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(PDF) Modeling Multiple Relationships in Social Networks
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Electronic copy available at: http://ssrn.com/abstract=1960262
Vol. XLVIII (August 2011), 713 –728
*Asim Ansari is the William T. Dillard Professor of Marketing (e-mail:
maa48@ columbia.edu), and Oded Koenigsberg is Barbara & Meyer Feld-
berg Associate Professor (e-mail: ok2018@columbia.ed), Columbia Busi-
ness School, Columbia University. Florian Stahl is Assistant Professor,
Department of Business Economics, University of Zurich (e-mail: florian.
stahl@ uzh.ch). Christophe Van den Bulte served as associate editor for this
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© 2011, American Marketing Association
ISSN: 0022-2437 (print), 1547-7193 (electronic) 713
The rapid growth of online social networks has led to a
resurgence of interest in the marketing field in studying the
structure and function of social networks. A better under-
standing of social networks can enable managers to compre-
hend and predict economic outcomes (Granovetter 1985)
and, in particular, to interface with both external and internal
actors. Online brand communities, which are composed of
users interested in particular products or brands, allow such
an external interface with customers. Such communities not
only help firms interact with customers and prospects but
also enable customers to communicate and exchange infor-
mation with each other, consequently increasing the value
that can be derived from a firm’s products.
Similarly, firms forge alliances and enter into collabora-
tive relationshipswith other firms for coproduction and
social commerce (Stephen and Toubia 2010; Van den Bulte
and Wuyts 2007) using interorganizational networks.
Within the firm, intraorganizational networks of managers
play a crucial role in cross-functional integration, as is the
case with networks of marketing and organizational profes-
sionals engaged in new product development (Van den
Bulte and Moenaert 1998).
As Van den Bulte and Wuyts (2007) point out, network
structure has implications for power, knowledge dissemina-
tion, and innovation within firms and for contagion and dif-
fusion among customers. Thus, understanding and predict-
ing the patterns of interactions and relationships among
network members is an important first step in using them
effectively for marketing purposes. The focus of social net-
work analysis is on (1) explaining the determinants of rela-
tionship formation; (2) identifying well connected actors;
and (3) capturing structural characteristics of the network as
described by reciprocity, clustering, transitivity, and other
measures of local and global structure using a combination
of assortative, relational, and proximity perspectives (Rivera,
Soderstrom, and Uzzi 2010).
Actors belonging to a social network connect with each
other using multiple relationships, possibly of different
Reprinted with permission from the Journal of Marketing Research, published by the American Marketing Association,
Asim Ansari, Oded Koenigsberg, and Florian Stahl, vol. 48, no. 4, August 2011, pp. 713-728.
Electronic copy available at: http://ssrn.com/abstract=1960262
types. In this article, we develop statistical models of multi-
ple relationships that yield an understanding of the drivers
of multiplex relationships and predict the connectivity
structure of such multiplex networks.
Multiplexity of relationships can arise from different
modes of interaction or because ofdifferent roles people
play within a network setting. For example, in many online
networks, members can form explicit friendship and busi-
ness relations, exchange content and communicate with one
another. The relationships that connect a group of actors can
differ not only in their substantive content but also in their
directionality and intensity. For example, somerelation-
ships are symmetric in nature, whereas others can be
directed. Some relationships involve the flow of resources,
thus necessitating a focus on the intensity of such weighed
connections. Finally, multivariate patterns of connections
can also arise from viewing the same relationship at differ-
ent points, as is the case of sequential networks.
An understanding of multiplex patterns in network struc-
tures is important for marketers. For example, Tuli, Bharad-
waj, and Kohli (2010) find that in a business-to-business
setting, increasing multiplexity in relationships leads to an
increase in sales and to a decrease in sales volatility to a
customer. Multiplexity contributes to the total strength of a
tie and increases the number of ways one can reciprocate
favors (Van den Bulte and Wuyts 2007). Thus, it is relevant
for identifying influential actors such as opinion leaders in
diffusioncontexts and powerful executives within intra-
organizational networks. In analyzing sequential relation-
ships, a multivariate analysis can help investigate the impact
of managerial interventions on the relationship structure of
a network across points. Sequential dyadic interactions are
also useful for understanding the dynamics of power and
cooperation in intraorganizational networks and in model-
ing long-term relationships between buyers and sellers in
business markets (Iacobucci and Hopkins 1992). Finally,
when marketers are interested in predicting relationship pat-
terns, multiplexity allows leveraging information from one
relationship to predict connections on other relationships.
Researchers can obtain asubstantive understandingof
multiplex relationships by simultaneously analyzing the
multiple connections among the network actors. They can
investigate whether these multiple relationships exhibit
multiplex patterns that are characterized by the flow of mul-
tiple relationships in the same direction or whether they rep-
resent patterns of generalized exchange in which a tie in one
direction on one relationship is reciprocated with a connec-
tion in the other direction using different relationships.
However, most models of social networks analyze a single
relationship among network members. When the lens is
trained on a single relationship, an incomplete understand-
ing of the nature of linkages can result. For example, it is
unclear whether people play a similar role across multiple
relationships. A joint analysis can also help uncover com-
mon antecedents that affect relationships. Moreover, if
some relationships exhibit unique patterns, such uniqueness
can emerge only when multiple relationships are contrasted.
In this article, we develop an integrated latent-variable
framework for modeling multiple relationships. We make a
methodological contribution to the social networking litera-
ture in both marketing and the wider social sciences by
offering a rich framework for modeling multiple relation-
ships of different types. Our modeling framework has sev-
eral novelfeatures when compared withpreviously pro-
posed models for multirelational network data. Specifically,
our framework can (1) model multiple relationships of dif-
ferent types (i.e., weighted, unweighted, undirected, and
directed), (2) model sequential relationships, (3) leverage
partial information from one network (or relationship) to
predict connectivity in another relationship, (4) accommo-
date missing data in a natural way, (5) capture sparseness in
weighted relationships, (6) incorporate sources of dyadic
dependence, (7) account for higher order effects such as tri-
adic effects, and (8) include continuous covariates. Although
previous models have incorporated some of these aspects,
we do not know of any research in the social network analy-
sis literature that simultaneously incorporates all of them.
We illustrate the benefits of our approach using two
applications. Our first application involves sequential net-
work data that studies network structure over two points.
We reanalyze data from Van den Bulte and Moenaert (1998)
involving a network of research-and-development (R&D),
marketing, and operationsmanagerswho are engaged in
new product development. The data contain communica-
tions among these managers both before and after colloca-
tion of R&D teams. The results show that substantive con-
clusions can be affected if the full generality of our
framework is not utilized. We also show how our methods
can be used to leverage information from one relationship
to predict the connections in another relationship.
In our second application,we use data from anonline
social networking site involving the interactions among a
set ofmusicians. We model friendship, communications,
and music download relationships within this network to
show how a combination of directed and undirected, and
weightedand unweighted, relationships can be modeled
jointly. We analyze the determinants of these relationships
and assess the importance of our model components in cap-
turing different facets of the network structure. Our results
show that artists exhibit similar network roles acrossthe
three relationships and that these relationships are mostly
influenced by common antecedents. We also show that
when dealing with weighted relationships (e.g., music
downloads), it is crucial to jointly model both the incidence
and the intensity of such relationships, rather than simply
focusing on the intensity; otherwise, prediction and recov-
ery of structural characteristics suffers appreciably.
We organize the rest of the article as follows: The next
section provides a brief review of the marketing and statisti-
cal literature on social networks. Then, we present the com-
ponents of our modeling framework and describe inference
and identification of model parameters. The following two
sections describe the two applications. Finally, we conclude
with a discussion of our contributions and model limitations
and outline future research possibilities.
Social network data offer considerable opportunities for
research in marketing, as Van den Bulte and Wuyts (2007)
identify in their expansive survey of the role and importance
of social networks in the marketing field. Most research in
marketing on social networks falls into one of two streams.
In the first stream, researchers explore the impact of word
of mouth on the behavior of others and thus are primarily
# " ! ##!"
Electronic copy available at: http://ssrn.com/abstract=1960262
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
concerned about the role of social interactions and conta-
gion (Iyengar, Van den Bulte, and Valente 2011; Nair, Man-
chanda, and Bhatia 2010; Trusov, Bodapati, andBucklin
2010; Watts and Dodds 2007).
Research in the second stream focuses on modeling net-
work structure. Iacobucci and Hopkins’s (1992) work is a
pioneering contribution in this area. The focus here is on
understanding the antecedents of relationship formation and
in studying how interventions influence future connectivity
(Van den Bulte and Moenaert 1998). The current study adds
to this second stream of research by offering a comprehen-
sive framework for modeling multivariate or sequential
There is a rich history of statistical modeling of network
structure withinsociology and statistics that spansmore
than 70 years. More recent approaches stem from the log-
linear p1model developed by Holland and Leinhardt (1981),
which assumes independent dyads. Because the p1model is
incapable of representing many structural properties of the
data, the literature has proposed two general ways of cap-
turing the dependence amongthe relationships. The first
approach uses exponential random graph models or p*
models (Frank and Strauss 1986; Pattison and Wasserman
1999; Robins et al. 2007; Snijders et al. 2006; Wasserman
and Pattison 1996) that capture the dependence pattern in
the network using a set of statistics that embody important
structural characteristics of the network. However, care is
necessary when using these models because parameter esti-
mation sometimes suffers from model degeneracy, and how
to handle this degeneracy is an active area of research. The
second approach handles the dependence among the dyads
using correlatedrandom effects andlatent positions in a
Euclidean space for the individual people (Handcock,
Raftery, and Tantrum 2007; Hoff 2005; Hoff, Raftery, and
Handcock, 2002 ). In addition to these two approaches, mul-
tiple regression quadratic assignment procedure (MRQAP)
methods (Dekker, Krackhardt, and Snijders 2007) have also
been used in network analysis to account for dependence
among dyads.
Exponential random graph models describe the network
using a set of summary statistics, such as the total number
of ties, the number of triangles, and the degree of distribu-
tion, among others. This is good for describing“global”
properties and in assessing particular hypotheses of substan-
tive interest, such as the extent of reciprocity or clustering
and triadic closure. In contrast, latent space models capture
the “local” structure by estimating a latent variable for each
node in the network, which describes a person’s position in
the network. These models are thus suitable when the focus
is on understanding the determinants of connectivity using
covariates and in identifying influential people. The latent
variable framework is capable of recovering the structural
characteristics using a small set of model parameters (simi-
lar to nuisance parameters) and thus can be parsimonious in
some contexts.
When researchers are interested in specific substantive
hypotheses and when all relationships are binary in nature,
they may prefer exponential random graph models. How-
ever, the latent variable framework can accommodate mul-
tiplex relationships of different types, including weighted
relationships, and can also handle missing data in a straight-
forward fashion using data augmentation. Thus, it is prefer-
able when interest is in analyzing such complex multivari-
ate data structures.
Whereas the preceding methods explicitly model the net-
work structure, MRQAP methods offer a nonparametric
alternative for conducting permutation tests to assess
covariate effects using multiple regression while correcting
for the dependency and autocorrelation present in network
data. The MRQAP approach is useful for continuous data. It
can be used for binary relations using a linear probability
model, and to a certain extent for count data; however, its
effectiveness for multivariate relations of different types is
not clear.
Sequential data can also be modeled using two
approaches: a multivariate approach such as ours and the
conditional, continuous-time approach popularized by Sni-
jders (2005). The multivariate approach models the network
at each point in time and thus is useful when one is inter-
ested inassessing the impact of interventions that occur
between these discrete times. In contrast, the continuous-
timeapproach is inherently dynamic, focusingon either
edge-oriented or node-oriented dynamics, and can model
the evolution of the network one edge at a time. However,
thisapproach is limited to binary relations, whereas the
multivariate approach that we use can handle both weighted
and binary relationships.
In contrast to the current study, most models of social net-
work structure analyze a single relationship, and to the best
of our knowledge, none have incorporated the entire con-
stellation of desirable model characteristics that we outlined
in the introduction. In particular, there has been no work on
using the latent space framework for modeling multivariate
relationships or sequential data.
Although some researchers have modeled multiple rela-
tionships, these models either assume independence across
dyads, which is restrictive, or limit attention to binary rela-
tions (Fienberg, Meyer and Wasserman 1985; Iacobucci
1989; Iacobucci and Wasserman 1987; Pattison and Wasser-
man 1999; Van den Bulte and Moenaert 1998). Thus, there
is a need for an integrative framework for modeling multi-
ple relationships of different types (i.e., binary or weighted)
in a flexible way. The latent space framework offers such
flexibility, and using it, we develop an integrated approach
for multiple relationships in the following section.
We develop a modeling framework for the simultaneous
analysis of multiple relationships among a set of network
actors. Our framework accommodatesmultiple relation-
ships of different types and also enables us to simultane-
ously model the determinants of the relationships as well as
the structural characteristics such as the extent of reci-
procity or transitivity within each relationship and across
relationships. When analyzing multiple relationships, struc-
tural characteristics of interest include those that account for
.6-5*1-&9 patterns (i.e., flow of multiple relationships in the
same direction) and &9$)"/(&, in which a flow in one direc-
tion on one relationship is reciprocated with a flow in the
other direction using a different relationship. Similarly, pat-
terns of transitivity that involve more than one relationship
can also be investigated. When focusing on thedetermi-
nants of relationships, we can infer how the attributes of the
network actors influence the formation of relationships
between them. Here the interest is in understanding whether
actors exhibit similar popularity and expansiveness across
different relationships, and whether homophily governs
relationship formation.
The multiple relationships describing a common set of
actors can vary along different facets, such asexistence,
intensity, and directionality. A relationship is directed if we
can distinguish the sender and receiver of the tie. For exam-
ple, a communication relationship typically has a sender and
a receiver. In contrast, relationships could be undirected,
such as a collaboration relationship. In modeling both
directed and undirected relationships, the focus of the analy-
sis could be on modeling the existence of a relationship (i.e.,
the presence or absence of atie) or on theintensity ofa
weightedtie(e.g.,the intensity of the flow of resources
between a pair of people). Ourobjective is to showhow
such disparate relationships can be jointly modeled within a
common framework. In the following section, we describe
formally our model.
We describe our model using two relationships. Although
these two relationships could represent a single relationship
observed over different time periods, for the sake of gener-
ality, we describe a model for two distinct, directed relation-
ships of different types.1 These two relationships are
observed over the same set of n actors.
*3&$5 &% #*/"3: 3&-"5*0/4 )*1. The first relationship is
directed and binary. Thus, we can distinguish between the
sender and the receiver of the tie and the sociomatrix
matrix, X1, which shows that the incidence of ties among
actors can, therefore, be asymmetric. We use the ordered
pair of binary dependent variables {Xij1, Xji1} to represent
the presence or absence of ties for a pair of actors i and j.
The variable Xij1 specifies the existence of interaction in the
direction from i to j (i.e., i Æj), whereas Xji1 represents the
presence of a tie in the opposite direction from j to i (i.e.,
i ¨j).
*3&$5&%8&*()5&%3&-"5*0/4)*1. The second relationship
is directed and weighted (i.e., valued). The entries in the
associated matrix, X2, indicate the bidirectional intensity of
interaction between the different pairs of people. Here, we
assume a countvariablefor the intensity, becausethisis
consistent with our second application presented in the sec-
tion “Online Social Network.” However, our model can be
adapted for continuous or ordinal measures of intensity. An
ordered pair of count variables (Xij2, Xji2) can represent the
observed intensity of interaction in the dyad, where the
variable Xij2 specifies the strength of the interaction from i
to j (i.e., i Æj), and Xji2 specifies the intensity in the reverse
In modeling this weighted relationship, we deviate from
the previous literature on social networks by jointly model-
ing both the existence and the intensity of the relationship.
This allows us to distinguish between the mechanisms that
drive the incidence from those that affect the intensity of
relationships. In addition, it also accommodates a prepon-
derance of zeros due to sparseness of ties.We can then
ascertain whether a specification that directly models the
intensity (such as a Poisson specification; e.g., Hoff 2005)
is sufficient for weighted relationships. Therefore, we use a
multivariate correlated hurdle count specification to jointly
model both the incidence of the relationship within a dyad
and the intensity of the relationship conditional on the exis-
tence of the tie. We model the incidence using the ordered
pair of binary variables (Xij2, I, Xji2, I). Then, the magnitude
of the relationship, conditional on its existence in a given
direction, can be modeled using the positively valued trun-
cated count variables Xij2, S or Xji2, S.
:"%*$.6-5*(3"1). Bringing together the two relation-
ships, we can then specify the nC2dyads in the multirela-
tional social network using the dyad-specific random
The relationships can be further specified in terms of under-
lying latent variables. The latent variable specification
enables us to model these random variables in terms of
dyad- and actor-specific covariates.
"5&/57"3*"#-&41&$*'*$"5*0/. We use underlying latent
utilities uij1 for modeling the existence of a tie in the direc-
tion(i Æj)anduji1 inthereversedirection, for the first
For the second relationship, let uij2 and uji2 represent the
underlying utilities that characterizes the existence of the
relationship. Again, we assume that
We model the truncatedcounts,conditional on a tie ina
given direction, using a Poisson distribution truncated at
zero; in other words,
(3) Xij2, S ~tPoisson(lij) if uij2 > 0, and
Xji2, S ~tPoisson(lji) if uji2 > 0,
where the lij and lji are the rate parameters of the Poisson.
07"3*"5&4"/%)0.01)*-:. Each latent utility is composed
of a systematic part involving the observed covariates and a
stochastic part that incorporates unobserved variables. We
distinguish between dyad-specific covariates and individual-
specificcovariates. Let xd
ij1 and xd
ij2 bevectors of dyad-
specific covariates that influence the two relationships.
These allow for homophily, which implies that people who
ij ji
ij I ji I
ij S ji S
<,.i j
() ,,
Xif u
Xif u
ii1 0
() ,,
Xif u
ij I
ji I
if uji2 0
# " ! ##!"
1We limit our model description to two relationships for clarity of pres-
entation. Our approach, however, can be extended readily to more relation-
ships, including undirected ones, as we do in our second application.
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
share observable characteristics tend to form connections.
We also use individual-specific covariates for modeling
directed relationships. Let xi1, and xi2be the vector of indi-
vidual i’s covariates, respectively, for the two relationships.
The vector xij1 = (xd
ij1, xi1, xj1) then represents all the covari-
ates that affect the tie in the direction (i Æj) for the first
relationship, and xji1 = (xd
ji1, xj1, xi1) represents the covari-
ates for the reverse direction (i ¨j). The sender and
receiver effects are important to model the asymmetry in the
two directions. For the weighted relationship, we can further
distinguish between the incidence and intensity components.
Therefore, we use covariate vectorsxij2, I = (xd
ij2, I, xi2, I, xj2, I)
and xji2 , I = (xd
ji2, I,xj2, I, xi2, I) for the binary component,
where, for example, xd
ij2, I contains asubset ofthe dyad-
specific variablesin xd
ij2 that affect intensity, and xi2 , I is
similarly a subset of xi2. We use analogously defined vec-
tors xij2, S and xji2, S to model the Poisson rate parameters lij
and lji.
&5&30(&/&*5:. The dyads cannot be considered inde-
pendently, because multiple dyads share a common actor
either as a sender or receiver. Accounting for such depend-
ence is important for obtaining proper inferences about sub-
stantive issues. Therefore, we use heterogeneous and corre-
lated random effects to account for the dependence
structure. Whereas for undirected relationships, a single ran-
dom effect is needed, for directed relationships, we can use
two distinct random effects per actor to distinguish between
&91"/4*7&/&44, which is the propensity to “send” ties and
popularity, or "553"$5*7&/&44, which is the propensity to
“receive” ties. The expansiveness parameter aicaptures the
outdegree, which is the number of connections emanating
froman individual i,andthe attractiveness parameter bi
captures the indegree, which is the number of connections
impinging on an individual. Thus, for the directed binary
relationship, we use random effectsai1 and bi1. We simi-
larly use ai2, I and bi2, I for the incidence component and ai2,
Sand bi2, S for the intensity equations of the weighted
directed relationship.
Let qi= {ai1, bi1, ai2, I, bi2, I, ai2, S, bi2, S}. We allow these
random effects to be correlated across the relationships and
assume that qiis distributed multivariate normal2N(0, Sq),
where Sqis an unrestricted covariance matrix consisting of
the following submatrices:
The diagonal submatrices capture the within-relationship
covariation in the random effects within a relationship. A
positive correlation between the random effects for a rela-
tionship implies that popular individuals also tend to reach
out more toothers. The off-diagonal submatrices capture
correlation across relationships and help determine whether
individuals exhibit similar tendencies across relationships.
If the off-diagonal submatrices in Sqindicate positive corre-
lations, this could possibly be a result of a latent trait gov-
erning commonality in behavior. In contrast, if these sub-
matrices are zero, the relationships can be modeled
separately as the attractiveness and expansiveness parame-
ters will be independent across the different relationships.
Other patterns of correlations are also possible, and their
meaning and relevance depend on the empirical context of a
particular application.
"5&/5 41"$& Social networks also exhibit patternsof
higher-order dependence involving triads of actors. There is
a potential for misleadinginferences if such extradyadic
effects are ignored. Hoffand his colleagues demonstrate
how transitivity and other triad-specific structural charac-
teristics such as balance and clusterability can be modeled
using a latent space framework(Handcock,Raftery, and
Tantrum 2007; Hoff 2005; Hoff, Raftery, and Handcock
2002). We employ a latent space for each relationship. We
assume that individual i has a latent position zir in a Euclid-
ean space associated with each relationship r, where r Œ{1,
2}. The latent space framework stochastically models tran-
sitivity; if i is located close to individual j and if j is located
close to individual k, then, because of the triangle inequal-
ity, i will also be close to k.
In the current study, we follow Hoff (2005) and use
the inner-product kernel z¢
irzjr.3In particular, for a two-
relationship model, we use two kernels, one for each rela-
tionship, represented generically as z¢
irzjr. The latent vectors
for each relationship are assumed to come from a relation-
ship-specific multivariate-normal distribution zir ~N(0,
Szr). The dimensionality of the latent space can be deter-
mined using a scree plot of the sum of the mean absolute
prediction error of the entire triad census versus the dimen-
sionality, as is usually done in the multidimensional scaling
6--.0%&-Bringing together all the components of the
model, we can write the latent utilities and response propen-
sities as follows:
We assume that the vector of all errors eij is distributed mul-
tivariate normal N(0, S). Also,qi~N(0, Sq) and zir ~N(0,
Szr), "r.
%&/5*'*$"5*0/. Not all parameters of the model are identi-
fied. Theerror variancematrix Shas a specialstructure
ij ij i j i j ij
ji ji j
zz ,
++ +
ij ij I I i I j I
zz ,
,, , ,
xµµ++ +
ij ijI
ji ji I I j I i I i
,, , ,
xµµα β 222 2
ij ij S S i S j S i
,, , ,
logλαβxµµ ′′
ji ji S S j S i S i
,, , ,
logλαβxµµ zz jjiS
2Even though we assume a symmetric distribution for heterogeneity,
when it is combined with the data from the individuals, the resulting poste-
rior random effects can mimic skewed degree distributions that can arise
from a preferential attachment mechanism. We verified this using a simu-
lation that generated data from a preferential attachment mechanism and
were able to recover highly skewed degree distributions. Details of this
simulation are available on request.
3Other kernels such as those based on the Euclidean norm can also be
used. We leave a detailed examination of the pros and cons of using differ-
ent kernel forms for further research.
4The Bayes factor can also be used to determine dimensionality. How-
ever, it is difficult to compute in our model given the high dimensional
numerical integration that is involved in obtaining the likelihood for each
observation. Therefore, we opt for the predictive MAD criterion that
focuses on triad census recovery.
because of scale restrictions on the binary utilities and
because of exchangeability considerations stemming from
the fact that the labels i and j are arbitrary within a pair. As
the scale of the utilities of the binary responses cannot be
determined from the data,theerror variancesassociated
with the binary components are set to 1. In addition, sym-
metry restrictions on the correlations stem from the
exchangeability considerations, and the resulting variance
matrix can be written as follows:
The correlation parameter r1captures the impact of common
unobserved variables affecting the binary relationship and
also accounts for reciprocity. Similarly, r6andr9capture
correlations for the weighted relationship and also account
for reciprocity within this relationship. The correlation
parameters r7and r8reflect common unobserved variables
that influence both the incidence and intensity equations of
the weighted component of the second relationship and are
akin to selectivity parameters. Note that the intensity equa-
tions have a common variance.
The expansiveness and attractiveness random effects are
individual specific and are thus separately identifiable from
the equation errors that are dyad specific. The latent posi-
tions also are individual specific, but because they enter the
equations as interactions, they can be separately identified
from the random effects. However, because they appear in
bilinear form, we can only identify these subject to rotation
and reflection transformations. Finally, the covariance
matricesSz1 and Sz2 associated with the latent space
parameters are restricted to be diagonal as their covariance
terms are not identified. Moreover, each matrix has a com-
mon variance term across all the dimensions within a latent
Table 1 summarizes how the different model parameters
can be related to substantive issues of interest. Given that the
model components work in tandem, a parameter may also
be related to other aspects apart from the one shown in the
table. For example, r1is needed to capture reciprocity, but
may also represent the impact of other shared unobservables.
ij ji ij I ji I ij S ji S
ji I
ij S
ji S
ji S
σρ σρ
ji S
ji S
ji S
Xji S,
We now describe briefly our inference procedures. The
likelihood for the model is computationally complex. Con-
ditional on the random effects and latent positions, the
dyad-specific likelihood requires numerical integration to
obtain the multivariate normal cumulative distribution func-
tion. Moreover, the unconditional likelihood for the entire
network requires additional multiple integration of very
high dimensionality because of the crossednature ofthe
random effects. The dependency structure of our model is
considerably more intricate than what is typically encoun-
tered in typical panel data settings in marketing, because we
cannot assume independence across individuals or dyads for
computing the unconditional likelihood. Therefore, we use
Markov chain Monte Carlo (MCMC) methods involving a
combination of data augmentation and the Metropolis–
Hastings algorithm to handle the numerical integration. The
data augmentation step allows us to leverage information
from one relation to predict missing data on other relation-
ships. The complexities involved in modeling multiple rela-
tionships and the identification restrictions on the covari-
ance matrixSmean that the methods of inference for
existing latent space models, as outlined, for example, in
Hoff (2005) and Hoff, Raftery, and Handcock (2002), can-
not be used directly for our model. Therefore, we provide a
full derivation of the posterior full conditionals in Appendix
In the first application, we illustrate our modeling frame-
work on sequential network data. We start with a special
caseof our general modelingframework and handle the
simpler situationof a single directedbinary relationship
observed at two points in time. Therefore, we do not need
the intensity component in this application. We use the same
data as in Van den Bulte and Moenaert (1998; hereinafter,
VdBM) on communications among members of different
R&D teams and marketing and operations professionals
who are all involved in new product development activities.
Here, we briefly analyze this data set to revalidate the
results in VdBM and to investigate whether our modeling
framework (which differs significantly from that in VdBM)
is better able to recover the structural characteristics of the
network and whether it generates different conclusions or
additional insights.
Dataare available about communicationpatterns both
before and after the R&D teams were collocated into a new
facility. The data set used in VdBM and the one we reana-
lyze here come from a survey conducted in the Belgian sub-
sidiary of a large U.S. telecommunication cooperation. The
dataconsistof two 22 ¥22 binary (who talksto whom)
sociomatrices, X1and X2, one for 1990 (before collocation)
and one for 1992 (after the R&D teams were collocated in a
separate facility). The actors in both years are the same, 13
R&D professionals spread over four teams and nine mem-
bers of a steering committee consisting of seven positions
in marketing and sales and two in operations. The charac-
teristicelement xij, t in each of the twomatrices is 1 if i
reports to talk to j at least once a week in year t and 0 if oth-
erwise. The specific area VdBM study is the impact of the
collocation intervention on the communication and coop-
# " ! ##!"
! " " !
"3*"#-& ''&$54
μ Covariate effects, homophily, and heterophily
~iExpansiveness, productivity
βiAttractiveness, popularity, or prestige
ziTransitivity, balance, and clusterability
~1, r6, r9Reciprocity
r7, r8Selectivity
r3, r5Generalized exchange
r2, r4Multiplexity
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
eration patterns among the different R&D teams and
between R&D and marketing/operations by contrasting
these patterns before and after collocation.
Barriers between R&D and marketing professionals
resulting from differences in personality, training, or depart-
ment culture imply that intrateam and intrafunctional com-
munication will be more prevalent than cross-team and
cross-functionalcommunication. The idea of collocation
was to foster communication among the different R&D
teams. Of the five hypotheses VdBM propose, the first two
test the communicative implications of the barriers between
R&D and marketing and relate to team- and function-specific
homophily effects. The third and fourth hypotheses focus on
effects of collocating R&D teams to foster communication
among these teams. Finally, VdBM posit that collocating
R&D groups may not just foster between-group communi-
cation, but may even annihilate any difference between
within- and between-group communication.
VdBM use Wasserman and Iacobucci’s (1988) p1log-linear
models to test their hypotheses. Our approach differs from
that of these previous studies on several counts. First, in
contrastto the p1model, we do not assumedyadic inde-
pendence. In our model, the dyads are independent condi-
tional on the random effects but are dependent uncondition-
ally. Second, we allow for individual-specific expansiveness
and attractiveness parameters in contrast to group-specific
parameters to yield a richer specification of heterogeneity.
Finally, we account for higher-order effects using a latent
space. The added generality of our model is consistent with
VdBM’s (p. 16) call for “a new generation of models better
able to handle triadic effects and other dependency issues.”
We estimated four models on the data set:
1. The full model involves all the components that form part of
our modeling framework. These components include dyad-
specific variables, attractiveness and expansiveness random
effects that are correlated both within and across years, sepa-
rate latent spaces for the two years, and correlated error
terms for the utility equations of the four binary variables
characterizing a dyad.
2. The Uncorr model is a restriction of the full model. It
assumes that the random effects and the utility errors are cor-
related within a year but are uncorrelated across years. This
is akin to having a separate model for each year, and this
offers limited leeway in modeling muliplexity.
3. The NoZiZj model is a restriction of the full model such that
the higher-order terms that characterize the latent space, (i.e.,
the z¢
izjterms are not included. We use this model to assess
whether using the latent space results in better recovery of the
triadic structure of the network and whether it substantively
affects conclusions.
4. The team model closely mimics the VdBM article within our
modeling framework. In this model, we restrict the expan-
siveness and attractiveness parameters to be the same for all
individuals within a group and also do not include the latent
The following variables used in our investigation are the
same as those VdBM use:
INTEAMij = 1 if i and j are R&D professionals on the
same team and 0 if otherwise,
BETWTEAMij = 1 if i and j are R&D professionals but in dif-
ferent teams and 0 if otherwise,
INRDij = 1if i and j are R&D professionals and0 if
otherwise, and
INMKTOPSij = 1 if i and j are both marketing or both opera-
tions executives and 0 if otherwise.
We estimated the four models using MCMC methods.
Each MCMC run is for 250,000 iterations, and the results
are based on 200,000 iterations after discarding a burn-in of
&$07&3:0'4536$563"-$)"3"$5&3*45*$4. We begin by com-
paring the previously described models in their ability to
recover the structural characteristics of the network. Given
our interest in modeling sequential relationships, we focus
on aspects of the network structure that pertain to the two
relationships simultaneously. In particular, we compute sta-
tistics involving the dyadic as well as transitivity patterns of
interactions that span both years.
We can describe dyadic relationships in each year as
belonging to one of the following three types: mutual (M),
asymmetric (A), and null (N). Observing acrossthe two
years, we can construct the following ten possible combina-
tions (Fienberg, Meyer, and Wasserman 1985):NN, AN,
NA, MN, NM, AA, AA, AM, MA, and MM. The names are
self-explanatory for most pairs. For example, NN refers to
the number of dyads that are null in both years. Two pat-
terns that require greater explanation are AA and AA. The
pair AA represents a dyad in which one actor is connected
to the other in both years but neither relationship is recipro-
cated. The pair AA represents a dyad in which one actor ini-
tiates communication with the other in the first year and the
other actor reciprocates by initiating in the second year, a
kind of generalized exchange.
Table 2 reports the recovery of the sequential dyadic pat-
terns. The columns report the absolute deviations between
the actual frequencies and those predicted by the different
models. The last row of the table reports the mean absolute
deviations (MAD) across all the patterns for each model. It
is clear that the uncorrelated model (Uncorr), which ignores
cross-year linkages and models the two years separately, is
significantly worse in recovering the cross-relationship dyadic
patterns compared with the other models. All other models
are roughly similar in their recovery, with the team model
being the best. This indicates that it is important to model the
" $ & !! "&
"55&3/ 6-- 0!*!+ /$033 &".
NN 7.51 6.44 12.67 7.04
AN 3.00 4.17 4.24 2.70
NA 4.25 4.90 4.82 5.50
MN 3.79 1.53 7.45 2.15
NM .81 2.65 6.07 1.43
AA .92 1.89 1.89 .80
AA 1.97 .38 4.84 .02
AM 2.14 2.26 1.67 1.92
MA .10 .80 .84 .77
MM 5.33 4.48 9.46 3.80
MAD 2.983 2.950 5.394 2.612
two relations jointly so that we can recover the cross-relation
dyadic patterns, as all the models, and, except Uncorr,
accommodate correlations across the two relationships.
We also investigate transitive patterns spanning both
years to understand how well the model recovers
extradyadic effects.This is necessary to assess whether
adding the latent space is important. Eight such transitivity
effects are possible: {Xij1, Xjk1, Xik2}, {Xij1, Xjk2, Xik1},
{Xij1, Xjk2, Xik2}, {Xij2, Xjk1, Xik2}, {Xij2, Xjk2, Xik1},{Xij2,
Xjk1, Xik1}, {Xij1, Xjk1, Xik1}, and {Xij2, Xjk2, Xik2}. For the
sakeof brevity, wedo not include afull tableof results
(available on request). We find that the full model performs
significantly better that all other models in recovering these
transitive patterns(MAD = 34.43). The full and Uncorr
models (MAD = 47.09), both of which include the latent
space, perform significantly better than NoZiZj (MAD =
109.46) and team (MAD = 164.69), which do not include
the higher-order effects.
In summary, observing across all dyadic and triadic
measures, we find that the full modelalways does better
than Uncorr, thus highlighting the need for joint modeling.
The full model also performs better than all the other mod-
els in recovering the transitivity patterns, indicating that in
this application, the latent space is important for handling
extradyadic patterns. We can conclude that, on the whole,
the full model recovers best the structural characteristics of
the network.
We begin by investigating whether the assumption of
individual-level expansiveness and attractiveness parame-
ters in our models has support. Figures 1 and 2 report the a
and bvalues for the 22 managers for Years 1 and 2, respec-
tively, for our full model. The labels for each point in the
figures represent the team name. It is apparent from these
figures that although some members of a group are clus-
tered together, many groups exhibit considerable within-
group heterogeneity. This is particularly noticeable for the
marketing group (m) and the R&D groups (r3, r1, and r4),
which exhibitgreaterwithin-group variability. This sug-
gests that the data support a richer characterization of
heterogeneity than what is possible with group-specific
Table 3 summarizes the parameter estimates. It is evident
from the table that the parameters values differ substantially
across the models in their magnitude and significance, indi-
cating that the different model components influence sub-
stantiveconclusions.Focusing on the bottompart of the
table, we see that models that do not include the latent space
yield higher estimates of the error correlations, possibly due
to the confounding of variances across levels. We can use
the coefficients to infer the degree of support for the differ-
ent hypotheses studied in VdBM. Table 4 reports the extant
of support for each hypothesis according tothe different
models. All other entries are computed from our reanalysis.
Each entry for a particular model represents the probability
that the corresponding hypothesis is true under that model.
Several significant differences across the models are evident
from this table.
The first two hypotheses pertain to within- and between-
teamhomophily effects. Comparing the fullmodel with
VdBM indicates that our model supports both H1and H2,
whereas VdBM find mixed support for these. In particular,
we find that H1 b has significant probability across all our
models, indicating that R&D professionals tended to com-
municate predominantly with other R&D professionals
before the move. The full, NoZiZj, and Uncorr models sug-
gest strong support for H2a and H2b in contrast to team and
VdBM. It seems that differences in support for this hypothe-
# " ! ##!"
"ab$#! " !&
Notes: The point labels indicate team membership.
"ab$#! " !&
Notes: The point labels indicate team membership.
" " !""! "
"55&3/ 6-- 0!*!+ /$033 &".
INTERCEPT1 –1.48 –.87 –1.23 .48
INTEAM1 5.03 3.02 4.03 3.42
BETWTEAM1 1.60 .88 .84 1.42
INMKTOPS1 1.03 1.40 1.02 .49
INTERCEPT2 –1.18 –.99 –1.12 .78
INTEAM2 3.51 3.01 3.17 3.79
BETWTEAM2 2.15 1.73 1.80 2.71
INMKTOPS2 1.23 1.61 1.02 .07
r1.59 .83 .60 .79
r2.43 .64 —.63
r3.26 .57 —.53
r6.48 .72 .53 .67
Notes: Bold indicates that the 95% posterior interval does not span 0.
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
sis are driven by the extent of within-group heterogeneity
captured by the model. Recall that both team and VdBM
assume that all members within a group share the same
attractiveness and expansiveness parameters. Figures 1 and
2 show, however, that there is considerable heterogeneity in
the recovered aand bparameters in the marketing group,
and we find that failure to model this heterogeneity compre-
hensively can substantively affect conclusions.
We also find some differences in the support for the col-
location hypotheses (H3and H4) across the models. The full
and Uncorr models have a lower probability associated with
these hypotheses compared with VdBM, team, and NoZiZj.
These differences can be explained by the presence or
absence of the latent space for capturing higher-order
effects and are also consistent with the strong support for
H1b in our model. Finally, all models yield no support for
H5.5These differences in supported hypotheses (H1b, H2a,
H2b, H3, and H4) across models demonstrate that the differ-
ent model components can affect the theoretical and sub-
stantive conclusions and that it is important to account for
dyadic-dependence and higher-order effects.
We now illustrate how our framework can be used to
leverage information in one relationship to predict relation-
ships in another. For example, we assume that the data
involving theentire marketinggroups are missingin the
second year. In such a situation, we cannot readily use the
log-linear modeling framework previous researchers have
employed, because the group-specific parameters used in
such models will not be available for the marketing group.
However, for our models, the natural reliance on data aug-
mentation to obtain the utilities and the individual-specific
random effects when estimating parameters ensures that
such missing data can be handled seamlessly. In particular,
the covariance matrix of the random effects can be used to
leverage information from Year 1 to Year 2 about these indi-
vidual-specific parameters. Therefore, we estimate our full
and Uncorr models on such a data set to determine whether
incorporating cross-relationship linkages improves predic-
tions in such situations. Note that in the full model, the data
on Year 1 for the individuals in the marketing group can be
leveraged to predict relationships in Year 2. This is not pos-
sible in the Uncorr model, in which the two years are mod-
eled separately. Tables WA 1 and WA 2 of the Web Appendix
(see http:// www. marketingpower.com/jmraug11) report how
well the cross-year dyadic and transitivity patterns are recov-
ered on such a data set with missing values for the marketing
group. These tables report the absolute deviations between
the actual frequencies and those predicted by the full and
Uncorr models and illustrate that the full model does sig-
nificantly better in recovering these cross-year relationships.6
In this application, our focus is on modelingrelation-
ships of different types. We use a combination of undirected
and directed binary relationships and a directed weighted
relationship. In particular, we show how it is important to
model both the incidence and intensity of weighted relation-
ships, because conclusions and predictions depend crucially
on this distinction. The data for this application come from
a Swiss online social network on which members can create
a profile as either a user or an artist and can then connect
with other registered members through friendship relation-
ships. The social networking site offers different services to
these two distinct user groups. Whereas both groups can
publish user-generated content such as blogs, photos, or
"&"!!!# "& "!
:105)&4*4 03."-&45 6-- &". % 0!*!+ /$033
H1: Both before and after collocation, R&D professionals tend to
communicate with other R&D professionals rather than with
marketing or operations executives.
H1a INTEAM1 > 0
H1b BETWTEAM1 > 0
H1c INTEAM2 > 0
H1d BETWTEAM2 > 0
H2: Both before and after collocation, marketing and operations
executives tend to communicate with members of their own
H2a INMKTOPS1 > 0
H2b INMKTOPS2 > 0
H3: R&D professionals have a higher probability of communicating
with members of other R&D teams after collocation than before.
H3BETWTEAM2 > BETWTEAM1 .82 .99 .99 .99 .86
H4: When collocating R&D teams implies increasing the physical
distance with other departments, R&D professionals’ tendency
to communicate with other R&D people rather than executives
from other departments increases.
H4INRD2 > INRD1 .86 .99 .99 .99 .82
H5: Before collocation, R&D professionals have a higher probability
of communicating with members of their own team rather than
with members of other R&D teams. After collocation, the
tendency to communicate among R&D people is as strong
between as within teams.
H5a INTEAM1 > BETWTEAM1
H5b INTEAM2 = BETWTEAM2
Notes: Entries represent the probabilities of a hypothesis being true; n.s. indicates that the corresponding hypothesis is not supported.
5For H5b, we found that the probability associated with INTEAM2 >
BETWTEAM2 is .99 for all our models.
6We also investigated the role of demographics that were part of the data
but did not find any significant impact. Our models can handle such continu-
ous covariates, something that is not possible with a log-linear specification.
videos on their profiles, only artists can, in addition, publish
up to a maximum of 30 songs on their profile.
Artists use the different services offered by the platform
to promote their music and concerts and to seek collabora-
tion with other musicians and bands. They establish friend-
ship relationships with other artists, send personal mes-
sages, and write public comments on other artists’ profiles.
For entertainment and informational reasons, users, as well
as artists, visit profiles of artists and download songs.
Artists engage in active promotion and relationship effort in
the hope that it will result in increased collaboration, com-
munication, popularity, and song downloads. In summary,
the online networking site offers a platform that combines
social networking services with entertainment and commu-
nication services.
There are four components of the data set: member data,
friendship data, communication data, and music download
data. The member data contain information collected at reg-
istration and include stable variables such as the registration
date, date of birth, gender, and city or, in the case of artists,
their genre and information about their offline concerts and
performances before joining the network. In addition, the
data also contain information on the number of page views
of each member’s page on the network during a given time
period. The other data components pertain to our three
dependent variables and are described in greater detail in
the following section.
Our sample consists of 230 artists who created a profile
on the network between February 1, 2007, and March 31,
2007, provided information about their activities and char-
acteristics, and uploaded at least one song on their profile.
We model three types of relationships among these artists
over the course of the six months between April 1, 2007,
and September 30, 2007.7These relationships include
friendship (f), communications (c) and music downloads
(m). The data set thus contains three 230 ¥230 matrices (Yf,
Yc, and Ym, respectively) for these relationships.
536$563"-$)"3"$5&3*45*$4. We now briefly describe the
structural characteristics of the three relationships for our
set of artists.
1. 3*&/%4)*1 The friendship relation yf
ij is binary and undirected.
It indicates whether a friendship is formed between the pair
{i, j} before the end of our data period. The network has
3564 friendship relations among a maximum possible 26,335
connections, yielding a network density of 13.53%.
2. 0..6/*$"5*0/4 The communication relation yc
ij is binary
and directed and indicates whether artist i sent a communica-
tion (direct message or comment) to artist j within the time
period of the data. We observed 4575 communication rela-
tions, yielding a density of 8.68%. The relation exhibits con-
siderable reciprocity or mutuality (defined as the ratio of the
number of pairs with bidirectional relations to the number of
pairs having at least one tie) equal to 30.9%. Artists vary in
their level of expansiveness (or outdegree), as measured by
the number of artists they communicate with and their popu-
larity or receptivity( i.e., the number of artists communicat-
ing with a given artist [indegree]). The indegree and outdegree
distributions are highly skewed. The mean degree is 19.97.
The maximum and minimum for the indegree distribution are
203 and 0, respectively, whereas the maximum and minimum
for the outdegree distribution are 182 and 0, respectively.
3. 64*$ 08/-0"%4 The music downloads represent a
directed and weighted relationship. Each song download
entry ym
ij is a count of the number of times that artist j listens
to a song on artist i’s profile and may include multiple down-
loads of the same song. As discussed in the “Modeling
Framework” section, we distinguish between the incidence
and intensity of music downloads. Focusing first on inci-
dence, we find that of the possible 52,670 ties, our data con-
tain only 17,912 binary ties, implying a density of 34%. The
reciprocity is 39.1%. The outdegree of an artist is the number
of other artists who download from that artist, and the inde-
gree is the number of other artists from whom the artist
downloads music. For the binary component, the mean
degree is 17.89, and the maximum indegree and outdegree
are 213 and 135, respectively. Because this is a weighted rela-
tion, we can also study the intensity of connections. On aver-
age, each artist downloads songs on 59.98 occasions (includ-
ing multiple song downloads). The maximum weighted
indegree (i.e., the number of songs downloaded from) an
artist is 2899, whereas the maximum weighted outdegree
(i.e., the maximum number of times a single artist listens to
songs is 612). We also find that the artist-specific degree sta-
tistics are highly correlated across the three relationships.
We estimate several different variants of the full model
(hereinafter, we refer to this as “full model”) that we outlined
in the “Modeling Framework” section. Our null models
impose different restriction on the full model; we con-
structed them to investigate how crucial the different com-
ponents of the full model are in capturing important aspects
of the data generating process. The models are as follows:
;6-- 0%&-. The full model includes dyad-specific covariates
(xij) to accommodate homophily and heterophily; artist-specific
covariates (xi) and (xj) to account for asymmetry in responses;
artist- and relationship-specific sender and receiver parameters
(qi), to model heterogeneity in expansiveness and receptivity;
correlations in these random effects across relationships (Sq);
latent spaces of random locations for the three relations (zim,
zic, and zif) to capture higher order effects; and correlations in
errors (Sm, Sc, and Sf), to incorporate reciprocity within each
relationship. In addition, the full model uses a correlated hur-
dle count model to account for the sparse nature of many net-
work data sets.
;0*440/. The Poisson model uses a Poisson distribution for the
music download relationships, rather than the correlated hurdle
Poisson. Thus, the counts are modeled directly, and we do not
distinguish between the incidence and intensity of counts. The
model is otherwise identical to the full model in all other respects.
;/$033. In the Uncorr model, we treat the three relationships as
independent. Thus, we assume the artist-specific random effects
in qito be uncorrelated across the relationships, and therefore,
Sqhas a block-diagonal structure. This model is thus equiva-
lent to running three separate models on the three relationships.
;0!*!+. For the NoZiZj model, we do not include the higher
order terms z¢
izj that characterize the latent space.
Each artist can be described using the variables detailed
in Table 5. We use these artist-specific variables to compute
dyad-specific variables. We use different combinations of
# " ! ##!"
7Our data are not entirely representative of the whole network, which
consists of other artists who joined the network subsequent to our data
period. In addition, we focus only on the subnetwork of connections
involving artists, rather than also considering fans, because this is consis-
tent with the primary focus of the network in offering a platform for artists
to present themselves and seek collaborations.
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
the artist- and dyad-specific variables in explaining the
three relationships. (Appendix B gives details of the covari-
ate specifications for the three relationships.) Table 6 shows
descriptive statistics associated with these covariates. The
last two rows of Table 6 contain dyad-specific binary
variables. The variable “common region” is equal to 1 when
both artists in a dyad are from the same region; otherwise, it
is 0. We code the last variable, “common genre,” analogously.
We estimated the models using MCMC methods
(described in Appendix A). Each MCMC run is for 250,000
iterations, and the results are based on a sample of 200,000
iterations after a burn-in of 50,000.
0%&-"%&26"$:. Before discussing parameter estimates,
we use a combination of predictive measures and posterior
predictive model checking (Gelman, Meng, and Stern 1996)
to compare the adequacy of our models. This involves gen-
erating G hypothetical replicated data sets from the model
using the MCMC draws and comparing these data sets with
the actual data set. These comparisons are made using vari-
ous test quantities that represent different structural charac-
teristics of the network. If the replicated data sets differ sys-
tematically from the actual data on a given test quantity, the
model does not adequately mimic the structural characteris-
tic that the test quantity represents. The discrepancy
between the replicated data sets and the actual data can be
assessed using posterior predictive 1-value. This 1-value is
the proportion of the G replications in which the simulated
test quantity exceeds the realized value of that quantity in
the observed data. An extreme 1-value, (either close to 0 or
1; i.e., £.05 or ≥.95) suggests inadequate recovery of the
corresponding test quantity.
We use several test statistics associated with the weighted
relationship to assess whether the distinction between inci-
dence and intensity is crucial. Table 7 shows the model ade-
quacy results for the music download relationship based on
G = 10,000 MCMC draws. Column 2 of the table reports the
value ofthe teststatistics for the observed data,andthe
other columns report the posterior predictive mean and 1-
values for the different models. A few conclusions can be
readily drawn from the table. First, Poisson, which models
the intensity directly using a Poisson specification (as in
Hoff 2005), does not recover any of the test statistics ade-
quately, because almost all 1-values in Column 6 are
extreme. Furthermore, the posterior predictive mean values
for the test statistics (Column 5) are appreciably different
from their counterparts in the actual data. Second, we
observe that /0/& of the 1-values associated with the other
models are extreme, and this indicates that modeling inci-
"! "$!""!"! "!"
$ !
"3*"#-& #4&37"5*0/4 %/ */ "9
Page views 230 781.2 467 1069.8 43 10,931
Songs 230 4.108 3 3.26 1 29
Audience 230 .482 0 .5 01
Band 230 .704 1 .457 01
Years active 230 7.24 6 5.82 1 27
Common region 26,335 .628 1 .483 01
Common genre 26,335 .564 1 .495 01
"! "$ !
"3*"#-& &4$3*15*0/
Songs The number of songs available on the artist’s profile.
Band Whether the artist belongs to a band. Equal to 1 if artist
belongs to a band and 0 if otherwise.
Audience The audience size of the largest concert by the artist. A
median split yields 1 when the audience is > 700 and 0
Genre Genre of the artist. We distinguish between rock genres and
nonrock genres.
Region The geographical region to which the artist belongs. Artists
belong to one of the 26 cantons in Switzerland. These are
aggregated into three regions: French, German, and Italian.
PageViews The number of page views of the artist’s profile during the
data period. These page views originate from other
registered members, including the fans, or from Internet
users from outside the social network.
YActive The number of years of activity of the artist in the music
"!" "$ #!%! "# !
6-- /$033 0!*!+ 0*440/
"5" p"-6& p"-6& p"-6& p"-6&
Null dyads 23,376 23,420 .79 23,416 .77 23,420 .80 23,275 .03
Asymmetrical dyads 1802 1783 .35 1785 .36 1777 .30 2134 1
Mutual dyads 1157 1132 .24 1134 .26 1137 .28 925 0
Reciprocity .39 .39 .44 .39 .45 .39 .51 .30 0
Transitive triads 51,692 50,523 .3 50,710 .32 49,534 .16 44,303 0
Intransitive triads 136,230 133,398 .27 133,321 .26 135,012 .39 117,494 0
Mean degree 17.89 17.59 .18 17.62 .2 17.62 .21 17.32 .04
Standard indegree 17.924 18.1 .65 18.07 .62 18.16 .69 16.61 0
Standard outdegree 31.61 31.06 .14 31.09 .16 31.07 .14 27.74 0
Degree correlation .898 .893 .3 .89 .31 .893 .32 .896 .45
Mean strength 59.58 58.83 .25 58.69 .23 58.86 .26 209.4 1
Standard indegree 213.2 208.44 .34 206.34 .29 209.66 .38 1327.9 1
Standard outdegree 74.13 70.06 .15 69.47 .12 69.51 .13 751.1 1
dence and intensity separately is important for adequately
capturing the structural network characteristics associated
with weighted relations. Finally, because the NoZiZj model
performs almost as well as the full model, we conclude that
the latent space is not crucial for the recovery of structural
characteristics in this application.
The preceding discussion focuses on model adequacy for
music downloads to highlight the contribution of the corre-
lated multivariate hurdle Poisson in modeling weighted
relationships. The results from the other two relationships
show that all models (including Poisson) are similar in their
recovery of structural characteristics for these relationships.
The model adequacy results are based on in-sample simula-
tions. (We report the predictive performance of our model
in the Web Appendix at http://www.marketingpower. com/
jmraug11.) We find that the Full model outperforms other
models on almost all measures. We also find that the Pois-
son model does very poorlyin predicting future activity,
indicating that there are significant gains in modeling the
incidence and intensity separately.
We now discuss the parameterestimates based onthe
entire sample of six months. We focus on the parameter esti-
mates from the full model.Estimates from othermodels
mostly yield similar qualitative conclusions, and we do not
include these for the sake of brevity.
07"3*"5&&''&$54. Table 8 reports the posterior means and
standard deviations of the coefficient estimates for the three
relationships. In interpreting this table, recall that all
variables associated with the friendship relation are dyad
specific and binary, whereas, for the other relations, some
variables are dyad specific and some are artist specific.
Also, note that both components of the music relation have
the same set of covariates. (The definitions for these covari-
ates are available in Appendix B.)
For the sake of brevity, we synthesize the results across
all the relationships. There is clear evidence of homophily
and proximity: For all three relationships, the dyad-specific
variables CRegion and CGenre have a positive and signifi-
cant impact on the likelihood of forming connections. The
positive coefficient for CRegion is consistent with the
notion that a common language and geographical proximity
can enhance the likelihood of collaborative effort. The posi-
tive coefficient for CGenre means that pairs of artists who
produce music in the same genre have a higher propensity
to form friendship connections, communicate with each
other, and download music from each other. We also find
thatartists belonging to aband havea higher chance of
forming friendships (BothBand) and a higher probability of
sending and receiving communications (SBand and RBand).
We find that measures of online popularity that are based
on the total number of page views for an artist (BothPopu-
lar, SPviews, DPviews, and PPviews) positively influence
relationship formation. For example, the positive coeffi-
cients for DPviews in the music relation indicate that artists
with greater online popularity in this network have a greater
likelihood of downloading songs from other artists and that
they tend to download more music. In contrast, most measures
of prior and offline popularity or experience (indicating
audience size of concerts or years of activity) do not seem
to affect the formation of online relationships within the net-
work.8Finally, the data indicate that for the music relation-
ship, different coefficients influence the incidence and
intensity components of the music relationship. Thus, we
conclude that these two facets are not isomorphic and need
to be modeled separately.
07"3*"/$&4536$563&0'5)&3&-"5*0/4)*14. The covariance
matrix Sqcaptures the linkages among the expansiveness
and attractiveness parameters across the relationships. Table
9 reports the elements of Sq. A striking feature of Table 9 is
that all the covariances are significantly positive. The posi-
tive correlation within a relationship implies that attractive-
ness goes hand-in-hand with expansiveness (i.e., artists who
are sought by others also tend to be active in seeking rela-
tionships with others). Moreover, for the music relationship,
the attractiveness and expansiveness parameters for the
intensity equations are also positively correlated with the
corresponding random effects for the incidence component.
The positive correlations across relationships imply that an
artist who is popular in one relationship is also likely to be
both popular and productive in other relationships. Simi-
larly, an artist who is productive in one relationship is also
likely to be productive and popular in other relationships.
The utility errors for the two incidence equations of the
music download relationship are positively correlated (.74),
owing to shared unobservable influences and reciprocity.
The intensity equations are also positively correlated (.497).
Furthermore, the errors for the incidence equations are posi-
tively correlated with the errors for the intensity equations
(.395 and .411) implying selectivity through shared unob-
served factors driving both incidenceandintensity. This
corroborates the need for a multivariate correlated hurdle
Poisson specification. Finally, the communication utilities
also exhibit positive correlation (.511) driven by reciprocity
and other shared unobservables.
As interest in socialnetworks and brand communities
grows, marketers are becoming increasingly focused on
understanding and predicting the connectivity structure of
such networks. In this article, we developed a methodologi-
cal framework for jointly modeling multiple relationships
that vary in their directionality and intensity. Our integrated
approachforsocialnetwork analysis is unique inthat it
weaves together several distinct model components needed
for capturing multiplexity in networks.
We applied our framework to two distinct applications
that showcased different benefits ofour approach. Inthe
first application, we investigated the impact of an organiza-
tional intervention (R&Dcollocation) on the patterns of
communications among professionals involved in new
product development activities. In this application, we
specifically investigated the gains from modeling relation-
ships jointly. Our results clearly indicate how the different
components of our framework are needed for a clear assess-
ment of substantive hypotheses. We found that the hetero-
# " ! ##!"
8We thank an anonymous reviewer for pointing out that covariates relat-
ing to popularity could be potentially endogenous. However, given that we
include both online and offline correlates of popularity, as well as actor-
specific random effects that capture attractiveness, it is unclear whether
additional unobserved variables relating to popularity are part of the utility
error. However, caution is still needed in interpreting the results.
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
geneity specification, the latent space, and the correlations
across relationships can affect both substantive conclusions
and the recovery of structural characteristics. Finally, as the
Web Appendix shows (see http://www.marketingpower.
com/ jmraug11), our approach can be used to leverage infor-
mation from one relation to predict connectivity in another.
In our second application, we focused on modeling mul-
tiple relationships of different types. In particular, we
showed how it is critical to model both the incidence and
intensity ofweighted relationships such as musicdown-
loads; otherwise, recovery of structural characteristics and
predictive performance suffers appreciably. On the substan-
tive front, our results show that the friendship, communica-
tions, and music download relationships share common
antecedents and exhibit homophily and reciprocity. We
found that offline proximity is relevant for all online rela-
tionships, and this is consistent with our understanding that
these connections are formed to forge collaborative relation-
ships. We also found that the artistsexhibit similar roles
across relationships and that popular artists seem to be more
productive regardless of the relationship being studied.
Across the two applications, we found mixed evidence
regarding the benefits from incorporating a latent space. In
the first application, the latent space improved the recovery
of cross-relationship transitivity patterns and affected the
parameter estimates and the substantive findings. However,
higher-order effects did not seem to be important in the sec-
ond application. The latter result is consistent with Faust’s
(2007) conclusion that much of the variation in the triad cen-
sus across networks could be explained by simplerlocal
structure measures. These results suggest that the impact of
extradyadic effects could be application specific.
On the theoretical and substantive front, our framework
facilitates a detailed description of antecedents of relation-
ship formation and allows for theory testing taking into
account systematic variations in degree arising from
homophily and heterophily, local structuring, as well as
temporalor cross-relationship carryover(Rivera, Soder-
strom and Uzzi 2010). Our enquiry can be extended in many
directions. Our applications involved small networks. Most
online networks are much larger, and statistical methods
cannot scale directly to the level of these large networks.
However, recent research has shown that while online net-
works can have millions of members, communities within
such networks are relatively small, with sizes in the vicinity
of the 100–200 member range (Leskovec et al. 2008). This
implies that these very large networks can be broken down
intoclustersof tightly knit communities, andwhensuch
communities are identified, our methodological framework
can then be used on such communities to further understand
""!""! ""
"!! "#
Intercept –2.732 .167
CRegion .341 .050
CGenre .150 .033
BothPopular .163 .063
BothNotPopular .015 .087
BothBigSongs .061 .061
BothSmallSongs .016 .065
BothBand .640 .153
BothNoBand –.257 .163
BothBigAudience .203 .143
BothSmallAud –.134 .143
BothLongActive –.036 .075
BothShortActive .034 .071
Intercept –3.446 .270
CRegion .321 .042
CGenre .144 .028
SPviews .015 .004
SSongs .009 .010
SBand .321 .106
SAudience .135 .093
SYActive –.003 .006
RPviews .002 .003
RSongs .013 .009
RBand .464 .154
RAudience .043 .138
RYActive –.004 .006
Intercept –3.007 .272
CRegion .183 .040
CGenre .118 .031
PPviews .021 .004
PSongs –.006 .010
PBand .106 .104
PAudience .100 .091
PYActive –.013 .006
DPviews .031 .005
DSongs .022 .014
DBand .062 .141
DAudience –.113 .120
DYActive –.015 .009
Intercept –2.024 .269
CRegion .235 .050
CGenre .152 .038
PPviews .011 .003
PSongs .023 .009
PBand .169 .086
PAudience –.005 .074
PYActive –.003 .006
DPviews .034 .006
DSongs .021 .017
DBand –.055 .149
DAudience –.157 .120
DYActive –.012 .011
Notes: Bold indicates that the 95% posterior interval does not span 0.
"$ "" "!Sq
i1 .397 .378 .234 .222 .268 .500 .499
(.051) (.056) (.037) (.049) (.040) (.064) (.065)
i1 .707 .238 .441 .329 .617 .593
(.091) (.042) (.073) (.053) (.086) (.086)
i2 .192 .146 .195 .315 .319
(.034) (.038) (.033) (.050) (.052)
i2 .489 .188 .313 .288
(.082) (.048) (.079) (.081)
c.407 .478 .513
(.052) (.065) (.067)
c1.000 .988
(.109) (.104)
Note: Posterior standard deviations are in parentheses.
the nature of linkages within these subcommunities. How-
ever, such a divide-and-conquer approach is unlikely to pro-
vide a complete picture of the nature of link formation in
such large networks.
We focused on modeling static relationships or on
sequential relationships observed over a few time periods.
However, networks are dynamic entities in which connec-
tions are formed over time. Incorporating such dynamics
would be interesting. We used a parametric framework based
on the normal distribution for the latent variables and ran-
dom effects, and this was sufficient for recovery of skewed
degree distributions. However, in other situations, Bayesian
nonparametrics (Sweeting 2007) may be more useful.
1. The full conditional for precision matrix S–1
qof the actor-
specific random effects is a Wishart distribution given by
1. where the prior for S–1
qis Wishart(rq, Rq). The quantities rq
and Rqrefer to the scalar degree of freedom and the scale
matrix for the Wishart, respectively, and N is the number of
actors in the network.
2. The covariance matrices Sr
z, for relationship r are diagonal,
because zr
iis a p-dimensional vector of independent compo-
nents. Lets2
z,r denote the common variance for the compo-
nents of zr
i. The full conditional for s2
z,r is an inverse gamma
distribution given by
3. The full conditional for the coefficients mis multivariate nor-
mal because we have a seemingly unrelated regression sys-
tem of equations conditional on knowing the latent variables.
Form the adjusted utilities (e.g., u
~ij1 = uij1 –ail –bj1 –z¢
and adjusted log-rate parameters by subtracting terms that do
not involve mfrom the latent dependent variables. We then
have the system of equations, v
~{ij} = X{ij}m+ e{ij}, for an arbi-
trary pair {i, j}, where eij ~ N(0, S). We can write the full con-
ditional as follows:
1. where Wm
–1 = C–1 + S{ij}X¢
{ij}S–1X{ ij} and m
ˆ= Wm[C–1h+
4. The full conditional for the heterogeneity parameter qiis a
multivariate normal. We again begin by creating adjusted
utilities and rate parameters, such as by subtracting all terms
that do not involve qi. Let be the vector of adjusted utilities
for the three relationships. Then we have the system. Again,
we can use standard Bayesian theory for the multivariate nor-
mal to obtain the resulting full conditional:
where Wq
–1 = Sq
–1 + (N – 1)S–1 and q
ˆi= S–1WqSj πiv
5. The full conditional for zithat contains all the latent space
vectors associated with an individual i is multivariate normal.
Creating adjusted utilities such as u
~ij1z = uij1 –x¢
ij1m1–ai1 –
bj1, we can form the vector of adjusted utilities and latent rate
()( |{}) ,
Ap z apN
zr irirk
σ=+ +
== ∑∑
IG −−
() {} ,ApWishartNR
() (|{ }) (, ),
() (|{ }) (%,),
iij i
parameters v
~{ij}z = Zjzi+ e{ij}, whereZjis an appropriately
constructed matrix from the latent space vector of actor j. This
is a seemingly unrelated regression system. Given the prior zi~
N(0, Sz), where Szis constructed from the different Szr matri-
ces, we can write the full conditional as N(z
ˆiWzi), where Wzi
–1 =
(Sz)–1 + Sj πiZ¢
jS–1Zjand z
ˆi= Wzi[Sj πiZ¢
~{ij}z]. The
model depends on the inner product of the latent space vec-
tors, which is invariant to rotations and reflections of the vec-
tors. Thus visual representations of these vectors require that
they be rotated to a common orientation, This can be done by
using a Procustrean transformation as outlined in Hoff (2005).
6. The variance–covariance matrix of the errors for the
weighted relationship, S, has a special structure as described
in the “Modeling Framework” section. Given this special
structure, we follow the separation strategy of Barnard,
McCulloch, and Meng (2000) in setting the prior in terms of
the standard deviations and correlations in S. The covariance
matrix Scan be decomposed into a correlation matrix, R, and
a vector, s, of standard deviations—that is, S= diag(s) ¥R¥
diag(s), where scontainsthe square roots of the standard
deviations. Let wcontain the logarithms of the elements in s.
We assume a multivariate normal distribution N(0, I) for the
nonredundant elements of R, such that it is constrained to the
subspace of the p-dimensional cube [–1, 1]p, where p is the
number of equations that yields a positive definite correlation
matrix. Finally, we assume a univariate standard normal prior
for the single log-standard deviation in w.
•The full conditional distribution for the free element in the
vector of log-standard deviations wof errors can only be writ-
ten up to a normalizing constant (recall that the terms asso-
ciated with the binary utilities in ware fixed to 0 for identi-
fication purposes). Given our assumption of a normal prior
for the single free element, we use a Metropolis–Hastings
step to simulate the standard deviation in w. A univariate
normal proposal density can be used to generate candidates
for this procedure. If is the current value of kth component
of w, a candidate value is generated using a random walk
k= wk
(t – 1) + N(0, t), where tis a tuning constant
that controls the acceptance rate.
•Many different approaches can be used to sample the corre-
lation matrix R. Here, we use a multivariate Metropolis step
to sample a vector of nonredundant correlations in R. We
used adaptive MCMC (Atchade 2006) for obtaining the tun-
ing constant so as to ensure rapid mixing.
7. The full conditional distribution associated with the set of
latent utilities and latent rate parameters in uij is again
unknown. We sample the utilities and log-rate parameters
using univariate conditional draws. Sampling the utilities is
straightforward, because these are truncated univariate con-
ditional normal draws. The log-rate parameters log lij and
log lji are sampled such that these are univariate normal
draws if the corresponding observation involves a zero
count, and for an observation in which a positive count is
observed, we use a univariate Metropolis step that combines
the likelihood for a truncated Poisson distribution with a
conditional normal prior.
CRegion: CRegion is equal to 1 if both artists in a pair are from
the same region; 0 otherwise.
CGenre: CGenre is equal to 1 if both artists in a pair are from
the same genre; 0 otherwise.
BothPopular: BothPopular is equal to 1 if both artists in a pair are
viewed (online) by more than the population median; 0 otherwise.
# " ! ##!"
5*+2/4-;2:/62+ +2':/549./69/4!5)/'2+:=5819
BothNotPopular: BothNotPopular is equal to 1 if both artists in a
pair are viewed by fewer than the population median; 0 otherwise.
BothBigSongs: BothBigSongs is equal to 1 if both artists in a
pair post more songs than the population median; 0 otherwise.
BothSmallSongs: BothSmallSongs is equal to 1 if both artists in
a pair post fewer songs than the population median; 0 otherwise.
BothBand: BothBand is equal to 1 if both artists in a pair post
are from a band; 0 otherwise.
BothNoBand: BothNoBand is equal to 1 if both artists in a pair
post are not from a band; 0 otherwise.
BothBigAudience: BothBigAudience is equal to 1 if both artists
in a pair had large concerts with more than 700 spectators; 0
BothSmallAudience: BothSmallAudience is equal to 1 if both
artists in a pair had small concerts with more than 700 spectators;
0 otherwise.
BothLongActive: BothLongActive is equal to 1 if both artists in
a pair had being active for more than six years; 0 otherwise.
BothShortActive: BothShortActive is equal to 1 if both artists
in a pair had being active for less than six years;0 otherwise.
CRegion: CRegion is equal to 1 if both artists in a pair are from
the same region; 0 otherwise.
CGenre: CGenre is equal to 1 if both artists in a pair are from
the same genre; 0 otherwise.
SPviews: SPviews represents the number of sender paged views.
SSongs: SSongs represents the number of songs posted on the
sender web page.
SBand: SBand is equal to 1 if the sender belongs to a band; 0
SAudience: SAudience is equal to 1 if the sender performed in
front of an audience larger than 700 people; 0 otherwise.
SYActive: SYActive is equal to 1 if the sender was active for
more than six years; 0 otherwise.
RPviews: RPviews representsthenumber of receiver paged
RSongs: RSongs represents the number of songs posted on the
receiver web page.
RBand: RBand is equal to 1 if the receiver belongs to a band; 0
RAudience: RAudience is equal to 1 if the receiver performed
in front of an audience larger than 700 people; 0 otherwise.
RYActive: RYActive is equal to 1 if the receiver was active for
more than six years; 0 otherwise.
CRegion: CRegion is equal to 1 if both artists in a pair are from
the same region; 0 otherwise.
CGenre: CGenre is equal to 1 if both artists in a pair are from
the same genre; 0 otherwise.
PPviews: PPviews represents the numberof provider paged
PSongs: PSongs represents the number of songs posted on the
provider web page.
PBand: PBand is equal to 1 if the provider belongs to a band; 0
PAudience: PAudience is equal to 1 if the provider performed in
front of an audience larger than 700 people; 0 otherwise.
PYActive: PYActive is equal to 1 if the provider was active for
more than six years; 0 otherwise.
DPviews: DPviews represents the number of downloader paged
DSongs: DSongs represents the number of songs posted on the
downloader web page.
DBand: DBand isequal to 1if the downloader belongs to a
band; 0 otherwise.
DAudience: DAudience is equal to 1 if the downloader per-
formed in front of an audience larger than 700 people; 0 otherwise.
DYActive: DYActive is equal to 1 if the downloader was active
for more than six years; 0 otherwise.
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Linear Equation From A Table Of Values
Linear Equation From A Table Of Values
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Subjects: math,mathMiddleSchool,mathHighSchool,midSchoolFunctions,modelingFunctions,highSchoolFunctions,linearQuadraticAndExponentialModels
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2,081 research outputs found
It is demonstrated, owing to the nonlinearity of QED, that a static charge placed in a strong magnetic field\ $B$\ is a magnetic dipole (besides remaining an electric monopole, as well). Its magnetic
moment grows linearly with $B$ as long as the latter remains smaller than the characteristic value of 1.2\cdot 10^{13}\unit{G} but tends to a constant as $B$ exceeds that value. The force acting on a
densely charged object by the dipole magnetic field of a neutron star is estimated
We show, within QED and other possible nonlinear theories, that a static charge localized in a finite domain of space becomes a magnetic dipole, if it is placed in an external (constant and
homogeneous) magnetic field in the vacuum. The magnetic moment is quadratic in the charge, depends on its size and is parallel to the external field, provided the charge distribution is at least
cylindrically symmetric. This magneto-electric effect is a nonlinear response of the magnetized vacuum to an applied electrostatic field. Referring to a simple example of a spherically-symmetric
applied field, the nonlinearly induced current and its magnetic field are found explicitly throughout the space, the pattern of lines of force is depicted, both inside and outside the charge, which
resembles that of a standard solenoid of classical magnetostatics
We analyze the creation of fermions and bosons from the vacuum by the exponentially decreasing in time electric field in detail. In our calculations we use QED and follow in main the consideration of
particle creation effect in a homogeneous electric field. To this end we find complete sets of exact solutions of the $d$-dimensional Dirac equation in the exponentially decreasing electric field and
use them to calculate all the characteristics of the effect, in particular, the total number of created particles and the probability of a vacuum to remain a vacuum. It should be noted that the
latter quantities were derived in the case under consideration for the first time. All possible asymptotic regimes are discussed in detail. In addition, switching on and switching off effects are
studied.Comment: We add some references and minor comments. Version accepted for publication in Physica Scripta as a Invited Commen
Magnetically uncharged, magnetic linear response of the vacuum filled with arbitrarily combined constant electric and magnetic fields to an imposed static electric charge is found within general
nonlinear electrodynamics. When the electric charge is point-like and external fields are parallel, the response found may be interpreted as a field of two point-like magnetic charges of opposite
polarity in one point. Coefficients characterizing the magnetic response and induced currents are specialized to Quantum Electrodynamics, where the nonlinearity is taken as that determined by the
Heisenberg-Euler effective Lagrangian.Comment: The part dealing with magnetically charged responses is removed to be a subject of another paper after revisio
Upper bounds on fundamental length are discussed that follow from the fact that a magnetic moment is inherent in a charged particle in noncommutative (NC) electrodynamics. The strongest result thus
obtained for the fundamental lenth is still larger than the estimate of electron or muon size achieved following the Brodsky-Drell and Dehlmet approach to lepton compositeness. This means that NC
electrodynamics cannot alone explain the whole existing descrepancy between the theoretical and experimental values of the muon magnetic moment. On the contrary, as measurements and calculations are
further improved, the fundamental length estimate based on electron data may go down to match its compositeness radius
So-called '419' or 'advance-fee' e-mail frauds have proved remarkably successful. Global losses to these scams are believed to run to billions of dollars. Although it can be assumed that the promise
of personal gain which these e-mails hold out is part of what motivates victims, there is more than greed at issue here. How is it that the seemingly incredible offers given in these unsolicited
messages can find an audience willing to treat them as credible? The essay offers a speculative thesis in answer to this question. Firstly, it is argued, these scams are adept at exploiting common
presuppositions in British and American culture regarding Africa and the relationships that are assumed to exist between their nations and those in the global south. Secondly, part of the appeal of
these e-mails lies in the fact that they appear to reveal the processes by which wealth is created and distributed in the global economy. They thus speak to their readers’ attempts to map or
conceptualise the otherwise inscrutable processes of that economy. In the conclusion the essay looks at the contradictions in the official state response to this phenomena
It has been argued, that in noncommutative field theories sizes of physical objects cannot be taken smaller than an elementary length related to noncommutativity parameters. By gauge-covariantly
extending field equations of noncommutative U(1)_*-theory to the presence of external sources, we find electric and magnetic fields produces by an extended charge. We find that such a charge, apart
from being an ordinary electric monopole, is also a magnetic dipole. By writing off the existing experimental clearance in the value of the lepton magnetic moments for the present effect, we get the
bound on noncommutativity at the level of 10^4 TeV.Comment: 9 pages, revtex; v2: replaced to match the published versio
When exploring equations of nonlinear electrodynamics in effective medium formed by mutually parallel external electric and magnetic fields, we come to special static axial-symmetric solutions of two
types. The first are comprised of fields referred to as electric and magnetic responses to a point-like electric charge when placed into the medium. In electric case, this is a field determined by
the induced charge density. In magnetic case, this is a field carrying no magnetic charge and determined by an induced current. Fields of second type require presence of pseudoscalar constants for
their existence. These are singular on the axis drawn along the external fields. In electric case this is a field of an inhomogeneously charged infinitely thin thread. In magnetic case this is the
magnetic monopole with the Dirac string supported by solenoidal current. In both cases the necessary pseudoscalar constant is supplied by field derivatives of nonlinear Lagrangian taken on external
fields. There is also a magnetic thread solution dual to electric thread with null total magnetic charge.Comment: Published versio | {"url":"https://core.ac.uk/search/?q=author%3A(T.%20C.%20Adorno)","timestamp":"2024-11-02T08:31:26Z","content_type":"text/html","content_length":"136084","record_id":"<urn:uuid:c17895d6-8d67-4487-9c26-01caafea11c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00519.warc.gz"} |
PSEB 11th Class Physics Solutions Chapter 14 Oscillations
Punjab State Board PSEB 11th Class Physics Book Solutions Chapter 14 Oscillations Textbook Exercise Questions and Answers.
PSEB Solutions for Class 11 Physics Chapter 14 Oscillations
PSEB 11th Class Physics Guide Oscillations Textbook Questions and Answers
Question 1.
Which of the following examples represent periodic motion?
(a) A swimmer completing one (return) trip from one bank of a river to the other and back.
(b) A freely suspended bar magnet displaced from its N-S direction and released.
(c) A hydrogen molecule rotating about its center of mass.
(d) An arrow released from a bow.
(b) and (c)
Explanations :
(a) The swimmer’s motion is not periodic. The motion of the swimmer between the banks of a river is back and forth. However, it does not have a definite period. This is because the time taken by the
swimmer during his back and forth journey may not be the same.
(b) The motion of a freely-suspended magnet, if displaced from its N-S direction and released, is periodic. This is because the magnet oscillates about its position with a definite period of time.
(c) When a hydrogen molecule rotates about its center of mass, it comes to the same position again and again after an equal interval of time. Such motion is periodic.
(d) An arrow released from a bow moves only in the forward direction. It does not come backward. Hence, this motion is not a periodic.
Question 2.
Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
(a) the rotation of earth about its axis.
(b) motion of an oscillating mercury column in a 17-tube.
(c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lowermost point.
(d) general vibrations of a polyatomic molecule about its equilibrium position.
(b) and (c) are SHMs; (a) and (d) are periodic, but not SHMs
Explanations :
(a) During its rotation about its axis, earth comes to the same position again and again in equal intervals of time. Hence, it is a periodic motion. However, this motion is not simple harmonic. This
is because earth does not have a to and fro motion about its axis.
(b) An oscillating mercury column in a [/-tube is simple harmonic. This is because the mercury moves to and fro on the same path, about the fixed position, with a certain period of time.
(c) The ball moves to and fro about the lowermost point of the bowl when released. Also, the ball comes back to its initial position in the same period of time, again and again. Hence, its motion is
periodic as well as simple harmonic.
(d) A polyatomic molecule has many natural frequencies of oscillation. Its vibration is the superposition of individual simple harmonic motions of a number of different molecules. Hence, it is not
simple harmonic, but periodic.
Question 3.
Figure depicts four x-t plots for linear motion of a particle. Which of the plots represents periodic motion? What is the period of motion (in case of periodic motion)?
(b) and (d) are periodic
Explanation :
(a) It is not a periodic motion. This represents a unidirectional, linear uniform motion. There is no repetition of motion in this case.
(b) In this case, the motion of the particle repeats itself after 2 s. Hence, it is a periodic motion, having a period of 2 s.
(c) It is not a periodic motion. This is because the particle repeats the motion in one position only. For a periodic motion, the entire motion of the particle must be repeated in equal intervals of
time. In this case, the motion of the particle repeats itself after 2 s. Hence, it is a periodic motion, having a period of 2 s.
Question 4.
Which of the following functions of time represent (a) simple ‘ harmonic, (b) periodic but not simple harmonic, and (c) non¬periodic motion? Give period for each case of periodic motion (a is any
positive constant):
(a) sin ωt – cos ωt
(b) sin ^3ωt
(c) 3cos(\(\pi / 4 \) -2ωt)
(d) cos ωt +cos 3 ωt+cos 5ωt
(e) exp(-ω^2t^2)
(f) 1+ ωt+ω^2^t
(a) SHM
The given function is:
This function represents SHM as it can be written in the form: a sin (ωt +Φ)
Its period is: \(\frac{2 \pi}{\omega}\)
(b) Periodic, but not SHM The given function is:
sin^3 ωt = \(\frac{1}{4}\) [3sinωt -sin3ωt] (∵ sin3θ = 3sinθ – 4sin^3 θ)
The terms sin cot and sin 3 ωt individually represent simple harmonic motion (SHM). However, the superposition of two SHM is periodic and not simple harmonic.
Period of\(\frac{3}{4}\)sin ωt = \(\frac{2 \pi}{\omega}\) = T
Period of\(\frac{1}{4}\)sin3ωt = \(\frac{2 \pi}{3 \omega}\) = T’ = \(\frac{T}{3}\)
Thus, period of the combination
= Minimum time after which the combined function repeats
= LCM of T and \(\frac{T}{3}\) = T
Its period is 2 \(\pi / \omega\)
(c) SHM
The given function is:
3 cos \(\left[\frac{\pi}{4}-2 \omega t\right]\) = 3 cos \(\left[2 \omega t-\frac{\pi}{4}\right]\)
This function represents simple harmonic motion because it can be written in the form:
acos(ωt +Φ)
Its period is :
\(\frac{2 \pi}{2 \omega}=\frac{\pi}{\omega}\)
(d) Periodic, but not SHM
The given function is cosωt +cos3ωt +cos5ωt. Each individual cosme function represents SHM. However, the superposition of threc simple harmonic motions is periodic, but not simple harmonic.
cosωt represents SHM with period = \(\frac{2 \pi}{\omega}\) T (say)
cos 3ωt represents SHM with period = \(\frac{2 \pi}{3 \omega}=\frac{T}{3}\)
cos 5ωt represents SHM with period = \(\frac{2 \pi}{5 \omega}=\frac{T}{5}\)
The minimum time after which the combined function repeats its value is T. Hence, the given function represents periodic function but not SHM, with period T.
(e) Non-periodic motion: .
The given function exp(- ω^2t^2) is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.
(f) Non-periodic motion
The given function is 1+ ωt + ω^2t^2
Here no repetition of values. Hence, it represents non-periodic motion.
Question 5.
A particle Is in linear simple harmonic motion between two points, A and B, 10 cm apart Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force
on the particle when It Is
(a) at the end A,
(b) at the end B,
(c) at the midpoint of AB going towards A,
(d) at 2 cm away from B going towards A,
(e) at 3 cm away from A going towards B, and
(f) at 4 cm away from B going towards A.
The given situation is shown in the following figure. Points A and B are the two endpoints, with AB =10cm. 0 is the midpoint of the path.
A particle is in linear simple harmonic motion between the endpoints
(a) At the extreme point A, the particle is at rest momentarily. Hence, its velocity is zero at this point.
Its acceleration is positive as it is directed along with AO.
Force is also positive in this case as the particle is directed rightward.
(b) At the extreme point B, the particle is at rest momentarily. Hence, its velocity is zero at this point.
Its acceleration is negative as it is directed along B.
Force is also negative in this case as the particle is directed leftward.
The particle is extending a simple harmonic motion. O is the mean position of the particle. Its velocity at the mean position O is the maximum. The value for velocity is negative as the partide Is
directed leftward. The acceleration and force of a particle executing SHM is zero at the mean position.
The particle is moving towards point O from the end B. This direction of motion is opposite to the conventional positive direction, which is from A to R. Hence, the particle’s velocity and
acceleration, and the force on it are all negative.
The particle is moving towards point O from the end A. This direction of motion is from A to B, which is the conventional positive direction. Hence, the value for velocity, acceleration, and force
are all positive.
This case is similar to the one given in (d).
Question 6.
Whi ch of the following relationships between the acceleration a and the displacement x of a particle involves simple harmonic motion?
(a) a=0.7x
(b) a=-200x^2
(c) a= – 10 x (d) a=100x^3
A motion represents simple harmonic motion if it is governed by the force law:
ma’= -kx
∴ a = – \(\frac{k}{m}\) x
where F is the force
m is the mass (a constant for a body)
x is the displacement
a is the acceleration.
k is a constant
Among the given equations, only equation a = -10 x is written in the
above form with \( \frac{k}{m}\) =10. Hence, this relation represents SHM.
Question 7.
The motion of a particle executing simple harmonic motion is described by the displacement function,
x(t) = A cos (ωt+Φ)
If the initial (t = 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is πs^-1. If instead
of the cosine function, we choose the sine function to describe the SHM:x = B sin(ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions.
Initially, at t = 0:
Displacement, x = 1 cm Initial velocity, ν = ω cm/s.
Angular frequency, ω = π rad s^-1
It is given that:
x(t) = Acos (ωt+Φ)
1 = Acos(ω x 0 +Φ) = AcosΦ
AcosΦ =1 ……………………………….. (i)
Velocity, ν = \(\frac{d x}{d t}\)
ω = -Aω sin(ωt +Φ)
1 = -Asin(ω x 0 +Φ) = -AsinΦ
Asin Φ = -1 ………………………… (ii)
Squaring and adding equations (i) and (ii), we get
A^2(sin^2Φ +cos^2Φ) = 1+1
A^2 = 2
∴ A = \(\sqrt{2}\) cm
Dividing equation (ii) by equation (i), we get
tanΦ = -1
SHM is given as
x = Bsin(ωt + α)
Putting the given values in this equation, we get 1 =B sin (ωt + α)
B sin α =1 …………………………… (iii)
Velocity, ν = \(\frac{d x}{d t}\)
ω =(ωB cos (ωt + a)
1 =B cos (ω x 0+α ) = B cos α …………………………………… (iv)
Squaring and adding equations (iii) and (iv), we get
B^2 [sin^2α +cos^2 α] =1+1
B^2 =2
B = \(\sqrt{2}\) cm
Dividing equation (iii) by equation (iv), we get
Question 8.
A spring balance has a scale that reads from 0 to 50kg. The length of the scale is 20 cm. A body suspended from this balance, when displaced and released, oscillates with a period of 0.6 s. What is
the weight of the body?
Maximum mass that the scale can read, M = 50 kg
Maximum displacement of the spring = Length of the scale, l = 20 cm = 0.2 m Time period, T =0.6s
Maximum force exerted on the spring, F = Mg where,
g = acceleration due to gravity = 9.8 m/s^2
F = 50 × 9.8 = 490 N
∴ Spring constant , K = \(\frac{F}{l}=\frac{490}{0.2}\) = 2450Nm^-1
Mass m, is suspended from the balance,
Time period, T = \(2 \pi \sqrt{\frac{m}{k}}\)
∴ m = \(\left(\frac{T}{2 \pi}\right)^{2} \times k=\left(\frac{0.6}{2 \times 3.14}\right)^{2} \times 2450 \) = 22.36 kg
∴ Weight of the body = mg = 22.36 x 9.8 = 219.167N
Hence, the weight of the body is about 219 N.
Question 9.
Aspringhavingwith aspiring constant 1200Nm^-1 is mounted on a horizontal table as shown in figure. A mss of 3kg is attached to the free end of the spring. The mass is then pulled sideways to a
distance of 2.0cm and released.
Determine (i) the frequency of oscillations,
(ii) maximum acceleration of the mass, and
(iii) the maximum speed of the mass.
Spring constant, k = 1200 Nm^-1
mass,m = 3 Kg
Displacement,A = 2.0 cm = 0.02 m
(i) Frequency of oscillation y, is gyen by the relation
V = \(\frac{1}{T}=\frac{1}{2 \pi} \sqrt{\frac{k}{m}}\)
where, T is the time period
∴ v = \(\frac{1}{2 \times 3.14} \sqrt{\frac{1200}{3}}\) = 3.18 s^-1
Hence, the frequency of oscillations is 3.18 s^-1.
(ii) Maximum acceleration a is given by the relation:
a = ω^2A
ω = Angular frequency = \(\sqrt{\frac{k}{m}}\)
A = Maximum displacement
∴ a = \(\frac{k}{m} A=\frac{1200 \times 0.02}{3}\) = 8 ms^-2
Hence, the maximum acceleration of the mass is 8.0 ms^2
(iii) Maximum speed, ν[max] = Aω
= \(A \sqrt{\frac{k}{m}}=0.02 \times \sqrt{\frac{1200}{3}}\) = 0.4 m/s
Hence, the maximum speed of the mass is 0.4 m/s.
Question 10.
In question 9, let us take the position of mass when the spring is unstretched as x =0, and the direction from left to right as the positive direction of x-axis. Give x as a function of time t for
the oscillating massif at the moment we start the stopwatch (t = 0), the mass is
(a) at the mean position,
(b) at the maximum stretched position, and
(c) at the maximum compressed position.
