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A Literature Review Comparing Experts’ and Non-Experts’ Visual Processing of Graphs during Problem-Solving and Learning
Department of Electrical and Computer Engineering, RPTU Kaiserslautern, 67663 Kaiserslautern, Germany
Faculty of Physics, LMU Munich, 80539 Munich, Germany
Institute of Medical Education, University Hospital, LMU Munich, 80336 Munich, Germany
Department of Psychology, LMU Munich, 80539 Munich, Germany
Author to whom correspondence should be addressed.
Submission received: 23 December 2022 / Revised: 10 February 2023 / Accepted: 16 February 2023 / Published: 19 February 2023
The interpretation of graphs plays a pivotal role in education because it is relevant for understanding and representing data and comprehending concepts in various domains. Accordingly, many studies
examine students’ gaze behavior by comparing different levels of expertise when interpreting graphs. This literature review presents an overview of 32 articles comparing the gaze behavior of experts
and non-experts during problem-solving and learning with graphs up to January 2022. Most studies analyzed students’ dwell time, fixation duration, and fixation count on macro- and meso-, as well as
on micro-level areas of interest. Experts seemed to pay more attention to relevant parts of the graph and less to irrelevant parts of a graph, in line with the information-reduction hypothesis.
Experts also made more integrative eye movements within a graph in terms of dynamic metrics. However, the determination of expertise is inconsistent. Therefore, we recommend four factors that will
help to better determine expertise. This review gives an overview of evaluation strategies for different types of graphs and across various domains, which could facilitate instructing students in
evaluating graphs.
1. Introduction
Interpreting data presented in graphs is essential to understanding concepts across domains [
], especially for learning mathematics [
], to interpret and represent data [
], as well as to use media [
]. Therefore, graph interpretation was highlighted as a valuable skill in PISA and as a 21st-century workforce skill [
]. Graph-comprehension skills differ across individuals and depend on multiple factors: (1) graphical literacy, meaning the ability to interpret information represented in graphical form, for
instance, identifying relevant features in any context [
]; (2) domain knowledge about the represented topic [
]; (3) prior knowledge about the underlying mathematical concepts of the graph [
]; (4) task knowledge, such as using a graph to solve a problem or identifying specific data points [
]. It is reasonable to assume that experts should have higher levels of graph-comprehension skills than non-experts. However, the determination of expertise can differ (see section Determination of
Expertise). This is an important aspect to keep in mind, as the interpretation of differences in the visual behavior of experts and non-experts may depend on how expertise is determined. This holds
true for this review when comparing the visual behavior of experts and non-experts during problem-solving and learning with graphs.
Visual processing of the graph is very important for graph comprehension. We use the term visual processing to emphasize that not only seeing the relevant information, but also actively processing is
important for comprehending the depicted information. There is evidence that the visual processing of external representations changes with increasing expertise [
]. The underlying assumption is that people mentally process the information they look at [
] (eye-mind hypothesis).
There are various theories why the way we distribute attention might change with increasing expertise. Several of those theories have been supported by eye-tracking studies and literature reviews.
For example, the holistic model of image perception states that experts can process an image more efficiently than non-experts [
]. This is explained by the enhanced parafoveal processing of experts [
]. Experts can analyze an entire image and fixate relevant information earlier than non-experts [
]. Furthermore, experts seem to process information faster than non-experts, as evidenced by shorter fixation durations (see the meta-analysis of Gegenfurtner et al. [
]). This supports the theory of long-term working memory. This theory states that experts learn how to store and retrieve information more effectively, which results in enhanced short-term memory
processing [
]. Additionally, the findings of Gegenfurtner et al. [
] support the information-reduction hypothesis [
]. With increasing practice, participants focused more on task-relevant information and less on information that was not relevant to the task [
]. This is called selective attention [
]. These results suggest differences in the visual behavior between experts and non-experts when viewing external representations, such as graphs.
The difference between experts and non-experts’ viewing behaviors can be important in the context of education. For example, experts’ eye movements could be used as visual instructions to help
learners make sense of external representations [
]. Knowledge about how experts read graphs could also be used to facilitate students’ information processing [
] or to identify student difficulties in problem-solving or learning with graphs. However, the theories mentioned above use various eye-tracking metrics, such as time to first fixation [
], the fixation count [
], total viewing time [
], or fixation duration [
]. There are similarities between different metrics, e.g., a correlation between total viewing times and fixation count [
] (see also [
] for similar results), but there are also conflicting relations between theoretical models and eye-tracking metrics. For instance, the theory of long-term memory predicts a shorter fixation duration
for experts. This, however, is only consistent with the information-reduction hypothesis if experts fixate shorter on irrelevant areas, as this hypothesis predicts more fixations on task-relevant
areas for experts than for non-experts [
]. Such possible inconsistencies make it more difficult to interpret how these metrics relate to the differences between experts and non-experts in viewing graphs or diagrams. Furthermore, the way
experts and non-experts are defined should be acknowledged, especially in the context of education.
There have been previous literature reviews of eye-tracking in education with various research foci, for example, to summarize the eye-tracking research in physics education [
], to review the scenarios of eye tracking in mathematics education research [
], to compare experts and novices’ gaze behavior in sports and medical education research [
], to present a summary of eye-tracking research within the ”Psychology of Mathematics Education” conference [
], to investigate the relation between eye movements and cognitive processes during multimedia learning [
], or to provide an overview of the applications of eye tracking in education [
]. None of these review articles focuses on a single type of representation, and regarding the pivotal role of graphs in education, we intend to fill this gap with our review.
We hence aim to (1) provide an overview of eye-tracking metrics that have been used to compare the visual processing of experts and non-experts during problem-solving and learning with graphs. We
also (2) summarize the previously found differences in visual behavior between experts and non-experts during learning or problem-solving with graphs.
Knowing how experts view graphs can provide guidelines to support students’ visual processing of graphs. For instance, it allows the identification of suitable eye movement modelling examples [
] or relevant areas for signaling support [
]. Moreover, such knowledge can be used to evaluate students’ fluency in the visual processing of graphs [
]. In this literature review, we provide an overview of the domains, the types of graphs, the eye-tracking metrics, and how experts are distinguished in the studies.
2. Materials and Methods
A literature review typically consists of three parts. First, the literature search. This is followed by the data extraction, which is then analyzed in the third part of a literature review. In the
following we first present the search process of our literature review and then continue with the method used for data extraction. The results based on the data extraction are shown in the Results
2.1. Literature Review
The literature search aimed to find articles that analyzed visual behavior when looking at graphs in the context of problem-solving and learning in Science, Technology, Engineering, and Math (STEM)
subjects. All included articles should fulfil the following criteria:
• Comparison of experts vs. non-experts (population)
• STEM subject (domain)
• Learning or problem-solving with graphs, diagrams, or functions (intervention)
• Analysis of visual behavior via eye-tracking metrics (outcome)
• Empirical study
• Full text available in English
This resulted in the following categories and terms (see
Table 1
). In the search string, categories were linked with the Boolean operator AND and terms with the Boolean operator OR.
To identify relevant articles, we searched for titles and abstracts in the databases ERIC, Scopus, Pedocs, and SpringerLink. One possible search string for Scopus would be (“eye tracking” OR “viewing
behavior” OR “visual attention”) AND (“graph” OR “diagram” OR “function”). As search algorithms differed between databases, key terms in the search string were sometimes replaced with corresponding
adjectives or adverbs to include alternative phrasings. This search was conducted in February 2022. Therefore, the publication deadline for relevant publications was 31 January 2022. After the
screening process, 24 empirical studies met the inclusion criteria and were included. We then conducted a backwards snowball search using Google Scholar for all included articles and found eight more
articles. In total, 32 articles were included in this review.
2.2. Data Extraction
Once the search was completed, relevant data were extracted. Based on our research focus on the differences in visual behavior between experts and non-experts during problem-solving or learning with
graphs, we extracted the following data:
• Year of publication
• STEM subject in which the study was conducted
• Type of graph
• Eye-tracking metrics
• Areas of interest (AOIs) used for the analysis of eye-tracking metrics
• Expertise determination
• Key findings
To analyze differences in visual behavior between experts and non-experts, we coded the way authors determined expertise. Furthermore, we coded the domain (STEM subject) and type of graph, as well as
the analyzed eye-tracking metrics. To analyze eye-tracking data, the stimuli are split into areas of interest (AOIs). This is useful to investigate the distribution of eye movements across relevant
and irrelevant areas. The distribution of eye movements can give insights into the relevance of a representation’s components. Depending on the research aim, an AOI can consist of an entire
representation, such as a graph, or smaller components, for example, the axes. Furthermore, the analysis of eye-tracking metrics depends on the granularity of the AOIs.
In this review, we differentiate between macro- and meso-level AOIs and micro-level AOIs when analyzing the gaze behavior of experts and non-experts [
]. We used this distinction to code AOIs based on descriptions in the included studies. Macro-level AOIs consist of an entire graph. These AOIs can be useful to research how graphs are embedded in
the learning material, e.g., between questions and answers. Meso-level AOIs divide the graph into large components, such as dividing the graph from the x- and y-axes. This means that more than one
AOI covers the graph area, but there are separate information sources, such as single-axis values that are included in the same AOI. Micro-level AOIs split a comprehensive representation into
particular elements, that can be based, for example, on specific information that is relevant to study specific sections of a graph, such as an axis with separate numbers on it.
3. Results
We identified 32 articles in our review, that analyzed the visual behavior of experts and non-experts when looking at graphs in the context of problem-solving and learning. An overview of all
included studies can be found in
Table 2
. This table surveys authors, publication years, subjects, graph types, measurements to determine expertise and analyzed eye-tracking metrics.
An overview of the analyzed variables can be seen in the graphs depicted in
Figure 1
. The included experiments are described in more detail regarding the individual variables in the following sections, starting with the publication period of the included studies.
3.1. Publication Period
Of the 32 included articles, the first study was published in 2003 (see
Figure 1
, top left). In the first decade starting from 2003, only a limited number of six studies were published, whereas most studies (
= 26) were published after 2013. The number of studies in our review did not increase uniformly, as we identified six years between 2003 and 2013 in which no studies with eye tracking that examined
the visual behavior of experts and non-experts during problem-solving and learning with graphs were published. After 2014, we could see an increase in the number of publications about visual behavior
when looking at graphs, with five publications in 2014 and four each in 2016, 2020, and 2021. Starting in 2018, a more constant number of studies comparing experts and non-experts when learning or
solving problems with graphs were published.
This distribution is comparable to other reviews about eye tracking in education. Before 2006, only a few eye-tracking studies were published in math education research [
], increasing until the year 2018. The authors stated that this increase could be due to the technical advances in eye-tracking technology and therefore easier usage [
]. Correspondingly, more terms related to eye tracking (“eye[-]tracking”, “eye[-]movement”, “gaze[-]tracking”, “gaze[-]movement”) were identified via content analysis in the proceedings of the
International Group for the Psychology of Mathematics Education, indicating an increased relevance of eye-tracking technology in education research [
3.2. Domains and Types of Graphs
In education research, eye-tracking studies about experts and non-experts learning and solving problems with graphs have been conducted in various STEM subjects (see
Figure 1
, top right). Out of 32 studies, 16 presented graphs based on the subject of physics, for example, works by Dzsotjan et al. [
] or Kozhevnikov et al. [
]. Out of these, three articles compared physics with economics graphs [
]. Following physics and economics, the second most studies were conducted in medicine [
], mathematics [
], and biology [
] with three published experiments per subject.
Most of the studies (
= 25) used line graphs (
Figure 1
, middle left). This finding holds when looking at specific STEM subjects. For example, 13 out of the 16 physics studies presented line graphs. This corresponds to the common topic of kinematics [
]. Studies on visual behavior in graphs in a biological domain also used line graphs exclusively [
]. Studies in a mathematics and medical domain also mostly used line graphs (math: [
]; medicine: [
]). However, Okan et al. [
] analyzed the visual processing of line and bar graphs in a medical domain. Likewise, bar graphs in combination with line graphs were the focus of studies in a geoscience domain [
]. Furthermore, bar graphs were used in combination with radar graphs [
]. Studies using only bar graphs ranged in topic from economics [
] to data analysis [
3.3. Determination of Expertise
To compare the visual behavior of participants of various expertise levels during problem-solving and learning with graphs, researchers classified their participants according to different measures.
An overview of the measures used for expertise determination across all experiments can be seen in
Figure 1
(middle right). An overview of the expertise determination in individual studies can be found in
Table 2
. Please note that we cannot identify potential differences and overlaps between the measures used to determine expertise because not all test materials were publicly available. In the Introduction
we presented four factors that are often used to determine expertise: (1) graphical literacy, (2) domain knowledge, (3) mathematical prior knowledge, and (4) task knowledge. However, some of the
measures used to determine expertise in the studies examined in this review cannot be categorized as one of these four. Clear discrimination between these factors may not always be possible and
mapping them with the indicators of expertise used in the studies is complex. For example, an item in which students solve a problem with a graph may contain information about students’ prior
knowledge in both domain and math contexts as well as a certain level of graphical literacy skills and task knowledge. In such a case, the performance when solving the item would measure all four
factors. Similarly, learning gain [
], teacher’s opinion [
], level of study [
], comparison with professionals [
] and pretest score (e.g., in graph understanding [
]) may all cover the four factors. In contrast, working memory [
], spatial abilities [
], and dyslexia [
] do not address any of these factors, whereas the remaining determinators cover only parts of the factors, although one might argue that the latter contains the factor of task knowledge, as dyslexic
participants had trouble with reading.
Most researchers determined expertise post-hoc based on participants’ performance in the learning or problem-solving task (e.g., [
]). Determining expertise a priori based on their domain of study was performed when comparing students of different subjects (e.g., [
]) or science with non-science students [
Moreover, some authors used multiple measures, for example, working memory capacity and subjective assessments of visualization experience [
Although there was a clear preference to use performance and domain to determine experts, other—sometimes unusual—measures were also employed. Many studies compared experts and non-experts via
students’ performance on specific tasks, where expertise might be located on a continuous scale, instead of comparing groups with clear distinctions. The variety of ways expertise was determined
should be kept in mind when interpreting the eye-tracking metrics and comparing experts and non-experts as described in the next sections.
3.4. Eye-Tracking Metrics
Previous studies used various eye-tracking metrics to compare the visual processes of experts and non-experts during problem-solving and learning with graphs. In the following, we aim to provide an
overview of the analyzed eye-tracking metrics in the included studies (research aim 1).
Figure 1
(lower left) shows the eye-tracking metrics that the authors of the 32 included studies used to compare the visual behavior of experts and non-experts. Eye-tracking metrics can be grouped into static
and dynamic metrics. The sum of static metrics or the average of eye movements over time, for example, attained by calculating the duration someone fixated on a stimulus for the entire time the
stimulus, is shown. Dynamic metrics include information about the change in visual attention over time, e.g., the number of eye-movement switches from one part of the stimulus to another (gaze
transitions) or the duration between two fixations (saccadic duration). Static eye-tracking metrics in the included studies were based on fixations. These metrics were evaluated by most studies,
e.g., mean fixation duration (e.g., [
]) or the fixation count. Another popular static metric was dwell time, which describes the sum of total fixation durations and the total duration of saccades within an AOI [
]. However, definitions of dwell time in the articles differ. Whereas some defined it as the “viewing time” [
] (p. 4), others used more specific definitions, such as “eye movements below an acceleration of 8500°/s2 and a velocity below 30°/s” [
] (p. 5). In some cases, we coded metrics as dwell time based on the description in the papers (e.g., “gaze duration”, p. 335, [
]), although, in general, we classified the used eye-tracking metrics based on the terms the authors used. Dwell time was also used to calculate new metrics, such as the so-called domain-relative
attention, which is defined by dividing the relative dwell time of an AOI by the relative area of the AOI [
]. Other static eye-tracking metrics were the mean time to first fixation on an AOI [
], the pupil size (e.g., [
]), and the number of revisits on AOIs [
]. Dynamic eye-tracking metrics that were used to distinguish the visual processing of experts and non-experts during problem-solving and learning with graphs included transitions (e.g., [
]), and saccades (gaze jumps between two fixations, e.g., saccade duration [
]; absolute saccadic direction [
]). One study qualitatively analyzed heat maps without specifying on what metric they were constructed [
Since there are noticeable differences in the type of metric between studies, we also analyzed how many eye-tracking metrics were used in each study. We found that most studies examined more than one
eye-tracking metric (M = 1.92, SD = 0.9) but this value differed across domains (see
Figure 1
, lower right). An exception was one study that used five metrics (fixation duration, fixation count, initial gaze, pupil size, and saccade counts [
]). Three studies used four eye-tracking metrics, e.g., for analyzing individual user characteristics when evaluating student performance (fixation count, fixation duration, saccades, and transitions
As physics is the most common domain in this review (
= 16, see section Domains and Types of Graphs), we wanted to take a closer look at the eye-tracking metrics used in physics studies. An overview of the metrics used to compare experts and
non-experts’ visual behaviors when looking at graphs in the domain of physics can be seen in
Figure 2
. As studies usually collected several eye-tracking metrics (e.g., [
]), the reported number of metrics exceeds the actual number of studies. In all these studies, participants were supposed to solve problems. One exception was a study that analyzed differences in
gaze behavior between experts and non-experts before walking the shape of a graph [
]. Static metrics were used to analyze differences in the visual attention of experts and non-experts on relevant and irrelevant areas [
]. Comparable to the overall distribution, most studies analyzed dwell time, often comparing physics and non-physics students [
]. Both studies found that physics students looked longer at the graph (see section Gaze Behavior below for a closer analysis). Dynamic metrics, such as transitions, were used to predict the
performance of students solving the Test of Understanding Graphs in Kinematics (TUG-K [
3.5. Gaze Behavior of Experts and Non-Experts
To summarize the previously found differences in visual behavior between experts and non-experts during problem-solving or learning with graphs (research aim 2), we differentiated the analysis of
eye-tracking metrics, whether static or dynamic, depending on the granularity of the AOIs. We therefore consider results based on the way AOIs are defined: at macro- or meso-level and at micro-level
(see also section Data Extraction). We first present the results based on the bigger macro- and meso-level AOIs and then go on to the smaller micro-level AOIs.
3.5.1. Macro- and Meso-Level
Macro- and meso-level AOIs consist of an entire graph or analyze mid-sized sections of a graph, such as the axes and the graph. Results of studies using meso- and micro-level AOIs can be seen in
Table 3
Regarding the analysis of meso- and macro-level AOIs, there were varying results when looking at fixation duration, the fixation count, transitions, and dwell time (see
Table 3
). First, we look at the static metrics that many studies analyzed: fixation duration, fixation count and dwell time. In general, it seems as if experts pay more attention to relevant areas than
non-experts (experiment 1 [
], [
]) and less attention to irrelevant areas [
]. Experts might also attend less to the graph than non-experts [
], although this finding is unclear, as other studies found no differences [
] or concluded that experts look longer at the graph than non-experts ([
]; [
], experiment 2, only for conflicting graphs).
One study with results that contradict other studies in several instances is the one by Huang and Chen [
]. In this case, expertise was based on gender under the assumption that the gender difference in spatial working memory might influence the integration between text and diagram [
]. However, the authors did not find gender differences in this task. Additionally, only one of the three diagrams analyzed together was a graph [
]. The operationalization of expertise could also not be categorized based on the four factors. The results of this experiment might not match the others due to differences in determining expertise.
Similarly, another experiment compared the expertise as determined by the teacher [
]. The authors also concluded that the teacher’s opinion was not well suited for grouping students according to performance [
]. The same might hold true for using dyslexia as a determinator of expertise [
]. The reasons for the varying results of the other studies are less clear. Some compared science and non-science students [
]. Brückner et al. [
] compared physics and economics students, whereas Susac et al. [
] compared physics and psychology students. Although these student groups had different domain knowledge, one could assume that economics students might have more experience with reading graphs
(factor graph literacy) as well as more experience with math lectures (factor math prior knowledge). Economics students might have been more similar to physics students than psychology students in
this regard, leading to varying results. Tai et al. [
] compared biology, chemistry, and physics students. Besides the differences in expertise determination, the sample sizes might also play a role in the results (e.g.,
= 6 [
There were not as many experiments analyzing dynamic eye-tracking metrics as there were for static eye-tracking metrics (see
Table 2
). Since transitions were the most used dynamic eye-tracking metric, we will take a closer look at them. Two studies found that experts transitioned less often than non-experts between graphs and
text [
], whereas others found the opposite [
]. An explanation could be that the transitions of experts were more strategic during problem-solving [
], which could lead to experts making the same relative number of transitions as non-experts, taking the total number of transitions into account [
] (experiment 1).
3.5.2. Micro-Level
In contrast to macro- and meso-level AOIs, AOIs at the micro-level are very small and include specific parts of the graph, for example, certain sections of the x-axis. In this section, we will
consider experts’ strategies solely on the graph area (i.e., without the question or answer choices). To get an understanding of experts’ strategies at this level, a finer classification of AOIs in
the graph domain is warranted, typically considering individual values separately. The results of studies using these types of AOIs can be seen in
Table 4
Similarly, to meso- and macro-level AOIs, regarding static eye-tracking metrics, experiments analyzing micro-level AOIs also found that experts paid more attention to relevant AOIs [
], including graph information [
]. Furthermore, experts looked at the entire graph [
]. Moreover, experts seemed to systematically distribute their gaze not only spatially but also temporally [
]. In one example, a faculty member analyzed a graph and the authors showed that efficient information processing meant specifically evaluating graph information and related data at the beginning of
viewing. In contrast, inexperienced students jumped between information sources and especially back to the task and the answer choices in no particular order [
Few experiments analyzed dynamic eye-tracking metrics in a micro-level analysis (see
Table 4
). It is difficult to draw a conclusion from such a small sample. In the following section we therefore aim to summarize the visual strategies of experts and non-experts during problem-solving and
learning with graphs over the bigger (meso- and macro-level) and smaller (micro-level) AOIs.
4. Discussion
The aim of the present literature review was twofold: (1) We wanted to give an overview of the eye-tracking metrics used to compare experts and non-experts when problem-solving and learning with
graphs. Furthermore, we focused on the visual strategies of experts and non-experts guided by the research foci of the identified research articles (2). We further categorized AOIs based on their
size, as it might influence the analysis of whether the AOIs are at the bigger meso- and macro-level or at the smaller micro-level.
4.1. Summary of Experts’ and Non-Experts’ Visual Strategies
To analyze the visual strategies of experts and non-experts during problem-solving and learning with graphs, we first summarize the eye-tracking metrics used in the studies and the according
experiments included in this literature review (research aim 1). As there were differences between meso-/macro- and micro-level eye-movement analyses of eye-tracking metrics, we examine those
separately before summarizing the visual strategies of experts and non-experts (research aim 2). Finally, we discuss the various ways expertise was determined and how this might influence the
interpretation of eye-tracking results.
4.1.1. Overview of Eye-Tracking Metrics
Most experiments compared static metrics, such as dwell time, and fixation duration or fixation count, to analyze visual behavior (n = 39). In comparison, only 15 experiments analyzed dynamic
eye-tracking metrics, such as transitions and saccades. Static metrics are useful to analyze the visual behavior over the entire time participants looked at stimuli (e.g., see section Eye-Tracking
Metrics). Dynamic metrics can be used to analyze the (temporal) strategy of participants when looking at a stimulus. Although many studies only measured one metric (n = 18), researchers analyzed two
eye-tracking metrics on average. Four out of 32 experiments used four or more eye-tracking metrics.
Fixation duration and fixation count were useful for both small and large AOIs. Using two or more (uncorrelated) metrics might give researchers more insight into the visual behavior, especially in a
combination of static and dynamic metrics. Regarding transitions between AOIs, we recommend a micro-level analysis, because it is very sensitive to differences between experts and non-experts in more
detail. As was common in most studies, we also recommend distinguishing between task- or conceptually relevant and irrelevant AOIs.
4.1.2. Meso-and Macro- vs. Micro-Level AOIs
The distinction between relevant and irrelevant AOIs was quite common in the experiments included in the literature review. However, there might be differences when taking the size of the AOIs into
In general, the findings between macro- or meso-level AOIs and micro-level AOIs were very similar (e.g., for fixation duration and fixation count, see
Table 3
Table 4
), but there were contrary findings when analyzing transitions at different levels. At the meso- and macro-level, experts seemed to make fewer transitions than non-experts (see
Table 3
). In contrast, at the micro-level, experts made more transitions than non-experts between conceptually relevant AOIs (see
Table 4
). On a micro-level analysis, experts transitioned more between AOIs, whereas experts seemed to make fewer transitions between AOIs when looking at macro- and meso-level AOIs. One reason could be
that experts seemed to pay closer attention to the relevant details of the graph (e.g., [
]). However, only one experiment analyzed transitions at the micro-level [
]; consequently, it will be necessary to confirm these results before a conclusion can be drawn. We nevertheless can make some statements taking previous theories into account.
As mentioned in the Introduction, there are several theories why there are differences in the visual behavior of experts and non-experts. The results of some experiments included in the literature
review support several of those theories. For example, at macro- and meso-level AOIs, Okan et al. [
] demonstrated the so-called information-reduction hypothesis [
] in a comparison of participants with high and low graph literacy as investigated in a pre-test. In two studies, AOIs were defined at the meso-level and experts were classified using a graph
comprehension test [
]. Consistent with the information-reduction hypothesis, the authors observed that experts were better at identifying task-relevant areas in a graph, which allowed them to spend a greater relative
amount of time evaluating relevant information. Specifically, the authors showed that participants with high graph understanding reviewed axes’ labels and scaling more frequently to avoid errors [
] (experiment 1). This corresponds to results by Rouinfar and colleagues [
], who found that participants who solved the problem correctly paid more attention to relevant areas of a diagram than incorrect solvers. Rouinfar et al. compared the influence of color highlighting
on information extraction with 80 physics students and they stressed the importance of the ability to organize and integrate information to solve a problem correctly. This result confirmed that the
improved performance was caused by a learned automatism in task performance (automatism hypothesis) and not by increased awareness of the relevant domains [
] (priority hypothesis). Similarly, Okan et al. [
] observed that the highest number of transitions seemed to occur between the graph region and the question and between the graph region and the axes [
] (experiment 1), which are relevant areas as well. At the micro-level, experts also paid more attention to relevant AOIs, which is in line with the results at the meso- and macro-level and the
information-reduction hypothesis [
There were not enough experiments to conclusively identify distinct differences between experts and non-experts for specific measures. However, taken together the results of these experiments are in
line with existing hypotheses. We therefore believe that we can make some statements about the visual strategies of experts and non-experts during problem-solving and learning with graphs that we
will present in the following.
