabilities and the empirical probability of the intersection. Compare the results.
The arrival time X of the A train is uniformly distributed on the interval (0,30), while the arrival time Y of the B train is uniformly distributed on the interval (15,30). (The arrival times are in minutes, after 8:00 AM). Moreover, the arrival times are independent. Find the probability of each of the following events:
The A train arrives first.
Both trains arrive sometime after 20 minutes.
Details:
1. 3 4
2. 2 9
Reliability
Recall the simple model of structural reliability in which a system is composed of n components, where n∈N+. Suppose in addition that the components operate independently of each other. As before, let X i denote the state of component i, where 1 means working and 0 means failure. Thus, our basic assumption is that the state vector X=(X 1,X 2,…,X n) is a sequence of independent indicator random variables. We assume that the state of the system (either working or failed) depends only on the states of the components. Thus, the state of the system is an indicator random variable Y=f(X 1,X 2,…,X n) where f:{0,1}n→{0,1} is the structure function. Generally, the probability that a device is working is the reliability of the device. Thus, we will denote the reliability of component i by p i=P(X i=1) so that the vector of component reliabilities is p=(p 1,p 2,…,p n). By independence, the system reliability r is a function of the component reliabilities: r(p 1,p 2,…,p n)=P(Y=1) Appropriately enough, this function is known as the reliability function. Our challenge is usually to find the reliability function r, given the structure function f. When the components all have the same probability p then of course the system reliability r is just a function of p. In this case, the state vector X=(X 1,X 2,…,X n) forms a sequence of Bernoulli trials as discussed above.
Comment on the independence assumption for real systems, such as your car or your computer.
Recall that a series system is working if and only if each component is working.
The state of the system is U=X 1 X 2⋯X n=min{X 1,X 2,…,X n}.
The reliability is P(U=1)=p 1 p 2⋯p n.
Recall that a parallel system is working if and only if at least one component is working.
The state of the system is V=1−(1−X 1)(1−X 2)⋯(1−X n)=max{X 1,X 2,…,X n}.
The reliability is P(V=1)=1−(1−p 1)(1−p 2)⋯(1−p n).
Recall that a k out of n system is working if and only if at least k of the n components are working, where k∈{1,2,…,n}. So a parallel system is a 1 out of n system and a series system is an n out of n system. For k∈N+, a k out of 2 k−1 system is a majority rules system. The reliability function of a general k out of n system is a mess. However, if the component reliabilities are the same, the function has a reasonably simple form.
For a k out of n system with common component reliability p, the system reliability is r(p)=∑i=k n(n i)p i(1−p)n−i
Consider a system of 3 independent components with common reliability p=0.8. Find the reliability of each of the following:
The parallel system.
The 2 out of 3 system.
The series system.
Details:
1. 0.992
2. 0.896
3. 0.512
Consider a system of 3 independent components with reliabilities p 1=0.8, p 2=0.8, p 3=0.7. Find the reliability of each of the following:
The parallel system.
The 2 out of 3 system.
The series system.
Details:
1. 0.994
2. 0.902
3. 0.504
Consider an airplane with an odd number of engines, each with reliability p. Suppose that the airplane is a majority rules system, so that the airplane needs a majority of working engines in order to fly.
Find the reliability of a 3 engine plane as a function of p.
Find the reliability of a 5 engine plane as a function of p.
For what values of p is a 5 engine plane preferable to a 3 engine plane?
Details:
1. r 3(p)=3 p 2−2 p 3
2. r 5(p)=6 p 5−15 p 4+10 p 3
3. The 5-engine plane would be preferable if p>1 2 (which one would hope would be the case). The 3-engine plane would be preferable if p<1 2. If p=1 2, the 3-engine and 5-engine planes are equally reliable.
The graph below is known as the Wheatstone bridge network and is named for Charles Wheatstone). The edges represent components, and the system works if and only if there is a working path from vertex a to vertex b.
Find the structure function.
Find the reliability function.
The Wheatstone bridge netwok
Details:
1. Y=X 3(X 1+X 2−X 1 X 2)(X 4+X 5−X 4,X 5)+(1−X 3)(X 1 X 4+X 2 X 5−X 1 X 2 X 4 X 5)
2. r(p 1,p 2,p 3,p 4,p 5)=p 3(p 1+p 2−p 1 p 2)(p 4+p 5−p 4,p 5)+(1−p 3)(p 1 p 4+p 2 p 5−p 1 p 2 p 4 p 5)
A system consists of 3 components, connected in parallel. Because of environmental factors, the components do not operate independently, so our usual assumption does not hold. However, we will assume that under low stress conditions, the components are independent, each with reliability 0.9; under medium stress conditions, the components are independent with reliability 0.8; and under high stress conditions, the components are independent, each with reliability 0.7. The probability of low stress is 0.5, of medium stress is 0.3, and of high stress is 0.2.
Find the reliability of the system.
Given that the system works, find the conditional probability of each stress level.
Details:
1. 0.9917. Condition on the stress level.
2. 0.5037 for low, 0.3001 for medium, 0.1962 for high. Use Bayes' theorem and part (a).
Suppose that bits are transmitted across a noisy communications channel. Each bit that is sent, independently of the others, is received correctly with probability 0.9 and changed to the complementary bit with probability 0.1. Using redundancy to improve reliability, suppose that a given bit will be sent 3 times. We naturally want to compute the probability that we correctly identify the bit that was sent. Assume we have no prior knowledge of the bit, so we assign probability 1 2 each to the event that 000 was sent and the event that 111 was sent. Now find the conditional probability that 111 was sent given each of the 8 possible bit strings received.
Details:
Let X denote the string sent and Y the string received.
| y | P(X=111∣Y=y) |
--- |
| 111 | 729/730 |
| 110 | 9/10 |
| 101 | 9/10 |
| 011 | 9/10 |
| 100 | 1/10 |
| 010 | 1/10 |
| 001 | 1/10 |
| 000 | 1/730 |
Diagnostic Testing
Recall the previous discussion of diagnostic testing. To review, we have an event A for a random experiment whose occurrence or non-occurrence we cannot observe directly. Suppose now that n∈N+ and that we have n tests for the occurrence of A, labeled from 1 to n. We will let T i denote the event that test i is positive for A.
We assume that the tests are independent in the following sense:
If A occurs, then (T 1,T 2,…,T n) are (conditionally) independent and test i has sensitivity a i=P(T i∣A).
If A does not occur, then (T 1,T 2,…,T n) are (conditionally) independent and test i has specificity b i=P(T i c∣A c).
Note that unconditionally, it is not reasonable to assume that the tests are independent. For example, a positive result for a given test presumably is evidence that the condition A has occurred, which in turn is evidence that a subsequent test will be positive. In short, we expect that T i and T j should be positively correlated.
We can form a new, compound test by giving a decision rule in terms of the individual test results. In other words, the event T that the compound test is positive for A is a function of (T 1,T 2,…,T n). The typical decision rules are very similar to the reliability structures in discussed above. A special case of interest is when the n tests are independent applications of a given basic test. In this case, a i=a and b i=b for each i.
Consider the compound test that is positive for A if and only if each of the n tests is positive for A.
T=T 1∩T 2∩⋯∩T n
The sensitivity is P(T∣A)=a 1 a 2⋯a n.
The specificity is P(T c∣A c)=1−(1−b 1)(1−b 2)⋯(1−b n)
Consider the compound test that is positive for A if and only if each at least one of the n tests is positive for A.
T=T 1∪T 2∪⋯∪T n
The sensitivity is P(T∣A)=1−(1−a 1)(1−a 2)⋯(1−a n).
The specificity is P(T c∣A c)=b 1 b 2⋯b n.
More generally, we could define the compound k out of n test that is positive for A if and only if at least k of the individual tests are positive for A where k∈{1,2,…,n}. The series test in is the n out of n test, while the parallel test in is the 1 out of n test. The k out of 2 k−1 test is the majority rules test.
Suppose that a woman initially believes that there is an even chance that she is or is not pregnant. She buys three identical pregnancy tests with sensitivity 0.95 and specificity 0.90. Tests 1 and 3 are positive and test 2 is negative.
Find the updated probability that the woman is pregnant.
Can we just say that tests 2 and 3 cancel each other out? Find the probability that the woman is pregnant given just one positive test, and compare the answer with the answer to part (a).
Details:
1. 0.834
2. No: 0.905.
Suppose that 3 independent, identical tests for an event A are applied, each with sensitivity a∈[0,1] and specificity b∈[0,1]. Find the sensitivity and specificity of the following tests:
1 out of 3 test
2 out of 3 test
3 out of 3 test
Details:
1. sensitivity 1−(1−a)3, specificity b 3
2. sensitivity 3 a 2, specificity b 3+3 b 2(1−b)
3. sensitivity a 3, specificity 1−(1−b)3
In a criminal trial, the defendant is convicted if and only if all 6 jurors vote guilty. Assume that if the defendant really is guilty, the jurors vote guilty, independently, with probability 0.95, while if the defendant is really innocent, the jurors vote not guilty, independently with probability 0.8. Suppose that 70% of defendants brought to trial are guilty.
Find the probability that the defendant is convicted.
Given that the defendant is convicted, find the probability that the defendant is guilty.
Comment on the assumption that the jurors act independently.
Details:
1. 0.5148
2. 0.99996
3. The independence assumption is not reasonable since jurors collaborate.
Genetics
Refer to the previous discussion of genetics if you need to review some of the definitions in this section. Recall first that the ABO blood type in humans is determined by three alleles: a, b, and o. Furthermore, a and b are co-dominant and o is recessive. Suppose that in a certain population, the proportion of a, b, and o alleles are p, q, and r respectively. Of course we must have p>0, q>0, r>0 and p+q+r=1.
Suppose that the blood genotype in a person is the result of independent alleles, chosen with probabilities p, q, and r as above.
The probability distribubtion of the geneotypes is given in the following table:
| Genotype | a a | a b | a o | b b | b o | oo |
| Probability | p 2 | 2 p q | 2 p r | q 2 | 2 q r | r 2 |
The probability distribution of the blood types is given in the following table:
| Blood type | A | B | A B | O |
| Probability | p 2+2 p r | q 2+2 q r | 2 p q | r 2 |
Details:
Part (a) follows from the independence assumption and basic rules of probability. Even though genotypes are listed as unordered pairs, note that there are two ways that a heterozygous genotype can occur, since either parent could contribute either of the two distinct alleles. Part (b) follows from part (a) and basic rules of probability.
Exercise is related to the Hardy-Weinberg model of genetics. The model is named for the English mathematician Godfrey Hardy) and the German physician Wilhelm Weiberg
Suppose that the probability distribution for the set of blood types in a certain population is given in the following table:
| Blood type | A | B | A B | O |
| Probability | 0.360 | 0.123 | 0.038 | 0.479 |
Find p, q, and r.
Details:
p=0.224, q=0.084, r=0.692
Suppose next that pod color in certain type of pea plant is determined by a gene with two alleles: g for green and y for yellow, and that g is dominant and o recessive.
Suppose that 2 green-pod plants are bred together. Suppose further that each plant, independently, has the recessive yellow-pod allele with probability 1 4.
Find the probability that 3 offspring plants will have green pods.
Given that the 3 offspring plants have green pods, find the updated probability that both parents have the recessive allele.
Details:
1. 987 1024
2. 27 987
Next consider a sex-linked hereditary disorder in humans (such as colorblindness or hemophilia). Let h denote the healthy allele and d the defective allele for the gene linked to the disorder. Recall that h is dominant and d recessive for women.
Suppose that a healthy woman initially has a 1 2 chance of being a carrier. (This would be the case, for example, if her mother and father are healthy but she has a brother with the disorder, so that her mother must be a carrier).
Find the probability that the first two sons of the women will be healthy.
Given that the first two sons are healthy, compute the updated probability that she is a carrier.
Given that the first two sons are healthy, compute the conditional probability that the third son will be healthy.
Details:
1. 5 8
2. 1 5
3. 9 10
Laplace's Rule of Succession
Suppose m∈N+ that we have m+1 coins, labeled 0,1,…,m. Coin i lands heads with probability i m for each i∈{0,1,…,m}. The experiment is to choose a coin at random (so that each coin is equally likely to be chosen) and then toss the chosen coin repeatedly.
For n∈N+, the probability that the first n tosses are all heads is p m,n=1 m+1∑i=0 m(i m)n
p m,n→1 n+1 as m→∞
The conditional probability that toss n+1 is heads given that the previous n tosses were all heads is p m,n+1 p m,n
p m,n+1 p m,n→n+1 n+2 as m→∞
Details:
Part (a) follows by conditioning on the chosen coin. For part (b), note that p m,n is an approximating sum for ∫0 1 x n d x=1 n+1. Part (c) follows from the definition of conditional probability, and part (d) is a trivial consequence of (b), (c).
In exercise , note that coin 0 is two-tailed, the probability of heads increases with i, and coin m is two-headed. The limiting conditional probability in part (d) is called Laplace's Rule of Succession, named after Simon Laplace). This rule was used by Laplace and others as a general principle for estimating the conditional probability that an event will occur on time n+1, given that the event has occurred n times in succession.
Suppose that a missile has had 10 successful tests in a row. Compute Laplace's estimate that the 11th test will be successful. Does this make sense?
Details:
11 12. No, not really.
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Chip-firing clocks
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Let G be some outdegree-regular directed graph with n vertices and let H be the Laplacian of G, so that the rows of H correspond to chip-firing moves. I’m interested in linear functions f from Zn to Z/kZ with the property that performing a chip-firing move on a vector in Zn increases the value of f by 1 mod k (I call such a function f a chip-firing “clock”). Such an f will exist only for certain values of k. This feels like it’s related to the structure of the cokernel of the Laplacian but I’m having trouble seeing exactly what's going on. I’m interested in knowing about all such functions f for a given G.
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edited Apr 4, 2021 at 23:12
James Propp
asked Apr 4, 2021 at 18:56
James ProppJames Propp
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Another way to phrase this question in terms of lifts of group homomorphisms: we have the linear map H:Zn→Zn, and the linear map α:Zn→Z/kZ for which α(v) is the sum of its coordinates modulo k; then the f in question are the lifts of α along H. But this doesn't really answer your question because I think in general questions about lifts of group homomorphisms are hard/might involve group cohomology.
– Sam Hopkins
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Apr 4, 2021 at 19:18
If α were instead the zero morphism, then indeed these f would just be maps from coker(H) to Z/kZ.
– Sam Hopkins
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Apr 4, 2021 at 19:19
One more (rather trivial) observation: we need that α(v)=0 for any v∈ker(H). You say "outdegree-regular graph" which maybe suggests you are thinking of directed graphs. But if G is undirected, then for instance one element of the kernel is the all 1's vector 1. And α(1)=0 if and only if k divides n. If G is moreover connected then 1 generates the kernel of H so this is the only requirement that we get from consideration of the kernel. But I don't see any reason why if k divides n then such an f must exist.
– Sam Hopkins
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Apr 4, 2021 at 21:56
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Is this the same as a solution to Hx≡1(modk)?
– lambda
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Apr 5, 2021 at 14:07
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@Josh: If H=(1−1−11) then, with k=2, (v1,v2)↦v1 is such a function f.
– Sam Hopkins
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Apr 5, 2021 at 15:16
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For two such functions f and g, the difference f−g is invariant under chip-firing, i.e. it factors through a function from the cokernel of the Laplacian to Z/k. Conversely, for such a function f, adding any linear function from the cokernel of the Laplacian to Z/k produces another such function.
So if there are any such functions, they are classified by linear functions from the cokernel of the Laplacian to Z/k.
Such a function exists if and only if there do not exist integer vectors v,w∈Zn where kv=Λw, for Λ the Laplacian, and the sum of the entries of w is nonzero mod k.
The "only if" direction is straightforward. Given such v,w, we would have 0=kf(v)=f(kv)=f(Λw) equal to the sum of entries of w.
For the "if" direction, write the cokernel of the Laplacian as a product of cyclic groups Z/ni generated by vectors vi.
Then nivi lies in image of the Laplacian, so we can write nivi=Λwi for some vector wi. We can choose f(vi) such that f(nivi) is the sum of entries of wi. We can do this as long as the some of entries of wi is divisible by gcd(ni,k), which it is because Λwik/gcd(ni,k)=k(vini/gcd(ni,k)) and vini/gcd(ni,k) is integral so the sum of entries of wik/gcd(ni,k) is divisible by k by construction.
Having made this chose, we can define f for an arbitrary v by writing it as a integer linear combination of vi plus a vector of the form Λw, and taking the appropriate linear combination of f(vi) and the sum of the entries of w. This is well-defined because the only ambiguity consists of adding ni to the coefficients of vi, which we checked is consistent, and adding something in the kernel of the Laplacian, whose consistency follows from the assumption.
Here is a simple example of a chip-firing clock. Consider the graph with vertices 1,…,n and edges from vertex i to i+1 and n to 1.
Then the function of a vector a1,…,an given by f(a1,…,an)=∑iiaimodn is a chip-firing clock.
In this case, the cokernel of the Laplacian is Z, which is torsion-free, so there is no obstruction, and there is a unique stationary probability distribution, so the longest period of a chip-firing clock is just the least common denominators of the probabilties, which is n.
For a more complicated example, consider a graph with vertices 1,…n where vertex i has edges to i+1 and n except for n which just connects to n+1.
Then the function of a vector a1,…an given by f(a1,…,an)=∑ni=12iaimod(2n−1) is a chip-firing clock. Again this is an example where the cokernel of the Laplacian is just Z.
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edited Apr 6, 2021 at 3:11
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Is the f(x) you defined in the 6th paragraph linear though?
– Sam Hopkins
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Apr 5, 2021 at 2:58
@SamHopkins Oh, sorry, I missed the "linear" thing. Will revise...
– Will Sawin
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Apr 5, 2021 at 2:58
Indeed I think the linear requirement makes it a tricky question...
– Sam Hopkins
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Apr 5, 2021 at 2:59
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@SamHopkins I put in the correct criterion one gets from thinking about group cohomology (whose expression doesn't really need group cohomology in this case). This reverses the problem - instead of checking existence by giving an explicit list of n numbers, one can check nonexistence by giving an explicit list of 2n numbers - but doesn't give a completely satisfying answer. I don't know if a more satisfying answer is possible.
– Will Sawin
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Apr 5, 2021 at 3:19
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@SamHopkins Just take ni=0 : )
– Will Sawin
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Apr 5, 2021 at 3:23
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This is just to record the observation of lambda from the comments.
I'll keep your convention that the rows of H determine the chip-firing moves, so that for a sequence of firings v∈Zn the result of caring out these firings (starting from the zero configuration) is vH. (But this means that we should really be talking about the kernel and cokernel of Ht everywhere...)
For x∈Zn, the linear function f(v)=⟨v,x⟩modk is a clock if and only if Hx=1modk. (Here ⟨⋅,⋅⟩ is the usual inner product, and 1 is the all ones vector.) This is because the requirement we have to satisfy to be a clock is that f(vH)=⟨v,1⟩modk for all v∈Zn, so we need f(vH)=⟨vH,x⟩=⟨v,Hx⟩ to be equal moulo k to ⟨v,1⟩ for all v∈Zn, which happens if and only if Hx=1modk.
It still is not totally clear how given a graph G to find the (finite!) list of k for which a clock exists, but for a fixed k at least this makes it clear that the question of whether such a k exists is a "linear algebra" problem (if k is prime then indeed we're talking about linear algebra over a field).
[ By the way, here is the argument that the set of such k is finite. Ht is a singular M-matrix, so there is some vector v∗∈ker(Ht) with v∗≠0 but all entries of v∗ nonnegative. Hence in particular ⟨v∗,1⟩>0. But, as mentioned, if there is a k-clock we need that ⟨v,1⟩=0modk for any v∈ker(Ht), so as long as k>⟨v∗,1⟩ then there cannot be a k-clock. ]
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If there exist clocks modulo coprime numbers k and ℓ then there is also one mod kℓ by the chinese remainder theorem, and clearly a clock mod k will also work mod any divisor of k. So the finiteness actually implies there is some m such that the integers for which clocks exist are exactly the divisors of m. Your argument shows that m divides ⟨v,1⟩ for all v∈KerHT. So I guess the optimistic conjecture would be that it equals the gcd of these. (For undirected graphs this is just the number of vertices.)
– lambda
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Apr 5, 2021 at 21:18
@lambda: yes, that would be the ideal situation, if the maximum k for which a k-clock exists was the gcd of ⟨v,1⟩ for v∈ker(Ht). I think it's possible to give a combinatorial description of the generators of ker(Ht) (and by the way if G is strongly connected then this kernel should be generated by a single primitive vector).
– Sam Hopkins
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Apr 5, 2021 at 21:23
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On that note, I of course really meant it equals the number of vertices for a connected undirected graph.
– lambda
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Apr 5, 2021 at 21:31
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@lambda This is not necesarily true. For example, if you take any directed graph and double every edge, i.e. double every entry in H, it is not possible to solve Hx=1 mod 2, so k cannot be even, but the kernel of HT is unchanged. It is true when the cokernel of the Laplacian Ht is torsion-free.
– Will Sawin
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Apr 5, 2021 at 22:04
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Okay, I'm no longer optimistic: the undirected 4-cycle has a clock mod 2 but not mod 4.
– lambda
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Apr 6, 2021 at 0:18
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How to show that a function satisfying certain inequalities exist? [closed]
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Let N be the set of Natural numbers, and let b,s∈N satisfy 1≤b≤s−3. I want to know whether there exists a Real-valued function f:N→R satisfying
f(s)>0f(s)>f(s+1)(s−b)⋅f(s)>(t−b)⋅f(t)∀t s.t. t≥b+2 and t<s
I have tried to construct one such function, but I haven't succeeded, as all of them have failed the third condition. Then, I have tried to prove that one such function does not exist, but I haven't succeeded. I do not need any specific function, I simply need an argument (constructive or otherwise) that establishes the existence of one such function.
functions
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edited Jan 13, 2024 at 0:26
EoDmnFOr3q
asked Jan 13, 2024 at 0:02
EoDmnFOr3qEoDmnFOr3q
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Define f′(s).
– Kavi Rama Murthy
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Jan 13, 2024 at 0:09
Thank you for your comment. Here, f′(s) is the derivative of f.
– EoDmnFOr3q
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Jan 13, 2024 at 0:10
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From your first sentence, it looks like s is fixed. But in the requirements, you have "∀s⩾4".
– user1266745
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Jan 13, 2024 at 0:10
Thank you for your comment. Here, s is not fixed: it can be anything bigger or equal than b+3.
– EoDmnFOr3q
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Jan 13, 2024 at 0:11
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Derivative is not defined for functions with domain N.
– Kavi Rama Murthy
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Jan 13, 2024 at 0:12
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Let f(n)=1/n for natural n.
Certainly f(s)>0, and certainly f(s)=1/s>1/(s+1)=f(s+1).
(s−b)f(s)=s−bs=1−bs
For s>t we have 1/s<1/t so −1/s>−1/t. Hence,
1−bs>1−bt=t−bt=(t−b)f(t)
and we win.
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answered Jan 13, 2024 at 0:58
Gerry MyersonGerry Myerson
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6611 | https://pmc.ncbi.nlm.nih.gov/articles/PMC7325104/ | A systematic review of validated screening tools for anxiety disorders and PTSD in low to middle income countries - PMC
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. 2020 Jun 30;20:338. doi: 10.1186/s12888-020-02753-3
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A systematic review of validated screening tools for anxiety disorders and PTSD in low to middle income countries
Anisa Y Mughal
Anisa Y Mughal
1 The University of Pittsburgh School of Medicine, 3550 Terrace Street, Pittsburgh, PA 15213 USA
Find articles by Anisa Y Mughal
1, Jackson Devadas
Jackson Devadas
2 Department of Health Behavior, Gillings School of Global Public Health, University of North Carolina at Chapel Hill, 135 Dauer Dr, Chapel Hill, NC 27599 USA
Find articles by Jackson Devadas
2, Eric Ardman
Eric Ardman
3 University of Miami Miller School of Medicine, 1600 NW 10th Ave #1140, Miami, FL 33136 USA
Find articles by Eric Ardman
3, Brooke Levis
Brooke Levis
4 Department of Epidemiology, Biostatistics and Occupational Health, McGill University, Montreal, Quebec H3A 1A2 Canada
5 Centre for Prognosis Research, School of Primary, Community and Social Care, Keele University, Staffordshire, ST5 5BG UK
Find articles by Brooke Levis
4,5, Vivian F Go
Vivian F Go
2 Department of Health Behavior, Gillings School of Global Public Health, University of North Carolina at Chapel Hill, 135 Dauer Dr, Chapel Hill, NC 27599 USA
Find articles by Vivian F Go
2, Bradley N Gaynes
Bradley N Gaynes
6 Department of Psychiatry, University of North Carolina School of Medicine, 101 Manning Dr, Chapel Hill, NC 27514 USA
Find articles by Bradley N Gaynes
6,✉
Author information
Article notes
Copyright and License information
1 The University of Pittsburgh School of Medicine, 3550 Terrace Street, Pittsburgh, PA 15213 USA
2 Department of Health Behavior, Gillings School of Global Public Health, University of North Carolina at Chapel Hill, 135 Dauer Dr, Chapel Hill, NC 27599 USA
3 University of Miami Miller School of Medicine, 1600 NW 10th Ave #1140, Miami, FL 33136 USA
4 Department of Epidemiology, Biostatistics and Occupational Health, McGill University, Montreal, Quebec H3A 1A2 Canada
5 Centre for Prognosis Research, School of Primary, Community and Social Care, Keele University, Staffordshire, ST5 5BG UK
6 Department of Psychiatry, University of North Carolina School of Medicine, 101 Manning Dr, Chapel Hill, NC 27514 USA
✉
Corresponding author.
Received 2019 Jul 22; Accepted 2020 Jun 23; Collection date 2020.
© The Author(s) 2020
Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit The Creative Commons Public Domain Dedication waiver ( applies to the data made available in this article, unless otherwise stated in a credit line to the data.
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PMCID: PMC7325104 PMID: 32605551
Abstract
Background
Anxiety and post-traumatic stress disorder (PTSD) contribute significantly to disability adjusted life years in low- to middle-income countries (LMICs). Screening has been proposed to improve identification and management of these disorders, but little is known about the validity of screening tools for these disorders. We conducted a systematic review of validated screening tools for detecting anxiety and PTSD in LMICs.
Methods
MEDLINE, EMBASE, Global Health and PsychINFO were searched (inception-April 22, 2020). Eligible studies (1) screened for anxiety disorders and/or PTSD; (2) reported sensitivity and specificity for a given cut-off value; (3) were conducted in LMICs; and (4) compared screening results to diagnostic classifications based on a reference standard. Screening tool, cut-off, disorder, region, country, and clinical population were extracted for each study, and we assessed study quality. Accuracy results were organized based on screening tool, cut-off, and specific disorder. Accuracy estimates for the same cut-off for the same screening tool and disorder were combined via meta-analysis.
Results
Of 6322 unique citations identified, 58 articles including 77 screening tools were included. There were 46, 19 and 12 validations for anxiety, PTSD, and combined depression and anxiety, respectively. Continentally, Asia had the most validations (35). Regionally, South Asia (11) had the most validations, followed by South Africa (10) and West Asia (9). The Kessler-10 (7) and the Generalized Anxiety Disorder-7 item scale (GAD-7) (6) were the most commonly validated tools for anxiety disorders, while the Harvard Trauma Questionnaire (3) and Posttraumatic Diagnostic Scale (3) were the most commonly validated tools for PTSD. Most studies (29) had the lowest quality rating (unblinded). Due to incomplete reporting, we could meta-analyze results from only two studies, which involved the GAD-7 (cut-off ≥10, pooled sensitivity = 76%, pooled specificity = 64%).
Conclusion
Use of brief screening instruments can bring much needed attention and research opportunities to various at-risk LMIC populations. However, many have been validated in inadequately designed studies, precluding any general recommendation for specific tools in LMICs. Locally validated screening tools for anxiety and PTSD need further evaluation in well-designed studies to assess whether they can improve the detection and management of these common disorders.
Trial registration
PROSPERO registry number CRD42019121794.
Keywords: Anxiety, Post-traumatic stress disorder, Screening tool, Validation, Low-to-middle income countries
Background
Mental health disorders, including anxiety and post-traumatic stress disorder (PTSD) are among the leading contributors to global disability adjusted life years, comprising five of the top twenty contributing disorders . The World Health Organization International Classification of Disease (ICD-11) defines anxiety as a disorder in which there is an extreme and excessive focus on an “anticipated threat” and defines PTSD as a disorder that results from exposure to one or more “horrific events”, both of whose symptoms include apprehension, motor tension and autonomic overactivity . In 2017, it was estimated that over 264 million people experienced an anxiety disorder, with the global prevalence for both anxiety disorders and PTSD ranging from 2.5 to 7% by country [2–4]. Both anxiety and PTSD are widespread common mental disorders (CMDs) that have been shown to cause significant negative health outcomes within various populations and contribute to a large portion of the global disease burden [5, 6]. There are noteworthy discrepancies in quality of life between people diagnosed with anxiety and/or PTSD and those who are not diagnosed with either, such as increased years lived with disability and decreased life expectancy [7–9]. Additionally, there is evidence suggesting that the presence of an anxiety disorder or PTSD increases the likelihood of comorbidity with other severe health conditions, such as major depressive disorder and substance use disorder [10, 11].
Anxiety and PTSD in low to middle income countries (LMICs) are highly prevalent and require further study given that access to care is hindered by availability and stigma [12–14]. Prevalence of these disorders is higher within LMICs; roughly 83% of people with mental illnesses globally are living within LMICs . In many LMICs, there is no robust mental healthcare system in place and the number of mental health professionals is sparse . Assessment and diagnosis of psychiatric illnesses thus often falls to primary care and general practitioners who have little training in mental health . Use of brief screening tools have been proposed as a way to improve identification and management of mental health problems, and may be useful in LMICs, especially among populations with elevated risk (e.g., pregnant women, refugees/displaced persons, and youth) within LMIC communities [17–19].
Despite multiple screening instruments for CMDs, there are significantly fewer screening instruments for anxiety and PTSD that have been validated in LMIC populations. Screening instruments that have been validated exclusively in high-income countries may not perform equivalently in LMIC populations, as anxiety and PTSD often present differently in different cultural contexts. For example, in sub-Saharan Africa, anxiety and PTSD are described through somatic symptoms as well as spiritual descriptions . Furthermore, differences in clinical presentation may render screening tools less accurate in LMICs. Thus, optimum cut-off scores validated in high income populations may not apply in LMIC populations. For instance, in a sample of 75 participants from Tajikistan , the optimal cut-off of 1.88 for the Harvard Trauma Questionnaire (HTQ), a measure of PTSD, was substantially lower than the standard cut-off score of 2.5 that has been recommended in previous studies in high-income countries . Failure to apply suitable cut-off scores may lead to an imbalance of positive and negative screening results. If chosen cutoffs are too high, actual cases of anxiety and PTSD may not reach the threshold for further assessment and diagnosis; thus, cases will be missed. Conversely, if chosen cutoffs are too low, there may a very large number of positive screens requiring substantial resources for further assessment, and healthcare systems may not be able to manage the load.
Although there has been an increasing interest in studying mental health within LMICs, there are still large gaps related to screening tools to assess mental health disorders, especially anxiety and PTSD. The most recent systematic review investigating screening tools for CMDs in LMICs was published in 2016 . Of the 273 validations included, 236 were validated tools for CMDs or depressive disorders while only 24 and 13 validated tools for anxiety and PTSD, respectively. Therefore, the objective of this study was to conduct a systematic review of screening tools for anxiety and PTSD within LMIC populations.
Methods
Aim: To validate screening tools for anxiety disorders and PTSD in LMICs.
We published a study protocol in advance in the PROSPERO registry (CRD42019121794).
Search strategy and study selection
We systematically searched four databases (MEDLINE, EMBASE, Global Health and PsychINFO) from inception to April 22, 2020 (see Fig.1).
Fig. 1.
Open in a new tab
Search strategy
Inclusion criteria
Our eligibility criteria required that studies: (1) screen specifically for anxiety (generalized anxiety disorder or anxiety disorders not otherwise specified) and/or PTSD; (2) provide estimates of sensitivity and specificity for a given cut-off value for one of the included disorders; (3) were conducted in a LMIC (based on the World Bank Classification) ; and (4) compare screening results to a validated reference standard. Reference standards included unstructured clinical diagnostic interviews as well as structured clinical interviews including the Mini International Neuropsychiatric Interview (MINI and MINI-KID) , Structured Clinical Interview for DSM (SCID, SCID-1 and NetSCID) [26, 27], Composite International Diagnostic Interview (CIDI and CIDI-PHCV) , Clinical Interview Schedule-Revised (CIS-R) , Psychiatric Assessment Schedule (PAS) , Kiddie Schedule for Affective Disorders and Schizophrenia (K-SADS and K-SADS-PL) and Clinician-Administered PTSD Scale (CAPS and CAPS-5) [32, 33]. LMIC populations residing in a LMIC at the time of study were included. No search restrictions were put on age, gender or comorbidities.
Exclusion criteria
We excluded papers that did not report sensitivity, specificity and cut-off value; that were not published in English; and that involved populations originally from an LMIC residing outside a LMIC at the time of the study. Persons from an LMIC residing in another LMIC at the time of the study were included (e.g., refugee populations and displaced persons).
Literature review
Abstracts returned from the search were reviewed separately by two independent reviewers for inclusion, with any discrepancies resolved by discussion and use of a third senior reviewer as needed. For abstracts meeting inclusion criteria, full-text articles were retrieved and reviewed by two separate reviewers for final inclusion, with discrepancies resolved by discussion and use of a third senior reviewer as needed. We also searched the reference lists of relevant systematic reviews for additional articles to add to our full-text review.
Quality appraisal
To assess study quality, we used a modified version of Greenhalgh’s ten item checklist previously used in a study by Ali et al. Elements of the quality checklist are provided in Fig.2. Credit was given for translation if a previously validated translated version of the tool or reference standard was used, or if the tool was administered in English. Studies of ‘very good’ quality fulfilled all the quality criteria. Studies deemed ‘good’ quality fulfilled criteria 1 through 3 in addition to at least one other criterion from 4 to 5. ‘Fair’ quality studies did not avoid work-up bias and ‘acceptable’ quality studies did not perform receiver operating characteristic curve (ROC) analysis to determine a normal range from the results. ‘Unblinded’ studies include studies that reported the interviewers were not blinded to the screening results; if the study did not specify whether the screening tool administrators and interviewers were blinded to each other’s results, we considered it unblinded but clarified this designation was unconfirmed.
Fig. 2.
Open in a new tab
A modified Greenhalgh’s ten item checklist, adapted from Ali et al.
Data abstraction and analysis
Numerical data was abstracted by one reviewer and checked by a separate reviewer to ensure quality extraction. Data abstraction sheets included extraction of the screening tool and disorder, number of participants, DSM version, screening tool administrator, language, region, population study characteristics and age, country, gold standard, area under the curve (AUC), cut-off score, sensitivity and specificity. If multiple screening tools and/or cut-offs were used, data was extracted for each cutoff, for each tool, separately. If values were split by population, the value most representative of the total was chosen (e.g., community values for data split by hospital inpatient unit). If multiple cut-offs were given without AUC, we extracted the set of values for the cutoff that maximized Youden’s J . Results were presented separately by disorder, screening tool and cut-off value. As anxiety and depression were combined in many screening tools, a third category of mixed anxiety and depression was included.
For validations of screening tools for the same disorder that used identical cut-off values, bivariate random-effects meta-analytic models were fitted to provide estimates of pooled sensitivity and specificity for the cut-off value.
Results
Study selection
Of 6322 unique citations identified from the database search, 6188 were excluded after title and abstract review and five additional papers from the reference lists of relevant systematic reviews were added. Of 140 included for full-text review, 81 were excluded, leaving 59 eligible articles inclusive of 77 screening tools (see Fig.3). The most common reasons for exclusion were not screening for the disorder of interest, not comparing to a gold standard, and failing to provide either sensitivity/specificity data or a threshold for screening.
Fig. 3.
Open in a new tab
Flow chart of study selection
Quality appraisal
Two studies met all the criteria of the modified Greenhalgh’s ten item checklist and deemed ‘very good’ quality while 20 studies were deemed to be ‘good’ quality, due to lack of reporting the confidence intervals for sensitivity, specificity or AUC. Two studies were ‘fair’ quality for not avoiding work-up bias and five were deemed ‘acceptable’ for failing to perform ROC analysis. A total of 29 studies were labelled ‘unblinded’ for failing to specify if they blinded the researchers or for explicitly stating they were not blinded (see Table1).
Table 1.
Quality rating statistics
| Quality Rating | Number of Studies |
--- |
| Very good | 2 |
| Good | 20 |
| Fair | 2 |
| Acceptable | 5 |
| Unblinded | 29 |
| Total | 58 |
Open in a new tab
Description of included studies
The final 59 studies selected included a total of 77 screening tools. There were 46 validations of screening tools for anxiety disorders, 19 for PTSD and 12 for anxiety and depression (see Table2).
Table 2.
Screening tool validation by disorder category
| Disorder Category | Specific disorders | Total |
---
| Anxiety Disorders | Generalized Anxiety Disorder | 46 |
| Panic Disorder |
| Social Anxiety Disorder |
| Anxiety Disorder NOS |
| PTSD | PTSD | 19 |
| Anxiety and Depression | Generalized Anxiety Disorder | 12 |
| Major Depressive Disorder |
| Total | | 77 |
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A minority of studies accounted for children and adolescent validations (10) despite a relatively young demographic present in LMICs . The majority of validations studied adults (36), with a select few including adolescents and adults (6) (see Table3). Particularly well-represented groups included the general population and clinical outpatients (13), perinatal populations (6), psychiatric patients (7) and those with another psychiatric comorbidity (7) (see Table3). Of the 19 validations for PTSD, only four studied children and adolescents.
Table 3.
Distribution by age a population characteristic
| Population Descriptors | | Number of Studies |
---
| Adults (36) | Outpatients | 5 |
| General Population | 7 |
| HIV | 4 |
| Psychiatric patients | 7 |
| Conflict area/refugee | 4 |
| Other or unspecified | 9 |
| Perinatal (6) | HIV | 1 |
| Other | 5 |
| Adolescents and Adults (6) | Survivors of natural disaster | 2 |
| Other | 4 |
| Children and/or Adolescents (10) | Psychiatric Patient | 2 |
| Survivor of natural disaster | 2 |
| Other | 6 |
Open in a new tab
The majority of screening tool validations were in Asia (35) followed by Africa (20), the Americas (5) and Europe (1) (see Table4). The best represented regions include South and West Asia, as well as South and East Africa, with a noticeable gap in Middle and Northern Africa. There were no studies from the Oceanic region.
Table 4.
Number of Studies by Region and Country
| Continent | Region | Country (Number of Studies) | LMICs with no studies |
--- --- |
| Africa (20) | North | None | 6 (Sudan, Algeria, Egypt, Libya, Morocco, Tunisia) |
| Middle | None | 9 (Angola, Cameroon, Central African Republic, Chad, Congo, Democratic Republic of the Congo, Equatorial Guinea, Gabon, Sao Tome and Principe) |
| East (8) | Zimbabwe (2), Somalia (1), Uganda (1), Burundi (1), Tanzania (1), Zambia (1), Ethiopia (1) | 10 (Comoros, Djibouti, Eritrea, Kenya, Madagascar, Malawi, Mauritius, Mozambique, Rwanda, South Sudan) |
| West (2) | Nigeria (2) | 14 (Benin, Burkina Faso, Cabo Verde, Cote dIvoire, Gambia, Ghana, Guinea-Bissau, Liberia, Mali, Mauritania, Niger, Senegal, Sierra Leone, Togo) |
| South (10) | South Africa (10) | 4 (Botswana, Lesotho, Namibia, Swaziland) |
| Asia (35) | East (7) | China (7) | 2 (North Korea, Mongolia) |
| South (11) | Pakistan (2), India (3), Nepal (3), Afghanistan (1), Iran (2) | 4 (Bangladesh, Bhutan, Maldives, Sri Lanka) |
| South East (7) | Vietnam (3), Malaysia (2), Indonesia (1), Thailand (1) | 4 (Cambodia, Laos, Philippines, Timor-Leste) |
| West (9) | Kuwait (1), Lebanon (3), Turkey (4), Iraq (1) | 7 (Armenia, Azerbaijan, Georgia, Jordan, Palestine, Syria, Yemen) |
| Central (1) | Tajikistan (1) | 4 (Kazakhstan, Kyrgyzstan, Turkmenistan, Uzbekistan) |
| America (5) | South (4) | Brazil (2), Peru (2) | 6 (Bolivia, Colombia, Ecuador, Guyana, Paraguay, Suriname) |
| Central (1) | Mexico (1) | 7 (Belize, Costa Rica, El Salvador, Guatemala, Honduras, Nicaragua, Panama) |
| Caribbean | None | 6 (Cuba, Dominica, Dominican Republic, Grenada, Haiti, Jamaica) |
| Europe (1) | Southern (1) | Bosnia and Herzegovina (1) | 4 (Albania, Macedonia, Montenegro, Serbia) |
| Eastern | None | 5 (Belarus, Bulgaria, Moldova, Romania, Ukraine) |
| Oceania | | None | 2 (Melanesia, Micronesia) |
| Total (61 a) | | | |
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a The country total is 61 instead of 58 as one study involved four countries (Mexico, China, Brazil and Pakistan)
The most commonly used tools to screen for generalized anxiety disorder were the Kessler-10 (K-10) and the Generalized Anxiety Disorder-7 item scale (GAD-7), totaling seven and six validations respectively. The Hopkins Symptom Checklist-25 item scale (HSCL-25), Hospital Anxiety and Depression Scale (HADS) and Hospital Anxiety and Depression Scale anxiety subscale (HADS-A) were validated almost equally while the majority of tools only had one validation (see Table5). PTSD had far fewer validations (19) with a wide range of tools receiving between one and three validations, similar to the screening tools validated for both anxiety and depression.
Table 5.
Screening Tool by Disorder and Number of Validations
| Disorder | Screening Tool | Number of Validations |
---
| Anxiety disorders | HADS-A | 3 |
| HADS | 3 |
| DASS-A | 1 |
| Zung SAS | 2 |
| STAI | 1 |
| EPDS | 2 |
| HAM-A | 1 |
| K10 | 7 |
| K6 | 3 |
| PHQ-4 | 1 |
| GAD-7 | 6 |
| HDRS | 1 |
| HSCL-25 | 4 |
| MINI-SPIN | 1 |
| PHC | 1 |
| GHQ-12 | 2 |
| SCARED/SCARED-C/−P | 1/1/1 |
| PASS | 1 |
| RCADS-GAD scale | 1 |
| BAI | 2 |
| Total | 46 |
| PTSD | HTQ/−R | 1 |
| HTQ | 3 |
| K10 | 2 |
| PDS | 3 |
| PCL-C/−5 | 2/2 |
| CPSS | 2 |
| TSSC | 1 |
| UCLA PTSD Index | 1 |
| PTSD Screening Tool | 2 |
| Total | 19 |
| Anxiety and Depression | HSCL-25 | 2 |
| Independently developed (Zambia) | 1 |
| YSR | 1 |
| HADS | 1 |
| AKUADS | 1 |
| SRQ-20 | 1 |
| AYMH | 1 |
| HEI | 1 |
| K10/K6 | 1/1 |
| PHQ-4 | 1 |
| Total | 12 |
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Abbreviations: HADS Hospital Anxiety and Depression Scale, HADS-A Hospital Anxiety and Depression Scale Anxiety subscale, DASS Depression Anxiety Stress Scales, Zung SAS Zung Self-Rating Anxiety Scale, STAI State Trait Anxiety Inventory, EPDS Edinburgh Postnatal Depression Scale, HAM-A Hamilton Anxiety Rating Scale, K10/K6 Kessler 10/6, GAD Generalized Anxiety Test, HDRS Hamilton Depression Rating Scale, HSCL Hopkins Symptom Checklist, MINI-SPIN Mini-Social Phobia Inventory, PHC Primary Health Care Screening Tool, GHC General Health Questionnaire, SCARED Screen for Child Anxiety Related Disorders, PASS Perinatal Anxiety Screening Scale, RCADS Revised Children’s Anxiety and Depression Scales, BAI Beck Anxiety Inventory, HTQ Harvard Trauma Questionnaire, PDS Posttraumatic Diagnostic Scale, PCL-C PTSD Checklist-Clinician Version, PHQ-4 Patient Health Questionnaire, CPSS Child PTSD Symptom Scale, TSSC Traumatic Stress Symptom Scale, YSR Youth Self-Report, AKUADS Aga Khan University Anxiety and Depression, SRQ Self-Reporting Questionnaire, AYMH Arab Youth Mental Health Scale, HEI Huaxi Emotional-Distress Index
Each included study is listed in Table6 by region, screening tool and study quality with the respective sensitivity, specificity and cut-off for each disorder. Continentally, Asia had the most validations (35) and the majority of studies were considered unblinded (29). Due to incomplete reporting, we could meta-analyze results from only two studies, which involved the GAD-7; using a cut-off ≥10; sensitivity = 76%, specificity = 64%.
Table 6.
Included studies listed by continent, sub-region, screening tool/disorder and quality
| Author (year) | Screening tool/disorder | Gold Standard | Subregion | Country | Population | Study Quality | No. Participants | Prevalence (%) | DSM Version | AUC | Cut-Off Score (≥) | Sensitivity (%) | Specificity (%) |
--- --- --- --- --- --- --- |
| Africa |
| Ventevogel et al. (2014) | CPSS/PTSD | K-SADS-PL | Africa East | Burundi | Children aged 10–15 | good | 65 | 23 | DSM 4 | 0.78 | 26 | 71 | 83 |
| Chibanda et al. (2016) | GAD-7/GAD | SCID | Africa East | Zimbabwe | Adults except pregnant women | good | 264 | 3 | DSM 4 | 0.9 | 10 | 89 | 73 |
| Kaaya et al. (2002) | HSCL-25/Anxiety and depression | SCID | Africa East | Tanzania | Pregnant women with HIV | good | 903 (100 for SCID) | 3.3 | DSM 4 | 0.86 | 1.06 | 89 | 80 |
| Verhey et al. (2018) | PCL-5/PTSD | CAPS-5 | Africa East | Zimbabwe | Adults except perinatal women | very good | 204 | 19.6 | DSM 5 | 0.78 | 33 | 74.5 | 70.6 |
| Odenwald et al. (2007) | PDS/PTSD | CIDI | Africa East | Somalia | Patients with trauma exposure | good | 135 (62 for CIDI) | 16.1 | DSM 4 | 0.874 | 14 | 90 | 79 |
| Ertl et al. (2011) | PDS/PTSD | CAPS | Africa East | Uganda | Adults and adolescents aged 12–25 | good | 68 | 32.4 | DSM 4 | 0.79 | 16 | 82 | 70 |
| Mbewe et al. (2013) | self-made/Anxiety and depression | Interview | Africa East | Zambia | Adults with epilepsy | good | 575 | 53.7 | DSM 4 | x | 17 | 56.5 | 68.1 |
| Geibel et al. (2016) | YSR/anxiety and depression | Interview | Africa East | Ethiopia | Vulnerable teens assisted by two aid organizations | good | 134 | 64.6 | DSM 4 | 0.729 | 6.5 | 75 | 63.1 |
| Saal (2019) | Beck Anxiety Inventory/GAD | SCID | Africa South | South Africa | Adults undergoing HIV testing | unblinded | 500 | 3.4 | DSM 5 | 0.86 | 21.5 | 82 | 80 |
| van Heyningen et al. (2018) | EPDS/anxiety | MINI | Africa South | South Africa | Adult women in the antenatal period | unblinded | 376 | 23 | DSM 4 | 0.69 | 5 | 67 | 59 |
| Marsay et al. (2017) | EPDS/anxiety | NetSCID | Africa South | South Africa | Adult women pregnant for 22–28 weeks | unblinded | 145 | 14.5 | DSM 5 | x | 7 | 54.8 | 81.6 |
| van Heyningen et al. (2018) | GAD-2/anxiety | MINI | Africa South | South Africa | Adult women in the antenatal period | unblinded | 376 | 23 | DSM 4 | 0.73 | 2 | 64 | 74 |
| Seedat et al. (2007) | HADS-A/anxiety | MINI | Africa South | South Africa | Adult schizophrenic patients | unblinded | 70 | 22.9 | DSM 4 | x | 11 | 37.5 | 72.2 |
| Seedat et al. (2007) | HAM-A/anxiety | MINI | Africa South | South Africa | Adult schizophrenic patients | unblinded | 70 | 22.9 | DSM 4 | x | 22 | 31.3 | 90.7 |
| Myer et al. (2008) | HTQ/PTSD | MINI | Africa South | South Africa | HIV-positive adults | good | 465 | 5 | DSM 4 | 0.74 | 62 | 74 | 70 |
| Spies et al. (2009) | K-10/Agoraphobia | MINI | Africa South | South Africa | HIV-positive adults | unblinded | 429 | 18.4 | DSM 7 | 0.69 | 26 | 65 | 67 |
| van Heyningen et al. (2018) | K10/anxiety | MINI | Africa South | South Africa | Adult women in the antenatal period | unblinded | 376 | 23 | DSM 4 | 0.77 | 11 | 76 | 70 |
| Andersen et al. (2011) | K-10/Anxiety and Depression | CIDI | Africa South | South Africa | Adults | unblinded | 4077 | x | DSM 4 | 0.73 | 16 | 70 | 67 |
| Spies et al. (2009) | K10/GAD | MINI | Africa South | South Africa | HIV-positive adults | unblinded | 429 | 18.4 | DSM 4 | 0.78 | 30 | 72 | 80 |
| Spies et al. (2009) | K-10/GAD | MINI | Africa South | South Africa | HIV-positive adults | unblinded | 429 | 18.4 | x | 0.78 | 30 | 72 | 80 |
| Spies et al. (2009) | K-10/Panic disorder | MINI | Africa South | South Africa | HIV-positive adults | unblinded | 429 | 15.3 | DSM 6 | 0.77 | 28 | 76 | 73 |
| Spies et al. (2009) | K-10/PTSD | MINI | Africa South | South Africa | HIV-positive adults | unblinded | 429 | 21.5 | DSM 8 | 0.77 | 29 | 75 | 78 |
| Spies et al. (2009) | K-10/PTSD | MINI | Africa South | South Africa | HIV-positive adults | unblinded | 429 | 21.5 | x | 0.77 | 29 | 75 | 78 |
| Spies et al. (2009) | K-10/Social anxiety | MINI | Africa South | South Africa | HIV-positive adults | unblinded | 429 | 12.3 | DSM 5 | 0.9 | 30 | 92 | 80 |
| van Heyningen et al. (2018) | K6/anxiety | MINI | Africa South | South Africa | Adult women in the antenatal period | unblinded | 376 | 23 | DSM 4 | 0.77 | 8 | 69 | 76 |
| Andersen et al. (2011) | K-6/Anxiety and Depression | CIDI | Africa South | South Africa | Adults | unblinded | 4077 | x | DSM 4 | 0.72 | 10 | 70 | 62 |
| Martin et al. (2009) | PDS/PTSD | CIDI | Africa South | South Africa | HIV-positive adults | unblinded | 85 | x | DSM 4 | 0.74 | 15 | 68.6 | 65 |
| van der Westhuizen (2016) | SRQ-20/Anxiety/Depression | MINI | Africa South | South Africa | Adults with assault-related injury or accidents | unblinded | 200 | x | ICD 10 | 0.87 | 5 | 83.3 | 76 |
| Seedat et al. (2007) | STAI/anxiety | MINI | Africa South | South Africa | Adult schizophrenic patients | unblinded | 70 | 22.9 | DSM 4 | x | 40 | 75 | 48.1 |
| Makanjuola et al. (2014) | GHQ-12/anxiety | CIDI | Africa West | Nigeria | Adult patients of general practices | unblinded | 1590 | x | DSM 4 | 0.61 | 3 | 59 | 63.3 |
| Abiodun et al. (1994) | HADS/Anxiety and Depression | Interview | Africa West | Nigeria | Adult patients in non-psychiatric wards and community | unblinded | 1078 | Various† | ICD 9 | x | 8 | 87.5 | 90.6 |
| Makanjuola et al. (2014) | K6/anxiety | CIDI | Africa West | Nigeria | Adult patients of general practices | unblinded | 1590 | x | DSM 4 | 0.58 | 4 | 65 | 55 |
| Asia |
| Hollander et al. (2007) | HSCL-25/anxiety | Interview | Asia Central | Tajikistan | Adult patients at outpatient clinics | acceptable | 75 | x | DSM 4 | x | 1.6 | 84 | 60 |
| Hollander et al. (2007) | HTQ-R/PTSD | Interview | Asia Central | Tajikistan | Adult patients at outpatient clinics | acceptable | 75 | x | DSM 4 | x | 1.73 | 97 | 65 |
| Tong et al. (2016) | GAD-7/Generalized anxiety | MINI | Asia East | China | Adults with epilepsy who were Chinese citizens | unblinded | 213 | 23.5 | DSM 4 | 0.974 | 6 | 94 | 91.4 |
| Sheng et al. (2010) | HADS-A/anxiety | MINI | Asia East | China | Adult psychiatric outpatients | unblinded | 70 | 25.5 | DSM 4 | 0.805 | 6 | 86 | 79 |
| Yang et al. (2014) | HADS-A/anxiety | MINI | Asia East | China | Adult cardiac outpatients | unblinded | 100 | 15 | DSM 4 | 0.81 | 6 | 81.6 | 75.8 |
| Wang et al. (2017) | HEI/Anxiety and depression | MINI | Asia East | China | Hospitalized patients aged 15+ | unblinded | 763 | 7.11 | DSM 4 | 0.88 | 11 | 88 | 76.6 |
| Liu et al. (2008) | PTSD screening tool/PTSD | DSM-IV PTSD criteria | Asia East | China | Survivors of a flood aged 16+ | unblinded | 27,267 | 9.5 | DSM 4 | 0.858 | 3 | 87.9 | 97.9 |
| Liu et al. (2007) | PTSD screening tool/PTSD | DSM-IV PTSD criteria | Asia East | China | Child survivors of a flood aged 7–15 | unblinded | 6073 | 4.6 | DSM 4 | x | 3 | 96.9 | 99 |
| Ali et al. (1998) | AKUADS/GAD and MDD | Interview | Asia South | Pakistan | Residents aged 16–60 in Karachi squatter settlement | unblinded | 487 | x | DSM 3 | x | 19 | 74 | 81 |
| Kohrt et al. (2003) | BAI/anxiety | DSM-IV criteria | Asia South | Nepal | Adults with psychiatric illness and controls | acceptable | 363 | Various† | DSM 4 | x | 14 | 91 | 89 |
| Thapa et al. (2005) | PCL-C/PTSD | CIDI | Asia South | Nepal | Adults residing in conflict areas | unblinded | 290 | 53.4 | DSM 4 | 0.81 | 50 | 80 | 80 |
| Kohrt et al. (2011) | CPSS/PTSD | K-SADS | Asia South | Nepal | Adolescents aged 11–14 | good | 162 | 6.4 | DSM 4 | 0.77 | 20 | 68 | 73 |
| Chaturvedi et al. (1994) | HADS/anxiety | Interview | Asia South | India | Cancer patients of all ages | unblinded | 70 | not specified | DSM 3 | x | 7 | 87 | 79 |
| Ventevogel et al. (2007) | HSCL/anxiety | PAS | Asia South | Afghanistan | Clinic patients aged 15+ | good | 116 | 24.1 | x | 0.61 | 2 | 75 | 43 |
| Ventevogel et al. (2007) | HSCL/depression and anxiety | PAS | Asia South | Afghanistan | Clinic patients aged 15+ | good | 116 | 24.1 | x | 0.61 | 2 | 69 | 67 |
| Housen et al. (2018) | HSCL-25/anxiety | MINI | Asia South | India | Adult general medical outpatients | good | 290 | 3.5 | DSM 4 | 0.81 | 1.75 | 73 | 81 |
| Thapa et al. (2005) | HSCL-25/anxiety | CIDI | Asia South | Nepal | Adults residing in conflict areas | unblinded | 290 | 80.7 | DSM 4 | 0.76 | 1.75 | 77 | 58 |
| Ahmadi (2020) | PHQ-4/anxiety | SCID | Asia South | Iran | Adults with coronary heart disease | unblinded | 279 | not specified | DSM 5 | 0.94 | 7 | 80 | 94 |
| Ahmadi (2020) | PHQ-4/Anxiety and depression | SCID | Asia South | Iran | Adults with coronary heart disease | unblinded | 279 | not specified | DSM 5 | 0.94 | 7 | 86 | 90 |
| Russell et al. (2013) | SCARED/anxiety | K-SADS-PL | Asia South | India | Adolescents aged 11–19 | unblinded | 500 | x | DSM 4 | 0.9 | 21 | 84.6 | 87.36 |
| Namazi et al. (2013) | UCLA PTSD (PTSD) | Interview | Asia South | Iran | Children aged 7–12 after earthquake | unblinded | 50 | 56 | 4-R | x | 38 | 96 | 50 |
| Tran et al. (2013) | DASS-A/anxiety | SCID | Asia South East | Vietnam | Adult perinatal women | good | 221 | 10.9 | DSM 4 | 0.806 | 10 | 79.2 | 67 |
| Sidik et al. (2012) | GAD-7/anxiety | CIDI | Asia South East | Malaysia | Adult females | good | 895 | 7.8 | DSM 4 | x | 8 | 76 | 94 |
| Yahya et al. (2015) | HDRS/anxiety | DSM-IV | Asia South East | Malaysia | Patients with existing psychiatric disorder and controls | unblinded | 120 | x | DSM 4 | 0.917 | 8 | 90 | 86.2 |
| Silove et al. (2007) | HTQ/PTSD | SCID | Asia South East | Thailand | Cambodian population in Thailand | good | 118 | 20.3 | DSM 4 | 0.71 | 2 | 63 | 61 |
| Tran et al. (2019) | K-10/anxiety | MINI-KID | Asia South East | Indonesia | Adolescents age 16–18 | unblinded | 196 | x | DSM 4 | 0.82 | 18 | 87.1 | 70.9 |
| Tran et al. (2019) | K-6/anxiety | MINI-KID | Asia South East | Indonesia | Adolescents age 16–19 | unblinded | 196 | x | DSM 4 | 0.8 | 12 | 83.9 | 73.3 |
| Tran et al. (2011) | Zung SAS/anxiety | Interview | Asia South East | Vietnam | Adult perinatal women | good | 364 | 11.8 | DSM 4 | 0.79 | 38 | 67.9 | 75.3 |
| Tran et al. (2012) | Zung SAS/anxiety | Interview | Asia South East | Vietnam | Men who are partners of pregnant or perinatal women | good | 231 | 5.2 | DSM 4 | 0.775 | 36 | 70.7 | 79 |
| Mahfoud et al. (2011) | AYMH/Anxiety and depression | Interview | Asia West | Lebanon | Socioeconomically disadvantaged children aged 10–14 | good | 153 | 17.6 | DSM 4 | 0.71 | 39 | 63 | 79 |
| Sawaya et al. (2016) | GAD-7/anxiety | Interview | Asia West | Lebanon | Adult psychiatric outpatients | acceptable | 176 | x | DSM 4 | 0.57 | 10 | 57 | 53 |
| Senturk et al. (2007) | GHQ-12/anxiety | CIDI-PHCV | Asia West | Turkey | Adult leprosy patients | unblinded | 65 | 12.3 | ICD 10 | 0.69 | 5 | 71 | 57 |
| Malasi et al. (1991) | HADS/anxiety | Interview | Asia West | Kuwait | Adult psychiatric outpatients and controls | acceptable | 135 | x | DSM 3 | | 13 | 45 | 47 |
| Senturk et al. (2007) | HADS/anxiety | CIDI-PHCV | Asia West | Turkey | Adult leprosy patients | unblinded | 65 | x | ICD 11 | 0.75 | 11 | 66 | 58 |
| Yazici et al. (2018) | PASS/anxiety | SCID-1 | Asia West | Turkey | Adult women in perinatal period | unblinded | 312 | 19.2 | DSM 4 | 0.94 | 16 | 95 | 84 |
| Ibrahim et al. (2018) | PCL-5/PTSD | DSM 5 interview | Asia West | Iraq | Adults living in a camp for displaced people in Iraq | good | 206 | 37.75 | DSM 5 | 0.82 | 23 | 82 | 70 |
| Gormez et al. (2017) | RCADS-GAD scale/GAD | K-SADS | Asia West | Turkey | Child psychiatry outpatients aged 8–17 | unblinded | 483 | not specified | DSM 4 | x | 7.5 | 70 | 71 |
| Hariz et al. (2013) | SCARED-C/anxiety | Interview | Asia West | Lebanon | Child and adolescent psychiatric patients | good | 82 | 40.2 | DSM 4 | 0.63 | 26 | 66 | 56 |
| Hariz et al. (2013) | SCARED-P/anxiety | Interview | Asia West | Lebanon | Child and adolescent psychiatric patients | good | 82 | x | DSM 4 | 0.7 | 24 | 67 | 55 |
| Başoglu et al. (2001) | TSSC/PTSD | CAPS | Asia West | Turkey | Survivors of 1999 August earthquake aged 16–70 | acceptable | 130 | 49 | DSM 4 | x | 2 | 76 | 73 |
| Europe |
| Oruc et al. (2008) | HTQ/(PTSD) | SCID | Europe Southern | Bosnia and Herzegovina | Adults enrolled in primary care clinic | very good | 180 | 26 | DSM 4 | 0.98 | 2.06 | 99.9 | 93.9 |
| South America |
| Zhong et al. (2015) | GAD-7/GAD | CIDI | South America | Peru | Pregnant women aged 18–49 who speak Spanish | unblinded | 946 | 33.3 | DSM 4 | 0.75 | 7 | 73.3 | 67.3 |
| de Lima Osório et al. (2007) | MINI-SPIN/Social anxiety disorder | SCID | South America | Brazil | University students | fair | 2320 | 10.4 | DSM 4 | 0.81 | 6 | 94 | 46 |
| Gelaye et al. (2017) | PCL-C/PTSD | CAPS | South America | Peru | Perinatal women | very good | 3289 | 3 | DSM 4 | 0.75 | 26 | 86 | 63 |
| Multiple Countries |
| Goldberg et al. (2017) | PHC/current anxiety | CIS-R | South America, Asia South, Asia East, Central America | Brazil, Pakistan, China, Mexico | Primary care patients | fair | 1488 (all countries) | Brazil: 26.5; Pakistan: 13; China: 18.9; Mexico: 23 | ICD 11 | 0.77 | 3 | 75 | 68 |
| Meta-analyzed GAD-7 Values |
| Chibanda et al. (2016) and Sawaya et al. (2016) | GAD-7/anxiety | NA | NA | NA | NA | NA | NA | NA | NA | NA | ≥10 | 76 | 64 |
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Quality: ranges from highest to lowest (very good, good, fair, acceptable, unblinded, unblinded (unblinded [unconfirmed so considered unblinded]); x: value not specified; various†: multiple values specified, see Appendix file; Abbreviations: HADS Hospital Anxiety and Depression Scale, HADS-A Hospital Anxiety and Depression Scale Anxiety subscale, DASS Depression Anxiety Stress Scales, Zung SAS Zung Self-Rating Anxiety Scale, STAI State Trait Anxiety Inventory, EPDS Edinburgh Postnatal Depression Scale, HAM-A Hamilton Anxiety Rating Scale, K10/K6 Kessler 10/6, GAD Generalized Anxiety Test, HDRS Hamilton Depression Rating Scale, HSCL Hopkins Symptom Checklist, MINI-SPIN Mini-Social Phobia Inventory, PHC Primary Health Care Screening Tool, GHC General Health Questionnaire, SCARED Screen for Child Anxiety Related Disorders, PASS Perinatal Anxiety Screening Scale, RCADS Revised Children’s Anxiety and Depression Scales, BAI Beck Anxiety Inventory; HTQ Harvard Trauma Questionnaire, PDS Posttraumatic Diagnostic Scale, PCL-C PTSD Checklist-Clinician Version, PHQ-4 Patient Health Questionnaire, CPSS Child PTSD Symptom Scale, TSSC Traumatic Stress Symptom Scale, YSR Youth Self-Report, AKUADS Aga Khan University Anxiety and Depression, SRQ Self-Reporting Questionnaire, AYMH Arab Youth Mental Health Scale, HEI Huaxi Emotional-Distress Index
Discussion
This review aimed to examine the screening tools that have been validated to detect anxiety and PTSD in LMICs. The most commonly validated tools were the K-10 and GAD-7 for anxiety and the HTQ and the Posttraumatic Diagnostic Scale (PDS) for PTSD. It is difficult to recommend one screening tool for anxiety and PTSD respectively, as various tools and cut-off values were tested, and sensitivities and specificities varied based on region, country and screening tool. Indeed, only two studies tested the same tool using the same cut-off value and reported sufficient information to allow us to quantitatively synthesize the results. Locally validated screening tools for anxiety and PTSD need further evaluation in well-designed studies to assess whether they can improve the detection and management of these common disorders.
A total of 46 validated screening tools were found for anxiety disorders. The most common tool used to screen for anxiety disorders was the Kessler-10 followed by the GAD-7, which had wide ranges of sensitivities (57–94%) and specificities (53–94%) varying by region and sample size. While previously the HADS-A was recommended , our updated review found that it was not as widely validated as the GAD-7 and Kessler-10, although it had consistent specificities (72–79%) with a range of sensitivities (38–86%). The Kessler may have an added time-efficiency component, as it is possible to screen for multiple common mental disorders, whereas screening tools such as the HADS-A target anxiety specifically. The GAD-7 reported some of the highest sensitivities for detection of generalized anxiety disorder. Other anxiety disorders, including agoraphobia, panic disorder and social anxiety disorder were less commonly validated. Our results are consistent with a previous systematic review and indicate using the GAD-7, K-10 or HAD-A yield good sensitivities and specificities while taking population-specific characteristics into account. Future research is needed to validate screening tools for these anxiety disorders in more regions.
The number of validations for PTSD increased from 10 to 19 since 2013 . The HTQ and PDS were the most commonly validated tools for PTSD, and sensitivities were generally high. Our findings add that in addition to the previously recommended HTQ, the PDS should be considered in screening for PTSD . Unfortunately, many tools were validated only once, preventing our combining them for analytic purposes. Only four PTSD validations describe children and adolescents, despite recent events that have displaced thousands of youth . The prevalence of PTSD remains high in LMICs and is expected to rise given increasing civil unrest and war [19, 94]. The year 2018 saw the highest recorded number of displaced persons globally leading the authors to emphasize more attention into detection and treatment of PTSD .
Anxiety and depression had the fewest validations across our search though were not the target of our validation given the existing literature on depression alone . All tools with the exception of the HSCL-25 had only one validation. The only independently developed screening tool of all the studies was for anxiety and depression, developed in Zambia. These disorders commonly occur together, and further research is needed to determine which tools are best suited to a region’s mental health screening needs.
We searched four databases with a robust library of psychiatric publications available. We also placed minimal exclusion criteria on our searches so as to maximize the number of studies returned, and we additionally reviewed relevant systematic reviews for additional relevant papers. At every stage of the process from title/abstract screen to data abstraction, two reviewers assessed each article and numerical data point to reduce human error. Our search strategy and protocol were published in PROSPERO and were not altered from the time of submission, with the exception that we did not calculate diagnostic odds ratios (DORs), as they provide no guidance to clinicians on what screening tool and cut-off threshold would be most appropriate to use in clinical practice. Rather, we reported sensitivity and specificity of each screening tool and cutoff separately, to better describe the accuracies of individual tools and cut-offs.
Our extraction was limited by the individual papers’ specific data reporting. Varying prevalence of an individual study may affect the cut-off score, sensitivity and specificity of screening tools, and some studies did not publish prevalence. Providers should reference the prevalence of each specific disorder to ascertain whether the cut-off is applicable to their respective population. The majority of studies did not provide sensitivities and specificities for multiple cut-off values. Reporting multiple cut-off values and their respective sensitivity and specificity estimates would allow providers to decide which cut-off they would choose to optimize screening for their setting. A lower cut-off with a higher sensitivity may be desired if cases are not to be missed and false negatives reduced. A higher cut-off with a higher specificity may be desired if false positives are to be minimized. Furthermore, reporting multiple cut-off values and their respective sensitivity and specificity estimates would also allow researchers to better synthesize accuracy results across multiple studies in meta-analysis. In the present study, only two validations with identical cut-off scores for the GAD-7 could be combined via meta-analysis as no other validations of the same disorder with identical cut-off values provided sufficient information to conduct a meta-analysis (i.e., 2 × 2 table numbers). Studies used various versions of the DSM and ICD. While the symptomatology for psychiatric diagnoses have not changed significantly, providers should reference which version was used when conducting the validation of the screening tool (see Table6).
Our review was also limited by the available publications on mental health screenings in LMICs. The entire region of Middle and North Africa, constituting over 300 million people, was not represented by a single validation while other regions such as South-East Asia were fairly well-represented. Cultural and linguistic factors may influence screening tool validation yet further discussion may be best served for individual validation papers. Most studies were rated in the lowest quality category of the modified Greenhalgh scale as they were unblinded, or downgraded to unblinded due to incomplete reporting. This is a severe limitation in the design of studies that may impact validation results; future studies should ensure adequate blinding in addition to the remainder of the quality checklist.
Our study did not look at CMDs or depression specifically, although we did consider anxiety and depression when screened for together. We chose to focus on anxiety and PTSD as they are less well-represented in the realm of LMIC validated screening tools. Additionally, anxiety and PTSD are becoming more important with the current displacement of millions of people due to civil unrest, socioeconomic upheaval and war.
The number of validated screening tools for mental health disorders as a whole has increased since 2013 . However, no large increase in the number of validations for specific disorders was seen, and most screening tools from our search were validated only once. We advise researchers and providers to refer to Table6 for a summary of validations for locations and disorders of interest and to use this table to identify their region of interest, find their disease focus of interest, and then identify what tools have been identified by the highest quality evidence.
Conclusions
Mental health disorders are highly prevalent yet are frequently stigmatized and disregarded as medical diseases. Validated screening tools for anxiety and PTSD in LMIC have made considerable progress, with validations for both disorders almost doubling since the prior systematic review completed in December 2013 . The increase in validated screening tools generally followed a regional pattern, with more emerging in countries already represented. For example, more tools have been validated in South Africa without an increase in validations in Botswana, Lesotho, Namibia or Swaziland. Middle and Northern Africa were also not well-represented by either anxiety or PTSD screening tools. The authors recognize that it may be near impossible to validate screening tools in areas of intense conflict and instability but acknowledge the need to evaluate screening tools in these areas.
The age distribution among screening tools was heavily biased towards the adult population. Children and adolescents accounted for only four of 19 validations for PTSD and six of 58 for anxiety and anxiety and depression. Given that age is skewed towards a younger population in LMICs , it is imperative that more research focuses on identifying anxiety and PTSD disorders in a pediatric population, especially in areas of increased civil war and conflict.
Use of brief screening instruments can bring much needed attention and research opportunities to various at-risk populations in LMICs. Many screening tools for anxiety and PTSD have been validated in LMICs, but there remain regions and subgroups of individuals for which more research is needed. Locally validated screening tools for anxiety and PTSD should be further evaluated in clinical trials to determine whether their use can reduce the burden of disease.
Supplementary information
Additional file 1. Appendix (76.7KB, xlsx)
Acknowledgements
Rebecca McCall (search strategy aid), UNC Chapel Hill.
Abbreviations
PTSD
Post-traumatic stress disorder
LMICs
Low to middle income countries
CMDs
Common mental disorders
MINI and MINI-KID
Mini International Neuropsychiatric Interview
SCID, SCID-1 and NetSCID
Structured Clinical Interview for DSM
CIDI and CIDI-PHCV
Composite International Diagnostic Interview
CIS-R
Clinical Interview Schedule-Revised
PAS
Psychiatric Assessment Schedule
K-SADS and K-SADS-PL
Kiddie Schedule for Affective Disorders and Schizophrenia
CAPS and CAPS-5
Clinician-Administered PTSD Scale
AUC
Area under the curve
ROC
Receiver operating characteristic curve
DORs
Diagnostic odds ratios
HADS
Hospital Anxiety and Depression Scale
DASS
Depression Anxiety Stress Scales
Zung SAS
Zung Self-Rating Anxiety Scale
STAI
State Trait Anxiety Inventory
EPDS
Edinburgh Postnatal Depression Scale
HAM-A
Hamilton Anxiety Rating Scale
K10/K6
Kessler 10/6
GAD
Generalized Anxiety Test
HDRS
Hamilton Depression Rating Scale
HSCL
Hopkins Symptom Checklist
MINI-SPIN
Mini-Social Phobia Inventory
PHC
Primary Health Care Screening Tool
GHC
General Health Questionnaire
SCARED
Screen for Child Anxiety Related Disorders
PASS
Perinatal Anxiety Screening Scale
RCADS
Revised Children’s Anxiety and Depression Scales
BAI
Beck Anxiety Inventory
HTQ
Harvard Trauma Questionnaire
PDS
Posttraumatic Diagnostic Scale
PCL-C
PTSD Checklist-Clinician Version
CPSS
Child PTSD Symptom Scale
TSSC
Traumatic Stress Symptom Scale
CAPS
Clinician-Administered PTSD Scale
YSR
Youth Self-Report
AKUADS
Aga Khan University Anxiety and Depression
SRQ
Self-Reporting Questionnaire
AYMH
Arab Youth Mental Health Scale
HEI
Huaxi Emotional-Distress Index
Authors’ contributions
All authors listed below have read and approved the manuscript. AYM: Design of systematic review, search criteria and primary reviewer, wrote manuscript. JD: Primary reviewer, aided in introduction and editing of manuscript. EA: Primary reviewer, aided in manuscript writing and editing. BL: Data analysis, aided in manuscript writing and editing. VFG: Manuscript writing and editing. 6. BNG: Design of systematic review, development of methods, manuscript writing and editing.
Funding
The Doris Duke Clinical Research Foundation funded the primary author, Ms. Mughal, for the design of the study and collection, analysis, and interpretation of data and in writing the manuscript through a Doris Duke International Clinical Research Fellowship. Dr. Levis was supported by a Canadian Institutes of Health Research (CIHR) Frederick Banting and Charles Best Canada Graduate Scholarship doctoral award, and a Fonds de recherche du Québec - Santé (FRQS) Postdoctoral Training Fellowship.
Availability of data and materials
All data generated or analysed during this study are included in this published article [and its supplementary information files].
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Footnotes
Publisher’s Note
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Contributor Information
Anisa Y. Mughal, Email: AYM6@pitt.edu
Eric Ardman, Email: eaardman@gmail.com.
Brooke Levis, Email: brooke.levis@gmail.com.
Vivian F. Go, Email: vgo@live.unc.edu
Bradley N. Gaynes, Email: bradley_gaynes@med.unc.edu
Supplementary information
Supplementary information accompanies this paper at 10.1186/s12888-020-02753-3.
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6612 | https://www.brookings.edu/articles/effects-of-income-tax-changes-on-economic-growth/ | Home
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Effects of Income Tax Changes on Economic Growth
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Effects of Income Tax Changes on Economic Growth
William G. Gale and William G. Gale The Arjay and Frances Fearing Miller Chair in Federal Economic Policy, Senior Fellow - Economic Studies, Urban-Brookings Tax Policy Center Andrew A. Samwick Andrew A. Samwick Dartmouth College
February 1, 2016
8 min read
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Abstract
This paper examines how changes to the individual income tax affect long-term economic growth. The structure and financing of a tax change are critical to achieving economic growth. Tax rate cuts may encourage individuals to work, save, and invest, but if the tax cuts are not financed by immediate spending cuts, they will likely also result in an increased federal budget deficit, which in the long-term will reduce national saving and raise interest rates. The net impact on growth is uncertain, but many estimates suggest it is either small or negative. Base-broadening measures can eliminate the effect of tax rate cuts on budget deficits, but at the same time, they reduce the impact on labor supply, saving, and investment and thus reduce the direct impact on growth. They may also reallocate resources across sectors toward their highest-value economic use, resulting in increased efficiency and potentially raising the overall size of the economy. Results in the literature suggest that not all tax changes will have the same impact on growth. Reforms that improve incentives, reduce existing distortionary subsidies, avoid windfall gains, and avoid deficit financing will have more auspicious effects on the long-term size of the economy, but may also create trade-offs between equity and efficiency.
Introduction
Policy makers and researchers have long been interested in how potential changes to the personal income tax system affect the size of the overall economy. In 2014, for example, Representative Dave Camp (R-MI) proposed a sweeping reform to the income tax system that would reduce rates, greatly pare back subsidies in the tax code, and maintain revenue levels and the distribution of tax burdens across income classes (Committee on Ways and Means 2014).
In this paper, we focus on how tax changes affect economic growth. We focus on two types of tax changes – reductions in individual income tax rates and “income tax reform.” We define the latter as changes that broaden the income tax base and reduce statutory income tax rates, but nonetheless maintain the overall revenue levels and the distribution of tax burdens implied by the current income system. Our focus is on individual income tax reform, leaving consideration of reforms to the corporate income tax (for which, see Toder and Viard 2014) and reforms that focus on consumption taxes for other analyses.
By “economic growth,” we mean expansion of the supply side of the economy and of potential Gross Domestic Product (GDP). This expansion could be an increase in the annual growth rate, a one-time increase in the size of the economy that does not affect the future growth rate but puts the economy on a higher growth path, or both. Our focus on the supply side of the economy in the long run is in contrast to the short-term phenomenon, also called “economic growth,” by which a boost in aggregate demand, in a slack economy, can raise GDP and help align actual GDP with potential GDP.
The importance of the topics addressed here derive from the income tax’s central role in revenue generation, its impact on the distribution of after-tax income, and its effects on a wide variety of economic activities. The importance is only heightened by concerns about the long-term economic growth rate (Gordon 2016; Summers 2014) and concerns about the long-term fiscal status of the federal government (Auerbach and Gale 2016).
We find that, while there is no doubt that tax policy can influence economic choices, it is by no means obvious, on an ex ante basis, that tax rate cuts will ultimately lead to a larger economy in the long run. While rate cuts would raise the after-tax return to working, saving, and investing, they would also raise the after-tax income people receive from their current level of activities, which lessens their need to work, save, and invest. The first effect normally raises economic activity (through so-called substitution effects), while the second effect normally reduces it (through so-called income effects).
The financing of tax cuts significantly affects its impact on long-term growth. Tax cuts financed by immediate cuts in unproductive government spending could raise output, but tax cuts financed by reductions in government investment could reduce output. If they are not financed by spending cuts, tax cuts will lead to an increase in federal borrowing, which in turn, will reduce long-term growth. The historical evidence and simulation analyses suggest that tax cuts that are financed by debt for an extended period of time will have little positive impact on long-term growth and could reduce growth.
Tax reform is more complex, as it involves tax rate cuts as well as base-broadening changes. There is a theoretical presumption that such changes should raise the overall size of the economy in the long-term, though the effect and magnitude of the impact are subject to considerable uncertainty. One fact that often escapes unnoticed is that broadening the tax base by reducing or eliminating tax expenditures raises the effective tax rate that people and firms face and hence will operate, in that regard, in a direction opposite to rate cuts and mitigate their effects on economic growth. But base-broadening has the additional benefit of reallocating resources from sectors that are currently tax-preferred to sectors that have the highest economic (pre-tax) return, which should increase the overall size of the economy.
A fair assessment would conclude that well-designed tax policies have the potential to raise economic growth, but there are many stumbling blocks along the way and certainly no guarantee that all tax changes will improve economic performance. Given the various channels through which tax policy affects growth, a tax change will be more growth-inducing to the extent that it involves (i) large positive incentive (substitution) effects that encourage work, saving, and investment; (ii) small or negative income effects, including a careful targeting of tax cuts toward new economic activity, rather than providing windfall gains for previous activities; (iii) reductions in distortions across economic sectors and across different types of income and consumption; and (iv) minimal increases in, or reductions in, the budget deficit.
The remainder of the paper is organized as follows. Section II provides a conceptual framework by discussing the channels through which tax changes can affect economic performance, including the many ways in which a positive substitution effect in response to a tax rate cut might be dissipated or even reversed by other factors.
Section III provides an empirical starting point. We show that growth rates over long periods of time in the United States have not changed in tandem with the massive changes in the structure and revenue yield of the tax system that have occurred. We also report findings from Piketty, Saez and Stantcheva (2014) that, across advanced countries, even large changes in the top marginal income tax rate over time do not appear to be strongly correlated with rates of growth.
Section IV explores empirical evidence on taxes and growth from studies of major income tax changes in the United States. Consistent with the discussion in Section III, the studies find little evidence that tax cuts or tax reform since 1980 have impacted the long-term growth rate significantly.
Section V examines the new “narrative” approach to identifying tax changes that are exogenous to current economic conditions, stemming from the seminal work of Romer and Romer (2010). The literature, which generally uses vector autoregression (VAR) models, finds that tax cuts that meet the exogeneity criteria raise short-term output and other economic activity. The narrative literature does not speak to the long-term effects, though.
Section VI discusses the results from the literature on simulation models, which has generated two main results. First, debt-financed tax cuts will tend to boost short-term growth (as in standard Keynesian models and in the literature using the narrative approach), but also tend to reduce long-term growth, if they are financed eventually by higher taxes. Second, revenue-neutral income tax reform can provide a modest boost to economic growth.
Section VII concludes.1
There are a number of related issues that are both interesting and important, but beyond the scope of the paper – including, for example, the elasticity of taxable income, the relationship between inequality (especially as it is affected by the tax system) and growth, the effects of corporate income tax reform on growth and the detailed literatures on the effects of taxes on labor supply, saving, and investment.
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6613 | https://ocw.mit.edu/courses/16-522-space-propulsion-spring-2015/resources/mit16_522s15_lecture6/ | Session 6 : Analytical Approximations for Low Thrust Maneuvers
As mentioned in the previous lecture, solving non-Keplerian problems in general requires
the use of perturbation methods and many are only solvable through numerical integration.
However, there are a few examples of low-thrust space propulsion maneuvers for which we
can find approximate analytical expressions. In this lecture, we explore a couple of these
maneuvers, both of which are useful because of their precision and practical value: i) climb
or descent from a circular orbit with continuous thrust, and ii ) in-orbit repositioning, or
walking .
Spiral Climb/Descent
We start by writing the equations of motion in polar coordinates,
� �2
d2r dθ μ
− r + = ar (1)
dt 2 dt r2
d2θ 2 dr dθ aθ
= (2)
dt 2 r dt dt r
We assume continuous thrust in the angular direction, therefore ar = 0. If the acceleration
force along θ is small, then we can safely assume the orbit will remain nearly circular and the
semi-major axis will be just slightly different after one orbital period. Of course, small and
slightly are vague words. To make the analysis rigorous, we need to be more precise. Let us
say that for this approximation to be valid, the angular acceleration has to be much smaller
than the corresponding centrifugal or gravitational forces (the last two terms in the LHS of
Eq. (1)) and that the radial acceleration (the first term in the LHS in the same equation) is
negligible. Given these assumptions, from Eq. (1),
(3)
dt r3 dt 2 2 r5 dt
Substituting into Eq. (2), we obtain a differential equation for r, which can be integrated
directly for an initial radius r0 and time t0,
r
dr t 2aθ 1 1 aθ
≈ √ dt → − ≈ √ (t − t0) (4) 3/2 1/2 1/2r μ r μr0 t0 r0
Re-arranging the integrated expression and setting t0 = 0,
r0
r ≈ (5) (1 − aθt/v 0)2
where the velocity of the initial circular orbit is v0 = μ/r 0. From the definition of Δv, we
also notice that,
t
� �
μ μ
Δv = aθdt = aθt ≈ − (6)
r0 r0
dθ ≈
� μ → d2θ ≈ − 3 � μ dr
1� �� �
From Eq. (5) we observe that the trajectory will be a climbing, or descending spiral, de pending on whether aθ is positive or negative. Eq. (6) shows that the Δv defined in terms of
the perturbation acceleration is equal to the change in velocity between the initial and final
orbits. To note that this change of velocity is not equal to the ideal rocket Δv, we compare
it with the corresponding Hohmann transfer,
2μr μ μ 2μr 0
ΔvH = ( vp − v0) + ( v − va) = − + − (7)
r0(r + r0) r0 r r(r + r0)
By definition, this is an impulsive maneuver, not susceptible to losses from gravitational, or
any other externally applied force. In addition, these impulses are provided at both ends of
the apsidal line resulting in the optimal Δv to change the altitude and circularize the orbit.
Assume the final and initial orbital radii are related by r = nr 0 and calculate the ratio
Δv/ ΔvH ,
� �−1√
Δv 2 n
= 2 1 + − 1 (8)
ΔvH n + 1
This ratio is, as expected, always larger than unity, meaning that Hohmann transfers always
require a lower Δv. Eq. (8) is shown in the figure below. The logarithmic plot is symmetric
about n = 1, accounting for descending (n < 1) and ascending (n > 1) trajectories. 10 −1 10 0 10 1 10 2
11.2 1.4 1.6 1.8 22.2
n = r/r 0
∆V/ ∆VH
The spiral approximation holds as long as the orbit remains near-circular, and within our
assumptions this will be true if the angular acceleration is small. We need to be careful
when applying these results, since in many instances we are interested in constant acceler ation maneuvers for which the strength of the angular acceleration relative to gravity and
centrifugal forces will decrease for n < 1, and increase for n > 1. Therefore, descending
spirals could safely be analyzed with these tools whereas the approximation to an ascending
spiral will eventually fail.
2
√ √[ ( )In any event, low thrust spiral maneuvers are not optimal in the sense that work is always done against the gravitational field. In this particular case, the thrust vector is not perfectly aligned along the “circular” trajectory and a small eccentricity will be introduced, which will increase with time. It is interesting to note that the same result in Eq. (5) could be obtained with alterna-tive methods. For instance, considering again that the orbit remains near circular, we can calculate the rate of change of the orbital energy, and make this equal to the thrust power delivered to the vehicle,
~dE T F
= · ~v d μ μ
= aθ (9)
dt m → dt
(
−2r
) √
r
Eq. (9) is identical to Eq. (4) and therefore its solution is the same. The spiral trajectory appears to be a trivial solution, but there are some subtleties. Notice that the velocity increment ∆ v is actually equal to the decrease in orbital velocity. The rocket is pushing forward, but the velocity is decreasing. This is because in a r−2 force field, the kinetic energy is equal in magnitude but opposite in sign to the total energy, 1 2 μ μ μ μv = ET + = + =2 r − (10) 2r r 2r
Eq. (5) suggests that, in principle, escape conditions will be reached at t = v0/a θ when
r → ∞ . But of course, the orbit is no longer near-circular when approaching escape, so we cannot expect this result to be precise. We could obtain a more precise determination of escape conditions (∆ vesc ) in the following way. The radial velocity r˙ can be calculated from Eq. (5) by differentiation. Notice that this is in the nature of an iteration, since r¨ was neglected in the energy balance which led to Eq. (9). We then obtain, 2aθr0/v 0
r˙ = (1 − aθt/v 0)3 (11) ˙The angular component vθ = rθ is approximately the orbital velocity, i.e., ˙rθ =
√μ a
=
√ μ θt
− aθt = v0 1
r r0
(
− v0
)
(12) The overall kinetic energy is therefore,
22
v2 1 ( 2 2 2
) v20
[( aθt) 4 ( aθr /v 2
˙r˙ + r θ = − 0
= 1 + 0 )2 2 2 v0 (1 − aθt/v 0)6
]
(13) Escape conditions are reached when the total energy vanishes, i.e.,
2
1 vv2 μ 1 r
− 0
= 0 or 2 r
(
v0
− = 0 (14) 2
)
r
3Substituting, 1 ( 2
aθtesc
)2 0/v 2 2
2 ( aθr 0 ) a
− θ
12 − v0 (1 − aθtesc /v 0)6
( t
1 − esc
= 0
v0
)
so we have,
1/41/4
aθtesc 2aθr0 2a
− θ
1 = = = (2 ε)1/4 (15)
v 2 20
(
v0
) (
μ/r 0
)
where ε = aθ/(μ/r 20 ) is the ratio of thrust to gravitational accelerations, and as before should be small for the approximation to hold. Since ∆ vesc = aθtesc ,∆vesc ≈ v 1/40
This
[1 − (2 ε) ] (16) result is useful in obtaining a preliminary determination of escape conditions, but because of the assumptions and the eventual increase of the eccentricity, we do not expect that Eq. (16) will converge to the exact result, even for very small values of ε. To evaluate this model, the equations of motion are solved numerically and the total energy is tracked until it vanishes. At that point, we compute the quantities shown in the table below.
ε (dr ) ∆vesc sesc resc 1 ∆v
ε − esc /v 0
ds 1esc v0r0r0
√
ε/4
10 −2 0.5327 0.7615 51.13 0.8518 0.7541 10 −3 0.5346 0.8657 503.6 0.8535 0.7552 10 −4 0.5348 0.9245 5011.4 0.8538 0.7549 10 −5 0.5347 0.9575 50036 0.8534 0.7554
~In the table, s is the distance along the orbital trajectory. In particular, ~s · F is the work done to reach escape and should be equal to the orbit’s initial energy,
μ/ 2r0 v2
sesc = = 0 (17)
aθ 2aθ
The factor 2 1/4 = 1.19 in Eq. (16) is definitely larger than the values in the last column of the table above. In consequence, a better expression to use would be, ∆v 1
esc
≈ v0
[1 − 0.754 ε /4 (18) Finally, we observe that the escape radius will be given by
]
,0.85
resc ≈ r0 √ (19)
ε
and the rate of climb compared to the distance traveled at escape is,
4� �
� � � �
dr ≈ 0.53 (20)
ds esc
which is very far from a circular orbit, and closer to a trajectory that starts to move away
from the orbit focus in almost a straight line.
The numerical results would be slightly different if instead of using angular thrust we use
tangential thrust. The analysis, however, is not as straightforward and closed analytical
expressions like those shown above are more difficult to obtain.
Re-positioning in Orbits: Walking
Suppose now that we want to move a satellite in a circular orbit to an angular position Δθ
apart in the same orbit, in a time Δt (assumed to be several orbital times at least). The
general approach is to transfer to a lower (for Δθ > 0) or higher (for Δθ < 0) nearby orbit,
then drift in this faster (or slower) orbit for a certain time, then return to the original orbit.
The analysis is similar for low and high thrust, because in either case the satellite is nearly
in the same orbit even during thrusting periods, and as we found out for spiral transfers,
the Δv for orbit transfer is equal to the magnitude of the difference between the beginning
and ending orbital speeds. In detail, of course, if done at high thrust the maneuver involves
a two-impulse Hohmann transfer to the drift orbit and one other two-impulse Hohmann
transfer back to the original orbit. For the low-thrust case, continuous thrusting is used
during both legs, with some guidance required to remove the very slight radial component
of iv picked up during spiral flight (and ignored here).
We will do the analysis for the low-thrust case only, then adapt the result for high-thrust.
Let δθ be the advance angle relative to a hypothetical satellite remaining in the original orbit
and left undisturbed. The general shape of the maneuver is sketched below: coasting forward thrust backwards thrust
The orbital angular velocity is Ω = μ/r 3, and its variation with orbit radius is,
3 dr d(δθ )
dΩ = − Ω = (21)
2 r dt
The radial variation can be computed through the power balance,
id μ F · iv μ 1 dr r
− = = ac → = 2 ac (22)
dt 2r m r r dt μ
5The rate of change of the angular velocity is,
dΩ d2(δθ ) 3= =
dt dt 2 − Ω2
( r
2ac
√ 0
μ
) 3a
= − c (23)
r0
in which we have made r ≈ r0 as an approximation, since we do not expect the radius to change significantly during the maneuver,
d(δθ ) 3ac
= tdt − + A (24)
r0
where A is a constant. Starting from t = 0, δθ = 0, then d(δθ )/dt = 0, we have,
d(δθ ) 3a a
= − c 3 c
t → δθ = − t2 (t < t 1) (25)
dt r0 2r0
After t = t1, we continue to drift at a constant rate,
d(δθ ) 3a
=
dt − c t1
r0
and since we start from, 3ac
δθ (t1) = − t2
2r 10
the relative angle δθ during the coasting phase is, 3ac 3a
− 2 − c 3act1 t1
δθ coast = t )2 1 t1(t 1
r0 r0
− t = − r0
(
t − 2
)
(26) At the end of coasting , and we have, 3a t θ(∆ t − c 1 3t1
δ t1) = − r0
(
∆t − 2
)
(27) and, after a second period t1 of reversed thrust, we return to the initial orbit with d(δθ )/dt =0, and with δθ as in Eq. (27), plus a further δθ (t1). The total ∆ θ is, 3act1 3t1 3act2 3act1
∆θ = −
(
∆tr0
−
)
− 1
2r0
→ ∆θ =2 − (∆ t t1) (28)
r0
−
Clearly, the mission (given ∆ θ and ∆ t) can be accomplished with different choices of thrust-ing time t1 (but notice that t1 < ∆t/ 2 in any case). The required acceleration ac and ∆v = 2 |ac|t1 depend on this choice,
rac = − 0∆θ (29) 3t1(∆ t − t1)
62 r0Δθ
Δv = (30)
3 (Δ t − t1)
Not surprisingly, we find again that low thrust ends up as a penalty on Δv, so that the
thrusting time should be selected as short as possible within the available on-board power.
In the limit of impulsive thrust, we realize that Δt cannot really be any less than the
Hohmann transfer time. A more detailed analysis of this case confirms that, for the high
thrust case, Eqs. (29-30) are indeed valid with t1 = π/ Ω.
The power per unit mass required is,
P 1 |F |c r0Δθ c
= = (31)
m 2η m 6ηt 1(Δ t − t1)
Finally, some analyses might benefit from expressing the results in terms of the coasting
time tc = Δ t − 2t1, so that,
Δt − tc Δt + tc Δt2 − t2
c
t1 = , Δt − t1 = and t1(Δ t − t1) =
2 2 4
We then have,
4r0Δθ
= − (32) ac 3(Δ t2 − t2
c
)
4 r0Δθ
Δv = (33)
3 (Δ t + tc)
P 2r0Δθ c
= (34)
m 3η(Δ t2 + t2
c
)
coasting reduces Δv, but increases P/m (not much if tc/Δt is small).
7MIT OpenCourseWare
16.522 Space Propulsion
Spring 20 15
For information about citing these materials or our Terms of Use, visit: . |
6614 | https://cran.r-project.org/web/packages/doremi/vignettes/first-order.html | first-order
first-order
library(doremi)
FIRST ORDER DIFFERENTIAL EQUATIONS
The linear first order differential equation with constant coefficients model is given by the following equation:
τ d y d t+(y−y e q)=k u(t)τ d y d t+(y−y e q)=k u(t)
or
d y d t+γ(y−y e q)=γ k u(t)d y d t+γ(y−y e q)=γ k u(t)
with the initial condition:
y(t=0)=y 0 y(t=0)=y 0
Where:
y(t)y(t) is the signal to be analyzed
d y d t d y d t is its first derivative
u(t)u(t) is the excitation term perturbing the dynamics of y(t)y(t)
And regarding the coefficients:
γ γ is the decay rate.Its inverse, τ τ represents the characteristic response time of the solution of the differential equation (i.e.the time it takes to the system to go back to equilibrium)
k k is the gain. It indicates the proportionality between the stationary increase of y y and the constant excitation increase that caused it.
y e q y e q is the signal equilibrium value. Value reached when the excitation term is 0.
It is common to find this equation with the excitation term set to 0, u(t)=0 u(t)=0:
d y d t+γ(y−y e q)=0 d y d t+γ(y−y e q)=0
The dynamics in this case follow and exponential decay of the form y(t)=(y 0−y e q)e−γ t+y e q y(t)=(y 0−y e q)e−γ t+y e q and are provoked either by a previous excitation that is no longer present or by the displacement of the system from its equilibrium position (i.e.an initial condition different from 0)
Two steps analysis procedure
The estimation of the fixed effect coefficients of the differential equation is performed in two steps. The first step consists in estimating the first derivative, using one of the three methods proposed.
The second step consists in performing the following linear mixed-effects regression:
y˙i j∼(b 0+b 0 j)+(b 1+b 1 j)y i j+(b 2+b 2 j)U i j+e i j y˙i j∼(b 0+b 0 j)+(b 1+b 1 j)y i j+(b 2+b 2 j)U i j+e i j
where:
i accounts for the time
j accounts for the different individuals
y˙i j y˙i j is the derivative estimated through one of the derivative estimation methods available in the package (gold, glla and fda) calculated on embedding points (gold, glla) or estimated on the time points provided (fda)
y i j y i j and U i j U i j are the variable and the excitation averaged on embedding points (gold, glla) or the original values provided (fda)
e i j e i j is the error term (residuals)
Note that random effects are estimated for the intercept (b 0+b 0 j b 0+b 0 j), variable (b 1+b 1 j b 1+b 1 j) and excitation terms (b 2+b 2 j b 2+b 2 j), so that individuals can have different coefficients (initial condition, damping time, gain and equilibrium value).
The fixed effect coefficients estimated from the regression are:
Coefficient g a m m a=b 1 g a m m a=b 1
Coefficient y e q g a m m a=b 0 y e q g a m m a=b 0
Coefficient $kgamma = b_{2} $
The coefficients of the differential equation can thus be calculated as:
Decay time: t a u=τ=1 b 1 t a u=τ=1 b 1
Gain: k=b 2 b 1 k=b 2 b 1
Equilibrium value: y e q=b 0 b 1 y e q=b 0 b 1
The estimation is performed using the function lmer if there are several individuals or lm if there is a single individual.
First order differential equation with no excitation term
Simulating data
Two functions are available to simulate data in the package: generate.1order simulates the solution of the differential equation for a given vector of time, and the parameters tau = τ τ, y0 = y(t=0)y(t=0),yeq = y e q y e q, k = k k and the vector of excitation u(t)u(t). The function create a data.table with a column t for the time, y for the signal, and exc for the excitation.
test <- generate.1order(0:100,y0 = 10,tau = 10)
plot(test$t,test$y)
The function generate.panel.1order uses generate.1order to generate a panel of nind individuals, with measurement noise and inter-individual noise.
Example 1a - Generating signals with no noise
In this example we will generate data for 4 individuals, with a decay time of 10s. That is, the signal follows exactly the differential equation (no measurement noise) with no variation of the decay time, the gain and the equilibrium value across individuals (no interindividual noise). As we are considering no excitation, then, as mentioned, the system should be out of its equilibrium value to observe the dynamics of return to equilibrium.
res1a <- generate.panel.1order(time = 0:99,
y0 = 1,
tau = 10,
nind = 4)
It can be specified to the function whether there is an excitation or not by either setting the input to NULL or directly ignoring it in the input arguments. The function returns a data.table with the class ‘doremidata’. As any data.table, it can be visualized using the str or head functions or entering the variable name.
```
res1a
> id time signalraw signal
> 1: 1 0 1.000000e+00 1.000000e+00
> 2: 1 1 9.048378e-01 9.048378e-01
> 3: 1 2 8.187322e-01 8.187322e-01
> 4: 1 3 7.408199e-01 7.408199e-01
> 5: 1 4 6.703215e-01 6.703215e-01
> ---
> 396: 4 95 7.485298e-05 7.485298e-05
> 397: 4 96 6.773005e-05 6.773005e-05
> 398: 4 97 6.128494e-05 6.128494e-05
> 399: 4 98 5.545313e-05 5.545313e-05
> 400: 4 99 5.017628e-05 5.017628e-05
```
Where:
id is the identifier of the individual
time is the time vector introduced as input
signalraw is the signal without noise
signal is the signal with noise (in this case, the values of this column are the same as those of the signalraw column and in the plot, and thus both lines are overlapped in the plot below).
Plotting the data with the plot method available in the package for doremidata objects is quite straight forward:
plot(res1a)
Example 1b - Changing initial condition, gain and equilibrium value
In this example, we show the flexibility of the functions to add changes on the mentioned parameters. For this, we simplify the form of the exitation to a simple step:
timevec <- 0:49
set.seed(123)
res1b <- generate.panel.1order(time = timevec,
excitation = as.numeric(timevec > 20),
y0 = 3,
tau = 5,
yeq = 1.5,
nind = 4)
plot(res1b) +
ggplot2::scale_y_continuous(limits = c(0, 3))
Example 2a - Generating signals with noise
The call to the function remains almost the same, with two additional arguments: a noise to signal ratio, set at 0.2, and an interindividual noise set at 40%. We can then visualize the different trajectories for the different individuals:
```
Generation of signals with intra and inter-noise
set.seed(123)
res2a <- generate.panel.1order(time = 0:49,
y0 = 1,
tau = 5,
nind = 6,
internoise = 0.4,
intranoise = 0.1)
```
plot(res2a)
Analyzing data
Example 3 - Analyzing data when the excitation is unknown, with some inter- and intraindividual noise
We will test how the analysis functions fit a decreasing exponential, as is the case when there is no excitation, when it is unknown, or when it is constant over time.
```
Simulating data with these hypothesis
set.seed(123)
data3 <- generate.panel.1order(time = 0:50,
y0 = 0.5,
tau = 10,
nind = 3,
internoise = 0.2,
intranoise = 0.1)
```
```
Analyzing
res3 <- analyze.1order(data = data3,
id = "id",
time = "time",
signal = "signal",
dermethod = "fda",
derparam = 0.7)
> WARN [2021-01-19 14:27:57] No excitation signal introduced as input. Input was set to 0.
```
Note that the input parameter has been omitted when calling the function. The result display the summary of the fixed effect coefficients:
```
res3
> id gamma gamma_stde yeqgamma yeqgamma_stde tau yeq
> 1: All 0.08519334 0.008371114 -0.0009016283 0.0003712671 11.73801 -0.01058332
> R2
> 1: 0.8072038
```
The data.table result_id gives the coefficients estimated for each individual:
```
res3$resultid
> id gamma yeqgamma tau yeq
> 1: 1 0.07182963 -0.0009016283 10.14641 -0.009148288
> 2: 2 0.08437508 -0.0009016283 11.62634 -0.010482635
> 3: 3 0.09937532 -0.0009016283 14.08225 -0.012696957
```
It is possible to use the plot function directly on the result of the analysis:
```
Plotting
plot(res3)
```
First order differential equation with an excitation term
Analyzing the signals with noise generated above, we can verify that the parameters were the one introduced in the simulation function and that the estimated signals generated match the simulated ones.
Simulating data
Example 1a - Generating signals with no noise
To generate the excitation process perturbing the system we want to simulate, we can use the generate.excitation function, which generate a sequence of square pulses of a given length randomly distributed:
set.seed(123)
U <- generate.excitation(nexc = 3, # number of square pulses
duration = 10, # pulse duration
deltatf = 1, # time spacing between points
tmax = 100, # maximum time
minspacing = 20) # minimum spacin between square pulses
plot(U$t,U$exc)
Note that the time vector and excitation vector can be of any form and created by the user.
In this example we will use the excitation vector we just generated and the generate.panel.1order function to generate the variable following the first order differential equation for 4 individuals with a decay time of 10s. The signal follows exactly the differential equation (no measurement noise) with no variation of the decay time, the gain and the equilibrium value across individuals (no interindividual noise).
res1a <- generate.panel.1order(time = U$t,
excitation = U$exc,
tau = 10,
k = 1,
nind = 4)
plot(res1a)
Example 1b - Changing initial condition, gain and equilibrium value
Using the same excitation process, but changing the initial condition, the gain the the equilibrium value:
res1b <- generate.panel.1order(time = U$t,
excitation = U$exc,
y0 = 5,
tau = 10,
k = 3,
yeq = 2,
nind = 4)
plot(res1b)
And it can be observed how the system tends to equilibrium (t e q=2 t e q=2) from the initial condition (y 0=3 y 0=3) after being perturbed by the excitations.
Example 2 - Generating signals with noise
The call to the function remains almost the same, this time we have added a Noise to Signal ratio of 2 (the variance of noise is twice the variance of the underlying signal) and a 20% inter-individual noise (that is the parameters gain, equilibrium value, the initial value y 0 y 0 and τ τ of each individual are sampled on a gaussian centered on the parameter value with a standard deviation of 20% its values) to visualize the different trajectories for the different individuals:
```
Generation of signals with intra and inter-noise
res2a <- generate.panel.1order(time = U$t,
excitation = U$exc,
tau = 10,
k = 1,
nind = 4,
yeq = 2,
y0 = 2,
internoise = 0.2,
intranoise = 2)
```
plot(res2a)
Analyzing data
Example 3 - Analyzing data from a single individual
In this case, we can reuse one of the signals from the previous example
data3 <- res2a[id==1]
As the table contains data from a single individual, the id parameter has been omitted when calling the function.
```
Analyzing
res3 <- analyze.1order(data = data3,
input = "excitation",
time ="time",
signal = "signal",
verbose=T)
> INFO [2021-01-19 14:28:01] Input data contains single individual.
> INFO [2021-01-19 14:28:01] One or several excitations. Linear regression calculated
> INFO [2021-01-19 14:28:01] Linear mixed-effect model had no errors.
> INFO [2021-01-19 14:28:01] One or several excitation terms. Calculation of estimated signal with deSolve
res3
> id gamma gamma_stde yeqgamma yeqgamma_stde tau yeq
> 1: All 0.06526137 0.08681774 0.168252 0.2511207 15.323 2.578126
> excitation_kgamma excitation_kgamma_stde excitation_k R2
> 1: 0.08469348 0.05704505 1.297758 0.342613
```
```
Plotting
plot(res3)
```
Example 3a - Analyzing data with several individuals and some inter and intra-individual noise
Analyzing the data generated in the example 2 above (table res2a), the user must specify the name of the columns containing the id of the participants, the excitation, and the signal. As several methods are available for the estimation of the derivatives, the user needs to specify which method to use (gold is the default) and the embedding dimension by modifying the parameter derparam (see the package pdf manual for more details).
res3a <- analyze.1order(data = res2a,
id = "id",
input ="excitation",
time ="time",
signal = "signal",
dermethod = "gold",
derparam = 3)
Now let’s take a look at the result. When calling the variable, the fixed effect coefficients of the regression with their associated standard error, the derived coefficient characterizing the signal shape (the decay time tau, the equilibrium value yeq and the gain k), and the R2 are displayed:
```
res3a
> id gamma gamma_stde yeqgamma yeqgamma_stde tau yeq
> 1: All 0.006561924 0.0239843 -0.006529568 0.05778508 152.3943 -0.9950691
> excitation_kgamma excitation_kgamma_stde excitation_k R2
> 1: 0.06712488 0.0227174 10.22945 0
```
The signal analyzed was generated with tau = 10, yeq = 0, and k = 1.
This is a simplified view of the results. The analysis function supplies an object of class “doremi” that contains in fact several lists. It is possible to explore the full result values by using the function “summary” for doremi objects (see the section on methods created for doremi objects at the end of this vignette).
```
summary(res3a)
> Derivative and mean calculation ($data):
> id excitation time signal signal_rollmean signal_derivate1 time_derivate
> 1: 1 1 0 1.548851 1.834129 0.181351993 1
> 2: 1 1 1 2.041981 2.114180 0.173511265 2
> 3: 1 1 2 1.911555 2.170934 0.150343864 3
> 4: 1 1 3 2.389003 2.335365 0.007922849 4
> 5: 1 1 4 2.212243 2.300707 0.036393743 5
> ---
> 400: 4 0 96 1.487556 1.477797 -0.106773663 97
> 401: 4 0 97 1.671826 1.412553 -0.189999930 98
> 402: 4 0 98 1.274009 1.502412 0.333695804 99
> 403: 4 0 99 1.291826 NA NA NA
> 404: 4 0 100 1.941400 NA NA NA
> excitation_rollmean totalexc totalexcroll signal_estimated
> 1: 1 10.22945 10.22945 1.785409
> 2: 1 10.22945 10.22945 1.834129
> 3: 1 10.22945 10.22945 1.882530
> 4: 1 10.22945 10.22945 1.930615
> 5: 1 10.22945 10.22945 1.978385
> ---
> 400: 0 0.00000 0.00000 1.948274
> 401: 0 0.00000 0.00000 1.929023
> 402: 0 0.00000 0.00000 1.909898
> 403: NA 0.00000 NA 1.890899
> 404: NA 0.00000 NA 1.872023
>
> Mixed-effects regression results ($regression):
>
> Linear mixed model fit by REML. t-tests use Satterthwaite's method [
> lmerModLmerTest]
> Formula: paste0("signal_derivate1 ~ signal_rollmean + (1 +", paste(doremiexc,
> "rollmean ", collapse = "+", sep = "_"), " + signal_rollmean |id) + ",
> paste(doremiexc, "rollmean ", collapse = "+", sep = "_"))
> Data: intdata
> Control: lmerControl(calc.derivs = FALSE, optimizer = "nloptwrap")
>
> REML criterion at convergence: -138.8
>
> Scaled residuals:
> Min 1Q Median 3Q Max
> -3.1701 -0.6832 -0.0177 0.6163 4.2287
>
> Random effects:
> Groups Name Variance Std.Dev. Corr
> id (Intercept) 1.718e-09 4.145e-05
> input1_rollmean 1.013e-09 3.183e-05 -0.26
> signal_rollmean 2.207e-10 1.486e-05 -0.99 0.10
> Residual 3.963e-02 1.991e-01
> Number of obs: 396, groups: id, 4
>
> Fixed effects:
> Estimate Std. Error df t value Pr(>|t|)
> (Intercept) -0.006530 0.057785 392.960102 -0.113 0.91009
> signal_rollmean -0.006562 0.023984 392.971440 -0.274 0.78454
> input1_rollmean 0.067125 0.022717 392.940144 2.955 0.00332
> ---
> Signif. codes: 0 '' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1
>
> Correlation of Fixed Effects:
> (Intr) sgnl_r
> signl_rllmn -0.977
> inpt1_rllmn -0.018 -0.110
>
>
> $id
> (Intercept) input1_rollmean signal_rollmean
> 1 4.579499e-10 1.196365e-08 -1.078989e-09
> 2 3.937595e-09 -7.368405e-09 -8.931788e-10
> 3 -1.325579e-10 -9.584872e-09 7.779417e-10
> 4 -4.262987e-09 4.989627e-09 1.194226e-09
>
> with conditional variances for "id"
>
>
> Mean coefficients of the differential equation ($resultmean):
> id gamma gamma_stde yeqgamma yeqgamma_stde tau yeq
> 1: All 0.006561924 0.0239843 -0.006529568 0.05778508 152.3943 -0.9950691
> excitation_kgamma excitation_kgamma_stde excitation_k R2
> 1: 0.06712488 0.0227174 10.22945 0
>
> Coefficients per individual ($resultid):
> id gamma yeqgamma tau yeq excitation_kgamma
> 1: 1 0.006561923 -0.006529568 152.3943 -0.9950689 0.06712489
> 2: 2 0.006561924 -0.006529564 152.3943 -0.9950684 0.06712487
> 3: 3 0.006561925 -0.006529568 152.3943 -0.9950692 0.06712487
> 4: 4 0.006561926 -0.006529573 152.3943 -0.9950699 0.06712488
> excitation_k
> 1: 10.22945
> 2: 10.22945
> 3: 10.22945
> 4: 10.22945
>
> Derivative estimation method used ($dermethod):
> "calculate.gold"
>
> Embedding number/smoothing parameter ($derparam):
> 3
```
The first object of the output contains the original data with some columns added. These columns contain intermediate variables necessary for the mixed-effect regression:
```
head(res3a$data)
> id excitation time signal signal_rollmean signal_derivate1 time_derivate
> 1: 1 1 0 1.548851 1.834129 0.181351993 1
> 2: 1 1 1 2.041981 2.114180 0.173511265 2
> 3: 1 1 2 1.911555 2.170934 0.150343864 3
> 4: 1 1 3 2.389003 2.335365 0.007922849 4
> 5: 1 1 4 2.212243 2.300707 0.036393743 5
> 6: 1 1 5 2.404849 2.666189 0.451919842 6
> excitation_rollmean totalexc totalexcroll signal_estimated
> 1: 1 10.22945 10.22945 1.785409
> 2: 1 10.22945 10.22945 1.834129
> 3: 1 10.22945 10.22945 1.882530
> 4: 1 10.22945 10.22945 1.930615
> 5: 1 10.22945 10.22945 1.978385
> 6: 1 10.22945 10.22945 2.025842
```
Where:
signal_rollmean contains the zeroth order derivative of the input signal in embedding points provided by gold or glla, or the smoothing performed by fda.
signal_derivate1 contains the first derivate of signal, calculated by using one of the derivative estimation methods available (gold, glla, fda).
time_derivate contains the values of time in which the derivative was evaluated.
excitation_rollmean contains the roll mean of the excitation signal in embedding points (gold, glla) or in the initial time points provided (fda).
If we want to visualize the summary of the mixed-effect regression:
```
res3a$regression
>
> Linear mixed model fit by REML. t-tests use Satterthwaite's method [
> lmerModLmerTest]
> Formula: paste0("signal_derivate1 ~ signal_rollmean + (1 +", paste(doremiexc,
> "rollmean ", collapse = "+", sep = "_"), " + signal_rollmean |id) + ",
> paste(doremiexc, "rollmean ", collapse = "+", sep = "_"))
> Data: intdata
> Control: lmerControl(calc.derivs = FALSE, optimizer = "nloptwrap")
>
> REML criterion at convergence: -138.8
>
> Scaled residuals:
> Min 1Q Median 3Q Max
> -3.1701 -0.6832 -0.0177 0.6163 4.2287
>
> Random effects:
> Groups Name Variance Std.Dev. Corr
> id (Intercept) 1.718e-09 4.145e-05
> input1_rollmean 1.013e-09 3.183e-05 -0.26
> signal_rollmean 2.207e-10 1.486e-05 -0.99 0.10
> Residual 3.963e-02 1.991e-01
> Number of obs: 396, groups: id, 4
>
> Fixed effects:
> Estimate Std. Error df t value Pr(>|t|)
> (Intercept) -0.006530 0.057785 392.960102 -0.113 0.91009
> signal_rollmean -0.006562 0.023984 392.971440 -0.274 0.78454
> input1_rollmean 0.067125 0.022717 392.940144 2.955 0.00332
> ---
> Signif. codes: 0 '' 0.001 '' 0.01 '' 0.05 '.' 0.1 ' ' 1
>
> Correlation of Fixed Effects:
> (Intr) sgnl_r
> signl_rllmn -0.977
> inpt1_rllmn -0.018 -0.110
>
>
> $id
> (Intercept) input1_rollmean signal_rollmean
> 1 4.579499e-10 1.196365e-08 -1.078989e-09
> 2 3.937595e-09 -7.368405e-09 -8.931788e-10
> 3 -1.325579e-10 -9.584872e-09 7.779417e-10
> 4 -4.262987e-09 4.989627e-09 1.194226e-09
>
> with conditional variances for "id"
```
Where we have the random and fixed effects and the residuals calculated by the function lmer or lm depending on if the sample had several or one individual respectively.
Beware that, as known in most two-steps procedures, the estimation of derivatives reduces variance, and thus the error terms provided by the regression are not final.
The following table contains the fixed effect coefficients (result displayed by default when calling a doremi variable, as mentioned before):
```
res3a$resultmean
> id gamma gamma_stde yeqgamma yeqgamma_stde tau yeq
> 1: All 0.006561924 0.0239843 -0.006529568 0.05778508 152.3943 -0.9950691
> excitation_kgamma excitation_kgamma_stde excitation_k R2
> 1: 0.06712488 0.0227174 10.22945 0
```
It can be observed that the decay time is close to the value introduced to the simulation function (6), the equilibrium value close to its true value (0), and the gain close to its true value (1). For each individual we have:
```
res3a$resultid
> id gamma yeqgamma tau yeq excitation_kgamma
> 1: 1 0.006561923 -0.006529568 152.3943 -0.9950689 0.06712489
> 2: 2 0.006561924 -0.006529564 152.3943 -0.9950684 0.06712487
> 3: 3 0.006561925 -0.006529568 152.3943 -0.9950692 0.06712487
> 4: 4 0.006561926 -0.006529573 152.3943 -0.9950699 0.06712488
> excitation_k
> 1: 10.22945
> 2: 10.22945
> 3: 10.22945
> 4: 10.22945
```
Where:
tau is the decay time.
yeq is the equilibrium value.
excitation_k is the gain associated to the excitation “excitation”. As we will see later, there is the possibility to consider several excitations, each one of which will have a different effect on the dynamics of the signal studied and thus, a different gain associated to them.
If we graphically wish to verify how the estimated signal fits the initial signal, we can once again call the function plot, that has been adapted to handle doremi objects as well:
plot(res3a)
Similarly to the print function, plot applied to a doremi object plots by default the first six individuals contained in the result. If we wish to visualize a single individual or a specific set of individuals, we can specify them by changing the “id” input parameter of the function:
plot(res3a, id = 3)
plot(res3a, id = c(1,4))
Finally, the results contain the derivative method used and the embedding number/smoothing parameter used for further reference.
Example 3b - Enhancing the fit by changing the embedding number/ smoothing parameter.
As mentioned before, the estimation of derivatives before model fit is a source of error underestimation. It is possible to enhance the quality of the fit in three ways:
By changing the embedding number/smoothing parameter
By changing the order of the derivative
By changing the derivation method
In the following example, we will use the function optimum_param to find the embedding number that provides the closest R 2 R 2 to 1. The other ways can be tested “manually” by calling the other functions, for instance, in a simulation study. Let us reuse the data from res2a:
```
res3b <- optimum_param (data = res2a,
id = "id",
input = "excitation",
time = "time",
signal = "signal",
model = "1order",
dermethod = "gold",
pmin = 3,
pmax = 21,
pstep = 2)
res3b$summary_opt
res3b$d
> 13
```
And can be seen that from the range provided, an embedding number of {r} res3b$d produces the best fit and the coefficients are closer to their true values than the ones estimated in the previous example.
The optimum_param function generates an object of class “doremiparam”. If we want to graphically see the evolution of the coefficients according to the embedding number, we can easily plot the results with the plot function too:
plot(res3b)
Example 4 - Analyzing data when the signal is subject to several excitations
In this example, we construct a signal having different gain for different excitation sources. For the sake of simplicity, we generate this example signal without noise.
```
Simulating data with these hypothesis
Generating the three excitation signals:
t <- 0:100
u1 <- as.numeric(t>20 & t<40)
u2 <- as.numeric(t>50 & t<70)
u3 <- as.numeric(t>80 & t<100)
Arbitrarily choosing a = 1, b = 2 and c = 5 for the first individual
et1 <- u1 + 3 u2 + 5 u3
y1 <- generate.1order(time = t,
excitation = et1,
tau = 10,
k = 1)$y
as we are using the $y argument of the object generated
Signals for the second individual;
Arbitrarily choosing a = 1, b = 2.5 and c = 4 for the second individual
et2 <- u1 + 2.5 u2 + 4 u3
y2 <- generate.1order(time = t,
excitation = et2,
tau = 10,
k = 1)$y
Generating a table with the signals
data4 <- data.frame(id = rep(c(1, 2), c(length(et1), length(et2))),
time = c(t, t),
excitation1 = c(u1, u1),
excitation2 = c(u2, u2),
excitation3 = c(u3, u3),
signalcol = c(y1, y2))
```
We have in data4 a signal, whose dynamics was created by three excitations with different gains: k = 1, 3 and 5 for id = 1, k = 1, 2.5 and 4 for id = 2. We can see in the graph below that each excitation is identical but generates a different change of the signal:
```
Plotting signals
ggplot2::ggplot( data = data4) +
ggplot2::geom_line(ggplot2::aes(time,signalcol, colour = "Signal-no noise"))+
ggplot2::geom_line(ggplot2::aes(time,excitation1,colour = "excitation 1"))+
ggplot2::geom_line(ggplot2::aes(time,excitation2,colour = "excitation 2"))+
ggplot2::geom_line(ggplot2::aes(time,excitation3,colour = "excitation 3"))+
ggplot2::facet_wrap(~id)+
ggplot2::labs(x = "Time (s)",
y = "Signal (arb. unit)",
colour = "")
```
It is possible in analyze.1order to give a vector of column as input. In this case, a gain will be estimated for each excitation column.