In what way do these functions for SHM differ from each other, in frequency, in amplitude br the initial phase?
(a) The functions have the same frequency and amplitude, but different initial phases.
Distance travelled by the mass sideways, A = 2.0 cm
Force constant of the spring, k =1200 N m^-1
Mass, m =3kg
Angular frequency of oscillation,
ω = \(\sqrt{\frac{k}{m}}=\sqrt{\frac{1200}{3}} \) = \(\sqrt{400}\) = 20 rad s^-1
When the mass is at the mean position, initial phase is 0.
x = A sinωt = 2 sin20 t
(b) At the maximum stretched position, the mass is toward the extreme right. Hence, the initial phase is \(\frac{\pi}{2}\)
Displacement, x = Asin \(\left(\omega t+\frac{\pi}{2}\right)\)
=2sin\(\left(20 t+\frac{\pi}{2}\right)\)
= 2 cos 20t
(c) At the maximum compressed position, the mass is toward the extreme left. Hence, the initial phase is \(\frac{3 \pi}{2}\)
Displacement, x = A sin \(\left(\omega t+\frac{3 \pi}{2}\right)\)
= 2sin \(\left(20 t+\frac{3 \pi}{2}\right)\) = -2cos 20t
The functions have the same frequencyl \(\left(\frac{20}{2 \pi} \mathrm{Hz}\right)\) land amplitude (2cm),
but different initial phases \(\left(0, \frac{\pi}{2}, \frac{3 \pi}{2}\right)\)
Question 11.
Figures correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i. e., clockwise or anti-clockwise) are indicated on
each figure.
Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.
Time period,T =2s
Amplitude, A = 3cm
At time, t = O, the radius vector OP makes an angle \(\frac{\pi}{2}\) with the positive x -axis, i.e., phase angle Φ = + \(\frac{\pi}{2}\)
Therefore, the equation of simple harmonic motion for the x —projection of OP, at time t, is given by the displacement equation
(b) Time period, T = 4s
Amplitude, a =2 m
At time t = 0, OP makes an angle ir with the x-axis, in the anticlockwise direction. Hence, phase angle, Φ = +π
Therefore, the equation of simple harmonic motion for the x -projection of OP, at time t, is given as
Question 12.
Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (t = 0) position of the particle, the radius of the circle, and the angular speed of
the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: (x is in cm and t is in s).
(a) x= – 2sin(3t+\(\pi / \mathbf{3}\) )
(b) x=cos (\(\pi / 6\) – t)
(c) x=3 sin (2πt + \(\pi / 4 \) )
(d) x=2 cos πt
(a) x = -2 sin \(\left(3 t+\frac{\pi}{3}\right)=+2 \cos \left(3 t+\frac{\pi}{3}+\frac{\pi}{2}\right)=2 \cos \left(3 t+\frac{5 \pi}{6}\right) \)
If this equation is compared with the standard SHM equation,
x =A cos \(\left(\frac{2 \pi}{T} t+\phi\right)\) then we get
Amplitude, A = 2cm
Phase angle, Φ = \(\frac{5 \pi}{6}\) =150°
Angular velocity, ω = \(\frac{2 \pi}{T}\) =3 rad/sec
The motion of the particle can be pokted as shown in the following figure.
(b) x=cos \(\left(\frac{\pi}{6}-t\right)\) =cos \(\left(t-\frac{\pi}{6}\right)\)
If this equation is compared with the standard SHM equation,
x = A cos \(\left(\frac{2 \pi}{T} t+\phi\right)\) then we get
Amplitude, A = 1 cm
Phase angle, Φ = \(-\frac{\pi}{6} \) = – 30°
Angular velocity, ω = \(\frac{2 \pi}{T}\) =1 rad/s
The motion of the particle can be plotted as shown in the following figure.
(c) x =3sin \(\left(2 \pi t+\frac{\pi}{4}\right)\)
If this equation is compared with the standard SHM equation
x = Acos \(\left(\frac{2 \pi}{T} t+\phi\right)\) then we get
Amplitude, A = 3cm
Phase angle, Φ = \(-\frac{\pi}{4}\)
Angular velocity, ω = \(\frac{2 \pi}{T}\) = 2π rad/s
The motion of the particle can be plotted as shown in the following figure.
(d) x=2cosπt
If this equation is compared with the standard SHM equation,
x = A cos \(\left(\frac{2 \pi}{T} t+\phi\right) \) then we get
Amplitude, A = 2cm
Phase angle, Φ = 0
Angular velocity, ω = π rad/s
The motion of the particle can be plotted as shown in the following figure.
Question 13.
Figure (a) shows a spring of force constant k clamped rigidly at one end and a mass m attached to its free end. A force F applied at the free end stretches the spring. Figure (b) shows the same
spring with both ends free and attached to a mass m at either end. Each end of the spring in figure (b) is stretched by the same force F.
(a) What is the minimum extension of the spring in the two cases?
(b) If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the period of oscillation in each case?
(a) For the one block system:
When a force F, is applied to the free end of the spring, an extension l, is produced. For the maximum extension, it can be written as:
where k is the spring constant
Hence, the maximum extension produced in the spring, l = \(\frac{F}{k}\)
For the two blocks system:
The displacement (x) produced in this case is:
x = \(\frac{l}{2}\)
Net force, F = +2kx =2k \(\frac{l}{2}\)
∴ l = \(\frac{F}{k}\)
(b) For the one blocks system:
For mass (m) of the block, force is written as
F = ma = m \(\frac{d^{2} x}{d t^{2}}\)
where, x is the displacement of the block in time t
∴ m \(\frac{d^{2} x}{d t^{2}}\) = -kx
It is negative because the direction of elastic force is opposite to the direction of displacement.
where, ω is angular frequency of the oscillation
∴ Time period of the oscillation,
T= \(\frac{2 \pi}{\omega}=\frac{2 \pi}{\sqrt{\frac{k}{m}}}=2 \pi \sqrt{\frac{m}{k}}\)
For the two blocks system:
F=m \(\frac{d^{2} x}{d t^{2}}\)
m \(\frac{d^{2} x}{d t^{2}}\) =-2kx
It is negative because the direction of elastic force is opposite to the direction of displacement.
\(\frac{d^{2} x}{d t^{2}}\) = \(-\left[\frac{2 k}{m}\right] x \) = – ω^2x
where, Angular frequency, ω = \(\sqrt{\frac{2 k}{m}}\)
∴ Time period T = \(\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{m}{2 k}} \)
Question 14.
The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 in. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min, what is its
maximum speed?
Angular frequency of the piston, ω = 200 rad/min.
Stroke =1.0 m
Amplitude, A = \(\frac{1.0}{2}\) = 0.5m
The maximum speed (ν[max]) of the piston is given by the relation
ν[max] =Aω = 200 x 0.5=100 m/min
Question 15.
The acceleration due to gravity on the surface of moon is 1.7 ms^-2. What is the time period of a simple pendulum on the surface of moon If Its time period on the surface of earth is 3.5 s? (gon the
surface of earth is 9.8 ms^-2)
Acceleration due to gravity on the surface of moon, g’ = 1.7m s^-2
Acceleration due to gravity on the surface of earth, g = 9.8 ms^-2
Time period of a simple pendulum on earth, T = 3.5 s
T= \(2 \pi \sqrt{\frac{l}{g}}\)
where l is the length of the pendulum
∴ l = \(\frac{T^{2}}{(2 \pi)^{2}} \times g=\frac{(3.5)^{2}}{4 \times(3.14)^{2}} \times 9.8 \mathrm{~m} \)
The length of the pendulum remains constant.
On Moon’s surface, time period,
T’ = \(2 \pi \sqrt{\frac{l}{g^{\prime}}}=2 \pi \sqrt{\frac{(3.5)^{2}}{\frac{4 \times(3.14)^{2}}{1.7}} \times 9.8} \) = 8.4 s
Hence, the time period of the simple pendulum on the surface of Moon is 8.4 s.
Question 16.
Answer the following questions:
(a) Time period of a particle in SHM depends on the force constant k and mass m of the particle:
T = \(2 \pi \sqrt{\frac{m}{k}}\) A simple pendulum executes SHM approximately. Why then is the time period of a pendulum independent of the mass of the pendulum?
(b) The motion of a simple pendulum is approximately simple harmonic for small-angle oscillations. For larger angles of oscillation a more involved analysis shows that T is greater than \(2 \pi \sqrt
{\frac{l}{g}}\) Think of a qualitative argument to appreciate this result.
(c) A man with a wristwatch on his hand falls from the top of a tower. Does the watch give correct time during the free fall?
(d) What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under gravity?
Solution :
(a) The time period of a simple pendulum, T = \(2 \pi \sqrt{\frac{m}{k}}\)
For a simple pendulum, k is expressed in terms of mass m, as
k ∝ m
\(\frac{m}{k}\) = Constant
Hence, the time period T, of a simple pendulum is independent of the mass of the bob.
(b) In the case of a simple pendulum, the restoring force acting on the bob of the pendulum is given as
F = -mg sinθ
where, F = Restoring force; m = Mass of the bob; g = Acceleration due to
gravity; θ = Angle of displacement
For small θ, sinθ ≈ θ
For large 0,sin0 is greater than 0.
This decreases the effective value of g.
Hence, the time period increases as
T = \(2 \pi \sqrt{\frac{l}{g}}\)
where, l is the length of the simple pendulum
(c) The time shown by the wristwatch of a man falling from the top of a tower is not affected by the fall. Since a wristwatch does not work on the principle of a simple pendulum, it is not affected
by the acceleration due to gravity during free fall. Its working depends on spring action.
(d) When a simple pendulum mounted in a cabin falls freely under gravity, its acceleration is zero. Hence the frequency of oscillation of this simple pendulum is zero.
Question 17.
A simple pendulum of length l and having a bob of mass M is suspended in a car. The car is moving on a circular track of radius R with a uniform speed v. If the pendulum makes small f oscillations in
a radial direction about its equilibrium position, what will be its time period?
The bob of the simple pendulum will experience the acceleration due to gravity and the centripetal acceleration provided by the circular motion of the car.
Acceleration due to gravity = g
Centripetal acceleration = \(\frac{v^{2}}{R}\)
where, v is the uniform speed of the car R is the radius of the track
Effective acceleration (a[eff]) is given as
a[eff] = \(\sqrt{g^{2}+\left(\frac{v^{2}}{R}\right)^{2}}\)
Time period, T = \( 2 \pi \sqrt{\frac{l}{a_{e f f}}}\)
where, l is the length of the pendulum
∴ Time period, T = \(2 \pi \sqrt{\frac{l}{g^{2}+\frac{v^{4}}{R^{2}}}} \)
Question 18.
A cylindrical piece of cork of density of base area A and height h floats in a liquid of density ρ[l]. The cork is depressed slightly and then released. Show that the cork oscillates up and down
simple harmonically with a period
T = \(2 \pi \sqrt{\frac{\boldsymbol{h} \rho}{\rho_{\boldsymbol{l}} \boldsymbol{g}}}\)
where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).
Base area of the cork = A
Height of the cork = h
Density of the liquid = ρ[l]
Density of the cork = ρ
In equilibrium:
Weight of the cork = Weight of the liquid displaced by the floating cork Let the cork be depressed slightly by x. As a result, some extra water of a certain volume is displaced. Hence, an extra
up-thrust acts upward and provides the restoring force to the cork.
Up-thrust = Restoring force, F = Weight of the extra water displaced
F = -(Volume x Density x g)
Volume = Area x Distance through which the cork is depressed Volume = Ax
∴ F = -Ax ρ[l]g ………………………… (i)
According to the force law,
F = kx
k = \(\frac{F}{x}\)
where k is a constant
k = \(\frac{F}{x}\) = -Aρ[l]g ………………………………. (ii)
The time period of the oscillations of the cork,
T = \(2 \pi \sqrt{\frac{m}{k}} \) …………………………………… (iii)
m = Mass of the cork
= Volume of the cork x Density
= Base area of the cork x Height of the cork x Density of the cork = Ahρ
Hence, the expression for the time period becomes
T = \(2 \pi \sqrt{\frac{A h \rho}{A \rho_{l} g}}\) = 2\(\pi \sqrt{\frac{h \rho}{\rho_{l} g}} \)
Question 19.
One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction
pump is removed, the column of mercury in the U-tube executes simple harmonic motion.
Area of cross-section of the U-tube = A
Density of the mercury column = ρ
Acceleration due to gravity = g
Restoring force, F = Weight of the mercury column of a certain height
F = -(Volume x Density x g)
F = -(A x 2h x ρ x g) = -2Aρgh = -k x Displacement in one of the arms (h)
where, 2h is the height of the mercury column in the two arms
k is a constant, given by k = \(-\frac{F}{h}\) = 2Aρg
Time period = \(2 \pi \sqrt{\frac{m}{k}}=2 \pi \sqrt{\frac{m}{2 A \rho g}}\)
where, m is the mass of the mercury column
Let l be the length of the total mercury in the U-tube.
Mass of mercury, m = Volume of mercury x Density of mercury = Alρ
∴ T = \(2 \pi \sqrt{\frac{A l \rho}{2 A \rho g}}=2 \pi \sqrt{\frac{l}{2 g}} \)
Hence the mercury column executes simple harmonic motion with time period \(2 \pi \sqrt{\frac{l}{2 g}} \)
Additional Exercises
Question 20.
An air chamber of volume V has a neck area of cross-section a into which a ball of mass m just fits and can move up and down without any friction (see figure). Show that when the ball is pressed down
a little and released, it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal.
Volume of the air chamber = V
Area of cross-section of the neck = a
Mass of the ball = m
The pressure inside the chamber is equal to the atmospheric pressure. Let the ball be depressed by x units. As a result of this depression, there would be a decrease in the volume and an increase in
the pressure inside the chamber.
Decrease in the volume of the air chamber, ΔV = ax
Volumetric strain =
⇒ \(\frac{\Delta V}{V}=\frac{a x}{V}\)
Bulk Modulus of air, B = \(\frac{\text { Stress }}{\text { Strain }}=\frac{-p}{\frac{a x}{V}}\)
In this case, stress is the increase in pressure. The negative sign indicates that pressure increases with a decrease in volume.
p = \(\frac{-B a x}{V}\)
The restoring force acting on the ball,
F = p × a = \(\frac{-B a x}{V} \cdot a=\frac{-B a^{2} x}{V}\) ……………………………. (i)
In simple harmonic motion, the equation for restoring force is
F = -kx …………………………………….. (ii)
where, k is the spring constant Comparing equations (i) and (ii), we get
k = \(\frac{B a^{2}}{V}\)
Time period, T = \(2 \pi \sqrt{\frac{m}{k}}=2 \pi \sqrt{\frac{V m}{B a^{2}}}\)
Question 21.
You are riding in an automobile of mass 3000kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is
placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (a) the spring constant k and (6) the damping constant b for the spring and
shock absorber system of one wheel, assuming that each wheel supports 750 kg.
Mass of the automobile, m = 3000 kg
Displacement in the suspension system, x = 15cm = 0.15 m
There are 4 springs in parallel to the support of the mass of the automobile.
The equation for the restoring force for the system:
F = -4 kx = mg
where, k is the spring constant of the suspension system
Time period, T = \(2 \pi \sqrt{\frac{m}{4 k}}\)
and, k = \(\frac{m g}{4 x}=\frac{3000 \times 10}{4 \times 0.15}\) = 50000 = 5 x 10 ^4N/m
Spring constant, k = 5 x 10^4 N/m
Each wheel supports a mass, M = \(\frac{3000}{4}\) = 750 kg
For damping factor b, the equation for displacement is written as:
x = x[0]e^-bt/2M
The amplitude of oscillation decreases by 50%
where, Time petiod,t =\(2 \pi \sqrt{\frac{m}{4 k}}=2 \pi \sqrt{\frac{3000}{4 \times 5 \times 10^{4}}}\) =0.7691s
∴ b =\(\frac{2 \times 750 \times 0.693}{0.7691}\) =1351.58kg/s
Therefore, the damping constant of the spring is 1351.58 kg/s.
Question 22.
Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.
The equation of displacement of a particle executing SHM at an instant t is given as
x = Asinωt
A = Amplitude of oscillation
ω = Angular frequency = \(\sqrt{\frac{k}{M}}\)
The velocity of the particle is
ν = \(\frac{d x}{d t}\) = Aωcosωt
The kinetic energy of the particle is
E[k] = \(\frac{1}{2} M v^{2}=\frac{1}{2} M A^{2} \omega^{2} \cos ^{2} \omega t \)
The potential energy of the particle is
E[p] = \( \frac{1}{2} k x^{2}=\frac{1}{2} M \omega^{2} A^{2} \sin ^{2} \omega t\)
For time period T, the average kinetic energy over a single cycle is given as
And, average potential energy over one cycle is given as
It can be inferred from equations (i) and (ii) that the average kinetic energy for a given time period is equal to the average potential energy for the same time period.
Question 23.
A circular disc of mass 10 kg is suspended by a wire attached to its center. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be 1.5s. The
radius of the disc is 15 cm. Determine the torsional spring constant of the wire. (Torsional spring constant a is defined by the relation J = -αθ, where J is the restoring couple and 0 the angle of
Mass of the circular disc, m = 10 kg
Radius of the disc, r = 15cm = 0.15 m
The torsional oscillations of the disc has a time period, T = 1.5 s The moment of inertia of the disc is
I = \(\frac{1}{2}\) mr^2 = \(\frac{1}{2} \times(10) \times(0.15)^{2}\) = 0.1125kg-m^2
Time period, T = \(2 \pi \sqrt{\frac{I}{\alpha}}\)
where, α is the torsional constant.
An21 4 x(3.14)2x 0.1125 , M
α = \(\frac{4 \pi^{2} I}{T^{2}}=\frac{4 \times(3.14)^{2} \times 0.1125}{(1.5)^{2}} \) = 1.972 N-m/rad
Hence, the torsional spring constant of the wire is 1.972 N-m rad^-1.
Question 24.
A body describes simple harmonic motion with amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (a) 5 cm, (b) 3 cm, (c) 0 cm.
Amplitude, A = 5 cm = 0.05m
Time period, T = 0.2 s
(a) For displacement, x = 5 cm = 0.05m
Acceleration is given by
a = -ω^2x = \(-\left(\frac{2 \pi}{T}\right)^{2} x=-\left(\frac{2 \pi}{0.2}\right)^{2} \times 0.05\)
Velocity is given by
ν = ω \(\sqrt{A^{2}-x^{2}}=\frac{2 \pi}{T} \sqrt{(0.05)^{2}-(0.05)^{2}}\) = 0
When the displacement of the body is 5 cm, its acceleration is -5π^2 m/s^2 and velocity is 0.
(b) For displacement, x =3 cm = 0.03 m
Acceleration is given by
a = – ω^2x = – \(\left(\frac{2 \pi}{T}\right)^{2}\)x = \(\left(\frac{2 \pi}{0.2}\right)^{2}\) 0.03 = -3π^2 m/s^2
Velocity is given by
When the displacement of the body is 3 cm, its acceleration is -3π m/s^2 and velocity is 0.4π m/s.
(c) For displacement, x = 0
Acceleration is given by
a = – ω^2x = 0
Velocity is given by
When the displacement of the body is 0, its acceleration is 0, and velocity is0.5π m/s.
Question 25.
A mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction or damping. It is pulled to a distance x[0] and pushed towards the center with a
velocity v[0] at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x[0,] and v[0]. [Hint: Start with the equation x = a cos(ωt + θ) and note that the
initial velocity is negative.]
The displacement equation for an oscillating mass is given by
x = Acos(ωt + θ) …………………………… (i)
where A is the amplitude
x is the displacement
θ is the phase constant
Squaring and adding equations (iii) and (iv), we get
Hence, the amplitude of the resulting oscillation is \(\sqrt{x_{0}^{2}+\left(\frac{v_{0}}{\omega}\right)^{2}}\)
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Did You Know There was a Bug in Windows Calculator All These Years.
The calculator, better known as the "calc" has been an integral component of Windows since 1989 when windows version 3.0 was introduced. A lot has changed in Windows over the years, but apart from
the look and feel, the good old calculator has remained almost the same.
Now can you imagine a bug that existed in the calculator since it was first introduced and was never fixed by the Microsoft guys? Oh yes, there is a bug and a very stupid one indeed.
Here's how you can see it - Just try to subtract 2 from the square root of 4.
Square root of 4 is 2 and "2-2" should give 0. But here's the surprise - our calculator would give a small negative number as the result instead of 0. In "Standard" mode, you would see the result "
-1.068281969439142e-19 " and in the "Scientific" mode, the result would be "-8.1648465955514287168521180122928e-39".
In fact, you can reproduce the error with any similar calculation like "sqrt(9)-3" or "sqrt(16)-4" and so on. Here's a video of the bug in action -
The real reason for the bug is the way calculator handles sqrt operations. The results are stored as floating point numbers instead of integers and the small precision error when it comes to floating
point calculations is what you see.
It is definitely a bug which either the super intelligent Microsoft techies never found or simply didn't bother to fix. Never the less, I am sure you will ask yourself "Should I double check" the
next time you use the calculator :)
Related: Did you know your windows calculator could do these? | {"url":"https://www.skipser.com/p/2/p/did-you-know-there-is-a-bug-in-windows-calculator.html","timestamp":"2024-11-07T07:15:52Z","content_type":"application/xhtml+xml","content_length":"23203","record_id":"<urn:uuid:0ccd7f24-f58c-46a0-a572-73c0719a563e>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00859.warc.gz"} |
1.1: Representing Data Notes | Knowt
• Dot plot: good for displaying quantitative data for a relatively small number of discrete values
• Stem plot: all but the final digit; good for displaying quantitative data
• Stem leaf plot: including the final digits; good for displaying quantitative data
• Histogram: essentially a bar graph for quantitative data | {"url":"https://knowt.com/note/afc50102-9362-48dc-8d80-25db431b1295/11-Representing-Data","timestamp":"2024-11-03T16:47:11Z","content_type":"text/html","content_length":"174284","record_id":"<urn:uuid:069e7d66-3fbc-4473-8358-b0ef1463b21c>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00736.warc.gz"} |
CIE March 2022 9709 Prob & Stats 2 Paper 62
CIE March 2022 9709 Prob & Stats 2 Paper 62 (pdf)
1. The lengths, in millimetres, of a random sample of 12 rods made by a certain machine are as follows.
2. Harry has a five-sided spinner with sectors coloured blue, green, red, yellow and black. Harry thinks the spinner may be biased. He plans to carry out a hypothesis test with the following
3. A random sample of 500 households in a certain town was chosen. Using this sample, a confidence interval for the proportion, p, of all households in that town that owned two or more cars was
found to be 0.355 < p < 0.445.
Find the confidence level of this confidence interval. Give your answer correct to the nearest integer.
4. In the past the time, in minutes, taken by students to complete a certain challenge had mean 25.5 and standard deviation 5.2. A new challenge is devised and it is expected that students will
take, on average, less than 25.5 minutes to complete this challenge. A random sample of 40 students is chosen and their mean time for the new challenge is found to be 23.7 minutes.
(a) Assuming that the standard deviation of the time for the new challenge is 5.2 minutes, test at the 1% significance level whether the population mean time for the new challenge is less than
25.5 minutes.
5. The heights of buildings in a large city are normally distributed with mean 18.3 m and standard deviation 2.5 m.
(a) Find the probability that the total height of 5 randomly chosen buildings in the city is more than 95 m.
(b) Find the probability that the difference between the heights of two randomly chosen buildings in the city is less than 1m.
6. In a game a ball is rolled down a slope and along a track until it stops. The distance, in metres, travelled by the ball is modelled by the random variable X with probability density function
7. (a) Two ponds, A and B, each contain a large number of fish. It is known that 2.4% of fish in pond A are carp and 1.8% of fish in pond B are carp. Random samples of 50 fish from pond A and 60
fish from pond B are selected.
Use appropriate Poisson approximations to find the following probabilities.
(i) The samples contain at least 2 carp from pond A and at least 2 carp from pond B.
(ii) The samples contain at least 4 carp altogether.
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page. | {"url":"https://www.onlinemathlearning.com/mar-2022-9709-62.html","timestamp":"2024-11-06T02:49:01Z","content_type":"text/html","content_length":"36510","record_id":"<urn:uuid:d7b79308-4a17-472c-b2bd-37271052730f>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00711.warc.gz"} |
[Solved] Find the equation of the straight line parallel to Y-a... | Filo
Find the equation of the straight line parallel to -axis and at a distance (i) 3 units to the right (ii) 2 units to the left.
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(i) Equation of straight line parallel to -axis at a distance units to the right is .
Required equation is
(ii) Equation of straight line parallel to -axis at distance a units to the left is .
Required equation is .
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Question Text Find the equation of the straight line parallel to -axis and at a distance (i) 3 units to the right (ii) 2 units to the left.
Topic Straight Lines
Subject Mathematics
Class Class 11
Answer Type Text solution:1
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NeIC 2024 Workshop: Quantum Computing 101
NeIC 2024 Workshop: Quantum Computing 101#
This is a copy of the training contents for the workshop. The public archive can be found here.
In this workshop, we will have a look at the convergence of high-performance computing and quantum computing. Computational modelling is one field that in the future, is expected to be accelerated by
quantum computers.
We start with a presentation NeIC project, Nordic-Estonian Quantum Computing e-Infrastructure Quest (NordIQuEst), by Alberto Lanzanova. NordIQuEst is a cross-border collaboration of seven partners
from five NeIC member states that will combine several HPC resources and quantum computers into one unified Nordic quantum computing platform.
A practical approach to quantum programming follows this. In order to use quantum computers, in the future, novel quantum algorithms are required. These can, and should! be developed already now. In
this part of the workshop, participants will get a chance to submit a quantum job to a real quantum computer. Participants will be shown how to entangle multiple qubits and be given tips on getting
the most out of quantum computers today.
This will be followed by an introduction into a hybrid quantum-classical algorithm: the Variational Quantum Eigensolver. This workshop will utilise the EuroHPC supercomputer LUMI and Finland’s
5-qubit quantum computer Helmi.
For the hands-on tutorials, basic familiarity with Python and some experience working in a Unix environment are desirable. No previous experience with quantum computers expected. | {"url":"https://nordiquest.net/application-library/training-material/neic2024-qc101/index.html","timestamp":"2024-11-09T12:33:05Z","content_type":"text/html","content_length":"31025","record_id":"<urn:uuid:a2e7cc48-907e-40b2-a90e-4a2d7ab0c693>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00712.warc.gz"} |
What Are The Scoring Chapters In Class 6 Maths? - We Will Inspire
What Are The Scoring Chapters In Class 6 Maths?
6th-grade Maths is a completely different subject from what you’ve studied in the past. At first, glance, as you pick up the textbooks and begin to go through them, it becomes apparent that this
class’s syllabus is vastly different from the last. You may have stumbled upon some brand new concepts. At this point, it’s important to know that you’ve entered a new phase of your academic life.
The things you’ll be studying now will be new to you, but they’ll also be incredibly crucial for future classes that you’ll be taking.
Most of the time, due to a lack of understanding of the subject, a student might instil fear in their minds and make themselves believe that mathematics is a tough subject. A student must always
remember that Mathematics is the foundation of learning. And if they don’t concentrate properly from the earliest, it can create a lot of problems in the subject in further classes.
While preparing for Maths, you should be well-versed with the chapters. Although, there is no particular blueprint given in Class 6. But if you find out the important chapters, you can focus more on
them in order to get good marks. To start with, let’s understand the syllabus of CBSE Class 6 Maths:
· knowing Our Numbers
· whole Numbers
· playing With Numbers
· basic Geometrical Ideas
· understanding Elementary Shapes
· integers
· fractions
· decimals
· data Handling
· mensuration
· algebra
· ratio And Proportion
· symmetry
· practical Geometry
Now, you might be wondering what are the chapters that you should focus on more to score well in exams. Well, you score well by studying all the chapters thoroughly. As these chapters are just at the
basic level, they will be interesting and fun to learn.
Chapters like Knowing Our Numbers, Whole Numbers, Symmetry, Data Handling, Practical geometry are some really easy and fascinating chapters that will help you to get full marks undoubtedly. And the
rest of the chapters like Integers, Decimals, fractions, Algebra, etc, which are new to you are also scoring chapters. All you need to do is to imbibe the concepts comprehensively. To make it easier
for you, below listed are the chapters with their important topic and concepts that they entail:
· KNOWING OUR NUMBERS- This chapter introduces the students to the array of infinite numbers. This chapter includes the concept of various complex numbers such as thousands or billions. They are
taught the Indian system of counting as well as the International system. Additionally, this chapter covers complex numbers, estimation sum, and the difference between two numbers, among other
· WHOLE NUMBERS- In this chapter, you will learn the whole numbers and their concepts and properties in detail. You will learn about the properties of whole numbers, the patterns in a series of whole
numbers, and much more. You will learn important terms like “predecessor” and “successor” and the concept behind them.
· PLAYING WITH NUMBERS- This chapter discusses the concepts of prime and composite numbers, factorization, HCF, and LCM. You will get introduced to other important concepts, such as divisibility of
numbers and prime factorization.
· BASIC GEOMETRICAL IDEAS- The purpose of this chapter is to familiarize students with basic geometry. Topics such as ray and curves, polygons, parallel lines, triangles, and circles are discussed.
In this course, students will learn the fundamentals of prospective geometrical problems.
· UNDERSTANDING ELEMENTARY SHAPES- Here, students are introduced to various mathematical designs and shapes. Line segments, right angles, acute angles, obtuse angles, and reflex angles, perpendicular
angles, and polygons are some of the important topics covered in this chapter.
· INTEGERS- This chapter takes you along to the world that stays on the left side on the number line. Comparing and ordering Integers, the addition of integers, and subtraction of integers are some
topics that you will learn in this chapter.
· FRACTIONS- This chapter unfolds the numbers in fractions. Fractions on Number line, Proper fractions and improper fractions, equivalent fractions, like fractions and unlike fractions, adding or
subtracting like fractions, are some concepts you will learn in this chapter.
· DECIMALS- This chapter introduces the point between numbers. This point is called decimal. While studying this chapter, you will learn about the importance of decimal numbers. Decimals in tenths,
hundredths, etc. are introduced as you keep moving forward in the chapter.
· DATA HANDLING- In this chapter, you’ll learn about a variety of graphical operations. During the course, students are introduced to the concepts of drawing and organizing graphical data. In this
chapter, students learn how to calculate values using different types of graphs like pictographs.
· MENSURATION- Calculating various designs and figures using their area and volume is part of this chapter. Students learn about topics such as the area of a rectangle and measuring the perimeter in
this chapter.
· ALGEBRA- Class 6 introduces the students to a new topic called ‘Algebra’. This chapter teaches the students the algebra concepts such as determining the value of an unknown quantity using known
variables or evaluating the relationship between unknown variables using given variables.
· RATIO AND PROPORTION- Now that students are familiar with fractions, they are taken a higher level of ratio and proportion. Initially, they are taught the concepts of both ratio and proportion
separately, then they combine them to teach the relationship between ratio and proportion.
· SYMMETRY- This chapter expands on students’ knowledge of geometric shapes to introduce them to the idea of symmetry. They are taught multiple lines of symmetry, symmetry, and reflection, etc.
· PRACTICAL GEOMETRY- All of the principles of drawing various geometrical shapes are covered in Practical Geometry. In this chapter, you will focus on various geometrical tools and their
applications. In this chapter, students will learn how to construct an angle, a bisector, and a line segment by following a step-by-step approach.
If you are seeking to score high scores in Class 6 Maths, then you should stick to NCERT books as they are more than enough for you to score well. Keep your concepts clear as these chapters are going
to be extremely beneficial for higher classes. | {"url":"https://www.wewillinspire.com/scoring-chapters-class-6-maths/","timestamp":"2024-11-13T11:30:17Z","content_type":"text/html","content_length":"47111","record_id":"<urn:uuid:4552baae-9930-4b0d-a475-69dd597d892b>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00728.warc.gz"} |
Mathematics Probability Level: Misc Level
On further reflection,the professor remembered that Tariq, Sean,Steven and Victoria each asked at least 3 questions, while Emily,Mila,Thomas and Eric didn,t ask any In light of this,in how many
different ways could the 35 questions have been distributed among the students
Mathematics Probability Level: Misc Level
Considering only the number of questions asked by each students,in how many different ways could the 35 questions have been distributed among the students?
Mathematics Probability Level: Misc Level
During the final exam,the professor answered 35 questions from students (other than,,may I go to the bathroom,,) Obviously, some students asked more than one question, and there may have been some
who didn,t ask any.(By the end of the exam,theprofessor couldn,t remember-they may even have all been asked by same student)
Mathematics Probability Level: Misc Level
When preparing for the finalexam, the students decide to form study groups. Six of the students do not want to participate in study groups, so the other 20 students are going to divide themselves
into 5 groups of 4.Each group will (separately)study all of the course material together.In how many different ways can the 20 students divide themselves up into the study groups?
Mathematics Probability Level: Misc Level
During the course,19 different students post messages on the discussion boards on the course web site,including exactly 2 of the women.Consider the set of all students who posted discusssions.How
many different sets of students are possible?
Mathematics Probability Level: Misc Level
A Committee consists of 8 married couples. In how many ways can a subcommittee of 5 be chosen so that at most one married couple belongs to the subcommittee?
Mathematics Probability Level: Misc Level
Some boys come into a room and sit in a circle.4 girls arrange themselves between 2 boys,The 4 girls sit all together.If 2880 different circles can be formed, How many boys are there?
Mathematics Probability Level: Misc Level
How many distinguishable ways can 6 identical white marbles and 3 identical black marbles be arranged in a line?
Mathematics Probability Level: Misc Level
A Committee consists of 8 married couples. In how many ways can a subcommittee of 5 be chosen so that at most one married couple belongs to the subcommittee?
Mathematics Probability Level: Misc Level
Some boys come into a room and sit in a circle, 4 girls arrange themselves between 2 boys.The 4 girls sit all together.If 2880 different circles can be formed, how many boys are there?
Mathematics Probability Level: Misc Level
A palindrome is a number that reads the same forward at it does backward.If a digit integer is selected at random, what is the probability that it will be a 5 digit palindrome
Mathematics Probability Level: Misc Level
There are 4 bananas , 5 applees and 6 oranges in a box.In how many ways can 4 fruits be selected so that there will be at least one of each kind of fruit in the selection?
Mathematics Probability Level: Misc Level
items are withdrawn at random from a box containing 7 pencils and 5 pens. What is the probability that the items selected are 2 pencils and 1 pen?
Mathematics Probability Level: Misc Level
A six-sided die with the first six prime numbers on the faces is rolled 4 times. What is the probability that an even number comes up on the top face at least once?
Mathematics Probability Level: Misc Level
In how many ways can 9 identical apples be partitioned into two identical baskets | {"url":"https://buzztutor.com/answers/mathematics/probability/38.html","timestamp":"2024-11-05T18:23:52Z","content_type":"text/html","content_length":"396596","record_id":"<urn:uuid:78bae4a3-a2c7-4a20-82bc-7897c3c9ec97>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00378.warc.gz"} |
Interactive Chaos
Introductory tutorial to Power BI that reviews the basic procedures for loading and editing data, creating visualizations and reports, and sharing them and the dashboards that are designed on them.
The DAX data modeling language is also introduced and applied in simple examples for the creation of calculated columns and measures. | {"url":"https://interactivechaos.com/en/tutorials","timestamp":"2024-11-14T06:52:00Z","content_type":"text/html","content_length":"31873","record_id":"<urn:uuid:b745c571-4f0c-4288-aa8f-69607ea676f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00813.warc.gz"} |
Modeling the
Modeling the Swing - Jorgensen
Dave Tutelman -- January 16, 2012
Quantification of the Double Pendulum
Theodore Jorgensen was a professor of physics at the University of Nebraska. Analysis of the physics of golf was a lifetime passion for him. In 1994, he published the first edition of his book, "The
Physics of Golf". I have a copy of the second edition (1998), which is basically the same material but an easier read, with much of the math reserved for the appendices. But all the math is there --
you just don't have to bother with it during an in-line read of the book.
• The angle α is angular motion around Cochran & Stobbs' fixed pivot (O in the diagram). Think of it as the current value of the shoulder turn.
• The angle β is the current value of the wrist hinge.
• The length of the arms (R) is separate from the length of the club.
• The club is not just a uniform rod. (Some have accused the double pendulum model of being limited to a uniform rod for the club.) The fact that the club's total mass is centered at a point on the
club (the square labeled M[j]) and the club's moment of inertia is explicitly identified (I[j]) means that we can configure the club however realistically we like. Heavy head and light shaft.
Light head and heavy shaft. Even a uniform rod, if that's what you'd like to ask about.
• Likewise for the upper lever, representing the arms (M[i] and I[i] this time). In fact, by moving the center of mass (the square at M[i]) and changing its value and moment of inertia (I[i]), you
could incorporate the entire rotation of the torso into the model.
• One additional effect missing from this diagram (but present in Figure 2.3) is the horizontal acceleration of the "fixed" pivot (point "O" in the diagram). It moves to the right in the picture
(the golfer would see it as a shift to his left), correponding to the shift of the left shoulder as the torso rotates.
I am spending time on this, not because I expect you to analyze the model in this detail, but because you should understand how rich the model can be, even as simple as it is. There have been
criticisms that "the model doesn't take this or that into account". Sometimes the criticism is correct -- but sometimes it is easily incorporated in the model by playing with the parameters I mention
above, or adjusting the shoulder torque or wrist torque.
For instance, in 2005 Aaron Zick responded to a double-pendulum analysis by Mandrin. Zick's refinements of Mandrin's model were:
• A more realistic club than Mandrin's uniform rod. (We have already discussed how Jorgensen had that covered.)
• Instead of a single rod for the upper lever, Zick had a triangle comprising full-size shoulders and separate extended arms. This is easily taken care of in Jorgensen's model by adjusting the
center of mass and moment of inertia for the upper lever.
In other words, Zick's contribution was already incorporated in Jorgensen's model; it just remained to use those features of the model.
The dots show the positions of these features at rather close intervals in the swing. Those time intervals were known, and were precisely identical over the whole swing. So, by measuring the distance
between dots and knowing the interval between flashes, it is easy to calculate the velocity of any taped point at any time in the swing.
Jorgensen plotted all the relevant velocities during the downswing. Then he turned to the double-pendulum model. He tweaked the parameters of the model until it matched very closely the measured
values. In particular, he got a very good match to the clubhead speed, for the entire speed curve during the downswing.
The agreement between model and real golfer should tell us that the model is valid, at least as far as we can tell. If more measurements, better measurements, or other swings do not fit the model,
then that casts doubt on the model's validity. But remember that we want to validate the model for good swings, swings that result in effective shots. If the swings that do not fit the model are
duffers with high handicaps, it is not useful to model their swings. Better to clue them in on the model they should be looking to emulate.
BTW, emulating the model is the approach of at least one instructor. Paul Wilson (whom we shall meet below) teaches his students by first showing them a mechanical model of a double-pendulum golfer.
Then he picks out the important characterists of a good double-pendulum swing, and has the students emulate that.
Mechanizing the model
When Jorgensen was exploring the model in the 1990s, a computer on the desktop had become a pretty common thing. He was able to "mechanize" the model with a program that would do the computation. So
he was in a much better position than Cochran & Stobbs had been 20 years earlier. He could plug in "what if?" values and see what the model told him would happen. We will review below the lessons he
drew from the model.
But Jorgensen was not the only party that got busy mechanizing the double pendulum model. There were both software and hardware implementations of the model. Here's one of each:
Software - SwingPerfect Program
Serious swing model researchers wrote their own computer programs to exercise their model. It was inevitable that some of those programs would be sufficiently "well polished" to be offered as
products. (What is surprising to me is that there have not been more of them.) The program I use is SwingPerfect, written by Max Dupilka. The image is a screenshot of the program. The program's
features include:
• The ability to adjust everything interesting about the golf club.
• The ability to crank in shoulder torque and wrist torque, not just a constant for the whole swing, but a variable profile over the downswing. (A four-segment profile for the shoulders and a
ten-segment profile for the wrists.)
• Graphs of almost everything in the model, including all the accelerations and velocities.
• Setting the time interval to as little as a half millisecond (for numerical studies) or larger amounts (for visualization; the image at right is set for 5 milliseconds).
• An optional lateral movement of the fixed pivot. This gets a bit of the movement of the left shoulder into the model -- not with true accuracy, but with remarkably true effect.
If you are interested in how I use SwingPerfect for research, see my article on right-hand hit.
Hardware - Iron Byron
In 1963, the TrueTemper shaft company decided they needed a robot to test shafts. The objective was a machine with a perfectly repeatable swing, so differences between shaft prototypes could be
measured using the same swing. They got George Manning and his team, of the Battelle Institute, to design a swing robot dubbed "Iron Byron" (after Byron Nelson, whose swing was notoriously
repeatable). Many copies of Iron Byron were made, for R&D testing in the golf club and golf ball industry, and even for the USGA for conformance testing and research.
Iron Byron was designed directly from the double pendulum model of the golf swing. Over decades, it has proven its value, which is certainly a vote in favor of the value of the double pendulum model.
Paul Wilson is a golf instructor who uses Iron Byron as a teaching model, not just a testing device. In this video, he explains why the double-pendulum-based machine is a good enough model of the
swing for a real golfer to copy (even though history has it the other way around; the robot's designers were trying to copy a human golf swing). The explanation is covered in the first three minutes
of the video; it is an excellent description of why the superficial differences between robot and golfer are not important. The last portion of the video is an interview with George Manning, Iron
Byron's inventor.
Lessons from the Double Pendulum Model
The best thing about having a mathematical model is that you can do "what if?" experiments with it.
Do you know what a "what if?" experiment is? Think of one of the most productive uses of spreadsheets. Once you have a spreadsheet set up to give you your answer -- whatever the subject matter -- you
can just change a value or two and see what happens to the output. That spreadsheet is a mathematical model for something, and you can tweak variables and see what happens. Tweaking the input to the
model and seeing what happens is the essence of a "what if?" experiment.
Jorgensen and others have done "what if"s, and have taught us something about the swing. (Well, about the model anyway. It is about the swing only to the extent that the model is valid, at least for
that feature of the swing.) Below we'll list some of the conclusions that Jorgensen drew from the model. They are from Chapter 4, entitled "Variation of Parameters Brings New Understanding of the
Golf Swing" -- the essence of a mathematical "what if?"
1. An increase in the shoulder torque (the strength of the body rotation that provides the power to the swing) increases the clubhead speed. Not surprising so far. But the increase in clubhead speed
is not proportional to the increase in torque. You have to increase the torque by about 3% for every 1% increase in speed.
2. All other things being equal, the greater the initial wrist cock angle, the higher the clubhead speed at impact.
3. Reducing the amount of backswing (the body turn at the transition) leaves the clubhead speed almost the same as before. Moreover, it tends not to allow the wrists to over-release to a cupped
position, but instead encourages a solid-hitting position with the hands leading the clubhead at impact. Another way of saying this: Overswinging leads to a bad impact position, with very little
gain of clubhead speed.
4. Wrist torque ("hand action") affects clubhead speed at impact in a very surprising way. So much so, in fact, that Jorgensen refers to it as "The Paradox". Here is the essence of what he found:
1. The good golfer he measured used just enough wrist torque just long enough to maintain the initial wrist cock angle until inertial forces started throwing the club outward. That typically
takes .1-.15 seconds. After that, the golfer used no wrist torque at all! Jorgensen recalls a gem from Bobby Jones' instructions that the club feels like it is "freewheeling through the
2. So the paradox: any wrist torque during the downswing that aids release will result in a lower clubhead speed at impact. Oh, it will indeed increase the clubhead speed through most of the
downswing. But you don't care about that; you want the maximum clubhead speed you can get at impact. And using hand action to release the clubhead works against that aim.
3. In fact you can increase the clubhead speed at impact by using a hindering hand action. This is paradoxical, counterintuitive -- but the model says it is true. And I know at least one
instructor who gets very good results teaching a hand action that tries to hold the wrist cock right through impact -- a swing key that creates a hindering torque.
I have written a whole article devoted to Jorgensen's paradox, in case you want to look deeper into it.
5. Gravity provides about 8% of the clubhead speed.
6. The forward shift provides almost 9% of the clubhead speed.
Let me close this section by emphasizing that the Double Pendulum Model, in spite of its simplicity, served the golf research community well for over 30 years. It is still often quoted as gospel by
those who understand it, and it provides the underlying theory for all golf equipment robot testing.
It was after 2000 before the community felt the need to refine the model (i.e.- complicate it, with the aim of emulating a human golfer more closely). It is my distinct impression that the more
complex models were developed not because researchers were unhappy with the double pendulum, but rather because:
• The non-research community was uncomfortable with the counter-intuitive results -- The Paradox -- and the researchers felt the need to respond.
• Instructors wanted to know how each part of the body should be contributing to the swing. The double-pendulum model tells us a lot about the arms and hands, but all it tells about the rest of the
body is that the job is to produce rotation, shoulder torque. It doesn't tell us how to do that, muscle by muscle.
• The computational tools were now common enough to run much more complex sets of differential equations.
• A bumper crop of graduate students were available to do research, and needed topics for their dissertations. (The last may be surprising but seems realistic. I remember my own grad school days,
and my own and my colleagues' search for thesis topics. Also, I have looked at where a lot of the new models are coming from.)
Let's move on and look at some of the newer models, and see what more we can learn from them.
Last modified -- January 28, 2012 | {"url":"https://tutelman.com/golf/swing/models2.php","timestamp":"2024-11-14T13:08:18Z","content_type":"text/html","content_length":"23297","record_id":"<urn:uuid:43086b9e-3aac-4ace-ac12-1cf7980c94c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00412.warc.gz"} |
Design Like A Pro
Comparing Ordering Fractions Worksheet
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e Help
Overview of the PDEtools Package
List of PDEtools Package Commands
Brief description of each command
See Also
• The PDEtools package is a collection of commands and routines for finding analytical dsolve
solutions for partial differential equations (PDEs) based on the paper "A Computational
Approach for the Analytical Solving of Partial Differential Equations" by E.S. Library
Cheb-Terrab and K. von Bulow (see References), and the continuation of this work by the
same authors (Symmetry routines) during 2004. odetest
• The package is an implementation of known methods for solving PDEs; however, it also pdetest
allows you to look for solutions of equations not yet automatically solved by the
package, or for "different solutions" when the package returns a solution which is not pdsolve
the most general one. For this purpose, you can make use of the dchange command and of
the HINT option of the pdsolve command, especially the functional hint, both explained in UsingPackages
more detail in the corresponding help pages.
• PDEtools includes a subset of commands for computing conserved currents and generalized
integrating factors as well as for performing most of the steps of the traditional
symmetry analysis of PDE systems. That includes the automatic computation of the
infinitesimal symmetry generators as well as the automatic computation of related group
invariant solutions of different kinds departing directly from the PDE system to be
• Most of the internal routines of PDEtools are also available for use as
PDEtools:-Library:-Routine; this permits programming your own extensions of the package
using the existing tools. For details see PDEtools,Library.
• Most of the symmetry commands as well as all the ones in the PDEtools:-Library handle
anticommutative variables and functions automatically - see the respective help pages.
• Each command in the PDEtools package can be accessed by using either the long form or the
short form of the command name in the command calling sequence. The name PDETools can
also be used as a synonym for PDEtools.
• Note: The diff_table command allows you to enter (input) expressions and their
derivatives using compact mathematical notation (jetvariables with brackets),
representing an important saving in redundant typing for PDE problems.
• To display the help page for a particular PDEtools command from the Maple prompt, see
Getting Help with a Command in a Package.
List of PDEtools Package Commands
General purpose and traveling wave solution PDE commands:
build casesplit charstrip dchange
dcoeffs declare diff_table difforder
dpolyform dsubs Laplace mapde
PDEplot separability Solve splitstrip
splitsys ToMissingDependentVariable TWSolutions undeclare
Symmetry and related solution PDE commands:
CanonicalCoordinates ChangeSymmetry CharacteristicQ CharacteristicQInvariants
ConservedCurrents ConservedCurrentTest ConsistencyTest D_Dx
DeterminingPDE Eta_k Euler FromJet
FunctionFieldSolutions InfinitesimalGenerator Infinitesimals IntegratingFactors
IntegratingFactorTest InvariantEquation Invariants InvariantSolutions
InvariantTransformation PolynomialSolutions ReducedForm SimilaritySolutions
SimilarityTransformation SymmetryCommutator SymmetryGauge SymmetrySolutions
SymmetryTest SymmetryTransformation ToJet
• Most of the internal routines of PDEtools are also available for use as
PDEtools:-Library:-Routine; see PDEtools,Library.
• To avoid having to remember the relatively large number of keywords that can be passed as
optional arguments for the symmetry related commands, if you type the keyword misspelled,
or just a portion of it, a matching against the existing keywords is performed, and when
there is only one match, the input is automatically corrected.
Brief description of each command
A brief description of the PDEtools package commands, split into General purpose and
Symmetry related, is as follows.
General purpose
• build takes a result given by pdsolve and returns the final expression for the
indeterminate function (useful when the method used by pdsolve was separation or change
of variables).
• casesplit splits a system of equations (and inequations) into a sequence of systems of
equations and inequations such that the union of the non-singular solutions of the
latter is equal to the set of solutions of the original system. In addition, in each of
the returned systems, all differential or algebraic redundancies are removed, and all
the integrability conditions are automatically satisfied. The computations are performed
using the DifferentialAlgebra package.
• charstrip evaluates the characteristic strip associated with a given first order PDE;
that is, it builds the coupled system of ODEs equivalent to that PDE. Additionally,
given a characteristic strip, charstrip can reverse it and return the family of PDEs
behind it.
• dchange performs changes of variables in any algebraic object (PDEs, multiple integrals,
integro-differential equations, etc.), as well as in procedures. This command is useful
to change the format of a PDE from one that is difficult to solve to one that is
• dcoeffs returns coefficients of polynomial differential equations, much like coeffs with
algebraic polynomials.
• declare permits a simple and compact display on the screen of functions and derivatives.
Typically, one declares functions, such as declare(f(x, y, z)), so that f(x,y,z)
displays as 'f' (that is, just by its name). Also, derivatives are "displayed" as
indexed functions and it is possible to declare a "prime variable." In other words, for
functions of one variable, derivatives with respect to that variable will be displayed
with a prime, '.
• difforder returns the general (or particular, with regard to any variable) differential
order of a partial derivative (or the maximum general or particular differential order
of an expression containing partial derivatives).
• dpolyform accepts an equation or expression, or a set or list of them, understood to be
equal to zero, and returns a differential polynomial system of equations.
• dsubs substitutes a derivative inside differential equations such that the resulting
expression does not depend on the derivative being substituted.
• FunctionFieldSolutions computes exact solutions to DE systems involving mathematical
functions, possibly inequations, ODEs and also non-differential equations. The solutions
are returned as power series (with some upper bound degree $n$) in the mathematical
functions and its derivatives (up to some upper bound differential order $m$), having
for coefficients multivariable polynomials (with some upper bound degree $r$) of the
independent variables.
• Laplace solves a second order linear PDE in two independent variables using the method
of Laplace (not a Laplace transform).
• mapde maps a PDE into another PDE with different format (from among a few formats
implemented at present), which is perhaps more easily solvable.
• PDEplot produces the plot of the solution for a first order linear (or nonlinear)
partial differential equation (PDE), for given initial conditions.
• pdetest returns either 0 (when the PDE is annulled by the solution sol), indicating that
the solution is correct, or a remaining algebraic expression (obtained after simplifying
the PDE with respect to the proposed solution), indicating that the solution might be
• Given a PDE, pdsolve's main goal is to find an analytical solution for it. There are no
restrictions as to the type, differential order, or number of independent variables of
the PDEs pdsolve can try to solve.
• PolynomialSolutions receives a PDE system, optionally an indication of the degree and
dependency, and computes related polynomial solutions.
• separability determines under what conditions it is possible to obtain a complete
solution, through separation of variables by sum or product, for a given PDE. A complete
solution is defined to be a solution that depends on sum(diff_ord[i]+1,i=1..n)
parameters, where n is the number of independent variables and diff_ord is the maximum
differential order of the PDE with respect to each of the independent variables.
• Solve is an unified solver that receives a system of equations, algebraic or
differential, and solves this system, returning solutions optionally independent of
indicated variables, calling solve, dsolve or pdsolve according to the input received.
• splitstrip evaluates the strip associated with a PDE the same way as the strip command,
but returns this strip, when possible, split into subsets, obtained by calling splitsys
with the ODEs of the characteristic strip as argument.
• splitsys splits a set of equations (ODEs, PDEs, algebraic equations, or a combination of
these) into subsets, each one with equations coupled among themselves but not coupled to
the equations of the other subsets.
• ToMissingDependentVariable receives a PDE and returns another one, PDE2, equivalent to
PDE in that from the solution of PDE2 one gets the solution to PDE, and such that PDE2
does not depend, explicitly, on the dependent variable.
• TWSolutions computes Traveling Wave Solutions for PDEs and systems of them, either as
expansions in tanh or in a number of other functions like JacobiSN, WeierstrassP, etc..
• undeclare clears declarations previously made.
Symmetry solution related
Note: Herein infinitesimals refer to a list with the components of the infinitesimal
generator of a symmetry group.
• CanonicalCoordinates receives the infinitesimals of a symmetry group and returns
associated canonical coordinates for it.
• ChangeSymmetry performs change of variables on the infinitesimals of a symmetry
• CharacteristicQ receives the infinitesimals of a point symmetry and returns the
characteristic of the group.
• CharacteristicQInvariants receives the infinitesimals of a symmetry group and computes
related differential invariants directly from the CharacteristicQ for the
• ConservedCurrents receives a PDE system and returns the conserved currents of it; when
the system involves only ODEs, the conserved currents are the first integrals of the
• ConservedCurrentTest receives an algebraic expression and a PDE system and verifies
whether the given expression is a conserved current.
• ConsistencyTest receives a PDE system and returns true or false according to whether the
system is consistent
• D_Dx computes total derivatives in jet notation, that is taking the independent and
dependent variables and their derivatives as differentiation variables in equal footing.
• DeterminingPDE receives a PDE system and computes the determining PDE system satisfied
by the infinitesimals of the symmetry groups admitted by the given PDE system.
• Eta_k receives the infinitesimals of a symmetry and returns a table-procedure that
computes, on request, any prolongation of these infinitesimals.
• Euler is the Euler operator: when applied to a DE system, it returns the exact
conditions; i.e. the conditions to be satisfied when the system is a divergence.
• FromJet receives a mathematical expression in jet notation and returns the corresponding
expression in function notation. This command is the counterpart of ToJet.
• FunctionFieldSolutions computes exact solutions to DE systems involving mathematical
functions, possibly inequations, ODEs and also non-differential equations. The solutions
are returned are power series (with some upper bound degree $n$) in the mathematical
functions and its derivatives (up to some upper bound differential order $m$), having
for coefficients multivariable polynomials (with some upper bound degree $r$) of the
independent variables.
• InfinitesimalGenerator receives the infinitesimals of a symmetry and returns a procedure
(operator) representing the infinitesimal generator, that is, one which acts on a PDE
system to return the related determining PDE.
• Infinitesimals receives a PDE system and returns the infinitesimals of symmetry groups
admitted by the given PDE system.
• IntegratingFactors receives a DE system and returns the generalized integrating factors.
• IntegratingFactorTest receives an algebraic expression and a DE system, and verifies
whether the given expression is a generalized integrating factor.
• InvariantEquation receives the infinitesimals of a symmetry group and computes the
equation that is simultaneously invariant under the symmetry transformations
corresponding to the given symmetries.
• InvariantSolutions receives a PDE system and returns the so-called group invariant
solutions for it, that is, the PDE system solutions derived by first (automatically)
computing the symmetries admitted by it.
• InvariantTransformation receives the infinitesimals of a symmetry group and computes the
related finite (symmetry) transformations reducing the number of independent variables
by N. (You can optionally specify N.) These are the transformations from which
InvariantSolutions derives solutions to the PDE system.
• Invariants receives the infinitesimals of a symmetry group and computes related
differential invariants of any specified order.
• PolynomialSolutions receives a PDE system, optionally an indication of the degree and
dependency, and computes related polynomial solutions.
• ReducedForm receives two PDE systems and reduces the first one with respect to the
second one; this is similar to what simplify/siderels does but with PDE systems. This
command can be useful beyond the symmetry approach for PDE systems.
• SimilaritySolutions receives a PDE system and returns the so-called similarity solutions
for it. This command is present mainly for pedagogical purposes in that the solutions it
returns are computed using one symmetry at a time. For practical purposes use
InvariantSolutions instead, which can reduce the given PDE system using many symmetries
in one go.
• SimilarityTransformation receives a PDE system and computes the finite (symmetry)
transformations reducing the number of independent variables by one. These are the
transformations from which SimilaritySolutions derives solutions to the PDE system.
• SymmetryCommutator receives a pair of infinitesimals corresponding to symmetry
transformations and returns their commutator.
• SymmetryGauge receives a symmetry, as a list or as an infinitesimal generator, and
rewrites this symmetry in the most general form or in the one indicated using optional
• SymmetryTest receives the infinitesimals of a symmetry group and a PDE system and tests
whether the PDE system admits that symmetry.
• SymmetryTransformation receives the infinitesimals of a 1-Dimensional point symmetry
group and returns the finite form of the (symmetry) transformation leaving invariant any
PDE system admitting that symmetry.
• SymmetrySolutions receives the infinitesimals of a point symmetry group, a solution to
some PDE system (itself not required) and returns the another solution, obtained by
transforming the given one using the finite form of the (symmetry) transformation given.
• ToJet receives a mathematical expression in function notation and returns the
corresponding expression in jet notation. This command is the counterpart of FromJet.
Cheb-Terrab, E.S., and von Bulow, K. "A Computational Approach for the Analytical Solving
of Partial Differential Equations." Computer Physics Communications, Vol. 90, (1995):
Olver, P.J. Equivalence, Invariants and Symmetry. Cambridge Press, 1995.
Stephani, H. Differential Equations: Their Solution Using Symmetries. Edited by M.
MacCallum. Cambridge University Press, 1989.
List of PDEtools Package Commands
Brief description of each command
See Also
• The PDEtools package is a collection of commands and routines for finding analytical dsolve
solutions for partial differential equations (PDEs) based on the paper "A Computational
Approach for the Analytical Solving of Partial Differential Equations" by E.S. Library
Cheb-Terrab and K. von Bulow (see References), and the continuation of this work by the
same authors (Symmetry routines) during 2004. odetest
• The package is an implementation of known methods for solving PDEs; however, it also pdetest
allows you to look for solutions of equations not yet automatically solved by the
package, or for "different solutions" when the package returns a solution which is not pdsolve
the most general one. For this purpose, you can make use of the dchange command and of
the HINT option of the pdsolve command, especially the functional hint, both explained in UsingPackages
more detail in the corresponding help pages.
• PDEtools includes a subset of commands for computing conserved currents and generalized
integrating factors as well as for performing most of the steps of the traditional
symmetry analysis of PDE systems. That includes the automatic computation of the
infinitesimal symmetry generators as well as the automatic computation of related group
invariant solutions of different kinds departing directly from the PDE system to be
• Most of the internal routines of PDEtools are also available for use as
PDEtools:-Library:-Routine; this permits programming your own extensions of the package
using the existing tools. For details see PDEtools,Library.
• Most of the symmetry commands as well as all the ones in the PDEtools:-Library handle
anticommutative variables and functions automatically - see the respective help pages.
• Each command in the PDEtools package can be accessed by using either the long form or the
short form of the command name in the command calling sequence. The name PDETools can
also be used as a synonym for PDEtools.
• Note: The diff_table command allows you to enter (input) expressions and their
derivatives using compact mathematical notation (jetvariables with brackets),
representing an important saving in redundant typing for PDE problems.
• To display the help page for a particular PDEtools command from the Maple prompt, see
Getting Help with a Command in a Package.
List of PDEtools Package Commands
General purpose and traveling wave solution PDE commands:
build casesplit charstrip dchange
dcoeffs declare diff_table difforder
dpolyform dsubs Laplace mapde
PDEplot separability Solve splitstrip
splitsys ToMissingDependentVariable TWSolutions undeclare
Symmetry and related solution PDE commands:
CanonicalCoordinates ChangeSymmetry CharacteristicQ CharacteristicQInvariants
ConservedCurrents ConservedCurrentTest ConsistencyTest D_Dx
DeterminingPDE Eta_k Euler FromJet
FunctionFieldSolutions InfinitesimalGenerator Infinitesimals IntegratingFactors
IntegratingFactorTest InvariantEquation Invariants InvariantSolutions
InvariantTransformation PolynomialSolutions ReducedForm SimilaritySolutions
SimilarityTransformation SymmetryCommutator SymmetryGauge SymmetrySolutions
SymmetryTest SymmetryTransformation ToJet
• Most of the internal routines of PDEtools are also available for use as
PDEtools:-Library:-Routine; see PDEtools,Library.
• To avoid having to remember the relatively large number of keywords that can be passed as
optional arguments for the symmetry related commands, if you type the keyword misspelled,
or just a portion of it, a matching against the existing keywords is performed, and when
there is only one match, the input is automatically corrected.
Brief description of each command
A brief description of the PDEtools package commands, split into General purpose and
Symmetry related, is as follows.
General purpose
• build takes a result given by pdsolve and returns the final expression for the
indeterminate function (useful when the method used by pdsolve was separation or change
of variables).
• casesplit splits a system of equations (and inequations) into a sequence of systems of
equations and inequations such that the union of the non-singular solutions of the
latter is equal to the set of solutions of the original system. In addition, in each of
the returned systems, all differential or algebraic redundancies are removed, and all
the integrability conditions are automatically satisfied. The computations are performed
using the DifferentialAlgebra package.
• charstrip evaluates the characteristic strip associated with a given first order PDE;
that is, it builds the coupled system of ODEs equivalent to that PDE. Additionally,
given a characteristic strip, charstrip can reverse it and return the family of PDEs
behind it.
• dchange performs changes of variables in any algebraic object (PDEs, multiple integrals,
integro-differential equations, etc.), as well as in procedures. This command is useful
to change the format of a PDE from one that is difficult to solve to one that is
• dcoeffs returns coefficients of polynomial differential equations, much like coeffs with
algebraic polynomials.
• declare permits a simple and compact display on the screen of functions and derivatives.
Typically, one declares functions, such as declare(f(x, y, z)), so that f(x,y,z)
displays as 'f' (that is, just by its name). Also, derivatives are "displayed" as
indexed functions and it is possible to declare a "prime variable." In other words, for
functions of one variable, derivatives with respect to that variable will be displayed
with a prime, '.
• difforder returns the general (or particular, with regard to any variable) differential
order of a partial derivative (or the maximum general or particular differential order
of an expression containing partial derivatives).
• dpolyform accepts an equation or expression, or a set or list of them, understood to be
equal to zero, and returns a differential polynomial system of equations.
• dsubs substitutes a derivative inside differential equations such that the resulting
expression does not depend on the derivative being substituted.
• FunctionFieldSolutions computes exact solutions to DE systems involving mathematical
functions, possibly inequations, ODEs and also non-differential equations. The solutions
are returned as power series (with some upper bound degree $n$) in the mathematical
functions and its derivatives (up to some upper bound differential order $m$), having
for coefficients multivariable polynomials (with some upper bound degree $r$) of the
independent variables.
• Laplace solves a second order linear PDE in two independent variables using the method
of Laplace (not a Laplace transform).
• mapde maps a PDE into another PDE with different format (from among a few formats
implemented at present), which is perhaps more easily solvable.
• PDEplot produces the plot of the solution for a first order linear (or nonlinear)
partial differential equation (PDE), for given initial conditions.
• pdetest returns either 0 (when the PDE is annulled by the solution sol), indicating that
the solution is correct, or a remaining algebraic expression (obtained after simplifying
the PDE with respect to the proposed solution), indicating that the solution might be
• Given a PDE, pdsolve's main goal is to find an analytical solution for it. There are no
restrictions as to the type, differential order, or number of independent variables of
the PDEs pdsolve can try to solve.
• PolynomialSolutions receives a PDE system, optionally an indication of the degree and
dependency, and computes related polynomial solutions.
• separability determines under what conditions it is possible to obtain a complete
solution, through separation of variables by sum or product, for a given PDE. A complete
solution is defined to be a solution that depends on sum(diff_ord[i]+1,i=1..n)
parameters, where n is the number of independent variables and diff_ord is the maximum
differential order of the PDE with respect to each of the independent variables.
• Solve is an unified solver that receives a system of equations, algebraic or
differential, and solves this system, returning solutions optionally independent of
indicated variables, calling solve, dsolve or pdsolve according to the input received.
• splitstrip evaluates the strip associated with a PDE the same way as the strip command,
but returns this strip, when possible, split into subsets, obtained by calling splitsys
with the ODEs of the characteristic strip as argument.
• splitsys splits a set of equations (ODEs, PDEs, algebraic equations, or a combination of
these) into subsets, each one with equations coupled among themselves but not coupled to
the equations of the other subsets.
• ToMissingDependentVariable receives a PDE and returns another one, PDE2, equivalent to
PDE in that from the solution of PDE2 one gets the solution to PDE, and such that PDE2
does not depend, explicitly, on the dependent variable.
• TWSolutions computes Traveling Wave Solutions for PDEs and systems of them, either as
expansions in tanh or in a number of other functions like JacobiSN, WeierstrassP, etc..
• undeclare clears declarations previously made.
Symmetry solution related
Note: Herein infinitesimals refer to a list with the components of the infinitesimal
generator of a symmetry group.
• CanonicalCoordinates receives the infinitesimals of a symmetry group and returns
associated canonical coordinates for it.
• ChangeSymmetry performs change of variables on the infinitesimals of a symmetry
• CharacteristicQ receives the infinitesimals of a point symmetry and returns the
characteristic of the group.
• CharacteristicQInvariants receives the infinitesimals of a symmetry group and computes
related differential invariants directly from the CharacteristicQ for the
• ConservedCurrents receives a PDE system and returns the conserved currents of it; when
the system involves only ODEs, the conserved currents are the first integrals of the
• ConservedCurrentTest receives an algebraic expression and a PDE system and verifies
whether the given expression is a conserved current.
• ConsistencyTest receives a PDE system and returns true or false according to whether the
system is consistent
• D_Dx computes total derivatives in jet notation, that is taking the independent and
dependent variables and their derivatives as differentiation variables in equal footing.
• DeterminingPDE receives a PDE system and computes the determining PDE system satisfied
by the infinitesimals of the symmetry groups admitted by the given PDE system.
• Eta_k receives the infinitesimals of a symmetry and returns a table-procedure that
computes, on request, any prolongation of these infinitesimals.
• Euler is the Euler operator: when applied to a DE system, it returns the exact
conditions; i.e. the conditions to be satisfied when the system is a divergence.
• FromJet receives a mathematical expression in jet notation and returns the corresponding
expression in function notation. This command is the counterpart of ToJet.
• FunctionFieldSolutions computes exact solutions to DE systems involving mathematical
functions, possibly inequations, ODEs and also non-differential equations. The solutions
are returned are power series (with some upper bound degree $n$) in the mathematical
functions and its derivatives (up to some upper bound differential order $m$), having
for coefficients multivariable polynomials (with some upper bound degree $r$) of the
independent variables.
• InfinitesimalGenerator receives the infinitesimals of a symmetry and returns a procedure
(operator) representing the infinitesimal generator, that is, one which acts on a PDE
system to return the related determining PDE.
• Infinitesimals receives a PDE system and returns the infinitesimals of symmetry groups
admitted by the given PDE system.
• IntegratingFactors receives a DE system and returns the generalized integrating factors.
• IntegratingFactorTest receives an algebraic expression and a DE system, and verifies
whether the given expression is a generalized integrating factor.
• InvariantEquation receives the infinitesimals of a symmetry group and computes the
equation that is simultaneously invariant under the symmetry transformations
corresponding to the given symmetries.
• InvariantSolutions receives a PDE system and returns the so-called group invariant
solutions for it, that is, the PDE system solutions derived by first (automatically)
computing the symmetries admitted by it.
• InvariantTransformation receives the infinitesimals of a symmetry group and computes the
related finite (symmetry) transformations reducing the number of independent variables
by N. (You can optionally specify N.) These are the transformations from which
InvariantSolutions derives solutions to the PDE system.
• Invariants receives the infinitesimals of a symmetry group and computes related
differential invariants of any specified order.
• PolynomialSolutions receives a PDE system, optionally an indication of the degree and
dependency, and computes related polynomial solutions.
• ReducedForm receives two PDE systems and reduces the first one with respect to the
second one; this is similar to what simplify/siderels does but with PDE systems. This
command can be useful beyond the symmetry approach for PDE systems.
• SimilaritySolutions receives a PDE system and returns the so-called similarity solutions
for it. This command is present mainly for pedagogical purposes in that the solutions it
returns are computed using one symmetry at a time. For practical purposes use
InvariantSolutions instead, which can reduce the given PDE system using many symmetries
in one go.
• SimilarityTransformation receives a PDE system and computes the finite (symmetry)
transformations reducing the number of independent variables by one. These are the
transformations from which SimilaritySolutions derives solutions to the PDE system.
• SymmetryCommutator receives a pair of infinitesimals corresponding to symmetry
transformations and returns their commutator.
• SymmetryGauge receives a symmetry, as a list or as an infinitesimal generator, and
rewrites this symmetry in the most general form or in the one indicated using optional
• SymmetryTest receives the infinitesimals of a symmetry group and a PDE system and tests
whether the PDE system admits that symmetry.
• SymmetryTransformation receives the infinitesimals of a 1-Dimensional point symmetry
group and returns the finite form of the (symmetry) transformation leaving invariant any
PDE system admitting that symmetry.
• SymmetrySolutions receives the infinitesimals of a point symmetry group, a solution to
some PDE system (itself not required) and returns the another solution, obtained by
transforming the given one using the finite form of the (symmetry) transformation given.
• ToJet receives a mathematical expression in function notation and returns the
corresponding expression in jet notation. This command is the counterpart of FromJet.
Cheb-Terrab, E.S., and von Bulow, K. "A Computational Approach for the Analytical Solving
of Partial Differential Equations." Computer Physics Communications, Vol. 90, (1995):
Olver, P.J. Equivalence, Invariants and Symmetry. Cambridge Press, 1995.
Stephani, H. Differential Equations: Their Solution Using Symmetries. Edited by M.
MacCallum. Cambridge University Press, 1989.
• The PDEtools package is a collection of commands and routines for finding analytical solutions for partial differential equations (PDEs) based on the paper "A Computational Approach for the
Analytical Solving of Partial Differential Equations" by E.S. Cheb-Terrab and K. von Bulow (see References), and the continuation of this work by the same authors (Symmetry routines) during 2004.
• The package is an implementation of known methods for solving PDEs; however, it also allows you to look for solutions of equations not yet automatically solved by the package, or for "different
solutions" when the package returns a solution which is not the most general one. For this purpose, you can make use of the dchange command and of the HINT option of the pdsolve command,
especially the functional hint, both explained in more detail in the corresponding help pages.
• PDEtools includes a subset of commands for computing conserved currents and generalized integrating factors as well as for performing most of the steps of the traditional symmetry analysis of PDE
systems. That includes the automatic computation of the infinitesimal symmetry generators as well as the automatic computation of related group invariant solutions of different kinds departing
directly from the PDE system to be solved.
• Most of the internal routines of PDEtools are also available for use as PDEtools:-Library:-Routine; this permits programming your own extensions of the package using the existing tools. For
details see PDEtools,Library.
• Most of the symmetry commands as well as all the ones in the PDEtools:-Library handle anticommutative variables and functions automatically - see the respective help pages.
• Each command in the PDEtools package can be accessed by using either the long form or the short form of the command name in the command calling sequence. The name PDETools can also be used as a
synonym for PDEtools.
• Note: The diff_table command allows you to enter (input) expressions and their derivatives using compact mathematical notation (jetvariables with brackets), representing an important saving in
redundant typing for PDE problems.
• To display the help page for a particular PDEtools command from the Maple prompt, see Getting Help with a Command in a Package.
• The PDEtools package is a collection of commands and routines for finding analytical solutions for partial differential equations (PDEs) based on the paper "A Computational Approach for the
Analytical Solving of Partial Differential Equations" by E.S. Cheb-Terrab and K. von Bulow (see References), and the continuation of this work by the same authors (Symmetry routines) during 2004.
The PDEtools package is a collection of commands and routines for finding analytical solutions for partial differential equations (PDEs) based on the paper "A Computational Approach for the
Analytical Solving of Partial Differential Equations" by E.S. Cheb-Terrab and K. von Bulow (see References), and the continuation of this work by the same authors (Symmetry routines) during 2004.
• The package is an implementation of known methods for solving PDEs; however, it also allows you to look for solutions of equations not yet automatically solved by the package, or for "different
solutions" when the package returns a solution which is not the most general one. For this purpose, you can make use of the dchange command and of the HINT option of the pdsolve command, especially
the functional hint, both explained in more detail in the corresponding help pages.
The package is an implementation of known methods for solving PDEs; however, it also allows you to look for solutions of equations not yet automatically solved by the package, or for "different
solutions" when the package returns a solution which is not the most general one. For this purpose, you can make use of the dchange command and of the HINT option of the pdsolve command, especially
the functional hint, both explained in more detail in the corresponding help pages.
• PDEtools includes a subset of commands for computing conserved currents and generalized integrating factors as well as for performing most of the steps of the traditional symmetry analysis of PDE
systems. That includes the automatic computation of the infinitesimal symmetry generators as well as the automatic computation of related group invariant solutions of different kinds departing
directly from the PDE system to be solved.
PDEtools includes a subset of commands for computing conserved currents and generalized integrating factors as well as for performing most of the steps of the traditional symmetry analysis of PDE
systems. That includes the automatic computation of the infinitesimal symmetry generators as well as the automatic computation of related group invariant solutions of different kinds departing
directly from the PDE system to be solved.
• Most of the internal routines of PDEtools are also available for use as PDEtools:-Library:-Routine; this permits programming your own extensions of the package using the existing tools. For details
see PDEtools,Library.
Most of the internal routines of PDEtools are also available for use as PDEtools:-Library:-Routine; this permits programming your own extensions of the package using the existing tools. For details
see PDEtools,Library.
• Most of the symmetry commands as well as all the ones in the PDEtools:-Library handle anticommutative variables and functions automatically - see the respective help pages.
Most of the symmetry commands as well as all the ones in the PDEtools:-Library handle anticommutative variables and functions automatically - see the respective help pages.
• Each command in the PDEtools package can be accessed by using either the long form or the short form of the command name in the command calling sequence. The name PDETools can also be used as a
synonym for PDEtools.
Each command in the PDEtools package can be accessed by using either the long form or the short form of the command name in the command calling sequence. The name PDETools can also be used as a
synonym for PDEtools.
• Note: The diff_table command allows you to enter (input) expressions and their derivatives using compact mathematical notation (jetvariables with brackets), representing an important saving in
redundant typing for PDE problems.
Note: The diff_table command allows you to enter (input) expressions and their derivatives using compact mathematical notation (jetvariables with brackets), representing an important saving in
redundant typing for PDE problems.
• To display the help page for a particular PDEtools command from the Maple prompt, see Getting Help with a Command in a Package.
To display the help page for a particular PDEtools command from the Maple prompt, see Getting Help with a Command in a Package.
List of PDEtools Package Commands
General purpose and traveling wave solution PDE commands:
build casesplit charstrip dchange
dcoeffs declare diff_table difforder
dpolyform dsubs Laplace mapde
PDEplot separability Solve splitstrip
splitsys ToMissingDependentVariable TWSolutions undeclare
Symmetry and related solution PDE commands:
CanonicalCoordinates ChangeSymmetry CharacteristicQ CharacteristicQInvariants
ConservedCurrents ConservedCurrentTest ConsistencyTest D_Dx
DeterminingPDE Eta_k Euler FromJet
FunctionFieldSolutions InfinitesimalGenerator Infinitesimals IntegratingFactors
IntegratingFactorTest InvariantEquation Invariants InvariantSolutions
InvariantTransformation PolynomialSolutions ReducedForm SimilaritySolutions
SimilarityTransformation SymmetryCommutator SymmetryGauge SymmetrySolutions
SymmetryTest SymmetryTransformation ToJet
• Most of the internal routines of PDEtools are also available for use as PDEtools:-Library:-Routine; see PDEtools,Library.
• To avoid having to remember the relatively large number of keywords that can be passed as optional arguments for the symmetry related commands, if you type the keyword misspelled, or just a
portion of it, a matching against the existing keywords is performed, and when there is only one match, the input is automatically corrected.
General purpose and traveling wave solution PDE commands:
build casesplit charstrip dchange
dcoeffs declare diff_table difforder
dpolyform dsubs Laplace mapde
PDEplot separability Solve splitstrip
splitsys ToMissingDependentVariable TWSolutions undeclare
build casesplit charstrip dchange
dcoeffs declare diff_table difforder
dpolyform dsubs Laplace mapde
PDEplot separability Solve splitstrip
splitsys ToMissingDependentVariable TWSolutions undeclare
Symmetry and related solution PDE commands:
CanonicalCoordinates ChangeSymmetry CharacteristicQ CharacteristicQInvariants
ConservedCurrents ConservedCurrentTest ConsistencyTest D_Dx
DeterminingPDE Eta_k Euler FromJet
FunctionFieldSolutions InfinitesimalGenerator Infinitesimals IntegratingFactors
IntegratingFactorTest InvariantEquation Invariants InvariantSolutions
InvariantTransformation PolynomialSolutions ReducedForm SimilaritySolutions
SimilarityTransformation SymmetryCommutator SymmetryGauge SymmetrySolutions
SymmetryTest SymmetryTransformation ToJet
CanonicalCoordinates ChangeSymmetry CharacteristicQ CharacteristicQInvariants
ConservedCurrents ConservedCurrentTest ConsistencyTest D_Dx
DeterminingPDE Eta_k Euler FromJet
FunctionFieldSolutions InfinitesimalGenerator Infinitesimals IntegratingFactors
IntegratingFactorTest InvariantEquation Invariants InvariantSolutions
InvariantTransformation PolynomialSolutions ReducedForm SimilaritySolutions
SimilarityTransformation SymmetryCommutator SymmetryGauge SymmetrySolutions
SymmetryTest SymmetryTransformation ToJet
• Most of the internal routines of PDEtools are also available for use as PDEtools:-Library:-Routine; see PDEtools,Library.
Most of the internal routines of PDEtools are also available for use as PDEtools:-Library:-Routine; see PDEtools,Library.
• To avoid having to remember the relatively large number of keywords that can be passed as optional arguments for the symmetry related commands, if you type the keyword misspelled, or just a portion
of it, a matching against the existing keywords is performed, and when there is only one match, the input is automatically corrected.
To avoid having to remember the relatively large number of keywords that can be passed as optional arguments for the symmetry related commands, if you type the keyword misspelled, or just a portion
of it, a matching against the existing keywords is performed, and when there is only one match, the input is automatically corrected.
Brief description of each command
A brief description of the PDEtools package commands, split into General purpose and Symmetry related, is as follows.
General purpose
• build takes a result given by pdsolve and returns the final expression for the indeterminate function (useful when the method used by pdsolve was separation or change of variables).
• casesplit splits a system of equations (and inequations) into a sequence of systems of equations and inequations such that the union of the non-singular solutions of the latter is equal to the
set of solutions of the original system. In addition, in each of the returned systems, all differential or algebraic redundancies are removed, and all the integrability conditions are
automatically satisfied. The computations are performed using the DifferentialAlgebra package.
• charstrip evaluates the characteristic strip associated with a given first order PDE; that is, it builds the coupled system of ODEs equivalent to that PDE. Additionally, given a characteristic
strip, charstrip can reverse it and return the family of PDEs behind it.
• dchange performs changes of variables in any algebraic object (PDEs, multiple integrals, integro-differential equations, etc.), as well as in procedures. This command is useful to change the
format of a PDE from one that is difficult to solve to one that is solvable.
• dcoeffs returns coefficients of polynomial differential equations, much like coeffs with algebraic polynomials.
• declare permits a simple and compact display on the screen of functions and derivatives. Typically, one declares functions, such as declare(f(x, y, z)), so that f(x,y,z) displays as 'f' (that is,
just by its name). Also, derivatives are "displayed" as indexed functions and it is possible to declare a "prime variable." In other words, for functions of one variable, derivatives with respect
to that variable will be displayed with a prime, '.
• difforder returns the general (or particular, with regard to any variable) differential order of a partial derivative (or the maximum general or particular differential order of an expression
containing partial derivatives).
• dpolyform accepts an equation or expression, or a set or list of them, understood to be equal to zero, and returns a differential polynomial system of equations.
• dsubs substitutes a derivative inside differential equations such that the resulting expression does not depend on the derivative being substituted.
• FunctionFieldSolutions computes exact solutions to DE systems involving mathematical functions, possibly inequations, ODEs and also non-differential equations. The solutions are returned as power
series (with some upper bound degree $n$) in the mathematical functions and its derivatives (up to some upper bound differential order $m$), having for coefficients multivariable polynomials
(with some upper bound degree $r$) of the independent variables.
• Laplace solves a second order linear PDE in two independent variables using the method of Laplace (not a Laplace transform).
• mapde maps a PDE into another PDE with different format (from among a few formats implemented at present), which is perhaps more easily solvable.
• PDEplot produces the plot of the solution for a first order linear (or nonlinear) partial differential equation (PDE), for given initial conditions.
• pdetest returns either 0 (when the PDE is annulled by the solution sol), indicating that the solution is correct, or a remaining algebraic expression (obtained after simplifying the PDE with
respect to the proposed solution), indicating that the solution might be wrong.
• Given a PDE, pdsolve's main goal is to find an analytical solution for it. There are no restrictions as to the type, differential order, or number of independent variables of the PDEs pdsolve can
try to solve.
• PolynomialSolutions receives a PDE system, optionally an indication of the degree and dependency, and computes related polynomial solutions.
• separability determines under what conditions it is possible to obtain a complete solution, through separation of variables by sum or product, for a given PDE. A complete solution is defined to
be a solution that depends on sum(diff_ord[i]+1,i=1..n) parameters, where n is the number of independent variables and diff_ord is the maximum differential order of the PDE with respect to each
of the independent variables.
• Solve is an unified solver that receives a system of equations, algebraic or differential, and solves this system, returning solutions optionally independent of indicated variables, calling solve
, dsolve or pdsolve according to the input received.
• splitstrip evaluates the strip associated with a PDE the same way as the strip command, but returns this strip, when possible, split into subsets, obtained by calling splitsys with the ODEs of
the characteristic strip as argument.
• splitsys splits a set of equations (ODEs, PDEs, algebraic equations, or a combination of these) into subsets, each one with equations coupled among themselves but not coupled to the equations of
the other subsets.
• ToMissingDependentVariable receives a PDE and returns another one, PDE2, equivalent to PDE in that from the solution of PDE2 one gets the solution to PDE, and such that PDE2 does not depend,
explicitly, on the dependent variable.
• TWSolutions computes Traveling Wave Solutions for PDEs and systems of them, either as expansions in tanh or in a number of other functions like JacobiSN, WeierstrassP, etc..
• undeclare clears declarations previously made.
Symmetry solution related
Note: Herein infinitesimals refer to a list with the components of the infinitesimal generator of a symmetry group.
• CanonicalCoordinates receives the infinitesimals of a symmetry group and returns associated canonical coordinates for it.
• ChangeSymmetry performs change of variables on the infinitesimals of a symmetry generator.
• CharacteristicQ receives the infinitesimals of a point symmetry and returns the characteristic of the group.
• CharacteristicQInvariants receives the infinitesimals of a symmetry group and computes related differential invariants directly from the CharacteristicQ for the infinitesimals.
• ConservedCurrents receives a PDE system and returns the conserved currents of it; when the system involves only ODEs, the conserved currents are the first integrals of the system.
• ConservedCurrentTest receives an algebraic expression and a PDE system and verifies whether the given expression is a conserved current.
• ConsistencyTest receives a PDE system and returns true or false according to whether the system is consistent
• D_Dx computes total derivatives in jet notation, that is taking the independent and dependent variables and their derivatives as differentiation variables in equal footing.
• DeterminingPDE receives a PDE system and computes the determining PDE system satisfied by the infinitesimals of the symmetry groups admitted by the given PDE system.
• Eta_k receives the infinitesimals of a symmetry and returns a table-procedure that computes, on request, any prolongation of these infinitesimals.
• Euler is the Euler operator: when applied to a DE system, it returns the exact conditions; i.e. the conditions to be satisfied when the system is a divergence.
• FromJet receives a mathematical expression in jet notation and returns the corresponding expression in function notation. This command is the counterpart of ToJet.
• FunctionFieldSolutions computes exact solutions to DE systems involving mathematical functions, possibly inequations, ODEs and also non-differential equations. The solutions are returned are
power series (with some upper bound degree $n$) in the mathematical functions and its derivatives (up to some upper bound differential order $m$), having for coefficients multivariable
polynomials (with some upper bound degree $r$) of the independent variables.
• InfinitesimalGenerator receives the infinitesimals of a symmetry and returns a procedure (operator) representing the infinitesimal generator, that is, one which acts on a PDE system to return the
related determining PDE.
• Infinitesimals receives a PDE system and returns the infinitesimals of symmetry groups admitted by the given PDE system.
• IntegratingFactors receives a DE system and returns the generalized integrating factors.
• IntegratingFactorTest receives an algebraic expression and a DE system, and verifies whether the given expression is a generalized integrating factor.
• InvariantEquation receives the infinitesimals of a symmetry group and computes the equation that is simultaneously invariant under the symmetry transformations corresponding to the given
• InvariantSolutions receives a PDE system and returns the so-called group invariant solutions for it, that is, the PDE system solutions derived by first (automatically) computing the symmetries
admitted by it.
• InvariantTransformation receives the infinitesimals of a symmetry group and computes the related finite (symmetry) transformations reducing the number of independent variables by N. (You can
optionally specify N.) These are the transformations from which InvariantSolutions derives solutions to the PDE system.
• Invariants receives the infinitesimals of a symmetry group and computes related differential invariants of any specified order.
• PolynomialSolutions receives a PDE system, optionally an indication of the degree and dependency, and computes related polynomial solutions.
• ReducedForm receives two PDE systems and reduces the first one with respect to the second one; this is similar to what simplify/siderels does but with PDE systems. This command can be useful
beyond the symmetry approach for PDE systems.
• SimilaritySolutions receives a PDE system and returns the so-called similarity solutions for it. This command is present mainly for pedagogical purposes in that the solutions it returns are
computed using one symmetry at a time. For practical purposes use InvariantSolutions instead, which can reduce the given PDE system using many symmetries in one go.
• SimilarityTransformation receives a PDE system and computes the finite (symmetry) transformations reducing the number of independent variables by one. These are the transformations from which
SimilaritySolutions derives solutions to the PDE system.
• SymmetryCommutator receives a pair of infinitesimals corresponding to symmetry transformations and returns their commutator.
• SymmetryGauge receives a symmetry, as a list or as an infinitesimal generator, and rewrites this symmetry in the most general form or in the one indicated using optional arguments.
• SymmetryTest receives the infinitesimals of a symmetry group and a PDE system and tests whether the PDE system admits that symmetry.
• SymmetryTransformation receives the infinitesimals of a 1-Dimensional point symmetry group and returns the finite form of the (symmetry) transformation leaving invariant any PDE system admitting
that symmetry.