4.1.3. Visual Strategies of Experts and Non-Experts during Problem-Solving and Learning with Graphs
Based on our results, we can make a statement about what distinguishes visual expertise in problem-solving and learning with graphs. Experts systematically looked at relevant information, such as
scales as well as labels (e.g., experiment 1 [
]), and performed more integrative eye movements within a graph in terms of dynamic metrics (see
Table 3
, transitions, revisits, saccades). Therefore, in addition to the formation of chunks [
], information reduction [
] is central to expertise related to graphs.
There were some conclusions regarding differences between experts and non-experts viewing specific AOIs. First, experts seemed to spend a relatively short amount of time on the task and answer
choices during problem-solving [
], which might also be attributed to the fact that experts did not (or hardly) perform comparisons between answer choices [
]. Instead, experts paid more relative attention to axis scaling, axis labels, and graph progression [
], as well as to conceptually relevant AOIs [
Moreover, at least in data extraction, an order of information extraction appeared by comparing several works [
]. The most efficient order of information extraction seemed to emerge when participants looked at the given variables early on (if this indication existed) and directly identified them in the graph
]. Thereby, a recognition of the respective axis and its scaling could take place (experiment 1 [
]; [
]), followed by a jump back to the task [
] to identify the target variable, which is then looked for directly in the graph [
]. Depending on cognitive abilities and the task difficulty, one may jump back to variable information [
]. The expertise seems easily transferable to other styles of graphs (e.g., linear vs. radial) but not (or only by further training) to other types of graphs (e.g., line and bar graphs) [
However, possible deviations from this strategy at high expertise have not been identified yet. Furthermore, influences or trade-offs that lead to deviation from this optimal strategy in experts
remain unclear. In addition, the optimal temporal sequence for more complex tasks was not determined. A complex task would be, for example, determining the slope or the area underneath a graph. So
far, in two tasks, it seemed that students with correct solutions looked longer along the graph (when determining the slope) and into the areas below and above the graph (when determining the area) [
There have also been some inconsistencies in our results (see Gaze Behavior of Experts and Non-Experts). These might be due to the determination of expertise in individual studies. As mentioned in
the beginning, four factors are important when determining expertise in this area: (1) graphical literacy [
]; (2) domain knowledge [
]; (3) math prior knowledge [
]; (4) task knowledge [
In our review, performance, learning gain, level of study, comparison with professionals, and a pretest were measures used to determine expertise that may have fulfilled all four factors of
graph-comprehension skills. A teacher’s opinion may also consider all four factors. However, this did not prove to be a good indicator of expertise. Of these measures, performance was the most common
one (
Figure 1
, middle right). Learning gain, level of study, comparison with professionals, and pretest were only used to determine expertise in one study, respectively (see
Table 2
). A direct comparison between studies using the same expertise determinator is generally possible, but the nine studies using performance vary strongly regarding AOI sizes and eye-tracking metrics,
which makes them unsuitable for direct comparison. However, there are no conflicts in the findings. In sum, we recommend using objective measures for determining expertise and using tests that
explicitly address all four factors to allow for replicability and comparability.
4.2. Limitations
Our review of the literature about visual processing comparing experts and non-experts during problem-solving and learning with graphs has several limitations. First, we did not concentrate on one
specific definition of expertise determination. Therefore, studies used various measures to define and compare groups of varying expertise. This could be one reason for the contrasting results. It
also made drawing overarching conclusions difficult.
Second, there were some inconsistencies in using terms for eye-tracking metrics. For example, the difference between dwell time and viewing time was not always clear. In one case, the basis for the
calculation of heat maps was not reported [
Third, in analyzing the various articles on eye tracking during learning and problem-solving with graphs, the resolution of the eye-tracking systems was not considered. This means that the accuracy
with which the results were reported may be subject to variation. An increase in spatial and temporal resolution, as well as accuracy, over the period studied may well be expected due to
technological advancements in eye-tracking devices.
We do not claim completeness for the studies included in our review. Our search process was not entirely systematic, which might have led to an incomplete list of included studies. We also did not
include grey literature, which might have resulted in a publication bias towards positive and significant results. Moreover, results were only coded by the first author; we could therefore not assess
the validity of our codes. However, the codes were straightforward, apart from the eye-tracking metrics concerning dwell time, which made coding relatively easy.
4.3. Future Research
We aimed to examine relevant articles that investigated gaze behavior during problem-solving and learning with graphs. One of the main limitations of this literature review was the differing
definitions of expertise determination. We therefore suggest the consideration of the four factors (1) graphical literacy, (2) domain knowledge, (3) mathematical prior knowledge, and (4) task
knowledge. For example, expertise is sometimes established only based on the study progress [
]. This leaves it unclear to the reader to what extent participants are truly experts. Ideally, a criterion based on an assessment that tests the four factors would be established. In addition to
these four factors, efficiency in visual processing, if applicable, may also be used as a criterion of expertise determination [
In general, it is probably the best idea to find a field consensus for the definition of experts. In the case of graphs, it might be difficult to identify the specific field to which graphs belong,
and to find a consortium of researchers that represents all relevant fields. Therefore, we suggest an iterative empirical approach: Due to the lack of consensus for the definition of experts, we
propose a research-informed and domain-independent identification of a group of experts. As a next step, it is necessary to verify and consequently to refine such identification of experts, which in
turn needs to be tested again.
In the case of graphs, we believe that the most important variables are the AOIs that experts used to solve the task for various types of graphs and domains, how long they need to focus on it, and
how they connect these areas (in terms of gaze transitions). Once there is such validated definition of experts, the visual processes of those experts would be a great implementation for teaching the
understanding and efficient processing of graphs, how to approach graphs in unknown fields, i.e., to transfer the skills to other domains, how to best implement information in graphs, and how to
design graphs.
We assumed that the articles identified in this review would be largely limited to stationary eye-tracking systems, as graphs in experiments in education research are primarily presented digitally on
a computer screen. In fact, only three studies examined gaze behavior during problem-solving or learning with mobile eye-tracking systems [
]. This observation could be expected given the more diverse technological solutions and easier feasibility of stationary eye-tracking studies. As most studies with mobile eye tracking were published
recently, we believe that their number will increase in the future. In terms of analysis of eye-tracking metrics, graphs mainly analyze spatial distributions of gaze. We could identify only one paper
] that evaluated a temporal sequence of attention in problem-solving with graphs. However, others made the first steps, such as looking at the total fixation time on an AOI vs. the fixation time in
the first two seconds in an AOI [
]. Accordingly, the evidence on expert strategies is also limited only to the spatial distribution of gaze. It would be interesting to see whether there are also temporal differences between experts
and non-experts during problem-solving or learning with graphs.
We found two papers that depicted an evolution in subjects’ gaze behavior while problem-solving or learning with graphs [
]. In both cases, there was no specific instruction to influence gaze behavior. Accordingly, the extent to which learning gains in graph comprehension are associated with changes in gaze behavior is
currently under research. Furthermore, studying whether the results of problem-solving activities are transferable to learning would be very valuable. In this way, it would also be interesting to
analyze the various phases of problem-solving separately. As mentioned above, there could be an ideal strategy to extract information from graphs and a closer look at these phases could be
Visual processing during problem-solving and learning might also depend on the education level of the participants. Most studies were conducted with college or university students; there are
currently only three studies that investigate the gaze behavior of high school students during graph viewing [
]. Consequently, most papers have investigated an advanced stage of gaze behavior in graphs; there were no studies that analyzed the gaze behavior of children just learning about graphs. An account
of the gaze behavior of students, who are just acquiring the understanding of graphs, and appropriate instructional suggestions based on this, are therefore currently missing. Our sample might also
be biased towards physics because half of the included experiments (
= 16) used graphs in this domain. Although some studies compared various STEM contexts (e.g., biology, chemistry, and physics [
]), future research would benefit from comparisons in more domains as well as more types of graphs, since most experiments analyzed line graphs. Due to our limited sample, replication studies of the
experiments presented here, for example with differing eye-tracking metrics or in other domains, might further strengthen the current evidence base.
5. Conclusions
Experts and non-experts differ in the way they interpret graphs. We reviewed 32 articles about experts and non-experts solving problems and learning with graphs. Most commonly examined eye-tracking
metrics were static, such as fixation duration and fixation count. Experts seemed to focus longer on relevant areas and to identify the relevant variables in the graphs faster than non-experts. Their
visual processing also seemed to be more systematic than that of non-experts: first identifying the given variables and then directly looking for the target variable in the task and the graph.
Regarding dynamic process metrics, we suggest studying transitions between small areas of interest, and we encourage considering temporal metrics in future research. Furthermore, expertise was
determined in different ways across studies, which are partially not in line with previous determinators of expertise in graph comprehension, limiting the replicability and comparability of findings.
As a starting point for future research, we therefore recommend a clear definition of expertise and propose four factors of graph-comprehension skills as a starting point for consideration: (1)
graphical literacy, (2) domain knowledge, (3) mathematical prior knowledge, and (4) task knowledge.
Author Contributions
Conceptualization, J.K. and S.K.; methodology, S.K. and V.R.; formal analysis, S.K. and V.R.; investigation: S.K. and V.R., validation: M.B., F.F., M.R.F., A.H., S.I.H., J.K., S.K., V.R. and J.M.Z.;
data curation, V.R.; writing—original draft preparation, S.K. and V.R.; writing—review and editing, M.B., F.F., M.R.F., A.H., S.I.H., J.K., S.K., V.R. and J.M.Z.; visualization: V.R.; supervision,
J.K. and S.K.; project administration, S.K. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Data is contained within the article.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1. Overview of the number of studies related to the visual behavior of experts and non-experts’ during learning and problem-solving with graphs per year (top left); number of studies using
graphs of a certain subject (multiple mentions are possible, top right); types of graphs used in the studies (middle left); overview of the measure for determining expertise (multiple mentions are
possible, middle right); overview of eye-tracking metrics used in the studies included in the literature review (low left); number of eye-tracking metrics used for analyzing visual behavior when
looking at graphs (low right).
Figure 2. The number and types of eye-tracking metrics used in studies investigating the visual behavior of experts and non-experts learning or problem-solving with physics graphs.
Categories Terms
Visual behavior “eye tracking”, “viewing behavior”, “visual attention”
Graphs “graph”, “diagram”, “function”
Table 2. Overview over studies included in the literature review, including eye-tracking metrics (FD: fixation duration, FC: fixation count; DT: dwell time; S: saccades; FG: first gaze; PS: pupil
size; T: transitions; NRV: number of revisits; AOI: area of interest; SD: standard deviation).
Reference Year of Subject Graph Type Determination of Expertise Eye-Tracking Metrics
Ahmed et al. 2021 Engineering Line graphs Professionals FD (average, total), FC (average, total)
Atkins and McNeal 2018 Geoscience Line and bar graphs Pre-test FD (normalized, total)
Brückner et al. 2020 Physics, Economics Line graphs Domain DT (total, on relevant AOIs)
Dzsotjan et al. 2021 Physics Line graphs Learning gain Multiple features including DT (total, mean; SD of both)
Harsh et al. 2019 Biology Line graphs, diagrams Level of study FC (normalized), DT (normalized), S (normalized)
Huang and Chen 2016 Physics Diagram Spatial working memory DT (average), FC (total stimulus, on AOIs), FG, PS, S
Ho et al. 2014 Biology Line graphs Prior knowledge FD (total), T, NRV
Kekule 2014 Physics Line graphs Performance Heat maps based on FC
Keller and Junghans 2017 Medicine Line graphs Numeracy FD (relative), FC (relative)
Kim et al. 2014 Math Line graphs Dyslexia DT, FG.
Kim and Wisehart 2017 Math Bar graphs Dyslexia DT, T
Klein et al. 2019 Physics, Finance Line graphs Domain DT (total; AOI and entire stimulus), FC (average; AOI), S
Klein et al. 2020 Physics Line graphs Performance DT
Kozhevnikov et al. 2007 Physics Line graphs Spatial ability FD (relative)
Küchemann et al. 2020 Physics Line graphs Performance DT
Küchemann et al. 2021 Physics Line graphs Performance DT (total, relative), T
Madsen et al. 2012 Physics Diagrams, line graphs Performance FD (normalized; overall, first two seconds)
Okan et al. 2016a Medicine Line and bar graphs Graph literacy FD (total)
Okan et al. 2016b Medicine Line and bar graphs Graph literacy FD
Peebles and Cheng 2003 Economics Line graphs NA ^† Not applicable
Richter et al. 2021 Economics Line graphs Prior knowledge DT, FG, T, PS
Rouinfar et al. 2014 Physics Diagram Performance Domain relative ration (relative dwell time /relative area of
Skrabankova et al. 2020 Physics Line graphs Teacher’s opinion T, FC
Strobel et al. 2019 Various topics Bar graphs Working memory capacity FD (total)
Susac et al. 2018 Physics, Finance Line graphs Domain DT
Tai et al. 2006 Various topics Line graphs Domain FD, DT, S
Toker et al. 2013 Evaluating student performance Bar and radar graphs Working memory capacity, visualization FD (total, relative, mean, SD), FC (total, relative), S,T
Toker and Conati 2014 Data analysis Bar graphs Perceptual speed, working memory FC, FD, S
Viiri et al. 2017 Physics Line graphs Performance Heat maps
Vila and Gomez 2016 Economics Bar graphs Performance DT
Yen et al. 2012 Physics, various topics Line graphs Domain DT (normalized), FC
Zhu and Feng 2015 Math Line graphs Performance T
Viiri et al. 2017 Physics Line graphs Performance Heat maps
Vila and Gomez 2016 Economics Bar graphs Performance DT
Yen et al. 2012 Physics, various topics Line graphs Domain DT (normalized), FC
Zhu and Feng 2015 Math Line graphs Performance T
^† Comparison with a scanpath assumed optimal for the task.
Table 3. Overview of findings of studies analyzing eye-tracking metrics based on meso- and macro-level AOIs.
Dependent Variable Findings and References
Experts have longer average fixation durations, but spend a shorter time on the graph than non-experts [50]
Experts have the same fixation duration on a graph as non-experts [55,58]
Experts fixate less on seductive details [54]
Fixation duration Experts pay more attention to trends than non-experts, but non-experts pay more attention to the title and the axes [23]
Experts look longer at the graph than non-experts ([42]; [10], experiment 2, only for conflicting graphs)
Experts look longer at relevant areas (experiment 1 [10]; [59])
Experts look less at irrelevant axes’ labels [54,55]
On average, experts fixate less often on graphs than non-experts [43,58]
Fixation count Experts and non-experts make the same number of fixations [49]
Experts look less often at irrelevant regions [55]
Experts transitioned less often between a graph and text [39,51]
Experts switch more often between graphs and between graphics and text than non-experts [42]
Transitions Experts made “more strategic transitions among AOI triples” [40] (p. 1)
Experts made fewer transitions than non-experts on harder tasks [48]
Experts made the same relative number of transitions as non-experts (experiment 1 [10])
First gaze/fixation Experts initially spend more time on the graph than non-experts [58]
Experts look at the graph data later than non-experts [60]
Non-experts spend more time on the graph than experts [36,38]
Dwell time There are no differences in total dwell time between experts and non-experts [11]
Experts look longer at the correct answer [45]
Experts (i.e., students without dyslexia) paid less attention to the x-axis [39]
Saccades Experts make fewer saccades than non-experts [43]
Revisits Experts visit the graph more often than non-experts [42]
Dependent Variable Findings and References
Experts spend more time on graph information (such as title and variables) than non-experts [41,46]
Fixation duration Experts look at the entire graph [1]
Experts spend more time on relevant areas [1,37,47]
Experts fixate on the axes more often [35]
Fixation count Experts visit graph information (such as title and variables) more often than non-experts [41]
Experts fixate more often on task-relevant AOIs [37]
Transitions Experts transition more often between conceptually relevant areas [53]
Revisits Experts study the axes, axes labels and line segments more often [35]
Experts look longer at conceptually relevant areas [52,53,56]
Dwell time Experts spend less time on areas that can be used to calculate the solution [53]
Experts spend less time on areas found relevant for non-experts [56]
Saccades Experts look along the graph slope [1]
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Ruf, V.; Horrer, A.; Berndt, M.; Hofer, S.I.; Fischer, F.; Fischer, M.R.; Zottmann, J.M.; Kuhn, J.; Küchemann, S. A Literature Review Comparing Experts’ and Non-Experts’ Visual Processing of Graphs
during Problem-Solving and Learning. Educ. Sci. 2023, 13, 216. https://doi.org/10.3390/educsci13020216
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Ruf V, Horrer A, Berndt M, Hofer SI, Fischer F, Fischer MR, Zottmann JM, Kuhn J, Küchemann S. A Literature Review Comparing Experts’ and Non-Experts’ Visual Processing of Graphs during
Problem-Solving and Learning. Education Sciences. 2023; 13(2):216. https://doi.org/10.3390/educsci13020216
Chicago/Turabian Style
Ruf, Verena, Anna Horrer, Markus Berndt, Sarah Isabelle Hofer, Frank Fischer, Martin R. Fischer, Jan M. Zottmann, Jochen Kuhn, and Stefan Küchemann. 2023. "A Literature Review Comparing Experts’ and
Non-Experts’ Visual Processing of Graphs during Problem-Solving and Learning" Education Sciences 13, no. 2: 216. https://doi.org/10.3390/educsci13020216
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LaTeX - MAA Mathematical Communication
LaTeX is the “industry standard” for producing readable mathematics (and is also in common, but not universal, use in other technical and scientific fields). As a result, being able to use LaTeX to
produce and modify documents is an important element of mathematical communication.
The following resources can help students to get started with LaTeX.
LyX: a user-friendly front-end for learning and using LaTeX
Much of the pain of learning LaTeX is removed, without reducing its functionality, by using the free/open-source cross-platform LyX front-end for LaTeX.
LaTeX for Presentation Slides
To make slides using LaTeX, students can use these packages:
• slides – using “documentclass{slides}” in any standard LaTeX installation,
• beamer – the most common presentation package,
• powerdot– another similar presentation package,
• TexPoint – Include LaTeX code in PowerPoint presentations.
All of these options have extensive documentation online (found using the ever-helpful Google).
LaTeX for Figures
A page about ways to include LaTeX labels on figures is here.
LaTeX for MSWord and other word-processing programs
Students who need the power of LaTeX for formatting equations but are still using word-processing programs can use an online LaTeX equation editor to make an image of an equation to place in a
document. Students who don’t know LaTeX can use the online examples as models: click “examples” & then click on an example. For MSWord, the image should be saved as a png file at 300 dpi.
Tips and Tricks
Finally, it is probably worth pointing out to your students the techniques most commonly used by mathematicians who need to learn how to do some new trick in LaTeX:
• Searching: many universities and other institutions host LaTeX FAQs or other similar resources, and standard search engines tend to be very effective at finding some of these sites. (Some care
needed when searching, of course.)
• Stealing: looking at the .tex source of a document with some desireable effect is often the quickest way to figure out how to get the same result. Caveat: mathematicians are not always
intelligent TeXers, so this can lead to some bad habits.
Teaching LaTeX
Lectures may be a relatively ineffective way to teach LaTeX; students can instead teach themselves LaTeX with some guidance and support from the instructor.
In M.I.T.’s communication-intensive offering of Real Analysis, students are directed to various LaTeX resources and are given writing assignments that require progressively more complicated LaTeX
(first just text with statement environments and a little notation, then a table, then text with figures, then figures with LaTeX labels, then slides.)
At Carnegie Mellon University, Clive Newstead introduced his students to LaTeX via a handout and in-class workshops, and then gave them an assignment of handwritten text to typset.
Page content licensed by MAA MathDL Mathematical Communication under the license:
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Calculate expected rate of return in excel
The expected return on an investment is the expected value of the probability This gives the investor a basis for comparison with the risk-free rate of return. Download the free Excel template now to
advance your finance knowledge! 19 Feb 2019 The expected return is the average probability distribution of possible returns. Investors, even in the same stock, assign different expected returns This
expected return template will demonstrate the calculation of expected return This gives the investor a basis for comparison with the risk-free rate of return,
(.30 x .20) + (.50 x .10) + (.20 x .05) = Expected Rate of Return. Step. Calculate each piece of the expected rate of return equation. The example would calculate as the following:.06 + .05 + .01 =
.12. According to the calculation, the expected rate of return is 12 percent. Real Rate of Return in Excel (with excel template) Let us now do the same example above in Excel. This is very simple.
You need to provide the two inputs of Nominal Rate and Inflation Rate. You can easily calculate the real rate of return in the template provided. The formula for expected return for an investment
with different probable returns can be calculated by using the following steps: Step 1: Firstly, the value of an investment at the start of the period has to be determined. Step 2: Next, the value of
the investment at the end of the period has to For example, if you had five rows of cash flows and dates, starting in cell A1, your command should say "=XIRR(A1:A5,B1:B5)." The cell shows the average
annual rate of return after Excel finishes calculating it.
These items represent an initial investment of $100,000 and payouts in the amounts that follow. Excel calculates the average annual rate of return as 9.52%. Remember that when you enter formulas in
Excel, you double-click on the cell and put it in formula mode by pressing the equals key (=). When Excel is in formula mode, type in the formula.
Calculating Total Expected Return in Excel First, enter the following data labels into cells A1 through F1: Portfolio Value, Investment Name, Investment Value, Investment Return Rate, Investment The
internal rate of return (IRR) is the discount rate providing a net value of zero for a future series of cash flows. The IRR and net present value (NPV) are used when selecting investments Calculate
the Total Expected Return Add the expected returns under the different outcomes to derive the total expected return for the investment. Continuing with the example, add cells B3 to E3 in cell F3. The
result is an expected return of 14.5 percent. These items represent an initial investment of $100,000 and payouts in the amounts that follow. Excel calculates the average annual rate of return as
9.52%. Remember that when you enter formulas in Excel, you double-click on the cell and put it in formula mode by pressing the equals key (=). When Excel is in formula mode, type in the formula.
Expected Return Formula – Example #1. Expected Return for Portfolio = 50% * 15% + 50% * 7%. Expected Return for Portfolio = 7.5% + 3.5%. Expected Return for Portfolio = 11%. In short, the higher the
expected return, the better is the asset. Recommended Articles. This has been a guide to the Expected Return Formula. Here we learn how to calculate Expected Return of a Portfolio Investment using
practical examples and downloadable excel template. You can learn more about financial analysis from the following articles – Then, the rate of return will be: Rate of Return = (Current Value –
Original Value) * 100 / Original Value Rate of Return Apple = (1200 – 1000) * 100 / 1000 Rate of Return Apple = 200 * 100 / 1000 Rate of Return Apple = 20% He also invested $2000 in Google stocks in
2015 and sold his stock in 2016 at $2800.
Internal Rate of Return IRR is a metric for cash flow analysis, used often investments, IRR takes an "investment view" of expected financial results. The Excel function takes two arguments: Firstly,
it provides the range of cash flow events
Internal Rate of Return IRR is a metric for cash flow analysis, used often investments, IRR takes an "investment view" of expected financial results. The Excel function takes two arguments: Firstly,
it provides the range of cash flow events 7 Dec 2019 To calculate the expected Cash-on-Cash (CoC) return in 2020 for this our Excel real estate financial models to see the Cash-on-Cash return in
practice. Equity Multiple, Yield-on-Cost, Debt Service Coverage, Debt Yield,
Definition of expected value & calculating by hand and in Excel. Expected value is exactly what you might think it means intuitively: the return you can In our example, if we won, we'd be up $15,000
(less the $10 cost of the raffle ticket).
22 May 2019 We first determine the excess return over a benchmark (the alpha) then determine the regularity of the excess returns by calculating the standard
Excel's IRR function calculates the internal rate of return for a series of cash flows, assuming equal-size payment periods. Using the example data shown above, the IRR formula would be =IRR
(D2:D14,.1)*12, which yields an internal rate of return of 12.22%.
The internal rate of return (IRR) is the discount rate providing a net value of zero for a future series of cash flows. The IRR and net present value (NPV) are used when selecting investments
Calculate the Total Expected Return Add the expected returns under the different outcomes to derive the total expected return for the investment. Continuing with the example, add cells B3 to E3 in
cell F3. The result is an expected return of 14.5 percent. These items represent an initial investment of $100,000 and payouts in the amounts that follow. Excel calculates the average annual rate of
return as 9.52%. Remember that when you enter formulas in Excel, you double-click on the cell and put it in formula mode by pressing the equals key (=). When Excel is in formula mode, type in the
Microsoft has a useful page of instructions on Excel's 'built-in' Internal Rate of Return function. It's available at: * IRR function - Office Support. The XIRR function can figure it out easily.
Calculate rate of return for a share of stock in Excel. Office Tab Enable Tabbed Editing and Browsing in Office, and Make Calculating Expected Portfolio Returns. A portfolio's expected return is the
sum of the weighted average of each asset's expected return. The Internal Rate of Return calculation has very real problems. Excel offers a practical solution. | {"url":"https://bestbinlmzqtzw.netlify.app/muth74512he/calculate-expected-rate-of-return-in-excel-copy.html","timestamp":"2024-11-06T01:10:24Z","content_type":"text/html","content_length":"34540","record_id":"<urn:uuid:1c92ef19-0d3d-4b21-8b91-654b60b07be2>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00148.warc.gz"} |
Codeforces Round #912 (Div. 2) Editorial - Codeforces
Problems A, B, C, D1 and E were authored by me and Adam_GS authored D2 and F.
Code (C++)
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Code (C++)
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Code (C++)
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1903D1 - Максимум и запросы (простая версия)
Code (C++)
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1903D2 - Максимум и запросы (сложная версия)
Code (C++)
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Code (C++)
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Code (C++)
jeroenodb's solution O(M log N)
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Great contest! Solved A and C, i guess i should practice bit operations
• »
» I'm too, I can accept some hard DP problems but, bit operations I can't, it is so much trick. :((
11 months ago, # |
← Rev. 2 → +8
Interesting problem specially C and D
But C is similar to an old problem I solved before called
(Array Splitting) 1900
» 11 months ago, # |
Tutorial came too early :D
Mij = ai | aj should be the statement in editorial of B
• »
» Yes , a typo from their side. Other than this , the solution was brilliant.