```
Analyzing signals
res4 <- analyze.1order(data = data4,
id = "id",
input = c("excitation1", "excitation2", "excitation3"),
time = "time",
signal = "signalcol",
dermethod = "fda",
derparam = 0.1)
Looking for the calculation of the coefficients of the excitation
res4$resultid
```
Here, we recover the estimated gains for each individual in the resultid table. They are a good approximation of the coefficients introduced, using this small sample of 2 individuals.
```
Plotting signals
plot(res4)
```
Example 5 - Analyzing data with missing values in the signal
We examine how the analysis function still manages to retrieve an accurate fit from signals with missing measurements:
t <- 0:200
set.seed(123)
data5 <- generate.panel.1order(time = t,
excitation = as.numeric(t>50 & t<100),
tau = 10,
k = 1,
nind = 6,
internoise = 0.4,
intranoise = 0.1)
plot(data5)
```
Keeping one third of the rows selected randomly from the full data set
set.seed(123)
data5rd <- data5[sample(nrow(data5), nrow(data5)/3), ]
data5rd <- data5rd[order(id,time)]
```
In the next plot we show the selection of random points made on the full data set, and also represent the signal without noise. The signal with noise will be used to carry out the analysis.
plot(data5rd)
Then analyzing and visualizing the random points selection:
res7 <- analyze.1order(data = data5rd,
id = "id",
input = "excitation",
time ="time",
signal = "signal")
plot(res7)+
ggplot2::geom_line(data = data5,
ggplot2::aes(time,signalraw, colour = "Original signal"))
C - Applications
Heart rate during effort tests
Using the cardio dataset present in this package, we will apply the model to real heart frequency measurements. The data fields are described with detail in the pdf manual of the package, together with the source from which the data was obtained. The data consists in heart rate measurements from 21 individuals carrying out an effort test on a cycle ergometer. Participants pedal on a bicycle with increasing resistance. The resistance load is measured in watts, with higher load forcing the individual to greater effort. His/her heart frequency will then vary according to the effort supplied and the participant’s fitness. According to this, the cardio data includes, for each individual, the time since the beginning of the test (seconds), the bicycle load (Watts) and the heart rate (1/min). Here is an example of these data:
ggplot2::ggplot(cardio[id == 2],
ggplot2::aes(x = time))+
ggplot2::geom_point(ggplot2::aes(y = hr,color = "heart rate (beat/s)"))+
ggplot2::geom_line(ggplot2::aes(y = load,color = "effort load (W)"))
We can perform the analysis using analyze.1order:
resc1a <- analyze.1order(data = cardio,
id = "id",
input = "load",
time ="time",
signal = "hr",
dermethod = "glla",
derparam = 5)
And use the plot function to plot the analysis results for all 21 participants. Note that omitting the argument id would lead to only the first 6 participants being plotted by default.
plot(resc1a, id = 1:21) +
ggplot2::facet_wrap(~id,ncol=3,scales="free")
As can be seen, the model reproduces reasonably well the variation of the heart rate, and can be used to fully characterize the heart rate dynamics with three simple parameters: the resting heart rate (the equilibrium value), the characteristic time of heart rate change (the decay time), and the increase of heart rate for a given effort (the gain).
Such analysis has been used in (Mongin et al., 2020a) and Mongin et al., 2020b.
Measuring heart frequency during effort tests by considering several excitations
In the previous example, each one of the steps in the resistance load can be considered as a separated perturbation having an effect on heart rate. In this case, the functions will estimate different gains as explained, to account for each one of the contributions of the load to the total heart rate dynamics. Let us first create a new column for each step of effort load:
```
mydata <- cardio[id %in% 1:10]
create a index indicate which step of the exercise test
mydata[,load_idx := data.table::rleid(load),by = id]
transforming to large format, to have one column per workload step
mydata_large <- data.table::dcast(id + time + hr ~ paste0("load_",load_idx),data = mydata,value.var = "load")
replacing NAs by 0s
load_cols <- paste0("load_",1:max(mydata$load_idx))
mydata_large[,c(load_cols) := lapply(.SD,function(col) data.table::fifelse(is.na(col),0,col)),.SDcols = load_cols]
head(mydata_large)
> id time hr load_1 load_2 load_3 load_4 load_5 load_6 load_7
> 1: 1 4 118 0 0 0 0 0 0 0
> 2: 1 14 119 0 50 0 0 0 0 0
> 3: 1 24 122 0 50 0 0 0 0 0
> 4: 1 34 120 0 50 0 0 0 0 0
> 5: 1 44 122 0 50 0 0 0 0 0
> 6: 1 54 125 0 50 0 0 0 0 0
```
To consider each effort step as an independent excitation associated with its own gain, we can use analyze.1order and give the vector of the excitation column as id argument:
```
analyzing
resc1b <- analyze.1order(data = mydata_large,
id = "id",
input = load_cols,
time = "time",
signal = "hr",
dermethod = "gold",
derparam = 5)
> WARN [2021-01-19 14:28:22] Excitation signal load_2 is constant and is removed.
```
Here we will estimate a gain per excitation column
```
resc1b
> id gamma gamma_stde yeqgamma yeqgamma_stde tau yeq
> 1: All 0.007693634 0.002857514 0.6122927 0.2212309 129.9776 79.58433
> load_2_kgamma load_2_kgamma_stde load_2_k load_3_kgamma load_3_kgamma_stde
> 1: 0.006661281 0.001369214 0.8658172 0.005590196 0.001688833
> load_3_k load_4_kgamma load_4_kgamma_stde load_4_k load_5_kgamma
> 1: 0.7266002 0.006671328 0.003565423 0.8671231 0.003535968
> load_5_kgamma_stde load_5_k load_6_kgamma load_6_kgamma_stde load_6_k
> 1: 0.001867571 0.4595966 0.0004774536 0.002649282 0.06205827
> load_7_kgamma load_7_kgamma_stde load_7_k R2
> 1: -0.003151768 0.1569559 -0.4096592 0.9575932
```
In the plot, each workload step of the exercise test appear in one different column.
plot(resc1b,id=1:10)+ ggplot2::facet_wrap(~id,ncol=3,scales="free")
This analysis has been the main analysis in Mongin et al.,2020c and used in Mongin et al.,2020d.
Measurements of response time of individuals when carrying out mental rotation tasks
Using the rotation dataset present in this package, we will apply the model to a case in which the excitation signal is not clearly defined. In brief, the data is from 17 individuals that carried out mental rotation tasks (identify if two figures, one of which is rotated, are the same or a mirror image) and the response time (in milliseconds) was measured (Courvoisier et al., 2013). Each individual was measured every day for 60 days, though there can be missing data. To account for this, the time has been represented by the number of days since the beginning of the experiment (variable days). The signal is the mean response time (variable meanRT) over the ~200 stimuli done every day. The purpose of the study was to compare the performance between men and women.
```
dermethod<- "fda"
pmin = 0.1
pmax = 1
pstep = 0.1
restemp <- optimum_param(data = rotation,
id = "id",
time ="days",
signal = "meanRT",
dermethod = dermethod,
model = "1order",
pmin = pmin,
pmax = pmax,
pstep = pstep)
> WARN [2021-01-19 14:28:31] No excitation signal introduced as input. Input was set to 0.
> WARN [2021-01-19 14:28:31] Some of the decay times calculated were negative and thus, the estimated signal was not generated for these.
> decay times can be negative for some individuals for the following reasons: 1. The signal of
> the individual doesn't go back to equilibrium. 2.The linear mixed-effects model estimating the random
> effect showed some error messages/warnings. 3.Model misspecification.
> WARN [2021-01-19 14:28:31] No excitation signal introduced as input. Input was set to 0.
> WARN [2021-01-19 14:28:31] Some of the decay times calculated were negative and thus, the estimated signal was not generated for these.
> decay times can be negative for some individuals for the following reasons: 1. The signal of
> the individual doesn't go back to equilibrium. 2.The linear mixed-effects model estimating the random
> effect showed some error messages/warnings. 3.Model misspecification.
> WARN [2021-01-19 14:28:31] No excitation signal introduced as input. Input was set to 0.
> WARN [2021-01-19 14:28:31] Some of the decay times calculated were negative and thus, the estimated signal was not generated for these.
> decay times can be negative for some individuals for the following reasons: 1. The signal of
> the individual doesn't go back to equilibrium. 2.The linear mixed-effects model estimating the random
> effect showed some error messages/warnings. 3.Model misspecification.
> WARN [2021-01-19 14:28:31] No excitation signal introduced as input. Input was set to 0.
> WARN [2021-01-19 14:28:32] Some of the decay times calculated were negative and thus, the estimated signal was not generated for these.
> decay times can be negative for some individuals for the following reasons: 1. The signal of
> the individual doesn't go back to equilibrium. 2.The linear mixed-effects model estimating the random
> effect showed some error messages/warnings. 3.Model misspecification.
> WARN [2021-01-19 14:28:32] No excitation signal introduced as input. Input was set to 0.
> WARN [2021-01-19 14:28:32] No excitation signal introduced as input. Input was set to 0.
> WARN [2021-01-19 14:28:32] No excitation signal introduced as input. Input was set to 0.
> WARN [2021-01-19 14:28:32] No excitation signal introduced as input. Input was set to 0.
> WARN [2021-01-19 14:28:33] No excitation signal introduced as input. Input was set to 0.
> WARN [2021-01-19 14:28:33] No excitation signal introduced as input. Input was set to 0.
restemp$summary_opt
resc2a <- analyze.1order(data = rotation,
id = "id",
time ="days",
signal = "meanRT",
dermethod = dermethod,
derparam = restemp$d)
> WARN [2021-01-19 14:28:33] No excitation signal introduced as input. Input was set to 0.
```
Plotting analysis results:
plot(resc2a, id = 1:17) |
6615 | https://www.ehd.org/dev_article_unit3.php | 2 to 3 Weeks | Prenatal Overview
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Home › Prenatal Overview
Prenatal Form and Function – The Making of an Earth Suit
[Help]
Unit 3:2 to 3 Weeks
Germ Layers | Organ & Body System Formation | Brain & Spinal Cord | Circulatory System
Closer Look:
Applying the Science:
00:00
00:00
Movie 3.1 - Amnion and Germ Layers
Play Movie|See Snapshots
Germ Layers
By about 15 days (2 weeks, 1 day) following fertilization, stem cells have divided and differentiated into three differentgerm layers called ectoderm, endoderm, and mesoderm.Each gives rise to major components of specific body structures and organs.1 (See Figure 3.2 below.)
Ectoderm derivatives include the skin, nails, hair follicles, sweat glands, and nerves within the lungs.2 Another specialized layer of cells appearing at this time is the neuroectoderm, which gives rise to the brain, spinal cord, and peripheral nerves, as well as many of the muscles and bones in the face.3
Endoderm forms the lining of the respiratory4 and gastrointestinal tracts and gives rise to major portions of internal organs including the lungs, liver, pancreas, and intestines.5
Mesoderm derivatives include the heart, kidneys, bones, muscles, and blood vessels as well as portions of the reproductive and urinary systems.6
Mesoderm also gives rise to specialized cells calledsomites (so’mits).7 These cells form most of the skull and ribs as well as the vertebral column or backbone.8
All of these cell layers and cell types work in concert forming the increasingly complex embryo.
Figure 3.1 - Upper Skeleton
Somites give rise to the skull and upper skeleton.
LifeART
Copyright © 2002 Lippincott, Williams & Wilkins.
All rights reserved.
These young ballerinas dance in part thanks to their bones and muscles originally formed from their mesoderm.
The cells of the skin, nails, and hair follicles are all derived from ectoderm.
Figure 3.2 - Dorsal View of Presomite Embryo (18 Days) [PDF version of FIG 3.2]
From Gasser RF, 1975, 19. Atlas of Human Embryos. Copyright © 1975 RF Gasser, PhD.
All rights reserved.
Figure 3.3 - Organ & Body System Formation
Copyright © 2007 EHD, Inc. All rights reserved.
Figure 3.4 - Right Half of the Late Trilaminar Blastocyst (16-17 Days) [PDF version of FIG 3.4]
From Gasser RF, 1975, 14. Atlas of Human Embryos. Copyright © 1975 RF Gasser, PhD. All rights reserved.
Figure 3.5 - Brain and Spinal Cord
LifeART
Copyright © 2002 Lippincott, Williams & Wilkins.
All rights reserved.
By day 17 the central portion of the thyroidgland appears.9 This important gland will soon regulate the rate of metabolism throughout the rest of the human life cycle.
The embryo’s respiratory system begins to develop by 3 weeks with an outgrowth of the foregut which will form the windpipe or trachea (tra’ke-a).10
The Brain & Spinal Cord - Early Emergence
The site of future brain development is first recognizable with the appearance of the neural plate by 2 weeks, 4 days. By 3 weeks the neural plate thickens first at the head end of the embryo and folds into the neural tube which will form the brain and the spinal cord.11 By 3 weeks, the 3 primary sections of the brain are identifiable. These sections are called the forebrain, midbrain,and hindbrain.12
Vital Systems – Circulation
The complexity achieved by the embryo in just the first 3 weeks of development is incredible. Considering the importance of distributing nutrients to the emerging brain and spinal cord, as well as the rest of the embryo, the early development of the circulatory system is not surprising.
Figure 3.6 - Early Blood Cells
Early red blood cell precursors are present in the yolk sac just three weeks after fertilization!
LifeART
Copyright © 2002 Lippincott, Williams & Wilkins.
All rights reserved.
Movie 3.2 - Heart and Circulation
Play Movie|See Snapshots
Yet the early completion of this body system – the first system to begin functioning – is remarkable.13 By 3 weeks early blood cell precursors appear in the yolk sac.14 This process of blood cell formation is called hematopoiesis. Also by 3 weeks, early blood vessels form throughout the embryo as the network of the early circulatory system begins to take shape.15
In the middle of week 3, only 18 days after fertilization, the embryo’s heart appears.
Intro
The First Week
1 to 2 Weeks
2 to 3
Weeks
3 to 4 Weeks
4 to 5 Weeks
5 to 6 Weeks
6 to 7 Weeks
7 to 8 Weeks
8 to 9 Weeks
9 to 10 Weeks
10 to 11 Weeks
11 to 12 Weeks
3 to 4 Months
4 to 5 Months
5 to 6 Months
6 to 7 Months
7 to 8 Months
8 to 9 Months
9 Months to Birth
Appendix
Sources
Footnotes
1Gasser, 1975. 7; Moore et al., 2000. 13; O'Rahilly and Müller, 2001. 14, 135.
2Moore et al., 1994. 9-10.
3DiFiore and Wilson, 1994. 221.
4DiFiore and Wilson, 1994. 221.
5Grand et al., 1976. 796; O'Rahilly and Müller, 1996. 123, 226, 243.
6Moore and Persaud, 2003. 60; Moore and Persaud, 2003. 80, 83; Sadler, 2005. 9.
7Moore and Persaud, 2003. 69.
8Fuse, 1996. 1.
9Fuse, 1996. 1.
10DiFiore and Wilson, 1994. 221; Fowler et al., 1988. 793.
11Moore and Persaud, 2003. 67.
12Bartelmez, 1923. 236; Moore and Persaud, 2003. 439-440; Müller and O'Rahilly, 1983. 419-420, 429; O'Rahilly and Gardner, 1979. 123, 129; O'Rahilly and Müller, 1984. 422; O'Rahilly and Müller, 1987. 90; O'Rahilly and Müller, 1999a. 47, 52.
13Moore and Persaud, 1993. 55; Moore and Persaud, 2003. 70.
14Carlson, 2004. 117; Guyton and Hall, 2000. 958; Metcalf and Moore, 1971. Table 4.1, 173; Moore and Persaud, 2003. 70; Palis and Yoder, 2001. 932.
15Gilmour, 1941. 28; Navaratnam, 1991. 147-148; O'Rahilly and Müller, 1987. 86.
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6616 | https://www.ck12.org/flexi/chemistry/titration-experiment/the-normality-of-a-solution-in-a-titration-can-be-calculated-using-the-formula-n1v1-n2v2-true-by-false/ | Flexi answers - The normality of a solution in a titration can be calculated using the formula N1V1 = N2V2. True/False | CK-12 Foundation
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Titration Experiment
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The normality of a solution in a titration can be calculated using the formula N1V1 = N2V2. True/False
Flexi Says:
Yes, that's correct. In a titration, the formula N 1 V 1 = N 2 V 2 is used to calculate the normality of a solution. Here, N 1 and N 2 are the normalities of the two solutions, and V 1 and V 2 are their respective volumes. This equation is based on the principle of conservation of equivalents, which states that the equivalents of the substance in the solution before the reaction are equal to the equivalents of the substance in the solution after the reaction.
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6617 | https://math.stackexchange.com/questions/4939865/counting-words-that-agree-at-some-place-with-each-of-their-cyclic-permutations | combinatorics - Counting words that agree at some place with each of their cyclic permutations - Mathematics Stack Exchange
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Counting words that agree at some place with each of their cyclic permutations
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Let a(m,n)a(m,n) be the number of words W W of length m m in an alphabet of n n letters, which have the property that each cyclic permutation of W,W, has the same letter as W W in some place.
For example, a a b c b a a b c b counts (for m=5 m=5 and n≥3 n≥3), since its cyclic permutations b a a b c,b a a b c,c b a a b,c b a a b,b c b a a,b c b a a,a b c b a,a b c b a, agree with it at places 2, 5, 3, 1. But a a b b c a a b b c does not count, because it disagrees with b c a a b b c a a b everywhere.
Some values of a(m,n)a(m,n) for small m m are as follows:
a(1,n)=a(2,n)=n a(1,n)=a(2,n)=n
a(3,n)=n(3 n−2)a(3,n)=n(3 n−2)
a(4,n)=n(4 n−3)a(4,n)=n(4 n−3)
a(5,n)=n+15 n(n−1)2 a(5,n)=n+15 n(n−1)2
a(6,n)=n+3 n(n−1)2(4 n+1)a(6,n)=n+3 n(n−1)2(4 n+1)
a(7,n)=n+7 n(n−1)(2 n 3+5 n 2−29 n+31)a(7,n)=n+7 n(n−1)(2 n 3+5 n 2−29 n+31)
a(8,n)=n+4 n(n−1)(24 n 3−66 n 2+37 n+27)a(8,n)=n+4 n(n−1)(24 n 3−66 n 2+37 n+27)
a(9,n)=n+3 n(n−1)(87 n 4−369 n 3+292 n 2+811 n−1145)a(9,n)=n+3 n(n−1)(87 n 4−369 n 3+292 n 2+811 n−1145)
The formulae for m≥5 m≥5 are due to Fabius Wiesner (see comments).
It thus appears that a(m,n)a(m,n) is a polynomial in n n for each m.m. (This is clear in fact - see below.)
Is a general formula for a(m,n)a(m,n) known?
Incidental remark: When m m is prime, the action of the cyclic group of order m m on the set of counted words shows that (a(m,n)−n)/m(a(m,n)−n)/m is an integer. When m=5 m=5 this ratio is 3 n(n−1)2,3 n(n−1)2, which is also the number of 2 x 2 planar subsets of a 3 x 3 cube (OEIS sequence A270205). This is probably a coincidence, but I mention it in case there is a nice reason.
Based on Fabius Wiesner's notes (see comments below) it may be more enlightening to write
a(m,n)=∑r=1 n−1 b(m,r)(n r)a(m,n)=∑r=1 n−1 b(m,r)(n r)
where b(m,r)=∑r s=1(−1)r−s(r s)a(m,s)b(m,r)=∑s=1 r(−1)r−s(r s)a(m,s) is the number of words with the given property, that use exactly all the first r r letters in the alphabet. The expression for b(m,r)b(m,r) is the inclusion-exclusion formula. Then the formulae above become:
a(1,n)=a(2,n)=(n 1)a(1,n)=a(2,n)=(n 1)
a(3,n)=(n 1)+6(n 2)a(3,n)=(n 1)+6(n 2)
a(4,n)=(n 1)+8(n 2)a(4,n)=(n 1)+8(n 2)
a(5,n)=(n 1)+30(n 2)+90(n 3)a(5,n)=(n 1)+30(n 2)+90(n 3)
a(6,n)=(n 1)+54(n 2)+306(n 3)+288(n 4)a(6,n)=(n 1)+54(n 2)+306(n 3)+288(n 4)
a(7,n)=(n 1)+126(n 2)+1428(n 3)+3864(n 4)+1680(n 5)a(7,n)=(n 1)+126(n 2)+1428(n 3)+3864(n 4)+1680(n 5)
a(8,n)=(n 1)+232(n 2)+3912(n 3)+14400(n 4)+11520(n 5)a(8,n)=(n 1)+232(n 2)+3912(n 3)+14400(n 4)+11520(n 5)
a(9,n)=(n 1)+510(n 2)+16470(n 3)+126432(n 4)+305640(n 5)+187920(n 6)a(9,n)=(n 1)+510(n 2)+16470(n 3)+126432(n 4)+305640(n 5)+187920(n 6)
a(10,n)=(n 1)+990(n 2)+45630(n 3)+460200(n 4)+1506000(n 5)+1605600(n 6)+403200(n 7)a(10,n)=(n 1)+990(n 2)+45630(n 3)+460200(n 4)+1506000(n 5)+1605600(n 6)+403200(n 7)
This formulation makes it clear why a(m,n)a(m,n) is a polynomial in n,n, and shows that it would be enough to determine b(m,r)b(m,r) for 1≤r<m.1≤r<m. It would be interesting to know just the degree of the polynomial, i.e. to understand when b(m,r)=0.b(m,r)=0. See comments for some remarks on this question.
Remark: Factorizing the b(m,r)b(m,r) The group C m×S r C m×S r acts on the words counted by b(m,r).b(m,r). For r>1,r>1, the orbits are mostly regular; this must hold when m m is prime, so that b(m,r)b(m,r) is divisible by m⋅r!m⋅r! for m m prime and r>1;r>1; for example b(7,4)=3864=7⋅4!⋅23.b(7,4)=3864=7⋅4!⋅23. But for example, b(6,2)b(6,2) is not divisible by 12 12 and b(6,3)b(6,3) is not divisible by 36;36; the words a a b a a b a a b a a b and a a b a a c a a b a a c have nontrivial stabilizers in C 6×S 2 C 6×S 2 and C 6×S 3 C 6×S 3 respectively.
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edited Jul 28, 2024 at 12:03
Tom WIldeTom WIlde
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1 Hopefully without errors, I have computed a(6,n)−n=3 n(n−1)2(4 n−1)a(6,n)−n=3 n(n−1)2(4 n−1), a(7,n)−n=7 n(n−1)(2 n 3+5 n 2−29 n+31)a(7,n)−n=7 n(n−1)(2 n 3+5 n 2−29 n+31), a(8,n)−n=4 n(n−1)(24 n 3−66 n 2+37 n+27)a(8,n)−n=4 n(n−1)(24 n 3−66 n 2+37 n+27), a(9,n)−n=3 n(n−1)(87 n 4−369 n 3+292 n 2+811 n−1145)a(9,n)−n=3 n(n−1)(87 n 4−369 n 3+292 n 2+811 n−1145). Therefore the degree is 5≠m−2 5≠m−2 for m=8 m=8 and 6≠m−2 6≠m−2 for m=9 m=9, unless I made mistakes.Fabius Wiesner –Fabius Wiesner 2024-06-30 17:55:06 +00:00 Commented Jun 30, 2024 at 17:55
1 Yes you are right it is 4 n+1 4 n+1, sorry. Regarding the degrees of a(m,n)−n a(m,n)−n they agree with A064726 "Sum of primes dividing the partitions of n into distinct parts (with repetition)" but probably a coincidence.Fabius Wiesner –Fabius Wiesner 2024-06-30 19:36:01 +00:00 Commented Jun 30, 2024 at 19:36
1 Don't know if you know already, and it is not much, but since a word that has at least half letters equal must agree somewhere for any permutation, we have a (not tight) lower bound (m⌈(m+1)/2⌉)n(n−1)⌊(m−1)/2⌋(m⌈(m+1)/2⌉)n(n−1)⌊(m−1)/2⌋ and thus a lower bound for the degree is ⌈(m+1)/2⌉⌈(m+1)/2⌉. However, e.g. for m=5 m=5 we need to count also words of the form a a a a a a a a a a (n n) and a b c c b a b c c b (5 n(n−1)2 5 n(n−1)2) to get the correct result, and for m>5 m>5 there are probably much more forms to evaluate. I don't know if there is a way to enumerate all of them.Fabius Wiesner –Fabius Wiesner 2024-07-02 12:13:43 +00:00 Commented Jul 2, 2024 at 12:13
1 I have computed that, in addition to the words that have at least half letters equal, there are 0,1,9,36,141 0,1,9,36,141 classes of isomorphic words satisfying the requirement (up to a letter replacement and a cyclic permutation) for m=4,5,6,7,8 m=4,5,6,7,8 respectively. Among these, those contributing a term with maximum degree are a a b c b a a b c b (≡a b c c b≡a b c c b) for m=5 m=5, a a b a c d a a b a c d and a a b c a d a a b c a d for m=6 m=6, a a b a c d e a a b a c d e and a a b c d a e a a b c d a e for m=7 m=7.Fabius Wiesner –Fabius Wiesner 2024-07-03 10:11:36 +00:00 Commented Jul 3, 2024 at 10:11
1 If we count for a binary word of length m m the minimum number of ones satisfying the requirement, apparently we get OEIS A283297 and this mathoverflow question. If the connection is correct, from there we get a lower bound for the degree of: m−2 m−−−√+1 m−2 m+1 (except for m=4 m=4).Fabius Wiesner –Fabius Wiesner 2024-07-04 07:49:55 +00:00 Commented Jul 4, 2024 at 7:49
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1.26: Pyramids and Cones
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Table of contents
1. Pyramids
1. Exercises
2. Exercises
Cones
Exercises
Exercises
You may use a calculator throughout this module.
Pyramids
The pyramid complex at Giza, Egypt. Photo by Ricardo Liberato on Wikipedia
A pyramid is a geometric solid with a polygon base and triangular faces with a common vertex (called the apex of the pyramid). Pyramids are named according to the shape of their bases. The most common pyramids have a square or another regular polygon for a base, making all of the faces identical isosceles triangles. The height, h, is the distance from the apex straight down to the center of the base. Two other measures used with pyramids are the edge length e, the sides of the triangular faces, and the slant height l, the height of the triangular faces.
Volume of a Pyramid
In general, the volume of a pyramid with base of area B and height h is
V=1 3Bh(or)V=Bh÷3
If the base is a square with side length s, the volume is
V=1 3s 2h(or)V=s 2h÷3
Interestingly, the volume of a pyramid is 1 3 the volume of a prism with the same base and height.
Exercises
A pyramid has a square base with sides 16.0 centimeters long, and a height of 15.0 centimeters. Find the volume of the pyramid.
2. The Great Pyramid at Giza has a height of 137 meters and a square base with sides 230― meters long. Find the volume of the pyramid.
The lateral surface area (LSA) of a pyramid is found by adding the area of each triangular face.
Lateral Surface Area of a Pyramid
If the base of a pyramid is a regular polygon with n sides each of length s, and the slant height is l, then
LSA=1 2nsl(or)LSA=nsl÷2
If the base is a square, then
LSA=2sl
The total surface area (TSA) is of course found by adding the area of the base B to the lateral surface area. If the base is a regular polygon, you will need to use the techniques we studied in Module 23.
Total Surface Area of a Pyramid
TSA=LSA+B
If the base is a square, then
TSA=2sl+s 2
Exercises
A pyramid has a square base with sides 16.0 centimeters long, and a slant height of 17.0 centimeters. Find the lateral surface area and total surface area of the pyramid.
2. The Great Pyramid at Giza has a slant height of 179 meters and a square base with sides 230― meters long. Find the lateral surface area of the pyramid.
Cones
A large cone-shaped pile of grain in Eastern Oregon.
A cone is like a pyramid with a circular base. You may be able to determine the height h of a cone (the altitude from the apex, perpendicular to the base), or the slant height l (which is the length from the apex to the edge of the circular base). Note that the height, radius, and slant height form a right triangle with the slant height as the hypotenuse. We can use the Pythagorean theorem to determine the following equivalences.
The slant height l, height h, and radius r of a cone are related as follows:
l=r 2+h 2
h=l 2−r 2
r=l 2−h 2
Just as the volume of a pyramid is 1 3 the volume of a prism with the same base and height, the volume of a cone is 1 3 the volume of a cylinder with the same base and height.
Volume of a Cone
The volume of a cone with a base radius r and height h is
V=1 3πr 2h(or)V=πr 2h÷3
Exercises
The base of a cone has a radius of 5.0 centimeters, and the vertical height of the cone is 12.0 centimeters. Find the volume of the cone.
2. The base of a cone has a diameter of 6.0 feet, and the slant height of the cone is 5.0 feet. Find the volume of the cone.
For the surface area of a cone, we have the following formulas.
Surface Area of a Cone
LSA=πrl
TSA=LSA+πr 2=πrl+πr 2
It’s hard to explain the LSA formula in words, but here goes. The lateral surface of a cone, when flattened out, is a circle with radius l that is missing a wedge. The circumference of this partial circle, because it matched the circumference of the circular base, is 2πr. The circumference of the entire circle with radius l would be 2πl, so the part we have is just a fraction of the entire circle. To be precise, the fraction is 2πr 2πl, which reduces to r l. The area of the entire circle with radius l would be πl 2. Because the partial circle is the fraction r l of the entire circle, the area of the partial circle is πl 2⋅r l=πrl.
Exercises
The base of a cone has a diameter of 6.0 feet, and the slant height of the cone is 5.0 feet. Find the lateral surface area and total surface area of the cone.
2. The base of a cone has a radius of 5.0 centimeters, and the vertical height of the cone is 12.0 centimeters. Find the lateral surface area and total surface area of the cone.
en.Wikipedia.org/wiki/Great_Pyramid_of_Giza ↵
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5.3
Exponential Growth & Decay Functions - Teacher's Edition
Written by:Sean Regan | CK-12 Foundation
Fact-checked by:The CK-12 Editorial Team
Last Modified: Sep 01, 2025
This is our newest and most engaging Algebra 2 content! If you would like to revisit the previous version of Algebra 2, please CLICK HERE.
Common Core Standards
Focus Standards: F.BF.3, F.BF.1b, A.SSE.4, A.CED.1
Additional Standard: F.LE.4
Lesson Objectives
Understand the characteristics of the general exponential growth function, where r is growth rate.
Solve problems involving interest and exponential growth.
Derive the formula for the sum of a finite geometric series when the common ratio is not one.
Use the formula for the sum of a finite geometric series to solve problems.
Solve problems involving exponential growth.
Write an exponential growth function given a context.
Write an exponential decay function given a context.
Agenda
| | |
--- |
| Introduction: Compound Interest | 5 minutes |
| Activity 1: Compound Interest Continued | 10 minutes |
| Activity 2: Paying Off a Loan | 10 minutes |
| Activity 3: Euler's Number | 10 minutes |
| Extension: Continuous Interest | |
| Activity 4: Exponential Decay | 10 minutes |
| Wrap-Up: Review Questions | 5 minutes |
(Students cannot see the purple text)
Related modalities include: 2 videos (avg 1 min each), 2 text-based lessons (30 mins each), real-world application (15 mins), for a total of approximately 1 hr and 15 mins of scaffolding and extension content.
This lesson will take the knowledge obtained in the previous lesson and extend it to growth and decay functions. The students will begin by looking at the compound interest formula. This will then lead to the derivation of the sum of the geometric series formula. The math and connections in this proof are difficult, be sure to take enough time to explain each step appropriately. This will be followed by the introduction of Euler's number and then finally a discussion of exponential decay.
Introduction: Compound Interest
Simple and compound interest are topics commonly introduced in Algebra I, and explored more deeply in Algebra II.
The principle of a loan is the original loaned amount. Interest is a percentage of the principle that is added as a cost to pay for the privilege of borrowing the money. Compound interest occurs when interest is added to principle and further interest is then calculated on previous interest as well as the principle.
Use the interactive below to review how compound interest works.
Activity 1: Compound Interest Continued
The formula for compound interest can be written as A=P(1+rn)nt. The variables are used to represent the following:
A is the total amount owed
P is the principal, which is the starting amount
r is the interest rate
n is the number of times the principal is compounded in a year
t is the number of years
The compound interest formula comes from the exponential growth formula. Exponential growth can be expressed as a percent of the starting amount. The function f(x)=2x represents a quantity that repeatedly doubles. Another way to think of doubling is an increase of 100%. This can be rewritten as follows:
f(x)=(200%)x=(100%+100%)x=(1+100%)x
Everything is being taken and added to itself, resulting in the general exponential growth equation: f(x)=a(1+r)x where a is the starting amount and r is the growth rate, written as a decimal.
Example
A population of 10 mice increases by 300% every month.
a. Write a function model that expresses the number of mice in population P as a function of x months.
This problem highlights a challenge when it comes to wording. If a sentence reads "increases by 300%," this means that the growth rate is 300%, and the quantity is quadrupling. Keep everything and add 300%. The common ratio is 4 or 400%. The growth rate is the rate at which an amount increases; the common ratio is the rate at which an amount is multiplied. In this case, the starting population is a=10, and thegrowth rate is r=300% or 3. Substituting this into the exponential growth formula will give the following:
P(x)=10(1+3)x or P(x)=10(4)x
Answer: P(x)=10(4)x
A graph of this function is displayed below.
As portrayed in the graph of this function above, the population of 10 is increasing exponentially. Every month, add triple the amount of mice, which is the same as quadrupling the total.
b. How many mice will there be after six months?
To determine the population of mice after 6 months, set up and solve the equation:
P(x)=10(1+3)6=10(4)6=10×4,096=40,960
Answer: There will be 40,960 mice after six months.
Inline Questions
(Fill in the Blank) Suppose you deposit $500 into a savings account. The account grows at a 1.2% annual interest rate, compounded monthly. How much will be in the account after 3 years?
$518.32 (round to the nearest cent)
(Fill in the Blank) If you wanted to know how much was in your savings account after 9 months, what value should you substitute for t?
t = 0.75 years
(Fill in the Blank) The population of bacteria in a dish increases by 200% every hour. The growth rate of bacteria is 2 and the common ratio is 3.
Tom put $5000 into a savings account that earns 2.4% annual interest, compounded monthly. Every month, he also puts $100 into a piggy bank. Which function models the overall amount Tom has after t years?
A(t)=5000(1.024)12t+1200t
A(t)=5000(1.002)12t+1200t
A(t)=5000(0.002)12t+1200t
A(t)=5000(1.024)t+100t
Which of the following expressions represents the interest earned on an investment?
P(1+rn)nt−P
P(rn)nt
P(1+rn)nt−1
Discussion Question:
Suppose you deposit $50 into a savings account. The account grows at a 1.2% annual interest rate, compounded monthly. Assuming you do not deposit or withdraw money from the account, how long will it take for there to be $70 in the account?
Discussion Question Answer: It will take between 28 and 29 years. Use this problem to highlight the challenges faced by students finding the input to produce a given output. The students will address these difficulties later in this chapter when learning logarithms.
Activity 2: Paying off a Loan
Compound interest is used in nearly all scenarios where money is borrowed. However, as you pay off your loan, the interest is applied to smaller and smaller amounts.
Example
You take out a 30-year mortgage on a house for $350,000 at a monthly interest rate of 0.4% per month. How much will you pay per month?
First consider the idea that you pay nothing until the 30 year period is over, at which point you pay the full amount. If this was the case, you would owe over $1,000,000 for the house, including interest. Paying this much for a $350,000 house seems pretty ridiculous, but it goes to show how quickly interest can build. The reason this seems like a high number is because the interest is only charged to the amount left on the loan as you pay it off. To determine your monthly mortgage payment, you will need to take it month by month. Use the variable m to represent the monthly payment, p to represent the principal loan amount, and r to represent the monthly interest.
| | | |
---
| Pay Period | Amount Due | Amount Due Simplified |
| Starting Amount Owed | p | p |
| Amount Owed After Month 1 | p(1+r)−m | p(1+r)−m |
| Amount Owed After Month 2 | (p(1+r)−m)(1+r)−m | p(1+r)²−m(1+r)−m |
| Amount Owed After Month 3 | (p(1+r)−m)(1+r)−m)(1+r)−m | p(1+r)³−m(1+r)²−m(1+r)−m |
Looking at the simplified amount owed, there is a pattern emerging. This pattern can be extended to write an expression representing the amount owed at the end of the loan:
p(1+r)n−m(1+r)n−1−m(1+r)n−2−...−m(1+r)0
Since you will owe $0 at the end of the loan, you can use the expression above to write the following equation:
p(1+r)n−m(1+r)n−1−m(1+r)n−2−...−m(1+r)0=0
Add each term with an m coefficient to the right side of the equation:
p(1+r)n=m(1+r)n−1+m(1+r)n−2+...+m(1+r)0
Factor out an m from each term on the right side of the equation.
p(1+r)n=m[(1+r)n−1+(1+r)n−2+...+(1+r)0]
Finally, solve for the monthly payment by dividing the expression in the brackets to both sides of the equation.
p(1+i)n(1+i)n−1+(1+i)n−2+...+(1+i)0=m
While this does represent a formula that can be used to find the monthly payment, the denominator will take a long time to calculate. The expression in the denominator represents a geometric sequence.
Ideally, this formula to be easy to calculate. To accomplish this, write an expression to represent the sum of a geometric sequence. Recall from Exponential Functions that a geometric series can be written as terms of a geometric sequence:
Sn=a1+a1r+a1r2+a1r3+…+a1rn−2+a1rn−1
Now, factor out a1 to get a1(1+r2+r3+…+rn−2+rn−1). By isolating what is in the parenthesis and multiplying it by (1−r) as shown below, the sum can be simplified:
(1−r)Sn=a1(1−r)(1+r+r2+r3+…+rn−2+rn−1)=a1(1+r+r2+r3+…+rn−2+rn−1−r−r2−r3−r4−…−rn−1−rn)=a1(1+r+r2+r3+…+rn−2+rn−1−r−r2−r3−r4−…−rn−1−rn)=a1(1−rn)
Multiplying (1+r2+r3+…+rn−2+rn−1) by 1−r resulted in the middle terms being canceled, leaving only (1−rn). However, both sides of the equation must be multiplied by (1−r) to preserve the equation. The final step is to solve for Sn by dividing (1−r) on both sides of the equation. The sum of a finite geometric series can be written as follows:
Sn=a1(1−rn)1−r
Sn is the sum of the finite geometric series.
a1 is the first term in the series.
n is the number of terms in the series.
r is the common ratio.
Use this equation to rewrite or monthly mortgage payment formula above. The geometric sequence that needs to be rewritten is (1+i)n−1+(1+i)n−2+...+(1+i)0. The value of a1 is 1, the value of r is i, and the value of n is n.
(1+i)n−1+(1+i)n−2+...+(1+i)0=1(1−(1+i)n)1−(1+i)=(1+i)n−1(1+i)−1=(1+i)n−1i
This substitution will look as follows:
m=p(1+i)n(1+i)n−1+(1+i)n−2+...+(1+i)0=p(1+i)n(1+i)n−1i
From here, simplify by multiplying p(1+i)n by the reciprocal of (1+i)n−1i:
m=p(1+i)n1⋅i(1+i)n−1=i⋅p(1+i)n(1+i)n−1
Now solve for the monthly payment given the following:
p=350,000
i=0.004
n=12⋅30=360
m=0.004⋅350000(1+0.004)360(1+0.004)360−1=1,836.33
Answer: The monthly mortgage payment will be $1,836.33.
Answer the questions below to practice finding the sum of a geometric series.
Inline Questions
1. (Short Answer) Use the formula Sn=a1(1−rn)1−r to find the sum of the first 10 numbers in the geometric sequence: 5, 15, 45, ...
147620
2. (Short Answer) A geometric series with 5 terms has the sum Sn=186 and the common ratio r=12. What is the first term of the sequence?
96
Discussion Question: How can the formula for the sum of a geometric series be useful when working with the exponential function f(x)=3x?
Discussion Question Answer: This formula can be used to find the sum of any exponential function. Think of y=3x as 3,9,27,81,... using x=1 as the first term in the sequence. In order to use the geometric sum formula, you will need to identify the first term, a1, to identify the common ratio, r, and identify the location of the term that you want to find the sum to, n, which will equal x. You can see from this pattern that the first term is 3 and the common ratio is 3. Substitute this information into the formula Sn=a1(1−rn)1−r to find the sum of the sequence at any x value. For example, you know that S3=39 because 3+9+27=39. Calculate this with the geometric sum formula as follows: S3=3(1−33)1−3=3(−26)−2=39.
Activity 3: Euler's Number
In the previous activity, an example was presented of how interest could be compounded. This process is important because there are many examples of exponential growth in nature that compound. Continuous growth can be seen in bacteria, populations, and many more examples. However, these examples do not compound yearly or quarterly; they grow continuously. Continuous growth means that an object is constantly compounding. The growth is growing. In activity 1, you explored how the function f(x)=2x grows as a pattern of doubling. However, while this function doubles an amount, the doubled quantity is not doubling until after each iteration. Use the interactive below to explore this visually.
When a quantity is growing continuously, the growth of the quantity is growing. One way to think of this relationship is that the interest is earning interest. Swiss mathematician Jacob Bernoulli first studied continuous growth in the late 1600s. Bernoulli wanted to better understand what was happening during compound interest so he focused on what would happen to 1 single dollar as it was compounded at an interest rate of 100%. Recall from above that interest rate of 100% means that every year, the dollar quantity will double. Bernoulli was especially focused on what would happen to the dollar over the first year. Use the interactive below to visually explore what happens as the dollar is compounded at varying frequencies: annually, bi-annually, tri-annually, etc.
Bernoulli saw that the value of the dollar increased as the frequency. However, the value of the dollar increased toward a certain number, which he was never able to determine. It was not until the early 1700s that another Swiss mathematician, Leonhard Euler, discovered this value to be an irrational constant with the value 2.71828... Euler used the letter e to refer to the constant, and it is now known as Euler's Number.
Recall that continuous growth happens constantly, which represents compounding with infinite frequency. Use the function f(x)=2.71828...x or f(x)=ex to model continuous growth. The value of x represents the number of doubling periods, and in the context of continuous growth, x represents the number of years. This function assumes a starting value of 1. If, for example, $3 was compounded continuously, the function f(x)=3ex could be used. Imagine the $1 above undergoing continuous growth 3 times. This function can be transformed in a variety of ways. Changing the starting amount will stretch or shrink the function vertically. Multiplying the input, or x, by a value to cause a horizontal stretch or shrink. This multiplication is performed to alter the rate of doubling. For example, if the interest rate is 50% instead of 100%, the function will increase at a slower rate. This can be accomplished by multiplying the input value by 0.5 or 50%. he total amount of continuous growth can be modeled as a function of time using the following function:
f(t)=a⋅ert
f(t) is the amount of total growth.
a is the starting amount.
e is the constant 2.71828...
r is the rate of growth
t is the number of periods, often years.
Example
A $2,500 loan earns interest continuously at a rate of 6% interest.
a. Write a function to model the total amount owed on the loan as a function of time, in years.
Since the interest rate is 6%, the r value will be 0.06 and will horizontally stretch the parent continuous growth function. The a value is $2,500 because this is the starting amount of the loan. Imagine Taking $1 and continuously growing it 2,500 times simultaneously.
Answer: f(t)=2,500e0.06t
b. How much will be owed on the loan in 12 years?
To find the amount owed on the loan in 12 years, find f(12).
f(12)=2,500e0.06×12=2,500e0.72=5,136.08302...
Answer: $5,136.08
Use the interactive below to explore how various transformations of the parent function f(x)=ex will look.
Inline Questions
(Fill in the Blank) Population growth can be modeled using the formula P=P0ert where P0 is the initial population, r is the rate of growth, and t is the time in years. The growth rate of the human population is approximately 1.1%. If the world population is 7.6 billion, what will the population be in 5 years (in billions)?
8.03 billion (round to the nearest hundredths place)
(Fill in the Blank) Recall that the world population, in billions, can be modeled by the equation P=7.6e0.011t where t is time in years. Use the interactive to approximate how long it will take for the world population to reach 10 billion.
25 years (round to the nearest whole number)
3. (Fill in the Blank) The world population will double after approximately 63 years. Round to the nearest whole number.
True or False: The world will take 63 years to double, regardless of population.
TRUE
FALSE
Discussion Question: How can you explain the answers to the previous 2 questions?
Discussion Question Answer: Determine the amount of time it will take for any population, p, to double using the formula P=P0ert. The starting population will be p and the ending population will be 2p.
2P=P⋅e0.011t÷P÷P2=e0.011tln2=lne0.011tln2=0.011t÷0.011÷0.011ln20.011=tt≈63.01
Extension: Continuous Interest
Use the interactive below to explore continuous interest further.
Activity 4: Exponential Decay
As you saw in the previous section, Exponential Functions, exponential decay behaves the same as exponential growth, with the exception that the common ratio is between 0 and 1. With a slight change to the exponential growth formula, the exponential decay formula can be written: f(x)=a(1−r)x where a is the starting amount and r is the rate of decay, written as a decimal.
Example
When businesses calculate the value of their assets, they consider depreciation. Depreciation is the decrease in the value of an object over time. Brandon runs a construction company and bought a new excavator for $60,000, which depreciates 20% per year.
a. Write a function to model this scenario?
The rate of decay is the rate at which an amount decreases. In this case, the rate of decay is 20%. Substituting this into the exponential decay formula will give the following:
f(x)=60,000(1−0.20)x=60,000(0.8)x
Answer: f(x)=60,000(0.8)x
Since the excavator loses 20% of its value every year, this means that it retains 80% of its value. The graph of this model can be seen below:
b. How much will the excavator be worth in 5 years?
To find the value in five years, reduce the value by 20% five times, which can be done finding f(5).
f(x)=60,000⋅0.8(5)=60,000⋅0.32768=19,660.8
Answer: $19,660.80
In the section Modeling With Quadratic Functions, a piecewise function was used to represent the height of a bouncing ball. However, this could also be represented using an exponential function since the ball only bounces to a percentage of its previous height. Use the interactive below to explore this further.
Inline Questions
Which of the following scenarios can be represented by the graph? A freshly brewed cup of coffee is 160°F. You set the cup on a table to cool where the room temperature is 70°F.
| | |
--- |
| | correct |
Recall the scenario from the previous question: A freshly brewed cup of coffee is 160°F. You set the cup on a table to cool where the room temperature is 70°F. Which of the following functions can be used to model the temperature of the coffee, where t is time in minutes.
f(t)=90(0.89)t+70
f(t)=160(0.89)t+70
f(t)=160(0.89)t
f(t)=90(1,12)t+70
A new car loses 10% of its value as soon as it leaves the lot. After that, the value depreciates by 15% each year. Which of the following functions models the value of a new car worth $25,000?
f(x)=22500(0.15)x
f(x)=22500(0.85)x−2500
f(x)=25000(1.15)x
f(x)=22500(0.85)x
(Fill in the Blank) Recall the car from the previous question whose value could be modeled by the function f(x)=22500(0.85)x, where x is years after purchase.
After 8 years, the car will be worth about $6131 (round to the nearest dollar).
Recall the car from the previous questions whose value could be modeled by the function f(x)=22500(0.85)x.
True or False: The value of the car will eventually depreciate to $0.
True
False
Wrap-Up: Review Questions
Remember This!
The exponential growth function can be written as f(x)=a(1+r)x, where r is the growth rate.
The function f(x)=ex can be used to model continuous growth with e≈2.718…
The function f(t)=a⋅ert can be used to model continuous growth as a function of time.
The function P=P0ert can be used to model population growth where P0 is the initial population, r is the growth rate, and t is time.
The exponential decay function can be written as f(x)=a(1−r)x where a is the starting amount and r is the rate of decay.
| Image | Reference | Attributions |
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| | | Credit: CK12 Source: CK12 License: CC BY-NC 3.0 |
| | | Credit: CK12 -Vera Source: CK12 License: CC BY-NC 3.0 |
| | | Credit: CK12 Source: CK12 License: CC BY-NC 3.0 |
| | | Credit: ck-12 Source: Ck-12 License: CC BY-NC 3.0 |
| | | Credit: ck-12 Source: Ck-12 License: CC BY-NC 3.0 |
| | | Credit: ck-12 Source: Ck-12 License: CC BY-NC 3.0 |
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6621 | https://www.chegg.com/homework-help/questions-and-answers/decomposition-reaction-n2o5-2-n2o5-g-4-no2-g-o2-g-rate-k-n2o5-overall-reaction-order-q116966415 | Solved For the decomposition reaction of N2O5, 2 N2O5(g) ⟶ | Chegg.com
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For the decomposition reaction of N2O5, 2 N2O5(g) ⟶ 4 NO2(g) + O2(g) Rate = k[N2O5] What is the overall reaction order?
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Question: For the decomposition reaction of N2O5, 2 N2O5(g) ⟶ 4 NO2(g) + O2(g) Rate = k[N2O5] What is the overall reaction order?
For the decomposition reaction of N2O5, 2 N2O5(g) ⟶ 4 NO2(g) + O2(g) Rate = k[N2O5] What is the overall reaction order?
There are 2 steps to solve this one.Solution 100%(1 rating) Share Share Share done loading Copy link Step 1 Rate of reaction: Change in concentration of reactant (or) product per unit time. View the full answer Step 2 UnlockAnswer Unlock Previous questionNext question
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6622 | https://proofwiki.org/wiki/Method_of_Truth_Tables | Method of Truth Tables - ProofWiki
Method of Truth Tables
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Contents
1 Proof Technique
2 Proof of Tautology
3 Proof of Interderivability
3.1 Express two statements as a single WFF
3.2 Compare two WFFs in the same table
4 Proof of Logical Implication
4.1 Express two statements as a single WFF
4.2 Compare two WFFs in the same table
5 Indirect Technique
6 Notational Convenience
7 Also known as
8 Comment
9 Sources
Proof Technique
The method of truth tables is a technique for determining the validity of propositional formulas with respect to boolean interpretations.
In particular, for discerning if a propositional formula is a tautology for boolean interpretations.
To start with, we establish the characteristic truth tables of the various logical connectives.
We write one row for each boolean interpretation of the set of variables that we are concerned with.
From Count of Rows of Truth Table this amounts to 2 n 2 nrows if there are n nvariables.
There are therefore two rows in the truth table for the only non-trivial unary connective:
p F T¬p T F p¬p F T T F
and four rows in the truth tables for the binary connectives (the usual subset of which is given below):
p F F T T q F T F T p∧q F F F T p∨q F T T T p⟹q T T F T p⟺q T F F T p⟸q T F T T p⊕q F T T F p↑q T T T F p↓q T F F F p q p∧q p∨q p⟹q p⟺q p⟸q p⊕q p↑q p↓q F F F F T T T F T T F T F T T F F T T F T F F T F F T T T F T T T T T T T F F F
There are various sorts of proof this technique can be put to, as follows.
These will be illustrated by various examples.
Proof of Tautology
This is used to establish whether or not a given propositional formula is a tautology for boolean interpretations; that, is valid in all boolean interpretations.
Let P P be a propositional formula we wish to validate.
Subsequently, determine its truth table.
In the column under the main connective of P P itself can be found the truth value of P P for each boolean interpretation.
If this contains nothing but T T, then P P is a tautology.If this contains nothing but F F, then P P is a contradiction.If this contains T T for some boolean interpretations and F F for others, then P P is a contingent statement.
Proof of Interderivability
Suppose we have two propositional formulasP P and Q Q and we wish to see whether P⊣⊢Q P⊣⊢Q or not.
Example: Let P P be p↑q p↑q and let Q Q be ¬(p∧q)¬(p∧q).
There are two things we can do.
Express two statements as a single WFF
We express P⊣⊢Q P⊣⊢Q as a single WFF(P⟺Q)(P⟺Q) and perform the method of truth tables on that.
Demonstrating this with the example given:
p F F T T q F T F T(p F F T T↑T T T F q)F T F T⟺T T T T¬T T T F(p F F T T∧F F F T q)F T F T p q(p↑q)⟺¬(p∧q)F F F T F T T F F F F T F T T T T F F T T F T T F T T T F F T T T F T T F T T T
As can be seen, the column under the main connective⟺⟺ of (P⟺Q)(P⟺Q) is all T T, so ((p↑q)⟺¬(p∧q))((p↑q)⟺¬(p∧q)) is a tautology.
Hence from Equivalences are Interderivable, ((p↑q)⊣⊢¬(p∧q))((p↑q)⊣⊢¬(p∧q)) and the two formulas are interderivable.
■◼
Compare two WFFs in the same table
Alternatively, we can place the two WFFs side by side in the same truth table:
p F F T T q F T F T(p F F T T↑T T T F q)F T F T¬T T T F(p F F T T∧F F F T q)F T F T p q(p↑q)¬(p∧q)F F F T F T F F F F T F T T T F F T T F T T F T T F F T T T F T F T T T
This time, we need to make sure that the truth values in the columns under the main connectives of both formulae are the same.
Note that this is exactly the same as putting an instance of ⟺⟺ between the two and making a WFF out of the pair of them.
■◼
Proof of Logical Implication
Suppose we wish to prove that P⊢Q P⊢Q for two propositional formulasP P and Q Q.
Example: Let P P be ¬p¬p and Q Q be p⟹q p⟹q.
Express two statements as a single WFF
We express P⊢Q P⊢Q as a single WFF(P⟹Q)(P⟹Q) and perform the method of truth tables on that expression.
Demonstrating this with the example given:
p F F T T q F T F T(¬T T F F p)F F T T⟹T T T T(p F F T T⟹T T F T q)F T F T p q(¬p)⟹(p⟹q)F F T F T F T F F T T F T F T T T F F T T T F F T T F T T T T T
As can be seen, the column under the main connective⟹⟹ of (P⟹Q)(P⟹Q) is all T T.
Hence ((¬p)⟹(p⟹q))((¬p)⟹(p⟹q)) is a tautology.
Hence from Equivalence of Logical Implication and Conditional, ¬p⊢p⟹q¬p⊢p⟹q.
■◼
Compare two WFFs in the same table
Alternatively, we can place the two WFFs side by side in the same truth table:
p F F T T q F T F T¬T T F F p F F T T p F F T T⟹T T F T q F T F T p q¬p p⟹q F F T F F T F F T T F F T T T F F T T F F T T F T T T T
This time, we need to make sure that when the truth values in the columns under the first main connectives is T T, then it is also T T under the second.
Note that this is exactly the same as putting a ⟹⟹ between the two and making a WFF out of the pair of them.
■◼
Indirect Technique
The conventional technique of proving a tautology by means of a truth table can be unwieldy when there are many variables.
There is an indirect method of proving a tautology which is shorter.
It is recognised that the truth table for a tautology contains all T T down the column containing the main connective.
Hence it follows that if there exist any instances of F F in that column, the proposition under investigation is not a tautology.
Suppose we make the hypothesis that the proposition under investigation is not a tautology.
Then there exists an instance of F F under the main connective.
We construct the heading of the truth table in the same way as for the conventional technique: one column for each variable and one for each connective.
Under the main connective we place F F.
Examining the conditions under which the consequence is F F for that connective, we place truth values under the elements of the proposition which are in the scope of the main connective so as to allow F F.
Similarly we work backwards through each of the components of the proposition.
If it is impossible to complete the truth table row consistent with the behaviour of the connectives, then no F F can appear under the main connective.
Therefore the proposition under investigation is a tautology.
If it is possible to complete the truth table row consistently, then the proposition under investigation is not a tautology.
■◼
Notational Convenience
It is not actually necessary to include the truth values of the variables themselves (as we have done in the leftmost columns).
One is equally justified to write this:
((p F F T T⟹T T F T q)F T F T⟹F F T T p)F F T T⟹T T T T p F F T T((p⟹q)⟹p)⟹p F T F F F T F F T T F F T F T F F T T T T T T T T T T T
and it serves just as well.
However, it can help to clarify the derivation, as well as making the truth table easier to construct, if they are included. It's a matter of personal taste.
Also known as
The method of truth tables is also sometimes referred to as the method of matrices.
However, it can be argued that the term truth table is more specific than matrix and hence preferable.
Comment
Note that solution by truth table is valid only for Aristotelian logic, as it takes for granted the Law of Excluded Middle and the Principle of Non-Contradiction.
Within that context, it is a completely mechanical procedure and about as exciting as a strip-tease artist who starts the performance naked.
Sources
1946:Alfred Tarski: Introduction to Logic and to the Methodology of Deductive Sciences(2nd ed.)... (previous)... (next): §II.13§II.13: Symbolism of sentential calculus
1959:A.H. Bassonand D.J. O'Connor: Introduction to Symbolic Logic(3rd ed.)... (previous)... (next): §3.3§3.3: The Construction and Application of Truth-Tables
1964:Donald Kalishand Richard Montague: Logic: Techniques of Formal Reasoning... (previous)... (next): II II: 'AND', 'OR', 'IF AND ONLY IF': §6§6
1965:E.J. Lemmon: Beginning Logic... (previous)... (next): Chapter 2 2: The Propositional Calculus 2 2: 3 3 Truth-Tables
1973:Irving M. Copi: Symbolic Logic(4th ed.)... (previous)... (next): 2 2 Arguments Containing Compound Statements: 2.3 2.3: Argument Forms and Truth Tables
1980:D.J. O'Connorand Betty Powell: Elementary Logic... (previous)... (next): §I:4§I:4: Using the Definitions
1988:Alan G. Hamilton: Logic for Mathematicians(2nd ed.)... (previous)... (next): §1§1: Informal statement calculus: §1.2§1.2: Truth functions and truth tables
1996:H. Jerome Keislerand Joel Robbin: Mathematical Logic and Computability... (previous)... (next): §1.6§1.6: Truth Tables and Tautologies
2000:Michael R.A. Huthand Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems... (previous)... (next): §1.4.1§1.4.1: The meaning of logical connectives
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Genius Mind
Human Intelligence Tasks
Mental Computation
Cognitive Abilities
Math Proficiency
Mathematical Aptitude
5
How do geniuses multiply large numbers in their head?
Harald Overbeek
Loves mental calculation
·
Author has 185 answers and 626.6K answer views
·
9y
I cannot speak for other people. However; I can explain how I multiply large numbers in my head.
1: Learn the criss cross multiplication.
2: Start from left to right. This is counter intuitive, since you learned it the other way in school. However; if I see 3,123 X 4,321 I start by saying to myself three thousands times four thousands make 12 million.
3: Since you work from left to right you can stop if you have enough numbers. Sometimes you only need the 3-5 most significant digits. If you have them you stop.
4: Make a guess first. In the case of 3,123 X 4,321 you can see that it starts with 12
I cannot speak for other people. However; I can explain how I multiply large numbers in my head.
1: Learn the criss cross multiplication.
2: Start from left to right. This is counter intuitive, since you learned it the other way in school. However; if I see 3,123 X 4,321 I start by saying to myself three thousands times four thousands make 12 million.
3: Since you work from left to right you can stop if you have enough numbers. Sometimes you only need the 3-5 most significant digits. If you have them you stop.
4: Make a guess first. In the case of 3,123 X 4,321 you can see that it starts with 12 million and then in the next step we get 13 hundred thousands. So around 13.5 million is a good guess. The guess puts your brain in the ball park where the answer is.
5: In order to keep track of where you are in the calculation, do this. Again, my example is 3,123 X 4,321. It starts with 12 million. After the millions come the hundred thousands. How do we calculate them? Well, hundred thousand has 5 zeroes. So the thousands (three zeroes) from the first number multiplied by the hundreds (two zeros) give me this and the hundreds from the first number multiplied by the thousands from the second number give me this. Tere are no other combinations I can make. So the 3 from the first number multiplied with the 3 from the second number added to the 1 from the first number multiplied by the 4 from the second. This makes 13 (hundred thousands). My subtotal is now 13 million 3 hundred thousand.
There are 3 combinations that make up the ten thousands: 3-1, 2-2, 3-1.
There are 4 combinations that make up the thousands: 3-0, 2-1, 1-2, 0-3.
There are 3 combinations that make up the hundreds: 2-0, 1-1, 0-2.
There are 2 combinations that make up the tens: 1-0, 0-1.
There is 1 combination that make up the ones: 0-0.
Effectively this is a way of circumventing all the subtotals (with the huge amount of zeroes) that you have in the long multiplication you learned in school.
Related questions
How do some people do multiplication with very large numbers quickly in their head?
How do you multiply large numbers together in your head?
Is there a correlation between multiplying big numbers in your head and IQ?
How can I multiply numbers really fast with my mind?
How can people multiply 4-digit numbers in their heads?
Tom Groves
Painter, mnemonist, reader, writer, ex HF partner, Sox fan
·
Author has 1.4K answers and 13.2M answer views
·
9y
I think there are a few ways to answer this.
First of all, most geniuses multiply large numbers the same way everyone else does, with a calculator.
But specifically geniuses who focus on calculations use a number of different techniques. Cross multiplying, as Harald Overbeek mentions, is actually not that difficult and non-geniuses like myself can use that technique to multiply large numbers reasonably quickly - I used to be able to do 8x8 multiplications this way without too much difficulty, albeit not very quickly. Essentially you break the entire problem down from one large multiplication t
I think there are a few ways to answer this.
First of all, most geniuses multiply large numbers the same way everyone else does, with a calculator.
But specifically geniuses who focus on calculations use a number of different techniques. Cross multiplying, as Harald Overbeek mentions, is actually not that difficult and non-geniuses like myself can use that technique to multiply large numbers reasonably quickly - I used to be able to do 8x8 multiplications this way without too much difficulty, albeit not very quickly. Essentially you break the entire problem down from one large multiplication to a bunch of trivial additions. It really works and it's a very simple method to learn!
Art Benjamin is very good at this kind of thing, and I suggest you watch his stuff. Here he explains how he squares a 5 digit number (a mixture of (a lot of) practice and mnemonics and breaking down the problem in an efficient way) (starting at just past the 11 min mark in case the embedding doesn't work).
I also highly recommend The Great Mental Calculators by Steven Smith if you can find a copy at a reasonable price. It discusses in detail how some of the greatest calculators have done some absolutely incredible things. People like Alex Aitken, who really was a genius calculator. This is how he would do 123 x 456
I do this in two moves: I see at once that 123 times 45 is 5535 and that 123 times 6 is 738; I hardly have to think. Then 55350 plus 738 gives 56088. Even at the moment of registering 56088, I have checked it by dividing by 8, so 7011, and this by 9, 779. I recognize 779 as 41 by 19; and 41 by 3 is 123, 19 by 24 is 456. A check you see; and it passes by in about one second. (Hunter 1978:341)
More details on Aitken here
Steven McGlinn
Former Selective High School Teacher
·
Author has 5K answers and 5.4M answer views
·
7y
The tricks below work with 3 to 5 digits but 8 digits or more especially when dividing or raising to powers of multiple digits, it won't work.
I have seen a 17th root of an 8 digit number written out to 6 digits quicker than it could be determined on a calculator. The trick is to memorise 8 digit log tables. Students with supreme memories are able to do this. I have had two students who could do this.
I am a maths teacher and I know many teachers who have never seen it. But these are elites, best of the best in a selective high school. With calculators these days few still try to master but was
The tricks below work with 3 to 5 digits but 8 digits or more especially when dividing or raising to powers of multiple digits, it won't work.
I have seen a 17th root of an 8 digit number written out to 6 digits quicker than it could be determined on a calculator. The trick is to memorise 8 digit log tables. Students with supreme memories are able to do this. I have had two students who could do this.
I am a maths teacher and I know many teachers who have never seen it. But these are elites, best of the best in a selective high school. With calculators these days few still try to master but was common half a century ago when calculators were still a pipe dream and computers still have valves!
David Frigault
Novelist at Self-Employment (2016–present)
·
Author has 1.6K answers and 19.7M answer views
·
2y
Related
How do some people do multiplication with very large numbers quickly in their head?
What most people who have never engaged in mental arithmetic do not understand, is that good mathematicians do not calculate high numbers — or at least, I don’t.
If somebody was to ask me what 4627 x 3908 is, would I calculate these two numbers at once and come up with the sum, or would I find a way to simplify it?
For myself, it is always the latter.
Just like a cartoonist is going to look at an image in steps, the same applies when doing mathematics without a paper or calculator.
During my high school days I invented my own method for calculating such questions by doing the following:
46 x 39 = 1
What most people who have never engaged in mental arithmetic do not understand, is that good mathematicians do not calculate high numbers — or at least, I don’t.
If somebody was to ask me what 4627 x 3908 is, would I calculate these two numbers at once and come up with the sum, or would I find a way to simplify it?
For myself, it is always the latter.
Just like a cartoonist is going to look at an image in steps, the same applies when doing mathematics without a paper or calculator.
During my high school days I invented my own method for calculating such questions by doing the following:
46 x 39 = 1,794
Add four zeroes — representing the numbers not used in step one
27 x 39 = 1,053
Add two zeroes — representing the two smaller numbers not used in step three
46 x 8 = 368
Add two zeroes — representing the two smaller numbers not used in step five
27 x 8 = 216
17,940,000 + 105,300 + 36,800 + 216 = 18,082,316.
What if I wanted to calculate 6543 x 4517?
65 x 45 = 2,925 + four zeroes = 29,250,000
43 x 45 = 1,935 + two zeroes = 193,500
65 x 17 = 1,105 + two zeroes = 110,500
43 x 17 = 731
ANSWER
29,554,731
I hope this helps.
David Frigault's answer to How do you do multiplication in your head?
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Related questions
What qualifies a person as being a genius?
Is there a way to mentally multiply large numbers under no conditions?
How did Ryan Fitzpatrick's son multiply two large numbers so fast?
Can you multiply random numbers in your head?
What are some tricks for multiplying large numbers in your head?
Assistant
Bot
·
1y
Geniuses often use a combination of techniques to multiply large numbers in their head efficiently. Here are some common strategies:
Breakdown Method (Distributive Property)
: This involves breaking down numbers into smaller, more manageable parts. For example, to multiply 23 by 47:
Break 23 into 20 and 3.
Calculate
and
.
Add the results:
.
Using Patterns and Properties
:
Recognizing patterns in numbers can simplify calculations. For example, multiplying by 9 can be done by multiplying by 10 and subtracting the original number
Geniuses often use a combination of techniques to multiply large numbers in their head efficiently. Here are some common strategies:
Breakdown Method (Distributive Property)
: This involves breaking down numbers into smaller, more manageable parts. For example, to multiply 23 by 47:
Break 23 into 20 and 3.
Calculate
and
.
Add the results:
.
Using Patterns and Properties
:
Recognizing patterns in numbers can simplify calculations. For example, multiplying by 9 can be done by multiplying by 10 and subtracting the original number:
.
Vedic Math Techniques
: Some people use techniques from Vedic mathematics, which include specific patterns and shortcuts for multiplication, such as the "vertically and crosswise" method.
Memorization of Squares and Cubes
: Many mentally adept individuals memorize squares and cubes of numbers, which can help in quickly estimating products.
Estimation
: Geniuses often use estimation to get close to the answer quickly, rounding numbers to simpler figures before refining the answer.
Practice and Mental Training
: Regular practice with mental math improves speed and accuracy. Many experts engage in mental math exercises to enhance their skills.
Visualization
: Some people visualize numbers and their relationships in a spatial manner, allowing them to manipulate and combine them more easily.
By combining these techniques, individuals can effectively multiply large numbers in their heads with speed and accuracy.
Kyle Certiorari
Practical Perfectionist
·
Author has 97 answers and 195.6K answer views
·
11y
Related
How do you multiply large numbers together in your head?
Have the power to add like a computer. When I say this, I mean that you must be able to do additions like 72937620 + 38296000000 in your head within less than 5 seconds.
Know the 0-9 multiplication table better than you know the ABCs.
However, these are both relatively easy for anyone to master. Most fourth graders know the time table and most people in high school can add.
The problem is literally holding the numbers in your head while you manipulate them. You have to do 127 (64 multiplication and 63 addition) to multiply two 8 digit numbers. I can personally do this amount of problems extr
Have the power to add like a computer. When I say this, I mean that you must be able to do additions like 72937620 + 38296000000 in your head within less than 5 seconds.
Know the 0-9 multiplication table better than you know the ABCs.
However, these are both relatively easy for anyone to master. Most fourth graders know the time table and most people in high school can add.
The problem is literally holding the numbers in your head while you manipulate them. You have to do 127 (64 multiplication and 63 addition) to multiply two 8 digit numbers. I can personally do this amount of problems extremely quickly with 100 percent accuracy. The problem is that you go through a few hundred numbers in your head and you must always remember every single one of them (until you are no longer using it) including the 16 that you're multiplying together. So come up with a visual way to see the numbers (some people use flashes of colors) and you could easily do this for two 3 digit numbers. Multiplying two 8 digit numbers is one of those things that only I seemed to do when I was in middle school (I don't blame other people for that. But I could get a little pissed off at these grown, capable adolescents when they literally entered 9-12 on their calculator.). Here is a hint. It looks like 12-9 but is multiplied by negative one.
Here's a better idea. Learn critical thinking and problem solving.
Learn to always make sure that you first observe all that you need to. No one can solve a problem when they have no idea what the problem is! Simply make sure that you have read or defined the problem clearly and know what the surface level details are.
Learn to brainstorm. What would solve the problem? What are you after? How would you even know that the problem is finished? What's the goal here? What resources are their? What questions have I not asked yet?
Learn to have good judgement and to double check things. This is the idea of calculation. You know that the goal is to do something, but what is the best way to do so? What are the pros and cons of all your choices? What will work and what will not? Simply come up with the best way to meet your goal.
Develop skills and book smarts. Find out what stuff your future job needs and then go after that for a month. Look up a college. Click on undergraduate degrees or whatever. Look for your job or degree. If you notice stuff like needing calculus for things like engineering and physics, I suggest Khan Academy.com or some free online calc courses from MIT (didn't try the mit yet. The khan acad had 135, 10 minute videos on calculus concepts. All of them completely different. Work on that if you like physics, math, or engineering!).
Yogesh Chandna Dr
Author (2019–present)
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Author has 597 answers and 1.3M answer views
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5y
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How do you mentally multiply big numbers?
In Vedic Mathematics it is possible.
We can use Urthva Sutra to solve a four digit numbers or five numbers.
In specific cases we can directly solve., let us consider multiplication of 3/4 digit number with 11
Vedic mathematics, simplifies arithmetic and algebraic operations, and has increasingly found acceptance the world over.
Vedic Mathematics helps find solutions in a single or a few steps where as conventional methods require multiple steps.
Vedic Mathematics eases and simplifies multiplication, divisibility, complex numbers, squaring, cubing, square and cube roots. Even recurring decimals and
In Vedic Mathematics it is possible.
We can use Urthva Sutra to solve a four digit numbers or five numbers.
In specific cases we can directly solve., let us consider multiplication of 3/4 digit number with 11
Vedic mathematics, simplifies arithmetic and algebraic operations, and has increasingly found acceptance the world over.
Vedic Mathematics helps find solutions in a single or a few steps where as conventional methods require multiple steps.
Vedic Mathematics eases and simplifies multiplication, divisibility, complex numbers, squaring, cubing, square and cube roots. Even recurring decimals and auxiliary fractions can be handled by Vedic Mathematics.
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Kyle Seipp
Studied at California Institute of Technology (Caltech)
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11y
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How do you multiply large numbers together in your head?
Normal person:I don't. I use a calculator.
Mathematician:It depends on what Field I'm working in. What's a number?
Physicist: Ehh use the nearest power of ten.
Computer Scientist: Ehh use the power of two
Computer Programmer: Well, if I write mumbles this algorithm is faster starts typing on phone
Engineer: Guess.
Philosopher: But what do numbers "mean?"
Why do you want this skill? Sure it's a neat party trick, but unless you are finding a dearth of batteries in your area, you shouldn't have too much issue with the calculator.
If you mean truly large numbers (like Grahm's number) then you
Normal person:I don't. I use a calculator.
Mathematician:It depends on what Field I'm working in. What's a number?
Physicist: Ehh use the nearest power of ten.
Computer Scientist: Ehh use the power of two
Computer Programmer: Well, if I write mumbles this algorithm is faster starts typing on phone
Engineer: Guess.
Philosopher: But what do numbers "mean?"
Why do you want this skill? Sure it's a neat party trick, but unless you are finding a dearth of batteries in your area, you shouldn't have too much issue with the calculator.
If you mean truly large numbers (like Grahm's number) then you probably want to use more abstract techniques like, have you considered logarithms?
Alex Eustis
Ph.D. in Mathematics, University of California, San Diego (Graduated 2013)
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Author has 4.6K answers and 23.8M answer views
·
6y
Related
How difficult is it to mentally multiply two ten digit numbers?
Well, I learned something today. Apparently the world record holder for multiplying two 20-digit numbers mentally without error is Granth Thakkar of India, with a time of 2 minutes 48 seconds.
Memoriad
Others listed on that page were able to do it less quickly than that.
However we're talking world records here. The answer to your question of how difficult it is to mentally multiply two 10-digit numbers, is still: extremely difficult. This is something you might never be able to learn even after 20,000 hours of dedicated practice. You’d probably have an easier time learning to juggle 6–7 balls.
In
Well, I learned something today. Apparently the world record holder for multiplying two 20-digit numbers mentally without error is Granth Thakkar of India, with a time of 2 minutes 48 seconds.
Memoriad
Others listed on that page were able to do it less quickly than that.
However we're talking world records here. The answer to your question of how difficult it is to mentally multiply two 10-digit numbers, is still: extremely difficult. This is something you might never be able to learn even after 20,000 hours of dedicated practice. You’d probably have an easier time learning to juggle 6–7 balls.
In college my thesis advisor actually performed a show in Las Vegas where he would impress the audience with mental arithmetic and other forms of “Mathemagics.” He had a special system for converting the intermediate calculations into “words” so that he could store and retrieve them accurately in memory. I've never met anyone before or since who came remotely close to that skill level of mental arithmetic.
Still, on a good day, I think he could multiply two five digit numbers in his head after a very long pause.
He was an actual mathematician, not a performance artist. So, perhaps not the best of the best of the best, but you should consider yourself lucky if you can even learn to multiply three or four digits.
You'll need some kind of system like his, of course. I can't remember what it was exactly, but there was a mapping of numbers to sounds, something like
0 -> T
1 -> S
2 - > D
3 -> K
etc., and so if he needed to store 3231 (say), he'd remember some “word” that used those sounds, like “Cadacus”. For more information I'd start with the website linked at the top and try to learn what some of their tips and tricks are.
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Joshua P. Hill
EE, Worked in HDTV Research
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Author has 24.6K answers and 98.7M answer views
·
6y
Related
Is there a correlation between multiplying big numbers in your head and IQ?
A correlation, yes, because as Bruno Campello de Souza points out, the ability to do this is correlated with working memory, which is one of the components measured by an IQ test. Quite simply, people with higher IQ’s can hold more digits in their head, allowing them to remember longer numbers and more intermediate results.
However, it has to be remembered that crystallized intelligence plays an important role here as well — there are a number of tricks that facilitate mental arithmetic, and these can be learned or intuited, increasing one’s ability to perform such calculations. It’s also true
A correlation, yes, because as Bruno Campello de Souza points out, the ability to do this is correlated with working memory, which is one of the components measured by an IQ test. Quite simply, people with higher IQ’s can hold more digits in their head, allowing them to remember longer numbers and more intermediate results.
However, it has to be remembered that crystallized intelligence plays an important role here as well — there are a number of tricks that facilitate mental arithmetic, and these can be learned or intuited, increasing one’s ability to perform such calculations. It’s also true that working memory is only one contributor to one’s overall IQ score, so two people of equal general intelligence might differ substantially in their native ability to do this.
It’s a statistical relationship rather than something that allows one to asses, in and of itself, someone’s IQ.
Really. Prof. de Souza has done such a good job of answer this that I wouldn’t bother if not for the A2A and to draw attention to his answer.
David Frigault
Writer
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Author has 1.6K answers and 19.7M answer views
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5y
Related
How do you multiply large numbers together in your head?
Here is an answer I wrote a couple of years back that should answer your question:
Since I do not know how good you are at multiplying off the top of your head, and because I am also not so certain as to the context of your question — that you cannot do multiplication at all, or that you have difficulty memorising the steps in your head and crunching the numbers together: I will start at the single digit number, which is where most learners of multiplication will start — and knowing what 2 x 2 is has been one of the most common examples of a first time learner.
2 x 2 is the only number in multip
Here is an answer I wrote a couple of years back that should answer your question:
Since I do not know how good you are at multiplying off the top of your head, and because I am also not so certain as to the context of your question — that you cannot do multiplication at all, or that you have difficulty memorising the steps in your head and crunching the numbers together: I will start at the single digit number, which is where most learners of multiplication will start — and knowing what 2 x 2 is has been one of the most common examples of a first time learner.
2 x 2 is the only number in multiplication that has the same answer in addition. Knowing the first four digits of multiplication, you can start off with a 2, then add a 1 to get 3 and another 1 to get 4.
Now, let’s get to a harder level — and believe me, things are going to get far more advanced as this response continues — by trying two single digits.
Let’s do the times table starting at 4 — the lowest number you can get that results in a two digit number while adding the same number — and work our way up to 9.
You can do 4 x 4 is 16. But if you want to know how to do the harder numbers later on, I recommend that what I am about to show next is remembered first, as it will save you a lot of grief once we start multiplying 4 or 5 digit numbers.
As we had discussed earlier, 2 x 2 is 4. How do we get 16? Well, if we were to double one of the 2’s and change the number to 4, we would get 8. Now, times the same number by 2 again when upgrading the other 2 to a 4 and you get 16. It may seem straightforward and easy right now — but if you do not know how to multiple 4 or 5 digit numbers, knowing the tricks and practicing them at the earliest stage will help you master the harder numbers later on.
4 x 4 is 16
5 x 5 is 25
6 x 6 is 36
7 x 7 is 49
8 x 8 is 64
9 x 9 is 81
We have completed our first segment of this tutorial. I am sure most of you reading this remembered how to do this simple mathematics in elementary school — but I did notice there were some tricks in this multiplication which none of my teachers pointed out — presumably because they did not know the trick themselves, which can be applied to any multiplication question, regardless of the number of digits involved. In fact, the number does not even need to start at 2 x 2 but can even start at 1 x 1. Allow me to demonstrate this trick.
Say you are a total novice at multiplication. You want to know how to figure out what 2 x 2 is. A good way to do this is to know what 1 x 1 is. If you know that the answer to 1 x 1 is 1, all you need to do is add an additional 1 for each 1 in the question, and then add a third 1 on top of the other two 1’s. Knowing the answer of 1 x 1 is 1, simply go 1+1+1+1 and get 4. Sounds easy? Now let’s figure out how we would figure out what 3 x 3 is by doing 2 x 2.
The answer is the same as the first question. Knowing that 2 x 2 is 4, one need only add both 2’s and then add 1. 4+2+2+1 = 9.
If we want to know what the answer to 4 x 4 is, the answer is the same. 3 x 3 =9. 9+3+3+1 = 16.
If we want to try a much harder number the same rule would apply. 15 x 15 is 225. How do I know? Knowing that 5 x 5 is 25, I could easily triple one of the 5’s to 15 and get 75, and then triple the other 5 to 15 and get 225. If I wanted to be able to figure out 16 x 16 by using the method we used before, as opposed to starting from an earlier step, I simply need to do 225+15+15+1 and get 256.
And here is a cooler trick. Say I don’t want to know 16 x 16. Suppose I want to know what 20 x 20 is right away without needing to figure out the steps in between — I could apply the same trick, but with an extra trick that can help save you time. Because numbers go up by a margin of 2 numbers higher when ever one multiples two identical numbers — and knowing that it took a number increase of 31 to go from 225 to 256 — I would know that it takes 33 numbers in addition to 256 to know that 17 x 17 is 289.
If there are 5 numbers leading from 15 to 20, simply repeat the process 5 times over. 31+33+35+37+39 = 175 + 225 = 400. We now know that 20 x 20 is 400. Neat, eh?
If you have remembered the earlier steps, let’s move to the next level — mastering 2 digit multiplication.
If we had to multiply two digits that were not identical — such as 17 x 39 — how would we calculate this?
Assuming you have never gone this far in doing multiplication in your head, let’s break the answer into smaller numbers, first. If we rounded 39 to 40 and then divided the 40 by 10, we would have 4. If we rounded up to 17 to 20 and then we added the 4 to multiply, we would have 80. Since 20 - 17= 3, you can simply do 3 x 4= 12. Subtracting 80 from 12, we would have 68.
Since we used the two numbers in figuring out 17 and only one number in figuring out 39, you need to only add one zero to the 68 and get 680. Subtracting 17 from 680 will get you 663. Once you become more skilled later on in your quest to do multiplication in your head you will be able to simply to 17 x 40 -17 and get the same answer. But for those who may need to practice using smaller breakdowns — this approach cannot hurt, either.
Here is a different question, and this one does involve using two identical numbers: How would I figure out 36 x 36? We could attempt to solve this question by using a similar method as the one demonstrated above — but I think there is an easier solution which I would like to show.
We know that 6 x 6 is 36, right? The numbers above are both a match and we know that doing 6 by 6 again would give 36 for the second number in the question. By doing what is called stacking the numbers, you could do 36 x 6 and get 216. How do we know it is 216? If you know that 6 x 30 will leader to 180, you will have 6 left out of 36. If you times the 6 by doing 6 x 6 and then add 180, you get 216.
Once you know the answer of 6 x 36 is 216, you could simply remove the 200 if you are less experienced and get a flat 1200 as the answer to the next step in solving the question. With 16 left out of 216 while multiplying by 6, you can use 16 times 6 and get 96. Add 1200 + 96 and get 1296.
Congratulations! You have just figured out your first four digit answer. Don’t get too excited, however, as we still have 5, 6, 7 and even 8+ digit answers to go. Each additional digit will be several times harder and more time consuming than the last. If you think going any higher might be too distressing at this moment, take a breather and practice with the easier steps before coming back to the harder problem solving. Do not take too long to come back, however, since you might eventually lose your confidence in going any higher up if you get too comfortable at where you are at.
Three Digit Multiplication
Welcome back! If you had decided to take a break, I am glad you have chosen to return. If you felt confident enough in yourself to continue on without stopping, then all the best for you!
Since 3 digit multiplication is something which even many mathematics professors struggle with doing at the top of their head, I will not start off with a question that is too overwhelming. I will, however, state that if you did not learn the steps from the previous lesson that you should do so before trying to tackle three digit multiplications.
How would we know what the answer to 100 x 100 is? Some of you might know that 100 x 100 is 10,000 by simply memorising the answer. But how would you prove it?
There are several ways in which it is possible to know what 100 x 100 is. If you know that 1 x 1 is 1, and you know that 100 has two 0’s — the most effective approach would be to take the four 0’s from the two 100’s and put those four to the right of the 1. 1+0+0+0+0 and the crunched together makes 10,000.
If you happened to already know what 99 x 99 is (if you want to learn this trick, please refer to previous lesson) you could also prove that 100 x 100 is 10,000 by knowing what 99 x 99 is and applying the common steps. 99 x 99 is 9801. Simply do 9801 + 99 +99 + 1 and you will receive the number 10,000 as your answer.
Let’s try 110 x 110. How would we solve this? If we know that 10 x 10 is 100, one could figure out what 11 x 11 is by doing 100 + 10 + 10 + 1 just as I have been using since the outset of this course. If we add 100 + 10 + 10 + 1 we get 121. Since we used both 11’s in the two 110’s in the question, only the two 0’s need to be add. 121 + 0 + 0 crunched together comes with an answer of 12,100.
What if one wanted to do 160 x 160? If we go back to doing basic multiplication, we know that 4 x 4 is 16. Any number that is a multiple of an identical number can be measured this way. If we were to times the additional 16 by 4, we would have 64. And then, quadrupling the 64 a second time, we would have 160. Using the two 0’s that have not been used, and crunching 256+0+0 together gives us a total of 25,600.
Now that we have the basics of the three digit multiplication problem solving skills written down, we must try and make the questions a bit more complicated.
How would we do 173 x 264? Go back to your two digit multiplication questions. If we are able to solve 17 x 26 is 442 by doing 17 x 2 and getting 34 and then add a 0 to get 340 — we can then solve the first step by then multiplying 17 x 6 and getting 102 and then adding 102 + 340 to get 442. We need to then add the two 0’s and crunch them together. 442 + 0 + 0 is 44,200
Our answer is not yet solve, however. And one way to know that is by looking at the two numbers furthest to the right — 3 and 4 — we can know from knowing 3 x 4 is 12 that the last number of 44,200 will have to be a 2.
To solve this question, we now need to use an additional step which two digit multiplication does not require. Take a look at the 3, and then look at the number 264. To make the problem solving a bit more convenient, put a 0 on the 4 and multiply 3 by 260. The answer is 780. Now look at the 4 and the number 173. Like the 4 on 264, remove the 3 and replace it with a 0 and get 170. Multiplying 4 by 170, you will get 680 as your answer — assuming you got the answer correctly.
Adding 680 + 780, you get 1,460. Look back to 44,200 and add the 1,460 to it by using simple additions. Since there are five digits in 44,200 and only four in 1,460, the number 1 in 1,460 will be added to the second four in 44,200.
44,200 + 1,460 = 45,660. All you need to do now is know the obvious — that 4 x 3 is 12 — and then you just have to add the 12 to the number 45,660 and get 45,672.
Let’s try a second question and see how we would solve this problem.
596 x 498 — How do we solve this?
If you remember our previous question, simply do 59 x 49. If you want to make it easier on yourself, you also have the option to round the numbers up at any time it feels convenient for you to do so.
If I wanted to round 59 to 60 and times it by 49, I could solve 49 times 60 in two ways. I could convert the 9 to a 0 and reduce the number to 40 — giving me 2400 as an answer and then doing 9 times 60 to get 540 and then add the 2,400 to get 2,940. If you subtract 49, you get 2,891.
Since 49 is close to 50, however, in this case you would probably be better off rounding the 49 to a 50 instead of a 40. If 60 x 50 is 3,000, we could subtract the 3,000 by 60 and get 2,940. If you subtract 49, you get 2,891.
Both solutions come up with the same answer. The same steps can be taken if we wanted to times 49 by 59.
49 x 50 is 2,450. And 49 x 60 is 3,140.
If we used 49 x 9 we could figure out the answer is 441 by knowing that 9 x 4 is 36, which in turn requires an additional 0, due to the 9 in 49 not being used to solve 9 x 4. If we then used the 9 from 49, as well as the single 9, and did 9 x 9, we would know the answer to be 81. Add 360 + 81 and you get 441.
Add 2,450 + 441 and you get 2,891 — the same as what you would get if you decided to use the 59 instead of the 49 for solving the first step.
Since two numbers have yet to be used — the 6 in 596 as well as the 8 in 498 — we can crunch the number of 2,891 with the other two 0’s by doing 2,891 + 0 + 0 and get 289,100.
The problem is not yet solved, however. If you remember the last question we solved, look at the 6 and then look at 498 and round it off to 490. Simply do 6 x 400 to get 2,400. Then do 6 x 90 to get 540. You will have 2,940 as your answer.
Now, let’s take a look at 8 and look at the 596. Round the 596 to 590. 59 x 8 is 472. Add a 0 and get 4,720. 2,940 + 4,720 is 7,660. 8 x 6 is 48 + 7,660 = 7,708 + 289,100 = 296,808.
The answer is now solved. I advice you not be tired, stressed, distracted or hungry when doing the higher problems. No matter how experienced you are at doing high multiplications in your head, I recommend you be fresh and alert and not be distracted by other things while practicing, or even being tested by others.
Four Digit Multiplication
Here we come to four digit multiplication. If you have been following the early steps, you will do far better when trying to learn numbers ranging anywhere from a million up to 99,980,001 (the highest number you can get in four digit multiplication.)
What do we do if we want to figure out what 5476 x 2152 is?
The steps are the same as three digit multiplications. Except that rather than putting two 0’s behind the first step, you would add four 0’s. Multiply 54 x 21. The answer is 1134. We can figure this out by simply rounding 20 x 54 to get 1080 and then add 54; or we can also divide the 21 into 7 and get 378 and times that by 3 if you are less experienced and need to break down the numbers further. As stated above, you will be able to figure these higher numbers without the need to break down once you are able to naturally calculate them as easily in your sleep as when you are awake.
Once we get 1134, we add four 0’s. 1134+0+0+0+0 crunched together makes 11,340,000.
Now, take the 76 from 5476 and times the 21 from 2152 and get 1596. In this step you are adding only two more 0’s (if you forget this step, just remember that the numbers behind those being calculated in the first step are to be converted into zeroes).
Now we get 159,600 + 0 + 0, crunched together and make 159,600.
Do the same for 54 in 5476 and 52 in 2152. 54 x 52 is 2808. Use the two 0’s (in this case, the 76 from 5476) and convert them and crunch them. 2808 + 0 + 0 is 280,800.
Last multiplication step is 76 and 52. The answer is 3,952.
Now add all your numbers up.
11,340,000 + 159,600 + 280,800 + 3,952 = 11,784,352
Let’s try a second four digit multiplication question.
9476 x 6593 How do we solve it?
94 x 65 = 6110 + 0 + 0 + 0 +0 = 61,100,000
76 x 65 = 4940 + 0 + 0 = 494,000
94 x 93 = 8,742 + 0 + 0 = 874,200
93 x 76 = 7,068
61, 100, 000 + 494,000 + 874,200 + 7,068 = 62,475,268
Let’s try another
7828 x 5913
78 x 59 = 4,602 + 0 + 0 + 0 + 0 = 46,020,000
59 x 28 = 1,652 + 0 + 0 = 165,200
78 x 13 = 1,014 + 0 + 0 = 101,400
28 x 13 = 364
46,020,000 + 165,200 + 101,400 + 364 = 46,286,964
Let’s try once more
8564 x 9172
91 x 85 = 7,735 + 0 + 0 + 0 + 0 = 77,350,000
91 x 64 = 5,824 + 0 + 0 = 582,400
85 x 72 = 6,120 + 0 + 0 = 612,000
72 x 64 4,608
77,350,000 + 582,400 + 612,000 + 4,608 = 78,549,008
Again
9475 x 3420
94 x 34 = 3,196 + 0 + 0 + 0 + 0 = 31,960,000
94 x 20 = 1,880 + 0 + 0 = 188,000
75 x 34 = 2,550 + 0 + 0 = 255,000
75 x 20 = 1,500
31,960,000 + 188,000 + 255,000 + 1,500 = 32,404,500
Note: Numbers which end in zero are typically easier to solve than numbers that don’t
Let’s try one more
8236 x 6458
82 x 64 = 5,248 + 0 + 0 + 0 + 0 = 52,480,000
64 x 36 = 2,304 + 0 + 0 = 230,400
82 x 58 = 4,756 + 0 + 0 = 475,600
58 x 36 = 2,088
52,480,000 + 230,400 + 475,600 + 2,088 = 53,188,088
If you keep practicing you will get better at this. When you are new and/or have not done much multiplication recently, try not to overwhelm yourself by attempting to solve too many questions at one time. You will be much more accurate and much quicker in the long run if you also make sure you get the answers right by taking your time and not coming back with an answer unless you are sure that answer is correct. Odds are nobody is watching you, so you can only hurt yourself in the long run if you do not wish to heed this warning.
Five Digit Multiplication
This is the level where it gets so difficult that even those who might be good at four digit multiplication either struggle or are incapable of doing it altogether. At five digits, the numbers are far bigger — meaning that even one wrong digit is a wrong answer.
The good news, however, is that by this time there is not much else to learn that you have not in previous steps, as far as problem solving goes. There is only one small difference which you will probably get used to later on.
How would I solve 53876 x 45918?
Look at the three digits furthest to the left in both numbers. We will work on 538 and 459. But first, let’s look at 53 and 45 before look at the 8 and 9.
53 x 45 = 2385
Convert the 459 to 450 and times it by 8 to get 3,600
Convert the 538 to 530 and times it by 9 to get 4,770
Now do 9 x 8 and get 72
Add them together
2385 + 0 + 0 + = 238,500 + 3,600 + 4,770 + 72 = 246,942
Because the 76 and 18 have not been used, they must be converted into 0’s
2,469,420,000
459 x 76 = 34,884
Because the 18 was not used in this step, convert it into two 0’s
34,884 + 0 + 0 = 3,488,400
538 x 18 = 9,684 Convert the 76 into two 0’s
9,684 + 0 + 0 = 968,400
76 x 18 = 1,368
2,469,420,000 + 3,488,400 + 968,400 + 1,368 = 2,473,878,168
Let’ try another
75492 x 53761
754 x 537 = 404,898
404,898 + 0 + 0 + 0 + 0 = 4,048,980,000
754 x 61 = 45,994 + 0 + 0 = 4,599,400
537 x 92 = 49,404 + 0 + 0 = 4,940,400
92 x 61 = 5,612
4,048,980,000 + 4,599,400 +4,940,400 +5,612 = 4,058,525,412
Let’s try again
67913 x 58952
679 x 589 = 399,931 + 0 + 0 + 0 + 0 = 3,999,310,000
589 x 13 = 7,657 + 0 + 0 = 765,700
679 x 52 = 35,308 + 0 + 0 = 3,530,800
52 x 13 = 676
3,999,310,000 + 765,700 + 3,530,800 + 676 = 4,004,607,176
Note: It is easier to add the digits furthest to the left and work your way to the right
71456 x 94582
714 x 945 = 674,730
The way this could be done, was by first multiplying 71 x 94 and getting 6,674 and then adding two extra 0’s behind the 4 and getting 667,400. Then, once you complete that step, simply do 5 times 700 and add the extra 3,500 to the 667,400 and get 670,900 before doing 5 x 10 and adding the extra 50 and making the number 670,950. Afterwards, you want to take the 4 from 714 and do 4 x 900 and get 3,600 and add that to the 670,950 to get 674,550 and then do 4 x 40 to get 160 and add that to the 674,550 and get 674,710. The last part of this step is to do 4 x 5 to add 20 to the 674,710 and get 674,730.
Since the 56 in 71,456 was never used and the 82 in 945 was also never used, you will want to add 4 extra 0’s to the 674,730 to get 6,747,300,000.
Now, what you do is break down the 56 if you want. For example, an easier way to multiply 56 times 945 is to do 945 times 7 and get 6,615 and times it by 8 to get 52,920.
714 x 82 is 58,548 and this can be solves by doing 7 x 82 is 574 and then adding two 0’s to get 57,400 before then doing 10 x 82 and getting 820 and adding it to the 57,400 to get 58,220. The last part then requires 82 x 4, which is 58,548.
52,920 + 58,548 is 111,468 + two 0’s makes 11,146,800
56 x 82 = 4,592
6,747,300,000 + 11,146,800 + 4,592 = 6,758,451,392
And that’s the answer
Six Digit Multiplication
Let’s go to the final level of this process: doing 6 digits by 6 digits
The steps are simple and straightforward enough and are almost identical to the previous steps. In order to solve six digit multiplication more efficiently and decrease the number of times you make a mistake while calculating, you want to know how to break down your steps into simpler chunks, as opposed to treating two or more high numbers as six digits.
745234 x 659617 = 491,569,015,378
Needless to say, it takes far more brainpower to do six digit multiplication, and unless you are experienced and are practicing consistently, even veteran players of this type of challenge are going to have a difficult time getting the answer right.
While you are doing all the additional steps, you are forced to also memorise what the answers for the earlier steps were for when you add them all together. To make matters more difficult, if you mess up even a single number in any of the following steps, you could end up coming up with a skewed answer, or worse, an answer where all the numbers are correct, save for one.
Depending on your comfort and preference, you can do any of the steps in any order you feel like.
Step 1: 745 x 659 = 490,955.
This can be solved by doing 74 x 65 which is 4,810. You then add two zeroes and convert it to 481,000.
Then you do 5 x 650 which is 3,250 and add it to the 481,000.
Then you take the 9 and times it by 740, which is 6,660.
Lastly, you do 5 x 9, which is 45 and add them all together. 481,000 + 3,250 + 6,660 + 45 = 490,955. You then add six zeroes and you get 490,955,000,000.
Step 2: 659 x 234 = 154,206.
You then add three zeroes and you get 154,206,000.
You can solve this answer by doing 65 x 23 and getting 1,495. Add two zeroes and you get 149,500.
Next, you want to add 9 times 230 to get 2,070.
Then, you want to add 4 times 650 to get 2,600.
Lastly, you do 9 x 4 and that is 36.
149,500 + 2,070 + 2,600 + 36 = 154,206.
Step 3: 745 x 617 = 459,665.
When doing this, you do the same thing as step two, and similarly as to step one. You take 74 x 61, and that is 4,514. Add two zeroes, and now you have 451,400.
Then, you do 5 times 610, and that is 3,050.
7 times 740 is 5,180.
7 times 5 is 35.
451,400 + 3,050 + 5,180 + 35 is 459, 665.
Lastly, add three zeroes and get 459,665,000.
Step 4:
617 x 234 = 144,378.
Since there are no numbers behind the 617 and the 234, there is no need to add zeroes. In this case, 144,378 is simply 144,378.
Now, the final step is adding them all together. If you made a single error at any point during the calculation process, the numbers are going to end up being wrong when it comes to adding them. Furthermore, make sure you don’t make any mistakes when adding, either.
490,955,000,000 + 459,665,000 + 154,206,000 + 144,378 = 491,569,015,378
FINAL NOTE:
When doing multiplications, make sure to take your time and not to rush, especially if you are a beginner, have not done this in a while, or if your mind is not clear and thus need more time to concentrate and process everything.
As a bonus before I depart and leave the aspiring readers to their devices, I will teach a trick on how multiplication can also help with certain percentage questions.
If you want to know, for example, what 34 percent of 21 is, for instance, you can do 34 times 21 and get 714. Divide that number by 100, and you get 7.14 and that should be the answer. The three numbers can also be used for any of the other three numbers present.
For instance, to know what 21 percent of 34 is, you could also use 7.14 and that would be the same answer.
This logic would be applicable to any number, regardless of how high the number is or what numbers are used.
For instance, 65 x 47 is 3,055. Divide that number by 100 and get 30.55.
To know what 47% of 65 is, you simply need to know that 65 times 47 is 30.55, and 30.55 would be the answer. In contrast, if you want to know 65% of 47, you would also only need to know that 65 times 47 is 30.55.
ENCORE BONUS:
For the more dedicated readers that have stuck with me the entire time, I present to you yet another math trick that you can learn, and this one is probably one of the easiest you could learn, yet it is probably also very helpful, especially when trying to figure out some of the more complicated arithmetic problems out there.
When you were in school, you probably learned the basic multiplication table. However, one thing that I noticed teachers never taught (possibly because it has never occurred to them) is that there is also a very easy way to figure out some fairly complicated math problems when both numbers are the same, especially if you already know the answer to the number below it.
For instance, 12 times 12 is 144.
However, if you wanted to know what 11 times 11 was, you could figure out the answer by doing 12+12–1 which is 23 and subtracting that number from 144 and get 121 as your answer.
In contrast, if you wanted to know what 13 times 13 was, you could figure it out by doing 12+12+1 and adding that to 144 and get 169 as your answer.
Even at the higher numbers, this trick can be applied.
If, for example, you knew that 213 x 213 is 45,369 and you wanted to know what 212 x 212 was, you simply need to do 213 + 213 -1 which is 425 and subtract that from 45,369 and get 44,944.
As well, if you want to know 214 times 214, simply do 213 + 213 + 1 and add that to the number 45,369 and get 45,796.
Even at 1000 x 1000 which is 1,000,000 you can figure out 999 x 999 by doing 1000 + 1000 - 1 which is 1,999 and subtracting that from 1,000,000 and getting 998,001 as your answer.
If you want to know 1001 x 1001, simply do 1000 + 1000 + 1 and get 2001 and add that to the 1,000,000 and get 1,002,001 as your answer.
I hope this helped. Remember to not go hard on yourselves if you mess up at any point. I do as well, especially when it comes to the higher numbers or when my mind is distracted. This was probably one of the hardest things I ever had to write, and it was more a result to the amount of frustration and brain processing needed to write this project, which unsurprisingly gave me quite a number of headaches.
Keep practicing, and eventually you will get better.
D. J. Hoskins
Lives in Washington, DC (2023–present)
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Author has 332 answers and 5.3M answer views
·
8y
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How does a genius person learn Math?
“One step at a time.”
Imagine math as a Rubiks cube.
It might look daunting at first, however you learn it step by step. Now what most people do is get shoved into learning higher mathematics than they have actually prepared themselves for. This is usually because of the public school system.
Now instead of starting with that evil looking puzzle, lets start with something more manageable.
Ah, much better.
As I was saying you learn math in steps. The goal is to understand the concepts of mathematics. Its not about getting it shoved down your throat and then quickly discarding it. When solving a rubi
“One step at a time.”
Imagine math as a Rubiks cube.
It might look daunting at first, however you learn it step by step. Now what most people do is get shoved into learning higher mathematics than they have actually prepared themselves for. This is usually because of the public school system.
Now instead of starting with that evil looking puzzle, lets start with something more manageable.
Ah, much better.
As I was saying you learn math in steps. The goal is to understand the concepts of mathematics. Its not about getting it shoved down your throat and then quickly discarding it. When solving a rubiks cube most people think that you solve it one side at a time. This is not true; you solve it layer by layer, step by step. Just like baking a cake.
Math is a similar concept. You start small with say basic arithmetic, then move onto pre-algebra and so on and so forth.
Here’s a complete list of the order through which you should learn it:
Basic Math
Pre-Algebra
Algebra
Algebra II
Trigonometry
Geometry - Most people stop here, or a little farther back.
Calculus - Few get here.
Calculus II - Very few get here.
Calculus III - Starting to be considered a genius
Linear Algebra
Calculus IV
After Calculus IV or Linear Algebra you can take your math skills in many different directions. Remember though, it all starts with building a structurally sound foundation before you begin to move through the ranks. Skipping steps, and leaving holes in your math armor will make it more difficult to learn it at the higher levels as you move forward.
Math geniuses all have a solid foundation from which the build upon.
Bruno Campello de Souza
Professor at Federal University of Pernambuco (UFPE) (2004–present)
·
Author has 1.3K answers and 22.7M answer views
·
6y
Related
Is there a correlation between multiplying big numbers in your head and IQ?
Actually, there is.
IQ tests are basically sets of mental tasks designed to assess intelligence. The mechanisms and phenomena underlying intelligence are basically information-processing and working memory, which are managed by executive functions (monitoring, control) so as to perform intellectual tasks. The better any of these things are, the better the intellectual performance tends to become.
Working memory is the mechanism or set of mechanisms that holds data and information “in one’s head” for logical processing, being somewhat analogous to a computer’s RAM or a video card’s VRAM. The larg
Actually, there is.
IQ tests are basically sets of mental tasks designed to assess intelligence. The mechanisms and phenomena underlying intelligence are basically information-processing and working memory, which are managed by executive functions (monitoring, control) so as to perform intellectual tasks. The better any of these things are, the better the intellectual performance tends to become.
Working memory is the mechanism or set of mechanisms that holds data and information “in one’s head” for logical processing, being somewhat analogous to a computer’s RAM or a video card’s VRAM. The larger that capacity, the “bigger” are the numbers that one can deal with in one’s “head”.
Of course, working memory is just one of the elements involved in the basic mechanisms and processes, and there are many additional psychological, social and cultural elements that intervene between those and the actual performance on a given task.
Also, there are diminishing returns or “saturation effects” to things like speed of processing and working memory, so that, beyond a certain point, an increase in them will translate into lesser and lesser impacts on performance. Using the computational metaphor again, if, say, a laptop has only 2GB of RAM, increasing it to 4GB will greatly improve performance, and if one goes from 4GB to 8GB there will also be a substantial improvement, though perhaps not as dramatic as the initial one. Going from 8GB to 16GB will still get you a moderate, but relevant, enhancement in performance. Going from 16GB to 32GB, however, will produce mostly negligible improvements in performance (except if you work with exceptionally large files, which is fairly rare).
Thus, one can say that you can expect that those with a higher IQ tend to, on average, be able to do mental multiplications involving larger numbers than those of individuals with a lower IQ and vice-versa. However, if one attempts to estimate one of these variables through observation of the other, the margin of error is quite substantial and the accuracy of the results will tend to be very poor (you can expect to predict only something like 10–20% of the variance).
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Mathematics
Proofs by Contradiction
Square Root Index
Rational Numbers
Logical Contradiction
Prime Number Theory
Logic and Proof
Integers
Square Roots (numbers)
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How do you prove by contradiction that sqrt ((n^2) +1) is not an integer for all n in N?
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Alireza Shariati
Math major college student · Author has 1.4K answers and 972.6K answer views
·4y
Suppose √n 2+1=a,a∈N n 2+1=a,a∈N for a natural number n n :
√n 2+1=a⟹n 2+1=a 2 n 2+1=a⟹n 2+1=a 2
⟹n 2−a 2=−1⟹n 2−a 2=−1
⟹(n+a)(n−a)=−1⟹(n+a)(n−a)=−1
If the product of two integers is −1−1, then one is 1 1 and the other one is −1−1 (why?) :
Assume n+a=1 n+a=1 and n−a=−1 n−a=−1 ( it doesn’t matter which one is 1 1 and which one is −1−1 ):
{n+a=1 n−a=−1⟹n+a+n−a=0{n+a=1 n−a=−1⟹n+a+n−a=0
⟹2 n=0⟹2 n=0
⟹n=0⟹n=0
So n=0 n=0, contradicting the fact that n n is a natural number.
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9 3
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Eleftherios Argyropoulos
B.S. in Mathematics&Physics, Northeastern University (Graduated 2002) · Author has 2K answers and 2.5M answer views
·4y
First proof:
Let us assume that there exists positive integer k, such that:
√n 2+1=k…(1)n 2+1=k…(1)
Then, by (1)(1), we take:
√n 2+1=k=>n 2+1=k 2=>k 2−n 2=1=>n 2+1=k=>n 2+1=k 2=>k 2−n 2=1=>
(k−n)(k+n)=1…(2)(k−n)(k+n)=1…(2)
By (2)(2), we take:
k−n=1…(3)k−n=1…(3)
and
k+n=1…(4)k+n=1…(4)
By adding (3)(3) and (4)(4) by parts, we obtain:
2 k=2=>k=1 2 k=2=>k=1
By subtracting (3)(3) and (4)(4) by parts, we obtain:
2 n=0=>n=0 2 n=0=>n=0
and since n>0 n>0 we have the contradiction.
Second proof:
et us assume that there exists positive integer k, such that:
√n 2+1=k…(1)n 2+1=k…(1)
Then, by (1)(1), we take:
√n 2+1=k=>n 2+1=k 2=>n 2+1 2=k 2…(2)n 2+1=k=>n 2+1=k 2=>n 2+1 2=k 2…(2)
By (2)(2), the positive integers 1 1, n n
Continue Reading
First proof:
Let us assume that there exists positive integer k, such that:
√n 2+1=k…(1)n 2+1=k…(1)
Then, by (1)(1), we take:
√n 2+1=k=>n 2+1=k 2=>k 2−n 2=1=>n 2+1=k=>n 2+1=k 2=>k 2−n 2=1=>
(k−n)(k+n)=1…(2)(k−n)(k+n)=1…(2)
By (2)(2), we take:
k−n=1…(3)k−n=1…(3)
and
k+n=1…(4)k+n=1…(4)
By adding (3)(3) and (4)(4) by parts, we obtain:
2 k=2=>k=1 2 k=2=>k=1
By subtracting (3)(3) and (4)(4) by parts, we obtain:
2 n=0=>n=0 2 n=0=>n=0
and since n>0 n>0 we have the contradiction.
Second proof:
et us assume that there exists positive integer k, such that:
√n 2+1=k…(1)n 2+1=k…(1)
Then, by (1)(1), we take:
√n 2+1=k=>n 2+1=k 2=>n 2+1 2=k 2…(2)n 2+1=k=>n 2+1=k 2=>n 2+1 2=k 2…(2)
By (2)(2), the positive integers 1 1, n n and k k form a Pythagorean triple, but this is a contradiction, since the smallest possible Pythagorean triple is:
3 3, 4 4, 5 5.
Upvote ·
Amitabha Tripathi
five decades of high school Algebra · Author has 4.7K answers and 13.9M answer views
·4y
Since n<√n 2+1<n+1 n<n 2+1<n+1, √n 2+1 n 2+1 lies between consecutive integers and so can’t be an integer. ■◼
Upvote ·
9 9
Mohammad Afzaal Butt
B.Sc in Mathematics&Physics, Islamia College Gujranwala (Graduated 1977) · Author has 24.6K answers and 22.9M answer views
·4y
Let√n 2+1=m where m∈Z Let n 2+1=m where m∈Z
⟹n 2+1=m 2⟹n 2+1=m 2
⟹m 2−n 2=1⟹m 2−n 2=1
⟹(m−n)(m+n)=1⟹(m−n)(m+n)=1
⟹⟹ m - n and m + n are multiplicative inverse of each other. Which is not true. Hence our assumption is wrong.
√n 2+1≠m where m∈Z n 2+1≠m where m∈Z
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1y
To prove by contradiction that √n 2+1 n 2+1 is not an integer for all n∈N∗n∈N∗ (the set of positive integers), we can follow these steps:
Step 1: Assume the contrary
Assume that there exists some n∈N∗n∈N∗ such that √n 2+1 n 2+1 is an integer. Let’s denote this integer by k k. Therefore, we have:
√n 2+1=k n 2+1=k
Step 2: Square both sides
Squaring both sides of the equation gives us:
n 2+1=k 2 n 2+1=k 2
Step 3: Rearranging the equation
Rearranging this equation, we get:
k 2−n 2=1 k 2−n 2=1
Step 4: Factor the difference of squares
The left side can be factored using the difference of squares:
(k(k
Continue Reading
To prove by contradiction that √n 2+1 n 2+1 is not an integer for all n∈N∗n∈N∗ (the set of positive integers), we can follow these steps:
Step 1: Assume the contrary
Assume that there exists some n∈N∗n∈N∗ such that √n 2+1 n 2+1 is an integer. Let’s denote this integer by k k. Therefore, we have:
√n 2+1=k n 2+1=k
Step 2: Square both sides
Squaring both sides of the equation gives us:
n 2+1=k 2 n 2+1=k 2
Step 3: Rearranging the equation
Rearranging this equation, we get:
k 2−n 2=1 k 2−n 2=1
Step 4: Factor the difference of squares
The left side can be factored using the difference of squares:
(k−n)(k+n)=1(k−n)(k+n)=1
Step 5: Analyze the factors
Now, since k−n k−n and k+n k+n are both integers and their product is 1, the only pairs of integers that multiply to 1 are (1,1)(1,1) and (−1,−1)(−1,−1).
Case 1: k−n=1 k−n=1 and k+n=1 k+n=1
From k−n=1 k−n=1, we have k=n+1 k=n+1.
From k+n=1 k+n=1, substituting k k gives us:
(n+1)+n=1⟹2 n+1=1⟹2 n=0⟹n=0(n+1)+n=1⟹2 n+1=1⟹2 n=0⟹n=0
However, n n must be a positive integer (n∈N∗n∈N∗), so this case leads to a contradiction.
Case 2: k−n=−1 k−n=−1 and k+n=−1 k+n=−1
From k−n=−1 k−n=−1, we have k=n−1 k=n−1.
From k+n=−1 k+n=−1, substituting gives:
(n−1)+n=−1⟹2 n−1=−1⟹2 n=0⟹n=0(n−1)+n=−1⟹2 n−1=−1⟹2 n=0⟹n=0
Again, this leads to a contradiction since n n cannot be zero.
Conclusion
In both cases, we arrive at the contradiction that n n must be zero, which is not allowed since n n is a positive integer. Thus, our initial assumption that √n 2+1 n 2+1 is an integer must be false.
Therefore, we conclude that √n 2+1 n 2+1 is not an integer for all n∈N∗n∈N∗.
Upvote ·
Paul Hudson
More than 40 years of programming experience · Author has 3.8K answers and 2.4M answer views
·4y
if it’s an integer, then n^2 +1 = m^2 for some other integer m.
n^2 +1= m^2
m^2 -n^2 = 1
But the difference between two squares a and b, a < b is never 1 (because the difference is 2a + 1 to (a+1)^2, except for 0 and 1.
So it’s true for for n == 0 and nothing else.
Upvote ·
Deivy Petrescu
Author has 54 answers and 12.2K answer views
·Updated 4y
Assuming you mean for n∈N+n∈N+, take it as true and square it.
n 2+1=z 2 n 2+1=z 2 where both z and n are integers.
Then z 2−n 2=(z+n)(z−n)=1 z 2−n 2=(z+n)(z−n)=1 .
In other words both (z+n) and (z-n) divide 1, thus (z-n)=(z+n)=1 which implies z=1 and n=0. A contradiction since n∈N+n∈N+
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Gram Zeppi
Weary of Quora. · Upvoted by
Alon Amit
, Lover of math. Also, Ph.D. and
Vladimir Novakovski
, silver medals, IOI 2001 and IPhO 2001 · Author has 667 answers and 1.8M answer views
·9y
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Let x n=√2+3√3+…+n√n x n=2+3+…+n n 3. Show that for any natural n≥2 n≥2, x n+1−x n<1 n!x n+1−x n<1 n!.
Thanks for the A2A. Well, let's try.
Let's first introduce double indexed variables:
x n,m:=m√m+m+1√m+1+…+n√n x n,m:=m+m+1+…+n n m+1 m for all 2≤m≤n.2≤m≤n.
Note that x n,2=x n.x n,2=x n. Furthermore, x n,m>1 x n,m>1, x n,n=n√n x n,n=n n and x n,m x n,m are increasing in n n, i.e. x n+1,m>x n,m x n+1,m>x n,m for all m m, and x n,m x n,m is decreasing in m.m.
Moreover, recall that a k−b k=(a−b)(a k+a k−1 b+…+b k)a k−b k=(a−b)(a k+a k−1 b+…+b k)() for a,b∈R a,b∈R and n∈N.n∈N.
Then [Math Processing Error]x_{n+1}-x_{n}=x_{n+1,2}-x_{n,2}=\dfrac{x_{n+1,2}^2-x_{n,2}^2}{x_{n+1,2}+x_{n,2}}=\dfrac{x_{n+1,3}-x_{n,3}}{x_{n+
Continue Reading
Thanks for the A2A. Well, let's try.
Let's first introduce double indexed variables:
x n,m:=m√m+m+1√m+1+…+n√n x n,m:=m+m+1+…+n n m+1 m for all 2≤m≤n.2≤m≤n.
Note that x n,2=x n.x n,2=x n. Furthermore, x n,m>1 x n,m>1, x n,n=n√n x n,n=n n and x n,m x n,m are increasing in n n, i.e. x n+1,m>x n,m x n+1,m>x n,m for all m m, and x n,m x n,m is decreasing in m.m.
Moreover, recall that a k−b k=(a−b)(a k+a k−1 b+…+b k)a k−b k=(a−b)(a k+a k−1 b+…+b k)() for a,b∈R a,b∈R and n∈N.n∈N.
Then x n+1−x n=x n+1,2−x n,2=x 2 n+1,2−x 2 n,2 x n+1,2+x n,2=x n+1,3−x n,3 x n+1,2+x n,2.x n+1−x n=x n+1,2−x n,2=x n+1,2 2−x n,2 2 x n+1,2+x n,2=x n+1,3−x n,3 x n+1,2+x n,2.
Continuing in this manner and using () for k=3 k=3 we have on the next step
x n+1,3−x n,3 x n+1,2+x n,2=x 3 n+1,3−x 3 n,3(x n+1,2+x n,2)(x 2 n+1,3+x n,3 x n+1,3+x 2 n,3)=x n+1,4−x n,4(x n+1,2+x n,2)(x 2 n+1,3+x n,3 x n+1,3+x 2 n,3).x n+1,3−x n,3 x n+1,2+x n,2=x n+1,3 3−x n,3 3(x n+1,2+x n,2)(x n+1,3 2+x n,3 x n+1,3+x n,3 2)=x n+1,4−x n,4(x n+1,2+x n,2)(x n+1,3 2+x n,3 x n+1,3+x n,3 2).
Finally we get:
x n+1−x n=n+1√n+1 x n+1,2+x n,2⋅1 x 2 n+1,3+x n,3 x n+1,3+x 2 n,3⋅…⋅1 x n−1 n+1,n+x n−2 n+1,n x n,n+…+x n−1 n,n.x n+1−x n=n+1 n+1 x n+1,2+x n,2⋅1 x n+1,3 2+x n,3 x n+1,3+x n,3 2⋅…⋅1 x n+1,n n−1+x n+1,n n−2 x n,n+…+x n,n n−1.
Just by replacing n+1 n+1 by n n we get the corresponding expression for x n−x n−1.x n−x n−1.
Thus their ratio is given by:
x n+1−x n x n−x n−1=n+1√n+1 n√n x n,2+x n−1,2 x n+1,2+x n,2⋅x 2 n,3+x n−1,3 x n,3+x 2 n−1,3 x 2 n+1,3+x n,3 x n+1,3+x 2 n,3⋅…⋅1 x n−1 n+1,n+x n−2 n+1,n x n,n+…+x n−1 n,n.(∗∗)x n+1−x n x n−x n−1=n+1 n+1 n n x n,2+x n−1,2 x n+1,2+x n,2⋅x n,3 2+x n−1,3 x n,3+x n−1,3 2 x n+1,3 2+x n,3 x n+1,3+x n,3 2⋅…⋅1 x n+1,n n−1+x n+1,n n−2 x n,n+…+x n,n n−1.(∗∗)
Now recall the standard fact that the sequence n√n n n is monotone decreasing for n≥3.n≥3. So n+1√n+1 n√n<1 n+1 n+1 n n<1 for n≥3.n≥3.
Moreover, since x n+1,m>x n,m x n+1,m>x n,m and x n,m>x n−1,m x n,m>x n−1,m, all fractions in () are less than 1.1.
So we get, at least for n≥3,n≥3,x n+1−x n x n−x n−1<1 x n−1 n+1,n+x n−2 n+1,n x n,n+…+x n−1 n,n<1 n.x n+1−x n x n−x n−1<1 x n+1,n n−1+x n+1,n n−2 x n,n+…+x n,n n−1<1 n.
The latter inequality holds since x i,j>1.x i,j>1.
Thus we conclude that x n+1−x n<1 n(x n−x n−1)x n+1−x n<1 n(x n−x n−1) () for all n≥3.n≥3.
So one should merely prove this inequality for n=2.n=2. This is left to the reader. Then the required inequality follows inductively from ().
Upvote ·
99 53
9 5
9 4
Yves Daoust
PhD from FPMs · Author has 3.4K answers and 277.5K answer views
·10mo
If this is true, then n 2+1=m 2 n 2+1=m 2, and 1=(m−n)(m+n)1=(m−n)(m+n), or m−n=m+n=1 m−n=m+n=1. but n=0 n=0 is not allowed.
Upvote ·
Kostyantyn Mazur
PhD in Mathematics, New York University (Graduated 2018) · Upvoted by
Richard Zhou
, B.S./M.S. Mathematics, Yale University (2022) · Author has 293 answers and 964.9K answer views
·8y
Related
If n n is a positive integer such that 2+2√28 n 2+1 2+2 28 n 2+1 is an integer, how do I show that 2+2√28 n 2+1 2+2 28 n 2+1 is the square of an integer?