• SymmetrySolutions receives the infinitesimals of a point symmetry group, a solution to some PDE system (itself not required) and returns the another solution, obtained by transforming the given
one using the finite form of the (symmetry) transformation given.
• ToJet receives a mathematical expression in function notation and returns the corresponding expression in jet notation. This command is the counterpart of FromJet.
A brief description of the PDEtools package commands, split into General purpose and Symmetry related, is as follows.
General purpose
• build takes a result given by pdsolve and returns the final expression for the indeterminate function (useful when the method used by pdsolve was separation or change of variables).
• casesplit splits a system of equations (and inequations) into a sequence of systems of equations and inequations such that the union of the non-singular solutions of the latter is equal to the set
of solutions of the original system. In addition, in each of the returned systems, all differential or algebraic redundancies are removed, and all the integrability conditions are automatically
satisfied. The computations are performed using the DifferentialAlgebra package.
• charstrip evaluates the characteristic strip associated with a given first order PDE; that is, it builds the coupled system of ODEs equivalent to that PDE. Additionally, given a characteristic
strip, charstrip can reverse it and return the family of PDEs behind it.
• dchange performs changes of variables in any algebraic object (PDEs, multiple integrals, integro-differential equations, etc.), as well as in procedures. This command is useful to change the
format of a PDE from one that is difficult to solve to one that is solvable.
• dcoeffs returns coefficients of polynomial differential equations, much like coeffs with algebraic polynomials.
• declare permits a simple and compact display on the screen of functions and derivatives. Typically, one declares functions, such as declare(f(x, y, z)), so that f(x,y,z) displays as 'f' (that is,
just by its name). Also, derivatives are "displayed" as indexed functions and it is possible to declare a "prime variable." In other words, for functions of one variable, derivatives with respect
to that variable will be displayed with a prime, '.
• difforder returns the general (or particular, with regard to any variable) differential order of a partial derivative (or the maximum general or particular differential order of an expression
containing partial derivatives).
• dpolyform accepts an equation or expression, or a set or list of them, understood to be equal to zero, and returns a differential polynomial system of equations.
• dsubs substitutes a derivative inside differential equations such that the resulting expression does not depend on the derivative being substituted.
• FunctionFieldSolutions computes exact solutions to DE systems involving mathematical functions, possibly inequations, ODEs and also non-differential equations. The solutions are returned as power
series (with some upper bound degree $n$) in the mathematical functions and its derivatives (up to some upper bound differential order $m$), having for coefficients multivariable polynomials (with
some upper bound degree $r$) of the independent variables.
• Laplace solves a second order linear PDE in two independent variables using the method of Laplace (not a Laplace transform).
• mapde maps a PDE into another PDE with different format (from among a few formats implemented at present), which is perhaps more easily solvable.
• PDEplot produces the plot of the solution for a first order linear (or nonlinear) partial differential equation (PDE), for given initial conditions.
• pdetest returns either 0 (when the PDE is annulled by the solution sol), indicating that the solution is correct, or a remaining algebraic expression (obtained after simplifying the PDE with
respect to the proposed solution), indicating that the solution might be wrong.
• Given a PDE, pdsolve's main goal is to find an analytical solution for it. There are no restrictions as to the type, differential order, or number of independent variables of the PDEs pdsolve can
try to solve.
• PolynomialSolutions receives a PDE system, optionally an indication of the degree and dependency, and computes related polynomial solutions.
• separability determines under what conditions it is possible to obtain a complete solution, through separation of variables by sum or product, for a given PDE. A complete solution is defined to be
a solution that depends on sum(diff_ord[i]+1,i=1..n) parameters, where n is the number of independent variables and diff_ord is the maximum differential order of the PDE with respect to each of
the independent variables.
• Solve is an unified solver that receives a system of equations, algebraic or differential, and solves this system, returning solutions optionally independent of indicated variables, calling solve,
dsolve or pdsolve according to the input received.
• splitstrip evaluates the strip associated with a PDE the same way as the strip command, but returns this strip, when possible, split into subsets, obtained by calling splitsys with the ODEs of the
characteristic strip as argument.
• splitsys splits a set of equations (ODEs, PDEs, algebraic equations, or a combination of these) into subsets, each one with equations coupled among themselves but not coupled to the equations of
the other subsets.
• ToMissingDependentVariable receives a PDE and returns another one, PDE2, equivalent to PDE in that from the solution of PDE2 one gets the solution to PDE, and such that PDE2 does not depend,
explicitly, on the dependent variable.
• TWSolutions computes Traveling Wave Solutions for PDEs and systems of them, either as expansions in tanh or in a number of other functions like JacobiSN, WeierstrassP, etc..
• undeclare clears declarations previously made.
• build takes a result given by pdsolve and returns the final expression for the indeterminate function (useful when the method used by pdsolve was separation or change of variables).
build takes a result given by pdsolve and returns the final expression for the indeterminate function (useful when the method used by pdsolve was separation or change of variables).
• casesplit splits a system of equations (and inequations) into a sequence of systems of equations and inequations such that the union of the non-singular solutions of the latter is equal to the set
of solutions of the original system. In addition, in each of the returned systems, all differential or algebraic redundancies are removed, and all the integrability conditions are automatically
satisfied. The computations are performed using the DifferentialAlgebra package.
casesplit splits a system of equations (and inequations) into a sequence of systems of equations and inequations such that the union of the non-singular solutions of the latter is equal to the set of
solutions of the original system. In addition, in each of the returned systems, all differential or algebraic redundancies are removed, and all the integrability conditions are automatically
satisfied. The computations are performed using the DifferentialAlgebra package.
• charstrip evaluates the characteristic strip associated with a given first order PDE; that is, it builds the coupled system of ODEs equivalent to that PDE. Additionally, given a characteristic
strip, charstrip can reverse it and return the family of PDEs behind it.
charstrip evaluates the characteristic strip associated with a given first order PDE; that is, it builds the coupled system of ODEs equivalent to that PDE. Additionally, given a characteristic strip,
charstrip can reverse it and return the family of PDEs behind it.
• dchange performs changes of variables in any algebraic object (PDEs, multiple integrals, integro-differential equations, etc.), as well as in procedures. This command is useful to change the format
of a PDE from one that is difficult to solve to one that is solvable.
dchange performs changes of variables in any algebraic object (PDEs, multiple integrals, integro-differential equations, etc.), as well as in procedures. This command is useful to change the format
of a PDE from one that is difficult to solve to one that is solvable.
• dcoeffs returns coefficients of polynomial differential equations, much like coeffs with algebraic polynomials.
dcoeffs returns coefficients of polynomial differential equations, much like coeffs with algebraic polynomials.
• declare permits a simple and compact display on the screen of functions and derivatives. Typically, one declares functions, such as declare(f(x, y, z)), so that f(x,y,z) displays as 'f' (that is,
just by its name). Also, derivatives are "displayed" as indexed functions and it is possible to declare a "prime variable." In other words, for functions of one variable, derivatives with respect
to that variable will be displayed with a prime, '.
declare permits a simple and compact display on the screen of functions and derivatives. Typically, one declares functions, such as declare(f(x, y, z)), so that f(x,y,z) displays as 'f' (that is,
just by its name). Also, derivatives are "displayed" as indexed functions and it is possible to declare a "prime variable." In other words, for functions of one variable, derivatives with respect to
that variable will be displayed with a prime, '.
• difforder returns the general (or particular, with regard to any variable) differential order of a partial derivative (or the maximum general or particular differential order of an expression
containing partial derivatives).
difforder returns the general (or particular, with regard to any variable) differential order of a partial derivative (or the maximum general or particular differential order of an expression
containing partial derivatives).
• dpolyform accepts an equation or expression, or a set or list of them, understood to be equal to zero, and returns a differential polynomial system of equations.
dpolyform accepts an equation or expression, or a set or list of them, understood to be equal to zero, and returns a differential polynomial system of equations.
• dsubs substitutes a derivative inside differential equations such that the resulting expression does not depend on the derivative being substituted.
dsubs substitutes a derivative inside differential equations such that the resulting expression does not depend on the derivative being substituted.
• FunctionFieldSolutions computes exact solutions to DE systems involving mathematical functions, possibly inequations, ODEs and also non-differential equations. The solutions are returned as power
series (with some upper bound degree $n$) in the mathematical functions and its derivatives (up to some upper bound differential order $m$), having for coefficients multivariable polynomials (with
some upper bound degree $r$) of the independent variables.
FunctionFieldSolutions computes exact solutions to DE systems involving mathematical functions, possibly inequations, ODEs and also non-differential equations. The solutions are returned as power
series (with some upper bound degree $n$) in the mathematical functions and its derivatives (up to some upper bound differential order $m$), having for coefficients multivariable polynomials (with
some upper bound degree $r$) of the independent variables.
• Laplace solves a second order linear PDE in two independent variables using the method of Laplace (not a Laplace transform).
Laplace solves a second order linear PDE in two independent variables using the method of Laplace (not a Laplace transform).
• mapde maps a PDE into another PDE with different format (from among a few formats implemented at present), which is perhaps more easily solvable.
mapde maps a PDE into another PDE with different format (from among a few formats implemented at present), which is perhaps more easily solvable.
• PDEplot produces the plot of the solution for a first order linear (or nonlinear) partial differential equation (PDE), for given initial conditions.
PDEplot produces the plot of the solution for a first order linear (or nonlinear) partial differential equation (PDE), for given initial conditions.
• pdetest returns either 0 (when the PDE is annulled by the solution sol), indicating that the solution is correct, or a remaining algebraic expression (obtained after simplifying the PDE with
respect to the proposed solution), indicating that the solution might be wrong.
pdetest returns either 0 (when the PDE is annulled by the solution sol), indicating that the solution is correct, or a remaining algebraic expression (obtained after simplifying the PDE with respect
to the proposed solution), indicating that the solution might be wrong.
• Given a PDE, pdsolve's main goal is to find an analytical solution for it. There are no restrictions as to the type, differential order, or number of independent variables of the PDEs pdsolve can
try to solve.
Given a PDE, pdsolve's main goal is to find an analytical solution for it. There are no restrictions as to the type, differential order, or number of independent variables of the PDEs pdsolve can try
to solve.
• PolynomialSolutions receives a PDE system, optionally an indication of the degree and dependency, and computes related polynomial solutions.
PolynomialSolutions receives a PDE system, optionally an indication of the degree and dependency, and computes related polynomial solutions.
• separability determines under what conditions it is possible to obtain a complete solution, through separation of variables by sum or product, for a given PDE. A complete solution is defined to be
a solution that depends on sum(diff_ord[i]+1,i=1..n) parameters, where n is the number of independent variables and diff_ord is the maximum differential order of the PDE with respect to each of the
independent variables.
separability determines under what conditions it is possible to obtain a complete solution, through separation of variables by sum or product, for a given PDE. A complete solution is defined to be a
solution that depends on sum(diff_ord[i]+1,i=1..n) parameters, where n is the number of independent variables and diff_ord is the maximum differential order of the PDE with respect to each of the
independent variables.
• Solve is an unified solver that receives a system of equations, algebraic or differential, and solves this system, returning solutions optionally independent of indicated variables, calling solve,
dsolve or pdsolve according to the input received.
Solve is an unified solver that receives a system of equations, algebraic or differential, and solves this system, returning solutions optionally independent of indicated variables, calling solve,
dsolve or pdsolve according to the input received.
• splitstrip evaluates the strip associated with a PDE the same way as the strip command, but returns this strip, when possible, split into subsets, obtained by calling splitsys with the ODEs of the
characteristic strip as argument.
splitstrip evaluates the strip associated with a PDE the same way as the strip command, but returns this strip, when possible, split into subsets, obtained by calling splitsys with the ODEs of the
characteristic strip as argument.
• splitsys splits a set of equations (ODEs, PDEs, algebraic equations, or a combination of these) into subsets, each one with equations coupled among themselves but not coupled to the equations of
the other subsets.
splitsys splits a set of equations (ODEs, PDEs, algebraic equations, or a combination of these) into subsets, each one with equations coupled among themselves but not coupled to the equations of the
other subsets.
• ToMissingDependentVariable receives a PDE and returns another one, PDE2, equivalent to PDE in that from the solution of PDE2 one gets the solution to PDE, and such that PDE2 does not depend,
explicitly, on the dependent variable.
ToMissingDependentVariable receives a PDE and returns another one, PDE2, equivalent to PDE in that from the solution of PDE2 one gets the solution to PDE, and such that PDE2 does not depend,
explicitly, on the dependent variable.
• TWSolutions computes Traveling Wave Solutions for PDEs and systems of them, either as expansions in tanh or in a number of other functions like JacobiSN, WeierstrassP, etc..
TWSolutions computes Traveling Wave Solutions for PDEs and systems of them, either as expansions in tanh or in a number of other functions like JacobiSN, WeierstrassP, etc..
Symmetry solution related
Note: Herein infinitesimals refer to a list with the components of the infinitesimal generator of a symmetry group.
• CanonicalCoordinates receives the infinitesimals of a symmetry group and returns associated canonical coordinates for it.
• ChangeSymmetry performs change of variables on the infinitesimals of a symmetry generator.
• CharacteristicQ receives the infinitesimals of a point symmetry and returns the characteristic of the group.
• CharacteristicQInvariants receives the infinitesimals of a symmetry group and computes related differential invariants directly from the CharacteristicQ for the infinitesimals.
• ConservedCurrents receives a PDE system and returns the conserved currents of it; when the system involves only ODEs, the conserved currents are the first integrals of the system.
• ConservedCurrentTest receives an algebraic expression and a PDE system and verifies whether the given expression is a conserved current.
• ConsistencyTest receives a PDE system and returns true or false according to whether the system is consistent
• D_Dx computes total derivatives in jet notation, that is taking the independent and dependent variables and their derivatives as differentiation variables in equal footing.
• DeterminingPDE receives a PDE system and computes the determining PDE system satisfied by the infinitesimals of the symmetry groups admitted by the given PDE system.
• Eta_k receives the infinitesimals of a symmetry and returns a table-procedure that computes, on request, any prolongation of these infinitesimals.
• Euler is the Euler operator: when applied to a DE system, it returns the exact conditions; i.e. the conditions to be satisfied when the system is a divergence.
• FromJet receives a mathematical expression in jet notation and returns the corresponding expression in function notation. This command is the counterpart of ToJet.
• FunctionFieldSolutions computes exact solutions to DE systems involving mathematical functions, possibly inequations, ODEs and also non-differential equations. The solutions are returned are power
series (with some upper bound degree $n$) in the mathematical functions and its derivatives (up to some upper bound differential order $m$), having for coefficients multivariable polynomials (with
some upper bound degree $r$) of the independent variables.
• InfinitesimalGenerator receives the infinitesimals of a symmetry and returns a procedure (operator) representing the infinitesimal generator, that is, one which acts on a PDE system to return the
related determining PDE.
• Infinitesimals receives a PDE system and returns the infinitesimals of symmetry groups admitted by the given PDE system.
• IntegratingFactors receives a DE system and returns the generalized integrating factors.
• IntegratingFactorTest receives an algebraic expression and a DE system, and verifies whether the given expression is a generalized integrating factor.
• InvariantEquation receives the infinitesimals of a symmetry group and computes the equation that is simultaneously invariant under the symmetry transformations corresponding to the given
• InvariantSolutions receives a PDE system and returns the so-called group invariant solutions for it, that is, the PDE system solutions derived by first (automatically) computing the symmetries
admitted by it.
• InvariantTransformation receives the infinitesimals of a symmetry group and computes the related finite (symmetry) transformations reducing the number of independent variables by N. (You can
optionally specify N.) These are the transformations from which InvariantSolutions derives solutions to the PDE system.
• Invariants receives the infinitesimals of a symmetry group and computes related differential invariants of any specified order.
• PolynomialSolutions receives a PDE system, optionally an indication of the degree and dependency, and computes related polynomial solutions.
• ReducedForm receives two PDE systems and reduces the first one with respect to the second one; this is similar to what simplify/siderels does but with PDE systems. This command can be useful
beyond the symmetry approach for PDE systems.
• SimilaritySolutions receives a PDE system and returns the so-called similarity solutions for it. This command is present mainly for pedagogical purposes in that the solutions it returns are
computed using one symmetry at a time. For practical purposes use InvariantSolutions instead, which can reduce the given PDE system using many symmetries in one go.
• SimilarityTransformation receives a PDE system and computes the finite (symmetry) transformations reducing the number of independent variables by one. These are the transformations from which
SimilaritySolutions derives solutions to the PDE system.
• SymmetryCommutator receives a pair of infinitesimals corresponding to symmetry transformations and returns their commutator.
• SymmetryGauge receives a symmetry, as a list or as an infinitesimal generator, and rewrites this symmetry in the most general form or in the one indicated using optional arguments.
• SymmetryTest receives the infinitesimals of a symmetry group and a PDE system and tests whether the PDE system admits that symmetry.
• SymmetryTransformation receives the infinitesimals of a 1-Dimensional point symmetry group and returns the finite form of the (symmetry) transformation leaving invariant any PDE system admitting
that symmetry.
• SymmetrySolutions receives the infinitesimals of a point symmetry group, a solution to some PDE system (itself not required) and returns the another solution, obtained by transforming the given
one using the finite form of the (symmetry) transformation given.
• ToJet receives a mathematical expression in function notation and returns the corresponding expression in jet notation. This command is the counterpart of FromJet.
Note: Herein infinitesimals refer to a list with the components of the infinitesimal generator of a symmetry group.
• CanonicalCoordinates receives the infinitesimals of a symmetry group and returns associated canonical coordinates for it.
CanonicalCoordinates receives the infinitesimals of a symmetry group and returns associated canonical coordinates for it.
• ChangeSymmetry performs change of variables on the infinitesimals of a symmetry generator.
ChangeSymmetry performs change of variables on the infinitesimals of a symmetry generator.
• CharacteristicQ receives the infinitesimals of a point symmetry and returns the characteristic of the group.
CharacteristicQ receives the infinitesimals of a point symmetry and returns the characteristic of the group.
• CharacteristicQInvariants receives the infinitesimals of a symmetry group and computes related differential invariants directly from the CharacteristicQ for the infinitesimals.
CharacteristicQInvariants receives the infinitesimals of a symmetry group and computes related differential invariants directly from the CharacteristicQ for the infinitesimals.
• ConservedCurrents receives a PDE system and returns the conserved currents of it; when the system involves only ODEs, the conserved currents are the first integrals of the system.
ConservedCurrents receives a PDE system and returns the conserved currents of it; when the system involves only ODEs, the conserved currents are the first integrals of the system.
• ConservedCurrentTest receives an algebraic expression and a PDE system and verifies whether the given expression is a conserved current.
ConservedCurrentTest receives an algebraic expression and a PDE system and verifies whether the given expression is a conserved current.
• ConsistencyTest receives a PDE system and returns true or false according to whether the system is consistent
ConsistencyTest receives a PDE system and returns true or false according to whether the system is consistent
• D_Dx computes total derivatives in jet notation, that is taking the independent and dependent variables and their derivatives as differentiation variables in equal footing.
D_Dx computes total derivatives in jet notation, that is taking the independent and dependent variables and their derivatives as differentiation variables in equal footing.
• DeterminingPDE receives a PDE system and computes the determining PDE system satisfied by the infinitesimals of the symmetry groups admitted by the given PDE system.
DeterminingPDE receives a PDE system and computes the determining PDE system satisfied by the infinitesimals of the symmetry groups admitted by the given PDE system.
• Eta_k receives the infinitesimals of a symmetry and returns a table-procedure that computes, on request, any prolongation of these infinitesimals.
Eta_k receives the infinitesimals of a symmetry and returns a table-procedure that computes, on request, any prolongation of these infinitesimals.
• Euler is the Euler operator: when applied to a DE system, it returns the exact conditions; i.e. the conditions to be satisfied when the system is a divergence.
Euler is the Euler operator: when applied to a DE system, it returns the exact conditions; i.e. the conditions to be satisfied when the system is a divergence.
• FromJet receives a mathematical expression in jet notation and returns the corresponding expression in function notation. This command is the counterpart of ToJet.
FromJet receives a mathematical expression in jet notation and returns the corresponding expression in function notation. This command is the counterpart of ToJet.
• FunctionFieldSolutions computes exact solutions to DE systems involving mathematical functions, possibly inequations, ODEs and also non-differential equations. The solutions are returned are power
series (with some upper bound degree $n$) in the mathematical functions and its derivatives (up to some upper bound differential order $m$), having for coefficients multivariable polynomials (with
some upper bound degree $r$) of the independent variables.
FunctionFieldSolutions computes exact solutions to DE systems involving mathematical functions, possibly inequations, ODEs and also non-differential equations. The solutions are returned are power
series (with some upper bound degree $n$) in the mathematical functions and its derivatives (up to some upper bound differential order $m$), having for coefficients multivariable polynomials (with
some upper bound degree $r$) of the independent variables.
• InfinitesimalGenerator receives the infinitesimals of a symmetry and returns a procedure (operator) representing the infinitesimal generator, that is, one which acts on a PDE system to return the
related determining PDE.
InfinitesimalGenerator receives the infinitesimals of a symmetry and returns a procedure (operator) representing the infinitesimal generator, that is, one which acts on a PDE system to return the
related determining PDE.
• Infinitesimals receives a PDE system and returns the infinitesimals of symmetry groups admitted by the given PDE system.
Infinitesimals receives a PDE system and returns the infinitesimals of symmetry groups admitted by the given PDE system.
• IntegratingFactors receives a DE system and returns the generalized integrating factors.
IntegratingFactors receives a DE system and returns the generalized integrating factors.
• IntegratingFactorTest receives an algebraic expression and a DE system, and verifies whether the given expression is a generalized integrating factor.
IntegratingFactorTest receives an algebraic expression and a DE system, and verifies whether the given expression is a generalized integrating factor.
• InvariantEquation receives the infinitesimals of a symmetry group and computes the equation that is simultaneously invariant under the symmetry transformations corresponding to the given
InvariantEquation receives the infinitesimals of a symmetry group and computes the equation that is simultaneously invariant under the symmetry transformations corresponding to the given symmetries.
• InvariantSolutions receives a PDE system and returns the so-called group invariant solutions for it, that is, the PDE system solutions derived by first (automatically) computing the symmetries
admitted by it.
InvariantSolutions receives a PDE system and returns the so-called group invariant solutions for it, that is, the PDE system solutions derived by first (automatically) computing the symmetries
admitted by it.
• InvariantTransformation receives the infinitesimals of a symmetry group and computes the related finite (symmetry) transformations reducing the number of independent variables by N. (You can
optionally specify N.) These are the transformations from which InvariantSolutions derives solutions to the PDE system.
InvariantTransformation receives the infinitesimals of a symmetry group and computes the related finite (symmetry) transformations reducing the number of independent variables by N. (You can
optionally specify N.) These are the transformations from which InvariantSolutions derives solutions to the PDE system.
• Invariants receives the infinitesimals of a symmetry group and computes related differential invariants of any specified order.
Invariants receives the infinitesimals of a symmetry group and computes related differential invariants of any specified order.
• ReducedForm receives two PDE systems and reduces the first one with respect to the second one; this is similar to what simplify/siderels does but with PDE systems. This command can be useful beyond
the symmetry approach for PDE systems.
ReducedForm receives two PDE systems and reduces the first one with respect to the second one; this is similar to what simplify/siderels does but with PDE systems. This command can be useful beyond
the symmetry approach for PDE systems.
• SimilaritySolutions receives a PDE system and returns the so-called similarity solutions for it. This command is present mainly for pedagogical purposes in that the solutions it returns are
computed using one symmetry at a time. For practical purposes use InvariantSolutions instead, which can reduce the given PDE system using many symmetries in one go.
SimilaritySolutions receives a PDE system and returns the so-called similarity solutions for it. This command is present mainly for pedagogical purposes in that the solutions it returns are computed
using one symmetry at a time. For practical purposes use InvariantSolutions instead, which can reduce the given PDE system using many symmetries in one go.
• SimilarityTransformation receives a PDE system and computes the finite (symmetry) transformations reducing the number of independent variables by one. These are the transformations from which
SimilaritySolutions derives solutions to the PDE system.
SimilarityTransformation receives a PDE system and computes the finite (symmetry) transformations reducing the number of independent variables by one. These are the transformations from which
SimilaritySolutions derives solutions to the PDE system.
• SymmetryCommutator receives a pair of infinitesimals corresponding to symmetry transformations and returns their commutator.
SymmetryCommutator receives a pair of infinitesimals corresponding to symmetry transformations and returns their commutator.
• SymmetryGauge receives a symmetry, as a list or as an infinitesimal generator, and rewrites this symmetry in the most general form or in the one indicated using optional arguments.
SymmetryGauge receives a symmetry, as a list or as an infinitesimal generator, and rewrites this symmetry in the most general form or in the one indicated using optional arguments.
• SymmetryTest receives the infinitesimals of a symmetry group and a PDE system and tests whether the PDE system admits that symmetry.
SymmetryTest receives the infinitesimals of a symmetry group and a PDE system and tests whether the PDE system admits that symmetry.
• SymmetryTransformation receives the infinitesimals of a 1-Dimensional point symmetry group and returns the finite form of the (symmetry) transformation leaving invariant any PDE system admitting
that symmetry.
SymmetryTransformation receives the infinitesimals of a 1-Dimensional point symmetry group and returns the finite form of the (symmetry) transformation leaving invariant any PDE system admitting that
• SymmetrySolutions receives the infinitesimals of a point symmetry group, a solution to some PDE system (itself not required) and returns the another solution, obtained by transforming the given one
using the finite form of the (symmetry) transformation given.
SymmetrySolutions receives the infinitesimals of a point symmetry group, a solution to some PDE system (itself not required) and returns the another solution, obtained by transforming the given one
using the finite form of the (symmetry) transformation given.
• ToJet receives a mathematical expression in function notation and returns the corresponding expression in jet notation. This command is the counterpart of FromJet.
ToJet receives a mathematical expression in function notation and returns the corresponding expression in jet notation. This command is the counterpart of FromJet.
Cheb-Terrab, E.S., and von Bulow, K. "A Computational Approach for the Analytical Solving of Partial Differential Equations." Computer Physics Communications, Vol. 90, (1995): 102-116.
Olver, P.J. Equivalence, Invariants and Symmetry. Cambridge Press, 1995.
Stephani, H. Differential Equations: Their Solution Using Symmetries. Edited by M. MacCallum. Cambridge University Press, 1989.
Cheb-Terrab, E.S., and von Bulow, K. "A Computational Approach for the Analytical Solving of Partial Differential Equations." Computer Physics Communications, Vol. 90, (1995): 102-116.
Stephani, H. Differential Equations: Their Solution Using Symmetries. Edited by M. MacCallum. Cambridge University Press, 1989. | {"url":"https://fr.maplesoft.com/support/help/addons/view.aspx?path=PDEtools&L=F","timestamp":"2024-11-07T15:40:25Z","content_type":"text/html","content_length":"210639","record_id":"<urn:uuid:b4270241-92b6-4772-9a5d-615c19bb49f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00224.warc.gz"} |
How antenna traps work - IW5EDI Simone - Ham-Radio
Article by VE3GK
I was assembling a HY/GAIN, TH6DXX, a 10 – 15 and 20 meter beam the other day and thought there might be some interest in the function of the traps. I have also included some practical information on
to build traps for a multi-band, 10-15-20 meter antenna.
When a certain amount of capacitance is connected in parallel with a certain amount of inductance it results in a very high resistance at a certain frequency. In other words this combination of
inductance and capacitance results in a special high resistance resistor designed to work at a narrow band of frequencies. Other frequencies above and below this frequency exhibit the opposite
condition, a very low resistance.
When a Tri-band 10, 15 and 20 meter beam is excited with 21 MHZ energy only the length of element to the 21 MHZ trap is used.
The trap is situated 1/4 wave length from the boom of a yagi and is similar to that of an open switch.
The same electric circuit is present on 10 meters. Conversely, the 14 MHZ signal perceives the 21 and 28 MHZ traps as a closed switches at 14 MHZ. The 20 meter signal and uses the whole length of the
TYPICAL TRAP FOR 20–15–10–METER BEAM
Please refer to the following drawings for the practical application of the parallel tuned traps that are used in multi band beams and wire dipoles. The capacitor is made by inserting one aluminum
tube inside another with an insulating sleeve in between serving as the dielectric.
The capacitance should be around 25 pf. for the 15 and 10 meter bands when the tubes over lap about 5 inches.
The dimensions of the coil determine the frequency of operation. The coil is air wound from one tube to the other.
The diameter and length of the coil should be about 3 inches. Allow 7 turns for 15 meters and 5 turns for 10 meters.
Approximately number 8 aluminum wire should be used.
The final tuning should be done on both networks with a grid dip meter. Adjustments should made to the coil and capacitor to resonate the network at the lower edge of the designated band.
TYPICAL TRAP FOR 20–15–10–METER WIRE DIPOLE
The same values of inductance and capacitance apply. It is important that the voltage rating of the capacitor be in the 2 to 3 KV range. Several lower voltage capacitors could be assembled in series
to make up the required voltage rating. The higher the power the higher the KV rating should be.
Remember when identical capacitors are installed in series you have to divide the value of one of them by the total number of capacitors in the string to get the total value of capacitance.
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Who was Socrates? Where did he live?
Hint: Socrates invented the famous style of teaching referred to as the ‘Socratic Method’ which involved a form of cooperative argumentative dialogue between a student and teacher. He was also the
motivation for Plato, the founder of the Western philosophical tradition.
Complete answer:
Socrates was a Greek philosopher and is believed to be the founding figure of modern Western philosophy. There were philosophers before Socrates (known as the pre-Socratics) but none of them, before,
touched upon the diversity of topics that Socrates did. Socrates used to live in the city-state of Athens. He spent the majority of his life there until he was given the death sentence in 399 BCE.
Socrates had not written anything and whatever we know of him is only because of the dialogues of his student named Plato, and through the other ‘Sokratikoi Logoi’ (discourses of Socrates) written by
other students like Xenophon. He was the motivation for Plato, the founder of the Western philosophical tradition. Plato, in turn, was the teacher of Aristotle, thus forming the popular triad of the
ancient Greek philosophers: Socrates, Plato, and Aristotle.
Note: Among the primary sources about his trial and death, the Apology of Socrates is the dialogue that depicts the trial, and is one of four Socratic dialogues, along with Euthyphro, Phaedo, and
Crito, by means of which the final days of the philosopher Socrates were described by Plato. He presents the philosopher as a martyr for philosophy and rational inquiry in general. | {"url":"https://www.vedantu.com/question-answer/socrates-where-did-he-live-class-8-social-science-cbse-60a65436d13f1000c94bd570","timestamp":"2024-11-03T13:55:42Z","content_type":"text/html","content_length":"148815","record_id":"<urn:uuid:250eff7e-a869-4132-8539-6ced6fa2eb1d>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00583.warc.gz"} |
circuit solver with steps
Reduce the original circuit to a single equivalent resistor, re-drawing the circuit in each step of reduction as simple series and simple parallel parts are reduced to single, equivalent resistors.
After that, all that must be done is to run the simulation. By using this website, you agree to our Cookie Policy. This button does not save changes. Once transformed into a series circuit, the
analysis can be conducted in the usual manner. Simply tap this button and you will be sent to the document page. With CircuitEngine, you will gain the intuitive grasp of circuits needed to be
competitive. After writing super-node KCL equation, the variable that the dependent source depends on should be written in terms of the node voltages. During the design portion of the circuit, One
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must be done is to run the simulation. Solve the system of simultaneous equations for the independent variables (voltages or currents). Previously in Lesson 4, the method for determining the
equivalent resistance of parallel are equal, then the total or equivalent resistance of those branche… Ohm’s law is a key device equation that relates current, voltage, and resistance. If you are
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Truth table, & Quizes Charge inductors and discharge them through other inductors. If you wish to revert back simply press the undo icon. Boolean Algebra simplifier & solver. You may even find
mistakes in your textbook. Once a circuit simulation is complete, the user can save the schematic image by tapping this button and giving the image a name and saving it. Circuit Solver doesn't
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variables, make a new table column f… Circuit Simulation and Resources Solve circuits instantly! Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators
step-by-step This website uses cookies to ensure you get the best experience. Launch it instantly with one click. Likewise, no components may be taken off the grid. CircuitEngine was originally a web
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become editable. Steps. Step 1:Assess which resistors in a circuit are connected together in simple series or simple parallel. If you see in step 4, 8kΩ resistor is in series with a parallel
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process, no other components may be added. Tap it multiple times to rotate the elements orientation. One can set the timestep between 1ns and 1Gs to view the waveform slowly or faster. Part of the
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for a website that will let me enter a problem and offers detailed step by step solution; basically it must walk me through the entire thing. Another way is to introduce a voltage source of zero
volts (i.e., a short circuit), and its concomitant node. Graphs are zoomable and resizable. Do not use the voltage for the whole circuit… This button changes some schematic symbols to be drawn as
US-based symbols or International Electrotechnical Commission-based symbols. This icon is to help create such circuits. No installation required! Graph voltage, charge, current and magnetic flux. Tap
the property in the edit menu and provide the new value of the property and hit "enter". In step 2, The Circuit is Open, Hence, Current will not flow in … In particular, consider the previous
circuit: We would like to find the current shown. Inside the menu bar you will find the components page, the help icon, the undo and redo icon, as well as other icons. In order to view the waveforms
for AC and Transient Analysis try to figure out the period and time constant respectively. How to specify and install CircuitSolver domestic hot water balancing valves - Literature, IMIs, submittals,
drawings, and other specs. To open the menu bar, simply hit this button. With the principle of superposition you can simplify the analysis of circuits with multiple inputs. The user must simply drag
the elements from the tool box onto the screen. One way would be to find each current into and out of node 1 and solve for the unknown current. To view the waveforms for AC and Transient analysis try
to figure out the period and constant! Is a place where two or more circuit elements meet poor and I have decided do! And install CircuitSolver domestic hot water balancing valves - Literature, IMIs,
submittals, drawings, and resistance feature! The min-terms once the items are placed unto the grid system it 's always adviced save. Agreeing to allow a support representative from Digi-Key to view
all the components available, navigate through the documents on! Tool box onto the screen graph voltage, and its concomitant node or.! Super-Node KCL equation, the analysis of circuits with multiple
inputs, open the resistance the! And magnetic flux process will inevitably simplify the problem US-based symbols or International Electrotechnical symbols! Determine the current shown by Thevenin ’ s
law is a key device that... Provide the new value of the design portion of the screen current I. In particular, consider the previous circuit: we would like to find each current into and out node., R
) Calculator, electrical quantities, relationships, and other specs two or more elements. Concomitant node the others to control the amount of magnification that is to... Undone with the undo button
Digi-Key to view your browser remotely the of! 5 and 6 Band resistor Color Code Calculators equation for each component describes... ( using a single equivalent resistor we talk about fundamental
circuit theories, ohm 's is. To utilize more of the resistance, open the resistance from the document menu, the can... The other sources by a short circuit ), and power dissipations in a circuit
connected... Values and press `` calculate '' to solve a circuit on paper theorem, need... This being the case OpAmps are usually drawn either with the principle superposition... Kirchhoff ’ s
theorem multiple times to rotate and find this icon, tap on the element and look this. ) Calculator press `` calculate '' to solve for the voltages and currents in loop! Together in simple series or
simple parallel go back to the circuit building. Same steps mentioned for dependent source with an extra step selected the icon will allow the user travel! The tool box onto the screen specify and
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save your work before as... Java 1.6 and a two button mouse are required Co-Browse window opens, give the session ID that is in! 60-Second video about the future of domestic hot water balancing
valves - Literature IMIs. This process will inevitably simplify the problem during the design process involves eliminating components little. Interest in CircuitEngine calculations you spend hours on
are correct … Boolean Algebra simplifier & solver simply press the button! Demo video → circuit Simulation and Resources Thank you for your interest in.... The documents page on the element and
search for this icon on the far right hand side of the and. Way is to run the Simulation the dependent source depends on should be while! Follow some steps or steps included are: 50 %, 75 % 75.
Screen display the toolbar to the right hand side of the screen while creating circuits I. The usual manner approach to building circuits of how use Thevenin 's theorem to reduce a complex to. Rotate
and find this icon to the right side of the circuit edit the properties, on! Toggle this button button mouse are required submittals, drawings, and other specs or leaving behind internal. It apart
solving a circuit with Thevenin theorem, we have to pick node... Resistor circuitanalysis is to be drawn as US-based symbols or International Electrotechnical Commission-based symbols 1ns and 1Gs to
all... Circuit and take it apart the image can be accessed through the menu bar and tap this icon, on! Theorem, we have to follow some steps or steps has been opened them... Properties, tap the icon
and you will gain the intuitive grasp of circuits with multiple inputs simply this! Certain orientations be written in terms of the node voltages are placed the! Buttons, simply hit this button
allows the user to control the amount of magnification that is located the. The flow of wires Algebra simplifier & solver node to be drawn as US-based symbols or International Electrotechnical
symbols. Matrix is defined based on all the components inside the circuit and take it apart certain orientations load. Zooming options included are: 50 %, 75 %, 75,., currents, and theorems taking
your time and going through this,! Circuitengine was originally a web applet that ran in this webpage far hand! Selected the icon will allow the user to travel from point to point any! Value of the
screen the resistance from the tool box onto the screen only used! 3 ) Mesh current method circuit fonts and calculate voltage between its terminals close the menu bar access... Tools icon for AC and
Transient analysis try to figure out the period and time respectively! On should be written in terms of the screen back simply press the undo icon add a meter simply..., 5 and 6 Band resistor Color
Code Calculators, currents, plots! Eliminating components of little use and provide the new value of resistor LED... The analysis can be accessed through the menu bar, simply tap this button look for
icon. A standalone program ( using a single equivalent resistor power, voltage, current & resistance ( P V. A parallel connection of 4kΩ resistor and 12k Ω resistor the usual manner, ohm 's is...
Undo button a circuit are connected together in simple series or simple parallel the board webpage... Reduce a complex circuit to a simply circuit items are placed unto the grid system it 's
selected, theorems... Meter, simply tap and drag them on to the board s law is a place two! Back simply press the undo button for this icon to the right hand of. You wo n't have to pick one node to
be drawn as US-based symbols or International Electrotechnical Commission-based symbols wish! 1: Assess which resistors in a circuit with Thevenin theorem, we need equation... Is located in the icon
a circuit on paper this icon, tap on the menu. Period and time constant respectively always adviced to save your work before leaving it... A web applet that ran in this webpage launch CircuitLab or
watch quick... That the dependent source with an extra step to view the waveform slowly or.... The circuits, a matrix is defined based on all the components available navigate! Solver uses a simple
drag and drop approach to building circuits the far right hand side the... No components may be taken during Simulation in order to view the waveforms for AC and analysis! Concepts circuit solver
with steps electrical physics them and edit their values Assess which resistors in a circuit goal... Once the items are placed unto the grid system it 's always adviced to save your documents
voltage,. Solving parallel circuits is an easy process once you know the basic formulas and principles behind. Circuit and take it apart on should be careful while placing the.! Meter and search for
this icon, tap on the element and for! Simplifier & solver circuits instantly find each current into and out of 1! Agreeing to allow a support representative from Digi-Key to view the waveform or.
Enter any two known values and press `` calculate '' to solve circuits! To reduce a complex circuit to a simply circuit to view your browser.! 2: Disconnect the load, and other specs, ohm 's law is
the most fundamental them... Solving a circuit find each current into and out of node 1 and solve for the voltages and in. Ohm 's law is a key device equation that relates current, voltage, current
and magnetic flux to move. Components may be added circuit solver with steps voltages are applied to the examples/documents pencil icon to the.. With 8kΩ are: 50 %, and power dissipations in a
simulator... Defined based on all the components available, navigate through the documents page on the and. Button to initiate the solving algorithm a single equivalent resistor inside the circuit
one! Superposition you can wire and edit their values can set the timestep between 1ns and 1Gs to the! Amplifiers can behave differently when voltages are applied to the right side of the screen: we
like. Serves as a way to quickly move to the right side of the screen while creating circuits,... The flow of wires are usually drawn either with the principle of you! Whether the countless pages of
calculations you spend circuit solver with steps on are correct of magnification that is applied to right!, LC, and its concomitant node that ran in this case, we need the equation each. Process once
you know the basic formulas and principles be drawn as US-based symbols or International Electrotechnical Commission-based.... Must simply drag the elements from the tool box onto the screen and
RLC.. Were made re-save your circuit if any changes were made that web applets are deprecated you..., wiring diagrams, and 100 % variable that the dependent source depends on be...
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Warnsdorff - Knight's Tour & Other ProblemsWarnsdorff
Exhaustive Analysis of Warnsdorff's Rule for Closed Tours
Introduction :
To the best of my knowledge, no exhaustive study of Warnsdorff’s rule for Knight’s Tour of the chessboard has been published. This paper presents the results for the work done on closed (re-entrant)
solutions (tours) obtainable on a standard 8 x 8 chessboard. Since an exhaustive study was intended, no attempt has been made to resolve a tie, if it occurs along the way. Instead, all the
alternative paths have been followed, without assigning priority thereto.
While conducting this study, it was noticed that strictly following the rule does not necessarily take the knight to a square with truly minimum further outlets. Some variations of the rule were
hence thought of, without violating the rule in its true spirit. In doing so, some new terms such as “Corner Dash (CD)” (full or semi), “End Corner Dash (ECD)” and “Extended End Corner Dash (XCD)”
have been coined and defined. All the computations have been carried out for all the 10 (Ten) unique starting points and their corresponding 53 (Fifty-three) ending points to make the tour closed.
Warnsdorff’s Rule :
Warnsdorff’s Rule[1] for Knight’s Tour states that at every step, a move should be made to a square from where there are fewest outlets to unvisited cells. This suggests that before making a move,
one should work out available number of moves from each target square. All such available number of moves should be compared and the target square from where minimum further moves are available,
should be chosen for making the move to that target cell.
This immediately raises a question. If there are two or more such target squares from where equal number of further moves are available and this number is minimum when compared with other remaining
target squares, which target square should be chosen for the move? Many people have devised methods for resolving this “Tie”. Literature is full of rules for resolving the tie. The present work has
not bothered with resolving any tie. The reason for this is already given above and will appear again later.
Previous Work Done :
The problem of Knight’s Tour of the chessboard, being at least 5 (perhaps 11 to 12) centuries old, has been widely worked on. A simple search using the Google® Search Engine easily detects several
thousand significant results (web-pages). I have visited several hundred such web-pages. I have not found any work similar to the one presented here.
I have published a book[2] in which a full chapter is devoted to this subject. All the material in this paper has been extracted from the book.
Background :
A little bit of background about the present work is in order. A computer program was made in 1978 for working out Knight’s Tour solutions. It followed a so called “brute-force” method.
Moves were numbered from 1 to 8. Move number 1 was two squares to right and one square down. Going clockwise, move number 2 was two squares down and one square right, and so on, till move number 8
was two squares to right and one square up.
This scheme is shown in Figure – 1.
At each move, the program worked out all the available moves for the knight from where it stood. Naturally, the moves that took the knight outside the board or the moves that took the knight to an
already visited square were omitted. These available moves were stored in a matrix called an Options Matrix against the move number to be made on the board. After storing all available moves from the
particular square (cell), the program made the numerically highest move and deleted the move being made from the Options Matrix.
This procedure was continued, hoping that it would result in a solution. The aim was to work out all available solutions, of course, a very naïve approach. If a solution was reached, the program
would store the solution and back-track by one move. Again, the numerically largest of the remaining available moves would be made and the process would continue. If at any stage the knight is
stuck-up and without a further move, it would back-track by one move. Once again, the process would continue with the numerically highest move and so on.
This program did not produce even a single solution despite providing lots of CPU time on DEC-1077 mainframe computer. This is when a friend suggested using Warnsdorff’s rule. In addition to the
evaluation part (of Warnsdorff’s rule) the only modification necessary in the available program was that instead of storing all the available moves at each stage, the available options were stored
only when there was a tie between (among) two (or more) moves for “minimum outlets” condition. Since all the alternative paths were to be tested one-by-one, no need was felt of resolving the tie. All
the options for equally minimum outlets were indeed tried out. This is the reason for labeling this work as an exhaustive study of Warnsdorff’s rule. There is one more reason for calling this as
exhaustive study but that will be discussed later in appropriate context.
If there is no tie for any particular position (only a single move has minimum further outlets), then no entry is made into the Options Matrix.
As mentioned earlier, back-tracking would be invoked under two circumstances. One is when the knight is stuck-up and there is no move left. Second is when a full solution is reached. In the modified
program, instead of back-tracking by one move only, it is by several moves, till the previous tie position is reached. This is easily detected by looking for a non-zero entry in the Options Matrix.
The program was modified and run in 1981. It was a pleasant surprise to find out that solutions started pouring in. Six thousand solutions were printed out on 250 pages, 24 solutions per page, with
five sets of 1,200 solutions per set of 50 pages. Each set of 1,200 solutions took only between 52 and 58 seconds of CPU run time on the mainframe computer. The 6,000 solutions were studied
carefully. It was found that the first 25 moves were identical in all the solutions. This meant that the solutions would be so many that it may not be feasible to record all of them. Therefore, it
was decided to concentrate on getting closed solutions only. The program was once again modified to get only closed solutions. This attempt was also successful in 1981. In order to get a closed
solution, the program first defines the starting point and ending point on the board. Obviously, to have a closed tour, the ending point and starting points are at a knight’s move apart. Since
several sets of starting and ending points exist on the board, there are as many sets of different number of solutions in each set.
Number of Sets (Algorithms) Worked Out :
Let us first understand the co-ordinate system used. Standard notation used for recording moves of a game of chess labels the columns from left to right as “a to h”. White pieces side is at the
bottom of the board, therefore, the rows are numbered from bottom to top as “1 to 8”. This system is depicted in Figure – 2.
There are only 10 unique starting points on the board of 64 cells. Other 54 cells are isomorphic with these 10. These starting points are:
1) Cell a8. There are 3 more (total 4) isomorphic points on the board.
2) Cell b8. There are 7 more (total 8) isomorphic points on the board.
3) Cell c8. There are 7 more (total 8) isomorphic points on the board.
4) Cell d8. There are 7 more (total 8) isomorphic points on the board.
5) Cell b7. There are 3 more (total 4) isomorphic points on the board.
6) Cell c7. There are 7 more (total 8) isomorphic points on the board.
7) Cell d7. There are 7 more (total 8) isomorphic points on the board.
8) Cell c6. There are 3 more (total 4) isomorphic points on the board.
9) Cell d6. There are 7 more (total 8) isomorphic points on the board.
10) Cell d5. There are 3 more (total 4) isomorphic points on the board.
These starting points are shown in Figure – 3.
The corresponding ending points for a closed tour are:
1) cells c7 or b6; i.e., 2 ending points
2) cells a6, c6 or d7; i.e., 3 ending points
3) cells a7, b6, d6 or e7; i.e., 4 ending points
4) cells b7, c6, e6 or f7; i.e., 4 ending points
5) cells a5, c5, d6 or d8; i.e., 4 ending points
6) cells a8, a6, b5, d5, e6 or e8; i.e., 6 ending points
7) cells b8, b6, c5, e5, f6 or f8; i.e., 6 ending points
8) cells b8, a7, a5, b4, d4, e5, e7 or d8; i.e., 8 ending points
9) cells c8, b7, b5, c4, e4, f5, f7 or e8; i.e., 8 ending points
10) cells c7, b6, b4, c3, e3, f4, f6 or e7; i.e., 8 ending points
There are thus 53 combinations in total for all the starting and ending points, giving us 53 algorithms for working out all the solutions with Warnsdorff’s rule.
Working of the Computer Program :
With the background as above, we are ready to understand the working of the program and how all the options from Warnsdorff’s rule are tried out one after another. We will generate the first solution
along with its options matrix. The starting point selected is the top left corner of the board, the cell “a8”. This gets the number 1. From here, we start evaluating each move before actually making
any move. From square marked 1, we have two moves available. Move type 1 takes us to cell “c7” and move type 2 takes us to cell “b6”. Since we are aiming to make the tour closed, the cell “c7″is
arbitrarily reserved for move number 64. Hence, the only target cell now left is b6. We write the number 2 in this cell, i.e., move the knight to this square b6.
From the cell marked 2, we have five moves available. Move type 1 takes us to cell “d5”, type 2 takes us to cell “c4”, type 3 takes us to cell “a4”, type 7 takes us to cell “c8” and type 8 takes us
to cell “d7”. From d5 and c4, there are 7 outlets available. From d7, there are 5 outlets. From a4 and c8, there are 3 outlets. The minimum number of outlets is 3 and these are from two target cells,
a4 and c8. There is thus a tie for this move. Any one move can be chosen, reserving the other for later evaluation. As per the program logic, the numerically higher move (type 7) is chosen, taking us
to c8. At the same time, Options Matrix stores the information that from cell numbered 2 move type 3 was also having minimum outlets. This is achieved by storing the number 3 at location MO(2,1).
Here, MO is the options matrix with DIMENSION (64,8).
Thus, we write the number 3 in the cell c8 and proceed further. The further working is continued in similar manner till we reach the cell a5 at move number 20. From here, there are three target cells
by move types 1, 2 and 8; reaching c4, b3 and c6 respectively. When we evaluate the number of outlets from each of these target cells, we find that from each target cell, there are exactly 4 outlets.
Thus, there is a triple tie. As before, per program logic, we select the last direction (8) and move to cell c6 and write the number 21 in that cell. Now, we have here two more target cells to be
remembered within the Options Matrix, having equal number of outlets to the selected option. This is done by storing (move type) number 1 at MO(20,1) and storing (move type) number 2 at MO(20,2).
Further working is not discussed in detail. By proceeding further as per above algorithm, we complete the tour as per Figure 4. This is the first solution obtained with this method.
There was no need to use back tracking of moves even once. While generating the solution, there are further entries in the options matrix MO whenever there are two or more target cells with equal
number of minimum outlets. Having obtained the first solution as above, now let us see how we can generate more solutions from the first one. First, let us review the nonzero entries in the options
matrix MO. These are at (2,1), (20,1), (20,2), (41,1), (41,2), (43,1), (47,1), (48,1) and (49,1) being the last nonzero entry. Hence, we come back to the cell e6 marked 49, erasing the numbers 50
through 64 working manually, or storing 0 (zero, signifying that the square is available [unvisited] for the knight) into those locations when the computer program back tracks to the cell e6 marked
49. At this point, we see that MO(49,1) had the value 2. So, we make move type 2 and write number 50 in cell f4. At the same time, we set the value of location MO(49,1) to 0 (zero) because we have
now used up the option. The further working is normal upto the solution obtained in Figure – 5 as the second solution by this method. Here, at move 54, there is a tie and a nonzero entry MO(54,1) = 5
gets generated.
For getting the next (third) solution, we only back track to cell marked 54 at f2 location, because that happens to be the last non-zero entry in the options matrix MO. Instead of making move type 6
(in figure – 5) we now make move type 5 as per MO(54,1) = 5 and go to cell d3 and write 55 in the cell. As usual, we set MO(54,1) to 0 (zero), having used up that option from the options matrix.
Along this route for the third solution, we notice that at 55, there is again a tie and another non-zero entry MO(55,1) = 4 gets generated.
For getting the fourth solution, we only back track to cell marked 55 at “d3” location, because that happens to be the last nonzero entry in the options matrix MO.
The process continues, exhausting all the ties recorded along the way from the last tie backwards. When the options matrix has no single non-zero entry left, the program comes to an end with a
message “All Solutions Exhausted” and gives the solution number for the last solution. For the starting point and ending point given above, total 14,258 solutions were obtained back-tracking and
moving forward then on.
The computer program was run repeatedly for all 53 combinations of starting and ending points. Just under a million, 989,931 closed solutions were obtained as per details given in Table – 1.
Corner Dash (CD) Concept :
It is quite obvious that in order to complete the tour, the four corners also have to be covered in the tour. There is a peculiarity here. Each corner can be visited only from one of the two approach
cells. If the knight lands on one of them, it is imperative that the knight immediately visits the corner and then comes out from the other approach cell. This, in a way, restricts the choices
available at the approach cells.
For the starting point a8 and ending point c7 discussed above for generating first few solutions, consider the position when the knight reaches the cell marked number 6, as depicted in Figure – 6.
However, the cell c2 is one of the two cells for approaching the corner at cell a1, the other being the cell b3. Hence, availability of 5 moves from c2 is illusory, there being an urgent need to
visit the corner at a1. In reality, there is only 1 move (and not 5 moves) available from c2. Thus, there is a strong case for jumping from a3 to c2 rather than from a3 to b1 (This is not a violation
of Warnsdorffs rule but we are applying it in its spirit rather than applying it by its word). What if we over-ride our usual criterion for this move only and force the knight to jump to cell c2
instead of to b1? From c2, the knight can visit the corner at a1 (even as per Warnsdorff’s rule and the program logic), come out to cell b3 and continue the tour as per Warnsdorffs rule. This forced
jump that temporarily overrides the normal program logic has been termed as making a Dash to the corner or simply “Corner Dash (CD)”.
This over-riding of normal program logic, or corner dash, can be done at all four corners, near cells a1, a8, h1 and h8. However, in the first set, the corner at cell a8 is not available to us for
corner dash because we have made it the starting point for our tour. With three corners available for corner dash, total eight algorithms can be created as follows –
1) Algorithm with no corner dash (original program).
2) Algorithm with only one corner dash
a) Top Right corner (cell h8) dash
b) Bottom Left corner (cell a1) dash
c) Bottom Right corner (cell h1) dash
3) Algorithm with two corners dash
a) Except Bottom Right, other corners (cells a1 and h8) dash
b) Except Bottom Left, other corners (cells h1 and h8) dash
c) Except Top Right, other corners (cells a1 and h1) dash
4) Algorithm with all the three corners (cells a1, h1 and h8) dash
Now we have total 8 algorithms instead of only 1 (additional 7). We stand to get many more different solutions for the tour. When we take the starting point away from the corner, e.g., b8 or c8, all
the four corners are available for corner dash. We then get following 16 algorithms instead of just 1 (additional 15) –
1) Algorithm with no corner dash (original program).
2) Algorithm with only one corner dash
a) Top Left corner (cell a8) dash
b) Top Right corner (cell h8) dash
c) Bottom Left corner (cell a1) dash
d) Bottom Right corner (cell h1) dash
3) Algorithm with two corners dash
a) Top Left and Top Right corners (cells a8 and h8) dash
b) Top Left and Bottom Left corners (cells a8 and a1) dash
c) Top Left and Bottom Right corners (cells a8 and h1) dash
d) Top Right and Bottom Left corners (cells a1 and h8) dash
e) Top Right and Bottom Right corners (cells h1 and h8) dash
f) Bottom Left and Bottom Right corners (cells a1 and h1) dash
4) Algorithm with three corners dash
a) Top Left, Top Right and Bottom Left corners (cells a8, h8 and a1) dash
b) Top Left, Top Right and Bottom Right corners (cells a8, h8 and h1) dash
c) Top Left, Bottom Left and Bottom Right corners (cells a8, a1 and h1) dash
d) Top Right, Bottom Left and Bottom Right corners (cells h8, a1 and h1) dash
5) Algorithm with all the four corners dash
When we apply Corner Dash concept to all the 53 combinations of starting and ending points, we get total of 736 algorithms to work with. The computer program was modified for each run and solutions
were worked out. The total number of solutions jumped to 5,201,841.
Semi Corner Dash Concept :
In the example for explaining the concept of corner dash (Figure – 6), we saw how the move evaluation process as per Warnsdorff’s rule could be modified for selecting the cell c2 for the move instead
of moving the knight to b1. Remembering that there are two approach cells for any corner, depending upon how the tour has progressed, it could have been the square b3 where the selection process
could have to be modified. Therefore, the corner dash logic inside the computer program is such that if the target square for the knight is either b3 or c2, then consider that it has only one further
exit and select that target square for the next move.
Now, suppose that we wish to allow the corner dash only if the square involved is b3 and not if it is c2. Alternatively, we wish to allow the corner dash only if the square involved is c2 and not if
it is b3. In these cases, each of the two conditions is termed as a “Semi Corner Dash” because it involves only one out of two approach cells for the corner.
What is the advantage or sense in this semi corner dash? Surprisingly, many times entirely different solutions (tours) are found by employing this idea. The number of solutions is also quite large.
If we label one approach cell as A and the other as B, for a single corner, we get 3 sets of solutions. We may call them set A, set B or set C where, C means either A or B, i.e., “full” corner dash
(the one we learned before semi corner dash). Since the case C has been already included in the 736 algorithms, we have two extra sets of solutions, A and B.
For two corners dash, there will be total 32 = 9 sets out of which one set will be with both full corner dashes. We then get 8 extra sets of solutions. Going further, for three corners dash, 33 = 27
and for four corners dash, 34 = 81 sets are present. In both these cases there is one set with all full corner dash, leaving us with 26 and 80 extra sets of solutions respectively. For the 736
algorithms with 0, 1, 2, 3 or 4 corners dashes, there are total 10,144 semi corner dash algorithms. All those runs were done to obtain 40,639,273 extra solutions. Adding the solutions for the 736
algorithms, the total solutions obtained were 45,841,114. This is 46.3 times the number of solutions obtained by strictly following Warnsdorff’s rule by word. Following the rule in spirit has paid
huge dividends!
The details about the number of solutions obtained from each algorithm, with full or semi corner dash, are given in the Table AlgoMain by following the link.
Concept of End Corner Dash (ECD) :
Observe set 3(b), i.e., starting point c8 (getting the number 1) and ending at b6 (getting the number 64). The cell b6 happens to be the access cell for the corner at a8. Therefore, the tour cannot
be completed unless the cell a8 has number 63 and naturally the other access cell c7 must be 62. The corner is hence visited in the end. This is hence called end corner dash (ECD). In this case,
there is no option but to cover the corner in the end only. Hence, such an end dash is called compulsory. Starting point and last few moves of this compulsory end corner dash are shown in Figure – 7.
One must distinguish this from a case when the end dash is optional. For example take the set 2(a) with starting point at b8 and ending point at a6. Here the normal full and semi corner dashes exist
and algorithms numbered 17 to 32 have been included already. An interesting observation is that the ending cell a6 is just one move away from the corner access cell c7. Why not force the program to
visit the a8 corner in the end only and not during the course of normal tour? The cells b6, a8, c7 and a6 will then get the numbers 61, 62, 63 and 64 respectively. This is the end corner dash. This
end dash is of optional (not compulsory) type. Starting point and last few moves of this optional end corner dash are shown in Figure – 8.
Once again, the justification for making this kind of variation in knight’s tour is for getting more solutions! All the 53 combinations of starting and ending points were carefully checked for
possibility of ECD. Total 32 such ECD were noted.
As one corner is being reserved for ECD, only 3 corners are available for our usual full and semi corners dash. That gives total 8 algorithms with 0, 1, 2 or 3 full corners dashes and 56 semi corner
dashes. Total of 256 algorithms for full dashes and 1,792 algorithms for semi dashes are obtained. The program was suitably modified and run 2,048 (256 + 1,792) times. For the 256 algorithms, 570,079
solutions and for 1,792 semi corner dashes, 2,366,427 solutions (tours) were obtained.
The details about the number of solutions obtained from each algorithm, with full or semi corner dash, are given in the Table AlgoECD by following the link.
As usual, there are other three corners (at a8, h1 and h8) available for full or semi corner dashes, giving 8 full corner dashes and 56 semi corner dashes.
Program changes for forcing extended end corner dash are again surprisingly simple. These runs were taken for all starting/ending points not missing any extended end dash possibility. There were
1,224 algorithms with or without full corner dashes, giving 1,091,095 solutions. There were 8,280 algorithms with semi corner dashes, giving 4,366,840 solutions.
The details about the number of solutions obtained from each algorithm, with full or semi corner dash, are given in the Table AlgoXCD by following the link.
What next? Should one look for “ultra-extended” end dash involving last six moves and what about “super-extended” end dash with last seven moves? All these are possible and each idea will have many
more solutions. I have decided to rest the work with the above grand total (sparing the reader!) and leave some scope for other enthusiasts.
Inclusion of corner dashes, semi corner dashes, ECD and XCD (the last two also with full and semi corner dashes) has made this work truly exhaustive indeed. This is one more reason for calling out
the work “exhaustive”.
The grand total of all solutions with Main, ECD and XCD algorithms comes to 54,234,604 including semi-corner dashes. It is interesting to note that to get these solutions, the computer program was
run 22,432 times with some modification or the other each time it was run.
Results for the Full Board :
The discussion so far has considered only 10 starting points as shown earlier in Figure – 3. To get the grand total number of tours for the full board, one needs to take into account the other 54
cells as well. To get this number, the results need to be expressed in a different manner, according to the starting point. Such tabulation is presented as Table – 2.
The 64 cells on the board are grouped into isomorphic sets. For starting point numbers 1, 5, 8 and 10, there are only four (each) isomorphic positions on the board. The total number of solutions
obtained with these four starting points will have to be multiplied by 4 to get corresponding total solutions for the sixteen cells on the full board. For the rest of the six starting points, each
has eight isomorphic positions on the board. The total number of solutions obtained with these six starting points will have to be multiplied by 8 to get corresponding total solutions for forty-eight
Making these calculations, the total (closed) solutions for the board works out to be 324,850,856. This number includes all types of full and semi corner dashes, end dashes and extended end dashes.
If one has to go strictly by the word of Warnsdorff’s rule, this number comes to 5,732,056. To enable this computation, table – 2 gives a column in which number of solutions without any corner dash
is given. By multiplying these numbers by number of isomorphic points and adding the products will give the total number.
References :
[1] Rouse Ball, W.W., Mathematical Recreations and Essays, 11th Edition, Reprinted 1940, p 174 – 185.
[2] Phadke, Pramod S., Computer Programs for Solving Mathematical Puzzles, Self-Published, 2007. Chapter – 11, p 86 – 114. | {"url":"https://knightstour.in/warnsdorff/","timestamp":"2024-11-04T02:36:52Z","content_type":"text/html","content_length":"130313","record_id":"<urn:uuid:a41027db-4190-4b5f-a0f0-608875c0e938>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00711.warc.gz"} |
Newton's Forward Interpolation Formula with MATLAB Program
In everyday life, sometimes we may require finding some unknown value with the given set of observations. For example, the data available for the premium, payable for a policy of Rs.1000 at age x, is
for every fifth year. Suppose, the data given is for the ages 30, 35, 40, 45, 50 and we are required to find the value of the premium at the age of 42 years, which is not directly given in the table.
Here we use the method of estimating an unknown value within the range with the help of given set of observation which is known as interpolation.
Definition of Interpolation
Given the set of tabular values (x0, y0), (x1, y1),…,(xn, yn) satisfying the relation y=f(x) where the explicit nature of f(x) is not known, it is required to find a simpler function say ?(x), such
that f(x) and ?(x) agree at the set of tabulated points. Such a process is called as interpolation.
If we know ‘n’ values of a function, we can get a polynomial of degree (n-1) whose graph passes through the corresponding points. Such a polynomial is used to estimate the values of the function at
the values of x.
We will study two different interpolation formula based on finite differences when the values of x are equally spaced. The first formula is:
Newton’s forward difference interpolation formula:
The formula is stated as:
Where ‘a+ph’ is the value for which the value of the function f(x) is to be estimated. Here ‘a’ is the initial value of x and ‘h’ is the interval of differencing.
The table gives the distance between nautical miles of the visible horizon for the given height in feet above the earth surface. Find the value of y when x= 218 feet.
MATLAB Program for Newtons Forward Interpolation Formula
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for future updates. Subscribe to our newsletter to get notifications about our updates via email. If you have any queries, feel free to ask in the comments section below. Have a nice day! | {"url":"https://www.myclassbook.org/2017/03/newton-forward-interpolation-formula.html","timestamp":"2024-11-11T02:10:30Z","content_type":"application/xhtml+xml","content_length":"260847","record_id":"<urn:uuid:a1e55c09-bd8a-4ca9-9bba-c466ed128654>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00430.warc.gz"} |
David Bachman, Saul Schleimer, and Henry Segerman
Saul Schleimer
Reader in Mathematics
Mathematics Institute, University of Warwick
Coventry, United Kingdom
David Bachman is a professor at Pitzer College in Claremont, CA. David studies geometry and topology, and enjoys creating 3D sculptures that illustrate these ideas. Saul Schleimer is a geometric
topologist, working at the University of Warwick. His other interests include combinatorial group theory and computation. He is especially interested in the interplay between these fields and
additionally in the visualization of ideas from these fields. Henry Segerman is an Associate Professor in the Department of Mathematics at Oklahoma State University. His interests include geometry
and topology, 3D printing, virtual reality, and spherical video.
Cannon-Thurston Maps
82 x 82 cm
Four digital prints
Four "views" of the inside of finite-volume hyperbolic three-manifolds. Each three-manifold contains a surface; when a “light ray” leaves your eye it becomes darker when it crosses the surface in the
positive sense and lighter when it crosses in the negative sense. The elaborate patterns come from the complicated geometry and topology of the ambient three-manifold. The four-manifolds (in reading
order) are m004, s227, s776, and s000 from the SnapPy census. | {"url":"https://gallery.bridgesmathart.org/exhibitions/2019-icerm-illustrating-mathematics/henrys-0","timestamp":"2024-11-05T14:04:10Z","content_type":"text/html","content_length":"30808","record_id":"<urn:uuid:fbe0407d-2d94-4de1-9c77-08f9eb1a8759>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00071.warc.gz"} |
Pendulum Animation
# Pendulum Animation # Install ffmpeg (allows us to create videos) !apt update -y !apt install ffmpeg -y # Normal Python Mathy Code # Import Libraries: import numpy as np import matplotlib.pyplot as
plt from matplotlib.animation import FFMpegWriter from matplotlib.patches import Circle # Pendulum Parameters g = 9.8 L = 1 # We need an array of time points from 0 to 10 in increments of 0.01
seconds dt = 0.001 t_vec = np.arange(0,10,dt) # Initialize a vector of zeros theta_vec = np.zeros(len(t_vec)) dtheta_vec = np.zeros(len(t_vec)) # Set our initial condition theta_vec[0] = np.pi/4 #
initial angle dtheta_vec[0] = 0 # initial angular velocity # Loop through time # Euler's Method (approximately integrates the differential equation) for i in range(1, len(t_vec)): theta_vec[i] =
theta_vec[i-1] + dtheta_vec[i-1]*dt dtheta_vec[i] = dtheta_vec[i-1] + (-g/L*np.sin(theta_vec[i-1]))*dt plt.plot(t_vec,theta_vec) plt.show() # Set up our Figure for drawing our Pendulum fig, ax = plt.
subplots() # Create a plot on those axes, which is currently empty p, = ax.plot([],[], color='cornflowerblue') #initializes an empyt plot ax.axis('equal') ax.set_xlim([-3, 3]) # x limits ax.set_ylim(
[-3, 3]) # y limits ax.set_xlabel('X') ax.set_ylabel('Y') ax.set_title('Pendulum Simulation') video_title = "simulation" # Now we want to plot stuff on those axes c = Circle((0, 0), radius=0.1, color
='cornflowerblue') ax.add_patch(c) # Define information for our Animation FPS = 20 sample_rate = int(1/(FPS*dt)) dpi = 300 # Quality of the video writerObj = FFMpegWriter(fps=FPS) # Now we're putting
the rod and bob in the right place # Initialize an array containing the # positions of the pendululm over time simulation_size = len(t_vec) # number of sim time points x_pendulum_arm = np.zeros(
simulation_size) y_pendulum_arm = np.zeros(simulation_size) for i in range(0, simulation_size): x_pendulum_arm[i] = L*np.sin(theta_vec[i]) y_pendulum_arm[i] = -L*np.cos(theta_vec[i]) # We've computed
all the pendulum positions # Now we need to update the plot in a loop and store each frame in a video # Plot and Create Animation: with writerObj.saving(fig, video_title+".mp4", dpi): # We want to
create video that represents the simulation # So we need to sample only a few of the frames: for i in range(0, simulation_size, sample_rate): # Update Pendulum Arm: x_data_points = [0, x_pendulum_arm
[i]] y_data_points = [0, y_pendulum_arm[i]] # We want to avoid creating new plots to make an animation (Very Slow) # Instead lets take the plot we made earlier and just update it with new data. p.
set_data(x_data_points, y_data_points) # Update plot with set_data # Update Pendulum Patch: patch_center = x_pendulum_arm[i], y_pendulum_arm[i] # ^ note: the commas without brackets create a LIST c.
center = patch_center # Same idea here, instead of drawing a new patch update its location # ^ updates the circle # Update Drawing: fig.canvas.draw() # Update the figure with the new changes # Grab
and Save Frame: writerObj.grab_frame() # Iport the video you just made and run it on the notebook from IPython.display import Video Video("/work/simulation.mp4", embed=True, width=640, height=480) #
You can find the video's path by right clicking on it in Notebooks & Files # and selecting "Copy path to clipboard" | {"url":"https://deepnote.com/app/john-sweet/Pendulum-Animation-2e878b67-4995-4021-95c5-2869041ac98f","timestamp":"2024-11-09T01:20:30Z","content_type":"text/html","content_length":"145837","record_id":"<urn:uuid:2ed73d2d-046b-4250-9b70-6004b4295646>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00869.warc.gz"} |
A method and a system for determining a product vector for performing Dynamic Time Warping - Patent 2854134
The present invention relates to the field of Dynamic Time Warping of signals, especially related to increasing the speed of performing Dynamic Time Warping, and particularly to a method and a system
for determining a product vector for performing Dynamic Time Warping of signals.
Modern day signal processing applications, such as Dynamic Time Warping, Data Compression, Data Indexing, Image Processing, et cetera, involve tremendous amounts of data processing. The different
signals involved therein are normally represented as matrices, which in turn comprise a vast multitude of vectors. The data processing involved thereof includes mathematical computations and
mathematical transformations, such as matrix additions, matrix multiplications, matrix inversions, determination of Fast Fourier Transforms, et cetera. Signal processing applications that involve
matrix multiplications and dot product computations, especially when the matrices are of immense dimensions and/or orders, can be both time consuming and resource intensive, because of the number of
multiplicative and additive operations that are required to be performed for the determination of one or more intermediate results and/or the final result.
For example, in the domain of Dynamic Time Warping, one or more Euclidean distances need to be determined for two input signals, prior to the computation of a Dynamic Time Warping Score for the two
input signals. The computation of the Euclidean distances in turn involves the determination of a product of the two input signals. Therefore, the speed of performing Dynamic Time Warping on the two
input signals is dependent on the speed of determination of the product of the two signals. Therewith, the speed of performing Dynamic Time Warping can be enhanced by reducing the time required for
the determination of the product of the two input signals.
Currently, the product of two matrices, wherein the matrices represent signals, is determined by direct multiplication of the matrices. However, the direct multiplication of the matrices is expensive
in terms of both time and the resources required to determine the product thereof. Thus, the current technique poses impediments, especially for very high speed and highly data intensive
applications, because latency is introduced in the determination of the final result.
Therefore, a need exists to increase the speed of determination of the product of the signals, therewith increasing the speed of the different mathematical computations involved therein, and the
performance of the signal processing applications thereof. For example, in the context of Dynamic Time Warping, an increase in the speed of determination of the product of the two signals, also
increases the speed of determination of the Euclidean distances associated therewith, and thereby leading to a reduction in the time required for performing Dynamic Time Warping.
It is an object of the present invention to provide an enhanced solution for increasing the speed of determination of the product of the two signals, especially in cases where the signals are
represented as matrices.
The aforementioned object is achieved by a method for determining a product vector according to claim 1, and a system thereof according to claim 8.
The underlying object of the present invention is to simplify the determination of a product of two signals, viz. a test signal vector and a template signal, especially when the two signals are
expressed as matrices. A simplified determination of the product of the two signals is beneficial in reducing the time and resources required, especially in time and resource intensive signal
processing applications, such as performing Dynamic Time Warping of the two signals, in which the Euclidean Distance of the two signals is required to be determined based on the product of the two
In the present invention, a method is proposed to determine a product vector of a test signal vector and a template signal vector. The test signal vector is a collection of vectorized values of a
portion of a test signal. The template signal vector is a collection of vectorized values of a template signal. The test signal vector is factorized, whereby a first and a second test signal
factorized vectors are obtained. Ranks of both the first and the second test signal factorized vectors are less than a rank of the test signal vector. The template signal vector is thereafter
multiplied with the first test signal factorized vector, wherewith an intermediate template signal vector. The intermediate template signal vector is thereafter multiplied with the second test signal
factorized vector, wherewith the product vector is obtained.
The low-rank factorization of the test signal vector simplifies the determination of the product of the test signal and the test signals, because the number of computations that are required to
determine the product vector is reduced. Furthermore, the low-rank test signal factorized vectors consume lesser memory space for storage as compared to the complete test signal vector, because of
the diminished ranks of the first and the second test signal factorized vectors as compared to the test signal vector.
In accordance with an embodiment of the present invention, a product of the first and the second test signal factorized vectors is an approximation of the test signal vector. Herewith, the memory
requirements for storing the first and the second test signal factorized vectors are further reduced, because the storage of accurate vectorized values of the test signal vector mandates more memory
In accordance with yet another embodiment of the present invention, a random signal is multiplied with the test signal vector, and a quasi product vector is therewith obtained. The random signal is a
collection of vectorized values of a random signal. The quasi product vector is factorized, wherewith a first and a second quasi product factorized vectors are obtained. Thereafter, the first quasi
product factorized vector is multiplied with an inverse random signal, wherewith the first test signal factorized vector is obtained. In a further embodiment, the low-rank factorization of the quasi
product vector is such that the second quasi product factorized vector is the second test signal factorized vector. Herewith, an alternate embodiment for the purpose of factorization of the test
signal vector can be realized.
In accordance with another embodiment of the present invention, the test signal vector and/or the quasi product vector is factorized into low-rank factors by means of performing Singular Value
Decomposition on the same. Singular Value Decomposition is a well-known method and follows a simple implementation of the same for the purpose of obtainment of the low-rank factors of the test
A method for performing Dynamic Time Warping of the test signal vector and the template signal vector, based on the product vector obtained in accordance any of the aforementioned embodiments is
herein disclosed. The product vector is processed along with the test signal vector and the template signal vector, wherewith a Euclidean distance between the test signal vector and the template
signal vector is obtained. Thereafter, the Euclidean Distance is processed to obtain a global distance between the test signal vector and the template signal vector, wherewith a Dynamic Time Warping
Score is obtained. The Dynamic Time Warping Score is a measure of the similarity between the test signal vector and the template signal vector.
A system disclosed herein, for the purpose of determination of the product vector of the test signal vector and the template signal vector, comprises a factorization module, a first multiplication
module and a second multiplication module. The factorization module is operably coupled to the first multiplication module, and the first multiplication module is operably coupled to the second
multiplication module. The test signal is factorized by the factorization module, wherewith the first and the second test signal factorized vectors are obtained. Thereafter, the first test signal
factorized vector and the template signal vector are multiplied by the first multiplication module, wherewith the intermediate template signal vector. The intermediate template signal vector is
thereafter multiplied with the second test signal factorized vector by the second multiplication module, wherewith the product vector is obtained.
In accordance with an embodiment of the present invention, the test signal vector is factorized by the factorization module such that the product of the first and the second template signal
factorized vectors yield at least an approximation of the template signal vector.
In accordance with another embodiment of the present invention, the factorization of the template signal by the factorization module is accomplished by performing Singular Value Decomposition of the
template signal vector.
In accordance with yet another embodiment of the present invention, a third multiplication module is provided therein. The multiplication of the random signal and the test signal vector is
facilitated by the third multiplication module for the purpose of obtainment of the quasi product vector.
In accordance with yet another embodiment of the present invention, the quasi product vector is factorized by the factorization module, therewith obtaining the first and the second quasi product
factorized vectors. The factorization is such that the second quasi product factorized vector is the second template signal factorized vector.
In accordance with yet another embodiment of the present invention, a fourth multiplication module is provided therein. The multiplication of the inverse random signal and the first quasi product
factorized vector is facilitated by the fourth multiplication module.
In accordance with yet another embodiment of the present invention, the first multiplication module is configured to perform the multiplication of the first test signal factorized vector and the
second quasi product factorized vector. Therewith, the product vector is obtained.
In accordance with yet another embodiment of the present invention, a memory unit is provided therein. The memory unit is beneficial for storing the test signal vector, the template signal vector,
the product vector, the first test signal factorized vector, and/or the second test signal factorized vector.
A Dynamic Time Warping Block for performing Dynamic Time Warping of the test signal vector and the template signal vector is disclosed herein. The Dynamic Time Warping Block comprises the system
according to any of the aforementioned embodiments, a Euclidean Distance Matrix Computation module, and a Dynamic Time Warping Score computation module. The Euclidean distance between the test signal
vector and the template signal vector is computed by the Euclidean Distance Computation module. The Euclidean distance is provided to the Dynamic Time Warping Score computation module, wherewith the
Euclidean Distance is processed and the global distance between the test signal vector and the template signal vector is determined. The global distance represents the Dynamic Time Warping Score for
the test signal vector and the template signal vector. The Dynamic Time Warping Score represents a similarity between the test signal vector and the template signal vector.
The aforementioned and other embodiments of the present invention related to a method and a system for determining a product vector for performing Dynamic Time Warping will now be addressed with
reference to the accompanying drawings of the present invention. The illustrated embodiments are intended to illustrate, but not to limit the invention. The accompanying drawings herewith contain the
following figures, in which like numbers refer to like parts, throughout the description and drawings.
The figures illustrate in a schematic manner further examples of the embodiments of the invention, in which:
FIG 1
depicts an overview of a system for determining a product vector of a test signal vector and a template signal vector according to one or more embodiments of the present invention,
FIG 2
depicts an exemplary embodiment of the system referred to in FIG 1,
FIG 3
depicts another embodiment of the system referred to in FIG 1,
FIG 4
depicts a Dynamic Time Warping Block comprising the system referred to in FIG 1 for determining a Dynamic Time Warping Score of the test signal vector and the template signal vector,
FIG 5
depicts a flowchart of a method for determining the product vector referred to in FIG 1,
FIG 6
depicts certain steps of the method referred to in FIG 5 with reference to another embodiment of the present invention,
FIG 7
depicts a flowchart of a method for performing Dynamic Time Warping of the test signal vector and the template signal vector referred to in FIG 1.
An overview of a system 10 for determining a product vector 40
from a test signal vector 30
and a template signal vector 20
in accordance with one or more embodiments of the present invention is depicted in FIG 1.
A plurality of test signal vectors ('n' number of exemplary test signal vectors 30
) 30 is depicted in FIG 1. It may be noted herein that each test signal vector 30
comprises vectorized values of at least a portion of a test signal (not shown), i.e. the vectorized values of the test signal vector 30
can correspond to respective discrete-time sampled values of the portion of the test signal. Herein, the test signal can correspond to a discrete-time signal, such as a discrete-time speech signal, a
discrete-time video signal, a discrete-time image signal, a discrete-time temperature signal, et cetera.
An exemplary manner of obtainment of the aforementioned 'n' number of exemplary test signal vectors 30
is elucidated herein. The aforementioned test signal can be windowed in time domain, wherein a certain time domain window of the test signal corresponds to the aforementioned portion of the test
signal, and thereafter the respective discrete-time sampled values that correspond to the portion of the test signal can be arranged accordingly to obtain the corresponding test signal vector 30
. Thus, it may be noted herein that sequentially arranged test signal vectors 30
correspond to respective vectorized values of sequential portions of the test signal, i.e. respective collections of sequential discrete-time sampled values of sequential time-domain windowed
portions of the discrete-time test signal.
Furthermore, it may be noted herein that if a number of discrete-time sampled values in each of the time domain windows is 'd', then a length of each test signal vector 20
is construed to be 'd'. Therefore, each test signal vector 20
is representable as a '1 X d' matrix, i.e. '1' row, and 'd' number of columns. Herein, each column is construed to represent respective vectorized values of the respective portions of the test
Herein, for the purpose of visualization, the plurality of 'n' number of test signal vectors 30 as depicted in FIG 1 are arranged in the form of an 'd X n' dimensional matrix, i.e. 'd' number of
rows, and 'n' number of columns. Herein, each column represents a particular test signal vector 30
, and each row represents a contiguous collection the corresponding vectorized values of respective test signal vectors 30
Hereinafter, the plurality of test signal vectors 30 which is arranged in the form of an 'd X n' dimensional matrix will be referred to as "the 'd X n' test signal matrix 30".
Herein, if 'n' is greater than 'd', then a rank of the 'd X n' test signal matrix 30 cannot exceed 'd'. Similarly, if 'n' is lesser than 'd', then the rank of the 'd X n' test signal matrix 30 cannot
exceed 'n'. Preferably, 'n' is greater than 'd'.
A plurality of template signal vectors ('m' number of exemplary template signal vectors 20
) 20 is depicted in FIG 1. It may be noted herein that each template signal vector 20
comprises vectorized values of at least a portion of a template signal (not shown). The vectorized values of the portion of a template signal refer to the respective discrete-time values of the
portion of the template signal.
It may be noted herein that the template signal vectors 20
serve as model signals for the purpose of comparison of the test signal vector 30
with one or more template signal vectors 20
for the purpose of determination of respective degrees of similarity between the test signal vector 30
and the respective template signal vectors 20
. The template signal vector 20
that is approximately similar to the test signal vector 30
can be thereafter selected. This is useful for performing certain signal processing applications such as Dynamic Time Warping, Data Compression, Data Indexing, et cetera.
Furthermore, it is construed that a length of each template signal vector 20
is also 'd', i.e. 'd' number of samples is comprised in each of the template signal vector 30
. However, herein each template signal vector 30
is representable as a 'd X 1' matrix, i.e. 'd' number of rows, and '1' column. Herein, each row is construed to represent the respective vectorized values of the respective portion of the template
Herein, for the purpose of visualization, and for the purpose of facilitation of the processing of a particular template signal vector 20
and the plurality of test signal vectors 30
, the plurality of template signal vectors 20 is arranged in a columnar manner, which is representable in the form of a 'm X d' dimensional matrix, i.e. 'm' number of rows and 'd' number of columns.
The columnar arrangement of the template signal vectors 20
as the 'm X d' matrix is beneficial for matrix multiplication of the 'm X d' template signal vectors 20
and the 'd X n' test signal matrix 30.
Hereinafter, the plurality of 'm' number of 'd X 1' template signal vectors 20
which is arranged in the form of an 'm X d' dimensional matrix will be referred to as "the 'm X d' template signal matrix 20".
A plurality of product vectors ('m X n' number of exemplary product vectors 40
) 40 is depicted in FIG 1. Herein an exemplary product vector 40
is to be construed as a vector-based dot product of an exemplary template signal vector 20
and an exemplary test signal vector 30
. It may be noted herein that a respective product vector 40
is determined as a dot product of a respective template signal vector 20
and a respective test signal vector 30
. The plurality of product vectors 40
is construed to be an ordered arrangement of the corresponding dot products of the respective plurality of template signal vectors 20 and the respective plurality of test signal vectors 30.
Therefore, 'm X n' number of product vectors 40
can be determined, because of the presence of 'm' number of template signal vectors 20
and 'n' number of test signal vectors 30
It may be noted herein that the length 'd' of the aforementioned exemplary template signal vector 20
and the length 'd' of the aforementioned exemplary test signal vector 30
are identical, for the purpose of determination of the dot product of the template signal vector 20
and the test signal vector 30
. Therewith, an entire length of the aforementioned exemplary product vector (i.e. the corresponding dot products of the respective exemplary template signal vectors 20
and the respective exemplary test signal vectors 30
) 40
is also 'm X n'.
Herein, for the purpose of visualization, the plurality of product vectors 40
is arranged in the form of an 'm X n' dimensional matrix, i.e. 'm' number of rows and 'n' number of columns.
Hereinafter, the plurality of product vectors 40
which is arranged in the form of an 'm X n' dimensional matrix will be referred to as "the 'm X n' product vector matrix 40".
Herein, if 'm' is greater than 'n', then a rank of the 'm X n' product vector matrix 40 cannot exceed 'n'. Similarly, if 'm' is lesser than 'n', then the rank of 'm X n' product vector matrix 40
cannot exceed 'm'.
In the subsequent paragraphs, it may be noted herein that the present invention will be elucidated specifically with respect to an exemplary template signal vector 20
and the 'd X n' test signal matrix 30 (i.e. the plurality of test signal vectors 30
) 30 for the purpose of determination of an exemplary product vector 40
. However, without loss of generality, it may be noted herein that the teachings of the present invention can be utilized and extended thereon to determine the product vectors 40
corresponding to the remaining template signal vectors 20
, should there be a scenario, which is usually the case in practical signal processing applications, such as Dynamic Time Warping, wherein a multitude of test signal vectors is present.
The system 10 of FIG 1, along with the various embodiments thereof, is configured to receive each of the template signal vectors 20
and the 'd X n' test signal matrix 30, and process the same for the determination of the respective product vectors 40
thereof. The processing of the template signal vectors 20
and the 'd X n' test signal matrix 30 involves the determination of the dot products thereof. FIG 1 is only a high level depiction of the system 10, and the various embodiments thereof are elucidated
with reference to FIG 2 and FIG 3.
It may be noted herein that the system 10 comprises a processing unit 15 to receive the template signal vector 20
and the plurality of test signal vectors 30 and to process the same to determine the respective product vector 40
thereof. The various components (which will be elucidated in the subsequent paragraphs) of the processing unit 15 can be implemented using one or more hardware modules, software modules, or
combinations thereof. For example, if the system 10 is implemented using a hardware unit, then the processing unit 15 may be realised by means of a processor of a General Purpose Computer, an
Application Specific Integrated Circuit, a Field Programmable Gate Array Device, a Complex Programmable Logic Device, et cetera.
In accordance with an embodiment of the present system 10, a memory unit 50 is provided thereunto, and the memory unit 50 is operably coupled to the processing unit 15 for enabling data transfer
between the processing unit 15 and the memory unit 50. The memory unit 50 facilitates the storage of one or more template signal vectors 20
, one or more test signal vectors 30
, and/or one or more product vectors 40
, et cetera. The memory unit 50 is preferably realizable as a database capable of being queried for obtaining data therefrom, whereby the template signal vectors 20
and/or the test signal vectors 30
can be provided to the processing unit 15 for the determination of the corresponding product vectors 40
. The coupling between the processing unit 15 and the memory unit 50 can be wired, wireless, or a combination thereof. Furthermore, according to an aspect of the present invention, the memory unit 50
can be internal to the processing unit 15 and the entire system 10 can be the processing unit 15 comprising the memory unit 50 itself - for example the memory unit 50 can be an internal cache memory
of the processing unit 15. Alternatively, according to another aspect of the present invention, the memory unit 50 can also be located external to the processing unit 15 - for example the memory unit
50 can be remotely located as compared to the processing unit 15.
Herein, the aforementioned matrix-arrangements 20,30,40 - which can correspond to the plurality of template signal vectors 20, the plurality of test signal vectors 30, and/or the plurality of product
vectors 40 - are depicted for illustrative purposes. The actual manner in which the aforementioned matrix-arrangements 20,30,40 are stored in the memory unit 50 and/or processed by the processing
unit 15 of the system 10 depends on the architecture of the system 10 and/or the architecture of the memory unit 50.
In the subsequent paragraphs, two exemplary embodiments of the system 10 are elucidated. The exemplary embodiments of the system 10 are utilised for the determination of the product vector 40
by processing each of the template signal vectors 20
and the 'd X n' test signal matrix 30. A first exemplary embodiment is elucidated with reference to FIG 2 and a second exemplary embodiment is elucidated with reference to FIG 3.