» 11 months ago, # |
» 11 months ago, # |
who can explain why C greedy works ?
• » 11 months ago, # ^ |
» +12
not_ahmed_hamed If your suffix sum is positive, it's more optimal to make an extra group on your current position since making another group will double the values on the suffix.
□ » 11 months ago, # ^ |
» +4
thank you ! this explanation was way better than in editorial, or only i think so.
☆ »
» Definitely
□ » 7 months ago, # ^ |
» +1
Very nice explanation buddy. Thank You.
□ »
» Nice explanation but an extra group will make another group +1 times the values of the suffix.
☆ »
» yup
○ »
» Thanks
• 11 months ago, # ^ |
» Consider scanning from front to back, deciding at each position whether to open a new subarray.
-Cap1taL- If I start a new subarray from the current position, then according to the question, the multiplier of this subarray will be 1 more than the previous one, which also means that the
multipliers of all subsequent numbers will also be "pulled high" at the same time. (It can be found that we do not need to know how the following numbers are divided)
Obviously, it is not advisable to "increase" the multiplier if the suffix is negative, as that will make the answer smaller.
□ »
» thank you too friend !
□ »
» Very nice explanation. Well deserved CM.
• » Any subdivision can always be written as a sum of suffix sums, which should always include the very first suffix sum, i.e. 1*(subarray starting at 1) + 2*(subarray starting at i)+3*(subarray
» starting at j).. = suff[1] + suff[i] + suff[j]+. Once you are convinced, there is one to one correspondence between these 2 functions, you can just pick suff[1] and rest positive suffix
□ »
» thanks a lot :)
□ »
» Better than editorial
• Greedy or DP is just a construct. What if I told you that the editorial's solution is just refactored DP solution?
Here's a question for you: You might have heard of Kadane's algorithm. We define $$$dp[i]$$$ to be the maximum sum subarray ending at index $$$i$$$. When written in this form
dp[i] = max(a[i], a[i] + dp[i - 1])
» You can confidently say that it's DP.
adaptatron But, notice that $$$a[i] + dp[i - 1] > a[i]$$$ iff $$$dp[i - 1] > 0$$$. So if you do a space optimization, and keep a max_so_far and max_ending_here, you can see that max_ending_here
would be a[i] + max_so_far if max_so_far is positive. And intuitively, that makes sense since you should keep the left extension if it gives you a net positive value. Would you still
call this approach Greedy or DP since now the decisions make sense intuitively? I also talk more about Greedy vs DP for this problem here
In case anyone's interested,I've added hints and thought process for problems:
on CF Step
□ »
» Hey, Thanks for the dp approach, i was actually able to solve the problem but wasn't sure why it would work, Really appreciate the efforts you put in the explanation!!
11 months ago, # |
← Rev. 3 → +77
I feel like these are indeed solutions, but sometimes there are no... proofs. Which is especially important for first problems so that beginners can understand the difference between a
peltorator correct solution and an incorrect solution.
P.S. And the comment above perfectly illustrates my point.
• » 11 months ago, # ^ |
» +10
Venugopal_Reddy20 Yeah, there is no intuition and approach. Just reading the code. What's the point in making editorials if they can't be understood by a learner.
why it's not truth that |(mat[0][0..n-1]) = |(mat[1][0..n-1]) = ... = |(mat[n-1][0..n-1]) in problem B?
I checked it and it crashed my solution! lets look on example with 3x3 matrix:
a_i — ith element of our array then:
1st row: 0 | (a_1|a_2) | (a_1|a_3) = a_1 | a_2 | a_3
2nd row: (a_2|a_1) | 0 | (a_2|a_3) = a_1 | a_2 | a_3
3rd row: (a_3|a_1) | (a_3|a_2) | 0 = a_1 | a_2 | a_3
help me pls
• »
» I found mistake..
11 months ago, # |
← Rev. 2 → +2
Here's how I solved E:
So, first we realise that checking if the sum of integers is even or odd should motivate reducing everything $$$\pmod{2}$$$. We note that $$$\text{distance}^2((x_1,y_1),(x_2,y_2)) \equiv
(x_1-y_1) - (x_2-y_2) \pmod{2}$$$
(this follows from casework on the possible values of $$$x_i$$$ and $$$y_i$$$).
Nice, so we can reduce this 2D problem to a 1D problem. Create a new array $$$a$$$ satisfying $$$a_i = x_i - y_i \pmod{2}$$$
Then, the parity of the squared distance between points $$$i$$$ and $$$j$$$ is captured simply as $$$a_j - a_i \equiv a_i - a_j \pmod{2}$$$. (Doesn't matter the order, we'll see why this is
Let's give the starting value its own special place in the array, Start $$$\to a_0$$$.
Now, let's make a couple moves. Suppose we choose to go to $$$(x_i,y_i)$$$ from Start. This is equivalent to going to $$$a_i$$$ from $$$a_0$$$. If the squared distance between $$$(x_i,y_i)$$$
» and Start happens to be even, then it's necessarily true that $$$a_i - a_0 \equiv 0 \pmod{2}$$$. If the distance happens to be odd, then $$$a_i - a_0 \equiv 1 \pmod{2}$$$
Chiral After we make our move, our new Start is $$$a_i$$$. Let's choose another point $$$(x_j,y_j)$$$. What will the parity of the squared distance between points $$$j$$$ and $$$i$$$ be? Well, it'll
just be $$$a_j - a_i \pmod{2}$$$. Note, that we care only about the sum of the parity of the squared distances, and what that's going to be. So if we keep a running tally of our score so far,
we have:
$$$(a_i - a_0) + (a_j - a_i) + \dotsc \pmod{2}$$$. Isn't it weird that the $$$a_i$$$'s cancel?
Turns out that this pattern continues. It telescopes so that only the $$$0^{\text{th}}$$$ and the last index of the game, let's call that $$$k$$$, are the only ones remaining, i.e.
$$$a_k - a_0 \pmod{2}$$$
Now, we see that if we go first, we want to ENSURE that $$$a_k$$$ is the same value as $$$a_0$$$. This is a lot simpler to think about. Let's count the number of zeroes and ones in $$$a$$$
(disregarding $$$a_0$$$) and enumerate their indices into two sets, the zero set and the one set.
An optimal adversary will want to prevent us from ensuring the last element selected is the same value as $$$a_0$$$, so on each turn, they will try to use them up. That means, they will choose
the points with those values. We can see that the only possible way for us to win if we go FIRST is if there are at least as many "$$$a_0$$$" values as non-"$$$a_0$$$" values. This is because
we'll take the non-$$$a_0$$$ values and as long as there are at least as many $$$a_0$$$ values, the non-$$$a_0$$$ values will deplete first, guaranteeing a victory for us.
If there are strictly more non-$$$a_0$$$ values, we would want to go second, because then we'll just take the $$$a_0$$$ values and ensure the total sum parity is $$$1$$$, aka odd.
235128893 (Code is really bad, I went through each case individually because I was rushing to get it in before contest ended).
• »
» It's pretty easy if just expand the formula, let the points chosen be 1,2,....,n+1, then the formula expands to s1^2+2*s2^2+.....+sn^2,-2*s1*s2.... You can eliminate all the terms with
2 coefficient for even sum , so the answer is just dependant on the first and last point and the starting point is fixed
□ » Hey, I used the same logic. I counted the even and odd parity coordinates and parity of the $$$s_x^2 + s_y^2 = v$$$. Then, I claim that if $$$v \oplus (\text{#odd} > \text{#even})
» $$$ equals 1, I will go second, or else I'll go first. If I go first, I will try to use all coordinates with parity $$$(1-v)$$$ (where v was the parity of the starting point).
But unfortunately, my code is giving WA. Can you please help me out?
☆ »
» There are 4 cases not 2 look at my submission
○ »
» Can you explain your approach in short? I am not getting which cases you are talking about?
■ »
» There are 2 cases if n is even and n is odd to determine who plays last further if the other player(who isnt playing last) can play all the points the last player needs
» to play on his last turn we go with the other player otherwise the last player
• » 2 months ago, # ^ |
» 0
Lucifer_30 beautiful explanation !
Why is this solution giving WA for test case 4 although I feel I have done what is mentioned in the editorial. Submission- 235137669
• »
» could be integer overflow cum*count in your code (i have not looked very carefully)
□ »
» Thanks for the reply but I dont think the reason can you please look into the following--> I discarded the cum variable cause its unnecessary 235118044
Can someone explain C in more detail? Didn’t understand the editorial(
• 11 months ago, # ^ |
» If you open a new subarray at an index i, then it is like you are adding the suffix sum from index i to n to your answer. So, you would start first subarray at index 1, so you
would add whole array sum to your answer, now if you start a second subarray at index i, then you only need to add the sum of second array one more time because you have already
jainaditya8464 added it once when you added suffix sum of index 1. For the k-th subarray, you have already added it's sum (k-1) times, so you would add it just once.
Now, it is beneficial to add the suffix sum only when it is positive, so we will do only that
• this is how I did it.
assume a b c d
we have two choices at after a, divided it or don't divide. "cuts" is how many subarrays we already made.
» divide it=> (a*cuts) + ((b+c+d)*(cuts+1)) -->DI
don't divide=> (a+b+c+d)*cuts -->DD
now let's see why ((b+c+d)*(cuts+1)) will never decrease, after placing a cut after "a" we have this subproblem to solve the "b c d" similarly, we will repeat the process and only
place a cut again if it is increasing the overall sum of our array hence it won't decrease.
my solution 235124903
for B, is there any way to confirm that solution doesn't exist without going through entire solution array?
11 months ago, # |
← Rev. 2 → +35
I have another solution to the problem C (maybe more intuitive):
Let $$$dp_e$$$ denote the maximum division value for the array $$$[a_e, a_{e+1} ... a_n]$$$.
Then if we add some block $$$[s...e)$$$ in front the division value gets updated by $$$\sum_{s}^{e-1}{a_i} + \sum_e^{n}{a_i} = \sum_{s}^{n}{a_i}$$$ (that is because every element after
$$$e$$$ adds $$$a_i$$$ to its corresponding block).
Thus we have $$$dp_i = max(dp_j) + \sum_{i}^{n} a_i$$$, where $$$j > i$$$.
• » 11 months ago, # ^ |
» +3
Chiral This is ridiculously slick. Nice solution!
• » 11 months ago, # ^ |
» +3
NerfThis Such an elegant solution! Thank you for sharing!
• » 11 months ago, # ^ |
» +8
manan_chhajed That is a brilliant solution! Hope someday even I can think of such solutions.
• »
» If we try to do this from front instead of back, we would face the problem of computing dp i = max(dpj + summation of a's from j to i) for j < i. Thus this would take O(n^2) time to
compute. Is this the reason we compute the dp from behind or I am wrong?
• »
» I think editorial in C should include that sum of suffix sums, results in what the problem is asking to maximize, i.e 1*(1st subarray sum) + 2*(2nd subarray sum)+..... This 'trick' is arguably
the hardest part in the problem to figure out. The editorial just assumes that everyone knows this.
• »
» True. I was trying to figure out what was going on until I saw your comment.
• »
» Thanks for the comment
• »
» A great explaination! Although I solve it with greedy,but still have some trouble about correctness.Help me a lot.
Could somebody explain to me why the solution of problem C in the editorial works, like a solid proof ?
• Let $$$w_i$$$ be the id of subarray that contains $$$a_i$$$.
» It is not hard to see that $$$w$$$ is non decreasing sequence where each adjacent element can differ by at most 1. ($$$a_i + 1 = a_{i+1}$$$ or $$$a_i = a_{i+1}$$$ for $$$i \in
» [1..n-1]$$$
sleepntsheep Initially let $$$w_i$$$ be 1 for all i.
For each element (excluding first element) $$$a_i$$$ we can either do nothing or increase $$$w_j$$$ by 1 for every $$$j >= i$$$, doing so will increase the cost by $$$suff_i$$$ so if
$$$suff_i$$$ is nonnegative we can get not-worse answer by increasing $$$w_j$$$
• »
Can anyone explain the binary operation of question B? It’s not suitable for understanding.
• »
» Check my submission, you can get another approach, somewhat similar but easy to undersatnd
□ »
» thans,it is good for me
Sad. Int overflow in D
• »
» Same, I wasted time trying to find mistake in my code. I used __int128 to deal with overflow.
» Great problem Indeed!! Enjoyed very much..should have able to solve C and D. I even calculate the suffix array for C and still not able find suf[i]>0 contribute in the answer..Very
How greedy works on problem C?Can someone explain me?
» 11 months ago, # |
F is absolutely brilliant. Amazing problem!
Can anyone tell me why we are doing AND operation in B problem what we get from common set bit
• » We use AND to unset bits.
• If the $$$k$$$-th bit is set in $$$M_{ij}$$$, then that bit should be set in either $$$a_i$$$ or $$$a_j$$$ or both.
N1664 • If the $$$k$$$-th bit is unset in $$$M_{ij}$$$, then that bit must be unset in both $$$a_i$$$ and $$$a_j$$$.
• 0 7 7 5 5 000 111 111 101 101
7 3 0 3 7 ---> 111 011 000 011 111
» solution array: 5 2 3 0 4 ---> 101 010 011 000 100
gvne Now, if you compare every elements of M that ai contributes. Their bits(bitwise) are gonna be either(ignoring i == j):
all 1: column 1 -> 11(1), 11(1), 10(1), 10(1)
or contains 0: column 2 -> 11(1), 01(1), 01(0), 11(0)
in latter case ai's corresponding bit has to be 0. Otherwise it could be 1. And AND gives us this operation.
What's the purpose of putting incorrect outputs in E's example testcases? I think it had confused people who didn't read problem statements carefully enough.
• »
» They were correct, but then you'd have to pray for the interactions to go your way hahahahah
» 11 months ago, # |
As Dr. Emix I can say that all of you helped in solving 1903C - Кошмар Теофаниса and now he can finally sleep peacefully!
• » 11 months ago, # ^ |
» +5
Abito As a solver, I want you to pay me for the help!
can anyone explain DP approach for problem-C?
• »
□ »
» ohh I see, thanks
» 11 months ago, # |
Can anyone explain the solution of D2?
• Although I solved D2, I found the editorial quite confusing as well, so my explanation might differ a bit from the post above.
First, we are still using a similar greedy approach from D1 to iterate through all of the bits from most significant to least and turning it on when we can. However, because N * Q is no
longer <= 10^5, we are motivated to come up with a way to check the cost of turning on a bit in the final answer faster than O(N).
We can see that we want to increment each element by 1 until it has the bit we want on. If the bit is already on, the cost is 0. Otherwise, there are two cases we need to consider:
First, if an earlier bit was already turned on. This means that the cost to turn on this bit will always be the bit itself. (proof: to turn on a more significant bit that was considered
earlier, the number was incremented just enough so that that bit is on, meaning that any smaller bits are off). Since the value of the bit will be constant for all of the numbers, we can
simply count how many numbers have already been turned on in a previous bit.
Second, if this is the first bit that has to be turned on. This is a little harder to compute all at once, since the cost will be bit - (num % bit).
» We can also observe that if the bit that we want to turn on is greater than 2^20, all of the numbers will fall into the same category. So, we can compute the cost for bits greater than 2^
20 quite quickly.
If a bigger bit has been turned on: (sum of all the numbers mod bit is equal to zero) cost = bit * n
If a bigger bit has not been turned on: (sum of all the numbers mod bit is equal to the sum of all the numbers) cost = bit — sum of numbers
So we simply need to keep track of whether a bit greater than 2^20 has previously been turned on to deal with these cases. However, we still do not have a way to calculate the cost when we
turn on bits smaller than 2^20. That's when SOS DP comes into play. Since we only need to keep track of which of the 20 bits has been turned on, we can turn this into a bitmasking DP
problem, where we calculate the cost needed to turn on a bit when a set of bits has already been calculated already. Something like dp[20][1<<20]. However, to calculate this, we run into a
problem, in that this will require us to iterate through all of the subsets of each number, which makes our runtime O(N * 2^20 * 20). This is obviously too big to compute, so we need to
use a DP Optimization aforementioned called SOS DP. I will not explain this trick here since it is already better explained in other CF Blogs (ex. https://mirror.codeforces.com/blog/entry/
45223). This allows us to calculate the cost to turn on bits when a set of bits has already been turned on in O(1) after O(2 ^ 20 * 20 * 20) precalculation.
Hope this helps!
» Why in b we don't have to remove anything else ? Since there just one of ai or aj need to have it . so ai can have it and ai can don't obtain that bit too if aj have it. Why we do not have to
remove anything else? If there a situation that ai doesn't have a bit then solution exists but we assume that ai need to have it then it seems not work properly. I understand that i may
Calista misunderstand or miss sth. Can anyone give me some explanations plz. Thx alot.
• I had the same question during the contest and took quite some time to figure out why we don't have to care about this thing.
Suppose we had $$$x^{th}$$$ bit turned on in $$$a[i]$$$. Now, you are asking how about we just turn it in from a[i] and leave it turned on in the corresponding $$$a[j]$$$.That should be
equivalent isn't it ?
Well, we calculated a[i] as the cumulative bitwise AND of the entire row $$$M[i]$$$. This means $$$x^{th}$$$ bit of $$$a[i]$$$ got turned on because it was turned on in all cells of the row
» $$$M[i]$$$. That means, if you turn off $$$x^{th}$$$ bit from $$$a[i]$$$, then you will have to turn on the $$$x^{th}$$$ bit in all columns from $$$1$$$ to $$$n$$$ except the $$$i^{th}$$$
» column. Why ? Because remember $$$x^{th}$$$ bit got turned on only because every cell of $$$M[i]$$$ required that to be turned on. (thats what taking the AND of the entire row means,
Now, if all the elements $$$a[1],a[2]...,a[n]$$$ except a[i] have the $$$x^{th}$$$ bit turned on, then the entries in matrix $$$M[1][i], M[2][i], M[3][i],..., M[n][i]$$$ will have to get
the $$$x^{th}$$$ bit turned on. But, isn't that exactly what would happen if we did not turn off $$$x^{th}$$$ bit of $$$a[i]$$$ ? Only the entry $$$M[i][i]$$$ will not have this constraint
but it is anyways guaranteed to be zero in problem statement.
Thus, conclusion, if you wish to turn off the $$$x^{th}$$$ bit from some $$$a[i]$$$, then you will have to turn this bit on in not just the corresponding $$$a[j]$$$, but in all of $$$a[1],a
[2],...,a[n]$$$ (except $$$a[i]$$$). This will ultimately have such an effect which is exactly same as just leaving the $$$x^{th}$$$ bit on in a[i] in the first place.
Hope you understood.
Can anyone explain D1
In problem B you wrote "We check if Mij = ai & aj holds for all element" Here it will be Mij = ai | aj. Isn't it?
• »
» Yes sorry. It will be fixed.
» In problem F, how is it possible to only use one segment tree structure? Won't you need two (also for the negation nodes — whose edges go to the parents, instead of towards the
» Interesting problem specially C and D
Flix_00_ But C is similar to an old problem I solved before called (Array Splitting) 1900
In the solution to problem C, I think it should be suff[i] >= 0 in the if condition.
• »
» It doesn't change anything.
□ »
» Hello Theo830 Can you please tell me how you avoid making TWO segment tree graphs in F? I could only do 2, one which has down orientation and one which has up. source code
» here: https://mirror.codeforces.com/contest/1903/submission/235239350 Thank you in advance!
• »
» So in that case of 0 we are adding 0 only which is same ultimately.
Can someone explain D in detail I could get the core idea :\
11 months ago, # |
← Rev. 2 → +10
I think problem E has a simpler explanation.
After doing the math and finding out that going from ($$$x_0$$$, $$$y_0$$$) to ($$$x_1$$$, $$$y_1$$$) changes your overall parity by $$$x_0 \oplus y_0 \oplus x_1 \oplus y_1$$$. You can do some
more math and find out that going from ($$$x_0$$$, $$$y_0$$$) to ($$$x_1$$$, $$$y_1$$$) to ($$$x_2$$$, $$$y_2$$$) changes your overall parity by $$$(x_0 \oplus y_0 \oplus x_1 \oplus y_1) \oplus
» (x_1 \oplus y_1 \oplus x_2 \oplus y_2) = x_0 \oplus y_0 \oplus x_2 \oplus y_2$$$.
Hagry Then we can easily see that this extends as follows: going from ($$$x_0$$$, $$$y_0$$$) to ($$$x_1$$$, $$$y_1$$$) to ($$$x_2$$$, $$$y_2$$$) to ... to ($$$x_n$$$, $$$y_n$$$) changes your overall
parity by $$$x_0 \oplus y_0 \oplus x_n \oplus y_n$$$. This means that only the parity of the first and last points matter, all other points get cancelled out by the XOR. Additionally, the first
point is fixed, so we only care about the last point.
Let's call points of the same parity as the starting point good, and the other points bad. If you are the first player, you want to make the last point be good, you can do this if the number
good points are not less than the the number of bad points. You can do this by repeated choosing the bad points to deplete them until your last turn, then choosing (or forcing the other player
to choose) a good point. Similarly, if you are the second player and the number of good points is greater than the number of bad points, you can just keep choosing the good points until the
last turn, then choose (or force the other player to choose) a bad point.
[Doubt][Problem D] — Can someone explain how the formula 2^b — ai(mod)2^b comes in the editorial.
• Consider $$$bin(num) = 111001*0101$$$. I ask you to apply bit manipulation to clear all the bits to the left of * (eventually converting it to $$$000000*0101$$$).
One way to do it is to iterate over higher order bits and manually turn them off one by one. But if you want a fancy way, you can just do $$$num \% 2^4$$$. This works because any
» number can be represented as $$$\sum 2^{set\_bit\_index}$$$, and all the set bits that we want to clear have index $$$\geq 4$$$. So their modulo is zero and they would be unset. The
» lower order bits won't be touched because $$$z \% 2^4 = z$$$ if $$$z < 2^4$$$.
adaptatron You already know a trick to unset the $$$i^{th}$$$ bit. Now you've learnt about a trick to unset all bits with indices $$$\geq i$$$.
Why exactly do we want to unset the higher order bits is something specific to the problem. I'll leave it upto you to figure that out, feel free to respond to the comment if you are
not able to.
□ »
» Thankyou so much adaptatron. I got the clear understanding.
anyone with binary search solution for D1?
Great problemset! I liked E a lot, since we don't get game theory that often, and this one was an especially cool problem.
Problem F can be solved using dfs on implicit graph. As mentioned in the editorial, the problem reduces to be able to visit all vertices in range $$$[l,r]$$$ from vertex $$$v$$$, but without
creating actual edges. We can store unvisited vertices in the set. In the dfs we first visit all vertices in the main graph, and after that use lower_bound to find next vertex in range $$$
[l,r]$$$ which is still not visited.
The code looks something like this
void dfs1(int v) {
if (v % 2) {
else {
for (auto to : g[v]) {
» if (oddV.count(to) || evenV.count(to)) {
Skeef79 }
for (auto[l, r] : g1[v]) {
while (true) {
auto it = oddV.lower_bound(l);
if (it == oddV.end() || *it > r) {
I used odd and even vertices, because in 2-sat graph we only have edges from even vertices to odd vertices and vice versa
» I'm not able to undersand any part of the editorial for problem F.
ramaaa Can anyone help me understand the logic for problem F?
Great Editorial.learn useful trick.Thank you..
» I am trying D using binary search but getting TLE, I don't think the time complexity of my code is that bad. Here is my submission. https://mirror.codeforces.com/contest/1903/submission/
My solution for Problem E: say the first point is $$$(x_0,y_0)$$$ and the remaining points (in the order in which they are chosen) be $$$(x_i,y_i)$$$ $$$(1 \le i \le n)$$$. The overall
expression comes out to:
$$$( 2 \cdot \sum_{n=0}^n x_{i}^2$$$ + $$$2 \cdot \sum_{i=0}^n y_{i}^2$$$ - $$$2 \cdot \sum_{i = 0}^{n-1} (x_i \cdot x_{i+1} + y_i \cdot y_{i+1})$$$ ) - $$$(x_{n}^2 + y_{n}^2 + x_{0}^2 + y_
Clearly the parity of the final sum is decided by the first (given) point and the last selected point. Since the parity of first point is fixed we can only try and select the ideal last
point. If
» $$$x_{i}^2 + y_{i}^2$$$
bedupako is odd, the first player will try to select
$$$(x_n , y_n)$$$
such that the parity changes i.e
$$$ x_{n}^2 + y_{n}^2$$$
is odd. Hence he will try to eliminate all even sum of squares if possible. Hence we can choose the winner for each case based on the parity of the sum of square of the first point and the
number of points with odd or even sum of squares of their respective components. My Solution: https://mirror.codeforces.com/contest/1903/submission/235405790
wtf the editorial is totally a piece of shit
Is $$$O(M \log(n))$$$ solution to F added?
» In the editorial code for problem D, what does p+=ans^(p&ans) does??
Suplex why it has been used ?
» 11 months ago, # |
The_Eureka I think in the editorial for D1, the two lines of p+=ans^(p&ans); are probably redundant? Because $$$ans$$$ would start with a series of '1's $$$p$$$ would start with the same series of
'1', too.
7 months ago, # |
» 0
AkashSonar i did'nt expected that i would get stuck on this problem for 3hr . i was trying to actually reverse the array for the elements where there is decreasing order and repeat this step n times
, but when i read hint i felt very dumb that we can sort any array of size k = 2 .
7 months ago, # |
for c problem (let's take all elements are positive ) how the summation is the suffix sum array elements is equal to summation of i*a[i]
suppose elements are 1 2 5 6 8 suf sum =22 21 19 14 8 summetion of suf sum =84
also if we do 1*1 +2*2+ 3*5 +4*6 + 5*8 =84
how it is doing any proof please Theo830
• Start by having only one time each element (only one big group). Then as you add another suffix group you will have two groups. The first group/subarray with one time for each element and
the second group/subarray with two times for each element.