Let n n be a positive integer such that 2+2√28 n 2+1 2+2 28 n 2+1 is an integer. Let x=2+2√28 n 2+1 x=2+2 28 n 2+1. Then, x−4=2√28 n 2+1−2 x−4=2 28 n 2+1−2 is also an integer. Both are positive, as, since n≥1 n≥1, 2√28 n 2+1−2≥2√28+1−2>0 2 28 n 2+1−2≥2 28+1−2>0, x−4 x−4 is positive, which makes x x also positive.
x(x−4)=(2+2√28 n 2+1)(2√28 n 2+1−2)x(x−4)=(2+2 28 n 2+1)(2 28 n 2+1−2)
=(2√28 n 2+1+2)(2√28 n 2+1−2)=(2 28 n 2+1+2)(2 28 n 2+1−2)
=(2√28 n 2+1)2−2 2=(2 28 n 2+1)2−2 2
=4(28 n 2+1)−4=4(28 n 2+1)−4
=112 n 2=112 n 2
112 112 is divisible by 16 16 (as 112=(7)(16)112=(7)(16)). Thus, x(x−4)x(x−4) is also divisib
Continue Reading
Let n n be a positive integer such that 2+2√28 n 2+1 2+2 28 n 2+1 is an integer. Let x=2+2√28 n 2+1 x=2+2 28 n 2+1. Then, x−4=2√28 n 2+1−2 x−4=2 28 n 2+1−2 is also an integer. Both are positive, as, since n≥1 n≥1, 2√28 n 2+1−2≥2√28+1−2>0 2 28 n 2+1−2≥2 28+1−2>0, x−4 x−4 is positive, which makes x x also positive.
x(x−4)=(2+2√28 n 2+1)(2√28 n 2+1−2)x(x−4)=(2+2 28 n 2+1)(2 28 n 2+1−2)
=(2√28 n 2+1+2)(2√28 n 2+1−2)=(2 28 n 2+1+2)(2 28 n 2+1−2)
=(2√28 n 2+1)2−2 2=(2 28 n 2+1)2−2 2
=4(28 n 2+1)−4=4(28 n 2+1)−4
=112 n 2=112 n 2
112 112 is divisible by 16 16 (as 112=(7)(16)112=(7)(16)). Thus, x(x−4)x(x−4) is also divisible by 16 16. This means that x x is divisible by 4 4. (If x x is odd, then x−4 x−4 is odd, so their product is odd. If x x is twice an odd number, then x−4 x−4 is also twice an odd number, so x(x−4)x(x−4) is 4 4 times an odd number, and thus is not divisible by 16 16).
Let x=4 u x=4 u. It was just shown that u u is an integer, and u u is positive because 4 u=x 4 u=x is positive. u−1 u−1 is also a positive integer. (u−1>0 u−1>0, as 4 u−4=x−4>0 4 u−4=x−4>0, as was shown earlier.) If u u can be shown to be a perfect square, that would make x x (and hence also 2+2√28 n 2+1 2+2 28 n 2+1, which x x was defined to be) a perfect square. Thus:
4 u(4 u−4)=112 n 2 4 u(4 u−4)=112 n 2
16 u(u−1)=112 n 2 16 u(u−1)=112 n 2
u(u−1)=7 n 2 u(u−1)=7 n 2
Now, u u and u−1 u−1 have no positive-integer common factors between them other than 1 1. (If they did, 1 1, which is the difference between them, would also have this positive-integer common factor other than 1 1, but it does not.) Thus, for any prime factor that u u has, u−1 u−1 does not have that prime factor.
Thus, any prime factor of u u, other than possibly 7 7, must appear an even number of times in the prime factorization of u u. The same holds for u−1 u−1: any prime factor of u−1 u−1, other than possibly 7 7, must appear an even number of times in the prime factorization of u−1 u−1.
As for 7 7 itself, it appears an odd number in total in the prime factorizations of u u and u−1 u−1. Thus, in the prime factorization of one of u u and u−1 u−1, 7 7 appears an odd number of times, and in the other, 7 7 appears an even number of times. In whichever of u u and u−1 u−1 the prime factor 7 7 appears an even number of times, that number (u u or u−1 u−1) is a perfect square, because each of its prime factors appears an even number of times. In whichever of u u and u−1 u−1 the prime factor 7 7 appears an odd number of times, that number (u u or u−1 u−1) is 7 7 multiplied by a perfect square, because dividing this u u or u−1 u−1 by 7 7 yields a positive integer each of whose prime factors appears an even number of times in its prime factorization (that is, of u 7 u 7 or of u−1 7 u−1 7, as the case may be).
Thus, there are two cases:
a) u=v 2 u=v 2 and u−1=7 w 2 u−1=7 w 2
and
b) u=7 v 2 u=7 v 2 and u−1=w 2 u−1=w 2
where u u and v v are positive integers. Case a) makes u u a perfect square, which is enough to make 2+2√28 n 2+1 2+2 28 n 2+1 a perfect square. Thus, the only case that needs to be examined is b).
In case b), w 2=u−1=7 v 2−1 w 2=u−1=7 v 2−1. Now, the remainder when 7 v 2−1 7 v 2−1 is divided by 7 7 is 6 6, but, if u=7 q+r u=7 q+r, where q q is an integer and r=0 r=0, 1 1, 2 2, 3 3, 4 4, 5 5, or 6 6, then u 2=(7 q+r)2=49 q 2+14 q r+r 2=7(7 q 2+2 q r)+r 2 u 2=(7 q+r)2=49 q 2+14 q r+r 2=7(7 q 2+2 q r)+r 2, and, since r 2=0 r 2=0, 1 1, 4 4, 9 9, 16 16, 25 25, or 36 36, it follows that the remainder when w 2 w 2 is divided by 7 7 is, correspondingly, 0 0, 1 1, 4 4, 2 2, 2 2, 4 4, or 1 1, which is never 6 6. However, w 2 w 2 and 7 v 2−1 7 v 2−1 are the same number, so there is a contradiction. Thus, case b) cannot happen.
Thus, case a) must happen, but it was shown that 2+2√28 n 2+1 2+2 28 n 2+1 a perfect square in that case. Thus, 2+2√28 n 2+1 2+2 28 n 2+1 is the square of an integer.
Upvote ·
99 22
9 1
9 5
Enrico Gregorio
Associate professor in Algebra · Upvoted by
David Joyce
, Ph.D. Mathematics, University of Pennsylvania (1979) · Author has 18.4K answers and 16M answer views
·1y
Related
How do I prove that for any natural numbers n it is true that sqrt {n +[ sqrt {n} + 1/2} ]} is irrational?
Suppose that
√n+√n+1 2 n+n+1 2
is rational. Then also
n+√n+1 2 n+n+1 2
is rational and therefore so is √n.n. This has the well-known consequence that n n is a perfect square, say n=m 2.n=m 2. Now let’s go back to the start: we have that
√m 2+m+1 2=1 2√4 m 2+4 m+2 m 2+m+1 2=1 2 4 m 2+4 m+2
is rational and this requires √4 m 2+4 m+2 4 m 2+4 m+2 to be rational as well, hence
4 m 2+4 m+2=(2 m+1)2+1 4 m 2+4 m+2=(2 m+1)2+1
should be a perfect square. But there aren’t nonzero squares differing by 1,1, so we have reached the desired contradiction.
However you might be asking whether
√n+⌊√n+1 2⌋n+⌊n+1 2⌋
can be
Continue Reading
Suppose that
√n+√n+1 2 n+n+1 2
is rational. Then also
n+√n+1 2 n+n+1 2
is rational and therefore so is √n.n. This has the well-known consequence that n n is a perfect square, say n=m 2.n=m 2. Now let’s go back to the start: we have that
√m 2+m+1 2=1 2√4 m 2+4 m+2 m 2+m+1 2=1 2 4 m 2+4 m+2
is rational and this requires √4 m 2+4 m+2 4 m 2+4 m+2 to be rational as well, hence
4 m 2+4 m+2=(2 m+1)2+1 4 m 2+4 m+2=(2 m+1)2+1
should be a perfect square. But there aren’t nonzero squares differing by 1,1, so we have reached the desired contradiction.
However you might be asking whether
√n+⌊√n+1 2⌋n+⌊n+1 2⌋
can be rational, where the brackets represent the largest integer function.
Let’s consider the largest square less than or equal to n>0,n>0, that is, suppose m m is a positive integer and
m 2≤n<(m+1)2 m 2≤n<(m+1)2
Therefore m<√n+1/2<m+3/2,m<n+1/2<m+3/2, so there are two possible cases
⌊√n+1/2⌋=m⌊n+1/2⌋=m
⌊√n+1/2⌋=m+1⌊n+1/2⌋=m+1
If n=m 2+r,n=m 2+r, we have 0≤r<2 m+1.0≤r<2 m+1. In the first case in order for the given expression to be rational we need
m 2+r+m=a 2 m 2+r+m=a 2
for some positive integer a.a. Noticing that m 2+m+r<m 2+3 m+1<m 2+4 m+4=(m+2)2 m 2+m+r<m 2+3 m+1<m 2+4 m+4=(m+2)2 we can only have a=m+1 a=m+1 and therefore r=m+1.r=m+1. Thus we must have
√m 2+m+1+1 2<m+1 m 2+m+1+1 2<m+1
that becomes
m 2+m+1<m 2+m+1 4 m 2+m+1<m 2+m+1 4
which is a contradiction.
Thus we must have ⌊√n+1/2⌋=m+1⌊n+1/2⌋=m+1 and so
m 2+r+m+1=b 2 m 2+r+m+1=b 2
for some positive integer b.b. Again
m 2+r+m+1<m 2+3 m+2<(m+2)2 m 2+r+m+1<m 2+3 m+2<(m+2)2
so b=m+1 b=m+1 and we have r=m.r=m. Now the inequality becomes
√m 2+m+1 2≥m+1 m 2+m+1 2≥m+1
hence
m 2+m≥m 2+m+1 4 m 2+m≥m 2+m+1 4
which is another contradiction.
Upvote ·
99 10
Jonathan Devor
PhD in Astronomy, Harvard University (Graduated 2008) · Author has 3.7K answers and 19.4M answer views
·4y
Related
How do you prove that 2^(n+1) >(n+2).sin(n), for all positive integers n?
A2A
The OP is asking us to prove the inequality: 2 n+1>(n+2)sin(n)2 n+1>(n+2)sin(n) for all n∈N n∈N
The first step is getting rid of the sin(n)sin(n) function. Because sin(n)≤1 sin(n)≤1, it’s sufficient to prove 2 n+1>n+2 2 n+1>n+2
The next step is generalizing this for all the positive real numbers.
2 x+1>x+2 2 x+1>x+2
By increasing the domain it may seem like we’re just making the problem harder, but in fact we can now put into play many powerful tools, that require a continuous function.
The plot below compares:
Blue: y=2 x+1 y=2 x+1
Black: y=x+2 y=x+2
Red: y=(x+2)sin(x)y=(x+2)sin(x)
It should be clear that the exponential (left hand side) grows faster asy
Continue Reading
A2A
The OP is asking us to prove the inequality: 2 n+1>(n+2)sin(n)2 n+1>(n+2)sin(n) for all n∈N n∈N
The first step is getting rid of the sin(n)sin(n) function. Because sin(n)≤1 sin(n)≤1, it’s sufficient to prove 2 n+1>n+2 2 n+1>n+2
The next step is generalizing this for all the positive real numbers.
2 x+1>x+2 2 x+1>x+2
By increasing the domain it may seem like we’re just making the problem harder, but in fact we can now put into play many powerful tools, that require a continuous function.
The plot below compares:
Blue: y=2 x+1 y=2 x+1
Black: y=x+2 y=x+2
Red: y=(x+2)sin(x)y=(x+2)sin(x)
It should be clear that the exponential (left hand side) grows faster asymptotically, however the question remains to determines the point x 0 x 0, from where all x>x 0 x>x 0 make the inequality true. To this end we can look at the difference:
f(x)=2 x+1−(x+2)f(x)=2 x+1−(x+2)
d f d x=2 x+1 ln(2)−1=2 x⋅(2 ln(2))−1 d f d x=2 x+1 ln(2)−1=2 x⋅(2 ln(2))−1
Since 2 ln(2)>1 2 ln(2)>1, we know the differential must be strictly positive for all x>0 x>0. In other words, f(x)f(x) is strictly monotonically growing for all positive x x. And because f(0)=0 f(0)=0, we can conclude that:
2 x+1>x+2 2 x+1>x+2 for all x>0 x>0
And that is sufficient to prove the OP’s assertion.
Upvote ·
9 6
Amitabha Tripathi
five decades of high school Algebra · Upvoted by
Terry Moore
, M.Sc. Mathematics, University of Southampton (1968) · Author has 4.7K answers and 13.9M answer views
·8y
Related
How do I solve this proof? Let n be an integer. Show that (n+2) (n+1) n(n-1) + 1 is the square of an integer.
(n+2)(n+1)n(n−1)+1=((n+2)(n−1))((n+1)n)+1(n+2)(n+1)n(n−1)+1=((n+2)(n−1))((n+1)n)+1
=(n 2+n−2)(n 2+n)+1=(n 2+n−2)(n 2+n)+1
=((n 2+n−1)−1)((n 2+n−1)+1)+1=((n 2+n−1)−1)((n 2+n−1)+1)+1
=(n 2+n−1)2.■=(n 2+n−1)2.◼
Upvote ·
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Informatics Olympiad Questions and Color Interval?
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This is an 2013 Local INOI Informatics Olympiad Question.
Suppose we have 1393 points on real axis. we want to color this points such that for each arbitrary interval [a, b], if at least one point of 1393 points be in interval, at least one point inside the interval has different color with other points in this interval. with how many colors this can be done? (best answer is 11).
INOI Commit say short answer with 11. How we can calculate this question?
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asked Mar 27, 2015 at 10:43
user4591951 user4591951
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The question looks incomplete. As far as I understand it 2 colors are enough. Just alternate the colors of the points.Dialecticus –Dialecticus 2015-03-27 10:54:16 +00:00 Commented Mar 27, 2015 at 10:54
1 @Dialecticus: That's exactly what I thought at first. Then I read the question more carefully: for each interval, there must be a point whose colour is different from all other points in the interval. For instance, this is not the case for an interval containing four points coloured B,W,B,W.TonyK –TonyK 2015-03-27 10:56:55 +00:00 Commented Mar 27, 2015 at 10:56
1 What does this have to do with programming?Mark Dickinson –Mark Dickinson 2015-03-27 10:57:32 +00:00 Commented Mar 27, 2015 at 10:57
2 I'm voting to close this question as off-topic.David Eisenstat –David Eisenstat 2015-03-27 11:53:05 +00:00 Commented Mar 27, 2015 at 11:53
2 @MarcGlisse: You have a set S of N points on the real line. Each point of S gets assigned one of k colors. This in such a way that whenever the intersection of S and some [a,b] is not empty, there is one color so that only and exactly one point of that color is in the intersection. Obviously, if every point has its own color, k=N, this is satisfied. Task: For N=1393 determine the minimal number k with that property.Lutz Lehmann –Lutz Lehmann 2015-03-27 14:37:21 +00:00 Commented Mar 27, 2015 at 14:37
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Claim
The most points you can colour with k colours is 2^k-1.
Proof
If we have 0 colours, then clearly we can colour no points, so this is true for k==0.
For k>0, we use induction.
Consider the interval containing all the points. There must be 1 colour that appears exactly once. Consider the points to the left of this point. These points are coloured with k-1 colours, so there can be at most 2^(k-1)-1 of them.
Similarly, there are at most 2^(k-1)-1 to the right.
So in total there can be at most 1+2^(k-1)-1+2^(k-1)-1=2^k-1 points with k colours.
2^10-1=1023, 2^11-1=2047, so this shows a lower bound of 11 colours for your case, and TonyK has given a construction showing that this can be achieved.
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edited Mar 27, 2015 at 12:13
answered Mar 27, 2015 at 11:51
Peter de RivazPeter de Rivaz
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TonyK
TonyKOver a year ago
This answer almost makes mine redundant -- it immediately suggests the construction that I described.
2015-03-27T13:43:47.367Z+00:00
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user4591951Over a year ago
@TonyK there are at most 2^(k-1)-1 to the right. means there are 2^(k-1)-1 points?
2015-03-27T15:24:24.64Z+00:00
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user4591951Over a year ago
there are at most 2^(k-1)-1 to the right. means there are 2^(k-1)-1 points?
2015-03-27T17:00:13.317Z+00:00
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If the colour of the nth point is determined solely by the number of trailing zeroes in the binary expansion of n, then the condition is satisfied. We can do this with 11 colours for up to 2047 points.
The converse -- can we do it with fewer points -- is another question.
Example for 15 points:
```cpp
Binary Trailing zeroes Colour
1 0 blue
10 1 red
11 0 blue
100 2 green
101 0 blue
110 1 red
111 0 blue
1000 3 yellow
1001 0 blue
1010 1 red
1011 0 blue
1100 2 green
1101 0 blue
1110 1 red
1111 0 blue
```
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edited Mar 27, 2015 at 11:08
answered Mar 27, 2015 at 11:00
TonyKTonyK
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user4591951
user4591951Over a year ago
I couldn't get it. is it possible make an example for clarify?
2015-03-27T11:03:16.757Z+00:00
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user4591951
user4591951Over a year ago
@Dialecticus, please you add this nice idea as an answer.
2015-03-27T11:11:24.1Z+00:00
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user4591951
user4591951Over a year ago
+1 Tonyk, but what do u mean by "The converse -- can we do it with fewer points -- is another question." please add more detail to your answer.
2015-03-27T11:11:50.713Z+00:00
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TonyKOver a year ago
@MaryamGhizhi: I have shown that 11 colours are enough. But I haven't shown that 10 colours are not enough.
2015-03-27T11:14:02.01Z+00:00
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TonyK
TonyKOver a year ago
@Maryam: Just continue my example sequence until you reach 1393. The hardest part will be thinking up seven more colour names.
2015-03-27T18:24:16.063Z+00:00
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6626 | https://www.quora.com/What-is-the-relationship-between-the-eccentricity-of-an-ellipse-and-its-shape | What is the relationship between the eccentricity of an ellipse and its shape? - Quora
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What is the relationship between the eccentricity of an ellipse and its shape?
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Bill Hazelton
Peregrinating Geospatial Professor & Practitioner · Author has 3.9K answers and 8.7M answer views
·4y
The eccentricity of an ellipse, e, is a value that is less than 1 and greater than or equal to 0.
Using the above diagram, the ellipse is defined as the locus of points such that the ratio of the distances from any point T to F and from T to the line D is a constant and equal to e. the line D is the directrix, while F and F’ are the foci of the ellipse.
Further:
O D=a e O D=a e
OF = OF’ = a e
b 2=a 2(1−e 2)b 2=a 2(1−e 2)
The eccentricity can be calculated using:
e 2=a 2−b 2 a 2 e 2=a 2−b 2 a 2
As e gets closer to 1, the ellipse gets progressively flatter and narrower, and e =1 would be a straight line
Continue Reading
The eccentricity of an ellipse, e, is a value that is less than 1 and greater than or equal to 0.
Using the above diagram, the ellipse is defined as the locus of points such that the ratio of the distances from any point T to F and from T to the line D is a constant and equal to e. the line D is the directrix, while F and F’ are the foci of the ellipse.
Further:
O D=a e O D=a e
OF = OF’ = a e
b 2=a 2(1−e 2)b 2=a 2(1−e 2)
The eccentricity can be calculated using:
e 2=a 2−b 2 a 2 e 2=a 2−b 2 a 2
As e gets closer to 1, the ellipse gets progressively flatter and narrower, and e =1 would be a straight line segment. Meanwhile, e = 0 gives you a circle.
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James A Schiller
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·4y
POSTULATE. An ellipse requires by definition for any and every point to be the same total distance from the two focii. Example if a perimeter point is 5 inches from one focal and 8 inches from the other focal, then the point that is 3 inches from a focal will be 10 inches from the other focal. Another fact is though both diameter axis have only two end points, all other points on the perimeter are in FOUR-set copies, or respectively a TWO-set of short radius from F1 with long radius from F2, and a reverse short radius to F2 and long radius from F1. The two radius will always add up to the same
Continue Reading
POSTULATE. An ellipse requires by definition for any and every point to be the same total distance from the two focii. Example if a perimeter point is 5 inches from one focal and 8 inches from the other focal, then the point that is 3 inches from a focal will be 10 inches from the other focal. Another fact is though both diameter axis have only two end points, all other points on the perimeter are in FOUR-set copies, or respectively a TWO-set of short radius from F1 with long radius from F2, and a reverse short radius to F2 and long radius from F1. The two radius will always add up to the same constant as it does when you draw an ellipse with a fixed-length tether. The tether remains constant.
Absolutes (not postulates /foundation-definitions, but proven theorems) in ellipse are:
(1) The center of the ellipse will always be an equal distance from both focal points or it is not an ellipse.
(2) The short axis (radius from center tagged as ‘a’) is not a radius to the perimeter, but only a bisector. The short radius is actually from the closest focal point, and the long radius is not the long axis bisector, but rather the longer radius from the other focal point. Ellipse calculations are complex because they use the (axis) bisectors thru the center for the primary values when in fact the primary relation is on the ratio between the two radius from two focal points. The center only aids in the right-angle trigonometry.
(3) As with absolute (1) center between focals, so too the two extreme points of the ellipse (on the long axis bisector) are equal distance from their closest focal point. This is caused by the rule that the two radius from the two focals will always add up to a constant.
(4) The two end points of short axis (bisecting the ellipse) will have equal length radius from the two focal points.
(5) Each of the two end points of the long axis bisector on their own will have the same constant total of the two focal radii from the two focals which therefore will also equal the same total as the separation distance of (segment between) the two focals plus the equal two rays, the radius from closest focal, and the extended ray from the farther focal. This will also equal the opposite direction to the other periextent. By definition the two extents are periextent from their closest focal, but apoextent from their furthest focal. In the case of the SUN as a focal point, (a peri-focalpoint), the other focal only exists in mathematical method where there is no actual blackhole or force for each planet’s or comet’s apo-focalpoint (if ellipse). Thus the two distant ends of the ellipse will both be the same equal matching distance from their closest focal. It is NOT an ellipse if it cannot be divided into four equal quarters. Nature does it by the postulate of total of the two-radius ”tether”.
(6) From the two focal points, the two radius to any given point on the perimeter (the tether string) will total the b-axis (longest axis bisector), and this length will extend equal distance beyond focal points and between focal points to the center.
(7) The long axis-bisector radius-b converted into the multiple ratio of short axis bisector-a (as in increasing axis radius-a to be 2a=b, or 5a=b, 9a=b, or 2.34a=b will always prove the focal point to be a distance from the ellipse center of [mulitple-squared minus 1]. Example: if axis-radius-a equals 6, and axis-radius-b equals 18 of which 18/6 is a mulitple of 3 of ‘a’, so that 3a is the length of b. Then the focal point will be the square root of (3-squared minus 1) from the center of the ellipse, or that is to say that because b is 3 times longer than ‘a’ that square it to be 9, minus 1 is then 8, and focal point is √8 from the center or twice that times 2 (√8) from each other, the other focal point. This ratio is absolute for all true ellipses.
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Eric Hawk
Author has 3K answers and 3.9M answer views
·4y
Ellipse - Wikipedia
An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity:
e=c a=√1−b 2 a 2 e=c a=1−b 2 a 2.
e is the eccentricity, a is one-half of the longer axis (semi-major axis), b is one-half of the shorter axis (semi-minor axis), and c is the distance from the center to either of the foci, related by
Analytically, the equa
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Ellipse - Wikipedia
An ellipse may also be defined in terms of one focal point and a line outside the ellipse called the directrix: for all points on the ellipse, the ratio between the distance to the focus and the distance to the directrix is a constant. This constant ratio is the above-mentioned eccentricity:
e=c a=√1−b 2 a 2 e=c a=1−b 2 a 2.
e is the eccentricity, a is one-half of the longer axis (semi-major axis), b is one-half of the shorter axis (semi-minor axis), and c is the distance from the center to either of the foci, related by
Analytically, the equation of a standard ellipse centered at the origin with width 2 a and height 2 b is:
x 2 a 2+y 2 b 2=1.x 2 a 2+y 2 b 2=1.
Assuming a ≥ b, the foci are (±c, 0) for c=√a 2−b 2 c=a 2−b 2.
Hope this helps and does not mislead or confuse you.
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Richard Mentock
Studied geodesy and geophysics · Author has 1.6K answers and 705.6K answer views
·4y
You’ve heard that an ellipse is a squashed circle? If a is the length of the semi-major axis, then for an eccentricity e (e x a is the distance from the center to a focus), its semi-major axis b is √1−e 2 1−e 2 times a. So, it’s a circle “squashed” by a factor of e 2/2 e 2/2, approximately.
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·Aug 18
Which are the best AI tools for students?
There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you?
That’s where Grammarly comes in. It’s an all-in-one writing surface designed specifically for students, with tools that help you brainstorm, write, revise, and grow your skills—without cutting corners.
Here are five AI tools inside Grammarly’s document editor that are worth checking out:
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There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you?
That’s where Grammarly comes in. It’s an all-in-one writing surface designed specifically for students, with tools that help you brainstorm, write, revise, and grow your skills—without cutting corners.
Here are five AI tools inside Grammarly’s document editor that are worth checking out:
Docs – Your all-in-one writing surface
Think of docs as your smart notebook meets your favorite editor. It’s a writing surface where you can brainstorm, draft, organize your thoughts, and edit—all in one place. It comes with a panel of smart tools to help you refine your work at every step of the writing process and even includes AI Chat to help you get started or unstuck.
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Need to make sure your ideas land with credibility? Expert Review gives you tailored, discipline-aware feedback grounded in your field—whether you're writing about a specific topic, looking for historical context, or looking for some extra back-up on a point. It’s like having the leading expert on the topic read your paper before you submit it.
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More answers below
What is the eccentricity of a ellipse?
Are INFJ’s eccentric?
What is genuinely eccentric?
How eccentric are you?
Is eccentricity a luxury?
Filippo Malgarini
Studied at Politecnico Di Milano
·8y
Related
What is the eccentricity of a ellipse?
The eccentricity of an ellipse is defined as the ratio of the distance between it’s two focal points and the length of it’s major axis.
If the major and minor axis are a and b respectively, calling c the distance between the focal points and e the eccentricity we can write
e=c/a=√1−(b/a)2 e=c/a=1−(b/a)2, because by the definition of the ellipse one can easily show that a 2=b 2+c 2.a 2=b 2+c 2.
Now that you know how to c
Continue Reading
The eccentricity of an ellipse is defined as the ratio of the distance between it’s two focal points and the length of it’s major axis.
If the major and minor axis are a and b respectively, calling c the distance between the focal points and e the eccentricity we can write
e=c/a=√1−(b/a)2 e=c/a=1−(b/a)2, because by the definition of the ellipse one can easily show that a 2=b 2+c 2.a 2=b 2+c 2.
Now that you know how to calculate eccentricity, let’s talk about what it means. As you can see from the relations above, when a=b a=b we have e=0.e=0. Since a and b are the axis of the ellipse, when they are equal we have a circle. Therefore, an ellipse with zero eccentricity is a circumference.
Eccentricity is infact a measure of how much the el...
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Daniel Bamberger
Studied Mathematics at University of Marburg · Author has 963 answers and 3.1M answer views
·8y
Related
Geometry: What is the eccentricity of an ellipse, and how is it measured?
The eccentricity is a measure for how circle-like an ellipse is. The word literally means “out of the center” (ex-centric).
An ellipse with an eccentricity of e=0 is just a circle; an ellipse with an eccentricity of e=1 is a parabola. Inbetween there’s a continuum of possible shapes.
Ellipses with 0<e<1 have the curious property that for every ellipse, there are always two points F1 and F2 called the focal points of the ellipse such that the sum x+y of the distances between any point on the ellipse and the two focal points is constant (the same for all points of the ellipse). That constant dista
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The eccentricity is a measure for how circle-like an ellipse is. The word literally means “out of the center” (ex-centric).
An ellipse with an eccentricity of e=0 is just a circle; an ellipse with an eccentricity of e=1 is a parabola. Inbetween there’s a continuum of possible shapes.
Ellipses with 0<e<1 have the curious property that for every ellipse, there are always two points F1 and F2 called the focal points of the ellipse such that the sum x+y of the distances between any point on the ellipse and the two focal points is constant (the same for all points of the ellipse). That constant distance is called the major axis of the ellipse, and it is equal to 2a, where a is the semi major axis (“semi” meaning “half”).
The eccentricity is measured as the ratio of the distance between the two focal points and the major axis.
For e=0 (a circle) the two focal points coalesce into one point (imagine two points that are 0 distance apart). That point is the center of the circle, and the major axis becomes the diameter.
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Promoted by Betterbuck
Anthony Madden
Writer for Betterbuck
·Updated Aug 15
What are the weirdest mistakes people make on the internet right now?
Here are a couple of the worst mistakes I’ve seen people make:
Not using an ad blocker
If you aren’t using an ad blocker yet, you definitely should be.
A good ad blocking app will eliminate virtually all of the ads you’d see on the internet before they load.
No more YouTube ads, no more banner ads, no more pop-up ads, etc.
Most people I know use Total Adblock (link here) - it’s about £2/month, but there are plenty of solid options.
Ads also typically take a while to load, so using an ad blocker reduces loading times (typically by 50% or more). They also block ad tracking pixels to protect your pr
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Here are a couple of the worst mistakes I’ve seen people make:
Not using an ad blocker
If you aren’t using an ad blocker yet, you definitely should be.
A good ad blocking app will eliminate virtually all of the ads you’d see on the internet before they load.
No more YouTube ads, no more banner ads, no more pop-up ads, etc.
Most people I know use Total Adblock (link here) - it’s about £2/month, but there are plenty of solid options.
Ads also typically take a while to load, so using an ad blocker reduces loading times (typically by 50% or more). They also block ad tracking pixels to protect your privacy, which is nice.
More often than not, it saves even more than 50% on load times - here’s a test I ran:
Using an ad blocker saved a whopping 6.5+ seconds of load time.
Here’s a link to Total Adblock, if you’re interested.
Not getting paid for your screentime
Apps like Freecash will pay you to test new games on your phone.
Some testers get paid as much as £270/game. Here are a few examples right now (from Freecash's website):
You don't need any kind of prior experience or degree or anything: all you need is a smartphone (Android or IOS).
If you're scrolling on your phone anyway, why not get paid for it?
I've used Freecash in the past - it’s solid. (They also gave me a £3 bonus instantly when I installed my first game, which was cool).
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Jafar Mortadha
BSc in Mechanical engineering · Author has 1.4K answers and 3.5M answer views
·8y
Related
What is the eccentricity of a ellipse?
Eccentricity e e of an ellipse is the ratio of the distance between the focus F and a general point Park on the ellipse AND the distance between a general point P and the directrix. For an ellipse, 0 0<e<1<e<1. You can see below what eccentricity means graphically. The vertical and horizontal red dashed lines are the directrices of the ellipse.
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Eccentricity e e of an ellipse is the ratio of the distance between the focus F and a general point Park on the ellipse AND the distance between a general point P and the directrix. For an ellipse, 0 0<e<1<e<1. You can see below what eccentricity means graphically. The vertical and horizontal red dashed lines are the directrices of the ellipse.
Upvote ·
9 9
Robert Paxson
BSME in Mechanical Engineering, Lehigh University (Graduated 1983) · Author has 3.9K answers and 4M answer views
·4y
Related
How is eccentricity e=c/a derived for an ellipse?
Eccentricity is defined as the ratio of the distances from the center to a focus and from the center to a vertex. For an ellipse, this ratio varies between 0 and 1, and indicates the degree of departure from circularity.
This concept can be generalized to any conic section. Given a cone that has a generating angle of β β with its axis of symmetry, and a cutting plane that makes an angle α α with the same axis of the cone, then the eccentricity is:
e=cos α cos β e=cosα cosβ
This relationship can be derived from the geometry of the cone and cutting plane by inscribing a sphere that is
Continue Reading
Eccentricity is defined as the ratio of the distances from the center to a focus and from the center to a vertex. For an ellipse, this ratio varies between 0 and 1, and indicates the degree of departure from circularity.
This concept can be generalized to any conic section. Given a cone that has a generating angle of β β with its axis of symmetry, and a cutting plane that makes an angle α α with the same axis of the cone, then the eccentricity is:
e=cos α cos β e=cosα cosβ
This relationship can be derived from the geometry of the cone and cutting plane by inscribing a sphere that is tangent to both the cone and the cutting plane. This sphere is tangent to the cone at a circle of contact that defines a plane that is perpendicular to the cone’s axis of symmetry and intersects the cutting plane at the line, L L, which is the directrix of the curve produced by that cutting plane. The inscribed sphere is tangent to the cutting plane at a point, F F, the focus. For a cutting plane that produces an ellipse, two such inscribed spheres can be defined with one between the vertex of the cone and the cutting plane and the other on the opposite side of the cutting plane.
For any point P P on the curve produced by the cutting plane, we can project this point down to the plane of the tangent circle along a line parallel to the axis of the cone to define a point Q Q in the plane of the tangent circle. A second line from P P to the vertex of the cone intersects the tangent circle at point A A. Let P D P D be the line from P P that is perpendicular to the directrix L L and intersects L L at point D D. Since P F P F and P A P A are both tangent to the same sphere from the same point P, they must have the same length:
P A=P F P A=P F
Also, we have two right triangles, △P Q A△P Q A and △P Q D△P Q D. from these triangles we observe:
P Q=P A cos(β)P Q=P A cos(β)
and
P Q=P D cos(α)P Q=P D cos(α)
Therefore,
P Q=P A cos(β)=P D cos(α)P Q=P A cos(β)=P D cos(α)
P A P D=c o s(α)cos(β)P A P D=c o s(α)cos(β)
But P A=P F P A=P F
P F P D=c o s(α)cos(β)P F P D=c o s(α)cos(β)
e=P F P D=c o s(α)cos(β)e=P F P D=c o s(α)cos(β)
which is a more general definition for eccentricity that is still consistent with the definition given for an ellipse in the question. But, notice that it becomes the definition for a parabola when α=β α=β such that e=1 e=1.
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Are people who are eccentric narcissists?
What makes the moon's shape an ellipse?
How can we draw an ellipse?
What is the difference between an eccentric and someone who is not considered eccentric?
Are most geniuses eccentric?
Sagar Seth
Engineering Physics from Indian Institute of Technology, Bombay (IITB) (Graduated 2021)
·9y
Related
What is the diameter of an ellipse?
Any line passing through centre of an ellipse is called a diameter.
You might also want to note that they occur in conjugate pairs.
Conjugate diameters bisect the chords parallel to each other.
Bold lines with equation are a pair of conjugate diameters.
Interestingly, for a standard ellipse
x 2 a 2+y 2 b 2=1 x 2 a 2+y 2 b 2=1
Slopes of conjugate diameters say m1 and m2 are related by
m1m2 = −b 2 a−b 2 a
Hope this helps :)
Continue Reading
Any line passing through centre of an ellipse is called a diameter.
You might also want to note that they occur in conjugate pairs.
Conjugate diameters bisect the chords parallel to each other.
Bold lines with equation are a pair of conjugate diameters.
Interestingly, for a standard ellipse
x 2 a 2+y 2 b 2=1 x 2 a 2+y 2 b 2=1
Slopes of conjugate diameters say m1 and m2 are related by
m1m2 = −b 2 a−b 2 a
Hope this helps :)
Upvote ·
99 30
9 3
Joe McAuley
Studied Mechanical Design Engineering&Mathematics at The Ohio State University · Author has 13.1K answers and 7.2M answer views
·5y
Related
What is the eccentricity of an ellipse?
Eccentricity = c/a. It is a measure of how much an ellipse departs from a circle.
c is the distance from the center to one of the two focuses (foci).
a is the distance from one of the focuses to where the semi-minor axis meets the closest perimeter or circumference. Just for your information, the semi-major axis is the distance from the center to the farthest perimeter or circumference.
In a circle,
Continue Reading
Eccentricity = c/a. It is a measure of how much an ellipse departs from a circle.
c is the distance from the center to one of the two focuses (foci).
a is the distance from one of the focuses to where the semi-minor axis meets the closest perimeter or circumference. Just for your information, the semi-major axis is the distance from the center to the farthest perimeter or circumference.
In a circle, the focus is also the center of the circle. a would be the radi...
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Gary Ward
MaEd in Education&Mathematics, Austin Peay State University (Graduated 1997) · Upvoted by
Richard Daugherty
, MA Mathematics Mathematics & History, University of Arkansas, Fayetteville (1968) · Author has 4.9K answers and 7.6M answer views
·1y
Related
How do you calculate eccentricity from two points on an ellipse?
How do you calculate eccentricity from two points on an ellipse?
You don’t. Two points on an ellipse do not tell where the center is or give you the line that the foci lie upon. The possibilities are narrowed down from infinite to infinite.
You need more.
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Terry Moore
Former Statistics lecturer at Massey University (1978–2000) · Author has 16.5K answers and 29.3M answer views
·7y
Related
Why is the eccentricity of an ellipse between 0 and 1?
As Amrit Kumar said, it is the definition of an ellipse. However, you might be wondering why an eccentricity between 0 and 1 leads to a closed curve, and an eccentricity of 1 or greater leads to an open curve (parabola when the eccentricity is 1 or a hyperbola otherwise).
It is easy to see this. Draw two perpendicular lines, one a directrix (one of the two directrices in the case of the ellipse and hyperbola) and the other the axis of the conic. Mark the focus corresponding to the directix on the axis. Now it’s simple geometry to show that the conic crosses the axis at only one point when the e
Continue Reading
As Amrit Kumar said, it is the definition of an ellipse. However, you might be wondering why an eccentricity between 0 and 1 leads to a closed curve, and an eccentricity of 1 or greater leads to an open curve (parabola when the eccentricity is 1 or a hyperbola otherwise).
It is easy to see this. Draw two perpendicular lines, one a directrix (one of the two directrices in the case of the ellipse and hyperbola) and the other the axis of the conic. Mark the focus corresponding to the directix on the axis. Now it’s simple geometry to show that the conic crosses the axis at only one point when the eccentricity is 1 and two points otherwise. You will also see that the two crossing points are on the same side of the directrix when the eccentricity is between 0 and 1 (ellipse) and on opposite sides when the eccentricity is greater than 1 (hyperbola, which has two separate branches).
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Gary Ward
MaEd in Education&Mathematics, Austin Peay State University (Graduated 1997) · Author has 4.9K answers and 7.6M answer views
·11mo
Related
How can you calculate the eccentricity of an ellipse if you know its two semi-major axes?
How can you calculate the eccentricity of an ellipse if you know its two semi-major axes?
If you have a smart phone, then one of you is smart; the internet browser on that device can find an answer quickly.
Eccentricity of Ellipse - Formula, Definition, Derivation, Examples Eccentricity of ellipse is a value lying between 0 and 1. The eccentricity of an ellipse is the ratio of the distance of a point on the ellipse from the focus and from the directrix. Let us learn more about the definition, formula, derivation of eccentricity of ellipse.
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How can we draw an ellipse?
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Are most geniuses eccentric?
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6627 | https://jnm.snmjournals.org/content/59/supplement_1/398 | Relationship between FDG PETCT imaging and CA 125 levels in treated patients with Ovarian cancers - Can FDG PETCT define and predict the disease burden in clinically suspected recurrence ? | Journal of Nuclear Medicine
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Meeting Report Oncology, Clinical Diagnosis Track
Relationship between FDG PETCT imaging and CA 125 levels in treated patients with Ovarian cancers - Can FDG PETCT define and predict the disease burden in clinically suspected recurrence ?
Palaniswamy Shanmuga Sundaram and Subramanyam Padma
Journal of Nuclear Medicine May 2018, 59 (supplement 1) 398;
Palaniswamy Shanmuga Sundaram
1 Nuclear Medicine & PETCT, Amrita Institute of Medical Sciences, Amrita Vishwavidyapeetham Cochin India
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Subramanyam Padma
1 Nuclear Medicine & PETCT, Amrita Institute of Medical Sciences, Amrita Vishwavidyapeetham Cochin India
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Introduction: Cancer antigen 125 (CA-125) is used for monitoring disease status in post treatment patients of ovarian cancer as rising concentrations predict recurrence in most cases. Many studies have demonstrated that 18 F-FDG PET/CT imaging is sensitive & a better modality when compared to conventional anatomical imaging in detection of recurrent disease in patients with raised CA-125 levels.
Objectives: The retrospective study was designed to determine: 1) If FDG PETCT can serve as an accurate maker to define and predict disease burden 2) If there exists any linear relationship between raised CA-125 levels and degree of disease burden 3) A possible CA-125 cutoff value which could predict whether the active disease predominantly involves lymph nodal stations or intra abdominal soft tissue (omental/peritoneal/mesenteric) sites.
Methods: 42 patients (35-74 years, Median 52 years) with suspected disease recurrence based on raised CA-125 levels referred for PET/CT examination were studied. Patients were classified into three groups based on range of CA-125 levels: Group A (35-175), Group B (175-350) and Group C (more than 350). Number of lesions per patient was counted in the following manner: Each lymph nodal station was given a score of 1; Each segment of liver was given a score of 1 irrespective to of number of lesions present in that segment.; Peritoneal soft tissue deposit along the surface of an organ was given a score of 1 and omental, mesenteric and serosal deposits were assigned a score of 1 individually. Data analysis was done using appropriate statistical tests. To arrive at a possible cutoff value for CA-125, various levels of CA-125 were analysed using Chi square test.
Results: PET/CT detected a total of 57 lesions in Group A (n=18), 68 in Group B (n=14) and 65 lesions in Group C (n=10) respectively. No significant statistical difference was found in total no of lesions detected in all the three groups with different levels of CA-125 (One Way ANOVA: p value 0.071). When comparing total no of lesions detected on PET/CT in groups with marked difference in CA-125 levels - Group A (CA-125: 35-175) and Group C (CA-125: >350), no significant statistical difference was seen (Proportional test: p value 0.380). Also, when number of lesions detected on PET/CT in each patient was correlated with the respective CA-125 levels and data analysed using Spearman Correlation coefficient, no correlation (r = 0.342) between the CA-125 levels and disease burden was observed. Scatter diagram given below depicts X axis: number of PET/CT lesions and Y axis: Level of CA-125. Using Chi square test, a cut off CA-125 value of 350 was established to be significant (p value <0.001) in suggesting whether the active disease predominantly involves lymph nodal or intra abdominal soft tissue sites. In our study, lymph nodal metastases were more common in patients with CA-125 value less than/equal to 350 (n=32/42, 76.2%) & soft tissue deposits were more common in patients with CA-125 levels more than 350 (n=10/42, 23.8%).
Conclusions: Whole body FDG PETCT imaging serves as a reliable, accurate marker of disease recurrence in treated cases of Ca Ovary with raised CA 125 levels. There exists no relationship between CA-125 levels and degree of disease burden. This suggests that PETCT can detect higher degree of disease burden even in patients with low levels of raised CA-125 and vice versa. A cutoff of 350 of CA-125 may be useful to predict the predominant disease distribution pattern i.e. lymph nodal stations versus soft tissue deposits in these patients.
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Journal of Nuclear Medicine
Vol. 59, Issue supplement 1
May 1, 2018
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Relationship between FDG PETCT imaging and CA 125 levels in treated patients with Ovarian cancers - Can FDG PETCT define and predict the disease burden in clinically suspected recurrence ?
Palaniswamy Shanmuga Sundaram, Subramanyam Padma
Journal of Nuclear Medicine May 2018, 59 (supplement 1) 398;
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Relationship between FDG PETCT imaging and CA 125 levels in treated patients with Ovarian cancers - Can FDG PETCT define and predict the disease burden in clinically suspected recurrence ?
Palaniswamy Shanmuga Sundaram, Subramanyam Padma
Journal of Nuclear Medicine May 2018, 59 (supplement 1) 398;
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6628 | https://www.quora.com/How-do-you-determine-if-a-constraint-is-binding-or-not-when-solving-a-linear-programming-LP-problem-with-Excel-solver | How to determine if a constraint is binding or not when solving a linear programming (LP) problem with Excel solver - Quora
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How do you determine if a constraint is binding or not when solving a linear programming (LP) problem with Excel solver?
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In Excel Solver, a constraint is binding if at the optimal solution it holds with equality and therefore limits further improvement of the objective. To determine whether a constraint is binding, use the Solver results and the Solver Report (or inspect cell values) and apply these checks:
Check constraint slack / surplus
For ≤ constraints: compute slack = RHS − LHS. Slack = 0 → binding. Slack > 0 → non‑binding.
For ≥ constraints: compute surplus = LHS − RHS. Surplus = 0 → binding. Surplus > 0 → non‑binding.
For equality (=) constraints: if |LHS − RHS| ≈ 0 (within numerical tolera
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In Excel Solver, a constraint is binding if at the optimal solution it holds with equality and therefore limits further improvement of the objective. To determine whether a constraint is binding, use the Solver results and the Solver Report (or inspect cell values) and apply these checks:
Check constraint slack / surplus
For ≤ constraints: compute slack = RHS − LHS. Slack = 0 → binding. Slack > 0 → non‑binding.
For ≥ constraints: compute surplus = LHS − RHS. Surplus = 0 → binding. Surplus > 0 → non‑binding.
For equality (=) constraints: if |LHS − RHS| ≈ 0 (within numerical tolerance), the constraint is binding.
Use the Solver Sensitivity (Answer) report
If you generate the Sensitivity report, the “Allowable Increase/Decrease” and shadow price rows show which constraints are active:
A nonzero shadow price (dual value) typically indicates a binding constraint.
A zero shadow price usually means the constraint is non‑binding (assuming nondegeneracy).
Note: Degeneracy can cause zero duals even when constraint is active; always combine with slack check.
Inspect cells directly after Solver finishes
Evaluate the formula used in the left‑hand side for each constraint and compare to the RHS. Use an absolute tolerance (e.g., 1e−6 or 1e−8) because Solver reports limited precision.
Example test in a worksheet: =ABS(LHS_cell − RHS_cell) < 1E-6 → TRUE means binding.
Practical steps in Excel
Run Solver, check “Answer” and “Sensitivity” reports.
Add explicit slack variables to constraints (convert ≤ to equation by adding slack variable ≥0). If the slack variable = 0 at optimality, the original constraint is binding.
Format or create helper cells that show Slack = RHS − LHS (or vice versa) for quick visual flags.
Caveats
Numerical tolerance: treat very small differences as zero.
Degeneracy and multiple optima: zero shadow price does not always imply non‑binding; verify slack directly.
Integer problems: dual values and sensitivity are not valid for integer constraints; use slack/explicit check instead.
Summary checklist
Compute LHS − RHS (or RHS − LHS depending on sense) and see if it is (near) zero.
Confirm with Solver’s Sensitivity report: nonzero shadow price supports binding status.
Use slack variables or helper cells to make checks explicit and robust to numerical issues.
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More answers below
What is binding constraint in linear programming?
How do you solve fuzzy linear programming in an Excel solver?
What day to day problems can be solved with linear programming?
Are non-negativity constraints considered binding (linear programming)?
What method is used to solve linear programming models with Excel Solver?
Sripad Sambrani
Knows Sanskrit · Author has 6.8K answers and 2.9M answer views
·2y
Binding constraint is an equation in linear programming that satisfies the optimal solution through its value.
Finding the satisfactory optimal solution through the certain value by using the equation in linear programming is known as a binding constraint. It provides the optimized solution after applying few important programming elements of this equation. In case of nonbinding constraint, you can figure it out by seeing no effects on constraints to the final optimized solution.
A binding constraint is a constraint used in linear programming equations whose value satisfies the optimal solution
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Footnotes
Learn about Binding Constraint in Linear Programming
Binding constraint is an equation in linear programming that satisfies the optimal solution through its value.
Finding the satisfactory optimal solution through the certain value by using the equation in linear programming is known as a binding constraint. It provides the optimized solution after applying few important programming elements of this equation. In case of nonbinding constraint, you can figure it out by seeing no effects on constraints to the final optimized solution.
A binding constraint is a constraint used in linear programming equations whose value satisfies the optimal solution; any changes in its value changes the optimal solution.
Trust this helps.
Footnotes
Learn about Binding Constraint in Linear Programming
What Is Binding Constraint?
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There are a lot of AI tools out there right now—so how do you know which ones are actually worth your time? Which tools are built for students and school—not just for clicks or content generation? And more importantly, which ones help you sharpen what you already know instead of just doing the work for you?
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Pedro Henrique González Silva
I work with Linear Programming on a daily basis · Upvoted by
Einszwein
, Master Mathematics & Statistics, Osaka University (2017)
·8y
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What is binding constraint in linear programming?
Complementing Bill Bell’s idea…
I will define non binding constraints as constraints whose changes do not affect the optimal solution.
We can now somehow think about the binding definition.
Supose that you have a problem like the following one:
maximize 5x1 + 4x2 + 6x3
subject to 6x1 + 5x2 + 8x3 <= 16 (c1)
10x1 + 20x2 + 10x3 <= 35 (c2)
0 <= x1, x2, x3 <= 1
In this case we have two binding constraints c1 and c2, which means that in optimality the left hand side is equal to the right hand side (as Bill said). More than that, if we analyse the shadow price of each constraint we will get the benefit of
Continue Reading
Complementing Bill Bell’s idea…
I will define non binding constraints as constraints whose changes do not affect the optimal solution.
We can now somehow think about the binding definition.
Supose that you have a problem like the following one:
maximize 5x1 + 4x2 + 6x3
subject to 6x1 + 5x2 + 8x3 <= 16 (c1)
10x1 + 20x2 + 10x3 <= 35 (c2)
0 <= x1, x2, x3 <= 1
In this case we have two binding constraints c1 and c2, which means that in optimality the left hand side is equal to the right hand side (as Bill said). More than that, if we analyse the shadow price of each constraint we will get the benefit of increasing by one unit the right hand side of each constraint.
So one definition to binding constraint may be: A binding constraint is a constraint whose value satisfies the optimal solution and that any changes in its value changes the optimal solution.
Using this definition we can also conclude that Nonbinding constraints have a shadow price of zero, while binding constraints typically have other shadow prices than zero.
Hope it helped you.
Best Regards
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Virinchi
MS in Analytics, Uconn School of Business (Graduated 2018)
·7y
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What is binding constraint in linear programming?
In simple words, Changing the binding constraint changes the area of the feasible region.
Feasible Region: Graphically, it is the area under the curve. In LP, it is the area under curve or curves - which represent the constraint equations
Example:
You have a manufacturing unit, the monthly ceiling for the production hours becomes the binding constraint. There is always limited time !:)
Non Binding Constraints:
In the same example, the production limit of units manufactured becomes the non binding constraints. because u can manufacture infinite units!. Hence, even the optimality also changes
To Summa
Continue Reading
In simple words, Changing the binding constraint changes the area of the feasible region.
Feasible Region: Graphically, it is the area under the curve. In LP, it is the area under curve or curves - which represent the constraint equations
Example:
You have a manufacturing unit, the monthly ceiling for the production hours becomes the binding constraint. There is always limited time !:)
Non Binding Constraints:
In the same example, the production limit of units manufactured becomes the non binding constraints. because u can manufacture infinite units!. Hence, even the optimality also changes
To Summarize, changing the binding constraints changes the feasible regions not the optimality of the solution
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Related questions
More answers below
What is the difference between solving linear programming (LP) and integer programming (IP) problems using Excel Solver or AMPL?
What is the best way to solve linear programming problems by using Excel Solver? Are there any better alternatives?
When setting up linear programming models in Excel, may the right hand side of constraints be included?
How can you use Excel Solver to determine if a solution is acceptable for a linear programming problem, with a 5% tolerance?
How can we solve linear programming using Excel or MATLAB?
Karthick Gopal
linear programming geometry and algorithms
·9y
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In linear programming, what does "non-binding constraint?" mean?
Generally the constraints in LP can be written of the form
a11x1 + a12x2 +.. + a1nxn <= b1
a21x1 + a22x2 +.. + a2nxn <= b2
..
This is a system of equations and when the solution vector to this equations X is such that some of constraints are not strictly agreeing with equality , ie, X vector will satisfy the one of the above equation in the following manner,
a11x1 + a12x2 +.. + a1nxn < b1 (for given X as solution)
then this constraint is said to be non-binding with respect to this Basic Feasible Solution.
Eg.
3x1 + 4x2 <= 10 is the constraint
and the solution X = (1,1) , satisfies the above constraint
Continue Reading
Generally the constraints in LP can be written of the form
a11x1 + a12x2 +.. + a1nxn <= b1
a21x1 + a22x2 +.. + a2nxn <= b2
..
This is a system of equations and when the solution vector to this equations X is such that some of constraints are not strictly agreeing with equality , ie, X vector will satisfy the one of the above equation in the following manner,
a11x1 + a12x2 +.. + a1nxn < b1 (for given X as solution)
then this constraint is said to be non-binding with respect to this Basic Feasible Solution.
Eg.
3x1 + 4x2 <= 10 is the constraint
and the solution X = (1,1) , satisfies the above constraint bu note it is not strictly equal to 10. so this constraint is non-binding
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Matthew Saltzman
Published researcher and graduate course instructor · Author has 283 answers and 457.3K answer views
·6y
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Are non-negativity constraints considered binding (linear programming)?
If a variable that is constrained to be nonnegative has value zero in some solution, then the nonnegativity constraint is binding in that solution.
An algorithm may treat nonnegativity constraints implicity (by having rules for handling variables at zero) or explicitly (by including x≥0 x≥0 as an explicit row of the constraint matrix), but it’s still a constraint either way.
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Brad Yundt
Mechanical engineer and Excel aficionado · Author has 3.6K answers and 18.4M answer views
·4y
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In Excel, I'm using solver with simplex LP and getting an error that says "linearity conditions not satisfied". I think the problem is IFERROR (B7/$I$7,0). How can I get around this?
Is 0 a sensible value for IFERROR to return? A common use of IFERROR is to avoid DIV/0! error values when the denominator (cell I7) is 0. But I would argue that IFERROR should return a large positive or negative value rather than 0, because that would be the limit of B7/$I$7 as $I$7 approaches 0. In other words, you shot yourself in the foot by using that formula.
The “linearity conditions not satisfied” error message is discussed on pages 136 and 137 of the Reference Guide for Solver, available from its developer frontline systems at
Continue Reading
Is 0 a sensible value for IFERROR to return? A common use of IFERROR is to avoid DIV/0! error values when the denominator (cell I7) is 0. But I would argue that IFERROR should return a large positive or negative value rather than 0, because that would be the limit of B7/$I$7 as $I$7 approaches 0. In other words, you shot yourself in the foot by using that formula.
The “linearity conditions not satisfied” error message is discussed on pages 136 and 137 of the Reference Guide for Solver, available from its developer frontline systems at
The Reference Guide says the error only appears if you check the box for “Assume linear model”—but I don’t see that box in the version of Solver shipped with Microsoft 365. If you have that setting in the Solver Options dialog, try unchecking the box.
The discussion goes on to say that Excel functions like IF break the linearity of the problem. The Evolutionary solution method is better able to tolerate these discontinuities than Simplex LP.
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Matthew Saltzman
Published researcher and graduate course instructor · Author has 283 answers and 457.3K answer views
·6y
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In Excel Solver linear programming, is it possible to use the average of values as a constraint without DIV/0! errors or sacrificing linearity?
I haven’t worked with Excel Solver much, but the arithmetic mean of a bunch of numbers is a linear function. If the data are components of a vector x x then the arithmetic mean is (1/n)e T x(1/n)e T x where e e is a vector of ones. The only division is by n n, which is a parameter and is never zero unless there are no observations.
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Matthew Saltzman
Published researcher and graduate course instructor · Author has 283 answers and 457.3K answer views
·5y
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Can Excel Solver generate both solutions to a linear programming problem with a special "alternative optima" case?
OK I don’t know much about Excel Solver, but I can say this: Most LP solvers find one optimal solution (simplex solvers find a basic optimal solution), and stop. There’s no reason to think the Excel Solver is any different. But you can tell from a report of reduced costs if there are alternative optima and you can either force the solver to move (if you have a command to force a variable into the basis) or persuade the solver to move (by making a small modification to the objective function to ensure that a nonbasic variable with zero reduced cost is slightly attractive.
Update: I should note t
Continue Reading
OK I don’t know much about Excel Solver, but I can say this: Most LP solvers find one optimal solution (simplex solvers find a basic optimal solution), and stop. There’s no reason to think the Excel Solver is any different. But you can tell from a report of reduced costs if there are alternative optima and you can either force the solver to move (if you have a command to force a variable into the basis) or persuade the solver to move (by making a small modification to the objective function to ensure that a nonbasic variable with zero reduced cost is slightly attractive.
Update: I should note that if there is more than one optimal solution then there are infinitely many, although there are only a finite number of basic optimal solutions. The set of all optimal solutions is the subset of the original feasible set in which all variables with positive reduced cost (in standard, minimization form) are fixed at zero.
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Istvan Varga
I write software for living... · Author has 314 answers and 493.7K answer views
·6y
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What are some good linear programming solvers for optimization problems with a large number of constraints?
SCS or splitting cone solver is a first order solver and suitable for massive problems while penalising the quality of solution. You can rework the method making it suitable for MPI.
The best solvers considering quality are based on interior point method and, similarly to SCS, require you to cast the original problem into a SOCP or second order conic program. MOSEK, ECOS, Gurobi fall into this category. These ones are second order methods and may not be the best choice for very large problems.
As always ‘the best/optimal’ is a function of the cost/merit function, which in this case takes the opt
Continue Reading
SCS or splitting cone solver is a first order solver and suitable for massive problems while penalising the quality of solution. You can rework the method making it suitable for MPI.
The best solvers considering quality are based on interior point method and, similarly to SCS, require you to cast the original problem into a SOCP or second order conic program. MOSEK, ECOS, Gurobi fall into this category. These ones are second order methods and may not be the best choice for very large problems.
As always ‘the best/optimal’ is a function of the cost/merit function, which in this case takes the optimiser, the problem set, the platform you running it on as inputs and the output is a pareto front considering used resources: running time, memory, quality of answer.
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Bill Bell
Lives in Canada · Author has 4.1K answers and 6.2M answer views
·10y
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What is binding constraint in linear programming?
An inequality constraint is said to be binding if it holds with equality at the optimal point.
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Jan van Delden
MSc Math and still interested · Author has 4.8K answers and 6.5M answer views
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When formulating a linear programming problem on a spreadsheet, will the data cells show the optimal solution?
A spreadsheet is rather general. I can give you some hints when working with Excel.
You should set up your problem by modeling your inequalities in terms of the variables x k x k you may vary. For these you must specify constraints. Furthermore you might have linear functions y m y m that may depend on a subset of these x k x k for which you also might want to specify constraints. Lastly you’ll need a linear function z z depending on all of these x k x k for which you want to find a minimum or maximum, your goal.
I would specify the potential values of these x k x k in one row. The constraints, the minimum and maximu
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A spreadsheet is rather general. I can give you some hints when working with Excel.
You should set up your problem by modeling your inequalities in terms of the variables x k x k you may vary. For these you must specify constraints. Furthermore you might have linear functions y m y m that may depend on a subset of these x k x k for which you also might want to specify constraints. Lastly you’ll need a linear function z z depending on all of these x k x k for which you want to find a minimum or maximum, your goal.
I would specify the potential values of these x k x k in one row. The constraints, the minimum and maximum values in the rows below the first row, maybe the coefficients of the goal function z z below that one, and all coefficients of the linear functions y m y m in the rows underneath.
At the right hand side compute the value of z,y m z,y m with the command SUMPRODUCT. Maybe you also have a constraint on the sum of the x k x k, so compute that as well (same column).
To the right hand side of this column, include two columns in which you specify the bounds that belong to the constraints of x,y m x,y m. Preferably a lower and upper bound. You may always choose a ridiculous large upper or lower bound, if you don’t need those for a particular y m y m (It will give less errors than leaving these cells blank, which is actually a toptip).
Now you are good to go.
In Excel you may install an Addin with the name: Solver. It will appear in the menu-bar with the title: Data (if I’m not mistaken). Here you could specify which cells specify constraints (Subject to the constraints) and which cell specifies your objective (Set objective) and which cells must be changed in order to achieve this objective (By changing variable cells). There are a few options to adjust the process in finding a solution.
Note that my layout suggestion of your spreadsheet makes sure that you could specify your constraints by adding four rules. Two for the constraints on x k x k and two for all constraints on y m y m in one go.
After specification of the whole model, you may run your model. If you are very lucky, it will find a solution in one go. You may choose between different reports: Answer (which you may separately save), Sensitivity or Limits.
If you are unlucky, you won’t find a solution (for instance because the constraints are too strict or contradict each other), you have the option to study the feasibility report. This will actually show which constraint is causing the problem (if more than one constraint is problematic, you may have to increase the bounds iteratively on more than one constraint in order to find all potential constraints that are problematic).
Sometimes one is very unlucky and the feasibility report doesn’t show up at all. This could be due to a small error in the specification of the model, check carefully, did I use the correct decimal separator everywhere, do I always use the correct range in my formulas, etc. Or your model specification is fine, but one should use ‘common sense’ to see which constraint is the root cause. (This is not always easy to spot).
There are some lovely explanations on the Internet. A simple example, explaining the interface and installation of the Solver is here:
How to use Solver in Excel with examples
Don’t worry that it says Excel 2003. You’ll be fine.
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What is an equality constraint in linear programming? Can we have inequality constraints in non-linear programming problems as well?
Certainly. If an ad designer wants to “fit” an ad into a time slot, s/he can elect to have “dead” time (unused time before and/or after the ad), which is an inequality constraint. Or the designer may want to fill the time EXACTLY, with no time left over. Bryant Gumbel became a go-to “talking head”, in both sports and “hard news” categories, through his ability to fill a “sound bite” (30 seconds) w
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Certainly. If an ad designer wants to “fit” an ad into a time slot, s/he can elect to have “dead” time (unused time before and/or after the ad), which is an inequality constraint. Or the designer may want to fill the time EXACTLY, with no time left over. Bryant Gumbel became a go-to “talking head”, in both sports and “hard news” categories, through his ability to fill a “sound bite” (30 seconds) with positively NO time left over, either before or after. As for the definition of “constraint”, it’s a rule or restriction on the solution. In th...
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Oxidation Number v.s. Formal Charge – MCAT Content
Table of Contents Learning
8 minutes remaining - you got this!
I. Introduction to Oxidation Number and Formal Charge
Oxidation Number
Formal Charge
II. Calculating Oxidation Numbers and Formal Charges
Oxidation Number Calculation Rules
Formal Charge Calculation
III. Importance of Oxidation Number and Formal Charge in Reactions
IV. Comparing Oxidation Number and Formal Charge
V. Practical Applications
VI. Bridge/Overlap
A. Electron Configurations
B. Chemical Bonding
C. Reaction Mechanisms
VII. Wrap-Up and Key Terms
Key Terms
VIII. Practice Questions
Sample Practice Question 1
Sample Practice Question 2
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Understanding the concepts of oxidation number and formal charge is essential in general chemistry. These concepts help predict how molecules interact and react. They also help us understand the behavior of atoms in molecules during chemical reactions.
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I. Introduction to Oxidation Number and Formal Charge
Oxidation Number
An oxidation number (or oxidation state) represents the total number of electrons an atom gains or loses to form a chemical bond. It helps identify the electron density around an atom in a molecule. Knowing the oxidation number is important because it tells us about the atom's electron control in a molecule.
Formal Charge
The formal charge is the charge an atom would have if all the electrons in a covalent bond were shared equally between the bonded atoms. It helps to determine the most stable Lewis structure for a molecule. By knowing the formal charge, we can predict a molecule's most likely arrangement of atoms.
II. Calculating Oxidation Numbers and Formal Charges
To fully understand oxidation numbers and formal charges, it's important to know how to calculate them accurately. This section explains the rules and provides examples.
Oxidation Number Calculation Rules
Elemental Form: The oxidation number of an atom in its elemental form is always 0. For example, O₂ or N₂.
Monatomic Ions: The oxidation number is equal to the charge of the ion. For instance, Na⁺ has an oxidation number of +1.
Oxygen: Usually has an oxidation number of -2, except in peroxides (like H₂O₂) where it is -1.
Hydrogen: Generally has an oxidation number of +1 when bonded to non-metals and -1 when bonded to metals.
Fluorine: Always has an oxidation number of -1 in its compounds.
Sum Rule: The sum of oxidation numbers in a neutral molecule is 0; in an ion, it equals the ion's charge.
Example:Calculate the oxidation number of sulfur in H₂SO₄.
Hydrogen (H) =+1, so 2 H = +2
Oxygen (O) =-2, so 4 O = -8
Let sulfur be x:
The equation:
2(+1) + x + 4(-2) = 0
Solving:
2 + x - 8 = 0
x = +6
Formal Charge Calculation
Formal Charge = Valence electrons - ½ (Bonding electrons) — Non-bonding electrons
Example: Determine the formal charge of oxygen in H₂O.
Valence electrons of O = 6
Bonding electrons (2 bonds with H) = 4
Non-bonding electrons (2 lone pairs) = 4
Formal charge:
6-4/2-4 = 0
III. Importance of Oxidation Number and Formal Charge in Reactions
Understanding how oxidation numbers and formal charges work is crucial in predicting the outcome of chemical reactions. This section explores their roles in different types of reactions.
Oxidation Numbers in Redox Reactions: Oxidation numbers help identify which species are oxidized and reduced in redox reactions. Oxidation involves an increase in oxidation number, while reduction involves a decrease.
Example:Consider the reaction:
2Mg (s) + O 2 (g) → 2MgO (s)
Mg goes from 0 (elemental) to +2 (MgO).
O₂ goes from 0 (elemental) to -2 (MgO).
Mg is oxidized, O₂ is reduced.
Formal Charges in Lewis Structures: Formal charges help determine molecules' most stable Lewis structure. This is crucial for predicting molecular geometry and reactivity.
Example:Compare possible Lewis structures for CO₂.
Structure 1: O=C=O, where each O is 0 and C is 0.
Structure 2: O≡C-O, where the left O is -1, C is +1, and the right O is 0.
The first structure is more stable as the formal charges are minimized.
IV. Comparing Oxidation Number and Formal Charge
While oxidation numbers and formal charges serve different purposes, they often lead to similar conclusions about the electron distribution in molecules. However, oxidation numbers are used mainly in redox chemistry, while formal charges are used for drawing and evaluating Lewis structures.
Example:In sulfate ion (SO₄²⁻):
The oxidation number of sulfur is +6.
Formal charges: Each oxygen typically has a formal charge of -1, and sulfur has +2.
V. Practical Applications
Oxidation numbers and formal charges are not just theoretical concepts. They have practical applications in various fields of chemistry, and this section highlights some of these applications.
Oxidation Numbers in Industrial Chemistry: Used in processes like refining metals and manufacturing chemicals. For example, understanding oxidation states is essential for efficient conversion in the production of sulfuric acid.
Formal Charges in Organic Synthesis: Crucial for predicting organic compounds' most likely structures and reactivity. For example, minimizing formal charges in designing pharmaceuticals helps ensure stability and effectiveness.
VI. Bridge/Overlap
Understanding oxidation numbers and formal charges is fundamental for grasping more complex topics in chemistry. These include:
A. Electron Configurations
Knowledge of how electrons are distributed in atoms and molecules helps predict oxidation states and formal charges.
B. Chemical Bonding
Understanding these concepts aids in learning about covalent, ionic, and metallic bonds, as well as molecular geometry.
C. Reaction Mechanisms
Helps in understanding how and why chemical reactions occur, which is essential for fields like biochemistry and pharmacology. For example, in nucleophilic substitution and electrophilic addition reactions.
VII. Wrap-Up and Key Terms
Oxidation numbers help us track electron transfer in reactions, which is key for redox processes. Formal charges help us determine the most stable structures of molecules. Both concepts are vital for understanding chemical reactivity and stability. They are essential tools for anyone studying or working in chemistry.
Key Terms
Oxidation Number: Indicates electron loss or gain.
Formal Charge: Indicates electron sharing in bonds.
Redox Reactions: Involve changes in oxidation states.
Lewis Structures: Help visualize molecular structure and stability.
VIII. Practice Questions
Sample Practice Question 1
What is the oxidation number of nitrogen in NO₃⁻?
A. +3
B. +4
C. +5
D. +6
Click to reveal answer
Ans. C
The total oxidation state of the nitrate ion (NO₃⁻) must equal -1. Oxygen has an oxidation number of -2. With three oxygens, the total is -6. Nitrogen must have an oxidation number of +5 (-6 + 5 = -1) to balance this.
Sample Practice Question 2
Calculate the formal charge of chlorine in ClO₄⁻.
A. -1
B. 0
C. +1
D. +3
Click to reveal answer
Ans. D
Chlorine has 7 valence electrons. In ClO₄⁻, it shares electrons with four oxygens (8 bonding electrons), and there are no lone pairs in chlorine. Formal charge: 7−8/2−0=+3.
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6630 | https://www.brainkart.com/article/Condition-for-the-line-y---mx---c-to-be-a-tangent-to-the-circle-and-finding-the-point-of-contact_39159/ | | | |
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Home | | Maths 12th Std | Condition for the line y = mx + c to be a tangent to the circle and finding the point of contact
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Formula, Solved Example Problems - Condition for the line y = mx + c to be a tangent to the circle and finding the point of contact | 12th Mathematics : UNIT 5 : Two Dimensional Analytical Geometry II
Posted On : 15.05.2019 12:18 pm
Chapter: 12th Mathematics : UNIT 5 : Two Dimensional Analytical Geometry II
Condition for the line y = mx + c to be a tangent to the circle and finding the point of contact
Condition for the line y = mx + c to be a tangent to the circle and finding the point of contact
Condition for the line y= mx + c to
be a tangent to the circle x2+ y2 = a2and
finding the point of contact
Let the line y = mx + c touch the circle x2
+ y2 = a2 . The centre and radius of the
circle x2 + y2
= a2 are (0, 0) and a respectively.
(i)Condition for a line to be tangent
Then the perpendicular distance of the line y − mx −
c = 0 from (0, 0) is
This must be equal to radius .Therefore = a or c2
= a2 (1+ m2) .
Thus the condition for the line y = mx + cto
be a tangent to the circle x2+ y2= a2 is c2= a2
(1 + m2).
(ii) Point of contact
Let (x1 , y1 ) be the the
point of contact of y = mx + c with the circle x2
+ y2 = a2,
Then y1 = mx1 + c ……….(1)
Equation of tangent at (x1 , y1
) is xx1 +
yy1= a2
.
yy1= -xx1+ a2 ... (2)
Equations (1) and (2) represent the same line and hence the coefficients are proportional.
Then the points of contact is either
Note
The equation of tangent at P to a circle is y = mx ±
a √[1 +m2]
Theorem 5.4
From any point outside the circle x2 + y2
= a2 two tangents can be drawn.
Proof
Let P(x1 , y1 ) be a point outside the
circle. The equation of the tangent is
It passes through (x1, y1) . Therefore
Squaring both sides, we get
( y1 – mx1)2 = a2 (1+ m2 )
y12 + m2 x12 - 2mx1y1- a2
- a2m2 = 0
m2 (x12
- a2 ) - 2mx1y1+ ( y12
- a2 ) = 0 .
This quadratic equation in m gives two values for m .
These values give two tangents to the circle x2
+ y2 = a2.
Note
(1) If (x1, y1 ) is a point outside the circle, then both the
tangents are real.
(2) If (x1, y1 ) is a
point inside the circle, then both the tangents are imaginary.
(3) If (x1, y1 ) is a
point on the circle, then both the tangents coincide.
Example 5.11
Find the equations of the tangent and normal to the circle x2
+ y2 = 25 at P(−3, 4) .
Solution
Equation of tangent to the circle at P(x1
, y1 ) is xx1 + yy1 = a2
That is, x(-3) + y(4) = 25
-3x + 4 y = 25
Equation of normal is xy1 - yx1
= 0
That is, 4x + 3y = 0 .
Example 5.12
If y = 4x + c is a tangent to the circle x2
+ y2 = 9 , find c.
Solution
The condition for the line y = mx + c to be
a tangent to the circle x2 + y2 = a2
is c2 = a2 (1+ m2).
Then, c = ± √[9(1+16)]
c = ±3 √17.
Example 5.13
A road bridge over an irrigation canal have two semi circular
vents each with a span of 20m and the supporting pillars of width 2m .
Use Fig.5.16 to write the equations that represent the semi-verticular vents.
Solution
Let O1 O2 be
the centres of the two semi circular vents.
First vent with centre O1 (12, 0) and radius r = 10 yields equation to first semicircle as
(x -12)2 + ( y - 0)2 = 102
Þ x2
+ y2 - 24x + 44 = 0 , y > 0 .
Second vent with centre O2 (34, 0) and radius r= 10 yields equation to second vent as
(x - 34)2 + y2 = 102
Þ x2
+ y2 - 68x +1056 = 0 , y > 0 .
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Tags : Formula, Solved Example Problems , 12th Mathematics : UNIT 5 : Two Dimensional Analytical Geometry II
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Mathematics : UNIT 5 : Two Dimensional Analytical Geometry II : Condition for the line y = mx + c to be a tangent to the circle and finding the point of contact | Formula, Solved Example Problems
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Related Topics
Maths 12th Std - TN 12th Maths (English Medium) | Questions with Answers, Solution
TN State Board School - All Subjects
12th Standard - All Subjects
12th Mathematics : UNIT 5 : Two Dimensional Analytical Geometry II
Introduction - Two Dimensional Analytical Geometry II
Circle - Definition
Equation of a circle in standard form - Formula, Solved Example Problems
Equations of tangent and normal at a point P on a given circle
Condition for the line y = mx + c to be a tangent to the circle and finding the point of contact - Formula, Solved Example Problems
Circle: Solved Example Problems - with Answers, Solution
Exercise 5.1: Circle - Problem Questions with Answer, Solution
Conics - Definition
The general equation of a Conic
Parabola - Equation, Definition, Example, Solution, vertex
Ellipse - Equation, Definition, Theorem, Proof, Example, Solution, Types
Hyperbola - Equation, Definition, Theorem, Proof, Types, Solved Example Problems, Solution
Exercise 5.2: Conics - Problem Questions with Answer, Solution
Conic Sections
Exercise 5.3: Conic Sections - Problem Questions with Answer, Solution
Parametric form of Conics
Tangents and Normals to Conics - Equation, Definition, Theorem, Proof, Solved Example Problems
Exercise 5.4: Tangents and Normals to Conics - Problem Questions with Answer, Solution
Real life Applications of Conics
Exercise 5.5: Real life Applications of Conics - Problem Questions with Answer, Solution
Solved Example Problems on Real life Applications of Conics
Choose the correct Answer - Two Dimensional Analytical Geometry II
Summary - Equation, Formula | Two Dimensional Analytical Geometry II
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6631 | https://www.ncbi.nlm.nih.gov/books/NBK582989/ | Topical Tretinoin - MotherToBaby | Fact Sheets - NCBI Bookshelf
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MotherToBaby | Fact Sheets [Internet]. Brentwood (TN): Organization of Teratology Information Specialists (OTIS); 1994-.
MotherToBaby | Fact Sheets [Internet].
Show details
Brentwood (TN): Organization of Teratology Information Specialists (OTIS); 1994-.
Search term
Topical Tretinoin
Published online: July 2024.
This sheet is about using topical tretinoin in pregnancy and while breastfeeding. This information is based on available published literature. It should not take the place of medical care and advice from your healthcare provider.
What is tretinoin?
Topical tretinoin is a medication that is applied to the skin to treat acne, prevent wrinkles, and help with other skin problems. Some brand names for tretinoin include Altreno®, Atralin®, Avita®, Retin-A®, Refissa®, Renova® and Tretin-X®. The amount of tretinoin in each brand can vary.
Tretinoin belongs to a group of medications called the retinoids. Retinoids are related to Vitamin A. Other medications in the retinoid family are isotretinoin (Accutane®, Claravis®), acitretin (Soriatane®) and adapalene (Differin®). MotherToBaby has a fact sheet on isotretinoin here:
Tretinoin is also available in an oral form (to take by mouth) for the treatment of leukemia. This sheet will discuss the topical (applied to skin) use of tretinoin.
My healthcare provider said that tretinoin is like isotretinoin. I’ve heard that it should not be used during pregnancy.
Tretinoin is related to a medication called isotretinoin. Isotretinoin is a medication known to cause birth defects involving the face, heart and brain. However, isotretinoin is taken by mouth and easily enters a person’s bloodstream at higher levels than with topical use of tretinoin. When tretinoin is applied to the skin, lower levels pass through the skin and get into the bloodstream than with oral (taken by mouth) isotretinoin.
In general, skin serves as a good barrier. Because of this, only a small amount of the tretinoin is likely to be absorbed with topical (skin) exposure when used as directed. More tretinoin could be absorbed into the person’s bloodstream if tretinoin is used on skin that is broken or irritated, or when it is used more than needed, or when used over a large area of the body. In general, the less tretinoin that is used on the skin, the less likely there will be risks to the fetus. However, because there might still be a small amount of tretinoin absorbed through the skin, the safest approach may be to avoid use of tretinoin during pregnancy.
I use topical tretinoin. Can it make it harder for me to get pregnant?
Studies have not been done to see if tretinoin can make it harder to become pregnant.
I am using tretinoin, but I would like to stop using it before becoming pregnant. How long does it stay in my body?
The time it takes the body to metabolize (to process) medication is not the same for everyone. In healthy non-pregnant adults, it takes up to 1 day, on average, for most of the tretinoin to be gone from the body. The makers of oral isotretinoin suggest that females stop using isotretinoin 1 month before trying to get pregnant. Based on this suggestion for isotretinoin, it may be suggested to stop using tretinoin 1 month before trying to get pregnant.
Does using topical tretinoin increase the chance of miscarriage?
Miscarriage is common and can occur in any pregnancy for many different reasons. The topical use of tretinoin is not expected to increase the chance for miscarriage.
Does taking tretinoin increase the chance of birth defects?
Birth defects can happen in any pregnancy for different reasons. Out of all babies born each year, about 3 out of 100 (3%) will have a birth defect. Several studies have tried to learn if use of topical tretinoin could harm a pregnancy. These studies have not found a greater chance for birth defects with proper application of tretinoin to the skin.
However, there have been case reports of babies born with birth defects after women used tretinoin on their skin during pregnancy. Usually, a few case reports do not cause healthcare providers to worry, but the birth defects reported in these case reports are similar to the birth defects seen in babies exposed to oral isotretinoin use during pregnancy. Since tretinoin and isotretinoin are related, it is possible that these two medications can affect the baby in the same way. Because many individuals have used tretinoin during pregnancy and have not had babies with a birth defect, the chance for birth defects is probably low. However, it has generally been recommended not to use tretinoin in pregnancy.
Does using tretinoin in pregnancy increase the chance of other pregnancy related problems?
Studies that have looked at this question have not reported a greater chance for preterm delivery (delivery before 37 weeks of pregnancy) with proper application of tretinoin to the skin.
If I stop using tretinoin in the first trimester, is it okay to start using it again later in my pregnancy?
During the first 3 months of pregnancy, the baby’s organs are forming. In months 4 through 9, the baby’s body and brain are growing. Tretinoin use in the 2nd and 3rd trimesters is less likely to cause a birth defect. However, until more information is available, avoiding this product throughout pregnancy might be the best course of action. For general information on the timing of exposures in pregnancy, see the MotherToBaby fact sheet on critical periods of development at
Does using tretinoin in pregnancy affect future behavior or learning for the child?
Studies have not been done to see if tretinoin can cause behavior or learning issues for the child.
Breastfeeding while using tretinoin:
Tretinoin use during breastfeeding has not been studied. However, when used on your skin, very little tretinoin passes into your body, and so the amount in breast milk would probably be small. Be sure to talk to your healthcare provider about all of your breastfeeding questions.
If a man used topical tretinoin, could it affect fertility or increase the chance of birth defects in a partner’s pregnancy?
Studies have not been done to see if topical tretinoin use could affect male fertility (ability to get a woman pregnant) or increase the chance of birth defects. In general, exposures that fathers or sperm donors have are unlikely to increase risks to a pregnancy. For more information, please see the MotherToBaby fact sheet Paternal Exposures at
Selected References:
Camera G and Pregliasco P. 1992. Ear malformation in baby born to mother using isotretinoin cream. Lancet, 339:687. [PubMed: 1347386]
De Wals P. et al. 1991. Association between holoprosencephaly and exposure to topical retinoids: results of the EUROCAT survey. Paediatr Perinat Epidemiol, 5:445-447. [PubMed: 1754503]
Jick S, et al. 1993. First trimester topical tretinoin and congenital disorders. Lancet, 341:1181-1182. [PubMed: 8098078]
Kaplan YC, et al. 2015. Pregnancy outcomes following first-trimester exposure to topical retinoids: A systematic review and meta-analysis. Br J Dematol, 173:1132-1141. [PubMed: 26215715]
Loureiro KD, et al. 2005. Minor malformations characteristic of the retinoic acid embryopathy and other birth outcomes in children of women exposed to topical tretinoin during early pregnancy. American Journal of Medical Genetics, 136A:117-121. [PubMed: 15940677]
Lipson AH, et al. 1993. Multiple congenital defects associated with maternal use of topical tretinoin. Lancet, 341:1352-1353. [PubMed: 8098487]
MacDonald SC, et al. 2019. Identifying pregnancies in insurance claims data: Methods and application to retinoid teratogenic surveillance. Pharmacoepidemiol Drug Saf 28(9):1211-1221. [PMC free article: PMC6830505] [PubMed: 31328328]
Panchaud A, et al. 2012. Pregnancy outcome following exposure to topical retinoids: a multicenter prospective study. J Clin Pharmacol, 52:1844-1851. [PubMed: 22174426]
Ross SA, et al. 2000. Retinoids in embryonal development. Physiol Rev, 80(3):1021-1054. [PubMed: 10893430]
Selcen D, et al. 2000. Otocerebral anomalies associated with topical tretinoin use. Brain Dev 22:218-220. [PubMed: 10838106]
Shapiro L, et al. 1997. Safety of first trimester exposure to topical tretinoin: prospective cohort study. Lancet, 350:1143-1144. [PubMed: 9343507]
Veraldi S, et al. 2016. Are topical retinoids teratogenic? G Ital Dermatol Venereol. 151(6):700-705. [PubMed: 27598619]
Williams AL et al. 2020. Teratogen update: Topical use and third-generation retinoids. Birth Defects Res, 112(15):1105-1114. [PubMed: 32643315]
Copyright by OTIS, July 1, 2024.
This work is available under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported license (CC BY-NC-ND 3.0)
Bookshelf ID: NBK582989 PMID: 35952261
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6632 | https://stats.stackexchange.com/questions/187335/validation-error-less-than-training-error | Skip to main content
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I found two questions here and here about this issue but there is no obvious answer or explanation yet.I enforce the same problem where the validation error is less than training error in my Convolution Neural Network. What does that mean?
machine-learning
mathematical-statistics
neural-networks
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edited Apr 13, 2017 at 12:44
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asked Dec 17, 2015 at 22:04
BidoBido
1,32333 gold badges99 silver badges55 bronze badges
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I don't think this question can be answered without knowing the absolute number of training (cv) and test cases as well as the variance observed for MSE for both cross validation and test.
cbeleites
–
cbeleites
2015-12-21 10:47:51 +00:00
Commented
Dec 21, 2015 at 10:47
1
shuffle the data
Samuel Weisenthal
–
Samuel Weisenthal
2017-05-04 22:16:58 +00:00
Commented
May 4, 2017 at 22:16
What do we infer from this? Yes, its generated from a dense network with dropout and batchnorm layers.
Srinath
–
Srinath
2019-06-18 21:20:56 +00:00
Commented
Jun 18, 2019 at 21:20
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It is difficult to be certain without knowing your actual methodology (e.g. cross-validation method, performance metric, data splitting method, etc.).
Generally speaking though, training error will almost always underestimate your validation error. However it is possible for the validation error to be less than the training. You can think of it two ways:
Your training set had many 'hard' cases to learn
Your validation set had mostly 'easy' cases to predict
That is why it is important that you really evaluate your model training methodology. If you don't split your data for training properly your results will lead to confusing, if not simply incorrect, conclusions.
I think of model evaluation in four different categories:
Underfitting – Validation and training error high
Overfitting – Validation error is high, training error low
Good fit – Validation error low, slightly higher than the training error
Unknown fit - Validation error low, training error 'high'
I say 'unknown' fit because the result is counter intuitive to how machine learning works. The essence of ML is to predict the unknown. If you are better at predicting the unknown than what you have 'learned', AFAIK the data between training and validation must be different in some way. This could mean you either need to reevaluate your data splitting method, adding more data, or possibly changing your performance metric (are you actually measuring the performance you want?).
EDIT
To address the OP's reference to a previous python lasagne question.
This suggests that you have sufficient data to not require cross-validation and simply have your training, validation, and testing data subsets. Now, if you look at the lasagne tutorial you can see that the same behavior is seen at the top of the page. I would find it hard to believe the authors would post such results if it was strange but instead of just assuming they are correct let's look further. The section of most interest to us here is in the training loop section, just above the bottom you will see how the loss parameters are calculated.
The training loss is calculated over the entire training dataset. Likewise, the validation loss is calculated over the entire validation dataset. The training set is typically at least 4 times as large as the validation (80-20). Given that the error is calculated over all samples, you could expect up to approximately 4X the loss measure of the validation set. You will notice, however, that the training loss and validation loss are approaching one another as training continues. This is intentional as if your training error begins to get lower than your validation error you would be beginning to overfit your model!!!
I hope this clarifies these errors.
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edited Nov 21, 2017 at 16:15
Abhinav Upadhyay
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answered Dec 18, 2015 at 13:51
cdetermancdeterman
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Nice answer. There is also the possibility that there is a bug in the code which makes it possible that training has not converged to the optimal soluion on the training set. Or,if the training objective is non convex and the training algorith converges to a local minimum that happens to be good for the validation set.
Sobi
–
Sobi
2015-12-18 14:49:46 +00:00
Commented
Dec 18, 2015 at 14:49
@cdeterman thanks.I use RMSE as a performance metric. I've divided my data into 20% for test and 80% for training and validation (20% of training data is cross validated to compute the validation error ). Actually, the Validation error is low, slightly lower than the training error. The test error is higher than training and validation errors. We can find a similar case in MNISTdataset for handwriting recognition stats.stackexchange.com/questions/178371/…
Bido
–
Bido
2015-12-18 22:34:06 +00:00
Commented
Dec 18, 2015 at 22:34
Great explanation, if you could add a few graphs - it would be the best one possible
taras
–
taras
2019-02-28 14:09:39 +00:00
Commented
Feb 28, 2019 at 14:09
Can cross validation such problem?
Elbek
–
Elbek
2019-11-20 04:58:00 +00:00
Commented
Nov 20, 2019 at 4:58
2
Consider this brilliant article about two other possible reasons for minor underfitting. It also features the desired Images. Regularization applied during training, but not during validation/testing and Training loss is measured during each epoch while validation loss is measured after each epoch from here: pyimagesearch.com/2019/10/14/…
Cadoiz
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Cadoiz
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Aug 10, 2020 at 3:08
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One possibility: If you are using dropout regularization layer in your network, it is reasonable that the validation error is smaller than training error. Because usually dropout is activated when training but deactivated when evaluating on the validation set. You get a more smooth (usually means better) function in the latter case.
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answered Apr 6, 2016 at 14:02
D-KD-K
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What a simple, sensible answer!
rajb245
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rajb245
2017-03-06 15:12:18 +00:00
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Mar 6, 2017 at 15:12
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Yes this should be marked as correct answer indeed.
Simanas
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Simanas
2017-06-30 11:40:01 +00:00
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Jun 30, 2017 at 11:40
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I removed my dropout layer, but still see the validation loss lower than the training loss initially! (I am not specifying any regularization on the layers, either!)
Josiah Yoder
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Josiah Yoder
2018-07-27 18:43:53 +00:00
Commented
Jul 27, 2018 at 18:43
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@MiloMinderbinder That's an interesting observation. No, I haven't figured anything else out on this yet.
Josiah Yoder
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Josiah Yoder
2018-12-13 21:07:29 +00:00
Commented
Dec 13, 2018 at 21:07
1
If its the dropout that is really the culprit, is the resulting model really viable ? Is it advisable to reduce the dropout such that this phenomenon disappears ?
Kanmani
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Kanmani
2019-09-08 22:46:46 +00:00
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Sep 8, 2019 at 22:46
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I don't have enough points to comment on @D-K's answer, but this is now answered as a FAQ on Keras' documentation:
Why is my training loss much higher than my testing loss?
A Keras model has two modes: training and testing. Regularization mechanisms, such as Dropout and L1/L2 weight regularization, are turned off at testing time. They are reflected in the training time loss but not in the test time loss.
Besides, the training loss that Keras displays is the average of the losses for each batch of training data, over the current epoch. Because your model is changing over time, the loss over the first batches of an epoch is generally higher than over the last batches. This can bring the epoch-wise average down. On the other hand, the testing loss for an epoch is computed using the model as it is at the end of the epoch, resulting in a lower loss.
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edited Mar 3, 2021 at 0:20
tuomastik
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answered Jan 29, 2018 at 12:33
dterdter
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This does not entirely answer the question either. With dropout disabled, I still see the validation loss about half of the training loss for several epochs in a row!
Josiah Yoder
–
Josiah Yoder
2018-07-27 18:54:36 +00:00
Commented
Jul 27, 2018 at 18:54
Is your training data representative of the dev data?
dter
–
dter
2018-09-17 10:10:52 +00:00
Commented
Sep 17, 2018 at 10:10
I randomly split the dataset into training and testing. It visually appeared to be a good sample. I was working on a regression problem where the best classifiers were only slightly better than always predicting the mean value.
Josiah Yoder
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Josiah Yoder
2018-09-17 13:58:26 +00:00
Commented
Sep 17, 2018 at 13:58
Your answer does not talk about training loss being greater than the validation loss which is the question that was asked. You are more focused on Training loss and test loss
nzy
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nzy
2019-05-19 22:06:11 +00:00
Commented
May 19, 2019 at 22:06
this answers the wrong question
cosmosa
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cosmosa
2022-11-12 18:23:15 +00:00
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Nov 12, 2022 at 18:23
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my 2 cents:
I also had the same problem even without having dropout layers.
In my case - batch-norm layers were the culprits.
When I deleted them - training loss became similar to validation loss.
Probably, that happened because during training batch-norm
uses mean and variance of the given input batch, which might be different
from batch to batch. But during evaluation batch-norm uses running mean and variance, both of which reflect properties of the whole training set much better than mean and variance of a single batch during training.
At least, that is how batch-norm is implemented in pytorch
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answered Aug 27, 2018 at 11:36
Mans007Mans007
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Thank @Mans007, this had happened to me and I was using Keras. The batch-norm layers were the cause.
Roei Bahumi
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Roei Bahumi
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Another possibility that combines both the answer of @cdeterman and @D-K in some way is if you're using some data augmentation mechanism. Infact data augmentation is usually done only on training set and not on validation set (as for the dropout regularization), and this may lead to a validation set containing "easier" cases to predict than those in the training set.
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answered Sep 26, 2018 at 11:47
se7entyse7ense7entyse7en
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That's what I am experiencing now.
Imran Kocabiyik
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Imran Kocabiyik
2021-04-06 16:43:20 +00:00
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Apr 6, 2021 at 16:43
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At this time, stochastic gradient based methods are almost always the algorithm of choice for deep learning. This means that data comes in as batches, gradients are computed and parameters are updated. This means you can also compute the loss over the data as each batch is selected. Under this framework, there are two ways in how the loss is computed that I can think of which can lead to this phenomenon that the training error is greater than the validation error. Below, I show that Keras does, in fact, appear to compute the in-sample errors in these ways.
1.) Training error is averaged over whole epoch, rather all at once at the end of the epoch, but validation error is only at end of epoch. As we sample our training data to compute gradients, we might as well compute the loss over them as well. But since the validation data is not used during the computation of gradients, we may decide to only compute the loss after the end of the epoch. Under this framework, the validation error has the benefit of being fully updated, while the training error includes error calculations with fewer updates. Of course, asymptotically this effect should generally disappear, since the effect on the validation error of one epoch typically flattens out.
2.) Training error is computed before batch update is done. In a stochastic gradient based method, there's some noise the gradient. While one is climbing a hill, there's a high probability that one is decreasing global loss computed over all training samples. However, when one gets very close to the mode, the update direction will be negative with respect to the samples in your batch. But since we are bouncing around a mode, this means on average we must be choosing a direction that is positive with respect to the samples out of batch. Now, if we are about to update with respect to the samples in a given batch, that means they have been pushed around by potentially many batch updates that they were not included in, by computing their loss before the update, this is when the stochastic methods have pushed the parameters the most in favor of the other samples in your dataset, thus giving us a small upward bias in the expected loss.
Note that while asymptotically, the effect of (1) goes away, (2) does not!
Below I show that Keras appears to do both (1) and (2).
(1) Showing that metrics are averaged over each batch in epoch, rather than all at once at the end. Notice the HUGE difference in in-sample accuracy vs val_accuracy favoring val_accuracy at the very first epoch. This is because some of in-sample error computed with very few batch updates.
```
model.fit(Xtrn, Xtrn, epochs = 3, batch_size = 100,
... validation_data = (Xtst, Xtst))
Train on 46580 samples, validate on 1000 samples
Epoch 1/3
46580/46580 [==============================] - 8s 176us/sample
- loss: 0.2320 - accuracy: 0.9216
- val_loss: 0.1581 - val_accuracy: 0.9636
Epoch 2/3
46580/46580 [==============================] - 8s 165us/sample
- loss: 0.1487 - accuracy: 0.9662
- val_loss: 0.1545 - val_accuracy: 0.9677
Epoch 3/3
46580/46580 [==============================] - 8s 165us/sample
- loss: 0.1471 - accuracy: 0.9687
- val_loss: 0.1424 - val_accuracy: 0.9699
```
(2) Showing error is computed before update for each batch. Note that for epoch 1, when we use batch_size = nRows (i.e., all data in one batch), the in-sample error is about 0.5 (random guessing) for epoch 1, yet the validation error is 0.82. Therefore, the in-sample error was computed before the batch update, while the validation error was computed after the batch update.
```
model.fit(Xtrn, Xtrn, epochs = 3, batch_size = nRows,
... validation_data = (Xtst, Xtst))
Train on 46580 samples, validate on 1000 samples
Epoch 1/3
46580/46580 [==============================] - 9s 201us/sample
- loss: 0.7126 - accuracy: 0.5088
- val_loss: 0.5779 - val_accuracy: 0.8191
Epoch 2/3
46580/46580 [==============================] - 6s 136us/sample
- loss: 0.5770 - accuracy: 0.8211
- val_loss: 0.4940 - val_accuracy: 0.8249
Epoch 3/3
46580/46580 [==============================] - 6s 120us/sample
- loss: 0.4921 - accuracy: 0.8268
- val_loss: 0.4502 - val_accuracy: 0.8249
```
Small note about the code above: an auto-encoder was built, hence why the input (Xtrn) is the same as the output (Xtrn).
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edited Mar 16, 2020 at 14:35
answered Oct 18, 2019 at 4:46
Cliff ABCliff AB
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I got similar results (test loss was significantly lower than training loss). Once I removed the dropout regularization, both the loss became almost equal.
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answered Dec 10, 2017 at 6:45
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@cdeterman and @D-K have good explanation. I would like to one more reason - data leakage. Some part of your train-data are "closely related" with the test-data.
Potential example: imagine you have 1000 dogs and 1000 cats with 500 similar pictures per pet (some owners love to take pictures of their pets in very similar positions), say on the background. So if you do random 70/30 split, you'll get data leakage of train data into the test data.
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answered May 24, 2018 at 18:00
Danylo ZherebetskyyDanylo Zherebetskyy
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A lower validation than training error can be caused by fluctuations associated with dropout or else, but if it persists in the long run this may indicate that the training and validation datasets were not actually drawn from the same statistical ensembles. This could happen if your examples come from a series and if you did not properly randomize the training and validation datasets.
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answered Aug 27, 2019 at 9:37
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Simply put, if training loss and validation loss are computed correctly, it is impossible for training loss to be higher than validation loss. This is because back-propagation DIRECTLY reduces error computed on the training set and only INDIRECTLY (not even guaranteed!) reduces error computed on the validation set.
There must be some additional factors that are different while training and while validating. Dropout is a good one, but there can be others. Make sure to check the documentation of whatever library that you are using. Models and layers can usually have default settings that we don't commonly pay attention to.
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answered Jul 16, 2019 at 23:53
Adam WilsonAdam Wilson
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It is actually not impossible. Your second part of the answer points into a good direction. Please consider pyimagesearch.com/2019/10/14/…
Cadoiz
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6633 | https://www.englishalex.com/post/prepositions-of-time-at-on-in-audio-lesson-included | top of page
ENGLISH WITH ALEX
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Prepositions of Time: At, On, In (Audio Lesson Included)
Alex
Aug 18, 2023
7 min read
Updated: Aug 19, 2023
Elementary English
For an easy-to-study reference chart, scroll to the bottom of the page.
Quick Reference
Use "at" with precise times (at 3 o'clock, at midnight, at the moment, etc.)
Use"on" with days and dates (on Monday, on January 14th, etc.)
Use "in" with months, years, decades, centuries, historical periods, and longer periods of time (in January, in 1999, in the 1950s, in the Middle Ages, etc.)
Use "in" for most times of day (in the morning, in the afternoon, in the evening).
Use "at" for "at night." "In the night" is also used, however, but mostly in literature and poetry.
Use "on" for days of the week combined with a time of day (on Friday morning, on Thursday afternoon, on Sunday evening, on Tuesday night)
People in North America typically use "on the weekend." People in the British Isles typically use "at the weekend."
Mastering the three major English prepositions of time—at, on, and in—is essential for accurate communication. If you want to organize a meeting, invite someone for dinner, tell someone your date of birth, you'll need to use prepositions. In this article, we'll learn how to use at, on, and in effectively.
To begin, test yourself. Which preposition should you use in these five cases?
"I'll see you _____ Wednesday."
"The meeting started _____ 9 o'clock."
"I was born _____ April."
"Halloween is _____ October 31st."
"She was born _____ 2015."
We'll come back to these in a little bit. For now, let's continue learning.
So, are there any general rules for at, on, and in? Luckily, yes! However, please note that I am stressing the phrase general rules. So...
In general, at is used for exact times, on is used for individual days, and in is used for longer periods of time. To explain most cases, you can think of it as a pyramid which becomes more precise as you move closer to the point at the top.
So, what does this mean? Let's look at things one at a time.
At
Use at with exact times. This means the hours on a clock. Let's look at some examples:
"I'll see you at 3 o'clock."
"The meeting ended at 10:30."
"We had lunch at noon."
"I fell asleep at midnight."
Okay, so far, so good, right? But there are other cases and common phrases which don't explicitly refer to a specific time on a clock. Here are some examples:
at the moment
at the time
at this/that time
at sunrise
at sunset
at lunch
at dinnertime
at bedtime
at the same time
at night
at Christmas
at the weekend
For the first nine cases, we can still say that the times are precise. "At the moment" means "right now," or "at this exact time." "At the time" means "during the precise moment or period we are discussing." "At sunrise" and "At sunset" both refer to the precise times that sunrise happens in the morning, and sunset happens in the evening. You can use a search engine right now to learn the precise time that the sun will rise in Sydney, Australia tomorrow, for instance. Lunch, dinnertime, and bedtime are also fairly specific events which happen at a particular time. Finally, "at the same time" means "simultaneously," or "more than one action or event happening at exactly the same time or moment." All of these fit with the idea of at being used for exact (or nearly exact) times.
However, "at night" is an irregularity. I suppose we can think of night as a period when most people are asleep, and in that way, we can try to make it feel logical that nighttime is an exact time when many people are doing the same thing, but we really know it's not the same as saying "at 5 o'clock," for example.
"At Christmas" is even weirder because we don't necessarily mean Christmas Day. In fact, at is often used to refer to multi-day holidays. It's close in meaning to during in this way. So, you can say "I'll see you at Christmastime," which means "I'll see you on one of the days of Christmas." If you want to mention a specific day that happens during Christmas--specifically Christmas Eve or Christmas Day--you need to say "I'll see you on Christmas Day" or "We saw each other on Christmas Eve." Fortunately, this isn't a situation that you need to worry about too often, and I'll describe the on part shortly.
Before I do that, I need to address "at the weekend," which is typically used to discuss the weekend in England and its surrounding English-speaking areas (or "in British English," if you prefer). In North America, "on the weekend" is what is preferred by the vast majority of speakers. As with many languages, phrases and linguistic preferences can change depending on the region.
Since we've been talking about on a little bit in the last two paragraphs, let's take a detailed look at it.
On
On is used with individual days. This means days and dates. Essentially, if you can point to a day on a calendar, you can use on to talk about that day. Pay attention to the following examples:
"I'll see you on Saturday."
"She was born on August 10th, 2001."
"Are you going to be busy on Valentine's Day?"
"We're going to a party on the 4th of July."
"My birthday is on June 11th."
Some people have problems with on because while they know it is used with days and dates, they don't know that it is also used for single-day holidays. Remember: If you can point to a single day on a calendar, use on.
As we have already discussed above, you can also use on to mention what you did during or on the weekend in North America.
Finally, if you want to mention what you did during a specific part of the day on a specific day of the week, use on. Here is what I mean:
"We saw each other on Tuesday morning."
"He's going to meet me on Saturday afternoon."
"Are you free on Thursday evening?"
"I didn't do anything on Friday night."
Not bad, right? Just make sure you don't say "at Friday night" just because you know that you're supposed to say "at night." The day of the week is the most important part of time phrases like "Friday night," and as we know, days of the week need on, which means it should be "on Friday night."
Wow. This section was much easier than the at section, wasn't it? There's only one more preposition to go, and then we can review those 5 questions you did at the start of the page...
In
In is used with longer periods of time. This means months, years, decades (10 years), centuries (100 years), millennia (1000 years), historical periods, and seasons. Basically, if the period is one month or more, use in. Are you ready for some examples? Here we go:
"I graduated in September."
"My family moved to Ireland in 2016."
"Grunge music was really popular in the 1990s."
"World War 2 happened in the 20th century." / "...in the 1900s."
"Do you think we'll master space travel in this millennium?"
"Would you want to live in the Middle Ages?"
"The best time to buy a new barbecue is in the fall." (Bonus note: Seasons can be written with or without the article "the." You can say "in the fall" or "in fall." It's your call.)
To practice using in with years, answer this question: In what year were you born? (Answer: "I was born in _____.") Remember, if you mention the date on which you were born, you must use on instead. To illustrate this, you can say "I was born in 1995," or "I was born on July 10th, 1995."
You can also use in to talk about specific times of day. Namely:
in the morning
in the afternoon
in the evening
While you can say "in the night," at night is what is typically used. Here are some examples of in with times of day.
"Can we talk about this in the morning?"
"I'll see you in the afternoon."
"Do you usually stay at home in the evening?"
"I've been having a hard time falling asleep at night."
Finally, you can use in to talk about the length of time that will pass before something happens in the future. For instance, you can say "I'll see you in 30 minutes," or "My father is going to visit us in 3 days." You can even say "He's going to graduate in 2 years."
And these are all the most important points about how to use at, on, and in. How do you feel? Are you ready to test your knowledge? Let's go back to the 5 questions at the top of this page. Answer them then check the answers directly under them.
"I'll see you _____ Wednesday."
"The meeting started _____ 9 o'clock."
"I was born _____ April."
"Halloween is _____ October 31st."
"She was born _____ 2015."
Answers: I'll see you on Wednesday. / The meeting started at 9 o'clock. / I was born in April. / Halloween is on October 31st. / She was born in 2015.
One more thing...
So, you now have a better understanding of how to use at, on, and in. This is good news! However, as you probably know, there are more ways to talk about time.
For instance, if you don't want to give a precise time for something, you can use around or about instead of at. Around and about mean approximately in these cases. For instance, "I should be there around 5 o'clock," or "What time is it?" "I don't know. It must be about 2 o'clock or something."
Next, if you want to talk about the duration of time which something takes, use for. For example, "I worked there for 6 years," or "We waited for 45 minutes."
Finally, you can use since to talk about specific periods in the past. For instance, "We haven't seen each other since high school," or "I haven't seen that movie since I was 12 years old."
And that's it! I hope you feel better informed about when to use at, on, and in, so you can start planning your next party, work meeting, or dinner reservation with confidence.
Prepositions Chart
| At | On | In |
| Hours at 6 o'clock at 5:15 at noon at midnight | Days on Monday on Tuesday on Friday on Sunday | Years in 1995 in 2015 in 1789 in 2001 |
| Precise times / events at the moment ("now") at sunrise at sunset at lunch at dinnertime at bedtime at the start of the day at the end of the day | Dates on July 14th on February 23rd, 1998 on the 19th of August on March 19th | Decades in the 1990s in the 1970s in the 2010s in the 1850s |
| "Time" expressions at the time at this/that time | Holidays / Individual days on a calendar on my birthday on Valentine's Day on Christmas Day on New Year's Eve | Centuries in the 1900s in the 20th century in the 1800s in the 19th century |
| Other at night at the weekend (British English) | Days with times of day on Tuesday morning on Wednesday afternoon on Thursday evening on Saturday night | Historical periods in the Middle Ages in the Victorian era in the Renaissance in the early modern period |
| | Other on the weekend (North American English) | in (the) spring in (the) summer in (the) fall in (the) winter |
| | | Times of day in the morning in the afternoon in the evening in the night (Possible, especially in literature and poetry, but "at night" is much more common in everyday English.) |
| | | Length of time before a future event in 2 hours in 30 minutes in a year |
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6634 | https://www.ijrcog.org/index.php/ijrcog/article/download/6262/4365/22314 | April 201 9 · Volume 8 · Issue 4 Page 1322
International Journal of Reproduction, Contraception, Obstetrics and Gynecology
Sorakayalapeta MR et al. Int J Reprod Contracept O bstet Gynecol. 201 9 Apr ;8(4): 1322 -1326
www.ijrcog.org pISSN 2320 -1770 | eISSN 2320 -178 9
Original Research Article
Maternal and perinatal outcome in placenta previa: an observational
study at a tertiary care hospital in Mysore, Karnataka, India
Maunica Reddy Sorakayalapeta , Nandish S. Manoli
INTRODUCTION
Placenta previa is a major cause of vaginal bleeding in
the late 2 nd and 3 rd trimester.
It clinically presents as causeless, painless and recurrent
bleeding of varying amount. It occurs in 2.8/1000 and
3.9/1000 in singleton and twin pregnancies respectively. 1
Classification of placenta previa :2
• Type 1 (Low lying): Major part of the placenta is in
the upper segment while only the lower margin
encroaches onto the lower uterine segment but not up
to the OS .
• Type 2 (Marginal): Placenta r eaches the internal os
bot does not cover it .
ABSTRACT
Background: 2-5% of the pregnancies are complicated by antepartum haemorrhage. About one third of them are due
to placenta previa thus contributing to a significant amount of maternal and perinatal morbidity and mortality. In the
present times with liberally increasing caesarean section rates, there is a changing trend in the incidence and
complications of placenta previa. The objective of this study was to evaluate the obstetr ical characteristics and
maternal and perinatal outcome of cases of placenta previa .
Methods: This was a prospective observational study conducted in the Department of Obstetrics and Gynaecology of
JSS Medical College and Hospital, Mysore during the period January 2017 to June 2018 .
Results: Out of total 13,150 deliveries during this period, placenta previa was observed in 131 cases with an
incidence of 1%. Majority belonged to the age group 25 -29 years (48.8%). 66% of the cases presented with painless
vaginal bleeding as their chief complaint. The major risk factor was previous caesarean delivery seen in 29.8% cases
followed by history of abortion in 18.3%. 67% cases had major degree placenta previa. Remaining 33% cases had
minor degree. One ca se was complicated by placenta accrete. Peripartum hysterectomy was performed in 3.1%.
Preterm deliveries amounted to 29.8%. Maternal and perinatal mortality were 0.76% and 3.05% respectively. 10.7%
cases had Postpartum haemorrhage and 3.8% required ICU ad mission. 25% neonates required NICU admissions and
10% had RDS .
Conclusions: Placenta previa is a prime contributor to substantial maternal and perinatal morbidity and mortality.
Early referral to tertiary care centres, anticipation of clinical complicatio ns and appropriate measures can avoid grave
consequences. Such cases must always be managed at a higher centre with good NICU services and round the clock
operation theatre and blood bank facility .
Keywords: Perinatal mortality , Placenta previa, Postpartum haemorrhage
Dep artmen t of Obstetrics and Gynecology, JSS Academy of Higher Education and Research ( deemed to be
University), Mysore, Karnataka , India
Received: 04 February 201 9
Accepted : 25 February 201 9
Correspondence:
Dr. Maunica Reddy Sorakayalapeta ,
E-mail: maunica473@gmail.com
Copyright: © the author(s), publisher and licensee Medip Academy . This is an open -access article distributed under
the terms of the Creative Commons Attribution Non -Commercial License, which perm its unrestricted non -commercial
use, distribution, and reproduction in any medium, provided the original work is properly cited .
DOI: -1770. ijrcog20190978 Sorakayalapeta MR et al. Int J Reprod Contracept O bstet Gynecol. 201 9 Ap r;8(4): 1322 -1326
International Journal of Reproduction, Contraception, Obstetrics and Gynecology Vol ume 8· Issue 4Page 1323
• Type 3 (Incomplete or partial central): Placenta
covers the internal os when closed but not when fully
dilated .
• Type 4 (central or total): Placenta completely covers
the internal OS even when fully dilated .
Several predisposing factors have been identified for
placenta previa like multiparity, multifetal gestation,
advancing maternal age, previous caesarean section,
trauma, dilatation and curettage, etc. 3 The risk of adherent
placenta increases to 11 %, 40% and 61% with previous
one, two or three caesarean sections. Maternal and foetal
morbidity and mortality from placenta previa are
considerably associated with high demand on health care
services. With the increasing incidence of caesarean
section rat es and increasing maternal age, the number of
cases of placenta previa and its complications including
placenta accreta will continue to increase. The main
diagnostic modality is by transvaginal ultrasound. Early
diagnosis will help us in reducing maternal morbidity and
mortality by being cautious about the need for blood
transfusions and arranging an efficient team of surgeon,
anaesthetist and paediatrician. Over the years, the
maternal and perinatal morbidity due to placenta previa
greatly decreased. This is mainly due to the early
diagnosis that is made possible by the Ultrasound
diagnostic modality, increasingly accessible health care,
avoidance of internal examination, presence of blood
transfusion facilities, 24 hours available operation theatre
facili ty for caesarean section and the integrated
management by the surgeon, anaesthetist and
neonatologist. The better perinatal outcome is also
ensured through the presence Neonatal intensive care unit
and appropriate management.
However, the incidence of placenta previa is on the rise.
This is due to the wider accessibility of Ultrasound.
Another significant cause is also due to the caesarean
section rate which is on a rise. The risk attributed by this
cause to placenta previa can be brought down by
decreasing the primary caesarean sections for non -
recurrent indications .
METHODS
This was a prospective observational study conducted in
the Department of Obstetrics and Gynaecology of JSS
Medical College and Hospital, Mysore during the period
Jan uary 2017 to June 2018.
The relevant details of all the women who were operated
for placenta previa and satisfied the inclusion and
exclusion criteria were recorded and the data was
analyzed statistically using Descriptive statistics
(frequency and percen tage) .
Inclusion criteria
• Cases of placenta previa diagnosed on routine
ultrasound examination
• Placenta previa cases with scarred/non -scarred uteri .
Exclusion criteria
• Other causes of antepartum haemorrhage
• Other causes of painless vaginal bleeding like
carcinoma cervix, cervical polyp, local trauma, etc .
RESULTS
Total number of deliveries during the study period were
13,150. Total number of live births were 13,087. Total
number of twin deliveries were 36. Total number of
placenta previa cases were 131. The incidence of placenta
previa thus obtained in present stud y was 1% (Table 1) .
Table 1: Overview of statistics .
Parameters St atistics
Total number of deliveries 13,150
Total number of live births 13,087
Incidence of placenta previa 1%
Number of maternal deaths due to placenta
previa 01
Number of perinatal deaths due to placenta
previa 04
Perinatal mortality rate due to placenta
previa 30.5/1000
Number of NICU admissions 33
In our present study, the incidence of placenta previa was
found to be the highest among the age group 25 -29 years
with 48.8%, followed in a decreasing fashion among the
age group 20 -24 years with 29%, 13% among 30 -34
years and the least among those with a ge <20 years and
≥35 years with 4.6%. The above incidence could be
partly influenced by the increasing adaptation of late
marriages during the present times. The mean maternal
age obtained in present study was 25.6 years (Figure 1) .
Figure 1: Distribution of age groups among PP cases .
Unlike numerous other studies, in present study authors
came across higher incidence of placenta previa among
the primigravids. This could be explained by the
increasing age at first pregnancy and the possibility of
hidden history of abortions with curettage. Previous
0
10
20
30
40
50
60
70
<20 20-24 25-29 30-35 >35
No. of patients
Age groups Sorakayalapeta MR et al. Int J Reprod Contracept O bstet Gynecol. 201 9Ap r;8(4): 1322 -1326
International Journal of Reproduction, Contraception, Obstetrics and Gynecology Vol ume 8· Issue 4Page 1324
caesarean section was seen to be a major risk factor for
placenta previa complicating 29.8% of the cases. Second
most important risk factor was previous abortion with
curettage amounting to 18.3% (Tab le 2).
Table 2: Obstetric profile of cases .
Obstetric profile No. %
Primigravida 70 53.4
Multigravida 61 46.6
Previous LSCS 39 29.8
Previous abortion 24 18.3
Malpresentations (Breech, transverse lie,
oblique lie, face ) 13 9.9
Adherent placenta 1 0.76
Among the 4 cases of peripartum hysterectomy, 3 cases
were performed due to failure to control atonic PPH with
medical and conservative surgical techniques. The fourth
case was a case of two previous LSCS with morbidly
adherent placenta.
Among the 131 cas es included in present study spinal
anaesthesia was administered to 127 women while in the
rest of the four cases general anaesthesia was
administered (Table 3) .
Table 3: Different modes of delivery among PP cases .
Mode of delivery Frequency %
LSCS 87 66.4
Repeat LSCS 39 29.8
Hysterotomy 01 0.8
Peripartum hysterectomy 04 3.1
Majority of the cases i.e. 67.9% cases had major degree
placenta previa. The most common type was type 2b
(45.8%). From the results it can be noticed that Type 2B
placenta previa was the most frequent in occurrence with
45.8% followed by Type 1, Type 2A and 3 and Type 4
with 19.8%, 12.2%, 12.2% and 9.9% respectively (Table
4) .
Table 4: Occurrence of different types of placenta
previa .
Types of placenta previa Frequency %
1 26 19.8
2a 16 12.2
2b 60 45.8
3 16 12.2
4 13 9.9
25.2% cases needed NICU admissions. The criteria for
admission included prematurity, low birth weight
Respiratory distress syndrome and neonatal anaemia.
Respiratory distress syndrome was seen in 9.9%. Low
birth weight babies contributed to 20.6% and extre mely
low birth weight babies amounted to 5.3% thus
contributing to significant neonatal morbidity and
mortality (Table 5) . Among the 131 cases studied, 3.8%
required ICU admissions.
Table 5: Neonatal outcome .
Factors No. %
Maturity (in weeks) <30 weeks 2 1.5
30 -33 weeks 12 9.2
34 -37 weeks 25 19.1
37 weeks 91 70.2
preterm 39 29.7
Birth weights (in kg) <1.5 7 5.3
1.5 -2.49 27 20.6
2.5 -3.0 39 29.8
3 58 44.3
Still born 4 3.05
Respiratory distress syndrome 13 9.9
Postpartum Haemorrhage occurred in 10.7% cases. Blood
transfusions were needed in 18.3% of the cases (Table 6) .
Table 6: Obstetric complications .
Complications No. %
Postpartum Haemorrhage 14 10.7
Blood transfusions 24 18.3
Peripartum hysterectomy 4 3.05
ICU admissions 5 3.8
DISCUSSION
In present study , the incidence of placenta previa among
the total number of cases delivered at JSS Hospital,
Mysore during the period January 2017 to June 2018 was
1%. Total number of deliveries during this period were
13,150 out of which 131 cases were complicated by
placenta previa. The incidence of 1% obtained in present
study is higher in comparison to study by Martina
Kollman et al.,1993 -2012 which showed 0.15% and study
by Sarojini et al that showed 0.64%. 4,5
The high in cidence obtained could be due to the reason
that this hospital is a tertiary referral center. However,
few other studies showed a much greater incidence of
placenta previa like study by Sunil Kumar Samal et al.
showing 2.9% and Meenakshi et al.,2014 -2016 showing
1.8%. 6 Among the cases studied, greater number of
placenta previa were among the primigravidas when
compared to multiparas.
This could partly be reasoned out by the late marriages
and elderly primigravidas, increasingly diagnosed cas es
of minor degree placenta previa through ultrasound.
However, this aspect could be studied in greater detail in
the future through other studies.
In present study , it can also be noted that major number
of cases fell into the age group 25 -29 years, mean age Sorakayalapeta MR et al. Int J Reprod Contracept O bstet Gynecol. 201 9 Ap r;8(4): 1322 -1326
International Journal of Reproduction, Contraception, Obstetrics and Gynecology Vol ume 8· Issue 4Page 1325
being 25.6 years. This observation is similar to the study
conducted by Raja Rajeshwari et al. at Thanjavur,
wherein placenta previa cases were the highest in 20 -29
years. 7 Similar results were also obtained in study by VN
Kurudu et al A lot of other s tudies showed that risk of
placenta previa increases with age. 8 In present study , one
fifth of the women were of the age 30 and above. 57% of
the women in present study had antepartum hemorrhage,
which led to them being diagnosed with placenta previa
with the help of ultrasound. No attempt was made to
confirm the ultrasonographic findings, even in
asymptomatic cases with double set up vaginal
examination.
Shortcomings of USG could be overcome with a gentle
transvaginal USG and MRI. Antepartum hemorrhage wa s
more prevalent in major degree placenta previa (52.8%)
in comparison to minor degree (45.2%). Study by Atsuko
Sekiguchi et al. also showed a greater occurrence of
antepartum hemorrhage of 59% in complete type when
compared to 17.6% in incomplete type. 9 29.8% cases in
present study had a history of prior caesarean section and
18.3% cases had history of one or more abortions.
Similar results were noted in study carried out by CH
Nirmal et al. Mustafa et al. reported previous scar of
uterus as a major risk factor (56.5%). 10 No twin
pregnancies with coexisting placenta previa were noted in
present study . Caesarean section is the method of choice
for delivery in cases of major degree placenta previa.
Four cases underwent subtotal hysterectomy in present
study following atonic PPH. One of them had a morbidly
adherent placenta. The study presented includes one
maternal death which resulted from Disseminated
intravascular coagulation that resulted following atonic
PPH and multiorgan dysfunction. Numerous studies
conducted over the years showed a wide variation in the
incidence of PPH among the pregnant women with
placenta previa. In present study , PPH was seen in 10.7%
of the cases.
The incidence of PPH as low as 3.56% was reported by
Ogueh et al. and as high as 58.68% was reported by
Zlatnik et al. The wide variation in the incidence could be
due to the geographical differences, multiparity, prior
caesarean section and other potential characteristics. 11
The prime factor responsible for neonatal morbidity and
mortality in cases of placenta previa is prematurity.