The system 10 in accordance with the first exemplary embodiment of the present invention is depicted in FIG 2.
FIG 1 is also referred to herein for the purpose of elucidation of FIG 2. The system 10 comprises a factorization module 60, a first multiplication module 70 and a second multiplication module 80,
for the determination of the product vector 40
. Herein, it may be noted that the factorization module 60, the first multiplication module 70 and the second multiplication module 80 can be realized as hardware modules, software modules, or
combinations thereof. The functioning of the aforementioned modules 60,70,80 is elucidated in the subsequent paragraphs.
In accordance with one or more aspects of the first exemplary embodiment of the invention, the factorization module 60 is configured to receive the 'd X n' test signal matrix 30, in order to
factorize the 'd X n' test signal matrix 30, viz. into a first test signal factorized vector 64 and a second test signal factorized vector 66. Herein, to achieve the purpose of faster and more
efficient computation of the product vector 40
, 'd X n' test signal matrix 30 is factorized by the factorization module 60 in a manner such that respective ranks of the first and second test signal factorized vectors 64,66 are both lower than a
rank of the 'd X n' test signal matrix 30. This aspect is termed as low-rank factorization of the 'd X n' test signal matrix 30.
It may be observed herein that according to one aspect of the aforementioned low-rank factorization, one or more individual dimensions of both the first test signal factorized vector 64 and
dimensions of the second test signal factorized vector 64 are reduced as compared to dimensions of the 'd X n' test signal matrix 30. For example, the 'd X n' test signal matrix 30 can be factorized
into a 'd X d-k' dimensional first test signal factorized vector 64 and a 'd-k X n' dimensional second test signal factorized vector 66. Herein 'k' is preferably less than both 'd' and 'n', and it
may be therewith observed that d-k < d < n.
Hereinafter, "the 'd X d-k' dimensional first test signal factorized vector 64" will be referred to as "the 'd X d-k' first test matrix 64", and "the 'd-k X n' dimensional second test signal
factorized vector 66" will be referred to as "the 'd-k X n' second test matrix 66".
Since 'd-k' is less than 'd', it may be noted herein that the rank of 'd X d-k' first test matrix 64 cannot exceed 'd-k'. Similarly, since 'd-k' is also less than 'n', it may be noted herein that the
rank of 'd-k X n' second test matrix 66 cannot exceed 'd-k', and the same is again less than 'd'. Therefore, the 'd X d-k' first test matrix 66 and the 'd-k X n' second test matrix 66 are both
low-rank factors of the 'd X n' test signal matrix 30.
Herein, according to an aspect of the present invention, the 'd X d-k' first test matrix 64 and the 'd-k X n' second test matrix 66 are factors such that, if the 'd X d-k' first test matrix 64 and
the 'd-k X n' second test matrix 66 were to be synthesized, then at least an approximation of the 'd X n' test signal matrix 30 is obtained, and the degree of approximation can be for example 80% of
the 'd X n' test signal matrix 30. This aspect is beneficial in reducing the memory space required for the storage of 'd X n' test signal matrix 30, because only 'd X d-k' first test matrix 64 and
the 'd-k X n' second test matrix 66 are required to be stored, which consume lesser memory space as compared to storing the accurate values of the test vectors 30
comprised in the 'd X n' test signal matrix 30.
Therewith, it is possible to factorize the 'd X n' test signal matrix 30 into a lower rank 'd X d-k' first test matrix 64 and a lower rank 'd-k X n' second test matrix 66. The aforementioned
factorization of the 'd X n' test signal matrix 30 into two matrices 64,66 of lower ranks as compared to the rank of 'd X n' test signal matrix 30 can be achieved using well-known low-rank matrix
approximation techniques. Certain well-known low-rank approximation techniques include Singular Value Decomposition, Principal Component Analysis, Factor Analysis, Total Least Squares Method, et
cetera. Singular Value Decomposition simplifies the task of factorizing the 'd X n' test signal matrix 30 into the aforementioned low-rank factors 64,66, and the same is preferably used for low-rank
factorization of the 'd X n' test signal matrix 30 in accordance with an embodiment of the present invention. The aforementioned low-rank approximation techniques are well-known in the art and the
same are not elucidated in detail herein for the purpose of brevity.
To summarize, the functioning of the factorization module 60 is such that the factorization module 60 receives any matrix as an input and provides at least two lower rank factors of the input matrix.
Additionally, the lower rank factors that are therewith obtained are such that the lower rank factors upon synthesis result in at least an approximation of the input matrix.
The first multiplication module 70 of the system is herein operably coupled to the factorization module 60, thereby enabling data transfer between the factorization module 60 and the first
multiplication module 70. Herein, the first multiplication module 70 is configured to receive the aforementioned 'd X d-k' first test matrix 64 , and the '1 X d' exemplary template signal vector 20
. Furthermore, the first multiplication module 70 is configured to multiply the '1 X d' exemplary template signal vector 20
and the 'd X d-k' first test matrix 64, whereby an intermediate template signal vector 75 is obtained. Dimensions of the intermediate template signal vector 75 obtained therewith is '1 X d-k', i.e.
the intermediate template signal vector 75 comprises '1' row and 'd-k' number of columns.
Hereinafter, the intermediate template signal vector 75 comprising '1' row and 'd-k' number of columns will be referred to as '1 X d-k' intermediate vector 75.
The second multiplication module 80 is operably coupled to the first multiplication module 70, thereby enabling data transfer between the second multiplication module 80 and the first multiplication
module 70. Herein, the second multiplication module 80 is configured to receive the aforementioned '1 X d-k' intermediate vector 75 and the 'd-k X n' second test matrix 66. Furthermore, the second
multiplication module 80 is configured to multiply the '1 X d-k' intermediate vector 75 and the 'd-k X n' second test matrix 66, wherewith a single row, for example the product vectors 40
, of the aforementioned 'm X n' product vector matrix 40 is obtained. It may be noted herein that dimensions of the aforementioned single row 40
of the 'm X n' product vector matrix 40 is '1 X n'.
It may be noted herein that subsequent rows 40
to 40
of the 'm X n' product vector matrix 40 can be obtained by providing subsequent template signal vectors 20
to the first multiplication module 70. Each of these template signal vectors 20
is thereafter respectively multiplied with the 'd X d-k' first test matrix 64, wherewith respective subsequent '1 X d-k' intermediate vectors 75 are obtained. The respective subsequent '1 X d-k'
intermediate vectors 75 are thereafter provided to the second multiplication module 80, wherein the respective '1 X d-k' intermediate vectors 75 are multiplied with the 'd-k X n' second test matrix
66, wherewith the respective subsequent rows 40
of the 'm X n' product vector matrix 40 are obtained.
Furthermore, the memory unit 50 can be configured to store the 'd X d-k' first test matrix 64, the '1 X d-k' intermediate vector 75, and/or the 'd-k X n' second test matrix 66. The operable coupling
of the memory unit 50 with the processing unit 15 enables data transfer between the memory unit 50 and the processing unit 15. Furthermore, the 'd X d-k' first test matrix 64 and the 'd-k X n' second
test matrix 66 can be fetched by the processing unit 15 from the memory unit 50 for processing the same and for additional purposes such as the determination of the 'm X n' product vector matrix 40.
Herein, according to another aspect of the present invention, the memory unit 50 can be configured to provide the 'd X d-k' first test matrix 64 to the first multiplication module 70 for the purpose
of computation of the '1 X d-k' intermediate vector 75. Similarly, the memory unit 50 can be configured to provide the '1 X d-k' intermediate vector 75 and the 'd-k X n' second test matrix 66 for the
purpose of determination of the 'm X n' product vector matrix 40.
The system 10 in accordance with the second exemplary embodiment of the present invention is depicted in FIG 3.
The preceding figures are also referred to herein for the purpose of elucidation of FIG 3. The second embodiment elucidates an alternate implementation of the system 10 for obtaining the
aforementioned 'm X n' product vector matrix 40. According to the second exemplary embodiment, the processing unit 15 comprises a third multiplication module 100. The third multiplication module 100
is configured to receive a random signal 90 and the 'd X n' test signal matrix 30, and to multiply the random signal 90 and the 'd X n' test signal matrix 30.
Herein, the random signal 90 is a plurality of 'p' number of '1 X d' dimensional random row vectors (not shown). The 'p' number of '1 X d' dimensional random row vectors are arranged in a row-wise
manner, thereby resulting in a 'p X d' matrix. Preferably, 'p' is equal to 'd', thereby resulting in a square matrix.
According to an alternate aspect of the present invention, the random signal 90 can also comprise a multitude of randomly selected template signal vectors 30
from the plurality of template signal vectors 30
Hereinafter, the 'p' number of '1 X d' dimensional random row vectors will be referred to as 'p X d' random signal matrix 90.
It may be noted herein that, if 'p' is greater than 'd', then a rank of the 'p X d' random signal matrix 90 cannot exceed 'd'. Similarly, if 'p' is lesser than 'd', then the rank of the 'p X d'
random signal matrix 90 cannot exceed 'p'.
By the multiplication of the 'p X d' random signal matrix 90 and the 'd X n' test signal matrix 30, a quasi product vector 110 is obtained. The quasi product vector 110 is represented as a 'p X n'
dimensioned matrix, and will be hereinafter referred to as 'p X n' quasi product matrix 110. The 'p X n' quasi product matrix 110 is an intermediate matrix that will be beneficial in the
determination of the 'm X n' product vector matrix 40.
It may be noted herein that, if 'p' is greater than 'n', then a rank of the 'p X n' quasi product matrix 110 cannot exceed 'n'. Similarly, if 'p' is lesser than 'n', then the rank of the 'p X n'
quasi product matrix 110 cannot exceed 'p'.
In accordance with this embodiment of the present invention, the factorization module 60 is configured to receive the 'p X n' quasi product matrix 110, and to factorize the 'p X n' quasi product
matrix 110 to obtain low-rank factors of the 'p X n' quasi product matrix 110. The 'p X n' quasi product matrix 110 is factorized to obtain at least two low-rank factors of the same, viz. a first
quasi product factorized vector 114 and a second quasi product factorized vector 116.
Low-rank factorization of the 'p X n' quasi product matrix 110 is achieved by performing any of the aforementioned low-rank factorization techniques on the 'p X n' quasi product matrix 110, for
example, by performing Singular Value Decomposition of the 'p X n' quasi product matrix 110.
It may be observed herein that according to one aspect of the aforementioned low-rank factorization, one or more individual dimensions of both the first quasi product factorized vector 114 and
dimensions of the second quasi product factorized vector 116 are reduced as compared to dimensions of the 'p X n' quasi product matrix 110. For example, the 'p X n' quasi product matrix 110 can be
factorized into a 'p X p-k' dimensional first quasi product factorized vector 114 and a 'p-k X n' dimensional second quasi product factorized vector 116. Herein 'k' is preferably less than both 'p'
and 'n'.
Hereinafter, "the 'p X p-k' dimensional first quasi product factorized vector 114" will be referred to as "the 'p X p-k' first quasi matrix 114", and "the 'p-k X n' dimensional second quasi product
factorized vector 116" will be referred to as "the 'p-k X n' second quasi matrix 116".
It may be noted herein that the rank of 'p X p-k' first quasi matrix 114 cannot exceed 'p-k', and the same is less than 'p'. Similarly, the rank of 'p-k X n' second quasi matrix 116 cannot exceed
'p-k', which is again less than 'p'. Therefore, the 'p X p-k' first quasi matrix 114 and the 'p-k X n' second quasi matrix 116 are both low-rank factors of the 'p X n' quasi product matrix 114.
Herein, according to an aspect of the present invention, the 'p X p-k' first quasi matrix 114 and the 'p-k X n' second quasi matrix 116 are factors such that, if the 'p X p-k' first quasi matrix 114
and the 'p-k X n' second quasi matrix 116 were to be synthesized, then at least an approximation of the 'p X n' quasi product matrix 110 is obtained, and the degree of approximation can be for
example 80% of the 'p X n' quasi product matrix 110. This aspect is beneficial in reducing the memory space required for the storage of 'p X n' quasi product matrix 110, because only 'p X p-k' first
quasi matrix 114 and the 'p-k X n' second quasi matrix 116 are required to be stored, which consume lesser memory space as compared to storing the 'p X n' quasi product matrix 110.
An inversion module 120 comprised in the system 10 is configured to receive the 'p X d' random signal matrix 90 for the purpose of inverting the 'p X d' random signal matrix 90. An inverse random
signal matrix 125 is therewith obtained wherein the inverse random signal matrix 125 comprises 'd' number of rows and 'p' number of columns.
Hereinafter the inverse random signal matrix 125 will be referred to as 'd X p' inverse matrix 125. It may be observed herein that the 'd X p' inverse matrix 125 can also be a pseudo-inverse of 'p X
d' random signal matrix 90, if 'p' and 'd' are unequal.
A fourth multiplication module 130 comprised in the system is configured to receive the 'd X p' inverse matrix 125 and the 'p X p-k' first quasi matrix 114, and configured to multiply the 'd X p'
inverse matrix 125 and the 'p X p-k' first quasi matrix 114. By the multiplication of the 'd X p' inverse matrix and the 'p X p-k' first quasi matrix, a first intermediate quasi matrix 134 is
obtained. The first intermediate quasi matrix 134 comprises 'd' number rows and 'p-k' number of columns.
Hereinafter, the first intermediate quasi matrix 134 comprising 'd' number rows and 'p-k' number of columns will be referred to as 'd X p-k' first intermediate quasi matrix 134.
The 'd X p-k' first intermediate quasi matrix 134 can also be the 'd X d-k' first test matrix 64, if the multiplication of the 'd X p' inverse matrix 125 and the 'p X p-k' first quasi matrix 114 were
to annul the effect of the multiplication of the 'p X d' random signal matrix 125 and the 'd X n' test signal matrix 30, and the subsequent factorization of the 'p X n' quasi product matrix 110 into
the 'p X p-k' first quasi matrix 114 and the 'p-k X n' second quasi matrix 116.
In accordance with the present embodiment, the first multiplication module 70 is herein configured to receive the exemplary 'm X d' template signal matrix 20 and the 'd X p-k' first intermediate
quasi matrix 134, and also configured to multiply the 'm X d' template signal matrix 20 and the 'd X p-k' first intermediate quasi matrix 134. By the multiplication of the 'm X d' template signal
matrix 20 and the 'd X p-k' first intermediate quasi matrix 134, a second intermediate quasi matrix 136 is therewith obtained. The second intermediate quasi matrix 136 comprises 'm' number of rows
and 'p-k' number of columns.
Hereinafter, the second intermediate quasi matrix 136 comprising 'm' number of rows and 'p-k' number of columns will be referred to as 'm X p-k' second intermediate quasi matrix 136.
In furtherance to the aforementioned, in accordance with the present embodiment, the second multiplication module 80 is configured to receive the 'm X p-k' second intermediate quasi matrix 136 and
the 'p-k X n' second quasi matrix 116, and also configured to multiply the 'm X p-k' second intermediate quasi matrix 136 and the 'p-k X n' second quasi matrix 116. By the multiplication of the 'm X
p-k' second intermediate quasi matrix 136 and the 'p-k X n' second quasi matrix 116, the 'm X n' product vector matrix 40 is therewith obtained. The 'm X n' product vector matrix 40 can be stored in
the memory unit 50 and retrieved later, for example for further processing of the 'm X n' product vector matrix 40 for any signal processing application.
The 'm X n' product vector matrix 40 determined in accordance with the aforementioned paragraphs is beneficial in the determination of respective Euclidean Distances between the respective 'm' number
of plurality of test signal vectors 20 and the respective 'n' number of plurality of template signal vectors 30. Thereafter, the Euclidean Distances can be used for performing Dynamic Time Warping of
the test signal vector 30
with the plurality of 'm' number of template signal vectors 20. These aspects are elucidated with reference to FIG 4 for exemplary and illustrative purposes.
Herein, the system 10 comprising the factorization module 60, the first multiplication module 70 and the second multiplication module 80 can be realised as a single hardware unit, wherein different
entities of the hardware unit are configured to perform the functions of the factorization module 60, the first multiplication module 70 and the second multiplication module 80. For example, the
system 10 depicted in FIG 2 can be realised on a Field Programmable Gate Array Device which comprises a plurality of Configurable Logic Blocks. A first set of the Configurable Logic Blocks can be
configured to perform one or more functions associated with the factorization module 60, and a second set of the Configurable Logic Blocks can be configured to perform one or more functions
associated with the first multiplication module 70, and a third set of the Configurable Logic Blocks can be configured to perform one or more functions associated with the second multiplication
module 80, et cetera.
A Dynamic Time Warping Block 150 comprising the system 10 in accordance with any of the aforementioned embodiments is depicted in FIG 4.
One or more of the preceding figures are also referred to herein for the purpose of elucidation of the Dynamic Time Warping Block 150 depicted in FIG 4. The Dynamic Time Warping Block 150 is
beneficial for determining a similarity between one or more of the plurality of the test signal vectors 30 and the plurality of template signal vectors 20. The Dynamic Time Warping Block 150
comprises the system 10 in accordance with any of the aforementioned embodiments, a Euclidean Distance Matrix Computation module 140 and a Dynamic Time Warping Score computation module 160. In FIG 1,
the system 10 is depicted to be located internal to the Dynamic Time Warping Block 150. However, according to an alternate aspect, and without loss of any generality, the system 10 may be also
located external to the Dynamic Time Warping Block 150.
The Euclidean Distance Matrix Computation module 140 is configured to receive the 'm X n' product vector matrix 40, the 'm X d' template signal matrix 20, and the 'd X n' test signal matrix 30 as
inputs. Herein, the Euclidean Distance Matrix Computation module 140 is configured to determine an 'm X n' Euclidean Distance Matrix (not depicted), which comprises a plurality of Euclidean Distances
(not depicted). Each Euclidean Distance thereby determined signifies a respective distance between a certain test signal vector (comprised in the 'd Xn' test signal matrix 30) 30
and a certain template signal vector (comprised in the 'm X d' template signal matrix 20) 20
. The collection of such respective Euclidean Distances between each of the respective test signals 30
and each of the respective template signals 20
constitutes the 'm X n' Euclidean Distance Matrix, which is the output provided by the 'm X n' Euclidean Distance Matrix Computation module. The determination of the 'm X n' Euclidean Distance Matrix
based upon the provision of the 'm X d' template signal matrix 20, the 'd X n' test signal matrix 30, and the 'm X n' product vector matrix 40 and the implementation of the Euclidean Distance Matrix
Computation module 140 are well-known in the art, and is not elucidated herein for the purpose of brevity.
Thereafter the 'm X n' Euclidean Distance Matrix is provided to the Dynamic Time Warping Score computation module 160 for performing Dynamic Time Warping of the plurality of test signals 30 and the
plurality of template signals 20. Herewith, an 'm X n' Global Distance Matrix (not depicted) is determined for the test signals 30
represented in the 'd X n' test signal matrix 30 and for the template signals 20
comprised in the 'm X d' template signal matrix 20, and thereby a Dynamic Time Warping Score purporting to the similarity of a certain test signal 30
with any of the template signals 20
is determinable. The determination of the Dynamic Time Warping Score, i.e. the determination of the 'm X n' Global Distance Matrix, by the performance of Dynamic Time Warping of the plurality of test
signals 30 and the plurality of template signals 20 based on the aforementioned 'm X n' Euclidean Distance Matrix is well-known in the art and is not elucidated herein for the purpose of brevity.
A flowchart 500 of an overview of a method for determining the product vectors 40
from the test signal vectors 30
and the template signal vector 20
in accordance with one or more embodiments of the present invention is depicted in FIG 5.
Reference is made to one or more of the preceding Figures for the purpose of elucidation of the aforementioned flowchart 500.
In steps 510 and 520, the test signal vector 30
and the template signal vector 20
are received respectively. Herein, the test signal vector 30
and the template signal vector 20
are represented as 'd X n' test signal matrix 30 and 'm X d' test signal matrix 20 respectively. According to one aspect, the 'm X d' template signal matrix 20 and 'd X n' test signal matrix 30 can
be stored in the memory unit 50, and the memory unit 50 can thereafter be queried by the processing unit 15 to receive the 'm X d' template signal matrix 20 and 'd X n' test signal matrix 30.
In step 530, the 'd X n' test signal matrix 30 is factorized into the aforementioned 'd X d-k' first test matrix 64, and the 'd-k X n' second test matrix 66, which are low-rank factors of the 'd X n'
test signal matrix 30. In accordance with an embodiment of the present method, the low-rank factors, viz. 'd X d-k' first test matrix 64 and the 'd-k X n' second test matrix 66, are obtained by
performing Singular Value Decomposition of the 'd X n' test signal matrix 30. The step 530 can be performed by providing the 'd X n' test signal matrix 30 to the factorization module 60 for the
purpose of low-rank factorization of the 'd X n' test signal matrix 30. For example, the low-rank factorization of the 'd X n' test signal matrix 30 can be achieved by performing Singular Value
Decomposition on the 'd X n' test signal matrix 30.
In step 540, the '1 X d' exemplary template signal vector 20
and the 'd X d-k' first test matrix 64 are multiplied, wherewith the intermediate template signal vector 75 is obtained. The step 540 can be performed by providing the '1 X d' exemplary template
signal vector 20
and the 'd X d-k' first test matrix 64 to the first multiplication module 70 for the purpose of multiplication of the '1 X d' exemplary template signal vector 20
and the 'd X d-k' first test matrix 64.
In step 550, the '1 X d-k' intermediate vector 75 and the 'd-k X n' second test matrix 66 are multiplied, wherewith the single row (of dimensions '1 X n') 40
of the aforementioned 'm X n' product vector matrix 40 is obtained. The step 550 can be performed by providing the '1 X d-k' intermediate vector 75 and the 'd-k X n' second test matrix 66 to the
second multiplication module 80 for the purpose of multiplication of the '1 X d-k' intermediate vector 75 and the 'd-k X n' second test matrix 66.
It may be noted herein that subsequent rows 40
to 40
of the 'm X n' product vector matrix 40 can be obtained by sequential repetition of the steps 540 and 550 for each of the subsequent template signal vectors 20
. Herein, different template signal vectors 20
are provided to the first multiplication module 70, wherein the 'd X d-k' first test matrix 64 remains the same. Therewith, respective subsequent '1 X d-k' intermediate vectors 75 are obtained, which
are thereafter provided to the second multiplication module 80 for the purpose of determination of the respective subsequent rows 40
to 40
of the 'm X n' product vector matrix 40. Furthermore, in the second multiplication module 80, the 'd-k X n' second test matrix 66 remains the same.
Thereafter, in step 560, the 'm X n' product vector matrix 40 obtained therewith is stored in the memory unit 50. The 'm X n' product vector matrix 40 can be provided to the processing unit 15 at a
subsequent stage for the purpose of processing the same in the context of a signal processing application, such as Dynamic Time Warping, Data Compression, Data Indexing, et cetera.
Certain steps comprised in the step 530, which is related to the factorization of the 'd X n' test signal matrix 30, in accordance with an alternate embodiment are depicted in FIG 6.
In step 531, the 'p X d' random signal matrix 90 and the 'd X n' test signal matrix 30 are multiplied, wherewith the aforementioned 'p X n' quasi product matrix 110 is obtained. The step 531 can be
performed by providing the 'p X d' random signal matrix 90 and the 'd X n' test signal matrix 30 to the third multiplication module 100 for the purpose of multiplication of the 'p X d' random signal
matrix 90 and the 'd X n' test signal matrix 30.
Thereafter, in step 532, 'p X n' quasi product matrix 110 is low-rank factorized into the 'p X p-k' first quasi matrix 114 and the 'p-k X n' second quasi matrix 116. The step 532 can be performed by
providing 'p X n' quasi product matrix 110 to the factorization module 60, and the low-rank factors of the same can be obtained by performing Singular Value Decomposition on the 'p X n' quasi product
matrix 110.
In step 533, 'd X p' inverse matrix 125 and the 'p X p-k' first quasi matrix 114 are multiplied, wherewith the 'd X p-k' first intermediate quasi matrix 134 is obtained.
Herein, the step 533 can be performed by providing the 'd X p' inverse matrix 125 and the 'p X p-k' first quasi matrix 114 to the fourth multiplication module 130 for the purpose of multiplication of
the 'd X p' inverse matrix 125 and the 'p X p-k' first quasi matrix 114.
The 'd X p' inverse matrix 125 can be obtained by providing the 'p X d' random signal matrix 90 to the inversion module 120 for the purpose of determination of the inverse of the 'p X d' random
signal matrix 90.
In step 534, the 'p X p-k' first quasi matrix 114 and the 'p-k X n' second quasi matrix 116 obtained therewith are stored in the memory unit 50. The 'p X p-k' first quasi matrix 114 and the 'p-k X n'
second quasi matrix 116 can be provided to the processing unit 115 at another subsequent stage for the purpose of processing the same for the determination of the 'm X n' product vector matrix 40.
The 'm X n' product vector matrix 40 obtained in accordance with the aforementioned steps can be used for the purpose of performing Dynamic Time Warping of the plurality of test signals 30
and the plurality of template signals 20
A flowchart 700 of a method for performing Dynamic Time Warping of the test signals 30
and the template signals 20
is depicted in FIG 7.
In step 710, the 'm X n' product vector 40 is received. In accordance with one aspect, the 'm X n' product vector 40 is stored in the memory unit 50, and the memory unit 50 can thereafter be queried
by the processing unit 15 to receive , 'm X n' product vector 40.
In step 720, the test signal vector ('d X n' test signal matrix 30) 30
and the template signal vector ('m X d' template signal matrix 20) 20
are received respectively. Herein, the memory unit 50 can be queried by the processing unit 15 to receive the 'm X d' template signal matrix 20 and 'd X n' test signal matrix 30.
Thereafter, in step 730, the 'm X n' Euclidean Distance Matrix is determined. The step 730 can be performed by providing the 'm X n' product vector matrix 40, the 'm X d' template signal matrix 20,
and the 'd X n' test signal matrix 30 to the aforementioned Euclidean Distance Matrix Computation module 140 for the purpose of determination of the 'm X n' Euclidean Distance Matrix.
In a subsequent step 740, the Dynamic Time Warping Score is determined. Herein, the step 740 can be performed by providing the 'm X n' Euclidean Distance Matrix to the aforementioned Dynamic Time
Warping Score computation module 160. Herewith, the 'm X n' Global Distance Matrix is determined, wherewith the Dynamic Time Warping Score for the plurality of test signals 30
and the plurality of template signals 20
is obtained.
Herein, in accordance with an aspect of the present invention, the aforementioned plurality of test signal vectors 30 can also be a concatenation of a plurality of groups of test signal vectors.
Herein, each group of test signal vectors comprises those test signal vectors that belong to a certain signal class. In such a scenario, the aforementioned 'm X n' product vector matrix 40 can be
determined on a per-class basis, i.e. corresponding product vector can be determined for the plurality of template signal vectors 20 and an individual group of test signal vectors. Herein, for
facilitating the determination of the product vector on a per-class basis, for each group comprising test signal vectors, respective low-rank factors are determined, and the plurality of template
signals 20 is multiplied with the respective low-rank factors corresponding to that particular group of test signal vectors in accordance with the aforementioned teachings of the present invention,
in order to obtain the corresponding product vector.
The aforementioned per-class based technique is beneficial for performing Dynamic Time Warping based classification of the plurality of test signals 30, if multiple classes of test signal vectors are
present. Herein, individual product vectors can be determined on a per-class basis, for the purpose of determination of the corresponding Euclidean Distance Matrices. The corresponding Euclidean
Distance Matrices are thereafter utilised for obtaining Dynamic Time Warping scores on a per-class basis, therewith increasing the speed and reliability of the aforementioned Dynamic Time Warping
Block 150. Furthermore, in the per-class based implementation of the Dynamic Time Warping Block 150, multiple processing units can be utilised, wherein each processing unit can be configured to
determine a certain product vector for a certain class of test signal vectors 20, the corresponding Euclidean Distance Matrix and the corresponding Dynamic Time Warping Score. Furthermore, the
multiple processing units of the Dynamic Time Warping Block 150 can be configured to operate in parallel, wherewith the speed of Dynamic Time Warping Block is further enhanced.
Though the invention has been described herein with reference to specific embodiments, this description is not meant to be construed in a limiting sense. Various examples of the disclosed
embodiments, as well as alternate embodiments of the invention, will become apparent to persons skilled in the art upon reference to the description of the invention. It is therefore contemplated
that such modifications can be made without departing from the embodiments of the present invention as defined.
A method for determining a product vector (40
) for determining a Euclidean distance between a test signal vector (30
) and at least a template signal vector (20
), wherein the test signal vector (30
) comprises vectorized values of at least a portion of a test signal, and wherein template signal vector (20
) comprises vectorized values of a template signal,
the method comprising:
- a step (530) of factorizing the test signal vector (30[1]-30[n]) for obtaining at least a first test signal factorized vector (64) and a second test signal factorized vector (66) of the test signal
vector (30[1]-30[n]), wherein respective ranks of the first test signal factorized vector (64) and the second test signal factorized vector (66) are both less than a rank of the test signal vector
- a step (540) of multiplying the template signal vector (20[1]-20[m]) and the first test signal factorized vector (64) for obtaining an intermediate template signal vector (75), wherein a rank of
the intermediate template signal vector (75) is less than or equal to a rank of the template signal vector (20[1]-20[m]), and
- a step (550) of multiplying the intermediate template signal vector (75) and the second test signal factorized vector (66) for determining the product vector (40[1,1]-40[m,n]).
2. The method according to claim 1, wherein a product of the first test signal factorized vector (64) and the second test signal factorized vector (66) is at least an approximation of the test signal
vector (30[1]-30[n]).
3. The method according to claim 1 or claim 2, wherein the first test signal factorized vector (64) and the second test signal factorized vector (66) are obtained by performing Singular Value
Decomposition of the test signal vector (30[1]-30[n]).
The method according to claim 1 or claim 2, wherein in the step (530) of factorizing the test signal vector (30
), the obtainment of the first test signal factorized vector (64) comprises:
- a step (531) of multiplying a random signal (90) with the test signal vector (30[1]-30[n]) for obtaining a quasi product vector (110), wherein the random signal (90) comprises a plurality of random
signal vectors, wherein each random signal vector of the plurality of the random signal vectors comprises a plurality of random values,
- a step (532) of factorizing the quasi product vector (110) for obtaining a first quasi product factorized vector (114) and a second quasi product factorized vector (116) for the quasi product
vector (110), wherein respective ranks of the first quasi product factorized vector (114) and the second quasi product factorized vector (116) are both less than a rank of the quasi product vector
(110), and
- a step (532) of multiplying the first quasi product factorized vector (114) with an inverse random signal (125) for obtaining the first test signal factorized vector (64), wherein the inverse
random signal (125) is an inverse of the random signal (90).
5. The method according to claim 4, wherein the second quasi product factorized vector (116) is the second test signal factorized vector (66) and/or wherein the first quasi product factorized vector
(114) and the second quasi product factorized vector (116) are obtained by performing Singular Value Decomposition of the quasi product factorized vector (110).
A method for performing Dynamic Time Warping between a test signal vector (30
) and at least a template signal vector (20
), wherein the test signal vector (30
) comprises vectorized values of at least a portion of a test signal, and wherein template signal vector (20
) comprises vectorized values of a template signal,
the method comprising:
- a step (510-550) of determining a product vector (40[1,1]-40[m,n]) of the test signal vector (30[1]-30[n]) and the template signal vector (20[1]-20[m]), wherein the product vector (40[1,1]-40[m,n])
is determined according to any of the claims 1 to 6,
- a step (730) of processing the product vector (40[1,1]-40[m,n]) for determining a Euclidean distance between the test signal vector (30[1]-30[n]) and the template signal vector (20[1]-20[m]),
- a step (740) of processing the Euclidean distance for determining a global distance between the test signal vector (30[1]-30[n]) and the template signal vector (20[1]-20[m]), wherein the global
distance represents a Dynamic Time Warping Score for the test signal vector (30[1]-30[n]) and the template signal vector (20[1]-20[m]), and wherein the Dynamic Time Warping Score represents a
similarity between the test signal vector (30[1]-30[n]) and the template signal vector (20[1]-20[m]).
A system (10) for performing the method according to any of the claims 1 to 5 for determining the product vector (40
) from the test signal vector (30
) and the template signal vector (20
the system (10) comprising:
- a factorization module (60) for factorizing the test signal vector (30[1]-30[n]) for obtaining the first test signal factorized vector (64) and the second test signal factorized vector (66) for the
test signal vector (30[1]-30[n]),
- a first multiplication module (70) for multiplying the template signal vector (20[1]-20[m]) and the first test signal factorized vector (64) for obtaining the intermediate template signal vector
(75), wherein the first multiplication module (70) is operably coupled to the factorization module (60) for receiving the first test signal factorized vector (64), and
- a second multiplication module (80) for multiplying the second test signal factorized vector (66) and the intermediate template signal vector (75) for obtaining the product vector (40[1,1]-40
[m,n]), wherein the second multiplication module (80) is operably coupled to the first multiplication module (70) for receiving the intermediate template signal vector (75).
8. The system (10) according to claim 7, wherein the factorization module (60) is configured to factorize the test signal vector (30[1]-30[n]), such that the product of the first test signal
factorized vector (64) and the second test signal factorized vector (66) is at least an approximation of the test signal vector (30[1]-30[n]).
9. The system (10) according to claim 7 or claim 8, wherein the factorization module (60) is configured to factorize the test signal vector (30[1]-30[n]) by performing Singular Value Decomposition of
the test signal vector (30[1]-30[n]) for obtaining the first test signal factorized vector (64) and the second test signal factorized vector (66).
10. The system (10) according to claim 9, further comprising a third multiplication module (100) for multiplying the random signal (90) and the test signal vector (30[1]-30[n]) for obtaining the
quasi product vector (110).
11. The system (10) according to claim 10, wherein the factorization module (60) is further configured to factorize the quasi product vector (110) for obtaining the first quasi product factorized
vector (114) and the second quasi product factorized vector (116) from the quasi product vector (110), wherein the second quasi product factorized vector (116) is the second test signal factorized
vector (66).
12. The system (10) according to claim 11, further comprising a fourth multiplication module (130) for multiplying the inverse random signal (125) and the first quasi product factorized vector (114)
for obtaining the first test signal factorized vector (64).
13. The system (10) according to claim 12, wherein the second multiplication module (80) is further configured to multiply the first test signal factorized vector (64) and the second quasi product
factorized vector (116) for obtaining the product vector (40[1,1]-40[m,n]).
14. The system (10) according to any of the claims 7 to 13, further comprising a memory unit (50) for storing at least one of the template signal vector (20[1]-20[m]), the test signal vector (30[1]
-30[n]), the product vector (40[1,1]-40[m,n]), the first test signal factorized vector (64), and the second test signal factorized vector (66).
A Dynamic Time Warping Block (150) for performing Dynamic Time Warping of a test signal vector (30
) and at least a template signal vector (20
), wherein the test signal vector (30
) comprises vectorized values of at least a portion of a test signal, and wherein template signal vector (20
) comprises vectorized values of a template signal,
the Dynamic Time Warping Block (150) comprising:
- the system (10) according to any of the claims 7 to 14 for determining a product vector (40[1,1]-40[m,n]) of the test signal vector (30[1]-30[n]) and the template signal vector (20[1]-20[m]),
- a Euclidean Distance Matrix Computation module (140) configured to process the test signal vector (30[1]-30[n]), the template signal vector (20[1]-20[n]) and the product vector (40[1,1]-40[m,n])
for determining a Euclidean Distance between the test signal vector (30[1]-30[n]) and the template signal vector (20[1]-20[m]) based on the product vector (40[1,1]-40[m,n]),
- a Dynamic Time Warping Score computation module (160) for processing the Euclidean Distance for determining a global distance between the test signal vector (30[1]-30[n]) and the template signal
vector (20[1]-20[m]), wherein the global distance represents a Dynamic Time Warping Score for the test signal vector (30[1]-30[n]) and the template signal vector (20[1]-20[m]), and wherein the
Dynamic Time Warping Score represents a similarity between the test signal vector (30[1]-30[n]) and the template signal vector (20[1]-20[m]). | {"url":"https://data.epo.org/publication-server/rest/v1.2/publication-dates/20150401/patents/EP2854134NWA1/document.html","timestamp":"2024-11-12T02:05:11Z","content_type":"text/html","content_length":"112348","record_id":"<urn:uuid:7eb05d4e-a38e-4ea7-bae4-44f5a4a8683d>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00596.warc.gz"} |
Lessons and tips for designing a machine learning study using EHR data | Journal of Clinical and Translational Science | Cambridge Core
The mandate to adopt electronic health records (EHRs) in 2009 under the Health Information Technology for Economic and Clinical Health Act has resulted in widespread electronic collection of health
data [Reference Blumenthal and Tavenner1]. With the increase in the use of electronic medical data comes an increase in the amount of healthcare data that are generated. Such EHR data, insurance
databases, and other available clinical data have the potential to aid researchers in solving clinical problems. Machine learning (ML) algorithms are quickly rising to the top of the list of tools
that can be used to address such problems. However, how to properly use ML methods in the clinical setting is not widely understood. Liu et al. have published an article educating clinicians how to
read papers utilizing ML techniques in the medical literature [Reference Liu2]. Nonetheless, understanding the use of ML approaches in healthcare is still needed, especially since precision and
accuracy are often key components of solutions to healthcare problems. The intended audience for this manuscript is clinicians and translational researchers interested in learning more about the
overall process of ML and predictive analytics, clinicians interested in gaining a better understanding of the working framework for ML so that they can better communicate with the individuals
creating such models, and analytics professionals that are interested in expanding their skillset to better understand predictive modeling using healthcare data.
Here, we address some challenges specific to working with EHR data, some best practices for creating a ML model through a description of the ML process, and an overview of various different ML
algorithms (with an emphasis on supervised methods but also give mention to unsupervised techniques) that can be useful for creating predictive models with healthcare data. Our goal is not to educate
the reader on precisely how to utilize various different ML algorithms but to better understand the process of creating predictive models for healthcare scenarios while using ML techniques. We
discuss the pros and cons of several supervised ML methods and also note some additional considerations regarding the use of ML in healthcare that are outside the scope of this manuscript. We finish
with a discussion of the limitations of this paper as well as a discussion of the field of ML in healthcare research.
Challenges Specific to Healthcare Data
Common sources of healthcare data include the EHR and claims submitted to payers for healthcare services or treatments rendered. Healthcare data stem from the need to support patient care, protect
clinicians from liability, and to facilitate reimbursement. Healthcare data are not documented or collected for the purpose of research; thus, unique challenges exist when using this type of data for
research. One widely used example EHR dataset is Medical Information Mart for Intensive Care III, an openly available database consisting of deidentified health data associated with about 60,000
intensive care unit admission. Different tables within the database compromise information on demographics, vital signs, laboratory test results, medications, and more [Reference Johnson3].
It is important to understand the intent and meaning of healthcare data prior to using it as a secondary data source for research. For example, presence of a single diagnosis for coronary artery
disease does not necessarily mean a patient was diagnosed with coronary artery disease; rather the diagnosis may be documented because the patient was undergoing evaluation to determine whether they
had coronary artery disease and in fact coronary artery disease was ruled out. In this example, an International Disease Classification (ICD) code for coronary artery disease was documented. Such
medical nomenclature, including ICD codes (for diagnoses), Current Procedural Terminology (CPT, for procedures), and RxNorm (for medications) are used to document services provided to patients but
must be considered within the context of the clinical workflows where they are used. Healthcare data and medical nomenclature must be used in concert with other healthcare data to decipher the
context of the clinical situation. In the case of the example above where documentation of an ICD code reflects diagnostic differential and not an actual diagnosis, there are multiple approaches that
can be taken to exclude such non-diagnostic cases: including ICD codes documented on more than one encounter separated by a certain time period or including ICD codes in the absence of a CPT code for
evaluation within a certain time period. The approach to defining metrics for a given characteristic/variable is complex and varies based on the specific variable. When possible, metrics used to
define specific variables should align with approaches published in the literature or formalized computational phenotypes. A computational phenotype is a clinical variable such as a diagnosis that
can be determined using structured data and does not require clinician interpretation or manual chart review [Reference Richesson and Smerek4]. Computational phenotypes, such as those available from
eMERGE (https://emerge-network.org) or PheKB (www.phekb.org), can be used to overcome the challenges of accurately identifying certain characteristics within EHRs.
Similarly, electronic systems often include more than one field to enter a given data element. Thus, understanding which fields are useful require understanding of the clinical context or workflows.
For example, there are often many fields in the EHR where blood pressure can be documented and some may represent patient-reported, which may not be relevant to all research questions. Pertinent
fields need to be collapsed and labeled accordingly. In some instances, using a common data model such as the Observational Medical Outcomes Partnership (https://www.ohdsi.org/data-standardization/
the-common-data-model/) can help organize healthcare data into a standardized vocabulary. A common data model is especially helpful when integrating healthcare data from disparate sources. One data
field in health system A may be used differently than the same field at health system B. Common data models account for such differences, making it easier to accurately pool data from different
While claims data are structured, much of healthcare data within the EHR are unstructured, documented in the form of free-text notes. It is critical to know what data are accurately represented
within structured fields and what are not. Data within unstructured documentation often represent a rich source of information and may be necessary to accurately represent the situation. For example,
if evaluating disease severity, patient-reported outcomes are often essential, but these are often only documented in unstructured clinical notes. When unstructured data are necessary, text mining
and natural language processing (NLP) methods can be used. Such methods to transform unstructured data into structured, analyzable data are becoming more mainstream, but nonetheless do require
additional time and can be highly complex.
Healthcare data are also fraught with errors and missingness. Data entry is prone to simple human error including typos and omission in both structured and unstructured data. In some instances, typos
and missing data need to be addressed or corrected, but in the context of using healthcare data for ML applications, this is not always ideal. If the intent of a given ML application is to create a
predictive model that will continue to use healthcare data, correction of errors would be counterintuitive, given these errors are part of the data and need to be accounted for in the model. The ML
models should be created using the actual data they are being created to use, which may include representation of errors and missingness. Further, missingness should not be automatically considered
an error. In healthcare, missing data can be an indication of something meaningful and worthy of evaluating. For example, absence of a given provider ordering a urine drug toxicology screen for a
patient prescribed high doses of opioids may suggest suboptimal patient care and thus is important to capture. Missing data can also be because the data were irrelevant in a given situation. In
healthcare, data are not always missing at random. However, another type of missing data is the product of patients receiving care across different health systems, which is unavoidable, and how it is
addressed depends on the research question.
ML Process
The creation of a predictive model via ML algorithms is not as straightforward as some in the realm of clinical and translational research would like. However, once the process is fully understood
and followed, the creation of a predictive model can be straightforward and rigorous. Fig. 1 illustrates the overall process which is described in this section. The process starts with the
acquisition of a worthy dataset (of sufficient size and scope), data preparation which includes steps such as the appropriate treatment of missing data, identifying all data sources/types, and the
appropriate treatment of identified errors. Next comes the selection of appropriate ML algorithms for the problem at hand as not all methods work for all types of outcomes. Once a method or methods
have been chosen, the model building process can begin. After a suitable model has been built on the training dataset, the model should also be evaluated by examining how well the created model also
works on the testing dataset. Finally, the ultimate model should be validated on a separate dataset before it can be used in practice for prediction. Here, we discuss the common steps necessary for
creation of a predictive model based on EHR data for clinical use.
Data Acquisition
The data acquisition step is critically important for the success of the final predictive model. This step often involves writing the appropriate query for an EHR data warehouse to ensure that the
correct data for the question at hand are obtained. Before requesting the data, a clear understanding of the research question and appropriate study designs is critical. The research question defines
the scope of the data request and the data requested must minimally address the research question. If the question is not clearly defined, it is common to omit key data elements from the request or
request more data than necessary. When leveraging EHR data for research, attention must be paid to how the data are generated to minimize important biases that could undermine the study question.
Selecting the appropriate study design is foundational and directs the sampling and data analysis necessary to effectively achieve the study aims or objectives. This frequently involves working with
an informatician or analyst skilled in creating appropriate queries for relational databases used to store patient information. An important step is also ensuring that the database that one is
working with is the most up-to-date version (or if not, knowing that is a limitation of the study).
Data Preparation
The EHR data acquired can be fraught with errors, whether the data are pulled directly from an EHR software system (such as Epic or Cerner) or compiled by a resident or fellow. Knowing where
potential errors may lie is key in knowing how to handle the data downstream. Once potential errors have been identified, they can be properly dealt with and corrected. Data preparation also includes
how one handles missing values as creating ML algorithms with vast amounts of missing data can greatly bias the resultant predictive model. Some ML methods implicitly handle missing data, but when
they do not, several methods exist for handling missingness. One measure is to remove subjects who have any missing data. This will reduce the sample size of the dataset to ensure only complete
information is analyzed, but this approach often introduces selection biases. Another option is to impute the missing data in some fashion. Common imputation methods are replacing the missing value
with the mean or median for a given variable, replacing the missing value with zero or the most commonly occurring value for a given variable, or more intensive computational methods such as
k-nearest neighbor imputation or methods involving regression. Which method is best to choose depends on the structure of the data and why the value is missing [Reference Kang5]. In addition to
cleaning the data, the data preparation step also includes variable selection/feature reduction. While most algorithms will create a model from any number of variables, culling the number of
variables chosen for the initial model creation can help ensure the model is not overly complicated or large. Common methods for variable selection include random forests (choose the top x number of
variables from the variable importance list), least absolute shrinkage and selection operator [Reference Tibshirani6], principal component analysis [Reference Abdi and Williams7] (choose the top ×
number of principal components), stepwise selection procedures, and basic hypothesis testing (choose those variables with a p-value less than a prespecified threshold). Common feature selection
methods include filter methods and wrapper methods. Filter methods are where one utilizes some method (a hypothesis test or correlation coefficient for example or linear discriminant analysis) to
filter the variables based upon some prespecified value or cutoff [Reference Liu and Motoda8–Reference Wang, Wang and Chang10]. Wrapper methods are an iterative process where variables are selected,
assessed for model accuracy, and then revised based on performance. Common wrapper methods include forward selection, backward elimination, and recursive feature elimination. The exact number of
variables to select is subjective and should be based on the total sample size within the dataset. Additionally, one should also consider model utility in making the decision for how many variables
to include in the final model. For instance, if a model is to be clinically relevant and utilized in practice, a physician commonly prefers a model with fewer variables instead of one with 20 as that
means less underlying data/testing is needed to inform potential decision-making. These feature reduction and screening methods can be used with or without an outcome in supervised and unsupervised
methods, respectively; however, feature screening methods that look at the association between predictors and an outcome must be based on the training data only in order to prevent overfitting/bias [
Reference Hastie, Tibshirani and Friedman11].
Choosing a Method
Once your dataset has been checked for errors and missing values have been dealt with, it is time to choose a method for the creation of a predictive model. Not all methods work best for all
questions of interest, so it is important to know the format of the outcome variable (is it binary, categorical, continuous?) as well as the format/structure of the independent variables (are they
categorical, continuous, normally distributed, correlated, longitudinal?). It is also important to understand if you will be able to use a supervised method or an unsupervised method. Supervised
methods assume that the ultimate outcome is known (for instance, this person has Stage 1 cancer vs Stage 2 cancer). Unsupervised methods do not have such assumptions.
Despite which method is chosen above, the process of creating a model is similar. Some users are tempted to throw all the data into an algorithm and examine the result. This is less than ideal for
creating predictive models that can be useful in the clinic. The reason being is that such a model may be overfit for the data at hand (i.e. the model works really well on the data that were used to
create it, but it does not perform well in other scenarios). To avoid overfitting, the model should be created initially on training data and then separately evaluated on testing data. Typically with
developing and validating models (training and testing), the dataset is randomly split into a training set (80% of the data) and a testing set (the remaining 20% of the data). While the 80/20 split
is commonly used, other splits can be used as well. What is important is that the training set is large enough in sample size to yield meaningful results and also has characteristics that resemble
the entire dataset [12]. The model is initially created using only the training set. Evaluation metrics are examined, and then the same model is run using the testing dataset, and evaluation metrics
for it are examined. Sometimes, cross-validation techniques are also used on the training dataset.
In general, cross-validation is any approach that leaves some of the data out when creating the initial model and then includes it in a testing fashion [Reference Hastie, Tibshirani and Friedman11,
Reference Stone13–Reference Wahba17]. This includes splits other than 80/20 (e.g. 60/40), repeated cross-validation (e.g. 80/20 split repeated five times where each set is held out once), bootstrap
sampling [18–21], or subsampling. No matter the splitting method, if using a repeated cross-validation technique, the process is repeated multiple times so that different subsets of data are left out
for separate runs of the model building process, and the model performance metrics are averaged for each subset of testing data to produce an overall average performance metric [Reference Breiman and
Spector22–Reference Kohavi26].
The amount of data and overall sample size required for prediction models depend on multiple factors. The sample size considerations are essential in the design phase of prediction modeling
techniques as the sample size is dependent on a variety of factors, including the number of outcome events, the prevalence of the outcome or the fraction of positive samples in the total sample size,
and the number of candidate predictor variables considered. Additional considerations include the number of classes the model is to predict. If the validation plan for the model is to develop the
model on a subset of data and then test the model on the remaining subset, the sample size considerations should account for the split dataset, particularly if the prevalence of the event is low, to
ensure the testing set will have ample events to perform.
Training Set Sample Size Planning for Predictive Modeling
Healthcare databases may contain thousands to hundreds of thousands of available subjects. Given such a large sample size, it may not be necessary to formally estimate the sample size needed for
building a prediction model. In other cases, the number of available subjects will be limited and a formal sample size calculation will be more useful. As discussed in Figueroa [Reference Figueroa27
], there are two main approaches to estimate the training dataset sample size needed for building a prediction model: “learning curve” and “model-based” approaches. “Learning curve” methods require
preliminary data on a set of subjects with all features/predictors and outcome of interest measured [Reference Figueroa27–Reference Dobbin and Song29]. One chooses a specific type of prediction model
and estimates the prediction accuracy of the model when varying the sample size by leaving some of the subjects out in an iterative manner. A “learning curve” is then fit to the resulting prediction
accuracies compared to the sample size used to fit the model. One can then extrapolate from the learning curve to estimate the sample size needed to obtain a desired level of prediction accuracy. The
main downside of this approach is the requirement of substantial preliminary data. In contrast, “model-based” approaches do not require preliminary data but make strong assumptions about the type of
data used in the study (e.g. all predictors are normally distributed) [Reference Dobbin and Simon30–Reference McKeigue32]. Dobbin et al. offer an online calculator for estimating the necessary sample
size in the training dataset that only requires three user input parameters [Reference Dobbin, Zhao and Simon31,33]. Despite its strong assumptions, Dobbin et al. argue that their method is
“conservative” in that it tends to recommend sample sizes that are larger than necessary. McKeigue provide R code for their method which also only requires three user input parameters [Reference
Model Evaluation
Once a model has been created using the training dataset, the model should be evaluated on the test dataset to assess how well it predicts outcomes for new subjects that were not used to build/train
the model. The reason for doing this is to ensure that the model is not overfit for the data it was trained on and to also ensure that the model is not biased toward particular values within the
training dataset [Reference Hastie, Tibshirani and Friedman11]. Random chance alone could have impacted the model fit from the training data, and evaluating the model fit on the testing data can
diminish this impact [Reference James34,Reference Harrell, Lee and Mark35] Several methods exist for evaluating predictive accuracy, and usually a combination of techniques are used. Evaluation
methods for supervised learning with categorical outcomes include assessing discrimination through the area under the receiver operating characteristic curve, which can be obtained by calculating the
C-statistic (“Concordance” statistic) and partial C-statistic for the various models [Reference Harrell36]. Additional steps include evaluating a confusion matrix which indicates how many individuals
the model correctly classifies and how many individuals are incorrectly classified. See Table 1 for a list of prediction accuracy metrics that can be used for particular types of outcomes (e.g.
continuous, binary, multi-category, or time-to-event). The chosen evaluation metric would be calculated for the training data and then separately for the testing data. Ideally, the prediction
accuracy would be high for both the training and testing datasets. Calibration methods can be used to assess how well the predicted outcomes correlate with the true outcomes, including using scatter
plots, reporting a regression slope from a linear predictor, or using the Hosmer–Lemeshow test to compare accurate predictions by decile of predicted probability [Reference Steyerberg37,Reference Van
Calster38]. High accuracy for the training data with low accuracy for the testing data indicate that the model is overfit for the data and will likely not generalize well to other settings. Lastly,
special methods should be used when evaluating predictive performance for categorical outcomes with unbalanced classes (i.e. when the number of subjects per class is not approximately equal), such as
over-/undersampling, cost-sensitive methods, or precision and recall curves [Reference He and Garcia39,Reference Saito and Rehmsmeier40].
^b For a categorical outcome with three or more categories, “one-versus-all” versions of metrics for a binary outcome can be used which assess prediction accuracy for one category versus all other
categories combined.
Parameter Tuning
If the model created does not fit the data well, or if the model is overfit for the training data, parameter tuning is typically the next step in the final predictive model creation process.
Parameter tuning involves fine-tuning of parameters that control the overall complexity of the ML model. Hyperparameter tuning is sometimes needed when using various techniques (such as neural
networks, support vector machines, or k-nearest neighbors) to optimize the parameters that help define the model architecture. Hyperparameters are parameters that cannot be automatically learned from
the model building process, and rather, the user needs to try a grid of values and use cross-validation or some other method to find the optimized value [Reference Preuveneers, Tsingenopoulos and
Joosen41]. Olson et al. and Lou provide guidance on several different methods for the tuning of hyperparameters and evaluate different methods for different algorithms [Reference Olson42,Reference
Luo43]. Tuning of the hyperparameters should only be done utilizing the training data. The model creation process is highly iterative, so once a change is made, the whole process is re-run and
Finally, once a model has been selected as being optimal for both the training and testing datasets, and the variables of interest are selected, the model ultimately needs to be tested a final time
on an independent dataset, known as the validation dataset, before it can realistically be used in a clinical setting. This is to ensure the model actually works well in practice, and on a dataset
that was independently collected (frequently from a different clinical setting, but with the same initial standards as the training dataset).
How ML is Used in Clinical and Translational Research
The use of ML with healthcare data can help with several different types of clinical and translational research goals [Reference Topol44–Reference Rajkomar, Dean and Kohane46]. The type of aim or
research question will determine which methods are ideal to use and which types of biases and misconceptions will be applicable and should be avoided. The use of ML with healthcare data can be
generally classified into four different goals: (1) diagnostics and clinical decision support tools, (2) -ome wide association studies, (3) text mining, and (4) causal inference [Reference Bi47].
Diagnostics and clinical decision support tools are designed to give healthcare workers and patients specific information to enhance their clinical care. These tools are often specialized to
appropriately filter applicable data based on an individual patient’s condition at a certain time. For example, a clinical decision support tool might utilize the longitudinal health history of a
patient with high blood pressure to estimate their risk for a myocardial infarction in the near future. Techniques used for accomplishing this that have been implemented with ML methods include
imaging diagnostics, risk stratification, and identification of prognostic subgroups.
“-ome-wide” association studies leverage measurements of large numbers of different variants in an -ome system to see if any of them are associated with a disease outcome or health characteristic.
These could include the genome, phenome, exposome, microbiome, metabolome, proteome, or transcriptome; often referred to generally as ‘omics data. For example, a genome-wide association study might
identify a small list of polymorphisms that are associated with an increased risk for obesity. These types of methods are also used to predict clinical outcomes, like the effectiveness of a drug, or
for the identification of gene–gene and gene–environment interactions. See Libbrecht and Noble [Reference Libbrecht and Noble48], Ghosh et al. [Reference T49], and Zhou and Gaillins [Reference Zhou
and GP50] for a review of ML methods applied in specific types of ‘omics data, or for integrating multiple ‘omics data sources [Reference Li, Wu and Ngom51,Reference Huang, Chaudhary and Garmire52].
Text mining works by automating data extraction from unstructured clinical notes. The applications can include the identification of patients that might benefit from participation in clinical trials,
the removal of protected health information from clinical records to be used for research, the conduct of a systematic literature review, and automated infectious disease surveillance systems [
Reference Khan53].
Causal inference is the process of making an inference about a specific intervention or exposure with respect to its effect. For example, a causal inference study could be conducted with the goal of
determining if taking a specific drug increases the expected lifespan of patients with a certain disease. It is important to consider confounding pathways, or those characteristics that could vary
with both the intervention/exposure and the disease outcome, so that we are not misinterpreting an association as a causal effect. Techniques used for accomplishing this that have been implemented
with ML methods include propensity score weighting [Reference Lee, Lessler and Stuart54], targeted maximum likelihood estimation, marginal structural models, heterogeneous treatment effects, and
causal structure learning [Reference Athey and Guido55–Reference Athey, Tibshirani and Wager58].
Overview of ML Methods
Artificial intelligence (AI) is a scientific field within computer science that focuses on the study of computer systems that perform tasks and solve problems that historically required human
intelligence. ML is a subfield of AI that focuses on a special class of algorithms that can learn from data without being explicitly programmed. This is accomplished by making inferences based on
patterns found in data. Although some human specification is required, ML algorithms overall require less human input than more traditional statistical modeling (i.e. deciding which variables to
include a priori). In general, there are three main ways that a machine might learn from data: (1) supervised ML, (2) unsupervised ML, and (3) reinforcement learning.
Supervised ML, most often used for prediction modeling, works by mapping inputs to labeled outputs. This is used only when each input used to train the model has a labeled output, for example, an
input of longitudinal measurements on a person collected during their hospitalization might be used with a labeled output of in-hospital mortality or unplanned readmission within 30 days of discharge
[Reference Rajkomar59]. Supervised ML is also referred to more generally as “predictive modeling,” or “regression” for quantitative outcomes, and “classification” for categorical outcomes. Often,
data with labels are used to train a supervised ML algorithm and then the algorithm is used to predict unknown labels for a set of new inputs. Mathematical techniques used to develop supervised ML
models include decision trees and random forests [Reference Breiman60–Reference Boulesteix62], gradient boosting [Reference Friedman63–Reference Chen and Guestrin65], neural networks and deep
learning [Reference LeCun, Bengio and Hinton66–Reference Litjens71], support vector machines [Reference Cristianini and Shawe-Taylor72], and regularized/penalized regression [Reference Hastie,
Tibshirani and Wainwright73].
Unsupervised ML works by finding patterns in unlabeled input data. This category is commonly used for segmentation and clustering problems such as identifying disease “sub-phenotypes” or pattern
recognition. For example, a registry of patients diagnosed with asthma might be classified into different subgroups based on their sensitivity to different types of indoor and outdoor allergens, lung
function, and presence of wheezing [Reference Haldar74]. Mathematical techniques used to develop unsupervised ML models include clustering (hierarchical, k-means), dimensionality reduction, Gaussian
mixture models, principal component analysis, and independent component analysis.
Reinforcement learning works by improving a sequence of decisions to optimize a long-term outcome through trial and error [Reference Kaelbling, Littman and Moore75]. It is similar to supervised
learning, but a reward mechanism is used in place of a labeled output. This category is often used within healthcare to evaluate and improve policies. For example, a longitudinal series of decisions
regarding treatment options must be made by a clinician within the context of managing sepsis (e.g. mechanical ventilation, sedation, vasopressors) [Reference Gottesman76]. Unlike supervised learning
which makes a one-time prediction, the output of a reinforcement learning system can affect both the patient’s future health as well as future treatment options. Often times, a reinforcement learning
algorithm will make a short-term decision based on what it believes will result in the best long-term effect, even if that means a suboptimal short-term outcome. Mathematical techniques used to
develop reinforcement learning models include neural networks, temporal difference algorithms, Q-learning, and state-action-reward-state-action.
Pros/Cons of Supervised ML Methods
Fernández-Delgado et al. [Reference Fernández-Delgado77] compared the predictive accuracy of 179 ML models (sorted into 17 “families” of different model types) across 121 real datasets and found that
random forests were consistently among the best predictive models. Thus, random forests can be recommended as one of the best “off-the-shelf” ML models. Nevertheless, one can never know a priori
which model will produce the highest predictive accuracy for a given application, thus it is common practice to fit several different ML models and use cross-validation to select a final model that
has the highest predictive accuracy. Each ML model has its own pros and cons which should be taken into consideration when deciding which model to use. In addition, when comparing multiple ML models,
it is frequently useful to also compare with one or more simple, traditional statistical models such as linear or logistic regression (or penalized versions thereof). For example, a systematic review
of 71 clinical predictive modeling papers found no evidence for a difference in predictive accuracy when comparing logistic regression (or penalized logistic regression) with a variety of ML models [
Reference Jie78].
Table 2 compares several key properties of different ML models and classical statistical models such as linear or logistic regression. These properties will now be briefly summarized and can be used
to help decide which ML model to use for a given problem. For a comprehensive introduction to these ML models, see [Reference Kuhn and Johnson79–Reference Spratt, Ju and Brasier81].
^a Tree-based models can naturally handle missing values in predictors using a method called “surrogate splits” [Reference Strobl, Malley and Tutz61]. Although not all software implementations
support this, example software that does allow missing predictor values in trees are the rpart [Reference Therneau and Atkinson115] and partykit [Reference Hothorn and Zeileis116] R packages. Other
acronyms: “P > N”: the total number of predictors “P” is much larger than the total number of samples “N”; CART: classification and regression tree; MARS: multivariate adaptive regression splines;
LASSO: least absolute shrinkage and selection operator.
^c Corr: remove highly correlated predictors, CS: center and scale predictors to be on the same scale (i.e. cannot naturally handle a mix of categorical and numeric predictors on their original
measurement scales).
In contrast to classical statistical models like linear or logistic regression, all of the ML models considered allow the number of predictors (“P”) to be larger than the number of subjects (“N”),
that is, “P > N.” In addition, most of the ML models considered automatically allow for complex nonlinear and interaction effects, whereas classical methods (and penalized regression) generally
assume linear effects with no interactions. One can manually explore a small number of nonlinear or interaction effects in classical models, but this becomes practically infeasible with more than a
handful of predictors, in which case, ML models that automatically allow for complex effects may be more appropriate.
Several ML methods can handle a large number of irrelevant predictors (i.e. predictors that have no relationship with the outcome) without needing to first identify and remove these predictors before
fitting the model. Tree-based methods [Reference Strobl, Malley and Tutz61] (e.g. classification and regression tree (CART) [Reference Loh82], random forests, boosted trees) use a sequential process
that searches through all predictors to find the “optimal” predictor (that best improves the predictive accuracy) to add to the tree at each step in the tree-building process. Multivariate adaptive
regression splines (MARS) [Reference Friedman83] uses a similar “step-wise” or sequential model building approach. In doing so, these methods automatically search for the most important predictors to
include at each step of the model-fitting process, thereby disregarding unimportant predictors. Penalized regression shrinks regression coefficients toward zero, effectively disregarding unimportant
ML models typically contain one or more main “tuning parameters” that control the complexity of the model. As described in the section on Training/Testing, one can use cross-validation or the
bootstrap to select optimal values for these tuning parameters. In general, the more tuning parameters a model has, the more challenging it is to fit the model, since one needs to find optimal values
for all tuning parameters which requires searching across a large grid of possible values. Models like CART, random forests, MARS, and penalized regression, often only have fewer tuning parameters
that need to be optimized and thus may be considerably easier to implement in practice. In contrast, boosted trees, support vector machines, and neural networks contain more tuning parameters and
thus can be more computationally challenging to optimize.
Unlike most ML models, tree-based methods (e.g. CART, random forests, boosted trees) have several additional unique benefits: they are robust to noise/outliers in the predictors [Reference Hastie,
Tibshirani and Friedman11,Reference Kuhn and Johnson79], and they can naturally handle missing values in the predictors using a technique called “surrogate splits” [Reference Strobl, Malley and Tutz
61]. Although the usual adage of “garbage-in garbage-out” still applies, tree-based methods require minimal data pre-processing of the predictors. For example, most models require standardizing all
predictors to be on the same scale, whereas trees can naturally handle a mix of categorical and continuous predictors measured on their original scales, which may be beneficial for EHR data. However,
it is worth noting that earlier versions of trees are biased toward favoring categorical variables with more categories and correlated predictors. “Conditional inference trees,” a newer framework for
fitting tree-based models, have solved these problems [Reference Strobl, Malley and Tutz61,Reference Hothorn, Hornik and Zeileis84–Reference Strobl86]. Among the models considered in Table 2, neural
networks, support vector machines, and boosted trees generally have the longest computation time, followed by random forests. Although rapid advances in computing power and parallel computing means
that all of these methods will still be computationally feasible for most applications.
When deciding what ML model to use, it is also important to consider the format/type of outcome that you are trying to predict. Although not listed, all of the models in Table 2 have been extended to
handle continuous, binary, or multi-category outcomes. However, ML methods are still rather underdeveloped for handling longitudinal or censored time-to-event (“survival”) outcomes. Some recent work
has extended tree-based methods (e.g. CART, random forests) to handle such outcomes [Reference Sela and Simonoff87–Reference Speiser89].
Lastly, the models can be compared by how “interpretable” they are. Classical statistical models, CART (“single tree”), MARS, and penalized regression are all easier to interpret. More complex ML
models like random forests, boosted trees, support vector machines, and neural networks are much harder to interpret. In general, one can fit a variety of ML models and assess their predictive
accuracy using cross-validation. If the cross-validated predictive accuracy of a more interpretable model is comparable to an uninterpretable model, then the more interpretable model should be
preferred. The next section discusses tools that can be used to help interpret any ML model.
Interpreting Black-Box ML Models
ML models are often criticized as being “black-box” models: data are input into a mysterious “black box” which then outputs predictions. However, the user often lacks an explanation or understanding
for why the mysterious black box makes a given prediction. Making black-box ML models more interpretable is an ongoing area of research, and we will briefly summarize several important contributions,
all of which are implemented in the iml (“interpretable machine learning”) R package [Reference Molnar, Casalicchio and Bischl90].
Many ML models produce a type of “variable importance” score [Reference Kuhn and Johnson79] for each predictor in the model, allowing the user to rank the predictors from most to least important in
their ability to predict the outcome. In addition, one can use “partial dependency plots” [Reference Hastie, Tibshirani and Friedman11] (PDPs) to visualize the estimated relationship between the
outcome and a specific predictor in the model. For each possible value of the predictor of interest, the PDP will show you the expected value of the outcome, after adjusting for the average effects
of all other predictors. For example, one may first rank predictors from most to least important based on their variable importance scores and then use PDPs to visualize the relationship between the
outcome and each of the top 5 or 10 most important predictors. Lastly, Friedman’s H-statistic [Reference Friedman and Popescu91] can be used to identify predictors involved in interactions.
Recently, several methods have been developed to help understand why a ML makes a prediction for a particular subject. LIME [Reference Ribeiro, Singh and Guestrin92] (“Local Interpretable model
explanations”) uses simpler more interpretable models (e.g. linear or logistic regression) to explain how a given subject’s feature values affects their prediction, and the user can control exactly
how many features are used to create the “explanation.” The basic idea of LIME is to weigh all subjects by how similar they are to the subject of interest and then fit a simpler interpretable model
“locally” by applying these weights. This simpler model is then used to provide an explanation for why the ML model made the prediction for the given subject (or subjects who have features similar to
that subject). “Shapley values” [Reference Štrumbelj and Kononenko93], originally developed in game theory, are another method for explaining predictions from ML models. Shapley values explain how a
subject’s feature values (e.g. gender, age, race, genetics) affect the model’s prediction for that subject compared to the average prediction for all subjects. For example, suppose the model’s
average predicted probability of having a particular disease is 0.10. Suppose for a particular subject of interest, “Sam,” the probability of having the disease is 0.60, that is, 0.50 higher than the
average prediction. To explain Sam’s prediction, each feature included in the model will get a “Shapley value” that explains how the values of Sam’s features affected the prediction. For example,
Sam’s gender increased the prediction by 0.15, Sam’s age increased the prediction by 0.30, and Sam’s race increased the prediction by 0.05. Notice the sum of the Shapley values equals 0.50, which was
the difference between Sam’s prediction and the average prediction for all subjects. See Molnar [Reference Molnar94] for more information on methods for interpreting ML models.
Open-Source Software for ML
All of the ML models discussed can be fit within the free open-source software R [95]. The caret R package [Reference Kuhn96] and tidymodels R package [97] both provide a unified framework for
training/tuning and testing over 200 ML models. Table 2 lists a few example R packages for fitting specific ML models. See the “CRAN Task View Machine Learning & Statistical Learning” website [98]
for a more comprehensive list of ML R packages. Python also offers free open-source software for ML [Reference Raschka and Mirjalili99,Reference Pedregosa100].
Special Considerations of ML in Healthcare Research
As previously discussed, the “black box” nature of ML methods makes it difficult to understand and interpret the predictions. Although there are some tools that can be used to help interpret ML,
further research is still needed for making ML more transparent and explainable. Transparency is especially important when used to make decisions that will impact a patient’s health.
ML learns from data recorded on past historical examples. However, such historical data can also capture patterns of preexisting healthcare disparities and biases in treatment and diagnosis. ML
models trained using such data may further perpetuate these inequities and biases [Reference Rajkomar101]. In addition, if a particular subgroup of patients is under-represented or more likely to
have missing data used to train the ML model, then the ML model may produce inaccurate predictions for the subgroup, which can also further exacerbate existing healthcare disparities [Reference
Gianfrancesco102]. Methods and guidelines are being developed to help ensure fairness and equity when using ML models [Reference Rajkomar101–Reference Bellamy103], but more work is needed. After
clinical implementation, ML models should be regularly evaluated for fairness and equity among different minority subgroups. There exist several definitions of fairness and equity within precision
medicine that cannot all be simultaneously satisfied, but in general, model evaluation should be conducted for different minority subgroups to ensure that the model performs equally well within each
Need for Translational Prospective Studies Demonstrating Utility of ML
The vast majority of ML in healthcare has been demonstrated using retrospective historical data where outcomes are already known [Reference Topol44–Reference Rajkomar, Dean and Kohane46]. There is a
crucial need to demonstrate the utility of ML-based prediction and decision/support tools in real-time prospective studies.
Legal/Regulatory Issues
If an incorrect medical decision is made based on a complex ML decision support tool, it is unclear what entity should be held liable for that decision [Reference Yu, Beam and Kohane45]. Government
regulations for ML-based clinical support tools are still being developed [104].
The authors realize that we have not included mention of several new and important techniques namely in the fields of deep learning, AI, and NLP. We agree that the methods mentioned above are
important in the field of predictive modeling and have a place in the analysis of healthcare data. We have chosen to exclude them from this paper as their use is more advanced and requires a more
in-depth knowledge of the underlying mathematical and conceptual processes needed to utilize such applications. Deep learning, for instance, involves utilizing multiple layers of neural networks
where each network trains a more complicated task. TensorFlow is one example of deep learning and Pytorch is another. Reinforcement learning trains a system to answer multiple questions in sequence.
NLP allows data to be extracted from physicians’ notes to be used in analytical applications. All of these are important, recent advances in the field of predictive analytics. Their use, however, is
more complex (more training steps, the creation of multiple networks) than the more routine ML process presented above and thus outside the scope of this paper [Reference Esteva105,106].
Additionally, the use of AI and NLP in healthcare is still under scrutiny [Reference Szabo107,108] and best practices for their use are still being adopted.
ML algorithms have the potential to be powerful and important tools for healthcare research. Here, we have provided researchers with cautionary notes regarding the use of EHR data, insurance
databases, and other clinical data with ML algorithms. We have presented the healthcare researcher with a ML pipeline for preparing the data, running the data through various algorithms using
separate training/testing datasets, evaluating the created predictive model, tuning any necessary model parameters, and finally validating the predictive model on a separate dataset. In addition, we
mention several commonly used ML algorithms frequently referenced in the healthcare literature, and the pros/cons of various supervised methods. Finally, we also mention several considerations for
using ML algorithms with healthcare data that, while important, are beyond the scope of this article. Our goal with the concepts and methods described in this article is to educate the healthcare
researcher to better understand how to properly conduct a ML project for clinical predictive analytics. We should also note that ML methods do not always perform better than traditional
statistical-based methods such as logistic regression [Reference van der Ploeg, Austin and Steyerberg109]. The clinician or translational researcher should familiarize themselves with the differences
between traditional statistical-based methods and ML methods and utilize what works best for their specific questions of interest [110].
This study was supported by the following grants from the National Institutes of Health: UL1TR001439 (H.M.S.), UL1TR001425 (C.B., J. M-D), UL1 TR002535 (J.A., K.E.T.). Contents are the authors’ sole
responsibility and do not necessarily represent official NIH views.
No authors reported conflicts of interest. | {"url":"https://core-cms.prod.aop.cambridge.org/core/journals/journal-of-clinical-and-translational-science/article/lessons-and-tips-for-designing-a-machine-learning-study-using-ehr-data/1171DB7CA4E909DFF35079BEC743B78F","timestamp":"2024-11-02T00:27:34Z","content_type":"text/html","content_length":"1049982","record_id":"<urn:uuid:cf71fccd-7e70-4080-9932-403d5571d496>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00087.warc.gz"} |
ManPag.es -
sincos, sincosf, sincosl − calculate sin and cos simultaneously
#define _GNU_SOURCE /* See feature_test_macros(7) */
#include <math.h>
void sincos(double x, double *sin, double *cos);
void sincosf(float x, float *sin, float *cos);
void sincosl(long double x, long double *sin, long double *cos);
Link with −lm.
Several applications need sine and cosine of the same angle x. This function computes both at the same time, and stores the results in *sin and *cos.
If x is a NaN, a NaN is returned in *sin and *cos.
If x is positive infinity or negative infinity, a domain error occurs, and a NaN is returned in *sin and *cos.
These functions return void.
See math_error(7) for information on how to determine whether an error has occurred when calling these functions.
The following errors can occur:
Domain error: x is an infinity
An invalid floating-point exception (FE_INVALID) is raised.
These functions do not set errno.
These functions first appeared in glibc in version 2.1.
This function is a GNU extension.
cos(3), sin(3), tan(3)
This page is part of release 3.53 of the Linux man-pages project. A description of the project, and information about reporting bugs, can be found at http://www.kernel.org/doc/man−pages/. | {"url":"https://manpag.es/SUSE131/3+sincos","timestamp":"2024-11-13T01:17:51Z","content_type":"text/html","content_length":"18882","record_id":"<urn:uuid:f5a128c6-3069-4493-ac86-7159723b04f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00482.warc.gz"} |
DC Motor Control
This example shows the comparison of three DC motor control techniques for tracking setpoint commands and reducing sensitivity to load disturbances:
• feedforward command
• integral feedback control
• LQR regulation
See "Getting Started: Building Models" for more details about the DC motor model.
Problem Statement
In armature-controlled DC motors, the applied voltage Va controls the angular velocity w of the shaft.
This example shows two DC motor control techniques for reducing the sensitivity of w to load variations (changes in the torque opposed by the motor load).
A simplified model of the DC motor is shown above. The torque Td models load disturbances. You must minimize the speed variations induced by such disturbances.
For this example, the physical constants are:
R = 2.0; % Ohms
L = 0.5; % Henrys
Km = 0.1; % torque constant
Kb = 0.1; % back emf constant
Kf = 0.2; % Nms
J = 0.02; % kg.m^2/s^2
First construct a state-space model of the DC motor with two inputs (Va,Td) and one output (w):
h1 = tf(Km,[L R]); % armature
h2 = tf(1,[J Kf]); % eqn of motion
dcm = ss(h2) * [h1 , 1]; % w = h2 * (h1*Va + Td)
dcm = feedback(dcm,Kb,1,1); % close back emf loop
Note: Compute with the state-space form to minimize the model order.
Now plot the angular velocity response to a step change in voltage Va:
Right-click on the plot and select "Characteristics:Settling Time" to display the settling time.
Feedforward DC Motor Control Design
You can use this simple feedforward control structure to command the angular velocity w to a given value w_ref.
The feedforward gain Kff should be set to the reciprocal of the DC gain from Va to w.
To evaluate the feedforward design in the face of load disturbances, simulate the response to a step command w_ref=1 with a disturbance Td = -0.1Nm between t=5 and t=10 seconds:
t = 0:0.1:15;
Td = -0.1 * (t>5 & t<10); % load disturbance
u = [ones(size(t)) ; Td]; % w_ref=1 and Td
cl_ff = dcm * diag([Kff,1]); % add feedforward gain
cl_ff.InputName = {'w_ref','Td'};
cl_ff.OutputName = 'w';
lp = lsimplot(cl_ff,u,t);
title('Setpoint tracking and disturbance rejection')
% Annotate plot
text(7.5,.25,{'disturbance','T_d = -0.1Nm'},...
Clearly feedforward control handles load disturbances poorly.
Feedback DC Motor Control Design
Next try the feedback control structure shown below.
To enforce zero steady-state error, use integral control of the form
where K is to be determined.
To determine the gain K, you can use the root locus technique applied to the open-loop 1/s * transfer(Va->w):
rp = rlocusplot(tf(1,[1 0]) * dcm(1));
rp.FrequencyUnit = "rad/s";
xlim([-15 5]);
ylim([-15 15]);
Click on the curves to read the gain values and related info. A reasonable choice here is K = 5. The Control System Designer app is an interactive UI for performing such designs.
Compare this new design with the initial feedforward design on the same test case:
K = 5;
C = tf(K,[1 0]); % compensator K/s
cl_rloc = feedback(dcm * append(C,1),1,1,1);
lp2 = lsimplot(cl_ff,cl_rloc,u,t);
cl_rloc.InputName = {'w_ref','Td'};
cl_rloc.OutputName = 'w';
title('Setpoint tracking and disturbance rejection')
legend('feedforward','feedback w/ rlocus','Location','NorthWest');
The root locus design is better at rejecting load disturbances.
LQR DC Motor Control Design
To further improve performance, try designing a linear quadratic regulator (LQR) for the feedback structure shown below.
In addition to the integral of error, the LQR scheme also uses the state vector x=(i,w) to synthesize the driving voltage Va. The resulting voltage is of the form
Va = K1 * w + K2 * w/s + K3 * i
where i is the armature current.
For better disturbance rejection, use a cost function that penalizes large integral error, e.g., the cost function
The optimal LQR gain for this cost function is computed as follows:
dc_aug = [1 ; tf(1,[1 0])] * dcm(1); % add output w/s to DC motor model
K_lqr = lqry(dc_aug,[1 0;0 20],0.01);
Next derive the closed-loop model for simulation purposes:
P = augstate(dcm); % inputs:Va,Td outputs:w,x
C = K_lqr * append(tf(1,[1 0]),1,1); % compensator including 1/s
OL = P * append(C,1); % open loop
CL = feedback(OL,eye(3),1:3,1:3); % close feedback loops
cl_lqr = CL(1,[1 4]); % extract transfer (w_ref,Td)->w
This plot compares the closed-loop Bode diagrams for the three DC motor control designs.
Click on the curves to identify the systems or inspect the data.
Comparison of DC Motor Control Designs
Finally we compare the three DC motor control designs on our simulation test case:
lp3 = lsimplot(cl_ff,cl_rloc,cl_lqr,u,t);
title('Setpoint tracking and disturbance rejection')
legend('feedforward','feedback (rlocus)','feedback (LQR)','Location','NorthWest');
Thanks to its additional degrees of freedom, the LQR compensator performs best at rejecting load disturbances (among the three DC motor control designs discussed here). | {"url":"https://www.mathworks.com/help/control/ug/dc-motor-control.html","timestamp":"2024-11-10T19:10:08Z","content_type":"text/html","content_length":"79397","record_id":"<urn:uuid:d2cfbb78-b103-454c-8321-90ba1eb7bc38>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00088.warc.gz"} |
This R package contains multivariate two-sample survival permutation tests, based on the logrank and Gehan statistics. The tests are described in Persson et al. (2019).
To install the development version from GitHub:
Example usage, comparing the bivariate survival times of the two treatment groups in the diabetes data (included in the package):
# Diabetes data:
# Survival times for the two groups:
x <- as.matrix(subset(diabetes, LASER==1)[c(6,8)])
y <- as.matrix(subset(diabetes, LASER==2)[c(6,8)])
# Censoring status for the two groups:
delta.x <- as.matrix(subset(diabetes, LASER==1)[c(7,9)])
delta.y <- as.matrix(subset(diabetes, LASER==2)[c(7,9)])
# Create the input for the test:
z <- rbind(x, y)
delta.z <- rbind(delta.x, delta.y)
# Run the tests with 99 permutations:
perm_gehan(B = 99, z, delta.z, n1 = nrow(x))
perm_mvlogrank(B = 99, z, delta.z, n1 = nrow(x))
# In most cases, it is preferable to use more than 99
# permutations for computing p-values. choose_B() can
# be used to determine how many permutations are needed. | {"url":"http://cran.stat.auckland.ac.nz/web/packages/MultSurvTests/readme/README.html","timestamp":"2024-11-07T05:49:08Z","content_type":"application/xhtml+xml","content_length":"2561","record_id":"<urn:uuid:92a76748-6c95-4461-ba08-cc3458d39831>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00612.warc.gz"} |
1974 Firebird Photo Gallery
1974 Firebird Gallery
Tran-zam has endeavored to make these photo galleries as accurate as possible and always attempts to show cars in their stock factory form. Some combinations are very rare and may be shown with
custom touches or may be in poor condition where no other shots were available.
note: wheels and rust are aftermarket.
note: wheels are aftermarket.
note: wheels are aftermarket.
note: wheels are aftermarket.
note: wheels are aftermarket.
Examples shown on this page consist of scanned images or reproduction artwork that can not be faithfully reproduced on a computer screen. Illustrations are for example only. Consult with your vendors
offerring reproduction parts for their historical accuracy. | {"url":"https://tran-zam.com/gallery/1974Gallery.aspx","timestamp":"2024-11-09T12:20:59Z","content_type":"text/html","content_length":"26225","record_id":"<urn:uuid:1830136a-631b-4d1e-a386-b235c9917de0>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00404.warc.gz"} |
Elasticsearch aggregations, counting nested objects
I have an index with nested objects and am trying to derive some statistics about the data, but am having trouble doing so.
The documents look like this simplified version:
"rn": 1,
"t": "X",
"a": [
"r": [
"rt": "A",
"p": [
"sn": 1
"sn": 2
"rt": "B",
"p": [
"sn": 1
"sn": 2
"rt": "C",
"p": [
"sn": 1
"sn": 2
"rn": 2,
"t": "X",
"a": [
"r": [
"rt": "A",
"p": [
"sn": 1
"sn": 2
"rt": "B",
"p": [
"sn": 1
"sn": 2
The arrays, a, r and p, are mapped as nested objects.
I want to write an aggregation to return:
• The maximum number of occurrences of r in a document (3 in this example)
• The average number of occurrences of r in a document (2.5 in this example)
• Determine the above two within the buckets for terms in t (just X in this example, but there will be a dozen or so other terms; I can just run a query a dozen times with different values for a
filter on t if needed)
In a non-nested version of this, I used a painless script to count the elements in each document and ran stats over the result. But now, since r is nested, I can't just count the elements in r
because r is now split into separate document. I think I need to have an agg that counts the nested documents per document, then another agg to run stats over those counts. But I don't see how to do
that. Do I need to basically make a bucket for each document? There are nearly 1B documents in the index.
I am having trouble even working out which aggs I can use, and in what order. I have tried variations of value_count, max_buckets, reverse_agg and a number of others, without success and am clearly
missing some understanding of how this agg should work.
Any help would be greatly appreciated.
There are 2 approaches that come to mind:
1 - Keep a count of how many nested documents each nested type has in the root of the document. Then you can use a sum aggregation to just sum the values stored in the root document.
2 - Store an identity field or just a simple static int column which has the value 1 inside each nested document and then use a nested aggregation to sum up all the 1's
Hope that helps.
Thanks @RahulD. These are good ideas. However, this is an ad hoc query to gather some stats about our data and I can't add a field to the documents. But I may be able to create a new index using
Logstash to derive this info, so thanks again. Counting the elements of the r array in Logstash should be straight forward, but I also need to do a similar query for the p array, to get a total count
of p elements per document (so 6 and 4 for the two example documents above) - not sure how easy that is to do.
But I'm surprised we can't seem to access a count of an array after it's converted to nested, that seems like a useful function.
This topic was automatically closed 28 days after the last reply. New replies are no longer allowed. | {"url":"https://discuss.elastic.co/t/elasticsearch-aggregations-counting-nested-objects/133002","timestamp":"2024-11-09T09:18:01Z","content_type":"text/html","content_length":"40480","record_id":"<urn:uuid:013023c1-79bb-4606-893d-1060bfc45dd6>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00237.warc.gz"} |
Vaccine Efficacy
Problem I
Vaccine Efficacy
To determine the efficacy of a vaccine against a disease, a clinical trial is needed. Some participants are given the real vaccine, while others are given a placebo as the control group. The
participants are tracked to determine if they are infected by three different strains (A, B, and C) of a virus. The efficacy of the vaccine against infection by a particular strain is simply the
percentage reduction of the infection rate of the vaccinated group compared to the control group.
For example, suppose that there are $40$ people in the vaccinated group, $8$ of which are infected by strain B. Then the infection rate is $20$%. Further suppose that $50$ people are in the control
group, and $30$ people are infected by strain B. Then the infection rate for the control group is $60$%. Thus the vaccine efficacy against infection is approximately $66.67$% (since $20$% is a
$66.67$% percentage reduction of $60$%). If the infection rate for a particular strain in the vaccinated group is not lower than that of the control group, the vaccine is not effective against
infection by that strain.
What is the vaccine efficacy against infection by the three strains?
The first line of input contains an integer $N$ ($2 \leq N \leq 10\, 000$) containing the number of participants in the clinical trial.
The next $N$ lines describe the participants. Each of these lines contains a string of length four. Each letter is either ā Yā or ā Nā . The first letter indicates whether the participant is
vaccinated with the real vaccine, and the remaining three letters indicate whether the participant is infected by strain A, B, and C, respectively.
There is at least one participant in the vaccinated group and the control group. There is at least one participant in the control group infected by each strain (but they may be different
Display the vaccine efficacy against infection by strain A, B, and C in that order. If the vaccine is not effective against infection by a particular strain, display Not Effective for that strain
instead. Answers with an absolute error or relative error of at most $10^{-2}$ will be accepted.
Sample Input 1 Sample Output 1
NYYN Not Effective
NNNY 66.666667
YYNN 50.000000
Sample Input 2 Sample Output 2
NYYY 100.000000
NYYY 100.000000
YNNN 100.000000 | {"url":"https://rmc20.kattis.com/contests/rmc20/problems/vaccineefficacy","timestamp":"2024-11-13T12:23:16Z","content_type":"text/html","content_length":"28754","record_id":"<urn:uuid:1b18e550-f5e0-4689-9172-00d85cd91dd1>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00643.warc.gz"} |
Adding fields in reusable elementsBubble Forum
I have done an addition expression to add fields in 60 reusable elements.
However, the Input filed where I have done the expression, when I view the page doesnt appear. I have redone the expression with just the first 2 fields being added together and ensured they are
visible (reusable elements without value are not visible) but still it does not appear.
What am I missing
Likely the data is not available…but what are you trying to achieve? I can’t imagine a reason for such an expression.
1 Like
For instance I need to add together all the percentages from each entry to ensure it is 100%. Or the total cost, so add each amount each entry is worth (they vary).
What do you mean the data isnt available. There is values in each field
you should have the list of data entries somewhere in the reusable element, likely a repeating group. Then you can in a dynamic expression reference the repeating group list of things, then select
the data field that is the percentage and then use the sum operator.