In the example that you are saying. We start with
» $$$[1,1,1,1,1]$$$ (This is how many times we take each element)
Theo830 Then it becomes $$$[1,2,2,2,2]$$$
Then $$$[1,2,3,3,3]$$$ etc.
In the end we will have $$$[1,2,3,4,5]$$$
□ »
» now it's better to understand.
5 months ago, # |
How can I figure out patterns and problems faster in different types of questions. Is there a certain common way of approaching certain category of problems like for greedy we have
lakshit.saini2022 to think in 1st way or for graphs in 2nd way???????
Like I couldn't think about Suffix in Problem C. I was thinking about something like Kadane. | {"url":"https://mirror.codeforces.com/blog/entry/122820","timestamp":"2024-11-05T07:15:46Z","content_type":"text/html","content_length":"501930","record_id":"<urn:uuid:c342f0a9-b521-4d25-8a8c-91a020e0f364>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00683.warc.gz"} |
A study of the transient fluid flow around a semi-infinite crack
Exadaktylos Georgios
URI: http://purl.tuc.gr/dl/dias/2047B9E6-E359-4B21-AC46-D639569795FA
Year 2012
Type of Item Peer-Reviewed Journal Publication
Bibliographic G. Exadaktylos, "A study of the transient fluid flow around a semi-infinite crack," Int. J. Solids Structur., vol. 49, no. 23-24, pp. 3323-3334, Nov. 2012. doi:10.1016/
Citation j.ijsolstr.2012.07.012 https://doi.org/10.1016/j.ijsolstr.2012.07.012
Appears in
Applying the implicit finite difference approximation of the time derivative term, the diffusion equation governing fluid-flow around a crack in a fluid-infiltrated undeformable porous medium is
transformed into a non-homogeneous modified Helmholtz’s equation. Then, Vekua’s theory regarding the solution of linear, second order, elliptical partial differential equations is employed for its
solution and the corresponding Riemann function is found. Subsequently, the general solution of the Dirichlet initial-boundary value problem for a prescribed arbitrary distribution of pressure acting
along a semi-infinite crack is found in the form of a Cauchy singular integral equation of the second kind. A numerical Gauss–Chebyshev quadrature scheme is proposed to solve this singular integral
equation that is first applied to the steady-state problem and then to the transient problem. It is shown that the density of the Cauchy integral of the transient problem View the MathML source bears
a simple similarity relationship with the steady-state problem View the MathML source of the form View the MathML source for View the MathML source, wherein View the MathML source, with D denoting
the diffusivity coefficient and t the time. This solution is the first step towards the solution of transient fluid flow around multiple cracks and then of the coupled problem of a crack or cracks in
deformable porous media and for the study of fluid-driven cracks in poroelastic media. | {"url":"https://dias.library.tuc.gr/view/61789","timestamp":"2024-11-10T05:56:42Z","content_type":"application/xhtml+xml","content_length":"17278","record_id":"<urn:uuid:a2dcf59b-3843-4d94-b24e-9c8964a25260>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00041.warc.gz"} |
IELTS Academic Reading # 4 - A Workaholic Economy
Last Updated: Friday, 29 July 2022 02:55
Written by IELTS Mentor
Hits: 179967
IELTS Academic Reading Sample.
You should spend about 20 minutes on Questions 27-38 which are based on Reading Passage 4 below.
A Workaholic Economy
For the first century or so of the industrial revolution, increased productivity led to decreases in working hours. Employees who had been putting in 12-hour days, six days a week, found their time
on the job shrinking to 10 hours daily, then finally to eight hours, five days a week. Only a generation ago social planners worried about what people would do with all this new-found free time. In
the US, at least it seems they need not have bothered.
Although the output per hour of work has more than doubled since 1945, leisure seems reserved largely for the unemployed and underemployed. Those who work full-time spend as much time on the job as
they did at the end of World War II. In fact, working hours have increased noticeably since 1970 — perhaps because real wages have stagnated since that year. Bookstores now abound with manuals
describing how to manage time and cope with stress.
There are several reasons for lost leisure. Since 1979, companies have responded to improvements in the business climate by having employees work overtime rather than by hiring extra personnel, says
economist Juliet B. Schor of Harvard University. Indeed, the current economic recovery has gained a certain amount of notoriety for its “jobless” nature: increased production has been almost entirely
decoupled from employment. Some firms are even downsizing as their profits climb. “All things being equal, we'd be better off spreading around the work," observes labour economist Ronald G. Ehrenberg
of Cornell University.
Yet a host of factors pushes employers to hire fewer workers for more hours and at the same time compels workers to spend more time on the job. Most of those incentives involve what Ehrenberg calls
the structure of compensation: quirks in the way salaries and benefits are organised that make it more profitable to ask 40 employees to labour an extra hour each than to hire one more worker to do
the same 40-hour job.
Professional and managerial employees supply the most obvious lesson along these lines. Once people are on salary, their cost to a firm is the same whether they spend 35 hours a week in the office or
70. Diminishing returns may eventually set in as overworked employees lose efficiency or leave for more arable pastures. But in the short run, the employer’s incentive is clear. Even hourly employees
receive benefits - such as pension contributions and medical insurance - that are not tied to the number of hours they work. Therefore, it is more profitable for employers to work their existing
employees harder.
For all that employees complain about long hours, they too have reasons not to trade money for leisure. “People who work reduced hours pay a huge penalty in career terms,” Schor maintains. “It's
taken as a negative signal’ about their commitment to the firm.’ [Lotte] Bailyn [of Massachusetts Institute of Technology] adds that many corporate managers find it difficult to measure the
contribution of their underlings to a firm’s well-being, so they use the number of hours worked as a proxy for output. “Employees know this,” she says, and they adjust their behaviour accordingly.
“Although the image of the good worker is the one whose life belongs to the company,” Bailyn says, “it doesn't fit the facts.’ She cites both quantitative and qualitative studies that show increased
productivity for part-time workers: they make better use of the time they have and they are less likely to succumb to fatigue in stressful jobs. Companies that employ more workers for less time also
gain from the resulting redundancy, she asserts. "The extra people can cover the contingencies that you know are going to happen, such as when crises take people away from the workplace." Positive
experiences with reduced hours have begun to change the more-is-better culture at some companies, Schor reports.
Larger firms, in particular, appear to be more willing to experiment with flexible working arrangements...
It may take even more than changes in the financial and cultural structures of employment for workers successfully to trade increased productivity and money for leisure time, Schor contends. She says
the U.S. market for goods has become skewed by the assumption of full-time, two-career households. Automobile makers no longer manufacture cheap models, and developers do not build the tiny bungalows
that served the first postwar generation of home buyers. Not even the humblest household object is made without a microprocessor. As Schor notes, the situation is a curious inversion of the
“appropriate technology” vision that designers have had for developing countries: U.S. goods are appropriate only for high incomes and long hours. --- Paul Walluh.
Questions 27-32
Do the following statements agree with the views of the writer in reading passage 4? In boxes 27-32 on your answer sheet write:
YES if the statement agrees with the writer
NO if the statement contradicts the writer
NOT GIVEN if it is impossible to say what the writer thinks about this
Example Answer
During the industrial revolution, people worked harder NOT GIVEN
27. Today, employees are facing a reduction in working hours.
28. Social planners have been consulted about US employment figures.
29. Salaries have not risen significantly since the 1970s.
30. The economic recovery created more jobs.
31. Bailyn’s research shows that part-time employees work more efficiently.
32. Increased leisure time would benefit two-career households.
Questions 33-34
Choose the appropriate letters A-D and write them in boxes 33 and 34 on your answer sheet.
33. Bailyn argues that it is better for a company to employ more workers because
A. it is easy to make excess staff redundant.
B. crises occur if you are under-staffed.
C. people are available to substitute for absent staff.
D. they can project a positive image at work.
34. Schor thinks it will be difficult for workers in the US to reduce their working hours because
A. they would not be able to afford cars or homes.
B. employers are offering high incomes for long hours.
C. the future is dependent on technological advances.
D. they do not wish to return to the humble post-war era.
Questions 35-38
The writer mentions a number of factors that have resulted, in employees working longer hours. Which FOUR of the following factors are mentioned? Write your answers (A-H) in boxes 35-38 on your
answer sheet.
List of Factors
A Books are available to help employees cope with stress.
B Extra work is offered to existing employees.
C Increased production has led to joblessness.
D Benefits and hours spent on the job are not linked.
E Overworked employees require longer to do their work.
F Longer hours indicate a greater commitment to the firm.
G Managers estimate staff productivity in terms of hours worked.
H Employees value a career more than a family.
Click the button to Show/ Hide Answers
For the answer explanation visit - Answer Explanation: A Workaholic Economy
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Hi, Could someone explain to me why in question 38 the answer is 'G'? I think that it cannot be deduced from the reading passage.
Graydon Augustus
The answer 'G' is very much in the passage. If you can't find the answer, it's because you are not able to understand paraphrase.
27 - N. In fact, working hours have increased noticeably since 1970 — perhaps because real wages have stagnated since that year.
29 - Y. In fact, working hours have increased noticeably since 1970 — perhaps because real wages have stagnated since that year.
30 - N. The current economic recovery has gained a certain amount of notoriety for its “jobless” nature: increased production has been almost entirely decoupled from employment. Some firms are even
downsizing as their profits climb.
31 - YES. She cites both quantitative and qualitative studies that show increased productivity for part-time workers: they make better use of the time they have and they are less likely to succumb to
fatigue in stressful jobs.
33 - C. The extra people can cover the contingencies that you know are going to happen, such as when crises take people away from the workplace.
34 - A. Automobile makers no longer manufacture cheap models, and developers do not build the tiny bungalows that served the first postwar generation of home buyers.
35 - B. Extra work is offered to existing employees. Companies have responded to improvements in the business climate by having employees work overtime rather than by hiring extra personnel.
36 - D. Benefits and hours spent on the job are not linked. Once people are on salary, their cost to a firm is the same whether they spend 35 hours a week in the office or 70.
37 - F. Longer hours indicate a greater commitment to the firm. People who work reduced hours pay a huge penalty in career terms,” Schor maintains. “It's taken as a negative signal’ about their
commitment to the firm.’
38 - G. Managers estimate staff productivity in terms of hours worked. But in the short run, the employer’s incentive is clear. Even hourly employees receive benefits - such as pension contributions
and medical insurance - that are not tied to the number of hours they work.
There are many reading samples like this one that doesn't have the 'answer explanation' section and I am afraid to say that this section is very important as it answers the question "why and where I
am wrong" or "why I am right" and also it guides us as a virtual teacher which is really useful. So as a student, it is my humble request to add this section to every reading practice sample if
Phuong Linh said :
Can anybody explain "two-career households"? Thanks.
Shruti said :
Please explain the answer to the question 33.
Extra people can cover the contingencies that you know are going to happen, such as when crises take people away from the workplace.
Umesh Satashia
It will be a great help to me if anyone could explain the answer to the question no. 34. Thank you in advance.
It means both husband and wife work.
HienDo said :
In the last part "questions 35-38"... I only see the "List of Factors" but doesn't shows the question 35-38 at all. Please help...
Emma Yu said :
Q.19 - this question is very tricky as it is listed whole paragraph 1 to describe work hours reduction. But the question is "Today". It is requested that candidates must pay attention on details.
Sometimes I feel really frustrated whenever I do reading practice. Because almost every time I do the YES/NO, TRUE/FALSE questions, it does take me lots of time to look as more details as
possible. Because sometimes the answer is obvious to find, then there maybe BUT, HOWEVER, to deny it. Sometimes, there is nothing to deny, I still have to spend time on looking through all just
in case. Certainly, I should not deny my English level is not good enough to take high bands.29. stagnated= have not risen.30. notoriety for "jobless" deny "more jobs"31. better use of time=
..... In fact, working hours have increased noticeably since 1970 — perhaps because real wages have stagnated since that year.
Phuong Linh
Can anybody explain "two-career households"? Thanks.
In the last part "questions 35-38"... I only see the "List of Factors" but doesn't shows the question 35-38 at all. Please help...
Mohsin Raza
Could someone please tell me what is the answer to the question no. 1?
Under the end of the answer.
Victor Chen
A) Bookstores now abound with manuals describing how to manage time and cope with stress. I don't see why this is not included in the answer.
Emma Yu
Q.19 - this question is very tricky as it is listed whole paragraph 1 to describe work hours reduction. But the question is "Today". It is requested that candidates must pay attention to details.
Sometimes I feel really frustrated whenever I do reading practice. Because almost every time I do the YES/NO, TRUE/FALSE questions, it does take me lots of time to look as more details as possible.
Because sometimes the answer is obvious to find, then there are "maybe" "but" & "however" to deny it. Sometimes, there is nothing to deny, I still have to spend time looking through all just in case.
Certainly, I should not deny that my English level is not good enough to take high bands.
29. stagnated= have not risen.
30. notoriety for "jobless" deny "more jobs".
31. better use of time= efficiency.
I don't know where it is stated that there is a reduction. Do you know?
Lee said :
Where are the hints for the answers 'F' and 'G' in the questions 35-38 ???? Can't seem to figure out why 'F' and 'G' are one of the answers!
sorry! F_“People who work reduced hours pay a huge penalty in career terms,” Schor maintains. “It's taken as a negative signal’ about their commitment to the firm.
Lee said :
Where are the hints for F and G in the questions 35-38 ???? Can't seem to figure out why F and G are one of the answers!
F _ Even hourly employees receive benefits - such as pension contributions and medical insurance - that are not tied to the number of hours they work.
Even hourly employees receive benefits - such as pension contributions and medical insurance - that are not tied to the number of hours they work. Therefore, it is more profitable for employers to
work their existing employees harder.
Razib said :
How is that a reason for employees working longer?
Because that is one of the factors resulted in employing workers for longer hours. The employers don't have to pay extra pension and medical insurance, those fixed fees are already exciting no matter
how many hours those workers have worked. If the employer hires another person to work, he/she has to pay for the new employee's pension and medical insurance, then it will be not a good deal for the
Where are the hints for 'F' and 'G' in the questions 35-38???? Can't seem to figure out why 'F' and 'G' are two of the answers!
How is that a reason for employees working longer?
Hemant said :
Can anybody explain the answer 'D' for question 35-38?
Even hourly employees receive benefits - such as pension contributions and medical insurance - that are not tied to the number of hours they work.
In the 2nd paragraph, 6th line, it says- in fact, working hours have increased noticeably since 1970. That means the answer is quite opposite to the question, right?
Can anybody explain the answers for question 35-38?
The utilisation of the present perfect shows that it continues nowadays.
Hi, please explain the answer to the question 33 and answer of 'F' and 'G' from questions 35 to 38.
Please explain the answer to the question 33.
Linh said :
Can somebody explain the question 29 and the answer D in the question from 35-38, please?
29. yes, because the real wage has stagnated. Stagnated means 'did not rise'.
I think you should notice the time period. In the text, it's been mentioned 'first century' for the time while the question was saying 'today'.
Can somebody explain the question 29 and the answer D in the question from 35-38, please?
"In fact, working hours have increased noticeably since 1970 — perhaps because real wages have stagnated since that year."
Totally agree with you. There is no exact opposite statement.
Answer #27 is wrong, in my opinion. The first two sentences state exactly the opposite. How do we know if the answers given are right?
Analysis answer for this passage multiple choice name question="Simran"]Where are the answers of this?
Right after the end of the reading passage, there is a tab/ button called "Show/ Hide Answers".
Where are the answers of this reading passages? | {"url":"https://ielts-mentor.com/reading-sample/academic-reading/29-a-workaholic-economy","timestamp":"2024-11-07T00:25:47Z","content_type":"text/html","content_length":"153314","record_id":"<urn:uuid:9bf35bc2-9aa5-4193-b465-d960be43d7df>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00640.warc.gz"} |
[GAP Forum] semidirect product
Alexander Hulpke hulpke at math.colostate.edu
Wed Jan 20 17:34:51 GMT 2010
Dear Forum,
On Jan 20, 2010, at 1/20/10 5:11, Alex Trofimuk wrote:
> How to construct a group G=[E_{3^2}\times E_{5^2}]A_4, where [A]B -- semidirect product with normal subgroup A, A\times B --- direct product of group A and B, E_{3^2} --- elementary abelian group of order 9, A_4 --- alternating group of degree 4.
We had a couple of such questions recently, so I'll be brief. Basically you will need to describe the action of A4 in the form of a homomorphism from A4 into the automorphism group of the direct product.
Lets construct this automorphism group first. For working with it, it is convenient to use a permutation representation instead:
gap> e1:=ElementaryAbelianGroup(3^2);
<pc group of size 9 with 2 generators>
gap> e2:=ElementaryAbelianGroup(5^2);
<pc group of size 25 with 2 generators>
gap> d:=DirectProduct(e1,e2);
<pc group of size 225 with 4 generators>
gap> au:=AutomorphismGroup(d);
<group with 8 generators>
gap> Size(au);
gap> auh:=IsomorphismPermGroup(au);
<action isomorphism>
gap> p:=Image(auh);
<permutation group of size 23040 with 8 generators>
As I can't think of an obvious action, lets see whether there are subgroups of the automorphism group isomorphic A4. We do this by calculating all subgroups up to conjugacy and picking the right ones.
gap> cl:=List(ConjugacyClassesSubgroups(p),Representative);;
gap> cl:=Filtered(cl,x->Size(x)=12);;
gap> IdGroup(AlternatingGroup(4)); # use to test isomorphism
[ 12, 3 ]
gap> cl:=Filtered(cl,x->IdGroup(x)=[12,3]);
[ ]
So there is no faithful action of A4. Lets try a factor group, C3:
gap> cl:=List(ConjugacyClassesSubgroups(p),Representative);;
gap> cl:=Filtered(cl,x->Size(x)=3);
[ <permutation group of size 3 with 1 generators>,
<permutation group of size 3 with 1 generators>,
<permutation group of size 3 with 1 generators> ]
This gives us three different products.
To create the first one, e.g. create the map form A4 to the corresponding subgroup of the automorphism group
gap> acthom:=GQuotients(a4,cl[1])[1];
[ (2,4,3), (1,3,2) ] ->
[ (3,7,17)(4,8,18)(12,38,22)(13,39,23)(14,24,40)(15,25,41)(16,26,42)(31,68,
gap> acthom:=acthom*InverseGeneralMapping(auh);
[ (2,3,4), (2,4,3), (1,2,3), (1,3,2), (1,3,4), (1,4,3) ] ->
[ [ f1, f1*f2, f3, f4 ] -> Pcgs([ f1, f2, f3, f4 ]),
[ f1, f1^2*f2, f3, f4 ] -> Pcgs([ f1, f2, f3, f4 ]),
[ f1, f1^2*f2, f3, f4 ] -> Pcgs([ f1, f2, f3, f4 ]),
[ f1, f1*f2, f3, f4 ] -> Pcgs([ f1, f2, f3, f4 ]),
[ f1, f1^2*f2, f3, f4 ] -> Pcgs([ f1, f2, f3, f4 ]),
[ f1, f1*f2, f3, f4 ] -> Pcgs([ f1, f2, f3, f4 ]) ]
Now we can from the SDP:
gap> s:=SemidirectProduct(a4,acthom,d);
<pc group of size 2700 with 7 generators>
Alexander Hulpke
-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
More information about the Forum mailing list | {"url":"https://www.gap-system.org/ForumArchive2/2010/002664.html","timestamp":"2024-11-04T10:59:34Z","content_type":"text/html","content_length":"5940","record_id":"<urn:uuid:5c3d90bd-16b2-4a43-9d9a-4714e4f97db9>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00639.warc.gz"} |
pst-ob3d – Three dimensional objects using PSTricks
The package uses PSTricks to provide basic three-dimensional objects. As yet, only cubes (which can be deformed to rectangular parallelipipeds) and dies (which are only a special kind of cubes) are
Sources /graphics/pstricks/contrib/pst-ob3d
Version 0.22 2020-03-24
Licenses The LaTeX Project Public License
Maintainer Denis Girou
Herbert Voß
Contained in TeXLive as pst-ob3d
MiKTeX as pst-ob3d
Topics 3D Graphics
Download the contents of this package in one zip archive (176.7k).
Community Comments
Maybe you are interested in the following packages as well. | {"url":"https://ctan.org/pkg/pst-ob3d","timestamp":"2024-11-06T21:49:42Z","content_type":"text/html","content_length":"16161","record_id":"<urn:uuid:ce0f11e6-5a8e-413c-865c-7dddbc7e2d69>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00510.warc.gz"} |
Transductive Conformal Inference With Adaptive Scores: Main Results | HackerNoon
This paper is available on arxiv under CC 4.0 license.
(1) Ulysse Gazin, Universit´e Paris Cit´e and Sorbonne Universit´e, CNRS, Laboratoire de Probabilit´es, Statistique et Mod´elisation,
(2) Gilles Blanchard, Universit´e Paris Saclay, Institut Math´ematique d’Orsay,
(3) Etienne Roquain, Sorbonne Universit´e and Universit´e Paris Cit´e, CNRS, Laboratoire de Probabilit´es, Statistique et Mod´elisation.
Table of Links
2 Main results
2.1 Setting
We denote integer ranges using JiK = {1, . . . , i}, Ji, jK = {i, . . . , j}. Let (Si)i∈Jn+mK be real random variables corresponding to non-conformity scores, for which (Sj )j∈JnK are the “reference”
scores and (Sn+i)i∈JmK are the “test” scores. We assume
Under (Exch), the p-values (1) have super-uniform marginals (see, e.g., Romano and Wolf, 2005). In addition, the marginal distributions are all equal and uniformly distributed on {ℓ/(n + 1), ℓ ∈ Jn +
1K} under the additional mild assumption:
While the marginal distribution is well identified, the joint distribution of the p-values is not well studied yet. In particular, we will be interested in the empirical distribution function of the
p-value family, defined as
Note that the p-values are not i.i.d. under (Exch), so that most classical concentration inequalities, such as DKW’s inequality (Massart, 1990), or Bernstein’s inequality, cannot be directly used.
Instead, we should take into account the specific dependence structure underlying these p-values.
2.2 Key properties
We start with a straightforward result, under the stronger assumption
Proof sketch. The conditional distribution of pi only depends on score ordering which is unambiguous due to (NoTies), and is thus invariant by monotone transformation of the scores by (1 − F).
Writing explicitly the cdf of pi from the uniformly distributed transformed scores yields (4). See Appendix C.1 for details.
In the literature, such a result is used to control the conditional failure probability P(p1 ≤ α| Dcal ) around its expectation (which is ensured to be smaller than, and close to, α) with
concentration inequalities valid under an i.i.d. assumption (Bates et al., 2023; Sarkar and Kuchibhotla, 2023).
Proposition 2.2. Assume (Exch) and (NoTies), then the family of p-values (pi , i ∈ JmK) given by (1) has joint distribution Pn,m, which is defined by (5)-(6) and is independent of the specific score
The next proposition is an alternative and useful characterization of the distribution Pn,m.
Proposition 2.3 is proved in Appendix A, where several explicit formulas for Pn,m are also provided. We also show that this generalizes the previous work of Marques F. (2023)
Comparing Proposition 2.1 and Proposition 2.2, we see that having i.i.d. scores is more favorable because guarantees are valid conditionally on Dcal (with an explicit expression for U = U(Dcal )).
However, as we will see in Sections 3 and 4, the class of exchangeable scores is much more flexible and includes adaptive scores, which can improve substantially inference sharpness in specific
situations. For this reason, we work with the unconditional distribution as in Proposition 2.2 in the sequel.
2.3 Consequences
We now provide a DKW-type envelope for the empirical distribution function (2) of conformal p-values. Let us introduce the discretized identity function
and the following bound:
Proof sketch. Use the representation (6), apply the DKW inequality separately to (U1, . . . , Un) and to (q1, . . . , qm) conditional to U, and integrate over U. See Appendix C.4 for details (a
slightly more accurate bound is also proposed). | {"url":"https://hackernoon.com/transductive-conformal-inference-with-adaptive-scores-main-results","timestamp":"2024-11-06T04:56:31Z","content_type":"text/html","content_length":"247288","record_id":"<urn:uuid:7cfdc710-eadc-44b8-85fe-0e18610a98c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00525.warc.gz"} |
Working with Sequences on a Graphing Calculator Q&As - Precalculus | HIX Tutor
Working with Sequences on a Graphing Calculator
Working with sequences on a graphing calculator introduces a dynamic dimension to mathematical exploration and problem-solving. Harnessing the power of modern technology, this practice enables users
to visualize, analyze, and manipulate sequences with unprecedented ease and efficiency. Through the intuitive interface of a graphing calculator, students and professionals alike can explore the
behavior of sequences, identify patterns, and make conjectures with confidence. By leveraging the computational capabilities of these devices, individuals can streamline calculations, test
hypotheses, and deepen their understanding of mathematical concepts. Thus, integrating sequences into graphing calculator workflows enhances learning experiences and empowers users to tackle
increasingly complex challenges.
• What are the next two numbers in the sequence 72, 86, 75, 89, 80?
• How do I do geometric sequences on an Nspire?
• How do I find the common difference of an arithmetic sequence on a calculator?
• How do I find the given term of an arithmetic sequence on a calculator?
• How do I find the #n#th partial sum of an arithmetic sequence on a calculator?
• How do I find the fifth term of a geometric sequence on a calculator?
• How do you use the graphing calculator to graph the first 10 terms of the sequence #a_n=0.2n+3#?
• How do you use the graphing calculator to graph the first 10 terms of the sequence given #a_n=10(1.2)^(n-1)#?
• How do you write a function to represent the arithmetic sequence 3, 7, 11, 15...?
• What is the next item in the sequence #1, 4, 16, 64 ,256#?
• Hi. I am stuck with “Write down the first term in the sequence given by T(n) = n2 ( squared)+4 ? Thank you
• If #h( n ) = 4n - 5# and #g ( n ) = 2n ^ { 2} - 1#, what is #( h \circ g ) ( n )#?
• How do you calculate the number of cans in this arithmetic progression?
• For the Visual Pattern #10. What are the next two patterns going to look like, write a generic rule for the pattern, and use your rule to determine how many objects are in the pattern when n is
• Help? Series and Sequences Question!
• If Tn=2-3n ?which term is -58
• If f(x)=2x+1, and g(x)=x-3, then g(f(x))= ( ?)x + (?)