Neonatal mortality in this study was 30.5 per 1000,
mainly due to prematurity and birth asphyxia.
Prematurity in present study contributed up to 2 9.8%. A
much greater number of cases (40.86%) delivered
preterm in Wilson Rodie et al., study. Meenakshi et al.
study showed 42.85% 12 prematurity with 24% needing
NICU admissions.
In present study 25.2% NICU admissions were recorded.
Perinatal morbidity in present study was 3%, lower in
comparison to study by Lavanya Kumari Sarella et al that
showed 6.6%. 13 This can be credited to the timely and
judicious antepartum and intrapartum care and the well -
equipped NICU and skilled neonatologists . The
shortcomi ngs of this study are that it is a single center
study and there is lack of controls.
CONCLUSION
The present study suggested that placenta previa
significantly contributes to maternal and perinatal
morbidity and mortality. Accurate diagnosis, appropriate
management and timely delivery can provide us with the
most desired outcome. The current study suggested that
there is an association between advancing age, previous
caesarean section and abortion as the risk factors for
placenta previa. Regional anesthesia can be safely
administered to cases of placenta previa.
Placenta previa as noted from the study was seen to be
associated with increased risk of maternal complications
like PPH and neonatal complications including
prematurity and low b irth weight. Hence it is advisable to
manage a case of placenta previa in a tertiary care center
with a good neonatal support, round the clock operation
theatre and blood bank facilities .
Funding: No funding sources
Co nflict of interest : None declared
Ethi cal approval: The study was approved by the
Institutional Ethics Committee
REFERENCES
Oppenheimer L, Society of obstetricians and
gynaecologists of Canada, Diagnosis and management
of placenta previa. JObstet Gynaecol
Canada.2007;29(3):261 -6.
Gupta B, Misra R. Antepartum haemorrhage. In. Renu
Misra. (ed.) Ian Donald’s Practical Obstetric problems.
Seventh edition; 2014.p.315 .
Sinha P, Kuruba N.An tepartum haemorrhage: an
update. J Obstet Gynecol .2008;28 (4) :377 -81 .
Kollmann M, Gaulhofer J, Lang U, Klaritsch P.
Placenta praevia: incidence, risk factors and outcome. J
Maternal -Fetal Neonat Med. 2016;29(9):1395 -8.
Sarojini, Malini KV, Radhika. Clinical study of
placenta previa and its effect on maternal health and
fetal outcome. Int JReproduct, Contracep, Obstet ,
Gynecol. 2016;5(10):3496 -9.
Meenakshi Devarmani, Patil Sanjana Tallur. Placenta
previa: Maternal and foetal outcome. JEvol Med
Dental Sci. 2016;5(40):2477 -80 .
Rajeshwari RR, Rubini M. Maternal and perinatal
outcome in placenta previa :one -year study in tertiary
care center in Tamil Nadu, India. Int J Reproduct,
Contracep, Obstet Gynecol. 2016;5(8):2819 -22 .
Kurude VN , Saha D. Study of maternal and perinatal
outcome in placenta praevia at tertiary care center.
Indian J Res.2016;6 (3):44 -5.
Sekiguchi A,Nakai A,Kawabata I,Hayashi M,
Takeshita T. Type and Location of Placenta Previa Sorakayalapeta MR et al. Int J Reprod Contracept O bstet Gynecol. 201 9Ap r;8(4): 1322 -1326
International Journal of Reproduction, Contraception, Obstetrics and Gynecology Vol ume 8· Issue 4Page 1326
Affect Preterm Delivery Risk Related to Antepartum
Hemorrhage. Int J Med Sci.2013;10(12):1683 -8.
Nirmala CH, Mounisha NV. Placenta Praevia - A Study
On Maternal and Perinatal outcome. IOSR Journal of
Dental and Medical Sciences (IOSR -
JDMS) .2017; 04 (7):4 -7.
Zlatnik MG, Cheng YW, Norton ME, Thiet MP,
Caughey AB. Placenta previa and the risk of preterm
delivery. J Maternal -Fetal Neonat Med.
2007;20(10):719 -23.
Meenakshi Devarmani, Patil Sanjana Tallur. Placenta
previa: Maternal and foetal outcome. J Evol Med Dent
Sci. 2016;5(40):2477 -80 .
Lavanyakumari S, Arunajyothi C. A Study On maternal
and Perinatal Outcome in Placenta Previa. Scholars J
App Medi Sci. 2014;2(5A):1555 -8.
Cite this article as: Sorakayalapeta MR, Manoli NS .
Maternal and perinatal outcome in placenta previa:
an observational study at a tertiary care hospital in
Mysore, Karnataka, India . Int J Reprod Contracept
Obstet Gynecol 201 9;8:1322 -6. |
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Table Of Contents
What are Even and Odd Functions?
How to Determine Even and Odd Functions?
What are Even and Odd Functions in Trigonometry?
Integral Properties of Even and Odd Functions
What are the Properties of Even and Odd Functions?
How to Represent Even and Odd Functions Graphically?
Common Mistakes of Even and Odd Functions and How to Avoid Them
Real-Life Applications of Even and Odd Functions
Solved Examples of Even and Odd Functions
FAQs of Even and Odd Functions
Explore More algebra
Summarize this article:
170 Learners
Last updated on September 18, 2025
Even and Odd Functions
Even functions produce the same output when the input sign is changed. For example, the function f(x) = x² is even because f(2) = 2² = 4 and f(–2) = (–2)² = 4, yielding the same result. A function is odd if substituting a number and its negative yields opposite results. For example, the function f(x) = x3 is odd because f(2) and f(−2) are opposites.
Even and Odd Functions forUSStudents
What are Even and Odd Functions?
Even Functions: An even function is one where the output value remains the same even when the input is negated. Replacing x with −x in the function, the result will not change. This shows that the function has symmetry about the y-axis. Functions like f(x) =x2 yields the same result, for positive or negative inputs, e.g., f(2) = f(–2) = 4.
Odd Functions: These functions exhibit symmetry about the origin. For example, an odd function satisfies the condition f(−x) = −f(x) for all x in its domain, indicating rotational symmetry about the origin.
How to Determine Even and Odd Functions?
To determine if a function is even or odd, follow these steps: Even Function: If f(−x) = f(x) for all x in its domain, the function is even. Odd Function: If f(−x) = −f(x) for all x in its domain, the function is odd.
Even Function Definition: A function f(x) is even if f(−x) = f(x) for all x in its domain.
Graphical Symmetry: Graph of an even function is symmetric about the y-axisExamples: f(x) = x2, f(x) = cos(x), f(x) = ∣x∣
Odd Function Definition: A function f(x) is odd if f(−x) = −f(x) for all x in its domain.
Graphical Symmetry: The graph of an odd function has rotational symmetry about the origin.Examples: f(x) = x³, f(x) = sin (x), f(x) = x
Both Even and Odd Zero Function: The only function that is both even and odd is f(x) = 0 since f(−x) = f(x) = −f(x) = 0
Neither Even Nor Odd Example: f(x) = (-x)³ + (-x) + 1 f(−x) = −x3−x+1
Neither f(−x) = f(x) nor f(−x) = −f(x), so it's neither even nor odd.
What are Even and Odd Functions in Trigonometry?
Trigonometric functions, such as sine, cosine, tangent, cotangent, secant, and cosecant, can be categorized based on their symmetry and behavior. In even functions, the condition f(−x) = f(x) means the graph is symmetric about the y-axis. In odd functions, the condition f(−x) = −f(x) , is such that the graph has rotational symmetry about the origin.
Even Functions A function f(x) is defined as even if it satisfies the condition: f(−x) = f(x) This tells us that the graph of the function is symmetric about the y-axis Trigonometric Examples:
Cosine Function: cos(−x) = cos(x)
Secant Function: sec(-x) = sec(x)
These functions exhibit symmetry about the y-axis; their values remain unchanged when the input angle is negated.
Odd Functions A function f(x) is defined as odd if it satisfies the condition: f(−x) = −f(x) This shows that the graph of the function has origin symmetry.
Trigonometric Examples:
Sine Function: sin(−x) = −sin(x)
Tangent Function: tan(−x) = −tan(x)
Cosecant Function: csc(−x) = −csc(x)
Cotangent Function: cot(−x) = −cot(x)
Integral Properties of Even and Odd Functions
Even and odd functions exhibit defined symmetry properties that clarify the evaluation of integrals.
Even Functions
For a continuous even function f(x) satisfying f(−x) = f(x), the integral above the symmetric interval [−a, a] can be simplified:
∫₋ₐᵃ f(x) dx = 2 ∫₀ᵃ f(x) dx" and "∫₋ₐᵃ f(x) dx = 0
This symmetry simplifies calculations by doubling the integral from 0 to a, as the function is identical on both sides of the y-axis.
Odd Functions
For a continuous odd function f(x) satisfying f(−x) = −f(x), the integral over the symmetric interval [−a, a] equals zero:
-aa f(x) dx=0
This is because the areas above and below the x-axis cancel each other out due to symmetry.
What are the Properties of Even and Odd Functions?
Addition & Subtraction
The sum of two even functions is even, as it retains symmetry about the y-axis.
Even + Even = Even
If you add two functions that have rotational symmetry about the origin, the result will also have rotational symmetry about the origin.
Odd + Odd = Odd
When you combine a function with y-axis symmetry and one with origin symmetry, it doesn't result in a function that is symmetric about the y-axis or origin.
Even + Odd = Neither
Multiplication & Division
Multiplying two functions that are symmetric about the y-axis results in a function that is also symmetric about the y-axis.
Even × Even = Even
Multiplying two functions with rotational symmetry about the origin results in a function that is symmetric about the y-axis.
Odd × Odd = Even
Multiplying a function symmetric about the y-axis with one having rotational symmetry about the origin results in a function with rotational symmetry about the origin.Even × Odd = Odd
Dividing two such functions does not necessarily preserve this symmetry.Even ÷ Even = Even
Multiplying functions symmetric about the y-axis keep symmetry, but dividing them doesn't necessarily do so if the quotient is undefined at certain points.Odd ÷ Odd = Even
Dividing a function symmetric about the y-axis by one with rotational symmetry about the origin results in a function with rotational symmetry about the origin.Even ÷ Odd = Odd
Composition
Composing two functions which are symmetric about the y-axis results in a function symmetric.
Even ∘ Even = Even
Composing two functions with rotational symmetry about the origin results in a function which has rotational symmetry about the origin.
Odd ∘ Odd = Odd
Composing a function symmetric about the y-axis with one having rotational symmetry about the origin results in a function which is symmetric about the y-axis.
Even ∘ Odd = Even
How to Represent Even and Odd Functions Graphically?
An even function exhibits symmetry about the y-axis, its graph remains unchanged when reflected across the y-axis, and the function's values are identical for every pair of opposite x-values.
Even Functions Graph
An even function is a type of mathematical function that behaves symmetrically around the y-axis. This means, reflecting its graph over the y-axis, the shape would remain unchanged. For every point (x, y) on the graph of an even function, the point (–x, y) is also on the graph, mirroring across the y-axis.
An odd function is a type of mathematical function that has a specific kind of symmetry. This symmetry means that if you rotate the graph of the function 180 degrees around the origin, the graph will look the same.
Common Mistakes of Even and Odd Functions and How to Avoid Them
Students get confused when identifying the even and the odd functions, especially while applying their definitions or interpreting their graphs. Here are some mistakes to help avoid them.
Mistake 1
Assuming All Functions with Even Powers Are Even
Not all functions with even powers will be even. It's a general mistake for students to assume that any function involving only even powers of x is an even function. However, this doesn't always happen. A function is considered even when it satisfies the condition f(−x) = f(x) in its domain, meaning its graph is symmetric about the y-axis.
Mistake 2
Incorrectly Assuming Sums of Functions Follow Parity Rules
The sum of two even functions is even, and the sum of two odd functions is odd. The sum of an even and an odd function is generally neither even nor odd.
To check the resulting function’s parity, it is necessary to check if each function is odd or even before the calculation.
For f(x) =x2(even) and g(x) =x3(odd),
Their sum is f(x) g(x) =x2+x3, which is neither even nor odd because it doesn't satisfy the symmetry conditions for either.
Mistake 3
Misunderstanding the Behavior of Derivatives
The derivative of an even function will be odd, and the derivative of an odd function is even. This is important in calculation, especially when examining the behavior of functions and their rates of change. Apply the differentiation rules carefully, considering the equality of the function being differentiated. For example, let h(x) = x5 (an odd function).
The derivative is h′(x) = 5x⁴, which is an even function. This shows that the derivative of an odd function can be even.
Mistake 4
Overlooking the Impact of Domain Restrictions
Whether a function is even or odd can be affected by its domain restrictions.
Mistake 5
Assuming All Functions with Odd Powers Are Odd
Not all functions with odd powers are odd. For example, f(x)=x3+2 is not odd because substituting −x gives f(−x)=(−x)3+2= −x3 + 2 −f(x). To find out if a function is odd, substitute −x for x in the function. If the resulting expression is the negative of the original function, it is odd.
Real-Life Applications of Even and Odd Functions
Even and odd functions are core concepts in mathematics, reflecting symmetry in various real-world situations. Understanding the real-life applications of even and odd functions have applications in patterns in nature, technology, and everyday life.
Electromagnetic Theory: In electromagnetic theory, odd functions model current distributions with rotational symmetry, such as f(x) = x for magnetic field behavior.
Symmetrical Sound Waves: When certain musical instruments produce sound, the waveforms can be symmetric. This symmetry in sound waves is analogous to even functions, where the pattern remains unchanged when reflected across the y-axis.
Optical Lenses: The design of some optical lenses ensures that light passing through them is symmetrically refracted. This symmetrical behavior is related to the properties of even functions.
Torque and Angular Displacement: The relationship between the torque applied to a wheel and its angular displacement is an example of an odd function: applying torque in one direction results in angular displacement in the same direction, and vice versa, satisfying f(−x) = −f(x).
Vibrating Guitar String: When a guitar string vibrates, the displacement of the string can be described by odd functions, where the displacement at one point is the negative of the displacement at the opposite point.
Hey!
Solved Examples of Even and Odd Functions
Problem 1
Is the function f(x) = x² even or odd?
Okay, lets begin
Even Function
Explanation
To check if a function is odd, substitute -x and see if the result equals -f(x).For f(x) = x²,f(-x) = (-x)² = x² = f(x).As f(-x) = f(x), the function is even.Here, the graph of f(x) = x² is symmetric about the y-axis
Well explained 👍
Problem 2
Is the function f(x) = x³ even or odd?
Okay, lets begin
Odd Function
Explanation
To check if a function is odd, we substitute -x into the function and see if it is equal to the negative of the original function.For f(x) = x³,f(-x) = (-x)³ = -x³ = -f(x).As f(-x) ≠ f(x) and f(-x) ≠ -f(x), the function is neither even nor odd.This means the graph of f(x) = x³ has rotational symmetry about the origin.
Well explained 👍
Problem 3
Is the function f(x) = x² + x even, odd, or neither?
Okay, lets begin
Neither
Explanation
To check if a function is even or odd, we substitute -x into the function and compare.For f(x) = x² + x,f(-x) = (-x)² + (-x) = x² - x.As f(-x) ≠ f(x) and f(-x) ≠ -f(x)The function is neither even nor odd.The graph f(x) = x² + x does not have symmetry about the y-axis or the origin.
Well explained 👍
Problem 4
Is the function f(x) = cos(x) even or odd?
Okay, lets begin
Even Function
Explanation
For f(x) = cos(x),f(-x) = cos(-x) = cos(x) = f(x).As f(-x) = f(x) It's even.This means the graph f(x) = cos(x) is symmetric about the y-axis.
Well explained 👍
Problem 5
Is the function f(x) = sin(x) even or odd?
Okay, lets begin
Odd Function
Explanation
For f(x) = sin(x),f(-x) = sin(-x) = -sin(x) = -f(x). As f(-x) = -f(x), the function is odd.This means the graph f(x) = sin(x) has rotational symmetry about the origin.
Well explained 👍
FAQs of Even and Odd Functions
1.What is an even function?
An even function is a mathematical function that satisfies the condition: If you fold its graph along the y-axis (the middle line), both sides match perfectly. For example, f(x) = cos(x). This function satisfies the condition f(x) = f(−x) for all x in its domain, making it an even function.
2.What is an odd function?
An odd function has a special symmetry. If you rotate its graph 180° around the origin (the center point), it looks the same. For instance, f(x) = x³ it is odd because f(-x) = -f(x).
3.How can I tell if a function is even or odd?
To check if a function is even, replace x with -x and see if the function stays the same; it's even. To check if it's odd, replace x with -x and see if the function becomes the opposite. If it does, it's odd.
4. Can a function be both even and odd?
Yes, but only one function f(x) = 0. This function is even and odd because f(-x) = f(x) = -f(x).
5.Are all functions either even or odd?
No, many functions are neither even nor odd. For example, f(x) = x² + x is neither even nor odd, as f(–x) = (–x)² + (–x) = x² – x, which satisfies neither f(–x) = f(x) nor f(–x) = –f(x).
6.How does learning Algebra help students in United States make better decisions in daily life?
Algebra teaches kids in United States to analyze information and predict outcomes, helping them in decisions like saving money, planning schedules, or solving problems.
7.How can cultural or local activities in United States support learning Algebra topics such as Even and Odd Functions ?
Traditional games, sports, or market activities popular in United States can be used to demonstrate Algebra concepts like Even and Odd Functions , linking learning with familiar experiences.
8.How do technology and digital tools in United States support learning Algebra and Even and Odd Functions ?
At BrightChamps in United States, we encourage students to use apps and interactive software to demonstrate Algebra’s Even and Odd Functions , allowing students to experiment with problems and see instant feedback for better understanding.
9.Does learning Algebra support future career opportunities for students in United States?
Yes, understanding Algebra helps students in United States develop critical thinking and problem-solving skills, which are essential in careers like engineering, finance, data science, and more.
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Previous to Even and Odd Functions
Sum of Odd Numbers|Simplifying Expressions|Onto Function|Interval Notation|Quadratic Expressions|Cardinality|Axis of Symmetry |Algebraic Equations |Explicit Formulas|Nilpotent Matrix |Division of Algebraic Expressions |Exponent and Power|Solving Linear Equations|Scalar Triple Product|Properties of Logarithms|Matrix Multiplication|Vertical Line Test |Cross Multiplication Method |Modulus of Complex Number|nth Term of AP
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Identity Matrix|Non-Singular Matrix|A Union B Complement |Substitution Property |Subtraction Property of Equality |Compound Inequalities |Homogeneous System of Linear Equations|The Rank of a Matrix|Sum and Product of Roots - Examples |Universal Set |Complement of a Set |Classification|Factor Theorem|Simplifying Radical Expression:|Antilog Table|Sets |Rational Expression |Algebraic Expressions|Dividing Polynomials|Covariance Matrix
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6636 | https://taylorandfrancis.com/knowledge/Medicine_and_healthcare/Anatomy/Cochlear_aqueduct/ | Skip to content
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Cochlear aqueduct
The cochlear aqueduct is a small channel that connects the subarachnoid cerebrospinal fluid (CSF) and the cochlear perilymph, extending from the basal turn of the cochlea to the jugular foramen. It allows for bi-directional flow or diffusion within the duct, which equalizes the pressure in the perilymph and CSF pressure.From: Multi-Detector CT Imaging , Anatomy of the Temporal Bone with Surgical Implications , Non-iatrogenic spontaneous acute spinal subdural haematoma after transforaminal lumbar interbody fusion
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Nonconventional Clinical Applications of Otoacoustic Emissions: From Middle Ear Transfer to Cochlear Homeostasis to Access to Cerebrospinal Fluid Pressure
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Published in Stavros Hatzopoulos, Andrea Ciorba, Mark Krumm, Advances in Audiology and Hearing Science, 2020
Blandine Lourenço, Fabrice Giraudet, Thierry Mom, Paul Avan
The first issue is that ICP monitoring through the ear rests on the hypothetical presence of connections between intralabyrinthine fluids and CSF. Several anatomical pathways have been identified that possibly transmit pressure between these two spaces: The cochlear aqueduct, the endolymphatic duct and sac, and the venous system (Fig. 8.2). The cochlear aqueduct seems to provide a straightforward pressure communication with the cochlea; however, its very small diameter and the fact that normally when the cochlea is opened during surgery no gusher is observed suggest that a normal cochlear aqueduct is not patent to CSF flow. Anatomical data suggesting that the cochlear aqueduct gets gradually blocked with increasing age (Wlodyka, 1978) have been later challenged (Gopen et al., 1997). It has been shown that normal variations in ICP such as those induced by heartbeats and breathing are filtered by the pathways connecting CSF and cochlea in humans (Traboulsi and Avan, 2007), such that transmission of pressure waves from the skull to labyrinth undergoes a delay of the order of 10 seconds. Anyway, all that the in-ear monitoring of ICP requires is that there exist canal(s) between CSF and endocochlear fluid spaces that are patent to hydrostatic pressure even if they are not patent to fluid flow.
Anatomy and Embryology of the External and Middle Ear
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Published in John C Watkinson, Raymond W Clarke, Christopher P Aldren, Doris-Eva Bamiou, Raymond W Clarke, Richard M Irving, Haytham Kubba, Shakeel R Saeed, Paediatrics, The Ear, Skull Base, 2018
Peter Valentine, Tony Wright
The mesenchyme enclosing the otocyst becomes chondrified to form the otic capsule. As the membranous labyrinth expands, the otic capsule remodels and in places undergoes dedifferentiation to form fluid-filled spaces that eventually become the perilymphatic spaces. This dedifferentiation does not occur where nerves enter the sensory cell regions. Elsewhere, the perilymphatic spaces become continuous and a communication with the cerebrospinal fluid is formed by the development of the cochlear aqueduct, which runs to the posterior cranial fossa from the scala tympani in the base of the cochlea.
Anatomy
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Published in Stanley A. Gelfand, Hearing, 2017
Stanley A. Gelfand
The cochlear aqueduct leads from the vicinity of the round window in the scala tympani to the subarachnoid space medial to the dura of the cranium. Although the aqueduct leads from the perilymph-filled scala to the cerebrospinal fluid-filled subarachnoid space, it is not patent in many humans. Thus, it is doubtful that there is any real interchange between these two fluid systems. The endolymphatic duct leads from the membranous labyrinth within the vestibule to the endolymphatic sac. The sac is located partially between the layers of the dura in the posterior cranial fossa and partly in a niche in the posterior aspect of the petrous portion of the temporal bone.
Non-iatrogenic spontaneous acute spinal subdural haematoma after transforaminal lumbar interbody fusion
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Published in British Journal of Neurosurgery, 2023
Ahmed Aly, Daniel D’Aquino, Eman Khedr, Olakunle Badmus, Masood Shafafy
A durotomy that remained occult at the time of surgery is a possible explanation of our case. The initial post-operative was uncomplicated with headache, photophobia and tinnitus developing on the third which may have been secondary to cerebrospinal fluid (CSF) leakage as has been suggested.13 They reported a female case, who presented with left sciatica and underwent a standard L4/5 discectomy in which no dural tear or CSF leak was noted at the time of operation. Six days after operation, the patient developed headache, photophobia and hearing loss. An association between hearing symptoms and a CSF leak is well-established.14 The pathophysiology involves inner ear hydromechanics. The cochlear aqueduct communicates between the subarachnoid CSF and the cochlear perilymph, and bi-directional flow or diffusion within the duct allows equalisation of pressure in perilymph and CSF pressure. A CSF leak causes a reactive change in the endolymph volume and reversible hearing disturbance.
Effect of ossicular chain deformity on reverse stimulation considering the overflow characteristics of third windows
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Published in Computer Methods in Biomechanics and Biomedical Engineering, 2022
Houguang Liu, Lin Xue, Jianhua Yang, Gang Cheng, Lei Zhou, Xinsheng Huang
The assumption that the fluid-filled cochlea is surrounded by bone with only two mobile windows, i.e., the oval window and the round window has been generally accepted under forward stimulation (Zhang and Gan 2013; Zhang et al. 2018). This assumption is supported by experimental studies (Kringlebotn 1995; Stenfelt et al. 2004), which found that the fluid volume displacement of the oval window and the round window are equal under forward stimulation. Whereas, the actual anatomy of the cochlea has other smaller-scale and longer sound pathways, mainly the vestibular aqueduct and cochlear aqueduct, which were named as third windows (Rosowski et al. 2018). By stimulating the round window with an actuator in human temporal bone experiment, Stieger et al. (2013) reported that there exists fluid flow through the third windows in the cochlea. Meanwhile, based on the measurement of intra-cochlear sound pressure and ossicular chain motion, Frear et al. (2018) also found that the third windows have volume velocities leakage during reverse stimulation. Moreover, our theoretical investigation found that the third windows have an important effect on the reverse stimulation at low frequencies (Xue et al. 2020). Therefore, incorporating the third windows is essential for investigating the influence of OCD during reverse stimulation.
Change of VOR gain and pure-tone threshold after single low-dose intratympanic gentamicin injection in Meniere’s disease
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Published in Acta Oto-Laryngologica, 2020
Jung-Yup Lee, Min-Beom Kim
The reason for failure of initial ITG is important because there could be several substantial factors which may influence on clear dose-response relationship. An Salt et al. reported that gentamicin may enter the inner ear more easily through the oval window than through the round window in guinea pigs study . Also they demonstrated a slow flow of perilymph in the scala tympani towards the apex of the cochlear in previous study . Another factor is anatomical difference around round window and permeability of round window membrane. In one study, 29 of the round windows were judged to be unobstructed, 7 were obstructed partially, and 5 were obstructed completely in 41 cases . A. Walsted suggested that if the internal orifice of cochlear aqueduct is obliterated, drugs delivered into the middle ear concentrate in the inner ear fluids behind the round window membrane after absorption, resulting in a prolonged and damaging ototoxic effect in the cochlea . In our study, 6 out of 16 patients needed further injection after initial ITG. We can expect that patients in multiple injections group might have anatomical obstruction through round window membrane or hindered cochlear aqueduct. In that cases, we can consider the method which can resolve impermeability or anatomical problems, such as exploratory tympanotomy and gentamicin application . |
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加法逆元（additive inverse）又称相反数（opposite）、反数，其定义是对于任意数 a{\displaystyle a}，存在相反数满足其与 a{\displaystyle a}的和为零（加法单位元）；a{\displaystyle a} 的加法逆元表示为 −a{\displaystyle -a}。
在实数中，数 a{\displaystyle a}的相反数−a{\displaystyle -a}，称为其加法逆元；相对地，数 a{\displaystyle a}的倒数1 a{\displaystyle {\frac {1}{a}}}或 a−1{\displaystyle a^{-1}}，则称为其乘法逆元。
一般定义
[编辑]
设“+”为一个交换性的二元运算，即对于所有 x{\displaystyle x}, y{\displaystyle y}, x+y=y+x{\displaystyle x+y=y+x}。若该集合中存在一个元素 0{\displaystyle 0}，使得对于所有 x{\displaystyle x}, x+0=0+x=x{\displaystyle x+0=0+x=x}，则此元素是唯一的。如果对于一个给定的 x{\displaystyle x}，存在一个 x′{\displaystyle x'}使得 x+x′=x′+x=0{\displaystyle x+x'=x'+x=0}，则称 x′{\displaystyle x'}是 x{\displaystyle x}的加法逆元。
特殊情况
[编辑]
定义
[编辑]
若“+”满足结合律，则任意数的加法逆元是唯一的。
证明
[编辑]
反证法： 设 x{\displaystyle x}有两个相异的加法逆元 x 1{\displaystyle x_{1}}、x 2{\displaystyle x_{2}}
有 x=x+0{\displaystyle x=x+0} 的关系。
⇒ 0=x+x 1=x+x 2{\displaystyle 0=x+x_{1}=x+x_{2}}
⇒ x 1=x 2{\displaystyle x_{1}=x_{2}}
产生矛盾，证讫。
例
[编辑]
向量空间：纯量乘法−1{\displaystyle -1}
欧几里得空间：以原点为中心的反演变换
参考文献
[编辑]
| 展开 查 论 编 二元运算的性质 |
| 闭包（封闭律） · 分配律 ·零因子 ·吸收律 ·幂等律 ·幂零律 |
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| 交错性 | 交换律 · 反交换律 |
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6638 | https://www.ck12.org/flexi/chemistry/determining-the-limiting-reactant/for-the-following-reaction-38-6-grams-of-sulfuric-acid-are-allowed-to-react-with-32-9-grams-of-calcium-hydroxide-sulfuric-acid-(-aq-)-plus-calcium-hydroxide-(-s-)-calcium-sulfate-(-s-)-plus-water-(-l-)-what-is-the-maximum-43cbea5d/ | Flexi answers - For the following reaction, 38.6 grams of sulfuric acid are allowed to react with 32.9 grams of calcium hydroxide. sulfuric acid ( aq ) + calcium hydroxide ( s ) calcium sulfate ( s ) + water ( l ) What is the maximum amount of calcium sulfate that can be formed? grams What is the FORMULA for the limiting reagent? | CK-12 Foundation
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Chemistry
Determining the Limiting Reactant
Question
For the following reaction, 38.6 grams of sulfuric acid are allowed to react with 32.9 grams of calcium hydroxide. sulfuric acid ( aq ) + calcium hydroxide ( s ) calcium sulfate ( s ) + water ( l ) What is the maximum amount of calcium sulfate that can be formed? grams What is the FORMULA for the limiting reagent?
Flexi Says:
The balanced chemical equation for the reaction is:
H 2 S O 4(a q)+C a(O H)2(s)→C a S O 4(s)+2 H 2 O(l)
From the balanced equation, we can see that one mole of H 2 SO 4 reacts with one mole of Ca(OH)2 to produce one mole of CaSO 4.
First, we need to calculate the number of moles of each reactant:
Moles of H 2 SO 4:
36.8 g 98.08 g/m o l=0.39 m o l
Moles of Ca(OH)2:
32.9 g 74.09 g/m o l=0.44 m o l
The stoichiometry of the reaction tells us that H 2 SO 4 and Ca(OH)2 react in a 1:1 ratio. Therefore, H 2 SO 4 is the limiting reagent because we have less of it.
The maximum amount of CaSO 4 that can be formed is determined by the amount of the limiting reagent. Since one mole of H 2 SO 4 produces one mole of CaSO 4, the maximum amount of CaSO 4 that can be formed is 0.39 mol.
To convert this to grams, we multiply by the molar mass of CaSO 4:
0.39 m o l×136.14 g/m o l=53.09 g
So, the maximum amount of calcium sulfate that can be formed is 53.09 grams and the formula for the limiting reagent is H 2 SO 4.
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Hydrochloric acid and sodium hydroxide react as follows: HCl + NaOH → H 2 O + NaCl. Calculate the mass of NaCl formed when 40.0 grams of HCI and 40.0 grams of NaOH react? Molar mass of HCl = 36.5 g/mol, Molar mass of NaOH = 40.0 g/mol, Molar mass of NaCl = 58.5 g/mol.For the following reaction, 5.47 grams of hydrogen gas are allowed to react with 9.60 grams of ethylene (). What is the maximum mass of ethane () that can be formed? Mass = g What is the FORMULA for the limiting reactant? What mass of the excess reagent remains after the reaction is complete?Use the equation for the production of ammonia by the Haber process (shown below): N 2(g) + 3H 2(g) → 2NH 3(g) In a certain reaction, you start with 4.0 moles of nitrogen and 10.0 moles of hydrogen. How many moles of the excess reactant will be left over after the reaction? a. 0.67 mol b. 1.5 mol c. 2.0 mol d. 3.3 mol-n2(g)-plus-3h2(g)-2nh3(g)-in-a-certain-reaction-you-start-with-4-0-moles-of-nitrogen-and-10-0-moles-of-hydrogen-how-many-moles-of-the-excess-reactan/ "
Use the equation for the production of ammonia by the Haber process (shown below):
N2(g) + 3H2(g) → 2NH3(g)
In a certain reaction, you start with 4.0 moles of nitrogen and 10.0 moles of hydrogen. How many moles of the excess reactant will be left over after the reaction?
a. 0.67 mol
b. 1.5 mol
c. 2.0 mol
d. 3.3 mol
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6639 | https://tiangroup.engr.tamu.edu/wp-content/uploads/sites/150/2020/01/mwiselong.pdf | On Top-k Selection from m-wise Partial Rankings via Borda Counting Wenjing Chen, Ruida Zhou, and Chao Tian Department of Electrical and Computer Engineering Texas A&M University {jj9754, ruida, chao.tian}@tamu.edu Cong Shen Electrical and Computer Engineering Department University of Virginia cong@virginia.edu Abstract—We analyze the performance of Borda counting algorithm on noisy m-wise ranking data to accurately select the top-k items from a total of n items. This generalizes a previous result of a similar nature reported by Shah et al. on the noisy pairwise comparison data. We show that the associated score separation ∆k between the k-th item and the (k+1)-th item plays an important role: if ∆k is greater than a threshold depending on (n, k) and the scoring system in Borda counting, then the top-k selection is accurate asymptotically almost surely; if ∆k is below a threshold, then the top-k selection will not be accurate with at least a constant probability. This separation between the two thresholds depends on m and the scoring systems in the Borda counting procedure.
I. INTRODUCTION The problem of rank aggregation has drawn considerable at-tention due to its diverse applications in information retrieval, recommending system, and social science; see e.g., –.
In such a system consisting of n items, given noisy partial rankings among subsets of the items, the goal is to recover the desired results regarding the full rank, such as the top-k items or the ranking among items in a given subset.
A well studied case in this setting is rank aggregation from pairwise comparisons. The Bradley-Terry-Luce (BTL) model was adopted by many –, where each item is assigned an underlying score, which the probabilities of the noisy comparisons are based on. Under this model, the rank aggregation problem can be reduced to an estimation problem.
However, algorithms based on the BTL model may perform poorly if there is a mismatch to the model. To tackle such difficulty, it is desirable to adopt a non-parametric model. In a recent work , Shah et al. considered a counting based method for pairwise comparison, which can be viewed as a special case of the Borda counting method . The procedure is similar to voting, where the votes to a candidate are counted.
It was shown that the method is robust and optimal, which was established with a pair of almost matching conditions on the error probability, for top-k and approximate top-k selection.
Pairwise comparisons can be viewed as a special case of m-wise partial ranking. For parametric models, there are past works considering the m-wise case. For example, studied the top-k selection items from m-wise partial rankings under the Plackett-Luce (PL) model ; see also . In this work, we extend the study of the non-parametric model to consider the m-wise case. The natural algorithm of choice is the Borda counting method, for which we establish an almost matching pair of the conditions in a similar manner as those given by Shah et al. . The critical quantity ∆k again plays an important role. The difference between the achievable bound and the converse bound depends on m, as well as the choice of the scoring systems in the Borda counting.
II. PROBLEM SETTING We consider the following scenario: there are a total of n items indexed by [n] = {1, 2, . . . , n}, and the noisy samples of the ranking among the items in the subsets of them are collected in r rounds. In round ℓ∈[r], each subset of m items, say A ⊆[n], where |A| = m, are ranked with probability p, and not ranked at all with probability 1 −p. When the items in set A are being ranked, the probability of observing the ranking ⃗ v = (v1, v2, ...vm) is denoted as Mv1v2...vm or M⃗ v.
Here ⃗ v is considered a vector and the sequence of the elements is important; we shall write ⃗ v .
= A if the items in the vector ⃗ v are those in the set A. It is clear that such M⃗ v must satisfy M⃗ v ≥0 for any ⃗ v with distinct elements, and moreover, X ⃗ v .
=A M⃗ v = 1.
(1) Denote the collection of probability distributions as M. From the r rounds of rankings, we wish to identify the top-k items.
A. The Borda Counting Procedure The mechanism we consider for selecting the top-k items is the well-known Borda counting procedure : in a competition among the items in A where |A| = m, the i-th item will receive score βi, where we assume 1 = β1 ≥β2 ≥... ≥ βm ≥0; we denote the vector as ⃗ β. The random variable X(ℓ) a,A−is used to denote the score a particular item-a receives in the ℓ-th round in the competition among the items in A = a ∪A−⊆[n], and thus it is distributed as follows Pr(X(ℓ) a,A−= β) = ( pRa,A−(t) β = βt 1 −p β = 0 , (2) where Ra,A−(t) is the probability that item-a ranks at the t-th position in the set {a} ∪A−, and thus Ra,A−(t) = X ⃗ v .
={a}∪A− vt=a M⃗ v.
(3) After r rounds, the total score item-a receives is thus Wa = X ℓ∈[r] X A−⊆[n]{a} X(ℓ) a,A−.
(4) The top-k among the items will be estimated in this procedure to be the items which receive the highest k scores empirically, which is denoted as ˜ Sk.
In the sequel, we use A to denote a set of cardinality m in [n] without specifying m, A−to denote that of cardinality m −1, and A−−that of cardinality m −2.
B. The Associated Score It is clear that E X(ℓ) a,A− = p m X t=1 βtRa,A−(t).
(5) We define the associated score of item-a as τa = 1 ρn,m X A−⊆[n]{a} m X t=1 βtRa,A−(t) , (6) where ρn,m = n −1 m −1 .
(7) The top-k set of the highest associated scores is denoted as S∗ k. The difference between the k-th highest associated score and (k + 1)-th highest associated score is denoted as ∆k = τ(k) −τ(k+1).
(8) This quantity is important because if it is large, then we would expect the top-k items are easy to find.
III. TOP-k SELECTION A. Main Result Our main result is a pair of almost matching conditions regarding the accuracy of the Borda-count-based method. To present the result, we first define the following set of partial ranking probabilities: Fk(α) = ( M ∈M : ∆k ≥α s log n rpρn,m ) .
(9) Theorem 1. For any α > 0, the probability of choosing incorrect top-k items using the score-based method for any items with M ∈Fk(α) is upper-bounded as sup M∈Fk(α) PM[ ˜ Sk ̸= S∗ k] ≤n−α2/4+2.
(10) Our next result is regarding the fundamental limit of the score-based method.
Theorem 2. Let n, k where 2k ≤n be chosen. If α ≤ ¯ α(g, m, ⃗ β) ≜ √ 2 7 g(n, m, ⃗ β) q 1 h(n)ρn,m , p ≥ log n 4rh(n), and n ≥7, then the error probability of any estimator ˆ Sk is lower bounded as sup M∈Fk(α) PM[ ˆ Sk ̸= S∗ k] ≥1 7, (11) 5 10 15 20 25 30 35 40 45 50 55 60 n 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 ̄ α(g, m, ⃗ β⃗ m = 2, k = 3 m = 3, k = 4 m = 4, k = 4 m = 5, k = 4 m = 6, k = 5 m = 8, k = 6 Fig. 1. The bound ¯ α(g, m, ⃗ β) for different m and k. The notch on the dotted line is where the value of q changes from m −n + k to 1. When n is fixed and q = 1, the increase of m will lead to the decrease of ¯ α(g, m, ⃗ β).
where g(n, m, ⃗ β) = n m!
q X t=1 (βt −βm−q+t)Aq−1 k−1Am−q−1 n−k−1 h(n) = 1 m q k −1 q −1 n −k m −q + k q n −k −1 m −q −1 , and q ≜max(1, m −n + k) and Ak n = n!/(n −k)!.
In Fig. 1 we plot the bound ¯ α(g, m, ⃗ β) when ⃗ β is chosen to be of uniform spacing and βm = 0. It is seen that the bound is lower for larger m in general.
B. Example Special Case: m = 2 Consider the special case m = 2, where (β1, β2) = (1, 0).
This was exactly the case considered in . For this case, pρn,m = p(n −1).
(12) 1) Theorem 1 can be simplified to produce the following set Fk(α) = ( M ∈M : ∆k ≥α s log n rp(n −1) ) .
(13) If we set α ≥8, then the bound becomes sup M∈Fk(α) PM[ ˜ Sk ̸= S∗ k] ≤n−α2/4+2 ≤n−14.
(14) The result in has a slightly different definition of ∆k, which can be translated into our notation, and it induces the following set F (1) k (α) F (1) k (α) = ( M ∈M : ∆k ≥α s log n rp(n −1) r n n −1 ) .
The eventual error probability bound given in is then of the same form as (14), except that Fk(α) is replaced by F (1) k (α). It is easily seen that F (1) k (α) ⊆Fk(α), and thus Theorem 1 is in fact slightly stronger than the corresponding result in for this special case.
2) When m = 2, the assumptions in Theorem 2 are reduced to α ≤ 1 7 q n n−1, and that p ≥ log n 2rn . It implies that the conditions and the bound given in Theorem 2 match precisely those given in the corresponding converse in , after taking into account the slightly different definition of ∆k between ours and theirs.
C. Proofs of Theorem 1 We will need to invoke Bernstein’s inequality in the proof of Theorem 1, which is stated below.
Lemma 1 (Bernstein’s inequality). Let Y1, ..., Yn be indepen-dent zero-mean random variables. Suppose that |Yi| ≤M almost surely, for all i. Then, for all positive t, we have that P n X i=1 Yi > t !
≤exp − 1 2t2 P E [Y 2 i ] + 1 3Mt .
Proof of Theorem 1. Consider any item a ∈S∗ k and b ∈[n] \ S∗ k, then define the event that Wb > Wa as Eba, i.e., Pr(Eba) = Pr(Wb −Wa > 0).
(15) We start by noting that Wb −Wa = X ℓ∈[r] X A−⊆[n]{b} X(ℓ) b,A−− X A−⊆[n]{a} X(ℓ) a,A− .
It follows that Wb −Wa = X ℓ∈[r] X A−⊆[n]{a,b} X(ℓ) b,A−−X(ℓ) a,A− − X ℓ∈[r] X A−−⊆[n]{a,b} X(ℓ) a,{b,A−−} −X(ℓ) b,{a,A−−} . (16) In order to apply Bernstein’s inequality, define the centralized scores. For any a ∈[n] ¯ X(ℓ) a,A−≜X(ℓ) a,A−−E X(ℓ) a,A− = X(ℓ) a,A−−p m X t=1 βtRa,A−(t), and the centralized cross-score ¯ X(ℓ) {a,b},A−−≜X(ℓ) a,{b,A−−} −X(ℓ) b,{a,A−−} −E X(ℓ) a,{b,A−−} + E X(ℓ) b,{a,A−−} = X(ℓ) a,{b,A−−} −X(ℓ) b,{a,A−−} − p m X t=1 βtRa,{b,A−−}(t) −p m X t=1 βtRt b,{a,A−−} !
. (17) Since all the centralized scores and centralized cross-scores are zero-mean and mutually independent, we can apply Bern-stein’s inequality on these random variables.
It follows after some simple algebra that Wb −Wa = ∆¯ Xa,b −rpρn,m(τa −τb), (18) where for notation simplicity we have defined ∆¯ Xa,b ≜ X ℓ∈[r] X A−⊆[n]{a,b} ¯ X(ℓ) b,A−−¯ X(ℓ) a,A− − X ℓ∈[r] X A−−⊆[n]{a,b} ¯ X(ℓ) {a,b},A−−.
(19) Because a is in the top-k and b is not in the top-k, we have rpρn,m(τa −τb) ≥rpρn,m∆k.
(20) Since Wb−Wa ≥0 is equivalent to ∆¯ Xa,b ≥rpρn,m(τa−τb), we have Pr(Wb −Wa ≥0) ≤Pr ∆¯ Xa,b ≥rpρn,m∆k .
In order to bound the probability on the right hand side, we need to bound the sum of the variances of the centralized random variables, for which we have the following lemma.
Lemma 2.
Σ ¯ Xa,b ≜ X A−⊆[n]{a,b} E ¯ X(ℓ) b,A− 2 + E ¯ X(ℓ) a,A− 2 + X A−−⊆[n]{a,b} E ¯ X(ℓ) {a,b},A−− 2 ≤2pρn,m −pρn,m∆k.
(21) The proof of this lemma can be found in the appendix. It is also clear that the scores are upper-bounded as | ¯ Xa,A−|, | ¯ Xb,A−| ≤β1 = 1, | ¯ X{a,b},A−−| ≤2β1 = 2.
(22) We are now ready to plug in the required quantities in Bernstein’s inequality, and obtain Pr(Wb −Wa > 0) ≤exp −1 2 (rpρn,m∆k)2 2rpρn,m −rpρn,m∆k + 2 3ρn,mrp∆k !
≤exp −(rpρn,m∆k)2 4rpρn,m !
≤exp −α2 log n 4 = n−α2 4 , (23) where in the last inequality we substituted the bound in (9).
Taking the union bound on the possible pairs (a, b) gives the desired result, and the proof is complete.
D. Proof of Theorem 2 For each a ∈{k, . . . , n}, denote the k-sized subset S∗[a] = {1, 2, . . . , k −1} ∪{a}. In order to prove the converse result, we construct a probability distribution M ∈Fk(α) such that it is difficult for any method to find the top-k items, which is: M a ⃗ v = (m!)−1(1 + δ) if C1 (m!)−1(1 −δ) if C2 (m!)−1 otherwise , (24) where δ ∈[0, 1] is a parameter to be specified later, and the conditions C1 and C2 are given as: C1: ⃗ v1:q ⊆S∗[a] and ⃗ vq+1:m ⊆[n] \ S∗[a]; C2: ⃗ v1:m−q ⊆[n] \ S∗[a] and ⃗ vm−q+1:m ⊆S∗[a].
The theorem is established through two propositions.
Proposition 1.
For the distribution M a ⃗ v , where a ∈ {k, . . . , n}, we have ∆k = g(n, m, ⃗ β) ρn,m δ.
(25) The proof of this proposition can be found in the appendix.
Proposition 2. For any distinct a, b ∈{k, k + 1, . . . , n}, DKL(Pa||Pb) ≤rph(n) 4δ2 (1 −δ2).
(26) For notation convenience, we shall also define µ(δ) ≜p [(1 + δ) log(1 + δ) + (1 −δ) log(1 −δ)] υ(δ) ≜p log(1 + δ)−1 + log(1 −δ)−1 , ωm,q ≜ m q −1 .
Proof of Proposition 2. Using the property of the KL-divergence, we can write DKL(Pa||Pb) = X ℓ∈[r] p X A⊆[n]:|A|=m DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )) = rp X A⊆[n]:|A|=m DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )), (27) where V (ℓ) A is the result of the comparison among the elements in A. It is clear that we need to bound the KL divergence for each subset A ⊆[n], which can be done as follows.
1) If a, b / ∈A, then clearly DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )) = 0; 2) If a ∈A but b / ∈A, a) If A ∩[k −1] = q −1, we have DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )) = q!(m −q)!
(m!) µ(δ) ≤2ωm,qpδ2 ≤2ωm,q pδ2 1 −δ2 .
(28) b) If A ∩[k −1] = q, we have DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )) = q!(m −q)!
m!
υ(δ) ≤ωm,qp −δ 1 + δ + δ 1 −δ = 2ωm,qp δ2 1 −δ2 . (29) c) If A ∩[k −1] ̸= q −1 and A ∩[k −1] ̸= q, then DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )) = 0.
(30) 3) If b ∈A but a / ∈A, a) If A ∩[k −1] = q, we have DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )) = q!(m −q)!
m!
µ(δ) ≤2ωm,qpδ2 ≤2ωm,qp δ2 1 −δ2 .
(31) b) If A ∩[k −1] = q −1, we have DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )) = q!(m −q)!
m!
υ(δ) ≤2ωm,qp δ2 1 −δ2 .
(32) c) If A ∩[k −1] ̸= q −1 and A ∩[k −1] ̸= q, then DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )) = 0.
4) If a, b ∈A, a) If A ∩[k −1] = q −1, and q > 1, DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )) = q!(m −q)!
m!
[µ(δ) + υ(δ)] ≤4ωm,qpδ2 ≤4ωm,qp δ2 1 −δ2 .
(33) If A ∩[k −1] = q −1, and q = 1, DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )) = (m −2)(m −2)!
(m!) [µ(δ) + υ(δ)] + (m −2)!
(m!) p[(1 −δ) log 1 −δ 1 + δ + (1 + δ) log 1 + δ 1 −δ ] ≤8ωm,qp δ2 1 −δ2 .
(34) Thus for all q, we have DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )) ≤8ωm,qp δ2 1 −δ2 .
(35) b) A∩[k−1] ̸= q−1, then DKL(Pa(V (ℓ) A )||Pb(V (ℓ) A )) = 0.
Summarizing the above bound, we arrive at DKL(Pa||Pb) = rp X A⊆[n]:|A|=m DKL(Pa(V (1) A )||Pb(V (1) A )) ≤r4δ2ωm,qp (1 −δ2) k −1 q −1 n −k −1 m −q + k −1 q n −k −1 m −q −1 + 2 k −1 q −1 n −k −1 m −q −1 ≤r4δ2ωm,qp (1 −δ2) k −1 q −1 n −k m −q + k q n −k −1 m −q −1 (36) which is the desired result.
Suppose the underlying distribution is drawn uniformly at random from the set {M a|a ∈[n] \ [k −1]}, and the true index is a∗. By Fano’s inequality, any estimator ˆ a must have an error probability lower-bounded by PM[ˆ a ̸= a∗] ≥1 − rph(n) 4δ2 (1−δ2) + log 2 log(n −k + 1) .
(37) Since α ≤ √ 2 7 g(n, m, ⃗ β) q 1 h(n)ρn,m and p ≥ log n 4rh(n), we can choose ∆k = √ 2 7 g(n, m, ⃗ β) q 1 h(n)ρn,m q log n rpρn,m . Combining these bounds with (25) in Proposition 1 gives δ ≤ √ 2 7 s log n rph(n) ≤2 √ 2 7 .
(38) Finally (37) and (38) give PM[ˆ a ̸= a∗] ≥1 − rph(n) 4( √ 2 7 q log n rph(n) ) 2 (1−( 2 √ 2 7 ) 2) + log 2 log n −log 2 ≥1 − 8 41 log n + log 2 log n −log 2 ≥1 7, (39) where we also used 2k ≤n and n ≥7, and this completes the proof.
IV. APPROXIMATE TOP-k SELECTION In the previous section, we considered the top-k selection problem. In many cases, it may be sufficient to identify the top-k set approximately, which we consider next. This was considered previously in for the case of m = 2.
A. Main Results First, define Fk,h(α) = ( M ∈M : ∆k,h ≥α s log n rpρn,m ) , (40) where ∆k,h := τ(k−h) −τ(k+h). Comparing Fk,h(α) with Fk(α) in (9), it is seen that the only difference is that ∆k in (9) is replaced by ∆k,h, i.e., it was for the case ∆k,0 = ∆k.
In the sequel, we denote the Hamming distance between two subsets A, B of [n] as DH(A, B) = | {A ∪B} \ {A ∩B} |.
(41) Theorem 3. For any α > 0, we have sup M∈Fk,h(α) PM[DH( ˜ Sk, S∗ k) > 2h] ≤n−α2/4+2.
(42) Theorem 4. Let ν1, ν2 ∈(0, 1) be two constants such that 2h ≤ 1 1+ν2 min n −k, k, n1−ν1 . In the regime p ≥ log n 4h(n)r, for any α ≤ √ 2 14 g(n, m, ⃗ β) q ν1ν2 h(n)ρn,m , any estimator ˆ Sk has an error probability lower bounded by sup M∈Fk,h(α) PM[DH( ˆ Sk, S∗ k) > 2h] ≥1 7, (43) for all n larger than a constant c(ν1, ν2), and q ≜max(1, m− n + k).
Consider again the special case m = 2: it can be verified that Theorem 3 is slightly stronger than the corresponding result in , and Theorem 4 is precisely the same.
B. Proofs of Theorem 3 and 4 Theorem 3 is in fact a direct consequence of Theorem 1. To see this, note that Theorem 1 implies with probability at least 1 −n−α2/4+2, ˆ Sk ranks every item in {1, ..., k −h} higher than every item in the set {k + h + 1, ..., n}. Thus, we have either ˆ Sk ⊆[k + h] or [k −h] ⊆ˆ Sk. Either case would lead to | ˆ Sk ∩[k]| ≥k −h, thereby proving Theorem 3.
To prove Theorem 4, we follow the approach in . The following lemma is instrumental in the proof.
Lemma 3 (Shah et al.
).
In the regime 2h ≤ 1 ν1+ν2 min n1−ν1, k, n −k for some constants ν1 ∈(0, 1) and ν2 ∈(0, 1), and when n is larger than a (ν1, ν2)-dependent constant, there exists a subset b1, ..., bL ⊆{0, 1}n/2 with cardinality L ≥L∗≜e 9 10 ν1ν2h log n, such that DH(bj, 0) = 2(1 + ν2)h, DH(bj, bk) > 4h for all j ̸= k ∈[L].
It was shown in that constructing L∗different probability distributions satisfying the two properties below would prove Theorem 4 when m = 2.
1) For every i ∈[L∗], let Si k ⊆[n] denote the set of top-k items under the i-th distribution. Then for every k-sized set S ∈[n], L X i=1 1(DH(S, Si k) ≤2h) ≤1.
(44) 2) If the underlying distribution is chosen uniformly at random from this set of L∗distributions, then any estimator that attempts to identify the underlying distribution i ∈[L∗] errs with probability at least 1 −1 7.
We follow the approach in to first construct L∗different k-sized subset of [n]. For each i ∈[L∗], let Bi denote a [2(1 + ν2)h]-sized subset of n 2 + 1, ..., n 2 . The items in Bi correspond to the 2(1 + ν2)h positions being 1 in the i-th string as specified by Lemma 3. Define the sets Ai = {1, ..., k −2(1 + ν2)h}. The k-sized subset Si k is then constructed as Si k = Ai ∪Bi, which is valid since 2h ≤ 1 1+ν2 k. By Lemma 3, for any distinct i1, i2 ∈[L], we have DH(Ai1 ∪Bi1, Ai2 ∪Bi2) ≥4h+1. This implies that the condition (44) is satisfied, since otherwise the existence of a set S that makes the LHS of (44) greater than 1 would make the two corresponding sets Si1 k and Si2 k differ by strictly less than (4h + 1).
We now construct L∗probability distributions. For every i ∈ [L∗] and the k-sized subset Si k given above, let the distribution M i ⃗ v be as in (24), with the conditions replaced by C1: ⃗ v1:q ⊆Si k and ⃗ vq+1:m ⊆[n] \ Si k; C2: ⃗ v1:m−q ⊆[n] \ Si kand ⃗ vm−q+1:m ⊆Si k, which leads to ∆k,h = ∆k = g(n, m, ⃗ β) ρn,m δ.
(45) For any i1, i2 ∈[L∗], an upper bound for the K-L divergence between different probability distributions is needed. For any A ⊆[n] where |A| = m, Pi1(V (ℓ) A ) ̸= Pi2(V (ℓ) A ) only if ∃j ∈A, such that j ∈Bi1 ∪Bi2. In Theorem 2, we have shown that DKL(Pi1(V (ℓ) A )||Pi2(V (ℓ) A )) ≤ m q (−1) 8δ2 1−δ2 , from which it follows that DKL(Pi1||Pi2) ≤4prh(n)(1 + ν2)h 8δ2 1 −δ2 .
(46) Under the assumptions on α and p, it can be verified straight-forwardly that δ ≤ √ 2 7 , from which it follows that DKL(Pi1||Pi2) ≤3 4ν1ν2h log n.
(47) Suppose that the underlying distribution is drawn uniformly at random from the constructed set, and the index is i∗. By Fano’s inequality, any estimator ˆ i∗must have error probability lower bounded by P( ˆ i∗̸= i∗) ≥1 − 3 4ν1ν2h log n + log 2 9 10ν1ν2h log n ≥1 7.
(48) This establishes property 2) for the distributions. The proof of Theorem 4 can be completed alone the same line as in .
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L. Baltrunas, T. Makcinskas, and F. Ricci, “Group recommendations with rank aggregation and collaborative filtering,” in Proceedings of the fourth ACM conference on Recommender systems, pp. 119–126, ACM, 2010.
A. Caplin and B. Nalebuff, “Aggregation and social choice: a mean voter theorem,” Econometrica: Journal of the Econometric Society, pp. 1–23, 1991.
R. A. Bradley and M. E. Terry, “Rank analysis of incomplete block designs: I. the method of paired comparisons,” Biometrika, vol. 39, no. 3/4, pp. 324–345, 1952.
F. Wauthier, M. Jordan, and N. Jojic, “Efficient ranking from pair-wise comparisons,” in International Conference on Machine Learning, pp. 109–117, 2013.
S. Negahban, S. Oh, and D. Shah, “Rank centrality: Ranking from pairwise comparisons,” Operations Research, vol. 65, no. 1, pp. 266– 287, 2016.
Y. Chen and C. Suh, “Spectral mle: Top-k rank aggregation from pair-wise comparisons,” in International Conference on Machine Learning, pp. 371–380, 2015.
N. B. Shah and M. J. Wainwright, “Simple, robust and optimal ranking from pairwise comparisons,” The Journal of Machine Learning Re-search, vol. 18, no. 1, pp. 7246–7283, 2017.
J. d. Borda, “M´ emoire sur les ´ elections au scrutin,” Histoire de l’Academie Royale des Sciences pour 1781 (Paris, 1784), 1784.
M. Jang, S. Kim, C. Suh, and S. Oh, “Optimal sample complexity of m-wise data for top-k ranking,” in Advances in Neural Information Processing Systems, pp. 1686–1696, 2017.
R. L. Plackett, “The analysis of permutations,” Journal of the Royal Statistical Society: Series C (Applied Statistics), vol. 24, no. 2, pp. 193– 202, 1975.
R. D. Luce, Individual choice behavior: A theoretical analysis. Courier Corporation, 2012.
APPENDIX Proof of Lemma 2. We first calculate the variances of the cen-tralized random variables as follows. For any A−∈[n]\ {a, b} and |A−| = m −1, we have for the centralized scores E ¯ X(ℓ) b,A− 2 = E X(ℓ) b,A− 2 −E h X(ℓ) b,A− i2 (49) ≤E X(ℓ) b,A− 2 = p m X t=1 β2 t Rb,A−(t), (50) and similarly E ¯ X(ℓ) a,A− 2 ≤p m X t=1 β2 t Ra,A−(t).
(51) Moreover, we have for any a, b, A−− E ¯ X(ℓ) {a,b},A−− 2 ≤E X(ℓ) a,{b,A−−} −X(ℓ) b,{a,A−−} 2 = p X j,k∈[m] (βj −βk)2 X ⃗ v .
=A−−∪{a,b} vj=a,vk=b M⃗ v ≤p X j,k∈[m] (β2 j + β2 k) X ⃗ v .
=A−−∪{a,b} vj=a,vk=b M⃗ v = p m X j=1 β2 j X ⃗ v .
=A−−∪{a,b} vj=a M⃗ v (52) + p m X k=1 β2 k X ⃗ v .
=A−−∪{a,b} vk=b M⃗ v = p m X t=1 β2 t Ra,{b,A−−}(t) + Rb,{a,A−−}(t) .
(53) Thus we have the following bound on the sum of the variances of centralized scores and the cross-scores Σ ¯ Xa,b ≤p X A−⊆[n]{a} m X t=1 β2 t Ra,A−(t) + p X A−⊆[n]{b} m X t=1 β2 t Rb,A−(t) = p X A−⊆[n]{a} m−1 X t=1 (β2 t −β2 m)Ra,A−(t) + pβ2 mρn,m + p X A−⊆[n]{b} m X t=1 β2 t Rb,A−(t).
(54) Because for any i ∈[m], we have βi ≤1, and thus β2 i ≤βi.
It follows that X A−⊆[n]{b} m X t=1 β2 t Rb,A−(t) ≤ X A−⊆[n]{b} m X t=1 βtRb,A−(t) ≤ X A−⊆[n]{a} m X t=1 βtRa,A−(t) −ρn,m∆k Since m X j=1 Rj a,A−= 1, (55) we further have Σ ¯ Xa,b ≤p(1 −β2 m)ρn,m + pβ2 mρn,m + p X A−⊆[n]{a} |A−|=m−1 m X t=1 βtRa,A−(t) −pρn,m∆k ≤pρn,m + pρn,m −ρn,m∆k = 2pρn,m −pρn,m∆k (56) The Proof of Proposition 1. Let us consider the associated score of item-k using the distribution M ∗, which can be seen as τ(k) = 1 m n−1 m−1 n m −1 m X t=1 βt (57) + δ n−1 m−1 1 m!
q X t=1 (βt −βm−q+t)Aq−1 k−1Am−q n−k (58) Similarly, we have τ(k+1) = 1 n−1 m−1 1 m n m −1 m X t=1 βt (59) + δ n−1 m−1 1 m!
m X t=q+1 (βt −βt−q)Aq kAm−1−q n−k−1 .
(60) It follows that ∆k = τ(k) −τ(k+1), (61) which leads to the quantity in (25), after the first terms in both τ(k) and τ(k+1) cancel out each other. |
6640 | https://www.youtube.com/watch?v=77LVxv05XKE | Criss-Cross Method for Writing Formulas for Ionic Compounds
Wayne Breslyn (Dr. B.)
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Posted: 15 Sep 2023
Here's a step-by-step guide on how to employ the Criss Cross Method:
Identify the Ions: Begin by identifying the cation (the positively charged ion) and the anion (the negatively charged ion) in the compound. It's crucial to know the charges of these ions, as they determine how many of each ion are needed to balance the compound.
Determine the Charges: Determine the numerical charges of both ions. These charges can typically be found on the periodic table or are given in the chemical name or formula. Cations are typically positively charged, while anions carry a negative charge.
Join this channel to get full access to Dr. B's chemistry guides:
Criss Cross: To write the formula, "criss cross" the numerical charges as subscripts. In other words, take the absolute value of the charge on one ion and make it the subscript for the other ion, and vice versa. This step ensures that the total positive charge equals the total negative charge, resulting in a neutral compound.
Simplify (if needed): If the subscripts can be further simplified by dividing both numbers by a common factor, do so to obtain the simplest, most reduced formula.
Check for Proper Ratios: Ensure that the subscripts represent the smallest whole number ratio of the ions in the compound. If necessary, multiply all the subscripts by a common factor to achieve this ratio.
Finalize the Formula: Write the final chemical formula with the cation listed first and the anion second, using the subscripts as indicated by the Criss Cross Method.
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Transcript:
Introduction this is Dr beat and we're going to look at the crisscross method today it's a real fast way to write the formulas for ionic compounds so let me show you one and then we'll do some examples in practice calcium nitride it's ionic we have a metal and a non-metal so we write CA and then we write n that's for the nitrogen there nitride is from nitrogen now we look the charges up to plus 3 minus I'll show you how to do that in just a second and now we crisscross them so we move the 3 here and the 2 here get rid of the charges here and that's it this is the formula for calcium nitride ca3n2 and the charges they all balance out so let's do some practice make sure you get this Calcium Fluoride so let's do three or four of these why don't we do calcium fluoride so we've got calcium we've got fluorine metal and non-metal this is an ionic compound so we have to think about the charge here this is the trend for charge so I'll write calcium c a and then fluoride that's from fluorine so f now for the charges calcium is in group two two plus fluorine is right here Group 17 these are one minus so we put our 1 minus and now we can just crisscross them take the one down here and the two here get rid of these charges here and we don't write the one by convention calcium fluoride CA F2 and the charges they all balance out let's try another one Calcium Sulfate so pause and give this one a try calcium sulfate and the trick here when you see ate you're not going to find it on the periodic table sulfate is one of those you have to memorize so4 2 minus so give it a try using the crisscross method so again we've got calcium C A and we know it's in group two so it's 2 plus then the sulfate because it says eight we know that we're not going to find it on the periodic table we just need to memorize this as the sulfate ion so4 and the whole thing has a 2 minus ionic charge and you can see that these charges here they balance out just get rid of them and you're done that's the formula for calcium sulfate so now I'm going to give you a little more difficult sulfate to write the formula for Aluminum Sulfate aluminum sulfate give it a try so we have aluminum right here that's our metal and then the sulfate that's a polyatomic ion it's an ionic compound we write Al for aluminum and then we write the sulfate so4 2 minus and it's always so4 2 minus now aluminum that's right here so it has a three plus ionic charge so we're going to crisscross them move the 2 here the 3 here let's get rid of this and we can't really have the four and three here together this 3 means we have three sulfate ions so we need to put parentheses around it that's the formula for aluminum sulfate and if you look at the charges 3 times 2 minus that's 6 minus 2 times the 3 plus that's six plus six plus six minus that balances out let's do another Iron III Oxide Iron III oxide and the three is because this is a transition metal that means it has a 3 plus ionic charge so write the formula for Iron III oxide iron that's f e oxide that's from oxygen right here that's o That's going to be 2 minus and we know this is 3 plus because it's called iron three let's just move that up there now we just need to crisscross 3 goes here 2 goes here and you've got the formula for Iron III oxide Fe2O3 all right one last one and we'll call it good carbon dioxide give it a try and be careful make sure you know what you have up here in terms of metals and nonmetals so carbon right here that's a nonmetal and oxygen that's a non-metal so in this case we're not going to worry about the charge because we have two non-metals we just look at the prefixes we have carbon that's C and then we know oxygen that's o the oxide and that die die means two like a paradise CO2 that's the formula for carbon dioxide so when you're writing formula and you see these prefixes here you're probably dealing with non-metals and it's a covalent or molecular compound that's it this is Dr B I hope this helped you with formula writing thanks for watching |
6641 | https://www.trinka.ai/blog/advisor-vs-adviser-a-guide-to-the-right-usage/ | Grammar Tips
Advisor vs. Adviser: A Guide to the Right Usage
By Trinka
Reading Time 4 minutes
The most ambiguous differences come through in language, when subtleties of meaning and usage are concerned. Many people have found themselves unsure whether to use the words “advisor” and “adviser,” each referring to a guiding force; the difference between them will depend on the connotations, usage, and context intended for a situation. In this blog post, we will discuss two similar words that tend to confuse people. Whether you are writing a resume or looking for professional assistance, it is good to know which term is best used in order not to miscommunicate. Let’s explore the definition and discover what makes the difference.
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Table of Contents
Advisor vs. Adviser |Definition, Difference & Examples
When discussing advisor vs. adviser, it’s helpful to start with definitions. An advisor is often considered a more formal term. It typically refers to someone in a certain professional capacity, providing expert advice on specific subjects, like financial or academic matters.
On the other hand, an adviser is broadly speaking a term considered somewhat less formal. The word commonly found in educational fields when one educator guides students through decisions and into professions.
While this all happens, both terms are largely considered acceptable interchangeably in most speaking or everyday conversation. Selection may actually have more to do with regional preference or individual choice over anything in grammar book law.
Definition of Advisor?
An advisor is a professional who advises or recommends on different issues. The term is used in various fields, including finance, education, and health. Advisors assist individuals or organizations in making the right decisions.
In business, an advisor might specialize in strategy or marketing. They analyze situations and offer solutions tailored to improve outcomes. Their expertise may be very valuable to clients who are facing complex challenges.
The role of an advisor often entails building relationships based on trust and understanding the specific needs of their clients. They work closely with those they advise to ensure that the advice given aligns with individual goals and circumstances.
Definition of Adviser?
An adviser is a professional who provides guidance or recommendations in a given field. This can be in finance, education, or even career development. An adviser usually helps a person make the right decision.
An adviser is someone who has special knowledge and expertise in a given area; they are usually able to share this with clients either individually or as a group, depending on the situation. The key purpose of an adviser, however, is to help individuals grow and become better people.
In many instances, advisers provide insights and strategies that further empower their clients toward meeting specific needs. This, in turn, develops trust between the parties, as well as constructive conversation regarding opportunities and challenges ahead.
The Difference Between Advisor and Adviser
Often the words “advisor” and “adviser” are confusing. They look pretty much alike, but there is a difference–mostly in usage and by preference. Many institutions prefer one spelling over the other, which accounts for the variation in job titles.
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“Advisor” is commonly used in business settings, especially when it comes to financial services. It gives a perception of authority or expertise. Conversely, “adviser” tends to bring into mind government roles or formal positions within educational institutions.
Both words have the same goal, which is to guide other people with expert knowledge. The main point is that context dictates the use of one over the other for clarity and professionalism.
Which Is Best, Advisor or Adviser?
The decision between the terms “advisor” and “adviser” usually depends on the situation. Both are often considered as synonyms for each other; however, some industry leaders maintain specific usage rules. Within financial services, “Advisor” is more popularly used for terms like “financial advisor.”
While many institutions in higher learning and other formal systems are more inclined to “adviser.” This might be due to the nature of the academic institution. This is where faculty often guides students.
Your decision may be based on regional predilections or particular professional standards. Whatever you pick, uniformity is what matters. Keep one word in mind to use across your entire work for easier and more professional reading.
Examples: Advisor and Adviser in Sentences
In this case, context defines the words. For example, you can state, “She consulted her financial advisor for investment advice.” This demonstrates a person who advises.
On the other hand, “The consultant suggested a new approach to increase efficiency,” showcases another more general role that may be applicable to such spheres as education or business. Both sentences demonstrate how each word can fit into distinct contexts while still expressing the core idea of guidance.
This understanding helps you choose the right word for your purpose. Whether it is an advisor or an adviser, there is a difference, and being clear in communication is key. To be precise about the type of assistance someone offers makes your conversation more enriching and information more accurate.
The terms “advisor” and “adviser” both refer to someone who provides advice, but there are subtle differences in their usage. “Adviser” is the more traditional and widely accepted spelling, particularly in formal and academic contexts. “Advisor,” on the other hand, is more commonly used in certain industries, especially in the United States, such as in financial advising or in educational settings like “academic advisor.” Choosing between these terms can depend on regional preferences or specific industry norms. A grammar checker like Trinka helps by offering suggestions for consistent and correct word choice, ensuring your writing aligns with the appropriate conventions and standards for your context, whether academic, professional, or informal.
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6642 | https://www.engineeringtoolbox.com/aisi-stainless-steels-corrosion-resistance-d_1843.html | Published Time: 2012-11-14T04:25:03Z
Stainless Steels - Corrosion Resistance
Engineering ToolBox - Resources, Tools and Basic Information for Engineering and Design of Technical Applications!
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Stainless Steels - Corrosion Resistance
Corrosion resistanse of AISI stainless steels in various environments.
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Corrosion resistance of AISI Stainless Steels in typical environments:
For full table with Oxidizing and Reducing Chemicals -rotate the screen!
Stainless Steels - Corrosion Resistance| AISI Type | Fresh Water and Mild Atmospheric Environments | Industrial Environments | Marine Environments | Mild Chemicals | Oxidizing Chemicals | Reducing Chemicals |
--- --- ---
| 201 | • | • | • | • | • | |
| 202 | • | • | • | • | • | |
| 205 | • | • | • | • | • | |
| 301 | • | • | • | • | • | |
| 302 | • | • | • | • | • | |
| 302B | • | • | • | • | • | |
| 303 | • | • | • | • | | |
| 303 Se | • | • | • | • | | |
| 304 | • | • | • | • | • | |
| 304L | • | • | • | • | • | |
| 304N | • | • | • | • | • | |
| 305 | • | • | • | • | • | |
| 308 | • | • | • | • | • | |
| 309 | • | • | • | • | • | |
| 309S | • | • | • | • | • | |
| 310 | • | • | • | • | • | |
| 310S | • | • | • | • | • | |
| 314 | • | • | • | • | • | |
| 316 | • | • | • | • | • | • |
| 316F | • | • | • | • | • | • |
| 316L | • | • | • | • | • | • |
| 316N | • | • | • | • | • | • |
| 317 | • | • | • | • | • | • |
| 317L | • | • | • | • | • | |
| 321 | • | • | • | • | • | |
| 329 | • | • | • | • | • | • |
| 330 | • | • | • | • | • | • |
| 347 | • | • | • | • | • | |
| 348 | • | • | • | • | • | |
| 384 | • | • | • | • | • | |
| 403 | • | | | • | | |
| 405 | • | | | • | | |
| 409 | • | | | • | | |
| 410 | • | | | • | | |
| 414 | • | | | • | | |
| 416 | • | | | | | |
| 416 Se | • | | | | | |
| 420 | • | | | | | |
| 420F | • | | | | | |
| 422 | • | | | | | |
| 429 | • | • | | • | • | |
| 430 | • | • | | • | • | |
| 430F | • | • | | • | | |
| 430F Se | • | • | | • | | |
| 431 | • | • | • | • | | |
| 434 | • | • | • | • | • | |
| 436 | • | • | • | • | • | |
| 440A | • | | | • | | |
| 440B | • | | | | | |
| 440C | • | | | | | |
| 442 | • | • | | • | • | |
| 446 | • | • | • | • | • | |
• considered as resistant to the environment category
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6643 | https://www.linguno.com/wordComparison/esp/gratuito-gratis/ | Sign up Log in
Gratuito vs. gratis
Word practice demo
The Spanish words gratuito and gratis share a similar meaning referring to that which costs nothing or is free. However, their usage differs slightly depending on the context in which they are used.
Gratuito
B1
Gratuito is often used to describe services, things, or actions that are provided without any charge or payment. It is typically used in formal contexts or when referring to more significant items or services that are being offered at no cost.
El servicio de salud es gratuito en este país.
(The health service is free in this country.)
Los museos ofrecen entrada gratuita los domingos.
(Museums offer free admission on Sundays.)
Proporcionamos capacitación gratuita a nuestros empleados.
(We provide free training to our employees.)
Gratis
A2
Gratis is a bit more informal and it's often used to denote minor items or additional bonuses that come with no cost. The term is commonly seen in advertising for promotional offers.
Compra uno y llévate otro gratis.
(Buy one and get one free.)
Descarga esta aplicación, es gratis.
(Download this app, it's free.)
Te regalan una copa gratis con tu entrada.
(You get a free drink with your ticket.)
Gratuito and gratis both refer to something being free or not costing anything in Spanish. However, they are used differently depending on the formality of the situation and whats being described as free. Gratuito usually refers to larger, more substantial things or services that are provided free of charge and is used in a more formal context. Meanwhile, gratis' often refers to smaller items or additional benefits and is usually seen in advertising. |
6644 | https://www.machinesl.com/trowel-or-float/ | Trowel or Float Master Concrete [Difference & Overlapping]
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Trowel or Float Master Concrete [Difference & Overlapping]
When it comes to working with concrete, two essential tools that often come into play are the float and the trowel. While they may seem similar at first glance, understanding their differences and unique characteristics is crucial for achieving the desired results in concrete finishing.
A float is a versatile tool used for initial surface leveling and creating a textured finish, while a trowel is designed for achieving a smooth and polished surface. In this post, we will explore the distinctions between a float and a trowel, their overlapping use cases, and the importance of selecting the right tool for specific concrete projects.
By gaining a comprehensive understanding of these tools, you can enhance your craftsmanship and achieve professional-quality concrete finishes.
Definition of a float:
A float is a concrete finishing tool with a flat, rectangular, or round blade.
It is typically made of wood, magnesium, or aluminum.
Floats are used for leveling and smoothing the surface of freshly poured concrete.
Definition of a trowel:
A trowel is a handheld tool with a flat, rectangular blade made of steel or other durable materials.
It is used for finishing and polishing concrete surfaces.
Trowels come in various sizes and are commonly used in small to medium-sized concrete projects.
Understanding the difference:
Understanding the difference between a float and a trowel is crucial for proper concrete finishing.
Each tool has its own specific purpose and application in the concrete construction process.
Knowing when to use a float and when to switch to a trowel ensures the desired result of a smooth and level concrete surface.
edging trowel machine, start from $150
Contents
Float
Description of a float
Purpose of a float
Common materials used for floats
Applications of a float
Examples of float variations
Trowel
Description of a trowel
Purpose of a trowel
Common materials used for trowels
Applications of a trowel
Examples of trowel variations
Key Differences between a Float and a Trowel
Design and shape differences
Functionality and purpose distinctions
Typical applications and use cases
Materials and construction variances
Overlapping Use Cases
Situations where both a float and a trowel be used
Complementary roles of a float and a trowel in certain tasks
Conclusions
Float
Description of a float
A float is a concrete finishing tool with a flat, rectangular, or round-shaped blade.
It typically features a handle attached to the top for easy maneuverability.
The blade of a float is usually made of wood, magnesium, aluminum, or other materials.
Purpose of a float
The primary purpose of a float is to level and smooth freshly poured concrete.
It helps to remove imperfections, fill in voids, and create a uniform surface.
Floats are particularly useful in the early stages of concrete finishing.
Common materials used for floats
Wood floats: These are often made of hardwood, such as maple or mahogany.
Magnesium floats: Lightweight and durable, these floats are commonly used for larger concrete areas.
Aluminum floats: These floats offer similar properties to magnesium floats but may be more affordable.
Applications of a float
Floats are used for leveling and smoothing concrete slabs, sidewalks, driveways, and other flat surfaces.
They are effective in removing excess water from the surface and consolidating the concrete.
Examples of float variations
Bull float: A large float, often with an extension handle, used for smoothing larger areas of concrete.
Magnesium float: A lightweight float made of magnesium, known for its strength and smooth finish.
Trowel
Description of a trowel
Atrowel is a handheld tool with a flat, pointed, or curved blade.
It is typically made of steel or stainless steel and has a handle for gripping and control.
Purpose of a trowel
The primary purpose of a trowel is to apply and finish mortar or concrete.
It is used for spreading, leveling, and shaping the material in various construction and masonry tasks.
Common materials used for trowels
Steel: Trowels made of steel are durable and suitable for heavy-duty applications.
Stainless steel: These trowels resist rust and corrosion, making them ideal for working with wet materials.
Applications of a trowel
Trowels are commonly used in masonry work, such as bricklaying, plastering, and tile installation.
They are essential for smoothing and shaping surfaces, filling joints, and achieving precise finishes.
Examples of trowel variations
Finishing trowel: A flat-edged trowel used for creating a smooth, polished surface on concrete or plaster.
Pointing trowel: A small, triangular-shaped trowel with a pointed tip, used for applying and shaping mortar joints.
Key Differences between a Float and a Trowel
Design and shape differences
Float: Typically, larger in size with a rectangular or square-shaped flat surface.
Trowel: Generally smaller in size with a flat, pointed, or curved blade.
Functionality and purpose distinctions
Float: Primarily used for initial concrete surface leveling, smoothing, and removing imperfections.
Trowel: Primarily used for applying, spreading, and finishing mortar or concrete.
Typical applications and use cases
Float: Used in early stages of concrete work, such as floating the surface after pouring and screeding.
Trowel: Used in later stages of concrete work, after floating, to further refine and finish the surface.
Materials and construction variances
Float: Often made of lightweight materials like wood, aluminum, or magnesium.
Trowel: Typically made of heavier and more durable materials like steel or stainless steel.
Overlapping Use Cases
Situations where both a float and a trowel be used
Finishing concrete surfaces: Both a float and a trowel may be employed in the final stages of concrete finishing to achieve the desired smoothness and texture.
Repair and patching work: When repairing or patching concrete surfaces, using a float to level and smooth the repaired area and a trowel to blend it with the surrounding surface can result in a seamless finish.
Complementary roles of a float and a trowel in certain tasks
Texture creation: While a float is typically used to achieve a consistent and level surface, a trowel can be employed to create unique textures and patterns on the concrete, adding decorative elements to the finished product.
Edge and corner work: Floats are commonly used for larger areas, but when it comes to working on edges and corners, a trowel’s smaller size and shape make it more suitable for precise control and maneuverability.
Conclusions
In conclusion, understanding the differences between a float and a trowel is essential for anyone involved in concrete work. While floats are primarily used for initial surface leveling and finishing, trowels excel in achieving a smooth and polished concrete surface.
By selecting the right tool for the job, whether it’s a float or a trowel, individuals can ensure optimal results and enhance the quality of their concrete work. Additionally, recognizing the overlapping use cases of floats and trowels allows for their complementary roles in certain tasks, providing versatility and efficiency.
Ultimately, being knowledgeable about these tools empowers individuals to make informed decisions, leading to successful and professional outcomes in their concrete projects.
Sources:
Trowel – Wiki
Float – Wiki
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This is a challenging problem-solving book in Euclidean geometry, assuming nothing of the reader other than a good deal of courage. Topics covered included cyclic quadrilaterals, power of a point, homothety, triangle centers; along the way the reader will meet such classical gems as the nine-point circle, the Simson line, the symmedian and the mixtilinear incircle, as well as the theorems of Euler, Ceva, Menelaus, and Pascal. Another part is dedicated to the use of complex numbers and barycentric coordinates, granting the reader both a traditional and computational viewpoint of the material. The final part consists of some more advanced topics, such as inversion in the plane, the cross ratio and projective transformations, and the theory of the complete quadrilateral.
The exposition is friendly and relaxed, and accompanied by over 300 beautifully drawn figures. The emphasis of this book is placed squarely on the problems. Each chapter contains carefully chosen worked examples, which explain not only the solutions to the problems but also describe in close detail how one would invent the solution to begin with. The text contains a selection of 300 practice problems of varying difficulty from contests around the world, with extensive hints and selected solutions.
This book is especially suitable for students preparing for national or international mathematical olympiads or for teachers looking for a text for an honor class.
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6646 | https://behcetsuk.org/how-is-behcets-diagnosed/ | How is Behçet's diagnosed?
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How is Behcets diagnosed
How is Behçet’s diagnosed?
There is no test for Behçet’s at the moment. It is diagnosed by specific patterns of symptoms and repeated outbreaks of them. Any other causes for these symptoms have to be ruled out first. The symptoms do not have to occur together but can have happened at any time.
Diagnostic Guidelines
There are levels of certainty for diagnosis (and additional knowledge since the original criteria):
NHS England Diagnosis – Diagnosing Behçet’s Disease (as of 20 November 2019)
The International Criteria for Behçet’s Disease (ICBD) (2013)
Suspected or possible diagnosis (incomplete pattern of symptoms)
Practical clinical diagnosis (generally agreed pattern but not so strict)
International Study Group diagnostic guidelines (very strict for research purposes) (1990)
1. NHS England Diagnosis – Diagnosing Behçet’s disease (as of November 2019)
The main symptoms of Behçet’s disease include:
genital and mouth ulcers
red, painful eyes and blurred vision
acne-like spots
headaches
painful, stiff and swollen joints
In severe cases, there’s also a risk of serious and potentially life-threatening problems, such as permanent vision lossandstrokes. Most people with the condition experience episodes where their symptoms are severe (flare-ups or relapses), followed by periods where the symptoms disappear (remission).
There’s no definitive test that can be used to diagnose Behçet’s disease. Several tests may be necessary to check for signs of the condition, or to help rule out other causes, including:
blood tests
urine tests
scans, such as X-rays, a computerised tomography (CT) scan or a magnetic resonance imaging (MRI) scan
a skin biopsy
a pathergy test – which involves pricking your skin with a needle to see if a particular red spot appears within the next day or two; people with Behçet’s disease often have particularly sensitive skin
Current guidelines state a diagnosis of Behçet’s disease can usually be confidently made if you’ve experienced at least three episodes of mouth ulcers over the past 12 months and you have at least two of the following symptoms:
genital ulcers
eye inflammation
skin lesions (any unusual growths or abnormalities that develop on the skin)
pathergy (hypersensitive skin)
Other potential causes also need to be ruled out before the diagnosis is made.
2. The International Criteria for Behçet’s Disease (ICBD) (2013)
An International Team for the Revision of the International Criteria for BD (from 27 countries (though not the UK)) submitted data from 2556 clinically diagnosed BD patients and 1163 controls with BD-mimicking diseases or presenting at least one major BD sign. The new proposed criteria derived from multinational data exhibits much improved sensitivity over the [original] ISG criteria while maintaining reasonable specificity. It is proposed that the ICBD criteria to be adopted both as a guide for diagnosis and classification of BD.
The final proposed criteria included oral aphthosis, genital aphthosis, ocular lesions (anterior uveitis, posterior uveitis, or retinal vasculitis), neurological manifestations, skin lesions (pseudofolliculitis, skin aphthosis, erythema nodosum) and vascular manifestations (arterial thrombosis, large vein thrombosis, phlebitis or superficial phlebitis). Oral aphthosis, genital aphthosis and ocular lesions were each given 2 points, whereas 1 point was assigned to each of skin lesions, vascular manifestations and neurological manifestations. A patient scoring 4 points or above was classified as having BD (Table 5).
Table 5 – International Criteria for Behçet’s Disease – point score system: scoring ≥4 indicates Behçet’s diagnosis.
Society note: While the criteria are useful in allowing additional symptoms to be considered – as they note, “…exhibits much-improved sensitivity over the [original]…” merely relying on Oral and Genital ulcers, thus scoring≥4, is too simplistic and can lead to over-diagnosis. In the UK population where Behçets Disease is rare, the use of the 2014 criteria results in a loss of specificity compared to the ISG (1990) criteria. The (multi-disciplinary) National Centre of Excellence supports this view.
Pathergy test is optional and the primary scoring system does not include pathergy testing. However, where pathergy testing is conducted one extra point may be assigned for a positive result.
The distribution of scores in cases and controls in both validation and training data sets along with a proposed plausibility scale of BD diagnosis is presented in Table 6.
Table 6
The PDF of the ICBD document can be viewed here
3. ‘Suspected’ or ‘possible’ diagnosis
This is usually given when someone does not have mouth ulcers or has mouth ulcers but does not have 1 of the 4 ‘hallmark’ symptoms but has other symptoms and signs of inflammation and other causes for these have been ruled out.
Symptom proportions for members of the Behçet’s UK (then Behçet’s Syndrome Society) in the UK (survey dated 1994)
Symptom% of people with symptom
Mouth ulcers 100
Arthritis/arthralgia 93
Genital ulcers 89
Skin lesions 86
Eye inflammation 68
Tissue reactivity (pathergy)32
Thrombophlebitis 32
Neurological problems 31
Inflammatory bowel disease 28
Deep vein thrombosis 22
(Other symptoms were not measured)
4. Practical clinical guidelines for patients not included in research cohorts
Must have:
mouth ulcers
Along with 1 out of the 4 ‘hallmark’ symptoms above
Along with 2 of the following symptoms:
arthritis/arthralgia
nervous system symptoms
stomach and/or bowel inflammation
deep vein thrombosis
superficial thrombophlebitis
cardiovascular problems
inflammatory problems in chest and lungs
problems with hearing and/or balance
extreme exhaustion
changes of personality, psychoses
any other member of the family with a diagnosis of Behçet’s
5. International Study Group strict research level guidelines for diagnosis (1990)
Must have:
mouth ulcers (any shape, size or number at least 3 times in any 12 months)
Along with 2 out of the next 4 ‘hallmark’ symptoms:
genital ulcers (including anal ulcers and spots in the genital region and swollen testicles or epididymitis in men)
skin lesions (papulo-pustules, folliculitis, erythema nodosum, acne in post-adolescents not on corticosteroids)
eye inflammation (iritis, uveitis, retinal vasculitis, cells in the vitreous)
pathergy reaction (papule >2 mm diameter, 24-48 hrs or more after needle-prick)
Criteria for diagnosis of Behcet’s disease, ISG (1990) The Lancet, 05 May 1990
What is Behçet’s?
What is a rare disease?
Similar Disorders
Behçet’s Medical Factsheets
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Calcium-storing organelle in muscle cells
The sarcoplasmic reticulum (SR) is a membrane-bound structure found within muscle cells that is similar to the smooth endoplasmic reticulum in other cells. The main function of the SR is to store calcium ions (Ca2+). Calcium ion levels are kept relatively constant, with the concentration of calcium ions within a cell being 10,000 times smaller than the concentration of calcium ions outside the cell. This means that small increases in calcium ions within the cell are easily detected and can bring about important cellular changes (the calcium is said to be a second messenger). Calcium is used to make calcium carbonate (found in chalk) and calcium phosphate, two compounds that the body uses to make teeth and bones. This means that too much calcium within the cells can lead to hardening (calcification) of certain intracellular structures, including the mitochondria, leading to cell death. Therefore, it is vital that calcium ion levels are tightly controlled and can be released into the cell when necessary, and then removed from the cell.
Structure
[edit]
The sarcoplasmic reticulum is a network of tubules that extend throughout muscle cells, wrapping around (but not in direct contact with) the myofibrils (contractile units of the cell). Cardiac and skeletal muscle cells contain structures called transverse tubules (T-tubules), which are extensions of the cell membrane that travel into the centre of the cell. T-tubules are closely associated with a specific region of the SR, known as the terminal cisternae in skeletal muscle, with a distance of roughly 12 nanometers, separating them. This is the primary site of calcium release. The longitudinal SR are thinner projections that run between the terminal cisternae/junctional SR, and are the location where ion channels necessary for calcium ion absorption are most abundant. These processes are explained in more detail below and are fundamental for the process of excitation-contraction coupling in skeletal, cardiac and smooth muscle.
Calcium absorption
[edit]
The SR contains ion channel pumps, within its membrane, that are responsible for pumping Ca2+ into the SR. As the calcium ion concentration within the SR is higher than in the rest of the cell, the calcium ions will not freely flow into the SR, and therefore pumps are required, which use energy, which they gain from a molecule called adenosine triphosphate (ATP). These calcium pumps are called Sarco(endo)plasmic reticulum Ca2+ ATPases (SERCA). There are various forms of SERCA, with SERCA 2a being predominantly found in cardiac and skeletal muscle.
SERCA consists of 13 structural elements (labelled M1-M10 α-helices in the transmembrane domain, and N, P, and A citosolic domains). Calcium ions bind to the M1-M10 transmembrane region, whereas ATP binds to the N domain. When 2 calcium ions, along with a molecule of ATP, bind to the cytosolic side of the pump (i.e., the region of the pump outside the SR), the pump opens. This occurs because ATP (which contains three phosphate groups) releases a single phosphate group (becoming adenosine diphosphate). The released phosphate group then binds to the pump (in the P domain), causing the pump to change shape. This shape change causes the cytosolic side of the pump to open, allowing the two Ca2+ to enter. The cytosolic side of the pump then closes and the sarcoplasmic reticulum side opens, releasing the Ca2+ into the SR.
A protein found in cardiac muscle, called phospholamban (PLB), has been shown to prevent SERCA from working. It does this by binding to the SERCA and decreasing its attraction (affinity) to calcium, therefore preventing calcium uptake into the SR. Failure to remove Ca2+ from the cytosol prevents muscle relaxation and therefore means that there is a decrease in muscle contraction too. However, molecules such as adrenaline and noradrenaline, can prevent PLB from inhibiting SERCA. When these hormones bind to a receptor, called a beta 1 adrenoceptor, located on the cell membrane, they produce a series of reactions (known as a cyclic AMP pathway) that produces an enzyme called protein kinase A (PKA). PKA can add a phosphate to PLB (this is known as phosphorylation), preventing it from inhibiting SERCA and allowing for muscle relaxation.
Calcium storage
[edit]
Located within the SR is a protein called calsequestrin. This protein can bind to around 50 Ca2+, which decreases the amount of free Ca2+ within the SR (as more is bound to calsequestrin). Therefore, more calcium can be stored (the calsequestrin is said to be a buffer). It is primarily located within the junctional SR/luminal space, in close association with the calcium release channel (described below).
Calcium release
[edit]
Calcium ion release from the SR occurs in the junctional SR/terminal cisternae through a ryanodine receptor (RyR) and is known as a calcium spark. There are three types of ryanodine receptor, RyR1 (in skeletal muscle), RyR2 (in cardiac muscle) and RyR3 (in the brain). Calcium release through ryanodine receptors in the SR is triggered differently in different muscles. In cardiac and smooth muscle, an electrical impulse (action potential) triggers calcium ions to enter the cell through an L-type calcium channel located in the cell membrane (smooth muscle) or T-tubule membrane (cardiac muscle). These calcium ions bind to and activate the RyR, producing a larger increase in intracellular calcium. In skeletal muscle, however, the L-type calcium channel is bound to the RyR. Therefore, activation of the L-type calcium channel, via an action potential, activates the RyR directly, causing calcium release (see calcium sparks for more details). Also, caffeine (found in coffee) can bind to and stimulate RyR. Caffeine makes the RyR more sensitive to either the action potential (skeletal muscle) or calcium (cardiac or smooth muscle), thereby producing calcium sparks more often (this is partially responsible for caffeine's effect on heart rate).
Triadin and Junctin are proteins found within the SR membrane that are bound to the RyR. The main role of these proteins is to anchor calsequestrin (see above) to the ryanodine receptor. At ‘normal’ (physiological) SR calcium levels, calsequestrin binds to the RyR, Triadin, and Junctin, which prevents the RyR from opening. If calcium concentration within the SR falls too low, there will be less calcium bound to the calsequestrin. This means that there is more room on the calsequestrin to bind to the junctin, triadin, and ryanodine receptor; therefore, it binds tighter. However, if calcium within the SR rises too high, more calcium binds to the calsequestrin, and therefore it binds to the junctin-triadin-RyR complex less tightly. The RyR can therefore open and release calcium into the cell.
In addition to the effects that PKA had on phospholamban (see above) that resulted in increased relaxation of the cardiac muscle, PKA (as well as another enzyme called calmodulin kinase II) can also phosphorylate ryanodine receptors. When phosphorylated, RyRs are more sensitive to calcium; therefore, they open more often and for longer periods. This increases calcium release from the SR, increasing the rate of contraction. Therefore, in cardiac muscle, activation of PKA, through the cyclic AMP pathway, results in increased muscle contraction (via RyR2 phosphorylation) and increased relaxation (via phospholamban phosphorylation), which increases heart rate.
The mechanism behind the termination of calcium release through the RyR is still not fully understood. Some researchers believe it is due to the random closing of ryanodine receptors (known as stochastic attrition), or the ryanodine receptors becoming inactive after a calcium spark, while others believe that a decrease in SR calcium, triggers the receptors to close (see calcium sparks for more details).
Role in rigor mortis
[edit]
The breakdown of the sarcoplasmic reticulum, along with the resultant release of calcium, is an important contributor to rigor mortis, the stiffening of muscles after death.
An increase in calcium concentration in the sarcoplasm can also cause muscle stiffness.
References
[edit]
^ Bronner, F. (2003) ‘Extracellular and intracellular regulation of calcium homeostasis’, TheScientificWorldJournal., 1, pp. 919–25.
^ Trump, B., Berezesky, I., Laiho, K., Osornio, A., Mergner, W. and Smith, M. (1980) ‘The role of calcium in cell injury. A review’, Scanning electron microscopy., pp. 437–62.
^ The anatomy of the sarcoplasmic reticulum in vertebrate skeletal muscle: Its implications for excitation contraction coupling, Zeitschrift für Naturforschung. Section C, Biosciences., 37, pp. 665–78.
^ Arai, M.; Matsui, H.; Periasamy, M. (1994-04-01). "Sarcoplasmic reticulum gene expression in cardiac hypertrophy and heart failure". Circulation Research. 74 (4): 555–564. doi:10.1161/01.RES.74.4.555. ISSN 0009-7330. PMID 8137493.
^ Periasamy, M. and Kalyanasundaram, A. (2007) ‘SERCA pump isoforms: Their role in calcium ion transport and disease’, Muscle & Nerve, 35(4), pp. 430–42.
^ Kekenes-Huskey, P.M., Metzger, V.T., Grant, B.J. and McCammon, A.J. (2012b) ‘Calcium binding and allosteric signaling mechanisms for the sarcoplasmic reticulum Ca2+ ATPase’, 21(10).
^ Akin, B., Hurley, T., Chen, Z. and Jones, L. (2013) ‘The structural basis for phospholamban inhibition of the calcium pump in sarcoplasmic reticulum’, The Journal of Biological Chemistry., 288(42), pp. 30181–91.
^ Beard, N. A.; Laver, D. R.; Dulhunty, A. F. (2004-05-01). "Calsequestrin and the calcium release channel of skeletal and cardiac muscle". Progress in Biophysics and Molecular Biology. 85 (1): 33–69. doi:10.1016/j.pbiomolbio.2003.07.001. ISSN 0079-6107. PMID 15050380.
^ Kobayashi, Y. M.; Alseikhan, B. A.; Jones, L. R. (2000): Localization and characterization of the calsequestrin-binding domain of triadin 1. Evidence for a charged beta-strand in mediating the protein-protein interaction. In The Journal of biological chemistry 275 (23), pp. 17639–17646. DOI: 10.1074/jbc.M002091200.
^ Cheng, H.; Lederer, W. J.; Cannell, M. B. (1993-10-29). "Calcium sparks: elementary events underlying excitation-contraction coupling in heart muscle". Science. 262 (5134): 740–744. Bibcode:1993Sci...262..740C. doi:10.1126/science.8235594. ISSN 0036-8075. PMID 8235594.
^ Lanner, J.T., Georgiou, D.K., Joshi, A.D. and Hamilton, S.L. (2010b) ‘Ryanodine receptors: Structure, expression, molecular details, and function in calcium release’, 2(11).
^ Cheng, H. and Lederer, W. (2008) ‘Calcium sparks’, Physiological Reviews., 88(4), pp. 1491–545.
^ Sitsapesan R, Williams AJ. Mechanisms of caffeine activation of single calcium-release channels of sheep cardiac sarcoplasmic reticulum. J Physiol (Lond) 1990;423:425– 439]
^ Zhang, L; Kelley, J; Schmeisser, G; Kobayashi, YM; Jones, LR (1997). "Complex formation between junctin, triadin, calsequestrin, and the ryanodine receptor: proteins of the cardiac junctional sarcoplasmic reticulum membrane". J Biol Chem. 272 (37): 23389–23397. doi:10.1074/jbc.272.37.23389. PMID 9287354.
^ Györke, I., Hester, N., Jones, L.R. and Györke, S. (2004) ‘The role of Calsequestrin, Triadin, and Junctin in conferring cardiac Ryanodine receptor responsiveness to Luminal calcium’, 86(4).
^ Bers, D.M. (2006) ‘Cardiac ryanodine receptor phosphorylation: Target sites and functional consequences’, 396(1).
^ Sham, J. S. K.; et al. (1998). "Termination of Ca2+ release by a local inactivation of ryanodine receptors in cardiac myocytes". Proc. Natl. Acad. Sci. USA. 95 (25): 15096–15101. Bibcode:1998PNAS...9515096S. doi:10.1073/pnas.95.25.15096. PMC 24581. PMID 9844021.
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6648 | https://proofwiki.org/wiki/Binomial_Theorem/General_Binomial_Theorem | Binomial Theorem/General Binomial Theorem
From ProofWiki
< Binomial Theorem
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Contents
1 Theorem
1.1 Convergence
2 Proof 1
3 Proof 2
4 Proof 3
5 Examples
5.1 Example: $\paren {1 - x}^{-3}$
5.2 Example: $\paren {1 + 2 x}^{-\frac 3 2}$
5.3 Example: $\paren {1 + 5 x}^{\frac 1 5}$
5.4 Example: $\paren {1 - 4 x}^{\frac 1 2}$
6 Historical Note
7 Sources
Theorem
Let $\alpha \in \R$ be a real number.
Let $x \in \R$ be a real number such that $\size x < 1$.
Then:
| | | | | | | | | | | | | | | |
--- --- --- --- --- --- ---
| | | | | | (\ds \paren {1 + x}^\alpha) | (=) | | | | (\ds \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n) | | | | |
| | | | | | (\ds ) | (=) | | | | (\ds \sum_{n \mathop = 0}^\infty \dbinom \alpha n x^n) | | | | |
| | | | | | (\ds ) | (=) | | | | (\ds \sum_{n \mathop = 0}^\infty \frac 1 {n!} \paren {\prod_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } x^n) | | | | |
| | | | | | (\ds ) | (=) | | | | (\ds 1 + \alpha x + \dfrac {\alpha \paren {\alpha - 1} } {2!} x^2 + \dfrac {\alpha \paren {\alpha - 1} \paren {\alpha - 2} } {3!} x^3 + \cdots) | | | | |
where:
: $\alpha^{\underline n}$ denotes the falling factorial
: $\dbinom \alpha n$ denotes a binomial coefficient.
Convergence
The above binomial series:
: converges when $\size x < 1$
: diverges when $\size x > 1$
For the special case where $x = 1$, the binomial series converges if $n > -1$.
For the special case where $x = -1$, the binomial series converges if $n > 0$.
Proof 1
Let $R$ be the radius of convergence of the power series:
: $\ds \map f x = \sum_{n \mathop = 0}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {n!} x^n$
Then:
| | | | | | | | | | | | | | | |
--- --- --- --- --- --- ---
| | | | | | (\ds \frac 1 R) | (=) | | | | (\ds \lim_{n \mathop \to \infty} \frac {\size {\alpha \paren {\alpha - 1} \dotsm \paren {\alpha - n} } } {\paren {n + 1}!} \frac {n!} {\size {\alpha \paren {\alpha - 1} \dotsm \paren {\alpha - n + 1} } }) | | | Radius of Convergence from Limit of Sequence | |
| | | | | | (\ds ) | (=) | | | | (\ds \lim_{n \mathop \to \infty} \frac {\size {\alpha - n} } {n + 1}) | | | | |
| | | | | | (\ds ) | (=) | | | | (\ds 1) | | | | |
Thus for $\size x < 1$, Power Series is Differentiable on Interval of Convergence applies:
: $\ds D_x \map f x = \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {n!} n x^{n - 1}$
This leads to:
| | | | | | | | | | | | | | | |
--- --- --- --- --- --- ---
| | | | | | (\ds \paren {1 + x} D_x \map f x) | (=) | | | | (\ds \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {\paren {n - 1}!} x^{n - 1} + \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {\paren {n - 1}!} x^n) | | | | |
| | | | | | (\ds ) | (=) | | | | (\ds \alpha + \sum_{n \mathop = 1}^\infty \paren {\frac {\prod \limits_{k \mathop = 0}^n \paren {\alpha - k} } {n!} + \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {\paren {n - 1}!} } x^n) | | | | |
| | | | | | (\ds ) | (=) | | | | (\ds \alpha + \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^n \paren {\alpha - k} } {\paren {n - 1}!} \paren {\frac 1 n + \frac 1 {\alpha - n} } x^n) | | | | |
| | | | | | (\ds ) | (=) | | | | (\ds \alpha + \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^n \paren {\alpha - k} } {\paren {n - 1}!} \frac \alpha {n \paren {\alpha - n} } x^n) | | | | |
| | | | | | (\ds ) | (=) | | | | (\ds \alpha \paren {1 + \sum_{n \mathop = 1}^\infty \frac {\prod \limits_{k \mathop = 0}^{n - 1} \paren {\alpha - k} } {n!} x^n}) | | | | |
| | | | | | (\ds ) | (=) | | | | (\ds \alpha \map f x) | | | | |
Gathering up:
: $\paren {1 + x} D_x \map f x = \alpha \map f x$
Thus:
: $\map {D_x} {\dfrac {\map f x} {\paren {1 + x}^\alpha} } = -\alpha \paren {1 + x}^{-\alpha - 1} \map f x + \paren {1 + x}^{-\alpha} D_x \map f x = 0$
So $\map f x = c \paren {1 + x}^\alpha$ when $\size x < 1$ for some constant $c$.
But $\map f 0 = 1$ and hence $c = 1$.
$\blacksquare$
Proof 2
From Sum over k of r-kt choose k by r over r-kt by z^k:
: $\ds \sum_n \dbinom {\alpha - n t} k \dfrac \alpha {\alpha - n t} z^n = x^\alpha$
where:
: $z = x^{t + 1} - x^t$
: $x = 1$ for $z = 0$.
Setting $t = 0$:
| | | | | | | | | | | | | | | |
--- --- --- --- --- --- ---
| | | | | | (\ds \sum_k \dbinom {\alpha - n \times 0} n \dfrac \alpha {\alpha - n \times 0} z^n) | (=) | | | | (\ds x^\alpha) | | | | |
| | | (\ds \leadsto \ \ ) | | | (\ds \sum_n \dbinom \alpha n \dfrac \alpha \alpha z^n) | (=) | | | | (\ds \paren {1 + z}^\alpha) | | | | |
| | | (\ds \leadsto \ \ ) | | | (\ds \sum_n \dbinom \alpha n z^n) | (=) | | | | (\ds \paren {1 + z}^\alpha) | | | | |
$\blacksquare$
Proof 3
The series is the Maclaurin series expansion of the function $\map f x = \paren {1 + x}^\alpha$.
The derivatives of $f$ are:
| | | | | | | | | | | | | | | |
--- --- --- --- --- --- ---
| | | | | | (\ds \map {f^{\paren 0} } x) | (=) | | | | (\ds \paren {1 + x}^\alpha) | | | | |
| | | | | | (\ds \map {f^{\paren 1} } x) | (=) | | | | (\ds \alpha \paren {1 + x}^{\alpha - 1}) | | | | |
| | | | | | (\ds \map {f^{\paren 2} } x) | (=) | | | | (\ds \alpha \paren {\alpha - 1} \paren {1 + x}^{\alpha - 2}) | | | | |
| | | | | | (\ds \map {f^{\paren n} } x) | (=) | | | | (\ds \alpha \paren {\alpha - 1} \cdots \paren {\alpha - n + 1} \paren {1 + x}^{\alpha - n}) | | | | |
| | | | | | (\ds ) | (=) | | | | (\ds \alpha^{\underline n} \paren {1 + x}^{\alpha - n}) | | | | |
Evaluated at $x = 0$, we have:
| | | | | | | | | | | | | | | |
--- --- --- --- --- --- ---
| | | | | | (\ds \map {f^{\paren 0} } x) | (=) | | | | (\ds \alpha^{\underline n} \paren {1 + 0}^{\alpha - n}) | | | | |
| | | | | | (\ds ) | (=) | | | | (\ds \alpha^{\underline n}) | | | | |
The Maclaurin series of $f$ is:
| | | | | | | | | | | | | | | |
--- --- --- --- --- --- ---
| | | | | | (\ds \map f x)) | (=) | | | | (\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0) | | | | |
| | | | | | (\ds ) | (=) | | | | (\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \alpha^{\underline n}) | | | substituting derivative at $0$ | |
| | | | | | (\ds ) | (=) | | | | (\ds \sum_{n \mathop = 0}^\infty \frac {\alpha^{\underline n} } {n!} x^n) | | | rearranging | |
$\blacksquare$
Examples
Example: $\paren {1 - x}^{-3}$
: $\paren {1 - x}^{-3} = 1 + 3 x + 6 x^2 + 10 x^3 + \cdots$
Example: $\paren {1 + 2 x}^{-\frac 3 2}$
: $\paren {1 + 2 x}^{-\frac 3 2} = 1 - 3 x + \dfrac {15} 2 x^2 - \dfrac {35} 2 x^3 + \cdots$
Example: $\paren {1 + 5 x}^{\frac 1 5}$
: $\paren {1 + 5 x}^{\frac 1 5} = 1 + x - 2 x^2 + 6 x^3 + \cdots$
Example: $\paren {1 - 4 x}^{\frac 1 2}$
: $\paren {1 - 4 x}^{\frac 1 2} = 1 - 2 x - 2 x^2 + 4 x^3 + \cdots$
Historical Note
The General Binomial Theorem was first conceived by Isaac Newton during the years $1665$ to $1667$ when he was living in his home in Woolsthorpe.
He announced the result formally, in letters to Henry Oldenburg on $13$th June $1676$ and $24$th October $1676$ but did not provide a proper proof (at that time the need for the appropriate level of rigor had not been recognised).
Leonhard Paul Euler made an incomplete attempt in $1774$, but the full proof had to wait for Carl Friedrich Gauss to provide it in $1812$.
This was, in fact, the first time anything about infinite summations was proved adequately.
Sources
1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{VI}$: On the Seashore
1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: The Binomial Theorem: Negative and fractional indices
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Binomial Series: $20.4$
1969: J.C. Anderson, D.M. Hum, B.G. Neal and J.H. Whitelaw: Data and Formulae for Engineering Students (2nd ed.) ... (previous) ... (next): $4.$ Mathematics: $4.2$ Series
1976: K. Weltner and W.J. Weber: Mathematics for Engineers and Scientists ... (previous) ... (next): $8$. Taylor Series and Power Series: Appendix: Table $8.2$: Power Series of Important Functions
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.6$: Binomial Coefficients: $\text{F} \ (15)$
1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.9$: Generating Functions: $(19)$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): binomial coefficients
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): binomial theorem
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): binomial coefficients
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): binomial theorem
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 22$: Taylor Series: Binomial Series: $22.4.$
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): binomial series (expansion)
2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): binomial series (expansion)
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1986 AHSME Problems/Problem 28
Contents
[hide]
1 Problem
2 Solution 1
3 Solution 2
4 See also
Problem
is a regular pentagon. and are the perpendiculars dropped from onto extended and extended, respectively. Let be the center of the pentagon. If , then equals
Solution 1
To solve the problem, we compute the area of regular pentagon in two different ways. First, we can divide regular pentagon into five congruent triangles.
If is the side length of the regular pentagon, then each of the triangles , , , , and has base and height 1, so the area of regular pentagon is .
Next, we divide regular pentagon into triangles , , and .
Triangle has base and height . Triangle has base and height . Triangle has base and height . Therefore, the area of regular pentagon is also Hence, which means , or . The answer is .
Solution 2
Now, we know that angle has measure . Since Therefore, . Therefore, . Recalling that gives a final answer of .
See also
1986 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 27Followed by
Problem 29
1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25•26•27•28•29•30
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6650 | https://pubchem.ncbi.nlm.nih.gov/compound/Sulfur-Hexafluoride | Sulfur Hexafluoride | F6S | CID 17358 - PubChem
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compound Summary
Sulfur Hexafluoride
PubChem CID
17358
Structure
Primary Hazards
Laboratory Chemical Safety Summary (LCSS) Datasheet
Molecular Formula
F 6 S
Synonyms
SULFUR HEXAFLUORIDE
Sulphur hexafluoride
2551-62-4
Elegas
Lumason
View More...
Molecular Weight
146.06 g/mol
Computed by PubChem 2.2 (PubChem release 2025.04.14)
Dates
Create: 2004-09-16
Modify: 2025-09-20
Description
Sulfur hexafluoride appears as a colorless odorless gas. Noncombustible. Shipped as a liquefied gas under own vapor pressure. Contact may cause frostbite. Under prolonged exposure to fire or heat the containers may rupture violently and rocket.
CAMEO Chemicals
Sulfur hexafluoride is a sulfur coordination entity consisting of six fluorine atoms attached to a central sulfur atom. It is the most potent greenhouse gas currently known, with a global warming potential of 23,900 times that of CO2 over a 100 year period (SF6 has an estimated lifetime in the atmosphere of between 800 and 3,000 years). It has a role as an ultrasound contrast agent and a member of greenhouse gas.
ChEBI
Sulfur hexafluoride is an ultrasound contrast agent indicated for use •in echocardiography to opacify the left ventricular chamber and to improve the delineation of the left ventricular endocardial border in adult patients with suboptimal echocardiograms •in ultrasonography of the liver for characterization of focal liver lesions in adult and pediatric patients
DrugBank
View More...
1 Structures
1.1 2D Structure
Structure Search
Get Image
Download Coordinates
Chemical Structure Depiction
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PubChem
1.2 3D Status
Conformer generation failed at generation step
PubChem
1.3 Crystal Structures
COD records with this CID as component
1526434
2106938
4101489
Crystallography Open Database (COD)
2 Names and Identifiers
2.1 Computed Descriptors
2.1.1 IUPAC Name
hexafluoro-λ 6-sulfane
Computed by Lexichem TK 2.7.0 (PubChem release 2025.04.14)
PubChem
2.1.2 InChI
InChI=1S/F6S/c1-7(2,3,4,5)6
Computed by InChI 1.07.2 (PubChem release 2025.04.14)
PubChem
2.1.3 InChIKey
SFZCNBIFKDRMGX-UHFFFAOYSA-N
Computed by InChI 1.07.2 (PubChem release 2025.04.14)
PubChem
2.1.4 SMILES
FS(F)(F)(F)(F)F
Computed by OEChem 2.3.0 (PubChem release 2025.04.14)
PubChem
2.2 Molecular Formula
F 6 S
Computed by PubChem 2.2 (PubChem release 2025.04.14)
Australian Industrial Chemicals Introduction Scheme (AICIS); CAMEO Chemicals; PubChem
F 6 S
ILO-WHO International Chemical Safety Cards (ICSCs)
2.3 Other Identifiers
2.3.1 CAS
2551-62-4
Australian Industrial Chemicals Introduction Scheme (AICIS); CAMEO Chemicals; CAS Common Chemistry; ChemIDplus; DrugBank; EPA Chemical Data Reporting (CDR); EPA Chemicals under the TSCA; EPA DSSTox; European Chemicals Agency (ECHA); FDA Global Substance Registration System (GSRS); Hazardous Substances Data Bank (HSDB); ILO-WHO International Chemical Safety Cards (ICSCs); New Zealand Environmental Protection Authority (EPA); NJDOH RTK Hazardous Substance List; Occupational Safety and Health Administration (OSHA)
2.3.2 Deprecated CAS
29267-82-1, 36575-03-8, 59109-69-2
ChemIDplus; EPA Chemicals under the TSCA
29267-82-1, 36575-03-8
EPA DSSTox
2.3.3 European Community (EC) Number
219-854-2
European Chemicals Agency (ECHA)
2.3.4 UNII
WS7LR3I1D6
FDA Global Substance Registration System (GSRS)
2.3.5 UN Number
1080 (SULFUR HEXAFLUORIDE)
CAMEO Chemicals; Emergency Response Guidebook (ERG)
1080
ILO-WHO International Chemical Safety Cards (ICSCs); The National Institute for Occupational Safety and Health (NIOSH)
2.3.6 ChEBI ID
CHEBI:30496
ChEBI
2.3.7 ChEMBL ID
CHEMBL1796998
ChEMBL
2.3.8 DrugBank ID
DB11104
DrugBank
2.3.9 DSSTox Substance ID
DTXSID8029656
EPA DSSTox
2.3.10 ICSC Number
0571
ILO-WHO International Chemical Safety Cards (ICSCs)
2.3.11 KEGG ID
D05962
KEGG
2.3.12 NCI Thesaurus Code
C84187
NCI Thesaurus (NCIt)
2.3.13 Nikkaji Number
J95.218D
Japan Chemical Substance Dictionary (Nikkaji)
2.3.14 RXCUI
1599276
NLM RxNorm Terminology
2.3.15 Wikidata
Q279055
Wikidata
2.3.16 Wikipedia
Sulfur hexafluoride
Wikipedia
2.4 Synonyms
2.4.1 MeSH Entry Terms
Sulfur Hexafluoride
Medical Subject Headings (MeSH)
2.4.2 Depositor-Supplied Synonyms
SULFUR HEXAFLUORIDE
Sulphur hexafluoride
2551-62-4
Elegas
Lumason
Hexafluorosulfur
Sulfur fluoride (SF6)
sulfur(VI) fluoride
Hexafluorure de soufre
hexafluoridosulfur
sulfur(6+) fluoride
Sulfur fluoride (SF6), (OC-6-11)-
WS7LR3I1D6
CHEBI:30496
DTXSID8029656
R 7146
SonoVue (for the microbubble formulation)
Hexafluoride, Sulfur
RefChem:6275
DTXCID009656
DTXSID901026628
219-854-2
Distearoylphosphatidylcholine, DL-; sodium 1,2-dipalmitoyl-SN-glycero-3-phospho-(1'-rac-glycerol); sulfur hexafluoride
SonoVue
Sulfur hexafluoride [USAN]
hexafluoro-lambda6-sulfane
MFCD00011447
BRI
Sulfur hexafluoride (USAN)
Esaflon
SF6
HSDB 825
Hexafluorure de soufre [French]
EINECS 219-854-2
Sulfur hexafluoride lipid microsphere
UN1080
UNII-WS7LR3I1D6
SF6 Microbubbles
Sulfur fluoride, (OC-6-11)-
Lumason (TN)
Sonovue (TN)
hexafluoro-l6-sulfane
EC 219-854-2
hexafluoro-$l^{6}-sulfane
SCHEMBL24171
CHEMBL1796998
SULFUR HEXAFLUORIDE [MI]
[SF6]
SULFUR HEXAFLUORIDE [HSDB]
Sulfur hexafluoride, >=99.75%
hexakis(fluoranyl)-$l^{6}-sulfane
SULFUR HEXAFLUORIDE [MART.]
SBB086453
SULFUR HEXAFLUORIDE [WHO-DD]
DB11104
OC-6-11
SULPHUR HEXAFLUORIDE [EMA EPAR]
UN 1080
F6S
Stabilized sulfur hexafluoride microbubbles
NS00078794
Sulfur hexafluoride, phospholipid microspheres
D05962
A817889
Q279055
Sulfur hexafluoride [UN1080] [Nonflammable gas]
SULFUR HEXAFLUORIDE LIPID-TYPE A MICROSPHERES [ORANGE BOOK]
PubChem
3 Chemical and Physical Properties
3.1 Computed Properties
Property Name
Property Value
Reference
Property Name
Molecular Weight
Property Value
146.06 g/mol
Reference
Computed by PubChem 2.2 (PubChem release 2025.04.14)
Property Name
XLogP3-AA
Property Value
4.2
Reference
Computed by XLogP3 3.0 (PubChem release 2025.04.14)
Property Name
Hydrogen Bond Donor Count
Property Value
0
Reference
Computed by Cactvs 3.4.8.18 (PubChem release 2025.04.14)
Property Name
Hydrogen Bond Acceptor Count
Property Value
6
Reference
Computed by Cactvs 3.4.8.18 (PubChem release 2025.04.14)
Property Name
Rotatable Bond Count
Property Value
0
Reference
Computed by Cactvs 3.4.8.18 (PubChem release 2025.04.14)
Property Name
Exact Mass
Property Value
145.96249015 Da
Reference
Computed by PubChem 2.2 (PubChem release 2025.04.14)
Property Name
Monoisotopic Mass
Property Value
145.96249015 Da
Reference
Computed by PubChem 2.2 (PubChem release 2025.04.14)
Property Name
Topological Polar Surface Area
Property Value
1 Ų
Reference
Computed by Cactvs 3.4.8.18 (PubChem release 2025.04.14)
Property Name
Heavy Atom Count
Property Value
7
Reference
Computed by PubChem
Property Name
Formal Charge
Property Value
0
Reference
Computed by PubChem
Property Name
Complexity
Property Value
62.7
Reference
Computed by Cactvs 3.4.8.18 (PubChem release 2025.04.14)
Property Name
Isotope Atom Count
Property Value
0
Reference
Computed by PubChem
Property Name
Defined Atom Stereocenter Count
Property Value
0
Reference
Computed by PubChem
Property Name
Undefined Atom Stereocenter Count
Property Value
0
Reference
Computed by PubChem
Property Name
Defined Bond Stereocenter Count
Property Value
0
Reference
Computed by PubChem
Property Name
Undefined Bond Stereocenter Count
Property Value
0
Reference
Computed by PubChem
Property Name
Covalently-Bonded Unit Count
Property Value
1
Reference
Computed by PubChem
Property Name
Compound Is Canonicalized
Property Value
Yes
Reference
Computed by PubChem (release 2025.04.14)
PubChem
3.2 Experimental Properties
3.2.1 Physical Description
Sulfur hexafluoride appears as a colorless odorless gas. Noncombustible. Shipped as a liquefied gas under own vapor pressure. Contact may cause frostbite. Under prolonged exposure to fire or heat the containers may rupture violently and rocket.
CAMEO Chemicals
Gas or Vapor
EPA Chemical Data Reporting (CDR)
Colorless, odorless gas. [Note: Shipped as a liquefied compressed gas. Condenses directly to a solid upon cooling.]; [NIOSH]
Haz-Map, Information on Hazardous Chemicals and Occupational Diseases
ODOURLESS COLOURLESS COMPRESSED LIQUEFIED GAS.
ILO-WHO International Chemical Safety Cards (ICSCs)
Colorless, odorless gas.
Occupational Safety and Health Administration (OSHA)
Colorless, odorless gas. [Note: Shipped as a liquefied compressed gas. Condenses directly to a solid upon cooling.]