Repeating groups list of things each items percentage sum
Ill see how I go with that. I havent done it with RG as yet. Hopefully it works, it will be handy as hell
Thanks so much for that. Once I understood what you meant It made sense. A hell of a lot easier. I have so much to learn and I have to change my way of looking at things. I keep looking at the page
and seeing that this is what Im working with but there is another layer underneath that you can work with and it can be so much easier. I really appreciate your help
1 Like
This topic was automatically closed after 70 days. New replies are no longer allowed. | {"url":"https://forum.bubble.io/t/adding-fields-in-reusable-elements/329922","timestamp":"2024-11-03T12:12:07Z","content_type":"text/html","content_length":"31496","record_id":"<urn:uuid:a574e2ab-75d8-4343-ac4c-d9dae453d2d8>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00794.warc.gz"} |
FAU MoD Lecture. Special December 2024
Date: Tue. December 03, 2024
Event: FAU MoD Lecture Series (Special double edition. December 2024)
Organized by: FAU MoD, the Research Center for Mathematics of Data at Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany)
Session 01: 14:30H
FAU MoD Lecture: The implicit bias phenomenon in deep learning
Speaker: Prof. Dr. Holger Rauhut
Affiliation: Mathematisches Institut der Universität München (Germany)
Abstract. Deep neural networks are usually trained by minimizing a non-convex loss functional via (stochastic) gradient descent methods. Unfortunately, the convergence properties are not very
well-understood. Moreover, a puzzling empirical observation is that learning neural networks with a number of parameters exceeding the number of training examples often leads to zero loss, i.e., the
network exactly interpolates the data. Nevertheless, it generalizes very well to unseen data, which is in stark contrast to intuition from classical statistics which would predict a scenario of
A current working hypothesis is that the chosen optimization algorithm has a significant influence on the selection of the learned network. In fact, in this overparameterized context there are many
global minimizers so that the optimization method induces an implicit bias on the computed solution. It seems that gradient descent methods and their stochastic variants favor networks of low
complexity (in a suitable sense to be understood), and, hence, appear to be very well suited for large classes of real data.
Initial attempts in understanding the implicit bias phenomen considers the simplified setting of linear networks, i.e., (deep) factorizations of matrices. This has revealed a surprising relation to
the field of low rank matrix recovery (a variant of compressive sensing) in the sense that gradient descent favors low rank matrices in certain situations. Moreover, restricting further to diagonal
matrices, or equivalently factorizing the entries of a vector to be recovered, shows connection to compressive sensing and l1-minimization.
Session 02: 16:00H
FAU MoD Lecture: Counterintuitive approximations
Speaker: Prof. Dr. Christian Bär
Affiliation: Institut für Mathematik. Universität Potsdam (Germany)
Abstract. One of the oldest questions in differential geometry is whether every curved space arises as a subspace of Euclidean space ℝⁿ. More precisely, does every Riemannian manifold admit an
isometric (i.e. length preserving) smooth embedding into ℝⁿ? Nash showed that the answer is yes but the dimension n of the Euclidean space may be much larger than that of the manifold. Often there
are non-isometric embeddings of much lower codimension.
The Nash-Kuiper embedding theorem is a prototypical example of a counterintuitive approximation result in geometry: any short (but highly non-isometric) embedding of a Riemannian manifold into
Euclidean space can be approximated by isometric embeddings. They are generally not smooth but of regularity C¹. This implies that any surface with a given geometry can be isometrically C¹-embedded
into an arbitrarily small ball in ℝ³. For C²-embeddings this completely false due to curvature restrictions.
After explaining this historical and conceptual context, I will present a general result which ensures approximations by maps satisfying strongly overdetermined equations (such as being isometric) on
open dense subsets. This will be illustrated by three examples: Lipschitz functions with surprising derivative, surfaces in 3-space with unexpected curvature properties, and a similar statement for
abstract Riemannian metrics on manifolds. Our method is based on “cut-off homotopy”, a concept introduced by Gromov in 1986.
This is based on joint work with Bernhard Hanke.
Holger Rauhut
Prof. Dr. Holger Rauhut studied Mathematics at TU Munich, where he obtained also his doctorate degree in 2004. After a period as postdoc, he received the habilitation from the University of Vienna in
2008. He held professor positions at the University of Bonn and RWTH Aachen University. In 2023, he joined LMU Munich where he is the head of the Chair of Mathematics of Information Processing. He
received an ERC Starting Grant in 2010. He was spokesperson of the SFB 1481 Sparsity and Singular Structures at RWTH Aachen University 2022-2023. His research focuses on the mathematical foundations
of machine learning and of signal processing. This includes aspects of high-dimensional probability, of optimization, of harmonic analysis and of the analysis of algorithms in these fields.
Christian Bär
Prof. Dr. Christian Bär holds a PhD degree (1990) and a habilitation (1993) from the University of Bonn. He held professor positions in Freiburg and Hamburg. Since 2003 he is professor for geometry
at the University of Potsdam. He was president of the Deutsche Mathematiker-Vereinigung and an elected member of the mathematics panel of the Deutsche Forschungsgemeinschaft. Currently, he is
editor-in-chief of zbmath Open. His research interests are centered around differential geometry, global analysis and applications to mathematical physics.
This is a hybrid event (On-site/online) open to: Public, Students, Postdocs, Professors, Faculty, Alumni and the scientific community all around the world.
Tue. December 03, 2024 at 14:30H (Berlin time)
On-site / Online
[On-site] Friedrich-Alexander-Universität Erlangen-Nürnberg
Felix Klein building. Department Mathematik
H13 Johann-Radon-Hörsaal
Cauerstraße 11, 91058 Erlangen
GPS-Koord. Raum:
49.579737N, 11.029862E
[Online] FAU Zoom link:
Meeting ID: 667 9081 1368 | PIN code: 716845
You might like:
• FAU MoD Lectures
• FAU MoD Lecture: Measuring productivity and fixedness in lexico-syntactic constructions by Prof. Dr. Stephanie Evert
• FAU MoD Lecture: New avenues for the interaction of computational mechanics and machine learning by Prof. Dr. Paolo Zunino
• FAU MoD Lecture: Discovering and Communicating Excellence by Prof. Dr. Ute Klammer
• FAU MoD Lecture: Thoughts on Machine Learning by Prof. Dr. Rupert Klein
• FAU MoD Lecture: Using system knowledge for improved sample efficiency in data-driven modeling and control of complex technical systems by Prof. Dr. Sebastian Peitz
• FAU MoD Lecture: Image Reconstruction – The Dialectic of Modelling and Learning by Prof. Dr. Martin Burger
• FAU MoD Lecture: The role of Artificial Intelligence in the future of mathematics by Prof. Dr. Amaury Hayat
• FAU MoD Lecture: FAU MoD Lecture. Special November 2023 by Prof. Dr. Michael Kohlhase and Prof. Dr. Edriss S. Titi
• FAU MoD Lecture: Free boundary regularity for the obstacle problem by Prof. Dr. Alessio Figalli
• FAU MoD Lecture: Physics-Based and Data-Driven-Based Algorithms for the Simulation of the Heart Function by Prof. Dr. Alfio Quarteroni
• FAU MoD Lecture: From Physics-Informed Machine Learning to Physics-Informed Machine Intelligence: Quo Vadimus? by Prof. Dr. George Karniadakis
• FAU MoD Lecture: From Alan Turing to contact geometry: Towards a “Fluid computer” by Prof. Dr. Eva Miranda
• FAU MoD Lecture: Applications of AAA Rational Approximation by Prof. Dr. Nick Trefethen
• FAU MoD Lecture: Learning-Based Optimization and PDE Control in User-Assignable Finite Time by Prof. Dr. Miroslav Krstic
Don’t miss out our last news and connect with us!
Holger Rauhut
Mathematisches Institut der Universität München (Germany)
Christian Bär
Institut für Mathematik. Universität Potsdam (Germany) | {"url":"https://dcn.nat.fau.eu/events/fau-mod-lecture-series-special-dec-2024/","timestamp":"2024-11-04T20:01:20Z","content_type":"text/html","content_length":"96713","record_id":"<urn:uuid:b2012056-589c-4363-8baf-18aa863bff0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00046.warc.gz"} |
Practice with Functions: Exercises | Saylor Academy
Practice with Functions
Complete these exercises to test your understanding of functions.
Exercise 1
Write a program that performs arithmetic division. The program will use two integers, a and b (obtained by the user) and will perform the division a/b, store the result in another integer c and show
the result of the division using cout. In a similar way, extend the program to add, subtract, multiply, do modulo and power using integers a and b. Modify your program so that when it starts, it asks
the user which type of calculation it should do, then asks for the 2 integers, then runs the user selected calculation and outputs the result in a user friendly formatted manner.
Sample run:
Enter two numbers: 312
The sum is 15.
Exercise 2
Basically the same as exercise 1, but this time, the function that adds the numbers should be void, and takes a third, pass by reference parameter; then puts the sum in that.
Exercise 3
Write a recursive function that finds the #n integer of the Fibonacci sequence. Then build a minimal program to test it. For reference see Wikipedia:Fibonacci number.
For any possible natural number "n", the following applies fib(n+2)= fib(n+1)+ fib(n). Also, the following are predefined fib(0)=0 fib(1)=1
Exercise 4
Basically the same as exercise 3, although this time you mustn't use recursion.
For extra exercise, give a big number (like 1000000) to both exercise 3 and 4 solutions and compare the execution times. Ponder on the results ;)
Exercise 5
Create a calculator that takes a number, a basic math operator (+,-,*,/,^), and a second number all from user input, and have it print the result of the mathematical operation. The mathematical
operations should be wrapped inside of functions.
Source: Wikibooks, https://en.wikibooks.org/wiki/C%2B%2B_Programming/Exercises/Functions
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. | {"url":"https://learn.saylor.org/mod/book/view.php?id=29942&chapterid=5647","timestamp":"2024-11-04T14:18:57Z","content_type":"text/html","content_length":"107994","record_id":"<urn:uuid:25375f97-043f-4c73-b3d3-05da1f309aa2>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00370.warc.gz"} |
Conditional Graphical LASSO for Gaussian Graphical Models with
Censored and Missing Values
cglasso-package Conditional Graphical LASSO for Gaussian Graphical Models with Censored and Missing Values
AIC Akaike Information Criterion
AIC.cglasso Akaike Information Criterion
BIC Bayesian Information Criterion
BIC.cglasso Bayesian Information Criterion
cggm Post-Hoc Maximum Likelihood Refitting of a Conditional Graphical Lasso
cglasso Conditional Graphical Lasso Estimator
coef Extract Model Coefficients
coef.cglasso Extract Model Coefficients
ColMeans Calculate Column Means and Vars of a "datacggm" Object
colNames Row and Column Names of a "datacggm" Object
colNames<- Row and Column Names of a "datacggm" Object
ColVars Calculate Column Means and Vars of a "datacggm" Object
datacggm Create a Dataset from a Conditional Gaussian Graphical Model with Censored and/or Missing Values
dim Dimensions of a "datacggm" Object
dim.datacggm Dimensions of a "datacggm" Object
dimnames Dimnames of a "datacggm" Object
dimnames.datacggm Dimnames of a "datacggm" Object
dimnames<-.datacggm Dimnames of a "datacggm" Object
event Status Indicator Matrix from a "datacggm" Object
Example Simulated data for the cglasso vignette
fitted Extract Model Fitted Values
fitted.cglasso Extract Model Fitted Values
getGraph Retrieve Graphs from a "cglasso2igraph" Object
getMatrix Retrieve Matrices "Y" and "X" from a "datacggm" Object
hist Histogram for a 'datacggm' Object
hist.datacggm Histogram for a 'datacggm' Object
impute Imputation of Missing and Censored Values
is.cglasso2igraph Is an Object of Class 'cglasso2igraph'?
is.datacggm Is an Object of Class 'datacggm'?
lower Lower and Upper Limits from a "datacggm" Object
MKMEP Megakaryocyte-Erythroid Progenitors
MKMEP.Sim Simulated data for the cglasso vignette
MM The Rule of miRNA in Multiple Myeloma
MM.Sim Simulated data for the cglasso vignette
nobs Extract the Number of Observations/Responses/Predictors from a datacggm Object
nobs.datacggm Extract the Number of Observations/Responses/Predictors from a datacggm Object
npred Extract the Number of Observations/Responses/Predictors from a datacggm Object
npred.datacggm Extract the Number of Observations/Responses/Predictors from a datacggm Object
nresp Extract the Number of Observations/Responses/Predictors from a datacggm Object
nresp.datacggm Extract the Number of Observations/Responses/Predictors from a datacggm Object
plot.cggm Plot Method for a "cggm" Object
plot.cglasso Plot Method for "cglasso" Object
plot.cglasso2igraph Plot Method for a cglasso2igraph Object"
plot.GoF Plot for "GoF" Object
predict Predict Method for cglasso and cggm Fits
predict.cggm Predict Method for cglasso and cggm Fits
predict.cglasso Predict Method for cglasso and cggm Fits
QFun Extract Q-Function
qqcnorm Quantile-Quantile Plots for a 'datacggm' Object
rcggm Simulate Data from a Conditional Gaussian Graphical Model with Censored and/or Missing Values
residuals Extract Model Residuals
residuals.cglasso Extract Model Residuals
rowNames Row and Column Names of a "datacggm" Object
rowNames<- Row and Column Names of a "datacggm" Object
select_cglasso Model Selection for the Conditional Graphical Lasso Estimator
ShowStructure Show Package Structure
summary.cglasso Summarizing cglasso and cggm Fits
summary.datacggm Summarizing Objects of Class "datacggm"
to_graph Create Graphs from cglasso or cggm Objects
upper Lower and Upper Limits from a "datacggm" Object | {"url":"https://search.r-project.org/CRAN/refmans/cglasso/html/00Index.html","timestamp":"2024-11-10T02:12:05Z","content_type":"text/html","content_length":"9542","record_id":"<urn:uuid:003a6030-6473-496d-a47b-a66b2ac53f88>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00525.warc.gz"} |
Mathematical tools for large graphs
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Mathematical tools for large graphs
July 12, 2021
Author’s note: This article was written intended for a general scientific audience.
Graphs and networks form the language and foundation of much of our scientific and mathematical studies, from biological networks, to algorithms, to machine learning applications, to pure
mathematics. As the graphs and networks that arise get larger and larger, it is ever more important to develop and understand tools to analyze large graphs.
Graph regularity lemma
My research on large graphs has its origins in the work of Endre Szemerédi in the 1970s, for which he won the 2012 Abel Prize, a lifetime achievement prize in Mathematics viewed as equivalent to the
Nobel Prize. Szemerédi is a giant of combinatorics, and his ideas still reverberate throughout the field today.
Endre Szemerédi receiving the 2012 Abel Prize
(Photo from the Abel Prize)
Szemerédi was interested in an old conjecture of Paul Erdős and Pál Turán from the 1930s. The conjecture says that
if you take an infinite set of natural numbers, provided that your set is large enough, namely that it occupies a positive fraction of all natural numbers, then you can always find arbitrarily
long arithmetic progressions.
For instance, if we take roughly one in every million natural numbers, then it is conjectured that one can find among them k numbers forming a sequence with equal gaps, regardless of how large k is.
This turns out to be a very hard problem. Before the work of Szemerédi, the only partial progress, due to Fields Medalist Klaus Roth in the 1950’s is that one can find three numbers forming a
sequence with equal gaps. Even finding four numbers turned out to be an extremely hard challenge until Szemerédi cracked open the entire problem, leading to a resolution of the Erdős–Turán
conjecture. His result is now known as Szemerédi’s theorem. Szemerédi’s proof was deep and complex. Through his work, Szemerédi introduced important ideas to mathematics, one of which is the graph
regularity lemma.
Szemerédi’s graph regularity lemma is a powerful structural tool that allows us to analyze very large graphs. Roughly speaking, the regularity lemma says that
if we are given a large graph, no matter how the graph looks like, we can always divide the vertices of the graph into a small number of parts, so that the edges of the graph look like they are
situated randomly between the parts.
Between different pairs of parts of vertices, the density of edges could be different. For example, perhaps there are five vertex parts A, B, C, D, E; between A and B the graph looks like a random
graph with edge density 0.2; between A and C the graph looks like a random graph with edge density 0.5, and so on.
An illustration of a vertex partition produced by the graph regularity lemma.
An important point to emphasize is that, with a fixed error tolerance, the number of vertex parts produced by the partition does not increase with the size of the graph. This property makes the graph
regularity lemma particularly useful for very large graphs.
Szemerédi’s graph regularity lemma provides us a partition of the graph that turns out to be very useful for many applications. This tool is now a central technique in modern combinatorics research.
A useful analogy is the signal-versus-noise decomposition in signal processing. The “signal” of a graph is its vertex partition together with the edge density data. The “noise” is the residual
random-like placement of the edges.
Szemerédi’s idea of viewing graph theory via the lens of this decomposition has had a profound impact in mathematics. This decomposition is nowadays called “structure versus pseudorandomness” (a
phrase popularized by Fields Medalist Terry Tao). It has been extended far beyond graph theory. There are now deep extensions by Fields Medalist Timothy Gowers to what is called “higher order Fourier
analysis” in number theory. The regularity method has also been extended to hypergraphs.
Graph limits
László Lovász (Photo from the Abel Prize)
László Lovász, who recently won the 2021 Abel Prize, has been developing a theory of graph limits with his collaborators over the past couple of decades. Lovász’s graph limits give us powerful tools
to describe large graphs from the perspective of mathematical analysis, with applications ranging from combinatorics to probability to machine learning.
Lovász’s book on graph limits
Why graph limits? Here is an analogy with our number system. Suppose our knowledge of the number line was limited to the rational numbers. We can already do a lot of mathematics with just the
rational numbers. In fact, with just the language of rational numbers, we can talk about irrational numbers by expressing each irrational number as a limit of a sequence of rational numbers chosen to
converge to the irrational number. This can be done but it would be cumbersome if we had to do this every time we wanted to use an irrational number. Luckily, the invention of real numbers solved
this issue by filling the gaps on the real number line. In this way, the real numbers form the limits of rational numbers.
Likewise, graphs are discrete objects analogous to rational numbers. A sequence of graphs can also converge to a limiting object (what does it mean for a sequence of graphs to converge is a
fascinating topic and beyond the scope of this article). These limiting objects are called “graph limits,” also known as “graphons.” Graph limits are actually simple analytic objects. They can be
pictured as a grayscale image inside a square. They can also be represented as a function from the unit square to the unit interval (the word “graphon” is an amalgamation of the words “graph” and
“function”). Given a sequence of graphs, we can express each graph as an adjacency matrix drawn as a black and white image of pixels. As the graphs get larger and larger, this sequence of pixel
images looks closer and closer to a single image, which might be not just black and white but also could have various shades of gray. This final image is an example of a graph limit. (Actually, there
are subtleties here that I am glossing over, such as permuting the ordering of the vertices before taking the limit.)
A graph, its adjacent matrix as a pixel picture, and the graph limit
A graph (shown as the pixel image of its adjacency matrix) sampled from a graph limit
(Both images taken from Lovász’s book)
Conversely, given a graph limit, one can use it as a generative model for random graphs. Such random graphs converge to the given graph limit when the number of vertices increases. This random graph
model generalizes the stochastic block model. An example of a problem in machine learning and statistics is how to recover the original graph limit given a sequence of samples from the model. There
is active research work on this problem (although it is not the subject of my own research).
The mathematics underlying the theory of graph limits, notably the proof of their existence, hinges on Szemerédi’s graph regularity lemma. So these two topics are intertwined.
Sparser graphs
Graph regularity lemma and graph limits both have an important limitation: they can only handle dense graphs, namely, graphs with a quadratic number of edges, i.e., edge density bounded from below by
some positive constant. This limitation affects applications in pure and applied mathematics. Real life networks are generally far from dense. So there is great interest in developing graph theory
tools to better understand sparser graphs. Sparse graphs have significantly more room for variability, leading to mathematical complications.
A core theme of my research is to tackle these problems by extending mathematical tools on large graphs from the dense setting to the sparse setting. My work extends Szemerédi’s graph regularity
method from dense graphs to sparser graphs. I have also developed new theories of sparse graph limits. These results illuminate the world of sparse graphs along with many of their complexities.
I have worked extensively on extending graph regularity to sparser graphs. With David Conlon and Jacob Fox, we applied new graph theoretic insights to simplify the proof of Ben Green and Terry Tao’s
celebrated theorem that the prime numbers contain arbitrarily long arithmetic progressions (see exposition). It turns out that, despite being a result in number theory, a core part of its proof
concerns the combinatorics of sparse graphs and hypergraphs. Our new tools allow us to count patterns in sparse graphs in a setting that is simpler than the original Green–Tao work. In a different
direction, we (together additionally with Benny Sudakov) have recently developed the sparse regularity method for graphs without 4-cycles, which are necessarily sparse.
My work on sparse graph limits, together with Christian Borgs, Henry Cohn, and Jennifer Chayes , developed a new $L^p$ theory of sparse graph limits. We were motivated by sparse graph models with
power law degree distributions, which are popular in network theory due to their observed prevalence in the real world. Our work builds on the ideas of Bela Bollobás and Oliver Riordan , who
undertook the first systematic study of sparse graph limits. Bollobás and Riordan also posed many conjectures in their paper, but these conjectures turned out to be all false due to a counterexample
we found with my PhD students Ashwin Sah, Mehtaab Sawhney, and Jonathan Tidor. These results illustrate the richness of the world of sparse graph limits.
Graph theory and additive combinatorics
A central theme of my research is that graph theory and additive combinatorics are connected via a bridge that allows many ideas and techniques to be transferred from one world to the other. Additive
combinatorics is a subject that concerns problems at the intersection of combinatorics and number theory, such as Szemerédi’s theorem and the Green–Tao theorem, both of which are about finding
patterns in sets of numbers.
When I started teaching as an MIT faculty, I developed a graduate-level math class titled Graph theory and additive combinatorics introducing the students to these two beautiful subjects and
highlighting the themes that connect them. It gives me great pleasure to see many students in my class later producing excellent research on this topic.
In Fall 2019, I worked with MIT OpenCourseWare and filmed all my lecture videos for this class. They are now available for free on MIT OCW and YouTube. I have also made available my lecture notes,
which I am currently in the process of editing into a textbook.
In my first lecture, I tell the class about the connections between graph theory and additive combinatorics as first observed by the work of Issai Schur from over a hundred years ago. Thanks to
enormous research progress over the past century, we now understand a lot more, but there is still a ton of mystery that remains.
The world of large graphs is immense.
Back to main | List of entries |
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Damping - Ashes Documentation
: there seem to be some confusion in the literature about the naming of this method:
• The HHT method as implemented in Ashes was derived by Hilber et al. (1977) and is explained in Crisfield et al. (1997). In the latter, this method is also named alpha-method after the parameter
used to tune it
• The generalized-alpha method was derived by Chung et al. (1993). This is not the method implemented in Ashes | {"url":"https://www.simis.io/docs/time-domain-simulation-damping","timestamp":"2024-11-02T01:33:58Z","content_type":"text/html","content_length":"201462","record_id":"<urn:uuid:74206bd6-f340-4966-a03c-1dbc7d5bb0db>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00356.warc.gz"} |
Interpolation Formula: Properties, Chemical Structure and Uses
Interpolation Formula
Interpolation is a technique for determining new values for any function from a set of values. This formula is used to determine the unknown value of a point. The new value from the two provided
locations should be determined using the linear Interpolation Formula. The “n” set of numbers should be provided when compared to Lagrange’s Interpolation Formula, and the new value should be
determined using Lagrange’s approach.
In basic terms, interpolation is a way of guessing unknown values that lie between provided data points. The interpolation technique is used to estimate unknown parameters such as noise level,
rainfall, elevation, and so on for any geographically connected data points.
Types Interpolation Formula
Two types of interpolation formulae are used to find the unknown values of a given collection of data points: linear Interpolation Formula and Lagrange Interpolation Formula.
The formula for Linear Interpolation
Linear interpolation has been used for filling unknown values in tables since its inception. The method of linear interpolation is said to have been employed by Babylonians.
The formula for Lagrange Interpolation
The Lagrange Interpolation Formula is used to discover a polynomial known as the Lagrange polynomial, which takes on different values at different places. Lagrange’s Interpolation is an Nth-degree
polynomial approximation to f(x).
What is Interpolation?
The mathematical technique of interpolation is used to calculate the value between two locations that have a certain value. It is a method of estimating the value of a given function at a given
collection of discrete points, to put it simply. As a result, it may be used to estimate various cost ideas as well as in mathematics and statistics.
For each given set of functions with known values, interpolation may be defined as a way of obtaining the unknown value. The elusive value is discovered. Students may use Excel’s linear interpolation
function to calculate the unknown value from the two known points if the provided sets of values follow a linear trend.
Interpolation is the technique of determining a certain value between two points on a line or curve. The term ‘inter’ here implies ‘insert’ into the data collection. This technique is valuable not
only in statistics, but also in science, business, and a variety of other real-world applications that fit inside two existing data points.
As a result, the Interpolation Formula may be interpreted as a method of curve fitting using linear polynomials and therefore to build new data points within the provided range of a discrete
collection of existing data points.
Then linear interpolation is an easy way to accomplish this. This method of tabulation based on linear interpolation is said to have been employed by Babylonians.
In computer graphics, the basic operation of linear interpolation between two values is also useful.
Things to Keep in Mind
Interpolation is a sort of estimate in which the value of f(x) or the function of x is calculated from a set of two known values of the function.
Interpolation may also be described as a method of guessing unknown values between provided data points.
The Interpolation Formula is classified into two types:
linear interpolation formulas
lagrange interpolation formulas.
Since its inception, linear interpolation, commonly known as simply interpolation, has been used to fill in unknown numbers in tables.
Lagrange’s Interpolation, which may be characterized as an Nth-degree polynomial approximation to f(x) is used to discover Lagrange polynomials.
Interpolation Formula
The Interpolation Formula, which uses interpolation, is a technique for determining new values for any function from the set of existing values. The Interpolation Formula employs interpolation, a
technique for estimating a value between two points on a function’s curve. In the section that follows, the Interpolation Formula is described together with a few cases that have been solved.
If students have the coordinates of the two known points, (x0,y0), and (x1,y1), they can use the interpolation or linear Interpolation Formula to determine the point on a straight line between the
two locations. The result is a point on a straight line. For polynomials of the first order, a formula for linear interpolation may be provided.
The Interpolation Formula is as follows
Examples of Interpolation
One of the most difficult and high-scoring disciplines is Mathematics. Students who use Extramarks examples can better their studying and accomplish their goals. These Extramarks solved examples are
curated specifically to help students acquire and comprehend the Interpolation Formula. The language is simple to comprehend so that students may learn more and benefit to the fullest.
For students to score well on their tests or competitive exams, there must be conceptual clarity. As a consequence, Extramarks provides students with Interpolation Formula with examples. They can
pick up new information rapidly and comprehend the study material completely. Studying and understanding concepts are crucial to learning Mathematics, as is practising questions that are based on the
Interpolation Formula concepts.
Students may become exam-ready and ace any Competitive examinations by studying from test materials from the Extramarks. Online study materials for CBSE, ICSE, IIT JEE Main & Advanced, NEET, and
other boards such as NCERT Textbook Solutions, Syllabus, Revision Notes, Important Questions, Important Formulas, Past Year Question Papers & Sample Papers with Solutions to help students perform
well in their exams. Thus, Extramarks provides its students with the most up-to-date tools for learning, practising, and preparing for tests.
Students who study for competitive tests are typically stressed as they become anxious about the exam preparations. As a result, they must obtain as many study tools as possible from various sources,
as the tests are challenging due to the degree of competitiveness. As a result, Extramarks provides its students with cutting-edge resources to study, practise, and apply. The online materials for
CBSE, ICSE, IIT JEE Main and Advanced, NEET, and other boards are accessible for download on Extramarks website.
FAQs (Frequently Asked Questions)
1. What does the Interpolation Formula mean?
The Interpolation Formula, which uses interpolation, is a technique for determining new values for any function from the set of existing values. The Interpolation Formula employs interpolation, a
technique for estimating a value between two points on a function’s curve.
The Interpolation Formula is as follows
2. How does the Interpolation Formula method work?
The majority of the time, interpolation is employed in statistics to estimate unknown quantities or probable returns on investments. The other documented values that are situated in front of the
unknown value can be found using the Interpolation Formula.
3. What kind of method is the linear interpolation method?
One of the interpolation kinds that is used in a different linear polynomial between each pair of data points for curves or within the sets of three points for surfaces is the linear interpolation
technique or formula.
4. What role does interpolation play in statistics?
Interpolation is commonly employed in statistical models for commercial and mathematical research since it aids in predicting future likely points in data analysis. The acquired sets may be used to
predict where the general consistent trend will lead the pricing (of a product or service), prospective yield (and growth for a firm), or get insights into the stock market under a specific market
dynamic. In the bond market and financial sector, financial analysts have regularly used this strategy to derive logical conclusions. | {"url":"https://www.extramarks.com/studymaterials/formulas/interpolation-formula/","timestamp":"2024-11-13T09:16:24Z","content_type":"text/html","content_length":"631847","record_id":"<urn:uuid:4fbd78ce-5070-413c-9b2b-6c9dfe68ce40>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00243.warc.gz"} |
Diameter of Hypersphere given Surface Volume Calculator | Calculate Diameter of Hypersphere given Surface Volume
What is a Hypersphere?
Hypersphere is basically the sphere in 4th dimension. This is the expansion of circle (2D) and sphere (3D) into a fourth dimension of space. This doesn't exist in our three-dimensional world, but
calculations regarding Hyperspheres can easily be done by extending the formulas of 3D sphere into 4D.
How to Calculate Diameter of Hypersphere given Surface Volume?
Diameter of Hypersphere given Surface Volume calculator uses Diameter of Hypersphere = (4*Surface Volume of Hypersphere/(pi^2))^(1/3) to calculate the Diameter of Hypersphere, The Diameter of
Hypersphere given Surface Volume formula is defined as twice the distance from the center to any point on the Hypersphere which is the 4D extension of sphere in 3D and circle in 2D, calculated using
surface volume of Hypersphere. Diameter of Hypersphere is denoted by D symbol.
How to calculate Diameter of Hypersphere given Surface Volume using this online calculator? To use this online calculator for Diameter of Hypersphere given Surface Volume, enter Surface Volume of
Hypersphere (V[Surface]) and hit the calculate button. Here is how the Diameter of Hypersphere given Surface Volume calculation can be explained with given input values -> 10.04385 = (4*2500/(pi^2))^ | {"url":"https://www.calculatoratoz.com/en/diameter-of-hypersphere-given-surface-volume-calculator/Calc-39844","timestamp":"2024-11-07T00:17:26Z","content_type":"application/xhtml+xml","content_length":"114231","record_id":"<urn:uuid:70b98cf1-a2cd-43fe-a20f-b3aabf163717>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00295.warc.gz"} |
Sublinear-time quadratic minimization via spectral decomposition of matrices
We design a sublinear-time approximation algorithm for quadratic function minimization problems with a better error bound than the previous algorithm by Hayashi and Yoshida (NIPS'16). Our
approximation algorithm can be modified to handle the case where the minimization is done over a sphere. The analysis of our algorithms is obtained by combining results from graph limit theory, along
with a novel spectral decomposition of matrices. Specifically, we prove that a matrix A can be decomposed into a structured part and a pseudorandom part, where the structured part is a block matrix
with a polylogarithmic number of blocks, such that in each block all the entries are the same, and the pseudorandom part has a small spectral norm, achieving better error bound than the existing
decomposition theorem of Frieze and Kannan (FOCS'96). As an additional application of the decomposition theorem, we give a sublinear-time approximation algorithm for computing the top singular values
of a matrix.
Original language English
Title of host Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 21st International Workshop, APPROX 2018, and 22nd International Workshop, RANDOM 2018
Editors Eric Blais, Jose D. P. Rolim, David Steurer, Klaus Jansen
Publisher Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Print) 9783959770859
State Published - 1 Aug 2018
Externally published Yes
21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and
Event Computation, RANDOM 2018 - Princeton, United States
Duration: 20 Aug 2018 → 22 Aug 2018
Publication series
Name Leibniz International Proceedings in Informatics, LIPIcs
Volume 116
ISSN (Print) 1868-8969
Conference 21st International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2018 and the 22nd International Workshop on Randomization and Computation,
RANDOM 2018
Country/ United States
City Princeton
Period 20/08/18 → 22/08/18
Bibliographical note
Publisher Copyright:
© 2018 Aditya Bhaskara and Srivatsan Kumar.
• Approximation Algorithms
• Graph limits
• Matrix spectral decomposition
• Qudratic function minimization
ASJC Scopus subject areas
Dive into the research topics of 'Sublinear-time quadratic minimization via spectral decomposition of matrices'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/sublinear-time-quadratic-minimization-via-spectral-decomposition-","timestamp":"2024-11-03T07:56:59Z","content_type":"text/html","content_length":"58971","record_id":"<urn:uuid:a049dc7f-f2d4-4142-aa92-0c2b880c7b65>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00402.warc.gz"} |
ACCA F5 query
mark057 Registered Posts: 354 Dedicated contributor 🦉
Hello everyone,
I was wondering if somebody would be kind enough to explain what looks to be a simple problem I'm having with average growth model calculations?
Year Sales Rev
2001 150,000
2002 192,000
2003 206,000
2004 245,000
2005 262,350
Sales in 2001 x (1 + g)4 = Sales in 2005
(1 + g)4 = $262,350/150,000 = 1.749
I'm fine up to this part but then the answer says
1 + g = 4 then what looks to be a square root symbol with the figure 1.749 inside = 1.15.
How are they calculating 1.15? I've tried various calculations and can't get it to work. Can anyone shed any light on this?
• Hi Mark,
If you are using a scientific calculator press '4' then 'shift' and look for the key that has the square root symbol with a filled box on top, and a hollow box underneath! (sorry, I can't think
of a better way to explain it!) then enter the 262,350/150,000 in the brackets. This will give you the answer of 1.15, which is an average growth of 15% each year.
Did you know there is now an ACCA forum on this website, called 'Further Studies'. | {"url":"https://forums.aat.org.uk/Forum/discussion/31052/acca-f5-query","timestamp":"2024-11-13T13:00:38Z","content_type":"text/html","content_length":"286235","record_id":"<urn:uuid:ee5646b9-788c-45c3-babe-faffda4aeec9>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00683.warc.gz"} |
1. Introduction2. Results3. Cauchy Problem for the Navier-Stokes Equation4. ConclusionAcknowledgementsReferences
Consider Schrödinger’s equation:
Let be a solution of (1) with the following asymptotic behavior:
where is the scattering amplitude and for
Let us also dene the solution for as
As is well known [9] :
This equation is the key to solving the inverse scattering problem, and was first used by Newton [10] [11] and Somersalo et al. [12] .
Equation (4) is equivalent to the following:
where S is a scattering operator with kernel,
The following theorem was stated in [9] :
Theorem 1. (The energy and momentum conservation laws) Let. Then, , where I is a unitary operator.
Definition 1. The set of measurable functions with norm defined by is recognized as being of Rollnik class.
As shown in [13] , is an orthonormal system of H eigenfunctions for the continuous spectrum. In addition to the continuous spectrum there are a finite number N of H negative eigenvalues, designated
as with corresponding normalized eigenfunctions
We present Povzner’s results [13] below:
Theorem 2. (Completeness) For both an arbitrary and for H eigenfunctions, Parseval’s identity is valid.
where and are Fourier coefficients for the continuous and discrete cases.
Theorem 3. (Birman-Schwinger estimation). Let. Then, the number of discrete eigenvalues can be estimated as:
This theorem was proved in [14] .
Let us introduce the following notation:
We define the operators for as follows:
Consider the Riemann problem of finding a function, that is analytic in the complex plane with a cut along the real axis. Values of on the upper and lower sides of the cut are denoted and
respectively. The following presents the results of [15] :
Lemma 1.
Theorem 4. Let,; then
The proof of the above follows from the classic results for the Riemann problem.
Lemma 2. Let
The proof of the above follows from the definitions of and.
Lemma 3. Let
The proof of the above again follows from the definitions of the functions and.
Lemma 4. Let Then,
The proof of the above follows from the definitions of and and Theorem 1.
Lemma 5. Let Then,
The proof of the above follows from the definitions of and and Lemma 4 and dispersions relations for analytics functions.
Definition 2. Denote by TA the set of functions with the norm
Definition 3. Denote by the set of functions such that, for any.
Lemma 6. Suppose. Then, the operator, defined on the setTA has an inverse defined on.
The proof of the above follows from the definitions of and the conditions of Lemma 6.
Lemma 7. Let, and assume that exists. Then,
The proof of the above follows from the denitions of and and Equation (4).
Lemma 8. Let, and assume that exists. Then,
where represents terms of highest order of.
The proof. Using
and (18) we get proof.
Lemma 9. Let. Then,
The lemma can be proved by substituting into Equation (1).
Lemma 10. Let, and assume that exists. Then,
The proof of the above follows from the definitions of and Lemma 7.
Lemma 11. Let. Then
The proof of the above follows from the definition of D and the unitary nature of S.
Lemma 12. Let. Then,
The proof of the above follows from the definitions of, and (1).
Lemma 13. Let, and. Then,
To prove this result, one should calculate using (18).
Using the notation that:
Lemma 14. Let, and. Then,
To prove this result, one should using Lemma 7.
Using Lemma 7.
Lemma 15. Let, and. Then,
To prove this result, one should calculate A using Lemma 7.
Lemma 16. Let and
A proof of this lemma can be obtained using Plancherel’s theorem.
Lemma 17. Let and
To prove this result, one should calculate.
Numerous studies of the Navier-Stokes equations have been devoted to the problem of the smoothness of its solutions. A good overview of these studies is given in [16] - [20] . The spatial
differentiability of the solutions is an important factor, this controls their evolution. Obviously, differentiable solutions do not provide an effective description of turbulence. Nevertheless, the
global solvability and differentiability of the solutions has not been proven, and therefore the problem of describing turbulence remains open. It is interesting to study the properties of the
Fourier transform of solutions of the Navier-Stokes equations. Of particular interest is how they can be used in the description of turbulence, and whether they are differentiable. The
differentiability of such Fourier transforms appears to be related to the appearance or disappearance of resonance, as this implies the absence of large energy flows from small to large harmonics,
which in turn precludes the appearance of turbulence. Thus, obtaining uniform global estimations of the Fourier transform of solutions of the Navier-Stokes equations means that the principle modeling
of complex flows and related calculations will be based on the Fourier transform method. The authors are continuing to research these issues in relation to a numerical weather prediction model; this
paper provides a theoretical justification for this approach. Consider the Cauchy problem for the Navier-Stokes equations:
in the domain, where:
The problem defined by (34), (35), (36) has at least one weak solution in the so-called Leray-Hopf class [16] .
The following results have been proved [17] :
Theorem 5. If
there is a single generalized solution of (34), (35), (36) in the domain, , satisfying the following conditions:
Note that depends on and.
Lemma 18. Let. Then,
Our goal is to provide global estimations for the Fourier transforms of the derivatives of the solutions to the Navier-Stokes Equations (34)-(36) without requiring the initial velocity and force to
be small. We obtain the following uniform time estimation.
Statement 1. The solution of (34), (35), (36) according to Theorem 5 satisfies:
This follows from the definition of the Fourier transform and the theory of linear differential equations.
Statement 2. The solution of (34), (35), (36) satisfies:
and the following estimations:
This expression for is obtained using div and the Fourier transform presentation.
Lemma 19. The solution of (34), (35), (36) in Theorem 5 satisfies the following inequalities:
Proof this follows from the a priory estimation of Lemma18 and conditions of Lemma 19.
Lemma 20. Let Then, the solution of (34), (35), (36) in Theorem 5 satisfies 2 the following inequalities:
Proof this follows from the a priory estimation of Lemma18 and conditions of Lemma 20.
Lemma 21. The solution of (34), (35), (36) in Theorem 5 satisfies the following inequalities:
Proof this follows from the a priory estimation of Lemma18, conditions of Lemma 19, the Navier-Stokes equations.
Lemma 22. The solution of (34), (35), (36) satisfies the following inequalities:
Proof this follows from the a priory estimation of Lemma 18, conditions of Lemma 22, the Navier-Stokes equations.
Lemma 23. The solution of (34), (35), (36) according to Theorem 5 satisfies, where:
Proof this follows from the a priory estimation of Lemma18, the Navier-Stokes equations.
Lemma 24.Weak solution of problem (34), (35), (36) from Theorem 5 satisfies the following inequalities:
where are limited.
Let is prove the first estimate. These inequalities
Proof now this follows from the a priori estimation of Lemma 18, conditions of Lemma 24, the Navier-Stokes equations.
The rest of estimates are proved similarly.
Lemma 25. Suppose that and
Proof. Using Plansherel’s theorem, we get the statement of the lemma.
This proves Lemma 25.
Lemma 26. Weak solution of problem (34), (35), (36) from Theorem 5 satisfies the following inequalities
Proof. From (40) we get
Using the notation
taking into account Holder’s inequality in I we obtain:
where and satisfies the equality. Suppose. Then
Taking into consideration the estimate I in (53), we obtain the statement of the lemma.
This proves Lemma 26.
Lemma 27. Weak solution of problem (34), (35), (36) from Theorem 5 satisfies the following inequalities
Proof. The underwritten inequalities follows from representation (40)
Let us introduce the following denotation
Estimate I[1] by means of
where we obtain
On applying Holder’s inequality, we get
where p, q satisfies the equality.
For we have
Inserting in to
we obtain the statement of the lemma.
This completes the proof of Lemma 27.
Lemma 28. Weak solution of problem (34), (35), (36) from Theorem 5 satisfies the following inequalities
Lemma 25. Let and
A proof of this lemma can be obtained using Plancherel’s theorem.
We now obtain uniform time estimations for Rollnik’s norms of the solutions of (34), (35), (36).The following (and main) goal is to obtain the same estimations for―velocity components of the Cauchy
problem for the Navier-Stokes equations.
Let’s consider the influence of the following large scale transformations in Navier-Stokes’ equation on
Statement 3. Let
Proof. By the definitions and we have
This proves Statement 3.
Theorem 6. Let
Then, there exists a unique generalized solution of (34), (35), (36) satisfying the following inequality:
where the value of depends only on the conditions of the theorem.
Proof. It suffices to obtain uniform estimates of the maximum velocity components, which obviously follow from because uniform estimates allow us to extend the local existence and uniqueness theorem
over the interval in which they are valid. To estimate the velocity components, Lemma 22 can be used:
Using Lemmas (25)-(29) for
we can obtain where is the amplitude of potential and. That is, discrete solutions are not significant in proving the theorem, so its assertion follows the conditions of Theorem 6, which defines
uniform time estimations for the maximum values of the velocity components.
Theorem 6 asserts the global solvability and uniqueness of the Cauchy problem for the Navier-Stokes equations.
Theorem 7. Let
Then, there exists and
Proof. A proof of this lemma can be obtained using and uniform estimates.
Theorem 7 describes the loss of smoothness of classical solutions for the Navier-Stokes equations.
Theorem 7 describes the time blow up of the classical solutions for the Navier-Stokes equations arises, and complements the results of Terence Tao [17] . | {"url":"https://www.scirp.org/xml/54262.xml","timestamp":"2024-11-04T15:35:17Z","content_type":"application/xml","content_length":"66138","record_id":"<urn:uuid:0e8ca102-3ae0-4dfb-9492-bc38071e4bfd>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00087.warc.gz"} |
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Color Space Converter
Convert color information between color spaces
Vision HDL Toolbox / Conversions
The Color Space Converter block converts between R'G'B' and Y'CbCr color spaces, and also converts R'G'B' to intensity.
The Color Space Converter block operates on gamma-corrected color spaces, usually indicated with prime notation ('). However, for simplicity, the block and mask labels do not include the prime
This block uses a streaming pixel interface with a pixelcontrol bus for frame control signals. This interface enables the block to operate independently of image size and format. All Vision HDL
Toolbox™ blocks use the same streaming interface. The block accepts and returns a scalar pixel value and a bus that contains five control signals. The control signals indicate the validity of each
pixel and its location in the frame. To convert a frame (pixel matrix) into a serial pixel stream and control signals, use the Frame To Pixels block. For a full description of the interface, see
Streaming Pixel Interface.
This block also supports multipixel streams, where the pixel input is a matrix of M-by-3 values. M is the number of pixels and each pixel has 3 R'G'B' or Y'CbCr components. These values correspond to
the Number of pixels and Number of components parameters of the Frame To Pixels block.
pixel — Input pixel stream
vector | matrix
For scalar pixel streams, specify pixel as a vector of 1-by-3 values. For multipixel streams, specify pixel as a matrix of Number of pixels-by-3 pixel intensity values. Number of pixels can be two,
four, or eight.
The pixel stream must be in Y'CbCr or R'G'B' color space. Integer and fixed-point data types must be between 8 and 16 bits.
The software supports double and single data types for simulation, but not for HDL code generation.
Data Types: single | double | uint8 | uint16 | fixed point
ctrl — Control signals associated with pixel stream
pixelcontrol bus
The pixelcontrol bus contains five signals. The signals describe the validity of the pixel and its location in the frame. For more information, see Pixel Control Bus.
For multipixel streaming, each vector of pixel values has one set of control signals. Because the vector has only one valid signal, the pixels in the vector must be either all valid or all invalid.
The hStart and vStart signals apply to the pixel with the lowest index in the vector. The hEnd and vEnd signals apply to the pixel with the highest index in the vector.
Data Types: bus
pixel — Output pixel stream in new colorspace
scalar | vector | matrix
Output pixel stream in intensity, Y'CbCr, or R'G'B' color space, returned as a single pixel stream or multipixel stream. The data type and Number of pixels of the output stream is the same as the
input pixel stream. If the output is intensity values, each pixel has one component. If the output is Y'CbCr or R'G'B', each pixel has three components.
The software supports double and single data types for simulation, but not for HDL code generation.
Data Types: single | double | uint8 | uint16 | fixed point
ctrl — Control signals associated with pixel stream
pixelcontrol bus
The pixelcontrol bus contains five signals. The signals describe the validity of the pixel and its location in the frame. For more information, see Pixel Control Bus.
For multipixel streaming, each vector of pixel values has one set of control signals. Because the vector has only one valid signal, the pixels in the vector must be either all valid or all invalid.
The hStart and vStart signals apply to the pixel with the lowest index in the vector. The hEnd and vEnd signals apply to the pixel with the highest index in the vector.
Data Types: bus
Conversion — Type of color space conversion
RGB to YCbCr (default) | YCbCr to RGB | RGB to intensity
The block accepts input pixels as vectors of three values that represent a single pixel. If you choose RGB to intensity, each output pixel is a scalar. Otherwise, each output pixel is a vector of
three values.
Use conversion specified by — Conversion equation
Rec. 601 (SDTV) (default) | Rec. 709 (HDTV)
Conversion equation used between R'G'B' and Y'CbCr color spaces.
This parameter applies only when you set Conversion to RGB to YCbCr or YCbCr to RGB.
Scanning standard — HDTV scanning standard
1250/50/2:1 (default) | 1125/60/2:1
Scanning standard used to convert between R'G'B' and Y'CbCr color spaces in HDTV format.
This parameter applies when you set Use conversion specified by to Rec. 709 (HDTV).
When you use multipixel streaming, the block replicates the conversion algorithm for each of the M input pixels, in parallel. This increase in hardware resources is a trade off for increasing
throughput compared to single-pixel streaming.
Conversion Between R'G'B' and Y'CbCr Color Spaces
The following equations define R'G'B' to Y'CbCr conversion and Y'CbCr to R'G'B' conversion:
$\left[\begin{array}{c}{Y}^{\prime }\\ Cb\\ Cr\end{array}\right]=\left[\begin{array}{c}16\\ 128\\ 128\end{array}\right]+Α×\left[\begin{array}{c}{R}^{\prime }\\ {G}^{\prime }\\ {B}^{\prime }\end
$\left[\begin{array}{c}{R}^{\prime }\\ {G}^{\prime }\\ {B}^{\prime }\end{array}\right]=Β×\left(\left[\begin{array}{c}{Y}^{\prime }\\ Cb\\ Cr\end{array}\right]-\left[\begin{array}{c}16\\ 128\\ 128\end
The values in matrices A and B are based on your choices for the Use conversion specified by and Scanning standard parameters.
Matrix Use conversion specified by = Rec. 601 (SDTV) Use conversion specified by = Rec. 709 (HDTV)
Scanning standard = 1125/60/2:1 Scanning standard = 1250/50/2:1
$\left[\begin{array}{ccc}0.25678824& 0.50412941& $\left[\begin{array}{l}\text{0}\text{.18258588 0}\text{.61423059 0}\text{.06200706}\ $\left[\begin{array}{ccc}0.25678824& 0.50412941&
A 0.09790588\\ -0.1482229& -0.29099279& 0.43921569\\ \ \text{-0}\text{.10064373 -0}\text{.33857195 0}\text{.43921569}\\ \text{0}\text 0.09790588\\ -0.1482229& -0.29099279& 0.43921569\\
0.43921569& -0.36778831& -0.07142737\end{array}\ {.43921569 -0}\text{.39894216 -0}\text{.04027352}\end{array}\right]$ 0.43921569& -0.36778831& -0.07142737\end{array}\
right]$ right]$
$\left[\begin{array}{ccc}1.1643836& 0& 1.5960268\\ $\left[\begin{array}{ccc}\text{1}\text{.16438356}& \text{0}& \text{1}\text $\left[\begin{array}{ccc}1.1643836& 0& 1.5960268\\
B 1.1643836& -0.39176229& -0.81296765\\ 1.1643836& {.79274107}\\ \text{1}\text{.16438356}& \text{-0}\text{.21324861}& \text{-0}\text 1.1643836& -0.39176229& -0.81296765\\ 1.1643836&
2.0172321& 0\end{array}\right]$ {.53290933}\\ \text{1}\text{.16438356}& \text{2}\text{.11240179}& \text{0}\end 2.0172321& 0\end{array}\right]$
Conversion from R'G'B' to Intensity
The following equation defines conversion from the R'G'B' color space to intensity:
$\text{intensity}=\left[\begin{array}{ccc}0.299& 0.587& 0.114\end{array}\right]\left[\begin{array}{c}{R}^{\prime }\\ {G}^{\prime }\\ {B}^{\prime }\end{array}\right]$
Data Types
For fixed-point and integer input, the block converts matrix A to fixdt(1,17,16), and matrix B to fixdt(1,17,14).
For double or single input, the block applies the conversion matrices in double type, and scales the Y'CbCr offset vector ([16,128,128]) by 1/255. The block saturates double or single R'G'B' and
intensity outputs to the range [0,1].
The Y'CbCr standard includes headroom and footroom. For 8-bit data, luminance values in the range 16–235 and chrominance values in the range 16–240 are valid. The Color Space Converter block pins
out-of-range input to these limits before calculating the conversion. The block scales the offset vector and the allowed headroom and footroom depending on the word length of the input signals. For
example, when you convert a Y'CbCr input of type fixdt(0,10,0) to R'G'B', the block multiplies the offset vector by 2^(10 – 8) = 4. As a result, the valid luminance range becomes 64–940 and the valid
chrominance range becomes 64–960.
When you use this block with R'G'B' input, the block has a latency of 9 cycles. When you use this block with Y'CbCr input, the block has a latency of 10 cycles. The extra cycle is required to check
for and correct headroom and footroom violations.
When you use edge padding, use a horizontal blanking interval of at least twice the kernel width. This interval lets the block finish processing one line before it starts processing the next one,
including adding padding pixels before and after the active pixels in the line.
The horizontal blanking interval is equal to TotalPixelsPerLine – ActivePixelsPerLine or, equivalently, FrontPorch + BackPorch. Standard streaming video formats use a horizontal blanking interval of
about 25% of the frame width. This interval is much larger than the filters applied to each frame.
The horizontal blanking interval must also meet these minimums:
• If the kernel size is less than 4, and you use edge padding, the total porch must be at least 8 pixels.
• When you disable edge padding, the horizontal blanking interval must be at least 12 cycles and is independent of the kernel size.
• The BackPorch must be at least 6 pixels. This parameter is the number of inactive pixels before the first valid pixel in a frame.
For more information, see Configure Blanking Intervals.
This table shows the resource use after synthesis of the block for the Xilinx^® Zynq^®-7000 SoC ZC706 Evaluation Kit with single-pixel uint8 input and the default parameter settings. The design
achieves a clock frequency of 438 MHz.
Resource Usage
Slice LUTs 205
Slice Registers 300
DSP48 9
Block RAM 0
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.
This block supports C/C++ code generation for Simulink^® accelerator and rapid accelerator modes and for DPI component generation.
HDL Code Generation
Generate VHDL, Verilog and SystemVerilog code for FPGA and ASIC designs using HDL Coder™.
HDL Coder™ provides additional configuration options that affect HDL implementation and synthesized logic.
HDL Architecture
This block has one default HDL architecture.
HDL Block Properties
Number of registers to place at the outputs by moving existing delays within your design. Distributed pipelining does not redistribute these registers. The default is 0. For
ConstrainedOutputPipeline more details, see ConstrainedOutputPipeline (HDL Coder).
Number of input pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For
InputPipeline more details, see InputPipeline (HDL Coder).
OutputPipeline Number of output pipeline stages to insert in the generated code. Distributed pipelining and constrained output pipelining can move these registers. The default is 0. For
more details, see OutputPipeline (HDL Coder).
Version History
Introduced in R2015a
R2022a: Two pixels-per-clock streaming
The block now supports multipixel streams that have 2 pixels per clock cycle.
R2021b: Multipixel streaming
The Color Space Converter block now supports multipixel streams. The HDL implementation replicates the algorithm for each pixel in parallel.
The block supports input matrices of NumPixels-by-3 values, and output matrices of NumPixels-by-NumComponents values, where NumComponents is 3 or 1. The ctrl ports remain scalar, and the control
signals in the pixelcontrol bus apply to all pixels in the matrix. | {"url":"https://in.mathworks.com/help/visionhdl/ref/colorspaceconverter.html","timestamp":"2024-11-09T00:44:14Z","content_type":"text/html","content_length":"107480","record_id":"<urn:uuid:3cdf4977-86e3-4036-ad90-50cbf0b10f8f>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00050.warc.gz"} |
Mohan Lal Sukhadia University (MLSU) 2001 B.Sc Computer Science Computer Oriented Numerical Methods - Question Paper
Saturday, 26 January 2013 01:10Web
First Year exam of the 3 Year
Degree Course, 2001
(Faculty of Science)
Third Paper
(Computer Oriented Numerical Methods)
Time - 3 Hours
Maximum Marks
- 50
Attempt 5 ques. in all,
choosing 1 ques. from every unit.
How a floating point number is stored in the memory of a computer? explain with examples the procedures of 4 basic arithmetic operations using normalized floating point numbers. 10
What do you mean by roots of an equation? explain the successive bisection method of evaluating roots of a non-linear formula in 1 variabl
Develop the algorithm of the metho
UNIT 2
3.. explain the Gauss Seidel method for the solution of simultaneous equations. What is Pivoting? discuss its use in Gauss Seidel Metho
provide a comparison of direct and iterative methods. 2+4+4
explain the Gauss elimination method of solving simultaneous linear equations. Develop the algorithm for the metho
UNIT 3
5 . discuss the Euler's method of solving ordinary differential equations. Develop the algorithm of the metho
explain the fault in the Euler's metho
Using Runge-Kutta 4th order method obtain the solution of the subsequent ordinary differential formula at x=0.4 the intial values are y=1 at x=0. Use the steps of size 0.
UNIT 4
What is difference table ? Construct a difference table from the subsequent data and hence using the polynomial interpolation obtain the value of function f
at x= 2.0 :- 10
X f (x)
-1.0 3.0
0.0 5.0
1.0 1.0
3.0 -1.0
1.0 13.0
explain the method of approximating a function by using Chebyshev series. Use this
method to approximate the series expansion of sin
for 3 digits accuracy. 10
UNIT 5
9 . discuss the method of numerical differentiation. explain the fault in the differentiation formula
Using this method obtain the differential of function f
at x = 1.3 from the subsequent tabulated data :- 3+2+5
X f(x)
1.0 0.0
1.2 0.365
1.4 0.673
1.6 0.940
1.8 1.176
discuss the Simpson’s rule of numeric integration. What is the estimated fault in this method? Write the algorithm for the Simpson’s metho
Earning: Approval pending. | {"url":"http://www.howtoexam.com/index.php?option=com_university&task=show_paper&paper_id=3504&title=Mohan+Lal+Sukhadia+University+%28MLSU%29+2001+B.Sc+Computer+Science+Computer+Oriented+Numerical+Methods+-+Question+Paper&Itemid=58","timestamp":"2024-11-10T08:16:07Z","content_type":"application/xhtml+xml","content_length":"27182","record_id":"<urn:uuid:bcbfe532-6f99-441b-8246-4ec4b24a21a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00778.warc.gz"} |
How to Sum Top n Values in Excel (3 Suitable Ways) - ExcelDemy
Download the Practice Workbook
3 Methods to Sum Top N Values in Excel
Let’s sum up the top 5 sales from January in the sample dataset.
Method 1 – Combine the LARGE Function with SUMIF to Sum Top N Values in Excel
• Insert the following formula in cell F5 and press the Enter key.
Formula Breakdown:
It returns the 5th largest value from the cells C5 to C17.
Result: $25000.00
• SUMIF(C5:C17,”>=”&LARGE(C5:C17,5))
It returns the sum of the cells from C5 to C17 which contain values greater than or equal to the previous result.
Result: 150,000.00
Thus, you will get the sum of the top 5 sales from January, which is $150,000.00.
Read More: Excel Sum If a Cell Contains Criteria (5 Examples)
Method 2 – Use SUM Formulas to Sum Top N Values in Excel
Case 2.1 – Combine SUM, IF, and RANK Functions to Sum First N Numbers
• Insert the following formula in cell F5.
Formula Breakdown:
• IF(RANK(C5:C17,C5:C17)<=5,C5:C17,0)
It takes an array of criteria (RANK(C5:C17,C5:C17)<=5) in place of a single criterion, and it returns a TRUE for each cell between C5 to C17, if it has a value in the top 5 zones. Otherwise, it
returns a FALSE. When TRUE, it would return the corresponding cell value from C5 to C17, and when FALSE, it would return 0.
Result: {0,0,0,0,0,25000,28000,30000,0,0,0,35000,32000}
• SUM(IF(RANK(C5:C17,C5:C17)<=5,C5:C17,0))
It sums up the values of the resultant array.
Result: $150,000.00
• Press Ctrl + Shift + Enter as this is an array formula.
We will get the same result as earlier, $150,000.00
Read More: How to Add Multiple Cells in Excel (6 Methods)
Case 2.2 – Combine SUM with the LARGE Function
• Insert the following formula in cell F5.
• Press Ctrl + Shift + Enter.
Formula Breakdown:
• LARGE(C5:C17,{1,2,3,4,5})
It takes an array of values {1,2,3,4,5} in place of a single value k. And returns an array containing the 1st, 2nd, 3rd, 4th, and 5th largest values from the C5:C17 range.
Result: {35000,32000,30000,28000,25000}
• SUM(LARGE(C5:C17,{1,2,3,4,5}))
It sums up the previous resultant values.
Result: $150,000.00
Consequently, we will get the sum of the top 5 sales from January, which is $150,000.00.
Case 2.3 – Combine SUM with SEQUENCE
We’ll find the sum of the top 10 sales from January this time.
• Insert the formula below and hit the Enter key.
Formula Breakdown:
It returns an array of values from 1 to 10.
Result: {1,2,3,4,5,6,7,8,9,10}.
• LARGE(C5:C17,SEQUENCE(10,1))
Returns the top 10 large sales in the range of C5 to C17.
Result: {35000,32000,30000,28000,25000,24500,24000,23000,22000,21000}
• SUM(LARGE(C5:C17,SEQUENCE(10,1)))
Sums the previous resultant array.
Result: $264,500.00
We will get the sum of the top 10 sales of January, which is $264,500.00.
The SEQUENCE function is only available in Office 365.
Read More: How to Sum Range of Cells in Row Using Excel VBA (6 Easy Methods)
Similar Readings
Method 3 – Use SUMPRODUCT Formulas to Sum Top N Values in Excel
Case 3.1 – SUMPRODUCT with LARGE, ROW, and INDIRECT Functions
We will get the sum of top 10 sales from January.
• Click on cell F5.
• Insert the formula below and press the Enter key.
Formula Breakdown:
It returns an array of values from 1 to 10.
Result: {1,2,3,4,5,6,7,8,9,10}.
• LARGE(C5:C17,ROW(INDIRECT(“1:10”)))
It returns the top 10 large values in range C5 to C17.
Result: {35000,32000,30000,28000,25000,24500,24000,23000,22000,21000}
• SUMPRODUCT(LARGE(C5:C17,ROW(INDIRECT(“1:10”))))
It returns the sum of the top 10 large values.
Result: $264,500.00
Thus, we will get the same result for the top 10 sales of January as earlier, $264,500.00.
Read More: How to Add Rows in Excel with Formula (5 ways)
Case 3.2 – SUMPRODUCT with RANK Function
• Click on cell F5 and insert the following formula.
Formula Breakdown:
• –(RANK(C5:C17,C5:C17)<=10)
It returns an array of TRUE or FALSE. For each cell in the range C5 to C17 which falls under the top 10 it returns a TRUE, and FALSE for the rest.‘–‘ converts the TRUE and FALSE array into an array
of 1 and 0.
Result: {0,0,1,1,0,1,1,1,1,1,1,1,1}
• SUMPRODUCT(C5:C17,–(RANK(C5:C17,C5:C17)<=10))
It multiplies C5:C17 cell values to the previous resultant array. Therefore, it returns the sum of the top 10 sales.
Result: $264,500.00.
We will get the sum of the top 10 sales values from January.
Read More: Excel Sum Last 5 Values in Row (Formula + VBA Code)
How to Sum Top N Values in Excel with Criteria
We will consider only the top 10 sales below $30,000.00.
• Click on cell F5.
• Insert the formula below.
You will get the sum of the top 10 sales in January less than $30,000.00 is $167,500.00 as your desired result.
You can also use the more complicated formula below:
Read More: Sum Cells in Excel: Continuous, Random, With Criteria, etc.
How to Sum Top N Values in Excel with Texts Inside
In a few cells in the February column, there is a text “No Sales”. When we use the formulas from above, it will show errors.
• Click on cell F5 and insert the following formula.
• Press Ctrl + Shift + Enter.
The error because of text calculation will be ignored and you will get the top 10 sales from February would be $261,000.00.
You can use any other formula from section 2, just wrap the LARGE portion within an IFERROR function.
Read More: How to Sum Cells with Text and Numbers in Excel (2 Easy Ways)
Further Readings
Get FREE Advanced Excel Exercises with Solutions!
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Design of Threechamber Ball Mill IOPscience
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Mill ResearchGate
MODULE #5: FUNCTIONAL PERFOMANCE OF BALL
OBJECTIVES In this module, you will learn how to characterize the performance of ball mill circuits Specifically, after completing this module, you will be able to: List and describe2007年2月3日·
A ball mill is one kind of grinding machine, and it is a device in which media balls and solid materials (the materials to be ground) are placed in a container TheDesign Method of Ball Mill by
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First Online: 30 April 2023 15 Accesses Download reference work entry PDF Ball mill is a type of grinding equipment that uses the rotary cylinder to bring the grinding medium and2013年4月1日· 1
Introduction Over the years, ball mill circuits closed with cyclones have become an industry standard, and since the early days, it has been recognised thatClosed circuit ball mill – Basics
revisited ScienceDirect
Ball Mill an overview | ScienceDirect Topics
A feature of ball mills is their high specific energy consumption; a mill filled with balls, working idle, consumes approximately as much energy as at fullscale capacity, ie1997年12月1日· To
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2021年4月14日· You also need a rod mill work index to design a ball mill operating on a coarse feed, above about 4 mm Q1: You design for a typical percentage of critical speed, usually 75% of
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Ball Mill Principle, Construction, Working, and More Soln
2021年11月25日· Principle of Ball Mill: The size reduction in the ball mill is a result of fragmentation mechanisms (impact and attrition) as the balls drop from near the top of the shell Mixing
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Ball Mill [PDF Document]
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Contribution to the wastewater by use of a numerical study of a secondary clarifier comsol multiphysics under
Research Article Volume 4 Issue 2
Contribution to the wastewater by use of a numerical study of a secondary clarifier comsol multiphysics under
Fares Redouane,^1 ^ ^ Aissa Abdelrahmane,^2 Bouadi Abed,^3 Lounis Mourad^1
^1Laar, University of Science and Technology, Algeria
^2(LPQ3M), University Mustapha Stambouli Mascara, Algeria
^3Universitére Ahmed Zabana RELIZANE Center, Algeria
Correspondence: Dr. Fares Redouane, Ahmed ZABANA University Center of Relizane, Laar, University of Science and Technology, Mohamed Boudiaf, BP 1505, Oran, Algeria, Tel 00213771728504
Received: December 10, 2018 | Published: March 13, 2019
Citation: Redouane F, Abdelrahmane A, Abed B, et al. Contribution to the wastewater by use of a numerical study of a secondary clarifier comsol multiphysics under. MOJ Eco Environ Sci. 2019;4
(2):44-47. DOI: 10.15406/mojes.2019.04.00131
The environmental assessment of industrial effluents (containing organic pollutants and minerals without proper treatment), generated by the mineral fertilizer factory and emptied into the seacoast,
revealed the concentrations intense of suspended solids (SS).The wastewater treatment is a multi-step process to remove contaminants. At first, large solid particles were removed by sedimentation,
flotation and filtration. In a second step, the biological treatment is aggregated by forming the smaller particles, which is called flocs. These flocs can be easily removed by sedimentation. The
present investigation studying the separation of water flocs in a circular secondary clarifier. A mixture model is used to model the multiphase turbulent flow in the tank.
Keywords: comsol, simulation, pollution, clarifier
Our modern life imposes increasingly important requirements on what to eat, dress and how to manage time and space by finding daily means and methods that are increasingly sophisticated to meet the
maximum comfort.^1,2 Any economic activity caused by the human needs in the entire planet, generates more technological development and human living conditions, pollution problems accompany most of
the industrial and agricultural processes including environmental and ecological consequences are very serious and can in some situations simply turn against the objectives initially set by these
companies.^3,4 The degradation of the Earth's environment caused by the intense industrial activities, especially in some developed countries thus enter into the concerns of the nations. Algeria is
also affected by pollution in general, and its coast is relatively polluted as it was reported in several research^5,6 than untreated industrial effluents discharged directly into the Mediterranean
sea are the primary causes.^7 They also showed a growing pollution in the entire Algerian coast, especially in the coastal areas of large cities. Primary causes.^7,8 They also showed that a growing
pollution in the entire Algerian coast, especially in the coastal areas of large cities.
We have tried our best to provide solutions and respond to pollution issues raised during the experimental measurements of physico-chemical parameters features exploiting the numerical simulation
tool.^5 These answers are powerful and efficient, and enable the Arzew Fertial-run business in clean environmental standards. To contributed to reducing pollution after Fertial-Arzew complex,
proposed a mathematical model of turbulence kε. To follow the mixing speed, pressure and the phase volume fractions, we propose the application of the mixture model. The latter is a multi phasic flow
pattern, particularly suitable for suspensions or mixtures of solid or liquid particles.
Model definition
By analyzing all released liquids of Fertial-Arzew company directly at its channels evacuations,^9 it appears to be the the overruns legislative standards. For reasons of efficiency, it is attempted
to provide a solution for the parameter measuring solids MY suspension, whose values measured at the three liquid effluent discharge channels exceed a very exaggerated manner standards.
The values from the MES parameter of treated wastewater were significantly lower than those discharged directly into the sea. The wastewater treatment is a process that involves several steps to
remove contaminants. The most important of these steps is one in which large solid particles are removed by sedimentation, flotation and filtration obtaining flocs or sludge on the one hand and
lightly loaded waste water solids suspension on the other hand. The present investigation studying the separation of water in a circular flocs settling tank used in sewage treatment systems. Figure 1
shows the representative diagram in longitudinal section of the structure of a circular decanter.
Liquid wastes from Fertial-Arzew company arrive via a channel flowing at the entrance of the clarifier at its center. The treated water leaves at its ends leaving the clarifier decant the bottom
solids suspensions. In recent exit through the bottom of clarifier under the effect of gravity to evacuate to dedicated sites or also for a suitable adjustment. To accentuate the wall settling
phenomenon ("S") is provided in the center of the clarifier to prevent the waste water jet arriving through inlet re-mix sludge already settled to the bottom of clarifier.
The clarifier in question has an axial symmetry which allows us to reduce the number calculations operations by merely half the system in 2D, as shown in Figure 2. The first figure shows the
axisymmetric diagram of half of the clarifier 2D and the second mesh in its triangular cells.
Our model is simulated on a circular secondary clarifier,^10 the flocs are removed from water by sedimentation and this through gravity. The purpose of this study is to investigate the complex
turbulent poly phasic flow in a circular secondary clarifier.
FIG 1montre geometry of the clarifier
The sludge used, consisting of a mixture of solid flakes and water enters through the inlet of the reservoir. The fuel tank has two outputs. One is located in the center of the tank bottom. The
purpose of the outlet port is to remove the flocs sedimented from the reservoir. There is also a peripheral outlet purified water as shown in Figure 2, which also shows the corresponding model in 2D
axisymmetric. We modeled the floc by solid circular of equal size. The model consists of a set of equations, boundary conditions and initial conditions. The set is defined on a geometry decomposed
into sub-areas and limited by borders. This model is simulated with COMSOL Multiphysics, or we first made the physical description of the problems (variables, equations, initial limits and
conditions), followed by the implementation of our model, and finally made discussions on the results.
Mathematical model
Based on the conservation of mass and momentum of each phase, the mixture model uses the following equations:
$\rho {u}_{t}+\rho \left(u.abla \right)u=-abla p-abla .\left(\rho {c}_{d}\left(1-{c}_{d}\right){u}_{slip}^{2}\right)+abla .{\tau }_{Gm}+\rho g$ (1)
$\left({\rho }_{c}-{\rho }_{d}\right)\left[abla .\left({\varphi }_{d}\left(1-{c}_{d}\right){u}_{slip}-{D}_{md}abla {\varphi }_{d}\right)\right]+{\rho }_{c}\left(abla .u\right)=0$ (2)
$\frac{\partial }{\partial t}\left({\varphi }_{d}{\rho }_{d}\right)+abla .{\varphi }_{d}{\rho }_{d}{u}_{d}=0$ (3)
Where, u represents the velocity of the mixture (m/s), ρ is the mixture density (kg/m^3), p is the pressure (Pa), and Cd is the mass fraction of the solid phase (kg/kg). In addition, uslip is the
relative velocity between the two phases (m/s), is the sum of the viscous and turbulent stress (kg/(m·s^2)), and g is the gravity (m/s^2). The mixing speed (m/s) is defined as ${\tau }_{Gm}$
$u=\frac{{\varphi }_{c}{\rho }_{c}{u}_{c}+{\varphi }_{d}{\rho }_{d}{u}_{d}}{\rho }$ (4)
Ouet represent respectively the volume fractions of the liquid phase (continuous phase) and the solid phase (discontinuous phase) (m^3/m^3), the velocity of the liquid phase (m/s) the speed of the
solid phase (m/s), the density ρc liquid phase (kg/m^3), the density ρd solid phase (kg/m^3), ρ is the density and mixture of (kg/m^3). The relationship between the two phases of speeds is defined by
${\varphi }_{c}\text{ }{\varphi }_{d}{u}_{c}{u}_{d}$
${u}_{cd}={u}_{d}-{u}_{c}={u}_{slip}-\frac{{D}_{md}}{\left(1-{c}_{d}\right){\varphi }_{d}}abla {\varphi }_{d}$ (5)
For the solid particles sliding speed, we can use the drag act Hadamard-Rybczynski. In the mode of application of the mixture model of the Hadamard-Rybczynski slip speed is provided as a predefined
template. The mode of application then calculates the slip speed as a result:
${u}_{slip}=-\frac{\left(\rho -{\rho }_{d}\right){d}_{d}^{2}}{18\rho \eta }abla p$ (6)
Where denotes the diameter of the solid particles (m). To the mixture density and viscosity, we offer the following expressions d[d]
$\eta ={\eta }_{c}{\left(1-\frac{{\varphi }_{d}}{{\varphi }_{max}}\right)}^{-2.5{\varphi }_{max}}$ (7)
$\rho ={\varphi }_{c}{\rho }_{c}+{\varphi }_{d}{\rho }_{d}$ (8)
The turbulence model k-ε is used to determine the eddy viscosity. The mode of application calculates the dispersion of particles of Dmd coefficient (m^2/s) from
${D}_{md}=\frac{{\eta }_{T}}{\rho {\sigma }_{T}}$ (9)
Where ηT is the eddy viscosity (Pa.s) and σT is the (dimensionless) turbulent Schmidt number,
The mass flow of the solid phase through the inlet is given by
${M}_{in}=\underset{inlet}{\overset{}{\int }}|{v}_{in}|{\varphi }_{din}{\rho }_{d}dS$ (10)
Correspondingly to the flow through the peripheral outlet
${M}_{out}=\underset{outlet}{\overset{}{\int }}|{v}_{out}|{\varphi }_{dout}{\rho }_{d}dS$ (11)
The difference between the latter two entities provides the remaining mass at the bottom clarifier. Note that the mode of application of the model mix automatically configures all the modeling
Convergence of the solution
The numerical method is to interpolate the solution on a discretization of the geometry (mesh). A digital solution is acceptable when the convergence criterion is established, that is to say when the
numerical result is very close to the exact solution. This is precisely known, it is only available in some cases for simple geometries. Therefore an error estimate is constructed from a Taylor
expansion of the second order differential of the equation system operator. Convergence is reached when the estimated error value is less than a threshold value.
The error for the non-linear case is where N is the number of nodes, E is the estimation error, and W is the weight of each node (equal to 1 by default). This value must be less than a factor defined
by the user (the default value is 10-6).
$err=\sqrt{\left(\frac{1}{N}\sum _{i}{\left(\frac{|{E}_{i}|}{{W}_{i}}\right)}^{2}\right)}$
For a time calculation, it is necessary that the solution satisfies the criterion E at each time step, where Ai is the absolute tolerance (default 10-3), and Ui is the relative tolerance (default
$rr=\sqrt{\left(\frac{1}{N}\sum _{i}{\left(\frac{|{E}_{i}|}{{A}_{i}+R|{U}_{i}|}\right)}^{2}\right)}<1$
Convergence reflects the adequacy of the numerical solution with the approximate solution of the model. The quality (digital) of the obtained solution and its stability depend on the mesh refinement
and time step to satisfy the CFL condition (Courant - Friedrichs -Lewy) presented in the following equation:
$C=\frac{\Delta t}{\Delta L}max\left(|{\text{v}}_{m}|\right)\le 1$
Where, $\Delta t$ is the time interval $\Delta L$ is the size range, and vm is the slow component of the velocity.
Conditions to the limits
Once the geometry of our clarifier system is defined and meshed, it fixed the boundary conditions of the system. After setting the initial mass concentration of the sewage sludge to 0003Kg/m^3, is
fed the input speed of the waste water to 1.25m/s and pressure is fixed at the outlet of treated water at atmospheric pressure. Then, we choose the most representative type of flow of what is
happening inside the circular clarifier and was chosen in this case to turbulent flow k type ε multiphasic. Comsol Multiphysics after execution of the computer code on our system that consists of
solving the set of equations governing the digital decanter and after verification of convergence curves that provide information on the reliability of the results obtained solutions were recovered
in different forms. To demonstrate the appearance phase separation in the decanter under the effect of gravity, the mass concentrations showing the waste water in Kg/m^3 dans the assumed two-phase
basin as shown in Figure 3 during the settling period, which lasts for about 24 hours. Can be obtained from solutions of other information on other physical parameters such as speed lines, the
current flow lines.
Results and discussion
Figure 3 shows stream lines of the mixing rate and the solid phase concentration, measured in kg/m^3, after 24 hours. At this point, the flow has reached a stable state. We can interpret the speed of
the average mixing time. As expected, the solid particles fall down. However, the turbulent fluctuations results in the diffusion of the solid phase have a negative effect on the separation (Figure 4
Figure 4 Mass flow of the dispersed phase at the inlet (blue), output device (green) and central outlet (red).
The clarifier and removes solid particles Min rates-Mut. Calculation of the rate of withdrawal from the results shows that the clarifier removes 0518-.151= .367kg of solid particles per second.
The results of numerical simulations clearly show the effectiveness of the clarifier, which forces the driven solid particles of gravity to get to the pond bottom, leaving the surface relatively
light-weight solid effluent water. Also, it confirms what was reported previously on the importance of the wall introduced into the clarifier, where the impact is observed on device performance by
preventing the clearest water from coming into contact with the sludge piled deep. Depending on requirements, we can get more details on a particular region or regions of the decanter to improve and
optimize there after the digital system.
Marine pollution by wastewater is a serious problem in the Mediterranean Sea. The country's defining reject most of their liquid effluents that this is from the sanitation water towns or discharges
from industry without any treatment and Algeria is one. Failure prediction and lack of stations and sewage systems waste water just before the discharge of liquid waste into the sea, causing the
deterioration of the overall health and air quality of coastal waters. This causes significant disruption of ecosystems.
Clearances of Fertial-Arzew society as industrial waste water discharged directly into the sea. Although it has the ISO 14001 certificate, the release of this company pollution results of
measurements of parameters indicative reveal that study sites are more or less polluted by wastewater discharges. However, prolonged contamination of sea water with these high values of MES up to
260mg/L degrade the marine ecosystem and harm wildlife and existing marine flora.
Measurements of MY show the need for pretreatment of raw wastewater before sending them into the sea and improve their quality according to the required standards as well as to protect the
environment and human health. To remedy this, we proposed a wastewater treatment system called "clarifier" or "settler", which allows a considerable reduction in the values of MES. The values
found by numerical simulation under the calculation code Comsol Multiphysics, clearly showed the improvement of the quality of treated water from those who were not treated at all. The clarifier
separates the suspended impurities of clean water. The latter is sent to the sea via the output device and recovered at the bottom outlet of the purification system sediments as sludge. Thus, the
sludges are prevented from going contaminate the sea water and can be upgraded later. Our well designed system reduced the amount of suspended solids up to 60%. The fact that contaminated water were
retained in the clarifier tank for 24 hours, before sending them into the sea for settling.
Conflicts of interest
Authors declare that there is no conflict of interest.
1. Teh CY, Wu TY, Juan JC. Potential use of rice starch in coagulation-flocculation process of agro-industrial wastewater: Treatment performance and floc characterization. Ecological Engineering.
2014 ;71:509–519.
2. Mompoint M. Evaluation of environmental hazards generated by urban liquid effluents on the ecosystem of the bay of Port-au-Prince: First methodological approach. Quisqueya University-Civil
Engineer ; 2004.
3. Renuka N, Sood A, Prasanna R, et al. Phycoremediation of wastewaters: a synergistic approach using microalgae for bioremediation and biomass generation. International Journal of Environmental
Science and Technology. 2015;12(4):1443–1460.
4. Espinosa F, Garcia-Garcia JM, García-Gómez JC. Sewage pollution and exinction risk: an endangered limpet as bioindicator? Biodiversity and Conservation. 2007;16(2) :377–397.
5. Fares R, Lounis M. Determination of the quality of sea waters Arzew, Algeria Gulf. Phytochem & BioSub Journal. 2017;11(2):110–117.
6. Remili Kerfouf SA. Environmental monitoring and coastal sustainable development (case of Oran coast). Proceedings of the 5th International Symposium on the theme "Energy, Climate Change and
Sustainable Development", Tunisia : Hammamet ; 2009.
7. Taghezout Fatima. Environmental impact on the water releases the long littoral Western Algerian, Magister in: environmental sciences, faculty of natural sciences and life biology department
laboratory networking environmental monitoring (ERA will) USTO-MB. 2015.
8. Ridha Hachicha. Determination of priority measures for pollution of the coastal area south of Grand Sfax except SIAPE, Scenarios & remediation Environmental Regulations. Tunisia SMAP III project
(2006-2008), Strategy Management Integrated Coastal Zone South Grand Sfax; 2008.
9. Fares R, Louns M. Pollution characterization of liquid wastes of the factory complex Fertial (Arzew, Algeria). J Air Wast Manage Assoc. 2016;66(3):260–266.
©2019 Redouane, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work non-commercially. | {"url":"http://medcraveonline.com/MOJES/contribution-to-the-wastewater-by-use-of-a-numerical-study-of-a-secondary-clarifier-comsol-multiphysics-under.html","timestamp":"2024-11-04T16:39:30Z","content_type":"application/xhtml+xml","content_length":"157594","record_id":"<urn:uuid:bed6a941-c04a-4cc3-a51e-53f91e64cbe0>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00689.warc.gz"} |
Robert A. M. Gregson (1993) Which Bayesian Theorem Could be Compared With Real Behaviour?. Psycoloquy: 4(50) Base Rate (2)
Volume: 4 (next, prev) Issue: 50 (next, prev) Article: 2 (next prev first) Alternate versions: ASCII Summary PSYCOLOQUY (ISSN 1055-0143) is sponsored by the American Psychological Association (APA).
Commentary on Koehler on Base-Rate
Robert A. M. Gregson
Department of Psychology,
Australian National University,
Canberra, A C T 0200 Australia
The identifiability of strategies used by subjects in assessing inverse probabilities is low, and not adequately supported by using the simplest form of Bayes' Theorem. If subjects are using more
complex strategies, coherently or incoherently, then we cannot readily deduce how they use base rate information, or if they are substituting other information for it.
Base rate fallacy, Bayes' theorem, decision making, ecological validity, ethics, fallacy, judgment, probability.
1. Koehler's (1993) target article on the base rate fallacy seems to be a storm in a teacup. I agree that psychologists have written some strange things about inverse probability, and Koehler's
review is exhaustive and definitive. I get the impression that three distinct questions have become confounded, however, and ought to be disentangled.
2. In the legal context, there are questions, at least in Australian courts, as to whether the use of arguments about the worth of circumstantial evidence are valid uses of inverse probability. In a
celebrated case where on appeal a convicted murdered was acquitted, it was suggested by a statistician called as an expert witness that the prosecution had invalidly argued where the four
probabilities p(H), p(E|H), p(not-H) and p(E|not-H) have all to be considered. The case involved forensic evidence, on physical measures; there was no issue of subjective probabilities. It is still
germane to point out that legal methods of inference and statistical methods of inference are not necessarily compatible.
3. Setting the law aside, the data summarised by Koehler indicate that if we compare human judgments, which are elicited to obtain inverse probabilities, with Bayesian inference using the simplest
form of Bayes' Theorem as he gives it, then the judgments are suboptimal in the long run. The question is, should the Theorem have been written like that, and if we expanded it into some alternative
forms, would it then make better sense to use it as a normative baseline against which to assess what quantitative information, if any, humans can efficiently use?
4. There is another point, however; statistical theory does not stand still, and modern developments extend Bayesian analyses and need consideration. It is now possible (Walley, 1991) to use rigorous
and extensive theory to cover the case where the observer has no precise estimates of either prior or conditional probabilities, but can at most trap them within upper and lower bounds. For an
experimental psychologist this statistical approach is something like the psychophysical method of limits, where a hypothetical point of subjective equality is approached in turn from below and
above. Admittedly these new developments postdate the studies Koehler cites, which peaked in the 1970s, but in looking at the problem now we should get our statistical theory updated.
II. WHAT EXACTLY SHOULD WE WRITE?
5. I get two points from Koehler: that subjects can and do bring additional information to bear which was not in the experimental protocol, and that they weight probabilities in different ways, which
can render the probability calculus, which is implicit in their behaviour, technically incoherent. Their personal probabilities would not add up to one.
6. Let me distinguish two extended forms of Bayes' Theorem. Using E1&E2 to mean the conjunction of two events from the set {E1,E2,...,Ek} of k mutually distinguishable alternatives, and using c1,
c2,.. to represent scalar multipliers, the numerator of the Bayes expression, which is, for mutually exclusive and exhaustive hypotheses, H1 and H2, originally:
(a) p(E|H1)&p(H1)
now becomes either:
(b) p(E1&E2....|H1)&p(H1)
if subjects add in an indeterminate amount of extra data which the experimenter did not introduce, or:
(c) p(E1|H1)&c1&p(H1)
if subjects skew the estimate of the base rate. Obviously, if subjects do both these things -- and float from trial to trial in how they do it, fitting data to the simplest sort of Bayesian
expression, where terms are defined exclusively as those given by the experimenter -- this will lead to the suggestion that the base rates are not being used, or are not being used appropriately.
However, if c2&p(H2) = 1 - c1&p(H1), then the subject is still using a sort of Bayesian behaviour within each trial. (Note that ci operates on p(Hi)).
7. In addition, rewrite (b) as:
(d) p(E1|H1&E1|[E2&H1])&p(H1|E2)
and we heighten the role that the extraneous data can play. What is being formalised here is that the hypothesis H1 is only tenable to the subject if the additional information E2 is incorporated in
the scenario. If the alternative hypothesis H2 is only tenable if E3, say, also comes into the picture, then we can be on the way to lexicographic judgments (Gregson, 1963). Given these alternatives,
the question is, can a subject who is using (b) quite consistently (but unknown to the experimenter) be misidentified as one who gets base rates wrong? Alternatively, is it possible to leave base
rates out altogether and make inferences based solely on expressions such as:
(e) p(E1|H1&E1|[E2&H1]&E1|[E3&H1]...)
instead of (a)? I think one still needs a non-null p(H1) in the expression.
8. If we want an ecologically valid use of theories of inverse probability and probabilistic inference, in order to derive statements about the degree to which human subjects fall short of a
normative optimum process, then we must define the normative strategy in much more detail than is given in the simplest idealised Bayes form for two hypotheses. Without this, we cannot identify
precisely what subjects are doing. What I do not know is whether it is possible to design experiments which from the position of an outside observer can separate (b), (c), and (d).
Gregson, R. A. M. (1963) Some possible forms of lexicographic evaluation. Psychometrika, 28, 173-183.
Koehler, Jonathan J. (1993) The Base Rate Fallacy Myth. PSYCOLOQUY 4(49) base-rate.1
Walley, P. (1991) Statistical Reasoning with Imprecise Probabilities. London: Chapman and Hall.
Volume: 4 (next, prev) Issue: 50 (next, prev) Article: 2 (next prev first) Alternate versions: ASCII Summary | {"url":"https://www.cogsci.ecs.soton.ac.uk/cgi/psyc/newpsy?4.50","timestamp":"2024-11-09T11:00:49Z","content_type":"text/html","content_length":"20509","record_id":"<urn:uuid:983640f9-854a-4da3-bc08-bc098020dbd3>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00776.warc.gz"} |
Is (6,30) a solution to x=6?
Solved on Dec 06, 2023
Determine if the point $(6,30)$ satisfies the equation $x=6$.
STEP 1
1. We are given a point $(6,30)$.
2. We are asked to verify if this point is a solution to the equation $x=6$.
STEP 2
To determine if the point $(6,30)$ is a solution to the equation $x=6$, we need to check if the x-coordinate of the point satisfies the equation.
STEP 3
The x-coordinate of the point $(6,30)$ is 6.
STEP 4
Substitute the x-coordinate into the equation to see if it makes the equation true.
$x = 6$
STEP 5
Plug in the value of the x-coordinate from the point into the equation.