• What is the 12th term of the sequence 5, 12, 19....? | {"url":"https://tutor.hix.ai/subject/precalculus/working-with-sequences-on-a-graphing-calculator","timestamp":"2024-11-03T07:54:21Z","content_type":"text/html","content_length":"551135","record_id":"<urn:uuid:2cdd50c6-a66e-476e-b7ec-46ee5d91ffe7>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00306.warc.gz"} |
9.1: Notice and Wonder: What Do You See? (5 minutes)
The purpose of this warm-up is to elicit the idea that graphs can be discrete or continuous, which will be useful when students interpret exponential functions and make sense of whether the graph
should be discrete or continuous in the associated Algebra 1 lesson. While students may notice and wonder many things about these graphs, the fact that one is a line and one is a discrete set of
points are the important discussion points.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or
imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Display the table and the 2 graphs for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1
minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.
Student Facing
Here is a table of values of data that was collected.
│\(x\) │0│1│2│3│4│5│6│
│\(y\) │6│5│4│3│2│1│0│
Here are two graphs of the data. What do you notice? What do you wonder?
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the graphs. After all responses
have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification,
or point out contradicting information.
If the situation represented does not come up during the conversation, ask students to discuss this idea. What are some situations that might be appropriate to represent with just points, versus a
connected line? (Students will see many examples in this lesson, so it’s not necessary to dwell on this question for long.)
9.2: Connect . . . or Not (20 minutes)
The purpose of this activity is to elicit the idea that graphs can be discrete or continuous based on the context, which will be useful when students interpret exponential functions and make sense of
whether the graph should be discrete or continuous in the associated Algebra 1 lesson. When students pay close attention to the appropriate domain and range, both restricted by the context, they are
engaging in work that is important in modeling with mathematics (MP4).
Arrange the students in groups of 4. Each member of the group should choose one of the questions and work individually. After quiet work time, ask students to share their responses within their group
and decide if they are correct. Follow with a whole-class discussion.
In order to find the table of values, students may choose to use graphing technology strategically (MP5), if available.
Student Facing
Here are descriptions of relationships between quantities.
• Make a table of at least 5 pairs of values that represent the relationship.
• Plot the points. Label the axes of the graph.
• Should the points be connected? Are there any input or output values that don’t make sense? Explain.
1. A cab charges \$1.50 per mile plus \$3.50 for entering the cab. The cost of the ride is a function of the miles, \(m\), ridden and is defined by \(c(m)=1.50m+3.50\).
2. The admission to the state park is \$5.00 per vehicle plus \$1.50 per passenger. The total admission for one vehicle is a function of the number of passengers, \(p\), defined by the equation \(a
(p) = 5 + 1.50p\).
3. A new species of mice is introduced to an island, and the number of mice is a function of the time in months, \(t\), since they were introduced. The number of mice is represented by the model \(b
(t)=16 \boldcdot (1.5)^t\).
4. When you fold a piece of paper in half, the visible area of the paper gets halved. The area is a function of number of folds, \(n\), and is defined by \(A(n)=93.5\left(\frac12\right)^n\).
Activity Synthesis
The goal of this activity is for students to interpret functions presented in a context and represent the function in a table and with a graph. The focus is on making sense of whether the graph
should be discrete or continuous. If it would be helpful to have words that refer to the ideas, introduce the terms continuous and discrete. (Students won’t be assessed on knowing these terms.)
9.3: Thinking Like a Modeler (15 minutes)
The purpose of this activity is to show how a context can impose restrictions on the domain and range of a function. Given a description of a context, students determine the restrictions on the
domain and range imposed by the context. Considering how a context restricts the domain or range of a function modeling it is an important part of mathematical modeling (MP4). This prepares students
to interpret exponential functions and make sense of whether the graph should be discrete or continuous in the associated Algebra 1 lesson.
Remind students that domain refers to possible values of the independent variable, and range refers to possible values of the dependent variable. Use a few examples from the previous activity to
illustrate. For example, for the cab ride, the domain is all numbers that are greater than 0, and the range is all numbers that are greater than 3.5. This presumes that a ride must have some distance
(it can’t be 0 miles long), and can be any number of miles long. A modeler might decide not to consider any cab rides that are longer than, say, 100 miles, in which case the domain would be all
numbers between 0 and 100. Sometimes restrictions on the domain are a decision made by the modeler. In contrast, for the function modeling the admission price to the park based on number of people,
the domain only contains whole, positive numbers, since the number of people must be a whole, positive number.
Arrange students in groups of 2. After a few minutes of quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they look different.
Follow with a whole-class discussion.
Student Facing
To make sense in a given context, many functions need restrictions on the domain and range. For each description of a function
• describe the domain and range
• describe what its graph would look like (separate dots, or connected?)
1. weight of a puppy as a function of time
2. number of winter coats sold in a store as a function of temperature outside
3. number of books in a library as a function of number of people who live in the community the library serves
4. height of water in a tank as a function of volume of water in the tank
5. amount of oxygen in the atmosphere as a function of elevation above or below sea level
6. thickness of a folded piece of paper as a function of number of folds
Activity Synthesis
Much discussion takes place between partners. Invite students to share how they determined the domain and range of the function representing the context. Ask if other groups have different responses.
Choose as many different groups as time allows. Attend to the language that students use to describe their pairing, giving them opportunities to describe their relationships more precisely. | {"url":"https://im-beta.kendallhunt.com/HS/teachers/4/5/9/index.html","timestamp":"2024-11-03T06:17:03Z","content_type":"text/html","content_length":"102310","record_id":"<urn:uuid:ae9593d3-5325-43c5-b47c-de08d93fcc04>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00781.warc.gz"} |
Adapting Proofs-as-Programs for the Synthesis of Imperative SML Programs
The Curry-Howard isomorphism says that intuitionistic logic can be presented as a constructive type theory in which proofs correspond to terms, formulae to types, logical rules to type inference and
proof normalization to term simplification. In order to represent intuitionistic proofs, terms of the constructive type theory contain constructive information used to prove formulae. This
information can be used to synthesize correct, error-free programs from proofs. Such approaches to program synthesis, based upon the Curry-Howard isomorphism, consistute the area referred to as the
proofs-as-programs paradigm. The advantage of proofs-as-programs techniques is that the task of programming a function is reduced to reasoning with domain knowledge. After more than 30 years of
research, proofs-as-programs constitutes a mature field with an established theory and set of best practices. State-of-the-art approaches to proofs-as-programs usually involve some form of
optimization and extraction strategy, transforming intuitionistic proofs to a commonly used functional programming language that can encode a simply typed lambda caclulus, such as SML, Scheme or
Work has been done in providing analogous results to the Curry-Howard isomorphism and proofs-as-programs for other logical systems and programming languages. However, little work has been done in
identifying a general framework that generalizes the form such analogies should take over arbitrary logical calculi and programming languages. Such a framework is useful because it can then be used
to guide how to go about adapting proofs-as-programs to new contexts.
This talk considers such a framework, which we call the Curry-Howard protocol. It requires an analogous property to the Curry-Howard isomorphism to hold between a given logic and type theory.
However, generalizing state-of-the-art approaches to proofs-as-programs, the protocol requires an optimization and extraction strategy from proofs represented in the logical type theory to programs
in a separate programming language. While program synthesis methods have been developed that conform to our protocol, such a framework has not been explicitly identified previously.
We then use the protocol to show how proofs-as-programs can be adapted to two different contexts:
1. Proofs-as-imperative-programs. The Hoare logic provides a method for the simultaneous development of imperative programs and proofs of their properties. We adapt proofs-as-programs to the Hoare
logic for the purpose of extending it to developing imperative programs with side-effect-free return values and views on state.
2. Structured proofs-as-programs. Structured algebraic specifications are an approach to the compositional design of software systems based on the development of data types. There are proof systems
that enable us to reason about structured specifications. We develop such a system and use proofs-as-programs-style techniques for the synthesis of programs from proofs about specifications, and
the eventual refinement of specifications into structured code.
These adaptations constitute an exemplary justification for the applicability of the protocol to different contexts.
Author Bio
Iman Poernomo's main research interests are computational logic for program synthesis and distributed component-based software engineering. He completed a PhD on adapting proofs-as-programs with John
Crossley at the School of Computer Science, Monash University, Australia, following Bachelor of Arts (Philosophy) and Bachelor of Science Honours (Pure Maths) degrees at Monash. From 2000 to 2003 he
was employed as a Senior Research Scientist and then Project Leader at the Distributed Systems Technology Centre (DSTC Pty Ltd). Currently he is employed as an independent Research Fellow at the
Faculty of Information Technology at Monash University. | {"url":"https://nuprl-web.cs.cornell.edu/KB/show.php?ID=615","timestamp":"2024-11-06T21:16:08Z","content_type":"text/html","content_length":"15789","record_id":"<urn:uuid:977ae81b-34af-49a8-932f-904d104ba547>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00660.warc.gz"} |
Nested sorting class of array sorting algorithms
02-16-2020, 04:27 AM
Post: #1
Namir Posts: 1,107
Senior Member Joined: Dec 2013
Nested sorting class of array sorting algorithms
Hi All,
I recently found the GeeksForGeeks.com web site and it had listed, among others, a long set of sorting algorithms. Moreover, that web site presents implementations for the sorting algorithms in
different programming languages, like C++ and Python 3. I decided to tinker with the subject and came out with a somewhat new general approach—nested sorting which is a divide-and-conquer scheme. The
general idea behind nested sorting is to divide the big array to sort into buckets (or pages). They have equal sizes (except for the last bucket in most cases). The general approach is:
1. Sort the elements in each bucket by applying a sorting algorithm. You end up with buckets, each containing a sorted sub-array.
2. Apply the same (or even different) sorting algorithms to the buckets. When comparing two buckets you compare the last element of the first bucket with the first element of the second bucket. If
they are in order, then the buckets are in order. Otherwise you do a merge-sort of the sub-arrays and write the ordered sub-arrays back in the source bucket. This phase involves comparison of
elements, but not traditional swapping, as the merge-sort step writes, in order, the elements from the two sub-arrays into a temporary array. Finally the values from the temporary array are written
back to the source sub-array memory spaces.
Nested sorting algorithms require fewer comparisons and swaps. The improvement brought by applying the nested sorting scheme is more effective with less efficient sorting algorithms.
The nested sorting scheme works very well (with the sub-arrays in the buckets and with the buckets) with algorithms like Bubble sort, Shell Sort, Comb Sort, and other algorithms that compare and swap
near/distant neighboring elements.
For algorithms, like Selection Sort, sorting the buckets requires a modified version of the basic sort algorithm. This difference is to the fact that this kind of algorithm attempts to scan the (sub)
array and locate the appropriate location of a targeted element. In other words, steps that involve scanning elements can impose a hurdle for implementing the nested sorting scheme with that basic
sorting algorithm.
I would not be surprised if the nested sorting scheme hits some walls where it cannot easily applied, or applied at all, with certain sorting algorithms. I have not explored this impasse yet. It’s
just a hunch for now!
I am developing some Matlab code for examples using the nested sorting scheme and plan to publish a report on my web site within a month. So stay tuned! I will publish the report on my web site and
provide a link on this site.
02-16-2020, 05:50 AM
Post: #2
ttw Posts: 287
Member Joined: Jun 2014
RE: Nested sorting class of array sorting algorithms
This is equivalent (in the general case) to one of the merge sorts. These are as efficient as it gets with comparison based sorting. The binary version sorts a bunch of short arrays (or buckets) by
whatever method is efficient for short arrays. Then the arrays are merged pairwise to grow their lengths. (If the starting arrays are differing size, merging from the smallest is best. This isn't
usually mentioned.)
02-16-2020, 07:20 AM
Post: #3
Paul Dale Posts: 1,849
Senior Member Joined: Dec 2013
RE: Nested sorting class of array sorting algorithms
I think this will fall out as O(n log n) time and O(n) storage. However, the constant terms will be large. The temporary buffers in the merge phase guarantee two copies for every item for every
merge. For the final merge of two half lists, it requires O(n) storage. On the other hand, quicksort requires O(log n) additional storage and takes O(n log n) time on average. It also doesn't double
The sorting wars were essentially resolved in the 1970s. There are some interesting parallel sorts but my favourite is
this one
If you've not watched
before, have fun.
02-16-2020, 12:42 PM
Post: #4
Albert Chan Posts: 2,767
Senior Member Joined: Jul 2018
RE: Nested sorting class of array sorting algorithms
(02-16-2020 04:27 AM)Namir Wrote: The general idea behind nested sorting is to divide the big array to sort into buckets (or pages). They have equal sizes (except for the last bucket in most
cases). The general approach is:
1. Sort the elements in each bucket by applying a sorting algorithm. You end up with buckets, each containing a sorted sub-array.
2. Apply the same (or even different) sorting algorithms to the buckets. When comparing two buckets you compare the last element of the first bucket with the first element of the second bucket.
If they are in order, then the buckets are in order. Otherwise you do a merge-sort of the sub-arrays and write the ordered sub-arrays back in the source bucket. This phase involves comparison of
elements, but not traditional swapping, as the merge-sort step writes, in order, the elements from the two sub-arrays into a temporary array. Finally the values from the temporary array are
written back to the source sub-array memory spaces.
Any difference to Timsort ?
02-16-2020, 05:08 PM
Post: #5
Namir Posts: 1,107
Senior Member Joined: Dec 2013
RE: Nested sorting class of array sorting algorithms
(02-16-2020 12:42 PM)Albert Chan Wrote: Any difference to Timsort ?
I will look at it. Thanls for the link.
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Naming Chemical Equations Worksheet - Equations Worksheets
Naming Chemical Equations Worksheet
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Understanding Continuity from the Perspective of AGT - Math Research of Victor PortonUnderstanding Continuity from the Perspective of AGT
Continuity and limits, as understood in traditional calculus, rely on infinitesimally small sets to arrive at limits for arbitrary functions. This idea, when translated into other definitions of
continuities, relies on ideas about convergence of sequences and the like.
Such an approach relegates analysis on discontinuous functions to the side-lines because these functions violate the principles of infinitesimal calculus and thus not worthy of consideration.
It’s understandable why mathematicians have given continuity and infinitesimally small sets this much import—calculations of a practical nature may seem much more relevant to our concerns. Even with
these convictions, it doesn’t disqualify analysis of discontinuous functions from mathematical theorization.
One of the outcomes of Algebraic Generalized Topology is the generalization of limits of discontinuous functions. This is a relatively novel path, considering the extensive literature that relies on
the fundamental theorem of calculus and the traditional understanding of continuity, but is no less relevant to mathematical analysis. It is a branch of mathematics that is capable of generalizing
continuity as well as discontinuity across different mathematical spaces, effectively making mathematical analysis much more powerful than it is currently.
The Building Blocks of AGT
Funcoids and Reloids are the fundamental components of mathematical analysis in AGT. For the purposes of the reader, I think it would be prudent to go through the basic definitions. These include:
A relation from a set A to a set B, such that it is a quadruple (A, B, α, β) where α ϵ F(B)^F^(^^A^), αϵ F(B)^F^(^^A^) such that:
∀XϵF(A), YϵF(B) : (Y∩α(X) ≠0 ⇔ X∩β(Y) ≠0).
F(A) here denotes the set of filters on a set A.
A Reloid is defined as:
A Reloid from a set A to a set B is a triple (A, B, F) where,
F is a filter on the Cartesian product A x B of two sets
A Definition of Continuity
I intend to go into deeper explanation of my work in later publications, but, for perspective, I think it is also worthwhile to think of the definition of continuity in terms of Funcoids and Reloids.
The definition is as follows:
f ϵ C(µ,ν) ↔ f ∘µ ≤ ν∘ f
This definition applies to topological spaces and also to uniform spaces, proximity spaces, graphs, etc. I also take the concept of funcoids to create an elegant generalization of limits for
arbitrary discontinuous functions.
I have also worked extensively with limits of discontinuous functions and mathematical nondifferentiable analysis in my math research monograph titled Algebraic General Topology. Anyone interested in
my work should get in touch with me directly. | {"url":"https://math.portonvictor.org/2020/01/06/understanding-continuity-from-the-perspective-of-agt/","timestamp":"2024-11-10T22:32:47Z","content_type":"text/html","content_length":"103107","record_id":"<urn:uuid:c1f080a0-8e1a-4136-93c3-e5099a1bfd3d>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00297.warc.gz"} |
Editorial comments: The letter to Weil that saw the birth of the \(L\)-group was written in January, 1967. Somewhat later that same year, Roger Godement asked Langlands to comment on the Ph. D.
thesis of Hervé Jacquet. His reply included a number of conjectures on Whittaker functions for both real and \(p\)-adic reductive groups. These were later to be proven, first in the \(p\)-adic case
by Shintani for \(\mathrm{GL}_n\) and Casselman Shalika in general, and much later in the real case by a longer succession of people.
Author's comments: This letter, a report on Jacquet's thesis, is undated, but a letter from Godement dated May 12th, 1967 asks that the report be submitted before the end of May. I assume it was sent
from Princeton so as to arrive in Paris before the date requested.
The notation may cause the reader some difficulties. Some symbols, for example \(\chi\), have meanings that change (sometimes explicitly but sometimes only implicitly) in the course of the letter.
There is a particularly dangerous lapse in regard to \(\xi\). Other symbols, sometimes the same, are employed in ways that have become uncommon. The symbol \(\pi\) appears, for example, as a
representation of a compact group. The notation \(\langle a, \alpha\rangle\) for the value of the multiplicative function \(\alpha\) at the group element \(a\) is particularly disconcerting.
References to pages either in Jacquet's thesis or in the handwritten letter have been allowed to stand.
The formula for Whittaker functions for unramified representations suggested in the letter was proved by Casselman and Shalika.
It appears from the Institute records that Godement visited Princeton early in March of 1967. It must have been then that I spoke to him. The lectures at Yale were given early in April of 1967 and
appeared later as the monograph Euler Products (included just above). | {"url":"https://publications.ias.edu/node/38","timestamp":"2024-11-06T01:23:13Z","content_type":"text/html","content_length":"16958","record_id":"<urn:uuid:3c301ce7-118d-4d6a-8c0f-7a977482863e>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00835.warc.gz"} |
An Exercise in Unplugging the Church-Turing Thesis [Part 2]
Submitted by egdaylight on
In my
previous blog post
I provided my latest work on the Church-Turing Thesis; that is, a paper which I submitted for peer review in December 2018 to a computer science journal. In my paper, I introduce the ADM model of
computation and claim that it is more powerful than the traditional Turing Machine (TM) model of computation. In the first week of February 2019, I received three potentially good reasons (from
colleagues, via email) why I might have made a mistake or overlooked something in my paper. Each received critique along with my rebuttal is presented in a separate section below.
Operationally, there is no difference between TM's and ADM's
The critique goes like this:
1.) Any ADM can be simulated by a standard TM. The data structure in Definition 11 and the subsequent description of how an ADM machine executes a move (step of computation) are clearly
implementable in any general purpose model of computation (such as Turing Machines).
2.) The argument outlined at the bottom of page 2, and which is used in the proof of Theorem 25, does not show that ADM's produce a strictly larger class of partial numerical functions f than are
computed by TM's.
I approve Statement 1 on page 4 and in the rest of my paper. Operationally, there is no difference between TM's and ADM's. Loosely speaking, Statement 1 is about the operational semantics of ADM's,
which is no different from the operational semantics of TM's. To be more precise, the “next move” relation between two instantaneous descriptions is (intrinsically) the same for both ADM's and TM's.
That is why Statement 1 is correct.
Does Statement 1 imply Statement 2? The answer is “yes” according to the person who made the above remark. In contrast, an underlying theme in my paper is, that:
Statement 1 (which is about the operation of a machine) does not imply Statement 2 (which is about the denotation f of a machine).
Indeed, in computer science it is perfectly possible, that:
• The denotations of machines M and N are functions f and g, respectively.
• Machine M operationally simulates machine N (and vice versa). That is, M and N have the same input/output behavior at the syntactic level of strings.
• Functions f and g are not equal to each other.
So, if there is only one message I can get across about my paper, then it is this:
I have deliberately ensured that no operational distinction can be made between ADM's and TM's. I have, however, arranged the denotational semantics of both kinds of machines to be different. The
operational semantics (of both ADM's and TM's) is incomplete: it excludes the interpretation (made by the human observer) of the raw syntax. The denotational semantics, in contrast, is complete
in this sense and thus sits in the driver's seat.
How, then, do the denotations between ADM's and TM's differ? Can I explain informally? Yes. The difference amounts to:
A.) In my ADM account, you can look at the output tape when the machine says it has produced a new output symbol. (In fact, you can always observe the output tape.)
B.) In the traditional TM story, you are prohibited from looking at the output tape if the machine has not (yet) halted.
The output tape (in both A and B) is a write-only tape, and output symbols are printed from left to right only.
To further understand the denotational difference between ADM's and TM's, here's how you can misuse an ADM machine so that it does no more than a TM:
1. Feed the finite input to the machine.
2. Press the “go” button.
3. Close your eyes until the machine says “I have halted.”
4. Then, and only then, open your eyes and read the finite output.
In Step 3 you, the user, are asked to consistently ignore the following kind of feedback from the machine: “I already have part of the final output for you to read.” Surely, humans use machines in
adherence to the ADM approach, not the TM way. That is, humans do look at the intermediate, definite output symbols and they do something with them while the machine continues to operate. Likewise,
an ADM can resemble a mathematician better than a TM. For, we look at the mathematician's intermediate output results (= lemmas), awaiting the mathematician's grand theorem.
At this point the critical reader might object that my non-standard denotational semantics has to comply with “locality” and “finite-in-time” constraints. I agree and my ADM model does exactly that.
After all, my ADM model is almost completely identical to the traditional TM model.
An important remark is, that, I use the words “TM model” to refer to the raw TM syntax (= quintuples, tape squares, ...), the next-move function (cf. operation semantics), and the
“implements-a-function” relation (cf. denotational semantics). Concerning the latter, function f is, in my paper, consistently called the “semantics” or the “functional semantics” of the TM in hand.
The distinction I make between “syntax” and “semantics” complies with standard terminology used in both linguistics and logic. And I deliberately do not use the terms “operational semantics” and
“denotational semantics” in my paper, for they are foreign terms to many recursive function theorists.
Coming now to the comment (made above in Statement 2) about my Theorem 25. My proof of that theorem is in character no different from any other classical diagonal argument in computer science, in
that I refer to both the denotation of a machine and its operation. The person who provided the above critique has chosen to only compare the operational semantics of the two kinds of machines used
in my proof. He has not scrutinized my complete mathematical argument, which explicitly refers to functions and the different ways in which ADM's and TM's implement them.
As with any classical diagonal argument, the denotation (semantics) of each machine has to be taken into account.
My remark thus also holds for the diagonal arguments presented in standard computer science textbooks. The whole point, after all, is to prove theorems about static, mathematical objects; that is,
the partial numerical functions f; that is, the denotations.
Begin of Intermezzo.
One could also use a machine M as a language acceptor. In this case the denotation of M is a set of natural numbers. I also discuss this second kind of denotation of my ADM's in my paper, and
contrast it with the one used in standard computability textbooks.
One could, like Alan Turing in 1936, use a machine M to calculate the infinite digits that represent a particular real number. In this case the denotation of M is a real number. I also discuss this
third kind of denotation in my paper.
My point is that it is tempting, and not always harmless, to ignore the distinction between abstract objects (denotations) and concrete representations (syntax) — as Turing came to realize (even
more) after submitting his 1936 paper (for publication), which led him to write his 1937 correction: a beautiful explanation is provided in [3]. Did Turing, in writing, explicitly distinguish between
the operation of a machine and its denotation when spelling out his diagonal argument? I take the answer to be “no,” nor was this necessary.
End of Intermezzo.
The person who provided the above critique is an expert in programming language semantics. So, here's a more refined way for me to rebut:
For most programming language experts, the operational semantics of a programming language L is given precedence over the denotational semantics of L, should there be a discrepancy between the
two [1, Section 4.2]. However, in computability theory, when experts (such as Hopcroft & Ullman) reason about computability per se, they are (often implicitly) giving priority to the denotation f
of a machine. The simulation of one machine's operations by another is merely a tool to prove something about partial numerical functions (or other denotations).
The implicit assumption, made by the person who approves both Statements 1 and 2, goes as follows: There is no discrepancy between the operational semantics of raw TM's and any conceivable,
realistic, denotational semantics of raw TM's. This assumption is false. In my paper I contrast the standard, functional denotation of a raw TM (cf. Hopcroft & Ullman) with a non-standard, functional
denotation (which brings us to my ADM story). They are not the same. And then I rigorously prove that ADM's do produce a strictly larger class of partial numerical functions f than are computed by
TM's — contrary to Statement 2.
My submitted paper has been rejected without peer review. (This comes as no surprise. Recently, I read an article by Paul Vitanyi in which he explains that it took about a decade for him to get some
of his ideas officially accepted [2].) I had a very brief correspondence with the editor of the journal. His professional feedback is well intended. I have paraphrased all of his technical remarks in
Statements 1 and 2 above. I take him to be reading my proof of Theorem 25 merely in terms of an operational semantics. From that restricted perspective, his reasoning makes sense to me.
Unfortunately, consistently eschewing the distinction between a machine's simulation and a machine's denotation is unwarranted, also when thoroughly studying classical textbooks.
I don't think somebody with the profile of Martin Davis will use Statements 1 and 2 to scrutinize my research findings. This brings me to the following critique of my work.
Aren't you merely shifting the problem on defining the possible values of computable functions?
The second critique again pertains to my informal explanation at the bottom of page 2, and it goes like this:
3.) The fact that ADM V, in case of non-termination, outputs 0 (i.e. bottom), only shifts the problem on defining the possible values of computable functions.
4.) In my view, machine V in this case simply does not return a value.
First of all, with regard to Statement 3, also terminating ADM's are allowed to output the string 0. Second, concerning Statement 4, what does “return” mean in the phrase “a machine that returns a
value”? Does the machine have to halt in order to “return” a value? The classical answer is “yes.” And my ADM's do not comply with this classical understanding of what an algorithm entails. But I
quote Moshe Vardi, Robin Milner et al. in my paper; that is, researchers who have embraced a non-classical understanding of the word algorithm (for several decades now). In this setting, my ADM's fit
the bill.