The National Institute for Occupational Safety and Health (NIOSH)
3.2.2 Color / Form
Colorless gas [Note: Shipped as a liquefied compressed gas. Condenses directly to a solid upon cooling].
NIOSH. NIOSH Pocket Guide to Chemical Hazards. Department of Health & Human Services, Centers for Disease Control & Prevention. National Institute for Occupational Safety & Health. DHHS (NIOSH) Publication No. 2010-168 (2010). Available from:
Hazardous Substances Data Bank (HSDB)
3.2.3 Odor
Odorless
O'Neil, M.J. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Cambridge, UK: Royal Society of Chemistry, 2013., p. 1659
Hazardous Substances Data Bank (HSDB)
3.2.4 Taste
Tasteless
Evans FE, Mani G; Fluorine Compounds, Inorganic, Sulfur. Kirk-Othmer Encyclopedia of Chemical Technology. (1999-2015). New York, NY: John Wiley & Sons. Online Posting Date: 4 Dec 2000.
Hazardous Substances Data Bank (HSDB)
3.2.5 Boiling Point
Sublimes (NIOSH, 2024)
CAMEO Chemicals
-63.8 °C (sublimes)
O'Neil, M.J. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Cambridge, UK: Royal Society of Chemistry, 2013., p. 1659
Hazardous Substances Data Bank (HSDB)
Sulfur hexafluoride is an unreactive substance. Sulfur hexafluoride is not attacked by water, acids, or bases, at room temperature. It is resistant to the action of carbon, copper or magnesium at red heat, and will not react with sodium below its boiling point. It reacts with sulfur vapor or hydrogen at 400 °C.
Braker W, Mossman A; Matheson Gas Data Book 6th Ed p.651 (1980)
Hazardous Substances Data Bank (HSDB)
sublimes
Occupational Safety and Health Administration (OSHA); The National Institute for Occupational Safety and Health (NIOSH)
3.2.6 Melting Point
-83 °F (Sublimes) (NIOSH, 2024)
CAMEO Chemicals
-50.8 °C
O'Neil, M.J. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Cambridge, UK: Royal Society of Chemistry, 2013., p. 1659
Hazardous Substances Data Bank (HSDB)
-51Â °C
ILO-WHO International Chemical Safety Cards (ICSCs)
-83 °F (sublimes)
Occupational Safety and Health Administration (OSHA); The National Institute for Occupational Safety and Health (NIOSH)
3.2.7 Solubility
0.003 % at 77 °F (NIOSH, 2024)
CAMEO Chemicals
In water, 31 mg/L at 25 °C
Giardino JE et al; J Air Pollut Control Assoc 38: 278-80 (1988)
Hazardous Substances Data Bank (HSDB)
Solubility in water at 25 °C at partial pressure of 101.325 kPa = 5.4 cu cm/kg water.
Braker W, Mossman A; Matheson Gas Data Book 6th Ed p.649 (1980)
Hazardous Substances Data Bank (HSDB)
SOLUBLE IN POTASSIUM HYDROXIDE AND ALCOHOL
Weast, R.C. (ed.) Handbook of Chemistry and Physics. 69th ed. Boca Raton, FL: CRC Press Inc., 1988-1989., p. B-135
Hazardous Substances Data Bank (HSDB)
0.297 mL dissolves in 1.0 mL transformer oil at 25 °C, 1 atm
O'Neil, M.J. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Cambridge, UK: Royal Society of Chemistry, 2013., p. 1659
Hazardous Substances Data Bank (HSDB)
For more Solubility (Complete) data for SULFUR HEXAFLUORIDE (6 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
Solubility in water: none
ILO-WHO International Chemical Safety Cards (ICSCs)
(77 °F): 0.003%
The National Institute for Occupational Safety and Health (NIOSH)
3.2.8 Density
6.5 g/L (gas); 1.67 (liquid)
Lewis, R.J. Sr.; Hawley's Condensed Chemical Dictionary 15th Edition. John Wiley & Sons, Inc. New York, NY 2007., p. 1192
Hazardous Substances Data Bank (HSDB)
Inert to nucleophilic attack; does not attack glass; one of heaviest known gases; density approx 5 times that of air; no fluorine exchange with anhydrous hydrogen fluoride; unchanged at 500 °C. .. thermodynamically unstable, but kinetically stable gas ... stable to silent electrical discharge
O'Neil, M.J. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Cambridge, UK: Royal Society of Chemistry, 2013., p. 1659
Hazardous Substances Data Bank (HSDB)
Specific volume: 2.5 cu ft/lb at 21.1 °C. Freezing point = -64 °C (sublimes); density, gas = 6.5 g/L; density, liquid = 1.67 g/L
Lewis, R.J. Sr.; Hawley's Condensed Chemical Dictionary 15th Edition. John Wiley & Sons, Inc. New York, NY 2007., p. 1192
Hazardous Substances Data Bank (HSDB)
Critical density: 0.736 kg/cu dm.
Braker W, Mossman A; Matheson Gas Data Book 6th Ed p.649 (1980)
Hazardous Substances Data Bank (HSDB)
Triple point = -49.596 °C; density = 5.970 g/L
Haynes, W.M. (ed.). CRC Handbook of Chemistry and Physics. 95th Edition. CRC Press LLC, Boca Raton: FL 2014-2015, p. 4-93
Hazardous Substances Data Bank (HSDB)
Relative density (water = 1): 1.9
ILO-WHO International Chemical Safety Cards (ICSCs)
5.11(relative gas density)
The National Institute for Occupational Safety and Health (NIOSH)
3.2.9 Vapor Density
5.11 (NIOSH, 2024) - Heavier than air; will sink (Relative to Air)
CAMEO Chemicals
5 (Air = 1)
CDC; International Chemical Safety Cards (ICSC) 2014. Atlanta, GA: Centers for Disease Prevention & Control. National Institute for Occupational Safety & Health (NIOSH). Ed Info Div. Available from, as of Sept 25, 2015:
Hazardous Substances Data Bank (HSDB)
Relative vapor density (air = 1): 5
ILO-WHO International Chemical Safety Cards (ICSCs)
5.11
Occupational Safety and Health Administration (OSHA)
3.2.10 Vapor Pressure
21.5 atm (NIOSH, 2024)
CAMEO Chemicals
VP: 1 Pa at -158 °C (solid), 10 Pa at -147 °C (solid), 100 Pa at -133.6 °C (solid), 1 kPa at -116.6 °C (solid), 10 kPa at -94.4 °C (solid), 100 kPa at -64.1 °C (solid)
Haynes, W.M. (ed.). CRC Handbook of Chemistry and Physics. 95th Edition. CRC Press LLC, Boca Raton: FL 2014-2015, p. 6-91
Hazardous Substances Data Bank (HSDB)
2367 kPa at 25 °C (1.78X10+4 mm Hg)
ECHA; Search for Chemicals. Sulfur Hexafluoride (CAS 2551-62-4) Registered Substances Dossier. European Chemical Agency. Available from, as of Sept 28, 2015:
Hazardous Substances Data Bank (HSDB)
21.5 atm
Occupational Safety and Health Administration (OSHA); The National Institute for Occupational Safety and Health (NIOSH)
3.2.11 LogP
log Kow = 1.68
Hansch, C., Leo, A., D. Hoekman. Exploring QSAR - Hydrophobic, Electronic, and Steric Constants. Washington, DC: American Chemical Society., 1995., p. 3
Hazardous Substances Data Bank (HSDB)
1.68
ILO-WHO International Chemical Safety Cards (ICSCs)
3.2.12 Henry's Law Constant
Henry's Law constant = 4.52 atm-cu m/mole at 25 °C
Giardino JE et al; J Air Pollut Control Assoc 38: 278-80
Hazardous Substances Data Bank (HSDB)
3.2.13 Stability / Shelf Life
Stable under recommended storage conditions.
Sigma-Aldrich; Safety Data Sheet for Sulfur hexafluoride. Product Number: 295701, Version 3.6 (Revision Date 06/08/2015). Available from, as of January 28, 2016:
Hazardous Substances Data Bank (HSDB)
Sonovue shelf-life: 2 years. Once reconstituted, chemical and physical stability has been demonstrated for 6 hours. From a microbiological point of view, the product should be used immediately. If not used immediately, in use storage times and conditions prior to use are the responsibility of the user.
European Medicines Agency (EMEA), The European Agency for the Evaluation of Medicinal Products, Summary of product characteristics: Sonovue (sulfur hexafluoride), Annex I p.6. Available from, as of April 2, 2007:
Hazardous Substances Data Bank (HSDB)
3.2.14 Decomposition
Sulfuryl and thionyl fluorides are the major decomposition products of sulfur hexafluoride.
TRUHAUT R ET AL; ARCH MAL PROF MED TRAV SECUR SOC 34 (10-11): 581-91 (1973)
Hazardous Substances Data Bank (HSDB)
When heated to decomposition it emits highly toxic fumes of /hydrogen fluoride and sulfur oxides/.
Lewis, R.J. Sr. (ed) Sax's Dangerous Properties of Industrial Materials. 11th Edition. Wiley-Interscience, Wiley & Sons, Inc. Hoboken, NJ. 2004., p. 3331
Hazardous Substances Data Bank (HSDB)
3.2.15 Viscosity
Gas: 150.29 micropoise at 25 °C; liquid: 0.277 centipoise at 25 °C
Yaws C; Matheson Gas Data Book 7th ed., Parsippany, NJ: Matheson Tri.Gas p. (2001)
Hazardous Substances Data Bank (HSDB)
3.2.16 Heat of Vaporization
9.6419 kJ/mol
Evans FE, Mani G; Fluorine Compounds, Inorganic, Sulfur. Kirk-Othmer Encyclopedia of Chemical Technology. (1999-2015). New York, NY: John Wiley & Sons. Online Posting Date: 4 Dec 2000.
Hazardous Substances Data Bank (HSDB)
3.2.17 Ionization Potential
19.30 eV
Occupational Safety and Health Administration (OSHA); The National Institute for Occupational Safety and Health (NIOSH)
3.2.18 Refractive Index
Index of refraction: 1.000783 (gas)
Evans FE, Mani G; Fluorine Compounds, Inorganic, Sulfur. Kirk-Othmer Encyclopedia of Chemical Technology. (1999-2015). New York, NY: John Wiley & Sons. Online Posting Date: 4 Dec 2000.
Hazardous Substances Data Bank (HSDB)
3.2.19 Kovats Retention Index
Standard non-polar
135 , 135
NIST Mass Spectrometry Data Center
3.2.20 Other Experimental Properties
Liquid at -50.5 °C
Weast, R.C. (ed.). Handbook of Chemistry and Physics. 64th ed. Boca Raton, Florida: CRC Press Inc., 1983-84., p. B-145
Hazardous Substances Data Bank (HSDB)
Upon being cooled, it condenses directly to a solid
Reynolds, J.E.F., Prasad, A.B. (eds.) Martindale-The Extra Pharmacopoeia. 28th ed. London: The Pharmaceutical Press, 1982., p. 543
Hazardous Substances Data Bank (HSDB)
The Bunsen lipid/gas partition coefficient of sulfur hexafluoride in the pure phospholipid bilayer at 25 °C was 0.24
PMID:7272292
Smith RA et al; Biochim Biophys Acta 645 (2): 327-38 (1981)
Hazardous Substances Data Bank (HSDB)
Critical volume: 1.356 cu dm/kg.
Braker W, Mossman A; Matheson Gas Data Book 6th Ed p.649 (1980)
Hazardous Substances Data Bank (HSDB)
For more Other Experimental Properties (Complete) data for SULFUR HEXAFLUORIDE (17 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
3.3 SpringerMaterials Properties
phonon wavenumber
dielectric constant
crystal structure
fusion temperature
dielectricity
diffusion
spectroscopic data
piezooptic effect
enthalpy
Gibbs energy
Wyckoff sequence
phase equilibrium
partial pressure
positional coordinate
crystallographic point group
kinetic properties
heat of solution
electron conductivity
space group
molecular structure
enthalpy change
phase diagram
unit cell parameter
melting temperature
heat capacity
electronic structure
adsorption
acentric factor
Schoenflies notation
sound velocity
photoemission spectroscopy
excitation energy
vapor-liquid equilibrium
fusion entropy
transition enthalpy
displacement parameter
symmetry axis
internuclear distance
structural transition temperature
sound propagation
heat flow rate
grain size
high frequency properties
lattice stiffness
sound absorption
phase transition
piezooptic constant
chemical diffusion
molar mass
self-diffusion
diffusion of impurities
impurity concentration
boiling point
Pearson symbol
band structure
formation entropy
virial coefficient
fluorescence
migration energy
preexponential factor
photoemission
chemical shift
atomic environment
transition entropy
core level transition
formation enthalpy
bond type
mixing enthalpy
reaction coordinate
density
surface tension
isotope mass-effect
creep
diamagnetic susceptibility
elastooptic coefficient
formation energy
formula unit
unit cell axes
chemical bond
gross formula
compressibility
latent heat
vapor pressure
activation energy
tricritical point
heat of sublimation
refractive index
elasticity
valence band
entropy
electrooptical constant
tracer diffusion
Raman spectroscopy
thermal conductivity
excess enthalpy
point group
heat transfer coefficient
site multiplicity
viscosity
nuclear magnetic resonance
diffusive flux
Raman effect
critical point
spin-lattice relaxation
SpringerMaterials
3.4 Chemical Classes
Toxic Gases & Vapors -> Simple Asphyxiants
Haz-Map, Information on Hazardous Chemicals and Occupational Diseases
3.4.1 Drugs
Pharmaceuticals -> Various
S92 | FLUOROPHARMA | List of ~340 ATC classified fluoro-pharmaceuticals | DOI:10.5281/zenodo.5979646
NORMAN Suspect List Exchange
3.4.1.1 Human Drugs
Breast Feeding; Lactation; Milk, Human; Contrast Media; Diagnostic Agents; Fluorides; Gases
Drugs and Lactation Database (LactMed)
Human drug -> Prescription
Drugs@FDA
Human drugs -> Contrast media -> Human pharmacotherapeutic group -> EMA Drug Category
European Medicines Agency (EMA)
4 Spectral Information
4.1 1D NMR Spectra
4.1.1 19F NMR Spectra
Instrument Name
Varian HA-100
Copyright
Copyright © 2002-2025 Wiley-VCH GmbH. All Rights Reserved.
Thumbnail
SpectraBase
4.2 Mass Spectrometry
4.2.1 GC-MS
1 of 3 items
View All
NIST Number
284499
Library
Main library
Total Peaks
12
m/z Top Peak
127
m/z 2nd Highest
89
m/z 3rd Highest
108
Thumbnail
NIST Mass Spectrometry Data Center
2 of 3 items
View All
NIST Number
21382
Library
Replicate library
Total Peaks
13
m/z Top Peak
127
m/z 2nd Highest
89
m/z 3rd Highest
108
Thumbnail
NIST Mass Spectrometry Data Center
4.3 IR Spectra
4.3.1 Vapor Phase IR Spectra
Instrument Name
DIGILAB FTS-14
Technique
Vapor Phase
Copyright
Copyright © 1980, 1981-2025 John Wiley & Sons, Inc. All Rights Reserved.
Thumbnail
SpectraBase
5 Related Records
5.1 Related Compounds with Annotation
Follow these links to do a live 2D search or do a live 3D search for this compound, sorted by annotation score. This section is deprecated (see the neighbor discontinuation help page for details), but these live search links provide equivalent functionality to the table that was previously shown here.
PubChem
5.2 Related Compounds
Mixtures, Components, and Neutralized Forms Count
54
Similar Compounds (2D)
View in PubChem Search
Similar Conformers (3D)
View in PubChem Search
PubChem
5.3 Substances
5.3.1 PubChem Reference Collection SID
481107615
PubChem
5.3.2 Related Substances
All Count
165
Same Count
77
Mixture Count
88
PubChem
5.3.3 Substances by Category
PubChem
5.4 NCBI LinkOut
NCBI
6 Chemical Vendors
PubChem
7 Drug and Medication Information
7.1 Drug Indication
Echocardiography: Sulfur hexafluoride is indicated for use in adult patients with suboptimal echocardiograms to opacify the left ventricular chamber and to improve the delineation of the left ventricularendocardial border. Ultrasonography of the Liver: Sulfur hexafluoride is indicated for use with ultrasound of the liver in adult and pediatric patients to characterize focal liver lesions.
DrugBank
This medicinal product is for diagnostic use only. SonoVue is for use with ultrasound imaging to enhance the echogenicity of the blood, or of fluids in the urinary tract which results in an improved signal to noise ratio. SonoVue should only be used in patients where study without contrast enhancement is inconclusive. EchocardiographySonoVue is a transpulmonary echocardiographic contrast agent for use in adult patients with suspected or established cardiovascular disease to provide opacification of cardiac chambers and enhance left ventricular endocardial border delineation. Doppler of macrovasculatureSonoVue increases the accuracy in detection or exclusion of abnormalities in cerebral arteries and extracranial carotid or peripheral arteries in adult patients by improving the Doppler signal to noise ratio. SonoVue increases the quality of the Doppler flow image and the duration of clinically useful signal enhancement in portal vein assessment in adult patients. Doppler of microvasculatureSonoVue improves display of the vascularity of liver and breast lesions during Doppler sonography in adult patients leading to more specific lesion characterisation. Ultrasonography of excretory urinary tractSonoVue is indicated for use in ultrasonography of the excretory tract in paediatric patients from newborn to 18 years to detect vesicoureteral reflux. For the limitation in the interpretation of a negative urosonography.
European Medicines Agency (EMA)
Open Targets
7.2 Drug Classes
Breast Feeding; Lactation; Milk, Human; Contrast Media; Diagnostic Agents; Fluorides; Gases
Drugs and Lactation Database (LactMed)
7.3 FDA National Drug Code Directory
National Drug Code (NDC) Directory
7.4 Drug Labels
Drug and label
DailyMed
View More...
7.5 Clinical Trials
7.5.1 ClinicalTrials.gov
ClinicalTrials.gov
7.5.2 EU Clinical Trials Register
EU Clinical Trials Register
7.6 EMA Drug Information
Medicine
SonoVue
Category
Human drugs
Therapeutic area
Ultrasonography; Echocardiography
Active Substance
sulphur hexafluoride
INN/Common name
sulphur hexafluoride
Pharmacotherapeutic Classes
Contrast media
Status
This medicine is authorized for use in the European Union
Company
Bracco International B.V.
Market Date
2001-03-26
European Medicines Agency (EMA)
7.7 Therapeutic Uses
Lumason is indicated for use in patients with suboptimal echocardiograms to opacify the left ventricular chamber and to improve the delineation of the left ventricular endocardial border.
NIH; DailyMed. Current Medication Information for LUMASON- sulfur hexafluoride (Revised: October 2014). Available from, as of August 2, 2015:
Hazardous Substances Data Bank (HSDB)
An inert gas used mainly as a test gas in respiratory physiology. Other uses include its injection in vitreoretinal surgery to restore the vitreous chamber ...
National Library of Medicine - Medical Subject Headings (2007)
Hazardous Substances Data Bank (HSDB)
The intraocular injection of sulfur hexafluoride appears to be useful contribution to the surgical treatment of superior bullous hemi-retinal detachment, allowing effective and durable internal tamponade, while avoiding prolonged bedrest.
J Fr Ophtalmol 6 (11): 889-93 (1983)
Hazardous Substances Data Bank (HSDB)
In this randomized clinical trial, 18 patients received treatment with silicone oil and 16 patients received SF6. The primary outcome was defined as successful anatomic attachment of the retina. Secondary outcomes included the time to retinal detachment, visual acuity, anatomic macular attachment, and any complications of surgery. RESULTS: The odds of successful reattachment with silicone oil were 50% greater than they were with SF6, but this difference was not statistically significant. There were no differences between the two groups in any of the secondary outcomes. ... This study did not have enough statistical power to detect a small but clinically important difference between the two treatment groups ...
PMID:9387180
Hammer M et al; Ophthalmic Surg Lasers 28 (11): 926-31 (1997)
Hazardous Substances Data Bank (HSDB)
For more Therapeutic Uses (Complete) data for SULFUR HEXAFLUORIDE (11 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
7.8 Drug Warnings
/BOXED WARNING/ WARNING: SERIOUS CARDIOPULMONARY REACTIONS Serious cardiopulmonary reactions, including fatalities, have occurred uncommonly during or following the injection of ultrasound contrast agents, including sulfur hexafluoride lipid microspheres. Most serious reactions occur within 30 minutes of administration. Assess all patients for the presence of any condition that precludes administration. Always have resuscitation equipment and trained personnel readily available.
NIH; DailyMed. Current Medication Information for LUMASON- sulfur hexafluoride (Revised: October 2014). Available from, as of August 2, 2015:
Hazardous Substances Data Bank (HSDB)
Serious cardiopulmonary reactions, including fatalities have occurred uncommonly during or shortly following administration of ultrasound contrast agents, including Lumason. These reactions typically occurred within 30 minutes of administration. The risk for these reactions may be increased among patients with unstable cardiopulmonary conditions (acute myocardial infarction, acute coronary artery syndromes, worsening or unstable congestive heart failure, or serious ventricular arrhythmias). Always have cardiopulmonary resuscitation personnel and equipment readily available prior to Lumason administration and monitor all patients for acute reactions. The reported reactions that may follow the administration of ultrasound contrast agents include: fatal cardiac or respiratory arrest, shock, syncope, symptomatic arrhythmias (atrial fibrillation, tachycardia, bradycardia, supraventricular tachycardia, ventricular fibrillation, and ventricular tachycardia), hypertension, hypotension, dyspnea, hypoxia, chest pain, respiratory distress, stridor, wheezing, loss of consciousness, and convulsions.
NIH; DailyMed. Current Medication Information for LUMASON- sulfur hexafluoride (Revised: October 2014). Available from, as of August 2, 2015:
Hazardous Substances Data Bank (HSDB)
Lumason is contraindicated in patients with: known or suspected right-to-left, bi-directional, or transient right-to-left cardiac shunts history of hypersensitivity reactions to sulfur hexafluoride lipid microsphere components or to any of the inactive ingredients in Lumason Do not administer by intra-arterial injection.
NIH; DailyMed. Current Medication Information for LUMASON- sulfur hexafluoride (Revised: October 2014). Available from, as of August 2, 2015:
Hazardous Substances Data Bank (HSDB)
In patients with right-to-left, bi-directional, or transient right-to-left cardiac shunts, some intravenously injected sulfur hexafluoride lipid containing microspheres may bypass filtering by the lung and directly enter the arterial circulation. Occlusion of the microcirculation by these microspheres may result in tissue ischemia. Lumason is only for intravenous administration; do not administer Lumason by intra-arterial injection.
NIH; DailyMed. Current Medication Information for LUMASON- sulfur hexafluoride (Revised: October 2014). Available from, as of August 2, 2015:
Hazardous Substances Data Bank (HSDB)
For more Drug Warnings (Complete) data for SULFUR HEXAFLUORIDE (19 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
8 Pharmacology and Biochemistry
8.1 Pharmacodynamics
Sulfur hexafluoride provides useful echocardiographic signal intensity for two minutes after the injection. Sulfur hexafluoride microspheres are destroyed and contrast enhancement decreases as the mechanical index increases (values of 0.8 or less are recommended). For ultrasonography of the liver, Sulfur hexafluoride provides dynamic patterns of differential signal intensity enhancement between focal liver lesions and liver parenchyma during the arterial, portal venous, and late phase of signal intensity enhancement of the microvasculature.
DrugBank
8.2 FDA Pharmacological Classification
FDA UNII
WS7LR3I1D6
Active Moiety
SULFUR HEXAFLUORIDE
Pharmacological Classes
Established Pharmacologic Class [EPC] - Contrast Agent for Ultrasound Imaging
Pharmacological Classes
Mechanisms of Action [MoA] - Ultrasound Contrast Activity
FDA Pharmacology Summary
Sulfur hexafluoride is a Contrast Agent for Ultrasound Imaging. The mechanism of action of sulfur hexafluoride is as an Ultrasound Contrast Activity.
FDA Pharm Classes
8.3 ATC Code
V08DA04
European Medicines Agency (EMA)
V08DA05
S76 | LUXPHARMA | Pharmaceuticals Marketed in Luxembourg | Pharmaceuticals marketed in Luxembourg, as published by d'Gesondheetskeess (CNS, la caisse nationale de sante, www.cns.lu), mapped by name to structures using CompTox by R. Singh et al. (2021) DOI:10.1021/acsenvironau.1c00008. List downloaded from Dataset DOI:10.5281/zenodo.4587355
NORMAN Suspect List Exchange
V - Various
V08 - Contrast media
V08D - Ultrasound contrast media
V08DA - Ultrasound contrast media
V08DA05 - Sulfur hexafluoride, phospholipid microspheres
WHO Anatomical Therapeutic Chemical (ATC) Classification
ATCvet Code
QV - Various
QV08 - Contrast media
QV08D - Ultrasound contrast media
QV08DA - Ultrasound contrast media
QV08DA05 - Sulfur hexafluoride, phospholipid microspheres
WHO ATCvet - Classification of Veterinary Medicines
8.4 Absorption, Distribution and Excretion
Absorption
The pharmacokinetic of the SF6 gas component of Lumason was evaluated in 12 healthy adult subjects (7 men and 5 women). After intravenous bolus injections of 0.03 mL/kg and 0.3 mL/kg of Lumason, corresponding to approximately 1 and 10 times the recommended doses, concentrations of SF6 in blood peaked within 1 to 2 minutes for both doses.
DrugBank
Route of Elimination
The SF6 component of Lumason is eliminated via the lungs.
DrugBank
Volume of Distribution
In a study of healthy subjects, the mean values for the apparent steady-state volume of distribution of SF6 were 341 L and 710 L for Lumason doses of 0.03 mL/kg and 0.3 mL/kg, respectively. Preferential distribution to the lung is likely responsible for these values.
DrugBank
SonoVue, is a new echo contrast agent based on stabilized sulfur hexafluoride (SF6) microbubbles ... The blood kinetics and pulmonary elimination of SF6 after intravenous bolus injection of two dosage levels (0.03 and 0.3 mL/kg) of SonoVue were evaluated in 12 healthy subjects (7 men, 5 women). In addition, safety and tolerability were evaluated by monitoring vital signs, adverse effects, discomfort, and physical examination and laboratory parameters associated with the SonoVue injection. The blood kinetics of SF6 was not dose dependent. SF6 was rapidly removed from the blood by the pulmonary route, with 40% to 50% of the injected dose eliminated within the first minute after administration and 80% to 90% eliminated by 11 minutes after administration; the elimination was similar in men and women and independent of dose. Both dosages were well tolerated. No adverse effects were observed immediately or during the 24-hour follow-up period. ... The route of SF6 elimination was by means of the lungs in the expired air. ...
PMID:10639039
Morel DR et al; Invest Radiol 35 (1): 80-5 (2000)
Hazardous Substances Data Bank (HSDB)
Sonovue (trade mark) is a new echo contrast agent made of microbubbles stabilized by phospholipids and containing sulphur hexafluoride (SF6), an innocuous gas. ... With regard to the gas contained in the bubbles, its pharmacokinetics have been assessed during a study in human volunteers. Following intravenous administration of 0.3 mL/kg of SonoVue (trade mark) (i.e., approximately ten times the imaging dose), the blood level curve showed a distribution half-life of about 1 minute and an elimination half-life of about 6 minutes. More than 80% of the administered gas is exhaled via the lungs after 11 minutes. ...
PMID:10639039
Schneider M; Echocardiography 16 (7, Pt 2): 743-746 (1999 Oct)
Hazardous Substances Data Bank (HSDB)
The pharmacokinetic of the SF6 gas component of Lumason was evaluated in 12 healthy adult subjects (7 men and 5 women). After intravenous bolus injections of 0.03 mL/kg and 0.3 mL/kg of Lumason, corresponding to approximately 1 and 10 times the recommended doses, concentrations of SF6 in blood peaked within 1 to 2 minutes for both doses. The terminal half-life of SF6 in blood was approximately 10 minutes for the 0.3 mL/kg dose. (At the 0.03 mL/kg dose, terminal half-life could not be estimated.) The area-under-the-curve of SF6 was dose-proportional over the dose range studied.
NIH; DailyMed. Current Medication Information for LUMASON- sulfur hexafluoride (Revised: October 2014). Available from, as of August 2, 2015:
Hazardous Substances Data Bank (HSDB)
In a study of healthy subjects, the mean values for the apparent steady-state volume of distribution of SF6 were 341 L and 710 L for Lumason doses of 0.03 mL/kg and 0.3 mL/kg, respectively. Preferential distribution to the lung is likely responsible for these values.
NIH; DailyMed. Current Medication Information for LUMASON- sulfur hexafluoride (Revised: October 2014). Available from, as of August 2, 2015:
Hazardous Substances Data Bank (HSDB)
For more Absorption, Distribution and Excretion (Complete) data for SULFUR HEXAFLUORIDE (6 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
8.5 Metabolism / Metabolites
SF6 undergoes little or no biotransformation; 88% of an administered dose is recovered unchanged in expired air.
NIH; DailyMed. Current Medication Information for LUMASON- sulfur hexafluoride (Revised: October 2014). Available from, as of August 2, 2015:
DrugBank; Hazardous Substances Data Bank (HSDB)
Under brief and prolonged exposures, sulfur hexafluoride was biologically inert and did not metabolize.
PMID:7250729
Mel'nikova LV, Rozova TA; Gig Tr Prof Zabol 6: 48-9 (1981)
Hazardous Substances Data Bank (HSDB)
8.6 Biological Half-Life
The terminal half-life of SF6 in blood was approximately 10 minutes for the 0.3 mL/kg dose. (At the 0.03 mL/kg dose, terminal half-life could not be estimated.)
DrugBank
The pharmacokinetic of the SF6 gas component of Lumason was evaluated in 12 healthy adult subjects (7 men and 5 women). After intravenous bolus injections of 0.03 mL/kg and 0.3 mL/kg of Lumason... The terminal half-life of SF6 in blood was approximately 10 minutes for the 0.3 mL/kg dose. (At the 0.03 mL/kg dose, terminal half-life could not be estimated.) The area-under-the-curve of SF6 was dose-proportional over the dose range studied.
NIH; DailyMed. Current Medication Information for LUMASON- sulfur hexafluoride (Revised: October 2014). Available from, as of August 2, 2015:
Hazardous Substances Data Bank (HSDB)
... Following intravenous administration of 0.3 mL/kg of SonoVue (trade mark) (i.e., approximately ten times the imaging dose), the blood level curve showed a distribution half-life of about 1 minute and an elimination half-life of about 6 minutes. ...
PMID:10639039
Schneider M; Echocardiography 16 (7, Pt 2): 743-746 (1999 Oct)
Hazardous Substances Data Bank (HSDB)
In a study of patients with pulmonary impairment, blood concentrations of SF6 peaked at 1 to 4 minutes following Lumason administration. The cumulative recovery of SF6 in expired air was 102 +/- 18% (mean +/- standard deviation), and the terminal half-life of SF6 in blood was similar to that measured in healthy subjects.
NIH; DailyMed. Current Medication Information for LUMASON- sulfur hexafluoride (Revised: October 2014). Available from, as of August 2, 2015:
Hazardous Substances Data Bank (HSDB)
8.7 Mechanism of Action
Within the blood, the acoustic impedance of Lumason microspheres is lower than that of the surrounding non-aqueous tissue. Therefore, an ultrasound beam is reflected from the interface between the microspheres and the surrounding tissue. The reflected ultrasound signal provides a visual image that shows a contrast between the blood and the surrounding tissues.
DrugBank
9 Use and Manufacturing
9.1 Uses
Sources/Uses
Used as a gaseous electric insulator and as a tracer gas to measure ventilation patterns; [ACGIH]
ACGIH - Documentation of the TLVs and BEIs, 7th Ed. Cincinnati: ACGIH Worldwide, 2020.
Haz-Map, Information on Hazardous Chemicals and Occupational Diseases
An inert gas used mainly as a test gas in respiratory physiology. Other uses include its injection in vitreoretinal surgery to restore the vitreous chamber and as a tracer in monitoring the dispersion and deposition of air pollutants.
National Library of Medicine's Medical Subject Headings. Sulfur Hexafluoride. Online file (MeSH, 2015). Available from, as of November 13, 2015:
Hazardous Substances Data Bank (HSDB)
Because of its inertness under normal conditions, SF6 has been used as a tracer for a variety of studies such as airflow patterns, underground pipe leak detection, and dispersion of air pollutants.
Evans FE, Mani G; Fluorine Compounds, Inorganic, Sulfur. Kirk-Othmer Encyclopedia of Chemical Technology (1999-2015). John Wiley & Sons, Inc. Online Posting Date: December 4, 2000
Hazardous Substances Data Bank (HSDB)
Sulfur hexafluoride is the main insulation for power transmission in high voltage coaxial lines which can move power above ground, underground, or underwater.
Evans FE, Mani G; Fluorine Compounds, Inorganic, Sulfur. Kirk-Othmer Encyclopedia of Chemical Technology (1999-2015). John Wiley & Sons, Inc. Online Posting Date: December 4, 2000
Hazardous Substances Data Bank (HSDB)
In electrical circuit interrupters. In electronic ultra-high frequency piping.
O'Neil, M.J. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Cambridge, UK: Royal Society of Chemistry, 2013., p. 1659
Hazardous Substances Data Bank (HSDB)
For more Uses (Complete) data for SULFUR HEXAFLUORIDE (10 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
9.1.1 Use Classification
Human drugs -> Contrast media -> Human pharmacotherapeutic group -> EMA Drug Category
European Medicines Agency (EMA)
9.1.2 Industry Uses
Not Known or Reasonably Ascertainable
Etching agent
Other (specify)
EPA Chemical Data Reporting (CDR)
9.1.3 Consumer Uses
Other (specify)
EPA Chemical Data Reporting (CDR)
9.2 Methods of Manufacturing
Several techniques are described in the literature that use simple sulfur melting pot or flame reactors. The heat of reaction must be removed by cooling the reactor jacket. The crude gas produced, consisting mainly of SF6, must be purified because it contains SOF2, SO2F2, and toxic lower fluorides such as S2F2, SF4, and S2F10. The gases are first washed in a KOH spray scrubber to remove the major portion of the hydrolyzable compounds (S2F2, SF4, HF, F2, SO2F2, etc.). The gas mixture is then passed through a metal tube at 400 °C to pyrolyze the S2F10 to SF4 and SF6. Another washing with an alkaline solution is necessary to remove the last portions of hydrolyzable compounds. The gases next pass through a tower containing a suitable drying material, such as alumina, phosphorus pentoxide, or barium oxide, before being purified over charcoal, distilled, compressed, and transferred to steel cylinders for storage.
Aigueperse J et al; Fluorine Compounds, Inorganic; Ullmann's Encyclopedia of Industrial Chemistry 7th ed. (1999-2015). NY, NY: John Wiley & Sons. Online Posting Date: June 15, 2000
Hazardous Substances Data Bank (HSDB)
Prepared by direct fluorination of sulfur or sulfur dioxide.
O'Neil, M.J. (ed.). The Merck Index - An Encyclopedia of Chemicals, Drugs, and Biologicals. Cambridge, UK: Royal Society of Chemistry, 2013., p. 1659
Hazardous Substances Data Bank (HSDB)
Sulfur hexafluoride is manufactured by combining sulfur vapor and pure elemental fluorine. It is then given a preliminary scrubbing with caustic. Any disulfur decafluoride, S2F10, formed is decomposed by heating the product to 400 °C, to give SF4 and SF6. The SF4 and any remaining lower fluorides of sulfur are removed by a second caustic scrubber. The gas is then dried in a sulfuric acid tower, distilled, and packaged. A routine manufacturing quality-control test establishes the absence of toxic impurities. In this test, mice are exposed to an atmosphere of 80% sulfur hexafluoride and 20% oxygen for 16-20 hr and must show no visible effects from the gas.
Evans FE, Mani G; Fluorine Compounds, Inorganic, Sulfur. Kirk-Othmer Encyclopedia of Chemical Technology (1999-2015). John Wiley & Sons, Inc. Online Posting Date: December 4, 2000
Hazardous Substances Data Bank (HSDB)
9.3 Impurities
... The only permissible impurities are traces of air, carbon tetrafluoride (0.05 wt% max), and water (9 ppm by wt max; dew point -45 °C max).
Evans FE, Mani G; Fluorine Compounds, Inorganic, Sulfur. Kirk-Othmer Encyclopedia of Chemical Technology (1999-2015). John Wiley & Sons, Inc. Online Posting Date: December 4, 2000
Hazardous Substances Data Bank (HSDB)
9.4 Formulations / Preparations
Lumason (sulfur hexafluoride lipid-type A microspheres) for injectable suspension is a 3-part kit comprised of: one Lumason vial containing 25 mg of lipid-type A sterile lyophilized powder with headspace filled with 60.7 mg of sulfur hexafluoride gas one prefilled syringe containing 5 mL Sodium Chloride 0.9% Injection, USP (Diluent) one Mini-Spike Following reconstitution with the provided diluent, Lumason suspension contains 1.5 to 5.6 x108 microspheres/mL with 45 mcg/mL of sulfur hexafluoride.
NIH; DailyMed. Current Medication Information for LUMASON- sulfur hexafluoride (Revised: October 2014). Available from, as of August 2, 2015:
Hazardous Substances Data Bank (HSDB)
Instrument grade - 99.9 wt% available; commercial grade - 99.9 wt% available
AIR PRODUCTS; SPECIALTY GASES AND EQUIPMENT p.60 (1982)
Hazardous Substances Data Bank (HSDB)
Sonovue sulfur hexafluoride microbubbles 8 uL/mL. Powder and solvent for dispersion of injection. 1 vial containing 25 mg of powder to be reconsitituted with 5 mL sodium chloride 9 mg/mL (0.9%) solution for injection.
European Medicines Agency (EMEA), The European Agency for the Evaluation of Medicinal Products, Sunovue (sulfur hexafluoride) labeling, p.15. Available from, as of April 2, 2007:
Hazardous Substances Data Bank (HSDB)
9.5 Consumption Patterns
APPROX 100% AS GASEOUS INSULATION MEDIUM (1977)
SRI
Hazardous Substances Data Bank (HSDB)
Minor uses of SF6 (about 5% of the production): insulation of nuclear particle accelerators and X-ray equipment, microwave and radar equipment, thermoacoustic insulation of windows ... meteorology and medical uses related to its inertness, treatment with a plasma containing SF6 for etching silicon or for increasing the water repellency of a wet strength of kraft paper, and protection of molten magnesium in the magnesium industry.
Aigueperse J et al; Fluorine Compounds, Inorganic; Ullmann's Encyclopedia of Industrial Chemistry 7th ed. (1999-2015). NY, NY: John Wiley & Sons. Online Posting Date: June 15, 2000
Hazardous Substances Data Bank (HSDB)
Because of the good dielectric properties of SF6 and its ability to trap electrons during arc blowing, 90-95% of the SF6 produced is used in electromechanical equipment ...
Aigueperse J et al; Fluorine Compounds, Inorganic; Ullmann's Encyclopedia of Industrial Chemistry 7th ed. (1999-2015). NY, NY: John Wiley & Sons. Online Posting Date: June 15, 2000
Hazardous Substances Data Bank (HSDB)
The electric power industry uses approximately 80% of the sulfur hexafluoride produced worldwide. The rest of ... production is used in aluminum and magnesium foundries, semiconductor technology, other manufacturing processes (e.g., plasma processing), or even specific tasks such as thermal insulation, scuba diving, sound insulation, torpedo propeller quieting, insulation for AWACS radar domes, atmospheric trace gas studies, leak testing and land permeability detection for nuclear waste deposition.
Dervos CT, Vassiliou P; J Air Waste Mange Assoc 50: 137-41 (2000)
Hazardous Substances Data Bank (HSDB)
9.6 U.S. Production
Aggregated Product Volume
2019: 1,000,000 - <10,000,000 lb
2018: 1,000,000 - <10,000,000 lb
2017: 1,000,000 - <10,000,000 lb
2016: 1,000,000 - <10,000,000 lb
EPA Chemical Data Reporting (CDR)
The annual world production of sulfur hexafluoride has increased to 7,000 tons per year in 1993, and it is predicted that, by the year 2010, ... production levels could reach 10,000 metric tons per year.
Dervos CT, Vassiliou P; J Air Waste Mange Assoc 50: 137-41 (2000)
Hazardous Substances Data Bank (HSDB)
Production volumes for non-confidential chemicals reported under the Inventory Update Rule.
Year
Production Range (pounds)
Year
1986
Production Range (pounds)
10 thousand - 500 thousand
Year
1990
Production Range (pounds)
No Reports
Year
1994
Production Range (pounds)
No Reports
Year
1998
Production Range (pounds)
No Reports
Year
2002
Production Range (pounds)
10 thousand - 500 thousand
US EPA; Non-confidential Production Volume Information Submitted by Companies for Chemicals Under the 1986-2002 Inventory Update Rule (IUR). Sulfur fluoride (SF6), (OC-6-11)- (2551-62-4). Available from, as of March 6, 2007:
Hazardous Substances Data Bank (HSDB)
Non-confidential 2012 Chemical Data Reporting (CDR) information on the production and use of chemicals manufactured or imported into the United States. Chemical: Sulfur fluoride (SF6). National Production Volume: 1,000,000 - 10,000,000 lb/yr.
USEPA/Pollution Prevention and Toxics; 2012 Chemical Data Reporting Database. Sulfur fluoride (SF6) (2551-62-4). Available from, as of August 31, 2015:
Hazardous Substances Data Bank (HSDB)
9.7 General Manufacturing Information
Industry Processing Sectors
Computer and Electronic Product Manufacturing
Utilities
Electrical Equipment, Appliance, and Component Manufacturing
EPA Chemical Data Reporting (CDR)
EPA TSCA Commercial Activity Status
Sulfur fluoride (SF6), (OC-6-11)-: ACTIVE
EPA Chemicals under the TSCA
Global Warming Potential (GWP): Chemical: Nitrous oxide; GWP: 298 (100-Year Time Horizon)
40 CFR 98 (USEPA); U.S. National Archives and Records Administration's Electronic Code of Federal Regulations. Available from, as of February 5, 2016:
Hazardous Substances Data Bank (HSDB)
Sulfur hexafluoride (SF6) is the electric power industry's preferred gas for electrical insulation and arc quenching/current interruption equipment used in the transmission and distribution of electrical energy. However, SF6 is a potent greenhouse gas and this has led to concerns regarding its long-range environmental impact. Besides the current efforts to curtail the releases of SF6 into the environment, a comprehensive and focused program is indicated to develop alternative gaseous insulators and alternative high voltage technology. Final rept.
Christophorou LG et al; Govt Reports Announcements & Index (GRA&I), Issue 11, 1998
Hazardous Substances Data Bank (HSDB)
Not covered by the Montreal Protocol or any other international agreement.
Oram DE et al; Geophys Res Lett 25: 35-8 (1998)
Hazardous Substances Data Bank (HSDB)
According to the Electric Power Research Institute (EPRI), SF6 /sulfur hexafluoride/ is a synthetic gas that was developed for use as an electrical insulating medium for the power industry. Originally, its outstanding insulation characteristics were used primarily to reduce power circuit breaker interruption time; reducing interruption time enhances power system stability. As concerns escalated regarding poly-chlorinated biphenyls (PCBs) that were found in insulating oil used in older breakers, the use of SF6 insulated breakers grew. SF6 is chemically inert, nonflammable, and nontoxic. Although SF6 gas is not detrimental to the ozone layer, it is a highly potent greenhouse gas. It is 23,900 times more effective at trapping infrared radiation than carbon monoxide and is stable in the atmosphere for some 3,200 years. Although the percentage of SF6 found in the atmosphere is relatively small, the rate of growth is alarming. Currently, there is no Federal legislation curtailing the use of SF6. However, the Environmental Protection Agency (EPA) sponsors and facilitates a program for a voluntary reduction of SF6 emissions within the electric power industry. This program provides a forum for EPA and the electric power industry to work together to reduce SF6 emissions to technically and economically feasible levels, thereby helping to reduce global climate change. Implied is that if voluntary programs are not successful, legislation may be required to restrict the use of SF6.
US Department of the Interior Bureau of Reclamation; Facilities Instructions, Standards, and Techniques. Vol 5-9: Management and Safe Handling Procedures for Sulfur Hexafluoride (SF6) (March 2004)
Hazardous Substances Data Bank (HSDB)
For more General Manufacturing Information (Complete) data for SULFUR HEXAFLUORIDE (9 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
10 Identification
10.1 Analytic Laboratory Methods
Method: NIOSH 6602; Procedure: gas chromatography (portable); Analyte: sulfur hexafluoride; Matrix: air; Detection Limit: 0.05 ppb.
CDC; NIOSH Manual of Analytical Methods, 4th ed. Sulfur Hexafluoride (2551-62-4). Available from, as of November 2, 2015:
Hazardous Substances Data Bank (HSDB)
Sulfur hexafluoride may be analyzed chromatographically using a molecular sieve or a Porapak QS column. Using an electron-capture detector, a sensitivity of 1X10-3 to 1X10-4 ppb is possible.
Evans FE, Mani G; Fluorine Compounds, Inorganic, Sulfur. Kirk-Othmer Encyclopedia of Chemical Technology (1999-2015). John Wiley & Sons, Inc. Online Posting Date: December 4, 2000
Hazardous Substances Data Bank (HSDB)
Phase-fluctuation optical heterodyne spectroscopy (a selective, nondestructive gas chromatographic detector which measures the infra-red absorption of the functional group of interest using a laser homodyne photothermal refractometer) has demonstrated an absorption sensitivity of 10-8/cm. A measured detection limit of less than 1 pg of injected sulfur hexafluoride was obtained. Primary utilization of the technique will be in atmospheric trace gas analysis.
Lin HB et al; J Chromatogr 206 (2): 205-14 (1981)
Hazardous Substances Data Bank (HSDB)
Method is given to prepare highly stable molecular sieve columns with high reproducibility of the separation efficiency and longer lifetime for trace analysis of sulfur hexafluoride and bromotrifluoromethane in ambient air samples. The simultaneous gas chromatographic determination of about 5 parts per trillion sulfur hexafluoride and 100 parts per trillion bromotrifluoromethane in a 1 ml air sample is possible.
Heggen R, Oehme M; J Chromatgr 243 (1): 168-72 (1982)
Hazardous Substances Data Bank (HSDB)
For more Analytic Laboratory Methods (Complete) data for SULFUR HEXAFLUORIDE (8 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
10.2 NIOSH Analytical Methods
SULFUR HEXAFLUORIDE by portable GC 6602
NIOSH Manual of Analytical Methods
11 Safety and Hazards
11.1 Hazards Identification
11.1.1 GHS Classification
1 of 4 items
View All
Pictogram(s)
Signal
Warning
GHS Hazard Statements
H280 (100%): Contains gas under pressure; may explode if heated [Warning Gases under pressure]
H336 (19.7%): May cause drowsiness or dizziness [Warning Specific target organ toxicity, single exposure; Narcotic effects]
Precautionary Statement Codes
P261, P271, P304+P340, P319, P403+P233, P405, P410+P403, and P501
ECHA C&L Notifications Summary
Aggregated GHS information provided per 239 reports by companies from 8 notifications to the ECHA C&L Inventory. Each notification may be associated with multiple companies.
Information may vary between notifications depending on impurities, additives, and other factors. The percentage value in parenthesis indicates the notified classification ratio from companies that provide hazard codes. Only hazard codes with percentage values above 10% are shown. For more detailed information, please visit ECHA C&L website.
European Chemicals Agency (ECHA)
11.1.2 Hazard Classes and Categories
Press. Gas (Liq.) (100%)
STOT SE 3 (19.7%)
European Chemicals Agency (ECHA)
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11.1.3 DOT Hazard Classification
Substance (Descriptions/Shipping Name)
Sulfur hexafluoride
DOT ID (UN/NA Number)
UN1080
Hazard Class/Label Code(s)
Div 2.2 Non-flammable compressed gas (49 eCFR § 173.115)
Placard/Label(s)
US Code of Federal Regulations, Hazardous Materials, 49 CFR Part 172
11.1.4 Health Hazards
Excerpt from ERG Guide 126 [Gases - Compressed or Liquefied (Including Refrigerant Gases)]:
Vapors may cause dizziness or asphyxiation without warning, especially when in closed or confined areas. Vapors from liquefied gas are initially heavier than air and spread along ground. Contact with gas or liquefied gas may cause burns, severe injury and/or frostbite. Fire may produce irritating, corrosive and/or toxic gases. (ERG, 2024)
2024 Emergency Response Guidebook,
CAMEO Chemicals
ERG 2024, Guide 126 (Sulfur hexafluoride)
· Vapors may cause dizziness or asphyxiation without warning, especially when in closed or confined areas.
· Vapors from liquefied gas are initially heavier than air and spread along ground.
· Contact with gas or liquefied gas may cause burns, severe injury and/or frostbite.
· Fire may produce irritating, corrosive and/or toxic gases.
Emergency Response Guidebook (ERG)
11.1.5 Fire Hazards
Excerpt from ERG Guide 126 [Gases - Compressed or Liquefied (Including Refrigerant Gases)]:
Some may burn but none ignite readily. Containers may explode when heated. Ruptured cylinders may rocket. CAUTION: Aerosols (UN1950) may contain a flammable propellant. (ERG, 2024)
2024 Emergency Response Guidebook,
CAMEO Chemicals
ERG 2024, Guide 126 (Sulfur hexafluoride)
· Some may burn but none ignite readily.
· Containers may explode when heated.
· Ruptured cylinders may rocket.
CAUTION: Aerosols (UN1950) may contain a flammable propellant.
Emergency Response Guidebook (ERG)
Not combustible. Gives off irritating or toxic fumes (or gases) in a fire.
ILO-WHO International Chemical Safety Cards (ICSCs)
11.1.6 Hazards Summary
. . . it seems unlikely that a gas as inert as sulfur hexafluoride would exert any toxic effect in concentrations as low as 1000 ppm. The chief hazard, as with other inert gases, would seem to be asphyxiation as a result of the displacement of air by this heavy gas. [ACGIH] Possible frostbite from contact with liquid; [NIOSH]
ACGIH - Documentation of the TLVs and BEIs, 7th Ed. Cincinnati: ACGIH Worldwide, 2020.
Haz-Map, Information on Hazardous Chemicals and Occupational Diseases
The major hazards encountered in the use and handling of sulfur hexafluoride stem from its toxicologic properties. Toxic by all routes (inhalation, ingestion, and dermal contact), exposure to this colorless, odorless gas may occur from its production, storage, transportation, or use as a gaseous insulating medium for a wide range of high voltage electrical and electronic equipment. Effects from exposure may include dizziness, headache, frostbite, and asphyxiation. OSHA has established an 8-hour time weighted average (TWA) exposure limit of 1000 ppm. In activities and situations where over-exposure may occur, wear a self-contained breathing apparatus and protective clothing. If contact should occur, irrigate eyes with copious amounts of tepid water for at least 15 minutes, and wash exposed skin thoroughly with soap and water. Contaminated clothing should be removed and left at the worksite for cleaning. While sulfur hexafluoride does not ignite easily, it may burn with the production of poisonous gases. Also, cylinders may explode in the heat of a fire. For fires involving sulfur hexafluoride, extinguish with dry chemical, CO2, Halon, standard foam, or water spray or fog. If water is used, apply from as far a distance as possible, and stay away from ends of tanks. Sulfur hexafluoride may be shipped via air, rail, road, and water, in containers bearing the label, "Nonflammable gas." Before implementing land disposal of sulfur hexafluoride waste, consult with environmental regulatory agencies for guidance.
Hazardous Substances Data Bank (HSDB)
11.1.7 Fire Potential
Noncombustible
Lewis, R.J. Sr.; Hawley's Condensed Chemical Dictionary 15th Edition. John Wiley & Sons, Inc. New York, NY 2007., p. 1192
Hazardous Substances Data Bank (HSDB)
11.2 Safety and Hazard Properties
11.2.1 Flammable Limits
Flammability
Nonflammable Gas
The National Institute for Occupational Safety and Health (NIOSH)
11.2.2 Critical Temperature & Pressure
Critical temperature: 318.723 deg K; Critical pressure: 3.77 MPa
Haynes, W.M. (ed.). CRC Handbook of Chemistry and Physics. 95th Edition. CRC Press LLC, Boca Raton: FL 2014-2015, p. 6-85
Hazardous Substances Data Bank (HSDB)
11.2.3 Physical Dangers
The gas is heavier than air and may accumulate in lowered spaces causing a deficiency of oxygen.
ILO-WHO International Chemical Safety Cards (ICSCs)
11.2.4 OSHA Standards
Permissible Exposure Limit: Table Z-1 8-hr Time Weighted Avg: 1000 ppm (6000 mg/cu m).
29 CFR 1910.1000 (USDOL); U.S. National Archives and Records Administration's Electronic Code of Federal Regulations. Available from, as of August 26, 2015:
Hazardous Substances Data Bank (HSDB)
11.2.5 NIOSH Recommendations
Recommended Exposure Limit: 10 Hour Time-Weighted Average: 1000 ppm (6000 mg/cu m).
NIOSH. NIOSH Pocket Guide to Chemical Hazards. Department of Health & Human Services, Centers for Disease Control & Prevention. National Institute for Occupational Safety & Health. DHHS (NIOSH) Publication No. 2010-168 (2010). Available from:
Hazardous Substances Data Bank (HSDB)
11.3 First Aid Measures
Inhalation First Aid
Fresh air, rest. Artificial respiration may be needed. Refer for medical attention.
ILO-WHO International Chemical Safety Cards (ICSCs)
Skin First Aid
ON FROSTBITE: rinse with plenty of water, do NOT remove clothes. Refer for medical attention .
ILO-WHO International Chemical Safety Cards (ICSCs)
Eye First Aid
First rinse with plenty of water for several minutes (remove contact lenses if easily possible), then refer for medical attention.
ILO-WHO International Chemical Safety Cards (ICSCs)
11.3.1 First Aid
Excerpt from NIOSH Pocket Guide for Sulfur hexafluoride:
Eye: FROSTBITE - If eye tissue is frozen, seek medical attention immediately; if tissue is not frozen, immediately and thoroughly flush the eyes with large amounts of water for at least 15 minutes, occasionally lifting the lower and upper eyelids. If irritation, pain, swelling, lacrimation, or photophobia persist, get medical attention as soon as possible.
Skin: FROSTBITE - If frostbite has occurred, seek medical attention immediately; do NOT rub the affected areas or flush them with water. In order to prevent further tissue damage, do NOT attempt to remove frozen clothing from frostbitten areas. If frostbite has NOT occurred, immediately and thoroughly wash contaminated skin with soap and water.
Breathing: RESPIRATORY SUPPORT - If a person breathes large amounts of this chemical, move the exposed person to fresh air at once. If breathing has stopped, perform artificial respiration. Keep the affected person warm and at rest. Get medical attention as soon as possible. (NIOSH, 2024)
CAMEO Chemicals
ERG 2024, Guide 126 (Sulfur hexafluoride)
General First Aid:
· Call 911 or emergency medical service.
· Ensure that medical personnel are aware of the material(s) involved, take precautions to protect themselves and avoid contamination.
· Move victim to fresh air if it can be done safely.
· Administer oxygen if breathing is difficult.
· If victim is not breathing:
-- DO NOT perform mouth-to-mouth resuscitation; the victim may have ingestedor inhaled the substance.
-- If equipped and pulse detected, wash face and mouth, then give artificial respiration using a proper respiratory medical device (bag-valve mask, pocket mask equipped with a one-way valve or other device).
-- If no pulse detected or no respiratory medical device available, provide continuouscompressions. Conduct a pulse check every two minutes or monitor for any signs of spontaneous respirations.
· Remove and isolate contaminated clothing and shoes.
· For minor skin contact, avoid spreading material on unaffected skin.
· In case of contact with substance, remove immediately by flushing skin or eyes with running water for at least 20 minutes.
· For severe burns, immediate medical attention is required.
· Effects of exposure (inhalation, ingestion, or skin contact) to substance may be delayed.
· Keep victim calm and warm.
· Keep victim under observation.
· For further assistance, contact your local Poison Control Center.
· Note: Basic Life Support (BLS) and Advanced Life Support (ALS) should be done by trained professionals.
Specific First Aid:
· In case of contact with liquefied gas, only medical personnel should attempt thawing frosted parts.
Emergency Response Guidebook (ERG)
(General first aid procedures)
Eye: Frostbite - If eye tissue is frozen, seek medical attention immediately; if tissue is not frozen, immediately and thoroughly flush the eyes with large amounts of water for at least 15 minutes, occasionally lifting the lower and upper eyelids. If irritation, pain, swelling, lacrimation, or photophobia persist, get medical attention as soon as possible.
Skin: Frostbite - Compressed gases may create low temperatures when they expand rapidly. Leaks and uses that allow rapid expansion may cause a frostbite hazard. Wear appropriate personal protective clothing to prevent the skin from becoming frozen.
Breathing: Respiratory support
The National Institute for Occupational Safety and Health (NIOSH)
11.4 Fire Fighting
Excerpt from ERG Guide 126 [Gases - Compressed or Liquefied (Including Refrigerant Gases)]:
Use extinguishing agent suitable for type of surrounding fire.
SMALL FIRE: Dry chemical or CO2.
LARGE FIRE: Water spray, fog or regular foam. If it can be done safely, move undamaged containers away from the area around the fire. Damaged cylinders should be handled only by specialists.
FIRE INVOLVING TANKS: Fight fire from maximum distance or use unmanned master stream devices or monitor nozzles. Cool containers with flooding quantities of water until well after fire is out. Do not direct water at source of leak or safety devices; icing may occur. Withdraw immediately in case of rising sound from venting safety devices or discoloration of tank. ALWAYS stay away from tanks in direct contact with flames. Some of these materials, if spilled, may evaporate leaving a flammable residue. (ERG, 2024)
2024 Emergency Response Guidebook,
CAMEO Chemicals
In case of fire in the surroundings, use appropriate extinguishing media. In case of fire: keep cylinder cool by spraying with water.
ILO-WHO International Chemical Safety Cards (ICSCs)
11.4.1 Fire Fighting Procedures
Suitable extinguishing media: Use water spray, alcohol-resistant foam, dry chemical or carbon dioxide.
Sigma-Aldrich; Safety Data Sheet for Sulfur hexafluoride. Product Number: 295701, Version 3.6 (Revision Date 06/08/2015). Available from, as of January 28, 2016:
Hazardous Substances Data Bank (HSDB)
Advice for firefighters: Wear self-contained breathing apparatus for firefighting if necessary.
Sigma-Aldrich; Safety Data Sheet for Sulfur hexafluoride. Product Number: 295701, Version 3.6 (Revision Date 06/08/2015). Available from, as of January 28, 2016:
Hazardous Substances Data Bank (HSDB)
Use water spray to cool unopened containers.
Sigma-Aldrich; Safety Data Sheet for Sulfur hexafluoride. Product Number: 295701, Version 3.6 (Revision Date 06/08/2015). Available from, as of January 28, 2016:
Hazardous Substances Data Bank (HSDB)
If material involved if fire: Extinguish fire using agent suitable for type of surrounding fire. (Material itself does not burn or burns with difficulty). Cool all affected containers with flooding quantities of water. Apply water from as far a distance as possible.
Association of American Railroads; Bureau of Explosives. Emergency Handling of Hazardous Materials in Surface Transportation. Association of American Railroads, Pueblo, CO. 2005, p. 850
Hazardous Substances Data Bank (HSDB)
11.4.2 Firefighting Hazards
Special hazards arising from the substance or mixture: Sulphur oxides, Hydrogen fluoride
Sigma-Aldrich; Safety Data Sheet for Sulfur hexafluoride. Product Number: 295701, Version 3.6 (Revision Date 06/08/2015). Available from, as of January 28, 2016:
Hazardous Substances Data Bank (HSDB)
11.5 Accidental Release Measures
Public Safety: ERG 2024, Guide 126 (Sulfur hexafluoride)
· CALL 911. Then call emergency response telephone number on shipping paper. If shipping paper not available or no answer, refer to appropriate telephone number listed on the inside back cover.
· Keep unauthorized personnel away.
· Stay upwind, uphill and/or upstream.
· Many gases are heavier than air and will spread along the ground and collect in low or confined areas (sewers, basements, tanks, etc.).
· Ventilate closed spaces before entering, but only if properly trained and equipped.
Emergency Response Guidebook (ERG)
Spill or Leak: ERG 2024, Guide 126 (Sulfur hexafluoride)
· Do not touch or walk through spilled material.
· Stop leak if you can do it without risk.
· Do not direct water at spill or source of leak.
· Use water spray to reduce vapors or divert vapor cloud drift. Avoid allowing water runoff to contact spilled material.
· If possible, turn leaking containers so that gas escapes rather than liquid.
· Prevent entry into waterways, sewers, basements or confined areas.
· Allow substance to evaporate.
· Ventilate the area.
Emergency Response Guidebook (ERG)
11.5.1 Isolation and Evacuation
Excerpt from ERG Guide 126 [Gases - Compressed or Liquefied (Including Refrigerant Gases)]:
IMMEDIATE PRECAUTIONARY MEASURE: Isolate spill or leak area for at least 100 meters (330 feet) in all directions.
LARGE SPILL: Consider initial downwind evacuation for at least 500 meters (1/3 mile).
FIRE: If tank, rail tank car or highway tank is involved in a fire, ISOLATE for 800 meters (1/2 mile) in all directions; also, consider initial evacuation for 800 meters (1/2 mile) in all directions. (ERG, 2024)
2024 Emergency Response Guidebook,
CAMEO Chemicals
Evacuation: ERG 2024, Guide 126 (Sulfur hexafluoride)
Immediate precautionary measure
· Isolate spill or leak area for at least 100 meters (330 feet) in all directions.
Large Spill
· Consider initial downwind evacuation for at least 500 meters (1/3 mile).
Fire
· If tank, rail tank car or highway tank is involved in a fire, ISOLATE for 800 meters (1/2 mile) in all directions; also, consider initial evacuation for 800 meters (1/2 mile) in all directions.
Emergency Response Guidebook (ERG)
11.5.2 Spillage Disposal
Evacuate danger area! Ventilation. NEVER direct water jet on liquid. Personal protection: chemical protection suit including self-contained breathing apparatus.
ILO-WHO International Chemical Safety Cards (ICSCs)
11.5.3 Cleanup Methods
ACCIDENTAL RELEASE MEASURES: Personal precautions, protective equipment and emergency procedures: Avoid breathing vapors, mist or gas. Ensure adequate ventilation. Evacuate personnel to safe areas; Environmental precautions: Do not let product enter drains; Methods and materials for containment and cleaning up: Clean up promptly by sweeping or vacuum.
Sigma-Aldrich; Safety Data Sheet for Sulfur hexafluoride. Product Number: 295701, Version 3.6 (Revision Date 06/08/2015). Available from, as of January 28, 2016:
Hazardous Substances Data Bank (HSDB)
11.5.4 Disposal Methods
SRP: Recycle any unused portion of the material for its approved use or return it to the manufacturer or supplier. Ultimate disposal of the chemical must consider: the material's impact on air quality; potential migration in air, soil or water; effects on animal, aquatic and plant life; and conformance with environmental and public health regulations. If it is possible or reasonable use an alternative chemical product with less inherent propensity for occupational harm/injury/toxicity or environmental contamination.
Hazardous Substances Data Bank (HSDB)
Product: Offer surplus and non-recyclable solutions to a licensed disposal company. Contact a licensed professional waste disposal service to dispose of this material; Contaminated packaging: Dispose of as unused product.
Sigma-Aldrich; Safety Data Sheet for Sulfur hexafluoride. Product Number: 295701, Version 3.6 (Revision Date 06/08/2015). Available from, as of September 3, 2015:
Hazardous Substances Data Bank (HSDB)
11.5.5 Preventive Measures
If material not involved in fire: Attempt to stop leak if without undue personnel hazard.
Association of American Railroads; Bureau of Explosives. Emergency Handling of Hazardous Materials in Surface Transportation. Association of American Railroads, Pueblo, CO. 2005, p. 850
Hazardous Substances Data Bank (HSDB)
Personnel protection: Avoid breathing vapors. Keep upwind. ... Do not handle broken packages unless wearing appropriate personal protective equipment.
Association of American Railroads; Bureau of Explosives. Emergency Handling of Hazardous Materials in Surface Transportation. Association of American Railroads, Pueblo, CO. 2005, p. 850
Hazardous Substances Data Bank (HSDB)
Appropriate engineering controls: Handle in accordance with good industrial hygiene and safety practice. Wash hands before breaks and at the end of workday.
Sigma-Aldrich; Safety Data Sheet for Sulfur hexafluoride. Product Number: 295701, Version 3.6 (Revision Date 06/08/2015). Available from, as of January 28, 2016:
Hazardous Substances Data Bank (HSDB)
Gloves must be inspected prior to use. Use proper glove removal technique (without touching glove's outer surface) to avoid skin contact with this product. Dispose of contaminated gloves after use in accordance with applicable laws and good laboratory practices. Wash and dry hands.
Sigma-Aldrich; Safety Data Sheet for Sulfur hexafluoride. Product Number: 295701, Version 3.6 (Revision Date 06/08/2015). Available from, as of January 28, 2016:
Hazardous Substances Data Bank (HSDB)
Handle in accordance with good industrial hygiene and safety practice. Wash hands before breaks and at the end of workday.
Sigma-Aldrich; Safety Data Sheet for Sulfur hexafluoride. Product Number: 295701, Version 3.6 (Revision Date 06/08/2015). Available from, as of January 28, 2016:
Hazardous Substances Data Bank (HSDB)
11.6 Handling and Storage
11.6.1 Nonfire Spill Response
Excerpt from ERG Guide 126 [Gases - Compressed or Liquefied (Including Refrigerant Gases)]:
Do not touch or walk through spilled material. Stop leak if you can do it without risk. Do not direct water at spill or source of leak. Use water spray to reduce vapors or divert vapor cloud drift. Avoid allowing water runoff to contact spilled material. If possible, turn leaking containers so that gas escapes rather than liquid. Prevent entry into waterways, sewers, basements or confined areas. Allow substance to evaporate. Ventilate the area. (ERG, 2024)
2024 Emergency Response Guidebook,
CAMEO Chemicals
11.6.2 Safe Storage
Fireproof if in building. Cool.
ILO-WHO International Chemical Safety Cards (ICSCs)
11.6.3 Storage Conditions
Keep container tightly closed in a dry and well-ventilated place. Contents under pressure.
Sigma-Aldrich; Safety Data Sheet for Sulfur hexafluoride. Product Number: 295701, Version 3.6 (Revision Date 06/08/2015). Available from, as of January 28, 2016:
Hazardous Substances Data Bank (HSDB)
11.7 Exposure Control and Personal Protection
Protective Clothing: ERG 2024, Guide 126 (Sulfur hexafluoride)
· Wear positive pressure self-contained breathing apparatus (SCBA).
· Wear chemical protective clothing that is specifically recommended by the manufacturer when there is NO RISK OF FIRE.
· Structural firefighters' protective clothing provides thermal protection but only limited chemical protection.
Emergency Response Guidebook (ERG)
Maximum Allowable Concentration (MAK)
5000.0 [ppm]
Haz-Map, Information on Hazardous Chemicals and Occupational Diseases
11.7.1 Recommended Exposure Limit (REL)
REL-TWA (Time Weighted Average)
1000 ppm (6000 mg/m³)
Occupational Safety and Health Administration (OSHA)
TWA 1000 ppm (6000 mg/m 3)
The National Institute for Occupational Safety and Health (NIOSH)
11.7.2 Permissible Exposure Limit (PEL)
1000.0 [ppm]
Haz-Map, Information on Hazardous Chemicals and Occupational Diseases
PEL-TWA (8-Hour Time Weighted Average)
1000 ppm (6000 mg/m³)
Occupational Safety and Health Administration (OSHA)
TWA 1000 ppm (6000 mg/m 3)
The National Institute for Occupational Safety and Health (NIOSH)
11.7.3 Immediately Dangerous to Life or Health (IDLH)
N.D.
See: IDLH INDEX
The National Institute for Occupational Safety and Health (NIOSH)
11.7.4 Threshold Limit Values (TLV)
1000.0 [ppm]
Haz-Map, Information on Hazardous Chemicals and Occupational Diseases
8 hr Time Weighted Avg (TWA): 1000 ppm.
American Conference of Governmental Industrial Hygienists TLVs and BEIs. Threshold Limit Values for Chemical Substances and Physical Agents and Biological Exposure Indices. Cincinnati, OH 2015, p. 54
Hazardous Substances Data Bank (HSDB)
Excursion Limit Recommendation: Excursions in worker exposure levels may exceed 3 times the TLV-TWA for no more than a total of 30 minutes during a work day, and under no circumstances should they exceed 5 times the TLV-TWA, provided that the TLV-TWA is not exceeded.
American Conference of Governmental Industrial Hygienists TLVs and BEIs. Threshold Limit Values for Chemical Substances and Physical Agents and Biological Exposure Indices. Cincinnati, OH 2015, p. 5
Hazardous Substances Data Bank (HSDB)
1000 ppm as TWA.
ILO-WHO International Chemical Safety Cards (ICSCs)
TLV-TWA (Time Weighted Average)
1000 ppm
Occupational Safety and Health Administration (OSHA)
11.7.5 Occupational Exposure Limits (OEL)
MAK (Maximale Arbeitsplatz Konzentration)
6100 mg/m
ILO-WHO International Chemical Safety Cards (ICSCs)
11.7.6 Emergency Response Planning Guidelines
Emergency Response: ERG 2024, Guide 126 (Sulfur hexafluoride)
· Use extinguishing agent suitable for type of surrounding fire.
Small Fire
· Dry chemical or CO2.
Large Fire
· Water spray, fog or regular foam.
· If it can be done safely, move undamaged containers away from the area around the fire.
· Damaged cylinders should be handled only by specialists.
Fire Involving Tanks
· Fight fire from maximum distance or use unmanned master stream devices or monitor nozzles.
· Cool containers with flooding quantities of water until well after fire is out.
· Do not direct water at source of leak or safety devices; icing may occur.
· Withdraw immediately in case of rising sound from venting safety devices or discoloration of tank.
· ALWAYS stay away from tanks in direct contact with flames.
· Some of these materials, if spilled, may evaporate leaving a flammable residue.
Emergency Response Guidebook (ERG)
11.7.7 Inhalation Risk
A harmful concentration of this gas in the air will be reached very quickly on loss of containment.
ILO-WHO International Chemical Safety Cards (ICSCs)
11.7.8 Effects of Short Term Exposure
Rapid evaporation of the liquid may cause frostbite.
ILO-WHO International Chemical Safety Cards (ICSCs)
11.7.9 Personal Protective Equipment (PPE)
Excerpt from NIOSH Pocket Guide for Sulfur hexafluoride:
Skin: FROSTBITE - Compressed gases may create low temperatures when they expand rapidly. Leaks and uses that allow rapid expansion may cause a frostbite hazard. Wear appropriate personal protective clothing to prevent the skin from becoming frozen.
Eyes: FROSTBITE - Wear appropriate eye protection to prevent eye contact with the liquid that could result in burns or tissue damage from frostbite.
Wash skin: No recommendation is made specifying the need for washing the substance from the skin (either immediately or at the end of the work shift).
Remove: No recommendation is made specifying the need for removing clothing that becomes wet or contaminated.
Change: No recommendation is made specifying the need for the worker to change clothing after the workshift.
Provide: FROSTBITE WASH - Quick drench facilities and/or eyewash fountains should be provided within the immediate work area for emergency use where there is any possibility of exposure to liquids that are extremely cold or rapidly evaporating. (NIOSH, 2024)
CAMEO Chemicals
Personnel protection: Wear appropriate chemical protective gloves and goggles.
Association of American Railroads; Bureau of Explosives. Emergency Handling of Hazardous Materials in Surface Transportation. Association of American Railroads, Pueblo, CO. 2005, p. 850
Hazardous Substances Data Bank (HSDB)
Compressed gases may create low temperatures when they expand rapidly. Leaks and uses that allow rapid expansion may cause a frostbite hazard. Wear appropriate personal protective clothing to prevent the skin from becoming frozen.
NIOSH. NIOSH Pocket Guide to Chemical Hazards. Department of Health & Human Services, Centers for Disease Control & Prevention. National Institute for Occupational Safety & Health. DHHS (NIOSH) Publication No. 2010-168 (2010). Available from:
Hazardous Substances Data Bank (HSDB)
Wear appropriate eye protection to prevent eye contact with the liquid that could result in burns or tissue damage from frostbite.
NIOSH. NIOSH Pocket Guide to Chemical Hazards. Department of Health & Human Services, Centers for Disease Control & Prevention. National Institute for Occupational Safety & Health. DHHS (NIOSH) Publication No. 2010-168 (2010). Available from:
Hazardous Substances Data Bank (HSDB)
Quick drench facilities and/or eyewash fountains should be provided within the immediate work area for emergency use where there is any possibility of exposure to liquids that are extremely cold or rapidly evaporating.
NIOSH. NIOSH Pocket Guide to Chemical Hazards. Department of Health & Human Services, Centers for Disease Control & Prevention. National Institute for Occupational Safety & Health. DHHS (NIOSH) Publication No. 2010-168 (2010). Available from:
Hazardous Substances Data Bank (HSDB)
For more Personal Protective Equipment (PPE) (Complete) data for SULFUR HEXAFLUORIDE (8 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
11.7.10 Respirator Recommendations
Important additional information about respirator selection
The National Institute for Occupational Safety and Health (NIOSH)
11.7.11 Inhalation Prevention
Use ventilation.
ILO-WHO International Chemical Safety Cards (ICSCs)
11.7.12 Skin Prevention
Cold-insulating gloves.
ILO-WHO International Chemical Safety Cards (ICSCs)
11.7.13 Eye Prevention
Wear safety goggles or face shield.
ILO-WHO International Chemical Safety Cards (ICSCs)
11.7.14 Ingestion Prevention
Do not eat, drink, or smoke during work.
ILO-WHO International Chemical Safety Cards (ICSCs)
11.8 Stability and Reactivity
11.8.1 Air and Water Reactions
No rapid reaction with air. No rapid reaction with water.
CAMEO Chemicals
11.8.2 Reactive Group
Not Chemically Reactive
CAMEO Chemicals
11.8.3 Reactivity Profile
This substance undergoes chemical reactions only under relatively severe circumstances. They are resistant to ignition, although they may become flammable at very high temperatures. They may be resistant to oxidation reduction, except in the most severe conditions. These materials may be nontoxic. They can asphyxiate. Contact of very cold liquefied gas with water may result in vigorous or violent boiling of the product and extremely rapid vaporization due to the large temperature differences involved. If the water is hot, there is the possibility that a liquid "superheat" explosion may occur. Pressures may build to dangerous levels if liquid gas contacts water in a closed container [Handling Chemicals Safely 1980].
CAMEO Chemicals
11.8.4 Hazardous Reactivities and Incompatibilities
Incompatible materials: Strong oxidizing agents
Sigma-Aldrich; Safety Data Sheet for Sulfur hexafluoride. Product Number: 295701, Version 3.6 (Revision Date 06/08/2015). Available from, as of January 28, 2016:
Hazardous Substances Data Bank (HSDB)
Disilane explodes violently in contact with sulfur hexafluoride.
National Fire Protection Association; Fire Protection Guide to Hazardous Materials. 14TH Edition, Quincy, MA 2010, p. 491-78
Hazardous Substances Data Bank (HSDB)
... Reacts with aluminum chloride at 200 °C to give sulfur chloride.
Cotton FA, Wilkinson G; Advanced Inorganic Chemistry 4th Ed p.521 (1980)
Hazardous Substances Data Bank (HSDB)
... /Reacts/ with sulfur trioxide at 250 °C to give sulfuryl fluoride.
Cotton FA, Wilkinson G; Advanced Inorganic Chemistry 4th Ed p.521 (1980)
Hazardous Substances Data Bank (HSDB)
... /Reacts/ with carbon or carbon disulfide at 500 deg and 400 atm.
Cotton FA, Wilkinson G; Advanced Inorganic Chemistry 4th Ed p.521 (1980)
Hazardous Substances Data Bank (HSDB)
11.9 Transport Information
11.9.1 DOT Emergency Guidelines
/GUIDE 126 GASES - COMPRESSED or LIQUEFIED (Including Refrigerant Gases)/ Fire or Explosion: Some may burn but none ignite readily. Containers may explode when heated. Ruptured cylinders may rocket.
U.S. Department of Transportation. 2012 Emergency Response Guidebook. Washington, D.C. 2012
Hazardous Substances Data Bank (HSDB)
/GUIDE 126 GASES - COMPRESSED or LIQUEFIED (Including Refrigerant Gases)/ Health: Vapors may cause dizziness or asphyxiation without warning. Vapors from liquefied gas are initially heavier than air and spread along ground. Contact with gas or liquefied gas may cause burns, severe injury and/or frostbite. Fire may produce irritating, corrosive and/or toxic gases.
U.S. Department of Transportation. 2012 Emergency Response Guidebook. Washington, D.C. 2012
Hazardous Substances Data Bank (HSDB)
/GUIDE 126 GASES - COMPRESSED or LIQUEFIED (Including Refrigerant Gases)/ Public Safety: CALL Emergency Response Telephone Number on Shipping Paper first. If Shipping Paper not available or no answer, refer to appropriate telephone number listed on the inside back cover. As an immediate precautionary measure, isolate spill or leak area for at least 100 meters (330 feet) in all directions. Keep unauthorized personnel away. Stay upwind. Many gases are heavier than air and will spread along ground and collect in low or confined areas (sewers, basements, tanks). Keep out of low areas. Ventilate closed spaces before entering.
U.S. Department of Transportation. 2012 Emergency Response Guidebook. Washington, D.C. 2012
Hazardous Substances Data Bank (HSDB)
/GUIDE 126 GASES - COMPRESSED or LIQUEFIED (Including Refrigerant Gases)/ Protective Clothing: Wear positive pressure self-contained breathing apparatus (SCBA). Wear chemical protective clothing that is specifically recommended by the manufacturer. It may provide little or no thermal protection. Structural firefighters' protective clothing will only provide limited protection.
U.S. Department of Transportation. 2012 Emergency Response Guidebook. Washington, D.C. 2012
Hazardous Substances Data Bank (HSDB)
For more DOT Emergency Guidelines (Complete) data for SULFUR HEXAFLUORIDE (8 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
11.9.2 DOT ID and Guide
1080 126
The National Institute for Occupational Safety and Health (NIOSH)
11.9.3 Shipping Name / Number DOT/UN/NA/IMO
UN 1080; Sulfur hexafluoride
Hazardous Substances Data Bank (HSDB)
IMO 2.2; Sulfur hexafluoride
Hazardous Substances Data Bank (HSDB)
11.9.4 Standard Transportation Number
49 045 75; Sulfur hexafluoride
Hazardous Substances Data Bank (HSDB)
11.9.5 Shipment Methods and Regulations
No person may /transport,/ offer or accept a hazardous material for transportation in commerce unless that person is registered in conformance ... and the hazardous material is properly classed, described, packaged, marked, labeled, and in condition for shipment as required or authorized by ... /the hazardous materials regulations (49 CFR 171-177)./
49 CFR 171.2 (USDOT); U.S. National Archives and Records Administration's Electronic Code of Federal Regulations. Available from, as of September 1, 2015:
Hazardous Substances Data Bank (HSDB)
The International Air Transport Association (IATA) Dangerous Goods Regulations are published by the IATA Dangerous Goods Board pursuant to IATA Resolutions 618 and 619 and constitute a manual of industry carrier regulations to be followed by all IATA Member airlines when transporting hazardous materials. Sulfur hexafluoride is included on the dangerous goods list.
International Air Transport Association. Dangerous Goods Regulations. 55th Edition. Montreal, Quebec Canada. 2014., p. 311
Hazardous Substances Data Bank (HSDB)
The International Maritime Dangerous Goods Code lays down basic principles for transporting hazardous chemicals. Detailed recommendations for individual substances and a number of recommendations for good practice are included in the classes dealing with such substances. A general index of technical names has also been compiled. This index should always be consulted when attempting to locate the appropriate procedures to be used when shipping any substance or article. Sulfur hexafluoride is included on the dangerous goods list.
International Maritime Organization. IMDG Code. International Maritime Dangerous Goods Code Volume 2 2012, p. 47
Hazardous Substances Data Bank (HSDB)
11.9.6 DOT Label
Non-Flammable Gas
CAMEO Chemicals
11.9.7 UN Classification
UN Hazard Class: 2.2
ILO-WHO International Chemical Safety Cards (ICSCs)
11.10 Regulatory Information
The Australian Inventory of Industrial Chemicals
Chemical: Sulfur fluoride (SF6)
Australian Industrial Chemicals Introduction Scheme (AICIS)
REACH Registered Substance
Status: Active Update: 11-04-2023
European Chemicals Agency (ECHA)
New Zealand EPA Inventory of Chemical Status
Sulphur hexafluoride: Non hazardous
New Zealand Environmental Protection Authority (EPA)
New Jersey Worker and Community Right to Know Act
The New Jersey Worker and Community Right to Know Act requires public and private employers to provide information about hazardous substances at their workplaces. (N.J.S.A. 34:5A-1 et. seq.)
NJDOH RTK Hazardous Substance List
11.11 Other Safety Information
Chemical Assessment
Evaluation - Chemicals that are unlikely to require further regulation to manage risks to human health
Australian Industrial Chemicals Introduction Scheme (AICIS)
11.11.1 History and Incidents
First manufactured in 1902, sulfur hexafluoride (SF6) is a fluorinated compound with an extremely stable molecular structure. This structure makes it far superior to oil or air in its insulation properties, dielectric strength, and arc quenching abilities. Electric equipment manufacturers and utilities therefore rely heavily on SF6 in transmission systems. In 2001, electric utilities and electrical equipment manufacturers, taken together, purchased over 80 percent of all SF6 produced worldwide. Under ideal conditions, SF6 would remain contained within transmission equipment. In reality, however, SF6 is inadvertently emitted into the atmosphere as leaks develop during various stages of the equipment's life cycle. SF6 can also be accidentally released during equipment installation, servicing, or de- commissioning. SF6 has been identified by the Intergovernmental Panel on Climate Change (IPCC) as a highly potent greenhouse gas that contributes to climate change. According to the IPCC, SF6 is 22,200 times more effective at trapping infrared radiation than an equivalent amount of carbon dioxide (CO2) over a 100-year period. Additionally, it has an atmospheric life of 3,200 years; its accumulation in the atmosphere is virtually irreversible. Measurements taken from 1978 through 1996 indicate that SF6 concentrations are increasing in the atmosphere at a rate of 7 percent per year. In 2002, SF6 emissions from the U.S. electric power industry totaled 589 metric tons. This paper discusses the SF6 Emissions Reduction Partnership for Electric Power Systems, a forum for the industry and EPA to share information on ways to reduce SF6 emissions. The efficiency of SF6 management can be improved through cost-effective operational improvements and equipment upgrades. These opportunities fall into four categories: SF6 inventory tracking systems, SF6 recycling, improved leak detection technology, and SF6 management training programs to improve operation and maintenance practices.
Blackman J and Kantamaneni R; Govt Reports Announcements & Index (GRA&I), Issue 19, 2005
Hazardous Substances Data Bank (HSDB)
12 Toxicity
12.1 Toxicological Information
12.1.1 Toxicity Summary
IDENTIFICATION AND USE: Sulfur hexafluoride (SF6) is a colorless gas. SF6 is used in several industrial applications. The insulation of electrical equipment, the magnesium casting processes and the semiconductor manufacturing are the main applications. Beside the industrial applications, SF6 is used in medical applications as an ultrasound contrast agent and as a tamponade gas in ophthalmology. It has been identified as one of the greehouse gases. HUMAN EXPOSURE AND TOXICITY: SF6 is a simple asphyxiant. its chemical inertness and its very low accumulation potential support the low concern for the toxicity of this substance. The possible formation of highly toxic breakdown products may occur when SF6 is subjected to high stress conditions; in particular electrical discharges occurring in the gas-insulated equipment may promote the formation of highly reactive species of toxicological concern. The degeneration products can be toxic, causing nausea and vomiting, pulmonary symptoms, and transient atelectasis. It may be contaminated with other fluorides of sulfur, such as sulfur pentafluoride and disulfur decafluoride, which are extremely toxic and are respiratory irritants. Inhaled as a mixture of 80% with 20% oxygen, it produces tingling, excitement, and altered hearing and is a mild anesthetic. Impairments of psychomotor, perceptual, and cognitive abilities were determined in nine male subjects exposed to inhaled SF6. In the clinical trials of SF6- based contrast agent Lumason, serious adverse reactions were observed in 2 subjects; one who experienced a hypersensitivity-type rash and near syncope symptoms and another who experienced anaphylactic shock shortly following Lumason administration. Other adverse events for European SF6-based contrast agent SonoVue included severe hypotension, bradycardia, cardiac arrest and acute myocardial infarction. Three fatalities occured in patients with pre-existing, severe coronary artery disease. No evidence of genotoxicity was found in the in vitro human lymphocyte chromosome aberration assay. European contrast agent Sonovue was not clastogenic in human lymphocytes in vitro. ANIMAL STUDIES: Commercial grade sulfur hexafluoride (SF6) showed no cytotoxicity in hamster lung cells (V-79) even after an exposure of 24-48 hr to a 95% SF6-5% air mixture. Sparked sulfur hexafluoride exhibited a strong cytotoxic effect and 1 hr of exposure killed 74% cells. The cytotoxicity increased with time of exposure and varied with the spark energy. In a 4 week study in rats cecitis, colitis, cecum erosion and ulcers were observed. The effects were reversible within 14 days after treatment cessation. Colon erosion was observed in one animal. These toxic effects were not observed in a confirmatory study in rats and in a similar study in cynomolgous monkeys. No impairment of fertility was observed in rats receiving SF6-based contrast agent Lumason at doses up to 8 times the human dose based on body surface area. No evidence of genotoxicity was found in the following studies conducted with Lumason: a bacterial mutagenesis (Ames) assay, and an in vivo mouse micronucleus assay. Similarly, Sonovue was neither mutagenic in Salmonella typhimurium nor in the mouse bone marrow micronucleus assay. ECOTOXICITY STUDIES: SF6 has been identified as a potent greenhouse gas.
Hazardous Substances Data Bank (HSDB)
12.1.2 Effects During Pregnancy and Lactation
◉ Summary of Use during Lactation
No published experience exists with sulfur hexafluoride during breastfeeding. The half-life of the agent is about 10 minutes and absorption of the drug by the infant is unlikely. If sulfur hexafluoride is required by the mother, it is not a reason to discontinue breastfeeding. Because of the lack of information, the American College of Radiology states that temporary (~24 hours) pumping and discarding of milk may be considered.
◉ Effects in Breastfed Infants
Relevant published information was not found as of the revision date.
◉ Effects on Lactation and Breastmilk
Relevant published information was not found as of the revision date.
Drugs and Lactation Database (LactMed)
12.1.3 Exposure Routes
The substance can be absorbed into the body by inhalation.
ILO-WHO International Chemical Safety Cards (ICSCs)
inhalation
The National Institute for Occupational Safety and Health (NIOSH)
12.1.4 Signs and Symptoms
Inhalation Exposure
Suffocation.
ILO-WHO International Chemical Safety Cards (ICSCs)
Skin Exposure
ON CONTACT WITH LIQUID: FROSTBITE.
ILO-WHO International Chemical Safety Cards (ICSCs)
Eye Exposure
See Skin.
ILO-WHO International Chemical Safety Cards (ICSCs)
asphyxia: increased breath rate, pulse rate; slight muscle incoordination, emotional upset; lassitude (weakness, exhaustion), nausea, vomiting, convulsions; liquid: frostbite
The National Institute for Occupational Safety and Health (NIOSH)
12.1.5 Target Organs
respiratory system
The National Institute for Occupational Safety and Health (NIOSH)
12.1.6 Adverse Effects
Other Poison - Simple Asphyxiant
Haz-Map, Information on Hazardous Chemicals and Occupational Diseases
12.1.7 Acute Effects
ChemIDplus
12.1.8 Interactions
In anesthetized cats ventilated with oxygen, 0.5 mL of the inert gas sulfur hexafluoride was substituted for vitreous. When the ventilating gas was changed to nitrous oxide 66%, balance oxygen, intraocular pressure incr from 14.4 to 30.3 mm Hg in 19.5 min. When the ventilating gas was changed back to oxygen, intraocular pressure decr from 29.1 to 12.0 mm Hg in 18.1 min. Nitrous oxide should be avoided in patients during and following intravitreal injection of sodium hexafluoride for up to 10 days.
PMID:6650912
Wolf GL et al; Anesthesiology 59 (6): 547-8 (1983)
Hazardous Substances Data Bank (HSDB)
To determine retinal toxicity after intravitreous balanced salt solution, sulfur hexafluoride gas, and perfluorocarbon liquid injection in rabbit eyes ... 22 eyes of 16 New Zealand albino rabbits were divided into groups: balanced salt solution (7 eyes); sulfur hexafluoride gas (4 eyes); and perfluorocarbon liquid (5 eyes). After the introduction of a needle through the sclera the vitreous was aspirated (0.3 mL), balanced salt solution, sulfur hexafluoride gas 100% and perfluorocarbon liquid (0.3 mL) were injected into rabbit vitreous cavity. The control group (6 eyes) was not submitted to any procedure. After three weeks the vitreous was aspirated and submitted to biochemical analysis and the eyes prepared for histological analysis. ... The eyes submitted to perfluorocarbon liquid and sulfur hexafluoride gas injection showed a greater L-glutamate increase in the vitreous compared to balanced salt solution and control groups (p<0.05). Histological results confirmed small changes in the sulfur hexafluoride group and important lesions in perfluorocarbon liquid group, such as external photoreceptor segment disruption, external and internal plexiform layer thinning, ganglionar and internal nuclear layer decrease of nucleus number, edema, and presence of macrophages in the superficial layers. No major histological changes were observed with balanced salt solution and in control groups. /The authors concluded that/ intravitreous injection of sulfur hexafluoride gas and perfluorocarbon are potentially toxic to rabbit retina, compared to control and balanced salt solution groups.
PMID:16322837
Souza EV et al; Arq Bras Oftalmol 68 (4): 511-5 (2005)
Hazardous Substances Data Bank (HSDB)
... Gross and histopathologic examinations of rats exposed for 6 hrs to a recirculated mixture of sulfur-hexafluoride in oxygen showed no evidence of effects attributable to the exposures. When adult males inhaled a mixture of sulfur-hexafluoride and oxygen, some effects were impairment of speech, sense of pervasion of the upper respiratory tract, and slight to marked vertigo. The possibility of the presence of small amounts of other fluorine compounds which are highly toxic caused suspension of studies on humans.
PMID:14913132
Specht H, Brubach HF; Science 114: 662-3 (1951)
Hazardous Substances Data Bank (HSDB)
... Modern vitreoretinal techniques often use /long acting inert gases, such as sulfur hexafluoride or perfluoropropane / as intraocular ... tamponading agents. These gases may persist in the eye for up to three months after surgery. During this period further anesthesia using nitrous oxide will cause the intraocular gas bubble to expand, which can result in sight threatening increases in intraocular pressure.
PMID:12217995
Full text:
Yang YF et al; BMJ 325 (7363): 532-533 (2002)
Hazardous Substances Data Bank (HSDB)
For more Interactions (Complete) data for SULFUR HEXAFLUORIDE (11 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
12.1.9 Antidote and Emergency Treatment
Immediate first aid: Ensure that adequate decontamination has been carried out. If patient is not breathing, start artificial respiration, preferably with a demand-valve resuscitator, bag-valve-mask device, or pocket mask, as trained. Perform CPR as necessary. Immediately flush contaminated eyes with gently flowing water. Do not induce vomiting. If vomiting occurs, lean patient forward or place on left side (head-down position, if possible) to maintain an open airway and prevent aspiration. Keep patient quiet and maintain normal body temperature. Obtain medical attention. /Simple asphyxiants and related compounds/
Currance, P.L. Clements, B., Bronstein, A.C. (Eds).; Emergency Care For Hazardous Materials Exposure. 3rd revised edition, Elsevier Mosby, St. Louis, MO 2007, p. 449
Hazardous Substances Data Bank (HSDB)
Basic treatment: Establish a patent airway (oropharyngeal or nasopharyngeal airway, if needed). Suction if necessary. Watch for signs of respiratory insufficiency and assist ventilations if necessary. Administer oxygen by nonrebreather mask at 10 to 15 L/min. Anticipate seizures and treat if necessary ... . Use rapid rewarming techniques if frostbite occurs ... . /Simple asphyxiants and related compounds/
Currance, P.L. Clements, B., Bronstein, A.C. (Eds).; Emergency Care For Hazardous Materials Exposure. 3rd revised edition, Elsevier Mosby, St. Louis, MO 2007, p. 439
Hazardous Substances Data Bank (HSDB)
Advanced treatment: Consider orotracheal or nasotracheal intubation for airway control in the patient who is unconscious or is in severe respiratory distress. Monitor cardiac rhythm and treat arrhythmias if necessary ... . Start IV administration of D5W /SRP: "To keep open", minimal flow rate/. Treat seizures with diazepam or lorazepam ... . /Simple asphyxiants and related compounds/
Currance, P.L. Clements, B., Bronstein, A.C. (Eds).; Emergency Care For Hazardous Materials Exposure. 3rd revised edition, Elsevier Mosby, St. Louis, MO 2007, p. 439-40
Hazardous Substances Data Bank (HSDB)
Immediate first aid: Ensure that adequate decontamination has been carried out. If patient is not breathing, start artificial respiration, preferably with a demand-valve resuscitator, bag-valve-mask device, or pocket mask, as trained. Perform CPR as necessary. Immediately flush contaminated eyes with gently flowing water. Do not induce vomiting. If vomiting occurs, lean patient forward or place on left side (head-down position, if possible) to maintain an open airway and prevent aspiration. Keep patient quiet and maintain normal body temperature. Obtain medical attention. /Choking Agents (Pulmonary/Lung-Damaging Agents)/
Currance, P.L. Clements, B., Bronstein, A.C. (Eds).; Emergency Care For Hazardous Materials Exposure. 3rd revised edition, Elsevier Mosby, St. Louis, MO 2007, p. 590
Hazardous Substances Data Bank (HSDB)
For more Antidote and Emergency Treatment (Complete) data for SULFUR HEXAFLUORIDE (6 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
12.1.10 Human Toxicity Excerpts
/HUMAN EXPOSURE STUDIES/ As a noninvasive treatment technique, ultrasound-guided high-intensity focused ultrasound (HIFU) has been considered as a routine treatment for uterine fibroids and adenomyosis in China. Contrast-enhanced ultrasound (CEUS) has been proposed as another option to assess the treatment efficacy during HIFU treatment. The aim of this investigation is to evaluate the adverse effects of HIFU ablation for benign uterine diseases in a group of patients studied with ultrasound contrast agent (UCA), in comparison with a group of patients not exposed to UCA. From November 2010 to December 2013, 2604 patients with benign uterine diseases were treated with HIFU. Among them, 1300 patients were exposed to an UCA, whereas 1304 patients were not.During HIFU procedure, the incidences of leg pain, sacral/buttock pain, groin pain, treatment area pain, and the discomfort "hot" sensation on skin were higher in the patients who were exposed to SonoVue (Bracco, Milan, Italy) than those who were not (20.5% vs 11.7%, 52.5% vs 42.3%, 6.5% vs 4.5%, 68.9% vs 55.4%, and 48.1% vs 42.9%, respectively). Among the postoperative adverse effects, the incidence of lower abdominal pain was significantly higher in patients who were exposed to an UCA than those who were not (51.2% vs 39.9%, P<0.05). Two patients who were exposed to an UCA had acute renal function failure. In conclusion, UCA may increase the incidences of some common HIFU-related adverse effects during HIFU treatment for benign uterine diseases, but most of which were acceptable and self-limited. After HIFU treatment, renal function should be monitored in patients with a history of hypertension or taking nonsteroidal anti-inflammatory drugs.
PMID:25906100
Full text:
Cheng CQ et al; Medicine (Baltimore) 94 (16): e729 (2015)
Hazardous Substances Data Bank (HSDB)
/HUMAN EXPOSURE STUDIES/ To evaluate the safety of intravesical administration of a second-generation ultrasound (US) contrast agent for the diagnosis of vesicoureteral reflux in children. One thousand and ten children (563 girls, 447 boys; mean age: 2.9 years, range: 15 days-17.6 years) with 2,043 pelvi-ureter-units underwent contrast-enhanced voiding urosonography (ce-VUS) to rule out vesicoureteral reflux. A second-generation US contrast agent (SonoVue, Bracco, Milan) was administered intravesically through transurethral bladder catheterization at a dose of 0.5 mL/bladder filling. Possible adverse events were monitored during the examination and followed up for 7 days after ce-VUS by phone calls. Urine analysis and culture were performed 3-5 days before ce-VUS in all children and 24-48 hr in any patient who reported with adverse events. No case of serious adverse event was recorded. Minor events were reported in 37 children (3.66% of the study population). These included dysuria (n=26, 2.57%), urinary retention (n=2, 0.2%), abdominal pain (n=2, 0.2%), anxiety (n=1, 0.1%) and crying (n=1, 0.1%) during micturition, blood and mucous discharge (n=1, 0.1%), increased frequency of micturition (n=1, 0.1%), vomiting (n=1, 0.1%), perineal irritation (n=1, 0.1%), and an episode of urinary tract infection 10 days after ce-VUS (n=1, 0.1%). Of these adverse events, 91.9% were subacute in onset and 8.1% were delayed. All events were self-limited and none required hospitalization. There were no serious adverse events. Only a few minor events were reported, most likely due to the catheterization process. Thus, ce-VUS with intravesical administration of the second-generation US contrast agent (SonoVue) for vesicoureteral reflux detection or exclusion had a favorable safety profile in our study group.
PMID:24442338
Papadopoulou F et al; Pediatr Radiol. 44 (6): 719-28 (2014)
Hazardous Substances Data Bank (HSDB)
/HUMAN EXPOSURE STUDIES/ The aim of the present retrospective study was to assess the incidence of adverse events (AE) of a second-generation ultrasound contrast agent in real clinical practice. A total of 28 Italian Centres provided data on the postmarketing use of SonoVue (Bracco Spa, Milan, Italy) in abdominal examination performed between December 2001 and December 2004. A total of 23 188 investigations were reported. No fatal event occurred. AEs were reported in 29 cases, of which only two were graded as serious; the rest, 27, were nonserious (23 mild, three moderate and one severe). The overall reporting rate of serious AE was 0.0086%. Overall, only four AEs required treatment (two serious, two nonserious including one moderate and one severe AEs). In conclusion, the present large-scale retrospective analysis showed that SonoVue has a good safety profile in abdominal applications, with an AE reporting rate lower than or similar to that reported for radiologic and magnetic resonance contrast agents.
PMID:16965977
Piscaglia F et al; Ultrasound Med Biol. 32 (9): 1369-75 (2006)
Hazardous Substances Data Bank (HSDB)
/HUMAN EXPOSURE STUDIES/ Safety concerns regarding the use of echo-contrast agents during baseline and SE in patients with recent chest pain have been raised. The purpose of the present study was to provide evidence regarding the safety of flash-replenishment contrast dipyridamole-atropine echocardiography (DASE) in such patients. Five hundred consecutive individuals who presented to the Emergency Department with chest pain, normal electrocardiograms (ECG) and troponin I were selected based on a less than 5 days interval between chest pain episode and performance of contrast flash-replenishment DASE. Analysis of myocardial perfusion with SonoVue infusion after dipyridamole was routinely added on top of standard wall motion assessment during DASE. Adverse events (AEs) were reported according to standardized terminology and then compared with a historical control group in which contrast was not used. No deaths, myocardial infarctions, sustained arrhythmias, or any other life-threatening events were observed. Adverse events were not significantly different between the study group and the control group. In the selected subgroup of patients (n = 149) who underwent coronary angiography, accuracy of DASE with additional perfusion assessment was higher (88%, 95% C.I. 83-93%) than without (72%, 95% C.I. 65-79%). DASE with SonoVue infusion for myocardial perfusion assessment was exceptionally safe even when routinely performed within the first 5 days following a chest pain episode of undetermined origin in subjects without ECG and troponin abnormalities.
PMID:19502619
Gaibazzi N ET AL; Eur J Echocardiogr. 10 (6): 726-32 (2009)
Hazardous Substances Data Bank (HSDB)
For more Human Toxicity Excerpts (Complete) data for SULFUR HEXAFLUORIDE (26 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
12.1.11 Non-Human Toxicity Excerpts
/LABORATORY ANIMALS: Acute Exposure/ Sonovue administered at a dosage up to 20 mL/kg (500 fold more than the expected dose in humans) in rats and monkeys using the IV route did not induce any adverse effects or animal deaths. The acute toxicity potential of Sonovue is probably very low.
European Medicines Agency (EMEA), The European Agency for the Evaluation of Medicinal Products, Scientific Discussion for the approval of Sonovue (sulfur hexafluoride), (2004) p.5. Available from, as of April 2, 2007:
Hazardous Substances Data Bank (HSDB)
/LABORATORY ANIMALS: Acute Exposure/ In vivo assessment of corneal endothelial toxicity of air and SF6 /was conducted/ in the feline model. This research was motivated by the increased use of air in anterior segment surgery in human subjects. This was a prospective masked study. The eyes of 16 healthy adult cats were randomly assigned for the injection of 0.7 mL air into the anterior chamber of one eye and SF6 in the contralateral eye. Daily examination included slit lamp photographs, pachymetry, and tonometry. Specular microscopy was performed before, 7 days after, and 10 days after injection. The animals were euthanatized, and the corneas were processed for alizarin red-trypan blue staining and for light and electron microscopy. SF6 remained in the anterior chamber significantly longer than air. Both groups showed postinjection inflammation, which on average was maximal at day 2 and more severe with SF6. No difference in intraocular pressure (IOP) was observed between the two groups. Specular microscopy showed significant endothelial cell loss in the SF6 group (mean postinjection cell loss, 132 +/- 50 cells/sq mm) but not in the group injected with air. Alizarin red staining revealed significant regional differences in cell density only in the SF6 group and more pronounced endothelial cell loss in the superior area. These results indicate that both air and SF6 injected into the anterior chamber of the eye can induce intraocular reaction in the feline model and that SF6 is more toxic than air in terms of endothelial cell loss and anterior chamber inflammation.
PMID:21178142
Landry H et al; Invest Ophthalmol Vis Sci 52 (5): 2279-86 (2011)
Hazardous Substances Data Bank (HSDB)
/LABORATORY ANIMALS: Subchronic or Prechronic Exposure/ In a 4 week study in SD rats (0; 0.2; 1; 5 mL/kg) cecitis, colitis, cecum erosion and ulcers were observed from 0.2 mL/kg (dose-related in females but not in males). The effects were reversible within 14 days after treatment cessation. Colon erosion was observed in one animal at the 5 mL/kg dose. These toxic effects were not observed in a confirmatory study in rats and in a similar study in cynomolgous monkeys. Nor was caecum inflammation observed in the fertility and general reproductive performance studies conducted in male and female rats. The mechanism of the toxicity to the gastrointestinal system observed in the one rat study was not elucidated. However, it must be noticed that it was observed only in one study and one species. In summary, Sonovue up to 5 mL/kg administered for 28 days by the IV route to rats and monkeys showed a low toxicity potential in rats and monkeys.
European Medicines Agency (EMEA), The European Agency for the Evaluation of Medicinal Products, Scientific Discussion for the approval of Sonovue (sulfur hexafluoride), (2004) p.5. Available from, as of April 2, 2007:
Hazardous Substances Data Bank (HSDB)
/LABORATORY ANIMALS: Developmental or Reproductive Toxicity/ On day 16 of the chick embryo, a catheter was implanted in the allantoic vein carrying arterialized blood, and a syringe was attached to the blunt end of the shell connecting to the air cell. This technique allowed for repetitive sampling and analysis of air cell gas and arterialized blood when these eggs were exposed to a He-O2 or SF6-O2 atmosphere. Exposure to SF6-O2 incr arterial carbon dioxide tension from 37 to 62 Torr and reduced the pH by 0.14 units. These responses were brought about by changes in the gas conductance of the shell, resulting in a diffusive hypercapnia and respiratory acidosis in SF6-O2. During a 4 hr exposure to these foreign gases the observed pH changes were smaller than predicted because of marked shifts of HCO3- into the blood (SF6-O2) or out of the blood (He-O2).
PMID:6790488
Tazawa H et al; J Appl Physiol 50 (4): 819-23 (1981)
Hazardous Substances Data Bank (HSDB)
For more Non-Human Toxicity Excerpts (Complete) data for SULFUR HEXAFLUORIDE (15 total), please visit the HSDB record page.
Hazardous Substances Data Bank (HSDB)
12.1.12 Populations at Special Risk
The use and the safety of radiographic, MR- or ultrasound contrast media in the diagnostic work-up of pregnant or lactating patients is a frequently discussed question. As only sparse clinical data is available, a careful benefit-risk assessment must contain physico-chemical properties, preclinical data including teratogenicity and embryotoxicity, as well as maternal and fetal exposure. With consideration to the individual risks, iodinated contrast media, macrocyclic MR contrast media with increased stability or sulphur hexafluoride ultrasound contrast media may, if clinically justified, be administered in the smallest possible doses throughout pregnancy. After parental administration of an iodinated contrast medium after the 12th week of pregnancy, the neonate's thyroidal function should be checked during the first week after birth. After parental administration of iodinated, stable macrocyclic, gadolinium or ultrasound contrast media, lactation can be continued normally. In any case, contrast media should be used with caution and only if the benefits outweigh the risk.
PMID:23108903
Frohlich JM, Kubik-Huch RA.; Rofo. 185(1):13-25 (2013)
Hazardous Substances Data Bank (HSDB)
Lumason is contraindicated in patients with: known or suspected right-to-left, bi-directional, or transient right-to-left cardiac shunts history of hypersensitivity reactions to sulfur hexafluoride lipid microsphere components or to any of the inactive ingredients in Lumason Do not administer by intra-arterial injection.
NIH; DailyMed. Current Medication Information for LUMASON- sulfur hexafluoride (Revised: October 2014). Available from, as of August 2, 2015:
Hazardous Substances Data Bank (HSDB)
SonoVue should not be administered to patients with known hypersensitivity to sulfur hexafluoride or to any of the components of Sonovue.
European Medicines Agency (EMEA), The European Agency for the Evaluation of Medicinal Products, Public Statement on Sonovue (sulfur hexafluoride): New contraindications in patients with heart disease. Restriction of use to non-cardiac imaging (2006) Available from, as of April 2, 2007:
Hazardous Substances Data Bank (HSDB)
12.2 Ecological Information
12.2.1 Ecotoxicity Excerpts
/OTHER TOXICITY INFORMATION/ Many chemical compounds present in Earth's atmosphere behave as greenhouse gases. These are gases which allow direct sunlight (relative shortwave energy) to reach the Earth's surface unimpeded. As the shortwave energy (that in the visible and ultraviolet portion of the spectra) heats the surface, longer-wave (infrared) energy (heat) is reradiated to the atmosphere. Greenhouse gases absorb this energy, thereby allowing less heat to escape back to space, and trapping it in the lower atmosphere. Many greenhouse gases occur naturally in the atmosphere, such as carbon dioxide, methane, water vapor, and nitrous oxide, while others are synthetic. Those that are man-made include the chlorofluorocarbons (CFCs), hydrofluorocarbons (HFCs) and Perfluorocarbons (PFCs), as well as sulfur hexafluoride (SF6). Atmospheric concentrations of both the natural and man-made gases have been rising over the last few centuries due to the industrial revolution.
NOAA; Greenhouse gases; Available from, as of September 1, 2015:
Hazardous Substances Data Bank (HSDB)
12.2.2 Environmental Fate / Exposure Summary
Sulfur hexafluoride's production and use as a gaseous insulator for high voltage generators, a component in electrical equipment and limited use in special applications ranging from medical applications to space research may result in its release to the environment through various waste streams. Its former use as a tracer gas for both indoor and outdoor source dissemination experiments as well as for measuring gas exchange coefficients in lakes resulted in its direct release to the environment. Sulfur hexafluoride may occur in fluorite and granite. If released into the atmosphere, sulfur hexafluoride will exist in the gas phase in the ambient atmosphere, based on a vapor pressure of 1.78X10+4 mm Hg at 25 °C. Sulfur hexafluoride is one of the heaviest known gases with a vapor density approximately five times greater than air. Therefore, when released to the atmosphere, it will tend to remain close to the ground and be transported to soil by wet deposition. Long-term observations of sulfur hexafluoride in the atmosphere revealed that the concentration of sulfur hexafluoride increased by 2 orders of magnitude between 1970 and 1992. Perfluoro compounds have very long atmospheric lifetimes and thus relatively high Global Warming Potentials (GWP). GWP of sulfur hexafluoride is 25,000 (GWP of CO2 is 1). The main process of removal of sulfur hexafluoride from the troposphere has been reported to be its advection into the mesosphere, where the compound may be subjected to degradation via the reaction with free electrons; a global atmospheric lifetime of 3200 years was calculated for sulfur hexafluoride. Using a two-dimensional transport and chemistry model, the predicted atmospheric lifetime after photolysis for sulfur hexafluoride was determined to be greater than or equal to 600 years. If released to soil, sulfur hexafluoride is expected to have very high mobility based upon an estimated Koc of 29. Experimental data reported no absorption of sulfur hexalfuoride in soil. Volatilization from moist soil surfaces is expected to be an important fate process based upon an experimental Henry's Law constant of 4.52 atm-cu m/mole. Sulfur hexafluoride may volatilize from dry soil surfaces based upon its vapor pressure. Biodegradation data were not available. If released into water, sulfur hexafluoride is not expected to adsorb to suspended solids and sediment based upon the estimated Koc. Volatilization from water surfaces is expected to be an important fate process based upon this compound's Henry's Law constant. Estimated volatilization half-lives for a model river and model lake are 3.5 hours and 4.8 days, respectively. However, experimental data indicate that sulfur hexafluoride may not volatilize as rapidly as the Henry's Law constant predicts. An estimated BCF of 6 suggests the potential for bioconcentration in aquatic organisms is low. While hydrolysis is energetically favorable, the fluorine groups effectively shield the sulfur atom and impede this reaction. Occupational exposure to sulfur hexafluoride may occur through inhalation and dermal contact with this compound at workplaces where sulfur hexafluoride is produced or used. Monitoring data indicate that the general population may be exposed to sulfur hexafluoride via inhalation of ambient air. (SRC)
Hazardous Substances Data Bank (HSDB)
12.2.3 Natural Pollution Sources
Sulfur hexafluoride may occur in fluorite and granite. It was reported that sulfur hexafluoride was present in fluorite and in two of eight granites analyzed. Sulfur hexafluoride was detected in hot springs from volcanic and igneous areas, and in pre-1940 ground waters from Maryland. These results indicated that small concentrations were present in waters that pre-dated industrial production of sulfur hexafluoride(1).
(1) USGS; Natural SF6. U.S. Geol Survey. Reston Groundwater Dating Laboratory. Available from, as of Jan 25, 2016:
Hazardous Substances Data Bank (HSDB)
12.2.4 Artificial Pollution Sources
Sulfur hexafluoride's production and use as a gaseous insulator for high voltage generators and other electrical equipment and radar wave guides(1); a component in various electrical equipment; and limited use in special applications ranging from medical applications to space research(2,3) may result in its release to the environment through various waste streams. Its former use as a tracer gas for both indoor and outdoor source dissemination experiments as well as for measuring gas exchange coefficients in lakes(2-5) resulted in its direct release to the environment(SRC).
(1) Lewis, RJ Sr; Hawley's Condensed Chemical Dictionary 15th ed., New York, NY: John Wiley & Sons, Inc., p. 1192 (2007)
(2) Evans FE, Mani G; Fluorine Compounds, Inorganic, Sulfur. Kirk-Othmer Encyclopedia of Chemical Technology. (1999-2015). New York, NY: John Wiley & Sons. Online Posting Date: 4 Dec 2000.
(3) Zweldinger RB et al; Environ Sci Technol 22: 956-62 (1988)
(4) Giardino NJ et al; J Air Pollut Control Assoc 38:278-80 (1988)
(5) Wanninkhof P et al; Science 227: 1224 (1985)
Hazardous Substances Data Bank (HSDB)
12.2.5 Environmental Fate
TERRESTRIAL FATE: Based on a classification scheme(1), an estimated Koc value of 29(SRC), determined from a log Kow of 1.68(2) and a regression-derived equation(3), indicates that sulfur hexafluoride is expected to have very high mobility in soil(SRC). Experimental data suggest that sulfur hexafluoride does not absorb to soil(4). Volatilization of sulfur hexafluoride from moist soil surfaces is expected to be an important fate process(SRC) given an experimental Henry's Law constant of 4.52 atm-cu m/mole(5). Sulfur hexafluoride is expected to volatilize from dry soil surfaces(SRC) based upon a vapor pressure of 1.78X10+4 mm Hg(6). Biodegradation data were not available(SRC, 2015).
(1) Swann RL et al; Res Rev 85: 17-28 (1983)
(2) Hansch C et al; Exploring QSAR. Hydrophobic, Electronic, and Steric Constants. ACS Prof Ref Book. Heller SR, consult. ed., Washington, DC: Amer Chem Soc p. 3 (1995)
(3) US EPA; Estimation Program Interface (EPI) Suite. Ver. 4.11. Nov, 2012. Available from, as of Sept 28, 2015:
(4) Bremner JM, Banwart WL; Soil Biol Biochem 8: 79-83 (1976)
(5) Giardino JE et al; J Air Pollut Control Assoc 38: 278-80 (1988)
(6) ECHA; Search for Chemicals. Sulfur Hexafluoride (CAS 2551-62-4) Registered Substances Dossier. European Chemical Agency. Available from, as of Sept 28, 2015:
Hazardous Substances Data Bank (HSDB)
AQUATIC FATE: Based on a classification scheme(1), an estimated Koc value of 29(SRC), determined from a log Kow of 1.68(2) and a regression-derived equation(3), indicates that sulfur hexafluoride is not expected to adsorb to suspended solids and sediment(SRC). Volatilization from water surfaces is expected(4) based upon the experimental Henry's Law constant of 4.52 atm-cu m/mole(5). Using this Henry's Law constant and an estimation method(3), volatilization half-lives for a model river and model lake are 3.5 hours and 4.8 days, respectively(SRC). However, study results indicate that the amount of sulfur hexafluoride volatilized may be much less than would be predicted from the Henry's Law constant(5). In these cases, equilibrium was not attained and the rate of volatilization was determined by mass-transport across the air-water interface(5). According to a classification scheme(6), an estimated BCF of 6(SRC), from an estimated log Kow(2) and a regression-derived equation(3), suggests the potential for bioconcentration in aquatic organisms is low(SRC). While hydrolysis is energetically favorable, the fluorine groups effectively shield the sulfur atom and impede this reaction(7). Because sulfur hexafluoride is inert over a wide range of environmental conditions, it has great potential as a geothermal tracer(8). An important aspect of sulfur hexafluoride is its extremely low aqueous solubility; this gives it the potential to delineate unsaturated zones underground(8). Biodegradation data were not available(SRC, 2015).
(1) Swann RL et al; Res Rev 85: 17-28 (1983)
(2) Hansch C et al; Exploring QSAR. Hydrophobic, Electronic, and Steric Constants. ACS Prof Ref Book. Heller SR, consult. ed., Washington, DC: Amer Chem Soc p. 3 (1995)
(3) US EPA; Estimation Program Interface (EPI) Suite. Ver. 4.11. Nov, 2012. Available from, as of Sept 28, 2015:
(4) Lyman WJ et al; Handbook of Chemical Property Estimation Methods. Washington, DC: Amer Chem Soc pp. 15-1 to 15-29 (1990)
(5) Giardino JE et al; J Air Pollut Control Assoc 38: 278-80 (1988)
(6) Franke C et al; Chemosphere 29: 1501-14 (1994)
(7) Eibeck RE, Mears W; Kirk-Othmer Enclycl Chem Tech 3rd ed New York, NY: Wiley 10: 799-811 (1980)
(8) Upstill-Goddard RC, Wilkins CS; Wat Res 29: 1065-68 (1995)
Hazardous Substances Data Bank (HSDB)
ATMOSPHERIC FATE: According to a model of gas/particle partitioning of semivolatile organic compounds in the atmosphere(1), sulfur hexafluoride, which has a vapor pressure of 1.78X10+4 mm Hg at 25 °C(2), is expected to exist solely as a gas in the ambient atmosphere. Sulfur hexafluoride is one of the heaviest known gases with a vapor density approximately five times that of air(3). Therefore, if released in the atmosphere, it will tend to remain close to the ground and be transported to earth by wet deposition(SRC). Long-term observations of sulfur hexafluoride in the atmosphere have revealed that the concentration of sulfur hexafluoride has increased by 2 orders of magnitude between 1970 and 1992(4). In addition, scientists at the National Oceanic and Atmospheric Administration have reported that perfluoro compounds have very long atmospheric lifetimes (on the order of millennia) and thus significant global warming potentials(4). The main process of removal of sulfur hexafluoride from the troposphere has been reported to be its advection into mesosphere, where the substance may be subjected to degradation via the reaction with free electrons(5); a global atmospheric lifetime of 3200 years was calculated for sulfur hexafluoride(5). Using a two-dimensional transport and chemistry model which includes removal by electrons and ions in the mesosphere and lower thermosphere, the predicted atmospheric lifetime after photolysis for sulfur hexafluoride was determined to be greater than or equal to 600 years(6). The main products of sulfur hexafluoride arc decomposition in the presence of air are thionyl fluoride, sulfur tetrafluoride, and sulfur tetrafluoromonoxide plus metal fluorides and sulfides(7).
(1) Bidleman TF; Environ Sci Technol 22: 361-367 (1988)
(2) ECHA; Search for Chemicals. Sulfur Hexafluoride (CAS 2551-62-4) Registered Substances Dossier. European Chemical Agency. Available from, as of Sept 28, 2015:
(3) O'Neil MJ, ed; The Merck Index. 15th ed., Cambridge, UK: Royal Society of Chemistry, p. 1659 (2013)
(4) Morris RA et al; J Geophysical Research 100: 1287-94 (1995)
(5) Ravishankara AR et al; Science 259: 194-199 (1993)
(6) Morris RA et al; J Geophysical Research 100: 1287-94 (1995)
(7) Evans FE, Mani G; Fluorine Compounds, Inorganic, Sulfur. Kirk-Othmer Encyclopedia of Chemical Technology. (1999-2015). New York, NY: John Wiley & Sons. Online Posting Date: 4 Dec 2000.
Hazardous Substances Data Bank (HSDB)
12.2.6 Environmental Abiotic Degradation
According to a model of gas/particle partitioning of semivolatile organic compounds in the atmosphere(1), sulfur hexafluoride, which has a vapor pressure of 1.78X10+4 mm Hg at 25 °C(2), is expected to exist solely as a gas in the ambient atmosphere. With few exceptions, sulfur hexafluoride is chemically inert at ambient temperature and atmospheric pressure(3). Its high resistance has been ascribed to its high S-F bond strength, coordinate saturation, steric hindrance, and nonpolarity(4). While hydrolysis is energetically favorable, the fluorine groups effectively shield the sulfur atom and impede this reaction(5). Sulfur hexafluoride is very resistant to attack and extreme conditions are required for reactions to occur(4). For example, it resists molten potassium hydroxide and steam at 500 °C(4). Thermodynamically sulfur hexafluoride is unstable and should react with many materials, including water, but these reactions are kinetically impeded by the fluorine shielding the sulfur(3). Exposed to 1000 °C temperatures, sulfur hexafluoride decomposes to thionyl fluoride and sulfur tetrafluoride to the extent of 10 mol percent. In spite of its decomposition, the dielectric strength of sulfur hexafluoride remains the same. The main products of sulfur hexafluoride decomposition in the presence of air are thionyl fluoride, sulfur tetrafluoride, and sulfur tetrafluoromonoxide plus metal fluorides and sulfides(3).