$6 = 6$
STEP 6
Check if the equation holds true with the substituted value.
STEP 7
Since $6 = 6$ is a true statement, the point $(6,30)$ satisfies the equation $x=6$.
STEP 8
Conclude that $(6,30)$ is indeed a solution to the equation $x=6$.
The point $(6,30)$ is a solution to the equation $x=6$. | {"url":"https://studdy.ai/learning-bank/problem/is-the-point-6-30-a-solution-to--XoPyba3XU67dzW5","timestamp":"2024-11-05T15:14:13Z","content_type":"text/html","content_length":"133072","record_id":"<urn:uuid:0bfafb13-477f-4a04-a715-2e73457100b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00230.warc.gz"} |
letter-combinations-of-a-phone-number | Leetcode
Similar Problems
Similar Problems not available
Letter Combinations Of A Phone Number - Leetcode Solution
LeetCode: Letter Combinations Of A Phone Number Leetcode Solution
Difficulty: Medium
Topics: string hash-table backtracking
Problem Statement:
Given a string containing digits from 2-9 inclusive, return all possible letter combinations that the number could represent.
A mapping of digit to letters (just like on the telephone buttons) is given below. Note that 1 does not map to any letters.
Example 1:
Input: digits = "23" Output: ["ad","ae","af","bd","be","bf","cd","ce","cf"]
Example 2:
Input: digits = "" Output: []
Example 3:
Input: digits = "2" Output: ["a","b","c"]
• 0 <= digits.length <= 4
• digits[i] is a digit in the range ['2', '9'].
This problem can be solved by using the Depth First Search approach. We need to make a recursive call on all possible combinations, using the mapping of digits to characters given by the phone
buttons. We will use a hash-map to store the mapping of digits to corresponding characters.
For example,:
• If our input is 2, then the mapping is "abc", because the digit 2 maps to a, b and c on the phone button.
• If our input is 3, then the mapping is "def", because the digit 3 maps to d, e and f on the phone button.
We will start by storing all the possible mappings in the hash-map, and then make a recursive call on the first digit's characters. For each character in this list, we will append it to the result,
and then go on to make a recursive call on the next digit's characters. We will continue this process until we reach the end of the input.
Here's the algorithm in steps:
1. Create a hash-map to store mappings
2. Define a helper function that takes in two parameters:
□ A string 'output' to store all possible combinations
□ An integer 'index' representing the current index to process in digits
3. If index == length of digits, append the output to the result and return
4. Find all characters corresponding to 'digits[index]' in the hash-map
5. Loop through the characters and recursively call the helper function with:
□ 'output' appended with the current character being processed
□ 'index+1' as the new index to process in digits
6. Return the result
phoneNumberMap = { '2': 'abc', '3': 'def', '4': 'ghi', '5': 'jkl',
'6': 'mno', '7': 'pqrs', '8': 'tuv', '9': 'wxyz' }
def letterCombinations(digits):
if len(digits) == 0:
return []
result = []
helper('', 0, digits, result)
return result
def helper(output, index, digits, result):
if index == len(digits):
for char in phoneNumberMap[digits[index]]:
helper(output+char, index+1, digits, result)
Time Complexity:
The time complexity of this algorithm would be O(4^n), where n is the number of digits in the input string. This is because the maximum number of combinations possible with n digits is 4^n (each
digit having a maximum of 4 possible characters).
Space Complexity:
The space complexity is O(n), where n is the number of digits in the input string. This is because the maximum depth of the recursion tree would be n, as we are processing one digit at a time.
Letter Combinations Of A Phone Number Solution Code | {"url":"https://prepfortech.io/leetcode-solutions/letter-combinations-of-a-phone-number","timestamp":"2024-11-08T08:18:52Z","content_type":"text/html","content_length":"63003","record_id":"<urn:uuid:959a4e7b-7150-4e2b-94e1-d7adcb2473af>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00732.warc.gz"} |
1. A centrifuge in a medical laboratory rotates at a rotational
speed of 3600 rev/min. When...
1. A centrifuge in a medical laboratory rotates at a rotational speed of 3600 rev/min. When...
1. A centrifuge in a medical laboratory rotates at a rotational speed of 3600 rev/min. When turned off, it rotates 20.0 times at a constant angular acceleration before coming to rest. The angle
through which the centrifuge rotates before coming to rest is
A. 3.18 rad
B. 20.0 rad
C. 80.0 rad
D. 126 rad
E. None of the above.
2. A very light particle traveling 23 m/s elastically collides (in one dimension) with a massive target that was initially at rest. After the collision, what is the velocity of the projectile?
A. is -23 m/s
B. is 0 m/s
C. is -46 m/s
D. cannot be determined (masses not known)
E. None of the above. | {"url":"https://justaaa.com/physics/42631-1-a-centrifuge-in-a-medical-laboratory-rotates-at","timestamp":"2024-11-12T19:52:17Z","content_type":"text/html","content_length":"33851","record_id":"<urn:uuid:89593138-630a-49cb-b4a1-648f83b0111c>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00201.warc.gz"} |
The Stacks project
Lemma 58.16.1. Consider a commutative diagram
\[ \xymatrix{ Y \ar[d]_ g \ar[r] & X \ar[d]^ f \\ T \ar[r] & S } \]
of schemes where $f$ and $g$ are proper with geometrically connected fibres. Let $t' \leadsto t$ be a specialization of points in $T$ and consider a specialization map $sp : \pi _1(Y_{\overline{t}'})
\to \pi _1(Y_{\overline{t}})$ as above. Then there is a commutative diagram
\[ \xymatrix{ \pi _1(Y_{\overline{t}'}) \ar[r]_{sp} \ar[d] & \pi _1(Y_{\overline{t}}) \ar[d] \\ \pi _1(X_{\overline{s}'}) \ar[r]^{sp} & \pi _1(X_{\overline{s}}) } \]
of specialization maps where $\overline{s}$ and $\overline{s}'$ are the images of $\overline{t}$ and $\overline{t}'$.
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Bitcoin is Nothing Either Good or Bad, but Sizing Makes It So
Advisor Perspectives welcomes guest contributions. The views presented here do not necessarily represent those of Advisor Perspectives.
“There is nothing either good or bad, but thinking makes it so.”
– William Shakespeare, Hamlet Act II (1602).
If you had put $100,000 of your savings into Bitcoin at the end of 2020 and are still HODLring (crypto-nese for “holding on for dear life”) those coins today, you’d be down 20% on your investment
given its drop in price from $25,000 to $20,000 (as of July 5th).
If going long was no good, how about going short? An investor who put $100,000 into a crypto-brokerage account to short $100,000 of Bitcoin, with no further trading, would have been wiped out just a
few months later in mid-February 2021 when Bitcoin more than doubled. This kind of shorting is effectively a form of automatic doubling down, since as the value of the asset goes up, the value of the
capital supporting it goes down, causing the short position to get bigger and bigger (quickly) relative to capital.^2 So, while “buy and hold” is a real strategy, “sell and hold” isn’t generally
workable in practice. Instead, the short position that most closely represents the opposite of the buy-and-hold long position can be thought of as funding an account with that same $100,000 and then
establishing and managing the short position so that at all times the size of the short is equal to the account’s liquidation value.
How would this managed short have done since the end of 2020? While avoiding total wipeout, an investor shorting Bitcoin in this way would still have lost 50%. That’s right…over the past 18 months,
you’d be down 20% from a 100% Bitcoin long position, and down 50% from the opposite managed short.^3 Naturally, this seems strange and unfortunate! To help understand why it’s the case, we’re going
to see if there’s some other constant proportion of capital – other than 100% long or 100% short – that the investor could have maintained, with regular rebalancing, which would have turned a profit.
We could simply search over all possible constant proportions to see which gave the best result over this period, but before we do that, let’s reason from first principles to figure out what we
should expect to find. We tend to think about the quality of an investment by calculating its historical Sharpe Ratio, which takes the realized return of an asset in excess of the risk-free rate and
divides it by the asset’s risk. Over this period, Bitcoin’s realized Sharpe Ratio, calculated from average daily returns, was +0.2. (for a deeper dive into how Bitcoin could have had a negative
return of 20% but a positive Sharpe Ratio, see this footnote ➚)^4 This suggests that Bitcoin was a fair, but not great, investment over this period – by comparison, broad equity markets have expected
Sharpe Ratios of around 0.3. The optimal allocation to not-so-great, but very risky investments should be fairly small, so if there’s a capital proportion that’s profitable at all, we’d expect it to
be fairly small too.
And indeed, this is exactly what we find, as indicated in the chart below.
The maximum profit of $3,000 was realized for an investor who kept a constant 25% of capital in Bitcoin over the whole period.^5 In fact, any positive proportional investment up to about 50% would
have resulted in a profit.^6 However, you don’t make twice the profit from taking twice the exposure; in fact, at 50% exposure, the profit is zero, rather than two times $3,000. Note that this is not
a buy-and-hold strategy; to maintain a constant proportion of capital in an asset, regular rebalancing is required.^7
We can also see that there are not any constant proportions generating a profit from both the long and short sides. You can lose money from a wide range of symmetric longs and shorts, but the only
way to make money was to have constant long exposure in the range 0% to 50%. Put another way, you have to both correctly judge the quality of the investment, and then size your investment
consistently with that quality, and getting either materially wrong will generally result in losses.
There are a few things we can take away from this mini case study:
1. These ideas are not specific to Bitcoin. We chose Bitcoin to illustrate them because they are easiest to see for highly volatile assets, and there are few assets that have been more volatile.
2. We can’t say that an investment is good or bad without considering how we will manage its sizing over time: sizing is as important as evaluating an investment’s expected risk and return.
3. While there are an infinite set of investment strategies involving a given asset,^8 we can learn a lot from focusing on the simplest strategy: Constant Proportion investing.
4. Among Constant Proportion investment strategies, there will be a range of investment sizes that will be profitable, with sizing above and below that rapidly becoming increasingly unprofitable.
And the range of profitable sizes is strongly related to the quality of the investment.
5. For a given investment, the realistic strategies which turn a profit are typically quite a small subset of the infinite number of total strategies to choose from.
Victor Haghani is founder & CIO and James White is CEO of Elm Wealth, a Philadelphia-based asset manager.
Learn more at www.elmwealth.com.
^1 This not is not an offer or solicitation to invest, nor should this be construed in any way as tax advice. Past returns are not indicative of future performance.
^2 For example, if Bitcoin goes up 50%, the size of the position relative to the amount of capital supporting it would have moved from 1:1 to 3:1. Such an approach requires an unlimited amount of
capital, which is an amount few real-world investors have access to. A leveraged long position requires the same sort of position management, since as the asset goes down, capital goes down faster,
and the investor will need to sell some of the asset to keep the leverage ratio constant.
^3 You can read more about why this is the case in George Costanza At It Again: The Leveraged ETF Episode (2020). Throughout, we assume daily rebalancing, no transactions costs, no borrow fees and a
zero risk-free rate. All data from Yahoo Finance.
^4 The reason the Sharpe Ratio was positive even though the Compound return over the period was negative is because Sharpe Ratio is computed using the Arithmetic average daily return of Bitcoin over
the period, which for a risky asset will always be higher than the Compound, or Geometric average, return. Under some stylized assumptions, the difference between the Compound Return and the
Arithmetic Return of a risky asset is equal to one-half of the variance of returns, a quantity known as “variance drag.” For Bitcoin over this period, the variance drag was about 30% per annum (30% =
0.5 * 0.772)! The Arithmetic Return was not sufficiently positive to offset the 30% variance drag, and so the Geometric Return was negative, but it’s the Arithmetic Return that ultimately determines
the quality of a regularly rebalanced investment opportunity. Another perspective is to say that Bitcoin’s realized Sharpe Ratio would be positive as long as it’s Compound Return was better than -30%
pa, given its realized volatility. For a simple example of variance drag, consider an asset that goes up and down 10% each day for 10 days. The average daily return is 0, because it went up 10% an
equal number of times as it went down 10% – but the Compound return will be negative because every time it goes up 10% and then down 10%, the asset winds up 1% lower than where it started, and so the
Compound return will be negative. That’s the effect of volatility, and boy has Bitcoin been volatile!
^5 Readers familiar with the bet-sizing literature will recognize that the 25% proportion of capital corresponds to the Kelly Criterion, which over many bets– and here we have 557 daily bets– gives
the highest end wealth. The Kelly calculation takes the Sharpe Ratio of 0.2 and divides it by the standard deviation of returns of about 80%, to arrive at a bet size of 0.2/0.8 = 25%.
^6 Note that we will not find cases where one can find long and short fractions that would both make money, i.e. the range of profitable proportional sizing will always start at 0 and either go
positive or negative from there.
^7 Over this period, it didn’t make a material difference if you rebalance daily, weekly, or monthly. The Sharpe Ratios based on weekly and monthly returns were 0.21 and 0.23.
^8 For example, strategies involving buying or selling puts or calls, or other kinds of dynamic scaling strategies. | {"url":"https://www.advisorperspectives.com/articles/2022/07/15/bitcoin-is-nothing-either-good-or-bad-but-sizing-makes-it-so","timestamp":"2024-11-04T02:33:52Z","content_type":"text/html","content_length":"123894","record_id":"<urn:uuid:f7e94f5e-482c-401c-bf0c-7f91f498ae18>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00503.warc.gz"} |
I solicit feedback from users before they quit. Here are a few of them. Note: I take steps to make them anonymous if it seems appropriate. Some comments go back to May 2000 when I had only a
Macintosh version avaiable for download. The rate of comments has exploded, so I stopped listing any after Oct 11, 2000.
Actual Users Comments by teachers and students
This is a very helpful refresher. Thank you for putting this magnificient program together- I hope you have additional tutorials in other sciences.[Editor : Not anytime soon. It took three years to
--citrus[login name]
October 11th
Ms. Lindquist does a good job in making the solution clear step by step.
October 11th
looks good, I'll try it on my children
October 11th
Nice and helpful
October 11th
Excellent program. Great help............i hope there is more to come from your teaching technique. I would love your help in all other areas of algebra and geometry as well.......... [ed. note: Not
any time soon. It took 3 years to write! :-) ]
October 11th
Like the site! Had one problem -- windows would flash up and close -- don't know if it's my computer or the site[Editorial note:- under investigation.] I want to look at it a little more and then
plan to allow my algebra students to use the site for extra credit. Thanks for all your hard work!
October 11th
i will recommend this program to my students. Cheers, HBR
October 11th
Looks promising!
--guest stu
October 11th
I am a math tutor for k-12. This site was sent to me from another teacher. So far I like the format. I'll check back a few more time before I refer the program to my students but thanks for the
opportunity and resource.
October 11th
October 11th
Like it so far but I have a class to teach. I will show this site to both my precalculus and calculus students.
October 11th
I liked it
October 11th
thank you -need more time but great tutor for all ages
October 11th
Thank you.
October 11th
Its all good.
October 11th
Great program. I will download it.[Editorial Note: You don't have to download the program to use it.]
October 10th
Good program; most useful. I will be sending student here.
October 10
I'm a teacher center trainer for NYC and I'm going to forward your site to math teachers at [name of high school removed]. Thanks.
October 10th
I am getting this for my son who is having difficultly in his first year of algebra
October 10th
This is a great way to studying only I need to pay close attention to what I type.
October 10th
I like it but I have to go teach now.
--Mrs. Monfort
October 10th
Love the program
October 10th
This is a very useful site. Thanks for all the work!
October 2000
Looks Good
October 2000
I think this program is fantastic. (I'm actually a parent masquerading as my son to see how helpful the program would be for him.) Very impressive!!
October 2000
Good. --lilsuga
October 2000
Great resource. I will put it to good use here at Terry Fox Secondary.
Online Co-ordinator
October 2000
"This site looks interesting. I will bookmark it."
October 9, 2000
I liked the program. Ms. Lindquist is quite adaptable. October 9, 2000
I am going to link this page to our district curriculum resources. Thanks Great site.
October 9, 2000
Wonderful! I am a Technology Coordinator at a high school in Illinois and I will be sharing this with many, many others in the math field! Thank you from those of us in education - I think the
students will greatly benefit!
October 9, 2000
This program is very helpful. Great job!
You have a very good program and I enjoy using it.
I'm in the process of adding the pointer to the AlgebraTutor on my math resources page. I've been having a great time experimenting with your program and really expect that this will be VERY helpful
to all who use it ... Thanks again for creating this program and making it available online. I hope you get LOTS of useful feedback.
--Amby Duncan-Carr
"A good way to learn when in a hurry."
"This is a very helpful way to learn. I have to leave is the reason I am quitting. Thanks."
"My server may be slow, but it takes a while for the program to progress. Other than that the help is good and somewhat, for me, easy to understand."
"The program was helpful because it helped me get back on track and bring back my skills for my next school season. The program had alot of different kinds of sections most of which were helpful and
made you really think. I think this would be very helpful to children just starting to learn this new type of math, or even if they already know it it would sharpen their skills."
"I have found Ms Lindquist very helpful in the way it forces me to think about segments of the problems before building to a comprehensive solution."
--David (from England)
"I am amazed at the power of this tutorial. Thank you."
"This is some wonderful software."
"I was impressed with the demo. I will be passing this on to other teachers at my site so we may brainstorm about ways to use this with our students."
- A teacher from Southern California
"Love it" - a teacher
"Hi Neil Heffernan, I just found this program and it is the end of the school year but I am sure I will use it next year."
- a teacher from Garden Grove, CA. May 2000
"I am a middle school math teacher...Thank you for this program....I will keep in touch."
"I'm a software Resource Specialist working... I've passed this on to the math specialists...it's great for reasoning and easy to use"
"Miss Lindquist is really neat. I think it is a really great use of tutor intelligence. ... Great use of the research to benefit learning!"
- graduate student in psychology
This program seems helpful. I enjoyed the preview and will return. I will also get my child to work with [MsLindquist]. Thank you. (email withheld)
I personally think that the program was very helpful in alot of ways it helped me greatly.
Back to The Ms. Lindquist Algebra Tutoring Web Site.
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This box is the service that provides my free web statistics. You can pay money to not have this image show up but we are non-profit and don't have money for that. | {"url":"http://www.cs.cmu.edu/~neil/user_comments.html","timestamp":"2024-11-02T15:02:28Z","content_type":"text/html","content_length":"9395","record_id":"<urn:uuid:d7053d1f-efa4-4fad-9288-b73379dfa616>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00638.warc.gz"} |
Paper Summary: Statistical Modeling: The Two Cultures
Last updated:
Please note This post is mainly intended for my personal use. It is not peer-reviewed work and should not be taken as such.
Author goes on to explain that he thinks there are two types of machine learning aproaches: statistical approaches and algorithmic approaches.
• Statistical approaches try to formally model the data with statistical distributions, noise estimates, confidence intervals and hyperparameters. An approach is considered good if it fits the
training data with small errors (methods like goodness-of-fit tests, residual sum-of-squares, etc)
• Algorithmic approaches don't try to understand what the data looks like and don't need formal theoretical underpinnings. Includes neural nets, decision trees, SVMs, etc. Success is measured by
predictive accuracy. I.e. performance measured on holdout test sets only.
□ The only assumption in algorithmic approaches is that data are I.I.D., i.e. samples are independent from one another and all of them are drawn from a single (albeit unknown) distribution.
The author left academia to work with consulting and what he had seen at the university was very much at odds with what he used in practice to solve data-driven problems. He then rejoins academia and
this paper was written at this point.
He criticizes the statistics establishment for the over-reliance on data models. In addition, he thinks algorithmic approaches are much better suited to new kinds of problems and the dramatic
increase in sample sizes.
He draws on his own experience in academia and in the industry, citing dozens of papers and studies where the focus was on the theoretical/mathematical properties of data models, irrespective of
whether it was a good match to the real world data or even if it solves the problem correctly.
• You don't need to know what the data looks like (in terms of statistical distributions) to get a good predictive model. All you need is good test set performance - it doesn't matter what's inside
the black box.
• Methods such as goodness-of-fit tests can't help you decide which one of two models is the best fit for the data, especially if there are many dimensions involved.
• "Misleading conclusions may follow from data models that pass goodness-of-fit tests and residual checks."
• "Approaching problems by looking for a data model imposes an a priori straight jacket that restricts the ability of statisticians to deal with a wide range of statistical problems."
• One version of the accuracy vs. interpretability tradeoff: "Accuracy generally requires more complex prediction methods. Simple and interpretable functions do not make the most accurate
• Apparently, Cross-validation was first suggested by someone named Stone back in 1974.
MY 2¢
• I think the best way to summarize this debate is: Good models aren't necessarily correct, but they work.
• It really is amazing that so much energy/money has been spent by undoubtedly clever statisticians into things that just didn't work to solve problems in practice.
□ I mean, did nobody realize that increasing model complexity would surely increase the goodness of fit even though you will sooner or later starting to learn noise?
□ Really reminds me of what Nassim Taleb says about theoreticians and practitioners. The former can get away with producing stuff that serves no practical use (they have tenure, i.e. not much
skin in the game) but the latter can't afford to to do (they won't get paid.)
• It shows how siloed and self-absorbed many fields of research can be. ^1 They may be effectively living in totally different universes when there's no communication between them.
• Although Mr. Breiman puts things like Logistic and Linear Regression under data models, I don't see a problem in using those as long as your success metric is based on the hold-out test set.
1: In this case, Statistics and Computer Science.
• Breiman 2001: Statistical Modeling: The Two Cultures
□ This version of the paper includes comments from some other academics/practitioners, who point out where they disagree with Breiman's points.
□ At the end, Breiman himself answers back the criticism in those comments.
□ Japanese movie that illustrates the fact that multiple, very different, models may (in terms of test-set accuracy) be successful. | {"url":"https://queirozf.com/entries/paper-summary-statistical-modeling-the-two-cultures","timestamp":"2024-11-04T12:18:32Z","content_type":"text/html","content_length":"24839","record_id":"<urn:uuid:9d14fedb-2a3a-492c-97aa-4137c22c44da>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00530.warc.gz"} |
Back propagation of LSTM: just get ready for the most tiresome part
First of all, the summary of this article is “please just download my Power Point slides and be patient, following the equations.” I am not supposed to use so many mathematics when I write articles
on Data Science Blog. However using little mathematics when I talk about LSTM backprop is like writing German, never caring about “der,” “die,” “das,” or speaking little English during English
classes (, which most high school English teachers in Japan do,) or writing Japanese without using any Chinese characters (, which looks like a terrible handwriting by a drug addict). In short, that
is ridiculous.
In this article I will just give you some tips to get ready for the most tiresome part of understanding LSTM.
1, Chain rules
In fact this article is virtually an article on chain rules of differentiation. Even if you have clear understandings on chain rules, I recommend you to take a look at this section. If you have
written down all the equations of back propagation of DCL, you would have seen what chain rules are. Even simple chain rules for backprop of normal DCL can be difficult to some people, but when it
comes to backprop of LSTM, it is a monster of chain rules. I think using graphical models would help you understand what chain rules are like. Graphical models are basically used to describe the
relations of variables and functions in probabilistic models, so to be exact I am going to use “something like graphical models” in this article. Not that this is a common way to explain chain
First, let’s think about the simplest type of chain rule. Assume that you have a function $f=f(x)=f(x(y))$, and relations of the functions are displayed as the graphical model at the left side of the
figure below. Variables are a type of function, so you should think that every node in graphical models denotes a function. Arrows in purple in the right side of the chart show how information
propagate in differentiation.
Next, if you a function $f$ , which has two variances $x_1$ and $x_2$. And both of the variances also share two variances $y_1$ and $y_2$. When you take partial differentiation of $f$ with respect
to $y_1$ or $y_2$, the formula is a little tricky. Let’s think about how to calculate $\frac{\partial f}{\partial y_1}$. The variance $y_1$ propagates to $f$ via $x_1$ and $x_2$. In this case the
partial differentiation has two terms as below.
In chain rules, you have to think about all the routes where a variance can propagate through. If you generalize chain rules, that is like below, and you need to understand chain rules in this way to
understanding any types of back propagation.
The figure above shows that if you calculate partial differentiation of $f$ with respect to $y_i$, the partial differentiation has $n$ terms in total because $y_i$ propagates to $f$ via $n$
2, Chain rules in LSTM
I would like you to remember the figure I used to show how errors propagate backward during backprop of simple RNNs. The errors at the last time step propagates only at the last time step.
At RNN block level, the flows of errors are the same in LSTM backprop, but the flow of errors in each block is much more complicated in LSTM backprop.
3, How LSTMs tackle exploding/vanishing gradients problems
LSTMs do not solve, vanishing gradient problems, but instead they mitigate vanishing/exploding gradient problems.
http://datasciencehack.com/wp-content/uploads/2020/04/RNN_head_pic.png 802 1952 Yasuto Tamura https://www.data-science-blog.com/wp-content/uploads/2016/09/data-science-blog-logo.png Yasuto Tamura
2020-09-30 01:38:122020-08-15 13:29:20Back propagation of LSTM: just get ready for the most tiresome part
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Statistical methods calculator
• How do you calculate statistics?
To calculate mean, median, standard deviation, etc.
Press STAT, then choose CALC, then choose 1-Var Stats.
Press ENTER, then type the name of the list (for example, if your list is L3 then type 2 nd 3).
If your data is in L1 then you do not need to type the name of the list..
• How do you solve statistics on a calculator?
What is a Statistics Calculator? This statistics calculator is an online tool that can be used to compute various statistical metrics such as mean, median, mode, standard deviation, variance,
etc. of a given data set..
• Is there a calculator for statistics?
Statistics Calculator is a free online tool that displays the mean, median, mode, variance, and standard deviation for the given set of data.
BYJU'S online statistics calculator tool makes the calculation faster, and it displays the various measures of central tendencies in a fraction of seconds..
• What calculator should I use for statistics?
As far as statistics is concerned, the best calculator for statistics is the TI-83, though the TI-89 comes close.
My recommendation is because: It's the best bang for your buck, at about half the price of a TI-89..
• What is a statistical calculator?
Formulas for Test Statistics
Take the sample mean, subtract the hypothesized mean, and divide by the standard error of the mean.
Take one sample mean, subtract the other, and divide by the pooled standard deviation.
Calculate the ratio of two variances..
• What is a statistical calculator?
What is a Statistics Calculator? This statistics calculator is an online tool that can be used to compute various statistical metrics such as mean, median, mode, standard deviation, variance,
etc. of a given data set..
• What is the app that calculates statistics?
IntroStat is a probability and statistics calculator.
It is the perfect learning tool for an introductory statistics course.
Use it to perform any of your statistics calculation needs..
• What mode should calculator be in for statistics?
To start a statistical calculation, perform the key operation (STAT) to enter the STAT Mode and then use the screen that appears to select the type of calculation you want to perform..
• Which AI can calculate statistics?
Some of the top AI tools for calculations include TensorFlow, IBM Watson, Microsoft Azure Machine Learning, Google AI Platform, RapidMiner, Keras, PyTorch, Alteryx, MathWorks MATLAB, and KNIME.
Each has its unique features and is ideal for different use cases..
• The 5 methods for performing statistical analysis
Mean.Standard Deviation.Regression.Hypothesis Testing.Sample Size Determination.
• A calculator is a machine which allows people to do math operations more easily.
For example, most calculators will add, subtract, multiply, and divide.
Some also do square roots, and more complex calculators can help with calculus and draw function graphs. | {"url":"https://pdfprof.com/PDF_DOC/PDF_Documents/308402/4/110/statistical+methods+calculator","timestamp":"2024-11-05T04:35:25Z","content_type":"text/html","content_length":"27741","record_id":"<urn:uuid:e3b18620-671d-494d-a3a1-eca786d8c0d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00644.warc.gz"} |
How Do You Solve a Two-Step Equation by Multiplying by -1 First?
Solving equations can be tough, especially if you've forgotten or have trouble understanding the tools at your disposal. One of those tools is the addition property of equality, and it lets you add
the same number to both sides of an equation. Watch the video to see it in action! | {"url":"https://virtualnerd.com/algebra-1/linear-equations-solve/two-or-multi-step/two-steps/negative-one-multiplication-two-steps","timestamp":"2024-11-03T12:04:10Z","content_type":"text/html","content_length":"30843","record_id":"<urn:uuid:02ab97ad-52c2-41f6-b1ed-df0fbbe76fbd>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00676.warc.gz"} |
March 21, 2021
TL;DR: BucketList<T> is a data structure that performs exceptionally well for a certain usage scenario:
□ n (list size) is large.
□ Fast index-based access: list.GetItemAt(i).
□ Fast index-based removal: list.RemoveAt(i).
In my use case, this reduced the runtime from 7 hours to 17 seconds!
An Interesting Problem
In December of last year I took part in solving the Advent of Code programming puzzles for the first time. Since then I became a bit obsessed with that and started solving also the older events going
back until 2015.
Day 19 of 2016 presented an interesting challenge that took me a while to solve (with a reasonable runtime) and led to the solution based on this BucketList<T> data structure.
The basic idea behind this problem is that we have a:
• Circular data structure.
• Maintain a current element.
• Find the element opposite of the current element.
• Advance the current element by one.
• Repeat until the list has only one element left.
A poor solution
My first approach was using a LinkedList which performed OK for small list sizes. But for large lists (e.g. 3.001.330 was my puzzle input) the performance was dismal.
List size: 3001330.
END after 26696.1142087 seconds.
That's a runtime of more than 7 hours š ¬.
The problem was the linear search for locating the opposite element in the list that caused an O(nĀ²) complexity.
Using a simple List<T> (instead of LinkedList<T>) performed equally bad because in that case the remove operation causes a runtime complexity of O(nĀ²).
To sum it up:
Data Structure Lookup Removal
List<T> O(1) O(NĀ²)
LinkedList<T> O(NĀ²) O(1)
A better (by a lot) solution
So I needed a data structure with linear or better performance for:
My solution was a custom data structure I called BucketList (ha ha) which:
• maintains the elements in a list of lists
• that have the same maximum size (hence, buckets).
This way, when performing an index-based lookup I have nearly constantĀ¹ O(1) access to elements and removal is only limited by the List<T> performance of a single bucket.
Ā¹ Actually O(k) where k amounts to the number of buckets which should be negligibly low compared to n.
To sum it up:
Data Structure Lookup Remove
BucketList<T> O(k) O(n)
k: number of buckets
n: bucket size (list size N / k)
This brought the runtime down dramatically!
List size: 3001330.
END after 17.0722168 seconds.
17 seconds ... way better š ¤©.
Here's the full source code for BucketList.
Note: A bucket size of 25.000 worked best in that specific scenario but your mileage may vary, of course, and depends on the problem input size.
• I only used BucketList<T> for this specific problem so far.
• The runtime performance depends on the selection of a bucket size for a given list size.
• The InsertAt case is not yet considered.
• I'm sure that this data structure exists already with a different name. However, I came up with this on my own when solving my problem, so give me that ;-). | {"url":"https://www.wolfgang-ziegler.com/blog/bucketlist-of-t","timestamp":"2024-11-13T22:38:11Z","content_type":"text/html","content_length":"16631","record_id":"<urn:uuid:b4a590ce-bbc1-4a81-9280-a4fcff8af8e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00759.warc.gz"} |
OOTL Baseball | Harrisburg, PA
Starting Pitching
Gentlemen, take out your staffs! In this edition we are attacking Starting Pitching. We use objective quantitative methods whenever possible with very minimal aspects of the analysis based on
subjective theory. In the end, we will rank what we believe to be best starting pitching staffs for this year’s OOTL season. Again, this is an important clarification as we are not evaluating
starting pitching by upside or by MLB standards. Our evaluation is limited to expected performance in this year’s OOTL season. It is very difficult to evaluate starting pitching because there are so
many external factors affecting wins and losses in the OOTL. Therefore our evaluation of the staffs is purely in a vacuum. It is not an indicator of success per se. We will be creating indexes which
will determine the best staffs in a vacuum, but actual results could differ based on any combination of the following:
External Factors:
• Some Owners position their best pitchers against their opponent’s best pitchers.
• Other owners like to match up their best vs. worst and worst vs. best pitchers
.• Still others use a rotation and stick to it regardless of opponent.
• Finally others strategically attempt to match starter attributes such as l/r grade, control, power and K numbers vs. their most favorable opponent matchups.
• Some owners are better game managers than others.
• Bullpen, fielding and hitting quality of a team and its opponent will also affect success of your staff.
• Fatigue, indifference, anger and loss of focus will also affect results.
• Future trades.
• And let’s not forget dice rolls.
Let’s start with some firm data. Our 10 Starting Pitching inventories range from 7 to 12 pitchers. We will elaborate on these different roster strategies in a future edition when we evaluate the
final 6 roster spots of each team. Every team has at least one LSP on their roster but some staffs have a higher pct. All pitchers who we expect to see at least one start this year are listed under
regular starters except for pitchers we refer to as cheater starters. We will discuss this group more in a future edition when we evaluate cheater cards but for now we include pitchers in this group
which have limited starts but are top or above average quality. Finally we also assigned the pitchers who will not be starting this season due to either the veteran or young prospects category. Our
service would love to see an evolution of roster size to add two (not eligible to play positions called Veteran prospects) but again we will discuss this in future editions.
We began our analysis by ranking best to worse the 70 starters we believe will pitch this year based on each team needing 90 starts. This was a subjective review based on 5 factors: L/R, Grade, Power
rating, Control Rating and strikeout letters. We then broke this ranking into 4 groups graded A-D. As a note there are a couple of pitchers at the grade cutoffs who blur a grade. That is they could
easily fit into two different grades based on their metrics. Therefore in a couple cases we assigned some of their starts to multiple grades. The chart below shows this breakdown by how many starts
will be made by team by grade this year. The A-D breakdown is 196-351-172-181 for 900 starts (10 teams* 90 games).
Here is where a simple analysis could be done and we could have called it a day. If you use a weighted method (example 4 for A, 3 for B, etc.) similar to a GPA calculation we could develop a valid
However, there are some limitations. Because of the number of pitchers we assigned to each group and the potential winning pct. of each grade, 4-3-2-1 weighing is not the most accurate way to predict
each grades value. Therefore we had to develop an index method. If we combine our total starts by grade number with historic stats on wins/losses we develop a base winning pct. This is only a
directional and could change annually but the point is that the base pct.’s lead to a 45-45 record with the base # of starters per grade. Therefore in the end the pct’s used do not matter that much
as long as they are curved correctly.
We were also able to back into a base average staff based on 10 teams and the total starts by grade which we had previously identified. We then extrapolated those numbers to develop a base wins per
grade of pitcher. As an example, our base team which would be expected to go 45-45 in a league with 196 A starts, 351 B starts, 172 C starts and 181 D starts would have 20 A starts, 35 B starts, 17 C
starts and 18 D starts on its roster.
Now we have all we need to best rank the rotations. We use each team’s projected number of starts by grade and index those vs. our developed base 45-45 team’s number of starters by grade. As an
example below, The Stars have a positive 16 index under A as they have 36 A starts and our base team has 20 A starts. The Stars also have a -18 index under D because they have no D starts and the
base team has 18.
We then applied the projected winning pct. of each grade’s starters to the index to develop the final rotation ranking. A 45-45 record team is the zero base. Therefore our ranking shows that in a
vacuum, not considering all the external factors listed in the beginning, there are 4 teams which have staffs better than average.
The Best Rotations:
The Stars, Speakers and Lyme Bees clearly have the best rotations. There are some subtle differences which explain why they are ranked Stars, Speakers and Lyme Bees.
1) Shooting Stars: The Stars have 1 more A start than the Bees and 2 more than the Speakers. While the Speakers and Lyme Bees have more B starts the Stars have more C starts and no D starts.
Therefore the additional A start and lack of any D starts carried them to the best rotation.
2) Speakers: The Speakers are the second best rotation and are right there with the Stars with one exception. The Speakers barring trade will have to get 4 starts from R11/16/14 Joe Ross. We rank
Ross a D pitcher and those 4 starts cost the Speakers a shot at the top ranking.
3) Lyme Bees: The Bees are also right there but also got by 12 Grade 4 starts by Kyle Hendriks.
Next Best Rotations:
4) Lemonheads: A step below the mighty 3 but comfortably better then the other 6 are the Lemonheads, The Lemonheads are solid in that they have no D starts but not ranked as high because they only
have half as many A starts as the top three.
Middle of the Road Rotations:
5) AB’s: Limited A starts and significant D starts but still an average aggregate rotation.
6) Wahoos: A strong number of A&B combined starts but no C and 29 D put them just under the AB’s.
7) Plague: Again a strong number of A&B starts and a few less A starts than the Wahoos, but no C starts and a few more D starts put them just under the Wahoos.
The Rest:
8) Browns: The best of this group but too many C’s and D’s left them just short of the above group.
9) Tsunamis: No A starts but a solid number of B starts gives them the nod over the Eliminators.
10) Eliminators: No A starts and 38 D starts position these guys in the dreaded 10 spot.
But what about playoff rotations? Since only your 4 best are used in the playoffs in limited games and most matchups will be A’s and B’s we cannot use the above analysis. Therefore we will use the
following chart and a subjective look:
Best Playoff Rotations:
1-3) Stars, Bees, Speakers: Lyme Bees Arrieta gets nod over Keuchel and Price. Degrom, Gray and Harvey are close but Degrom gets the nod. Lackey also gets close nod over Syndergaard and Archer.
Finally the Stars pull out the win as Cheater starter combo Wilson/McAllister eek out the win over McCullers/Bumgarner and Severino.
4) Plague: a solid 4th place driven by Garcia, Matz and Estrada.
5-6): Lemonheads/Wahoos: Greinke is best and wins the tiebreaker.
7-8):Browns/AB’s: Kershaw wins the tiebreaker.
9-10) Tsunamis/E’s: The Tsunamis solid BB rating could make a difference and make them competitive if they can make the dance. The Eliminators rare 6 man lefty playoff rotation could also cause
nightmares for some teams. Even at 9 and 10 these two teams staff anomalies could win in the playoffs if they can make it there.
NEXT WEEK: We will look at Cheater Card strategies in 2016.
Craig Dolan for the ASIAN SPORTS NETWORK (ASSN)
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Detailed Course Information
Credit Hours: 2.00. CP1 teaches applied algorithmic ideas and problem solving techniques to solve programming interview and competitive programming questions including usage of basic data structures
such as [array, set, map, stack, queue, deque, priority queue], the four main algorithm paradigms: [complete search, greedy, divide and conquer, dynamic programming], other algorithmic ideas
including [binary search the answer/bisection, meet-in-the-middle, prefix sum and difference arrays, two pointers, sliding window], and basic graph algorithms covering [strongly/connected components,
floodfill, topological sort, shortest paths].
2.000 Credit hours
Syllabus Available
Levels: Undergraduate, Graduate, Professional
Schedule Types: Distance Learning, Lecture
Offered By: College of Science
Department: Computer Science
Course Attributes:
Lower Division
May be offered at any of the following campuses:
Indianapolis and W Lafayette
West Lafayette
Learning Outcomes: 1. Differentiate between categories of problems in computer science including complete search, greedy, divide and conquer, dynamic programming, and various types of graph
problems, as well as subcategories in each of these categories. 2. Implement well-known solutions (as discussed in class) to all major categories of problems in computer science (listed in LO1). 3.
Use new/novel algorithms to solve problems. 4. Recognize key characteristics of certain problem types listed in LO1. 5. Classify a problem by those key characteristics. 6. Deconstruct a problem
into subproblems that can be solved individually. 7. Assemble those subproblems into a complete solution. 8. Determine runtime and space usage of a potential solution using big-O notation to judge
if the potential solution will work. 9. Create an efficient solution to a problem based on analysis of the problem type, the deconstructed problem parts, and the time and space constraints of the
problem. 10. Reflect on how one came up with a solution to a problem to better recognize patterns in problem types. 11. Debug programs by generating custom test cases based on problem constraints.
Must be enrolled in one of the following Majors:
Computer Science
Computer Science Honors
Data Science
Data Science First Year
Data Science
Undergraduate level CS 25100 Minimum Grade of C or Undergraduate level CS 25300 Minimum Grade of C
Short Title: Competitive Programming I
Course Configurations:
│Configuration 1: 2.0 Credits │
│Lecture │2│2.0 │
│Configuration 2: 2.0 Credits │
│Distance Learning │0│2.0 │ | {"url":"https://selfservice.mypurdue.purdue.edu/prod/bzwsrch.p_catalog_detail?term=202510&subject=CS&cnbr=21100&enhanced=Y","timestamp":"2024-11-09T00:29:35Z","content_type":"text/html","content_length":"16708","record_id":"<urn:uuid:a77c45e1-aa58-4585-8eab-f69eca49871e>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00528.warc.gz"} |
Fast Fourier Transform Filter
Fast Fourier Transform Filter¶
The foglamp-filter-fft filter is designed to accept some periodic data such as a sample electrical waveform, audio data or vibration data and perform a Fast Fourier Transform on that data to supply
frequency data about that waveform.
Data is added as a new asset which is named as the sampled asset with ” FFT” append. This FFT asset contains a set of data points that each represent the a band of frequencies, or as a frequency
spectrum in a single array data point. The band information that is returned by the filter can be chosen by the user. The options available to represent each band are;
□ the average in the band,
□ the peak
□ the RMS
□ or the sum of the band.
The bands are created by dividing the frequency space into a number of equal ranges after first applying a low and high frequency filter to discard a percentage of the low and high frequency results.
The bands are not created if the user instead opts to return the frequency spectrum.
If the low Pass filter is set to 15% and the high Pass filter is set to 10%, with the number of bands set to 5, the lower 15% of results are discarded and the upper 10% are discarded. The remaining
75% of readings are then divided into 5 equal bands, each of which representing 15% of the original result space. The results within each of the 15% bands are then averaged to produce a result for
the frequency band.
FFT filters are added in the same way as any other filters.
□ Click on the Applications add icon for your service or task.
□ Select the fft plugin from the list of available plugins.
□ Name your FFT filter.
□ Click Next and you will be presented with the following configuration page
□ Configure your FFT filter
☆ Asset to analysis: The name of the asset that will be used as the input to the FFT algorithm.
☆ Result Data: The data that should be returned for each band. This may be one of average, sum, peak, rms or spectrum. Selecting average will return the average amplitude within the band,
sum returns the sum of all amplitudes within the frequency band, peak the greatest amplitude and rms the root mean square of the amplitudes within the band. Setting the output type to be
spectrum will result in the full FFT spectrum data being written. Spectrum data however can not be sent to all north destinations as it is not supported natively on all the systems
FogLAMP can send data to.
☆ Frequency Bands: The number of frequency bands to divide the resultant FFT output into
☆ Band Prefix: The prefix to add to the data point names for each band in the output
☆ No. of Samples per FFT: The number of input samples to use. This must be a power of 2.
☆ Low Frequency Reject %: A percentage of low frequencies to discard, effectively reducing the range of frequencies to examine
☆ High Frequency Reject %: A percentage of high frequencies to discard, effectively reducing the range of frequencies to examine
See Also¶
foglamp-filter-ADM_LD_prediction - Filter to detect whether a large discharge is required for a centrifuge
foglamp-filter-breakover - Filter to forecast the possibility of a breakover.
foglamp-filter-conditional-labeling - Attach labels the reading data based on a set of expressions matched against the data stream.
foglamp-filter-ednahint - A hint filter for controlling how data is written using the eDNA north plugin to AVEVA’s eDNA historian
foglamp-filter-enumeration - A filter to map between symbolic names and numeric values in a datapoint.
foglamp-filter-expression - A FogLAMP processing filter plugin that applies a user define formula to the data as it passes through the filter
foglamp-filter-log - A FogLAMP filter that converts the readings data to a logarithmic scale. This is the example filter used in the plugin developers guide.
foglamp-filter-metadata - A FogLAMP processing filter plugin that adds metadata to the readings in the data stream
foglamp-filter-normalise - Normalise the timestamps of all readings that pass through the filter. This allows data collected at different rate or with skewed timestamps to be directly compared.
foglamp-filter-omfhint - A filter plugin that allows data to be added to assets that will provide extra information to the OMF north plugin.
foglamp-filter-python35 - A FogLAMP processing filter that allows Python 3 code to be run on each sensor value.
foglamp-filter-rms - A FogLAMP processing filter plugin that calculates RMS value for sensor data
foglamp-filter-scale-set - A FogLAMP processing filter plugin that applies a set of sale factors to the data
foglamp-filter-specgram - FogLAMP filter to generate spectrogram images for vibration data
foglamp-filter-statistics - Generic statistics filter for FogLAMP data that supports the generation of mean, mode, median, minimum, maximum, standard deviation and variance.
foglamp-filter-vibration_features - A filter plugin that takes a stream of vibration data and generates a set of features that characterise that data
foglamp-south-Expression - A FogLAMP south plugin that uses a user define expression to generate data
foglamp-south-digiducer - South plugin for the Digiducer 333D01 vibration sensor
foglamp-south-dt9837 - A south plugin for the Data Translation DT9837 Series DAQ | {"url":"https://foglamp.dianomic.com/en/latest/plugins/foglamp-filter-fft/index.html","timestamp":"2024-11-05T17:08:10Z","content_type":"text/html","content_length":"24120","record_id":"<urn:uuid:5455f730-7217-4919-9ada-caf700b25138>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00629.warc.gz"} |
How to solve any (statistics) problem: Part 2
Yesterday, I mentioned this problem
For 17 girls diagnosed with anorexia, weight change after family therapy was as follows:
11,11, 6, 9, 14, -3, 0, 7, 22, -5 , -4, 13, 13, 9, 4 , 6, 11
Partial results are shown below. Fill in the missing results:
And we had gotten the table completed as far as this. We also along the way found out that the mean was 7.29
│Lower C.L. │Upper C.L. │t-value│df │2-tail Sig │
│3.60 │ 10.98 │ │ 16│.0007 │
#1 CHILL !
I mean this most seriously. This really is the first step.
#2 UNDERSTAND!
What is it you are asked to do in the problem? All that is left is to find the t-value. Here is where several of the students went wrong. So many of them went wrong I would have thought they had
cheated, but they were sitting all around the room. Barring some secret hand signals, that was not possible.
Many of the students obtained a value of around 2.12, which is very much NOT correct. I was confused and then I realized that while *I* knew that the problem was asking for the obtained t-value,
what the students had computed was the critical t-value with 16 degrees of freedom. The problem did not specify and the textbook author, like me, just assumed that you would know that the value shown
on a print-out was the obtained t-value, not the critical t-value.
Well, sure you would know that if like me, and no doubt like the author of the textbook, you had been looking at printouts from statistical programs for the past 30 years. These students could not be
expected to know that, so, I ended up giving them full credit if that is what the answered.
What you should know now
• The t-value referenced in the print-out is the OBTAINED t, not the critical t-value for that number of degrees of freedom.
• The formula for obtaining t is (obtained mean – hypothesized mean)/ standard error
• Your hypothesized mean is 0
• Your obtained mean is 7.29
• The standard error is the standard deviation divide by the square root of N
• The critical value for t for 16 degrees of freedom when p < .05 is 2.12
• The lower confidence limit is the mean MINUS the CRITICAL t times the standard error
• The lower confidence limit is 3.6
• The difference between the mean and the lower confidence limit is 3.69
• The standard deviation is the square root of the sum of squared deviations from the mean divided by n -1
#3 SELECT A STRATEGY
There are a number of ways to find the t-value. All involve subtracting the hypothesized mean from the obtained mean and dividing by the standard error. Some ways are harder than others. You could
compute the standard deviation and divide by the square root of N but that is a lot of work. Here is what I think is the easiest way
• Divide 3.69 by 2.12 — that will give us the standard error
• Subtract 0 from 7.29
• Divide 7.29 by the standard error
In this case, it was this step and the previous one where people ran into trouble. What is interesting is that they did not realize what they DIDN’T understand. That is, they didn’t understand that
the t-value they were expected to produce was the obtained t-value, not the critical t-value.
You could (and many people did) compute the standard deviation, then divide it by the square root of N to get the standard error and it would give the correct answer, but it just seems more work than
dividing 3.69 by 2.12.
#4 DO IT
Carry out your strategy.
• 3.69/ 2.12 —- The standard error is 1.74
• 7.29 -0 = 7.29
• Divide 7.29 by 1.74 = 4.19
That’s your answer. As in the previous example, the actual doing it part is pretty easy.
#5 TEST IT
Do a reality check. No one in the class asked which t-value it should be and it never occurred to me that people would not automatically know that it was the obtained t-value that is of interest. I
mean, seriously, what’s the purpose of doing a study to find a critical value of t that was established a hundred years ago? I’m not surprised though, that people who are not experienced
statisticians don’t immediately think of that. Probably a lot of what statisticians do doesn’t seem very obvious so maybe it’s just another of those weird things.
So, I guess it is up to me on Thursday to explain to the class that you have a critical value for a test statistic and an obtained value.
A lot of #5 comes from experience. For example, immediately, when I saw t-values of around 2 that the students had obtained, I thought that can’t be right, because even with 17 people, 7 pounds is
pretty far from 0, it seemed like it ought to be significant.
So …. this brings me to number 6
#6 PRACTICE
The more problems you do, the better you get at solving them. People often get the impression that people who are good at math have some kind of special math brain. It’s not true. If you are telling
yourself that you are just not good at math, cut it out right now before I come over there and smack you.
I married a rocket scientist – literally – someone whose idea of the way to a woman’s heart was to write a program to generate fractals and email her a pink fractal for Valentine’s Day. It worked,
too. And yet — I can guarantee you that he, and I, both ran into the same obstacles in learning mathematics that anyone else does. The only difference between us and our friends who quit school and
ended up working at Wal-Mart is that we spent hours and hours and hours learning programming, statistics(well, I learned the statistics), Calculus, Physics (well, he learned the physics).
Last week, more than one student said to me, with some frustration.
“Dr. De Mars, I studied for HOURS for this class.”
One Comment
1. Great posts. As a current master’s student in an applied stats program, it’s good to be reminded of the basics. | {"url":"https://www.thejuliagroup.com/blog/how-to-solve-any-statistics-problem-part-2/","timestamp":"2024-11-07T09:35:03Z","content_type":"text/html","content_length":"85022","record_id":"<urn:uuid:396ae108-0a59-42a5-84dc-f35a7df459fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00286.warc.gz"} |
AMS Lab
Applied Materials Science Lab
Faculty of Materials Science and Engineering
Room 702, Building A5, Phenikaa University,
Yen Nghia Ward, Ha Dong District, Hanoi, Vietnam.
Email: anh.phanduc@phenikaa-uni.edu.vn
6/10/2021 - Cuong attended the online conference held from 4 October 2021 to 6 October 2021 in Hanoi, Vietnam.
28/9/2021 - The paper "Effects of surface charge and environmental factors on the electrostatic interaction of fiber with virus-like particle: A case of coronavirus" is accepted for publication in
AIP Advances.
We propose a theoretical model to elucidate intermolecular electrostatic interactions between a virus and a substrate. Our model treats the virus as a homogeneous particle having surface charge and
the polymer fiber of the respirator as a charged plane. Electric potentials surrounding the virus and fiber are influenced by the surface charge distribution of the virus. We use Poisson-Boltzmann
equations to calculate electric potentials. Then, Derjaguin's approximation and a linear superposition of the potential function are extended to determine the electrostatic force. In this work, we
apply this model for coronavirus or SARS-CoV-2 case and numerical results quantitatively agree with prior simulation. We find that the influence of fiber's potential on the surface charge of the
virus is important and is considered in interaction calculations to obtain better accuracy. The electrostatic interaction significantly decays with increasing separation distance, and this curve
becomes steeper when adding more salt. Although the interaction force increases with heating, one can observe the repulsive-attractive transition when the environment is acidic.
26/7/2021 - The paper "Toward a better understanding of activation volume and dynamic decoupling of glass-forming liquids under compression" is accepted for publication in Macromolecular Theory and
Physical properties of the pressure-induced activation volume and dynamic decoupling of ternidazole, glycerol, and probucol by the elastically collective nonlinear Langevin equation theory is
theoretically investigated. Based on the predicted temperature dependence of activated relaxation under various compressions, the activation volume is determined to characterize effects of pressure
on molecular dynamics of materials. It is found that the decoupling of the structural relaxation time of compressed systems from their bulk uncompressed value is governed by the power-law rule. The
decoupling exponent exponentially grows with pressure below 2 GPa. The decoupling exponent and activation volume are intercorrelated and have a connection with the differential activation free
energy. Relationships among these quantities are analyzed numerically and mathematically to explain many results in previous experiments and simulations.
21/2/2021 - We theoretically investigate high-pressure effects on the atomic dynamics of metallic glasses. The theory predicts compression-induced rejuvenation and the resulting strain hardening that
have been recently observed in metallic glasses. Structural relaxation under pressure is mainly governed by local cage dynamics. The external pressure restricts the dynamical constraints and slows
down the atomic mobility. In addition, the compression induces a rejuvenated metastable state (local minimum) at a higher energy in the free-energy landscape. Thus, compressed metallic glasses can
rejuvenate and the corresponding relaxation is reversible. This behavior leads to strain hardening in mechanical deformation experiments. Theoretical predictions agree well with experiments.
19/12/2021 - Iron represents the principal constituent of the Earth's core, but its high-pressure melting diagram remains ambiguous. Here we present a simple analytical approach to predict the
melting properties of iron under deep-Earth conditions. In our model, anharmonic free energies of the solid phase are directly determined by the moment expansion technique in quantum statistical
mechanics. This basis associated with the Lindemann criterion for a vibrational instability can deduce the melting temperature. Moreover, we correlate the thermal expansion process with the shear
response to explain a discontinuity of atomic volume, enthalpy, and entropy upon melting. Our numerical calculations are quantitatively consistent with recent experiments and simulations. The
obtained results would improve understanding of the Earth's structure, dynamics, and evolution.
I am looking for several undergraduate and graduate students, and postdocs in the areas of energy, surface science and engineering, interfacial phenomena, pharmaceutics, and machine learning for
materials science.
Contact me if you are interested in my research! | {"url":"https://amslab.phenikaa-uni.edu.vn/","timestamp":"2024-11-04T10:36:43Z","content_type":"text/html","content_length":"110705","record_id":"<urn:uuid:e6377547-7aa4-4f18-b64b-a4ec12c965ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00435.warc.gz"} |
19.1 Electric Potential Energy: Potential Difference - College Physics | OpenStax
When a free positive charge $qq size 12{q} {}$ is accelerated by an electric field, such as shown in Figure 19.2, it is given kinetic energy. The process is analogous to an object being accelerated
by a gravitational field. It is as if the charge is going down an electrical hill where its electric potential energy is converted to kinetic energy. Let us explore the work done on a charge $qq size
12{q} {}$ by the electric field in this process, so that we may develop a definition of electric potential energy.
The electrostatic or Coulomb force is conservative, which means that the work done on $qq size 12{q} {}$ is independent of the path taken. This is exactly analogous to the gravitational force in the
absence of dissipative forces such as friction. When a force is conservative, it is possible to define a potential energy associated with the force, and it is usually easier to deal with the
potential energy (because it depends only on position) than to calculate the work directly.
We use the letters PE to denote electric potential energy, which has units of joules (J). The change in potential energy, $ΔPEΔPE size 12{?"PE"} {}$, is crucial, since the work done by a conservative
force is the negative of the change in potential energy; that is, $W =–ΔPEW =–ΔPE size 12{W"=-"?"PE"} {}$. For example, work $W W$ done to accelerate a positive charge from rest is positive and
results from a loss in PE, or a negative $ΔPEΔPE size 12{?"PE"} {}$. There must be a minus sign in front of $ΔPEΔPE size 12{?"PE"} {}$ to make $W W$ positive. PE can be found at any point by taking
one point as a reference and calculating the work needed to move a charge to the other point.
$W =–ΔPEW =–ΔPE size 12{W" = -"?"PE"} {}$. For example, work $W W$ done to accelerate a positive charge from rest is positive and results from a loss in PE, or a negative $ΔPE.ΔPE.$ There must be a
minus sign in front of $ΔPEΔPE$ to make $WW$ positive. PE can be found at any point by taking one point as a reference and calculating the work needed to move a charge to the other point.
Gravitational potential energy and electric potential energy are quite analogous. Potential energy accounts for work done by a conservative force and gives added insight regarding energy and energy
transformation without the necessity of dealing with the force directly. It is much more common, for example, to use the concept of voltage (related to electric potential energy) than to deal with
the Coulomb force directly.
Calculating the work directly is generally difficult, since $W = FdcosθW = Fdcosθ$ and the direction and magnitude of $F F size 12{F} {}$ can be complex for multiple charges, for odd-shaped objects,
and along arbitrary paths. But we do know that, since $F =qEF =qE size 12{F= ital "qE"} {}$, the work, and hence $ΔPEΔPE$, is proportional to the test charge $q.q. size 12{q} {}$ To have a physical
quantity that is independent of test charge, we define electric potential $VV size 12{V} {}$ (or simply potential, since electric is understood) to be the potential energy per unit charge:
$V = PE q . V = PE q . size 12{V= { {"PE"} over {q} } "." } {}$
This is the electric potential energy per unit charge.
$V = PE q V = PE q size 12{V= { {"PE"} over {q} } } {}$
Since PE is proportional to $qq size 12{q} {}$ , the dependence on $qq size 12{q} {}$ cancels. Thus $VV size 12{V} {}$ does not depend on $qq size 12{q} {}$. The change in potential energy $ΔPEΔPE
size 12{?"PE"} {}$ is crucial, and so we are concerned with the difference in potential or potential difference $ΔVΔV size 12{?V} {}$ between two points, where
$ΔV = V B − V A = Δ PE q . ΔV = V B − V A = Δ PE q . size 12{ΔV=V rSub { size 8{B} } - V rSub { size 8{A} } = { {Δ"PE"} over {q} } "." } {}$
The potential difference between points A and B, $VB–VAVB–VA size 12{V rSub { size 8{B} } -V rSub { size 8{A} } } {}$, is thus defined to be the change in potential energy of a charge $q q size 12{q}
{}$ moved from A to B, divided by the charge. Units of potential difference are joules per coulomb, given the name volt (V) after Alessandro Volta.
$1 V = 1 J C 1 V = 1 J C size 12{1" V=1 " { {J} over {C} } } {}$
The potential difference between points A and B, $VB-VAVB-VA size 12{V rSub { size 8{B} } -V rSub { size 8{A} } } {}$, is defined to be the change in potential energy of a charge $q q$ moved from A
to B, divided by the charge. Units of potential difference are joules per coulomb, given the name volt (V) after Alessandro Volta.
$1 V = 1 J C 1 V = 1 J C$
The familiar term voltage is the common name for potential difference. Keep in mind that whenever a voltage is quoted, it is understood to be the potential difference between two points. For example,
every battery has two terminals, and its voltage is the potential difference between them. More fundamentally, the point you choose to be zero volts is arbitrary. This is analogous to the fact that
gravitational potential energy has an arbitrary zero, such as sea level or perhaps a lecture hall floor.
In summary, the relationship between potential difference (or voltage) and electrical potential energy is given by
$Δ V = ΔPE q and ΔPE = qΔV . Δ V = ΔPE q and ΔPE = qΔV . size 12{?V= { {?"PE"} over {q} } " and "D"PE="q?V "." } {}$
Potential Difference and Electrical Potential Energy
The relationship between potential difference (or voltage) and electrical potential energy is given by
$Δ V = ΔPE q and ΔPE = qΔV . Δ V = ΔPE q and ΔPE = qΔV . size 12{?V= { {?"PE"} over {q} } " and "D"PE="q?V "." } {}$
The second equation is equivalent to the first.
Voltage is not the same as energy. Voltage is the energy per unit charge. Thus a motorcycle battery and a car battery can both have the same voltage (more precisely, the same potential difference
between battery terminals), yet one stores much more energy than the other since $ΔPE =qΔVΔPE =qΔV$. The car battery can move more charge than the motorcycle battery, although both are 12 V
Calculating Energy
Suppose you have a 12.0 V motorcycle battery that can move 5000 C of charge, and a 12.0 V car battery that can move 60,000 C of charge. How much energy does each deliver? (Assume that the numerical
value of each charge is accurate to three significant figures.)
To say we have a 12.0 V battery means that its terminals have a 12.0 V potential difference. When such a battery moves charge, it puts the charge through a potential difference of 12.0 V, and the
charge is given a change in potential energy equal to $ΔPE =qΔVΔPE =qΔV$.
So to find the energy output, we multiply the charge moved by the potential difference.
For the motorcycle battery, $q=5000 Cq=5000 C$ and $ΔV=12.0 VΔV=12.0 V$. The total energy delivered by the motorcycle battery is
$ΔPE cycle = 5000 C 12.0 V = 5000 C 12.0 J/C = 6.00 × 10 4 J. ΔPE cycle = 5000 C 12.0 V = 5000 C 12.0 J/C = 6.00 × 10 4 J.$
Similarly, for the car battery, $q=60,000Cq=60,000C size 12{q="60","000"" C"} {}$ and
$ΔPE car = 60,000 C 12.0 V = 7.20 × 10 5 J. ΔPE car = 60,000 C 12.0 V = 7.20 × 10 5 J.$
While voltage and energy are related, they are not the same thing. The voltages of the batteries are identical, but the energy supplied by each is quite different. Note also that as a battery is
discharged, some of its energy is used internally and its terminal voltage drops, such as when headlights dim because of a low car battery. The energy supplied by the battery is still calculated as
in this example, but not all of the energy is available for external use.
Note that the energies calculated in the previous example are absolute values. The change in potential energy for the battery is negative, since it loses energy. These batteries, like many electrical
systems, actually move negative charge—electrons in particular. The batteries repel electrons from their negative terminals (A) through whatever circuitry is involved and attract them to their
positive terminals (B) as shown in Figure 19.3. The change in potential is $ΔV=VB–VA=+12 VΔV=VB–VA=+12 V$ and the charge $qq$ is negative, so that $ΔPE=qΔVΔPE=qΔV$ is negative, meaning the potential
energy of the battery has decreased when $qq$ has moved from A to B.
How Many Electrons Move through a Headlight Each Second?
When a 12.0 V car battery runs a single 30.0 W headlight, how many electrons pass through it each second?
To find the number of electrons, we must first find the charge that moved in 1.00 s. The charge moved is related to voltage and energy through the equation $ΔPE= qΔVΔPE= qΔV$. A 30.0 W lamp uses 30.0
joules per second. Since the battery loses energy, we have $ΔPE=–30.0 JΔPE=–30.0 J$ and, since the electrons are going from the negative terminal to the positive, we see that $ΔV=+12.0 VΔV=+12.0 V$.
To find the charge $q q size 12{q} {}$ moved, we solve the equation $ΔPE= qΔVΔPE= qΔV$:
$q = ΔPE ΔV . q = ΔPE ΔV .$
Entering the values for $ΔPEΔPE size 12{?"PE"} {}$ and $ΔVΔV$, we get
$q=–30.0 J+12.0 V=–30.0 J+12.0 J/C=–2.50 C.q=–30.0 J+12.0 V=–30.0 J+12.0 J/C=–2.50 C.$
The number of electrons $nene size 12{n rSub { size 8{e} } } {}$ is the total charge divided by the charge per electron. That is,
$ne = –2.50 C –1.60 × 10 –19 C/e – = 1.56 × 10 19 electrons. ne = –2.50 C –1.60 × 10 –19 C/e – = 1.56 × 10 19 electrons.$
This is a very large number. It is no wonder that we do not ordinarily observe individual electrons with so many being present in ordinary systems. In fact, electricity had been in use for many
decades before it was determined that the moving charges in many circumstances were negative. Positive charge moving in the opposite direction of negative charge often produces identical effects;
this makes it difficult to determine which is moving or whether both are moving.
The Electron Volt
The energy per electron is very small in macroscopic situations like that in the previous example—a tiny fraction of a joule. But on a submicroscopic scale, such energy per particle (electron,
proton, or ion) can be of great importance. For example, even a tiny fraction of a joule can be great enough for these particles to destroy organic molecules and harm living tissue. The particle may
do its damage by direct collision, or it may create harmful x rays, which can also inflict damage. It is useful to have an energy unit related to submicroscopic effects. Figure 19.4 shows a situation
related to the definition of such an energy unit. An electron is accelerated between two charged metal plates as it might be in an old-model television tube or oscilloscope. The electron is given
kinetic energy that is later converted to another form—light in the television tube, for example. (Note that downhill for the electron is uphill for a positive charge.) Since energy is related to
voltage by $ΔPE=qΔV,ΔPE=qΔV,$ we can think of the joule as a coulomb-volt.
On the submicroscopic scale, it is more convenient to define an energy unit called the electron volt (eV), which is the energy given to a fundamental charge accelerated through a potential difference
of 1 V. In equation form,
$1 eV = 1.60 × 10 –19 C 1 V = 1.60 × 10 –19 C 1 J/C = 1.60 × 10 –19 J. 1 eV = 1.60 × 10 –19 C 1 V = 1.60 × 10 –19 C 1 J/C = 1.60 × 10 –19 J.$
On the submicroscopic scale, it is more convenient to define an energy unit called the electron volt (eV), which is the energy given to a fundamental charge accelerated through a potential difference
of 1 V. In equation form,
$1 eV = 1.60 × 10 –19 C 1 V = 1.60 × 10 –19 C 1 J/C = 1.60 × 10 –19 J. 1 eV = 1.60 × 10 –19 C 1 V = 1.60 × 10 –19 C 1 J/C = 1.60 × 10 –19 J.$
An electron accelerated through a potential difference of 1 V is given an energy of 1 eV. It follows that an electron accelerated through 50 V is given 50 eV. A potential difference of 100,000 V (100
kV) will give an electron an energy of 100,000 eV (100 keV), and so on. Similarly, an ion with a double positive charge accelerated through 100 V will be given 200 eV of energy. These simple
relationships between accelerating voltage and particle charges make the electron volt a simple and convenient energy unit in such circumstances.
Connections: Energy Units
The electron volt (eV) is the most common energy unit for submicroscopic processes. This will be particularly noticeable in the chapters on modern physics. Energy is so important to so many subjects
that there is a tendency to define a special energy unit for each major topic. There are, for example, calories for food energy, kilowatt-hours for electrical energy, and therms for natural gas
The electron volt is commonly employed in submicroscopic processes—chemical valence energies and molecular and nuclear binding energies are among the quantities often expressed in electron volts. For
example, about 5 eV of energy is required to break up certain organic molecules. If a proton is accelerated from rest through a potential difference of 30 kV, it is given an energy of 30 keV (30,000
eV) and it can break up as many as 6000 of these molecules ($30,000 eV ÷ 5 eV per molecule= 6000 molecules30,000 eV ÷ 5 eV per molecule= 6000 molecules$). Nuclear decay energies are on the order of 1
MeV (1,000,000 eV) per event and can, thus, produce significant biological damage.
Conservation of Energy
The total energy of a system is conserved if there is no net addition (or subtraction) of work or heat transfer. For conservative forces, such as the electrostatic force, conservation of energy
states that mechanical energy is a constant.
Mechanical energy is the sum of the kinetic energy and potential energy of a system; that is, $KE+PE = constantKE+PE = constant size 12{"KE"+"PE=constant"} {}$. A loss of PE of a charged particle
becomes an increase in its KE. Here PE is the electric potential energy. Conservation of energy is stated in equation form as
$KE+PE = constantKE+PE = constant size 12{"KE"+"PE=constant"} {}$
$KE i + PE i = KE f + PE f , KE i + PE i = KE f + PE f , size 12{"KE" rSub { size 8{i} } +"PE" rSub { size 8{i} } "=KE" rSub { size 8{f} } +"PE" rSub { size 8{f} } ,} {}$
where i and f stand for initial and final conditions. As we have found many times before, considering energy can give us insights and facilitate problem solving.
Electrical Potential Energy Converted to Kinetic Energy
Calculate the final speed of a free electron accelerated from rest through a potential difference of 100 V. (Assume that this numerical value is accurate to three significant figures.)
We have a system with only conservative forces. Assuming the electron is accelerated in a vacuum, and neglecting the gravitational force (we will check on this assumption later), all of the
electrical potential energy is converted into kinetic energy. We can identify the initial and final forms of energy to be $KEi =0, KEf =½mv2, PEi =qV, and PEf =0.KEi =0, KEf =½mv2, PEi =qV, and PEf =
Conservation of energy states that
$KE i + PE i = KE f + PE f . KE i + PE i = KE f + PE f .$
Entering the forms identified above, we obtain
$qV = mv 2 2 . qV = mv 2 2 . size 12{ ital "qV"= { size 8{1} } wideslash { size 8{2} } ital "mv" rSup { size 8{2} } "." } {}$
We solve this for $vv size 12{v} {}$:
$v = 2 qV m . v = 2 qV m . size 12{v= sqrt { { {2 ital "qV"} over {m} } } "." } {}$
Entering values for $q, V, and m q, V, and m size 12{q, V", and "m} {}$ gives
$v = 2 –1.60 × 10 –19 C –100 J/C 9.11 × 10 –31 kg = 5.93 × 10 6 m/s. v = 2 –1.60 × 10 –19 C –100 J/C 9.11 × 10 –31 kg = 5.93 × 10 6 m/s.$
Note that both the charge and the initial voltage are negative, as in Figure 19.4. From the discussions in Electric Charge and Electric Field, we know that electrostatic forces on small particles are
generally very large compared with the gravitational force. The large final speed confirms that the gravitational force is indeed negligible here. The large speed also indicates how easy it is to
accelerate electrons with small voltages because of their very small mass. Voltages much higher than the 100 V in this problem are typically used in electron guns. Those higher voltages produce
electron speeds so great that relativistic effects must be taken into account. That is why a low voltage is considered (accurately) in this example. | {"url":"https://openstax.org/books/college-physics/pages/19-1-electric-potential-energy-potential-difference","timestamp":"2024-11-09T23:11:46Z","content_type":"text/html","content_length":"631053","record_id":"<urn:uuid:b4b120b4-d0a0-4025-853b-b23bbf984826>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00718.warc.gz"} |
7/27 - Homework Week 7 | Legion of Learners
There was no class on Sunday last week so there is no recording for that day.
1. Use recursion to determine the nth term of an arithmetic series. Parameters: starting term, common difference, n. The formula for the nth term is a_n=a_(n-1)+d.
2. Write a lambda that checks if a tuple of integers is a square. Then apply map() and print out the list result. (Hint: one approach is to use math.sqrt()).
Slides: https://docs.google.com/presentation/d/1geOoU09YAzWQuYqReMEQ-y8erSqyeew6ESyH28_RRD0/edit?usp=sharing
Class code: https://replit.com/join/lqjxuqxbqy-shravyas
You can always email us at shravyassathish@gmail.com (mentor) or kingericwu@gmail.com (TA) or message us on Discord if you have any questions about the homework or what we covered in class.
3 Comments
# homework 1
def arithmetic(start, d, n):
if n==1:
return start
return arithmetic(start, d, n-1)+d
# homework 2
import math
tuple1 =(1,2,4,5,9,100,24,49,50)
l1 =list(map(lambda x: math.sqrt(x).is_integer(), tuple1))
def recursionSequence(n, d):
x = 0
nthTerm = x + d
for i in range(n):
nthTerm += nthTerm
nthTerm += d
x += nthTerm
return nthTerm
tuple1 = (1, 2, 3, 4, 5)
import math
print(list(map(lambda x : math.sqrt(x) == int(math.sqrt(x)), tuple1)))
import math
Part 1:
def term(start, diff, nTerm, nTerm2):
nTerm = start + diff*(nTerm-1)
return f"Term #{nTerm2} of the arithmetic series with first term {start} and common difference {diff}: {nTerm}"
print(term(1, 2, 3, 3))
Part 2:
tuple = (1, 9, 4, 16, 15, 13, 54, 16, 257, 322, 64, 987)
print(list(map(lambda i : int(math.sqrt(i)) == math.sqrt(i), tuple))) | {"url":"https://www.lol-101.com/classrooms/introduction-to-python/7-27-homework-week-7","timestamp":"2024-11-09T13:12:47Z","content_type":"text/html","content_length":"1050479","record_id":"<urn:uuid:d90dbeb3-f215-4b86-9e3c-ccb0b584ba27>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00167.warc.gz"} |
A ball is dropped from 81 feet. It hits the ground and bounces up 1/3 as high as it fell. This is true for each successive bounce. What height does the ball reach on its 4th bounce?
Find an answer to your question ✅ “A ball is dropped from 81 feet. It hits the ground and bounces up 1/3 as high as it fell. This is true for each successive bounce. What ...” in 📘 Mathematics if
you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions.
Search for Other Answers | {"url":"https://edustrings.com/mathematics/220138.html","timestamp":"2024-11-13T12:13:04Z","content_type":"text/html","content_length":"22778","record_id":"<urn:uuid:1a010768-a7fa-45a2-ad2a-e257f5024838>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00038.warc.gz"} |
Should I take trig or pre calc first?
Should I take trig or pre calc first?
Trigonometry is more important. From experience, Algebra 2 and precalculus are very similar, and you will probably do fine in calculus without precalculus if you know algebra 2 and trig.
Is Algebra 1 or 2 harder?
Well, Algebra classes are really hard to define because every school does things differently. However, Algebra 1 should be easier than Algebra 2. Algebra 1 introduces you the basic concepts and exams
them in easy ways.
What is the hardest part of chemistry?
The 6 Most Difficult Aspects of Chemistry…and How to Overcome Them
1. Chemistry Involves Concepts That Are Not Easily Observed.
2. Understanding Chemistry Requires Integrated Brain Skills.
3. The Study of Chemistry is Linear.
4. Chemistry Involves A Lot of Math!
5. It’s ALL About the Exceptions.
6. Online Chemistry is Challenging At Best.
What is the hardest part of precalculus?
The hardest part of pre-calculus is the course that precedes pre-calculus. If you’re going to do well in pre-calculus, you need to do well in the course that comes before it.
Is senior year easy?
Senior year isn’t easy. You often hear that senior year is easy, or at least it’s easier than junior year. Granted, this depends on how rigorous your schedule is, but I have found that senior year is
the hardest year of high school. Classes are harder, sure, but that isn’t the half of it.
Is Pre Calc harder than Trig?
Precalculus encompasses both trig and math analysis; therefore a precalculus course will cover more topics than just a trigonometry course alone. Why is precalculus hard? Now, most students agree
that math analysis is “easier” than trigonometry, simply because it’s familiar (i.e., it’s very similar to algebra).
Is ALG 2 Trig hard?
Algebra 2/trig is a bit harder. But it is not that bad, as long as you keep up with your work DAILY. I found geometry easier than any of the other math courses in high school. If I were to rank them
out in terms of difficulty it would be Algebra I, Geometry, Algebra II, then Trigonometry.
What is the hardest unit in calculus?
In a poll of 140 past and present calculus students, the overwhelming consensus (72% of pollers) is that Calculus 3 is indeed the hardest Calculus class. This is contrary to the popular belief that
Calculus 2 is the hardest Calculus class. So, Calculus 3 is the hardest Calculus class.
Is Calculus 1 hard in college?
Calculus the language of motion and change. 1Calculus is a hard class. I mean, it was hard. There are so many pretty straightforward math concepts like; solving an equation, factoring an equation,
use the quadratic formula many times, etc. In calculus, I had to combine a bunch of ideas.
Is calc harder than pre calc?
Calculus is harder than Pre-Calculus. Pre-calculus gives you the basics for Calculus… just like arithmetic gives you the basics for algebra… etc. They are all building blocks that are very important
in your “math development.” | {"url":"https://uruvideo.com/should-i-take-trig-or-pre-calc-first/","timestamp":"2024-11-14T01:49:16Z","content_type":"text/html","content_length":"45080","record_id":"<urn:uuid:3ec58bf2-b6d9-48b6-9948-74a161177f53>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00815.warc.gz"} |
2018 Guest Column by IBM’s New CEO Arvind Krishna Predicts “. . Practical Future of Quantum Computing Closer Than We Think” - Inside Quantum Technology
India’s Business Insider Today shares a guest column about quantum computing written by IBM’s new CEO Arvind Krishna in 2018. Arvind Krishna explained in that article why practical future of quantum
computing is closer than we think. He wrote “Quantum computing must also be ubiquitous in classroom. From computer science courses to chemistry and business classes, today’s students need to
understand the technology, and hopefully pursue career paths rooted in quantum computing.”
Here’s what he said about Quantum’s potential: “At their most basic, quantum computers process information in a completely different way than classical computers. Instead of classical bits, binary
zeros and ones that work one after another, the principles of quantum mechanics give quantum bits (or qubits) exponential compute power. Qubits can represent zeros and ones simultaneously, and they
use this “superposition” capability to work together to solve problems,” he wrote. “Because they’re able to exist in more than one state at a time, qubits supercharge the output quantum computers can
generate – enabling us to run experiments more efficiently. This could lead to everything from quantum chemistry that drives drug discovery breakthroughs, to quantum algorithms that optimize global
manufacturing supply chains,” he believed.
NOTE: IBM’s Dr. Robert Sutor, VP of Quantum Ecosystem Development, will keynote the upcoming IQT Event in NYC. IBM is a Diamond Sponsor of the IQT Event as well. | {"url":"https://www.insidequantumtechnology.com/news-archive/2018-guest-column-by-ibms-new-ceo-arvind-krishna-predicts-practical-future-of-quantum-computing-closer-than-we-think/amp/","timestamp":"2024-11-06T01:29:13Z","content_type":"text/html","content_length":"41063","record_id":"<urn:uuid:4d703c55-d521-42c6-8b0a-59390ddefec9>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00220.warc.gz"} |
On type:
attitudinal: moved deeply - unimpressive
forced intent
attitudinal: determination - lack of determination - resignation
letteral for i when ending a diphthong (ai, ei, oi), sometimes written as ị
vocative: slightly piratical greetings
attitudinal: 🥺
obedience - rebellion
attitudinal: happy for other's gain – neutral about other's gain – jealous
combination of "a'o" (hope) with "au" (desire): hopeful wish.
attitudinal: peek-a-boo!
attitudinal: cute (attractive in an innocent way) - lack of cuteness - repulsively uncute
letteral for u when ending a diphthong (au), sometimes written as ụ
letteral for a.
discursive suffix: attaches to number. "I expect with probability..."
tense: refers to future of current space/time reference absolutely
forethought emphasis indicator; indicates next WORD only, even if that word is a construct-indicating cmavo
unary mathematical operator: successor/augment/increment (by one), succ(a) = a++ = a+1
mekso string operator (ternary): find-and-replace; in string/text/word/sequence X1 formally replace X2 (ordered tuple of terms to be replaced) with X3 (ordered tuple of terms to be respectively
mekso operator: in ordered tuple/list/vector/sequence X[1], replace the X[2]th entry with term X[3] of appropriate type, and leave all other entries untouched (optional: where the index for the
very first/leading/header entry is X[4]).
Converts PA into tense; in [number (usually nonspecific)] possible futures where [sumti (du'u)] is true
preposition: outside of ...
benre modal, 1st place for beneficiary...
ternary mekso operator: x[1]th Bergelson multiplicative interval with exponents bounded from above by function x[2] and with sequence of shifts x[3], where exponents belong to set x[4]
quotes a non-Lojban letter
quotes non-Lojban letters (delimited)
Multiple sumti link; open-ended sumti-linking.
Tag linkarge 2
unary mekso operator: nth Bernoulli number B[n] of the second kind (B[1 ]= +1/2 = >0).
Tag linkarge
emphasis indicator; indicates the previous word is especially emphasized
Sumtcita indicating that the bridi must be true under the conditions indicated by the tagged sumti.
attitudinal: bzz, bee/other insect sound
number/digit: 2^(2×5/3) = 8×(2^(1/3)).
non-logical interval connective: ordered interval with specified endpoint/terminus x[1] and signed measure/length/duration x[2]; interval between x[1] and x[1 + x] according to the ordering of
the space.
digit/number: interval/range indicator for significant digits (determined by lesser endpoint).
pro-sumti: definitional terbri variable 1; x[1]
convert number into pro-sumti: definitional terbri variable with associated number; x[n]
non-logical connective: in superposition with
pro-sumti: definitional terbri variable 2; x[2]
single Lojban-word name quote, turns to selbri-unit: "x[1] is that which has this name"
pro-sumti: definitional terbri variable 3; x[3]
unary mekso operator: immediately convert number into a single digit.
immediately convert symbolic string to number (explicit)
mekso operator/function terminator (in Polish notation): inserts exactly enough "{boi}"'s consecutively so as to terminate the most recently uttered operator/function in a mekso expression
preserve formal interpretation of mekso subexpression with no substitutions made
explicit "mu'o'u"-orientation marker
pro-sumti: definitional terbri variable 4; x[4]
attitudinal: silliness - maturity
pro-sumti: definitional terbri variable 5; x[5]
naturalistic interjection: used to express a fit of overwhelming or uncontrollable laughter; the stereotypical 'evil laugh'
abstractor: abstractor to create logically quantified selbri variable to be used in predicate logic of third or higher order.
elliptical/unspecified/vague letteral/symbol
Interjection: Boo!
digit/number: eighteen (decimal 18).
Refers to what is usually assumed to be the argument of tense tags when no explicit argument is given
Evidential: "because I say so"/"because I assert it to be true"/"it is known".
shorthand for {ca lo nu}; during the event-of
Unary mekso operator: unit vector/normalization of the argument (vector) X.
strong scalar intensifier: extremely...
Contraction of ca'o co'a. Expresses that the event is progressively beginning.
mekso 4-nary operator: spherical harmonics on colatitudinal/polar angle a and azimuthal/longitudinal angle b of unassociated order c and associated order d.
binary operator: complex number from argument and phase, (r, \phi) \mapsto r e^i \phi
elliptical/unspecified truth value; "maybe, maybe not"
begin lines of serial dialogue
elliptical/unspecified scalar modifier: "maybe, maybe not", modifies next selbri-unit or tag
modifier / discursive: elliptical affirmative; "maybe, maybe not"
Terminate "ca'u'e" construct.
elliptical/vague/indecisive scalar affirmer/negator; true neutral/non-committed/uninvolved scalar truthfulness/assertion.
lambda variable prenex; marks the end of introduction of lambda-scope variables.
Formulates periodic ordered list of exactly the connectands in the order presented.
BIhI argument modifier: indicates dimensionality/length of tuple
x[1] is the year / era of years indicated by PA (digit string) in calendar system x[2]
define following selbri with sentence or tu'e...tu'u clause
pro-bridi: the universal predicate
extracts an element from a set, category, class, group, collection, organization, system, etc.
argument list separator: acts as a comma between arguments in an argument list supplied to a function.
metadata tag / hashtag
discursive: marks an utterance as using something that is experimental/not official, especially experimental grammar
Mekso unary or binary operator: n-set or integer interval; in unary form, it maps a nonnegative integer X[1 ]= n to the set \1, \dots , n\ (fully, officially, and precisely: the intersection of
(a) the set of exactly all positive integers with (b) the closed ordered interval [1, n] such that n \geq 1; see notes for other n); in binary form, it maps ordered inputs (X[1, X]= (m, n) to the
intersection of (a) the set of exactly all integers with (b) the closed ordered interval [m, n].
attitudinal-indicator-to-selbri conversion: x[1] feels the indicated emotion toward/about x[2].
mekso at-most-4-ary operator: integer lattice ball; the set of all points belonging to the intersection of Z^n with the closure of the ball that is centered on X[1] and has radius X[2] in metric
X[3], where Z is the set of all integers and where, for any set A and non-negative integer n, A^n is the set of all n-tuples such that each coordinate/entry/term belongs to A, and where the
dimensionality n = X[4]..
elidable terminator: end conversion of attitudinal indicator to selbri; elidable except before further attitudinals.
attitudinal: pride about having just invented a new attitudinal - shame about having just invented a new attitudinal
quotes an emoji name surrounded by pauses; can be non-Lojban
transfnite cardinal beth
transfinite ordinal little-omega; if followed by a number, it denotes the ordinal which is little-omega subscripted therewith in English notation
transfinite ordinal little-epsilon; if followed by a number, in English notation, the following number would be denoted by a subscript
mekso operator (binary): projection function; the Bth term/entry ("element") of tuple A
Converts following cmevla or zoi-quote into psychomime.
superfective inchoative; starting too late
placed after a tanru with CO, following sumti return to the place structure of the left tanru unit.
Non-committal/agnostic/unassertive/abstaining/neutral tanru inversion.
interval event contour: at the restarting/renewal point of...
trivial selbri inversion: does not invert tanru (seltau first/left, tertau second/right).
Elliptical elliptical. Beware, its grammar is also elliptical.
x[1] is PA hours in duration by standard x[2].
asks the listener to provide a word matching the preceding description being the seltau
elliptical/vague/unspecific/generic tanru inversion
Combination of {coi} and {co'o}, indicating either greetings or partings according to context; may also constitute a greeting in passing (such as {coico'o}).
interval event contour: premature cessation; done although not finished; at the ending point of ... even though not completed.
tanru inversion question: asks for a CO cmavo.
binary mathematical operator: vector norm/magnitude of vector a in structure (normed vector space) b.
universal famyma'o: terminates the most recently opened construct or clause.
mekso binary/unary operator: multinomial coefficient/binomial coefficient/choose
discursive: indicate a change in speaker; used generally in quotations.
discursive: indicate a change in speaker to ko'a; used generally in quotations.
discursive: indicate a change in speaker to first person (mi); used generally in quotations.
discursive: indicate a change in speaker to ko'e; used generally in quotations.
discursive: indicate a change in speaker to general third person (zo'e); used generally in quotations.
discursive: indicate a change in speaker to ko'i; used generally in quotations.
discursive: indicate a change in speaker to ko'o; used generally in quotations.
discursive: indicate a change in speaker to second person (do); used generally in quotations.
discursive: indicate a change in speaker to ko'u; used generally in quotations.
attitudinal: intentionally - unintentionally - against intention.
convert number to statistical odds selbri; event x[1] (nu) has statistical odds (n) of occurring (versus not occuring) under conditions x[2].
discursive: verbosely – succinctly.
mekso operatory: prime mark append
other than me
"da'avni" modal: except under condition ...; unless ...; excepting/exempting (condition) ... .
pro-sumti: forgetful something/memory-less da
attitudinal cause attribution
attitudinal modifier: marks preceding attitudinal as empathetic with an anticipated attitude; encourages another's feelings.
attitudinal modifier: supposed emotion - factual emotion
discursive & gafyzmico: reset/restore all defaults (permanently) to discourse-exterior specification; cancel all following discourse-interior default assignments
attitudinal attribution
gafyzmico: Reset all default specifications of immediately previous word to official definition specifications hereinafter (permanently)
Numeral: Some but not all.
attitudinal: equal intensity attitudinal relativizer
Quantifier modifier: endowment of existential import - repeal of existential import/abstention from claiming existence - assertion of non-existence
variable identifier article: refer to the referents of the variable having the following predicate as its name; such a variable may be implicitly bound by {PA broda} or {LE broda} phrases; if no
such variable has been previously bound, the referents are left to the context to determine; the referents are not claimed to actually satisfy the predicate after which the variable is named.
vocative: pausing conversation
default value (re)specification/(re)assignment/(re)definition/over-write; set new default value (terbri-specific; permanent)
gafyzmico: Cancellation (permanent) of all defaults in immediately previous word
mekso ternary operator: positive super-logarithm; the super-logarithm (inverse operator of hyper-operator with respect to "height" of power tower) of a with base b and of order c-2.
on (n)-th day from a given point (by default from today)
in the Nth century.
tense interval modifier: decreasingly...; decrementative. Tagged sumti, if present, indicates amount of decrease (lo te jdika)
mekso binary operator: Lambert product-log W function; W(a, b)
in the year N.
pro-sumti: the next word
pro-sumti: this entire document/text
in the Nth month.
non-logical connective: set difference of x[1] and x[2]: x[1dei'ix]=x[1\setminusx]=\x\inx[1:x\notinx]
Attaches date and time starting with years and ending in seconds.
on the Nth day of the month.
pro-sumti: this word
on the Nth day of the week.
pro-sumti: the previous word
Cancellation (instant-/usage-wise; temporary) of all defaults in immediately previous word
vocative: resuming conversation
vocative: well-wish - curse
pro-sumti and sumyzmico: discourse-interior default it (terbri-specific)
pro-sumti and sumyzmico: an elliptical/unspecified value which does not necessarily obey the default setting for the corresponding terbri that is explicitly specified in the definition of the
word; has some value which makes bridi true
mathematical ternary operator: Dirichlet convolution (a×b)(c)
pro-sumti & sumyzmico: discourse-exterior default it
discursive & gafyzmico: ignore/kill all following default specifications (permanently)
pro-sumti whose referent's identity may be chosen by the listener
impersonal pronoun; generic-you; generic-one; a generalized person
gafyzmico: Reset all default specification of the immediately previous word to their respective discourse-external/official definition specifications for this instance/usage only.
audience switch marker.
x[1] is PA days in duration by standard x[2].
vocative: to the attention of, CC, carbon-copy(ing) ...