Two more detailed comments now follow in connection with both Statements 3 and 4. First of all:
In classical computability theory, the “undefined value” is not represented in the syntax; that is, on the tape of a Turing machine.
In my paper I consistently represent the “undefined value” with the empty string.
(And see Remark 9 on page 13 if one wishes to represent the “undefined value” with a non-empty string, although that would complicate some matters.)
In other words: the “undefined value” can be represented in, say, bits on the tapes of my machines. The caveat is that my machines will never halt with the empty string as output. This comes with an
increase, not a loss, of generality concerning the partial numerical functions (that are "algorithmic").
Second, if the critical reader insists that only “bottom” and not, say, the natural number 0, may be associated with non-terminating computation, then she is absolutely right that my ADM model of
computation should be rejected. However, there is no harm done in associating multiple natural numbers with non-terminating behavior. Nothing in computability theory precludes this.
How will you ever know?
The third critique is already addressed at length in my paper. It goes like this:
5.) How will you ever know that the “real” output of your ADM V (as in Theorem 25) is 0 or 00 or 01? Seeing just 0 on the output tape you will never know if this is the final output.
I counter the question posed in Statement 5 with another one:
How will you ever know that your favorite TM computation halts?
You don't know in the general case, and thus also not in a classical diagonal argument pertaining to TM's. (The point of diagonalization is that you don't know.) So, the critique in Statement 5 also
holds for TM's, not just ADM's. It is the infinite tape which is the source of this “problem,” not my ADM's. And both TM's and ADM's have infinite tapes.
Coming to another concern raised in Statement 5, a direct practical application of a Turing machine that diagonalizes does not exist either. The profit of diagonalizers (be they ADM's or TM's) is
that they help the pure mathematician to prove impossibility results — which, if used meticulously, can have industrial implications.
To recap, if you want to use ADM's in practice, then you will have to introduce constraints, just like we do with TM's. In practice, nobody waits forever to observe a computation.
Perhaps the following refined remarks help:
A.) For every TM which computes a partial function f, there is an ADM which computes the same function f.
B.) There is an ADM which computes a partial function g, which no TM computes.
Statement A can be strengthened:
For every TM which computes a partial function f, there is an ADM which computes the same f by halting on each (encoded) input x for which f(x) is well-defined.
So, anything you want to do with Turing machines, you can also do with well-chosen ADM's. Colloquially speaking: there are plenty of ADM's that don't have the annoying “printing later” effect that is
alluded to in Statement 5. For every halting TM there is an ADM that mimics the beautiful, finite behavior of that TM and partially computes the same function; that is, has the same denotation.
[1] R. Turner and N. Angius,
The Philosophy of Computer Science
, The Stanford Encyclopedia of Philosophy (Spring 2019 Edition), E.N. Zalta (ed.), forthcoming
URL = <https://plato.stanford.edu/archives/spr2019/entries/computer-science/>
[2] S. Haldar and P. Vitanyi,
Bounded Concurrent Timestamp Systems Using Vector Clocks
, Journal of the ACM, Vol. 49, No. 1, January 2002, pp. 101-126.
[3] G. Gherardi,
Alan Turing and the Foundations of Computable Analysis
, The Bulleting of Symbolic Logic, 17(3):394-430, 2011.
Hasty Afterthoughts
When you open Michael Sipser's 2006 textbook – Introduction to the Theory of Computation – on page 177: he discusses Cantor's diagonal argument.
• He carefully remarks that a real number x can be represented in more than one way (e.g. in decimal notation).
• So one cannot diagonalize out correctly without taking this constraint into account. Because the proof has to work for the denotations (= the real numbers), not merely their syntactic
• From a macro perspective, Sipser's remarks are similar in kind to Turing's insights in 1937, which are discussed in [3].
Coming to the Hopcroft & Ullman book from 1979, entitled Introduction to Automata Theory, Languages and Computation. The authors:
1. Make a remark similar to Turing 1937 w.r.t. numerical functions but not w.r.t. `fomal languages'.
2. Indeed, much to my surprise, Hopcroft & Ullman don't distinguish between strings and naturals when dealing with `formal languages' — that's, presumably, what “formal” means in `formal languages'.
Abiding by 2. is fine as long as the fusion of denotation and representation is justified. But no justification is given. Hopcroft & Ullman implicitly assume that their TM's work in the classical
input/output way with halting. That's perfectly reasonable, and then you can build a whole theory on top of that (which I call mainstream computer science). But the conflation is not warranted as
soon as we peruse more general forms of computation: where we look inside the machine a bit while it is operating. An operational semantics does not capture these nuances. That says someting about
the limitations of operational semantics, not of my paper.
If you, the reader, have made it up to here (in the year 2019), then chances are you are not a computer scientist.
1 Comment
Submitted by egdaylight on
Coming again to Section 1 in this blog post and the person X who provided the corresponding critique. Observe the following:
• In 1987, T. McCarthy & S. Shapiro proved that their "extended TM's " are recursive in the halting problem of ordinary TM's.
• If I follow X's line of reasoning, then we have this absurd result: since ordinary TM's can also operationally simulate the "extended TM's" (for the same reasons provided in Statement 1), it
follows that TM's can solve their own Halting Problem.
The critical reader will notice that:
• My ADM machines are more powerful than ordinary TM's and less powerful than the aforementioned "extended TM's".
• This makes perfect sense if the do's and dont's of each model of computation are compared with each other:
1. ADM's are restricted "extended TM's" in that they are not allowed to overwrite their output.
2. The "extended TM's" of McCarthy & Shapiro are allowed to overwrite their output (albeit only a finite number of times). | {"url":"https://dijkstrascry.com/CTT2?page=0%2C0%2C0","timestamp":"2024-11-10T01:23:53Z","content_type":"text/html","content_length":"55317","record_id":"<urn:uuid:9f9dbf82-26cf-4546-b36a-92750ad21634>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00195.warc.gz"} |
Solving Linear Quadratic Systems Algebraically - A Plus Topper
Solving Linear Quadratic Systems Algebraically
A linear quadratic system is a system containing one linear equation and one quadratic equation
which may be one straight line and one parabola,
or one straight line and one circle.
Algebraic Solutions
straight line: y = mx + b
parabola: y = ax^2 + bx + c; a ≠ 0
circle: (x – h)^2 + (y – k)^2 = r^2 ; center (h.k), radius r
Let’s look at how to solve a linear quadratic system of equations algebraically.
Example 1:
Solve this linear-quadratic system of equations algebraically and check your solution:
y = x^2– 6x + 3 (parabola)
y = -2x + 3 (straight line)
1. Solve for one of the variables in the linear equation.
Note: In this example, this process is already done for us, since y = -2x + 3.
y = -2x + 3
2. Substitute this value into the quadratic equation, and solve the resulting equation.
• Substitute -2x + 3 for y in the quadratic equation.
• Subtract 3 from both sides; then add 2x to both sides.
• Factor.
• Set each factor equal to zero and solve.
You now have TWO values for x. This tells you that there may be two possible solutions.
TWO SOLUTIONS.
3. Find the corresponding values for y. Substitute each value into the linear equation in place of x.
Yes, you could substitute in the quadratic equation, but substituting into the linear equation will be easier.
4. Check: Be sure to check BOTH solutions in both equations.
5. State the final solutions. The solutions may be stated as the set {(0, 3), (4, -5)}
Example 2:
Solve this linear-quadratic system of equations algebraically and check your solution:
y = x^2– 6x + 3 (parabola)
2x – y = 13 (straight line)
1. Solve for one of the variables in the linear equation.
2x – y = 13
y = 2x – 13
2. Substitute this value into the quadratic equation, and solve the resulting equation.
• Substitute 2x – 13 for y in the quadratic equation.
• Add 13 to both sides; then subtract 2x from both sides.
• Factor.
• Set each factor equal to zero and solve.
You now have ONE value for x. This tells you that there may be only one solution.
3. Find the corresponding value for y. Substitute the value into the linear equation in place of x.
Yes, you could substitute in the quadratic equation, but substituting into the linear equation will be easier.
4. Check: Be sure to check the solution in both equations.
5. State the final solution. The solution may be stated as (4, -5) or {(4, -5)}
Example 3:
Solve this linear-quadratic system of equations algebraically and check your solution:
x^2+ y^2 = 9 (circle)
x – y = 3 (straight line)
1. Solve for one of the variables in the linear equation.
y = x – 3
2. Substitute this value into the quadratic equation, and solve the resulting equation.
• Substitute x – 3 for y in the quadratic equation.
• Expand (x – 3)^2
• Combine terms.
• Factor.
• Set each factor equal to zero and solve.
You now have TWO values for x. This tells you that there may be two possible solutions.
3. Find the corresponding values for y. Substitute each value into the linear equation in place of x.
Yes, you could substitute in the quadratic equation, but substituting into the linear equation will be easier.
4. Check: Be sure to check BOTH solutions in both equations.
5. State the final solutions. The solutions may be stated as the set {(0, -3), (3, 0)} | {"url":"https://www.aplustopper.com/solving-linear-quadratic-systems-algebraically/","timestamp":"2024-11-14T15:19:55Z","content_type":"text/html","content_length":"48028","record_id":"<urn:uuid:16e09548-ed88-4ad5-91f8-cbd28581b2c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00378.warc.gz"} |
Numbers Are For More Than Pages
(This column is posted at www.StevenSavage.com and Steve’s Tumblr. Find out more at my newsletter.)
Being a writer, on the side or professionally, requires a lot of skills. A self-publisher wears many hats, but even authors with agents and support have to take on tasks other than writing. Of those
many skills, one stands out as very important and easy to miss – Math.
People have widely differing reactions to hearing “we’re going to talk about math.” Trust me, it’s worth it whatever your response is – because math is used everywhere in an author’s work.
A writer’s growth requires math to be measured – and improved. Comparing word counts lets you determine if your typing speed is improving. Time taken to edit a document helps you determine if your
grammar is improving. Becoming a better writer may mean being better at math.
But once you’re writing, math comes in again as you plot a schedule. How long will it take you to write this chapter for your pre-readers? How long until you need to get a cover from your artist?
Scheduling is all math – often made more challenging with timezones, calculating dates, and the like.
As a book progresses, math once again comes to the fore. How fast are you working? What’s the percentage of a book done? Do you have to change your schedule or speed up your pace? Scheduling is math
– but so is seeing how you’re doing.
When a book is done, there comes more math. How many pages is a book, and how does that affect cover size? What’s the ideal formatting with font sizes and margins? If you do self-publishing and don’t
outsource formatting and the like, get out your calculator.
Finally, a book launches. It’s out and . . . here comes more math. You have to calculate if your ad spends are paying off. Evaluating book sales requires math, often with complex date-time
calculations. Your newsletter opens and clicks need to be compared to past events – which means math.
It’s exhausting, isn’t it? When I first realized I had to write this column, I was overwhelmed with the realization of just how much math my own publishing involved. I was so used to it I didn’t see
it – until I wrote this.
If you like math like me, or don’t, this should be a helpful realization. Math is a skill you need to use in writing, and if your math skills are lacking you have a new motivation to improve them.
Math makes a better author.
Steven Savage | {"url":"https://www.stevensavage.com/blog/2021/05/numbers-are-for-more-than-pages.html","timestamp":"2024-11-04T23:48:33Z","content_type":"text/html","content_length":"45235","record_id":"<urn:uuid:518a0b83-6b38-493c-9fc5-3ab2699d7d2c>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00755.warc.gz"} |
[Solved] A population of N 57 scores has a mean of | SolutionInn
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A population of N 57 scores has a mean of m 5 13.
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The Problem of Measurement Output Control under Set-membership Uncertainty
A key problem of control theory is to construct feedback control laws based on available data and to determine how the level of uncertainty in the model and in the information arriving on-line,
through noisy measurements, would affect the system performance. In the absence of statistical description such problems of Measurement Output Feedback Control (MOFC) were investigated mostly within
the H(inf) setting, with soft-type integral costs, while problems with hard bounds on the unknowns were less developed. The present proposal is aimed to produce a fairly complete investigation of the
problem of output feedback control constructed through available measurements under unknown but bounded disturbances subject to hard bounds on the uncertain items. The proposal covers topics from
basic theoretical problems to computational methods. The novelty of the suggested solution schemes, applicable to nonlinear systems, with greater specifics for the linear case, lies in the
combination of dynamic programming, set-valued analysis and minmax approaches. It is well understood that the overall problem under consideration is a combination of two ? a finite-dimensional
problem of guaranteed state estimation and an infinite-dimensional problem of feedback control under set-membership uncertainty. The second problem is especially difficult to formalize and solve. In
the proposed solution, based on earlier work, the aim is to reduce the second problem to a finite-dimensional one, which would facilitate calculation. For systems with linear structure and convex
constraints the computational procedure is to be based on using and also generalizing the ellipsoidal calculus which proved effective for many problems and allows development of complementary
software. It is expected that such approach will produce solutions "to the end," with illuminating examples. For the nonlinear case calculations may be facilitated by using the specifics of the
problem and applying modifications of the earlier suggested comparison principles that allow one to relax the original equations or variational inequalities of the Hamilton-Jacobi type through
simpler, finite-dimensional relations.
The Measurement Output Feedback Control (MOFC) problem has been thoroughly studied in a stochastic setting as a combination of stochastic filtering theory within the theory of stochastic control.
However a considerable number of problems in control design have to deal with systems subjected to information conditions that are other than stochastic. Such problems of MOFC are increasingly
motivated by applied issues, given the progress in design of high-tech complex systems arising in automation, navigation and the cyber-physical field (including hybrid, impulsive, time-lag,
multi-agent, communication-oriented processes and the like). They naturally require new techniques, new types of models and their mathematical formalization, as well as new numerical methods,
algorithms and software. The present project is designed in response to the indicated demand.
Project Report
This research produced a complete theory for the problem of output feedback control based on available measurements under unknown but bounded disturbances subjected to hard bounds on the uncertain
items. The research results range from basic theoretical problems to computational methods. The novelty of the derived solution schemes lies in the combination of Hamiltonian methods in the form
dynamic programming with set-valued analysis and minmax approaches. The overall general solution for the considered problem is a combination of the solutions of a finite-dimensional problem of
guaranteed state estimation and an infinite- dimensional problem of feedback control under set-membership uncertainty. For the first problem new types of set-valued observers were introduced. For the
second problem which is especially difficult to formalize and solve, the achieved solution is reduced to a finite- dimensional one, which facilitates calculation. For systems with linear structure
and convex constraints the solution is more detailed. Here the computational procedure is based on new developments in ellipsoidal calculus, which proved to be effective and allowed to design
complemental software, producing solutions o the end", with examples including out put feedback for impulse control. For the nonlinear case the calculations are facilitated by using the specifics of
the problem in combination with comparison principles that allow to relax the original equations or variational inequalities of the Hamilton-Jacobi type through simpler, nite-dimensional relations.
The broader impact of this research indicates a class of important applied problems solvable to the end through mathematics of control; improves existing computational methods for feedback control
under complete and incomplete information and set-membership uncertainty; indicates new approaches to specific classes of systems (bilinear, with sampled measure- ments, impulse, etc.) and new
techniques for existing approaches like model-predictive control; triggers new development of observer-based control in complex systems (hybrid, time-lag, communication, multiagent, etc), which
typically operate under incomplete feedback and set-membership uncertainty. | {"url":"https://grantome.com/grant/NSF/DMS-0807771","timestamp":"2024-11-02T13:53:39Z","content_type":"text/html","content_length":"29331","record_id":"<urn:uuid:a1198804-0a74-433d-9a6d-83164493859a>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00353.warc.gz"} |
Efficient Tuition Fees, Examinations, and Subsidies (new title: Efficient tuition fees and subsidies)
An electronic version of the paper may be downloaded
• from the SSRN website: http://SSRN.com/abstract=551424
CESifo Working Paper No. 1189
We assume that students can acquire a wage premium, thanks to studies, and form a rational expectation of their future earnings, which depends on personal "ability". Students receive a private, noisy
signal of their ability, and universities can condition admission decisions on the results of noisy tests. We assume first that universities are maximizing social surplus, and contrast the results
with those obtained when they are profit maximizers. If capital markets are perfect, and if test results are public knowledge, then the optimal tuition fee is greater than marginal cost, and there is
no sorting on the basis of test scores. Students optimally self-select as a result of pricing only. If capital markets are perfect but asymmetries of information are bilateral, i.e., if universities
observe a private signal of each student's ability, or if there are borrowing constraints, then the optimal policy involves a mix of pricing and pre-entry selection on the basis of test scores.
Optimal tuition can then be set below marginal cost, and can even become negative, if the precision of the university's private assessment of students' abilities is high enough.
JEL classification: H42, I22, J24, D82.
Keywords: tuition fees, examinations, state subsidies, higher education, incomplete information.
Robert J. Gary-Bobo University de Paris I Pantheon-Sorbonne, TEAM Maison des Sciences Economiques
106-112, boulevard de l’Hopital 75647 Paris, cedex 13 France [email protected] Alain Trannoy EHESS, GREQAM 2, rue de la Charité 13002 Marseille France [email protected]
1. Introduction
The public universities' state of ¯nancial crisis is an acute problem in many countries. Pub-lic funds shortages being recurrent, the idea that tuition fees could be increased naturally arises, but
is understandably met with ¯erce resistance of citizens. In the United States, many public schools have faced the hard choice of either cutting educational spending and quality, or increasing price,
and tuition has gone up in the recent years (see Winston (1999), and his references). In Europe, the situation is almost tragic. A journalist recently described Britain's universities as
"depressingly threadbare, overcrowded and politicized"1; it seemed to us |as Frenchmen and university teachers|, that this statement is close to being an excellent description of the current state of
our country's (mostly public) univer-sities. To be more precise, we think that in French universities, poverty and bureaucracy are a cause of demoralization. Tony Blair's plans to let tuition fees
increase are hugely unpopular2[. In France, the question is still an absolute political taboo, although many]
feel that an evolution towards university decentralization and a form of pricing is almost inescapable. Tuition reform has already started in Germany, and is currently discussed in other countries as
well. Yet, it seems that the formal economic theory of university pricing, the question of the optimal balance of fees and subsidies has not been studied with enough precision.
The present article proposes an approach of optimal fees, as well as an analysis of university behavior. In doing so, we pay special attention to informational asymmetries. University policies are
examined under opposite assumptions; we assume ¯rst that univer-sities are non-pro¯t institutions, and contrast the results with those obtained under the assumption that they are rent-seeking, or
"for-pro¯t" organizations. The possibility of regulating, or providing incentives to rent-seeking universities, as well as the relevance of normative "marginal cost pricing" theories is also
considered in the latter case.
In our model, heterogeneous students can acquire a wage premium, thanks to studies. They form a rational expectation of their future earnings, and apply for higher education on the basis of this
forecast. The expected wage premium depends on the university's quality, the number of graduates, and on the student's personal "talent", or "ability". Informa-tion is incomplete, in the sense that
neither the university, nor the student, can directly observe talent. Students receive a private, noisy signal of their ability, and universities can condition admission decisions on the results of
tests, or entrance examinations. The cases in which test results are, and are not publicly disclosed are both analyzed. Higher educa-tion is costly, total cost depending on quality, quantity, and the
average ability of recruits (the well-known peer e®ect). A university has the right to set fees, and to set admission standards, in the form of a minimal grade or test score. It also chooses a
quality variable and total enrollment. Non-pro¯t universities are assumed to choose their policy in order to maximize social surplus. This provides us with a useful benchmark. The non-pro¯t managers
have no concern for equity, or no aversion for inequality, so that our results will depend on e±ciency considerations only. In contrast, the rent-seeking university simply
The Economist (2003a) 2 [The Economist (2003b)]
maximizes an expression of pro¯t, that is, tuition revenues minus costs, which provides a clear description of the other extreme.
We consider both the case of "perfect capital markets", and the situation in which there are borrowing constraints, with the consequence that talented students from low income families can be barred
from higher education by tuition.
Our results are the following. First, if capital markets are perfect, and if test or entrance examination results are public knowledge, and rational students condition their application decisions on
private as well as public signals, then, an optimal policy involves a positive fee, and no sorting on the basis of test scores. Students optimally self-select as a result of pricing only. In
addition, the optimal tuition fee is greater than marginal cost. Second, if capital markets are perfect but asymmetries of information are bilateral, in the sense that universities observe a signal
of each student's ability which is not disclosed or not taken into account by students, then, the optimal policy of a non-pro¯t university involves a mix of pricing and pre-entry selection on the
basis of grades or test scores. The optimal policy can then entail a direct student subsidy: optimal tuition can be set below marginal cost, and can even become negative, if the precision of the
university's private assessment of students' abilities is good enough. This result is not due to redistribution or equity motives on the part of the benevolent university; it is driven solely by
e±ciency considerations.
Third, the rent-seeking university's policies are ine±cient: tuition fees tend to be too high and admission standards tend to be too low. An incentive transfer schedule, depending on enrollment and
knowledge of the wage distributions, can fully correct the ine±ciencies due to rent-seeking or for-pro¯t behavior.
Fourth, when students face borrowing constraints, even if test scores are publicly dis-closed, the university's optimal policy must be a combination of pricing and selection, with the possibility
that, again, tuition be set below marginal cost, to alleviate the ine±ciencies due to the fact that some good students are deterred by price. Again, this result is not due to assumed redistribution
motives of the benevolent university manager; it follows from e±ciency, surplus maximization considerations only.
To sum up, we ¯nd that university pricing is a socially e±cient policy, but that it should be mixed with pre-entry selection of students, either because the university has private information on
students' abilities, or because, due to ¯nancial market imperfections, some students face borrowing constraints, or both. Now, the optimal tuition fee can be optimally set below marginal cost, and
will be a decreasing function of the entrance examination's precision as a signal of student ability. The more accurate entrance exams are, the larger the discount on tuition.
To arrive at these results, we had to choose a university objective function. This is problematic, and every choice is open to criticism3[. Winston (1999) remarkably ]
summa-rizes the intuitions, and provides a non-technical description, of university economics. We have kept his vision of the "industry" in mind and arrive at results which, we think, do not
contradict his observations. Our choice has been to "cut the Gordian knot" and to consider two extreme, probably equally unrealistic cases: the purely benevolent, and the
A few contributions have been devoted to a formal microeconomic analysis of universities, but the topic seems underdeveloped; see, for instance, Garvin (1980), Borooah (1994).
purely greedy, for-pro¯t (or rent-seeking) university managers4. The perfect competition case has been studied by Rothschild and White (1995), who emphasize the important idea that students are
inputs in the production of their own human capital. In the present contribution, the university is endowed with market power. This could represent a private university with a substantial market
share or a leadership position, but also a dominant network of public universities. The industrial organization of the higher education sec-tor remains di±cult to model, and in particular, the
various forms of price and non-price competition among universities are still very much unexplored5.
We assume the existence of peer e®ects, in the sense that the average ability of en-rolled students increases the quality of education for a given expenditure, or equivalently, decreases total cost
for a given quality objective. The theory of peer group e®ects, or local interactions in education has been studied, among other contributions, by Arnott and Rowse (1987), de Bartolome (1990),
Benabou (1993), and Epple and Romano (1998). There have been many attempts at testing for the presence of peer-group e®ects in educa-tion, from school to college, since the Coleman (1966) report;
e.g., among recent contribu-tions, Betts and Morell (1999), Hoxby (2000), Sacerdote (2001), Angrist and Lang (2002), Rothstein (2002), Zimmerman (2003). The common view is that these e®ects are
impor-tant in higher education, even if is di±cult to estimate their magnitude. All our results would be qualitatively the same if peer group e®ects were negligible. Yet, our analysis permits one to
see how they intervene in the determination of optimal university pricing and enrollment. They would probably become crucial in theories of strategic interactions among universities.
Our representation of university technology comprises a quality variable. Quality can be viewed as an index aggregating particular expenditures and teachers' e®orts, including the teacher's endeavor
to stimulate student e®ort. To keep the model relatively simple, we did not introduce a moral hazard (i.e., hidden e®ort incentives) problem explicitly in the analysis. Recent empirical studies show
the importance of education quality on future wages (e.g., Card and Krueger (1992), Angrist and Lavy (1999)), and the e®ect of teacher incentives on student achievement (e.g., Lavy (2002)).
Finally, our analysis of optimal pricing is not independent of an economic theory of examination procedures. Important pioneering work on the economic theory of exams is due to Costrell (1994) and
Betts (1998). Our philosophy is closer to that of Fern¶andez and Gal¶³ (1999), and Fern¶andez (1998). The latter work of Fern¶andez presents a simpli¯ed version of the model analyzed in the former
contribution. In this paper, student population is described by a joint distribution of ability and wealth. The problem is to allocate students to high quality, or to low quality schools, knowing
that high quality school capacity is ¯xed and that student ability and school quality are complementary inputs in the production of future earnings. For e±ciency reasons, high ability types should be
allocated to high quality schools. A costly (and socially wasteful) test technology can be used to decide which students will be admitted to high quality schools. Each student can produce a given,
deterministic test result at the personal cost of a given amount of e®ort (which varies with
4 [In our model, rent-seeking and pro¯t maximization behaviors are formally equivalent.] 5
However, Del Rey (2001) and De Fraja and Iossa (2001), study asymmetric duopoly models; Gary-Bobo and Trannoy (2002) contains an attempt at modelling monopolistic competition.
ability). Fern¶andez (1998) then compares a publicly regulated, test-based allocation system with a competitive equilibrium allocation in which schools set prices. Tests and prices are equivalent
when students can borrow against future income. We ¯nd the same equivalence of tests and fees as screening devices in Section 2 of the present paper, in a setting which is di®erent, but essentially
related. She then shows that markets and exams are not equivalent when students cannot borrow. Examination-led allocations dominate market equilibria because, under zero-borrowing, some able-but-poor
students take the place of some wealthy-but-less-able students, which improves the overall matching of students to schools, and therefore, aggregate output. The main di®erences of our work with that
of Fern¶andez consist in introducing incomplete information under more radical forms: (students observe noisy signals of their ability, test results are random), in comparing various forms of
informational asymmetry, (unilateral and bilateral), and studying variable enrollment and education quality. With the addition of these elements, we show under which circumstances optimal policies
should balance selection on the basis of grades and self-selection by means of the price mechanism; we compute the optimal tuition and show how the amount of subsidy (or tuition rebate) is related to
the informational properties of the examination technology; we ¯nally sketch the analysis of the role of borrowing constraints in the determination of tuition rebates.