(1) Bidleman TF; Environ Sci Technol 22: 361-367 (1988)
(2) ECHA; Search for Chemicals. Sulfur Hexafluoride (CAS 2551-62-4) Registered Substances Dossier. European Chemical Agency. Available from, as of Sept 28, 2015:
(3) Evans FE, Mani G; Fluorine Compounds, Inorganic, Sulfur. Kirk-Othmer Encyclopedia of Chemical Technology. (1999-2015). New York, NY: John Wiley & Sons. Online Posting Date: 4 Dec 2000.
(4) Cotton FA, Wilkenson G; Advanced Inorganic Chemistry. 4th ed New York,NY: Wiley p. 522 (1980)
(5) Eibeck RE, Mears W; Kirk-Othmer Enclycl Chem Tech 3rd ed New York, NY: Wiley 10: 799-811 (1980)
Hazardous Substances Data Bank (HSDB)
12.2.7 Environmental Bioconcentration
An estimated BCF of 6 was calculated in fish for sulfur hexafluoride(SRC), using an experimental log Kow of 1.68(1,SRC) and a recommended regression-derived equation(2). According to a classification scheme(3), this BCF suggests the potential for bioconcentration in aquatic organisms is low(SRC).
(1) Hansch C et al; Exploring QSAR. Hydrophobic, Electronic, And Steric Constants. ACS Prof Ref Book. Washington,DC: Amer Chem Soc (1995)
(2) US EPA; Estimation Program Interface (EPI) Suite. Ver. 4.11. Nov, 2012. Available from, as of Sept 28, 2015:
(3) Franke C et al; Chemosphere 29: 1501-14 (1994)
Hazardous Substances Data Bank (HSDB)
12.2.8 Soil Adsorption / Mobility
The Koc of sulfur hexafluoride is estimated as 29(SRC), using a log Kow of 1.68(1) and a regression-derived equation(2). According to a classification scheme(3), this estimated Koc value suggests that sulfur hexafluoride is expected to have very high mobility in soil. Experimental data suggests that sulfur hexafluoride does not absorb at all in soils(4). The adsorption of sulfur hexafluoride was studied in four soils that differed markedly in pH, texture, and organic carbon content by injecting 100 ppm of the gas into bottles containing 5 g of soil and following the concentration of the sulfur hexafluoride in the head space air for 15 days(4). The experiments were conducted using both air-dried soils and soils moistened to fifty percent of their water holding capacity. None of the soils tested adsorbed any sulfur hexafluoride. Sulfur hexafluoride's lack of adsorptivity is one characteristic that makes it an ideal tracer gas(4).
(1) Hansch C et al; Exploring QSAR. Hydrophobic, Electronic, and Steric Constants. ACS Prof Ref Book. Heller SR, consult. ed., Washington, DC: Amer Chem Soc p. 3 (1995)
(2) US EPA; Estimation Program Interface (EPI) Suite. Ver. 4.11. Nov, 2012. Available from, as of Sept 28, 2015:
(3) Swann RL et al; Res Rev 85: 17-28 (1983)
(4) Bremner JM, Banwart WL; Soil Biol Biochem 8: 79-83 (1976)
Hazardous Substances Data Bank (HSDB)
12.2.9 Volatilization from Water / Soil
The Henry's Law constant for sulfur hexafluoride is 4.25 atm-cu m/mole(1). This Henry's Law constant indicates that sulfur hexafluoride is expected to volatilize rapidly from water surfaces(2). Based on this Henry's Law constant, the volatilization half-life from a model river (1 m deep, flowing 1 m/sec, wind velocity of 3 m/sec)(2) is estimated as 3.5 hours(SRC). The volatilization half-life from a model lake (1 m deep, flowing 0.05 m/sec, wind velocity of 0.5 m/sec)(2) is estimated as 4.8 days(SRC). Sulfur hexafluoride's Henry's Law constant indicates that volatilization from moist soil surfaces may occur(SRC). However, experiments were conducted in which volatilization of sulfur hexafluoride from a shower was determined(1); results showed the amount of sulfur hexafluoride volatilized was much less than would be predicted from the Henry's Law constant(1). In these cases, equilibrium was not attained and the rate of volatilization was determined by mass-transport across the air-water interface(1). Sulfur hexafluoride is expected to volatilize from dry soil surfaces(SRC) based upon a vapor pressure of 1.78X10+4 mm Hg(3).
(1) Giardino JE et al; J Air Pollut Control Assoc 38: 278-80 (1988)
(2) Lyman WJ et al; Handbook of Chemical Property Estimation Methods. Washington, DC: Amer Chem Soc pp. 15-1 to 15-29 (1990)
(3) ECHA; Search for Chemicals. Sulfur Hexafluoride (CAS 2551-62-4) Registered Substances Dossier. European Chemical Agency; Available from, as of Sept 28, 2015:
Hazardous Substances Data Bank (HSDB)
12.2.10 Effluent Concentrations
Recent experimental data concerning sulfur hexafluoride suggest an early 1996 emission rate of 5.9 Gg/year and an interhemispheric exchange time of 1.3 years(1). The atmospheric burden of sulfur hexafluoride has increased from 5000 tons in 1970 to 90,000 tons in 1995, corresponding to emissions of <0.5 tons/yr in 1970 to 6,250 tons/yr in 1995(2). This amount increases by 7% every year. In 2002, emissions of sulfur hexafluoride to the atmosphere from the US electrical power industry totaled 589 metric tons(3).
(1) Dervos CT, Vassiliou P; J Air Waste Mange Assoc 50: 137-41 (2000)
(2) Maiss M, Brenninkmeijer CAM; Environ Sci Technol 32: 3077-86 (1998)
(3) Blackman J, Kantamaneni R; EPA's SF6 Emission Reduction Partnership: Maximizing the Benefits of SF6 Emission Reductions for Electric Utilities. Washington, DC: US EPA. p. 8 (2004). NTIS ADP019533. Govt Reports Announcements & Index (GRA&I), Issue 19, 2005. Available from, as of Oct 1, 2015:
Hazardous Substances Data Bank (HSDB)
12.2.11 Sediment / Soil Concentrations
SOIL: Soil air samples were collected at depths of 50, 100, and 200 cm in Central Park, NY (March 2001 to 2002), Riverside Park, NY (February 2001 to 2002), Pallisades, NY (March to April 2000), Sparkill, NY (May 2000 to February 2002), and Winthrop, ME (November 2000 to July 2001). Mean sulfur hexafluoride mixing ratio ranges at the 5 sampling sites were calculated as 15.0-15.8 parts per trillion, 9.0-10.7 parts per trillion, 8.2-8.7 parts per trillion, 8.6-8.9 parts per trillion, and 4.8 parts per trillion, respectively(1).
PMID:12680656
(1) Santella N et al; Environ Sci Technol 37: 1069-74 (2003)
Hazardous Substances Data Bank (HSDB)
12.2.12 Atmospheric Concentrations
URBAN/SUBURBAN: In a field study conducted along US Highway 70 near Raleigh, NC during May 1983, the concentration of sulfur hexafluoride emitted from automobiles ranged from 1.5 to 13.4 parts per billion(1). Air monitoring at sites that include urban, non-urban, and marine environments in California was conducted during 21 July 1976 to 21 July 1997. The average sulfur hexafluoride concentration in the northern hemisphere was determined to range from 0.24 to 0.9 parts per thousand(2).
(1) Zweidinger RB et al; Environ Sci Technol 22: 956-62 (1988)
(2) Singh HB et al; Fate of Halogenated Compounds in the Atmosphere - Interim Report 1977; Stanford Research Institute, Menlo Park, CA. US NTIS PB Report 1978, PB-278198, pp. 65 (1978)
Hazardous Substances Data Bank (HSDB)
RURAL/REMOTE: Above 25 km height, sulfur hexafluoride concentrations are lagging the tropospheric ones by several years: 4.5 years for the tropics, 6 years for mid-latitudes and up to 10 years for the arctic winter vortex(1). Due to is enormous global warming potential, sulfur hexafluoride is now being systematically monitored by a number of air sampling programs which may nclude high resolution latitude profiles over the Atlantic and Pacific oceans, weekly flask samples from remote, globally distributed sites, hourly in situ measurements in rural or industrial areas, and a series of archived air samples(1). The natural background of sulfur hexafluoride is lower than 0.4 parts per trillion(2). Observations from four background monitoring stations, Neumayer, Antarctica (1986-1994), Cape Grim, Tasmania (1978-1994), Izana, Canary Islands (1991-1994) and Alert, Canada (1993-1994), of sulfur hexaflouride were as follows: for the period from January 1978 to December 1994 the data confirm a stable and unbroken quadratic rise in tropospheric sulfur hexafluoride from 0.5 to 3.11 parts per trillion in the southern hemisphere and for July 1991 to December 1994 from 2.69 to 3.44 parts per trillion in the northern hemisphere(3). The global mean tropospheric increase rate in late 1994 was 0.225 parts per trillion/year (6.9%/yr)(3). The long term trend and interhemispheric gradients are due to industrial production and emission, rising approximately linearly with time and with 94% emissions located mainly in the northern atmosphere(3). Monitoring at a background site in Shangdianzi China during June 2009 to May 2011 detected a mean background level of sulfur hexafluoride of 7.22X10-12 mol/mol(4). Since 1995 two different sampling programs in The Halocarbons and other Atmospheric Trace Species (HATS) group have measured sulfur hexafluoride (SF6); the combined data set is calculated by taking weighted averages of co-located measurements from background NOAA/ESRL GMD air measurement program(5); the combined global mean concentration of sulfur hexafluoride has increased (almost linearly) from 3.1 ppt in 1995 to 8.6 ppt in 2015(5).
(1) Dervos CT, Vassiliou P; J Air Waste Mange Assoc 50: 137-41 (2000)
(2) Maiss M, Brenninkmeijer CAM; Environ Sci Technol 32: 3077-86 (1998)
(3) Maiss M et al; Atmos Environ 30: 1621-9 (1996)
(4) Yao B et al; J Environ Sci (China) 26(12): 2451-8 (2014)
(5) NOAA; Halocarbons and other Atmospheric Trace Gases, Sulfur hexafluoride (SF6) - Combined Data Set (August 2015); Available from, as of Sept 29, 2015:
Hazardous Substances Data Bank (HSDB)
Use of ... SF6 in the Netherlands (kiloton CO2 equivalents yr-1) and the Global Warming Potential (GWP) used. NOTE: Not covered by the Montreal Protocol
Application
GWP
2000
2005
2010
2020
Application
Fire Extinguishing
GWP
23900
2000
162
2005
175
2010
189
2020
2020
Application
Pre-Montreal Use
GWP
23900
2000
1577
2005
1697
2010
1816
2020
2199
Kroeze C, Matthijsen AJCM; Water Air Soil Pollut 107: 197-218 (1998)
Hazardous Substances Data Bank (HSDB)
12.2.13 Probable Routes of Human Exposure
According to the 2012 TSCA Inventory Update Reporting data, 2 reporting facilities estimate the number of persons reasonably likely to be exposed during the manufacturing, processing, or use of sulfur hexafluoride (2551-62-4) may be as low as <10 workers up to the range of 50-99 workers per plant; the data may be greatly underestimated due to confidential business information (CBI) or unknown values(1).
(1) US EPA; Chemical Data Reporting (CDR). Non-confidential 2012 Chemical Data Reporting information on chemical production and use in the United States. Available from, as of Sept 24, 2015:
Hazardous Substances Data Bank (HSDB)
NIOSH (NOES Survey 1981-1983) has statistically estimated that 9,282 workers (1,529 of these are female) are potentially exposed to sulfur hexafluoride in the US(1). Occupational exposure may occur through inhalation and dermal contact with this compound at workplaces where sulfur hexafluoride is produced or used(SRC). In the presence of an electrical discharge such as an arc, spark, or corona, a portion of sulfur hexafluoride decomposes into lower fluorides of sulfur that can react to form a number of chemically active byproducts (concentration, % by volume): SOF2 (SF4) (0.5); SOF4 (0.085); SF4 (0.085); S2F10 (0.025); SO2F2 (0.006); SO2 (0.002); HF (1.0). This is now well documented and accepted by workers under laboratory or practical operations in gas insulated switchgear (GIS) power systems(2). Monitoring data indicate that the general population may be exposed to sulfur hexafluoride via inhalation of ambient air(SRC).
(1) NIOSH; NOES. National Occupational Exposure Survey conducted from 1981-1983. Estimated numbers of employees potentially exposed to specific agents by 2-digit standard industrial classification (SIC). Available from, as of Sept 25, 2015:
(2) Dervos CT, Vassiliou P; J Air Waste Mange Assoc 50: 137-41 (2000)
Hazardous Substances Data Bank (HSDB)
13 Associated Disorders and Diseases
Associated Occupational Diseases with Exposure to the Compound
Asphyxiation, simple [Category: Acute Poisoning]
Haz-Map, Information on Hazardous Chemicals and Occupational Diseases
Open Targets
14 Literature
14.1 Consolidated References
PubChem
14.2 NLM Curated PubMed Citations
Medical Subject Headings (MeSH)
14.3 Springer Nature References
Springer Nature
14.4 Thieme References
Thieme Chemistry
14.5 Wiley References
Wiley
14.6 Chemical Co-Occurrences in Literature
PubChem
14.7 Chemical-Gene Co-Occurrences in Literature
PubChem
14.8 Chemical-Disease Co-Occurrences in Literature
PubChem
14.9 Chemical-Organism Co-Occurrences in Literature
PubChem
15 Patents
US5686060
US10232061
US10335502
US11723869
DrugBank
15.1 Depositor-Supplied Patent Identifiers
PubChem
Link to all deposited patent identifiers
PubChem
15.2 WIPO PATENTSCOPE
Patents are available for this chemical structure:
PATENTSCOPE (WIPO)
15.3 Chemical Co-Occurrences in Patents
PubChem
15.4 Chemical-Disease Co-Occurrences in Patents
PubChem
15.5 Chemical-Gene Co-Occurrences in Patents
PubChem
15.6 Chemical-Organism Co-Occurrences in Patents
PubChem
16 Interactions and Pathways
16.1 Molecular Imaging Information
16.1.1 MICAD Imaging Information
PMID
20641961
Abbreviated name
SF6 microbubbles
Agent Category
Lipid microbubbles
Target
Nontarget
Target Category
Nontarget filling of blood vessels and heart chambers
Method of detection
Ultrasound (US)
Source of signal
Microbubbles and sulfur hexafluoride
Activation
No
Studies
In vitro; Non-Primate Non-Rodent Mammals; Humans;
Use Classification
Imaging agents
Molecular Imaging and Contrast Agent Database (MICAD)
17 Biological Test Results
17.1 BioAssay Results
PubChem
18 Classification
18.1 MeSH Tree
Medical Subject Headings (MeSH)
18.2 NCI Thesaurus Tree
NCI Thesaurus (NCIt)
18.3 ChEBI Ontology
ChEBI
18.4 KEGG: ATC
KEGG
18.5 WHO ATC Classification System
WHO Anatomical Therapeutic Chemical (ATC) Classification
18.6 FDA Pharm Classes
FDA Pharm Classes
18.7 ChemIDplus
ChemIDplus
18.8 CAMEO Chemicals
CAMEO Chemicals
18.9 UN GHS Classification
GHS Classification (UNECE)
18.10 NORMAN Suspect List Exchange Classification
NORMAN Suspect List Exchange
18.11 EPA DSSTox Classification
EPA DSSTox
18.12 EPA TSCA and CDR Classification
EPA Chemicals under the TSCA
18.13 FDA Drug Type and Pharmacologic Classification
National Drug Code (NDC) Directory
18.14 PFAS and Fluorinated Organic Compounds in PubChem
PubChem
18.15 EPA Substance Registry Services Tree
EPA Substance Registry Services
18.16 MolGenie Organic Chemistry Ontology
MolGenie
18.17 Chemicals in PubChem from Regulatory Sources
PubChem
18.18 ATCvet Classification
WHO ATCvet - Classification of Veterinary Medicines
19 Information Sources
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ChEBISulfur hexafluoride ChEBI Ontology
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6651 | https://www.cenieh.es/en/press/news/study-shows-homo-erectus-used-both-olduvayense-and-achelense-industry | Skip to main content
A study shows that 'Homo erectus' used both Olduvayense and Achelense industry
The smallest Homo erectus cranium in Africa and diverse stone tools found at Gona, Ethiopia, indicate that human ancestors were more varied, both physically and behaviorally, than previously known.
Anterior
Posterior
A study published in Sicience Advances by Sileshi Semaw, from CENIEH (Burgos, Spain), and Michael Rogers, from SCSU (Connecticut, USA), together with their research team presents a nearly-complete hominin cranium estimated to 1.5 million years, and a partial cranium dated to 1.26 million years from the Gona study area, in the Afar State of Ethiopia. Both crania, assigned to Homo erectus, associated with both simple Oldowan-type (Mode 1) and more complex, Acheulian (Mode 2) stone tool assemblages.
The nearly-complete cranium was discovered at Dana Aoule North (DAN5) and the partial cranium at Busidima North (BSN12), sites that are 5.7 km apart. Remarkably, the DAN5 cranium has the smallest endocranial volume documented for Homo erectus in Africa, about 590 cubic centimeters, probably representing a female. This cranium bears some similarity to the small individuals discovered at Dmanisi (Republic of Georgia), and dated to 1.8 million years. The BSN12 partial cranium is robust and large (similar to OH 9 from Olduvai Gorge), while the DAN5 cranium is smaller and more gracile, suggesting that Homo erectus was probably a sexually dimorphic species.
This physical diversity is mirrored by the stone tool technologies exhibited by the artifacts found in association with both crania. Instead of only finding the expected large handaxes or picks (signature tools of Homo erectus), the Gona team found both well-made handaxes and plenty of less-complex Oldowan tools and cores. This suggests that Homo erectus had a degree of cultural/behavioral plasticity that has yet to be fully understood.
Scott. Simpson, a co-author on the study, commented that, “In the almost 130 years since its initial discovery in Java, Homo erectus has been recovered from many sites across Eurasia and Africa. The new remains from the Gona study area exhibit a degree of biological diversity in Africa that had not been seen previously, notably the small size of the DAN5 cranium. The BSN12 partial cranium also provides evidence linking the African and eastern Asian fossils demonstrating how successful Homo erectus was.”
Environmental setting and carnivory
The DAN5 and BSN12 sites at Gona are among the earliest examples of Homo erectus associated with both Oldowan and Acheulian stone assemblages. The toolmakers at both sites lived in close proximity to ancient rivers, in settings with riverine woodlands adjacent to open habitats. The low δ13C isotope value from the DAN5 cranium (from the right M1) is consistent with a diet dominated by C3 plants (trees, shrubs) or, alternatively, broad spectrum omnivory.
Homo erectus at Gona used locally-available stone cobbles to make their tools, which were accessed from the nearby riverbeds. Fossil fauna were abundant at BSN12, but cutmarks or hammerstone-percussed bones were not identified. At DAN5, an elephant phalanx was found with stone tool cutmarks, and a small antelope leg bone had a percussion notch, implying that H. erectus butchered both large and small mammals, though it is not clear whether they hunted or scavenged their prey.
Significance
There is a common view that early Homo (e.g. Homo habilis) invented the first simple (Oldowan) stone tools, but when Homo erectus appeared 1.8-1.7 million years ago, a new stone tool technology called the Acheulian, with purposefully-shaped large cutting tools such as handaxes, emerged in Africa. The timing, causes and nature of this significant transition to the Acheulian by 1.7 million years is not entirely clear, though, and is an issue debated by archaeologists.
Our investigations at DAN5 and BSN12 have clearly shown that Oldowan technology persisted much longer after the invention of the Acheulian, indicative of a particular behavioral flexibility and cultural complexity practiced by Homo erectus, a trait not fully understood or appreciated in paleoanthropology.
Semaw says, “Although most researchers in the field consider the Acheulian to have replaced the earlier Oldowan (Mode 1) by 1.7 million years, our research has shown that Mode 1 technology actually remained ubiquitous throughout the entire Paleolithic.” Rogers adds, “The simple view that a single hominin species is responsible for a single stone tool technology is not supported. The human evolutionary story is more complicated.”
It is important to note that the first hominins that migrated out of Africa probably left with Oldowan technology, traveling as far as Dmanisi in Georgia by 1.8 Ma. The Acheulian, then, was likely invented by a Homo erectus population that evolved in Africa. Semaw suggests that, “it’s still possible that the descendants of the hominins who migrated earlier to the Caucasus could have migrated back to Africa, as others also have argued that H. erectus evolved in Asia and possibly returned back to Africa.”
The evidence from Gona suggests a lengthy and concurrent use of both Oldowan and Acheulian technologies by a single long-lived species, Homo erectus
Interestingly, other than DAN5 and BSN12, the crania from Buia (in Eritrea) and Daka (from the Middle Awash study area) are the only two H. erectus cranial specimens known in the entire Afar in the 1.5-1.0 Ma critical time interval. Konso is the only other Ethiopian site that has produced H. erectus (from southern Ethiopia), and the Gona team has previously reported the first female Homo erectus pelvis known, dated to 1 million years. Thus, DAN5 is quite a rare discovery, and more hominins associated with stone artifacts are needed (2.0-0.5 million years) in order to better understand issues regarding Homo erectus stone technology, physiology, mobility across continents, etc.
Some argue that, in Africa multiple hominin species may have been responsible for the two distinct contemporary stone technologies (Oldowan and Acheulian). To the contrary, the evidence from Gona suggests a lengthy and concurrent use of both Oldowan and Acheulian technologies by a single long-lived species, Homo erectus, the variable expression of which deserves continued research. Rogers suggests that “one challenge in the future will be to understand better the stone tool attributes that are likely to be passed on through cultural tradition versus others that are more likely to be reinvented by different hominin groups.”
Gona sites
Gona is located in the Afar Triangle of Ethiopia, adjacent to the well-known Middle Awash and Hadar study areas, were the famous skeletons “Ardi” and “Lucy” were found, respectively. The research team has been investigating the Gona deposits since 1999, and the BSN12 partial cranium was discovered by Dr. N. Toth (Stone Age Institute, Indiana University) during the first season. The DAN5 cranium was found a year later in 2000 by the late Ibrahim Habib (local Afar colleague) on a camel trail.
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6652 | https://askfilo.com/user-question-answers-smart-solutions/5-find-the-measure-of-each-exterior-angle-of-a-regular-3132343132313135 | Question asked by Filo student
Views: 5,452 students
Updated on: Sep 8, 2025
Text SolutionText solutionverified iconVerified
Concepts:
Exterior angles, Regular polygon
Explanation:
The measure of each exterior angle of a regular polygon can be found using the formula: Exterior angle=n360° where n is the number of sides of the polygon.
Step by Step Solution:
Step 1
Identify the formula for the exterior angle of a regular polygon: Exterior angle=n360°.
Step 2
For part (a), substitute n=12 into the formula: Exterior angle=12360°.
Step 3
Calculate the exterior angle for the 12-sided polygon: Exterior angle=30°.
Step 4
For part (b), substitute n=18 into the formula: Exterior angle=18360°.
Final Answer:
The measure of each exterior angle of a regular polygon with 12 sides is 30°, and with 18 sides is 20°.
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| Question Text | 5. Find the measure of each exterior angle of a regular polygon with: (a) 12 sides (b) 18 sides |
| Updated On | Sep 8, 2025 |
| Topic | All topics |
| Subject | Smart Solutions |
| Class | Class 8 |
| Answer Type | Text solution:1 |
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6654 | https://www.researchgate.net/publication/371141561_FREQUENCY_OF_SELF_INFLICTED_INJURIES_IN_MEDICO-LEGAL_CASES_AND_ITS_DISTRIBUTION_AMONGST_MALES_AND_FEMALES_OF_DIFFERENT_AGE_GROUPS | FREQUENCY OF SELF INFLICTED INJURIES IN MEDICO-LEGAL CASES AND ITS DISTRIBUTION AMONGST MALES AND FEMALES OF DIFFERENT AGE GROUPS | Request PDF
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FREQUENCY OF SELF INFLICTED INJURIES IN MEDICO-LEGAL CASES AND ITS DISTRIBUTION AMONGST MALES AND FEMALES OF DIFFERENT AGE GROUPS
December 2016
Pakistan Postgraduate Medical Journal 27(1):10-15
DOI:10.51642/ppmj.v27i1.203
Authors:
MISHAL SALEEM
MISHAL SALEEM
This person is not on ResearchGate, or hasn't claimed this research yet.
HINA IRFAN
HINA IRFAN
This person is not on ResearchGate, or hasn't claimed this research yet.
Arif Rasheed Malik
Rahbar Medical & Dental College, Lhr
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Abstract
Objectives: This is a one-year study to find the frequency and pattern of self-inflicted injuries found on scrutiny of the record of medico-legal cases conducted during 01-01-2016 to 31-12-2016. Methodology: Out of a total of 3143 medico-legal cases conducted during the period of study, all the cases that fulfilled the inclusion criteria were selected. These were 77 cases where injury was caused by the victim himself. Analysis of these cases was made with respect to age, gender, type of injury, site of injury, nature of injury, number of injuries and employment status of the person. Results: Cases of self-inflicted injuries constituted 2.45 % of all the medico-legal cases conducted during 2016 at Mayo Hospital, Lahore. The incidence of incised wounds was highest being 29% while corrosive burns constituted 27%, firearm injuries constituted 21%, flame burns 19%, lacerated wounds 2.6% and bruises 1.3%. Victims were primarily in the 11-30 year age group (80.52%) with a male to female ratio of 2.67:1. Conclusion: Self-inflicted injuries are mostly incised wounds caused by sharp edged weapons. The most recurring site of injury is the forearm. Males are more commonly reported to be involved in such cases. Nature of injuries is predominantly non-suicidal. The usual pattern is a single injury in the low-income category males. However, among the life-threatening injuries, females outnumber males.
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Self-inflicted Cut Injury as Common Method of Deliberate Self Harm: A Retrospective Study from Nepal
Article
Full-text available
Sep 2017
Rabi Shakya
Sulochana Joshi
Pawan Sharma
Context Deliberate self-injury, a fairly common condition is present across all culture. It is defined as intentional, direct injuring of body tissue without suicidal intent. We continue to lack a sufficient understanding of this behavior in the context of South East Asian region. Aims This study aims to explore the characteristics of self-cutting, and gender differences in homologous group of youth applying for foreign employment in Nepal. Settings and Design Cross-sectional, single interview method in an out-patient setting. Subjects and Methods The youths applying for foreign employment were screened for self-harm attempt using single question. Screen positive individuals were applied with inclusion and exclusion criteria. The data were collected in a single interview by consultant psychiatrist. Results Males were more than females and the mean age (standard deviation) at the time of act and current presentation were 16.52 (3.13) and 24.70 (5.54) years, respectively. Various methods were used in self-inflicting cut, and certain amount of stress was present in the majority of the participants. There was no significant difference between male and female in all descriptive characteristics except for mean age at the time of presentation. Conclusions The descriptive results from Nepal are keeping with most of the published literature. The study is the first one from Nepal, and we expect that this will help in laying a good foundation for further studies with stronger methodologies.
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Epidemiology and Outcome of Self-Inflicted Burns at Pakistan Institute of Medical Sciences, Islamabad
Article
Full-text available
Jul 2014
Muhammad Saaiq
Bushra Ashraf
Self-inflicted burn injuries carry considerable mortality and morbidity among otherwise fit young individuals. This study assessed the epidemiologic pattern and outcome of these injuries in a burn care facility in Pakistan. The study was carried out at Pakistan Institute of Medical Sciences (PIMS) Burn Care Centre in Islamabad over a period of 2 years. It included all adult patients of either gender, aged over 14 years who presented as cases of burn suicides and attempted burn suicides during the study period. Convenience sampling technique was employed. The sociodemographic profile of the patients, motives underlying the act of self-immolation, any underlying psychiatric illness, alcohol abuse, total body surface area (TBSA) burnt, depth of burn injury, associated inhalation injury, duration of hospital stay, and mortality were all recorded. Seventy five patients (80.64%) were female while 18 patients (19.35%) were male. The overall mean age was 26.89±6.1 years (range=15-52 years). The affected TBSA ranged from 15%-100% with an overall mean of 69.30±25.42%. The hospital stay ranged from 1-37 days with a mean of 7.16±6.60 days. Marital conflicts constituted the most frequent motive underlying the suicidal attempts (n=57; 61.29%) followed by failed love affairs (n=9; 9.67%). There was an overall mortality of 84.95%. The most common sufferers of self inflicted burn injuries were young, married, illiterate housewives who were resident of rural area. Getting marriage was the most common triggering cause for such injuries. There is need to institute appropriate preventive measures to address the issue in a national perspective.
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International prevalence of non-suicidal self-injury and deliberate self-harm
Article
Full-text available
Mar 2012
Child Adolesc Psychiatr Ment Health
Jennifer J. Muehlenkamp
Laurence Claes
Lindsey Havertape
Paul L Plener
The behaviours of non-suicidal self-injury (NSSI) and deliberate self-harm (DSH) are prevalent among adolescents, and an increase of rates in recent years has been postulated. There is a lack of studies to support this postulation, and comparing prevalence across studies and nations is complicated due to substantial differences in the methodology and nomenclature of existing research. We conducted a systematic review of current (2005 - 2011) empirical studies reporting on the prevalence of NSSI and DSH in adolescent samples across the globe. Fifty-two studies fulfilling the inclusion criteria were obtained for analysis. No statistically significant differences were found between NSSI (18.0% SD = 7.3) and DSH (16.1% SD = 11.6) studies. Assessment using single item questions led to lower prevalence rates than assessment with specific behaviour checklists. Mean prevalence rates have not increased in the past five years, suggesting stabilization. NSSI and DSH have a comparable prevalence in studies with adolescents from different countries. The field would benefit from adopting a common approach to assessment to aide cross-cultural study and comparisons.
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Self inflicted burn; a high tide
Article
Full-text available
May 2010
J Pakistan Med Assoc
Tahir Syed
Abdul Razak
Mahesh Kumar
Syed Asad Ali
To highlight the demographic profile, various methods of self inflicted burn, and to probe into the problem by knowing various precipitating factors that may lead to self inflicted burn in our part of the country. This is a study of 154 cases of self-inflicted burns treated at the Burn Emergency Unit of Liaquat University of Medical and Health Sciences Jamshoro Pakistan, during a period of 08 years. These self inflicted burns, were divided into two groups, those with suicidal intent called suicide attempter and those who mutilate themselves as self immolator. Except that these patients throughout the period of their hospital stay were under surveillance of a psychiatrist, they were managed according to the routine protocol of our unit. The patients were analyzed with respect to age, sex, method used for self inflicted burns, place of burn, psychiatric history, interpersonal problems, total body surface area (TBSA) burnt, depth of burn and outcome. The continuous variables were compared using t-test while for categorical data chi-square test was used. SPSS 15 was used as statistical software. The prevalence of self inflicted burns was 9.80% with a consistent rise in number of self inflicted burns from 2001 through 2008. The self immolators were significantly younger then suicide attempters. Males dominated in self immolators, while in suicide attempters female outnumbered males. The mean TBSA affected was significantly higher in suicide attempters when compared to self immolators. The mean hospital stay did not differ significantly in both groups. The mortality for self immolators was 6.38%, in contrast to suicide attempter where it was 33.65%. The radical change in the socioeconomic condition of common people, traditional joint family system, political system and justice, has reflected as increased prevalence of self infliction both as self mutilators and suicide attempters.
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National Trends in Hospitalization of Youth With Intentional Self-Inflicted Injuries
Article
Full-text available
Aug 2005
Mark Olfson
Marc J Gameroff
Steven C Marcus
David Shaffer
The authors examined national trends from 1990 to 2000 in the utilization of community hospital inpatient services by young people (5-20 years of age) with intentional self-inflicted injuries. Discharge abstracts from a nationally representative sample of community hospitals were analyzed, with a focus on youth discharges (N=10,831) with a diagnosis of intentional self-inflicted injury (ICD-9-CM: E950-E959). Census data were used to derive national population-based rates of self-inflicted injuries requiring inpatient treatment. Overall population-based trends in hospitalizations for self-inflicted injury were calculated and stratified by gender and age. Among youths hospitalized with a self-inflicted injury, trends were also calculated for length of stay, inpatient costs, method of injury, and associated mental disorder diagnoses. The annual hospitalization rate of youths with self-inflicted injuries declined from 49.1 per 100,000 in 1990 to 44.9 per 100,000 in 2000, and the mean length of inpatient stay significantly declined from 3.6 days to 2.7 days. Among the hospitalized patients, there were increases in the rate of cutting (4.3% to 13.2%) and ingestion of acetaminophen (22.1% to 26.9%), antidepressants (10.0% to 14.0%), and opiates (2.3% to 3.3%) as a cause of injury, whereas there were decreases in the ingestion of salicylates (14.9% to 10.2%) and barbiturates (1.5% to 0.7%). There were significant increases in the proportion of subjects with primary mental disorder discharge diagnoses of depressive disorder (29.2% to 46.0%), bipolar disorder (1.3% to 8.2%), and substance use disorder (5.4% to 10.7%) and significant decreases in the rate of adjustment disorders (22.2% to 11.4%) and nonmental disorders (31.9% to 13.6%). After excluding cutting, which may be more closely related to self-mutilation than suicidal self-injury, the annual hospitalization rate of youths with self-inflicted injuries declined from 47.2 per 100,000 in 1990 to 39.4 per 100,000 in 2000. Over the decade of study, young people admitted to community hospitals with self-inflicted injuries tended to have more severe psychiatric diagnoses and to be treated during shorter inpatient stays. These trends suggest that the role of youth inpatient care has narrowed, becoming focused on those with severe psychiatric disorders.
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Parent–Child Interactions, Peripheral Serotonin, and Self-Inflicted Injury in Adolescents
Article
Full-text available
Feb 2008
Sheila E Crowell
Theodore Beauchaine
Elizabeth Mccauley
Adrianne L. Stevens
Self-inflicted injury in adolescence indicates significant emotional and psychological suffering. Although data on the etiology of self-injury are limited, current theories suggest that the emotional lability observed among self-injuring adolescents results from complex interactions between individual biological vulnerabilities and environmental risk. For example, deficiencies in serotonergic functioning, in conjunction with certain family interaction patterns, may contribute to the development of emotional lability and risk for self-injury. The authors explored the relation between peripheral serotonin levels and mother–child interaction patterns among typical (n = 21) and self-injuring (n = 20) adolescents. Findings revealed higher levels of negative affect and lower levels of both positive affect and cohesiveness among families of self-injuring participants. Peripheral serotonin was also correlated with the expression of positive affect within dyads. Furthermore, adolescents' serotonin levels interacted with negativity and conflict within dyads to explain 64% of the variance in self-injury. These findings underscore the importance of considering both biological and environmental risk factors in understanding and treating self-injuring adolescents.
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Deliberate Self-Harm
Article
Full-text available
Jan 2008
Nida Zakiullah
Sadia Saleem
Sameea Sadiq
Parvez Nayani
Introduction: Previous deliberate self-harm (DSH) is the strongest predictor of suicide. Although several studies exist in other countries, characteristics of DSH have not been well-studied in Pakistan. Aims: To identify characteristics of deliberate self-harm in a hospital population, building on previous work carried out on this subject at this center. Methods: 284 cases admitted to the Aga Khan University Hospital, Karachi, between January 1997 and August 2003 were studied by analyzing medical records. Information was collected regarding demographics, family history, personal history, suicidal ideation, current stressors, the act of DSH and its management. Results: The most common age group was 21-25 years of age. The majority were females (60.3%). 95.1% of patients used self-poisoning as a method of deliberate self-harm. 47.3% used benzodiazepines for self-poisoning. Precipitating factors included conflict with family, marital problems, chronic illnesses, and unemployment. Comorbid psychiatric disorders were present in 76.4% patients. Discussion: Despite the difference in socio-cultural background of our patients, many of our results were found to be consistent with Western data. Our findings suggest that problem-solving therapy, stress management and improved mental health care may be effective interventions in preventing DSH. Prescription of medications should be regulated. Further studies are suggested at the community level.
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Self-inflicted penetrating injuries at level I trauma center
Article
Jan 2010
INJURY
Marko Bukur
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Feb 2011
S. Taylor Williams
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Joseph Currier
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Asa Klitte
John D Gilbert
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National Study of US Emergency Department Visits for Attempted Suicide and Self-Inflicted Injury, 1997-2001
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Arpi Doshi
Edwin D Boudreaux
Nan Wang
Carlos A Camargo
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E David Klonsky
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Costs of Occupational Injuries and Illnesses. 1st ed. United States of America: The University of
Jan 2000
J P Leigh
M Fahs
P J Landrigan
S B Markowitz
Leigh J.P., Fahs M., Landrigan P.J., Markowitz S.B. Costs of Occupational Injuries and Illnesses. 1st ed. United States of America: The University of Michigan Press; 2000.
Parikh's Textbook of Medical Jurisprudence, Forensic Medicine and Toxicology
Jan 2011
Dck Parikh
Parikh DCK. Parikh's Textbook of Medical Jurisprudence, Forensic Medicine and Toxicology. 7th ed. Subrahmanyam B, editor. New Delhi: CBS Publishers; 2011.
Forensic Medicine and Toxicology: Theory, Oral & Practical
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Kamakar RN. Forensic Medicine and Toxicology: Theory, Oral & Practical. 5th ed. Kolkata: Academic Publishers; 2015.
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Davison A.M. The incised wound. In: Rutty G.N, editor. Essentials of autopsy practice. London: Springer; 2004. p. 187-220.
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October 2017 · Australian Journal of Forensic Sciences
Isabella Mercurio
Daniele Capano
Paride Minervini
Mario Gabbrielli
The authors describe a case in which a 21-year-old male committed suicide using a rifle equipped with a muzzle brake, placed in contact with his head. A muzzle brake for firearms is a device positioned at the mouth of a weapon, which dissipates the gases, that are generated subsequent to the gun being discharged. Generally, when a gun is pressed against an anatomical region, where the skin lies ... [Show full abstract] on the bone surface without interposition of other soft parts, the entrance wound consists of many lacerated branches, originating from a central lack of tissue. In this case report, the use of a rifle with a muzzle brake generated an unexpected wound, circular in shape, with a diameter of 0.8 cm. This lesion was surrounded by a bruised area, circular in shape, and an abrasion collar of the height of 0.5 cm. In addition, a muzzle imprint mark consisting of intradermal bruises, composed of a narrow red line concentrically encircling the entrance hole, was found. In order to analyze in detail if this particular entrance wound could be associated with a weapon equipped with the muzzle brake, ballistic tests – with weapons compatible with that used by the victim – were performed.
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Homicidal Deaths Analysis in Two Egyptian Governorates (Gharbia and Qalubia): A Five-Year Retrospect...
January 2014
Islam Mitwalli
Islam Mitwalli
M.D. Forensic Medicine Abdelmonem G. Madboly
Egypt
Violent deaths are examined to determine the cause and the manner of death. Homicidal victims referred to the two governorates departments of forensic medicine during a 5-year period from 2006 to 2010 were studied. Data of homicidal deaths was obtained from available medicolegal reports and statistically analyzed. The total number of postmortem examinations was 3235 in Gharbia, and 4233 in ... [Show full abstract] Qalubia; of these 9.2% and 8.8% were homicidal deaths respectively. The total incidence rate of homicidal deaths per 100,000 populations in Gharbia was (0.0092) and in Qalubia was (0.0088). The highest incidence of crimes in the two governorates occurred during summer months (41.4-48%), among family members (37.4-40.4%), took place outdoor (62.6-64.4%), and was among males (77.1-75.7%) in the age group 31-40 years (25.3-25.9%). Urban areas accounted for 61.5% in Gharbia and 68.2% in Qalubia. Victims were commonly unemployed in Gharbia (24.6%), while they were workers in Qalubia (24%). Sharp, blunt and firearm weapons were commonly used in the two governorates. Stabbing was the most common cause of death in both Gharbia and Qalubia (40.7-45%), and commonly occurred among males. Burn was more common among females while asphyxia was equally distributed in the two governorates. Defense wounds were recorded in 32.7% of homicidal deaths in Gharbia and 21% in Qalubia. In conclusion, the pattern of homicide in the two governorates was very similar
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Convexity of the variance of a mixture distribution
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X X is a random variable that is sampled from the mixture of uniform distributions. In other words: X∼N∑i=1 w i⋅U(x i,x i+1),
X∼∑i=1 N w i⋅U(x i,x i+1),
where U(x i,x i+1)U(x i,x i+1) denotes a random variable that follows a uniform distribution in [x i,x i+1][x i,x i+1]. For feasibility, we need w≥0,∑N i=1 w i=1 w≥0,∑N i=1 w i=1.
In an optimization problem my variables are w i w i for i=1,…,N i=1,…,N, and I would like to upper bound the variance of X X. According to Wikipedia, the variance of X X is: V a r(X)=N∑i=1 w i(σ 2 i+μ 2 i−μ 2)
V a r(X)=∑i=1 N w i(σ 2 i+μ 2 i−μ 2)
where σ 2 i σ 2 i and μ i μ i are the variance and mean of U(x i,x i+1)U(x i,x i+1), respectively (which are parameters), and μ μ is the mean of the mixture, which is μ=N∑i=1 w i x i+x i+1 2.
μ=∑i=1 N w i x i+x i+1 2.
Thus, if my derivation is not wrong: V a r(X)=N∑i=1 w i(σ 2 i+μ 2 i−(N∑j=1 w j x j+x j+1 2)2) which is very ugly and appears to be non-convex to upper bound this function (edit: I want to constrain V a r(X)≤c o n s t a n t).
My question is, is there any trick, or any other convex approximation of such a variance, such that I can include an upper bound on the variance constraint?
convexity
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edited Jul 31, 2020 at 16:40
independentvariableindependentvariable
asked Jul 29, 2020 at 23:47
independentvariableindependentvariable
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1 Just for completion: in terms of only w i and x i the variance is given by 1 3 N∑i=1 w i(x 2 i+x i x i+1+x 2 i+1)−1 4(N∑i=1 w i(x i+x i+1))2. Unfortunately using Cauchy-Schwarz on the last term yields a lower bound.TheSimpliFire –TheSimpliFire♦ 2020-07-30 06:39:20 +00:00 Commented Jul 30, 2020 at 6:39
@TheSimpliFire thanks! How do you not have w 3 i terms?independentvariable –independentvariable 2020-07-30 13:02:59 +00:00 Commented Jul 30, 2020 at 13:02
1 You can remove the w 3 i terms by taking μ out of the summation and using that ∑i w i=1. The Wikipedia link you provided makes the same step.Kevin Dalmeijer –Kevin Dalmeijer 2020-07-30 19:10:56 +00:00 Commented Jul 30, 2020 at 19:10
1 The Wikipedia formula for the variance looks wrong to me. Assuming the component random variables are independent, shouldn't the variance of X be ∑i w 2 i σ 2 i?prubin –prubin♦ 2020-07-30 21:58:09 +00:00 Commented Jul 30, 2020 at 21:58
1 @independentvariable Sorry, I was misled by your first formula. What I wrote was the variance of the weighted sum of the uniform variables. A mixture distribution is not a weighted sum of independent variables, though. Rather, you pick one of the variables randomly based on the weights and the get an observation of that variable.prubin –prubin♦ 2020-07-31 15:51:12 +00:00 Commented Jul 31, 2020 at 15:51
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In order to find the best upper bound for variance, for given input values of u i and σ 2 i, you should globally maximize variance with respect to the w i, subject to the constraints w i≥0,Σ w i=1.
This can be formulated as a convex QP (Quadratic Programming problem), i.e., maximizing a concave quadratic subject to linear constraints. Hence it is easy to solve, unless n is gigantic, which hardly seems likely for any reasonable mixture distribution. I leave to the OP as an exercise, whether the KKT conditions can yield a closed form solution.
The convex QP takes the form:
maximize (Σ n i=1 w i(σ 2 i+μ 2 i))−μ 2 with respect to μ,w i
subject to Σ n i=1 w i μ i=μ,w i≥0∀i,Σ n i=1 w i=1.
If all u i are equal to each other, this would be a Linear Programming problem with compact constraints. Therefore the optimum would be at a vertex of the constraints, and in this case, that vertex would be w i=1 for the i corresponding to the largest σ 2 i, and all other w i=0.
Edit: In response to edit to question: "I want to constrain Var(X) ≤ constant)"
If the naive approach of adding the constraint Var(X) ≤ constant to my above convex QP formulation were performed, that would add a non-convex quadratic constraint, making the problem a non-convex Quadratically-Constrained Quadratic Program (QCQP), which requires a global optimizer, such as Gurobi 9.x or BARON to solve to global optimality.
However, there is an easier, faster method: Solve the (pre-Edit) convex QP formulation. Then maximum variance, accounting for the constraint Var(X) ≤ constant), equals
min(optimal objective value of convex QP formulation,constant).
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edited Jul 31, 2020 at 17:08
answered Jul 31, 2020 at 5:30
Mark L. StoneMark L. Stone
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Many thanks for your reply! Before going to the direction of an upper bound, I want to make sure whether the Wikipedia formula is correct. I think user prubin's answer is also given here without any common mean assumption: theanalysisofdata.com/probability/….independentvariable –independentvariable 2020-07-31 09:52:48 +00:00 Commented Jul 31, 2020 at 9:52
2 @prubin is usually correct, but not on this. Your link, theanalysisofdata.com is just flat out wrong., Don't believe me? Try a simulation. Handy formula: Variance = mean of conditional variance + variance of conditional mean. The 2nd of these terms is affected by how common or not the μ i are. The linked (and prubin's) formula is incorrect even if n=2,w 1=w 2=1/2,μ 1=μ 2=0,σ 2 1=1,σ 2 2=1. In that case, variance = 1, but linked and prubin's formula gives variance = 1/2.Mark L. Stone –Mark L. Stone 2020-07-31 11:37:33 +00:00 Commented Jul 31, 2020 at 11:37
2 Yes, Wikipedia's formulas are correct. You can verify by simulation if you don't trust your analytical skills. I made use of Wikipedia's formulas in my QP formulation, but I cleverly didn't substitute everything into the objective rather, added linear constraint for μ.Mark L. Stone –Mark L. Stone 2020-07-31 11:45:29 +00:00 Commented Jul 31, 2020 at 11:45
I don't completely understand the easier, faster method. Also, I'm a bit sceptical because the Var(X) ≤ constant has a non-convex feasible region in w, which seems difficult to avoid.Kevin Dalmeijer –Kevin Dalmeijer 2020-07-31 18:28:52 +00:00 Commented Jul 31, 2020 at 18:28
If the optimal objective value of the convex QP is > "constant", the max variance constraint is non-binding. If "constant" ≤ optimal objective value of convex QP, then by continuity of the objective function with respect to w i, the optimal objective value (i.e., max variance) with the max variance constraint is "constant". QED. Continuity is the key to sneaking around the non-convexity.Mark L. Stone –Mark L. Stone 2020-07-31 20:31:24 +00:00 Commented Jul 31, 2020 at 20:31
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6656 | https://www.aimsciences.org/article/doi/10.3934/amc.2022009 | The zero-error capacity of binary channels with 2-memories
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2024,Volume 18,Issue 1:179-191. Doi: 10.3934/amc.2022009
This issuePrevious ArticleBalanced (Z 2 u×Z 38 v, {3, 4, 5}, 1) difference packings and related codesNext ArticleOn the generalised rank weights of quasi-cyclic codes
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The zero-error capacity of binary channels with 2-memories
Guofen Zhang1,2,,
Ping Li1,,
Jianfeng Hou1,3,,and
Bo Bai1,
1.
Theory Lab HK, Huawei Tech. Investment Co., Limited, Shatin, N. T., Hong Kong
2.
Department of Mathematics, Beijing Jiaotong University, Beijing, China, 100044
3.
Center of Discrete Mathematics, Fuzhou University, Fujian, China, 350108
Corresponding author: Jianfeng Hou, E-mail: jfhou@fzu.edu.cn
Corresponding author: Jianfeng Hou, E-mail: jfhou@fzu.edu.cn
Received: September 2021
Revised: January 2022
Early access: March 2022
Published: February 2024
Abstract / IntroductionFull Text(HTML)Figure(8)Related PapersCited by
Abstract
Abstract
The zero-error capacity of a noisy channel is defined as the least upper bound of rate at which it is possible to transmit information with zero probability of error. It was posed by Shannon and extended to channels with memories by Ahlswede, Cai and Zhang. In this paper, we give a first step towards the zero-error capacity problems of binary channels with 2-memories, and determine the zero-error capacity of at least 2 24 possible cases in all 2 28 cases. ### 1. Introduction
In information theory, the zero-error capacity of a channel, proposed by Shannon , is defined as the least upper bound of rates at which it is possible to transmit information with zero probability of error. It is closely related to graph theory: Shannon characterized a memoryless channel by a corresponding graph G whose the vertices represent the input letters in the channel. There is an edge between each pair of vertices, if and only if corresponding letters are indistinguishable in transmission. Denote by α(G k) the maximum number of k-letter messages which can be sent without danger of confusion (two k-letter words are c o n f o u n d a b l e if for each 1≤i≤k, their i-th symbol are confoundable or equal). Then the S h a n n o n c a p a c i t y of G, is
Θ(G)=lim k→∞α(G k)k.
Determining the Shannon capacity of a graph is one of the most important topic in classical information theory, quantum information theory and graph theory. Shannon first determined the Shannon capacity of graphs with at most 5 vertices except a cycle with 5 vertices C 5. A breakthrough in this field was given by Lovász who proved that Θ(G)=5. Despite many results on Shannon capacity of cycles with 7 vertices C 7 [14,15,17], the capacity problem for C 7 is still open. For disjoint union of two graphs, Alon showed that there exist graphs G and H such that Θ(G+H)>Θ(G)+Θ(H), which disproved a conjectured given by Shannon . Keevash gave a negative answer to the Alon's question that for a ε>0, whether there are two graphs G and H satisfying Θ(G),Θ(H)<ε and Θ(G+H)>1−ε. For general graphs, Hu, Tamo and Shayevitz gave an upper bound on the Shannon capacity of a graph via a linear programming variation. Further results on Shannon capacity and related problems can be found in [3,4,7,8,10,13,18,19].
Recall that a vertex in the corresponding graph of a memoryless channel with finite input alphabet Γ is a letter from Γ. It was extended to channels with memories by Ahlswede, Cai and Zhang , where a vertex in the corresponding graph is a sequence of the input letter. As a first step towards a systematic study of the zero-error capacity of channels with short memory, Cohen, Fachini and Körner investigated the binary channels with 1-memory. Cao, Cai, Guo and Yeung completely solved the remaining problem of zero-error capacity of the binary channels with 1-memory.
In this paper, we focus on the zero-error capacity of binary channels with 2-memories, and give the zero-error capacity of at least 2 24 graphs. Note that each vertex in the corresponding graph is a 3-length sequence, which means the corresponding graph has 8 vertices. There are 2(8 2) possible graphs, which is more complicated than the binary channels with 1-memory.
The remainder of this paper is organized as follows. In Section 2, the basic model is introduced and the results of some related works are reviewed. We give the zero-error capacity of several graphs in Section 3. Section 4 concludes the paper.
Keywords:
zero-error capacity,
binary channels,
channels with memories,
graph.
Mathematics Subject Classification:Primary: 94A24, 94A40; Secondary: 05C35, 05C90.
Citation: Guofen Zhang, Ping Li, Jianfeng Hou, Bo Bai. The zero-error capacity of binary channels with 2-memories. Advances in Mathematics of Communications, 2024, 18(1): 179-191. doi: 10.3934/amc.2022009
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Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles Nomenclature A CP CV COP d E e F g H h h K k k kt M M m N n η ηth P Pa Qnet qnet R Ru S s T U u V VG V v v X X x Z area (m2) specific heat at constant pressure (kJ/(kg-K)) specific heat at constant volume (kJ/(kg-K)) coefficient of performance exact differential stored energy (kJ) stored energy per unit mass (kJ/kg) force (N) acceleration of gravity ( 9 m/s2) enthalpy (H= U + PV) (kJ) specific enthalpy (h= u + Pv) (kJ/kg) convective heat transfer coefficient (W/(m2-K) Kelvin degrees specific heat ratio, CP/CV 103 thermal conductivity (W/(m-°C)) molecualr weight or molar mass (kg/kmol) 106 mass (kg) moles (kmol) polytropic exponent (isentropic process, ideal gas n = k) isentropic efficiency for turbines, compressors, nozzles thermal efficiency (net work done/ heat added) pressure (kPa, MPa, psia, psig) Pascal (N/m2) net heat transfer (∑Qin - ∑Qout) (kJ) Qnet /m, net heat transfer per unit mass (kJ/kg) particular gas constant (kJ/(kg K)) universal gas constant (= 8 kJ/(kmol K) ) entropy (kJ/K) specific entropy (kJ/(kg K)) temperature ( °C, K, °F, R) internal energy (kJ) specific internal energy (kJ/(kg K)) volume (m3 ) volume flow rate (m3/s) velocity (m/s) specific volume (m3/kg) molar specific volume (m3/kmol) distance (m) exergy (kJ) quality elevation (m) Chapter 1- 1 Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles Nomenclature (continued) Wnet net work done [(∑Wout - ∑Win)other + Wb] (kJ) where Wb = wnet Wt δ ε φ ρ ω z 2 1 PdV for closed systems and 0 for control volumes Wnet /m, net work done per unit mass (kJ/kg) weight (N) inexact differential regenerator effectiveness relative humidity density (kg/m^3) humidity ratio Subscripts, superscripts a b f g fg gen H HP L net other P REF rev s sat v 1 2 i e ⋅ actual boundary saturated liquid state satruated vapor state saturated vapor value minus saturated liquid value generation high temperature heat pump low temperature net heat added to system or net work done by system work done by shaft and electrical means constant pressure refrigerator reversible isentropic or constant entropy or reversible, adiabatic saturation value constant volume initial state finial state inlet state exit state per unit time REFERENCE Cengel, Yunus A. and Boles, Michael A., Thermodynamics: An Engineering Approach, 3rd ed., McGraw-Hill Book Co., New York, 1998. Chapter 1- 2 Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles Systems may be considered to be closed or open, depending on whether a fixed mass or a fixed volume in space is chosen for study. A closed system consists of a fixed amount of mass and no mass may cross the system boundary. The closed system boundary may move. Examples of closed systems are sealed tanks and piston cylinder devices (note the volume does not have to be fixed). However, energy in the form of heat and work may cross the boundaries of a closed system. An open system, or control volume, has mass as well as energy crossing the boundary, called a control surface. Examples of open systems are pumps, compressors, turbines, valves, and heat exchangers. Chapter 1- 4 Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles An isolated system is a general system of fixed mass where no heat or work may cross the boundaries. An isolated system is a closed system with no energy crossing the boundaries and is normally a collection of a main system and its surroundings that are exchanging mass and energy among themselves and no other system. Isolated System Boundary Heat = 0 Work = 0 Mass = 0 Across Isolated Boundary Work Mass Surr 1 Surr 4 System Surr 3 Mass Heat Surr 2 Since some of the thermodynamic relations that are applicable to closed and open systems are different, it is extremely important that we recognize the type of system we have before we start analyzing it. Energy G Consider the system shown below moving with a velocity, V at an elevation, Z, relative to the reference plane. Chapter 1- 5 Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles E U KE PE = + + m m m m G2 V = u+ + gZ 2 e= ( kJ ) kg where e = E/m is the specific stored energy, and u = U/m is the specific internal energy. The change in stored energy of a system is given by ∆E = ∆U + ∆KE + ∆PE ( kJ ) Most closed systems remain stationary during a process and, thus, experience no change in their kinetic and potential energies. The change in the stored energy is identical to the change in internal energy for stationary systems. If ∆KE = ∆PE = 0, ∆E = ∆U ( kJ ) Property: Any characteristic of a system in equilibrium is called a property. The property is independent of the path used to arrive at the system condition. Some thermodynamic properties are pressure P, temperature T, volume V, and mass m. Properties may be intensive or extensive. Extensive properties are those that vary directly with size---or extent---of the system. Some Extensive Properties a. mass b. volume c. total energy d. mass dependent property Chapter 1- 7 Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles Intensive properties are those that are independent of size. Some Intensive Properties a. temperature b. pressure c. age d. color e. any mass independent property Extensive properties per unit mass are intensive properties. For example the specific volume v defined as Volume V v= = mass m and density ρ defined as mass m ρ= = volume V are intensive properties. Chapter 1- 8 FG m IJ H kg K FG kg IJ Hm K 3 3 Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles Sometimes we use the mole number in place of the mass. In the SI units the mole number is in kilogram-moles, or kmol. Newton’s second law is often written as ma F = gc where gc is called the gravitational constant and is obtained from the force definition. In the SI System 1 Newton is that force required to accelerate 1 kg mass 1 m/s2. The gravitational constant in the SI System is m (1kg)(1 2 ) ma kg m s gc = = =1 F 1N N s2 In the USCS 1 pound-force is that force required to accelerate 1 pound-mass 32 ft/s2. The gravitational constant in the USCS is ft (1lbm)(32 2 ) ma lbm ft s gc = = = 32 F 1lbf lbf s2 In the slug system, the gravitational constant is ft (1slug)(1 2 ) ma lbm ft s gc = = =1 F 1 lbf lbf s2 Chapter 1-10 Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles Example 1-1 An object at sea level has a mass of 400 kg. a) Find the weight of this object on earth. b) Find the weight of this object on the moon where the local gravitational acceleration is one-sixth that of earth. (a) Wt = mg m 1N Wt = (400 kg )(9 2 )( ) m s kg s2 = 3922 N (b) Wt = mg 9 m 1 N )( ) = (400 kg )( 2 6 s kg m s2 . N = 6538 Chapter 1-11 Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles Process Any change from one state to another is called a process. During a quasi-equilibrium or quasi-static process the system remains practically in equilibrium at all times. We study quasi-equilibrium processes because they are easy to analyze (equations of state apply) and work-producing devices deliver the most work when they operate on the quasi-equilibrium process. In some processes one thermodynamic property is held constant. Some of these processes are Process isobaric isothermal isochoric isentropic Property held constant pressure temperature volume entropy (see Chapter 6) System Boundary F Water Constant Pressure Process We can understand the concept of a constant pressure process by considering the above figure. The force exerted by the water on the face of the piston has to equal the force due to the combined weight of the piston and the bricks. If the combined weight of the piston and bricks is constant then F is constant and the pressure is constant even when the water is heated. We often show the process on a P-V diagram as shown below. Chapter 1-13 Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles State Postulate As noted earlier, the state of a system is described by its properties. But by experience not all properties must be known before the state is specified. Once a sufficient number of properties are known, the state is specified and all other properties are known. The number of properties required to fix the state of a simple, homogeneous system is given by the state postulate: The thermodynamic state of a simple compressible system is completely specified by two independent intensive properties. Cycle A process (or a series of connected processes) with identical end states is called a cycle. Below is a cycle composed of two processes, A and B. Along process A, the pressure and volume change from state 1 to state 2. Then to complete the cycle, the pressure and volume change from state 2 back to the initial state 1 along process B. Keep in mind that all other thermodynamic properties must also change so that the pressure is functions of volume as described by these two processes. P 2 Process B Process A 1 Chapter 1-14 V Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles Pabs = Patm ± Pgage Where the +Pgage is used when Pabs > Patm and –Pgage is used for a vacuum gage. The relation among atmospheric, gage, and vacuum pressures is shown below. Small to moderate pressure differences are measured by a manometer and a differential fluid column of height h corresponds to a pressure difference between the system and the surroundings of the manometer. This pressure difference is determined from the manometer fluid displaced height as Chapter 1-16 Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles ∆P = ρ g h ( kPa ) Example 1-3 A vacuum gage connected to a tank reads 30 kPa at a location where the atmospheric pressure is 98 kPa. What is the absolute pressure in the tank? Pabs = Patm − Pgage = 98 kPa − 30 kPa = 68 kPa Chapter 1-17 Chapter 1 Lecture notes for Thermodynamics: An Engineering Approach, 3rd Ed by Cengel and Boles Example 1-5 Both a gage and a manometer are attached to a gas tank to measure its pressure. If the pressure gage reads 80 kPa, determine the distance between the two fluid levels of the manometer if the fluid is mercury whose density is 13,600 kg/m3. ∆P h= ρg 103 N / m3 80 kPa kPa h= 1N kg m 13600 3 9 2 m s kg m / s 2 = 0 m Temperature Although we are familiar with temperature as a measure of “hotness” or “coldness”, it is not easy to give an exact definition of it. However, temperature is considered as a thermodynamic property that is the measure of the energy content of a mass. When heat energy is transferred to a body, the body's energy content increases and so does its temperature. In fact it is the difference in temperature that causes energy, called heat transfer, to flow from a hot body to a cold body. Two bodies are in thermal equilibrium when they have reached the same temperature. If two bodies are in thermal equilibrium with a third body, they are also in thermal equilibrium with each other. This simple fact is known as the zeroth law of thermodynamics. The temperature scales used in the SI and the English systems today are the Celsius scale and Fahrenheit scale, respectively. These two scales are based on a specified number of degrees between the freezing point of water ( 0 °C or 32 °F) and the boiling point of water (100 °C or 212 °F) and are related by Chapter 1-19
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Chapter 1 Lect ure notes fo r Thermodynam ics: An E ngineering Approach, 3 rd Ed by Cengel and Boles
Ch apter 1- 1
Nomenclature
A area (m 2)
C P specific heat at consta nt pressure (kJ/(kg-K))
C V specific heat at cons tant volume (kJ/(kg-K))
COP coefficient of performance
d exact differential
E stored energy (kJ)
e stored energy per unit mass (kJ/kg)
F force (N)
g acceleration of gravity ( 9.807 m/s 2)
H enthalpy (H= U + PV) (kJ)
h specific enthalpy (h= u + Pv) (kJ/kg)
h convective heat transfer coefficient (W/(m 2-K)
K Kelvin degrees
k specific heat ratio, C P/C V
k 10 3
k t thermal conductivity (W/(m-°C))
M molecualr weight or molar m ass (kg/kmol)
M 10 6
m mass (kg)
N moles (kmol)
n polytropic exponent (isentropi c process, ideal gas n = k)
η isentropic efficiency for turbines, compressors, nozzles
η th thermal efficiency (net work done/ heat added)
P pressure (kPa, MPa, psia, psig)
Pa Pascal (N/m 2)
Q net net heat transfer (∑Q in - ∑Q out) (kJ)
q net Q net /m, net heat transfer per unit m ass (kJ/kg)
R particular gas constant (kJ/(kg K))
R u universal gas constant (= 8.314 kJ/(kmol K) )
S entropy (kJ/K)
s specific entropy (kJ/(kg K))
T temperature ( °C, K, °F, R)
U internal energy (kJ)
u specific internal energy (kJ/(kg K))
V volume (m 3 )
V volume flow rate (m 3/s)
G
V velocity (m/s)
v specific volume (m 3/kg)
v
molar specific volume (m 3/kmol)
X distance (m)
X exergy (kJ)
x quality
Z elevation (m)
Chapter 1 Lect ure notes fo r Thermodynam ics: An E ngineering Approach, 3 rd Ed by Cengel and Boles
Ch apter 1- 2
Nomenclature (continued)
W net net work done [(∑W out - ∑W in)other + W b] (kJ)
where W b = PdV
1
2
z for closed systems and 0 for control volumes
w net W net /m, net work done per unit mass (kJ/kg)
W t weight (N)
δ inexact differential
ε regenerator effectiveness
φ relative humidity
ρ density (kg/m^3)
ω humidity ratio
Subscripts, superscripts
a actual
b boundary
f saturated liquid state
g satruated vapor state
fg saturated vapor value minus saturated liquid value
gen generation
H high temperature
HP heat pump
L low temperature
net net heat added to system or net work done by system
other work done by shaft and electrical means
P constant pressure
REF refrigerator
rev reversible
s isentropic or constant entropy or reversible, adiabatic
sat saturation value
v constant volume
1 initial state
2 finial state
i inlet state
e exit state
⋅per unit time
REFERENCE
Cengel, Yunus A. and Boles, Michael A., Thermodynamics: An Engineering Approach,
3 rd ed., McGraw-Hill Book Co., New York, 1998.
Chapter 1 Lect ure notes fo r Thermodynam ics: An E ngineering Approach, 3 rd Ed by Cengel and Boles
Ch apter 1- 3
INTRODUCTION
The study of thermodynamics is concerned with the ways energy is stored within a body
and how energy transformations, which involve heat and work, may take place. One of
the most fundam ental laws of nature is the conservation of energy principle. It sim ply
states that during an energy interaction, energy can change from one form to another but
the total amount of energy remains constant. That is, energy cannot be created or
destroyed.
This review of thermodynamics is based on the m acroscopic approach where a large
number of particles, called molecules, makes up the substance in question. The
macroscopic approach to thermodynam ics doe s not require acknowledge of the behavior
of individual particles and is called classical ther modynamics. It provides a direct and
easy way to obtain the solution of engi neering problems without being overly
cumbersome. A more elaborate approach, base d on the average behavior of large groups
of individual particles, is cal led statistical thermodynamics. This microscopic approach
is rather involved and is not reviewed here and leads to the definition of the second law
of thermodynamics. W e will approach th e second law of thermodynam ics from the
classical point of view and will learn that the second law of thermodynamics asserts that
energy has quality as well as quantity, and actual processes occu r in the direction of
decreasing quality of energy.
Closed, Open, and Isolated Systems
Thermodynamic system, or sim ply system, is defined as a quantity of matter or a region
in space chosen for study. The region outside the system is called the surroundings.
The real or imaginary surface that separate s the system from its surroundings is called
the boundary. The boundary of a system may be fixed or movable.
Surroundings are physical space outs ide the system boundary.
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Chapter 1 Lect ure notes fo r Thermodynam ics: An E ngineering Approach, 3 rd Ed by Cengel and Boles
Ch apter 1- 4
Systems may be considered to be clo sed or open, depending on whether a fixed mass or
a fixed volume in space is chosen for study.
A closed system consists of a fixed amount of mass and no mass may cross the system
boundary. The closed system boundary may move.
Examples of closed systems are sealed tanks and piston cylinde r devices (note the
volume does not have to be fixed). However, energy in the form of heat and work m ay
cross the boundaries of a closed system.
An open system, or control volume, has mass as well as en ergy crossing the boundary,
called a control surface. Examples of ope n systems are pumps, com pressors, turbines,
valves, and heat exchangers.
Chapter 1 Lect ure notes fo r Thermodynam ics: An E ngineering Approach, 3 rd Ed by Cengel and Boles
Ch apter 1- 5
An isolated system is a general system of fixed mass where no heat or work may cross
the boundaries. An isolated system is a closed system with no energy crossing the
boundaries and is normally a collection of a main system and its surroundings that are
exchanging mass and energy among them selves and no other system.
Since some of the thermodynam ic relations that are applicable to closed and open
systems are different, it is extremely im portant that we recognize th e type of system we
have before we start analyzing it.
Energy
Consider the system shown be low moving with a velocity,
G
V at an elevation, Z, relative
to the reference plane.
Isolated Syste m Boundary
Mass
System
Surr 2
Surr 3
Surr 4
Mass
Heat
Work
Surr 1
Heat = 0
Work = 0
Mass = 0
Across
Isolated
Boundary
Chapter 1 Lect ure notes fo r Thermodynam ics: An E ngineering Approach, 3 rd Ed by Cengel and Boles
Ch apter 1- 6
The total energy, E, of a system is the sum of all forms of energy that can exist within
the system such as thermal, mechanical, ki netic, p otential, electric, magnetic, chem ical
and nuclear. The total energy of the system is normally thought of as the sum of the
internal energy, kinetic energy, and potential energy. The internal energy, U, is that
energy associated with the molecular stru ct ure of a system and the degree of the
molecular activity (see Section 1-4 of text). The kinetic ener gy, KE, exists as a result of
the system's motion relative to an external reference fram e. When the system m oves
with velocity,
G
V, the kinetic energy is expressed as
KE m V kJ=
G
2
2()
The energy that a system possesses as a resu lt of its e levation in a gravitational field
relative to the exte rnal reference frame is called potent ial energy, PE, and is expressed as
KE mgZ kJ
=
()
where g is the gravitational acceleration and z is the elevation of th e center of gravity of
a system relative to the reference frame. Th e total energy of the system is expressed as
EU K E P E k J=
+
+
()
or, on a unit mass basis,
Z
General
System
CM
Reference Plane, Z=0
G
V
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Chapter 1 Lect ure notes fo r Thermodynam ics: An E ngineering Approach, 3 rd Ed by Cengel and Boles
Ch apter 1- 7
e E
m
U
m
KE
m
PE
m
kJ
kg
u V gZ
==++
=++
()
G 2
2
where e = E/m is the specific stored energy, and u = U/m is the specific internal energy.
The change in stored energy of a system is given by
∆
∆
∆
∆
EU K E P E k J=
+
+
()
Most closed systems remain stationary dur ing a process and, thus, experience no change
in their kinetic and potential energies. The chan ge in the stored energy is identical to the
change in internal energy for stationary system s.
If ∆KE = ∆PE = 0,
∆
∆
EU k J
=
()
Property:
Any characteristic of a system in equilibrium is called a property.
The property is independent of the path used to arrive at the system condition.
Some thermodynamic properties are pressure P, tem perature T, volume V, and mass m.
Properties may be intensive or exten sive.
Extensive properties are those th at vary directly with size---or extent---of the system.
Some Extensive Properties
a. mass
b. volume
c. total energy
d. mass dependent property
1 out of 22
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6658 | https://masters.ecampus.com/introductory-nuclear-physics-3rd-krane/bk/9780471805533 | Introductory Nuclear Physics
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Introductory Nuclear Physics
byKrane, Kenneth S.
Edition: 3rd
ISBN13: 9780471805533
ISBN10: 047180553X
Format: Hardcover
Pub. Date: 1991-01-16
Publisher(s): Wiley
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Summary
This comprehensive text provides an introduction to basic nuclear physics, including nuclear decays and reactions and nuclear structure, while covering the essential areas of basic research and practical applications. Its emphasis on phenomonology and the results of real experiments distinguish this from all other texts available. Discussions of theory are reinforced with examples which illustrate and apply the theoretical formulism, thus aiding students in their reading and analysis of current literature. The text is designed to provide a core of material for students with minimal background in mathematics or quantum theory and offers more sophisticated material in separate sections.
Author Biography
Kenneth S. Krane is Professor of Physics at Oregon State University, where he has served on the faculty since 1974, including 14 years as Department Chair. He received the Ph.D. in nuclear physics from Purdue University in 1970 and held postdoctoral research positions at the Los Alamos National Laboratory and the Lawrence Berkeley National Laboratory before joining the faculty at Oregon State. His research involves nuclear structure and nuclear spectroscopy, and has led to more than 100 papers in refereed journals and 30 years of funding in experimental nuclear physics from NSF and DOE. He was selected to be a Fellow of the American Physical Society by the Division of Nuclear Physics. He is also involved in education research and curriculum development and has held numerous NSF grants supporting those activities. He has served as chair of the APS Committee on Education, the APS Forum on Education, and the AIP Advisory Committee on Physics Education. From 1995-2006 he was the director of the Workshop for New Faculty in Physics and Astronomy, a national mentoring program for college and university faculty. In 2004 he was awarded the Millikan Medal of the American Association of Physics Teachers in recognition of his contributions to physics teaching.
Table of Contents
UNIT I BASIC NUCLEAR STRUCTURE
Basic Concepts 2(7)
History and Overview 3(1)
Some Introductory Terminology 4(1)
Nuclear Properties 5(2)
Units and Dimensions 7(2)
Elements Of Quantum Mechanics 9(35)
Quantum Behavior 9(3)
Principles of Quantum Mechanics 12(3)
Problems in One Dimension 15(10)
Problems in Three Dimensions 25(9)
Quantum Theory of Angular Momentum 34(3)
Parity 37(2)
Quantum Statistics 39(1)
Transitions Between States 40(4)
Nuclear Properties 44(36)
The Nuclear Radius 44(15)
Mass and Abundance of Nuclides 59(6)
Nuclear Binding Energy 65(5)
Nuclear Angular Momentum and Parity 70(1)
Nuclear Electromagnetic Moments 71(4)
Nuclear Excited States 75(5)
The Force Between Nucleons 80(36)
The Deuteron 80(6)
Nucleon-Nucleon Scattering 86(10)
Proton-Proton and Neutron-Neutron Interactions 96(4)
Properties of the Nuclear Force 100(8)
The Exchange Force Model 108(8)
Nuclear Models 116(44)
The Shell Model 117(17)
Even-Z, Even-N Nuclei and Collective Structure 134(15)
More Realistic Nuclear Models 149(11)
UNIT II NUCLEAR DECAY AND RADIOACTIVITY
Radioactivite Decay 160(32)
The Radioactive Decay Law 161(4)
Quantum Theory of Radiative Decays 165(4)
Production and Decay of Radioactivity 169(1)
Growth of Daughter Activities 170(3)
Types of Decays 173(5)
Natural Radioactivity 178(3)
Radioactive Dating 181(3)
Units for Measuring Radiation 184(8)
Detecting Nuclear Radiations 192(54)
Interactions of Radiation with Matter 193(11)
Gas-Filled Counters 204(3)
Scintillation Detectors 207(6)
Semiconductor Detectors 213(4)
Counting Statistics 217(3)
Energy Measurements 220(7)
Coincidence Measurements and Time Resolution 227(3)
Measurement of Nuclear Lifetimes 230(6)
Other Detector Types 236(10)
Alpha Decay 246(26)
Why Alpha Decay Occurs 246(1)
Basic Alpha Decay Processes 247(2)
Alpha Decay Systematics 249(2)
Theory of Alpha Emission 251(6)
Angular Momentum and Parity in Alpha Decay 257(4)
Alpha Decay Spectroscopy 261(11)
Beta Decay 272(55)
Energy Release in Beta Decay 273(4)
Fermi Theory of Beta Decay 277(5)
The ``Classical'' Experimental Tests of the Fermi Theory 282(7)
Angular Momentum and Parity Selection Rules 289(3)
Comparative Half-Lives and Forbidden Decays 292(3)
Neutrino Physics 295(3)
Double Beta Decay 298(4)
Beta-Delayed Nucleon Emission 302(7)
Nonconservation of Parity 309(6)
Beta Spectroscopy 315(12)
Gamma Decay 327(51)
Energetics of Gamma Decay 327(1)
Classical Electromagnetic Radiation 328(3)
Transition to Quantum Mechanics 331(2)
Angular Momentum and Parity Selection Rules 333(2)
Angular Distribution and Polarization Measurements 335(6)
Internal Conversion 341(7)
Lifetimes for Gamma Emission 348(3)
Gamma-Ray Spectroscopy 351(10)
Nuclear Resonance Fluorescence and the Mossbauer Effect 361(17)
UNIT III NUCLEAR REACTIONS
Nuclear Reactions 378(66)
Types of Reactions and Conservation Laws 378(2)
Energetics of Nuclear Reactions 380(8)
Isospin 388(4)
Reaction Cross Sections 392(3)
Experimental Techniques 395(1)
Coulomb Scattering 396(9)
Nuclear Scattering 405(3)
Scattering and Reaction Cross Sections 408(5)
The Optical Model 413(3)
Compound-Nucleus Reactions 416(3)
Direct Reactions 419(5)
Resonance Reactions 424(7)
Heavy-Ion Reactions 431(13)
Neutron Physics 444(34)
Neutron Sources 445(2)
Absorption and Moderation of Neutrons 447(4)
Neutron Detectors 451(5)
Neutron Reactions and Cross Sections 456(6)
Neutron Capture 462(3)
Interference and Diffraction with Neutrons 465(13)
Nuclear Fission 478(50)
Why Nuclei Fission 479(5)
Characteristics of Fission 484(4)
Energy in Fission 488(5)
Fission and Nuclear Structure 493(8)
Controlled Fission Reactions 501(5)
Fission Reactors 506(6)
Radioactive Fission Products 512(4)
A Natural Fission Reactor 516(4)
Fission Explosives 520(8)
Nuclear Fusion 528(31)
Basic Fusion Processes 529(1)
Characteristics of Fusion 530(4)
Solar Fusion 534(4)
Controlled Fusion Reactors 538(15)
Thermonuclear Weapons 553(6)
Accelerators 559(43)
Electrostatic Accelerators 563(8)
Cyclotron Accelerators 571(10)
Synchrotrons 581(7)
Linear Accelerators 588(5)
Colliding-Beam Accelerators 593(9)
UNIT IV EXTENSIONS AND APPLICATIONS
Nuclear Spin and Moments 602(51)
Nuclear Spin 602(3)
Nuclear Moments 605(5)
Hyperfine Structure 610(9)
Measuring Nuclear Moments 619(34)
Meson Physics 653(48)
Yukawa's Hypothesis 653(3)
Properties of Pi Mesons 656(15)
Pion-Nucleon Reactions 671(8)
Meson Resonances 679(7)
Strange Mesons and Baryons 686(6)
CP Violation in K Decay 692(9)
Particle Physics 701(54)
Particle Interactions and Families 701(9)
Symmetries and Conservation Laws 710(8)
The Quark Model 718(3)
Colored Quarks and Gluons 721(4)
Reactions and Decays in the Quark Model 725(8)
Charm, Beauty, and Truth 733(9)
Quark Dynamics 742(4)
Grand Unified Theories 746(9)
Nuclear Astrophysics 755(33)
The Hot Big Bang Cosmology 756(4)
Particle and Nuclear Interactions in the Early Universe 760(4)
Primordial Nucleosynthesis 764(5)
Stellar Nucleosynthesis (A ≲ 60)769(7)
Stellar Nucleosynthesis (A > 60)776(4)
Nuclear Cosmochronology 780(8)
Applications of Nuclear Physics 788(27)
Trace Element Analysis 788(6)
Mass Spectrometry with Accelerators 794(2)
Alpha-Decay Applications 796(4)
Diagnostic Nuclear Medicine 800(8)
Therapeutic Nuclear Medicine 808(7)
Appendix A Special Relativity 815(3)
Appendix B Center-Of-Mass Reference Frame 818(4)
Appendix C Table Of Nuclear Properties 822(12)
Credits 834(1)
Index 835
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6659 | https://pmc.ncbi.nlm.nih.gov/articles/PMC3100749/ | Pituitary Magnetic Resonance Imaging for Sellar and Parasellar Masses: Ten-Year Experience in 2598 Patients - PMC
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J Clin Endocrinol Metab
. 2011 Apr 6;96(6):1633–1641. doi: 10.1210/jc.2011-0168
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Pituitary Magnetic Resonance Imaging for Sellar and Parasellar Masses: Ten-Year Experience in 2598 Patients
Pouyan Famini
Pouyan Famini
1 Departments of Medicine (P.F., S.M.) and Imaging (M.M.M.), Pituitary Center, Cedars Sinai Medical Center, Los Angeles, California 90048
Find articles by Pouyan Famini
1, Marcel M Maya
Marcel M Maya
1 Departments of Medicine (P.F., S.M.) and Imaging (M.M.M.), Pituitary Center, Cedars Sinai Medical Center, Los Angeles, California 90048
Find articles by Marcel M Maya
1, Shlomo Melmed
Shlomo Melmed
1 Departments of Medicine (P.F., S.M.) and Imaging (M.M.M.), Pituitary Center, Cedars Sinai Medical Center, Los Angeles, California 90048
Find articles by Shlomo Melmed
1,✉
Author information
Article notes
Copyright and License information
1 Departments of Medicine (P.F., S.M.) and Imaging (M.M.M.), Pituitary Center, Cedars Sinai Medical Center, Los Angeles, California 90048
✉
Address all correspondence and requests for reprints to: Shlomo Melmed, M.D., Cedars Sinai Medical Center, 8700 Beverly Boulevard, Room 2015, Los Angeles, California 90048. E-mail: melmed@csmc.edu.
✉
Corresponding author.
Series information
Extensive Clinical Experience
Received 2011 Jan 19; Accepted 2011 Mar 10; Issue date 2011 Jun.
Copyright © 2011 by The Endocrine Society
PMC Copyright notice
PMCID: PMC3100749 PMID: 21470998
We discuss the prevalence and diagnosis of sellar and parasellar masses by pituitary MRI.
Abstract
Context:
Sellar and parasellar masses present with overlapping clinical and radiological features ranging from asymptomatic incidental presentations and hormonal effects to compressive local mass effects. Pituitary masses are diagnosed with increased frequency with magnetic resonance imaging (MRI) advancements and availability, but indications and diagnostic outcomes of MRI screening for sellar lesions are not defined. Although pituitary adenomas are the most frequently encountered sellar mass lesions, other etiologies should be considered in the differential diagnosis of a sellar mass.
Setting:
The study was conducted at a tertiary pituitary center.
Patients:
This study was a retrospective review of 2598 subjects undergoing at least one pituitary MRI scan from 1999 to 2009.
Main Outcome Measure:
Prevalence and diagnosis of specific sellar and parasellar masses as screened by pituitary MRI.
Results:
The most common indications for pituitary imaging, excluding known mass follow-up, were for evaluation of hyperprolactinemia or hypogonadism. A normal pituitary gland was reported in 47% of subjects undergoing pituitary MRI. The most common pituitary adenomas initially identified by MRI included prolactinoma (40%), nonfunctioning adenoma (37%), and GH adenoma (13%). Nonadenomatous sellar masses accounted for 18% of visible lesions, of which the most common were Rathke's cleft cyst (19%), craniopharyngioma (15%), and meningioma (15%). Metastases accounted for 5% of nonpituitary lesions and breast cancer was the most common primary source.
Conclusions:
Half of all pituitary MRI scans performed in a large patient population yielded no visible lesion. Nonadenomatous pituitary lesions should be considered in the diagnosis of sellar masses observed on MRI, and a high clinical suspicion is required to exclude the presence of a nonfunctioning pituitary adenoma.
Pituitary tumors account for up to 15% of all intracranial masses (1), and pituitary adenomas are reported to account for 90% of sellar and parsellar lesions (2, 3). Clinically active pituitary adenomas occur at a prevalence of 1:1064 to 1:1288 to the general population (4, 5). Other sellar lesions include nonneoplastic cystic lesions, germ cell tumors, gliomas, lymphomas, meningiomas, metastatic tumors, vascular lesions, granulomatous and inflammatory lesions, and infections including bacterial abscesses as well as pituitary hyperplasia (2, 3, 6–8). Of the nonadenomatous masses reported in surgical series, Rathke's cleft cyst is the most commonly encountered (23%), metastatic cancer accounts for 12% of cases, and pituitary lymphoma occurred in one of 83 nonpituitary adenomatous masses observed (2). Because only single case reports of rare nonadenomatous lesions are usually published, it is difficult to assess the true prevalence of such lesions.
With advancements in imaging occurring over the past decade as well as the availability of refined endocrine testing techniques, pituitary masses are diagnosed with increased frequency. Increased incidence of pituitary adenomas observed over the second half of an 18-yr study period was due to a 3-fold increased frequency of incidentally discovered pituitary adenomas (9). These pituitary incidentalomas are discovered on computed tomography (CT), or magnetic resonance imaging (MRI) performed for evaluation of unrelated disorders such as head trauma or cancer staging or because of nonspecific symptoms such as headache. MRI scans of clinically normal subjects in the general population have visualized silent pituitary tumors sized 3 mm or greater in diameter in approximately 10% of subjects (10). CT scans of the sellar area in subjects examined for reasons unrelated to pituitary disease revealed sellar lesions sized 3 mm or greater in diameter in 3.7–20% of the subjects (11–13). Pituitary adenomas occur in 1.5–27% of individuals without prior suspected pituitary disease in autopsy series (14). The evaluation and treatment of pituitary incidentalomas are determined by their size and presence of hypersecretory syndromes and/or compressive central symptoms (15, 16). However, further evaluation of these small incidentalomas (<10 mm) may not be cost effective, especially if they are asymptomatic (17).
Specific anatomical landmarks of the sella turcica and the surrounding parasellar region may determine presenting symptoms of masses arising in this region. Nonadenomatous parasellar lesions may present with hypopituitarism, or mass effect with compressive symptoms including headache or visual symptoms, especially when endocrinological evaluation does not identify a hypersecretory syndrome (18). Precise imaging with high contrast and topographic resolution is critical in visualizing this small-volume area to determine both location and specific characteristics of masses, which are important for diagnosis (8, 19, 20).
To date, two surgical case series have reported the prevalence of nonadenomatous sellar and parasellar masses (2, 3). However, the utility of MRI imaging as a diagnostic tool for screening and diagnosis of sellar masses in a large patient population has not been reported. Pituitary MRI identifies sellar tumors and pituitary masses and offers high contrast and multiplanar, thin pituitary cuts enabling evaluation of small soft tissue changes (20). MRI also allows accurate visualization of mass effects on neighboring soft tissues. The objective of this study was to identify the frequency and diagnosis of pituitary masses in an identified patient population undergoing sellar imaging because diagnoses of pituitary-related masses are not all ultimately made by pathological analysis of surgical specimens. We also sought to identify indications for pituitary imaging and the prevalence of incidentalomas, hypopituitarism, and neuroophthalmic symptoms associated with nonadenomatous lesions, as identified by MRI.
Materials and Methods
This study was approved by the Institutional Review Board of the Cedars Sinai Medical Center. Dedicated pituitary MRIs performed at the Cedars Sinai Medical Center from January 1, 1999, to November 12, 2009, were reviewed to identify subjects who had undergone at least one pituitary MRI during this time period. Subjects were excluded from study if the indication for imaging was for purposes of another approved research project. Subjects meeting these criteria were identified, and one reviewer collected data from electronic medical records.
Medical charts were further reviewed for the primary clinical indication that led to a referral for pituitary MRI. In some instances in which indications were nonspecific for pituitary masses, as for evaluation of headache, a concurrent endocrinopathy was either not present or not yet diagnosed. Pituitary MRI scans with findings of questionable lesions of less than 3 mm, without evidence of a concurrent hyperfunctioning endocrinopathy, were considered normal scans. After pituitary MRI scans screened positively for sellar or parasellar masses were identified, the presence of these masses were also sought on prior nonpituitary imaging procedures, if present. Incidentalomas were defined as masses identified on imaging procedures before pituitary MRI for indications unrelated to an endocrinopathy or for visual symptoms consistent with a sellar mass. Endocrinopathies, including hyperfunctioning adenomas, or visual field loss not initially suspected at time of initial presentation was in some cases subsequently diagnosed after further evaluation.
The review of 2598 pituitary MRI records (Fig. 1) revealed a normal pituitary gland reported in 1231 subjects. Within the remaining 1367 subjects in whom a sellar, parasellar mass or postsurgical changes were reported on MRI, a final diagnosis could not be obtained in 159 cases. Reasons for lack of diagnosis included inaccessible outside records (n = 131), serial imaging follow-up (n = 13), diagnosis not pursued by the physician of record (n = 7), nondiagnostic pathologic result (n = 5), and patient lost to follow-up (n = 3). A diagnostic pathological report, considered the gold standard, was available in 435 cases.
Fig. 1.
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Screening results of pituitary MRI scans performed in 2598 patients between January 1999 and November 2009. After review and analysis of records, a normal pituitary was diagnosed in 1242 subjects, an abnormal pituitary in 1356 subjects, and 159 masses could not be assessed.
Results
Indications for Imaging
Clinical indications for performing a pituitary MRI are listed in Table 1. The most common indication was for follow-up of a previously identified mass including incidentalomas identified on prior x-ray, CT, or MRI scans. The most common endocrinological indication for performing a pituitary MRI was to evaluate for suspected prolactinoma, followed by evaluation of suspected Cushing's disease and hypogonadism. In 142 cases, a pituitary MRI was ordered for evaluation of neurological symptoms without the presence of a concurrent endocrinopathy. In such instances, headache was the most common indication followed by visual blurriness or loss [second cranial nerve (CN II)], and visual field defects.
Table 1.
Indications for initial pituitary MRI performed in 2598 patients from 1999 to 2009
| Indications | n |
:---: |
| Follow-up of known massesa | 924 |
| Follow-up of nonpituitary tumors | 4 |
| Suspected mass | |
| Prolactinoma | 455 |
| ACTH adenoma | 349 |
| Nonfunctioning adenoma/mass | 103 |
| GH adenoma | 39 |
| TSH adenoma | 20 |
| Metastases to pituitary | 18 |
| FSH/LH functioning adenoma | 11 |
| Sarcoidosis | 2 |
| Endocrinological work-up | |
| Hypogonadism | 247 |
| Hypopituitarism | 128 |
| GH deficiency | 41 |
| Hypothyroidism | 34 |
| Adrenal insufficiency | 32 |
| Diabetes insipidus | 18 |
| MEN1 syndrome | 17 |
| Postpartum | 8 |
| SIADH | 6 |
| Neurological symptoms | |
| Headache | 75 |
| Visual loss/blurring | 24 |
| Bitemporal hemianopsia | 15 |
| Altered mental status | 5 |
| Syncope | 5 |
| Homonymous hemianopsia | 5 |
| Diplopia | 3 |
| Seizure | 3 |
| Trauma | 3 |
| CVA/TIA | 2 |
| Meningitis | 1 |
| Cranial neuropathy (III, IV, VI) | 1 |
| Total | 2598 |
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MEN1, Multiple endocrine neoplasia 1; SIADH, syndrome of inappropriate antidiuretic hormone; CVA, cerebrovacular accident; TIA, transient ischemic attack.
a
Includes incidentalomas discovered on prior x-rays, CT, or MRI scans and undiagnosed masses.
Evaluation of endocrinopathies resulted in a variable percentage of positive pituitary MRI scans. Screening for acromegaly lead to a finding of a pituitary mass in 69% of pituitary MRIs ordered for this clinical indication. Evaluation of hyperprolactinemia with pituitary MRI resulted in 190 observed masses or 42% of 455 scans. A pituitary abnormality was found in 28% of pituitary MRIs ordered for evaluation of Cushing's disease. Evaluation of panhypopituitarism and hypogonadism resulted in identifying a pituitary lesion in 18 and 16% of pituitary MRIs, respectively.
Pituitary MRI
When comparing pituitary MRI results with the standard of pathological diagnosis, we found pituitary MRI to have a sensitivity and specificity of 99 and 29%, respectively. Four false-negative and 15 false-positive MRI reports were identified. Twenty-one diagnostic pathological reports demonstrated normal pituitary tissue.
Four hundred thirty-five diagnostic pathological reports were available in the electronic medical records. Subjects were excluded from this assessment if pituitary MRI was performed after surgical intervention, which resulted in complete resection of a known mass. When the findings of the first pituitary MRI performed for these patients were analyzed, 382 met such criteria, with 357 true positives, four false negatives, 15 false positives, and six true negatives.
Diagnosis
A pituitary mass was diagnosed in 1197 patients undergoing MRI as shown in Table 2. Of 2598 cases a normal pituitary was diagnosed in 1242 subjects, and 159 pituitary masses could not be diagnosed. Pituitary adenomas accounted for 981 of all diagnosed masses, with most due to prolactinomas and nonfunctioning adenomas. Rare cases of a functioning TSH adenoma, functioning LH/FSH adenoma, and other mixed hormone producing adenomas were also observed. Four hundred thirty-two adenomas were newly diagnosed after pituitary MRI scans. A total of 981 adenomas were identified and were confirmed in 303 cases with a diagnostic pathological report available at this institution (Table 2). Furthermore, 83 macroprolactinomas (>10 mm in diameter) were definitively diagnosed based on concomitant serum prolactin values of greater than 200 ng/ml.
Table 2.
Diagnoses in 2598 patients who underwent at least one pituitary MRI from January 1999 to November 2010
| Diagnoses | Total | Number of masses, [age range], and (average age), at initial pituitary MRIa | Newly diagnosedb | Pathologyc |
:---: :---:
| Males | Females |
| Anterior pituitary tumors | | | | | |
| Prolactinoma | 395 | 90 [17–81] (46) | 305 [15–89] (36) | 182 | 54 |
| Nonfunctioning adenoma | 364 | 162 [18–85] (54) | 202 [17–87] (47) | 166 | 164 |
| GH adenoma | 127 | 59 [19–84] (49) | 68 [18–82] (45) | 31 | 43 |
| ACTH adenoma | 84 | 18 [19–77] (43) | 66 [10–66] (38) | 47 | 34 |
| GH/prolactin mixed adenoma | 4 | 3 [24–42] (32) | 1 | 1 | 4 |
| Nelson's syndrome | 2 | 0 [24–42] (30) | 2 [44–66] (50) | 1 | 0 |
| Pituitary carcinomad | 2 | 2 [45–75] (60) | 0 | 2 | 2 |
| LH/FSH functioning adenoma | 1 | 1 | 0 | 0 | 1 |
| TSH adenoma | 1 | 0 | 1 | 1 | 1 |
| GH/TSH mixed adenoma | 1 | 0 | 1 | 1 | 0 |
| Cysts | | | | | |
| Rathke's cleft cyst | 42 | 12 [14–75] (40) | 30 [2–68] (40) | 35 | 25 |
| Craniopharyngioma | 33 | 13 [1–54] (29) | 20 [8–73] (45) | 12 | 20 |
| Arachnoid | 2 | 2 [33–56] (45) | 0 | 0 | 1 |
| Epidermoid | 1 | 0 | 1 | 1 | 1 |
| Pineal cyst | 1 | 1 | 0 | 1 | 0 |
| Nonadenomatous neoplasms | | | | | |
| Meningioma | 32 | 7 [28–81] (52) | 25 [33–89] (64) | 15 | 13 |
| Chordoma | 3 | 3 [24–69] (43) | 0 | 1 | 3 |
| Pituitary lymphoma | 2 | 1 | 1 | 2 | 2 |
| Chondrosarcoma | 1 | 0 | 1 | 1 | 1 |
| Embryonal rhabdomyosarcoma | 1 | 0 | 1 | 0 | 1 |
| Germinoma | 1 | 1 | 0 | 1 | 1 |
| Granular cell tumor | 1 | 0 | 1 | 0 | 1 |
| Hemangiopericytoma, malignant | 1 | 0 | 1 | 1 | 1 |
| Leiomyosarcoma | 1 | 0 | 1 | 1 | 1 |
| Mucoepidermoid carcinoma | 1 | 0 | 1 | 0 | 1 |
| Pituicytoma | 1 | 1 | 0 | 0 | 1 |
| Xanthogranuloma | 1 | 1 | 0 | 1 | 1 |
| Inflammatory and vasculitidies | | | | | |
| Lymphocytic hypophysitis | 3 | 1 | 2 [26–34] (30) | 1 | 2 |
| Hypophysitis, unspecified type | 2 | 0 | 2 [37–54] (46) | 0 | 0 |
| Lymphocytic infundibulitis | 1 | 0 | 1 | 0 | 1 |
| Amyloidosis, primary | 1 | 0 | 1 | 1 | 0 |
| Sarcoidosis | 1 | 1 | 0 | 0 | 0 |
| Wegener's granulomatosis | 1 | 1 | 0 | 1 | 1e |
| Infectious | | | | | |
| Pseudomonas aeruginosa | 1 | 1 | 0 | 1 | 1 |
| Syphilis | 1 | 1 | 0 | 1 | 0 |
| Metastases | | | | | |
| Breast | 3 | 0 | 3 [32–69] (52) | 3 | 1 |
| CNS lymphoma, to pituitary stalk | 1 | 1 | 0 | 1 | 1 |
| Nasopharyngeal lymphoma | 1 | 0 | 1 | 1 | 1 |
| Liver epitheliod hemangioendothelioma | 1 | 1 | 0 | 0 | 1 |
| Lung, adenocarcinoma | 1 | 1 | 0 | 1 | 0 |
| Pineal germinoma/dysgerminoma | 1 | 1 | 0 | 1 | 1 |
| Plasmacytoma | 1 | 1 | 0 | 1 | 1 |
| Prostate, adenocarcinoma | 1 | 1 | 0 | 1 | 1 |
| Sinusoidal squamous cell carcinoma | 1 | 0 | 1 | 1 | 1 |
| Vascular | | | | | |
| Apoplexy with masses | 16 | 9 [37–70] (54) | 7 [23–88] (50) | 13 | 9 |
| Carotid aneurysm | 4 | 2 [57–78] (68) | 2 [41–86] (64) | 4 | n/a |
| Hypothalamic cavernous angioma | 1 | 0 | 1 | 1 | 0 |
| Hypothalamic interpeduncular hematoma | 1 | 1 | 0 | 1 | 1 |
| Miscellaneous | | | | | |
| Empty sella | 21 | 5 [18–48] (36) | 16 [32–62] (47) | 16 | n/a |
| Hyperplasia | 14 | 0 | 14 [14–58] (40) | 10 | 11 |
| Ectopic pituitary gland | 4 | 2 [14 days–25] (12) | 2 [17–67] (42) | 4 | 0 |
| Fibrous dysplasia | 3 | 0 | 3 [26–67] (47) | 1 | 2 |
| Lipoma | 1 | 0 | 1 | 1 | 0 |
| Hypothalamic | | | | | |
| Astrocytoma | 2 | 0 | 2 [13–37] (25) | 1 | 1 |
| Germinoma | 1 | 1 | 0 | 0 | 0 |
| Hamartoma | 1 | 0 | 1 | 0 | 0 |
| Undiagnosed masses | 159 | 48 [15–93] (48) | 111 [10–89] (43) | n/a | 5f |
| Normal pituitary | 1242 | 436 [2 months–88] (44) | 806 [2–91] (39) | n/a | 21 |
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n/a, Not applicable.
a
Age specified in years, unless otherwise noted. Age range and average noted if applicable;
b
masses diagnosed after initial pituitary MRI; includes undiagnosed incidentalomas;
c
number of pathological reports available at this institution to confirm diagnoses;
d
two cases of pituitary carcinomas initially diagnosed as prolactinomas, which changed at time of biopsy after initial pituitary MRI;
e
pathology confirmed by serological tests;
f
five patients with nondiagnostic pathology results.
Two hundred sixteen pituitary masses identified by MRI were not adenomas, accounting for 18% of all observed lesions. One hundred twenty-four of these masses were newly diagnosed after pituitary MRI scans and confirmed with 111 pathological reports. The most common etiology encountered in this group was a pituitary cyst, with Rathke cleft cyst representing 19% of all nonadenomatous lesions. Other commonly observed masses were craniopharyngiomas (15%) and meningiomas (15%).
Eleven cases of pituitary metastases originating from extrapituitary primary cancers were observed. Breast cancer was the most common primary source (n = 3); lung and prostate cancer were also observed. Four cases were due to local tumor spread, including one primary central nervous system (CNS) lymphoma metastasis to the pituitary infundibulum and one nasopharyngeal lymphoma invading the parasellar region without a focal effect on the pituitary gland.
Two other cases of pituitary lymphoma were observed, one diagnosed with a primary diffuse large B cell pituitary lymphoma with lung metastases. Whole-body positron emission tomography/CT demonstrated a suprasellar mass and three pulmonary pleural masses. Biopsies of both the pituitary and pulmonary lesions demonstrated similar pathology. The second patient had stage IV large B cell lymphoma including infiltration of the anterior pituitary without CNS involvement and no primary lymphoma source identified.
Five patients were diagnosed with hypophysitis, three of whom were histologically confirmed as lymphocytic hypophysitis by pathology. Panhypopituitarism was the initial presentation in all five cases, two occurring after pregnancy without a history of significant peripartum bleeding. One patient underwent complete resolution of hypopituitarism without surgical intervention. Pituitary MRI demonstrated large masses (>10 mm in diameter) extending into the suprasellar space with radiographic evidence of chiasmatic compression. Three cases were initially diagnosed as macroadenomas based on MRI characteristics. Hypophysitis was suspected in the other two lesions due to atypical pituitary enhancement, consistent with inflammation. Serial pituitary MRI scans revealed spontaneous pituitary size regression in both cases of hypophysitis treated medically and in the case of lymphocytic hypophysitis diagnosed on surgical biopsy.
Two cases of pituitary infection were observed in immune-compromised patients with HIV infection. One pituitary abscess was due to Pseudomonas aeruginosa. The second patient presented with headache, fevers, and new-onset panhypopituitarism with serology positive for syphilis and extensive work-up negative for other infectious etiologies. MRI demonstrated mild pituitary enlargement and ring enhancement of the gland, mild infundibular thickening, and bony sellar expansion. After penicillin treatment and decreasing titers of serum rapid plasma reagin titers, serial pituitary MRIs demonstrated both decreasing pituitary gland size and differential enhancement consistent with decreasing inflammation.
Other notable lesions observed were pituitary hyperplasia (n = 14) and primary empty sella (n = 21). Five cases of hyperplasia were diffuse without pituitary hyperfunction, and nine others were pathologically proven corticotroph hyperplasias producing ACTH. An ACTH adenoma was initially suspected in all of nine cases with clinical and laboratory evidence of Cushing's disease. Primary empty sella was diagnosed based on MRI findings for evaluation of hypopituitarism, hyperprolactinemia, Cushing's syndrome, incidental findings of nonpituitary MRI scans, or follow-up of a known case.
Neuroophthalmic symptoms and endocrinopathies
Headache (57%) was the most common presenting symptom in patients with nonadenomatous masses identified by MRI, with a significant difference (P< 0.001) between nonfunctioning adenomas (38%) and functioning adenomas (27%) assessed by χ 2 test analysis. Furthermore, a significant difference (P< 0.01) was observed for headache prevalence between nonfunctioning and functioning adenomas. Twenty-nine cases of visual field deficits and 25 cases of decreased visual acuity or loss attributed to optic chiasm compression and CN II involvement, respectively, were also observed in nonadenomatous lesions. Invasion of the cavernous sinus leading to cranial nerve involvement was observed in 16 individuals with 11 CN III, six CN IV, and six CN VI palsies. In four individuals, more than one cranial nerve was involved.
Secondary hypogonadism and hyperprolactinemia were the most commonly identified endocrinopathies, both reported in 36% of patients, respectively. Other anterior pituitary dysfunction also occurred frequently with GH deficiency in 24%, central adrenal insufficiency in 21%, and central hypothyroidism in 19% of patients. Although central diabetes insipidus was identified in 11% of nonadenomatous lesions, the occurrence of this endocrinopathy with adenomas was observed far less frequently (0.01%).
Incidental masses
Masses were defined as incidentalomas if they were identified on imaging procedures before pituitary MRI for indications unrelated to an endocrinopathy or for visual symptoms consistent with a sellar mass. Overall, 282 incidental pituitary masses were discovered, with 104 initially discovered by head CT, 176 by brain MRI, and two cases by head or neck x-ray (Table 3).
Table 3.
Diagnoses of 282 incidentalomas discovered by imaging modalitya
| Diagnoses | n | Diagnoses | n |
:---: :---: |
| CT | MRI | CT | MRI |
| Anterior pituitary tumors | | | Metastases | | |
| Nonfunctioning adenoma | 51 | 66 | CNS lymphoma, to pituitary stalk | 0 | 1 |
| Prolactinoma | 9 | 13 | Liver epitheliod hemangioendothelioma | 1 | 0 |
| GH adenoma | 3 | 5 | Nasopharyngeal lymphoma | 0 | 1 |
| ACTH adenoma | 1 | 3 | Pineal germinoma/dysgerminoma | 0 | 1 |
| GH/prolactin mixed adenoma | 1 | 0 | Plasmacytoma | 1 | 0 |
| GH/TSH mixed adenoma | 1 | 0 | Prostate adenocarcinoma | 0 | 1 |
| TSH adenoma | 0 | 1 | Squamous cell carcinoma of sinus | 1 | 0 |
| Cysts | | | Infectious | | |
| Rathke's cleft cyst | 1 | 21 | Pseudomonas aeruginosa | 0 | 1 |
| Craniopharyngioma | 4 | 11 | | | |
| Epidermoid | 0 | 1 | Vascular | | |
| | | | Apoplexy with masses | 8 | 6 |
| Nonadenomatous neoplasms | | | Carotid aneurysm | 1 | 0 |
| Meningioma | 7 | 11 | Hypothalamic interpeduncular hematoma | 0 | 1 |
| Chordoma | 1 | 2 | | | |
| Chondrosarcoma | 0 | 1 | Miscellaneous | | |
| Germinoma | 0 | 1 | Hyperplasia | 4 | 0 |
| Granular cell tumor | 1 | 0 | Empty sella | 0 | 2 |
| Leiomyosarcoma | 1 | 0 | Lipoma | 1 | 0 |
| Mucoepidermoid carcinoma | 0 | 1 | | | |
| Xanthogranuloma | 0 | 1 | Undiagnosed masses | 5 | 22 |
| Inflammatory and vasculitidies | | | Total | 104 | 176 |
| Wegener's granulamatosis | 1 | 0 | | | |
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After screening pituitary MRI scans for sellar and parasellar masses, results were reviewed for identification of these masses on prior nonpituitary imaging procedures.
a
Not shown are two patients with diagnoses of nonfunctioning adenoma and empty sella identified on head and cervical spine x-rays, respectively.
One hundred eighteen of 282 incidental masses were ultimately identified as nonfunctioning pituitary adenomas. We also observed 37 functioning adenomas and 14 cases of pituitary apoplexy. Indications for imaging leading to incidental discovery of a sellar or parasellar mass are listed in Table 4, with headache (40%) the most common imaging indication. Other less common indications for imaging leading to findings of incidental sellar masses included evaluation for cerebrovascular accidents or transient ischemic attacks (9%), visual loss or blurring (9%), and syncope (7%).
Table 4.
Indications for imaging resulting in discovery of 282 incidentalomas
| Indications | CTa | MRIb | X-ray |
:---: :---: |
| Headache | 40 | 72 | |
| CVA/TIA | 10 | 16 | |
| Visual loss/blurring | 5 | 19 | |
| Syncope | 10 | 8 | |
| Altered mental status | 6 | 7 | |
| Diplopia | 6 | 7 | |
| Trauma | 8 | 4 | 1 |
| Dizziness | 3 | 6 | |
| Brain metastases | 3 | 6 | |
| Seizure | 4 | 4 | |
| Cranial nerve palsy | 3 | 5 | |
| Follow-up of nonpituitary brain mass | 1 | 7 | |
| Sinusitis | 2 | 2 | |
| Multiple sclerosis | 0 | 3 | |
| Tinnitus | 1 | 2 | |
| Memory loss | 1 | 1 | |
| Neck pain | 0 | 1 | 1 |
| Mastoiditis | 0 | 1 | |
| SIADH | 0 | 1 | |
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CVA/TIA, Cerebrovascular accident/transient ischemic attack; SIADH, syndrome of inappropriate antidiuretic hormone.
a
Indication for imaging was not identified in one case.
b
Indication for imaging was not identified in two cases.
Discussion
Dedicated pituitary MRI is the preferred diagnostic imaging modality for evaluation of sellar and parasellar tumors, including adenomas. In particular, when functioning adenomas are suspected, a dynamic pituitary MRI, which obtains images within seconds after gadolinium contrast injection, may be more useful because it has higher sensitivity than other imaging modalities for detecting small microadenomas (21). However, small incidental lesions of little or no clinical significance visualized on dynamic pituitary MRI may be misinterpreted as the pathological source of excess hormonal secretion during evaluation of patients for Cushing's disease or acromegaly given lower specificity vs. conventional MRI (21). However, our calculation of the specificity of pituitary MRI is likely limited due to underestimation of true negative values because there are few conditions in which clinicians would obtain pathology results of the pituitary mass when a normal pituitary gland is reported on MRI.
Variability in reported positive MRI results was observed for different clinical indications. Pituitary MRI ordered for the evaluation of patients with suspected acromegaly resulted in a mass reported in 69% of scans, the highest observed among all endocrinopathies. In contrast, 84% of MRI scans ordered for hypogonadism did not reveal a pituitary lesion. These results highlight that pituitary MRI is likely not helpful as a screening tool for patients with hypogonadism. Positive pituitary MRI scans observed with hypogonadotropic hypogonadism were typically observed in cases of severe testosterone deficiency in which total testosterone was less than 100 (normal 250-1000 ng/dl). Given the lack of definitive imaging changes in patients screened for hypogonadism, clinicians should use a higher judgment threshold before ordering pituitary imaging for these patients.
We aimed to identify the prevalence of clinically apparent pituitary masses as screened by MRI scans, and this aim was reflective of the clinical setting because not all pituitary masses are formally diagnosed with histological confirmation. Notably, some etiologies are diagnosed by imaging parameters alone, such as empty sella, or by combination of endocrine values and imaging, as for macroprolactinomas. We observed an 18% incidence of nonadenomatous lesions discovered by MRI, higher than previously reported for surgical series of sellar masses (2, 3). This difference could be attributed to our screening of masses based on imaging parameters, rather than a pathological diagnosis. Other causes for the increased incidence of nonadenomatous masses may be due to our reporting of miscellaneous masses including empty sella syndrome and ectopic pituitary tissue, which are diagnosed only by imaging and would not have been identified in a surgical or pathological series. Furthermore, due to the observational nature of this study, the observed population reflects a selected yet comprehensive group of patients referred for pituitary MRI, rather than the general population as would be encountered in an autopsy series.
Rathke cleft cysts (19%) and craniopharyngiomas (15%) were the most commonly observed nonadenomatous lesions, similar to previous reports. Although both are derived from Rathke's cleft remnants, they can be distinguished on imaging based on magnetic resonance signal appearance and specific characteristics, including homogenous hyperintense T1 signal intensity and midline anterior infundibular displacement in Rathke's cleft cysts, and presence of prominent cystic components and heterogenous hyperintense T1 signal intensity in craniopharyngiomas (22). Furthermore, MRI findings can be instrumental when determining the surgical approach for resection of craniopharyngiomas and likelihood of success in preventing reoccurrences (23).
Previously, large autopsy series have reported pituitary metastases in 1–3.6% of subjects harboring malignant tumors (24–26). Breast and lung cancer are the two most common malignancies that metastasize to the pituitary (27, 28). Breast cancer metastases occur at a high rate, observed in 29% of autopsies of women with disseminated breast cancer (29). Prostate adenocarcinoma may rarely metastasize to the pituitary, accounting for 3% of such cases (27, 28). Metastatic involvement of the pituitary has been reported to account for 12–25% of all nonadenomatous pituitary lesions (2, 3). However, we observed a lower rate (5%), with three cases of breast, one of lung, and one due to prostate cancer. This lower incidence may be due to our screening method and possibly having excluded patients with metastases if they had been imaged with brain MRI or positron emission tomography/CT. Furthermore, most cases of pituitary metastases are clinically silent and diagnosed at autopsy (27).
Although increased incidence of primary CNS or lymphoma has been reported (7, 30, 31), we observed only one primary pituitary lymphoma. This contrasted with the secondary stage IV lymphoma in which a primary source could not be identified, or the third case of a primary CNS lymphoma with pituitary stalk metastasis. These findings confirm that pituitary lymphoma is a rare entity and can be primary or secondary (7, 32).
Infections of the pituitary are rare and potentially life-threatening condition, yet few cases have been reported in literature (33). We observed two cases of infectious disease involving the pituitary gland accounting for 0.2% of all pituitary abnormalities with one due to an abscess caused by P. aeruginosa. Pituitary abscess has been estimated to account for 1% of clinically apparent pituitary disease and isolating an organism from gram stain and/or culture may be difficult with high occurrence of sterile cultures (6). Both cases of pituitary infection observed in this study occurred in immunocompromised patients with prior history of HIV/AIDS. Other risk factors besides an immunocompromised state include presence of preexisting lesions such as an adenoma or Rathke's cleft cyst, previous pituitary surgery, and irradiation of the pituitary (6, 33).
Acquired syphilis of the pituitary is a rare clinical entity, and despite an extensive PubMed literature search, we could not find a reported case in modern literature. However, we identified one case of syphilitic infection of the pituitary in a patient who presented with panhypopituitarism with concurrent systemic and neurosyphilis on serological studies. Pathological destruction of the anterior pituitary by syphilis was first noted in adults as early as 1858, and cases of acquired syphilis involving the pituitary body were reported even before widespread manufacturing of penicillin after World War II (34). Given earlier diagnosis of syphilis with availability of serological tests and accessibility of penicillin, tertiary syphilis has become even less prevalent. The diagnosis of congenital and acquired pituitary syphilis is most often made at autopsy, by identification of interstitial inflammation, gummatas, cystic degeneration, and fibrosis (33). Although a neurosurgical evaluation was not performed on this patient, diagnosis was made with serological studies and pituitary MRI scan consistent with infectious process.
We observed 282 incidentalomas, with nonfunctioning adenoma accounting for the most common incidentally discovered entity. This was not surprising because the most common indication for brain imaging leading to the finding of a pituitary incidentaloma included evaluation for headache, cerebrovascular accident, or transient ischemic attack and decreased visual acuity or visual loss. We found that 23% of pituitary masses were incidentally discovered before pituitary MRI scans. However, because we did not screen CT or MRI scans but rather reviewed them for the presence of incidentalomas after identification of these masses by dedicated pituitary MRI scans, this may not be reflective of the true overall frequency.
Headache is a common clinical indication for imaging leading to discovery of incidental pituitary masses (15) and was commonly observed as leading to a finding of an incidentaloma in this study. Pituitary tumor-related headaches may improve in up to 70% of patients after adenoma resection (35, 36). Furthermore, the presence of headaches does not necessarily correlate with the mass size. Nevertheless, a higher incidence of headache in pregnant women was observed harboring macroprolactinomas vs. microprolactinomas without previous surgical or irradiation therapy (37). Several mechanisms have been proposed for the cause of headaches in patients harboring pituitary masses (38, 39), although these have not been uniformly substantiated (35, 40). Regardless, the higher rate of headache occurrence observed for nonadenomatous lesions vs. both nonfunctioning and functioning adenomas suggests that nonadenomatous lesions are more likely to cause headache (P< 0.001).
Although the diagnosis of the etiology of a sellar or parasellar mass may be confirmed by endocrine, neurological, and radiological techniques, in many cases a pathological specimen is required. For example, as in the leiomyosarcoma and hemangiopericytoma cases observed in this study, a primary neoplasm was diagnosed by pathological specimen evaluation when a nonfunctioning adenoma was initially suspected on radiological imaging. However, as in the case of Wegener's granulomatosis observed in our series, a diagnosis suggested by a pathological result may require confirmation with further serological testing. Overall, given the compelling list of possible diagnoses, when a nonsecreting pituitary mass is observed by MRI, a high clinical suspicion and thorough endocrine and possible pathological assessment is required to exclude the presence of a nonfunctioning pituitary adenoma.
Acknowledgments
We thank Mr. James Mirocha for his help and guidance in the organization and statistical analysis of data.
This work was supported by National Institutes of Health Grant CA 75979.
Disclosure Summary: The authors have nothing to declare.
Footnotes
Abbreviations: CNS
Central nervous system
CT
computed tomography
MRI
magnetic resonance imaging.
References
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Prove that the product of three consecutive integers is never a perfect power (i.e., a perfect square or a perfect cube).
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The product of three consecutive integers is never a perfect power because it includes distinct prime factors with odd powers, contradicting perfect square requirements.
Explanation
Let's prove that the product of three consecutive integers is never a perfect power by contradiction.
Assume there exist three consecutive integers n, n+1,[/tex] and [tex]n+2 such that their product n(n+1)(n+2) is a perfect power. Let's assume it is a perfect square for simplicity.
So,n(n+1)(n+2)is a perfect square.
Now, consider the prime factorization of n, (n+1),[/tex] and(n + 2)). Since these integers are consecutive, one of them must be even, and another must be divisible by 3. Therefore[tex], (n), (n + 1\ and n+2 have distinct prime factors.
When multiplied together, these integers have an odd power for every prime factor because one of them is even and another is divisible by 3. A perfect square, however, requires each prime factor to have an even power.
Since the product n(n+1)(n+2) has at least one prime factor with an odd power, it cannot be a perfect square.
Therefore, the assumption that the product of three consecutive integers is a perfect square leads to a contradiction. Hence, the product of three consecutive integers is never a perfect power.
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The product of three consecutive integers cannot be a perfect power because it includes at least one prime factor with an odd exponent, violating the requirements for perfect squares and perfect cubes. Thus, it leads to the conclusion that this product is never a perfect power.
Explanation
To prove that the product of three consecutive integers is never a perfect power, we will use a logical approach involving prime factorization and the properties of perfect squares and perfect cubes.
Step 1: Define the integers
Let the three consecutive integers be represented as n, n+1, and n+2.
The product can be expressed as:
P=n(n+1)(n+2)
This product consists of three distinct integers.
Step 2: Investigate prime factorization
In order for a number to be a perfect square, all the prime factors in its prime factorization must have even exponents. For a perfect cube, all prime factors must have exponents that are multiples of three.
Since we have three consecutive integers, there are guaranteed to be different prime factors involved, thus disrupting any uniformity needed for even exponents or multiples of three.
Step 3: Analyze the properties of consecutive integers
At least one of the integers must be even, and at least one must be divisible by 3:
For instance, consider three consecutive integers: 2,3,4. Their product is 2×3×4=24.
The prime factorization of 24 is 2 3×3 1 which shows:
The exponent for 2 is 3 (odd) and for 3 is 1 (odd).
Hence, 24 is neither a perfect square nor a perfect cube.
The alternating nature of odd and even exponents continues for any set of three consecutive integers, meaning there will be at least one prime factor with an odd exponent.
Step 4: Conclusion
Since the product of three consecutive integers must always involve prime factors with at least one odd exponent, it cannot be a perfect square or cube. Therefore, the product of any three consecutive integers is never a perfect power.
Examples & Evidence
If we take the integers 1, 2, and 3, their product is 6, which factors as 2 1×3 1 ensuring both exponents are odd. Similarly, for integers 4, 5, and 6, we again end up with a product (120), which is not a perfect square or cube due to the same reasoning.
The fundamental theorem of arithmetic confirms that each integer greater than 1 can be represented uniquely as a product of prime factors. The necessity for even or multiple of three exponents for perfect squares and cubes leads to contradictions in the case of consecutive integers.
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6661 | https://math.stackexchange.com/questions/727286/calculate-arc-length-knowing-its-subtended-chord-and-circumference-diameter | geometry - Calculate arc length knowing its subtended chord and circumference diameter - Mathematics Stack Exchange
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Calculate arc length knowing its subtended chord and circumference diameter
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how do I calculate an arc length knowing only its subtended chord and the circumference diameter? I don't know the angle between OA and OB. Yesterday I did an experiment and calculated that the diameter / arc ratio is an exponential function which tends to 1 when lowering the numbers. My data: d / a 20 / 31.6 19.5 / 26.4 17.5 / 21.4 13.5 / 14.4 10 / 10.2 5 / 5 Thanks!
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edited Apr 1, 2014 at 15:43
Aaron Hall
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asked Feb 21, 2014 at 7:06
jackusCTB jackusCTB
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arc = CD asin(AB/CD)
(where the angle returned by asin is in radians)
EDIT:
I thought the derivation was obvious, but maybe I should explain it.