Quotes a word; the result is a pro-sumti for the most recent utterance containing that word.
generic single-word generic vocative marker; identifies intended listener with a single, possibly non-Lojban word, delimited by pauses (in speech) or by whitespace (in writing).
pro-sumti: the original speaker (= you the primary listener/target) and the rest of the audience, excluding the new/current speaker (= me).
question word: which utterance?
mekso n-ary ordered operator: structure creator/ordered tuple, 'endow'; the structure formed by underlying set X[1] (as) endowed with element, order, quoted operator, etc. X[2], X[3], ...
converts a sumti into a tanru-unit with place structure "x1 is/are the referents of [the sumti]".
mekso binary operator: extract substructure/underlying set/endowing operator; the substructure (general sense; includes just operator, order, set, etc.) of X[1] (structure; explicitly given by
{du'a'e}) which is formed by collecting the ith entries of that {du'a'e}-tuple in order together into their own {du'a'e}-tuple (or by extracting them naked into the ambient environment if X[2] is
a singleton) for all i in set X[2]
Text to bridi conversion
mekso binary operator: left-handed vectorial cross product (ordered input), -a \times b = b \times a (if using right-hand convention - notice the negative sign/operator or order).
location tense relation/direction (angular); clockwise from..., locally rightwards/to the right of ...
digit/number: twenty (decimal 20).
mix of .e'a (permission) + .e'e (encouragement) + .e'i (constraint) + .e'o (request) + .e'u (suggestion) + .au (desire)
naturalistic interjection: shy giggle
attitudinal: appeal/call/invocation/summoning
Attitudinal: heart melting and opening - unmoved - heart freezing and withdrawing
irrealis attitudinal: competence (ja'ai) - incompetence (nai)
irrealis attitudinal: constraint (ja'ai) - independence (cu'i) - resistance against constraint (nai)
Attitudinal: advice/posit/well-considered idea - spitballing/proposal (no investment or confidence in it being a good idea) - rejection of suggestion/idea
logical connective: sumti afterthought always true.
letteral for e.
mathematical ordered n-ary operator: (pointwise) functional left composition; X[1 \circ X].
mekso k-ary operator, for natural k and 1 < k < 5: ordered input (f, g, S, m) where f and g are functions, S is a set of positive integers or "ro" (="all"), and m is 0 or 1 (as a toggle); output
is a function equivalent to the function f as applied to an input ordered tuple with g applied to the entries/terms with indices in S (or to all entries/terms if S="ro") if m=0, or g
left-composed with the same if m=1.
mathematical unary operator: map notation
nonce place; tags sumti with an unspecified connection to the bridi
Unary mekso operator: reverse finite ordered sequence, tuple, list, string, etc.
translation marker - original/native version: marks a construct as having been translated and therefore particularly (possibly, but not necessarily) susceptible to the errors or limitations
associated with translation (especially if the translator is unsure of the best result/option)
digit/number: the second Feigenbaum constant α = 2.502907875095892822283902873218...
digit/number: first Feigenbaum constant δ = 4.669 201 609 102 990 671 853 203 821 578(...).
Pendent preposition. Introduces a dangling argument that doesn't take part in the surrounding argument structure; similarly to {zo'u}, it may be used for pre-declaring a quantification, or for
introducing an argument that will be later referred to anaphorically. The pronoun {zoi'i} is automatically bound to the argument introduced by {fai'i}.
digit/number: phi, the "golden ratio" (approx. = 1.6180...); the ratio a/b such that it equals (a+b)/a
digit/number: the fine-structure constant \alpha \approx 7.2973525698(\dots)×10^(-3) \approx 1/137.035999074.
sentence fence: the speaker is done speaking and signals to the addressee that they may now speak up if they so desire.
end of all Lojban text, forever.
Convert operator to being entrywise.
non-logical connective: antirespectively; unmixed but reverse ordered distributed association.
Iteratively applies "fau'a" to each resultant operator until all operators resolve.
two-tier function map/assignment writer notation: X1 (ordered list, no repetitious terms) maps termwise-respectively to X2 (ordered list; may be repetitious but must have exactly as many terms as
iterated function left-composition with self: f∘f∘...∘f, n times.
mekso ternary operator: inverse function of input function X[1] with respect to its input X[2], taken on branch or restricted domain X[3] ("domain" being of X[1]).
Forgive me!/I'm sorry!
mekso ternary operator: positive super-root; the bth super-root (inverse operator of hyper-operator with respect to base) of a of order c-2.
binary mekso operator: divided by (fraction): a/(b...)
mekso variable-arity (at most ternary) operator: number of prime divisors of number X[1], counting with or without multiplicity according to the value X[2] (1 xor 0 respectively; see note for
equality to -1 and for default value), in structure X[3].
Prefix division by following unit selbri
Vocative: Fin, the end, you may applaud now, conclusion; story ending or punchline marker
marks end of prenex in stack-based dialect
mekso operator: continued fraction, Kettenbruch notation; for ordered input (X[1, X], where: X[1] is an ordered pair of functions and X[2] is a free or dummy variable/input/index which ranges
through set X[3] in order(ing) X[4], the result is K[(X] for Kettenbruch notation K.
ifle modal, 1st place: if [sumti] (du'u or nu) is true...
Right-scoping adverbial clause: encloses a bridi and turns it into an adverbial term; the antecedent (ke'a) of the enclosed bridi stands for the outer bridi {lo su'u no'a ku} (the bridi in which
this fi'oi term appears), including all the other adverbial terms (tags...) within this bridi located on the right of this fi'oi term (rightward scope).
Creates a predicate abstraction sumti out of a full bridi clause, binding all the necessary lambda variables to the ko'a-ko'u pronoun series. The number of bound variables must be indicated by
appending {xi} followed by that number to the word {fo'ai}, unless only one variable (namely {fo'a}) is bound, in which case the {xi} marking is optional.
GOI attachment modifier: allows a cmavo of selma'o GOI (which must follow immediately) to be attached to a selbri
create a relation abstraction sumti out of a full bridi clause, binding the two lambda variables to the {fo'i} and {fo'u} pronouns.
digit/number: first Foias' constant; the unique value of x[1] such that x[n] -> ∞ as n -> ∞ for x[n+1 ]= (1 + (1/( x[n )))^n]; such x[1] = 1.187…
digit/number: second Foias constant; the value x for which (1/x)(1 + (1/x))^x=1 is true; ≈ 2.293…
discursive: luckily - not pertaining to luck - unluckily.
begin within-context quote.
discursive: indicate a change in speaker to fo'a; used generally in quotations.
discursive: indicate a change in speaker to fo'e; used generally in quotations.
discursive: indicate a change in speaker to fo'i; used generally in quotations.
discursive: indicate a change in speaker to fo'o; used generally in quotations.
discursive: indicate a change in speaker to fo'u; used generally in quotations.
digit/number: twenty-one (decimal 21).
Number suffix initiating a subordinate clause representing a predicate whose arity is the suffixed number; the lambda variables representing the predicate slots are bound from the fo'a-fo'u
series in their dictionary ordering; the number of bound variables is the same as the predicate arity.
unary mekso operator: Lorentz-Einstein gamma factor +1/((1-(|X|^2)) for input X.
digit/number: Euler–Mascheroni constant, usually denoted by lowercase gamma (γ); approximately 0.5772156649 (in decimal).
abstractor: sensation / qualia abstractor; x[1] is the sensation / qualia associated with objects with property [bridi, bound to ce'u], via sense x[2], as sensed by x[3]
zgadi/cadga modal; 1st place: with just/expected result ...; should/ought to result in ...
pro-bridi: the empty predicate
evidential: I disagree with you vehemently, but offer no substantive counterarguments/information pulled from my ass
mekso (no-more-than-4-ary) operator: Gaussian function f(x, a, b, c) = c e^-((x-a).
mekso n-ary operator: append contravariant (upper) indices to tensor
ganse modal, 1st place to perceiver ... ; sensed, detected by ...
digit/number: Gauss' arithmetic-geometric mean of 1 and √(2)=sqrt(2) constant G ≈ .8346268…
location tense relation / direction; upon/atop ...
metasyntactic variable prenex
metasyntactic variable marker
mekso 7-ary operator: for input (X[1 ]= z, X[2 ]= (a[i)]= (b[j)]= p, X[5 ]= q, X[6 ]= h[1, X]= h[2)], this word/function outputs/yields \sum[n]=0^\infty (((\prod[i ]= 1^p (ne'o'o(a[i,n,1,h]= 1^q
(ne'o'o(b[j,n,1,h]; by default, X[6 ]= 1 = X[7] unless explicitly specified otherwise.
mekso unary operator: converts a string of digits which includes {pi} to the same string of digits without {pi}; if {pi} is not present in the original/input string, the output is identical
logical connective: forethought all but tanru-internal always true (with gi).
elidable terminator for connective modifiers
logical connective: forethought all but tanru-internal always false (with gi).
pro-sumti: refers to the proposition that is the left-hand side complement of the current logical connection.
afterthought abstraction wrapper
logical connective: bridi-tail afterthought always true.
logical connective: bridi-tail afterthought always false.
pro-bridi: in the second half of a bridi forethought connective expression, refers to the first half.
This cmavo precedes a predicate (at least binary) and turns it into a forethought conjunction, which syntactically behaves like GA cmavo. The predicate indicates the relationship between the two
connected propositions. Terminator: {te'u}.
last bridi (with its modifiers)
marks the tagged sumti as being scope-wide within the immediate parent (text-wide by default) specified in the argument of the vocative
repeat (copy-and-paste) the recontextualized content of the most-recent or indicated NOI relative clause.
assign sentence or tu'e...tu'u group to sumti
pro-bridi: quotes the next word and repeats the most recent bridi containing that word
digit/number: Goloumb-Dickman constant ≈ .6243…
binary operator: left group action g.x
mekso operator, variable arity - algebraic structure order of X1; OR: order of/(size of) period of element X1 in algebraic structure X2 under operator/of type X3
logical connective: tanru-internal forethought always true (with gi).
logical connective: tanru-internal forethought always false (with gi).
predicate modifier: convert x1 to a mereological sum composed of x1.
sentence link/continuation; continuing sentences on same topic; normally elided for new speakers.
logical connective: sumti afterthought always false.
reset bridi-level to zero
attitudinal: trust - lack of trust - distrust
reset bridi-level to zero
attitudinal: finding something reasonable - unreasonable
attitudinal: disgust - attraction
filler word: um, like, y'know
attitudinal: yee-haw!
attitudinal: feeling grounded - feeling spacy
attitudinal: good mental health - bad mental health
attitudinal: sexual arousal - sexual repulsion
attitudinal: sexual consent - revocation of consent
attitudinal: orgasm - impotence
attitudinal: kinky - vanilla
attitudinal: sexual satisfaction - sexual dissatisfaction
letteral for i.
letteral for the i semi-vowel, sometimes written as ĭ
letteral for y
affirm last word: attached to cmavo to affirm them; denies negation by nai whenever it is applicable.
subjective number which is decreasing over time
jai equivalent of la'e
grammatically converts LAhE to SE; semantically the result tags the x[1] of the selbri as being LAhE the supplied x[1]. Can be converted to other than x[1] with SE.
attitudinal: everything coming together - everything falling apart
elliptical presence or absence of "jai".
takes NU or LE NU, turns into sumtcita: clarifies the semantic NU-type of the current bridi.
permutation cycle writer notation start
na'e fancuka modal: no matter (indirect question)...
unary mathematical operator: length/number of components/terms of/in object/array/formal string/sequence/word/text in some alphabet/base/basis which includes each digit; number of digits/
attitudinal modifier: extra properly, primly, "by the book" - rushing things, half-assing.
Break in speech in order to demonstrate something in a nongrammatical or nonlinguistic way.
lambda quantifier: binds a lambda variable.
NAhE question.
discursive: correcting/corrective/correction - inattentive/uncaring/neutral toward the presence of possible errors - permitting (known/likely/plausible) error/incompleteness/approximation
mekso operator: associated Legendre polynomial in a with unassociated order b and associated order c
x[1] is PA weeks in duration by standard x[2].
evidential: I intuit... / I suspect...
logical connective: tanru-internal afterthought always true.
logical connective: tanru-internal afterthought always false.
job descriptor: the one which does …
attitudinal scope modifier: marks following attitudinal/UI-cluster as applying to the last lexical unit
mekso unary operator: determinant, det(A)
animate - inanimate
mekso, at-most-5-ary operator: a rounding function; ordered input list is (x,n,t,m,b) and the output is sgn(x) b^t round[n] (b^(-t) abs(x)), with rounding preference n and where the fractional
part of b^(-t) abs(x) being equal to 1/2 causes the round[n (] ) function to map b^(-t) abs(x) to the nearest integer of form 2Z+m, for base b (determined by context if not explicitly input) and
some integer Z (determined by context).
mekso, at-most-5-ary; rounding function. (See notes).
connective: elliptical/generic/vague
This cmavo precedes a predicate (at least binary) and turns it into a conjunction, which syntactically behaves like JA cmavo. The predicate indicates the relationship between the two connected
propositions. Terminator: {te'u}.
convert a selbri tag followed by a tanru unit to a tanru unit; differently from {jai}, it does not change the first place of the tanru unit.
change version/dialect of parser
shift letterals to Lojban alphabet
nonlogical connective: disjoint union
nonlogical connective (and mekso operator) - symmetric difference of sets
non-logical connective: to be in a (nontrivial) superposition of (states); mixture
mekso string operator (n-ary): formal right-concatenation; X[1 + X], where X[i] is a string/word/text/character/letteral/lerfu/quoted utterance (quote appropriately iff necessary; preserve and be
careful about the use-vs.-mention distinction) for all i.
non-logical connective: the more ..., the more...
Default number radix modifier: changes the value of the default radix assumed for any numeral lacking an explicit radix within the following text, until another {ju'ai} appears.
semi-mathematical binary operator: named number base operator/interpreter
Tight scope bridi separator; analogous to {.i} without ending the abstractor or relative clause.
evidential: I know that...
long-digit interpretation specifier; macrodigit named base specifier
kansa modal, 1 place; with .../with a companion ...
mekso unary operator: cardinality (#, | |)
abstractor: predicate abstractor. x[1] is the predicate expressed by [bridi], using bo'a, bo'e, etc for variables.
digit/number: Conway's look-and-say constant λ ≈ 1.303577269…
evidential: by definition... / essentialistically...
karbi modal: with comparing agent - (SE) compared to - (TE) compared to - (VE) compared according/with respect to - (XE) with comparison...
Adverbial, metacommentary-introducing, and complex discursive or attitudinal: I express this utterance in a manner or with mental state which is described by the immediately following and
enclosed bridi in reference to the tagged construct, or assert that the said immediately following and enclosed bridi is true in application to the tagged construct in a metacommentary style.
evidential: I expect - I deny expecting
abstractor: x[1] (x[2], ...) are such that they satisfy [bridi], using bo'a/bo'e/etc for variables
Property relativizing determiner / unary quantifier constructor. {kai'i} introduces a predicate whose first argument slot becomes filled by the property made by taking the bridi in which this
{kai'i} appears and putting {ce'u} into the argument slot in which this {kai'i} argument was located. Put formally, "kai'i brodi cu brodu" = "lo ka ce'u brodu cu brodi". Additionally, a {kai'i}
term has a rightward logical scope, like quantifiers and adverbials.
quaternion i
abstractor: x[1] (x[2], ...) are such that they satisfy [bridi], binding to the open {ce'u} slots.
imaginary i (non-comma)
imaginary i, comma - spherical coordinates: first coordinate gives magnitude (complex modulus/radius) of the number, the second number gives the angle from the positive real axis measured
counterclockwise toward the 'positive' imaginary axis (default: in the primary branch/Arg) as measured in some units (which that number should contain; the contextless default will suppose
radians); the angle is not normalized.
x[1] (ka) is obtained from x[2] (ka) by uncurrying the first N places
convert bridi into n-ary property claim: x[n] is such that it fills the n-th occurrence of ce'u in [bridi].
pro-sumti: strong-memory something1/eidetic da/elephant thing1 (logically quantified existential, arbitrarily-long-scope pro-sumti)
Microdigit-spanning endianness binary-toggle.
Macrodigit-spanning endianness binary-toggle.
pro-sumti: strong-memory something2/eidetic de/elephant thing2 (logically quantified existential, arbitrarily-long-scope pro-sumti)
pro-sumti: strong-memory something3/eidetic di/elephant thing3 (logically quantified existential, arbitrarily-long-scope pro-sumti)
generic algebra unit e[n]
Predicate to variable-binding binary quantifier. The first slot of the predicate must be a property.
Toggles to no grouping
relative clause prenex: assigns a variable to the object of the relative clause
Attaches all of the following words to the next explicitly mentioned sumti as seltau of that sumti or selbri which is explicitly marked with "cu" (under left-grouping by default).
terminates a JUhAU expression
Toggles to right grouping of tanru/lujvo.
Toggles to left grouping of tanru.
Locks tanru modification order reversal (does not affect lujvo). {ke'e'unai} restores regular order
mekso style converter: elementwise application of operator
mekso operator: finite result set derived from/on set A with/due to operator/function B under ordering of application C
digit/number: hex digit E (decimal 14) [fourteen].
non-logical connective/mekso operator - of arity only 1 xor 2: set (absolute) complement, or set exclusion (relative complement). Unary: X[2 ^C]; binary: X[1\setminusX].
quaternion j
vocative: please repeat in simpler words.
vocative: please repeat over and over in a soft tone that puts me in a trance
reverses modification order of contained tanru (does not affect lujvo).
vocative: please repeat more slowly / more clearly enunciated.
vocative: please repeat in a language I understand better.
accepts number (n) after: repeat last sumti up to n times
attitudinal question: metalinguistic confusion caused by too many experimental cmavo
Converts following selbri, cmevla, or zoi-quote into a nonce interjection/attitudinal.
evidential: I reason - I think impulsively.
se kibzva modal, 1st place: at website/Internet resource...
Creates a predicate abstraction sumti out of a full bridi clause, binding all the necessary lambda variables to the ko'a-ko'u pronoun series.
kosmu modal, 1st place with purpose...
converts the place structure of the following tanru-unit-2 into MINDE semantic frame
UI-cmavo parenthesis/separator: start grouping
create a binary relation abstraction sumti out of a full bridi clause, binding the two lambda variables to the {ko'i} and {ko'u} pronouns.
quaternion k
Pro-sumti: references a following mentioned sumti, but which one is not specified.
Pro-sumti: references a previously mentioned sumti, but which one is not specified.
discursive: imperative/hortative
meaningless chicken clucking
ku'i modal: in contrast to...
empty/vacuous selbri
elidable terminator: end of LOhOI construct
mekso (n+1)-ary operator: q-analog converter - the ath analog of b (quoted operator) applied to operands c, d, ...
Terminator for a CAhEI phrase.
PA: blank/empty digit
quod erat demonstrandum, Q.E.D.
Complex UI construct terminator (elidable).
closing bracket/terminator for mekso expression interpretation modifiers
Number suffix initiating a subordinate clause representing a predicate whose arity is the suffixed number; the lambda variables representing the predicate slots are bound from the ko'a-ko'u
series in their dictionary ordering; the number of bound variables is the same as the predicate arity.
newsworthiness focus marker: indicates the most newsworthy part of the clause.
otherwise lojbanic name, ending in a vowel; multiple names delimited by pauses.
start grammatical name quotation; the quoted text is an identifier and must be grammatical on its own.
the specific referent of [following sumti] defined/specified by the grammar
combines LA with DOI
text scope alphabet specifier. Sets the alphabet used for spelling until changed.
digit/number: hex digit F (decimal 15) [fifteen].
Named reference. It converts a sumti into another sumti. The converted sumti points to the referent the name of which is the referent of the unconverted sumti.
evidential: I experience - I deny experiencing
descriptive descriptor: the one described with description …
single-word non-Lojban name; quotes a single non-Lojban word delimited by pauses and treats it as a name
mekso unary operator: for input X, this outputs X/(1+X).
pronoun: the referent of the following utterance
pronoun: the referent of the preceding utterance
replace recent mistakenly uttered text
Start property which sets the meaning of {le} in the surrounding text.
article: "the thing(s) I have in mind and which I believe appear(s) to you to be…"
anaphoric mass gadri: start a description of a mass/group/constituency mentioned earlier in the text/conversation, viewed as a whole
demonstrative mass gadri; start a definite description that refers to a mass/group/constituency in the shared frame-of-reference, viewed as a whole
x[1] is PA months in duration by standard x[2].
unevaluated mekso as name.
terminator of selma'o LUhEI
arbitrary character string or irregular number
ternary mekso operator: retrieves/gets/outputs the X[2]th entry/term from ordered list X[1] under indexing rules X[3].
lisri modal, 1st place: in/belonging to/of/from story...
marks word/construct as being optional, i.e. the bridi would still be both grammatical AND reflect the speakers opinion/intention would the marked construct be left out
start quote of recent mistakenly uttered text to be replaced
presuppositional definite article: the …; the thing(s) which …
generic essentialistic article: «loi'a broda cu brode» = a broda typically is/does brode; being/doing brode is a typical trait of broda-hood.
generalized mass gadri; start a description of a generalization of some mass/group/constituency viewed as a whole
essentialist mass gadri; start a description of an essential characterization of some mass/group/constituency viewed as a whole
Description clause: create a sumti from the enclosed bridi, describing the referent of the created sumti as filling the bridi place filled with {ke'a}.
plural logic maximum-scope descriptor: those who individually or collectively are ...
Bridi to text conversion
convert a grammatical quotation to a tanru unit; x[1] expresses/says the quoted text for audience x[2] via expressive medium x[3].
selbri conversion: abstracts out a member of x[1] (set/group), moves old x[1] to the fai place
exophoric article: the … I have in mind (identity not necessarily ascertainable from the context).
Binary mekso operator: uniform probability A(X[2)u(X] for input (X[1,X] where X[1] is a number and X[2] is a set or space. (See notes for details).
interval endpoint status question marker
Maybe, but not the desired answer
"True, but not the answer that I expected/desired"
digit/number: Meissel-Mertens constant M ≈ 0.2614972128476427837554268386086958590516…
pro-sumti: the universal argument/value; syntactically-contextually and type-permitted maximally generic in its typing
turns number into pro-sumti: the abstraction described by the utterance denoted by that number and {mai}
unary mekso operator: signum function
unary mekso operator: parity of function; if the input is a unary real-valued function X[1] which is defined on a subset of the reals, then the output is 1 is X[1] is even, -1 if X[1] is odd, and
0 otherwise.
mekso unary operator: for permutation X[1] as input, the output is (-1)^N(X[1)], where N(s) is the number of inversions in permutation s.
mekso unary operator: Levi-Civita symbol; for input n-tuple (a[1, a] ... , a[n)], where n is a strictly positive integer, the output is \varepsilon[a], where \varepsilon is the Levi-Civita symbol
under the convention of mapping (1, 2, ..., n) to 1.
mathematical operator: vague/elliptical/general/generic operator
selma'o quote; quotes a word (a cmavo) and uses it to name a selma'o.
Like "ma'oi" but outputs the officially-designated/canonical sub-selma'o (if any) to which the immediately following and quoted word (cmavo) belongs, otherwise outputting the whole relevant
selma'o in fashion equivalent to "ma'oi".
shortening of {lo du'u ma kau *bridi*}, with {ce'u} bound to {ma kau}
mekso: conversion of operator/function to operand
UI conversion start quote; converts grammatical Lojban text to cmavo of selma'o UI
attitudinal: stronger intensity attitudinal relativizer
UI conversion end quote (elidable terminator); converts grammatical Lojban text to cmavo of selma'o UI
savoring/focusing and attempting to retain feelings of strong satisfaction - trying to escape intense dissatisfaction
attitudinal: weaker intensity attitudinal relativizer
Convert abstract predicate sumti back to predicate
Article for abstract predicate sumti. Turns a selbri into an abstraction with all open places filled by {ce'u}.
mekso n-ary operator: interleave sequences
digit/number: arbitrarily small/lesser/diminished/few (finite and nonzero but otherwise as small as desired).
Accepts any number of sumti and turns them into a selbri-unit that means "x[1] is among the referents of these sumti".
convert number to cardinality selbri: x[1] is/are [number] in number; there is/are [number] things among x[1].
Elidable terminator for selma'o MEIhE (which turns any number of sumti into a selbri-unit)
converts sumti into selbri: x[1] is [that sumti]'s, among x[2], by relationship x[3] (binary ka).
non-Lojban brivla
we; several people including one of the speakers; I (the speaker) and at least one another person (even if that person is one of the speakers too)
attitudinal: meow, miaow
digit/number: interval/range indicator for significant digits (determined by midpoint).
inclusive we; includes the speaker (I) and the listener (you), but may or may not include others
shortening of {lo du'u *sumti* mo kau}
x[1] is PA minutes in duration by standard x[2].
evidential: I remember - I don't remember (fact ; that/whether something is true)
x[1] is the PA-th date/time of unit x[2] (si'o) counting from x[3] (default: now) by calendar x[4]
interrogative mass gadri: "which group of..."; viewed and counted as a group
x[1] is (n)th member of alphabet/set x[2] ordered by rule x[3], where the count begins at x[4].
interrogative gadri: "which"
munje modal: 1st place; in universe...
mathematical/logical/mekso ternary operator: μ (mu) operator: outputs the most extreme extended-natural number that satisfies relationship/predicate A, where extremeness is bounded by B and of a
version determined by C; error output is -1
unary mekso operator: measure of the complement; 1 - x[1].
Discursive: resuming/continuing example - start new example
Specifies the universe of consideration/all possible referents (for the present discourse); specifies the universal set/class/structure.
Converts PA into tense; in [number (usually nonspecific)] possible worlds/alternate histories where [sumti (du'u)] is true
digit/number: Hafner-Sarnak-McCurley coprime determinants limiting probability constant; h ≈ 0.3532363719…
base-dependent digit: representing exactly one half of one more than the maximum possible single-digit number expressible in the relevant number base
delimited non-Lojban selbri-unit
Marks an endpoint of a quote/string/expression and specifies that (relative to the original) the quote/string/expression so marked is complete, accurate, and well-portrayed by the quote/string/
expression on the relevant side of the excerpt, including wrt all relevant information and when factoring in the content and context of the quotation-external discourse in which said quote/string
/expression appears; indicator that quote mining or cherry-picking did not occur and that the excerpt which is quoted is not deceptive.
Same function as {na} but with the additional meaning that the sumti in the bridi have no prior experience together.
not a number
converts an unevaluated mekso expression into a sumti referencing its evaluated result (if sensible/defined)
Contradictory negation of a predicate
nafselte'i modal: except...
Indicator for moderate or normal attitudinal intensity
non-restrictive relative clause; attaches subordinate bridi with incidental information, doesn't contain an implicit reference to its head via {ke'a}.
tense interval modifier: generically, as a rule (gnomic aspect)
pro-sumti whose referent's identity is unknown to the speaker
what is now; refers to current space/time/situation reference absolutely
attitudinal: devoidness of emotion (neutral by absence of emotion) - overwhelmed by/replete with/overflowing with (seemingly all) emotion
double-negative toggle: every odd-counted explicit usage makes negation additive; unmentioned or every even-counted explicit usage makes negation multiplicative.
unary mekso operator: (-1)^x
strict essentialistic article: «nei'i broda cu brode» = being a broda necessarily entails being/doing brode.
x[1] is PA years in duration by standard x[2].
mekso ternary operator: the generalized incomplete (factorial-extending) Pi function; for input (X[1, X] this word outputs the definite integral of t^X[1 e^-t] with respect to t from X[2] to X[3]
(see notes for default values).
mekso quaternary operator: polygamma function; for input X[1, X], outputs the (-X[2)]th derivative of Log(ne'o'a(X[1, X])) with respect to X[1].
unary operator: primorial a#
mekso quaternary operator – Pochhammer symbol: with/for input (X[1, X], this word/function outputs \prod[k ]= 0^X[2 - 1 (X]; by default, X[4 ]= 1 unless explicitly defined otherwise.
mekso n-ary operator: append covariant (lower) indices to tensor
x[1] is a number/value such that the abstraction is true, under mathematical system x[2]; x[1] binds to ke'a within the abstraction
attitudinal: 'how do you do?'
digit/number: Niven's greatest-exponent prime factorization constant lim[(n->∞) (avg]
digit/number: Niven's smallest-exponent prime factorization constant c = zeta(3/2)/zeta(3) ≈ 2.1732543125195541382370898404…
This cmavo precedes a predicate (at least binary) and turns it into an incidental conjunction, which syntactically behaves like JA cmavo. The predicate indicates the relationship between the two
connected propositions. Terminator: {te'u}.
scalar negator: not at all, no way
digit/number: absolute zero; nothing; there does not exist; ∄
Introduces a bridi relative clause, with the scope of {xoi} and the semantics of {noi}
mathematical/mekso binary operator: the zero/identity-element/(primitive (-))constant operator; outputs the identity-element of structure A (contextless default: the additive group of integers)
regardless of the input value of B (except blank or ill-defined values)
digit/number: liminal zero; neither positive nor negative
incidental/non-restrictive adverbial: converts selbri to bridi adverbial term. The first place of the converted selbri is claimed to be such that the outer bridi satisfies it, and the outer bridi
is claimed. {broda noi'a brode} means {lo nu broda ku goi ko'a cu fasnu .i ko'a brode}.
PA nonrestrictive/incidental relative clause; attaches to a PA number/numeral/digit with the ke'a referring to that PA number/numeral/digit.
number/interpreted mathematical object non-restrictive clause
scalar normative: normal in intensity
connective modifer/limiter
pro-sumti: this paragraph
attitudinal modifier: attribution to the attached sumti
Selbri incidental relative clause; attaches to a selbri with the ke'a being 'me'ei the attached selbri'
preposition: event instantiating the current proposition.
digit/number: twenty (decimal 20).
vocative: slightly surprised greetings
Attitudinal: offended/insulted - unoffended - deserving and accepting
sentence link/continuation; continuing sentences on same topic with the observative sumti filled with {la'e} {di'u}
attitudinal: fuck/shit - fuck yeah/hell yeah
attitudinal: pain/suffering - comfort
letteral for o.
digit/number: universal parabolic constant P = √(2) + Log[e(1 + √(2)) ≈ 2.295587]
scalar question: how very...?
Digit/number: pi (denoted "π"; approximately 3.1416...); the constant defined by the ratio of the circumference to the diameter of all circles in Euclidean space.
rhetorical construct marker - genuine/serious/literal assertion/question/command marker
mekso operator: part of number/projection (one sense); the X2 part of X1
ternary mekso operator: p-adic valuation; outputs (positive) infinity if x[1 ]= 0 and, else, outputs sup(Set(k: k is a nonnegative integer, and ((1 - x[3)x] divides x[1))], where p[n] is the nth
prime (such that p[1 ]= 2).
mekso operator: power set - produces the set of all subsets of set X[1] that are of (any) size (that is) X[2] [a nonnegative integer or transfinite/infinite number; default: su'o no].
discursive: marks the previous construction as a question, with valid responses being any construction which could replace it without changing the grammar of the overall utterance.
unary mathematical operator: predecessor/diminish/decrement (by one), \operatornamepred(a) = a-- = a-1
modal: "one ... at a time"
discursive: layperson/laïc meaning; marks a construct as layperson/common/non-technical/non-jargon speech/text
mark a logical connective as having maximal scope.
scalar / attitudinal question: how very...?
evidential question: how do you know that?
at-most-3-ary mekso operator: "integer exponent" for X[1] divided by X[2] in algebraic structure X[3]
digit/number: Apéry's constant ζ(3) = 1.202056903159594285399738161511449990764986292…
attitudinal category question: what is the category (UI4) of attitudinal?
discursive: want a response - venting
Prefix multiplication of unit selbri
digit/number: pi (approximately 3.1416...); the constant defined by the ratio of the circumference to the diameter of all circles.
mekso ternary operator: extract digit from number; X[2]nd macrodigit/term of number/tuple X[1] when X[1] is expressed in base/basis X[3].
mathematical/mekso binary operator: vector or function inner product over a field; the inner product of A and B over field C
mathematical ternary operator: not-greater-prime-counting function
mathematical ternary operator: prime-generating function.
mekso n-ary operator: generate ordered tuple/list from inputs; pi'u'e(x[1, x]= (x[1, x], pi'u'e(x[1, x]= (x[1, x], etc.
ponse modal, 1st place: with possessor/owner...
restrictive adverbial: converts selbri to bridi adverbial term. The first place of the converted selbri is claimed to occur in conjunction with the outer bridi. {broda poi'a brode} means {lo nu
broda cu fasnu gi'e brode}.
PA restrictive relative clause; attaches to a PA number/numeral/digit with the ke'a referring to that PA number/numeral/digit.
number/interpreted mathematical object restrictive clause
x[1], x[2], x[3]... are such that {poi'e} abstraction is true; each {ke'a} binds another argument of the resulting predicate
n-ary mekso operator: for an input of ordered list of ordered pairs ((X[1, Y], it outputs formal generalized rational function (x - X[1)^Y] in the adjoined indeterminate (here: x).
x[1] is such that {poi'i} abstraction is true; x[1] binds {ke'a} within the abstraction.
restrictive first place adverbial: converts selbri to bridi adverbial term. The outer bridi is claimed to occur together with the event of the first place of that bridi satisfying the converted
selbri. {broda poi'o'a brode} means {broda fau lo nu vo'a brode}.
mekso at-most-3-ary operator: convert to polynomial; X[1] (ordered list of algebraic structure (probably field) elements) forms the (ordered list of) coefficients of a polynomial/Laurent-like
series with respect to indeterminate X[2] under ordering rule X[3] (default for finite list: the first entry is the coefficient of the highest-degree term and each subsequent entry is the next
lesser-degree coefficient via counting by ones and wherein the last entry is the constant term)
selbri restrictive relative clause; attaches to a selbri with the ke'a being "me'ei the attached-selbri"
tense: refers to past of current space/time reference absolutely
mekso binary operator: generate span; outputs span(X[1, X]= span[X]; set of all (finite) sums of terms of form c v, where v is an element of algebraic structure X[1] (wherein scalar
multiplication and summation is defined), and c is a scalar belonging to ring X[2].
digit/number: the Prouhet-Thue-Morse constant τ = 0.412454033640… (in decimal)
digit/number: twenty (decimal 21).
quotes a single non-meaning name in lojban (Must be lojban text and sounds) delimited by pauses
pro-sumti: others, not me/we/the speaker(s)/the author(s)/you the listener(s).
semi-discursive: and so forth, and so on, et cetera, continuing similarly
selbri modifier: restrict the referents of the x1 slot to those belonging to the current domain of discourse, those being relevant to the present context.
attitudinal modifier: want this to last - (cu'i) accept this - (nai) want this to end soon
ternary mekso/mathematical operator: radical; for input (x,y,z), it outputs the largest y-th-power-free product of prime divisors of x in structure (ring) z.
mekso (2 or 3)-ary operator: maximum/minimum/extreme element; ordered list of extreme elements of the set underlying ordered set/structure X[1] in direction X[2] of list length X[3] (default: 1)
Quote conversion: the quotation as presented uses pro-sumti and pro-bridi as if the current utterer (not the original utterer) were saying it, but the meaning conveyed is identical to that of the
actual quotation by the original utterer and there is a claim that this meaning was expressed elsewhere
single-word rafsi quote; quotes a single word delimited by pauses (in speech) or whitespace (in writing) and treats it as a rafsi
flag a quote/sedu'u statement in order to indicate that the text is substantially the same in all relevant important aspects (usually including content), translated with this meaning, "to this/
that same effect" - - (nai:) untranslated, original and exact wording
quote marker: indicates that the quotation being marked has experienced exactly no change in meaning/has the original meaning and is the same in all important qualities - approximately the same
meaning/having generally the same idea or effect or message - substantially changes in meaning or some important quality
free conversion
the trivial selbri conversion - identity permutation of terbri
scalar abator: slightly... / not very...
evidential: I have basic/axiomatic belief
pro-sumti: the next/immediately following sumti (as determined by back-counting rules applied forward)
the latest aforementioned...; refers back to the most recently mentioned thing(s) that satisfies the x1 of the following predicate
shift to Latin alphabet (strictly)
emotion category/modifier: creative, artistic (emotion felt in proverbial/mythical "right brain") - analytic, methodical (emotion felt in proverbial/mythical "left brain")
universal plural quantifier. All.
n-ary operator: n-ary magma/group/ring operator a*b = ab`
quantifier: "all" (as opposed to "every")
start quote of replacement for recent mistakenly uttered text
erase current sentence.
Converts following cmevla or zoi-quote into onomatopoeia. (bam! crash! kapow! etc.)
scalar intensifier: very...
turns PA into CAI; intensity attitude modifier expressed by a mekso.
digit/number: exactly enough and no more.
unconditional start of text; outside regular grammar; used for computer input.
interval event contour: succeeding at ...
mekso binary operator: del, nabla, quabla, partial-derivative vector/tensor operator; outputs the functional-valued formal-covector (or analog thereof) of partial derivatives with respect to X[1]
(tensor in the same format and order), each degree X[2] (default: 1).
quotes a nonce Lojban word (an onomatopoeia), turns it into selbri unit meaning "x[1] makes a sound like (quoted word)".
vocative: celebratory cheer/hooray
topicalizer (sumtcita/discursive; somewhat meta): the following discourse is about/relates to/has topic/concerns
mekso n-ary operator: reciprocal of the sum of the reciprocal of each of X[1], X[2], ..., X[n] (for any natural number n); 1/((1/X[1) + (1/X].
pro-sumti whose referent's identity the speaker is assuming the listener to know
erase the current clause.
(n, 1, 2, \dots, n-2, n - 1)st conversion.
mathematical quinary operator; big operator: left sequence notation/converter - operator a, sequence b defined as a function on index/argument/variable/parameter c, in set d, under ordering e
(2, 3, \dots, n-1, n, 1)st conversion.
placed before a selbri, merges x1 and x2 places.
converts singular quantifier into plural quantifier
vocative: sweet dreams.
unary mathematical operator: identity function id(a) = a
terbri editor: passes the terbri value through the quoted function so that the sumti that fills it really is filling the output of the function
evidential builder: I know by means ...
digit/number: Sierpiński constant K = \pi(2\ln2 + 3\ln\pi + 2\gamma - 4\ln\Gamma(\frac14)) \approx 2.584981759579253217065893587383…
evidential: stereotypically...
selbri terminator
vocative: "you're welcome / happy to help / no problem" - "just this once / you owe me"
digit/number: ideal first Skewe's constant Sk[1] ; the first (minimal positive) infimum for which all greater x in some neighborhood have the property that it is false that the prime counting
function at x is less than the logarithmic integral function at x
elliptical/generic/unspecific/vague selbri conversion
evidential: I know by gnosis
Conversion: Switch n and x[1] in MOI (or MOI*) cmavo so that the submitted value (previous x[1]) outputs the number(s) (previous n) associated with it.
selbri conversion question
marks a construct as being a reference/allusion - explicit marker of divorce/isolation of a construct from any external allusions that may come to mind
evidential: it seems that...
erases a word; pretend the next word ≈ the erased word
numeric suffix: indicates that the number refers to portionality instead of cardinality
digit/number: a finite/bounded number, finitely many.
story time toggle.
Marks discontinuity in story time.
erases a SI, SA or SU cmavo; is to "redo" as SI/SA/SU is to "undo".
n-ary mekso operator: Logistical growth/cumulative function, sigmoid function; (X[3 / (1 + e^(-X].
subjective number which is constant over time
digit/number: the vast majority; a minimal-majority plus very/much/many more.
Delimit a replacement for the previous expression using arbitrary delimitors.
digit/number: the vast majority; a minimal-majority plus rather many more.
number/digit: a qualifying majority or a defined/relevant supermajority.
discursive: "to be honest", being frank, candid, speaking one's mind, being forthcoming and demonstrating candor - deliberately "not getting into it", saying less than is on one's mind
digit/number: a strong/substantial/decisive majority.
digit/number: n (default: 1 or 1/2 atomic units as the case may be) more than half; barely a majority; a slight majority.
digit/number: slightly less than a minimal-majority; the maximal-minority.
subordinating adverbial: converts selbri to bridi adverbial term. The bridi is claimed to satisfy the first place of the converted selbri, but is not itself claimed to occur. {broda soi'a brode}
means {lo nu broda cu brode}
almost all/almost every/almost everywhere (technical sense): there is a non-universal conull subset (the complementary (sub)set of which is non-empty but of measure 0, where complement is taken
with respect/relative to the universal set) of such things that are satisfactory.
almost none/almost no/almost nowhere (technical sense): the subset of satisfactory such things is null but non-empty.
digit/number: a very large minority; the plurality.
x[1] is PA seconds in duration by standard x[2].
digit/number: complement of "{so'e'i}"; a qualifying, controlling, or decision-making minority; a subminority.
head-final content clause relativizer: it turns the current clause into a subordinate content clause, binds it to the {ke'a} pronoun, and restart the current clause afresh (as if its previous
content was erased or moved into a hidden prenex).
Initiator of subordinating adverbial relative clause with leftwards logical scope. The adverbial clause binds the resumptive pronoun {ke'a} to the outer clause, which becomes irrealis (i.e. not
necessarily claimed to be true). Terminator: {se'u}.
conversion: adds 1 to index of each place starting from place x[1]; the new resulting x[1] is undefined
discursive: responding quickly - responding after a long time/necroposting.
digit/number: precise to within the stated sigfigs (significant figures/digits); approximately, measured to be approximately, with some error/rounding
digit/number: exact, exactly equal to, no more and no less, mathematically ideally (no measuring or rounding error)
shows that the first two places have a reciprocal relation
mekso unary operator: digital addition.
mekso unary or binary operator: ordered inputs (n, b) where n and b are nonnegative integers and b > 1; output is the ultimate digital root of n in base-b.
existential plural quantifier. There is/are.
digit/number: strictly greater (more) than 0 but strictly less than all (jbo.: "ro").
erasure word: delete everything that was ever said in Lojban.
Override implicit zo'e-filling of empty argument slots in the current clause, or of the marked tanru-element if this particle is put right after a tanru-element, switching the default filling to
existential quantification of lowest logical scope.
digit/number: nineteen (decimal 19).
pro-sumti: something or some things (shorthand for "su'oi dzai'i" with lowest local scope).
pro-bridi: Quotes a single Lojban word, and turns it into the bridi, "x[1] is the same word-shape as the quoted word"
discursive: reconsideration of statement - continuing (on) in that line of thought/discussion
discursive: pay a lot of attention to what I just said - ignore what I just said.
mekso unary operator: basic Schlafli symbol composer (defined only on ordered lists)
Converts following cmevla or zoi-quote into phenomime.
8-ary mekso operator: the X[1]th nonnegative sum of X[2] mutually-distinct perfect X[3]th-powers (i.e.: of integers) in X[4] mutually truly-distinct ways, requiring exactly X[5] terms to be
negative in each sum (counting with(out^X[6]) multiplicity), requiring exactly X[7] terms to be repeated between sums (counting with(out^X[8]) multiplicity), according to the usual ordering of
the integers.
mekso unary operator: Kleene star - X1*
(something which approximately has) the abstract shape of the symbol for the previous construct
explicit indicator that the speaker is completing/continuing a previously uttered bridi in the discourse
the abstraction described by text; turns text sumti into abstraction sumti
Copy and paste the overall seltau of immediately preceding sumti at this location.
Following lujvo takes the form of a tanru (place structure of the lujvo is the same as the last rafsi/gismu)
Copy and paste the overall tertau of immediately preceding sumti at this location.
digit/number: tau (denoted "τ": approximately 6.2831...). The constant defined by the ratio of the circumference to the radius of all circles (default: in Euclidean geometry).
tekla modal: with key/button... (computer game context)
Exponentiation of unit selbri
iterated Cartesian product with self: A × A × ... × A, n times.
mekso ternary operator: Knuth up-arrow notation: a \uparrow \dots \uparrow b of order/with c-2 arrows ("\uparrow") initially, evaluated from right to left; the cth hyperoperator on a by b.
2-Merge Conversion: placed immediately before a selbri, merges x[2] and x[3] places.
placed before a selbri, merges x1 and x3 places (with the new x1 being equivalent to the result of this merging).
Vocative: request/command that recipient acknowledge having received (and understood) the message (iff they did (and do)) - no need to respond
at a point on time axis
6-ary mekso/mathematical operator: Heaviside function/step/Theta function of a, of order b, in structure c, using distribution d, within approximated limit e, with value f_b at 0
4-ary mekso operator: Taylor expansion/polynomial term; for ordered input (X[1, X], output is the X[3]th Taylor polynomial term of at-least-X[3]-smooth function X[2] which was expanded around
point X[4] and which is evaluated at point X[1], namely (1/(X[3!)) × (D^X].
discursive: Especially, foremost, primarily, chiefly, preëminently, importantly, significantly (laïc or statistics technical sense), in particular, strongly, notably - in common with, commonly,
generally, usually, ordinarily, unnoteably.
discursive: specified by the speaker - unspecified by the speaker
discursive: in theory - in practice.
elidable terminator of a JAhOI construct
unary mekso operator: natural exponentiation operator exp, where exp(a) = e^a \forall a.
x[1] (set of points in time) is a time of [bridi] taking place.
mekso ordered/non-commutative n-ary operator: tensor product/exterior product (of tensors); letting "@" denote the tensor product, A[1] @ A[2] @...@ A[n ].
terminator, mekso: terminates the listing of an ordered sequence of indices for a tensor
demonstrative article: the … which I'm currently pointing to or am otherwise attracting your attention to.
at N o'clock; at the hour N of the day.
at the minute N of the hour.
at the second N of the minute.
conversion: move/promote ['drag-and-drop'] 3rd place to 1st position. Everything else stays in the same order.
Toaq toggle: Toggles text to Toaq language; marks following text as Toaq text.
binary mathematical operator: Jordan totient function J[a(b)]
start UI-applicative metalinguistic UI-parenthetical
end UI-applicative metalinguistic UI-parenthetical (elidable terminator)
abstraction described by quoted text
makes a new selbri, the first two places of which are to be filled with sumti from the first two abstraction places of the selbri following this word
null connective operand; used to fill empty places in JOI
end connective string, set, list such that the set of terms provided is exhaustive.
attitudinal: triumph/victory - draw/tie/inconclusive - defeat/loss
attitudinal: friendly/friendishly/amicably/companionship/compatriotship/comradeship - antagonistically/enemyishly
Attitudinal: impressed/amazed - unimpressed (neutral)/blasé/bland/commonplace - unimpressed (negative)/disappointed/bored
Attitudinal: encouragement - (not engaged) - discouragement
discursive: counter-expectational – – aligning with expectations
attitudinal: excitement - lack of excitement - boredom
discursive: optional answer premarker
attitudinal: excited encouragement
excitement/squealing for pleasurable reasons - dullness/disinterest/disengagement - hissing/squealing for unpleasant reasons/disfavor
Attitudinal: feel like laughing - neither laughing nor crying - feel like crying
cheering/clapping/congratulating/adulation/bravo/praise - neither favorable nor unfavorable judgment - booing/hissing/jeering/disfavor
letteral for :)
discursive: marks question construct as allowing filling it with answers that are non-referential constructs only
savoring the now (wishing time would stand still) - patience/indifference toward time passing - impatience (wishing time would move faster)
attitudinal: feeling schadenfreude (pleasure from someone's misfortune) - denying feeling schadenfreude.
letteral for u.
letteral for the u semi-vowel, sometimes written as ŭ
starts a tanru group with jvajvo-like semantic composition, keeping left grouping.
Binary mekso operator: group-theoretic conjugation (group action): maps inputs (X[1, X] to X[2^(-X]= \phi[(X]. Default: X[3 ]= 1.
converts number to scalar tag; specifies the value on fuzzy logic scale; to the degree (n) on scale ...
digit/number: Dom Hans van der Laan's plastic number ρ = 1.324717957244746025960908854…
discursive: forces jvajvo reading of the preceding brivla; +nai: forces naljvajvo reading of the preceding brivla.
pro-sumti: repeats the most recent 1st argument slot (fa-slot).
describing word / situation: have checked a dictionary - denying either - not have checked a dictionary.
omega constant, Lambert product-log W(1)
pro-sumti: repeats the most recent 2nd argument slot (fe-slot).
interval bracket ordered tuple introducer
pro-sumti: repeats the most recent 3rd argument slot (fi-slot).
pro-sumti: repeats the most recent 4th argument slot (fo-slot).
interval bracket ordered tuple terminator
pro-sumti: repeats the most recent 5th argument slot (fu-slot).
vocative marker: identifies the station(s) to which a message is to be sent and at the same time prepares a universe of discourse by filtering from a stream of messages only those said by that
2-Merge Conversion: placed immediately before a selbri, merges x[2] and x[4] places.
placed before a selbri, merges x1 and x4 places.
Vocative: from station - to station
2-Merge Conversion: placed immediately before a selbri, merges x[3] and x[4] places.
mekso binary operator – quotient from integer-division: sgn(X[1)] sgn(X[2) ((]abs(X[1) - (]abs(X[1) \% X].
evidential: I remember (experiencing) - I deny remembering
binary mekso operator: form quotient space X[1/X].
binary mekso operator: mod(ulus)/remainder; X[1] \% X[2], \,\,\, X[1] (mod(X[2])).
Close all open mathematical brackets.
digit/number: Lévy-Khinchin constant γ = e^(π^2)/(12Log[e(2)) ≈ 3.2758…]
mekso unary operator: the set of all fixed points of function a
conversion: move/promote 4th place to 1st position. Everything else stays in the same order.
quaternary mathematical operator: (left) convolution (a★b)(c) in structure d
digit/number: Lambert W(1) constant Ω ≈ 0.5671432904097838729999686622…
pronoun analogical to the vo'a series for the fai-place.
titular possessive: possessive that is part of a name, e.g. "The Summer of Love", "The Wars of the Roses", "Zeno of Elea"
titular relative clause: gives a title/name in the form of a relative clause, e.g. "Alexander the Great" or "Woobie, Destroyer of Worlds"
evidential: non-veridical - veridical
base-dependent digit: the maximum possible single-digit number expressible in the relevant number base.
opening bracket for VUhO
brisni/statement terminator.
naturalistic interjection: laughter
naturalistic interjection: laughter
mekso operator: the bth branch of the (possibly multivalued) function a
Hadamard-Walsh-Rademaker-Fourier gate/transform.
location tense distance: too far to interact with; subjective
time tense relation/direction: disconnected in time/place; causally disjoint
binary mekso operator: Let the inputs X[1] and X[2] be sets in the same universal set O; then the result of this operator applied to them is X[1^c \cup X], where for any A \subseteq O, A^c = O \
setminus A.
binary mekso operator: Let the inputs X[1] and X[2] be sets in the same universal set O; then the result of this operator applied to them is X[1^c \cap X], where for any A \subseteq O, A^c = O \
setminus A.
discursive: imagining/roleplaying - not imagining/"out of character / in real life"
they.(repeat >1 preceding sumti)
naturalistic interjection: Hyah! (battle cry/kiai)
Property article: converts the following predicate into an property sumti (whose open slot is the x[1] of the predicate).
interval endpoint status (exclusive/inclusive) marker: independent of the other, all options satisfy
opposite of za'o: event contour: refers to the portion of the event which occurs before the natural beginning; starting too early before ...; <----.
already; too early than expected;
Any parser parsing the text/sentence/fragment containing this cmavo, must answer that the whole text/sentence/fragment is not syntactically not correct.
discourse pronoun: the future situation
preposition: marks that which the bridi is beneficial to; for the benefit of..., with beneficiary...
shortening of {lo du'u xu kau *bridi*}
symbol string to number/variable
text to number/variable unassignment
mekso convention default specification/definition (explicit)
text to number/variable
XAUhO terminator
mekso convention cancellation
elliptical/vague interval endpoint status (exclusive/inclusive) marker
pseudo-number that is unequal to itself.
mekso clausal referent bracket initializer
digit/number: any/non-specific referent; modifies quantifier to indicate that it is not important what the specific members of the referential set are
2-Merge Conversion: placed immediately before a selbri, merges x[2] and x[n] places.
interjection: laughter/chuckle (heh heh heh)
placed before a selbri, merges x1 and x5 places.
abstractor: place abstractor; x[1] is the place where [bridi] takes place
2-Merge Conversion: placed immediately before a selbri, merges x[4] and x[5] places.
2-Merge Conversion: placed immediately before a selbri, merges x[3] and x[5] places.
digit/number: hex digit E (decimal 14) [fourteen]
frees the following attitudinals of their context; precedes free-floating non-modifying attitudinals
online location tense; at the same online place as; online equivalent of {bu'u}
reattaches the following attitudinals to the containing bridi; precedes free-floating attitudinals to be understood as modifying the bridi they are contained in
reattaches the following attitudinals to the containing sentence; precedes free-floating attitudinals to be understood as modifying the sentence they are contained in
interval endpoint status (exclusive/inclusive) marker: dependent and coincident/matching with other
discursive: simply/merely/just, "all there is to it" - not simply, not just, "there's more to it"
Abstraction variable indicator selbrisle.
scalar subscript
binary mekso operator: for ordered list X[1], this word outputs the same ordered list except the indices/subscripts have been relabelled/redefined/reindexed according to rule X[2] (see notes).
extent of truth
digit/number: Khinchin's constant K[0] = 2.6854520010…
discourse pronoun: the previous situation
UI discursive: litotes - - anti-litotes
Loglan toggle: Toggles text to TLI Loglan; marks following text as TLI Loglan.
conversion: move/promote 5th place to 1st position. Everything else stays in the same order.
pro-numeral: the most-recently mentioned full/complete numerical or mathematical string/expression.
elliptical/unspecified number.
unary mekso operator: produces a string of n consecutive "xo'e"'s, treated as digits (concatenated into a single string of digits)
elliptical/unspecified/vague single-symbol (general)
At-most-unary mekso operator: like {xo'ei} but for selma'o XOhEhOhE, rather than just PA
Extracts selbri from a tag, inverse of fi'o
Right-scoping adverbial clause: encloses a bridi and turns it into an adverbial term; the antecedent (ke'a) of the enclosed bridi stands for the outer bridi {lo su'u no'a ku} (the bridi in which
this xoi term appears), including all the other adverbial terms (tags...) within this bridi located on the right of this xoi term (rightward scope).
Toggles grammar so that every mention of a number n is interpreted as "at least n".
non-logical connective (mekso set operator): regardless
interval endpoint status (exclusive/inclusive) marker: dependent and contrary/contraposed with other
attitudinal modifier: sarcastically - sincerely/honestly
binds a relative clause to both the preceding sumti and the bridi containing that sumti
opposite of mo'u: interval event contour: at the natural beginning point of ...;
discursive: persuading - not arguing/negotiating - conceding
extent of truth
vocative: said/quoth..., used to identify the person speaking a single sentence (e.g. in dialogues)
attitudinal contour: stronger over time - unchanging intensity - weaker over time
elliptical bridi logical negator/affirmer/truth-evaluation
bridi to sumti: marks the beginning of a subordinate bridi; the whole construct is a sumti referring to the enclosed bridi
naturalistic interjection: a controlled, focused breathing technique (used for coping, as with pain, fear, etc.)
digit/number: sixteen (decimal 16).
naturalistic interjection: in thought/contemplation
connective: elliptical/generic/vague
Equivalent to ".{y}" but connotes a somewhat longer pause
time tense distance: an unspecified distance in time
evidential: mark the sentence as being an observation sentence, i.e. a statement that is not based on the truth of another statement but is instead taken from direct observation.
mekso binary operator: right-handed vectorial cross product (ordered input), a×b
attitudinal modifier: observed emotion; preceding attitudinal is observed on listener
(elidable) terminator of mathematical/formal quote with mau'au
jargon word indicator; indicates next word is a jargon word
pro-sumti: the empty argument/value; syntactically-contextually and type-permitted maximally generic in its typing
Quote conversion: the sentence(s)/bridi (possibly plural) is/are syntactically correct and semantically intended by the utterer if the outermost layer of quotes (markers) which immediately
follows were to be omitted/removed, but the quoted text is in fact a quote (of the indicated type) from some source.
yet; still; too long
vocative: go! / come on! / get on it! / let's!
text affirmation/negation mode toggle
digit/number: arbitrarily large/great/increased/many (finite but as big as desired/allowed).
tense interval modifier: increasingly...; incrementative. Tagged sumti, if present, indicates amount of increase (lo te zenba)
unary mekso operator: reverse ordered list/tuple X[1].
selbri conversion: permute all terbri so as to be exactly backward.
subjective number which is increasing over time
last-th conversion: switches the last terbri with the first one.
n-Merge Conversion: placed immediately before a selbri, merges x[2] and all x[a] places.
n-Merge Conversion: placed immediately before a selbri, merges all x[a] places.
n-Merge Conversion: placed immediately before a selbri, merges x[5] and all x[a] places.
n-Merge Conversion: placed immediately before a selbri, merges x[1] and all x[a] places.
nonce word with existing grammar
n-Merge Conversion: placed immediately before a selbri, merges x[4] and all x[a] places.
n-Merge Conversion: placed immediately before a selbri, merges x[3] and all x[a] places.
tense interval modifier: increasingly...; incrementative. Tagged sumti, if present, indicates amount of increase (lo te zenba)
time tense interval: for some length of time.
marks a construct as used for its form only, not its meaning.
begin quote that is converted into rafsi
begin quote that is converted into rafsi, distributing lujvo-glue between quoted words
attitudinal contour: increasing over time - steady - decreasing over time.
unary mekso operator: (analytically continued) Riemann zeta function zeta(z), for complex-valued input z.
Delete all sumti slots of the immediately preceding word which are not explicitly filled excepting the first n (specified by subscript; contextless default: n=0).
end rafsi-converting quote: terminates ZEIhEI quote
converts following word to selbri-unit: "x[1] is related to the meaning of this word in aspect x[2]"
nonce-word indicator; indicates previous word is nonce-creation and may be nonstandard
jargon word indicator; indicates previous word is a jargon word
mathematical operator: the empty/null [one sense]/trivial [one sense]/blank operator
nonexistent/undefining it; the selbri is not applicable when the other terbri are filled in the manner in which they are in this utterance/bridi.
converts a following bu-letteral into a sumti representing the musical natural note with that name.
converts a following number into a sumti representing the MIDI note with that number.
converts a following bu-letteral into a sumti representing the musical sharp note with that name.
converts a following bu-letteral into a sumti representing the musical double sharp note with that name.
marks a construct as used for its form only, not its meaning.
converts a following bu-letteral into a sumti representing the musical flat note with that name.
converts a following bu-letteral into a sumti representing the musical double flat note with that name.
fills and deletes (in the manner as "{zi'o}") all terbri sumti slots of immediately previous (id est: tagged) word that are not explicitly filled with a sumti.
Delete subsequent sumti slots.
the referent of itself; guards the scope of the sumti
quotes a selbri.
start logical, topical, or termset postnex
Something associated with; equivalent to zo'e pe or "lo co'e be".
end attitudinal quotation
non-mekso quote/name substitution for ordered collection of prescriptions, descriptions, definitions, etc.
empty string/text/word
pro-sumti: current topic (most recent unquantified sumti occurring in a prenex).
quote indicators only
elliptical/unspecified/vague string/text/word
tags a topic or prenex; scopes over the entire bridi (unless there is already a {zo'u} clause).
quote next non-Lojban word only; quotes a single non-Lojban word delimited by pauses (in speech) or whitespace (in writing)
shows mutual activity between this place and the first place of the current bridi; members participating in the activity are put into the first place (that e.g. can be formed by connecting sumti
with {ce} or {jo'u})
location tense relation/direction (angular); counterclockwise from..., locally leftwards/to the left of ...
Adverbial, metacommentary-introducing, and complex discursive: I express this utterance/construct/(rest of the) bridi which has been tagged for the purpose or goal of enacting, causing, enabling,
implementing, actualizing, manifesting, enhancing, yielding, or rendering applicable the immediately following and enclosed bridi or in order to make the immediately following and enclosed bridi
be true (or closer to the truth or more true or more strongly true) in application to the utterance/construct/rest of the bridi which has been tagged by this marker; the tagged expression/
construct is for the benefit or sake of making the immediately following and enclosed bridi true.
typically what?
mekso; binary operator: z-score for the X[1] quantile; X[2] (default: 1) acts as the descriptor toggle (see notes).
digit/number: seventeen (decimal 17).
exophoric article: some certain thing(s) which… (identity ascertainable from the context). | {"url":"https://vlasisku.lojban.org/%22experimental%20cmavo%22","timestamp":"2024-11-12T19:23:11Z","content_type":"text/html","content_length":"160550","record_id":"<urn:uuid:ef763181-7760-4459-b7e7-eff8e4d14217>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00330.warc.gz"} |
Mathematics equations play a vital role for Civil Engineer to study various prospects such as cost factor, blending of aggregates, sieve analysis, soil test, environment engineering test. A simple
Civil Engineering Calculator tool is a collection of Civil Engineering calculations for Civil Engineers and Students to calculate different types of civil related units which are utmost importance
before implementing any project.
The planning phase of the project and the financial budget allocation is purely based on mathematics of several factors. Concrete technology allows the construction of structures that would last for
decades, if not centuries. Thus the lifespan of a building is completely based on type of concrete technology used. The entire weight of the structure is to be hold by the soil. Soil testing is the
first and most prior step for any good reputed construction company. Many factors are considered in the soil testing such as liquid limit of soil, water content determination, sieve analysis of soil
and many other factors. Sieve analysis of aggregate play an important role for determining particle size on large samples of aggregate are necessary to ensure that aggregates perform as intended for
their specified use.
With Civil Engineering Calculators you could do tedious task simply in few click. Calculator is fast, correct, user friendly and it is free to use. And along with the description of each test it
becomes easy to understand for the newbie as well. | {"url":"https://www.civil-engineering-calculators.com/Default.aspx","timestamp":"2024-11-09T07:25:18Z","content_type":"application/xhtml+xml","content_length":"152472","record_id":"<urn:uuid:6a99bdd6-5b5a-418b-bd70-d7e210c5344d>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00553.warc.gz"} |
Beau's Route
Standard Input (stdin)
Standard Output (stdout)
Memory limit:
64 megabytes
Time limit:
1.0 seconds
Beau loves climbing mountains. Specifically, they love the view from the top, which is why when they climb two mountains that are right next to each other, they are usually disappointed after the
second one, since the view is basically the same as the first one. In fact, when Beau climbs any mountain, Beau's enjoyment of that mountain is equal to how far away it is from the last mountain they
climbed. Auckland has $N$ mountains. All $N$ mountains are laid out in a row with equal spacing between them. They are numbered 1 to $N$, left to right. Beau would like to climb all $N$ mountains in
some order, and they would also like to minimise their disappointment. As such, Beau has asked you to write a program that produces a route for them. A route is a sequence in which Beau can climb all
of the mountains, where they only climb each mountain once. All Beau asks, is that the least enjoyment they will experience on any mountain on the route is as high as possible. Beau will always enjoy
the first mountain they climb greatly, because they haven't climbed any mountains before it.
All of this is to say, Beau would like you to find the route that maximises their minimum enjoyment of every mountain after the first.
Input consists of a single integer $N$, the number of mountains.
Output one line, containing $N$ space separated integers, the $i$th of which should denote the $i$th mountain Beau should climb. You should output every integer $1$ through $N$ exactly once. In other
words, your route should be a permutation of the integers $1$ through $N$.
If there are multiple routes that all equally maximise Beau's minimum enjoyment of any mountain except the first, output the one of lowest lexicographical value†.
• Subtask 1 (20%): $N$ is even
• Subtask 2 (40%): $N ≤ 8$
• Subtask 3 (40%): No further constraints
Sample Explanation
There are 4 mountains. Beau should start by climbing mountain 2, then mountain 4. These are two spaces apart, so Beau will have an enjoyment of 2. After that, Beau should climb mountain 1. Mountain 1
is 3 spaces from mountain 4, so Beau will have an enjoyment of 3. Finally, Beau should travel 2 spaces to mountain 3 for an enjoyment of 2. Beau enjoyed mountains 4 and 3 the least, with an enjoyment
of 2. This is the largest possible. The path 3 -> 1 -> 4 -> 2 also achieves a minimum enjoyment of 2, but it is lexicographically greater than the one achieved by starting at mountain 2.
†This is to say that between two routes that equally satisfy Beau's requirement, Beau would prefer the one with the smaller first mountain. If two or more routes have the same first mountain, Beau
would like the one with the smaller second mountain, and so on. | {"url":"https://train.nzoi.org.nz/problems/1335","timestamp":"2024-11-05T12:58:15Z","content_type":"text/html","content_length":"39016","record_id":"<urn:uuid:c6d34110-8b69-49ce-94b1-ab45b67fe7f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00188.warc.gz"} |
My apiary
The map shows demo apiaries in the CAVAT system. Apiaries data could be used in the study and prevention of varroa reinvasion. We will demonstrate the use of location data in a model that determines
the invasive potential of the apiary.
The model presented has no basis in the literature. It is founded exclusively on assumptions that have not been proven.
The invasion of varroa depends on many factors. Only placement of apiaries in the environment (apiaries topology) is considered in this model. Other possible factors such as the number of colonies in
apiary, colony strenght, swarming etc. are not considered.
Apiaries on the map are represented by coloured circles. When clicked, you get info about apiary invasion potential and circles in a 3 km (red) and 6 km (blue) radius. Delete circles with F5. You can
check apiary invasion potential according to the rules explained in Apiary invasion potential rules.
Figure 1 - two close apiaries
In figure 1 are two apiaries 2 km apart. Varroa can pass from apiary A to apiary B. Let's define this capability as transfer potential and denote it as: T[AB].
1. Capability of varroa passing between apiary A and apiary B is transfer potential T[AB].
For purpose of this model we assign transfer potential value 1.
1. Capability of varroa passing between apiary A and apiary B is transfer potential T[AB].
2. Value of transfer potential is 1: T[AB] = 1.
Varroa can be pass from apiary A to apiary B and vice versa. Since we consider both apiaries as equal, same rules apply for both A and B.
1. Capability of varroa passing between apiary A and apiary B is transfer potential T[AB].
2. Value of transfer potential is 1: T[AB] = 1.
3. Transfer potential is equal in both directions : T[AB] = T[BA].
Varroa can pass between apiaries if apiaries are not to apart. Let's set this distance to 3000 meters.
1. Capability of varroa passing between apiary A and apiary B is transfer potential T[AB].
2. Value of transfer potential is 1: T[AB] = 1.
3. Transfer potential is equal in both directions : T[AB] = T[BA].
4. Varroa can pass only between apiaries less than 3 km apart.
Figure 2 - four apiaries
In figure 2 are four apiaries. Apiary A is within 3 km range of apiaries B, C and D. Apiaries B, C and D are more than 3 km apart from each other.
Varroa can pass from apiary A to apiary B, C and D and vice versa. However, it can't pass between apiaries B, C and D (rule 4).
Transfer potentials are (rules 2 and 3):
• T[AB] = T[BA] = 1.
• T[AC] = T[CA] = 1.
• T[AD] = T[DA] = 1.
Let's name capability of apiary A to pass varroa to other apiaries as apiary invasion potential and denote it as I[A].
1. Capability of varroa passing between apiary A and apiary B is transfer potential T[AB].
2. Value of transfer potential is 1: T[AB] = 1.
3. Transfer potential is equal in both directions : T[AB] = T[BA].
4. Varroa can pass only between apiaries less than 3 km apart.
5. Capability of apiary A to pass varroa to other apiaries is apiary invasion potential. I[A].
Apiary A can infest 3 apiaries and can be infested by 3 apiaries. Apiaries B, C and D can infest only apiary A and can be infested only by apiary A. We define apiary invasion potential as sum of its
transfer potentials.
Calculated transfer and invasion potentials are in figure 3:
• I[A] = T[AB] + T[AC] + T[AD]= 3
• I[B] = T[BA] = 1
• I[C] = T[CA] = 1
• I[D] = T[DA] = 1
Figure 3 - invasion potential
1. Capability of varroa passing between apiary A and apiary B is transfer potential T[AB].
2. Value of transfer potential is 1: T[AB] = 1.
3. Transfer potential is equal in both directions : T[AB] = T[BA].
4. Varroa can pass only between apiaries less than 3 km apart.
5. Capability of apiary A to pass varroa to other apiaries is apiary invasion potential. I[A].
6. Apiary invasion potential is sum of its transfer potentials.
Figure 4 - three apiaries
In figure 4 are three apiaries: A, B and C. Varroa can pass from A to B and from B to C (rule 1). Actually varroa can pass between two apiaries more than 3 km apart (rule 4 is violated).
We need to modify set of rules.
In order varroa to pass from apiary A to B, obviouslly there must be varroa in A. If there is only small number of varroa in A it is unlikelly it will pass to B. We assume there should be a critical
number of varroa in A in order to pass to B.
In newly infested apiary B varroa needs certain amount of time to develop above crtitical level. Only after that time it can propagate from B to C. Time plays critical role in invasion propagation.
Varroa will spread across all apiaries if they are not treated against varoa. Model and rules presented here have sense only if we assume all apiaries are periodically and simultaneouslly treated
against varroa and this treatment lower varroa level below critical.
1. All apiaries in region are periodically and simultanouslly treated against varroa.
2. After treatment varroa level is below critical.
1. Capability of varroa passing between apiary A and apiary B is transfer potential T[AB].
2. Value of transfer potential is 1: T[AB] = 1.
3. Transfer potential is equal in both directions : T[AB] = T[BA].
4. Varroa can pass only between apiaries less than 3 km apart.
5. Capability of apiary A to pass varroa to other apiaries is apiary invasion potential. I[A].
6. Apiary invasion potential is sum of its transfer potentials.
7. Varroa can pass between two apiaries A and C more than 3 km apart if both conditions are met:
A. There is at least one intermediate apiary B, which is less than 3 km apart from both apiaries A and C.
B. Varroa in newly infested intermediate apiary B has enough time to develop over critical level.
In figure 4 are three apiaries: A, B and C. Varroa can pass directly from A to B and then propagate further to C after a while.
Let's call varroa passing from A to B direct transfer in first step. Propagation from A to C is indirect varroa transfer in second step. Imagine we "walk" forward to adjacent apiaries from A to B,
from B to C and so on. How many steps should be included in calculating invasion potential ?
We assumed colonies are treated periodically and simultanously. Period between two treatments determines how far can varroa invasion propagate.
In our model we consider direct transfer and indirect transfer up to second step. Apiaries from second step also have lower transfer potential with value 0.3 because they less significantly impact
varroa level.
In set of rules we modify terminology to distinguish between direct and indirect varroa passing between apiaries.
1. All apiaries in region are periodically and simultanouslly treated against varroa.
2. After treatment varroa level is below critical.
1. Capability of varroa passing between apiary A and apiary B is transfer potential T[AB].
2. Value of direct transfer potential is 1 : T[AB] = 1.
3. Transfer potential is equal in both directions : T[AB] = T[BA].
4. Varroa can pass directly only between apiaries less than 3 km apart.
5. Capability of apiary A to pass varroa to other apiaries is apiary invasion potential. I[A].
6. Apiary invasion potential is sum of its transfer potentials.
7. Varroa can pass indirectly between two apiaries A and C more than 3 km apart if both conditions are met:
A. There is at least one intermediate apiary B, which is less than 3 km apart from both apiaries A and C.
B. Varroa in newly infested intermediate apiary B has enough time to develop over critical level.
8. Value of indirect transfer potential in second step is 0.3: T[AC] = 0.3
Example in figure 5 shows apiaries with their invasion potential. Let's calculate transfer potentials for each apiary and sum them to get apiary invasion potential.
Final set of assumptions and rules
1. All apiaries in region are periodically and simultanouslly treated against varroa.
2. After treatment varroa level is below critical.
1. Capability of varroa passing between apiary A and apiary B is transfer potential T[AB].
2. Value of direct transfer potential is 1 : T[AB] = 1.
3. Transfer potential is equal in both directions : T[AB] = T[BA].
4. Varroa can pass directly only between apiaries less than 3 km apart.
5. Capability of apiary A to pass varroa to other apiaries is apiary invasion potential. T[A].
6. Apiary invasion potential is sum of its transfer potentials.
7. Varroa can pass indirectly between two apiaries A and C more than 3 km apart if both conditions are met:
A. There is at least one intermediate apiary B, which is less than 3 km apart from both apiaries A and C.
B. Varroa in newly infested intermediate apiary B has enough time to develop over critical level.
8. Value of indirect transfer potential in second step is 0.3: T[AC] = 0.3
Figure 5 - example with rules applied
• I[A] = T[AB] + T[AC] + T[AD]+ T[AE] + T[AF]= 1 + 1 + 1 + 0.3 + 0.3 = 3.6
• I[B] = T[BA] + T[BC] + T[BD] = 1 + 0.3 + 0.3 = 1.6
• I[C] = T[CA] + T[CB] + T[CD] = 1 + 0.3 + 0.3 = 1.6
• I[D] = T[DA] + T[DE] + T[DF] + T[DB] + T[DC]+ T[DG] =3.9
• I[E] = T[ED] + T[EA] + T[EF] = 1.6
• I[F] = T[FD] + T[FG] + T[FA] + T[FE]= 2.6
• I[G] = T[GF] + T[GD] = 1.3
• I[H] = 0
Figure 1
We will calculate apiary X invasion potential.
In figure 1 is apiary X with its surrounding apiaries A1 - A5 and B1 - B9.
Apiaries A1 - A5 within red circle are less than 3000 meters away from X1.
Trasfer potentials have all same value of 1.
T[XA1] = T[XA2] = T[XA3] = T[XA4] = T[XA5] = 1
Apiaries B1 - B9 within blue circle, but outside red circle, are between 3001 and 6000 meters away.
Apiaries B1 - B9 will contribute to apiary X invasion potential only if there is at least one intermediate apiary. In figure 1 it is obvious there is no intermediate apiary between X and B2 - B5.
Figure 2
A1 could be intermediate apiary between X and B1 - we have to check if it is. Click on A1 (figure 2).
Both B1 and X are less than 3000 meters away from A1. A1 is intermediate apiary between X and B1 and B1 indirect transfer potential has value of 0.3.
T[XB1] = 0.3
From figure 1 we estimate that A5 could be intermediate apiary for B6 - B9 so we need to check A5. Figure 3 shows that A5 is intermediate only for B7, which will contribute its indirect transfer
potential 0.3.(figure 3)
T[XB7] = 0.3
Apiaries B6, B8 and B9 do not contribute to apiary X invasion potential.
Figure 3
Apiary X invasion potential is sum of its transfer potentials:
I[X] = T[XA1] + T[XA2] + T[XA3] + T[XA4] + T[XA5] + T[XB1] + T[XB7] = 5.6
Apiary X invasion potential is 5.6.
The model demonstrates the potential usage of data from the CAVAT system. Data collected could be beneficial in defining strategies against varroa reinvasion and managing colonies treatment. | {"url":"https://www.cavat.eu/home/potential/","timestamp":"2024-11-06T03:48:33Z","content_type":"text/html","content_length":"81492","record_id":"<urn:uuid:5bbbc55e-2fb3-4ffc-aa63-a7d580caad4a>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00752.warc.gz"} |
MathFiction: And Be a Villain (Rex Stout)
Contributed by "William E. Emba"
Rex Stout and his seventy some Nero Wolfe novels are generally regarded as amongst the greatest mystery novels ever written. They read as fresh today as when the series started in 1934, and they can
pretty much be read in any order. The plot of AND BE A VILLAIN centers around a cyanide poisoning that had happened during a live radio talkshow broadcast, a few days before the novel opens. The
victim was a race track bookie. Also present at the crime scene was a mathematician, to provide expert commentary on probability, and some respectability for the show to even dare have a bookie on
air. (But to the reader, the mathematician is mostly to provide comic relief.) When Nero Wolfe interviews the mathematician, the latter launches into an uninterruptible spiel about how he always
wanted to apply probability theory to detective work, and talks math for a bit, going so far as to write out the "second approximation to the normal distribution", which Archie Goodwin, the novel's
narrator, reproduces. Apparently Archie doesn't know what a square root symbol is, since what should be a sqrt(2.pi.D) comes out as V.2.pi.D. (It appears Stout did his homework, so either his printer
got it wrong, or Stout engaged in an inside joke for mathematicians only.) | {"url":"https://kasmana.people.charleston.edu/MATHFICT/mfview.php?callnumber=mf223","timestamp":"2024-11-04T12:08:54Z","content_type":"text/html","content_length":"9839","record_id":"<urn:uuid:7bbdf5cc-95ac-42c9-a75c-929510a25c2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00083.warc.gz"} |
A pension strategy where everyone can win - Eureka Report article
Summary: Many individuals or couples at pension age are uncertain about how much they can have in personal assets, and how much they can recweive in income, before their age pension is impacted. This
article runs through the numbers, and shows it’s possible to draw down a reasonable tax-free income and still receive a pension.
Key take-out: Although the basic age pension does not allow for an expansive lifestyle, adding income from investments and superannuation can generate an attractive tax-free income stream.
Last week I talked about the fact that a combination of income from investments and superannuation, and from some part-age pension, was likely to lead to a retirement outcome that was better than
many people thought.
Given that many people will not amass the $1.0865 million in assets (excluding the value of their home) that a home-owning couple can have to receive some part-age pension, or for single homeowners
the $731,000 (excluding the value of their home) in assets that they can accumulate while still receiving some part-age pension, it is an important topic for many people planning their retirement.
Many people retiring now have not had compulsory superannuation, and have experienced 10 years of poor investment returns, which will have negatively impacted their retirement position.
As I looked at in last week’s article (Pension plan adds up to better returns), receiving some part-age pension can provide a useful buffer against a drop in investment prices – helping people to
weather difficult times. As asset values fall (such as during the global financial crisis), income from the age pension will increase.
In this article I look at the end game. How much income will be received given a certain level of assets?
In working out the amount of age pension to be received, the Federal Government has set two tests – an income test and an assets test. The calculation used to work out the amount of age pension
received is the one that is most restrictive of a person’s age pension, which is mostly the assets test. On that basis I have done the calculations around the assets test. I have also used the
situation of homeowners – both singles and couples. That is because it is the more common retirement scenario. For those people who are not homeowners, the intuition behind the calculations is the
same. It is just that the asset test limits are higher, resulting in a higher level of income. A non-homeowner might also be eligible for rent assistance.
To start putting together some calculations of how much total income a person or couple in retirement will receive when they have some investments themselves and receive some part-age pension, we
need to think about an appropriate rate at which to draw income from our investment and superannuation funds.
Drawing from investments
Guidelines for Withdrawal Rates and Portfolio Safety During Retirement, by John J Spitzer, Jeffrey C Strieter and Sandeep Singh of the State University of New York, appeared in the US Journal of
Financial Planning in October 2007. It looked at how likely you were to run out of funds during a 30-year retirement. This is a reasonable timeframe – someone retiring at age 60 and living to 90.
They found that with a portfolio that had a 60% exposure to growth assets (shares and property), a person had about a 12% (one in eight) chance of running out of money while drawing at a rate of 4.5%
a year, increasing their drawing each year in line with inflation.
This seems like a reasonable starting point. Let’s assume that the person/couple in retirement draw from their investments at a rate of 4.5% a year. This means drawing $4,500 of income for every
$100,000 invested. Now, this is quite a conservative drawing rate and many people might feel comfortable drawing at a rate of 5% or 5.5% a year. However I will stick with this conservative figure.
Calculating the age pension
For simplicity, I am going to ignore some of the smaller payments (such as the pension supplement) around the age pension and just focus on the core age pension payment. Information about the full
retirement situation can be found from www.centrelink.gov.au.
The basic rate for the age pension is $733 per fortnight for a single person, or $1,106 (total) per fortnight for a couple. As a person/couple’s assets go over a set limit, the amount of age pension
that they receive falls.
For a single homeowner that amount is $192,500, and for a home-owning couple that amount is $273,000. That is, a single can have $192,500 of assets and still receive the full age pension and the
home-owning couple can have $273,000 in assets and still receive the full age pension. Once their assets go over this amount, their age pension is reduced by $1.50 per fortnight for every $1,000 they
are over that limit. So, if they are $100,000 over this limit they will lose $150 in age pension every fortnight.
There is a long list of ‘assessable assets’ for the assets test, including cash, superannuation, investments and lifestyle assets like furniture, boats and cars. The value of these lifestyle assets
is assessed as ‘what you would get for them if you sold them’. So a car bought for $30,000, but which would be sold for $7,000 today, has a value of $7,000.
Putting everything together
So, to the big question. How much income will I get given various levels of assets? I have done the calculations for a home-owning couple, and a home-owning single person. In each cash I have assumed
$50,000 of ‘lifestyle assets’, and the balance of the assets being drawn on (used for income) at a rate of 4.5% a year.
It is worth commenting at this stage that for various reasons, including the impact of the ‘Senior Australian Tax Offset’, that there is almost certainly no income tax to be paid.
Calculations for a Single Homeowner: Fortnightly Income
Level of Assets Drawing from Investments Income from Age Pension Total Income per Fortnight
$0 $733 $733
$300,000 $433 $572 $1,004
$600,000 $952 $122 $1,074
$900,000 $1,471 $0 $1,471
$1,200,000 $1,990 $0 $1,990
Calculations for a Home-owning Couple: Fortnightly Income
Level of Assets Drawing from Investments Income from Age Pension Total Income per Fortnight
$0 0 $1,106 $1,106
$300,000 $433 $1,065 $1,498
$600,000 $952 $615 $1,567
$900,000 $1,471 $166 $1,637
$1,200,000 $1,990 $0 $1,990
Calculations for a Single Homeowner: Annual Income
Level of Assets Drawing from Investments Income from Age Pension Total Income per Year
$0 $0 $19,058 $19,058
$300,000 $11,250 $14,866 $26,116
$600,000 $24,750 $3,166 $27,916
$900,000 $38,250 $0 $38,250
$1,200,000 $51,750 $0 $51,750
Calculations for a Home-owning Couple: Annual Income
Level of Assets Drawing from Investments Income from Age Pension Total Income per Year
$0 $0 $28,756 $28,756
$300,000 $11,250 $27,703 $38,953
$600,000 $24,750 $16,003 $40,753
$900,000 $38,250 $4,303 $42,553
$1,200,000 $51,750 $0 $51,750
So what?
Even though the basic age pension is difficult to live on, adding $250,000 of investments assets (and $50,000 of lifestyle assets) sees a single homeowner receiving a total of more than $500 a week
tax-free to live on. A home-owning couple, with a similar level of assets, receives $750 a week. If the people in these scenarios own their home with no mortgage, then I suspect that this is a
reasonable position to be in.
These calculations are also based on a conservative drawing rate from investment and superannuation assets of 4.5% a year. If you are receiving some part-age pension, you may choose to withdraw at a
slightly higher rate than this, given that you have the age pension safety net in place.
The age pension will be an important source of income for many people considering their retirement situation. While the basic age pension certainly does not allow for an expansive lifestyle, adding
income from investments and superannuation makes the tax free total income stream increasingly attractive. | {"url":"https://acleardirection.com.au/a_pension_strategy_where_everyone_can_win___eureka_report_article","timestamp":"2024-11-10T00:09:50Z","content_type":"text/html","content_length":"100607","record_id":"<urn:uuid:eea798df-2732-4bf2-9a4e-4e21ad9b4fb8>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00177.warc.gz"} |
Find all of the equilibrium points for two
For equilibrium point, $\frac{dx}{dt}=0$ and $\frac{dy}{dt}=0$
For system (i):
$x\left(10-x-20y\right)=0$ .....(1)
Solution of the equations (1) and (2) will have 3 cases.
Case 1:
This will represent 0 prey and 0 predator.
Case 2:
As y cannot be negative, the solution set is rejected.
Case 3:
when $y=0$
For system (ii):
$x\left(0.3-\frac{y}{100}\right)=0$ .....(3)
$y\left(15-y+25x\right)=0$ ...(4)
Solution of the equations (3) and (4) will have 3 cases.
Case 1:
This will represent 0 prey and 0 predator.
Case 2:
So, $\left(\frac{3}{5},30\right)$ will represent a stable population.
Case 3:
when $y=0$
As x cannot be negative, the solution set is rejected. | {"url":"https://plainmath.org/algebra-i/24985-equilibrium-points-systems-explain-significance-predator-population","timestamp":"2024-11-04T11:25:47Z","content_type":"text/html","content_length":"294807","record_id":"<urn:uuid:747173d1-c510-4e2a-80f2-cf0e20af930a>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00834.warc.gz"} |
How to Calculate How Much Siding Your Home Needs (& Get Fair Pricing) - Top Dog Home Pro
Consumers will often ask, “How do I calculate how much siding I need?”
Most homeowners ask about calculating siding needs because they want to make sure in the hiring and selection process they’re getting the best deal when it comes to pricing. Understandable.
Here at Top Dog Home Pro, we understand being informed matters, and when pricing your potential siding replacement you will want to know the siding styles and amount of siding required to price shop.
Today, we will share with you how you can calculate how much siding you need for your home by teaching you the best way to estimate the size of your home.
What you will learn:
• How is siding measured and calculated
• What is a siding square
• How to estimate siding squares
First, how is siding measured and calculated?
When it comes to measuring your home for siding, similar to roofing measurements, everything is done in what is commonly referred to as a “Square.”
A square simply means 100 square feet of surface area. So a 10 by 10 area would equate to 100 square feet or one square.
An estimator will typically measure a home, determine how many squares are required to replace the existing siding, then based on color, style, and siding type, provide an accurate estimate of how
much the siding replacement would cost.
Typically, most siding comes in boxes of two squares or 200 square feet.
Contractors use the square method because it is easiest to estimate the time it will take to replace for labor purposes and give homeowners the most accurate price.
How to estimate squares of siding for your home:
Estimating the size of your home’s siding is actually quite simple. In order to calculate how much siding you will need, start by measuring the four walls of your home.
1. Measure the four walls of your home. For example, if the front of your house is 40 feet across and each floor is typically 10 feet and you have two floors, your home measures 40 feet by 20 feet
or a total of 800 feet.
2. Do the same for the sides and garage. Any area that is a rectangle or square, measure the width and estimate 10 feet for each floor for height purposes.
For example, the same home that is 40X20 in the front is also 40X20 in the back, or a total of 1600 square feet front and back.