In the following, Section 2 presents basic assumptions and an analysis of our model under complete information assumptions. Section 3 presents the results in the asymmetric information setting.
Section 4 is devoted to a variant of the model in which informational asymmetries are bilateral, and Section 5 analyzes the impact of borrowing constraints on the optimal policy.
2. Optimal Tuition Fee and Admission Policy under Complete Information
2.1. The Skill Premium
Our point of departure is a formalization of the skill premium which can be earned by means of higher education. We suppose to simplify the analysis that there exists a university (or college), with
a single department, exerting some form of monopoly or market power as a provider of higher education and skilled workers. Let the potential student population be of size N. Each student is
characterized by a "talent" or "ability" variable, denoted
µ. In the present section, ability is observed by the student, and in the next one, ability is not observed, neither by the university, nor by the student, but the student receives a noisy private
signal relative to her (his) own ability. The complete information setting of the present section can be viewed as a limiting case of the more realistic, incomplete information model to be studied
The university and public authorities are assumed able to estimate the probability distribution of ability. We suppose that there exist two categories of workers only, the skilled, who are graduates
from the university (or college), and the unskilled, who did not study. The unskilled workers' wage rate is a constant w0 and the university (or college)
graduates' wage isw. The future wagewof a skilled worker is random, because it depends on the individual's ability; we suppose that the following relation holds,
where ¢, the skill premium, is a function of the number of graduates, denotedb x, (it depends in fact on the ratio of skilled over unskilled workers in the economy), and of the quality of studies
provided by the university, denoted e.
Students do not exert any personal e®ort, and all enrolled students receive a diploma, for simplicity. It must however be clear that ^µ captures the fact that graduates are not all equal, due to
di®ering talents, which are re°ected in more or less brilliant grades, as well as more or less lucrative perspectives on the labor market. Therefore, the assumption that all students obtain a degree
does not preclude student heterogeneity. We assume that the skill premium is decreasing with respect to x, and increasing with respect to
e (and continuously di®erentiable with respect to both variables); these are reasonable assumptions, compatible with a general equilibrium model in which skilled and unskilled labor are used as
inputs in the production process.
With the help of these assumptions, we can derive the demand for higher education, that is, determine the number of candidates for registration. University policy is charac-terized by three
variables: the number of graduates x (equal to the number of enrolled students in equilibrium), the quality of studies e, and the tuition fee, denoted p. Potential students decide to apply for
registration in view of these three variables. An important, and probably strong |although very common| assumption is that all agents observe and (or) form correct expectations about the skill
premium¢ (for a discussion of this approach,b e.g. Manski (1993)).
The student's preferences are assumed to be represented by the same inter-temporal, in¯nite horizon, additively separable utility function. Higher education takes place in the ¯rst period of the
student's life-cycle, at time t = 0. Utility is assumed to be quasi-linear with respect to consumption att= 0. Formally, for a consumption pro¯lec= (c0; c1; c2; :::)
we de¯ne utility u(c) as follows,
u(c) =c0+ 1 X t=1 ®tln(ct); (2) where ®= 1 1 +r (3)
is the discount factor associated with a psychological interest rate r > 0, used by agents to discount future utility.
2.2. Number of Candidates for Registration (Demand)
To simplify the analysis, we assume that a worker's wage is constant during her (his) entire working life. Agents do not save, and consume their wages, which are expressed in real terms. With the
help of these assumptions, an agent who doesn't study remains an unskilled worker for life; his or her utility can be written, using (2) and (3),
u0 =w0+
Using (1), (2) and (3), if the agent chooses to study, her expected utility, conditional on ability µbcan be written,
uµ =¡p+ ln(w0) r + b ¢ r + b µ r; (5)
since the student is not working at time 0, and pays p as a tuition fee. Introducing a capitalized skill premium, as well as a capitalized value of ability,
¢(x; e) = ¢(b x; e)
r ; µ =
r; (6)
we get the following expression: uµ = ¡p+ ln(w0)=r+ ¢ +µ. An individual with ability
µ chooses to study if and only if uµ ¸u0, that is, if and only if,
µ [¸]y[´]p+w0¡¢; (7)
where y is an ability threshold above which agents do apply for registration at the univer-sity. This threshold depends on p and ¢ and thus on the values of (e; p; x) (the quality of studies, tuition
fee and total number of graduates), since ¢ is a function of (x; e).
Let us denote by F the cumulative probability distribution of µ. In the following, we assume that F is continuously di®erentiable with a density denoted f. With the help of these notations, the
demand facing the university isN (1[¡]F(y)).
Now, we want to allow for the possibility of student selection by the university. Let us assume for the time being that ability is observable, and that the university chooses an ability threshold,
denoted z, below which student applications are rejected, that is, only the µ [¸]z are accepted. Thus, "e®ective" demand, which is the combined product of screening and pricing, depends on p and z.
Formally, let ¯rst
t =max[f]y; z[g]: (8a)
E®ective demand, denoted q, is a function oft de¯ned as
q(t) =N(1[¡]F(t)): (8b)
2.3. University Teaching Technology and Peer Group E®ects
Our model will be fully speci¯ed if we describe higher education costs. We would like to capture the idea that the often discussed "peer group e®ects" can be present and a®ect the quality of
education for a given cost, or equivalently, a®ect the cost of providing a given quality of education. To do so, we assume that there exists a continuously di®erentiable university cost function,
denoted C, depending on chosen education quality e, on the number of enrolled students x, and on average student ability, denotedv. In other words, total university cost is given byC(x; e; v), wherev
is the expected value of ability, knowing that abilities µ [¸] t are enrolled by the university. We provide a formal expression for v
below. It is natural to assume that C is increasing with respect to x and e, and non-increasing with respect to v. This assumption, which is not the most general way of modelling the fact that
students are inputs in the production process of their own human capital (e.g., Rothschild and White (1995)), nevertheless captures the essential idea behind peer e®ects.
At this stage, it will probably clarify the discussion if we show that the model described above is formally equivalent to one in which the the peer group e®ect directly a®ects the skill premium
through average student ability v. To see this, assume that the job market values worker quality, denoted e, and that worker quality is produced with the help of teacher e®ort ´ and peer e®ects so
that e = Á(´; v), where Á is a kind of production function, which is increasing with respect to ´ for all v. Now, if the teaching technology is represented by a cost function depending on teacher
e®ort and the number of students only, that is, if C = ~C(x; ´), and if the skill premium depends on quality and the number of graduates only, that is, ¢ = ¢(x; e), then, the latter can be expressed
as a function of e®ort ´ as follows,
¢(x; ´; v)[´]¢(x; Á(´; v)):
This shows reasonable conditions under which the approach in which the skill premium would be a function of average peer ability is formally equivalent to one in which average peer ability is an
argument of the university cost function. It happens that the latter approach is much simpler than the former from the technical point of view, as will be seen below.
2.4. The Philanthropic View and Social Optimum
We de¯ne a higher education institution as "philanthropic" when its objective is to max-imize the social value, or social surplus, of its education activity. It is of course a strong assumption to
assume that a university is philanthropic, but this approach will provide us with a very clear benchmark, equivalent to the idea of Pareto optimum. Since utilities are quasi-linear, social surplus
maximization yields e±cient policies and allocations. In the following, we will of course contrast this view with other, less optimistic theories of the university objective, that will be called
"cynic views".
It is now possible to compute the optimal policy of a philanthropic university. The university will be ¯nanced by subsidies (or by donations), in the case of a de¯cit. The required amount of public
resources (or donations) is simply,
D=C(x; e)[¡]px: (9)
The amount D will be subtracted from social surplus to balance the university budget at time 0. We therefore suppose that the share of total cost that the students of a given cohort do not pay for in
the form of tuition fees will be paid in the form of (lump-sum) taxes, or contributions, made by the same or other other agents, such as alumni donations, etc. There are of course more subtle
relations between public pricing, public subsidies voluntary contributions, and the tax system, involving political economy and redistribution problems which will not be studied in the present
analysis. In particular, we assume that
the social justice problems are solved by means of other redistributive tools, in the hands of independent public authorities. More precisely, we assume that equality of opportunity problems are
solved in the sense that no student with the necessary talents is barred from studying because of a ¯nancial constraint. We are perfectly aware of the fact that this picture is a bit too rosy, and
that more sophisticated modelling work on the case in which imperfections of credit markets and family backgrounds di®erences create unequal opportunities is needed. We address the borrowing
constraints question in the ¯nal section of this paper. In the present section and the following one, our analysis aims at showing the simple structure of the optimal higher education pricing problem
in a pure e±ciency case, which can again be seen as a benchmark. Under the assumptions made above, the philanthropic objective (social surplus) can be de¯ned as follows,
W =xE(uµ jµ ¸y; µ¸z) + (N ¡x)u0+px¡C(x; e; v); (10)
and can be interpreted as the total sum of the x graduates' expected utilities, and of the remaining (N [¡]x) unskilled workers. If the expressions of utilities uµ and u0 given by
(4)-(6) are substituted in W, we get, after some straightforward simpli¯cations,
W =x(¢(x; e)[¡]w0+v(t))¡C(x; e; v(t)) +N u0; (11) where, by de¯nition, v(t) =E(µ [j]µ [¸]t) = R1 t µf(µ)dµ 1[¡]F(t) ; (12) is the mean ability of an individual, knowing that this individual is a
student, and t =
max(y; z). Thus, W is the di®erence between total student or skilled worker productivity,
x(¢ +v), and total opportunity costs xw0 plus total direct costs C of higher education.
To determine the optimum, the social surplus (11) must be maximized under the constraints that the number of enrolled students cannot be higher than the number of screened applications, that is, x[·]
q(t) with t=max(y; z) and y=p+w0¡¢.
The philanthropic faculty will always be able to close the gap betweenxandq(t) with the help of an increase of the tuition fee p or of the pass-threshold z ifx < q(t), because
q is a decreasing function of t. Thus x = q(t) at the optimum. To see this, suppose that
z > y and that x < q(z). Then, locally, dW=dz = (x[¡]Cv)v0(t), where Cv is the partial derivative of C with respect to v. But, by assumption, Cv <0 and
v0(t) = f(t)
1[¡]F(t)(v(t)¡t)¸0; (13) because v(t) =E(µ [j]µ [¸]t)[¸]t, with a strict inequality if t is not the highest value in the support off. It follows thatdW=dz >0 and the university should increasez.
Suppose now that y > z and x < q(y). Then locally, for the same reasons as above, dW=dp=dW=dy= (x[¡]Cv)v0(y) > 0, and the university should increase p. As a consequence, it is possible to set x =q
(t), and to invert the e®ective demand curve. We then obtain the expression of an ability-threshold t¤, or marginal ability, which is a function of x. Inverting (8b), we get,
t=t¤(x)[´]F¡1³1[¡] x
giving the value of t for which exactly x individuals with abilities greater than t apply for registration at the university.
It is then useful to remark that,
xv(t¤(x)) =x R1 t¤[(][x][)]µf(µ)dµ x=N =N Z 1 t¤[(][x][)] µf(µ)dµ; (15) and for future reference, using Leibniz's rule, we get
d dx[xv(t ¤[(][x][))] =][t]¤[(][x][)][:] [(16][a][)] and dv(t¤(x)) dx = µ 1 x ¶ [t¤(x)[¡]v(t¤(x))]<0 (16b) In order to derive the necessary conditions for an optimum, we now maximize the following
function with respect to (x; e),
W(x; e) =x(¢(x; e)[¡]w0) +xv(t¤(x))¡C[x; e; v(t¤(x))] +N u0; (17)
Let us denote Cx, ¢x, Ce, ¢e the partial derivatives of C and ¢ with respect to x and e respectively. The necessary conditions for optimality are,
x¢x+ ¢¡w0+t¤ =Cx+Cv µ 1 x ¶ (t¤[¡]v(t¤)); (18) x¢e =Ce: (19)
Conditions (18) and (19) are easily interpreted. Equation (18) says that the marginal social value of a graduate must be equal to its marginal cost, while equation (19) says that the marginal social
value of qualityx¢e must be equal to its marginal costCe, at the optimum. The marginal value of a graduate is the sum of two terms: the graduate's marginal ability
t¤, and the marginal "social revenue" of the skills produced by the university, that is,
x¢x + ¢¡w0: A negative term x¢x appears in the latter expression (since ¢x < 0 by assumption), and expresses the fact that an additional skilled worker lowers the wage of all the other graduates on
the labor market. At the optimum, the university must take this e®ect into account and should not °ood the market with too many skilled workers. The marginal cost is itself the sum of two terms: the
¯rst is the direct marginal cost
Cx, the second, which is also positive (being the product of two negative terms) is the marginal "peer e®ect". Increasing x by one unit reduces the average quality of students as shown by (16b);
hence, it reduces the peer group e®ect, which increases the cost by (dv=dt)(dt¤[=dx][)][C]
The optimal allocation is basically expressed in terms of x, e and t. The remaining part of the analysis is mainly a problem of implementation. Some form of student screening takes place at the
optimum, since the optimal t is not equal to its smallest possible value; then, do we wish the optimal screening level to be implemented by means of a "merit list"
(i.e., by the entrance admission-threshold z) or by "means of money" (i.e., the tuition p)? An increase of the tuition fee p will increase the average quality v of students equally well as an
increase of the admission threshold z, these tools are perfect substitutes. This is true if students and the university perfectly observe the ability levels. We will see that this conclusion does not
hold fully in an incomplete information version of the model, although money and merit may remain imperfect substitutes.
In welfare terms, given our utilitarian formulation of the social surplus, and the quasi-linearity of preferences, the tuition p has no e®ect on welfare, apart from its role in de-termining the
ability of the marginal enrolled student t¤[, that is, its role as a screening]
tool. The tuition has redistributive e®ects, but they do not matter under quasi-linearity. Again, this conclusion would not hold in a world in which informational and ¯nancial imperfections play an
important role, as will be seen below.
Formally, equations (18) and (19) determine the optimal values (x¤[; e]¤[), from which]
we derive t¤¤ [´]t¤(x¤), the optimal ability of the marginal student. In addition, we know that t¤¤[=] [max][(][y]¤[; z]¤[) for some pair (][y]¤[; z]¤[). The relation of] [y]¤ [with the optimal
tuition] p¤ is derived from (7) above, and given by the formula,
p¤ =y¤ + ¢(x¤; e¤)[¡]w0: (20)
Implementation of the optimum by means of tuition means choosing y¤ = t¤¤ > z¤ and choosing the tuitionp¤so that (20) holds withy¤ [´]t¤(x¤). Implementation of the optimum by "merit" is tantamount to
choosing z¤ = t¤¤ > y¤ and choosing the tuition p¤ so that (20) holds.
It follows that there are two possible pricing regimes in our complete information model: one in which tuition matters because it determines demand (locally) and one in which sorting according to
ability makes tuition ine®ective as a screening device. In the second regime, one can say that the usual interpretation of some observed facts holds: tuition is set at a deliberately low level to
create an excess demand, which aims at facil-itating the selection of good students by the university. We are now equipped to clarify the meaning of the often discussed application of "marginal cost
pricing" to universities.
2.5. "Marginal Cost" Pricing and Optimality
At this stage, it should be noted that implementation by means of a tuition fee does not require any observation of the admitted student's abilities: it would therefore also work under conditions of
asymmetric information, when ability µ is a private information of the applicants. Under these informational conditions, the only thing that the university authorities need to know is, as usual, the
distribution of abilities F. The university can then implement an optimum by setting p¤ appropriately (so that t¤¤=y¤ of course), and students will self-select according to the optimal screening
rule, since only abilities greater than t¤¤ will apply. In this case, introducing an entrance examination procedure would produce information on abilities, but would presumably be costly. Examination
costs can then be saved by the philanthropic university manager, insofar as redistribution e®ects do not matter, because pricing alone does the entire job of implementing the optimum.
Consider now the optimal tuition in this case. Using (18), (20), and y¤ =t¤(x¤), we easily derive, p¤ =Cx+Cv^vx¡x¤¢x: (21a) where, using (16b), ^ vx ´ t¤(x)[¡]v(t¤(x)) x <0: (21b)
Since x¢x · 0, Cx > 0, Cv < 0 and ^vx < 0, we ¯nd that p is unambiguously positive. Thus, we can state the following.
Result 1. If abilities are private information of the students applying for higher education, tuition alone can be used to screen applicants and to implement a social optimum. In this case, at any
interior optimum, tuition is greater than the university's marginal cost of a student.
When abilities are not observable, tuition fees are used to discourage the students whose abilities lie below the socially optimal threshold. These fees are not just a token, since they must then
cover the marginal cost of education (including the marginal peer e®ect). De¯ne now, the university ¯xed cost
K(e; v) = lim
x!0+C(x; e; v); (22)
which depends on the chosen quality of studies e, and possibly on the average ability of recruits v, and assume that marginal cost Cx is increasing with respect to x. From (21a), multiplying by x and
subtractingC, we get,
px[¡]C =xCx+xCvv^x¡x2¢x¡C:
De¯ne the university rent as R=px[¡]C (this does not mean that the university will earn this "rent" at the optimum, because the social planner can tax it). Then, from the above expression, we get,
R+K =[¡]x2¢x+xCvv^x+xCx¡C +K;
and sinceC is assumed to be convex (because marginal cost is assumed to be increasing), we haveK[¡]C+xCx ¸0. The assumptions ¢x <0 andCv <0 therefore yieldR+K >0, or equivalently px[¡]C+K >0, and we
can state the following result.
Result 2. If abilities are private information of the students and if the university cost is convex, then, a social optimum is implemented by means of a tuition fee, and university revenues px must
be greater than variable costs C[¡]K.
It must then be the case that part of the university ¯xed cost will be covered by a public subsidy, or another source of revenue. This being said, we know that the optimal policy can be implemented
by a combination of tuition and sorting of abilities, and in this case, the above pricing formula has no reason to apply. In practice, universities do combine the
tools of selection by money and selection by merit, they engage in the costly production of information on abilities (by means of tests and entrance examinations), and students are subsidized (e.g.
Winston (1999)). These facts are more appropriately captured in an incomplete information framework. Before we start the analysis of the incomplete information case, we now contrast the "cynic" and
"philanthropic views". To this end, and for the sake of completeness, we analyze the behavior of a rent-seeking (or purely for-pro¯t) university under conditions of complete information.
2.6. The Cynic View: Rent-Seeking and For-Pro¯t University
We now study the behavior of a deregulated rent-seeking or for-pro¯t university. The university is deregulated in the sense that it is free to choose total enrollment x, tuition fee p, quality e, and
the admission threshold z. The university is also endowed with market power; this is an approximation for a situation in which, say, a centrally governed public network of universities has a
quantitatively important share of the higher education market, but the model could as well capture the behavior of a private university with a dominant position.
The cynic view assumes that the university (the faculty) seeks to maximize their rent
R=px[¡]C(x; e; v(t)); (23) with respect to (e; x; p; z), subject to the constraints x [·] q(t), t = max(y; z) and y =
p+w0 ¡¢(x; e), e®ective demand q being de¯ned by (8b). This rent can be understood
as the amount of resources that are made available to ¯nance faculty activities other than teaching. In academic systems in which teacher's careers essentially depend on research achievement, it is
likely that research will be a major faculty objective (although there is of course no guarantee), so that R might as well stand for research. But R can of course easily be interpreted as pro¯t, and
our model is then that of a for-pro¯t university.
It can again be shown that x = q(t) at the optimum. Suppose that z < y, then
t = y, and an increase of p yields dR=dp = dR=dy = x[¡]Cvv0(t) > 0, so that p should be increased until q(t) equals x. Suppose now that z > y, then t = z, and an increase in z yields dR=dz = [¡]Cvv0
(t) > 0, so that z should be increased until q(t) equals x. Substituting q(t) = x and p = y + ¢[¡] w0 in the expression for rent (23) yields the
max(e;y;z)f(y+ ¢(q(t); e)¡w0)q(t)¡C[q(t); e; v(t)]g (24)
subject to t = max(y; z). Finally, z (and the constraint de¯ning t) can be eliminated from the optimization program as follows. To solve the problem, it is su±cient to choose
t and y subject to t [¸] y; if t > y at the optimum, then t = z and if t = y then z can be arbitrarily chosen so that y[¸]z at the optimum. The rent-seeking university therefore wants to maximize,
R(e; t; y) =q(t) [y+ ¢(q(t); e)[¡]w0]¡C(q(t); e; v(t)); (25)
subject to the constraint t[¸]y. Let now¸ be the constraint's Lagrange multiplier. Kuhn and Tucker's necessary conditions for constrained optimality can be written:
q(t)¢e=Ce; (26b)
q(t) =¸ and ¸(t[¡]y) = 0: (26c) Conditions (26) immediately yield an important qualitative result: since q(t) = ¸ > 0, they imply y =t, and the rent-seeking university will not use pre-enrollment
selection to sort students; the active screening variable is the tuition fee. Using q0=q = [¡]f =(1[¡]F) andv[t]0 =[¡](q0=q)(v[¡]t) to rearrange terms in (26), we get the following rent-maximization
conditions, x¢x+p=Cx+Cv µ y[¡]v(y) x ¶ + µ 1[¡]F(y) f(y) ¶ ; (27a) x¢e =Ce; with (27b) y =t¤(x); and p=t¤(x) + ¢[¡]w0: (27c)
A direct comparison of (27a)-(27c) with (18)and (19) above immediately shows that the rent-seeking university does not choose an optimal allocation (e; x; p; z). The rent-seeking university does not
enroll enough students. Since it behaves as if marginal cost was higher by (1[¡] F)=f, they will typically set enrollment x below its optimal level. It follows that they will choose a tuition level
which is too high with respect to the philanthropic optimum (conditional on chosen qualitye). Quality will be higher or lower than its socially optimal value, depending on the sign of the cross
second-order partial derivative ¢xe. Equations (19) and (26b) show that the rent-seeking university will choose an optimal level of quality e if and only if it chooses an optimal level of enrollment
x. These properties hold under complete information conditions, as well as if abilities are private information of the students. We can state the following result.
Result 3. When compared with the philanthropic university, the rent-seeking university (a), chooses a sub-optimal level of enrollment, and (b), tuition is their only screening device and they do not
sort students according to ability.
A comparison of the cynic objective (24) with the philanthropic objective (17) immediately shows that the cynic solution will typically not be socially optimal. The di®erence between the two
objectives stems from the fact that the rent-seeking university prices according to the last enrolled student's ability (or marginal ability t¤(x)), while the philanthropic man-ager takes the total
value of student abilities N R[t]1¤[(][x][)]µf(µ)dµ (or xv(t¤(x)) into account.
There is a close formal analogy between the complete information version of our model and the model of a monopoly choosing both quantity and quality, as studied by Spence (1975) and Sheshinski
The distortions caused by rent-seeking behavior can easily be corrected: it would be su±cient to tax the marginal value of ability and simultaneously, to pay a subsidy equal to the total value of
ability. We can state the following result.
Result 4. The rent-seeking university chooses a socially e±cient policy if it is subject to a public money transfer T(x), depending on enrollment only, and de¯ned as,
T(x) =[¡]xt¤(x) +N
Z [1]
µf(µ)dµ+constant; (28)
where t¤(x) is de¯ned by (14).
It is remarkable that the incentive money transfer T depends on the number of graduates
x, and on distributionF only | becauset¤ itself only depends on F. It is also remarkable that cost observations are not needed to regulate the rent-seeking university. Enrollment
x is in principle observable by the public regulator, and the ability distribution F can be estimated by an econometrician, by means of the Mincerian regression function (1), which was our point of
departure. If the cynic view is correct, a public regulator of universities should collect data on students' wages and careers and perform some econometric work, to correct the distortions caused by
rent-seeking behavior. Even though unbiased estimates of a regression function function like ¢(x; e), are more di±cult to obtain than it would seem at ¯rst glance (e.g. Card (1999), Harmon,
Oosterbek, and Walker (2003)), given the enormous amount of empirical work devoted to the topic, returns-to-education economet-rics is nowadays common practice. In our simpli¯ed model, it is easy to
see how two-step methods µa la Heckman (e.g. Heckman (1978), Willis and Rosen (1979)) can be used to estimateF and ¢. Assume that ¢ is linear andeis constant. We get the regression equa-tion,ln(wi=
w0) =a+bx+µi, wherei indexes individual observations. We neglect possible controls such as age and experience to simplify the discussion. Given that we observe students only, the expected value of µi
is not zero. We get instead E[µi j µi > y] = v(y), wherey=p+w0¡bx¡a. Two-step methods (or, of course, standard maximum likelihood
techniques) can be applied to estimate the regressionln(wi=w0) =a+bx+v(y) +´i, where
E(´i) = 0.
3. Asymmetric Information and Entrance Examinations
Let us now come back to the study of philanthropic and cynic views of university, but under less restrictive assumptions relative to the information of the students, public regulator, and Faculty.
Our fundamental assumption will now be that students do not observe their own ability µ; they are endowed with incomplete knowledge of their own talent, formed by means of noisy informative signals.
Students are assumed to observe a private signal of ability; more precisely, they observe s, where, by de¯nition,
s=µ+"; (29)
where abilityµ and", a zero-mean noise, are assumed to be independent random variables6. In addition, a costless examination technology provides an estimation of ability which is publicly observable.
The examination grade is a random variable denoted z, and is de¯ned as follows:
z =µ+º; (30)
where º has a zero mean and is independent from µ and ". To simplify the exposition, we assume that the support of µ, º and " is the entire real line, and thatµ has a ¯nite mean
The grade z is known to the student and to the Faculty (i.e., the university authori-ties). This examination can be interpreted as a national ¯nal high school exam, (such as baccalaur¶eat in France,
or Abitur in Germany), or z can be viewed as an entrance test score. The university can set a pass mark ¹z; a student is then admitted for registration only if his or her gradez is greater than ¹z.