Mark the midpoint of AB, call it E.
sin(AOE) = (AB/2)/r where r=CD/2
angle = 2 asin(AB/CD)
arc = r angle = CD asin(AB/CD)
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answered Feb 21, 2014 at 7:32
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I agree, probably the simplest solution.helb –helb 2014-02-21 09:36:48 +00:00 Commented Feb 21, 2014 at 9:36
I created an account on this site solely to thank "Beta" (first answer below question). However, being a new account, I haven't earned the "points"/right to comment or even Up-vote. So, this comment is to confirm for others the accuracy of his/her answer. I can confirm (using my CAD program) his/her arc length formula is accurate to at least 10 digits right of decimal point (far more precise than any tape-measure). I spent over an hour combing google/bing for a simplified answer. The vast majority of searches resulted in people seemingly wishing to demonstrate their superiority with overly com WroughtIronGuy –WroughtIronGuy 2018-01-22 23:56:09 +00:00 Commented Jan 22, 2018 at 23:56
plex responses. His/hers was the first I came across that didn't involve an extra 10 lines of code. Thank you, Beta! How I use it: I frequently have to fit rails to an existing layout. The only dimensions I can take on-site are chord and offset (in geometry terms: height). Previously, in order to quote the job (which involves arc length) I needed to use my CAD program. Thanks to Beta's condensed formula I can quickly total the equivalent linear lengths for quoting without creating a CAD file. (continued from the answer post)quid –quid 2018-01-23 00:38:53 +00:00 Commented Jan 23, 2018 at 0:38
@WroughtIronGuy: You made my day. And now my hat's too tight...Beta –Beta 2018-01-23 18:02:21 +00:00 Commented Jan 23, 2018 at 18:02
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The triangle O-A-B is defined since we know AB and OA = OB = 0.5CD. Calculate the angle using the Law of Cosines (see wikipedia):
Therefore
angle = arccos(((0.5CD)^2 + (0.5CD)^2 - AB^2) / (20.5CD0.5CD))
angle = arccos((0.25CD^2 - AB^2) / (0.5CD^2))
Then you can compute the arc length using the angle
arc = pi CD angle / 360°
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answered Feb 21, 2014 at 7:14
helb helb
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Thanks to both, it was pretty simple at the end...jackusCTB –jackusCTB 2014-02-21 13:53:59 +00:00 Commented Feb 21, 2014 at 13:53
Thanks to both, it was pretty simple at the end... draw an isosceles triangle with two sides of 124 and the smaller one of 32. Then divide the smaller side by two to get two right triangles. The hypotenuse is sqrt(124^2-16^2) = 122.96. The inner angle is arctan (122.96 / 16) = 7.42. Remember I divided the first triangle in two right triangles? Well, we should double 7.41 to get the angle of the arc, = 14.83. We can finally calculate the arc with [Arc = ( angle radius pi ) / 180 = 32.1! That's how I did thanks to your inputs!jackusCTB –jackusCTB 2014-02-21 14:15:49 +00:00 Commented Feb 21, 2014 at 14:15
There's a mistake in your algebra, going from the second to the third line.Beta –Beta 2014-02-22 00:07:05 +00:00 Commented Feb 22, 2014 at 0:07
Yes, the hypotenuse is 124, the longer chatetus is 122.96 and the shorter chatetus is 16. Sorry for this mistake.jackusCTB –jackusCTB 2014-02-22 10:31:57 +00:00 Commented Feb 22, 2014 at 10:31
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6662 | https://mathoverflow.net/questions/160811/what-is-exceptional-about-the-prime-numbers-2-and-3 | Stack Exchange Network
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What is exceptional about the prime numbers 2 and 3?
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$\begingroup$
Admittedly this question is vague. But I hope to convey my point. Feel free to downvote this.
Permit me to define prime number the following way:
A number $n>1$ is a prime if all integers $d$ with $1< d \leq \sqrt{n}$ give non-zero remainders while dividing $n$. This will expose the primes 2 and 3: they qualify as primes as this condition is vacuous (there are simply no integers in that interval to check divisibility).
Lot of exceptions to some theory (quadratic forms, elliptic curves, representation theory) happen at primes 2 and 3. Is it because 2 and 3 entered The Prime Club by this backdoor? It is my wild guess. Perhaps experts might be able to say if I am wrong. Or possibly there are more conceptual reasons. Kindly enlighten me.
prime-numbers
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edited Mar 21, 2014 at 23:03
P VanchinathanP Vanchinathan
asked Mar 19, 2014 at 9:51
P VanchinathanP Vanchinathan
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$\begingroup$ To nit-pick a bit: By this definition also 1 is prime. $\endgroup$
Dirk
– Dirk
2014-03-19 10:23:54 +00:00
Commented Mar 19, 2014 at 10:23
26
$\begingroup$ My wild guess is that $2$ is the degree of quadratic forms and $3$ is the degree of elliptic curves. Should you study forms/curves of degree $5$, I'm sure $5$ will become pretty special. $\endgroup$
Alex Degtyarev
– Alex Degtyarev
2014-03-19 10:23:59 +00:00
Commented Mar 19, 2014 at 10:23
5
$\begingroup$ This is very closely related to the question: mathoverflow.net/questions/915 $\endgroup$
Ben Webster
– Ben Webster
2014-03-19 11:35:29 +00:00
Commented Mar 19, 2014 at 11:35
2
$\begingroup$ @Dirk the inequality defining d is meaningless if n = 1 so I assume that there is an implicit definition that n>1 $\endgroup$
Chris
– Chris
2014-03-19 12:34:08 +00:00
Commented Mar 19, 2014 at 12:34
5
$\begingroup$ Define an odd number $n>1$ to be prime when for all odd $d$ with $1 $\endgroup$
Chris Wuthrich
– Chris Wuthrich
2014-03-19 14:27:24 +00:00
Commented Mar 19, 2014 at 14:27
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4 Answers 4
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$\begingroup$
I think that in different theories, there is often a "primitive" fact (which is hard to explain further) that lies at the heart of the complication you mention. Let me give examples.
As for the "2 is the oddest prime" credo in number theory, often it boils down to the fact that $\mathbb{Q}$ contains exactly the second roots of unity. Or equivalently, the unit group of $\mathbb{Z}$ is $2$-torsion. I do not know if this can be embedded in a conceptual explanation; maybe it's a fact one has to live with, with ever-occuring consequences.
In the theory of algebraic groups and Lie algebras, e.g. in Chevalley bases and related stuff, the coefficients will be (or have as prime factors) only $2$ or $3$. A consequence is that many integral structures are $p$-integral only for primes $\ge 5$, and this pops up again and again in the theory. See Dietrich Burde's answer for more. I think here an ultimate explanation for this occurrence of $2$ and $3$ is that they appear in the basic combinatorics of root systems. That is the "primitive" fact.
As for the characteristic $2$ exception for quadratic forms, it is the non-equivalence of quadratic and symmetric bilinear forms that causes trouble. This in turn seems to be "primitive", just try to show equivalence and see that you have to invert $2$. And of course one should expect that for something quadratic, the number $2$ plays a special role.
I guess if we were more interested in some tri-linear stuff, or more in things that can be given as $7$-tuples than in pairs, the cases of characteristic $3$ or $7$ would need more attention. So this translates the question into why bilinear things, and pairs, are often natural. (Remark that such a basic thing as multiplication, including Lie brackets and other non-associative stuff, is a bilinear map and thus will have a tendency to need special treatment in characteristic $2$. Same for any duality, pairings etc.)
As for $2$ and $3$ as bad primes for elliptic curves, the story seems to be a little different. The answers by jmc and Joe Silverman suggest the following view: there is a family of objects (abelian varieties) which can be parametrised roughly by certain numbers (dimension), and exceptional patterns are related to this parameter; and because elliptic curves are the ones where the parameter is small, there are small numbers that behave irregular. Now one would think that this is just a high-brow version of Alex Degtyarev's comment. But there is an interesting subtlety: It is not that in the general theory there are numbers different from $2, 3$ that misbehave (while these become nice), but there are more than just them. In other words: Granted that for every single number you might find some monstrosity somewhere in the general theory. But one might find it surprising that there are some numbers that always need care, even in the most specialised, well-behaved cases. For this, I have no better explanation than:
The strong law of small numbers: Small numbers (not necessarily primes) give exceptional patterns. Because naturally, there are so few of them, and they "have to satisfy too much at once". Maybe this is as far as one gets if one seeks after a common pattern between the "primitive" explanations above.
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edited Sep 8, 2014 at 12:18
answered Mar 19, 2014 at 10:44
Torsten SchoenebergTorsten Schoeneberg
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$\begingroup$ Thanks Torsten, for throwing light on this matter. I accept your answer. The answers by jmc and Dietrich are also helpful and I am voting them up. $\endgroup$
P Vanchinathan
– P Vanchinathan
2014-03-19 15:40:20 +00:00
Commented Mar 19, 2014 at 15:40
$\begingroup$ I downloaded RK Guy's article from maa.org on Strong law of small numbers and read a few pages. Thanks for pointing out that. It is a great article that can be used when I give popular talks to non-mathematics students. $\endgroup$
P Vanchinathan
– P Vanchinathan
2014-03-19 23:35:48 +00:00
Commented Mar 19, 2014 at 23:35
$\begingroup$ Just for fun : it is the only consecutive prime pair such that its sum is odd ... $\endgroup$
Duchamp Gérard H. E.
– Duchamp Gérard H. E.
2014-03-29 19:14:06 +00:00
Commented Mar 29, 2014 at 19:14
$\begingroup$ In the matter of the Collatz conjecture it looks like ${2,3}$ being the only primes less than $\pi$ is of critical importance. I suspect this has a close relation to the 2-torsion concept. Then $\pi<5<2\pi$ makes that the next special case. Does your experience in these fields suggest there is any hope for this idea to add resolution to your answer? $\endgroup$
Robert Frost
– Robert Frost
2018-08-28 12:35:34 +00:00
Commented Aug 28, 2018 at 12:35
$\begingroup$ No, my experience and common math sense strongly suggest that that is hogwash. $\endgroup$
Torsten Schoeneberg
– Torsten Schoeneberg
2018-08-29 09:05:32 +00:00
Commented Aug 29, 2018 at 9:05
Add a comment |
10
$\begingroup$
The reason that $2,3$ are special in the elliptic curve case has to do with the dimension of elliptic curves: $1$. Elliptic curves are precisely the $1$-dimensional abelian varieties. If $X$ is an abelian variety of dimension $d$, the primes $p > 2d + 1$ will behave well, and likely the primes $p \le 2d + 1$ will be bad. For elliptic curves, $d = 1$, and this gives you precisely $2,3$ as bad primes.
You can read about this in [ST], and more precisely Corollary 2. They remark on the page following this corollary:
[¦] Corrolary 2 suggests that, for abelian varieties of dimension $d$ (hence also for curves of genus $d$), it is the primes $p \le 2d+1$ which can play an especially nasty role. This is well-known for elliptic curves ($p = 2,3$), and the same set of bad primes seems to arise in other connections. For instance, a function field of one variable of genus $d$ is conservative if the characteristic $p$ is $> 2d + 1$.
[ST] Jean-Pierre Serre and John Tate, Good Reduction of Abelian Varieties. The Annals of Mathematics, Second Series, Volume 88, Issue 3, (Nov., 1968), 492517.
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edited Mar 19, 2014 at 12:28
answered Mar 19, 2014 at 12:17
jmcjmc
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$\begingroup$ I was going to add this as a separate answer, but I think it's really part of what you said: The lattice $\mathbb{Z^2}$ has automorphisms of order $2$ and $3$, but none of larger prime orders. The action of $\textrm{PSL}(2,\mathbb{Z})$ on the upper half-plane is free except for a point of order $2$ and a point of order $3$. So, the modular surface has cone points of angles $2\pi/2$ and $2\pi/3$. $\endgroup$
Douglas Zare
– Douglas Zare
2014-03-19 17:22:42 +00:00
Commented Mar 19, 2014 at 17:22
1
$\begingroup$ Is it related to tiling? It is also mentioned in Artin's textbook on Algebra why there is no covering by regular pentagons. $\endgroup$
P Vanchinathan
– P Vanchinathan
2014-03-20 14:42:34 +00:00
Commented Mar 20, 2014 at 14:42
Add a comment |
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$\begingroup$
A good example why $p=2$ and $p=3$ are exceptional for Lie theory is the classification of simple modular Lie algebras over algebraically closed fields of characteristic $p$. For $p=2$ a classification seems to be completely out of reach, and even for $p=3$ this might not be feasable. The reasons have been given already, but also trivial reasons from linear algebra play a role, e.g., $0=tr(A+A^t)=tr(A)+tr(A^t)=2tr(A)$ will not imply that $tr(A)=0$ for $p=2$. However, for $p\ge 5$ the classification is very nice: the simple Lie algebras fall into the following three classes:
$\bullet$ Finite dimensional graded Cartan type Lie algebras and their deformations.
$\bullet$ Classical Lie algebras $A_n,B_n,C_n,D_n,G_2,F_4,E_6,E_7,E_8$.
$\bullet$ The Melikyan algebras $M(m,n)$ on two parameters, for $p=5$.
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answered Mar 19, 2014 at 12:24
Dietrich BurdeDietrich Burde
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The quote from Serre-Tate in jmc's answer indicates that for an abelian variety $A$ of dimension $d$, the primes that have the most complicated behavior are those satisfying $p\le 2d+1$, but it doesn't explain WHY those primes appear. The answer has to do with the representation of $\hbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}$ on the Tate module, or more simply on the $\ell$-torsion $A[\ell]$. Things become more complicated, for example the computation of the conductor is quite complicated, if this representation is wildly ramified. And it is not that hard to prove (as is done in [ST]), by looking at the order of $\hbox{GL}_{2d}(\mathbb{Z}/\ell\mathbb{Z})$, that the representation can only be ramified at primes $p\le 2d+1$. For example, the gcd over all (sufficiently large) $\ell$ of $#\hbox{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z})$ is divisible only by 2 and 3, which is why they are the primes for elliptic curves that may be wildly ramified.
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answered Mar 20, 2014 at 11:30
Joe SilvermanJoe Silverman
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$\begingroup$ Thanks for adding more details. Yes, ramification is something easier to understand as culprit. Very benefitial answer. $\endgroup$
P Vanchinathan
– P Vanchinathan
2014-03-20 14:39:47 +00:00
Commented Mar 20, 2014 at 14:39
$\begingroup$ Do you know if the bound $p \le 2d + 1$ is sharp? $\endgroup$
Torsten Schoeneberg
– Torsten Schoeneberg
2014-03-21 01:05:40 +00:00
Commented Mar 21, 2014 at 1:05
1
$\begingroup$ @TorstenSchoeneberg My recollection is that it's not so hard to produce examples that are wildly ramified at any prime $p\le 2d+1$, but it's been a while since I've thought about it. OTOH, the maximum exponent of the conductor for such primes is trickier, especially if you work over a number field instead of over $\mathbb{Q}$. Lockhart, Rosen, and I got a fairly good upper bound, and Brumer and Kramer gave the best possible result. $\endgroup$
Joe Silverman
– Joe Silverman
2014-03-21 01:23:12 +00:00
Commented Mar 21, 2014 at 1:23
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Is there a high-concept explanation for why characteristic 2 is special?
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Circles>How to Make a Circle in GeoGebra
How to Make a Circle in GeoGebra
You can use GeoGebra to draw circles. GeoGebra has three built-in tools for drawing circles depending on the information you have about the circle:
1.
: If you have information on the center, and a point on the circle, choose Item 4.
2.
: If you have information on the center, and the radius of the circle, choose Item 4.
3.
: If you have information on three points on the circle, choose Item 3.
GeoGebra Instruction 1
Circle with Given Center and Boundary Point
1.
: Open Algebra View and Graphics View under View in Menu.
2.
: Click on Circle with Center through Point .
3.
: Click on a chosen point on the Graphics View. The chosen point will be the center of the circle.
4.
: Decide the length of the radius of the circle by moving the cursor away or toward the center.
5.
: Left-click when you have decided on the length of the radius. The size of the circle should not change when you move the cursor. The circle is now shown in Graphics View.
6.
: If you wish to know the length of the radius, click on Distance or length and click on the the point in the center and the point on the circular arc. The length of the radius will now be shown.
The circle will now appear in Graphics View. The equation of the circle will appear in a row in Algebra View as well.
GeoGebra Instruction 2
Circle with Given Center and Radius
1.
: Open Algebra View and Graphics View under View in Menu.
2.
: Click on Circle: Center & Radius which is the sixth button on the toolbar and the second tool in the tool list. Click on a chosen point on the Graphics View. The chosen point will be the center of the circle.
3.
: GeoGebra will now ask you to type in the radius of the circle. Type in the radius that you want. Click on OK.
The circle will now appear in Graphics View. The equation of the circle will appear in a row in Algebra View as well.
Example 1
Draw a circle with a radius of 5 in GeoGebra
You choose to use Circle: Center & Radius to draw the circle. The location of the center on the Graphics View does not matter.
GeoGebra Instruction 3
Circle Through Three Points
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: Open Algebra View and Graphics View under View in Menu.
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: Create the three points that you have been given by clicking on each of their positions.
3.
: Click on Circle through 3 Points which is the sixth button on the toolbar and the fourth tool in the tool list. Click on the three points that you made in the previous step. The order does not matter.
The circle will now appear in Graphics View. The equation of the circle will appear in a row in Algebra View as well.
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6664 | https://www.droracle.ai/articles/9749/complement-levels-in-post-strep-glomerulonephritis | What are the complement levels in post-streptococcal (post-strep) glomerulonephritis?
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What are the complement levels in post-streptococcal (post-strep) glomerulonephritis?
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From the Guidelines
Complement Levels in Post-Streptococcal Glomerulonephritis
Complement levels, specifically C3, are typically decreased during the acute phase of post-streptococcal glomerulonephritis (PSGN) 1, 2.
The degree of C3 depression can vary, but it is often more pronounced in patients with severe disease.
Serial measurements of C3 levels can be useful in monitoring disease activity and response to treatment.
Levels usually return to normal within 8-12 weeks2.
Persistently low C3 in serum beyond week 12 may be an indication for kidney biopsy to exclude complement C3 glomerulonephritis (C3GN) 2.
No specific medication, such as corticosteroids or immunosuppressants, is typically required for complement level management in PSGN.
C4 levels may also be assessed, but the primary focus is on C3 levels in the context of PSGN 1.
From the Research
Complement Levels in Post-Streptococcal Glomerulonephritis
Complement levels are decreased in patients with post-streptococcal glomerulonephritis (PSGN), with low serum C3, properdin, and C5 levels indicating aggressive complement activation 3, 4, 5.
The classical complement pathway is frequently activated in patients with PSGN, with elevated C4d/C4 ratios indicating C4 activation 6.
C3 and C5 levels are decreased in PSGN, with C3 levels being decreased in more than 90% of cases 5.
C4 levels are typically normal or only slightly decreased in PSGN, with the early return to normal of fourth component concentrations being a favourable prognostic sign 3, 6.
Other complement components, such as C6 and C8, may also be decreased in some patients with PSGN, particularly those with low C3 levels 6.
Patterns of Complement Activation
Complement activation in PSGN is thought to occur through the alternative pathway, with C3 splitting activity present before the clinical onset of nephritis 4.
The activation of the classical pathway also plays a role in PSGN, with C4 activation occurring in some patients 6.
The pattern of complement activation in PSGN is distinct from that seen in other types of glomerulonephritis, such as systemic lupus erythematosus 3.
Clinical Implications
Decreased complement levels, particularly C3, are a hallmark of PSGN and can be used to diagnose and monitor the disease 5, 7.
Measuring C3 and C4 levels may also be helpful in evaluating other causes of acute nephritis 5.
The management of PSGN is mainly supportive, with fluid restriction, anti-hypertensives, diuretics, and renal replacement therapy as needed 7.
References
1
Guideline
Guideline Directed Topic Overview
Dr.Oracle Medical Advisory Board & Editors, 2025
2
Guideline
Guideline Directed Topic Overview
Dr.Oracle Medical Advisory Board & Editors, 2025
3
Research
Serial complement component alterations in acute glomerulonephritis and systemic lupus erythematosus.
Clinical and experimental immunology, 1969
4
Research
Serum complement levels before and after the onset of acute post-streptococcal glomerulonephritis. A case report.
Pediatric nephrology (Berlin, Germany), 1994
5
Research
Acute poststreptococcal glomerulonephritis: the most common acute glomerulonephritis.
Pediatrics in review, 2015
6
Research
Complement profiles in acute post-streptococcal glomerulonephritis.
Pediatric nephrology (Berlin, Germany), 1988
7
Research
Management and outcomes of acute post-streptococcal glomerulonephritis in children.
World journal of nephrology, 2022
Related Questions
What is the management approach for acute poststreptococcal glomerulonephritis (APSGN) in pediatric patients?What is the most appropriate next diagnostic step for a 3-year-old girl presenting with progressive edema, recent sore throat, and laboratory findings indicating impaired renal function, including proteinuria and hematuria, with RBC casts?What are the causes of nephrotic syndrome in adults with low C3 (complement 3) and C4 (complement 4), low C3 and normal C4, and normal complement levels?What is the pathophysiology, laboratory exams, and treatment of Poststreptococcal Glomerulonephritis (PSGN) based on Kidney Disease: Improving Global Outcomes (KDIGO) guidelines and recent treatments?What is the prognosis of post-streptococcal glomerulonephritis?What labs are used to diagnose hypercoagulability?What are the minimum diagnostic criteria for Pelvic Inflammatory Disease (PID) in an adolescent female?What is the relationship between fatty liver disease and small fiber neuropathy?What is the interpretation of Doppler (Doppler ultrasound) pulses?What is the interpretation of Doppler (Doppler ultrasound) pulse waveforms that are monophasic, biphasic, and triphasic?What is the maximum daily dose of tramadol (tramadol hydrochloride)?
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6665 | https://www.ebsco.com/research-starters/chemistry/chlorine-cl | Research Starters Home
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Chlorine (CL)
Chlorine (Cl) is a yellowish-green gas and a member of the halogen group in the periodic table, known for its high reactivity and ability to combine with almost any element. With an atomic number of 17, chlorine has an atomic mass of 35.453 atomic mass units, and it exhibits oxidation states ranging from -1 to +7. Chlorine is found primarily in nature as part of compounds, most notably sodium chloride, or table salt. This element is instrumental in various applications, including water purification and disinfection, making it essential for public health.
Discovered in 1774 by Swedish chemist Carl Wilhelm Scheele, chlorine was initially misunderstood due to the prevailing phlogiston theory. It wasn't until 1810, when English chemist Humphry Davy recognized it as a distinct element, that chlorine's identity was clarified. Notably, chlorine gas has been used in historical contexts, such as World War I as a chemical weapon, although its strong odor allowed for some detection and mitigation. Due to its properties, chlorine is a crucial component in chemical reactions and compounds, influencing both industrial processes and everyday products.
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By: Gonzalez, David
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Chlorine (CL)
Part of the halogen group in the periodic table of the elements, chlorine (Cl) is a gas that can combine with almost any other element. It has an atomic number of 17 and an atomic mass of 35.453 atomic mass units (u). It melts at −103 degrees Celsius (−153 degrees Fahrenheit) and boils at −34 degrees Celsius (−29 degrees Fahrenheit). Its density at one atmosphere and 0 degrees Celsius (32 degrees Fahrenheit) is 3.214 grams per liter. Its oxidation states are −1, +1, +3, +5, and +7. Its electron configuration is 1s22s22p63s23p5.
Pure chlorine is yellowish green. It is considered dangerous to inhale, as it can irritate the mucus membranes of the lungs. An airborne dose of one part chlorine in one thousand can be fatal within minutes if inhaled. Chlorine gas was used as a chemical weapon during World War I.
Most chlorine found in nature is combined with something else. For instance, chlorine readily bonds with sodium to produce sodium chloride, or table salt. Scientists typically extract chlorine from a chloride substance through a highly specialized separation technique called electrolysis. Around the world, chlorine serves a variety of purposes and is found in products that most people use every day. It is used to clean swimming pools and even helps produce safe drinking water.
Brief History
Swedish chemist Carl Wilhelm Scheele (1742–86) discovered chlorine in 1774. After extracting and isolating the gas, Scheele noted that it was quite oppressive to the lungs and carried a suffocating smell, produced a yellowish-green color in the air, and made water taste acidic. After further research, Scheele discovered that this newly discovered gas was very reactive with other elements.
Despite his detailed observations of the gas, Scheele did not think that chlorine was its own element. This confusion was brought on by the phlogiston theory, which was accepted by nearly every chemist during the 1700s. By the early 1800s, the general consensus was that chlorine was an oxygen-containing compound. In 1810, English chemist Humphry Davy (1778–1829) concluded that the gas was, in fact, an element. He gave it the name chlorine, derived from the Greek term choloros, meaning “yellow-green.”
Overview
Chlorine is a relatively dense gas that smells similar to bleach. It is capable of bonding with almost all other elements and, as a liquid, can be used as a powerful oxidizing agent for disinfecting and cleaning purposes. Chlorine was part of the first chain reaction ever observed. It occurred in 1913 when German physical chemist Max Bodenstein (1871–1942) observed a mixture of hydrogen gas and chlorine; the two elements exploded when exposed to light. German physicist Walther Nernst (1864–1941) explained this phenomenon in detail in 1918.
Oceans contain a large amount of chlorine—so much that if all of the chlorine on Earth were simultaneously released as a gas, it would measure approximately five times larger than Earth’s current atmosphere. Chlorine gas is much heavier than air, and even low concentrations have been known to cause fatalities. During World War I, the Germans used chlorine for chemical warfare, although it ended up being less effective than they had originally hoped; chlorine had such a strong smell that soldiers from the opposing side were able to detect it and protect themselves by placing damp cloths over their faces.
Chlorine is reactive because it is unstable. Atoms of different elements all have a certain degree of stability, depending on the configuration of their electrons. In simple terms, an atom is said to be stable if its outermost, or valence, electron shell is filled to capacity with electrons, and it is said to be unstable if its valence electron shell is lacking electrons. Chlorine lacks a single electron in its valence shell. It therefore readily combines with elements that need to give up a single electron in order to become stable.
The most common chlorine-containing compound is sodium chloride, commonly known as table salt. Sodium and chlorine readily bond because sodium has only one electron in its valence electron shell and must give up that electron in order to become stable. When the two elements combine, a stable ionic compound is formed. Chlorine also bonds efficiently with aluminum to form a compound called aluminum trichloride (AlCl3). A single aluminum atom, missing three electrons in its outermost orbital, combines with three chlorine atoms to make a single molecule of this compound.
Depending on the elements involved, chlorine can form either a covalent bond, in which two atoms share electrons, or an ionic bond, in which one element gives up one or more electrons to another. The type of bond formed dramatically affects the behavior of the final molecule. For instance, when chlorine forms an ionic compound, as in the case of sodium chloride, the ions that make up the compound are held together by the attraction of their opposing electrical charges. This is why water becomes much more electrically conductive as the concentration of salt increases.
Bibliography
Blaszczak-Boxe, Agata. "Facts About Chlorine." Live Science, 8 Dec. 2021, www.livescience.com/28988-chlorine.html. Accessed 27 Dec. 2022.
Cairns, Donald. Essentials of Pharmaceutical Chemistry. 4th ed. London: Pharmaceutical, 2012. Print.
Henrickson, Charles H., Larry C. Byrd, and Norman W. Hunter. Chem Lab: Experiments in General, Organic and Biochemistry. 2nd ed. Dubuque: Kendall, 2002. Print.
Henrickson, Charles H., Larry C. Byrd, and Norman W. Hunter. A Laboratory Manual for General, Organic, & Biochemistry. 7th ed. New York: McGraw, 2011. Print.
Li, Tianling. "Chlorination in the Pandemic Times: The Current State of the Art for Monitoring Chlorine Residual in Water and Chlorine Exposure in Air." Science of The Total Environment, vol. 838, 10 Sept. 2022, doi.org/10.1016/j.scitotenv.2022.156193. Accessed 27 Dec. 2022.
Salem, Aliasger K. "Chlorine Is a Highly Useful Chemical That's Also Extremely Dangerous - Here's What to Know About Staying Safe Around It." The Conversation, 7 Nov. 2023, theconversation.com/chlorine-is-a-highly-useful-chemical-thats-also-extremely-dangerous-heres-what-to-know-about-staying-safe-around-it-213998. Accessed 15 Nov. 2024.
Schneider, Stefan. “Chlorine (Cl).” Encyclopedia Britannica. Encyclopedia Britannica, 27 Dec. 2013. Web. 3 Sept. 2014.
Tro, Nivaldo J. Chemistry: A Molecular Approach. 3rd ed. Upper Saddle River: Prentice, 2014. Print.
Tro, Nivaldo J. Introductory Chemistry Essentials. 5th ed. Upper Saddle River: Prentice, 2015. Print.
Zumdahl, Steven S., and Donald J. DeCoste. Basic Chemistry. 7th ed. Belmont: Brooks, 2010. Print.
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Math Courses / ELM: CSU Math Study Guide Course
Simplifying Square Root Expressions | Steps & Examples
Lesson
Transcript
Dawn Wells, Jennifer Beddoe
Author Dawn Wells
Dawn has over 15 years of math teaching and tutoring experience covering middle school, high school and dual enrollment classes. She has taught middle school math, Algebra, Geometry, Algebra II, college Algebra and Trigonometry. She has a Master's degree in Innovative Teaching in Mathematics from Nova Southeastern University and a Bachelor's degree in Mathematics from Edward Waters College. She is certified to teach math from middle school through high school.
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Instructor Jennifer Beddoe
Jennifer has an MS in Chemistry and a BS in Biological Sciences.
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Understand how to use different techniques for simplifying square root expressions and simplifying radicals, such as using a factor tree and fractional exponents. Updated: 11/21/2023
Table of Contents
Square Roots and Radicals
How to Simplify Square Root Expressions
Examples of Simplifying Square Root Expressions
How to Simplify Radicals
Simplifying Radicals
Lesson Summary
Show
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Table of Contents
Square Roots and Radicals
How to Simplify Square Root Expressions
Examples of Simplifying Square Root Expressions
How to Simplify Radicals
Simplifying Radicals
Lesson Summary
Show
Square Roots and Radicals -------------------------
A square root is a factor of a given number that multiplies times itself to produce the given number. The three is the square root of nine because three times itself produces nine; one can also write this as 3 2=9. The terms square and square root are opposite operations. The square root of nine is three and is written as 9=3.
The square root symbol is called a radical, while the value under the radical is called the radicand. A radical expression is an expression used to explain the value of a radical and its radicand. Square roots and radical expressions have real-life applications. The applications can be seen with pendulums, television screens, and visibility.
Real-Life Example #1: Pendulums
Pendulums demonstrate simple harmonic motion. The motion explains other natural behaviors that repeat periodically. There is a relationship between time and the length of the pendulum at sea level, expressed in the following equation: T=2 L. The variable T represents the time it takes the pendulum to make a full swing back and forth. The variable L represents the length of the pendulum. Knowing the time it takes the pendulum to make a full swing, one can calculate the pendulum's length and vice versa.
Real-Life Example #2: Television Screens
Television screens are described by the length of their diagonals. The relationship between the diagonal and area of a television is represented by d 2=25 12 A or d=5 2 3 A. The variable d is the length of the diagonal. The variable A is the area of the television screen. The equations stem from using the Pythagorean theorem and height-to-width ratio.
Real-Life Example #3: Visibility
Visibility is affected by nature and items at eyesight. One can see further when obstacles are removed or when the point of view is higher than the obstacles. There is a relationship that connects the distance one can see, visibility, and the height at which the person stands: v=1.225 a. The variable v is the distance measured in miles. The variable a is the altitude, height, measured in feet.
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0:06 Math Is Universal
0:25 The Language of Radicals
0:59 Simplifying Square…
2:40 Simplifying Square…
4:41 Put It All Together
6:26 Lesson Summary
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How to Simplify Square Root Expressions ---------------------------------------
Simplifying square root expressions is similar to simplifying fractions. One uses the greatest common factor of the numerator and denominator to simplify a fraction. Similarly, one uses the greatest perfect square factor to simplify a square root expression. A perfect square is the product of multiplying a whole number times itself. A perfect square can be a factor of the radicand and factored out from the radical.
The following steps should be followed in order to simplify a square root expression:
Identify factor pairs of the radicand.
Identify the factor pair with the greatest perfect square.
Break the factor pair into separate radicands.
Simplify possible radicands.
The start of simplifying a square root expression is with the factor pairs. The pairs can be displayed as part of a factor tree. Within the factor tree, one can identify the perfect squares.
The prime factorization of 24 displays all of its prime factors and its perfect square, 4, can be identified.
Using exponents, one can write the factors of 24 by grouping common factors together. Since simplifying square root expressions is about squares, one should group the common factors as pairs. The prime factorization of 24 written using exponents, grouped as pairs, looks like: 24=2 2⋅2⋅3. Having the factors written this way, the square root of 24 is:
24=2 2⋅2⋅3 24=4⋅6 24=2 6
Examples of Simplifying Square Root Expressions -----------------------------------------------
Square Root with Numbers and Variables: 18 x 4
18 x 4=3 2⋅2⋅x 2⋅x 2 18 x 4=9⋅x 2⋅x 2⋅2 18 x 4=3⋅x⋅x⋅2 18 x 4=3 x 2 2
The factors of radicand, simplified as perfect square factor pairs, are 3 2⋅2⋅x 2⋅x 2. The perfect square factors are square rooted or simplified and multiplied together.
Square Root with Only Variables: a 6 b 4 c
a 6 b 4 c=a 2⋅a 2⋅a 2⋅b 2⋅b 2⋅c a 6 b 4 c=a 2⋅a 2⋅a 2⋅b 2⋅b 2⋅c a 6 b 4 c=a⋅a⋅a⋅b⋅b⋅c a 6 b 4 c=a 3 b 2 c
As with numbers, variables are grouped together as perfect square factors for simplification.
Square Root that Cannot Be Simplified: 10 x y
10 x y=2⋅5⋅x⋅y
The value under the radicand does not have any perfect square factors. Since there are no perfect square factors, one can no longer simplify the value under the radicand.
How to Simplify Radicals ------------------------
Radicals represent numerical values; thus, one can perform mathematical operations with them. Radical expressions also represent more than just square roots of numbers. The expressions can be roots for cubes, fourths, fifths, two-thirds, and many others.
One can write radical expressions with exponents instead of a radical symbol. When simplifying radical expressions, the rules used for exponents are transferable. The rules transfer because radical expressions have equivalent exponent expressions. For instance, the 25 can also be written as 25 1 2. The denominator is the index of the radical, and the numerator is the power of the radicand. The reverse applies when given a number written with an exponent: 8 2 3=8 2 3 The value of the expression is 4.
Simplifying Radicals --------------------
Simplifying radicals involves knowing the value of the index, recognizing the roots of the radicand that match the index, and using the factors to rewrite the radicals in simplest form. The following sections illustrate how to simplify radicals using different methods.
Product Rule
The product rule is written as a b n=a n⋅b n, where n is the value of the index, the root of the radicand. The following steps should be followed when using the product rule:
Find the greatest factor of the radicand that represents the power of the given index.
Rewrite the radical to show the product of the original radicand.
Simplify the radicals.
Product Rule Examples
Simplify: 128 3
128 3=64 3⋅2 3 128 3=4 3 3⋅2 3 128 3=4 2 3
Simplify: 4 a 2 3⋅2 a 4 b 3 3
4 a 2 3⋅2 a 4 b 3 3=4 a 2 2 a 4 b 3 3 4 a 2 3⋅2 a 4 b 3 3=8 a 6 b 3 3 4 a 2 3⋅2 a 4 b 3 3=2 3 a 3 a 3 b 3 3 4 a 2 3⋅2 a 4 b 3 3=2 a 2 b
Quotient Rule
The quotient rule is written as x y n=x n y n. The following steps should be followed when using the quotient rule:
Simplify the value under the radicand like a fraction.
Rewrite the numerator and denominator using the quotient rule.
Simplify radicals.
Quotient Rule Example
Simplify: 35 80
35 80=7 16 35 80=7 4 2 35 80=7 4
Lesson Summary --------------
A square root is a perfect square factor of a given number. A perfect square is a product of multiplying a whole number times itself. The square root symbol is called a radical. The value under the radical is called the radicand. A radical expression is used to explain a radical's value and its radicand. When the index is written, a radical expression can represent values other than squares.
To simplify a square root expression, one should follow the following steps. First, identify factor pairs of the radicand. Then, identify the factor pair with the greatest perfect square. Next, split the factor pair into separate radicands. Finally, simplify radicands. Simplifying radicals involves knowing the value of the index, recognizing the roots of the radicand that match the index, and using the factors to rewrite the radicals in the simplest form.
Video Transcript
Math Is Universal
No matter where you live, the language of mathematics is the same. You might not be able to order dinner in a restaurant in South America, but by using the unique language that is math, you can work problems with anyone from any country. This is why it is very important that mathematical equations get treated the same no matter who is writing them.
The Language of Radicals
The term 'radical' is just another way to say 'square root.' When writing square roots in correct mathematical language, it is important that every radical is written in its simplest form. This applies to both the numbers and variables that are under the square root symbol.
Below you see some examples of radicals that contain both numbers and variables. Some of them are simplified and others aren't. Can you tell which examples need to be simplified further?
All of these are examples of radicals.
Of these four examples, numbers 1, 3 and 4 can all be simplified further.
Simplifying Square Roots of Numbers
Here are the steps to simplifying a square root with a number:
First, factor the number completely. An easy way to factor a number is by using a factor tree. A factor tree can be created by writing down the number you want to factor and drawing two lines coming down from that number. Then, write two factors of that number under the lines. Continue on in this manner until only prime numbers remain. A prime number is one that cannot be reduced any smaller. The purpose of the factor tree is to determine which numbers can be removed from under the square root symbol.
Match up pairs of the same number. Any numbers with a partner are perfect squares and you can take the square root of those numbers.
Numbers without a partner remain under the square root symbol. These numbers cannot be simplified further.
Let's return to the examples from earlier and look only at the number portion of some of them.
The first example is √81 x^4.
Looking at just the number portion, factor 81. 99 is 81, and then 33 is 9; therefore the factorization of 81 is 3333, which is two groups of 3. Each group means a 3 will be removed from underneath the radical, which means that the square root of 81 is 33, or 9.
The √13 xy is the second example.
Since 13 is a prime number, it cannot be factored and therefore is as simple as it can be and no changes can be made.
Simplifying Square Roots of Variables
Before we can talk about finding the square root of a variable, we should probably review what exactly the square root is. The square root operation is the opposite of squared. This works for both numbers and variables. So, xx = x^2, and by performing the opposite operation, √x^2 = x.
Simplifying square roots of variables works about the same way as it does with numbers. Just like you can factor numbers, variables with exponents can also be factored.
For example, x^4 is the same as xxxx.
Then, you can proceed the same way as simplifying the square root of numbers.
First, match up pairs of the same variable. Using our example from before, grouping pairs of x s gives us (xx)(xx) - two groups of x s.
Any letters with a partner are perfect squares, and you can take the square root of them. In this case, there are two pairs of x s, so there will be two x s removed from under the square root symbol.
Variables without a partner remain under the square root. They cannot be simplified further.
Going back again to our examples from earlier, let's look this time at the variable portion.
The first example is x^4.
If we factor x^4, we get xxxx.
Break that into pairs to get (xx)(xx). One x from each pair is taken out from under the square root symbol. This leaves us with an x^2 outside of the square root.
The final simplification of √81 x^4 is 9 x^2.
The second example is √13 xy, and as you can see, there is only one x and one y under the square root symbol. This means that it cannot be simplified any further; it is as simple as it can get.
Putting It All Together
Let's try the last two examples to put it all together. Example number 3 is √(13(x^6')y^2).
As before, the first step is to look at the number - in this case, 13. Since 13 is a prime number, it cannot be factored.
Next, we look at the variables. x^6 can be factored to (xx)(xx)(xx). Because there are three groups of x s, three x s will come out from underneath the radical symbol.
y^2 is yy, which is one group of y s. Therefore, one y can be removed from under the square root symbol.
So, the simplification of this problem is (x^3)y√13
Let's try the last one: √((8 x^2)(y^4)z).
Starting with the number, factor 8 to its smallest parts, which is 222. One pair of twos means that a 2 will come out of the square root symbol. The lone two will stay under the square root symbol.
Next, we move on to the variables. x^2 is xx, which is a pair of x s, meaning that one x will come out of the square root.
y^4 is yyyy. The two pairs of y s mean that 2 y s will come out of the square root.
Because the z is alone, it will stay underneath the square root symbol, which means that the simplification of √((8 x^2)(y^4)z) is 2 xy^2√(2 z).
Lesson Summary
In the international world of mathematics, it is important that there is a universal language so that problems can be understood all over the world. Part of this language means that radicals, or square roots, are simplified in a certain way. To simplify a square root, first factor the numbers or variables, then pair up the like terms. Finally, remove one of each pair from under the square root symbol. Any numbers or variables without a partner remain under the square root symbol.
Learning Outcome
You should be able to simplify radicals using ordered steps after watching this video lesson.
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Evaluating Square Roots of Perfect Squares 5:12
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Simplifying Square Root Expressions | Steps & Examples 7:03
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Family of ray-finned fishes From Wikipedia, the free encyclopedia
Esocidae is a family of ray-finned fish in the order Salmoniformes, which contains pike, pickerel, and mudminnows. While the family traditionally only contained the genus Esox, recent genetic and paleontological research have recovered Novumbra and Dallia as members of the family Esocidae, being closer related to Esox than Umbra. Fossil specimens from the Mesozoic in North America have been assigned as two additional genera in this family, although they may actually be more basal.
| Esocidae Temporal range: Campanian to present PreꞒ Ꞓ O S D C P T J K Pg N | |
--- |
| Thumb | |
| Scientific classification | |
| Kingdom: | Animalia |
| Phylum: | Chordata |
| Class: | Actinopterygii |
| Order: | Salmoniformes |
| Suborder: | Esocoidei |
| Family: | Esocidae G. Cuvier, 1817 |
| Genera | |
| Esox Linnaeus, 1758 Dallia Bean, 1880 Novumbra Schultz, 1929 †Estesesox Wilson et al., 1992 †Oldmanesox Wilson et al., 1992 †Archaeosiilik Brinkman et al., 2025 †Nunikuluk Brinkman et al., 2025 | |
Taxonomy
The family is classified as follows:
Classification
Esox is the youngest genus, with Dallia and Novumbra branching off further up the line. The cladogram below has been found by López et al., with the two additional fossil genera included at the base of the tree.
| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
--- --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- |
| Esociformes | | | | --- | | | Umbridae | | | | Esocidae | | | | --- | | | †Oldmanesox | | | | | †Estesesox | | | | | | | | --- | | | Dallia | | | | | | | | --- | | | Novumbra | | | | | | | | --- | | | Esox | | | | | | | | | | | | | | | |
| |
| | |
--- |
| | Umbridae |
| |
| Esocidae | | | | --- | | | †Oldmanesox | | | | | †Estesesox | | | | | | | | --- | | | Dallia | | | | | | | | --- | | | Novumbra | | | | | | | | --- | | | Esox | | | | | | | | | | | | |
| |
Umbridae
| | |
--- |
| | †Oldmanesox |
| |
| | †Estesesox |
| |
| | | | | --- | | | Dallia | | | | | | | | --- | | | Novumbra | | | | | | | | --- | | | Esox | | | | | | | | | |
| |
†Oldmanesox
†Estesesox
| | |
--- |
| | Dallia |
| |
| | | | | --- | | | Novumbra | | | | | | | | --- | | | Esox | | | | | | |
| |
Dallia
| | |
--- |
| | Novumbra |
| |
| | | | | --- | | | Esox | | | |
| |
Novumbra
| | |
--- |
| | Esox |
| |
Esox
The fossil genera Estesesox and Oldmanesox have been recovered in North America. Estesesox fossils have been described from the Lance, Hell Creek, Oldman, Foremost, and Milk River Formations. Oldmanesox fossils have been described from the Oldman Formation. In 2025, two more fossil esocid genera were described from the Prince Creek Formation of Alaska: Archaeosiilik and Nunikuluk.
Due to the fragmentary nature of Oldmanesox and Estesesox, little information on the exact relationships between the two genera and the rest of the species in the family has been published. More recent genetic studies which place Dallia and Novumbra in Esocidae may influence future studies on placement of Estesesox and Oldmanesox within the Esocidae and/or Esociformes family trees.
Distribution
Esocidae has a holarctic distribution. Species in the genus Esox can be found in Eurasia and North America, while Dallia has a more restricted range in Alaska and eastern Siberia. Of extant esocids, Novumbra has the most restricted range of all, being found only on the Olympic Peninsula in Washington state.
Behavior
Despite differing size and mouth shape, all extant species of esocids are sight-based ambush predators, taking any animal they can fit in their mouth. Both pike and blackfishes display cannibalistic tendencies. While the Olympic mudminnow is aggressively territorial to fish of the same size during the spawning season, it will generally leave its fry alone.
References
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6668 | https://math.stackexchange.com/questions/2681940/how-to-determine-half-life-t-1-2-in-terms-of-the-decay-constant-k-and-then | calculus - How to determine half life $T_{1/2}$ in terms of the decay constant $k$ and then solve for $P(t)$ in terms of $T_{1/2}$? - Mathematics Stack Exchange
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How to determine half life T 1/2 T 1/2 in terms of the decay constant k k and then solve for P(t)P(t) in terms of T 1/2 T 1/2?
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I am trying to figure out a radioactive decay problem, and I am a little stuck. I have the decay equation
d P d t=−k P d P d t=−k P
And I am trying to determine the half life T 1/2 T 1/2 in terms of the decay constant k k, and then solve for P(t)P(t) in terms of T 1/2 T 1/2
However I dont really know how to go about this...
Any help would be much appreciated Cheers
calculus
ordinary-differential-equations
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edited Mar 8, 2018 at 5:48
an4s
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asked Mar 8, 2018 at 5:39
FishingfonFishingfon
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Can you solve the differential equation? Then look up the definition of half-life.Ross Millikan –Ross Millikan 2018-03-08 05:43:05 +00:00 Commented Mar 8, 2018 at 5:43
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d P d t=−k P⟹d P d t=−k P⟹
P(t)=p(0)e−k t P(t)=p(0)e−k t
and the half life is found to be
T 1/2=ln 2/k T 1/2=ln2/k
Thus
k=ln 2/T 1/2 k=ln2/T 1/2
P(t)=p(0)e−k t=p(0)2−t T 1/2 P(t)=p(0)e−k t=p(0)2−t T 1/2
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answered Mar 8, 2018 at 6:08
Mohammad Riazi-KermaniMohammad Riazi-Kermani
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| | | | | | | |
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Inverse Laplace Transform Examples
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A Comprehensive Guide to the Inverse Laplace Calculator
1. An Overview of the Inverse Laplace Calculator
The Inverse Laplace Transform is a powerful mathematical technique that is used to translate functions that are in the complex frequency domain back into the time domain. Determining the solutions to differential equations, analyzing control systems, and gaining an understanding of many scientific and technological occurrences are all important aspects of this technique. Through the use of the Inverse Laplace Transform, we are able to observe the behavior of systems over time, which provides us with insights into stability, response characteristics, and other aspects.
The understanding of how systems evolve over time is essential to a wide variety of technological and scientific domains. Because the Laplace Transform and its inverse create a bridge between the time domain and the frequency domain, they make it possible to analyze and solve complex issues in a more straightforward manner. With the goal of providing readers with the knowledge and abilities necessary to make effective use of the Inverse Laplace Transform, this article will investigate the theory, applications, and practical applications of the instrument.
For example, consider a simple RC circuit with a resistor
R and a capacitor
C. The differential equation describing the voltage
V(t) across the capacitor is:
RCdtdV(t)+V(t)=V0
where V0 is the input voltage. Using the Laplace Transform, this equation can be converted into the frequency domain:
RC(sV(s)−V(0))+V(s)=sV0
Assuming initial conditions are zero, we get:
V(s)=s(RCs+1)V0
The Inverse Laplace Transform of V(s) gives us the time-domain response:
V(t)=V0(1−eRC−t)
This shows how the voltage across the capacitor changes over time.
2. What is Laplace and Inverse Laplace?
Laplace Transform
The Laplace Transform is an integral transform that converts a function belonging to the time domain into a representation belonging to the frequency domain. Due to the fact that it converts differential equations into algebraic equations that are easier to understand, it is particularly useful for solving linear differential equations that include initial conditions.
For example, the Inverse Laplace Transform of F(s)=s+a1 is:
L−1s+a1=e−at
Inverse Laplace Transform
Reversing this action is accomplished by the use of the Inverse Laplace Transform, which takes the function from the frequency domain and transfers it into the time domain. This is essential for an understanding of the consequences of Laplace Transforms in practical applications, which ultimately results in the provision of time-domain solutions that define the behavior of the system.
Expression that is both prompt and precise employing the Inverse Laplace Method for simplifying and utilizing the Inverse Laplace Transform facilitates the simplification of complicated expressions that are generated in the field of differential equations and control theory. By converting time-domain functions into frequency-domain expressions, engineers and scientists are able to evaluate system responses in a manner that is more accessible. The use of this instrument is very beneficial in fields such as electrical engineering, mechanical engineering, and physics, which need a comprehensive understanding of the behavior in the time domain.
4. Inverse Laplace as a Tool to Simplify Expressions Quickly and Accurately
By breaking down complex functions into more manageable components, the Inverse Laplace Transform makes it easier to analyze and comprehend the dynamics of a system. The Laplace Transform, for example, makes it possible to investigate the transfer function of a system in control systems. The Inverse Laplace Transform, on the other hand, is responsible for delivering the time-domain information. The stability, transient responsiveness, and steady-state behavior of the system are all shown by this technique.
For example, consider the transfer function of a control system.
H(s)=s2+3s+210
The Inverse Laplace Transform of H(s) gives the impulse response of the system:
h(t)=10e−t−10e−2t
This shows the system's response to an impulse input over time.
5. Features of Inverse Laplace Calculator
Students and professionals alike may benefit from using an inverse Laplace calculator due to the numerous characteristics that it has, including the following:
• Because it provides precise results for complex equations
• It ensures that analysis and design are carried out in a consistent manner.
• It calculates the inverse Laplace transform quickly, thereby saving both time and effort during the problem-solving process.
• The interface makes it accessible to a wide range of users since it is simple to use, even for individuals who have little understanding of mathematics.
• Its ability to handle a wide range of functions and expressions enables it to accommodate a wide range of applications and circumstances.
• Providing customers with comprehensive answers to aid them in comprehending the transformation process—enhancing their knowledge and comprehension— might be beneficial.
• Through the process of guiding people through the change, your precise solutions boost learning and comprehension.
• It is necessary to do the following actions to use an inverse Laplace calculator: In the calculator, type in the function that operates in the frequency domain. Check to see that the function is built in the appropriate manner and has all of the necessary components.
• If needed, kindly define the variables in question. This step ensures that the calculator understands the function's history.
• This guarantees that the calculator comprehends the context of the function. To get the function that operates in the time domain, hit the 'go' button. The calculator will perform the inverse Laplace transform and handle the input.
• To have a more profound understanding of the transformation process, you should study the step-by-step solution that the calculator provides. Particularly useful for the purpose of education, this function makes it possible for individuals to comprehend the fundamentals of mathematics.
Example 1 :
Let's figure out how to calculate the inverse Laplace transform of a simple unit step function.
G(s)=1/(s+2)
Solution:
L−1s(s+2)1
Take the partial fraction of
s(s+2)1
Create the partial fraction template using the denominator s(s+2)
For s add the partial fraction(s): sa0
For s+2 add the partial fraction(s): s+2a1
s(s+2)1=sa0+s+2a1
Multiply equation by the denominator
s(s+2)1⋅:s(s+2)=sa0s(s+2)+s+2a1s(s+2)
Simplify:
1=a0(s+2)+a1s
Solve the unknown parameters by plugging the real roots of the denominator: 0, -2
For the denominator root 0 : a0=1/2
Plug s=0 in into the equation
1=a0(0+2)+a1⋅0
Expand:
1=2a0
Switch sides
2a0=1
Divide both sides by 2
22a0=21
a0=21
For the denominator root -2 : a1=−21
Plug in s=-2 into the equation
1=a0((−2)+2)+a1(−2)
Expand
1=−2a1
a1=−21
a0=21,:a1=−21
Plug the solutions to the partial fraction parameters to obtain the final result
s21+s+2(−21)
2s1−2(s+2)1
=L−12s1−2(s+2)1
Use the linearity property of Inverse Laplace Transform:
For functions f(s), g(s) and constants a,b: L−1a⋅f(s)+b⋅g(s)=a⋅L−1f(s)+b⋅L−1g(s)
Use Inverse Laplace Transform table:
L−1s−a1=eat
L−1sa=aH(t)
=21H(t)−21e−2t
Example 2:
We are going to find the inverse Laplace transform of the cosine function.
F(s)=L−1(s2+a2)s
Solution:
L−1(s2+a2)s
Apply inverse transform rule:
if L−1F(s)=f(t)
then
L−1sF(s)=dtdf(t)+f(0)
For s2+a2s:F(s)=s2+a21
L−1s2+a21
=L−1a21⋅:s2+(a2)2a2
Use the constant multiplication property of Inverse Laplace Transform:
For function f(t) and constant a:L−1a⋅f(t)=a⋅L−1f(t)
=a21L−1s2+(a2)2a2
Use Inverse Laplace Transform table: L−1s2+a2a=sin(at)
6. Different kinds of inverse Laplace
a) The Inverse Laplace Standard Model
Typically, the process of translating a rational function back into a time-domain function is referred to as the inverse Laplace transform. The most common kind is this one, which is used in a wide variety of technological applications. In most cases, the standard form is made up of simple poles and zeros that are representative of exponential and sinusoidal functions in the time domain.
Example: Find the inverse Laplace transform of:
F(s)=s2+a2s
Using the standard formula:
L−1s2+a2s=cos(at)
Thus, the inverse Laplace transform is:
f(t)=cos(at)
b) Partial Fraction Decomposition
Before carrying out the inverse Laplace transform, the statement is simplified by the use of partial fraction decomposition. This is done in situations when the function in question is a ratio of polynomials. This strategy involves breaking the function down into a number of smaller components, each of which may be modified with minimum effort.
Example:
Through the use of a function, we are able to partition it into incomplete fractions. After determining the individual fractions, we now proceed to solve for the coefficients. By performing the inverse Laplace transform on each term, one may extract the function that operates in the time domain.
Example. Find the inverse Laplace transform of:
F(s)=(s+2)(s+3)1
Using partial fraction decomposition:
(s+2)(s+3)1=s+2A+s+3B
Solving for A and B, we get:
A = -1, B = 1
Thus,
F(s)=s+31−s+21
Using the inverse Laplace formulas:
L−1s+a1=e−at
we get:
f(t)=e−2t−e−3t
c) Convolution Theorem
Within the context of function products, the inverse Laplace transform is made possible by the convolution theorem. This theorem is very useful in the field of control systems and signal processing, where the output of a system is the convolution of the input and the impulse response of the system.
Example:
Any function may be expressed as a product of two other functions if we are given the function. We are able to determine the time-domain function with the use of the convolution theorem.
If F(s)=F1(s)⋅F2(s), then the inverse Laplace transform is given by the convolution integral:
L−1F1(s)F2(s)=(f1∗f2)(t)=∫0tf1(τ)f2(t−τ)dτ
Find the inverse Laplace transform of:
F(s)=s(s+1)1
We split this as:
F(s)=s1⋅s+11
Using known inverse transforms:
L−1s1=1,L−1s+11=e−t
Applying the convolution integral:
f(t)=∫0t1⋅e−(t−τ)dτ
Solving the integral gives:
f(t)=1−e−t
Conclusion:
It is essential for engineers and scientists to have access to the Inverse Laplace Calculator, which provides solutions to difficult mathematical problems in a quick and accurate manner. Because of its ability to condense sentences and deliver responses that are comprehensive, it is a beneficial tool that may be used in both educational and professional settings. Users are able to increase their problem-solving skills and get greater information of the behavior of dynamic systems by gaining an understanding of the properties and applications of the Inverse Laplace Calculator.
By bridging the gap between the frequency domain and the time domain, the Inverse Laplace Transform makes it possible to have a more intuitive understanding of the dynamics of the system. For the purpose of analyzing and developing systems that meet certain performance criteria, the Inverse Laplace Transform is a powerful tool that may be used in a variety of engineering fields, including control systems, electrical engineering, mechanical engineering, and other areas of study. It is possible for scientists and engineers to utilize this instrument in order to address challenging problems with reliability and precision.
Through the use of this instrument, engineers and scientists are able to confidently and accurately answer difficult problems.
Frequently Asked Questions (FAQ)
How do you find the inverse Laplace transforms of functions?
To find the inverse Laplace transform of a function, apply laplace transform properties and use tables of inverse Laplace transforms.
Is Laplace inverse linear?
The inverse Laplace transform is a linear operation.
Is there always an inverse Laplace transform?
A necessary condition for the existence of the inverse Laplace transform is that the function must be absolutely integrable, which means the integral of the absolute value of the function over the whole real axis must converge.
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f(x) = x^2 + 2x-1 over interval [0, 1], find the value of c that satisfies the mean value theorem.
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Question: f(x) = x^2 + 2x-1 over interval [0, 1], find the value of c that satisfies the mean value theorem.
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6671 | https://core.ac.uk/download/41439573.pdf | 《海外华文教育》 2010 年第3 期( 总第56 期) 现代汉字字形分析及 在对外汉语教学中的应用述评 郭 星 提要 自对外汉语教学事业开展以来， 汉字教学就一直占据着重要的地位， 汉字 因其结构的特殊性一直是教学中的难点及重点， 近年来， 众多学者从现代汉字字形角 度出发， 例如， 笔画、 部件， 不断解读汉字的形体特征， 并把相关成果运用在教学中。 本文尝试总结关于现代汉字字形分析的成果， 并提出自己的看法。 关键词 现代汉字 字形分析 对外汉语教学 一 现代汉字字形分析 自汉字产生起， 中国历代学者便不断分析、 总结其形体特征， 以便于汉字书写的教学。 当下， 各家虽然有不同的看法， 但汉字字形研究成果已基本趋于一致。下面笔者把一些学 者的观点以图解的形式展示给大家: 费锦昌先生在 《 现代汉字部件探究》 ( 1996) 中指出: 现代汉字字形 笔画基本笔画 {复合笔画 部件基础部件 {复合部件 整字单部件字 { 多部件字 苏培成先生在 《 二十世纪的现代汉字研究》 ( 2001) 中认为， 汉字字形分析有三个平 面: 汉字字形分析 造字法: 追溯字源( 古代) ， 汉字形体结构发展演变的规律。 造字法: 形、 音、 义( 内部结构) 。 构形法( 外部结构) 笔画 笔画数 笔形 笔顺 { 笔画组合 部件 分类 切分 规范 { …… 整字 独体字与合体字 合体字的部件数量 { 汉字的结构类型 · 4 7 · 邢红兵在 《 现代汉字特征分析与计算研究》 ( 2007) 中指出: 汉字字形特征 笔画 笔形 笔画数 {笔画趋减率 部件 类型 数量 成字及非成字部件 构字能力 典型位置 构形 构造 结构类型 结构层次 { 汉字的构造 从上述三位学者的划分体系中我们可以看到其异同点: 一、 在一级分类中都包括了笔 画和部件。它们的不同之处在于苏先生和费先生同时划分出了整字， 而邢先生另辟出构 造一类。笔者认为， 原因在于， 邢先生是从现代汉字字形的特点入手， 以在计算机中的应 用为目的展开分析讨论， 因此把 “构造” 作为一个字形特征是非常必要的。二、 在二级分 类中， 三家之说有交叉也有差异。费先生的划分整齐划一， 以对应的概念出现， 便于理解， 且简单明了， 一目了然。另外两位学者则划分较细， 在具体细分层次上有些出入， 可以看 做是一种相互补充。 综上所述， 大家出发点、 目的各异， 邢先生着重于计算机研究方面; 而费先生则是力求 统一现代汉字字形分析的层次; 苏先生则是站在前人的成果上， 多有补充和细化。对于对 外汉语教学来说， 汉字字形分的得太细， 太过于纠缠枝节， 不利于课堂教学的把握。笔者 认为对外汉字教学可以充分利用上述三位学者的研究成果， 三位学者的汉字分析理论， 各 有所长， 互相补充， 恰好可以运用于对外汉语教学的研究。 二 笔画及教学 既然汉字的字形构成是分层次的， 可以逐层进行分析的， 可以从整字分解出部件， 从 部件中分解出笔画， 那么我们首先就从笔画方面入手。在对外汉语教学界， 课堂讲解最多 的是笔形、 笔顺和整字的书写。 ( 一) 笔形及其教学 关于笔形的数量， 各家统计结果不同。例如陈仁凤、 陈阿宝 《一千高频度汉字的解析 及教学构想》 从1000 高频度汉字中归纳出6 种基本笔形和22 种派生笔形; 张静贤 《现代 汉字教程》 列出基本笔形6 种， 派生笔形25 种; 黄伯荣等主编的 《现代汉语》 ( 增订二版) 列基本笔形5 种， 变化笔形36 种。而且笔形的定名， 也存在着一些差异。比较笔形命名 的新旧两说， 不难看到说法虽不同， 却有共同的基点。所谓派生笔形( 也有称之为复合笔 形的) ， 是基本笔形的组合。既是组合， 必须可以连接过渡。这就涉及汉字笔画的连接方 式。习用汉字的人都知道汉字笔画的连接方式只有两种: 一是先从上到下再从左到右 ( L) ， 二是先从左到右然后再从上到下( 门) 。与单个笔画书写时的运笔方向是一致的。 其它方向上的连笔都是这两种方式的变化形式。旧名着眼于笔画连接处的形态描写 · 5 7 · ( 折: 方折; 弯: 弧形过渡) 而不说运笔方向， 新名只说笔画名称而不提连接位置， 却都能成 立而不致引起误解， 正是由于两者都是建立在众所周知的运笔方向和笔画连接过渡原则 上的。只是这种对于习用汉字的中国人来说 “ 不言而喻” 的约定， 在对外汉字教学中往往 容易被忽略。 目前， 对外汉语汉字教学的一般做法是， 在笔形数量上从一家之说。至于名称， 也多 从旧说。而且常常在教材的开头部分安排这些内容， 把基本笔形和派生笔形并名称逐一 介绍， 而初学汉语的学生很难记认 “横 ” 、 “竖 ” 、 “撇 ” 、 “捺” 这些名称， 往往只能机械记 忆， 增加了学习汉字的难度。万业馨认为: 在教学中我们不妨以简驭繁， 只将基本笔画和 连接方式告诉学生， 请他们自行找出例字中的派生笔形( 甚至给予冠名权， 由他们自己命 名) 。但笔者以为， 赋予学生冠名的权利， 是不利于他们日后系统的学习的， 也不利于老 师课上的示范讲解。如， 在老师引导学生学习 “ 木” 字时 ， “ 横、 竖、 撇、 捺” 几个名称必然会 让学生听得一头雾水。所以， 只有采用一些教学技巧引起学生的兴趣进而记住这些笔顺 才是唯一解决问题的方法。如 ， “横” 就用躺着的人的图片， 以刺激学生的视觉来达到记 忆的效果。 ( 二) 笔顺及其教学 笔顺是书写教学的重要内容之一。然而我们的笔顺教学却很难说取得了成功。究其 原因， 从主观上说， 是重视不够， 甚至以为无足轻重。因为笔顺错了， 不等于字一定写错。 在一些教师和学生的心目中， 把字写正确是第一位的， 笔顺只不过是为了书写便利， 书写 者有取舍的自主权。至于客观原因， 是教师很难甚至无法检查学生在写字时( 尤其是家 庭作业) 是否有正确的笔顺。因此， 往往到了高年级， 还有相当一部分学生的书写过程呈 无序状态， 不仅严重影响了书写速度乃至考试成绩， 而且导致书写错误概率的增长。 “ 从左到右 ” 、 “ 从上到下” 是指笔画书写时的运笔方向以及部件书写时的先后顺序; “ 先撇后捺 ” 、 “ 先横后竖” 讲的是不同笔形同时存在时的书写先后顺序。这些看来十分简 明的规则， 初学者在书写时却常会用错。如有的学生写 “王” 字时， 把所有的横画都写完 了才写竖画。问他为什么这样写， 回答是 “先横后竖” ; 有的学生写 “小” 字， 先写左边的 点， 理由是 “ 从左到右” 。可见， 什么时候用哪一条规则， 才是笔顺教学的难点。 对于这一问题， 万业馨做了很好的总结: 对先横后竖这一笔顺规则较为全面的理解是 横画者， 底部承托者后作， 被穿过者先作， 如 ， “ 土 ” 、 “ 丰” ; 竖画不穿过横画者先写， 穿过者 后写， 如 ， “ 日 ” 、 “ 申” ; 对 “ 先中间后两边” 的理解则是如果我们把包围、 三包围及类似的 结构统称之为封闭结构， 则凡有可能形成封闭结构的， 先定界画; 凡竖画穿过的部分为封 闭结构者， 后作， 如 ， “ 国” ; 不属于封闭结构者， 先作， 如 ， “ 包” 。 三 部件及其教学 部件及其教学在对外汉语教学中占据着及其重要的地位， 而部件内部本身就存在着 各种争议， 因此， 我们有必要对它有一个清晰的认识。 ( 一) 汉字部件的定义 二十多年来， 学者们先后提出了许多个部件的定义。这里略举一二: 国家语委 : “ 由笔画组成的具有组配汉字功能的构字单位。 ” 傅永和( 1991) 认为 : “汉 字部件是构成合体字的最小笔画结构单位， 其下限必须大于基本笔画， 上限小于复合偏 旁。从功能看， 部件并不一定具有音、 义; 从存在形式看， 它是一个独立的书写单位， 不管 · 6 7 · 笔画多么复杂， 凡是笔画串连在一起的， 都作为一个部件看待。 ” 晓东( 1994) 指出 ， “现代 汉字的部件应该是指: 一个汉字中， 由两笔以上( 含两笔) 笔画组合而成的、 相对独立的最 小笔画结构块。 ” 苏培成( 1995) 提到 ， “部件是汉字的基本构字单位， 介于笔画和整字之 间。它大于或等于笔画， 小于或等于整字。部件大于笔画是常例， 等于笔画是特例。在合 体字中， 部件小于整字; 在独体字中， 部件等于整字。部件组合为合体字， 具有层次性， 因 此从合体字中切分出部件来也有层次性。部件切分就是依据组合层次对合体字逐层进行 分解， 一直分到末级部件。末级部件是最小的构字部件， 再分只能分解为笔画。 ” 费锦昌 ( 1998) 将部件定义改为 : “ 部件是现代汉字字形中具有独立组字能力的构字单位， 它大于 或等于笔画， 小于或等于整字。 ” 邵敬敏( 2001) 的定义为 : “ 部件是构成汉字字形的基本单 位。 ” 黄伯荣、 廖序东( 2002) 提出 : “部件又称偏旁， 是由笔画组成的具有组配汉字功能的 构字单位。 ” 张静贤等( 2004) 则认为 “ 部件是由笔画组成的、 能独立运用的、 具有组配汉字 功能的构字单位。 ” 在众多学者中， 我比较赞同邵敬敏先生把部件与偏旁、 部首放在一起这样的定义比 较: 偏旁就是用二分法对合体字进行一次性切分而获得的结构单位。可以分为 “形旁” 和 “ 声旁” 两类。部件的概念可大可小， 有时部件是对合体字进行一次切分而得到的两个单 位， 这时的部件相当于偏旁; 有时部件是对合体字进行多次切分而得到的多个单位， 这时 的部件就要比偏旁小。部首是工具书为给汉字分类而专设的部目， 一般包括两类: 一是形 旁; 二是某些笔画。 笔者认为， 对研究者来说应该对偏旁、 部首和部件有一个清醒的认识， 唯有如此， 才能 在课堂上清晰地讲解。而对于学生来讲， 我们要避免在最开始就引导他们区分三者的区 别， 一方面会让学生产生畏难情绪， 另一方面会浪费课上时间。教学中， 我们导入的都是 常见的部件， 加入其意义的讲解， 对于那些特殊或不常用的部件， 我们应该在学生的不断 深入学习过程中个别讲解。把化零为整和化整为零结合起来， 不失为一个好办法。如， “ 财” 字中 “ 贝” 和 “ 才” ， 只需要告诉学生 “ 贝” 表示钱的意思， 所以是部件 ， “ 才” 表音， 不必 再解释深层原因。然后进一步列出以贝为部件的字加深学生印象 ， “ 贵、 贱、 赌、 贿” 等。 ( 二) 汉字部件的分类 很多学者对汉字部件的分类曾经有过论述， 例如: 费锦昌( 1996) 分为 “ 基础部件” 和 “复合部件” 两类: 把 “独体字、 合体字” 改称为 “单 部件、 多部件字” 。苏培成( 2001) 将部件分成若干类型: ( 1) 成字部件和非成字部件; ( 2) 基础部件和合成部件; 黄伯荣等( 2002) 指出部件 “按照不同标准可以分成不同的类型。 ” ( 1) 按照现在能否独立成字划分， 可以分为成字部件和非成字部件两类。( 2) 按照能否再 切分成小的部件划分， 可以分成单一部件和复合部件两类。( 3) 按照部件切分出的先后 划分， 可以分成一级部件、 二级部件、 三级部件等。 以上分类都是建立在纯形体角度之上的， 总的来说， 大同小异， 而且学术界和教学界 都比较这样的划分， 所以笔者认为这方面的研究对教学而言已经相对成熟了， 可以不再做 讨论。 ( 三) 汉字部件的切分 汉字本来是逐层组合的， 那么依据这种汉字本身的层次性进行逐层切分， 最后得出末 级部件， 也即基础部件。中间的层次依次对应一级部件、 二级部件等。那么， 如何切分视 为合理呢?
· 7 7 · 总的来说， 关于部件切分方法主要有两种意见， 即苏培成( 1995) 所提 ， “确定组合层 次， 有两个原则: 一个是单纯字形原则， 一个是构字原则。 ” 这就是通常所谓的 “据形切分” 和 “ 据理( 源) 切分” 。张晋对部件切分的原则作了研究， 提出 “从形切分 ” 、 “从义切分” 和 “ 形义兼顾” 三个选择。操作中， 从形切分有些合于构字理据， 有的则有悖于构字理据。 从汉字编码的角度看， 以从形切分的原则好。对于形体结构不明显可拆分的， 进一步加以 小类别处理。但是从识字教学角度来讲， 则以从义切分为好。然而， 对于某些情况， 亦为 不易。因为我们的切分面对的是现代汉字体系， 经过悠长历史的演变， 很多字的理据已为 一般拆字、 用字的人所不解。而形义兼顾的原则带来时而从形， 时而从义的结果， 很强的 主观随意性暴露无遗。 王宁先生( 1997) 对部件切分方法进行了更深层次的阐述: ( 1) 一部分部件， 属于既同 形、 又同源的部件， 由于同源， 它们必然同义， 如系成字部件， 又必然同音。这些部件归纳 为同一部件， 在形、 源两方面都是合理的。( 2) 一部分部件， 同源也同音义， 但由于书写部 位及结构环境的变化， 书写略有变异。这类情况， 形体相距不远的， 可独立用字的字形为 主形， 其余按其变体归纳: 形体相距较远的可以分立。( 3) 一部分部件， 同源而不同形， 变 异之后又与其他不同源的部件合流。这些部件异化后的形体都与主形距离较远， 按形归 纳与按源归纳发生矛盾。( 4) 既有同源异形而与不同源部件合流的现象， 也必然会有不 同源而同形的现象。 总之 ， “ 据形分析” 与 “据理切分” 各有利弊: 前者忽视汉字理据， 所得部件总量虽小， 但是很多是单纯的记号， 长于文字输入， 却不利于汉字教学; 后者重视文字字理和源流， 为 文字学界所广泛采用， 但由于汉字字形既有继承， 更有演化， 加上简化字推行等人为因素 的介入， 很难贯彻始终。如何在汉字教学的实际操作中把握二者的平衡， 值得进一步研 究。 ( 四) 汉字部件的数量 很多学者都对一定量的汉字进行过具体部件切分和统计， 从不同的目的及出发点各 家得出了不同的数字 : 《 信息处理用GB13000．1 字符集汉字部件规范》 以20902 个汉字为 范围拆分制定了 《 汉字基础部件表》 ， 包含有560 个基础部件; 费锦昌( 1996) 也就 “现代汉 语3500 常用字部件表” 拟定共有部件384 个， 包括 : “ 基础部件” 290 个( 含单笔部件8 个、 多笔部件282 个， 其中成字部件即单部件字162 个 ) 、 “常用复合部件” 94 个 。“常用复合 部件表” 是开放的， 可以根据实际需要增删; 崔永华( 1997) 针对国家对外汉语教学领导小 组办公室汉语水平考试部编制的 《汉语水平考试词汇等级大纲》 中的8822 个词， 重点分 析构成其中1033 个甲级词的801 个汉字和构成这801 个汉字的330 个部件; 1998 年提出 了 “ 基本部件+ 基本字” 汉字教学单位体系， 共537 个成员， 包括整字350 个、 表义部件42 个、 无义部件145 个。 以上切分的范围不同， 目的各异， 所得的部件数量自然不尽相同。但是总的来说， 如 此数量的部件， 是很难直接利用于教学的; 确定对外汉字教学中常用的几十个部件， 具有 十分重要意义。 ( 五) 部件教学 长期以来， 部件教学得到了广大从事汉语教学工作者的认同， 众多学者也不断有理论 及实践创新。 张旺熹( 1990) 从汉字的特点出发， 提出了汉字 “部件结构教学” 的基本原则和方法。 · 8 7 · 该方法立足于现行汉字的特点， 符合汉字本身的规律， 其有效性不断地得到教学实践的证 实。作为研究汉字的前期学者， 为后来的汉字教学方法研究做了巨大贡献。 崔永华( 1997) 通过统计分析， 说明了利用部件教学的可行性。后来， 他( 1999) 进步 提出了由 “ 汉字部件” 和部分 “ 基本字” 构成的汉字教学单位体系。其研究的深化和细化 不仅丰富和发展了" 部件教学" 的理论构想， 也为后来学者研究向纵深化发展提供了启 示。 万业馨肯定了 “ 部件教学” 的合理性， 对 “ 部件” 进行了重新界定—认为教学中可利用 的部件即 “ 字符” ( 1999) 。她还进步探讨了教学中 “ 先教部件还是先教整字” 的问题， 针对 声旁和形旁的不同特点， 提出了解决方案( 2000) 。万业馨对汉字部件的研究是逐渐深入 的， 他不停留在部件本身， 还尝试从文字学角度对部件教学进行挖掘， 在 《文字学视野中 的部件教学》 ( 2001) 一文中， 他指出可以给学生充当意符的字。这一系列关于汉字教学 的论文承接了研究想纵深化方向的发展。虽然" 字符" 这一术语在当下还没有引起广泛 关注， 但其对" 部件" 的重新界定的精神是值得肯定的; 其次， 声旁与形旁本来就是有交叉 的， 不是相对独立的， 所有对其分类处理是有利于教学的; 再次， 从文字学这样一个大的视 野下来谈部件教学又为汉字教学研究提供了新的出路。 李大遂( 2002) 则对汉字偏旁在教学中的作用进行了研究。对部件的研究还注意到 部件的区别特征对对外汉字教学的影响， 梁彦民( 2004) 就如何利用部件的区别性特征对对外汉字教学进行了研究。 这两位学者都抓住了利用部件的区别性特征， 从对比分析中形成了相对成熟的理论 并且在实际教学应用中的效果立竿见影。 值得一提的是， 最近两年 “ 部件赋音” 工作取得巨大进展， 程大琥及其学生龙瑛在大 量实验的基础上， 详细阐释了汉字部件赋音的依据及教学构想。部件赋音涉及到了部件 的称说问题， 进行了深入的探索， 虽然， 目前还没有形成统一的学术成果， 但这依然可以说 是部件教学的一大贡献。同时， 也有学者质疑" 部件赋音" 研究内容庞大， 许多生僻部件 用的很少， 那对于对外汉语教学来说， 我们只要选取常用的一部分来用， 并结合形、 义在课 堂上呈现给学生。 部件教学比之于由笔画直接过渡到整字教学在近年已经在对外汉语教学界达成共 识。原因在于部件教学符合汉字构形的基本特征: 汉字是具有层次性和组合性的; 其次， 成字部件具有一定的构字能力， 可以构成众多的合体字， 因此它们的总数必然少于汉字总 数。而对汉字教学而言更为重要的是， 充当部件的字原来是有音有义的， 较之笔画更有可 能在音义上对整字的识记起到提示作用。目前我国部件教学在理论与实践两个领域均已 取得重大成果， 但是仍有一部分细节问题需要解决， 比如讲解顺序、 讲解方式等方面。 四 小结 纵观现代汉字字形分析及其在对外汉语教学中的运用研究， 我们的确取得很多的成 绩， 推动了对外汉字教学及研究， 但同时也存在一些弊病和薄弱之处。 首先是对对外汉字教学中的汉字字形的分析研究逐步深入， 并在此基础上探究出很 多种教学方法， 充分利用了汉字字形上的特点。事实证明， 利用汉字字形进行教学的确可 以取得很好的效果。但是在形、 音、 义的结合方面， 研究工作还不是很充分， 迫切需要我们 去补充和完善。如上文提到， 有的学者注重形， 有的注重音、 义， 至今没有统一的研究成 · 9 7 · 果， 一线教师们也只是在实践中对三者有了一定的结合， 尚未归类形成系统成果。如， " 人" 字， 不仅要展示给学生甲骨文图形， 也要把其变形， 以及这一类字的含义进行讲解。 其次， 应该不断挖掘新的研究方法、 引入新的理论。新理论的引入给对外汉字教学研 究提供了理论支撑也开掘了新的教学方法， 新的研究方法也推动了对外汉字教学研究。 因此， 比较理想的做法是先进行一定的理论思考， 再进行教学实验， 最后拿出成果。汉字 教学实践的人多， 但成果零零星星， 问题在于实践之后没有反过来通过大量实验来验证理 论， 这就导致了大量的重复性研究。 最后， 在研究队伍方面， 前期有一批固定的学者进行了广泛而深入的理论及实践研 究， 但最近几年， 随着研究热点的不断消失， 研究成果也不如以前那么集中。现代汉字字 形分析及其在对外汉语教学中的应用， 始终是对外汉语教学中的一个重点， 一旦出现断 层， 对整个学科都是一种损失。 参考文献 邢红兵．现代汉字特征分析与计算研究［M］．商务印书馆．2007． 苏培成．二十世纪的现代汉字研究［M］．书海出版社．2001． 费锦昌．现代汉字部件探究［J］．语言文字应用．1996，( 2) ． 邵敬敏．现代汉语通论( 第二版) ［M］．上海教育出版社．2007． 黄伯荣等主编．现代汉语( 修订二版) ［M］．高等教育出版社．2000． 张静贤．现代汉字教程［M］．现代出版社．1992． 吕必松．汉字与汉字教学研究论文选［C］．北京大学出版社．1999． 陈仁凤，陈阿宝．一千高频度汉字的解析及教学构想［J］．语言文字应用．1998，( 1) ． 易洪川．折笔的研究与教学［J］．语言文字应用．2001，( 4) ． 万业馨．汉字字符分工与部件教学［J］．语言教学与研究．1999，( 4) ． 施正宇．现代汉字的几何性质及其在汉字教学中的意义［J］．语言文字应用．1998，( 4) ． 黄贞姬．HSK 等级大纲中的汉字与韩国教育用汉字的字形比较［J］．汉语学习．1998，( 1) ． 王建勤．汉语学习者汉字知识获得机制模拟研究［J］．语言文字应用．2008，( 1) ． 郝美玲．留学生汉字正字法意识的萌芽与发展［J］．世界汉语教学．2007，( 1) ． 江 新．外国学生形声字表音线索意识的实验研究［J］．世界汉语教学．2001，( 2) ． 张德鑫．关于汉字文化研究和汉字教学的几点思考［J］．世界汉语教学．1999，( 1) ． 陈 绂．谈对欧美留学生的字词教学［J］．语言教学与研究．1996，( 4) ． 王幼敏．日本人书写中文汉字差错规律的分析及思考［J］．华东师范人学学报．1998，( 2) ． 潘先军．对外汉字教学与小学识字教学［J］．汉字文化．1999，( 2) ． 李人遂．对外汉字教学回顾和展望［J］．渤海人学学报．2007，( 2) ． 万业馨．从汉字研究到汉字教学［J］．世界汉语教学．2004，( 2) ． 龙 瑛．汉字部件赋音及其教学构想［D］．湖南师范大学学士论文．2007． ( 郭 星 厦门大学海外教育学院) · 0 8 · |
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Difficult coin weighing puzzle: 14 coins, 1 fake (heavier or lighter), 3 pre-determined weighings
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This recent question reminds me of a coin weighing puzzle I learned many years ago. It is one of the hardest puzzles of this kind that I know. I will post my solution in a few days, and meanwhile hopefully someone may enjoy it. (My apologies if this is a repeat, but I searched and could not find this exact version.)
There are 14 14 suspect coins, 13 13 of which are good and have the same weight, and the last one is bad and have a different weight (heavier or lighter). In addition, you have a 15 15 th coin that is known to be good.
You want to find which suspect coin is bad, and as much as possible (see below), whether it is heavier or lighter. There are therefore 28 28 possible answers: 14 14 suspects ×{h e a v i e r,l i g h t e r}×{h e a v i e r,l i g h t e r}.
You are allowed 3 3 weighings on a balance. Now of course, 3 3 weighings only give you 3 3=27 3 3=27 possible outcomes, so you cannot fully distinguish all 28 28 answers. The requirement is that:
26 26 of the 27 27 outcomes must lead to a unique answer (which coin is bad and whether it is heavier or lighter)
while the last outcome must lead to knowing which coin is bad, but without knowing if it's heavier or lighter (i.e. it lumps together the 2 2 answers for that coin).
The above puzzle would be difficult enough, but here's the final twist: What coins to use in a weighing cannot depend on the results of previous weighings.
To be more precise, label the suspect coins ABCDEFGHIJKLMN and the known-to-be-good coin X. Before you begin, you must write down what two subsets of coins are involved in each of the 3 3 weighings, e.g. ABCDX-EFGHN, IJKL-MNAB, CDEFGH-IJKLMN. This way, your second weighing IJKL-MNAB is pre-determined and cannot depend on the result of the first weighing ABCDX >/=/< EFGHN, etc. (Indeed, you can now do the 3 3 weighings in any order.)
Can you find such a set of 3 3 pre-determined weighings that meets the requirement?
HINT #1: The outcome (=,=,=)(=,=,=), i.e. all 3 3 weighings being equal, can only happen if the bad coin is not used in any weighing at all. This corresponds to the 2nd bullet of the requirement. I.e. in any correct solution, there is exactly one coin that is unused in any weighing, and the outcome (=,=,=)(=,=,=) maps to this coin being bad, but without knowing if the coin is heavier or lighter.
HINT #2: Let the 28 28 answers be S={A+,A−,B+,B−,...,N+,N−}S={A+,A−,B+,B−,...,N+,N−} where ++ and −− mean heavier and lighter respectively. Meanwhile, the 27 27 outcomes form a 3×3×3 3×3×3 cube, which we can denote T={−1,0,+1}3 T={−1,0,+1}3, where −1,0,+1−1,0,+1 denote the left side of the balance being lighter, equal, or heavier. We need to find a mapping f:S→T f:S→T with these properties:
Hint #1 already shows that f(N+)=f(N−)=(0,0,0)f(N+)=f(N−)=(0,0,0).
The remaining 26 26 answers and 26 26 outcomes must map bijectively.
Pre-determined weighings ⟹f(A+)⟹f(A+) and f(A−)f(A−) are related in a certain way. How?
What other constraints do we need on f f?
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edited Aug 25, 2020 at 6:56
antkam
asked Aug 22, 2020 at 16:13
antkamantkam
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1 Is this even possible?user818490 –user818490 2020-08-24 09:34:07 +00:00 Commented Aug 24, 2020 at 9:34
5 I think that riddles are more suitable for Puzzling.Asaf Karagila –Asaf Karagila♦ 2020-08-24 14:30:51 +00:00 Commented Aug 24, 2020 at 14:30
@AsafKaragila - I understand your point of view, but I thought the solution (now that it's posted) shows a technique that might be more generally applicable. Also, I personally don't read puzzling.se. :) Anyway, thanks for indulging me and not voting to close this. Feel free to repost to puzzling.se if you judge that the people there might enjoy this.antkam –antkam 2020-08-25 14:09:32 +00:00 Commented Aug 25, 2020 at 14:09
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Suppose a triple of weighing results determines a coin. If a weighing result is "equal" then the coin did not appear in that weighing. Otherwise, the coin appeared on either the "less" side of each weighing or the "greater" side of each weighing depending on whether the coin was lighter or heavier.
For each coin, then, choose a distinct weighing result pattern that will determine that coin. (Weighing result patterns that are completely flipped must identify the same coin with the opposite weight, so we won't use these.)
A < = =
B = < =
C = = <
D < < =
E < = <
F = < <
G < > =
H < = >
I = < >
J < < <
K < < >
L < > <
M > < <
N = = =
Then we know exactly how to assemble each weighing (ie A appears in the first weighing only; G appears on opposite sides of the first two weighings; J appears on the same side of all weighings; etc) except that we don't know which side to put the coins on, but deciding the sides turns out to be easy, as we merely need to balance the number of coins in each weighing. Coin X (the known good coin) is needed because there are otherwise nine coins involved in each weighing. We will not be able to distinguish between coin N being lighter or heavier.
One solution is
AGJKL-DEHMX
BIJKM-DFGLX
CHJLM-EFIKX
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answered Aug 25, 2020 at 13:00
tehtmitehtmi
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Well done! Hope you enjoyed the puzzle. I did it in a slightly different way, but I like your approach better since the assignment from your first table (the mapping) to your second table (the weighings) indeed "turns out to be easy".antkam –antkam 2020-08-25 13:21:48 +00:00 Commented Aug 25, 2020 at 13:21
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There is a very simple description of an optimal predetermined weighing strategy for any number of coins n≥1 n≥1. This uses the balanced ternary system, which I describe now. Every positive integer n n can be written uniquely in the form
n=∑i=0∞b i 3 i,b i∈{−1,0,+1}for i∈N,only finitely many b i≠0.n=∑i=0∞b i 3 i,b i∈{−1,0,+1}for i∈N,only finitely many b i≠0.
For example, 25=1⋅3 3+0⋅3 2+(−1)⋅3 1+1.25=1⋅3 3+0⋅3 2+(−1)⋅3 1+1. Using ++ as the symbol for the digit 1 1 and −− for the digit zero, we would write 25 25 in balanced ternary, with infinitely many leading zeroes, as
25=⋯000+0−+25=⋯000+0−+
Now, consider the following transformation on this infinite sequence of ±±s and 0 0 s; negate each symbol which has an odd number of zeroes to its right. The result to the above example is
25⋈⋯000−0−+25⋈⋯000−0−+
I call this the twisted ternary representation of 25 25. Next, arrange all of these infinite sequences into an infinite matrix, where the digits which are were negated when converting to twisted ternary are highlighted in red.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14⋮⋈⋈⋈⋈⋈⋈⋈⋈⋈⋈⋈⋈⋈⋈⋈⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋮0 0 0 0 0 0 0 0 0 0 0 0 0 0+0 0 0 0 0+−+−+−+−+−0 0+−+−+−0 0 0+−+−⋮0+−0+−0+−0+−0+−0⋈⋯0 0 0 0 1⋈⋯0 0 0+2⋈⋯0 0+−3⋈⋯0 0−0 4⋈⋯0 0++5⋈⋯0+−−6⋈⋯0−+0 7⋈⋯0+−+8⋈⋯0−0−9⋈⋯0+0 0 10⋈⋯0−0+11⋈⋯0++−12⋈⋯0−−0 13⋈⋯0+++14⋈⋯+−−−⋮⋮⋮
To find the weighing strategy for n n coins, number the coins from 0 0 to n−1 n−1. For each column of that matrix, weigh the coins corresponding to row labels of the ++'s in that columns, against the coins corresponding to −−'s (ignoring the infinitely many leading columns whose entries 0 0 to n−1 n−1 are all zero). You may also need to add the reference coin to one side to equalize these groups.
For your problem of n=14 n=14, the weighings are (where R R denotes reference coin):
1,4,7,10,13 1,4,7,10,13 vs 2,5,8,11,R 2,5,8,11,R
2,4,6,11,13 2,4,6,11,13 vs 3,5,7,12,R 3,5,7,12,R
5,7,9,11,13 5,7,9,11,13 vs 6,8,10,12,R 6,8,10,12,R.
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edited Dec 3, 2020 at 20:22
answered Dec 3, 2020 at 7:11
Mike EarnestMike Earnest
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Now that @tehtmi has posted a valid solution, here's my slightly different approach.
As I alluded to in Hint #2, the interesting thing about pre-determined weighings is: f(A+)=−f(A−)f(A+)=−f(A−), i.e. the two answers A+,A−A+,A− must have opposite outcomes in all 3 3 weighings. (The opposite of "balance" aka "==" aka 0 0 is of course balance.) This is generally not true in a solution where a later weighing depends on the result of a previous weighing.
So anyway it becomes a matter of assigning 13 13++'s and 13 13−−'s to the 26 26 non-center outcomes in the overall 3×3×3 3×3×3 cube, such that:
Constraint 1: For any pair of outcomes y,z y,z which are reflections across the center, y,z y,z must have opposite signs.
In this cube, the 6 6 faces (3 3 pairs of faces) represent the 3 3 weighings. If we had access to an unlimited number of known-to-be-good coins (in fact 9 9 is sufficient), then Constraint 1 is sufficient. Say the top face has A+,B+,C+,D+,E+,F+,G+,H+,I+A+,B+,C+,D+,E+,F+,G+,H+,I+, then the bottom face has A−,B−,…,I−A−,B−,…,I− and the weighing would be those 9 9 coins vs 9 9 known-to-be-good coins.
But we only have 1 1 known-to-be-good coin, and this translates to:
Constraint 2: Each of the 6 6 faces (each face being 9 9 outcomes) must consist of 5 5 of one sign, and 4 4 of another. The weighing will be the 5 5 vs the 4 4 plus the known-good coin.
At this point, the problem becomes a small coloring puzzle that needs to be solved by trial and error. One solution is shown below (the three separate 3×3 3×3 squares represent the top, middle, bottom layers of the cube):
```
+ - +
- + +
+ - -
-
? -
+
-
+
-
```
and just for completeness, here is how to assign letters to them to match exactly tehtmi's solution:
```
J+ F- M+
E- C+ H+
L+ I- K-
D- B+ G-
A+ N? A-
G+ B- D+
K+ I+ L-
H- C- E+
M- F+ J-
```
where e.g. the left-face-right-face-pair is the weighing JLAGK-EDHMX, and the top-face-bottom-face-pair is the weighing LHCMJ-KIEFX, etc.
BTW, this result is equivalent to the following result:
If there were only 13 13 suspect coins (and 1 1 bad as usual), plus a single known-good coin, then in 3 3 pre-determined weighings we can find the bad coin and tell if it's heavier/lighter. After all, we did not even use the 14 14 th coin N in the solution above.
which is in turn strictly stronger than this classic result:
The classic 12 12-coin puzzle is often posed without the constraint of pre-determined weighings, but it can in fact be solved using pre-determined weighings. In this classic, there is no known-good coin. However, in our solution J (a suspect) and X (the known-good coin) appear in all 3 3 weighings and always on opposite sides. So Eliminating both of them solves the classic puzzle with 3 3 pre-determined weighings of 4 4-vs-4 4 each.
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edited Aug 25, 2020 at 14:46
answered Aug 25, 2020 at 14:01
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computing ln[(1+i)2]ln[(1+i)2]
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How to compute Natural logarithm of complex numbers?
and how to verify our answer?
in example: ln[(1+i)2]ln[(1+i)2]
complex-numbers
logarithms
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edited Dec 10, 2015 at 9:09
sajjadsajjad
asked Dec 9, 2015 at 19:50
sajjadsajjad
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This depends on which branch of the logarithm you're using, but note that (1+i)2=2 i(1+i)2=2 i, and in the most commonly used branches we have log i=π 2 i logi=π 2 i.Travis Willse –Travis Willse 2015-12-09 19:52:57 +00:00 Commented Dec 9, 2015 at 19:52
1 One doesn't "solve" a number, one "computes" a number.Najib Idrissi –Najib Idrissi 2015-12-10 09:07:52 +00:00 Commented Dec 10, 2015 at 9:07
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Solution 1
The point corresponding to the complex number 1+i 1+i is 2–√2 units from the origin, located π/4 π/4 radians (i.e. 45°) counter-clockwise from the real axis, and therefore it can be represented in polar coordinates as
1+i=2–√e i π/4 1+i=2 e i π/4
This representation is not unique, however, because you can add any multiple of 2 π 2 π radians (360°) to the angle. So more generally
1+i=2–√e i π/4+2 n π i 1+i=2 e i π/4+2 n π i
Squaring this is easy:
(1+i)2=2 e i π/2+4 n π i(1+i)2=2 e i π/2+4 n π i
Therefore the logarithm is:
ln(1+i)2=ln(2)+i π 2+4 n π i ln(1+i)2=ln(2)+i π 2+4 n π i
Notice that this is not a single value, but a collection of infinitely-many different values corresponding to different choices of n n. This is because the ln ln function on the complex plane is a multi-valued function; see here for more information.
Solution 2
Note that (1+i)2=2 i(1+i)2=2 i. So really the question is asking us to compute ln(2 i)ln(2 i). Since i=e i π/2 i=e i π/2, this is ln(2 e i π/2)=ln(2)+ln(e i π/2)=ln(2)+i π 2 ln(2 e i π/2)=ln(2)+ln(e i π/2)=ln(2)+i π 2. In contrast, this approach gives only the principal branch of the logarithm function.
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answered Dec 9, 2015 at 20:07
mweissmweiss
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ln((1+i)2)=ln((|1+i|e arg(1+i)i)2)=ln((1+i)2)=ln((|1+i|e arg(1+i)i)2)=
ln((2–√e π i 4)2)=ln(2 e π i 2)=π i 2+2 i π⌊−π 2+π 2 π⌋+ln(2)=ln(2)+π i 2 ln((2 e π i 4)2)=ln(2 e π i 2)=π i 2+2 i π⌊−π 2+π 2 π⌋+ln(2)=ln(2)+π i 2
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answered Dec 9, 2015 at 20:16
Jan EerlandJan Eerland
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To compute ln[(1+i)2]ln[(1+i)2] :
first we should answer : (1+i)2(1+i)2 as follows :
2–√c i s π 4=⎧⎩⎨r=1 2+1 2−−−−−−√=2–√tan θ=1 1=1=>θ=π 4 2 c i s π 4={r=1 2+1 2=2 tanθ=1 1=1=>θ=π 4
[2–√c i s π 4]2=2 C i s π 2[2 c i s π 4]2=2 C i s π 2
so we reach to this point : ln(2 C i s π 2)ln(2 C i s π 2) :
ln(2 C i s π 2)=ln(2)+i(π 2+2 k π)ln(2 C i s π 2)=ln(2)+i(π 2+2 k π)
assuming (k=0 k=0) =>0.693147181+1.57079633 i 0.693147181+1.57079633 i
but to proof our answer :
e ln(2)+i(π 2+2 k π)=2 c i s(π 2+2 k π)−−−−−−−−−−−−√=2–√c i s π 4 e ln(2)+i(π 2+2 k π)=2 c i s(π 2+2 k π)=2 c i s π 4
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edited Dec 10, 2015 at 9:04
answered Dec 9, 2015 at 19:50
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6676 | https://www.lew-port.com/cms/lib/NY19000328/Centricity/Domain/204/Ch5-2%20SimpleMachineUsesRead.pdf | C D 154 Unit: Motion and Forces VOCABULARY simple machine p. 154 lever p. 155 fulcrum p. 155 wheel and axle p. 156 pulley p. 156 inclined plane p. 158 wedge p. 158 screw p. 159 BEFORE, you learned • Machines help you work by changing the size or direction of a force • The number of times a machine multiplies the input force is the machine’s mechanical advantage NOW, you will learn • How six simple machines change the size or direction of a force • How to calculate mechanical advantage KEY CONCEPT Six simple machines have many uses.
EXPLORE Changing Forces How can you change a force?
PROCEDURE Lay one pencil on a flat surface. Place the other pencil on top of the first pencil and perpendicular to it, as shown. Place the book on one end of the top pencil.
Push down on the free end of the top pencil to raise the book.
Change the position of the bottom pencil so that it is closer to the book and repeat step 2. Then move the bottom pencil closer to the end of the pencil you are pushing on and repeat step 2.
WHAT DO YOU THINK?
• How did changing the position of the bottom pencil affect how much force you needed to lift the book?
• At which position is it easiest to lift the book? most difficult?
3 2 1 MATERIALS • 2 pencils • small book There are six simple machines.
You have read about how a ramp and a shovel can help you do work. A ramp is a type of inclined plane, and a shovel is a type of lever. An inclined plane and a lever are both simple machines.
are the six machines on which all other mechanical machines are based. In addition to the inclined plane and the lever, simple machines include the wheel and axle, pulley, wedge, and screw.
As you will see, the wheel and axle and pulley are related to the lever, and the wedge and screw are related to the inclined plane. You will read about each of the six simple machines in detail in this section.
Simple machines NOTE-TAKING STRATEGY As you read, remember to take notes about the main ideas and supporting details.
Page 1 of 9 C D reading tip The lengths of the arrows in the diagram represent the size of the force.
Chapter 5: Machines 155 Lever A is a solid bar that rotates, or turns, around a fixed point. The bar can be straight or curved. The fixed point is called the A lever can multiply the input force. It can also change the direction of the input force. If you apply a force downward on one end of a lever, the other end can lift a load.
The way in which a lever changes an input force depends on the positions of the fulcrum, the input force, and the output force in relation to one another. Levers with different arrangements have different uses.
Sometimes a greater output force is needed, such as when you want to pry up a bottle cap. At other times you use a greater input force on one end to get a higher speed at the other end, such as when you swing a baseball bat. The three different arrangements, sometimes called the three classes of levers, are shown in the diagram below.
check your reading What two parts are needed to make a lever?
fulcrum.
lever Levers can be classified according to where the fulcrum is.
Levers First-Class Lever The fulcrum is located between the input force and the output force. Use this type of lever to change the direction and size of a force.
fulcrum input force output force Second-Class Lever The output force is located between the input force and the fulcrum. Use this type of lever if you need a greater output force.
fulcrum input force output force Third-Class Lever fulcrum input force output force The input force is located between the output force and the fulcrum. Use this type of lever to reduce the distance over which you apply the input force or increase the speed of the end of the lever.
Page 2 of 9 Wheel and Axle A is a simple machine made of a wheel attached to a shaft, or axle. The wheels of most means of transportation—such as a bicycle and a car—are attached to an axle. The wheel and axle act like a rotating collection of levers. The axle at the wheel’s center is like a fulcrum. Other examples of wheels and axles are screwdrivers, steering wheels, doorknobs, and electric fans.
Depending on your purpose for using a wheel and axle, you might apply a force to turn the wheel or the axle. If you turn the wheel, your input force is transferred to the axle. Because the axle is smaller than the wheel, the output force acts over a shorter distance than the input force. A driver applies less force to a steering wheel to get a greater turning force from the axle, or steering column. This makes it easier to steer the car.
If, instead, you turn the axle, your force is transferred to the wheel.
Because the wheel is larger than the axle, the force acts over a longer distance. A car also contains this use of a wheel and axle. The engine turns the drive axles, which turn the wheels.
check your reading Compare the results of putting force on the axle with putting force on the wheel.
Pulley A is a wheel with a grooved rim and a rope or cable that rides in the groove. As you pull on the rope, the wheel turns.
A pulley that is attached to something that holds it steady is called a fixed pulley. An object attached to the rope on one side of the wheel rises as you pull down on the rope on the other side of the wheel. The fixed pulley makes work easier by changing the direction of the force.
You must apply enough force to overcome the weight of the load and any friction in the pulley system.
A fixed pulley allows you to take advantage of the downward pull of your weight to move a load upward. It does not, how-ever, reduce the force you need to lift the load. Also, the distance you pull the rope through is the same distance that the object is lifted. To lift a load two meters using a fixed pulley, you must pull down two meters of rope.
pulley wheel and axle Fixed Pulley 156 Wheel and Axle C D Page 3 of 9 C D In a movable pulley setup, one end of the rope is fixed, but the wheel can move. The load is attached to the wheel. The person pulling the rope provides the output force that lifts the load. A single movable pulley does not change the direction of the force. Instead, it multiplies the force. Because the load is supported by two sections of rope, you need only half the force you would use with a fixed pulley to lift it.
However, you must pull the rope through twice the distance.
check your reading How does a single fixed pulley differ from a single movable pulley?
A combination of fixed and movable pulleys is a pulley system called a block and tackle. A block and tackle is used to haul and lift very heavy objects. By combining fixed and movable pulleys, you can use more rope sections to support the weight of an object. This reduces the force you need to lift the object. The mechanical advantage of a single pulley can never be greater than 2. If engineers need a pulley system with a mechanical advantage greater than 2, they often use a block-and-tackle system.
What is the mechanical advantage of a pulley system?
PROCEDURE Hang the mass on the spring scale to find its weight in newtons. Record this weight as your output force.
Tie the top of one pulley to the ring stand.
Attach the mass to the second pulley.
Attach one end of the second pulley’s rope to the bottom of the first pulley. Then thread the free end of the rope through the second pulley. Loop the rope up and over the first pulley, as shown.
Attach the spring scale to the free end of the rope. Pull down to lift the mass. Record the force you used as your input force. Calculate the mechanical advantage of this pulley system. Hint: The mechanical advantage can be calculated by dividing the output force by the input force.
WHAT DO YOU THINK?
• How did your input force compare with your output force?
• What caused the results you observed?
CHALLENGE Explain what the mechanical advantage would be for a pulley system that includes another movable pulley.
5 4 3 2 1 Pulleys Pulleys SKILL FOCUS Inferring MATERIALS • 100 g mass • spring scale • 2 pulleys with rope • ring stand TIME 20 minutes Movable Pulley Chapter 5: Machines 157 Page 4 of 9 Inclined Plane Recall that it is difficult to lift a heavy object straight up because you must apply a force great enough to overcome the downward pull of the force of gravity.
For this reason people often use ramps. A ramp is an a simple machine that is a sloping surface. The photograph at the left shows the interior of the Guggenheim Museum in New York City.
The levels of the art museum are actually one continuous inclined plane.
Inclined planes make the work of raising an object easier because they support part of the weight of the object while it is being moved from one level to another. The surface of an inclined plane applies a reaction force on the object resting on it. This extra force on the object helps to act against gravity. If you are pushing an object up a ramp, you have to push with only enough force to overcome the smaller net force that pulls the object down parallel to the incline.
The less steep an inclined plane is, the less force you need to push or pull an object on the plane. This is because a less steep plane supports more of an object’s weight than a steeper plane. However, the less steep an inclined plane is, the farther you must go to reach a certain height. While you use less force, you must apply that force over a greater distance.
check your reading How do inclined planes help people do work? Your answer should mention force.
Wedge A is a simple machine that has a thick end and a thin end. Wedges are used to cut, split, or pierce objects—or to hold objects together. A wedge is a type of inclined plane, but inclined planes are stationary, while wedges often move to do work.
Some wedges are single, movable inclined planes, such as a doorstop, a chisel, or an ice scraper.
Another kind of wedge is made of two back-to-back inclined planes. Examples include the blade of an axe or a knife. In the photograph at the left, a sculptor is using a chisel to shape stone. The sculptor applies an input force on the chisel by tapping its thicker end with a mallet. That force pushes the thinner end of the chisel into the stone. As a result, the sides of the thinner end exert an output force that separates the stone.
wedge inclined plane, Inclined Plane 158 Unit: Motion and Forces Wedge C D Page 5 of 9 C D Chapter 5: Machines 159 The angle of the cutting edge determines how easily a wedge can cut through an object. Thin wedges have small angles and need less input force to cut than do thick wedges with large angles. That is why a sharp knife blade cuts more easily than a dull one.
You also can think of a wedge that cuts objects in terms of how it changes the pressure on a surface. The thin edges of a wedge provide a smaller surface area for the input force to act on. This greater pressure makes it easier to break through the surface of an object. A sharp knife can cut through an apple skin, and a sharp chisel can apply enough pressure to chip stone.
A doorstop is a wedge that is used to hold objects together. To do its job, a doorstop is pressed tip-first under a door. As the doorstop is moved into position, it lifts the door slightly and applies a force to the bottom of the door. In return, the door applies pressure to the doorstop and causes the doorstop to press against the floor with enough force to keep the doorstop—and the door—from moving.
Screw A is an inclined plane wrapped around a cylinder or cone to form a spiral. A screw is a simple machine that can be used to raise and lower weights as well as to fasten objects. Examples of screws include drills, jar lids, screw clamps, and nuts and bolts. The spiraling inclined plane that sticks out from the body of the screw forms the threads of the screw.
In the photograph at right, a person is using a screwdriver, which is a wheel and axle, to drive a screw into a piece of wood. Each turn of the screwdriver pushes the screw farther into the wood. As the screw is turned, the threads act like wedges, exerting an output force on the wood. If the threads are very close together, the force must be applied over a greater distance—that is, the screw must be turned many times—but less force is needed.
The advantage of using a screw instead of a nail to hold things together is the large amount of friction that keeps the screw from turning and becoming loose. Think of pulling a nail out of a piece of wood compared with pulling a screw from the same piece of wood. The nail can be pulled straight out. The screw must be turned through a greater distance to remove it from the wood.
Notice that the interior of the Guggenheim Museum shown on page 158 is not only an inclined plane. It is also an example of a screw.
The inclined plane is wrapped around the museum’s atrium, which is an open area in the center.
check your reading Explain how a screw moves deeper into the wood as it is turned.
screw Screw Page 6 of 9 C D The mechanical advantage of a machine can be calculated.
Recall that the number of times a machine multiplies the input force is the machine’s mechanical advantage. You can calculate a machine’s mechanical advantage using this formula: Mechanical Advantage = O In u p tp u u t t F F o o r r c c e e MA = F F o i u n t This formula works for all machines, regardless of whether they are simple machines or more complicated machines.
If a machine decreases the force you use to do work, the distance over which you have to apply that force increases. It is possible to use this idea to calculate the mechanical advantage of a simple machine without knowing what the input and output forces are. To make this calculation, however, you must assume that your machine is not losing any work to friction. In other words, you must assume that your machine is 100 percent efficient. The mechanical advantage that you calculate when making this assumption is called the ideal mechanical advantage.
Inclined Plane You can calculate the ideal mechanical advantage of an inclined plane by dividing its length by its height.
Ideal Mechanical Advantage = h le e n ig g h th t o o f f i i n n c c l l i i n n e e IMA = h l 160 Unit: Motion and Forces length height reading tip Scientists often consider the way in which an object will behave under ideal conditions, such as when there is no friction.
Page 7 of 9 C D Chapter 5: Machines 161 Be sure to use the length of the incline in your calculation, as shown in the diagram, and not the length of the base. If the mover in the photograph on page 160 increased the length of the ramp, he would increase the ramp’s mechanical advantage. However, he would also increase the distance over which he had to carry the box.
Wheel and Axle To calculate the ideal mechanical advantage of a wheel and axle, use the following formula: Ideal Mechanical Advantage = R R a a d d i i u u s s o o f f o i u n t p p u u t t IMA = R R o i u n t The Ferris wheel below is a giant wheel and axle. A motor applies an input force to the Ferris wheel’s axle, which turns the wheel. In this example, the input force is applied to the axle, so the radius of the axle is the input radius in the formula above. The output force is applied by the wheel, so the radius of the wheel is the output radius.
For a Ferris wheel, the input force is greater than the output force.
The axle turns through a shorter distance than the wheel does. The ideal mechanical advantage of this type of wheel and axle is less than 1.
Sometimes, as with a steering wheel, the input force is applied to turn the wheel instead of the axle. Then the input radius is the wheel’s radius, and the output radius is the axle’s radius. In this case, the input force on the wheel is less than the output force applied by the axle.
The ideal mechanical advantage of this type of wheel and axle is greater than 1.
SIMULATION CLASSZONE.COM Explore the mechanical advantage of an inclined plane.
radius of axle radius of wheel reminder The radius is the distance from the center of the wheel or axle to any point on its circumference.
Page 8 of 9 C D 162 Unit: Motion and Forces KEY CONCEPTS 1. Name the six simple machines and give an example of each.
2. Explain how a screw changes the size of the force needed to push it into wood.
3. To calculate mechanical advantage, what two things do you need to know?
CRITICAL THINKING 4. Synthesize How is a pulley similar to a wheel and axle?
5. Calculate What is the ideal mechanical advantage of a wheel with a diameter of 30 cm fixed to an axle with a diameter of 4 cm if the axle is turned?
CHALLENGE 6. Infer How can you increase a wedge’s mechanical advantage?
Draw a diagram to show your idea.
Lever The beam balance above is a lever. The beam is the solid bar that turns on a fixed point, or fulcrum. The fulcrum is the beam’s balance point. When you slide the weight across the beam, you are changing the distance between the input force and the fulcrum. The mechanical advantage depends on the distances of the input force and output force from the fulcrum. The output force is applied to balance the beaker.
To calculate the ideal mechanical advantage of a lever, use the following formula: Ideal Mechanical Advantage = IMA = d d o i u n t This formula applies to all three arrangements of levers. If the distance from the input force to the fulcrum is greater than the distance from the output force to the fulcrum, the ideal mechanical advantage is greater than 1. The beam balance is an example of this type of lever.
distance from input force to fulcrum distance from output force to fulcrum output force fulcrum input force dout din Page 9 of 9 |
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2021 AIME I Problems/Problem 9
Contents
1 Problem
2 Diagram
3 Solution 1 (Similar Triangles and Pythagorean Theorem)
4 Solution 2 (Similar Triangles and Pythagorean Theorem)
5 Solution 3 (Similar Triangles and Pythagorean Theorem)
6 Solution 4 (Cyclic Quadrilaterals, Similar Triangles, Pythagorean Theorem)
7 Solution 5 (Cyclic Quadrilaterals, Similar Triangles, Pythagorean Theorem)
8 Solution 6 (Similar Triangles and Trigonometry)
9 Solution 7 (Similar Triangles and Trigonometry)
10 Solution 8 (Heron's Formula)
11 Solution 9 (Three Heights)
12 Solution 10 (Area)
13 Video Solution
14 Video Solution
15 See Also
Problem
Let be an isosceles trapezoid with and Suppose that the distances from to the lines and are and respectively. Let be the area of Find
Diagram
~MRENTHUSIASM
Solution 1 (Similar Triangles and Pythagorean Theorem)
Let and be the perpendiculars from to and respectively. Next, let be the intersection of and
We set and as shown below. From here, we obtain by segment subtraction, and and by the Pythagorean Theorem.
Since and are both complementary to we have from which by AA. It follows that so or Since by vertical angles, we have by AA, with the ratio of similitude It follows that
Since by angle chasing, we have by AA, with the ratio of similitude It follows that
By the Pythagorean Theorem on right we have or Solving this system of equations ( and ), we get and so and Finally, the area of is from which
~MRENTHUSIASM
Remark
Instead of solving the system of equations and which can be time consuming, by noting that by AA, we could find out , which gives . We also know that by Pythagorean Theorem on . From we apply the Pythagorean Theorem to and obtain Substituting, we get from which
~Chupdogs
Solution 2 (Similar Triangles and Pythagorean Theorem)
First, draw the diagram. Then, notice that since is isosceles, , and the length of the altitude from to is also . Let the foot of this altitude be , and let the foot of the altitude from to be denoted as . Then, . So, . Now, notice that , where denotes the area of triangle . Letting , this equality becomes . Also, from , we have . Now, by the Pythagorean theorem on triangles and , we have and . Notice that , so . Squaring both sides of the equation once, moving and to the right, dividing both sides by , and squaring the equation once more, we are left with . Dividing both sides by (since we know is positive), we are left with . Solving for gives us .
Now, let the foot of the perpendicular from to be . Then let . Let the foot of the perpendicular from to be . Then, is also equal to . Notice that is a rectangle, so . Now, we have . By the Pythagorean theorem applied to , we have . We know that , so we can plug this into this equation. Solving for , we get .
Finally, to find , we use the formula for the area of a trapezoid: . The problem asks us for , which comes out to be .
~advanture
Solution 3 (Similar Triangles and Pythagorean Theorem)
Make perpendicular to ; perpendicular to ; perpendicular .
It's obvious that . Let . Then make perpendicular to , it's easy to get .
Since parallel to , , so . After drawing the altitude, it's obvious that , so . According to the property of similar triangles, . So, , or .
Now, we see the , pretty easy to find that , then we get , then express into form that we put the length of back to : . So, After calculating, we can have a final equation of . It's easy to find then . So, ~bluesoul
Solution 4 (Cyclic Quadrilaterals, Similar Triangles, Pythagorean Theorem)
Let the foot of the altitude from to be , to be , and to be .
Note that all isosceles trapezoids are cyclic quadrilaterals; thus, is on the circumcircle of and we have that is the Simson Line from . As , we have that , with the last equality coming from cyclic quadrilateral . Thus, and we have that or that , which we can see gives us that . Further ratios using the same similar triangles gives that and .
We also see that quadrilaterals and are both cyclic,it is clear that and ,which shows . we wish to find the ratio of similitude between the two triangles.
To do this, we use the one number we have for : we know that the altitude from to has length . As the two triangles are similar, if we can find the height from to , we can take the ratio of the two heights as the ratio of similitude. To do this, we once again note that . Using this, we can drop the altitude from to and let it intersect at . Then, let and thus . We then have by the Pythagorean Theorem on and : Then, . This gives us then from right triangle that and thus the ratio of to is . From this, we see then that and The Pythagorean Theorem on then gives that Then, we have the height of trapezoid is , the top base is , and the bottom base is . From the equation of a trapezoid, , so the answer is .
~lvmath
~Corrected by dongjiu0728(Former Ans made the wrong QH and wrong QR、RP but somehow came to the right answer,which trapped me for a weekend)
Solution 5 (Cyclic Quadrilaterals, Similar Triangles, Pythagorean Theorem)
Let and be the feet of the altitudes from to and , respectively.
Claim: We have pairs of similar right triangles: and .
Proof: Note that is cyclic. We need one more angle, and we get this from this cyclic quadrilateral: Let . We obtain from the similarities and .
By Ptolemy, , so .
We obtain , so .
Applying the Pythagorean theorem on , we get .
Thus, , and , yielding .
Solution 6 (Similar Triangles and Trigonometry)
Let . Draw diagonal and let be the foot of the perpendicular from to , be the foot of the perpendicular from to line , and be the foot of the perpendicular from to .
Note that , and we get that . Therefore, . It then follows that . Using similar triangles, we can then find that . Using the Law of Cosines on , We can find that the . Since , and each is supplementary to , we know that the . It then follows that . Then it can be found that the area is . Multiplying this by , the answer is .
~happykeeper
Solution 7 (Similar Triangles and Trigonometry)
Draw the distances in terms of , as shown in the diagram. By similar triangles, . As a result, let , then and . The triangle is which . By angle subtraction, . Therefore, and . By trapezoid area formula, the area of is equal to which .
~math2718281828459
Solution 8 (Heron's Formula)
Let the points formed by dropping altitudes from to the lines , , and be , , and , respectively.
We have and For convenience, let . By Heron's formula on , we have sides and semiperimeter , so so .
Then, and Finally, recalling that is isosceles, so .
~emerald_block
Solution 9 (Three Heights)
Let and be the perpendiculars from to and respectively. . Denote by the base of the perpendicular from to be the base of the perpendicular from to . Denote It is clear that the area of is equal to the area of the rectangle
The problem is reduced to finding .
In triangle all altitudes are known: We apply the Law of Cosines to and get We apply the Pythagorean Law to and get Required area is
vladimir.shelomovskii@gmail.com, vvsss
Solution 10 (Area)
Let be on such that . Let be on such that .
Let be the length of . Let be the length of .
The area of can be expressed in three ways: , , and .
Now, . We can substitute in to get
.
We have After a fairly straightforward algebraic bash, we get , and . By the Pythagorean Theorem on , , and .
Thus, . Therefore, . The requested answer is .
~ adam_zheng
Video Solution
~Mathproblemsolvingskills.com
Video Solution
See Also
2021 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 8Followed by
Problem 10
1•2•3•4•5•6•7•8•9•10•11•12•13•14•15
All AIME Problems and Solutions
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Verbal Ability
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Synonyms Examples
Understanding synonyms concepts and applying them is important to achieve good marks in placement exams. Synonyms examples are crucial to comprehend the basic principles of vocabulary building and contextual understanding. Individuals can improve their performance in the verbal ability section of placement exams by following synonyms tips and tricks.
Synonyms
Examples
Practice Questions
Understanding Synonyms with Examples
Synonyms are easy to learn and understand if you understand certain hacks to remember more words through a combination of suffixes, prefixes, and understanding root words to relate to more synonyms of a given word.
Synonyms can change completely according to the context. Two words can mean the same thing for one context but they might mean different things once the context changes. This happens because one word can have multiple meanings in different scenarios.
For Example:
The word liquid and fluid are synonyms when we are talking about a substance that can flow freely.
Although, when talking about finance and money, liquid means an asset that's in the form of cash or can be immediately converted into cash/money.
In this scenario, liquid and fluid are not synonyms anymore.
Hence, the synonyms for one word can change with the context.
Although there are no particular rules for synonyms to follow while answering questions, one can learn to crack such questions in placement exams with the help of examples and expert tips.