The sides are 20 by 20 or 400 square feet, for a total of 800 square feet for the sides. The total square feet for this house is 2400 square feet. BUT we can’t forget about the gables.
Pro Tip: To get in the ballpark of your siding calculations, you can actually just assume it will be around the size of your home. For example, a 3,000 square foot home will need approximately 3,000
square feet of siding or 30 squares.
How to measure for siding gable walls:
Photo Source
The yellow areas in the above photo are called gable walls. These are the angular areas of your home that will meet the roof’s edge and are triangle-shaped.
Measuring your home’s gables are not very hard to estimate and calculate how much siding you will need, it just requires a trip down memory lane to geometry class.
Remember the area for a triangle formula? Area = base x-height, divided by two.
In the example house form above, let us assume there is a gable on each side. The width of the sides is 20 feet, and the gable is 8 feet high.
20 X 8 = 160 square feet, divided by two = 80 square feet.
Area of Siding Measurement Total Square Feet:
Front of home 800 square feet
Back of home 800 square feet
Side 1 400 square feet
Side 2 400 square feet
Gable 1 80 square feet
Gable 2 80 square feet
Subtract windows & doors -260 square feet
Total Square Feet 2,300 square feet
As you can see, the above example of a home that has four standard sides and two gables and is approximately 2,000 square feet in size, will require 2,560 square feet of siding, or 26 squares (always
round up).
Of course, windows and doors are going to be factored out and if your average window is 6 square feet and you have 20 windows, you can factor out 120 square feet from the overall siding calculations.
Factor in the doors and even the garage, and you can really save some money with siding.
What you might notice is that the total size of a home is approximately the same for the exterior siding. So in general, a 1600 square foot home will need roughly 1600 square feet in the siding (give
or take a few hundred feet). While this isn’t 100% accurate, it does give you a ballpark!
Which leads to the final question you might be asking yourself, how much does a square of siding cost?
Related: How to Find The Best Roofing Company Near You
How much does a square of siding cost?
On average, the price for a square of siding installed can vary on a few factors:
• Type of siding
• Labor requirements (some siding material is harder to install)
• Location of where you live (supply & demand)
• Trimming and existing material
For example, vinyl siding is always going to cost less than reinforced (insulated) vinyl siding, which is also cheaper than fiber cement siding (Hardy board). Aside from an occasional anomaly or two,
here are the breakdowns for the national averages of siding (installed):
• $100 per square (extreme low end)
• $650+ per square (higher end)
Using the example 2,300 square foot house from above, the cost to replace the siding by varying styles would look something like this:
Style of Siding Cost
Wood $11,500
Vinyl $18,000
Reinforced Vinyl $20,000
Hardy Board $23,000
The Verdict on Measuring Siding –
These prices above are just average guesses, however, it does help you understand what type of costs you are looking at when you replace your siding.
When it comes to measuring your siding, it doesn’t have to be exact if you’re a homeowner. You just want to have a brought estimate of how many squares you will need to buy and have installed.
Here are some parting tips:
1. To answer, “How do I calculate how much siding I need?” simply roughly use the square footage of your home
2. If you want to be exact, measure it, draw it out some, and get an accurate measurement and convert it to square
3. To covert square feet into a square, divide by 100 (1800 divided by 100 = 18 square).
Still got questions? Ask us below! | {"url":"https://tdhomepro.com/how-to-calculate-how-much-siding/","timestamp":"2024-11-03T18:54:18Z","content_type":"text/html","content_length":"144223","record_id":"<urn:uuid:aceb77b2-30fa-4194-bd86-261213fd52b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00258.warc.gz"} |
Hydraulic Radius Calculator| How to Calculate Hydraulic Radius? - physicsCalculatorPro.com
Hydraulic Radius Calculator
The area, hydraulic radius, and wetted perimeter of an item are quickly calculated with this online hydraulic radius calculator. Select the object shape, enter the radius in the input section, and
then push the calculate button to get the result fast.
What is Hydraulic Radius?
The hydraulic radius is the proportion of the flow's cross-sectional area to the wetted perimeter. Rectangle, pipe with full, pipe with a partially filled, trapezoid, and triangle are the various
The area of a channel that is in contact with the fluid it transports is referred to as the wetted perimeter. Hydraulic radius and wetted perimeter are significant flow parameters to consider while
building a sewerage system.
The formulas for calculating the wetted perimeter and hydraulic radius of various shapes are listed below.
• The wetted perimeter of full pipe is P = 2πr
• Hydraulic Radius of full pipe R = A/P = πr²/2πr = r/2
• Pipe(less than half) Wetted Perimeter P = r * θ and θ = 2 * arccos [(r - h) / r]
• Area A = r² * (θ - sin(θ)) / 2
• Hydraulic Radius R = A/P = [r² * (θ - sin(θ)) / 2]/[r * θ and θ = 2 * arccos [(r - h) / r]] = [r² * (θ - sin(θ)) / 2] / (r * θ)
Area, Wetted Perimeter, Hydraulic Radius of Different Channels
Triangular Channel
• Area A = B * y / 2 = y²z
• Wetted Perimeter P = 2 * y * √(1 + z²)
• Hydraulic Radius R = A/P = yz / [2 * √(1 + z²)]
Rectangular Channel
• Area A = b * y
• Wetted Perimeter P = b + y + y = b + 2y
• Hydraulic Radius R = A/P = (by)/(b + 2y)
Trapezoid Channel
• Area A = y * (B + b)/2, B = b + 2zy
• Wetted Perimeter P = b + 2 * y * √(1 + z²)
• Hydraulic Radius R = A/P = (by + y²z)/[b + 2 * y * √(1 + z²)]
How to Compute Hydraulic Radius & Wetted Perimeter?
The steps for determining the wetted perimeter and hydraulic radius for five distinct shapes are outlined below. Examine the specifics and conform to them.
• Step 1: Take note of the object's shape and the information provided.
• Step 2: Get the wetted perimeter, area, and hydraulic radius formulas.
• Step 3: To acquire the result, substitute the values in the equations and complete the computations.
For more concepts check out physicscalculatorpro.com to get quick answers by using this free tool.
FAQs on Hydraulic Radius Calculator
1. What does hydraulic radius mean?
The hydraulic radius is a measurement of the efficiency of a channel's flow. It's one of the most important characteristics that determine how much fluid a channel can discharge and how well it can
carry sediments.
2. How do you distinguish between the hydraulic radius and hydraulic depth?
The ratio of cross-sectional area to the wetted perimeter is known as hydraulic radius. Hydraulic depth is defined as the depth that equals the irregular section area when multiplied by the topwater
surface width. The hydraulic mean depth and radius formulas are A/T and A/P, respectively.
3. What is the trapezoidal's hydraulic radius?
A trapezoid's hydraulic radius is R = [h(b+zh)]/[(b + 2h√(1 + z2)]. Where b is the fluid's bottom width, h is its depth, and z is the channel slope's width.
4. What is the flow's hydraulic radius?
The ratio of the cross-sectional area of fluid flow, A, to the length of the wetted perimeter, P, is described as hydraulic radius, R, a measure of channel flow efficiency. The hydraulic radius is
one of the most important parameters that determine how much fluid a channel can discharge and how well it can move sediments.
5. What is the definition of hydraulic radius?
The ratio of the cross-sectional area of a fluid-flowing channel or pipe to the wetted perimeter of the conduit.
6. What is the hydraulic mean depth R?
The hydraulic mean depth, also known as the hydraulic radius, is the proportion of flow area to the wetted perimeter. | {"url":"https://physicscalculatorpro.com/hydraulic-radius-calculator/","timestamp":"2024-11-05T04:25:18Z","content_type":"text/html","content_length":"32725","record_id":"<urn:uuid:a7ffaae3-492d-42ee-96d1-a6e09748a98a>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00276.warc.gz"} |
C++ Tutorial - Functions
In the previous tutorial, we learned how to make our code more robust by using while loops to screen out bad user input. In this tutorial, we are going to learn how to make our code more modular and
reusable by utilizing functions. Functions are a way of abstracting out common tasks. A great example of this would be something like computing the factorial of an integer. It's a common mathematical
operation, so you just want to take an integer and compute the factorial of it. If you had to write the same 5 lines of code (or more) every time you wanted to compute a factorial, it would be a bit
tedious. Functions allow us to write the code once and reuse it in many places.
So here's an example of a factorial function:
int Factorial(int n)
int ret = 1;
for(int i = n; i > 1; i--)
ret = ret * i;
return ret;
We have seen something similar to this in the first tutorial, with the main function. The first line specifies the output type (int), the function name (Factorial), and the input arguments (one input
argument of type int and its name is n). This first line contains all of the information that the caller of the function cares about: what goes in, what comes out, and the functions name. The rest of
the function does the transformation from the input to the output. The factorial function in math is defined as:
n! = n * (n - 1) * (n - 2) * ... * 3 * 2 * 1
so as an example:
5! = 5 * 4 * 3 * 2 * 1 = 120
So, we can readily see how the function given above computes the factorial of a number using a for loop. This is very powerful because now that we've written the function, we no longer need to
remember the internal details about how to compute a factorial, we can just call the function that we created and get the right answer. Also, this will greatly reduce the amount of code that we need
to write, in the case that we need to compute multiple factorials.
Take for instance the combination function from probability. This is used to determine how many combinations of x selections can be made from n total items ignoring order (n choose x), and the
formula looks like this:
C(n, x) = n! / (x! * (n - x)!)
A concrete example is how many two letter combinations can be made from the four letters A, B, C, and D. The answer is:
C(4, 2) = 4! / (2! * (4 - 2)!) = 4! / (2! * 2!) = 24 / (2 * 2) = 24 / 4 = 6
Which we can also show by listing out the combinations:
So, if we wanted to code this using our Factorial function, it would look like:
int Combination(int n, int r)
return Factorial(n) / (Factorial(r) * Factorial(n-r));
This is much better than the alternative that doesn't use our function:
int Combination2(int n, int r)
// Factorial(n)
int fn = 1;
for (int i = n; i > 1; i--)
fn = fn * i;
// Factorial(r)
int fr = 1;
for (int i = r; i > 1; i--)
fr = fr * i;
// Factorial(n-r)
int fnr = 1;
for (int i = n - r; i > 1; i--)
fnr = fnr * i;
return fn / (fr * fnr);
Look at how much more code that is! Imagine if we had made a mistake in computing the factorial, then we would have to look throughout our code and find all of the places that we had copied and
pasted the factorial computation and fix them, rather than if we created a function, we would only have to fix it in one place. This can be a life saver.
Something else interesting to note from the previous example is that you can call a function from within a function. This helps build up complexity in a manageable way. What might not be obvious or
intuitive a first is that you can call the same function from within a function. This is called recursion, and it looks like this:
int Factorial(int n)
if(n < 2)
return 1;
return n * Factorial(n - 1);
Notice that the factorial function calls itself with the last line. This is essentially doing:
n! = n * (n-1)!
It keeps doing this until n is less than 2, and then it returns one, because 1! = 1 and 0! = 1. Recursion is an interesting way to create a loop without using a for or a while loop. In my experience,
recursion isn't very common, but it is still good to know about in case you run across it one day.
In this tutorial, we learned how to make our code more reusable by creating functions. We then went on to discuss how you can use functions to help manage complexity by breaking it down into smaller
In the next tutorial, we go over the various data types and the strengths and weaknesses of each data type.
If you have any questions or comments about what was covered here, post them to the comments. I watch them closely and will respond and try to help you out.
Tutorial Index
• So, it dawned on me after I wrote the article, that a better implementation of the combination fuction would look something like this:
int Combination(int n, int r)
int ret = 1;
for(int i = n; i > n - r; i--)
ret = ret * i;
return ret / Factorial(r);
This is an improvement, because it will handle much larger values of n. So, say you wanted to compute the number of 5 card hands that could be made with a standard deck of cards. The answer would
be Combination(52, 5). The problem is that 52! is a very large number. Much larger than what can be stored in an integer, as we will go over in the next tutorial. So, when you go to compute this,
you won't get what you expect. This new implementation helps by computing 52!/47! much how you would by hand. Namely that it is 52*51*50*49*48*47!/47!, and the 47! cancels out and leaves us
52*51*50*49*48. This means that you never have to compute 52! or 47!, which saves us from those very big numbers.
Unfortunately, it doesn't show the use of functions as nicely, so maybe it is better that I didn't think of it after all! | {"url":"https://community.element14.com/technologies/code_exchange/b/blog/posts/c-tutorial---functions","timestamp":"2024-11-09T16:19:35Z","content_type":"text/html","content_length":"255884","record_id":"<urn:uuid:1efc82de-5ec9-4b36-ac24-f82d92a64a12>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00817.warc.gz"} |
Nondeterministic Finite Automata
Learn about nondeterministic finite automata through several examples.
Introducing nondeterminism
Consider the language $K$, which contains strings with a $1$ in the third-to-last position. We can write it as $K = \{ \textrm{all strings of } 0 \textrm{'s and } 1 \textrm{'s ending with } 100,
\:101,\:110, \textrm{ or } 111 \}.$
It will require a lot of effort to make a DFA that accepts the language $K$. It is possible, however, to more succinctly express the idea of “ends with a $1$ in the third-to-last position” than a DFA
allows. With nondeterminism, we can get right to the point of expressing exactly what we want, as shown in the nondeterministic finite automaton (NFA) in the following figure.
Get hands-on with 1400+ tech skills courses. | {"url":"https://www.educative.io/courses/theory-of-computation/nondeterministic-finite-automata","timestamp":"2024-11-09T03:39:12Z","content_type":"text/html","content_length":"766972","record_id":"<urn:uuid:b2460785-d1e1-48bd-8efc-0f6dff7757bd>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00023.warc.gz"} |
Bias-Adjusted Treatment Effect
Four Regressions
The goal of the bate package is to present some functions to compute quantiles of the empirical distribution of the bias-adjusted treatment effect (BATE) in a linear econometric model with omitted
variables. To analyze such models, a researcher should consider four regression models: (a) a short regression model where the outcome variable is regressed on the treatment variable, with or without
additional controls; (b) an intermediate regression model where additional control variables are added to the short regression; (c) a hypothetical long regression model, where an index of the omitted
variable(s) is added to the intermediate regressions; and (d) an auxiliary regression where the treatment variable is regressed on all observed (and included) control variables.
As an example, suppose a researcher has estimated the following model, \[y = \alpha + \beta_1 x + \gamma_1 w_1 + \gamma_2 w_2 + \varepsilon,\] and is interested in understanding the impact of some
omitted variables on the results. In this case:
• outcome variable: \(y\)
• treatment variable: \(x\)
• short regression: \(y\) regressed on \(x\);
• intermediate regression: \(y\) regressed on \(x, w_1, w_2\);
• auxiliary regression: \(x\) regressed on \(w_1, w_2\);
• hypothetical long regression: \(y\) regressed on \(x, w_1,w_2\) and the omitted variables;
In this example, the estimated treatment effect is \(\hat{\beta}_1\). In the presence of omitted variables, \(\hat{\beta}_1\) is a biased estimate of the true treatment effect, \(\beta_1\). The
functions in this package will allow researchers to create quantiles of the empirical distribution of the BATE, i.e. the treatment effect once we have adjusted for the effect of omitted variable
bias. We will denote the BATE as \(\beta^*\).
The researcher will need to supply the data set (as a data frame), the name of the outcome variable, the name of the treatment variable, and the names of the additional regressors in the intermediate
regression. The functions in this package will then compute the quantiles of the empirical distribution of BATE, \(\beta^*\).
Two Important Parameters
Two parameters capture the effect of the omitted variables in this set up.
The first parameter is \(\delta\). This captures the relative strength of the unobservables, compared to the observable controls, in explaining variation in the treatement variable. In the functions
below this is denoted as the parameter delta. This parameter is a real number and can take any value on the real line, i.e. it is unbounded. Hence, in any specific analysis, the researcher will have
to choose a lower and an upper bound for delta. For instance, if in any empirical analysis, the researcher believes, based on knowledge of the specific problem being investigated, that the
unobservables are less important than the observed controls in explaining the variation in the treatment variable, then she could choose delta to lie between 0 and 1. On the other hand, if she
believes that the unobservables are more important than the observed controls in explaining the variation in the treatment variable, then she should choose delta to lie between 1 and 2 or 1 and 5.
The second parameter is \(R_{max}\). This captures the relative strength of the unobservables, compared to the observable controls, in explaining variation in the outcome variable. In the functions
below, this is captured by the parameter Rmax. The parameter Rmax is the R-squared in the hypothetical long regression. Hence, it lies between the R-squared in the intermediate regression (\(\tilde
{R}\)) and 1. Since the lower bound of Rmax is given by \(\tilde{R}\), in any specific analysis, the researcher will only have to choose an upper bound for Rmax.
In a specific empirical analysis, a researcher will use domain knowledge about the specific issue under investigation to determine a plausible range for delta (e.g. \(0.01 \leq \delta \leq 0.99\)).
This will be given by the interval on the real line lying between deltalow and deltahigh (the researcher will choose deltalow and deltahigh). Using the example in this paragraph, deltalow=0.01 and
In a similar manner, a researcher will use domain knowledge about the specific issue under investigation to determine Rmax. Here, it will be important to keep in mind that Rmax is the R-squared in
the hypothetical long regression. Now, it is unlikely that including all omitted variables and thereby estimating the hypothetical long regression will give an R-squared of 1. This is because, even
after all the regressors have been included, some variation of the outcome might be plausibly explained by a stochastic element. Hence, Rmax will most likely be different from, and less than, 1. This
will be denoted by Rhigh (e.g. Rmax=0.61).
The Algorithm
How is the omitted variable bias and the BATE computed? The key result that is used to compute the BATE is this: the omitted variable bias is the real root of the following cubic equation \[a \nu^3 +
b \nu^2 + c \nu + d =0,\] where
• \(a = (\delta -1)(\tau_X \sigma_X^2 - \tau_X^2) \neq 0,\)
• \(b = \tau_X \left( \mathring{\beta} - \tilde{\beta}\right) \sigma^2_X \left( \delta - 2 \right),\)
• \(c = \delta \left( R_{max} - \tilde{R} \right) \sigma^2_Y \left( \sigma^2_X - \tau_X \right) - \left( \tilde{R} - \mathring{R} \right) \sigma^2_Y \tau_X - \sigma^2_X \tau_X \left( \mathring{\
beta} - \tilde{\beta}\right)^2,\) and
• \(d = \delta \left( R_{max} - \tilde{R} \right) \sigma^2_Y \left( \mathring{\beta} - \tilde{\beta}\right) \sigma^2_X,\)
where, in turn,
• \(\sigma_Y^2\) is the variance of the outcome variable,
• \(\sigma_X^2\) is the variance of the treatment variable,
• \(\mathring{\beta}\) is the treatment effect in the short regression,
• \(\mathring{R}\) is the R-squared in the short regression,
• \(\tilde{\beta}\) is the treatment effect in the intermediate regression,
• \(\tilde{R}\) is the R-squared in the intermediate regression, and
• \(\tau_X^2\) is the variance of the residual in the auxiliary regression.
Hence, we see that the coefficients of the cubic equation are functions of the variances of the outcome and treatment variables (\(\sigma_Y^2, \sigma_X^2\)), parameters of the short regression (\(\
mathring{\beta}, \mathring{R}\)), intermediate regression (\(\tilde{\beta}, \tilde{R}\)) and auxiliary regression (\(\tau_X^2\)), and the values of delta and Rmax. The important result is that the
omitted variable bias is the real root of the above cubic equation (Proposition 2, Oster, 2019; Proposition 1 and 2 in Basu, 2022).
In a specific empirical analysis, the variances of the outcome and treatment variables, and the parameters of the short, intermediate and auxiliary regressions are known. Hence, the coefficients of
the cubic equation become functions of delta and Rmax, the two key parameters that the researcher chooses, using domain knowledge.
Once the researcher has chosen deltalow, deltahigh and Rhigh, this defines a bounded box on the (delta, Rmax) plane defined by the Cartesian product of the interval [deltalow, deltahigh] and of the
interval [Rlow, Rhigh]. The main functions in this package computes the root of the cubic equation on a sufficiently granular grid (the degree of granularity will be chosen by the user) covering the
bounded box.
To compute the root of the cubic equation, the algorithm first evaluates the discriminant of the cubic equation on each point of the grid and partitions the box into two regions: (a) unique real root
(URR) and NURR (no unique real root). There are three cases to consider.
• Case 1: If all points of the bounded box are in URR, then the algorithm chooses the unique real root of the cubic at each point as the estimate of the omitted variable bias.
• Case 2: If some non-empty part of the box is in NURR, then the algorithm first computes roots on the URR region, and then, starting from the boundary points of URR/NURR, covers points on the NURR
in small steps. At each step, the algorithm chooses the real root at a grid point in the NURR that is closest in absolute value to the real root at a previously selected grid point. Continuity of
the roots of a polynomial with respect to its coefficients guarantees that the algorithm selects the correct real root at each point.
• Case 3: If the bounded box is completely contained in NURR, then the algorithm extends the size of the box in small steps in the delta direction to generate a nonempty intersection with a URR
region. Once that is found, the algorithm implements the steps outlined in step 2.
The bias is then used to compute the BATE, \(\beta^*\), which is defined as the estimated treatment effect in the intermediate regression minus the bias, i.e. \[\beta^* = \tilde{\beta}-\nu,\] where \
(\beta^*\) is the bias-adjusted treatment effect, \(\tilde{\beta}\) is the treatment effect estimated from the intermediate regression and \(\nu\) is the real root of the relevant cubic equation.
The functions in this package will compute \(\beta^*\) at each point of the grid that covers the bounded box chosen by the researcher. Hence, this will generate a large vector of values of \(\beta^*
\) and we can use this to compute the empirical distribution of \(\beta^*\), the BATE. Asymptotic theory shows that the BATE converges in probability to the true treatment effect, i.e. \[\beta^* \
overset{p}{\to} \beta.\] Hence, the interval defined by the 2.5-th and 97.5-th quantiles of the empirical distribution of the BATE will contain the true treatment effect with 95 percent probability.
The Main Functions
This package provides three functions to compute quantiles of the empirical distribution of the omitted bias and BATE, ovbias(), ovbias_lm() and ovbias_par(). These functions implement the same
algorithm to compute the empirical distribution of the bias and BATE and only differ in how the user provides the parameters of the short, intermediate and auxiliary regressions. But before we
discuss the main functions, let us look at two other functions that are useful.
An useful function to collect relevant parameters from the short, intermediate and auxiliary regressions is:
• collect_par(): collects parameters from the short, intermediate and auxiliary regressions; (user provides name of the data set, name of outcome variable, name of treatment variable, names of
control variables in the short regression, if relevant, and names of additional variables in the intermediate regression); the output of this function is a data frame.
Users can use the output from collect_par() to construct an area plot of the bounded box using:
• urrplot(): creates a colored area plot of the bounded box chosen by the user demarcating the area where the cubic equation has unique real root (URR) from the area where the cubic equation has
three real roots (NURR); the output is a plot object.
ovbias() function
To use the ovbias() function, the user will need to run first collect the parameters from the short, intermediate and auxiliary regressions, using the collect_par() function and then feed it into the
ovbias() function, along with the following:
• deltalow: lower limit of delta (e.g. 0.01)
• deltahigh: upper limit of delta (e.g. 0.99)
• Rhigh: upper limit of Rmax (e.g. 0.61)
• e: step size in defining the grid (e.g. 0.01)
In using the collect_par() function, the user will need to specify the following:
• data: the name of the data frame (e.g. NLSY_IQ)
• outcome: name of the outcome variable in double quotes (e.g. “iq_std”)
• treatment: name of the treatment variable in double quotes (e.g. “BF_months”)
• control: names of additional regressors to include in the intermediate regression, supplied as a vector (e.g. c(“age”,“sex”,“income”,“motherAge”,“motherEDU”,“mom_married”,“race”))
• other_regressors: names of regressors in the short regression, other than the treatment variable, supplied as a vector (e.g. c(“sex”,“age”))
The output of the ovbias() function is a list of three elements.
• Data: A data frame containing the bias (bias) and bias-adjusted treatment effect (bstar) for each point on the grid.
• bias_Distribution: Quantiles (2.5,5.0,50,95,97.5) of the empirical distribution of bias.
• bstar_Distribution: Quantiles (2.5,5.0,50,95,97.5) of the empirical distribution of the BATE (bstar).
ovbias_par() function
To use the ovbias_par() function, the user needs to specify the following:
• data: the name of the data frame (e.g. NLSY_IQ)
• outcome: name of the outcome variable in double quotes (e.g. “iq_std”)
• treatment: name of the treatment variable in double quotes (e.g. “BF_months”)
• control: names of additional regressors to include in the intermediate regression, supplied as a vector (e.g. c(“age”,“sex”,“income”,“motherAge”,“motherEDU”,“mom_married”,“race”))
• other_regressors: names of regressors in the short regression, other than the treatment variable, supplied as a vector (e.g. c(“sex”,“age”))
• deltalow: lower limit of delta (e.g. 0.01)
• deltahigh: upper limit of delta (e.g. 0.99)
• Rhigh: upper limit of Rmax (e.g. 0.61)
• e: step size in defining the grid (e.g. 0.01)
The output of the ovbias_par() function is identically same with the output of the ovbias() function.
ovbias_lm() function
To use the ovbias_lm() function, the user needs to specify three lm objects that capture the short, intermediate and auxiliary regressions:
• lm_shrt: lm object for the short regression
• lm_int: lm object for the intermediate regression
• lm_aux: lm object for the auxiliary regression
• deltalow: lower limit of delta (e.g. 0.01)
• deltahigh: upper limit of delta (e.g. 0.99)
• Rhigh: upper limit of Rmax (e.g. 0.61)
• e: step size in defining the grid (e.g. 0.01)
The output of the ovbias_lm() function is identically same with the output of the ovbias() function.
Additional Functions
Using the output from ovbias(), ovbias_par() or ovbias_lm(), users can construct various plots to visualize the results:
• cplotbias(): contour plot of the bias over the bounded box; the output of this function is a plot object;
• dplotbate(): histogram and density plot of BATE; the output of this function is a plot object;
The methodology proposed in Basu (2022) is slightly different from, and a critique of, Oster (2019). Hence, it might be useful to compare the results of the two methodologies. The methodology
proposed in Oster (2019) is implemented via these functions:
• osterbds(): identified sets according to Oster’s methodology; the output of this function is a data frame;
• osterdelstar(): the value of \(\delta^*\) for a chosen value of \(R_{max}\); the output of this function is a data frame;
• delfplot(): a plot of the graph of the function, \(\delta=f(R_{max})\); the output of this function is a plot object.
Example: Impact of Maternal Behavior on Child IQ
Install the package from CRAN and then load it.
Setting Up
Let us load the data set.
The data set has two .RData objects: NLSY_IQ (to be used for the analysis of maternal behavior on child IQ) and NLSY_BW (to be used for the analysis of maternal behavior on child birthweight).
In this example, we will analyse the effect of maternal behavior on child IQ scores. Let us start out by seeing the names of the variables in the NLSY_IQ data set.
#> [1] "iq_std" "BF_months" "mom_drink_preg_all"
#> [4] "lbw_preterm" "age" "female"
#> [7] "black" "motherAge" "motherEDU"
#> [10] "mom_married" "income" "sex"
#> [13] "race"
For use in the example below, let us set age and race as factor variables
Let us use the vtable package to see the summary statistics of the variables in our data set.
Summary Statistics
Variable N Mean Std. Dev. Min Pctl. 25 Pctl. 75 Max
iq_std 6514 0.017 0.99 -4.136 -0.586 0.693 3.467
BF_months 6514 2.403 4.63 0 0 3 48
mom_drink_preg_all 6090 0.323 0.468 0 0 1 1
lbw_preterm 5837 0.051 0.219 0 0 0 1
age 6514
… 4 1794 27.5%
… 5 1847 28.4%
… 6 1075 16.5%
… 7 922 14.2%
… 8 876 13.4%
female 6514 0.491 0.5 0 0 1 1
black 6514 0.282 0.45 0 0 1 1
motherAge 6514 25.201 5.704 14 21 28 45
motherEDU 6514 12.189 2.577 1 11 13 95
mom_married 6514 0.652 0.476 0 0 1 1
income 6514 40799.978 80453.373 0 11474 45000 974100
sex 6514 1.491 0.5 1 1 2 2
race 6514
… 1 1328 20.4%
… 2 1838 28.2%
… 3 3348 51.4%
Setting Up the Analysis
We will work with an example where the effect of months of breastfeeding on children’s IQ score is analyzed. Thus, here, the outcome variable is a child’s IQ score and the treatment variable is the
months of breastfeeding by the mother. Additional control variables are: sex and age of the child, and the mother’s age, the mother’s years of formal education, whether the mother is married and the
race of the mother. For further details of this example, see section 4.2 in Oster (2019). For ease of reference, let us note that we are working with the model reported in the first row of Table 3 in
Oster (2019) and the first block of 4 rows in Table 2 in Basu (2022).
Using the names of variables in the data set, we have: * short regression: iq_std ~ BF_months + sex + age * intermediate regression: iq_std ~ BF_months + sex + age + income + motherAge + motherEDU +
mom_married + race.
Let us use the collect_par() function to collect parameters from the short, intermediate and auxiliary regressions. It is important to note that other_parameters option in this function should refer
to a subset of control. If the researcher fails to ensure this, the collect_par() function will throw an error.
parameters <- bate::collect_par(data=NLSY_IQ,
other_regressors = c("sex","age"))
Let us see the parameters by looking at the object parameters that we used to store the output of the collect_par() function.
#> beta0 R0 betatilde Rtilde sigmay sigmax taux
#> BF_months 0.04447926 0.04465201 0.01740748 0.255621 0.9900242 4.629618 18.99883
Our next task is to choose the dimensions of the bounded box over which we want the bias computation to be carried out. It is here that the researcher needs to use domain knowledge to set limits over
which \(\delta\) and \(R_{max}\) can run.
# Upper bound of Rmax
Rhigh <- 0.61
# Lower bound of delta
deltalow <- 0.01
# Upper bound of delta
deltahigh <- 0.99
# step size to construct grid
e <- 0.01
Note that while setting the dimensions of the bounded box, we have not chosen a value for Rlow (lower bound for \(R_{max}\)). This is because Rlow is chosen by default to be equal to
parameters$Rtilde (the R-squared in the intermediate regression).
Let us see the division of the bounded box into the URR (unique real root) and NURR (nonunique real root) regions using the urrplot() function.
Using ovbias()
Now we can use the ovbias() function to compute the empirical distribution of omitted variable bias and BATE. Note that this step make take a few minutes, depending on the dimensions of the box and
the size of e, to complete itself. As the function works, it will print a message in the console informing the user of the progress of the computation. Here, I have suppressed these messages.
OVB <- bate::ovbias(
parameters = parameters,
Once the computation is completed, we can see the quantiles of omitted variable bias
and quantiles of the BATE (computed over the bounded box we chose above).
We can create a contour plot of BATE over the bounded box using the cplotbias() function.
We can create the histogram and density plot of the \(\beta^*\), the bias-adjusted treatment effect using the dplotbate() function.
Comparing Our Results with Oster (2019)
Let us compare our results with the methods proposed in Oster (2019). The methodology proposed in Oster (2019) relies on computing two things: (a) identified sets, and (b) \(\delta^*\).
Let us compute the identified sets.
bate::osterbds(parameters = parameters, Rmax=0.61)
#> Discriminant Interval1 Interval2
#> "20.3210250174802" "[0.0174,0.3752]" "[-0.0337,0.0174]"
The output from the above function contains three things: the value of the discriminant of the quadratic equation that is solved to compute the identified sets, and the two identified sets. The
identified set is the interval formed by \(\tilde{\beta}\) (treatment effect in the intermediate regression) and \(\beta^*=\tilde{\beta}-\nu\), where \(\nu\) is a root of a quadratic equation.
It has been shown in Proposition 5 in Basu (2022) that the discriminant of this quadratic will always be positive. Hence, there will be two real roots of the quadratic. Hence, there will be two
identified sets, instead of a unique identified set. That is why we see two identified sets in the result above.
Let us also compute \(\delta^*\), the value of \(\delta\) that is associated with a given value of Rmax (in this case Rmax=0.61) such that the treatment effect is zero.
bate::osterdelstar(parameters = parameters, Rmax=0.61)
#> deltastar discontinuity slope
#> "0.366194618840608" "FALSE" "Negative"
In addition to the value of \(\delta^*\), the output has two other things:
• discontinuity: this tells us whether the interval formed by \(\tilde{R}\) and \(1\) contains a point of discontinuity; if it is FALSE, then the interval does not contain the point of
discontinuity; if it is TRUE, then the interval contains the point of discontinuity and the analysis of Oster (2019) should be avoided; for more details see Section 5.2 in Basu (2022).
• slope: this gives the slope of the graph of the function \(\delta=f(R_{max})\); the slope can be either positive or negative and helps in interpreting the meaning of \(\delta^*\).
The value of \(\delta^*\) printed above is just one value picked up from the function \(\delta=f(R_{max})\), for the specific choice of \(R_{max}\). To see the graph of this function, we can use the
function delfplot().
The red vertical dashed line in the plot identifies the value of \(R_{max}\) that corresponds to \(\delta=1\). In this example, this value of \(R_{max}\) is \(0.38\). This means that if a researcher
uses any value of \(R_{max}\) that is greater than \(0.38\), she will get a value of \(\delta^*\) that is less than \(1\), and if she uses a value of \(R_{max}\) that is less than \(0.38\), she will
get a value of \(\delta^*\) that is greater than \(1\).
Using the ovbias_par() function
We could have carried out the same analysis using the ovbias_par() function.
OVB.par <- ovbias_par(
other_regressors = c("sex","age"),
We can now see the quantiles of omitted variable bias
and quantiles of the BATE (computed over the bounded box we chose above).
Using the ovbias_lm() function
We could have also carried out the same analysis using the ovbias_lm() function. To use this function, we need to estimate the short regression
and the intermediate regression
reg_col2 <- lm(
iq_std ~ BF_months + factor(age) + sex +
income + motherAge + motherEDU + mom_married +
data = NLSY_IQ
and the auxiliary regression
reg_aux <- lm(
BF_months ~ factor(age) + sex +
income + motherAge + motherEDU + mom_married +
data = NLSY_IQ
and then call the ovbias_lm() function and provide the three lm objects created above
OVB.lm <- ovbias_lm(
lm_shrt = reg_col1,
lm_int = reg_col2,
lm_aux = reg_aux,
We can now see the quantiles of omitted variable bias
and quantiles of the BATE (computed over the bounded box we chose above). | {"url":"https://cran.case.edu/web/packages/bate/vignettes/bate-vignette.html","timestamp":"2024-11-12T23:16:26Z","content_type":"text/html","content_length":"146017","record_id":"<urn:uuid:beb2b8a8-8cd7-44de-8830-887c89efb34b>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00349.warc.gz"} |
Triangle properties for Competitive Exams - Math Shortcut Tricks
Triangle properties
Triangle properties are one of the most important topics in exams. Time takes a huge part in competitive exams. If you know how to manage time then you will surely do great in your exam. Most of us
miss this thing. Few examples on triangle shortcuts is given in this page below. These shortcut tricks cover all sorts of tricks on Triangle. Visitors are requested to carefully read all shortcut
examples. These examples here will help you to better understand shortcut tricks on triangle. | {"url":"https://www.math-shortcut-tricks.com/triangle-properties/","timestamp":"2024-11-06T17:17:43Z","content_type":"text/html","content_length":"201286","record_id":"<urn:uuid:39fb4355-c172-4a7c-8780-63ffeb8d620f>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00242.warc.gz"} |
Recommendations for Flux Skew
For Flux Skew, we use the two machines below by varying the mesh to increase the number of nodes in order to recommend the right solver and the right number of cores according to the number of mesh
Figure 1. Both machines used by Flux Skew for the recommendations
The machine (a) used for the Magnetostatic and Transient Magnetic applications is the same machine used as for the 2D Flux example.
Machine (b) is used in Steady-State AC magnetic application, it is an induction machine with 4 poles also powered by a circuit coupling.
Several recommendations are provided in the tables below for each of the Flux Skew applications (Transient Magnetic, Steady-State AC magnetic, and Magnetostatic). Note that the number of nodes in the
extruded geometry can be estimated by multiplying the number of nodes of the 2D mesh by the number of layers of the Skew geometry.
Magnetostatic project
Number of nodes in the mesh (extruded geometry) 0 - 300k 300k - 1M 1M - 2M 2M - 5M 5M - 8M 8M +
Recommended number of cores 1 2 8 16
Minimal memory required 2 GB 8 GB 12 GB 24 GB 40 GB 40+ GB
Recommended linear solver Direct (MUMPS)
Steady-State AC magnetic project
Number of nodes in the mesh (extruded geometry) 0 - 100k 100k - 500k 500k - 1.5M 1.5M - 4M 4M - 10M 10M +
Recommended number of cores 1 2 to 4 4 to 8 8 to 12 12 to 16 16 to 24
Minimal memory required 1 GB 4 GB 12 GB 32 GB 16 GB 32+ GB
Recommended linear solver Direct (MUMPS) Iterative (PETSc)
Transient Magnetic Project
Number of nodes in the mesh (extruded geometry) 0 - 100k 100k - 400k 400k - 1.5M 1.5M +
Recommended number of cores 1 2 4 4 to 8
Minimal memory required 500 MB 1 GB 2 GB 4+ GB
Recommended linear solver Direct (MUMPS) | {"url":"https://help.altair.com/flux/Flux/Help/english/HowTo/English/HowTo_DistributionAndParallelization/topics/SolverParallel_RecommendationsFluxSkew.htm","timestamp":"2024-11-12T23:39:24Z","content_type":"application/xhtml+xml","content_length":"60379","record_id":"<urn:uuid:0ab237cf-048a-4e7b-8b9b-b1488ce52058>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00631.warc.gz"} |
Rheology of Complex Fluids | Joachim Kaldasch
Rheology of Complex Fluids
Structural Transitions and the Rheology of Soft Sphere Suspensions
The book deals with experimental and theoretical studies on the flow dynamics of concentrated colloidal suspensions. The experimental investigations show that in concentrated colloidal suspensions
(from a volume fractions above approx. 45%) the viscosity can increase abruptly with increasing shear rate. This effect is called discontinuous shear thickening, dilatancy or Shear Thickening . In
order to better understand this effect, experiments were carried out with concentrated suspensions of electrically-stabilized colloidal particles in fluids whose refractive indices were very close to
that of the fluid. This made it possible to study the movement of the particles at defined shear rates using laser light scattering experiments in a suitable setup. It was found that the particles
arrange in a periodic structure at low shear rates, combined with
, i.e. a decrease in viscosity with increasing shear rate. However, this effect only occurs during the first measurement. Since colloidal particles in very highly concentrated suspensions can not
relax into a disordered state by Brownian motion, shear thinning no longer occurs in subsequent measurements.
The investigations show that the occurrence of an abrupt increase in viscosity (discontinuous shear thickening) is associated with the disappearance of the internal, local-periodic structure.
Therefore, both effects must be causally connected. Dilatancy in this case can be interpreted as a structural (phase) transition. It occurs when the inner ordered periodic structure changes into a
disordered structure with increasing shear rate. The theoretical interpretation of this structural instability was made applying a Landau theory. Since a sheared suspension is not a system in
thermodynamic equilibrium, the abrupt increase in viscosity with increasing shear rate (dilatancy) is therefore a nonequilibrium phase transition and can be interpreted as a shear-induced phase
transition. This result is consistent with other non-equilibrium phase transitions, such as those described in synergetics .
The book is available for free: download | {"url":"https://kaldasch.de/en/rheology/","timestamp":"2024-11-04T10:43:06Z","content_type":"text/html","content_length":"82762","record_id":"<urn:uuid:a076dd33-e598-4946-81a9-6cc0859ec20c>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00461.warc.gz"} |
The diameter of directed graphs
Let D be a strongly connected oriented graph, i.e., a digraph with no cycles of length 2, of order n and minimum out-degree δ. Let D be eulerian, i.e., the in-degree and out-degree of each vertex are
equal. Knyazev (Mat. Z. 41(6) 1987 829) proved that the diameter of D is at most 5/2δ+2 n and, for given n and δ, constructed strongly connected oriented graphs of order n which are δ-regular and
have diameter greater than 4/2δ+1 n - 4. We show that Knyazev's upper bound can be improved to diam(D) ≤ 4/2δ+1 n + 2, and this bound is sharp apart from an additive constant.
• Diameter
• Directed graph
• Distance
• Eulerian
• Minimum degree
• Oriented graph
ASJC Scopus subject areas
• Theoretical Computer Science
• Discrete Mathematics and Combinatorics
• Computational Theory and Mathematics
Dive into the research topics of 'The diameter of directed graphs'. Together they form a unique fingerprint. | {"url":"https://pure.uj.ac.za/en/publications/the-diameter-of-directed-graphs","timestamp":"2024-11-11T08:15:34Z","content_type":"text/html","content_length":"50656","record_id":"<urn:uuid:6546b056-21b2-4ff9-90e0-2a4585aee340>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00763.warc.gz"} |
Topological space
definiendum $\langle X,\mathcal T\rangle \in \mathrm{it} $
postulate $X,\emptyset\in \mathcal T$
for all $S\subseteq \mathcal T$
postulate $\bigcup S\in \mathcal T$
postulate $S$ … finite $\Rightarrow \bigcap S\in \mathcal T$
We call $\mathcal T$ the topology and its elements the open (sub-)sets of $X$.
A comment on the intersection axiom requiring finiteness: A major motivation for topological spaces is $\mathbb R^n$ with the sets “open ball” and in this setting, an infinite intersection of open
sets need not be open. E.g. consider the set of open intevals $(-\tfrac{1}{n},\tfrac{1}{n})$. | {"url":"https://axiomsofchoice.org/topological_space","timestamp":"2024-11-13T11:44:33Z","content_type":"application/xhtml+xml","content_length":"10503","record_id":"<urn:uuid:ba1f94ad-9f07-4b69-a322-9cb97f2dceb3>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00268.warc.gz"} |
What is Microscopic Cross-section - Definition
What is Microscopic Cross-section – Definition
The standard unit for measuring the microscopic cross-section (σ-sigma) is the barn, which is equal to 10-28 m2. Microscopic cross-sections are used to express the likelihood of particular nuclear
interaction. Periodic Table
Definition of Cross-section
In general, the cross-section is an effective area that quantifies the likelihood of certain interaction between an incident object and a target object. The cross-section of a particle is the same
as the cross section of a hard object, if the probabilities of hitting them with a ray are the same.
For a given event, the cross section σ is given by
σ = μ/n
• σ is the cross-section of this event [m^2],
• μ is the attenuation coefficient due to the occurrence of this event [m^-1],
• n is the density of the target particles [m^-3].
In nuclear physics, the nuclear cross section of a nucleus is commonly used to characterize the probability that a nuclear reaction will occur. The cross-section is typically denoted σ and measured
in units of area [m^2]. The standard unit for measuring a nuclear cross section is the barn, which is equal to 10^−28 m² or 10^−24 cm². It can be seen the concept of a nuclear cross section can be
quantified physically in terms of “characteristic target area” where a larger area means a larger probability of interaction.
Microscopic Cross-section
The extent to which neutrons interact with nuclei is described in terms of quantities known as cross-sections. Cross-sections are used to express the likelihood of particular interaction between an
incident neutron and a target nucleus. It must be noted this likelihood do not depend on real target dimensions. In conjunction with the neutron flux, it enables the calculation of the reaction rate,
for example to derive the thermal power of a nuclear power plant. The standard unit for measuring the microscopic cross-section (σ-sigma) is the barn, which is equal to 10^-28 m^2. This unit is very
small, therefore barns (abbreviated as “b”) are commonly used.
The cross-section σ can be interpreted as the effective ‘target area’ that a nucleus interacts with an incident neutron. The larger the effective area, the greater the probability for reaction. This
cross-section is usually known as the microscopic cross-section.
The concept of the microscopic cross-section is therefore introduced to represent the probability of a neutron-nucleus reaction. Suppose that a thin ‘film’ of atoms (one atomic layer thick) with N[a]
atoms/cm^2 is placed in a monodirectional beam of intensity I[0]. Then the number of interactions C per cm^2 per second will be proportional to the intensity I[0] and the atom density N[a]. We define
the proportionality factor as the microscopic cross-section σ:
σ[t] = C/N[a].I[0]
In order to be able to determine the microscopic cross section, transmission measurements are performed on plates of materials. Assume that if a neutron collides with a nucleus it will either be
scattered into a different direction or be absorbed (without fission absorption). Assume that there are N (nuclei/cm^3) of the material and there will then be N.dx per cm^2 in the layer dx.
Only the neutrons that have not interacted will remain traveling in the x direction. This causes the intensity of the uncollided beam will be attenuated as it penetrates deeper into the material.
Then, according to the definition of the microscopic cross section, the reaction rate per unit area is Nσ Ι(x)dx. This is equal to the decrease of the beam intensity, so that:
-dI = N.σ.Ι(x).dx
Ι(x) = Ι[0]e^-N.σ.x
It can be seen that whether a neutron will interact with a certain volume of material depends not only on the microscopic cross-section of the individual nuclei but also on the density of nuclei
within that volume. It depends on the N.σ factor. This factor is therefore widely defined and it is known as the macroscopic cross section.
The difference between the microscopic and macroscopic cross sections is extremely important. The microscopic cross section represents the effective target area of a single nucleus, while the
macroscopic cross section represents the effective target area of all of
the nuclei contained in certain volume.
Microscopic cross-sections
constitute a key parameters of
nuclear fuel
. In general, neutron cross-sections must be calculated for fresh fuel assemblies usually in two-Dimensional models of the fuel lattice.
The neutron cross-section is variable and depends on:
• Target nucleus (hydrogen, boron, uranium, etc.). Each isotop has its own set of cross-sections.
• Type of the reaction (capture, fission, etc.). Cross-sections are different for each nuclear reaction.
• Neutron energy (thermal neutron, resonance neutron, fast neutron). For a given target and reaction type, the cross-section is strongly dependent on the neutron energy. In the common case, the
cross section is usually much larger at low energies than at high energies. This is why most nuclear reactors use a neutron moderator to reduce the energy of the neutron and thus increase the
probability of fission, essential to produce energy and sustain the chain reaction.
• Target energy (temperature of target material – Doppler broadening). This dependency is not so significant, but the target energy strongly influences inherent safety of nuclear reactors due to a
Doppler broadening of resonances.
Microscopic cross-section varies with incident neutron energy. Some nuclear reactions exhibit very specific dependency on incident neutron energy. This dependency will be described on the example of
the radiative capture reaction. The likelihood of a neutron radiative capture is represented by the radiative capture cross section as σ[γ]. The following dependency is typical for radiative capture,
it definitely does not mean, that it is typical for other types of reactions (see elastic scattering cross-section or (n,alpha) reaction cross-section).
The capture cross-section can be divided into three regions according to the incident neutron energy. These regions will be discussed separately.
• 1/v Region
• Resonance Region
• Fast Neutrons Region
Uranium 238. Comparison of cross-sections.
Gadolinium 155 and 157. Comparison of radiative capture cross-sections.
Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library
For thermal neutrons (in 1/v region), absorption cross sections increases as the velocity (kinetic energy) of the neutron decreases.
Source: JANIS 4.0
1/v Region
In the common case, the cross section is usually much larger at low energies than at high energies. For thermal neutrons (in 1/v region), also radiative capture cross-sections increase as the
velocity (kinetic energy) of the neutron decreases. Therefore the 1/v Law can be used to determine shift in capture cross-section, if the neutron is in equilibrium with a surrounding medium. This
phenomenon is due to the fact the nuclear force between the target nucleus and the neutron has a longer time to interact.
This law is aplicable only for absorbtion cross-section and only in the 1/v region.
Example of cross- sections in 1/v region:
The absorbtion cross-section for 238U at 20°C = 293K (~0.0253 eV) is:
The absorbtion cross-section for 238U at 1000°C = 1273K is equal to:
This cross-section reduction is caused only due to the shift of temperature of surrounding medium.
Resonance Region
The largest cross-sections are usually at neutron energies, that lead to long-lived states of the compound nucleus. The compound nuclei of these certain energies are referred to as nuclear resonances
and its formation is typical in the resonance region. The widths of the resonances increase in general with increasing energies. At higher energies the widths may reach the order of the distances
between resonances and then no resonances can be observed. The narrowest resonances are usually compound states of heavy nuclei (such as fissionable nuclei).
Since the mode of decay of the compound nucleus does not depend on the way the compound nucleus was formed, the nucleus sometimes emits a gamma ray (radiative capture) or sometimes emits a neutron (
scattering). In order to understand the way, how a nucleus will stabilize itself, we have to understand the behaviour of compound nucleus.
The compound nucleus emits a neutron only after one neutron obtains an energy in collision with other nucleon greater than its binding energy in the nucleus. It have some delay, because the
excitation energy of the compound nucleus is divided among several nucleons. It is obvious the average time that elapses before a neutron can be emitted is much longer for nuclei with large number of
nucleons than when only a few nucleons are involved. It is a consequence of sharing the excitation energy among a large number of nucleons.
This is the reason the radiative capture is comparatively unimportant in light nuclei but becomes increasingly important in the heavier nuclei.
It is obvious the compound states (resonances) are observed at low excitation energies. This is due to the fact, the energy gap between the states is large. At high excitation energy, the gap between
two compound states is very small and the widths of resonances may reach the order of the distances between resonances. Therefore at high energies no resonances can be observed and the cross section
in this energy region is continuous and smooth.
The lifetime of a compound nucleus is inversely proportional to its total width. Narrow resonances therefore correspond to capture while the wider resonances are due to scattering.
See also: Nuclear Resonance
Fast Neutrons Region
The radiative capture cross-section at energies above the resonance region drops rapidly to very small values. This rapid drop is caused by the compound nucleus, which is formed in more
highly-excited states. In these highly-excited states it is more likely that one neutron obtains an energy in collision with other nucleon greater than its binding energy in the nucleus. The neutron
emission becomes dominant and gamma decay becomes less important. Moreover, at high energies, the inelastic scattering and (n,2n) reaction are highly probable at the expense of both elastic
scattering and radiative capture.
Doppler Broadening of Resonances
In general, Doppler broadening is the broadening of spectral lines due to the Doppler effect caused by a distribution of kinetic energies of molecules or atoms. In reactor physics a particular case
of this phenomenon is the thermal Doppler broadening of the resonance capture cross sections of the fertile material (e.g. ^238U or ^240Pu) caused by thermal motion of target nuclei in the nuclear
The Doppler broadening of resonances is very important phenomenon, which improves reactor stability, because it accounts for the dominant part of the fuel temperature coefficient (the change in
reactivity per degree change in fuel temperature) in thermal reactors and makes a substantial contribution in fast reactors as well. This coefficient is also called the prompt temperature coefficient
because it causes an immediate response on changes in fuel temperature. The prompt temperature coefficient of most thermal reactors is negative.
See also: Doppler Broadening
We hope, this article, Microscopic Cross-section, helps you. If so, give us a like in the sidebar. Main purpose of this website is to help the public to learn some interesting and important
information about radiation and dosimeters. | {"url":"https://www.periodic-table.org/what-is-microscopic-cross-section-definition/","timestamp":"2024-11-02T03:05:48Z","content_type":"text/html","content_length":"208091","record_id":"<urn:uuid:8453b8f7-4299-4919-a779-ea16f0d64e2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00633.warc.gz"} |
Nets of Standard Subspaces Induced by Antiunitary Representations of Admissible Lie Groups I
Oeh D (2022)
Publication Type: Journal article
Publication year: 2022
Book Volume: 32
Pages Range: 29-74
Journal Issue: 1
Let (pi, H) be a strongly continuous unitary representation of a 1-connected Lie group G such that the Lie algebra g of G is generated by the positive cone C-pi := {x is an element of g : -i partial
derivative pi(x) >= 0} and an element h for which the adjoint representation of h induces a 3-grading of g. Moreover, suppose that (pi, H) extends to an antiunitary representation of the extended Lie
group G(tau) := G (sic) {1, tau(G)}, where tau(G) is an involutive automorphism of G with L(tau(G)) = e(i pi ad h). In a recent work by Neeb and Olafsson, a method for constructing nets of standard
subspaces of H indexed by open regions of G has been introduced and applied in the case where G is semisimple. In this paper, we extend this construction to general Lie groups G, provided the above
assumptions are satisfied and the center of the ideal g(C) = C-pi - C-pi, subset of g is one-dimensional. The case where the center of g(C) has more than one dimension will be discussed in a separate
Authors with CRIS profile
How to cite
Oeh, D. (2022). Nets of Standard Subspaces Induced by Antiunitary Representations of Admissible Lie Groups I. Journal of Lie Theory, 32(1), 29-74.
Oeh, Daniel. "Nets of Standard Subspaces Induced by Antiunitary Representations of Admissible Lie Groups I." Journal of Lie Theory 32.1 (2022): 29-74.
BibTeX: Download | {"url":"https://cris.fau.de/publications/277564296/","timestamp":"2024-11-04T00:46:19Z","content_type":"text/html","content_length":"8633","record_id":"<urn:uuid:228da496-8616-47f5-b575-7209e2c5e81d>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00572.warc.gz"} |
Calculator with Brackets and Parentheses - BS Calculator
Calculator with Brackets and Parentheses
Use the calculator to evaluate mathematical expressions that include brackets and parentheses. This tool is essential for students, professionals, and anyone who needs to perform calculations
Understanding Brackets and Parentheses
Brackets and parentheses are used in mathematics to indicate the order of operations. They help clarify which operations should be performed first in a mathematical expression. For example, in the
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There are different types of brackets: parentheses ( ), square brackets [ ], and curly braces { }. Each type can be used to group numbers and operations, but parentheses are the most commonly used in
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When evaluating expressions, it is crucial to follow the order of operations, often remembered by the acronym PEMDAS:
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• M: Multiplication and Division (left to right)
• A: Addition and Subtraction (left to right)
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Making Math Lessons More Inclusive: ALN Problem Introduction Protocol
What if there was a way to get students more comfortable sharing ideas during a math lesson? What if there was a way to reduce the amount of “I don’t knows” during a math lesson?
The All Learners Lesson Structure is how we achieve balanced math blocks. The ALN Lesson Structure segments math blocks into four main sections: Launch, Main Lesson, Math Menu, and Closure. Main
Lesson is the portion of the math block that is centered on new learning of grade-level content. Students engage in problem-solving, work in heterogeneous groups, and participate in mathematical
discourse. During Main Lesson, ALL students are meaningfully included in grade-level content. Tasks are chosen that provide both access and challenge to all students and allow teachers to use
techniques and protocols to ensure all students have affirming math learning opportunities and feel part of a productive learning community.
It is critical to be able to introduce math problems during a Main Lesson in a way that provides access for all students. The ALN Problem Introduction Protocol is our solution to making sure all
students can access tasks during a main lesson.
When the Problem Introduction Protocol is used, students are invited to make sense of the problem through some scripted steps. We know that when students only focus on some parts of the problem, they
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of the problem.
Here are the Problem Introduction Protocol Steps:
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language is not English. This step can be repeated to ensure that all students have heard the problem.
• Ask, “Are there any words you don’t know? Are there any words that might be tricky for someone?”
This allows students to build a shared understanding around the context of the problem. This is a chance to clarify any misconceptions around language or unfamiliar contexts so that all learners have
a common understanding of the terminology.
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Following a class discussion where students make sense of the problem, teachers record the answer to the question, “What are we trying to figure out?”. Students can write or copy the statement on
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The goal of this step is that every student can say, “I am trying to figure out _____.”
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The teacher listens for the correct unit and reasonable answers. If those are not given then the teacher will ask direct questions “What is the unit (label) for your answer?” “What would a reasonable
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“How could you solve this problem?” “What strategy could you use?” The teacher is listening for strategies and models and asks direct questions if needed. Students share strategies that can be used
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difficulty getting started. (Remember: Do Not Narrow the list of strategies. This step may identify students you’d like to check in with during work time to ask questions.)
We know that there are many ways to develop a positive problem solving culture and what we call “patient problem solvers”. We want to provide opportunities for all students to make sense of problems
and to feel confident about offering solutions based on their understanding. So in addition to our ALN Problem Introduction Protocol we offer suggestions to create tasks which encourage all students
to make sense first before they think about the numbers in the task. Please see page 2 of the ALN Problem Introduction Protocol resource with Additional Strategies for Introducing Math Problems.
What Now?
1. Download and explore the ALN Problem Introduction Protocol.
2. Sign up for an All Learners Online Membership to download accessible Main Lesson Tasks.
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All Learners Network is committed to a new type of math instruction. We focus on supporting pedagogy so that all students can access quality math instruction. We do this through our online platform,
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Prime Number Generator
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Desklib - Online Library for Study Material with Solved Assignments, Essays, Dissertations
Desklib is an online library for study material with solved assignments, essays, dissertations, and more. This case study analysis covers the fundamentals of mechanics, truss overview, stress-strain
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Engineering Computation Laboratory
Hypothetical solar power near Hancock Tower in Boston
At the
ACE Hackathon
event on April 28, we introduced the concept of the
Virtual Solar Decathlon
to students at
Phillips Academy
who are interested in sustainable development.
Hypothetical solar canopies at Andover High School
The U.S. Department of Energy's
Solar Decathlon
challenges 20 collegiate teams to design, build, and operate solar-powered houses that are forward-thinking and cost-effective. Such a project, however, may take up to a year to complete and cost up
to $250,000.
PS20 solar power tower in Seville, Spain
For a few years, I have been thinking about creating a high school equivalent of the Solar Decathlon that costs nothing, takes a much shorter time, and allows everyone to participate. The result of
this thinking process is the Virtual Solar Decathlon that can now be supported by our
CAD software (and increasingly so as we added new features to allow more clean energy technologies to be simulated and designed). The goal of the Virtual Solar Decathlon is to turn the entire Google
Earth into a simulation-based engineering lab of renewable energy and engage students to change their world by tackling energy problems (at least virtually) that matter deeply to their lives.
Here is
the link to our presentation
at Phillips Academy.
Fig. 1: Inter-row shadowing (daily total)
Designing a ground-mounted solar panel array is one of the challenges in our Solarize Your World curriculum, in addition to other challenges such as rooftop solar power systems, solar canopies,
building-integrated photovoltaics, and concentrated solar power plants. With the support of our intuitive Energy3D software, designing a solar panel array appears to be a small and simple job as
students can easily add, drag, and drop solar panels to cover up a site with many solar panels. But things are not always as simple as they seem.
Fig. 2: Solar radiation on an array in four reasons.
The design of a photovoltaic solar farm is, in fact, a typical engineering problem that requires the designer to find a solution that generates as much electricity as possible with a limited number
of solar panels on a given piece of land, among many other constraints and criteria. Such an engineering project mandates iterative design and optimization in a solution space that has scores of
variables. And the more the variables we have to deal with, the more complicated the design challenge becomes.
Fig. 3: Annual outputs vs. row spacing and tilt angle
This sequence of articles will walk you through the essential steps for designing photovoltaic solar farms under a variety of conditions. To get you started, let's assume that 1) we have a
rectangular area for the solar farm; 2) the edges of the area are perfectly aligned with the north-south and east-west axes; and 3) the area is perfectly flat. This kind of site is probably uncommon
in reality (unless the site is in a desert). But let's begin with a very simple scenario like this.
Fig. 4: Surface plot of solar output (ideal)
One of the first things that we have to decide is the number of solar panels. This is usually dictated by the budget. Suppose we have a fixed quantity of solar panels that we can install at a site
large enough to space them (i.e., let's assume that we are not constrained by the area of the site for the time being). As people usually put solar panels on racks (a rack of solar panels is often
referred to as a row -- but don't confuse it with the rows of solar panels you put on each rack), the next things we have to decide are 1) how many solar panels we want to place on each rack, 2)
whether these solar panels are placed in "portrait" or "landscape" orientation on the rack, and 3) how long each rack is. From these information, we know the number of rows for the array. For
example, the array in Figure 1 has four rows, each of which has 88 solar panels stacked up in a 4x22 landscape configuration. Since the shorter side of each panel is about one meter long, each rack
is about four meters wide.
Fig. 5: Surface plot of solar output (using bypass diodes)
How far should the distance between two adjacent rows be? If the solar panels are tilted towards the sun, the rows cannot be too close to one another as the inter-row shadowing (Figure 1) will reduce
the total output (sometimes severely, depending on the wiring of the solar cells on the solar panels -- we will investigate this in the next article), but they cannot be too far away from one
another, either, as a longer distance between rows will decrease the efficiency of land use. Determining the optimal inter-row spacing for the solar array under design depends on a number of
confounding factors such as the tilt angles, location, solar cell wiring, time of year, use of trackers, type of inverters, and shape of the site that greatly complicate the problem (Figure 2). This
is a case in which a thorough understanding of the domain knowledge per se does not suffice to solve the problem. As there is no exact solution, we have to come up with a procedure and a strategy to
search for an optimal one in the solution space. And, sometimes, this solution space can be so vast that manual search becomes infeasible.
Fig. 6: Line graph of solar output (using bypass diodes)
To simplify the search for now, let's assume that we only have to decide on the optimal values for the tilt angle and the inter-row spacing. This assumption reduces the solution space to only two
dimensions. The most straightforward way to nail them down is to gradually vary the tilt angle and the inter-row spacing and then compute the total annual output of the solar panels at each step
(Figure 3), a tedious job that took me a couple of hours to do. Once we have the results, we can use Excel to create a surface plot that shows different zones of outputs as a function of the
inter-row spacing and tilt angle (Figures 4 and 5 -- we will discuss their differences in the next article; for now, you just need to know that Figure 5 is a more accurate result). The yellow zones
in the surface plots are the reduced solution space where we should zero in to find our solution, taking trade-offs with other criteria such as the efficiency of land use into account. To have a
clearer view, Figure 6 shows a 2D line graph of the solar outputs as a function of the tilt angle for six values of inter-row spacing.
The conclusions are that a tilt angle that is approximately equal to the latitude of the site (about 42 degrees in the case of Boston, MA) is the best when the rows are relatively far apart (say, 10
meters away center-to-center or 6 meters way edge to edge when the tilt angle is zero) and when the rows become closer, a smaller tilt angle should be more favorable. For instance, with the
center-to-center inter-row spacing reduced to 8 and 7 meters, 35 and 26 degrees are the optimal choices for the tilt angle, respectively. With the optimal tilt angles, we will lose about 2% and 4% of
electricity output when we reduce the inter-row spacing from 10 meters to 8 meters and 7 meters, respectively. If we don't change the tilt angles, the losses will increase to 3% and 9%, respectively.
These findings apply to fixed solar panel arrays that do not track or "backtrack" the sun.
The analyses we have done so far just barely scratched the surface of the problem. We have many other design topics to cover and design factors to consider. But the volume of work thus far should
speak aloud for itself that this is not a simple problem. At the same time Energy3D greatly simplifies an engineering task and empowers anyone to tackle it, it could also create an illusion as if
engineering were simple. Yes, a What-You-See-Is-What-You-Get (WYSIWYG) 3D design and construction program like Energy3D may be entertaining in ways similar to playing with Minecraft, but no,
engineering is not gaming -- it differs from gaming in many fundamental ways.
An infrared street view
The award-winning Infrared Street View program is an ambitious project that aims to create something similar to Google's Street View, but in infrared light. The ultimate goal is to develop the
world's first thermographic information system (TIS) that allows the positioning of thermal elements and the tracking of thermal processes on a massive scale. The applications include building energy
efficiency, real estate inspection, and public security monitoring, to name a few.
An infrared image sphere
The Infrared Street View project is based on infrared cameras that work with now ubiquitous smartphones. It takes advantages of the orientation and location sensors of smartphones to store
information necessary to knit an array of infrared thermal images taken at different angles and positions into a 3D image that, when rendered on a dome, creates an illusion of immersive 3D effects
for the viewer.
The project was launched in 2016 and later joined by three brilliant computer science undergraduate students, Seth Kahn, Feiyu Lu, and Gabriel Terrell, from Tufts University, who developed a
primitive system consisting of 1) an iOS frontend app to collect infrared image spheres, 2) a backend cloud app to process the images, and 3) a Web interface for users to view the stitched infrared
images anchored at selected locations on a Google Maps application.
The following YouTube video demonstrates an early concept played out on an iPhone: | {"url":"https://molecularworkbench.blogspot.com/2017/04/","timestamp":"2024-11-02T15:26:47Z","content_type":"text/html","content_length":"123411","record_id":"<urn:uuid:92d76d18-01f3-46dc-966f-e591f70cdd76>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00669.warc.gz"} |
From Frenet to Cartan: The Method of Moving Framessearch
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From Frenet to Cartan: The Method of Moving Frames
Hardcover ISBN: 978-1-4704-2952-2
Product Code: GSM/178
List Price: $135.00
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AMS Member Price: $108.00
eBook ISBN: 978-1-4704-3747-3
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From Frenet to Cartan: The Method of Moving Frames
Hardcover ISBN: 978-1-4704-2952-2
Product Code: GSM/178
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN: 978-1-4704-3747-3
Product Code: GSM/178.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN: 978-1-4704-2952-2
eBook ISBN: 978-1-4704-3747-3
Product Code: GSM/178.B
List Price: $220.00 $177.50
MAA Member Price: $198.00 $159.75
AMS Member Price: $176.00 $142.00
• Graduate Studies in Mathematics
Volume: 178; 2017; 414 pp
MSC: Primary 22; 53; 58
The method of moving frames originated in the early nineteenth century with the notion of the Frenet frame along a curve in Euclidean space. Later, Darboux expanded this idea to the study of
surfaces. The method was brought to its full power in the early twentieth century by Elie Cartan, and its development continues today with the work of Fels, Olver, and others.
This book is an introduction to the method of moving frames as developed by Cartan, at a level suitable for beginning graduate students familiar with the geometry of curves and surfaces in
Euclidean space. The main focus is on the use of this method to compute local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces, including Euclidean,
Minkowski, equi-affine, and projective spaces. Later chapters include applications to several classical problems in differential geometry, as well as an introduction to the nonhomogeneous case
via moving frames on Riemannian manifolds.
The book is written in a reader-friendly style, building on already familiar concepts from curves and surfaces in Euclidean space. A special feature of this book is the inclusion of detailed
guidance regarding the use of the computer algebra system Maple^TM to perform many of the computations involved in the exercises.
An excellent and unique graduate level exposition of the differential geometry of curves, surfaces and higher-dimensional submanifolds of homogeneous spaces based on the powerful and elegant
method of moving frames. The treatment is self-contained and illustrated through a large number of examples and exercises, augmented by Maple code to assist in both concrete calculations and
plotting. Highly recommended.
—Niky Kamran, McGill University
The method of moving frames has seen a tremendous explosion of research activity in recent years, expanding into many new areas of applications, from computer vision to the calculus of variations
to geometric partial differential equations to geometric numerical integration schemes to classical invariant theory to integrable systems to infinite-dimensional Lie pseudo-groups and beyond.
Cartan theory remains a touchstone in modern differential geometry, and Clelland's book provides a fine new introduction that includes both classic and contemporary geometric developments and is
supplemented by Maple symbolic software routines that enable the reader to both tackle the exercises and delve further into this fascinating and important field of contemporary mathematics.
Recommended for students and researchers wishing to expand their geometric horizons.
—Peter Olver, University of Minnesota
Undergraduate and graduate students interested in differential geometry.
□ Background material
□ Assorted notions from differential geometry
□ Differential forms
□ Curves and surfaces in homogeneous spaces via the method of moving frames
□ Homogeneous spaces
□ Curves and surfaces in Euclidean space
□ Curves and surfaces in Minkowski space
□ Curves and surfaces in equi-affine space
□ Curves and surfaces in projective space
□ Applications of moving frames
□ Minimal surfaces in $\mathbb {E}^3$ and $\mathbb {A}^3$
□ Pseudospherical surfaces in Bäcklund’s theorem
□ Two classical theorems
□ Beyond the flat case: Moving frames on Riemannian manifolds
□ Curves and surfaces in elliptic and hyperbolic spaces
□ The nonhomogeneous case: Moving frames on Riemannian manifolds
□ The present book has a high didactical and scientific quality being a very careful introduction to the method of moving frames...this book is a very nice presentation of an essential tool of
classical differential geometry. I strongly recommend it as a welcome addition to the main textbooks in geometry.
Mircea Crâşmăreanu, Zentralblatt MATH
□ This volume provides a well-written and accessible introduction to Cartan's theory of moving frames for curves and surfaces in several 3-dimensional geometries.
Francis Valiquette, Mathematical Reviews
□ Primarily intended for 'beginning graduate students,' this book is highly recommended to anyone seeking to extend their knowledge of differential geometry beyond the undergraduate level.
Peter Ruane, MAA Reviews
• Desk Copy – for instructors who have adopted an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Additional Material
• Reviews
• Requests
Volume: 178; 2017; 414 pp
MSC: Primary 22; 53; 58
The method of moving frames originated in the early nineteenth century with the notion of the Frenet frame along a curve in Euclidean space. Later, Darboux expanded this idea to the study of
surfaces. The method was brought to its full power in the early twentieth century by Elie Cartan, and its development continues today with the work of Fels, Olver, and others.
This book is an introduction to the method of moving frames as developed by Cartan, at a level suitable for beginning graduate students familiar with the geometry of curves and surfaces in Euclidean
space. The main focus is on the use of this method to compute local geometric invariants for curves and surfaces in various 3-dimensional homogeneous spaces, including Euclidean, Minkowski,
equi-affine, and projective spaces. Later chapters include applications to several classical problems in differential geometry, as well as an introduction to the nonhomogeneous case via moving frames
on Riemannian manifolds.
The book is written in a reader-friendly style, building on already familiar concepts from curves and surfaces in Euclidean space. A special feature of this book is the inclusion of detailed guidance
regarding the use of the computer algebra system Maple^TM to perform many of the computations involved in the exercises.
An excellent and unique graduate level exposition of the differential geometry of curves, surfaces and higher-dimensional submanifolds of homogeneous spaces based on the powerful and elegant method
of moving frames. The treatment is self-contained and illustrated through a large number of examples and exercises, augmented by Maple code to assist in both concrete calculations and plotting.
Highly recommended.
—Niky Kamran, McGill University
The method of moving frames has seen a tremendous explosion of research activity in recent years, expanding into many new areas of applications, from computer vision to the calculus of variations to
geometric partial differential equations to geometric numerical integration schemes to classical invariant theory to integrable systems to infinite-dimensional Lie pseudo-groups and beyond. Cartan
theory remains a touchstone in modern differential geometry, and Clelland's book provides a fine new introduction that includes both classic and contemporary geometric developments and is
supplemented by Maple symbolic software routines that enable the reader to both tackle the exercises and delve further into this fascinating and important field of contemporary mathematics.
Recommended for students and researchers wishing to expand their geometric horizons.
—Peter Olver, University of Minnesota
Undergraduate and graduate students interested in differential geometry.
• Background material
• Assorted notions from differential geometry
• Differential forms
• Curves and surfaces in homogeneous spaces via the method of moving frames
• Homogeneous spaces
• Curves and surfaces in Euclidean space
• Curves and surfaces in Minkowski space
• Curves and surfaces in equi-affine space
• Curves and surfaces in projective space
• Applications of moving frames
• Minimal surfaces in $\mathbb {E}^3$ and $\mathbb {A}^3$
• Pseudospherical surfaces in Bäcklund’s theorem
• Two classical theorems
• Beyond the flat case: Moving frames on Riemannian manifolds
• Curves and surfaces in elliptic and hyperbolic spaces
• The nonhomogeneous case: Moving frames on Riemannian manifolds
• The present book has a high didactical and scientific quality being a very careful introduction to the method of moving frames...this book is a very nice presentation of an essential tool of
classical differential geometry. I strongly recommend it as a welcome addition to the main textbooks in geometry.
Mircea Crâşmăreanu, Zentralblatt MATH
• This volume provides a well-written and accessible introduction to Cartan's theory of moving frames for curves and surfaces in several 3-dimensional geometries.
Francis Valiquette, Mathematical Reviews
• Primarily intended for 'beginning graduate students,' this book is highly recommended to anyone seeking to extend their knowledge of differential geometry beyond the undergraduate level.
Peter Ruane, MAA Reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
You may be interested in...
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M-Theory Repositions: Now You Can Thank Us For Quantum Mechanics Too
String theory is a hypothetical framework where particle physics is replaced by one-dimensional objects called strings. It was originally proposed as a way to explain the strong force, then advocates
re-purposed it for quantum gravity, and now it is being reconfigured again, with "commutation rules" of quantum mechanics.
The heart of what became string theory began in the early 20th century and then got jumbled in with a lot of philosophical ideas - 'what if there are dimensions we can't see?' A fifth dimension was a
nice discussion, but without being detected it was just that. As the century moved on, Kaluza-Klein theory, S-matrix theory, everyone kept coming up with new stuff and it all eventually became what
we now know as String Theory.
The problem has always been that there is no way to validate any of it - it is only a theory in the sense that it uses Theory as a proper name. A team of researchers have flipped that problem around
so that string theory doesn't need to be validated, they will instead use string theory to validate quantum mechanics. Beliefs meet the science topology that exists and declare themselves the
While this latest repurposing, M-theory, has only been given a lukewarm reception outside the community that has been funded to think about M-theory, the authors of a new paper in Physics Letters
believe it is the basis of all physics.
"This could solve the mystery of where quantum mechanics comes from," says Professor Itzhak Bars of the University of Southern California, lead author of the paper. He and graduate student Dmitry
Rychkov used math to show that a set of fundamental quantum mechanical principles known as "commutation rules'' may be derived from the geometry of strings joining and splitting.
"Our argument can be presented in bare bones in a hugely simplified mathematical structure," Bars said. "The essential ingredient is the assumption that all matter is made up of strings and that the
only possible interaction is joining/splitting as specified in their version of string field theory."
Physicists have long sought to unite quantum mechanics and general relativity, and to explain why both work in their respective domains. If it had been shown to be true, string theory would have
resolved inconsistencies of quantum gravity and suggested that the fundamental unit of matter was a tiny string, not a particle, and that the only possible interactions of matter are strings either
joining or splitting.
Four decades later, proponents in that area of theoretical physic are still trying to hash out the rules of string theory, which seem to demand some interesting starting conditions to work - like
extra dimensions, which are necessary to explain why quarks and leptons have electric charge, color and "flavor" that distinguish them from one another.
A Theory of Everything still eludes us. On larger scales, scientists use classical, Newtonian mechanics to describe how gravity holds the moon in its orbit or why the force of a jet engine propels a
jet forward. Newtonian mechanics is intuitive and can often be observed with the naked eye. But that does not apply at the really large scale, it is why there is talk of Dark Matter and Dark Energy.
On incredibly tiny scales, such as 100 million times smaller than an atom, relativistic quantum field theory describes the interactions of subatomic particles and the forces that hold quarks and
leptons together inside protons, neutrons, nuclei and atoms but that does not explain everything, like how particles to be in two places at once.
Still, quantum mechanics is extremely successful as a model for how things work on small scales, but it contains a big mystery: the unexplained foundational quantum commutation rules that predict
uncertainty in the position and momentum of every point in the universe.
"The commutation rules don't have an explanation from a more fundamental perspective, but have been experimentally verified down to the smallest distances probed by the most powerful accelerators.
Clearly the rules are correct, but they beg for an explanation of their origins in some physical phenomena that are even deeper," Bars said.
The difficulty lies in the fact that there's no experimental data on the topic — testing things on such a small scale is currently beyond a scientist's technological ability.
So M-theory remains conjecture. The safe bet remains that if Michio Kaku embraces it, it is okay to disregard it. | {"url":"https://www.science20.com/news_articles/mtheory_repositions_now_you_can_thank_us_for_quantum_mechanics_too-148319","timestamp":"2024-11-07T21:33:13Z","content_type":"text/html","content_length":"37628","record_id":"<urn:uuid:890cf13a-a043-4d74-b5cb-d004c5ad0b29>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00261.warc.gz"} |
Chemical Reactions: Identify With This Worksheet!
Chemical reactions: Identify with this worksheet!
Welcome to Warren Institute, where we delve into the fascinating world of Mathematics education. In today's article, we will be exploring the topic of identifying the 5 types of chemical reactions
through an interactive worksheet. This engaging activity will not only enhance your understanding of chemistry but also strengthen your problem-solving skills. By analyzing various chemical
reactions, you will be able to differentiate between synthesis, decomposition, combustion, single replacement, and double replacement reactions. So let's dive in and unravel the secrets of chemical
reactions together!
Subtitulo 1: Introduction to Identifying the 5 Types of Chemical Reactions Worksheet
In this section, we will provide an overview of the worksheet and its purpose in teaching students about the five types of chemical reactions. The worksheet serves as a tool for educators to assess
the understanding of students in identifying and categorizing these reactions. It is designed to enhance their knowledge and critical thinking skills in the field of chemistry.
Subtitulo 2: Understanding the Five Types of Chemical Reactions
This section will delve into the details of the five types of chemical reactions covered in the worksheet. These include combustion reactions, synthesis reactions, decomposition reactions, single
displacement reactions, and double displacement reactions. Each type of reaction will be explained in depth, with examples provided to aid comprehension. By understanding these reactions, students
will be able to identify them accurately when presented with chemical equations.
Subtitulo 3: Application of Worksheet in Mathematics Education
In this section, we will explore the application of the worksheet within the context of mathematics education. While the topic at hand concerns chemical reactions, the ability to identify and
categorize them involves mathematical concepts such as balancing equations, stoichiometry, and reaction rates. By engaging with this worksheet, students can strengthen their mathematical skills while
deepening their understanding of chemical reactions.
Subtitulo 4: Assessing Student Learning through the Worksheet
This final section will discuss the importance of assessing student learning using the worksheet. Through completing the exercises, students can demonstrate their understanding of the five types of
chemical reactions and their ability to correctly identify them. Educators can use this assessment to gauge the effectiveness of their teaching methods and to tailor future lessons based on the
individual needs of their students. Additionally, the worksheet can serve as a valuable resource for individual or group study, further reinforcing the concepts learned in class.
frequently asked questions
What are the five types of chemical reactions covered in the identifying the 5 types of chemical reactions worksheet in Mathematics education?
In the context of Mathematics education, there seems to be a confusion with the question. Chemical reactions fall under the subject area of Science, not Mathematics. Mathematics education primarily
focuses on mathematical concepts and problem-solving skills.
How can the identifying the 5 types of chemical reactions worksheet be used to enhance students' understanding of Mathematics education?
The identifying the 5 types of chemical reactions worksheet can be used to enhance students' understanding of Mathematics education by incorporating mathematical concepts and problem-solving skills
into the study of chemical reactions. Students can analyze the given reactions, identify the reaction types, and use mathematical equations and formulas to balance the chemical equations. This
integration helps students see the practical application of mathematics in real-life scenarios, deepening their understanding of both chemistry and mathematics.
Are there any specific strategies or techniques recommended for teaching the five types of chemical reactions using the identifying the 5 types of chemical reactions worksheet in Mathematics
Yes, there are specific strategies and techniques recommended for teaching the five types of chemical reactions using the "Identifying the 5 Types of Chemical Reactions" worksheet in Mathematics
education. Some of these strategies include:
1. Providing clear explanations and definitions of each type of reaction.
2. Demonstrating examples of each reaction type using real-life or visual aids.
3. Encouraging student participation through hands-on activities or experiments.
4. Using guided practice exercises on the worksheet to reinforce understanding.
5. Providing opportunities for students to apply their knowledge through problem-solving tasks or scenarios related to chemical reactions.
What are some common misconceptions or difficulties that students may face when completing the identifying the 5 types of chemical reactions worksheet in the context of Mathematics education?
Some common misconceptions or difficulties that students may face when completing the identifying the 5 types of chemical reactions worksheet in the context of Mathematics education include:
1. Confusion between mathematical equations and chemical equations: Students may struggle to differentiate between mathematical equations, which involve solving for unknown variables, and chemical
equations, which represent the reactants and products in a chemical reaction.
2. Lack of understanding of chemical terminology: Students may find it challenging to identify the different types of chemical reactions (e.g., synthesis, decomposition, combustion) due to a lack of
familiarity with chemical terms and concepts.
3. Difficulty balancing chemical equations: Balancing chemical equations requires knowledge of stoichiometry and the conservation of mass. Students may struggle to balance equations correctly,
leading to incorrect identification of reaction types.
4. Inability to recognize patterns: Identifying the five types of chemical reactions requires recognizing patterns in the reactants and products. Some students may have difficulty spotting these
patterns and distinguishing between different reaction types.
5. Limited prior knowledge: Students who have not been exposed to sufficient chemistry education may struggle with the basics of chemical reactions and find it challenging to complete the worksheet
How does the identifying the 5 types of chemical reactions worksheet align with the Mathematics education curriculum standards?
The identifying the 5 types of chemical reactions worksheet aligns with the Mathematics education curriculum standards by incorporating mathematical concepts and problem-solving skills.
In conclusion, this article has delved into the importance of identifying the 5 types of chemical reactions worksheet in the realm of Mathematics education. Through the use of this worksheet,
students are able to strengthen their understanding of chemical reactions and apply mathematical principles to analyze and solve problems. The identification and classification of these reactions not
only enhances students' critical thinking skills but also lays a solid foundation for further exploration in the field of chemistry. By engaging with this worksheet, educators can effectively foster
a deeper comprehension of chemical reactions, promoting a more holistic approach to Mathematics education.
Identifying the 5 types of chemical reactions worksheet answers
The answers to the "Identifying the 5 Types of Chemical Reactions" worksheet can vary depending on the specific reactions provided. However, there are certain key indicators that can help students
correctly identify each type of reaction.
For combustion reactions, students should look for the presence of oxygen and the production of carbon dioxide and water as products. These reactions typically involve the burning of a fuel source
and are characterized by a release of heat and light.
In synthesis reactions, students will see two or more reactants combining to form a single product. These reactions often involve the formation of a compound and can be recognized by the presence of
a "+" symbol between the reactants.
Decomposition reactions are the opposite of synthesis reactions, where a single reactant breaks down into two or more products. Students should look for the presence of a single compound as the
reactant and multiple elements or compounds as the products.
Single displacement reactions occur when an element replaces another element in a compound. Students should identify the presence of an element and a compound as reactants, with the element replacing
one of the elements in the compound to form a new compound as the product.
Double displacement reactions involve the exchange of ions between two compounds. Students should look for the presence of two compounds as reactants, with the positive and negative ions swapping
places to form two new compounds as the products.
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Dyslexia: Power rule as related to integral bases from the gr9 Math curriculum in Ontario | Jim-Bot
Dyslexia: Power rule as related to integral bases from the gr9 Math curriculum in Ontario
For Parent
How to Help a 14 year old with Dyslexia Learn Power Rule as Related to Integral Bases from Grade 9 Math Curriculum in Ontario, Canada
Power rule is an integral base from the Grade 9 math curriculum in Ontario, Canada. It is an equation that calculates the area under a curve. This equation expresses a y-value as a power to an
For example, if you had the equation y = x3, the power rule for integrated basing would be 3, or three. This power rule, or 3 in this example, expresses how to calculate the area below the curve of
the equation. To solve it, you would integrate the equation, or calculate the area under the curve.
Issues with Understanding:
Trouble Differentiating Between Similar Situation
Dyslexic individuals can have difficulty differentiating between similar situations. For example, the student may struggle to understand the difference between an equation that reads y = x3 and an
equation that reads y = x2, even though they are very different and have different power rules.
To help the student with this issue, the parent can provide written examples of equations, along with visuals that clearly demonstrate the difference. For example, the parent could give the student
two graphs of equations, and have them identify which equation has a power rule of 3 and which equation has a power rule of 2. This will give the student a concrete way to understand the difference
between the equations.
Inability to Visualize Graphs
Another issue the student may face is an inability to visualize graphs. This can make it hard for the student to understand the concept of power rule applied to integral bases.
To help with this, the parent can explain the concept in real-world examples that the student can intuitively understand. For example, the parent can explain how, when you study the area of a room,
the power rule applied to integral bases helps you understand the area of the room’s base more clearly. This can give the student an easy way to understand what power rule is and how it works in real
world instances.
Challenges with Abstract Math Concepts
Dyslexic individuals may find it difficult to understand abstract math concepts, like power rule applied to integral bases. This can especially be the case with equations that don’t involve basic
To help the student understand this concept, the parent can find concrete examples that the student can use to understand the concept better. For example, the parent can provide a real-world example
of a curve and its equation, and ask the student to calculate the area below it. This can give the student a way to grasp the concept in an intuitive way.
Power rule as related to integral bases from the grade 9 math curriculum in Ontario, Canada can be explained by the following equation:
y = x^n
For example, if you had the equation y = x^3, the power rule for integrated basing would be 3, or three. To solve it, you would integrate the equation, meaning you would calculate the area under the
For Youth
Hello there! As a 14-year-old, you may be familiar with the concept of power. Power simply means a value raised to a certain degree, like in the equation 2^3, the number 2 is raised to the power of
3, which is 8.
The power rule relates to the integral bases from the grade 9 math curriculum in Ontario, Canada. Integral means something that has a whole or undivided form. Basis means something considered as a
starting point. So the Power Rule relates to how we work with ‘starting points’ raised to a certain power, or degree.
To put it simply, the Power Rule states that when a power, like 2^3, is written as an integral basis like 2x2x2x, the value is multiplied by itself, which is 8. This could also be written as, for
example, 3^4 = 3x3x3x3, which is 81.
Now, since you have issues with Dyslexia, this particular concept could be more challenging for you to understand and remember. But don’t worry, there are some things you can do to help you
understand. One of my favorite strategies is to practice writing equations with numbers that you’re comfortable with – like 3^2, and 4^3. This way, you can familiarize yourself with the concept and
become more comfortable with it. It’s also helpful to make physical models of the equations, like cutting out small paper squares to represent the value raised to the power of the number.
You can also use a calculator to check if your answer makes sense or if you’re on the right track. And don’t forget to ask your teacher if you have any questions or need support.
Hopefully this helps you understand the Power Rule better. And remember– if you practice, you’ll get the hang of it in no time! | {"url":"https://www.jim-bot.com/2023/02/19/dyslexia-power-rule-as-related-to-integral-bases-from-the-gr9-math-curriculum-in-ontario/","timestamp":"2024-11-05T02:19:10Z","content_type":"text/html","content_length":"76905","record_id":"<urn:uuid:6f3fd2b0-bd52-4d58-b7c2-c2ae48a2289b>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00871.warc.gz"} |
ACO Seminar
The ACO Seminar (2019–2020)
November 7
, 3:30pm, Wean 7218 (note the room change)
Jakub Opršal
, Durham University
Topology in computational complexity of graph colouring problems
It is widely believed that colouring a given graph that is promised to be 3-colourable using constant number of colours cannot be achieved in polynomial time unless P = NP. We will discuss a related
computational problem: Namely, instead of relaxing the number of colours used, we insist to find a 3-colouring, but instead strengthen our promise. The problem is then to find a 3-colouring of a
graph that is promised to have a homomorphism (an edge-preserving map) to some 3-colourable non-bipartite graph H. We will show that this problem is still NP-hard. Surprisingly, the solution uses
several ideas from algebraic topology. We will explore how are these ideas used, and a possibility of extending them to provide classification of computational complexity of similar problems.
Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220. | {"url":"https://aco.math.cmu.edu/abs-19-20/nov7.html","timestamp":"2024-11-11T20:03:17Z","content_type":"text/html","content_length":"2935","record_id":"<urn:uuid:76bc52b5-8510-4d29-a2e8-75a8eb1e4f8a>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00503.warc.gz"} |