Synonyms Examples
Read these examples of synonyms to understand and apply the concepts better:
| Word | Synonym 1 | Synonym 2 | Synonym 3 |
--- --- |
| Celebration | Festivity | Commemoration | Revelry |
| Flexibility | Adaptability | Versatility | Adaptation |
| Authenticity | Genuineness | Sincerity | Veracity |
| Serenity | Tranquility | Peacefulness | Equanimity |
| Curiosity | Inquisitiveness | Interest | Quest |
| Harmony | Unity | Agreement | Concord |
| Abundance | Plenty | Prosperity | Opulence |
| Motivation | Drive | Ambition | Initiative |
| Independence | Autonomy | Freedom | Self-reliance |
| Resilience | Adaptability | Endurance | Fortitude |
| Connection | Link | Relationship | Bond |
| Gratitude | Appreciation | Thankfulness | Acknowledgment |
| Exploration | Discovery | Investigation | Quest |
| Innovation | Invention | Creativity | Ingenuity |
| Simplicity | Minimalism | Uncomplicated | Elegance |
| Integrity | Honesty | Morality | Uprightness |
| Patience | Tolerance | Endurance | Perseverance |
| Empowerment | Strengthening | Liberation | Enfranchisement |
| Collaboration | Cooperation | Partnership | Synergy |
| Construction | Building | Creation | Fabrication |
| Transformation | Conversion | Metamorphosis | Transmutation |
| Communication | Interaction | Exchange | Conveyance |
| Reflection | Contemplation | Thoughtfulness | Introspection |
| Resilience | Persistence | Endurance | Grid |
| Curiosity | Inquisitiveness | Interest | Eagerness |
| Empowerment | Strengthening | Liberation | Enfranchisement |
| Optimization | Maximization | Enhancement | Streamlining |
| Motivation | Drive | Ambition | Initiative |
| Adaptation | Adjustment | Modification | Acclimatization |
| Persistence | Tenacity | Perseverance | Steadfastness |
| Efficiency | Effectiveness | Productivity | Competence |
| Integration | Incorporation | Assimilation | Fusion |
| Harmonization | Synchronization | Balancing | Alignment |
| Problem-solving | Troubleshooting | Resolution | Analytical thinking |
| Decision-making | Judgment | Determination | Deliberation |
| Teamwork | Cooperation | Collaboration | Cohesion |
| Strategic | Tactical | Calculated | Farsighted |
| Accountability | Responsibility | Answerability | Liability |
| Professionalism | Competence | Expertise | Demeanor |
| Adaptation | Adjustment | Modification | Acclimatization |
| Initiative | Proactiveness | Drive | Resourcefulness |
| Networking | Connection | Relationship | Engagement |
| Empowerment | Authorization | Delegation | Autonomy |
| Performance | Achievement | Effectiveness | Productivity |
| Development | Advancement | Growth | Progression |
| Mentoring | Coaching | Guidance | Support |
| Creativity | Imagination | Inventiveness | Originality |
| Diversity | Variety | Inclusivity | Multiculturalism |
| Entrepreneurship | Initiative | Risk-taking | Enterprise |
| Eloquent | Articulate | Fluent | Persuasive |
| Graceful | Elegant | Refined | Gracious |
| Charismatic | Magnetic | Compelling | Captivating |
| Ambitious | Driven | Aspiring | Enterprising |
| Compassionate | Empathetic | Sympathetic | Benevolent |
| Diligent | Hardworking | Industrious | Assiduous |
| Reliable | Dependable | Trustworthy | Steadfast |
| Vibrant | Lively | Dynamic | Spirited |
| Wise | Sagacious | Discerning | Prudent |
| Modest | Humble | Unassuming | Self-effacing |
| Enthusiastic | Passionate | Excited | Zealous |
| Resourceful | Ingenious | Clever | Inventive |
| Resilient | Tenacious | Strong | Indomitable |
| Genuine | Authentic | Sincere | Unfeigned |
| Gracious | Polite | Courteous | Considerate |
| Radiant | Bright | Glowing | Beaming |
| Imaginative | Creative | Innovative | Visionary |
| Harmonious | Balanced | Serene | Tranquil |
| Exuberant | Energetic | Spirited | Zestful |
| Meticulous | Thorough | Precise | Methodical |
| Beautiful | Gorgeous | Stunning | Exquisite |
| Intelligent | Smart | Clever | Ingenious |
| Brave | Courageous | Fearless | Valiant |
| Charming | Captivating | Enchanting | Alluring |
| Generous | Benevolent | Charitable | Magnanimous |
| Creative | Innovative | Imaginative | Inventive |
| Honest | Truthful | Sincere | Genuine |
| Kind | Compassionate | Caring | Benevolent |
| Confident | Self-assured | Assured | Poised |
| Funny | Humorous | Hilarious | Witty |
| Calm | Serene | Tranquil | Composed |
| Passionate | Enthusiastic | Fervent | Zealous |
| Talented | Gifted | Skilled | Accomplished |
| Patient | Tolerant | Enduring | Persevering |
| Friendly | Amiable | Sociable | Affable |
| Reliable | Dependable | Trustworthy | Steadfast |
| Empathetic | Understanding | Compassionate | Sympathetic |
| Adventurous | Daring | Bold | Audacious |
| Optimistic | Positive | Hopeful | Upbeat |
| Wise | Sagacious | Discerning | Prudent |
| Beautiful | Gorgeous | Stunning | Exquisite |
| Intelligent | Smart | Clever | Ingenious |
| Brave | Courageous | Fearless | Valiant |
| Adaptability | Flexibility | Versatility | Agility |
| Communication | Interaction | Exchange | Conveyance |
| Generous | Benevolent | Charitable | Magnanimous |
| Creative | Innovative | Imaginative | Inventive |
| Honest | Truthful | Sincere | Genuine |
| Kind | Compassionate | Caring | Benevolent |
| Confident | Self-assured | Assured | Poised |
| Funny | Humorous | Hilarious | Witty |
| Calm | Serene | Tranquil | Composed |
| Passionate | Enthusiastic | Fervent | Zealous |
| Talented | Gifted | Skilled | Accomplished |
| Patient | Tolerant | Enduring | Persevering |
| Friendly | Amiable | Sociable | Affable |
| Reliable | Dependable | Trustworthy | Steadfast |
| Empathetic | Understanding | Compassionate | Sympathetic |
| Adventurous | Daring | Bold | Audacious |
| Optimistic | Positive | Hopeful | Upbeat |
| Wise | Sagacious | Discerning | Prudent |
Tips and Tricks to Solve Synonyms Questions
Synonym questions can be easily solved by learning Synonym tips and tricks. These tips and tricks are very useful in finding the correct answer quickly and effectively, saving a lot of time.
Follow these tips and tricks while solving synonyms questions:
1. Grasp the Context
Assess the meaning of the word while reading. Then, select a synonym that fits the context of the sentence.
2. Tone and Emotion are the Key
The tone or emotion of the original word will share a similar mood with its synonyms. So, choose a synonym that retains the same feeling in the sentence.
3. Match the Part of Speech
The synonym and the original word will have the same part of speech/sentence. Analyze whether the given word is a noun, adjective, verb, adverb, etc. Pick a synonym that possesses the same part of speech.
4. Eliminate Wrong Choices
Rule out the obviously wrong answers that go against the flow or context of the sentence. This will make it easier to choose the correct synonym.
5. Practice and Learn
Maintain a synonym journal, read extensively, use the synonyms in sentences to retain new words. Form word associations and mnemonics to get better at recalling synonyms. Practice more through synonym exercises, quizzes, and other online resources.
Modal
Question 1 of 3
Coding journey
so far feels like...
Just starting out
Basics are clear
Ready to get a job
Next
Note : There are no wrong answers here, pick what feels closest.
FAQs FAQs
Faq - [x]
Why is understanding the concepts of synonyms important?
Understanding the concepts of synonyms is crucial because it forms a significant portion of the verbal ability section in various placement exams.
It is not only about understanding the word but also interpreting, analyzing, and performing word associations to identify the word that means the closest to the word in question.
Understanding these concepts helps to answer different types of verbal ability questions accurately, ranging from direct facts to complex inference-based questions.
Faq - [x]
How does understanding examples assist in solving synonyms questions more effectively?
Understanding examples is a powerful tool for effectively solving synonyms questions. Examples in synonyms often illustrate how a word can mean different things in different situations and contexts.
By studying the examples in the synonyms topic, learners can understand how to apply the concepts of vocabulary and root-word associations to easily identify the most appropriate answer during exams.
Faq - [x]
Do we have rules to solve questions on synonyms?
No, there are no specific rules that students need to learn to solve synonyms topic in verbal ability. However, there are certain tips and tricks for synonyms that students must know to solve the questions on this topic in a verbal ability exam.
Faq - [x]
What are some of the helpful tips for learning synonyms?
Here are some tips to improve and learn synonyms:
Understand word's meaning and choose fitting synonym.
Select an option with similar emotion as the given word
Eliminate wrong choices for easier selection.
Identify the given word's part of speech and find a synonym that matches.
Practice synonyms by forming sentences.
Know your reading abilities.
Read challenging texts.
Stay focused while reading to retain more words.
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6679 | https://artofproblemsolving.com/wiki/index.php/2014_AMC_12A_Problems/Problem_25?srsltid=AfmBOopUjuz59qWgh4afPHbPy_j70kZl5RsCN8wTahq3ea1hq37a5T98 | Art of Problem Solving
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2014 AMC 12A Problems/Problem 25
Contents
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1 Problem
2 Solution
3 Solution 2
3.1 Video Solution by Richard Rusczyk
4 See Also
Problem
The parabola has focus and goes through the points and . For how many points with integer coordinates is it true that ?
Solution
The parabola is symmetric through , and the common distance is , so the directrix is the line through and , which is the line Using the point-line distance formula, the parabola is the locus which rearranges to .
Let , . Put to obtain and accordingly we find by solving the system that and .
One can show that the values of that make an integer pair are precisely odd integers . For this is , so values work and the answer is .
(Solution by C-273)
You can do this by noting 2 has to divide , so is congruent to 3 modulo 2. Dividing by 3 (since 2 and 3 are coprime), is odd, so must be odd. - (Orion 2010)
Solution 2
The axis of is inclined at an angle relative to the coordinate axis, where . We rotate the coordinate axis by angle anti-clockwise, so that the parabola now has a vertical symmetry axis relative to the rotated coordinates. Let be the coordinates in the rotated system. Then and are related by In the rotated coordinate system, the parabola has focus at and the two points on it are at and . Therefore, the directrix is ; we can, WLOG, choose . For a point on the parabola, it is equidistant from the focus and directrix, so the equation of the parabola is From we have , so we need . Substituting in , we get For to be an integer must be a multiple of 5; setting we get Now we need to be odd, i.e. is an odd multiple of , in which case we get , which is also an integer. The values that satisfy the given conditions correspond to , and there are such numbers.
(Solution by Shaddoll)
Video Solution by Richard Rusczyk
See Also
2014 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 24Followed by
Last Question
1•2•3•4•5•6•7•8•9•10•11•12•13•14•15•16•17•18•19•20•21•22•23•24•25
All AMC 12 Problems and Solutions
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions.
1) The line of symmetry is NOT y= -x but 4x + 3y = 0
2) In the expression for x, it is NOT 8 but 8k.
With these minor corrections, the solution still holds good.
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If xcosθ+ysinθ=p is the normal form and y=mx+c is the slope-intercept form of the line x+2y+1=0, then tan−1(tanθ+m+c)= [9 Sep. 2020, Shift-1]
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| Question Text | If xcosθ+ysinθ=p is the normal form and y=mx+c is the slope-intercept form of the line x+2y+1=0, then tan−1(tanθ+m+c)= [9 Sep. 2020, Shift-1] |
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Problem
(Thailand) Find all functions such that for all . (Symbol denotes the set of all positive real numbers [sic].)
Solution
We will show that is the unique solution to this equation. To this end, let . The given condition then translates to or
Lemma 1. The function is injective.
Proof. Suppose . Then as desired.
Lemma 2. If , then .
Proof. Set , .
Lemma 3. For all , .
Proof. Pick an arbitrary positive real . Then by Lemma 2, Since is injective, it follows that . The lemma then follows.
Now, let be any positive real; pick some . Then by Lemmata 3 and 2, Hence and . Therefore the function is the only possible solution to the problem. Since this function evidently satisfies the problem's condition, it is the unique solution, as desired.
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6682 | https://en.wikipedia.org/wiki/Light_year_(disambiguation) | Light year (disambiguation) - Wikipedia
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1 Film and television
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2.1 Albums
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From Wikipedia, the free encyclopedia
Look up light year in Wiktionary, the free dictionary.
A light-year is the distance that light travels through a vacuum in one year.
Light year(s) and lightyear(s) may also refer to:
Film and television
[edit]
Buzz Lightyear, a main character in the animated film series Toy Story
Lightyear (film), a 2022 spin-off animated film featuring the character
Gandahar (film), a 1988 animated science fiction film known as Light Years in English
Light Years (2015 film), a British drama film
Light Years, an American TV series that has been renamed Life Unexpected
Light Years, a Singaporean drama produced by Mediacorp Channel 5
Music
[edit]
Lightyear (band), ska-punk band from the United Kingdom
Light Years (band), pop-punk band from Cleveland, Ohio
The Lightyears, pop-rock band based in London, England
Albums
[edit]
Light Years (Chick Corea album), 1987
Light Years, The Very Best of Electric Light Orchestra, 1997 compilation album by Electric Light Orchestra
Light Years (Glen Campbell album), 1988
Light Years (Kylie Minogue album), 2000
Light Year, a 2021 album by Emma Stevens
Light Years, a 2005 album by Kathy Valentine
Light Years, a 2012 album by Kora
Light Years, a 2014 album by Noah23 and David Klopek
Light Years, a 2008 album by The Shore
Lightyears, a 2014 album by David Ryan Harris
Lightyears, a 2012 EP by Mansions on the Moon
Lightyears, a 2007 album by Shin Terai, Bill Laswell and Buckethead
Lightyears, a 2001 album by Sunbeam
A series of 1995 compilation albums by Christian artists, including:
The Light Years by Allies
The Light Years by Andraé Crouch
The Light Years by Bryan Duncan
The Light Years by Resurrection Band
The Light Years by Sweet Comfort Band
The Light Years by Walter Hawkins
Songs
[edit]
"Light Years" (Jamiroquai song), 1994
"Light Years" (Pearl Jam song), 2000
"Light Years", 1994 song by Heather Nova from Oyster
"Lightyears", 1996 song by Eraserheads from Fruitcake (album)
"Lightyear", 1997 song by Mentallo & The Fixer from Burnt Beyond Recognition
"Light Years", 2019 song by The National from I Am Easy to Find
"Light Year", 2018 song by The Story So Far from Proper Dose
Other uses
[edit]
Lightyear (investment platform), a British fintech company
Lightyear Entertainment, a movie and music distributor
Light Years (book), a 1975 novel by James Salter
Light Years: An Investigation into the Extraterrestrial Experiences of Eduard Meier (1987), a non-fiction book by Gary Kinder
Lightyear 0, a Dutch electric car
See also
[edit]
Search for "lightyear" or "light-year" on Wikipedia.
All pages with titles containing light year
All pages with titles containing lightyear
Parsec (disambiguation)
Light (disambiguation)
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6683 | https://courses.lumenlearning.com/suny-potsdam-organicchemistry2/chapter/21-4-imine-formation/ | 21.4. Imine formation | Organic Chemistry II
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Organic Chemistry II
21. Nucleophilic Addition of Weak Nucleophiles
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21.4. Imine formation
Nucleophilic addition of amines: Imine and enamine formation
Objectives
After completing this section, you should be able to
write equations to describe the reactions that occur between aldehydes or ketones and primary or secondary amines.
identify the product formed from the reaction of a given aldehyde or ketone with a given primary or secondary amine.
identify the aldehyde or ketone, the amine, or both, required in the synthesis of a given imine or enamine.
write the detailed mechanism for the reaction of an aldehyde or ketone with a primary amine.
write the detailed mechanism for the reaction of an aldehyde or ketone with a secondary amine.
Key Terms
Make certain that you can define, and use in context, the key terms below.
2,4‑dinitrophenylhydrazone
enamine
imine
Study Notes
An imine is a compound that contains the structural unit
An enamine is a compound that contains the structural unit
Both of these types of compound can be prepared through the reaction of an aldehyde or ketone with an amine.
You may have the opportunity to observe the reaction of an aldehyde and ketone with 2,4‑dinitrophenylhydrazine (Brady’s reagent) to form a 2,4‑dinitrophenylhydrozone in the laboratory. This is a classical organic chemistry test to confirm the presence of a carbonyl group. The reaction produces very colourful and bright precipitates of yellow, orange and red.
If you can understand why the two reactions of imine and enamine formation are essentially identical, and can write a detailed mechanism for each one, you are well on the way to mastering organic chemistry. If you understand how and why these reactions occur, you can keep the amount of material that you need to memorize to a minimum.
Reaction with primary amines to form imines
The reaction of aldehydesand ketones with ammonia or 1º-amines forms imine derivatives, also known as Schiff bases (compounds having a C=N function). Water is eliminated in the reaction, which is acid-catalyzed and reversible in the same sense as acetal formation. The pH for reactions which form imine compounds must be carefully controlled. The rate at which these imine compounds are formed is generally greatest near a pH of 5, and drops at higher and lower pH’s. At high pH there will not be enough acid to protonate the OH in the intermediate to allow for removal as H 2 O. At low pH most of the amine reactant will be tied up as its ammonium conjugate acid and will become non-nucleophilic.
Converting reactants to products simply
Examples of imine forming reactions
Mechanism of imine formation
1) Nucleophilic addition
2) Proton transfer
3) Protonation of OH
4) Removal of water (nucleophile elimination)
5) Deprotonation
Reversibility of imine forming reactions
Imines can be hydrolyzed back to the corresponding primary amine under acidic conditions.
Reactions involving other reagents of the type Y-NH 2
Imines are sometimes difficult to isolate and purify due to their sensitivity to hydrolysis. Consequently, other reagents of the type Y–NH 2 have been studied, and found to give stable products (R 2 C=N–Y) useful in characterizing the aldehydes and ketones from which they are prepared. Some of these reagents are listed in the following table, together with the structures and names of their carbonyl reaction products. Hydrazones are used as part of the Wolff-Kishner reduction and will be discussed in more detail in section 21.6.
With the exception of unsubstituted hydrazones, these derivatives are easily prepared and are often crystalline solids – even when the parent aldehyde or ketone is a liquid. Since melting points can be determined more quickly and precisely than boiling points, derivatives such as these are useful for comparison and identification of carbonyl compounds. It should be noted that although semicarbazide has two amino groups (–NH 2) only one of them is a reactive amine. The other is amide-like and is deactivated by the adjacent carbonyl group.
Exercises
1)Please draw the products of the following reactions.
2) Please draw the structure of the reactant needed to produce the indicated product.
3) Please draw the products of the following reactions.
Answers
Show Answer
1)
2)
3)
Contributors
Prof. Steven Farmer (Sonoma State University)
William Reusch, Professor Emeritus (Michigan State U.), Virtual Textbook of Organic Chemistry
Reaction with secondary amines to form enamines
Most aldehydes and ketones react with 2º-amines to give products known as enamines. It should be noted that, like acetal formation, these are acid-catalyzed reversible reactions in which water is lost. Consequently, enamines are easily converted back to their carbonyl precursors by acid-catalyzed hydrolysis.
Mechanism
1) Nucleophilic addition
2) Proton transfer
3) Protonation of OH
4) Removal of water (nucleophile elimination)
5) Deprotonation
Reversibility of Enamines
Exercises
1) Please draw the products for the following reactions.
2) Please give the structure of the reactant needed to product the following product
Answers
Show Answer
1)
2)
Contributors
Dr. Dietmar Kennepohl FCIC (Professor of Chemistry, Athabasca University)
Prof. Steven Farmer (Sonoma State University)
Candela Citations
CC licensed content, Original
Imine formation. Authored by: Martin A. Walker. Provided by: SUNY Potsdam. Project: Organic chemistry: An open textbook. License: CC BY-SA: Attribution-ShareAlike
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19.8 Nucleophilic Addition of Amines: Imine and Enamine Formation. Authored by: Dr. Dietmar Kennepohl and Prof. Steven Farmer . Provided by: Chemistry LibreTexts. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
Organic Chemistry With a Biological Emphasis . Authored by: Tim Soderberg. Provided by: Chemistry LibreTexts. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
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CC licensed content, Original
Imine formation. Authored by: Martin A. Walker. Provided by: SUNY Potsdam. Project: Organic chemistry: An open textbook. License: CC BY-SA: Attribution-ShareAlike
CC licensed content, Shared previously
19.8 Nucleophilic Addition of Amines: Imine and Enamine Formation. Authored by: Dr. Dietmar Kennepohl and Prof. Steven Farmer . Provided by: Chemistry LibreTexts. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
Organic Chemistry With a Biological Emphasis . Authored by: Tim Soderberg. Provided by: Chemistry LibreTexts. Located at: License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike
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6684 | https://ozdic.com/ | OZDIC English Collocation Dictionary is designed to help language learners and users write and speak natural-sounding English.
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Functional equation
A functional equation, roughly speaking, is an equation in which some of the unknowns to be solved for are functions. For example, the following are functional equations:
Contents
1 Introductory Topics
1.1 The Inverse of a Function
2 Intermediate Topics
2.1 Cyclic Functions
2.2 Problem Examples
3 Advanced Topics
3.1 Functions and Relations
3.2 Injectivity and Surjectivity
4 See Also
Introductory Topics
The Inverse of a Function
The inverse of a function is a function that "undoes" a function. For an example, consider the function: . The function has the property that . In this case, is called the (right) inverse function. (Similarly, a function so that is called the left inverse function. Typically the right and left inverses coincide on a suitable domain, and in this case we simply call the right and left inverse function the inverse function.) Often the inverse of a function is denoted by .
Intermediate Topics
Cyclic Functions
A cyclic function is a function that has the property that:
A classic example of such a function is because . Cyclic functions can significantly help in solving functional identities. Consider this problem:
Find such that . Let and in this functional equation. This yields two new equations:
Now, if we multiply the first equation by 3 and the second equation by 4, and add the two equations, we have:
So, clearly,
Problem Examples
2006 AMC 12A Problem 18
2007 AIME II Problem 14
Advanced Topics
Functions and Relations
Given a set and , the Cartesian Product of these sets (denoted ) gives all ordered pairs with and . Symbolically,
A relation is a subset of . A function is a special time of relation where for every in the ordered pair , there exists a unique .
Injectivity and Surjectivity
Consider a function be a function from the set to the set , i.e., is the domain of and is the codomain of .
The function is injective (or one-to-one) if for all in the domain , if and only if . Symbolically, f(x)is injective⟺(∀a,b∈X,f(a)=f(b)⟹a=b).
The function is surjective (or onto) if for all in the codomain there exists a in the domain such that . Symbolically, f(x)is surjective⟺∀a∈Y,∃b∈X:f(b)=a.
The function is bijective (or one-to-one and onto) if it is both injective and subjective. Symbolically, f(x)is bijective⟺∀a∈Y,∃!b∈X:f(b)=a.
The function has an inverse function , where , if and only if it is a bijective function.
See Also
Functions
Cauchy Functional Equation
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6686 | https://www.youtube.com/watch?v=lRoFaux2V_g | Changes in POV and dramatic irony | Reading | Khan Academy
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When we analyze the perspectives of storytellers—whether that’s a point of view character, an omniscient narrator, or a narrator that attaches closely to multiple perspectives—we can understand the way that an author is controlling the flow of information. Who knows what, when? And what does that tell us?
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Attributions for Dramatic Irony:
some sounds licensed under a public domain Creative Commons 0 license
“Wet Splat” by JustInvoke licensed under an Attribution license
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Introduction hello readers today i'd like to talk about differences in point of view in literature when we analyze the perspectives of storytellers whether that's a point of view character an omniscient narrator or a narrator that attaches closely to multiple perspectives we can understand the way that an author is controlling the flow of information who knows what when and what does that tell us a writer can control which characters have access to information and they can control what access you the reader or the viewer have to information too and the information you have and the information the characters have helped develop our perspectives Conflict between characters within a story conflict between characters can come from a difference in perspective in avatar the last airbender the character of prince zuko is consumed by his desire to capture the avatar as he feels that this will end his exile and restore him to a place of prominence at his father's side his uncle iro however understands based on many years of hard experience that zuko's father is a cruel man who never intended to take his son back ira only wants to keep his nephew safe and this difference in perspective generates a lot of conflict between the two of them the show slowly reveals iro's perspective over time whereas we as viewers know what zuko wants and why almost immediately in fiction that kind of information asymmetry between the storyteller the characters and the audience leads to something called dramatic irony you're watching a play or a tv show or reading a book and you the audience the reader knows something that the characters don't this is possible because especially in a visual medium the audience has a different point of view than the characters do we can see things they don't see this difference in point of view is what allows tension suspense and jokes remember irony is just the difference between expectation and result it can be leveraged to achieve anything that uses the mechanism of surprise i don't want to belabor how jokes work literally nothing is more boring or unfunny than explaining jokes so let's use suspense as our example Lowstakes villain imagine a very low-stakes villain the cheezler who goes around pranking people by balancing buckets of cottage cheese on door frames you go through the door blam you get cheesed so i want you to imagine a scene in which we see the cheezler booby-trap the protagonist's door with a bucket of cottage cheese nobody else in the story is around to witness this after all the chiseler stands alone and now we see our hero in tamika she's walking down the hall with such confidence blindly unaware of the cheesy threat that awaits her when she opens that door we cut back to the cheese it's still there oh no oh no tamika [Music] Tension do you see what i'm doing here uh do you see how this generates suspense you the viewer knows something the character doesn't know the difference between her point of view and your point of view generates tension this is something that writers do on purpose this tension this dramatic irony draws us in because it's very satisfying to see it resolved we know that there's that possibility of tamika getting splatted with cottage cheese but what if tamika at the last moment kicks open the door the cheese bucket drops and tamika catches it neatly thank goodness she saw that the door was slightly ajar and put it together the tension is resolved as the gap closes between what tamika knows and what we know when characters resolve interpersonal differences it's through an alignment of their points of view that's what brings us to resolution the difference between two points of view can be reconciled and that too is a way of relieving story tension now i'd like to counsel you against making blanket statements about specific genres of work like sure a first person story could limit the flow of information because we're perceiving a story through the senses of the point of view character but in your analysis of that story be sure to get specific how does that specific character in that particular situation see events why does the author show us the readers information that that character doesn't get to see asking yourself questions about those differences in perspective between the narration and characters between two or more characters between the text and the reader will make it easy for you to analyze the impact of those decisions that the author made and now i'm going to go walk my dog i will see you all next time you can learn anything dave it out all right curse you cheeseler |
6687 | https://atcoder.jp/contests/abc286/tasks/abc286_c | C - Rotate and Palindrome
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Time Limit: 2 sec / Memory Limit: 1024 MiB
配点 : 300 300 3 0 0 点
問題文
長さ N N N の文字列 S S S が与えられます。S i(1≤i≤N)S_i\ (1\leq i \leq N)S i(1≤i≤N) を S S S の左から i i i 番目の文字とします。
あなたは以下の 2 2 2 種類の操作を好きな順番で 0 0 0 回以上好きな回数行うことができます。
A A A 円払う。 S S S の左端の文字を右端に移動する。すなわち、S 1 S 2…S N S_1S_2\ldots S_N S 1S 2…S N を S 2…S N S 1 S_2\ldots S_NS_1 S 2…S NS 1 に変える。
B B B 円払う。 1 1 1 以上 N N N 以下の整数 i i i を選び、 S i S_i S i を好きな英小文字で置き換える。
S S S を回文にするためには最低で何円必要ですか？
回文とは ある文字列 T T T について、 T T T の長さを ∣T∣|T|∣T∣ として、全ての整数 i i i (1≤i≤∣T∣1 \le i \le |T|1≤i≤∣T∣) について、 T T T の前から i i i 文字目と後ろから i i i 文字目が同じであるとき、またそのときに限って、 T T T は回文です。
制約
1≤N≤5000 1\leq N \leq 5000 1≤N≤5 0 0 0
1≤A,B≤1 0 9 1\leq A,B\leq 10^9 1≤A,B≤1 0 9
S S S は英小文字からなる長さ N N N の文字列
S S S 以外の入力は全て整数
入力
入力は以下の形式で標準入力から与えられる。
N N N A A A B B B
S S S
出力
答えを整数として出力せよ。
入力例 1 Copy
Copy
5 1 2
rrefa
出力例 1 Copy
Copy
3
最初に 2 2 2 番目の操作を 1 1 1 回行います。2 2 2 円払い、i=5 i=5 i=5 として S 5 S_5 S 5 を e で置き換えます。 S S S は rrefe となります。
次に 1 1 1 番目の操作を 1 1 1 回行います。1 1 1 円払い、S S S は refer となります。これは回文です。
よって 3 3 3 円払うことで S S S を回文にすることができました。 2 2 2 円以下払うことで S S S を回文にすることは不可能なので、これが答えです。
入力例 2 Copy
Copy
8 1000000000 1000000000
bcdfcgaa
出力例 2 Copy
Copy
4000000000
答えは 32 32 3 2 bit 整数に収まらない場合があることに注意してください。
Score : 300 300 3 0 0 points
Problem Statement
You are given a string S S S of length N N N. Let S i(1≤i≤N)S_i\ (1\leq i \leq N)S i(1≤i≤N) be the i i i-th character of S S S from the left.
You may perform the following two kinds of operations zero or more times in any order:
Pay A A A yen (the currency in Japan). Move the leftmost character of S S S to the right end. In other words, change S 1 S 2…S N S_1S_2\ldots S_N S 1S 2…S N to S 2…S N S 1 S_2\ldots S_NS_1 S 2…S NS 1.
Pay B B B yen. Choose an integer i i i between 1 1 1 and N N N, and replace S i S_i S i with any lowercase English letter.
How many yen do you need to pay to make S S S a palindrome?
What is a palindrome? A string T T T is a palindrome if and only if the i i i-th character from the left and the i i i-th character from the right are the same for all integers i i i (1≤i≤∣T∣1 \le i \le |T|1≤i≤∣T∣), where ∣T∣|T|∣T∣ is the length of T T T.
Constraints
1≤N≤5000 1\leq N \leq 5000 1≤N≤5 0 0 0
1≤A,B≤1 0 9 1\leq A,B\leq 10^9 1≤A,B≤1 0 9
S S S is a string of length N N N consisting of lowercase English letters.
All values in the input except for S S S are integers.
Input
The input is given from Standard Input in the following format:
N N N A A A B B B
S S S
Output
Print the answer as an integer.
Sample Input 1 Copy
Copy
5 1 2
rrefa
Sample Output 1 Copy
Copy
3
First, pay 2 2 2 yen to perform the operation of the second kind once: let i=5 i=5 i=5 to replace S 5 S_5 S 5 with e. S S S is now rrefe.
Then, pay 1 1 1 yen to perform the operation of the first kind once. S S S is now refer, which is a palindrome.
Thus, you can make S S S a palindrome for 3 3 3 yen. Since you cannot make S S S a palindrome for 2 2 2 yen or less, 3 3 3 is the answer.
Sample Input 2 Copy
Copy
8 1000000000 1000000000
bcdfcgaa
Sample Output 2 Copy
Copy
4000000000
Note that the answer may not fit into a 32 32 3 2-bit integer type.
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Nesbitt's Inequality
Nesbitt's Inequality is a theorem which, although rarely cited, has many instructive proofs. It states that for positive ,
,
with equality when all the variables are equal.
All of the proofs below generalize to prove the following more general inequality.
If are positive and , then
,
or equivalently
,
with equality when all the are equal.
Contents
[hide]
1 Proofs
1.1 By Smoothing
1.2 By Rearrangement
1.3 By Cauchy
1.3.1 By AM-GM
1.3.2 By Expansion and AM-GM
1.3.3 By AM-HM
1.4 By Substitution
1.5 By Normalization and AM-HM
1.6 By Weighted AM-HM
1.7 By Muirhead's and Cauchy
1.8 Another Interesting Method
1.9 By Muirhead's and expansion
Proofs
By Smoothing
We may normalize so that . Then we wish to showThe equality case is when so we can use smoothing. We wish to show that if thenIf this is true, then if the minimum occurs at a point where then we can replace and with their average and then the expression decreases, so that cannot be the minimum. Thus the minimum occurs where they are all equal, and we would be done.
Expanding both sides of the inequality and clearing denominators we get which is true because .
By Rearrangement
Note that and , , are sorted in the same order. Then by the rearrangement inequality,
.
For equality to occur, since we changed to , we must have , so by symmetry, all the variables must be equal.
By Cauchy
By the Cauchy-Schwarz Inequality, we have
,
or
,
as desired. Equality occurs when , i.e., when .
We also present three closely related variations of this proof, which illustrate how AM-HM is related to AM-GM and Cauchy.
By AM-GM
By applying AM-GM twice, we have
,
which yields the desired inequality.
By Expansion and AM-GM
We consider the equivalent inequality
.
Setting , we expand the left side to obtain
,
which follows from , etc., by AM-GM, with equality when .
By AM-HM
The AM-HM inequality for three variables,
,
is equivalent to
.
Setting yields the desired inequality.
By Substitution
The numbers satisfy the condition . Thus it is sufficient to prove that if any numbers satisfy , then .
Suppose, on the contrary, that . We then have , and . Adding these inequalities yields , a contradiction.
By Normalization and AM-HM
We may normalize so that . It is then sufficient to prove
,
which follows from AM-HM.
By Weighted AM-HM
We may normalize so that .
We first note that by the rearrangement inequality or the fact that ,
,
so
.
Since , weighted AM-HM gives us
.
By Muirhead's and Cauchy
By Cauchy, since by Muirhead as
Another Interesting Method
Let And And Now, we get Also by AM-GM; and
By Muirhead's and expansion
Let . Expanding out we get that our inequality is equivalent to This means So it follows that we must prove So it follows that we must prove which immediately follows from Muirhead's.
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6689 | https://unescollece.org/wp-content/uploads/2023/12/Aportes-Ciencias-LMCS.pdf | Aportes para la enseñanza de las Ciencias Naturales a partir del estudio ERCE 2019 Resumen Publicado en 2023 por la Organización de las Naciones Unidas para la Educación, la Ciencia y la Cultura, 7, place de Fontenoy, 75352 París 07 SP, Francia y la Oficina Regional de Educación para América Latina y el Caribe (OREALC/UNESCO Santiago), Enrique Delpiano 2058, 7511019 Santiago, Chile.
© UNESCO 2023 Esta publicación está disponible en acceso abierto bajo la licencia Attribution-ShareAlike 3.0 IGO (CC-BY-SA 3.0 IGO) ( Al utilizar el contenido de la presente publicación, los usuarios aceptan las condiciones de utilización del Repositorio UNESCO de acceso abierto
org/es/open-access/cc-sa.
Los términos empleados en esta publicación y la presentación de los datos que en ella aparecen no implican toma alguna de posición de parte de la UNESCO en cuanto al estatuto jurídico de los países, territorios, ciudades o regiones ni respecto de sus autoridades, fronteras o límites. Las ideas y opiniones expresadas en esta obra son las de los autores y no reflejan necesariamente el punto de vista de la UNESCO ni comprometen a la Organización. Autores: profesionales equipo LLECE de Unesco Santiago Coordinación General: Coordinación LLECE de Unesco Santiago Diseño de cubierta: Carolina Salas A.
Maquetación: Carolina Salas A.
Aportes para la enseñanza de las Ciencias Naturales a partir del estudio ERCE 2019 3 Tabla de contenidos 1. Introducción 5 2. ¿Qué se evaluó en la prueba de Ciencias del estudio ERCE 2019? 6 3. Niveles de desempeño de Ciencias 7 4. ¿Cómo avanzar entre los niveles de desempeño del estudio ERCE 2019 desde el aula? 8 Actividad científica escolar y alfabetización científica 8 Decisiones de las y los docentes centradas en desempeños de aprendizaje 8 5. Organización de las actividades para movilizar el aprendizaje 10 Transición de Nivel I a Nivel II 10 Transición de Nivel II a Nivel III 11 Transición de Nivel III a Nivel IV 11 6. ¿Cómo trabajar la brecha de género en el aula de Ciencias? 12 ¿Cómo incorporar la perspectiva de género en el aula de Ciencias? 12 Lista de tablas Tabla 1. Dominios evaluados en Ciencias, estudio ERCE 2019 6 Tabla 2. Habilidades evaluadas en Ciencias, estudio ERCE 2019 6 Tabla 3. Niveles de desempeño para 6° grado, estudio ERCE 2019 7 Tabla 4. Niveles de desempeño I y II de Ciencias Naturales en el estudio ERCE 2019 11 Tabla 5. Niveles de desempeño II y III de Ciencias Naturales en el estudio ERCE 2019 11 Tabla 6. Niveles de desempeño III y IV de Ciencias Naturales en el estudio ERCE 2019 11 Aportes para la enseñanza de las Ciencias Naturales a partir del estudio ERCE 2019 4 Agradecimientos La publicación Aportes para la enseñanza de Ciencias es una iniciativa del Laboratorio Latinoamericano de Evaluación de la Calidad de la Educación (LLECE), bajo el liderazgo de Claudia Uribe, directora de la Oficina Regional de Educación para América Latina y el Caribe (OREALC/UNESCO Santiago).
Este documento es un resumen de UNESCO del trabajo elaborado por el Centro de Medición MIDE UC de la Pontificia Universidad Católica de Chile.
Agradecemos los comentarios de Carlos Vargas, jefe de la Sección de Desarrollo Docente de la sede de la UNESCO en París; Zelmira May, especialista en educación de la oficina de la UNESCO en Montevideo; Rosa Wolpert, oficial de educación de la oficina de la UNESCO en México; Emilia Soto Echeverri y Meritxell Fernández, oficiales asociadas de Proyecto de la Teacher Task Force en la sede de la UNESCO en París. Esta iniciativa se llevó a cabo con la coordinación técnica del LLECE: Carlos Henríquez Calderón, coordinador general; Francisco Gatica y Álvaro Otaegui, especialistas, Luisa Muller y Álvaro Romero, consultores. El documento se realizó en el marco del Estudio Regional Comparativo y Explicativo (ERCE 2019), el cual es producto del trabajo de 19 países que conforman el LLECE. Damos las gracias por la retroalimentación y constantes aportes de los coordinadores nacionales y contrapartes de los países que participaron en este estudio: Argentina, Bolivia, Brasil, Chile, Colombia, Costa Rica, Cuba, Ecuador, El Salvador, Estado de Nuevo León (México), Guatemala, Honduras, Nicaragua, Panamá, Paraguay, Perú, República Dominicana, Uruguay y Venezuela.
Aportes para la enseñanza de las Ciencias Naturales a partir del estudio ERCE 2019 5 1. Introducción Acompañar el proceso de aprendizaje es una labor desafiante que, para ser exitosa y lograr aprendizajes, requiere prestar atención tanto a la guía que ofrecen los contenidos y habilidades curriculares en Ciencias como al diverso nivel de desempeños que muestran las y los estudiantes, para poder así ajustar la enseñanza a sus necesidades. Esto resulta retador para los docentes, pero a la vez muy gratificante, pues es posible constatar que atendiendo a las necesidades del alumnado sus trayectorias de aprendizaje se acercan a la progresión esperada para el logro de los objetivos.
El Laboratorio Latinoamericano de Evaluación de la Calidad de la Educación (LLECE), en coordinación con los países de América Latina y el Caribe, aplicó en 2019 el Estudio Regional Comparativo y Explicativo (ERCE 2019), con el que se obtuvo información de los logros de aprendizaje en estudiantes de 3° y 6° grado en las áreas curriculares de Lenguaje, que incluyó Lectura y Escritura, y Matemática para 3° y 6° grado y Ciencias para 6°, identificando los retos que aún se mantienen en la región. La evaluación ERCE 2019 en Ciencias visibilizó los desempeños de estudiantes de 6° grado de Primaria de diversos países de América Latina y el Caribe. Los resultados del estudiantado son preocupantes, pues se verificó que solo un 20,7 % de ellos y ellas alcanza o supera el Nivel III, considerado el mínimo esperable en esta evaluación. Estos resultados reflejan los bajos niveles de alfabetización en el área, situación que requiere de atención para la formación de ciudadanos responsables, críticos e informados, que utilizan su conocimiento científico en la toma de decisiones, y así disminuir las consecuencias negativas que la intervención humana pueda tener en el medio ambiente. Es por ello que se vuelve relevante enfocar la enseñanza en los desempeños de las y los estudiantes, para acompañarlos en su aprendizaje, avanzando hacia los niveles esperados para su etapa escolar.
Por otra parte, el ERCE 2019 reveló aspectos relacionados con la perspectiva de género en las aulas de Ciencias de Primaria. Contrario a lo esperado, considerando que históricamente las ciencias y la educación científica se han caracterizado por una visión positivista y centrada en el quehacer científico de hombres, los resultados indican que no existe una brecha de género significativa entre niños y niñas en nueve de los 16 países participantes, y que incluso se observan resultados que favorecen a las niñas en Ciencias en siete de esos países. Estos resultados son alentadores y nos motivan a promover en las aulas una educación científica no sexista, que busque la equidad en los aprendizajes científicos para todas y todos los estudiantes. Asimismo, para los países que presentan una brecha de género desfavorable para los niños varones, es un nuevo desafío lograr que no se queden rezagados en sus aprendizajes.
El Resumen de Aportes para la enseñanza de Ciencias se basa en los resultados de los países de América Latina y el Caribe participantes en la evaluación ERCE 2019 de Ciencias y se trata de una síntesis de un informe más extenso que contiene recursos teóricos y prácticos basados en evidencia. Tiene el objetivo central de contribuir a la labor de docentes en el acompañamiento de sus estudiantes hacia el logro de los aprendizajes, sumando recursos y ejemplos para que puedan transformarlos en un insumo para sus estrategias de enseñanza y la movilización de aprendizajes de alumnos y alumnas de acuerdo a la trayectoria educativa en la que se encuentren. Con este propósito se presenta un resumen de lo que compone la serie Aportes para la enseñanza de Ciencias hacia el logro de los aprendizajes considerados en el ERCE 2019, sugerencias generales y específicas para el trabajo en torno a prácticas científicas esenciales para alcanzar la alfabetización en el área y para promover la equidad de género en el aula.
Aportes para la enseñanza de las Ciencias Naturales a partir del estudio ERCE 2019 6 2. ¿Qué se evaluó en la prueba de Ciencias del estudio ERCE 2019 El Estudio Regional Comparativo y Explicativo (ERCE 2019) se realizó en 16 países de América Latina y el Caribe. La prueba de Ciencias se aplicó a estudiantes de 6° grado. Considerando los hallazgos en el análisis de los currículos de los países de la región, se desprenden temas que reflejan una relación entre la ciencia y los problemas reales de la sociedad actual, así como también La prueba ERCE 2019 evaluó tres habilidades de Ciencias: Tabla 2. Habilidades evaluadas en Ciencias, estudio ERCE 2019.
la importancia de desarrollar habilidades que promuevan el pensamiento científico (UNESCO/OREALC, 2020). Lo anterior se concretiza en la prueba a través de dominios de conocimiento y procesos cognitivos o habilidades, de cuyo cruce se infieren indicadores de evaluación que orientan la construcción de las preguntas. Tabla 1. Dominios evaluados en Ciencias, estudio ERCE 2019.
Cuerpo humano y salud Ciencia, tecnología y sociedad La Tierra y el sistema solar Energía y materia Seres vivos, ecología y medio ambiente Al conocimiento del cuerpo humano y a la comprensión de algunos aspectos vinculados con su desarrollo, crecimiento y equilibrio.
A la ciencia y la tecnología como una construcción humana, infuenciada por el contexto sociocultural, político y económico, donde no existe un método único para construir conocimientos. Incluye también contenidos sociocientífcos relativos al impacto del desarrollo de la ciencia y la tecnología en la salud y en el medio ambiente.
Al conocimiento y comprensión de las características físicas de la Tierra y de sus movimientos, así como a la comprensión de algunos fenómenos naturales. Incluye también la noción de sistema solar: la importancia del Sol para la vida en la Tierra, el conocimiento de los otros planetas del sistema solar y la comprensión de la Luna como satélite natural que tiene infuencias en la Tierra.
A los conocimientos relacionados con la noción de energía: qué es, de dónde viene, cómo se manifesta y para qué sirve. Específcamente se incluye el calor como una transferencia de energía en la naturaleza, la identifcación de algunas manifestaciones de energía en el planeta y la noción de fuerza. Considera, además, conocimientos relativos a la materia, como su estructura atómica y molecular (noción de átomo y molécula), y cómo se organizan ciertos estados de agregación (sólido, líquido y gaseoso) en relación con algunas de sus propiedades, como la masa, el volumen y la temperatura.
Al conocimiento de la noción de célula, particularmente de su estructura y función en los seres vivos. Este dominio contempla también las nociones y conceptos referentes a biodiversidad (animal y vegetal), ecosistema y equilibrio ecológico, concretamente a recursos naturales, como la energía del Sol, el agua, el suelo y el aire, y su importancia para la vida en el planeta, ciclo de vida en los seres vivos y la noción e importancia de la cadena trófca.
Dominio ¿A qué se refere?
Reconocimiento de información científca Análisis y aplicación del conocimiento científco Producción, transferencia y evaluación del conocimiento científco Al reconocimiento o identifcación de conceptos, principios, hechos, datos, hipótesis, modelos, teorías o leyes científcas.
Implica relacionar algunos fenómenos naturales con sus explicaciones científcas, la contrastación, análisis e interpretación de información y la aplicación de los conocimientos científcos necesarios para resolver una situación problemática simple.
A un dominio conceptual más amplio de los fenómenos naturales y a la construcción de conocimiento científco asociada a estos, a través de la elaboración o evaluación de propuestas, explicaciones y argumentos con base científca para la resolución de problemas de mayor complejidad.
Habilidad ¿A qué se refere?
Aportes para la enseñanza de las Ciencias Naturales a partir del estudio ERCE 2019 7 3. Niveles de desempeño de Ciencias En el ERCE 2019 se establecieron cuatro niveles de desempeño en cada una de las pruebas, con el fin de enriquecer el reporte de resultados. Cada nivel describe lo que las y los estudiantes saben y son capaces de hacer. Estos niveles ordenaron los logros de aprendizaje del alumnado en un continuo de creciente complejidad; así, los logros de los niveles inferiores son la base de los niveles más avanzados (UNESCO/OREALC, 2021). La descripción de los niveles de desempeño es información que puede utilizarse en el aula para comprender qué aprendizajes se espera que logren ellos y ellas en general y en los grados evaluados. Esta información puede ser importante para que cada docente diagnostique el logro de las habilidades descritas en las y los estudiantes de su grado, dando luces para el diseño de instrumentos de evaluación en el aula acordes con los aprendizajes claves.
Si bien el establecimiento de un nivel mínimo de competencias no señala la disciplina de Ciencias, el estudio ERCE 2019, también consideró la meta establecida por la Agenda 2030 para sexto grado Los niveles de desempeño: • Describen lo que las y los estudiantes saben y son capaces de hacer.
• Son progresivos. • Los niveles mayores incluyen las habilidades de los niveles menores.
correspondiente al Nivel III, pues entrega una orientación a los países de la región respecto de los desafíos existentes en el logro de los aprendizajes de las ciencias.
A continuación, se detallan los niveles de desempeño para Ciencias de 6° grado. Hay que tomar en cuenta que los niveles de desempeño son progresivos y la complejidad de las tareas que se realizan en cada nivel requiere de mayor habilidad conforme se avanza a los niveles superiores. La mayor parte de las y los estudiantes de este nivel mostró evidencia de ser capaces de: Tabla 3. Niveles de desempeño para 6° grado, estudio ERCE 2019.
• Reconocer acciones preventivas orientadas al cuidado de la salud en contextos cotidianos.
• Identifcar la función de diferentes partes o estructuras del cuerpo humano.
• Clasifcar seres vivos a partir de la observación o la descripción de sus características.
• Distinguir algunas relaciones de causa y efecto para el cuidado de la salud en contextos cotidianos.
• Identifcar la ubicación o función de una estructura o parte de un ser vivo en un sistema mayor.
• Reconocer similitudes en las necesidades vitales de los seres vivos.
• Identifcar posibles causas o efectos de fenómenos naturales en contextos cotidianos aplicando su conocimiento.
• Identifcar factores que infuyen en el funcionamiento de circuitos eléctricos simples.
• Discriminar recomendaciones apropiadas de cuidado de la salud según criterios o requerimientos específcos (ej., de balance de nutrientes).
• Reconocer la función específca de un elemento dentro de un sistema mayor o establecer relaciones entre sistemas (ej., el papel de un ser vivo en un sistema trófco o la relación entre el sistema digestivo y circulatorio).
• Identifcar los procesos que conforman ciclos naturales (ej., fotosíntesis, ciclo del agua), así como los factores que los afectan.
• Identifcar la conclusión que se deriva de la información presentada en un gráfco.
• Identifcar preguntas que son investigables científcamente o la hipótesis que orienta una investigación, o bien, evaluar la pertinencia de un diseño experimental o de una conclusión.
• Explicar un fenómeno demostrando su comprensión de las relaciones entre componentes de un sistema natural. • Utilizar su conocimiento científco para interpretar un gráfco en que se representa la variación de factores involucrados en un proceso natural.
• Analizar una actividad de investigación para seleccionar los datos que resultan pertinentes para responder la pregunta o para discriminar entre los distintos componentes de un diseño experimental (ej., distinguir entre objetivos e hipótesis o diferenciar cuál corresponde al diseño).
Nivel I Hasta 668 puntos Nivel II Entre 669 y 781 puntos Nivel III Entre 782 y 861 puntos Nivel IV Desde 862 puntos Aportes para la enseñanza de las Ciencias Naturales a partir del estudio ERCE 2019 8 4. ¿Cómo avanzar entre los niveles de desempeño del estudio ERCE 2019 desde el aula?
Actividad científica escolar y alfabetización científica Marcos curriculares alrededor del mundo concuerdan en la relevancia del aprendizaje de las ciencias a través del desarrollo de prácticas científicas en el aula (Osborne, 2014). Así, 10 de los 16 países participantes en el ERCE 2019 identifican en sus currículos la alfabetización científica y tecnológica como uno de sus pilares, conectando situaciones de aprendizaje de diversos dominios en Ciencias con fenómenos que ocurren en el entorno (UNESCO/OREALC, 2020). La enseñanza y el aprendizaje en torno a las prácticas científicas y, en consecuencia, a la alfabetización buscan que las y los estudiantes reproduzcan de manera similar las habilidades y formas de pensar utilizadas por los científicos al momento de buscar respuestas a preguntas investigables o resolver problemas. Así, desde el campo de la didáctica de las ciencias se ha reportado que las prácticas científicas involucran al alumnado en procesos de indagación, modelización y argumentación en Ciencias. Izquierdo et al. (1999) & Adúriz (2012), entre otros autores del campo de la didáctica de las ciencias, conciben las prácticas científicas en el aula como una actividad científica escolar (ACE) centrada en la construcción de representaciones de fenómenos del mundo natural que se usan para describir, predecir o explicar, involucrando al estudiantado en las formas de hablar, pensar y hacer de la ciencia en la escuela, de forma análoga al actuar de los científicos. Desde la ACE se promueve el desarrollo de prácticas cognitivas, discursivas y sociales propias de la ciencia, viendo el aprendizaje como un proceso en el que las y los estudiantes van construyendo representaciones propias, llamadas modelos, cada vez más sofisticadas y coherentes con los modelos científicos escolares que los docentes buscan promover.
Es por ello que la alfabetización científica se considera en los currículos de Ciencias como una dimensión relevante para el mejoramiento de la calidad de vida de los ciudadanos, al involucrar el entendimiento y la toma de decisiones en aspectos personales y sociales, como por ejemplo en contextos de salud, en el vínculo entre la ciencia y la tecnología o en el cuidado del medio ambiente. Así, las y los estudiantes alfabetizados científicamente pueden desarrollar «la capacidad de usar el conocimiento científico para identificar preguntas y para sacar conclusiones basadas en pruebas o evidencias, con el fin de entender y ayudar a tomar decisiones sobre el mundo natural y los cambios realizados en él a través de la actividad humana» (Harlen, 2012). La orientación de los currículos hacia el logro de la alfabetización científica es aún más relevante cuando el desarrollo de estas prácticas, acompañadas de un manejo orientado a la equidad en el aula, pueden ser claves para el desarrollo de la justicia social (González et al., 2009). Mediante el desarrollo de estas prácticas, los docentes pueden incentivar a sus estudiantes a tomar decisiones informadas ante cuestiones sociocientíficas relevantes para ellos, así como para su comunidad, promoviendo el desarrollo de la alfabetización en Ciencias (UNESCO/OREALC, 2020b). Al mismo tiempo, por ejemplo, es posible promover el desarrollo de un pensamiento crítico y la construcción de explicaciones de fenómenos con base en evidencias. Estas prácticas son las que se espera que las y los estudiantes alfabetizados científicamente sean capaces de llevar a cabo con éxito (Harlen, 2012).
Decisiones de las y los docentes centradas en desempeños de aprendizaje El logro de aprendizajes efectivos depende de diversos factores, entre los cuales puede encontrarse la flexibilidad de los docentes para ajustar la enseñanza según los logros y dificultades de sus estudiantes. Monitorear el aprendizaje del alumnado es una manera en la que les es posible recopilar información que posteriormente se utilice como evidencia para la toma de decisiones sobre la enseñanza y el aprendizaje. Una de Las prácticas científicas involucran a las y los estudiantes en procesos de indagación, modelización y argumentación en Ciencias.
Aportes para la enseñanza de las Ciencias Naturales a partir del estudio ERCE 2019 9 las maneras de llevar a cabo el monitoreo es analizando los resultados de evaluaciones sumativas; sin embargo, este no es el único momento en que es posible recopilar información para informar a los docentes. Acciones como la observación del actuar y las verbalizaciones de las y los estudiantes al resolver una tarea, el escuchar sus discusiones buscando respuestas a preguntas desafiantes, reunir evidencias de sus respuestas escritas en evaluaciones sumativas y formativas, se presentan como oportunidades para conocer cómo están aprendiendo y qué recursos o intervenciones se requieren para el logro de los objetivos.
En la literatura se han identificado ciclos que permiten a las y los docentes llevar a cabo una enseñanza centrada en sus estudiantes. Es así como las prácticas de evaluación formativa se presentan como una instancia en que es posible retroalimentar tanto la docencia como el aprendizaje. Los docentes pueden transitar durante una clase por varios ciclos de evaluación formativa, elicitando, focalizando su atención, interpretando respuestas y actuando con base en sus interpretaciones. La descripción de cada uno de estos pasos se presenta a continuación.
Elicitar Este primer paso implica que los docentes diseñen e implementen tareas que les permitan visibilizar el pensamiento de sus estudiantes. Solicitar un dibujo, proponer una tarea desafiante, realizar preguntas para obtener respuestas orales o escritas, son algunos ejemplos a través de los cuales es posible conocer la comprensión que presenta el alumnado, sus ideas sobre determinados fenómenos o conceptos.
Focalizar Cuando ellas y ellos están completando una tarea, como por ejemplo elaborando una pregunta investigable, los docentes pueden focalizar la atención en distintos elementos del desempeño de sus estudiantes, y preguntarse: ¿identifican las características de una pregunta investigable? ¿Están vinculando las variables dependientes e independientes? ¿Es correcta la relación entre las variables que proponen? En torno a estas preguntas, los docentes pueden identificar las fortalezas y Se usa la evaluación formativa porque es útil para el fortalecimiento de la construcción de conocimiento científico escolar, y para promover la sofisticación de prácticas claves en la ciencia.
aspectos por mejorar en la tarea. Así, les es posible focalizar la atención en un aspecto en específico y analizar qué está sucediendo con este desempeño. Es importante señalar que no siempre es dable abordar todos los aspectos por mejorar en una sola instancia. Por ello se recomienda implementar diversos ciclos de evaluación formativa o retomar tareas similares en clases siguientes.
Interpretar Una vez que los docentes han focalizado la atención en un aspecto específico del desempeño de sus estudiantes, se analizan sus respuestas o actuar y se interpreta qué tipo de dificultades enfrentan frente a esa tarea. Por ejemplo, puede suceder que las y los estudiantes identifiquen las características de una pregunta investigable, pero aún confundan las variables dependientes e independientes.
Actuar En este paso, una vez que se interpreta qué sucede con los desempeños de ellos y ellas, se toma una decisión sobre cómo acompañarlos para superar la dificultad identificada. Continuando con el mismo ejemplo de la pregunta investigable, una manera de actuar frente a esta dificultad es identificar primero si, por ejemplo, solo un grupo de estudiantes presenta este problema o si es necesario aclarar la diferencia entre las variables para toda la clase. Luego, se puede hacer un breve ejercicio, con el grupo específico o con la clase, para reforzar la identificación de variables dependientes e independientes. Si aún existen estudiantes que presentan confusión, los docentes pueden retomar este ejercicio en la clase siguiente y asegurarse de que todos han comprendido la diferencia. Este es un ejemplo de cómo abordar las dificultades en tiempo real y a corto plazo entre clases.
Fuente: Adaptado de Clinchot et al. (2017).
Focalizar Interpretar Elicitar Actuar Aportes para la enseñanza de las Ciencias Naturales a partir del estudio ERCE 2019 10 5. Organización de las actividades para movilizar el aprendizaje A continuación, se entregan sugerencias generales de planificación para que los docentes puedan promover en el aula de Primaria la construcción de aprendizaje en torno al desarrollo de prácticas análogas a la actividad científica, con elementos claves que son transversales a toda clase de ciencias, sin importar la estructura de clase utilizada.
• Entregar oportunidades de aprendizaje que apunten al desarrollo de la alfabetización científica. Se lleva a cabo mediante actividades que busquen desafiar a las y los estudiantes a pensar, hablar y actuar de manera análoga a los científicos. Por ejemplo, invitándolos a participar activamente en indagaciones con el propósito de levantar datos que les permitan llegar a conclusiones y explicaciones sobre cómo y por qué suceden ciertos fenómenos.
• Abrir momentos de diálogo para el aprendizaje. Con el propósito de promover la expresión e intercambio de ideas, la construcción conjunta de aprendizajes y el desarrollo de habilidades comunicativas, es necesario que las clases consideren momentos en que el alumnado piense individualmente sus respuestas, para luego discutir junto con sus compañeros, desafiando argumentos para robustecer sus ideas y acercarse hacia el conocimiento científico escolar que se busca desarrollar en esa clase.
• Construir nuevos aprendizajes con base en lo que ellas y ellos saben. Un aspecto clave para el desarrollo de los aprendizajes es lo que las y los estudiantes ya conocen sobre el tema. Este conocimiento puede tener como base su experiencia escolar, su interacción con la familia y amigos o elementos culturales. Ellas y ellos poseen un cierto grado de desarrollo de su conocimiento sobre el contenido, de sus habilidades científicas, así como de sus formas de razonar, lo que debe considerarse al momento de incorporar nuevas experiencias de aprendizaje.
• Promover la articulación de los modelos científicos escolares. Aprender ciencia implica, entre otras cosas, vincular componentes de un sistema y establecer relaciones entre sistemas, para comprender el funcionamiento de diversos procesos en el mundo circundante. Para ello, es necesario incluir una mirada integral que tome en cuenta aspectos de la física, la química y la biología. Por ejemplo, integrar los modelos de materia y energía para explicar cadenas tróficas.
• Planificar oportunidades de aprendizaje que consideren la transferencia y aplicación de los aprendizajes a situaciones fuera del contexto de aula. Por ejemplo, una vez que han estudiado los eslabones de una cadena trófica, el docente puede invitarlos a indagar sobre qué animales están en peligro de extinción en el área e identificar las repercusiones que esto ocasionaría en el ecosistema.
• Incorporar metodologías como el aprendizaje basado en proyectos. Con esto se busca transferir el aprendizaje a situaciones reales, estimulando con ello procesos cognitivos superiores, tales como la creatividad, el pensamiento crítico, la evaluación y análisis, habilidades centrales por desarrollar en el aprendizaje de las ciencias. A continuación, se hacen sugerencias didácticas específicas, acompañadas de ejemplos para apoyar las transiciones entre los niveles de desempeño identificados en la evaluación ERCE 2019. Es importante señalar que si bien estas transiciones muestran un avance lineal, en el aula estos avances pueden ser en espiral y no necesariamente lineales para todos. Para diseñar estas secuencias o transiciones, se han desagregado los indicadores de cada nivel de desempeño del ERCE 2019 en Ciencias, desarrollando la secuencia de aprendizaje asociada con el dominio Cuerpo humano y salud.
Para profundizar más en este tema revisar el siguiente link: Niveles_ciencias_2810.pdf Transición de Nivel I a Nivel II Esta secuencia considera el tránsito desde la Identificación de la función de diferentes partes o estructuras del cuerpo (Nivel I) hasta la Identificación de la ubicación o función de una estructura o parte de un ser vivo en un sistema mayor (Nivel II). Aportes para la enseñanza de las Ciencias Naturales a partir del estudio ERCE 2019 11 Tabla 4. Niveles de desempeño I y II de Ciencias Naturales en el estudio ERCE 2019 Tabla 5. Niveles de desempeño II y III de Ciencias Naturales en el estudio ERCE 2019 Transición de Nivel II a Nivel III Esta transición considera un aumento en la complejidad de la tarea respecto de establecer relaciones entre sistemas (Nivel III) y no solo identificar funciones o estructuras que componen un determinado sistema (Nivel II).
Transición de Nivel III a Nivel IV Con el objetivo de acercar a las y los estudiantes hacia el Nivel IV, la transición entre el Nivel III y el Nivel IV enfoca la atención específicamente en el desarrollo de habilidades científicas vinculadas a identificar preguntas investigables o hipótesis, • Reconocer acciones preventivas orientadas al cuidado de la salud en contextos cotidianos.
• Identifcar la función de diferentes partes o estructuras del cuerpo.
• Clasifcar seres vivos a partir de la observación o la descripción de sus características.
• Distinguir algunas relaciones de causa y efecto para el cuidado de la salud en contextos cotidianos.
• Identifcar la ubicación o función de una estructura o parte de un ser vivo en un sistema mayor.
• Reconocer similitudes en las necesidades vitales de los seres vivos.
• Identifcar posibles causas o efectos de fenómenos naturales en contextos cotidianos, aplicando su conocimiento.
• Identifcar factores que infuyen en el funcionamiento de circuitos eléctricos simples. Nivel I Nivel II Para lograr la transición esperada se recomienda comenzar con una tarea que le permita al docente diagnosticar en qué nivel se hallan sus estudiantes; es decir, monitorear si la mayoría de las y los estudiantes se encuentran en el Nivel II para facilitar la transición hacia el Nivel III. Con el propósito de comprobar el nivel en el que están, puede utilizar la actividad planteada en la transición I a II en la que se aumenta la complejidad de la tarea al incorporar estructuras de sistemas mayores.
así como evaluar la pertinencia de un diseño experimental en el Nivel III. En el Nivel IV se espera, por ejemplo, que ellos y ellas sean capaces de seleccionar datos pertinentes para dar respuesta a una pregunta investigable al analizar una actividad de indagación, o bien, distinguir entre los distintos componentes de un diseño experimental. • Distinguir algunas relaciones de causa y efecto para el cuidado de la salud en contextos cotidianos.
• Identifcar la ubicación o función de una estructura o parte de un ser vivo en un sistema mayor.
• Reconocer similitudes en las necesidades vitales de los seres vivos.
• Identifcar posibles causas o efectos de fenómenos naturales en contextos cotidianos, aplicando su conocimiento.
• Identifcar factores que infuyen en el funcionamiento de circuitos eléctricos simples.
• Discriminar recomendaciones apropiadas para el cuidado de la salud según criterios o requerimientos específcos (ej., de balance de nutrientes).
• Reconocer la función específca de un elemento en un sistema mayor o establecer relaciones entre sistemas (ej., el papel de un ser vivo en un sistema trófco o la relación entre el sistema digestivo y el circulatorio).
• Reconocer los procesos que conforman ciclos naturales (ej., fotosíntesis, ciclo del agua), así como los factores que los afectan.
• Identifcar preguntas que son investigables científcamente o la hipótesis que orienta una investigación, o bien, evaluar la pertinencia de un diseño experimental o de una conclusión.
• Identifcar la conclusión que se deriva de la información presentada en un gráfco.
Nivel II Nivel III Tabla 6. Niveles de desempeño III y IV de Ciencias Naturales en el estudio ERCE 2019 • Discriminar recomendaciones apropiadas para el cuidado de la salud según criterios o requerimientos específcos (ej., de balance de nutrientes).
• Reconocer la función específca de un elemento en un sistema mayor o establecer relaciones entre sistemas (ej., el papel de un ser vivo en un sistema trófco o la relación entre el sistema digestivo y el circulatorio).
• Reconocer los procesos que conforman ciclos naturales (ej., fotosíntesis, ciclo del agua), así como los factores que los afectan.
• Identifcar preguntas que son investigables científcamente o la hipótesis que orienta una investigación, o bien, evaluar la pertinencia de un diseño experimental o de una conclusión.
• Identifcar la conclusión que se deriva de la información presentada en un gráfco.
• Explicar un fenómeno demostrando su comprensión de las relaciones entre componentes de un sistema natural.
• Analizar una actividad de investigación para seleccionar los datos que resultan pertinentes para responder la pregunta o para discriminar entre los distintos componentes de un diseño experimental (ej., distinguir entre objetivos e hipótesis o diferenciar cuál corresponde al diseño). • Utilizar su conocimiento científco para interpretar un gráfco en que se representa la variación de factores involucrados en un proceso natural.
Nivel III Nivel IV Aportes para la enseñanza de las Ciencias Naturales a partir del estudio ERCE 2019 12 6. ¿Cómo trabajar la brecha de género en el aula de Ciencias?
El ámbito de las disciplinas científicas se ha caracterizado históricamente por la predominancia de una visión androcéntrica, en que se han llegado a cometer injusticias e invisibilización de la participación de mujeres en los procesos de construcción del conocimiento científico (Camacho, 2018). Uno de los ejemplos más conocidos es el de la científica Marie Curie, quien recibió el Premio Nobel de Física junto a su esposo, Pierre Curie, y el científico Henri Becquerel, por sus aportes en el ámbito de la investigación sobre la radiación; sin embargo, Marie Curie inicialmente iba a ser excluida de este reconocimiento por ser mujer. Gracias a los reclamos de su esposo, Pierre, ante la comunidad científica y de negarse a recibir el premio sin el reconocimiento al trabajo de su esposa, Marie fue reconocida.
Si bien esta larga tradición de sesgos de género en ciencias ha tenido como repercusión el desinterés de niñas y jóvenes por proseguir los estudios en carreras vinculadas con las ciencias, los resultados del ERCE 2019 en América Latina y el Caribe son alentadores, pues dan cuenta de cambios positivos en esta área. Aunque, como se ha mencionado, estos resultados dan luces de una educación científica cada vez menos sexista, no se puede olvidar que de los 2 a los 3 años de edad el desarrollo de estereotipos de género se hace presente en niños y niñas (Sullivan & Umaschi, 2018). Posteriormente, es posible que estos estereotipos se refuercen, por ejemplo, desde madres y padres hasta sus hijos e hijas. Entre esos aspectos se ha reportado que padres y madres suelen concebir a las ciencias y el quehacer científico como disciplinas y actividades principalmente del género masculino, mientras que áreas denominadas como humanistas (lenguaje, artes o ciencias sociales) presentan mayor asociación con el género femenino (Del Río, et al., 2019).
¿Cómo incorporar la perspectiva de género en el aula de Ciencias?
Ejemplos que facilitan la incorporación de una perspectiva de equidad de género en el aula de Ciencias: • Diseñar e implementar actividades que muestren el componente creativo del quehacer científico favoreciendo la incorporación de niños y niñas en el aula.
• Promover las prácticas cognitivas, discursivas y sociales de la ciencia para favorecer la participación de las y los estudiantes en una actividad científica escolar. • Proporcionar experiencias escolares diversas que coincidan con los distintos intereses del alumnado dentro de la ciencia.
• Visibilizar a mujeres en ciencias, tanto a nivel local como internacional. • Promover el aprendizaje más allá del aula de Ciencias. Los museos, las exposiciones, la naturaleza o distintos espacios urbanos ofrecen oportunidades para desarrollar habilidades científicas.
Preguntas para docentes y su reflexión para ser más conscientes de las interacciones que se implementan en el aula de Ciencias • ¿Las preguntas que planteo a las y los estudiantes de mi clase tienen el mismo grado de desafío cognitivo para niños y niñas o identifico algún sesgo de género en ellas? • ¿El lenguaje que utilizo en clases es neutro o inclusivo? • ¿Brindo espacio para que todas y todos los estudiantes tengan voz en mi clase? • ¿Utilizo ejemplos paritarios en mis clases en los que figuren referentes femeninos y masculinos del quehacer científico? • ¿Refuerzo positivamente a niñas y niños de manera equitativa en sus desempeños? Para profundizar más en este tema revisar el siguiente link: Genero_ciencias_2810.pdf Aportes para la enseñanza de las Ciencias Naturales a partir del estudio ERCE 2019 13 Referencias Agencia de Calidad de la Educación (2015). Buenas prácticas en la reducción de las brechas de género en resultados Simce de Comprensión de Lectura y Matemática II medio. Santiago: ACE. Disponible en Estudio_Brechas_de_Genero.pdf Alloway, N.; Freebody, P.; Gilbert, P., & Muspratt, S. (2002). «Boys, Literacy and Schooling: Expanding the Repertoires of Practice». Commonwealth Department of Education, Science and Training report. Disponible en school/cls/clearinghouse/ Anijovich, R., & Cappelletti, G. (2017). La evaluación como oportunidad. Buenos Aires: Paidós.
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6690 | https://www.sohu.com/a/413724166_120578689 | 【鸡兔同笼】有关鸡兔同笼问题经典解法及练习题50道
2020-08-18 17:35
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鸡兔同笼问题经典解法
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鸡兔同笼问题经典解法
鸡兔同笼问题经典解法第一种：列举法
鸡兔共有头10个，腿30个，求鸡和兔的数量？
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陶醉的香氛re
07月23日 07:32云南
我考试考到鸡兔同笼 直接懵了
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追寻的小鸽子P
06月10日 11:19
这是另一种解法。
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追寻的小鸽子P
06月10日 11:18
假设都是鸡，则每只兔手两只脚。
10×2＝20（只）
30-20=10（只）
鸡的数量：10÷2＝5（只）
兔的数量：10-5=5（只）
验算：5×4+2×5＝10（只）
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6691 | https://www.youtube.com/watch?v=glkkP0XxAhg | Gráficos de sistemas de inequações: problema - Khan Academy em português (9º ano)
Khan Academy em Português de Portugal
29100 subscribers
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Description
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Posted: 1 Jun 2018
Vídeo original: Graphs of systems of inequalities word problem
(
A Khan Academy Portugal disponibiliza explicações online de Matemática gratuitas desde o 1º até ao 12º ano de escolaridade.
Este vídeo foi produzido pela Khan Academy e traduzido para português pela Fundação Portugal Telecom (ver todos os vídeos disponíveis em
Transcript:
cada uma quer cortar o próprio destino horas numa competição de mora o mesmo número de segundos a cortar que a cenoura eo mesmo número dos humanos a cortar cada bloco o objectivo de cortar pelo menos 20 editais no tempo limite de quinhentos e quarenta segundos muito bem no referencial baixos estão representadas todas as combinações de cenouras sim e blocos b a encantar inequação ave apresenta o conjunto de todas as combinações que carla quer cortar e que ela quer cortar pelo menos 20 vegetais e skinny e custam a nos diz ela quer cortar pelo menos 20 editais então toda esta área pintada de azul incluindo a linha incluindo os pontos da linha apresentou cenários em que ela corta pelo menos 20 editais esta área azul incluindo a linha azul depois diz técnicos são de representou conjunto de combinações que ela pode cortar no limite de tempo portanto a inexecução bem isto representa todas as combinações cortadas dentro do limite de tempo em que não ultrapassa os 540 segundos qual é o número minutos nos olhos que carla pode cortar atingindo o objetivo lembrar é subjetiva e cortar pelo menos 20 vegetais portanto tem de estar na área azul tem de estar na áfrica apresenta em execução a ou seja a zona azul uma ea linha azul e também querer atingir o objetivo de cumprir um limite de tempo por tanque também tem de estar na área que representa é único são bi ou seja também tem de estar na área verde o não linha verde então a sobreposição dos duas em cumprir ambos do júri de itu será será tudo está área tudo esta área equipe este é sobreposição dos dois conjuntos das soluções então na área da sobreposição onde está ou não por mim senhoras qual é o número minutos nos que carla pode cortar atingindo o objetivo se olharmos por aqui o número mínimo de horas podemos ficar tentados a dizer que são 20 horas esse número está dentro da área sobreposta seriam 20 noroeste e 0 blocos mas conseguimos encontrar uma combinação com menos senhor ainda por ivo vieira ter este ponto por coimbra em suas pontos sobre a linha também fazem parte da área sobreposta por que são linhas contínuas e não três chamadas então este ponto aqui destino horas e 10 bróculos cumprir o objetivo dela deixe de escrever isso desde senor ans e 10 bróculos de cenouras idéias blocos esta é a quantidade mínima cortaz meninos do que destinou ans ensinar o alguma sobreposição por exemplo há alguma maneira de cortar e cenouras por exemplo sem agulhas aqui celebrarem aqui não há sobreposição dos dois conjuntos no ponto se iguala 7 então mínima é de facto o ponto de interseção destas duas linhas de cenouras idéias blocos essa combinação que permite cortar o número mínimo de cenouras atingindo o objetivo ambos os objetivos fazê lo dentro do limite de tempo e cortar pelo menos 20 editais |
6692 | https://www.cs.utexas.edu/~isil/cs311h/lecture1-6up.pdf | CS311H: Discrete Mathematics Intro and Propositional Logic Instructor: I¸ sıl Dillig Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 1/36 Course Staff ▶Instructor: Prof. I¸ sıl Dillig ▶TAs: Zeki Gurbuz, Letizia Fazzini, Sneha Ballabh, Noah Schell ▶Course webpage: ▶Contains syllabus, important information about HW policy, slides from lectures etc.
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 2/36 About this Course ▶Give mathematical background you need for computer science ▶Topics: Logic, proof techniques, number theory, combinatorics, graph theory, basic complexity theory . . .
▶These will come up again and again in higher-level CS courses ▶Master CS311H material if you want to do well in future courses!
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 3/36 Textbook ▶Textbook (optional): Discrete Mathematics and Its Applications by Kenneth Rosen ▶Textbook not a substitute for lectures: ▶Class presentation may not follow book ▶Skip many chapters and cover extra material Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 4/36 Ed Discussion ▶We will be using Ed Discussion for all course-related discussions ▶Make sure you can access Ed Discussion! (link available through Canvas + webpage) ▶Please post class-related questions on Ed Discussion instead of emailing instructor TA’s ▶You will get answers quicker, and it will benefit the whole class ▶Please use common sense when posting questions: Hints/ideas ok, but cannot post full solutions!!
▶If you have a more personal question, please send private message (also through Ed Discussion) Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 5/36 Discussion Sections and Office Hours ▶Discussion sections on Friday 12-1pm and Fri 1-2 pm ▶Please attend the section you were officially assigned to.
▶Discussion section will answer questions, solve new problems, and go over previous homework ▶Lots of office hours – times and location will be posted on Ed Discussion!
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 6/36 Requirements ▶Exams + problem sets + class attendance/participation ▶Three exams scheduled for Sep 26, Oct 31, Dec 5 (in person, closed-book + closed-notes) ▶Weekly problem sets Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 7/36 Grading ▶Exam: collectively 60% of final grade ▶Homework: 35% of final grade ▶Attendance/participation: 5% of final grade ▶Final grades will be curved Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 8/36 Homework Policy ▶Homework must be submitted by 1 pm on the due date ▶Late submissions not allowed, lowest homework score dropped when calculating grades ▶Homework must be done on your own, but allowed to ask conceptual (high-level) questions on Ed Discussion and during office hours ▶Not allowed to do problem sets in groups ▶Not allowed to check solutions with each other ▶Collaboration with other students on HW is considered cheating and will get you in very serious trouble Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 9/36 Homework Policy, cont.
▶You may not use AI Assistants like GPT in solving homework problems ▶You may also not do online search for solutions to similar homework problems ▶You may not discuss homework problems with each other through channels (e.g., Slack, Facebook, WhatsApp etc) that the course staffdoes not have access to.
▶Homework solutions must be typeset using Latex and submitted through Gradescope Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 10/36 Honor Code ▶Failing to adhere to the homework policy is a violation of the UT honor code ▶We take the honor code extremely seriously: people have failed the class in the past for violating the HW policy ▶In addition to failing the class, your case will be sent to the Dean of Students and placed on your file ▶Please don’t risk ruining your career for a slightly better grade on a problem set or exam...
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 11/36 More on Homework ▶Problem sets in this class will be much harder than what you are used to from high school!!
▶Normal to spend >30mins on a single HW question ▶Do not seek help from us unless you’ve spent at least one hour on each problem ▶Expect each problem set to take > 6 hours Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 12/36 Class Participation ▶Everyone expected to attend lectures and participate ▶5% of course grade for participation (attendance, asking/answering questions, being active on Ed Discussion) ▶Please ask questions!
▶Will make class more fun for everyone ▶Others also benefit from your questions Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 13/36 Let’s get started!
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 14/36 Logic ▶Logic: study of valid reasoning; fundamental to CS ▶Allows us to represent knowledge in a formal/mathematical way and automate some types of reasoning ▶Many applications in CS: AI, programming languages, databases, computer architecture, automated testing and program analysis, . . .
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 15/36 Propositional Logic ▶Simplest logic is propositional logic ▶Building blocks of propositional logic are propositions ▶A proposition is a statement that is either true or false ▶Examples: ▶” CS311 is a course in discrete mathematics” : True ▶” Austin is located in California” : False ▶” Pay attention” : Not a proposition ▶” x+1 =2” : Not a proposition Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 16/36 Propositional Variables, Truth Value ▶Truth value of a proposition identifies whether a proposition is true (written T) or false (written F) ▶What is truth value of ” Today is Friday” ?
▶Variables that represent propositions are called propositional variables ▶Denote propositional variables using lower-case letters, such as p, p1, p2, q, r, s, . . .
▶Truth value of a propositional variable is either T or F.
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 17/36 Compound Propositions ▶More complex propositions formed using logical connectives (also called boolean connectives) ▶Three basic logical connectives: 1. ∧: conjunction (read ” and” ), 2. ∨: disjunction (read ” or” ) 3. ¬: negation (read ” not” ) ▶Propositions formed using these logical connectives called compound propositions; otherwise atomic propositions ▶A propositional formula is either an atomic or compound proposition Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 18/36 Negation ▶Negation of a proposition p, written ¬p, represents the statement ” It is not the case that p” .
▶If p is T, ¬p is F and vice versa.
▶In simple English, what is ¬p if p stands for . . .
▶“Less than 80 students are enrolled in CS311” ?
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 19/36 Conjunction ▶Conjunction of two propositions p and q, written p ∧q , is the proposition ” p and q” ▶p ∧q is T if both p is true and q is true, and F otherwise.
▶What is the conjunction and the truth value of p ∧q for . . .
▶p =“It is Thursday” , q = ” It is morning”?
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 20/36 Disjunction ▶Disjunction of two propositions p and q, written p ∨q , is the proposition ” p or q” ▶p ∨q is T if either p is true or q is true, and F otherwise.
▶What is the disjunction and the truth value of p ∨q for . . .
▶p =“It is spring semester” , q = ” Today is Thursday” ?
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 21/36 Propositional Formulas and Truth Tables ▶Truth table for propositional formula F shows truth value of F for every possible value of its constituent atomic propositions ▶Example: Truth table for ¬p p ¬p T F F T ▶Example: Truth table for p ∨q p q p ∨q T T T T F T F T T F F F Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 22/36 Constructing Truth Tables Useful strategy for constructing truth tables for a formula F: 1. Identify F’s constituent atomic propositions 2. Identify F’s compound propositions in increasing order of complexity, including F itself 3. Construct a table enumerating all combinations of truth values for atomic propositions 4. Fill in values of compound propositions for each row Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 23/36 Examples Construct truth tables for the following formulas: 1. (p ∨q) ∧¬p 2. (p ∧q) ∨(¬p ∧¬q) 3. (p ∨q ∨¬r) ∧r Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 24/36 More Logical Connectives ▶∧, ∨, ¬ most common boolean connectives, but there are other boolean connectives as well ▶Other connectives: exclusive or ⊕, implication →, biconditional ↔ ▶Exclusive or: p ⊕q is true when exactly one of p and q is true, and false otherwise ▶Truth table: p q p ⊕q T T F T F T F T T F F F Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 25/36 Implication (Conditional) ▶An implication (or conditional) p →q is read ” if p then q”or ” p implies q” ▶It is false if p is true and q is false, and true otherwise ▶Exercise: Draw truth table for p →q ▶In an implication p →q, p is called antecedent and q is called consequent Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 26/36 Converting English into Logic Let p =”I major in CS”and q = ” I will find a good job” . How do we translate following English sentences into logical formulas?
▶” If I major in CS, then I will find a good job” : ▶” I will not find a good job unless I major in CS” : ▶” It is sufficient for me to major in CS to find a good job” : ▶” It is necessary for me to major in CS to find a good job” : Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 27/36 More English - Logic Conversions Let p =”I major in CS” , q = ” I will find a good job” , r = ” I can program” . How do we translate following English sentences into logical formulas?
▶”I will not find a good job unless I major in CS or I can program” : ▶”I will not find a good job unless I major in CS and I can program” : ▶” A prerequisite for finding a good job is that I can program” : ▶” If I major in CS, then I will be able to program and I can find a good job” : Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 28/36 Converse of a Implication ▶The converse of an implication p →q is q →p.
▶What is the converse of ” If I am a CS major, then I can program” ?
▶Note: It is possible for a implication to be true, but its converse to be false, e.g., F →T is true, but converse false Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 29/36 Inverse of an Implication ▶The inverse of an implication p →q is ¬p →¬q.
▶What is the inverse of ” If I get an A in CS311, then I am smart” ?
▶Note: It is possible for a implication to be true, but its inverse to be false. F →T is true, but inverse is false Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 30/36 Contrapositive of Implication ▶The contrapositive of an implication p →q is ¬q →¬p.
▶What is the contrapositive of ” If I am a CS major, then I can program” ?
▶Question: Is it possible for an implication to be true, but its contrapositive to be false?
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 31/36 Question ▶Given p →q, is it possible that its converse is true, but inverse is false?
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 32/36 Biconditionals ▶A biconditional p ↔q is the proposition ” p if and only if q” .
▶The biconditional p ↔q is true if p and q have same truth value, and false otherwise.
▶Exercise: Construct a truth table for p ↔q ▶Question: How can we express p ↔q using the other boolean connectives?
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 33/36 Operator Precedence ▶Given a formula p ∧q ∨r, do we parse this as (p ∧q) ∨r or p ∧(q ∨r)?
▶Without settling on a convention, formulas without explicit paranthesization are ambiguous.
▶To avoid ambiguity, we will specify precedence for logical connectives.
Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 34/36 Operator Precedence, cont.
▶Negation (¬) has higher precedence than all other connectives.
▶Question: Does ¬p ∧q mean (i) ¬(p ∧q) or (ii) (¬p) ∧q?
▶Conjunction (∧) has next highest predence.
▶Question: Does p ∧q ∨q mean (i) (p ∧q) ∨r or (ii) p ∧(q ∨r)?
▶Disjunction (∨) has third highest precedence.
▶Next highest is precedence is →, and lowest precedence is ↔ Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 35/36 Operator Precedence Example ▶Which is the correct interpretation of the formula p ∨q ∧r ↔q →¬r (A) ((p ∨(q ∧r)) ↔q) →(¬r) (B) ((p ∨q) ∧r) ↔q) →(¬r) (C) (p ∨(q ∧r)) ↔(q →(¬r)) (D) (p ∨((q ∧r) ↔q)) →(¬r) Instructor: I¸ sıl Dillig, CS311H: Discrete Mathematics Intro and Propositional Logic 36/36 |
6693 | https://www.wyzant.com/resources/answers/927877/over-the-interval-0-x-2pi-how-many-points-on-the-graph-of-the-function-y-co | Over the Interval 0 ≤x ≤2pi, how many points on the graph of the function y=cosx have a tangent line that has i)a zero slope ii) a minimum slope. Use diagrams to explain answer | Wyzant Ask An Expert
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Calculus
Mikayla D.
asked • 05/08/23
Over the Interval 0 ≤x ≤2pi, how many points on the graph of the function y=cosx have a tangent line that has i)a zero slope ii) a minimum slope. Use diagrams to explain answer
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NORMAN E.answered • 05/11/23
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The derivative of cos x is the negative of sin x. So cos x has zero slope whn sin x = 0 and cos x has a minium slope when sin x = +1. The interval 0 < x < 2 pi includes the end points. So there are three points at zero slope (0, pi, and 2 pi) and one minimum slope at 3 pi / 2.
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Raymond B.answered • 05/08/23
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y'=-sinx=0
y=cosx
x = sin^-1(0) = 0, pi, 2pi
three points on y=cosx, between x=0 and x=2pi,
have tangent lines with slopes = 0
(0,1), (pi,-1), (2pi,1)
minimum slope is when slope = -sinx =-1, sinx=1, x=sin^-1(1) = pi/2
the point is (pi/2, 0)
one "diagram" is the graph of cosx, use a graphing calculator
zero slope is when cosx is at a maximum or minimum
minimum slope is when the cosx curve is going down, the inflection point where it switches from concave down to concave up
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6694 | https://en.wikipedia.org/wiki/Talk%3ACircumcircle | Jump to content
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Contents
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1 Incomplete information 1 comment
2 Discussion: Merge with circumcircle
3 Triangles: Circumscribed circle is a circumcircle 8 comments
4 Incorrect Formula? 5 comments
5 Frustrating Merges 1 comment
6 Formula for calculating the length A 2 comments
7 Polar Coordinates 1 comment
8 Coordinates of circumcenter 1 comment
9 More Detail Please 2 comments
10 Circumcircle equations 4 comments
11 Wrong generalization to d-dimensional points 3 comments
12 splitting? 2 comments
13 Proof for circumcenter vector formula 1 comment
13.1 Step1.
13.2 Step2.
13.3 Step3.
14 External links modified 1 comment
15 Removed proof that every triangle is cyclic 2 comments
16 Horrible examples 2 comments
17 Alternative Construction diagram incorrect 1 comment
18 Article should be split 10 comments
Talk:Circumcircle
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Incomplete information
[edit]
The session "Cyclic n-gons" seems to be incomplete. The last sentence is cut off in the middle after the word "alte", which doesn't make any sense at all. Please help correct it. Thanks. User:Yuen.teddy —Preceding undated comment added 13:13, 5 August 2011 (UTC).[reply]
Discussion: Merge with circumcircle
[edit]
I assert that circumscribed circle and circumcircle not be merged. They are fundamentally different concepts with different definitions, and to merge them would show a profound lack of geometric sophistication. For example, if you draw a five-point star, this star has no circumscribed circle (since the star has 10 vertices), but it does indeed have a circumcircle. The circumcircle does not have to touch every vertex; instead, it merely has to be the unique circle with the smallest radius such that the entirety of the star's interior is bounded within it. User:KyleGoetz
Triangles: Circumscribed circle is a circumcircle
[edit]
A circumscribed circle for any triangle is the same circle as the circumcenter. Whoever wrote otherwise is absolutely wrong: User:KyleGoetz Just to clarify this with my comment above concerning the differences between a circumcircle and a circumscribed circle: for a triangle, they are the same; however, for other polygons this may not be true. User:KyleGoetz
I think it is very wrong to have a definition for triangles that is incompatible with the definition for polygons in general, but I also don't think the circumscribed circle and circumcircle are ever different. At best note the "smallest enclosing circle" version in passing as a variant usage rather than making it the main definition of the word. As source I cite this paper — it's my own work, so I'm not going to edit into the actual article, but it's a well respected survey from a number of years ago and clearly states (top of page 9, and theorem 1 page 13) that circumcircle = circumscribing circle for triangles, and that the smallest enclosing circle is something different. The only single word synonym I've seen in professional usage for smallest enclosing circle is minidisk or minidisc (or miniball in higher dimensions). —David Eppstein 04:49, 15 October 2006 (UTC)[reply]
: Yeah, there is no difference between circumscribed circle and a circumcircle. Any polygon has a circumcircle, and when polygon's vertices happen to be all on that circumcircle, it is called a circumscribed circle. Oleg Alexandrov (talk) 05:05, 15 October 2006 (UTC)[reply]
: : Er. There are polygons which have all vertices on a circle, but for which that circle is not the smallest enclosing circle. Polygons other than triangles, even. So I'm not sure which circle your comment is referring to. To me, "circumscribed circle" and "circumcircle" can only mean a circle passing through all vertices, even if that circle is not the smallest one enclosing the polygon. Anything else needs a different name. —David Eppstein 05:17, 15 October 2006 (UTC)[reply]
: OK, then my changes to the article were wrong. I still think that my merger of circumcircle into circumscribed circle was a good thing. I will try to fix my errors tomorrow. Oleg Alexandrov (talk) 05:34, 15 October 2006 (UTC)[reply]
: Thanks for fixing my mistake! Oleg Alexandrov (talk) 16:03, 15 October 2006 (UTC)[reply]
: David, I agree with you, and I did not mean to say anything different than what you did. It's now been so long since I edited this article, that I cannot remember exactly what brought about my mentioning of triangles specifically in the talk page, but I think it's because the article itself explicitly said that circumcenter and circumscribed circle were the same thing, and used a triangle as proof of the statement. I merely pointed out this was an erroneous proof that the two concepts are the same. KyleGoetz 23:25, 17 December 2006 (UTC)[reply]
: Is the property that all triangles can be circumscribed by a circle a defining property for triangles? What happens to a degenerate (colinear) triangle if that is the case? — Preceding unsigned comment added by Knservis (talk • contribs) 08:48, 2 May 2014 (UTC)[reply]
: In that case the circumscribing circle degenerates to the line through the three points. —David Eppstein (talk) 15:19, 2 May 2014 (UTC)[reply]
Incorrect Formula?
[edit]
is the formula under: "Coordinates of circumcenter" right, i'm not sure philb 14:53, 24 March 2007 (UTC)[reply] The formula is correct. Note that these barycentric coordinates are not "normalized" (as in areal coordinates, where λ1+λ2+λ3=1). If you want the barycentric coordinates that are most commonly used, divide the entire thing by the sum of the components 128.237.234.118 (talk) 19:37, 13 April 2008 (UTC)[reply] I don't think this is right. If you normalize the barycentric coordinates, the center will always be located inside the triangle, which is not correct. Also, barycentric coordinates of the circumcenter should be invariant with respect to scaling, which these are not. 131.246.191.182 (talk) 13:01, 22 January 2009 (UTC)[reply]
: What makes you think the given center will always be located inside the triangle? You have
: and that is negative in case the triangle has an obtuse angle between the sides of lengths a and b. A negative barycentric coordinate entails a point outside of the triangle, just as one would expect for an obtuse triangle. Michael Hardy (talk) 22:34, 22 January 2009 (UTC)[reply]
: : ...and now I've checked this and the given coordinates are correct. Michael Hardy (talk) 18:35, 31 January 2009 (UTC)[reply]
Frustrating Merges
[edit]
I agree with KyleGoetz. I spent time separating circumsphere and circumscribed sphere from eachother just to have someone come along and merge them. One must "touch 3 points and enclose the polygon" and the other "touch all the vertices of the polygon". As a software engineer who has worked on rigid body dynamics, there is a need for the former (circumscribed sphere) to define the collision area of ANY arbitrary polyhedron. Circumspheres (and circles) cannot be applied to any arbitrary polyhedra (or polygons) make the need for distincition clear. Coder0xff 19:24, 29 April 2007 (UTC)
: The article as it stands correctly distinguishes between two concepts, (1) touching all vertices, and (2) enclosing and smallest. As I understand it you are trying to introduce a distinction between these two and a third concept, (3) touching three vertices and enclosing. Circles satisfying this third definition always exist but are uniquely defined only when the convex hull of the polygon is cyclic. It's a reasonable definition, though I think not as useful as the other two. But can you provide reliable sources in the mathematical literature clearly stating that definition (3) is associated with one of the phrases "circumcircle" or "circumscribed circle" and that definition (1) is associated with the other phrase? Because I don't think I've ever seen such a definition in the papers I've read. This is not the place for making up distinctions that do not represent what is already in the literature. —David Eppstein 19:33, 29 April 2007 (UTC)[reply]
Formula for calculating the length A
[edit]
If I start out with the radius of a circle first, and inscribe an equilateral triangle inside the circle, how would I go about calculating the length A of any of the triangle's sides? If I had a notepad here, why, I guess I'd go ahead and figure it out myself, but my desk really doesn't have any room for that kind of stuff.
In any case, the main article contains a formula that lets you calculate the circumference of the circumscribed circle, if you start out with any of the sides of an equilateral triangle, but the article could be improved by including a way of figuring out the length of any of the triangle's sides, if you start out with a circle first. It makes a small difference which you start out with first, but one formula might be more easily understood than the other. 216.99.219.73 (talk) 01:07, 19 June 2009 (UTC)[reply]
: : If you start out with the length of the Radius of the circumscribing circle, R, you can calculate the length of any given side of the Triangle, A, by using the following formula:
: : A = 3R / square root of 3 Hope that helps. 216.99.198.254 (talk) 05:26, 21 June 2009 (UTC)[reply]
Polar Coordinates
[edit]
The main article uses Cartesian coordinates when it describes a circumcircle around an equilateral triangle. Are there any elegant equations when using Polar coordinates instead? Are the formulas any less elegant when the triangle is no longer equilateral? 216.99.219.73 (talk) 01:07, 19 June 2009 (UTC)[reply]
Coordinates of circumcenter
[edit]
Formula of circumcenter O for triangle AB'C' is given. Should it be mentioned that for triangle ABC circumcenter would be O + A ? Lost couple of hours figuring this out. Yatagarasu42 (talk) 12:48, 18 May 2010 (UTC)[reply]
More Detail Please
[edit]
The leap from the system of four equations for the vertices and an arbitrary point to the determinate form is for me too far to take without explanation. I don't recall enough linear algebra to follow the reductions. Can someone show the steps from the four equations for the lengths of the differences of the vectors being equal to the length of the radius squared, through the polarization identity transformation, to the matrix form, and then to the determinate expression?
It is the derivation of this matrix which, for the moment, is beyond me:
Lonniev (talk) 14:33, 3 June 2011 (UTC)[reply]
I support this. Which one of polarization identities is used? How is kernel related here? — Preceding unsigned comment added by 46.146.133.212 (talk • contribs) 7 July 2013
Lets see have a go
expanding the first gives
We have similar for the other equations. We can then right the four equations as a single matrix equation
So multiples of are the kernel vectors.--Salix (talk): 13:43, 7 July 2013 (UTC)[reply]
Circumcircle equations
[edit]
This content is clearly "owned" by David Eppstein because he quickly reverts any attempt to clarify the content. Admittedly, the new content was poor mathml but it was a work in progress on existing content which was itself less than perfect. So, I ask David to belabor the content in order to get from the four equations of the squared vector differences to the determinant. David, if you prefer not to dirty the article with the details, you can explain the steps here on the discussion page. My goal is not to get my content into the article but to understand the steps in this manner of determining the coordinates of the center and the radius of the circumscribed circle. — Preceding unsigned comment added by 97.122.243.92 (talk) 03:41, 25 July 2011 (UTC)[reply]
: Less than perfect? You had an equation with a matrix on the left hand side and a vector on the right hand side, and another equation with a matrix on the left hand side and a scalar on the right hand side. It made no sense mathematically. And I don't know what you're talking about re ownership: I've made a single edit to the article in the last two months (to undo your bad math), over which time seven other editors have been working on it. —David Eppstein (talk) 04:01, 25 July 2011 (UTC)[reply]
: David, your "single edit" was to revert away two paragraphs while I was in the middle of editing it. Yes, the [4x4] = [1x4] was "bad math" but I about to add the column vector. I've asked for clarity a month ago on the unstated steps taken to get from the equation set to the shown determinant. No one responded. At least by attempting to modify the presentation article, someone who cares enough about the article to try to protect it has responded. Again, it is not my intent to force my content into the article, I just want to know the steps taken and I want the article to at least reference the not-shown steps if not explicitly state them. Your contribution to that goal is greatly appreciated. — Preceding unsigned comment added by 65.102.235.174 (talk")) 13:44, 25 July 2011 (UTC)[reply]
: The preview button is your friend. It shouldn't be necessary to leave the article in a bad state in the middle of a sequence of edits. —David Eppstein (talk) 16:10, 25 July 2011 (UTC)[reply]
Wrong generalization to d-dimensional points
[edit]
The original article says:
Additionally, the circumcircle of a triangle embedded in d dimensions can be found using a generalized method. Let A, B, and C be d-dimensional points, which form the vertices of a triangle. We start by transposing the system to place C at the origin:
: ,
: .
The circumradius, r, is then
: ,
where θ is the interior angle between a and b. The circumcenter, p0, is given by
: .
For both the computation of and the cross product is used (this can be clearly seen in the simplification step from ). However, the cross product is not defined for any number of dimensions: it is in fact only defined for 3 and later for 7 dimensional spaces. This means that the above formula may be correct, but not for the general d-dimensional case. Probably who wrote this had the 3-dimensional formula in mind and thought that it could be extended to any number of dimensions.
Please correct me if I'm wrong, I'd be glad to learn about it. —Preceding unsigned comment added by Shirokuroneko (talk • contribs) 15:02, 24 October 2010 (UTC)[reply]
: Note how We start by transposing the system to place C at the origin: this plane is two dimensional so we can choose some 3D subspace including the plane and calculate the cross products in that space. It does not matter which 3D subspace we choose as long as it includes the plane. It would probably be better to reformulate to remove the use of cross product which will fall out. --Salix (talk): 17:32, 24 October 2010 (UTC)[reply]
: there is a problem with theese equation. because for the points [1,0,0][0,1,0][0,0,1] it gives wrong values of r and p0. For points [1,0,0] [0,1,0] [-1,0,0] result is correct 91.201.19.107 (talk")) 17:24, 25 April 2023 (UTC)[reply]
splitting?
[edit]
I was wondering should this article not be split in two seperate articles? one about Circumscribed circle (triangle) for all about the circumscribed circle of a triangle, and one about the general cirumscribed circle of polygons. Much of the article is only about triangles and other polygons are woefully under represented. WillemienH (talk) 09:56, 21 June 2015 (UTC)[reply]
: I disagree with the proposal to split. Most of the content is about triangles' circumcircles because that's what we know most about. It would look strange to have an article called "Circumscribed circle" that didn't say much about the triangle case. And anyway, there are undoubtedly a substantial number of articles about triangles that link specifically to this one -- all those links would be wrong if the triangle material were to be split off. Loraof (talk) 14:11, 13 July 2015 (UTC)[reply]
Proof for circumcenter vector formula
[edit]
The formula,
stated in the "Article" page, can be proved as following.
Step1.
[edit]
Let U be the circumcenter of ΔABC. If you draw lines from U to each of three vertices, three small triangles appear. As U is the circumcenter, lengths of now drawn lines should be equal. Let it be r. Each of three small triangles is isosceles, which also means related angles are equal. As the sum of inner angles of an arbitrary triangle is 180°,
: ∠BCU = (180°-∠CUB)/2
: ∠UCA = (180°-∠AUC)/2
: γ = ∠BCA (definition)
: = ∠BCU+∠UCA
: = (360°-∠CUB-∠AUC)/2
: = ∠BUA/2
: → ∠BUA = 2γ.
Adding a perpendicular line from U to each line, each of isosceles triangles is divided into two mutually congruent (though reflective) rectangular triangles. Thus angles are assigned as Figure 1.
Step2.
[edit]
Using r (distance between U and each vertex), and a, b, c (lengths of BC, CA, AB), areas of isosceles triangles SA, SB, SC are,
: SA = (a r cosα)/2
: SB = (b r cosβ)/2
: SC = (c r cosγ)/2
Configuration here is expressed in Figure 2. Area of original triangle is,
: S = SA + SB + SC = r(a cosα + b cosβ + c cosγ)/2.
Step3.
[edit]
Let's regard AB as a base line. As this is common to both the original triangle and ΔABU, proportion in their area is equal to proportion in their height. Adding a line that is parallel to AB and that passes U, inclined line CA (whose length is b) is divided into the same ratio as their height. So, over Figure 3,
: m/b = SC/S = c cosγ/(a cosα + b cosβ + c cosγ).
In the same way, regarding CA as a base line,
: n/c = SB/S = b cosβ/(a cosα + b cosβ + c cosγ).
Now using the parallelogram appeared at the below left corner, vector to U is got as the sum of three vectors,
: U = A + (C-A)m/b + (B-A)n/c
: = A + (C-A)c cosγ/(a cosα + b cosβ + c cosγ) + (B-A)b cosβ /(a cosα + b cosβ + c cosγ)
: = (A a cosα + B b cosβ + C c cosγ)/(a cosα + b cosβ + c cosγ)
Using the Law of cosines,
: U = { A a (b2 + c2 - a2)/2bc + B b (c2 + a2 - b2)/2ca + C c (a2 + b2 - c2)/2ab }/
: { a (b2 + c2 - a2)/2bc + b (c2 + a2 - b2)/2ca + c (a2 + b2 - c2)/2ab) }
: = { A a2 (b2 + c2 - a2) + B b2 (c2 + a2 - b2) + C c2 (a2 + b2 - c2) }/{ a2 (b2 + c2 - a2) + b2 (c2 + a2 - b2) + c2 (a2 + b2 - c2) }
Tsukitakemochi (talk) 14:09, 27 July 2015 (UTC) Tsukitakemochi (talk) 14:11, 27 July 2015 (UTC)[reply]
External links modified
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Removed proof that every triangle is cyclic
[edit]
The "Triangles" section previously contained this claimed proof of the fact that every triangle is cyclic:
: This can be proven on the grounds that the general equation for a circle with center (a, b) and radius r in the Cartesian coordinate system is
: Since this equation has three parameters (a, b, r) only three points' coordinate pairs are required to determine the equation of a circle. Since a triangle is defined by its three vertices, and exactly three points are required to determine a circle, every triangle can be circumscribed.
This proof is clearly flawed, because three collinear points do not determine a circle but the proof does not require that the points be noncollinear.
In general, there is no reason that n points must determine a unique member of a family of curves described by an equation with n parameters.
I've removed this flawed proof from the article. —Bkell (talk) 01:12, 14 October 2017 (UTC)[reply]
: I agree. The two-bisector construction provides a better proof. —David Eppstein (talk) 19:06, 16 October 2017 (UTC)[reply]
Horrible examples
[edit]
There are too few examples in the article where some points are inside the circle rather than on it. Especially examples are bad where there are many points and none of them inside the circle. Such examples make readers think that you can always draw a circle that goes through all points. — Preceding unsigned comment added by 2003:EC:9731:FD00:E519:7499:4D69:1592 (talk) 09:18, 24 May 2021 (UTC)[reply]
: You did see the part in the first sentence that the circumscribed circle "passes through all the vertices", right? If there are vertices inside, it is not a circumscribed circle and not an example. Maybe you are looking instead for smallest-circle problem? —David Eppstein (talk) 15:40, 24 May 2021 (UTC)[reply]
Alternative Construction diagram incorrect
[edit]
I believe the alt construction diagram has the incorrect angles labeled 90-α. The construction should proceed by drawing the 90-α angle from the side adjacent to the angle α. That way it's projection will intersect the opposite side in a right angle and so run through the circumcenter. — Preceding unsigned comment added by Rpnman (talk • contribs) 18:51, 7 December 2021 (UTC)[reply]
Article should be split
[edit]
This article's scope is too broad, and it ends up doing a poor job covering any of its included topics because the resulting organization is awkward. The circumcircle of a triangle, an essential topic linked to from all over is not given appropriate basic description in the lead section, even though it is what most readers are going to be looking for. The content under § Triangles dives right into a disorganized mishmash of specific details and does not do an adequate job of introducing or setting up the concept in a way that is legible and easy to follow for non-specialists.
In my opinion this article should be moved to circumcircle and should limit itself to describing the circumcircle of a triangle (and the perpendicular bisectors, circumcenter, etc.); everything in § Triangles can be moved to top level, which leaves more room for expansion, especially common proofs of some of the basics, a historical discussion, and so on. A section can be added about the circumcircle of triangles on the sphere and hyperbolic plane, a section can be added about the circumcircle in a pseudo-Euclidean plane of signature (1, 1), and so on.
The circumcircle of a general cyclic polygon should be discussed at a separate page cyclic polygon (we already have cyclic quadrilateral, which is an appropriate scope and title for an article).
The minimum bounding circle is already split to smallest-circle problem, again a fine scope for an article.
Thoughts? –jacobolus (t) 18:54, 15 December 2022 (UTC)[reply]
: I agree that splitting off the content about triangles from the content about cyclic polygons makes sense. I also agree that circumcircle and cyclic polygon are good names for the split-off articles. Maybe rather than redirecting to any one of these, the current circumscribed circle title could become a disambiguation or set-index article pointing to both of those topics + smallest-circle problem? —David Eppstein (talk) 19:31, 15 December 2022 (UTC)[reply]
: Having circumscribed circle as a disambiguation page sounds good to me. But if that's the result we decide on it should probably be accomplished by first moving this page to circumcircle, next copying relevant material over the redirect cyclic polygon (which was its own topic in 2003), and then finally creating a new article at circumscribed circle, to keep the edit history mostly intact. (Edit: or maybe the pages circumcircle and circumscribed circle could instead be swapped; both contain some relevant edit history.) –jacobolus (t) 19:53, 15 December 2022 (UTC)[reply]
: I support this plan, although I don't think the page history of circumcircle is particularly valuable. Apocheir (talk) 22:57, 15 December 2022 (UTC)[reply]
: Can someone with the appropriate technical privileges do this move? Splitting the text around shouldn’t be too tricky, but I think it takes some administrative privilege to move circumscribed circle → circumcircle when there is already a redirect there with some past history. –jacobolus (t) 18:32, 14 February 2023 (UTC)[reply]
: Moves that need admin assistance typically go to Wikipedia:Requested moves. Apocheir (talk) 23:27, 14 February 2023 (UTC)[reply]
: While we are at it, perhaps we can move Incircle and excircles of a triangle -> Incircle and excircles for concision. –jacobolus (t) 05:51, 18 December 2022 (UTC)[reply]
: I also did this one. –jacobolus (t) 17:45, 29 June 2023 (UTC)[reply]
: Okay, @UtherSRG helpfully did the swap for us. I merged the material about cyclic polygons in general into concyclic points, as it seems to have substantially overlapping scope. Ping @David Eppstein, @Apocheir. Still working on making an index page out of circumscribed circle. –jacobolus (t) 17:41, 29 June 2023 (UTC)[reply]
: David, can you help me properly format Circumscribed circle as an index/disambiguation page? –jacobolus (t) 18:15, 29 June 2023 (UTC)[reply]
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6695 | https://www.youtube.com/watch?v=D9QWlf4KzSs | Compensation Strategy for Multiplication 1
Susan Burke
759 subscribers
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Description
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Posted: 29 May 2014
A great demonstration of how to use the compensation strategy for multiplication.
Transcript:
we can use the compensation strategy to help us solve this multiplication problem 49 2 49 is close to the friendly number 50 so if we add one to 49 we'll get 50 now that we have the friendly number we can multiply it by two 50 2 = 100 now to compensate for the change that we made here we take that one that we added and multiply it by two and we get the answer two to find the answer to 49 2 we simply subtract this from this we need to subtract the two because we added the one in the first place so 100 take away 2 equals 98 our answer to 49 2 is 98 |
6696 | https://www.instagram.com/p/DCZfCNKOMnw/ | Instagram
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6697 | https://www.ck12.org/flexi/algebra/linear-programming/explain-non-negative-restrictions-in-linear-programming./ | Flexi answers - Explain Non-negative Restrictions in linear programming. | CK-12 Foundation
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Explain Non-negative Restrictions in linear programming.
Flexi Says:
In linear programming, non-negative restrictions are a set of constraints that ensure the decision variables do not take negative values. This is because in many real-world problems, negative values for decision variables do not make sense.
For example, you cannot produce a negative number of products, or assign a negative number of workers to a task.
Mathematically, these restrictions are represented as:
x i≥0∀i
Where x i represents the decision variables. The symbol ≥denotes 'greater than or equal to', and ∀i means 'for all i'. So, the above expression states that all decision variables must be greater than or equal to zero.
These non-negative restrictions are a fundamental part of the formulation of a linear programming problem, along with the objective function and other constraints.
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6698 | https://www.ck12.org/section/exterior-angles-%3A%3Aof%3A%3A-quadrilaterals-%3A%3Aof%3A%3A-ck-12-geometry/ | Exterior Angles | CK-12 Foundation
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HomeMathematicsExterior Angles
Exterior Angles
Difficulty Level: Basic | Created by: CK-12
Last Modified: Aug 22, 2014
Read Resources Details Attributions
Learning Objectives
Identify the exterior angles of convex polygons.
Find the sums of exterior angles in convex polygons.
Introduction
This lesson focuses on the exterior angles in a polygon. There is a surprising feature of the sum of the exterior angles in a polygon that will help you solve problems about regular polygons.
Exterior Angles in Convex Polygons
Recall that interior means inside and that exterior means outside. So, an exterior angle is an angle on the outside of a polygon. An exterior angle is formed by extending a side of the polygon.
[Figure 1]
As you can tell, there are two possible exterior angles for any given vertex on a polygon. In the figure above we only showed one set of exterior angles; the other set would be formed by extending each side in the opposite (clockwise) direction. However, it doesn’t matter which exterior angles you use because on each vertex their measurement will be the same. Let’s look closely at one vertex, and draw both of the exterior angles that are possible.
[Figure 2]
As you can see, the two exterior angles at the same vertex are vertical angles. Since vertical angles are congruent, the two exterior angles possible around a single vertex are congruent.
Additionally, because the exterior angle will be a linear pair with its adjacent interior angle, it will always be supplementary to that interior angle. As a reminder, supplementary angles have a sum of 180∘.
Example 1
What is the measure of the exterior angle∠O K L in the diagram below?
[Figure 3]
The interior angle is labeled as 45∘. Since you need to find the exterior angle, notice that the interior angle and the exterior angle form a linear pair. Therefore the two angles are supplementary—they sum to 180∘. So, to find the measure of the exterior angle, subtract 45∘ from 180∘.
180−45=135
The measure of ∠O K L is 135∘.
Summing Exterior Angles in Convex Polygons
By now you might expect that if you add up various angles in polygons, there will be some sort of pattern or rule. For example, you know that the sum of the interior angles of a triangle will always be 180∘. From that fact, you have learned that you can find the sums of the interior angles of any polygons with n sides using the expression 180(n−2). There is also a rule for exterior angles in a polygon. Let’s begin by looking at a triangle.
[Figure 4]
To find the exterior angles at each vertex, extend the segments and find angles supplementary to the interior angles.
[Figure 5]
The sum of these three exterior angles is:
150∘+120∘+90∘=360∘
So, the exterior angles in this triangle will sum to 360∘.
To compare, examine the exterior angles of a rectangle.
[Figure 6]
In a rectangle, each interior angle measures 90∘. Since exterior angles are supplementary to interior angles, all exterior angles in a rectangle will also measure 90∘.
[Figure 7]
Find the sum of the four exterior angles in a rectangle.
90∘+90∘+90∘+90∘=360∘
So, the sum of the exterior angles in a rectangle is also 360∘.
In fact, the sum of the exterior angles in any convex polygon will always be 360∘. It doesn’t matter how many sides the polygon has, the sum will always be 360∘.
We can prove this using algebra as well as the facts that at any vertex the sum of the interior and one of the exterior angles is always 180∘, and the sum of all interior angles in a polygon is 180(n−2).
Exterior Angle Sum: The sum of the exterior angles of any convex polygon is 360∘
Proof. At any vertex of a polygon the exterior angle and the interior angle sum to 180∘. So summing all of the exterior angles and interior angles gives a total of 180 degrees times the number of vertices:
(Sum of Exterior Angles)+(Sum of Interior Angles)=180∘n.
On the other hand, we already saw that the sum of the interior angles was:
(Sum of Interior Angles)=180(n−2)=180∘n−360∘.
Putting these together we have
180 n 360=(Sum of Exterior Angles)+(Sum of Interior Angles)=(180 n−360)+(Sum of Exterior Angles)=(Sum of Exterior Angles)
Example 2
What is m∠Z X Q in the diagram below?
[Figure 8]
∠Z X Q in the diagram is marked as an exterior angle. So, we need to find the measure of one exterior angle on a polygon given the measures of all of the others. We know that the sum of the exterior angles on a polygon must be equal to 360∘, regardless of how many sides the shape has. So, we can set up an equation where we set all of the exterior angles shown (including m∠Z X Q) summed and equal to 180∘. Using subtraction, we can find the value of X.
70∘+60∘+65∘+40∘+m∠Z X Q 235∘+m∠Z X Q m∠Z X Q m∠Z X Q=360∘=360∘=360∘−235∘=125∘
The measure of the missing exterior angle is 125∘.
We can verify that our answer is reasonable by inspecting the diagram and checking whether the angle in question is acute, right, or obtuse. Since the angle should be obtuse, 125∘ is a reasonable answer (assuming the diagram is accurate).
Lesson Summary
In this lesson, we explored exterior angles in polygons. Specifically, we have learned:
How to identify the exterior angles of convex polygons.
How to find the sums of exterior angles in convex polygons.
We have also shown one example of how knowing the sum of the exterior angles can help you find the measure of particular exterior angles.
Review Questions
For exercises 1-3, find the measure of each of the labeled angles in the diagram.
x=–––––,y=––––– [Figure 9]
w=–––––,x=–––––,y=–––––,z=––––– [Figure 10]
a=–––––,b=––––– [Figure 11]
Draw an equilateral triangle with one set of exterior angles highlighted. What is the measure of each exterior angle? What is the sum of the measures of the three exterior angles in an equilateral triangle?
Recall that a regular polygon is a polygon with congruent sides and congruent angles. What is the measure of each interior angle in a regular octagon?
How can you use your answer to 5 to find the measure of each exterior angle in a regular octagon? Draw a sketch to justify your answer.
Use your answer to 6 to find the sum of the measures of the exterior angles of an octagon.
Complete the following table assuming each polygon is a regular polygon. Note: This is similar to a previous exercise with more columns—you can use your answer to that question to help you with this one.
| Regular Polygon name | Number of sides | Sum of measures of interior angles | Measure of each interior angle | Measure of each exterior angle | Sum of measures of exterior angles |
--- --- --- |
| triangle | | | | | |
| | 4 | | | | |
| | 5 | | | | |
| | 6 | | | | |
| | 7 | | | | |
| octagon | | | | | |
| decagon | | | | | |
| | | 1,800∘ | | | |
| | n | | | | |
Each exterior angle forms a linear pair with its adjacent internal angle. In a regular polygon, you can use two different formulas to find the measure of each exterior angle. One way is to compute 180∘−(measure of each interior angle)… in symbols 180−180(n−2)n.
Alternatively, you can use the fact that all n exterior angles in an n−gon sum to 360∘ and find the measure of each exterior angle with by dividing the sum by n. Again, in symbols this is 360 n
Use algebra to show these two expressions are equivalent.
Review Answers
x=52∘, y=128∘
w=70∘,x=70∘,y=110∘,z=90∘
a=107.5∘,b=72.5∘
Below is a sample sketch. [Figure 12] Each exterior angle measures 120∘, the sum of the three exterior angles is 360∘
Sum of the angles is 180(8−2)=1080∘. So, each angle measures 1080 8=135∘
Since each exterior angle forms a linear pair with its adjacent interior angle, we can find the measure of each exterior angle with 180−135=45∘
45(8)=360∘
| Regular Polygon name | Number of sides | Sum of measures of interior angles | Measure of each interior angle | Measure of each exterior angle | Sum of measures of exterior angles |
--- --- --- |
| triangle | 3 | 180∘ | 60∘ | 120∘ | 360∘ |
| square | 4 | 360∘ | 90∘ | 90∘ | 360∘ |
| pentagon | 5 | 540∘ | 72∘ | 108∘ | 360∘ |
| hexagon | 6 | 720∘ | 60∘ | 120∘ | 360∘ |
| heptagon | 7 | 900∘ | 128.57∘ | 51.43∘ | 360∘ |
| octagon | 8 | 1,080∘ | 135∘ | 45∘ | 360∘ |
| decagon | 10 | 1,440∘ | 144∘ | 36∘ | 360∘ |
| dodecagon | 12 | 1,800∘ | 150∘ | 30∘ | 360∘ |
| n−gon | n | 180(n−2)∘ | 180(n−2)n∘ | 360 n∘ | 360∘ |
One possible answer. 180−180(n−2)n=180 n n−180(n−2)n=180 n−180(n−2)n=180 n−180 n+360 n=360 n
Notes/Highlights
| Color | Highlighted Text | Notes | |
--- --- |
| | Please Sign In to create your own Highlights / Notes |
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Description
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Difficulty Level
Basic
Tags
slope,Quadrilaterals,distance formula,biconditional statements,medians,rhombi,Base Angles Theorem,exterior angles,interior angles,counterexample,consecutive angles,Venn diagram,isosceles trapezoid,trapezoid,kite,square,rectangle,rhombus,parallelogram,convex polygons,CK.MAT.ENG.SE.1.Geometry.6, (18 more)
Subjects
mathematics
Grades
9,10,11
Standards Correlations
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Concept Nodes
License
CC BY NC
Language
English
| Cover Image | Attributions |
--- |
| | License:CC BY-NC |
Date Created
Feb 23, 2012
Last Modified
Aug 22, 2014
.
| Image | Reference | Attributions |
---
| | [Figure 1] | License:CC BY-NC |
| | [Figure 2] | License:CC BY-NC |
| | [Figure 3] | License:CC BY-NC |
| | [Figure 4] | License:CC BY-NC |
| | [Figure 5] | License:CC BY-NC |
| | [Figure 6] | License:CC BY-NC |
| | [Figure 7] | License:CC BY-NC |
| | [Figure 8] | License:CC BY-NC |
| | [Figure 9] | License:CC BY-NC |
| | [Figure 10] | License:CC BY-NC |
| | [Figure 11] | License:CC BY-NC |
| | [Figure 12] | License:CC BY-NC |
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6699 | https://physics.stackexchange.com/questions/307014/need-help-understanding-why-a-negative-work-done-is-caused-by-an-external-force | energy conservation - Need help understanding why a negative work done is caused by an external force in electrostatics - Physics Stack Exchange
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Need help understanding why a negative work done is caused by an external force in electrostatics
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I was working on a problem that included a electric dipole. I found the change in Potential energy that it took to move a proton from A to B in my diagram, however this was a negative value and I've been told that this work is due to an external force. Can someone please explain to my why its due to the "External Force"?
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asked Jan 23, 2017 at 11:07
Teyash ArjunTeyash Arjun
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1 Something has to "push" the proton, it's can't "roll uphill" on its own? Movement towards the + charge is "uphill" because the proton has a + charge. Is that the confusion?JMLCarter –JMLCarter 2017-01-23 11:23:28 +00:00 Commented Jan 23, 2017 at 11:23
That helped a bit but I'm wondering why negative work which is what I calculated tells us that the force is due to an external force.Teyash Arjun –Teyash Arjun 2017-01-23 12:07:51 +00:00 Commented Jan 23, 2017 at 12:07
The sign you get out depends on the signs of the inputs. Which is to say in some calculation there could be a positive value of work and it could still mean an external force was required. Examine your definitions and see if you can determine that energy must be input to the proton. Are you integrating from a force equation from original to final position - what is the force, that resisting or that causing motion?JMLCarter –JMLCarter 2017-01-23 12:15:32 +00:00 Commented Jan 23, 2017 at 12:15
I was just using the face that the negative change in potential energy is equal to the work done. The resisting force would be that created by the positive charge.Teyash Arjun –Teyash Arjun 2017-01-23 12:23:33 +00:00 Commented Jan 23, 2017 at 12:23
Whether it is negative or not depends on whether you move from high to low potential or in the other direction.JMLCarter –JMLCarter 2017-01-23 12:52:46 +00:00 Commented Jan 23, 2017 at 12:52
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To simplify matters suppose that there is a system which consists of two positive charges, one of which cannot move and that system of charges has some electric potential energy.
There are forces of repulsion between the two charges and these are internal forces – they are a Newton’s third law pair. You now apply a force on the mobile charge which is equal in magnitude but opposite in direction to the repulsive force on it, ie the net force on the mobile charge is zero.
The force that you apply on the mobile charge is an external force, it originates from outside the system of two charges. Now allow the mobile charge to move away from the other charge from an initial position of rest A A to a final position B B of rest.
You find that the electric potential energy of the system of two charges has decreased.
How can that happen?
It happens because the internal repulsive force does work and the work done is positive because the internal force and the displacement of the internal force are in the same direction.
As a result of that work done by the repulsive force, the electric potential energy of the system decreases.
The work done on the system by the external force exerted by you when the charge moved from position A A to position B B has the same value as the work done by the internal force except that it is negative.
It is negative because the external force and the displacement of the external force are in opposite directions.
So how does one interpret negative work done on the system by the external force exerted by you?
It is the positive work done by the system on you.
When you do positive work on a system, energy is transferred from you to the system and when you do negative work on a system, energy is transferred from the system to you.
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answered Jan 23, 2017 at 16:36
FarcherFarcher
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