The value of higher education is now expressed in expected terms, conditional on the two signals (s; z). Let u1 be the expected utility of
higher education,
u1 =¡p+
r + ¢ +E(µ js; z);
where variables, p, w,r, ¢, have the same meaning as in the above section. The utility of a non-educated worker is still
u0 =¡w0+
r :
An individual applies for higher education if and only if u1 ¸ u0, that is, equivalently, if
and only if,
E(µ [j]s; z)[¸]y¹[´]p+w0¡¢(x; e): (31)
Let us de¯ne
y[´]E[µ [j]s; z]: (32)
If we assumed thatµ,"andº are normal, then,ywould itself be a normal random variable. Let ¾[µ]2,¾["]2, ¾2[º], be the variances ofµ, ", andº, respectively. Some computations, using the normality
assumption would then yield the classic result,
E(µ [j]s; z) = s¾ 2 µ¾º2+z¾2µ¾"2+¹¾º2¾"2 ¾2 µ¾º2+¾µ2¾"2+¾º2¾"2 : (33)
This expression being linear with respect tos andz, we have,y[´]®s+¯z+°¹ where the values of ®, ¯, and° are de¯ned by identifying the latter expression with (33).
The random signal y can be interpreted as the expected ability of an individual, knowing her private signal and her test score or examination grade; it is also the student's rational expectation of
her own ability. With this speci¯cation, a student is enrolled if she is willing to apply and if she satis¯es the requirements of the entrance selection process based on z, that is, if and only if,
z [¸]z¹ and y[¸]y:¹ (34)
E®ective demand can then be written,
x=q(¹y;z¹) =NPr(y[¸]y; z¹ [¸]z¹); that is, q(¹y;z¹) =N Z 1 ¹ z Z 1 ¹ y ¹ Ã(y; z)dydz; (35) where ¹Ã is the joint normal density of (y; z).
3.1. The Philanthropic View: Optimal Examination and Tuition Fees
Expected social surplus can be written,
W =xE[u1 jy¸y; z¹ ¸z¹] + (N ¡x)u0+px¡C:
v(¹y;z¹)[´]E[µ [j]y[¸]y; z¹ [¸]z¹]: (36) Function vis the expected ability of enrolled students, knowing that their grade is greater than ¹z and that their private assessmentyis greater than ¹y.
After some simpli¯cations, we get,W =x[¢(x; e)[¡]w0+v(¹y;z¹)]¡C[x; e; v(¹y;z¹)] +N u0, and substituting the constraint
x=q(¹y;z¹) in the above expression yields the expression of social surplus, or philanthropic objective,
W(e;y;¹ z¹) =q(¹y;z¹) [¢(q(¹y;z¹); e)[¡]w0 +v(¹y;z¹)]¡C[q(¹y;z¹); e; v(¹y;z¹)] +N u0: (37)
W can be maximized with respect to (e;y;¹ z¹), instead of (e; p;z¹), given that ¹y=p+w0¡¢.
This maximization problem can be decomposed into two sub-problems. A ¯rst sub-problem is to maximize xv(¹y;z¹)[¡]C(x; e; v(¹y;z¹)) with respect to (¹y;z¹), for given (e; x). Given that
Cv · 0, this is tantamount to maximizing v(¹y;z¹) subject to x = q(¹y;z¹), with respect to (¹y;z¹), for ¯xed x. The necessary conditions for an optimal pair (¹y;z¹) (if it is ¯nite!) are simply vy¹
vz¹ = qy¹ qz¹ ; and x =q(¹y;z¹); (38) where subscripts denote partial derivatives. The interpretation of condition (38) is easy if it is reminded that, vy¹ =@v=@y¹= @v=@p, qy¹ =@q=@y¹=@q=@p; it says
that the marginal
rate of substitution between p and ¹z should equal its marginal rate of transformation, conditional on the ¯xed production targetx. The second sub-problem is then to maximize
W with respect to (x; e), given that the optimal (¹y;z¹) have been expressed as functions of
The determination of (¹y;z¹) is formally equivalent to the following problem. Assume that an examination procedure has two consecutive tests (say, Math and English), and that both tests are graded on
a numerical scale. Grades in Math and English are random variabless andz respectively. A student is admitted if a weighted averagey of both grades is greater than ¹y, and if | Math being considered
very important| the math grade is greater than ¹z. Our "¯rst sub-problem" above is formally equivalent to solving for the pair (¹y;z¹) which maximizes the expected ability v of admitted students,
given that a certain enrollment target x should be met.
To study the existence of solutions to the system of equations (38), we state two technical Lemmata. The ¯rst one is a key to what follows.
Lemma 1. y is a su±cient statistic for µ, and,
For proof, see the appendix
The next result will be useful for the study of equations (38).
Lemma 2. vy=qy ¸vz=qz is equivalent to hy(¹y;z¹)¸hz(¹y;z¹), where, by de¯nition,
hy(¹y;z¹)´E(µ jy= ¹y; z ¸z¹); (40a)
hz(¹y;z¹)´E(µ jy¸y; z¹ = ¹z); (40b)
For proof, see the appendix
Intuitively, Lemma 1 says that, being a statistically optimal combination of the two signals
s and z, y conveys all the useful (private and public) information about an individual's ability. This result has the following striking consequence.
Proposition 1. If the grade or test result z is publicly observed and s is privately observed by the students, then, the optimal (philanthropic) solution involvesz¹=[¡1], i.e., admission standards
are the lowest possible; optimal screening is performed by means of the tuition fee only. There does not exist a ¯nite solution of (38).
For proof, see the appendix
The meaning of Proposition 1 can be rephrased as follows. In a world in which economic agents are perfectly rational (i.e., if they are good enough statisticians), the university can safely rely on
student self-selection through the pricing mechanism only. An optimal tuitionp¤ is therefore the only useful tool, and selection by means of an admission standard is super°uous, provided that
students can assess their ability by conditioning on the publicly disclosed academic grade z.
Thus, according to the philanthropic view, social value maximization doesn't lead the university to make use of an optimal "policy mix" involving pricing and selection on the basis of test scores. We
study variants of our model leading to less radical conclusions in the following Sections. Let us now examine the pricing behavior of the philanthropic university, and the screening and pricing
policies of a rent-seeking, or pro¯t-maximizing, university.
Given Proposition 1, de¯ne ^ q(¹y) = lim ¹ z!¡1q(¹y;z¹) (41a) and ^ v(¹y) = lim ¹ z!¡1v(¹y;z¹) =E(µj y¸y¹); (41b)
and rewrite the philanthropic objective (37) as
to be maximized with respect to (e;y¹). The ¯rst-order conditions for the maximization of (41c) yield, ^q¢e = Ce with obvious notations, as a condition "determining" optimal quality e, and, after
some rearrangement of terms,
¢[¡]w0+ ^q¢ + ^v¡Cx = µ Cv¡q^ ^ q ¶ ³ ^ hy ¡^v ´ ; (42)
where ^hy = E(µ j y = ¹y), and ^v = E(µ j y ¸ y¹), and we make use of the relation ^
vy=q^y = (1=q^)(^hy¡v^). From this condition, with some reworking, we get the next result.
Proposition 2. If z is publicly observed, the optimal tuition fee p¤ of the philanthropic university is higher than marginal cost Cx. The optimal tuition is thus positive, i.e.,
students are not subsidized.
Proof: To prove Proposition 2, remark ¯rst that, using the de¯nition p = ¹y+ ¢[¡]w0,
equation (42) can be rewritten as,
p¤[¡]Cx =¡q^¢x+
Cv ^
q (^hy ¡v^) + (¹y¡h^y): (42b)
Remark then that the ¯rst term on the right-hand side of (42b) is positive since by assump-tion, ¢x ·0. We show next that the last term is zero. By de¯nition, ^hy =E(µ jy = ¹y). Using the properties
of conditional expectation, we get, E(µ [j] y) = E[E(µ [j] y; s; z) [j] y] =
E(y [j]y) =y. Thus ^hy = ¹y. Finally, we get, ^
hy = ¹y·E[yjy¸y¹] =E[E(µ jy)jy ¸y¹] =E(µ jy ¸y¹) = ^v;
and therefore, sinceCv ·0, the second term on the right-hand side of (42b) is non-negative. The right-hand side of (42b) is therefore a sum of nonnegative terms; we conclude that
p¤ > Cx >0. Q.E.D.
3.2. The Cynic Approach again: the Rent-Seeking University's Policy
Let us now study the rent-seeking university in the same asymmetric information frame-work, where z is publicly observed. The rent-seeking university will try to maximize
R=pq(¹y;z¹)[¡]C(q(¹y;z¹); e; v(¹y;z¹)), subject top= ¹y[¡]w0+ ¢(q(¹y;z¹); e). The fee pcan be
eliminated from the expression of rent, which becomes
q(¹y;z¹) [¹y+ ¢(q(¹y;z¹); e)[¡]w0]¡C(q(¹y;z¹); e; v(¹y;z¹)); (43)
and must be maximized with respect to (e;y;¹ z¹). It would be easy to prove that the rent-seeking university would not like to ration students (i.e.,choose to setx < q), for it would then always
bene¯t from a raise of the tuition p. The rent maximization problem can be decomposed into two steps. For ¯xed values of (x; e), the thresholds (¹y;z¹) can ¯rst be set
so as to maximize xy¹[¡]C(x; e; v(¹y;z¹)) subject to x = q(¹y;z¹). This yields the ¯rst-order conditions, x Cv =vy¡ vz qz qy: (44)
Given thatCv <0, due to peer e®ects, and by Lemma 2, (44) implies that the rent-seeking optimum should satisfy hy > hz, but, for ¯nite values of ¹z, this is again impossible. We can state,
Proposition 3. If z is publicly observed, the optimal rent-seeking (or for-pro¯t) university policy is to set z¹=[¡1], i.e., admission standards are the lowest possible and tuition does all the
screening job.
For proof, see the appendix
If the peer e®ects were negligible, i.e., Cv = 0, it would be easy to provide a proof of the latter result. It would then always be pro¯table to increase ¹y by dy >¹ 0 and to reduce ¹z
by dz¹= [¡](qy=qy)dy¹, keeping x = q(¹y;z¹) (and thus the cost C) constant, for that would increase the rent R by dR = xdp= xdy >¹ 0. The matter is slightly more complicated in the presence of a peer
e®ect, and the result is driven by the fact that hy < hz holds for every ¯nite value of ¹z.
With the help of de¯nitions (41a)-(41b), rewrite the rent as R = (¹y+ ¢(^q(¹y); e)[¡]
w0)^q(¹y)¡C(^q; e;v^(¹y)), to be maximized with respect to (e;y¹). A comparison of (41c) and
the above expression of rent obviously shows that the rent-seeking policy is not optimal in the philanthropic sense. But the rent-seekers become dedicated philanthropists if they are subjected to a
certain public incentive transfer.
Proposition 4. The rent-seeking university chooses a socially optimal policy if it is sub-jected to the following transfer T, de¯ned as,
T(e; x; p) =[¡]xy¹+xv^(¹y) +T0; (45)
where T0 is any constant.
The proof of this result is obvious since R+T [´] W + Constant. The result is more interesting because equation (45) shows that the transferT depends on ¹y,xand knowledge of ^v. Since ¹y =p+w0¡¢(x;
e), the transfer depends also one. So it seems that the public
authority or public regulator can only compute the transfer if they observex,p, ande. But it is in fact the knowledge of the Mincerian regression functionln(w) =ln(w0)+¢(x; e)+µ,
involving an estimation of distribution parameters ¹ and ¾2
µ which is required, with the addition of the covariance Cov(y; µ) (equal to V ar(y) here, under normality), which are needed to compute the average ability v. This is because
^ v(¹y) = R[1] ¹ y R[1] ¡1µÃ^(µ; y)dµdy R[1] ¹ y R[1] ¡1Ã^(µ; y)dµdy ;
where ^Ãis the joint density ofµ andy. Qualitye, intervening only through its e®ect on ¢, can be captured, in principle, as a ¯xed university e®ect. So the problem is to estimate the distribution of
y=E(µ [j] s; z). This latter distribution can in principle be estimated from regression work, where wages are regressed on higher education achievement, secondary education test scores, plus family
and social background control variables for a cross section of individuals. Again, a regulator which is also an (expert) econometrician can, in principle, compute and implement the incentive transfer
T, but the task is not really easy.
In practice, we know that the rent-seeking faculty will tend to underestimate the social value of educating students, because they take ¹y, the marginal student's value into account instead of ^v
(¹y), the average student's value, and that ¹y [·] ^v(¹y) (as shown by the proof of Proposition 2 above). It seems that the transfer function can be approximated by the sum of two terms, a per capita
subsidy, which aims at correcting the gap between marginal and average values, minus a lump sum tax, formally T [¼]s0x+T0, where s0 >0 and T0 <0.
The conclusions reached in this section are somewhat unpleasant, because the social optimum doesn't rely on exams or test scores, but only on money. But these results are probably less robust than it
seems, for they depend on the property that the university's information set is strictly included in that of the student. A balance between the two screening tools appears to be an optimum if we
assume a form of bilateral asymmetric information, in which the university knows something that the student doesn't take into account about his (her) own ability. We now turn to the study of such a
setting, in a variant of our model.
4. The Case of Bilateral Asymmetric Information
It seems reasonable to assume that the university is endowed with information about student abilities that students themselves do not have, while continuing to assume that students observe a noisy
private signal of ability s. If we interpretz as a private signal of the university about the student's ability, we can easily construct a variant of our model in which informational asymmetries are
bilateral. Assume then to ¯x ideas thatz is the result of an admission test which is compulsory, costless, and observed by academic authorities only, and that the admission pass-mark is ¹z. Students
do not know ¹z. Let ¼(s) be the student's subjective probability of admission. Using the same notation as above, unless speci¯ed otherwise, an individual will apply for registration if and only if u1
¸u0, where,
u1 =¼(s) · ¡p+ ln(w0) r + ¢ +E(µ js) ¸ + (1[¡]¼(s)) · w0+ ln(w0) r ¸ ; and u0 =w0+ ln(w0) r :
It follows that u1¸u0 if and only if E(µ js)¸w0+p¡¢´y¹. Rede¯ne now
y=E(µ [j]s): (46)
To simplify the analysis, let us now suppose thatµ,", andº are normally distributed. Conditional expectations are now linear, and coincide with the notion of theoretical re-gression. Due to
normality, we thus have,
y =®0s+ (1¡®0)¹; (47a) where ®0 = Cov(µ; s) V ar(s) = ¾2[µ] ¾2 µ +¾²2 : (47b)
The number of enrolled (admitted) students is still q(¹y;z¹) =NPr(y [¸]y; z¹ [¸]z¹).
4.1. Philanthropic Optimum under Bilateral Asymmetric Information
The philanthropic objective of the university is still the same as above, that is, W =
x(¢(x; e)[¡]w0 +v(¹y;z¹))¡C[x; e; v(¹y;z¹)] +N u0, where v = E[µ j y ¸ y; z¹ ¸ z¹], and of
course x=q(¹y;z¹).
It follows that the optimal (¹y;z¹) for given (x; e) are still a solution of system (38), i.e., vy=qy =vz=qz, and x = q, and by Lemma 2, which still applies, (38) is equivalent to
hy =hz, where the h functions are still de¯ned by (40a)-(40b).
There are di®erences with the above version of the model, starting from this point. Intuitively,z andynow play a symmetric informational part, and yis no longer a su±cient statistic for (y; z). We
can state,
Lemma 3. E[µ [j]y; z] =a0y+b0z; (48a) where, a0 = ¾2 º ¾2 º + (1¡®0)¾[µ]2 (48b)
and a0+b0 = 1, ®0 being de¯ned by (47).
For proof, see the appendix.
In this new setting, the system of equations (38) has a (¯nite) solution. Let Á(x) = (2¼)¡1=2e¡x2=2 denote the normal density and ©(x) = R[¡1]x Á(u)du denote the normal c.d.f. We can state the
Proposition 5. If z is a private information of the university and s is a private infor-mation of the student, the optimal policy of the philanthropic university involves a mix of non-trivial
admission standards z¹¤ [and a tuition] [p]¤[. Formally, equations (38) possess a]
solution (¹y¤;z¹¤) for every given x > 0. More precisely, under normality, this solution is fully characterized as follows:
where »¤ solves the equation, »= µ ©(»)[¡] ¾ 2 º ¾2 ² +¾º2 ¶ Á(») (1[¡]©(»))©(»); (49b) ¹ z¤ solves, x=q[(1[¡]®0)¹+®0(¹z¡¾0»¤);z¹]; (49c) and ¯nally, ¾0 = p ¾2 ² +¾º2.
For Proof, see the Appendix.
Proposition 5 provides us with a much more reasonable description of the world than Proposition 1. To clarify its meaning, assume for instance that¾2[²] =¾[º]2 (i.e., signals z and
s are "equally noisy"). Then, by (49b), we ¯nd that »¤ [= 0 is the only solution (because]
©(0) = 1=2), and by (49a), we get ¹y¤ = (1[¡]®0)¹+®0z¹¤: the ¯rst order condition in (38)
has provided us with a linear relationship between ¹y and ¹z. The "production level"x pins down the appropriate value of ¹z, as indicated by (49c).
The distance between ¹y and ®0z¹depends on the ratio ¾²2=¾º2. To see this, de¯ne,
¸ = ¾ 2 º ¾2 ² +¾2º :
Then, a simple application of the Implicit Function Theorem shows that
@»¤ @¸ <0;
i.e., »¤ decreases when¾2
º=¾²2increases. Using (49a), it is also easy to see that the distance ¹
y¤ [¡]®0z¹¤ increases when¸ increases, that is, formally,
@¸ =¡®0¾0 @»¤
@¸ >0:
This result is intuitive, if test scores z become more noisy than private signals s, then, less weight should be placed on selection by means of test scores, i.e., y¹¤ [¡]®0z¹¤ should
increase. This can be achieved if admission standards ¹z¤ are lowered and (or) tuition fees are raised conditional on (x; e). This is because optimal tuition (conditional on (x; e)) is of the form,
p¤ = ¹y¤+ ¢(x; e)[¡]w0: (49d)
Now, maximization of the philanthropic objective W with respect to (e;y;¹ z¹) yields | after some rearrangement of terms | the following ¯rst-order necessary conditions,
q¢e=Ce; hy =hz; (50a) q¢x+ ¢¡w0+v¡Cx = µ 1[¡] Cv q ¶ (v[¡]hy): (50b) But it is no longer possible to show that (49d), (50a) and (50b) jointly imply p¤ [>] [0. It]
happens that p¤ could be negative, a personal subsidy instead of a fee. More precisely, we get the following result.
Proposition 6. Assume that peer e®ects are negligible, i.e., Cv = 0. Then, in the
bi-lateral asymmetric information version of the model, the optimal tuition is smaller than marginal cost, or even negative if the test scorez is accurate enough as a measure of ability. Formally,
for su±ciently small ¾º, p¤ <0.
For Proof, see the Appendix
Proposition 6 captures in part the idea that some higher education institutions would at the same time be highly selective, and subsidize talented students to lure them into their classrooms. Winston
(1999) shows that top universities and colleges in the US do indeed at the same time seem to be those who o®er the highest subsidy to students, in view of unit cost information. Intuitively, if the
student's private signalssare very poor as indicators of talent, but if the university admission test technology is very precise, it could be optimal to select only the very best and to subsidize
them heavily, to be sure that no good element is deterred by the price. Remark that the result does not depend on a redistribution motive of the philanthropic university (because their objective W is
quasi-linear with respect to period 0 income, and utilitarian in nature). Proposition 6 is also independent of the existence of peer group e®ects since it is proved under Cv = 0. Proposition 6 shows
that negative fees can improve selection when the university can condition admission on su±ciently accurate information about student's talents. In the next section, we develop a version of the model
in which students also face a ¯nancial constraint: the presence of liquidity constrained students is then another motive for tuition fee subsidization. But before we turn to the study of this
question, let us compare the philanthropic and rent-seeking universities under the same bilateral asymmetric information assumptions.
4.2. The Cynic View under Bilateral Asymmetric Information
The rentRis stillR= (¹y+ ¢(x; e)[¡]w0)x¡C[x; e; v(¹y;z¹)], withx=q(¹y;z¹). Decomposing
the problem again, it is easy to see that for ¯xed (x; e), the rent-seeking faculty should choose (¹y;z¹) so as to maximize xy¹[¡]C[x; e; v(¹y;z¹)], subject to the constraint x =q(¹y;z¹). A necessary
condition is therefore obtained from the ¯rst-order conditions for this latter sub-problem. We must have (44) again, that is, x=Cv = vy ¡ (vz=qz)qy. This implies
vy=qy > vz=qz (recall that qy < 0 and Cv < 0), and, by Lemma 2, hy > hz. The screening policy of the rent-seeking university will therefore not be socially optimal, because optimality requires
equality of the latter two terms.
To see what happens in this case, use condition hy > hz, and with the help of the statement and proof of Proposition 5 above, it can be shown that the solution will be as that given by (49a) above,
except that »¤ is replaced with a value »r < »¤. It follows that ¹
y [¡]®0z¹ will be greater under rent-seeking than at the (philanthropic) optimum. From
this, and the above remark, we conclude,
Result 5. The screening policy of the rent-seeking university is not socially optimal. The rent-seeking university will set higher fees and (or) lower selection standards z¹ than the philanthropic
university, for any given value of (x; e).
5. Borrowing Constraints and Asymmetric Information
We now address the question of ¯nancial constraints in human capital accumulation, i.e., the question of poor talented students, who cannot convince a banker that they deserve credit. Would the
presence of this market imperfection, due to incomplete information, change the analysis presented above? We will see that it does, although not fundamen-tally, in an extension of our model taking
the distribution of student's "initial ¯nancial endowments" and borrowing constraints into account. Some students, due to the borrow-ing constraint, would not be able to pay the tuition fee, in spite
of havborrow-ing received very good signals relative to their future ability and earnings.
To perform the analysis, we come back to the informational assumptions of Section 3 (i.e. where z is publicly observed). For simplicity, we assume that the student's "initial ¯nancial endowment" (or
"asset") is a normal random variableawith mean ¹aand variance
¾[a]2. Assume that a is independent of µ, ² and º (in fact, assume a independent of every other random source in the model). Assume that if p > a, a student must borrow, and that the lender attaches
a "score" to each student, where the score is de¯ned as (´+ ¢), where ¢ is de¯ned by (1) and (6) above, and ´ is an independent normal random noise with mean¹=E(µ) and variance¾´. This random noise
re°ects the errors of appreciation made by the banker. Now, we assume that the lender will ¯nance the student's education project if and only if
p[·]a+·(¢ +´); (51)
where · is a coe±cient satisfying 0[·] ·[·]1. De¯ning a new random variable t = a+·´, the liquidity constraint (51) can be expressed as,
t [´]a+·´ [¸]p[¡]·¢[´]¹t; (52) or simply t [¸]t¹. Assume that, as in the asymmetric information model above, t, z and s
are observed by the students and that the university observesz only. Given independence, the e®ective demand for education is now
q(¹y;z;¹ ¹t) =NPr(y[¸]y; z¹ [¸]z¹) Pr(t[¸]¹t) =q(¹y;z¹) Pr(t[¸]¹t): (53) With the addition of the borrowing constraint, because of independence, the average ability of students does not change, that
v(¹y;z¹) =E(µ [j]y [¸]y; z¹ [¸]z; t¹ [¸]t¹) =E(µ [j]y[¸]y; z¹ [¸]z¹): (54) It follows that the philanthropic objective W can still be expressed as W = x(¢(x; e)[¡]
w0+v) +N u0 ¡C(x; e; v), where x = ~q(¹y;z;¹ ¹t). Another di®erence with the analysis of
Section 4 above is that ¹t depends on ¹y. To see this, recall that p= ¹y+ ¢[¡]w0, so that
t [´]y¹+ (1[¡]·)¢[¡]w0; (55)
The optimal screening policy (¹y;z¹) maximizesxv[¡]C(x; e; v), for ¯xed (e; x), subject to x= ~q(¹y;z;¹ ¹t). Introducing a Lagrange multiplier¿ for the latter constraint leads to the following system
of ¯rst order optimality conditions,
(x[¡]Cv)vy =¡¿(~qy + ~qt) (x[¡]Cv)vz =¡¿q~z:
Eliminating ¿ from the system and using the fact that ~qy =qyPr(t¸t¹) yields,
µ ~ qy ~ qy + ~qt ¶ vy qy = vz qz ;
Using then the result of Lemma 2, that isvz=qz = (1=q)(hz¡v), etc., yields the equivalent expression, µ ~ qy ~ qy + ~qt ¶ hy + µ ~ qt ~ qy+ ~qt ¶ v=hz (56)
Since by Lemma 1 we havehy = ¹y when z is publicly observed, we get the still equivalent, ~
qt ~
(v[¡]hz) =hz ¡y;¹
In this case, by Lemma 1 again, we also have
hz =E(µ jy¸y; z¹ = ¹z) =E(yjy ¸y; z¹ = ¹z):
Let g(t; ¹a; ¾t) be the Gaussian density of t, and G(t; ¹a; ¾t) its c.d.f (recall thatt » N(¹a+
·¹; ¾[t]2)). Using ~qy = (1¡G(¹t; ¹a; ¾t))qy, and ~qt = ¡g(¹t; ¹a; ¾t)q, we can rewrite again (56) as follows, g(¹t; ¹a; ¾t) (1[¡]G(¹t; ¹a; ¾t)) ¡ v(¹y;z¹)[¡]hz(¹y;z¹) ¢ = ¡qy(¹y;z¹) q(¹y;z¹) ¡ hz
(¹y;z¹)¡y¹ ¢ : (57)
There is no obvious impossibility to solve (57) for a ¯nite (¹y;z¹), because, even if z is publicly observed, the borrowing constraint is binding for some students. In the proof of Proposition 1, the
optimality condition hy = hz boils down to ¹y = E(y j y ¸ y; z¹ = ¹z) which holds only asymptotically, i.e., if ¹z =[¡1] (or if ¹y = +[1]).
Equation (57) is a generalization of (38) above. To see this, assume that the weight of liquidity constraints vanishes, i.e., formally, assume that ¹a[!] +[1] and ¾t !0. Then, in (57), the ratio g=(1
[¡]G)[!]0 and it follows that the equation boils down to hz = ¹y which is impossible, except asymptotically, if ¹z =[¡1].
Equation (57) is quite complex and hard to study, but it cannot be trivially solved with ¹z =[¡1]. | {"url":"https://1library.net/document/z1rnlepq-efficient-tuition-fees-examinations-subsidies-efficient-tuition-subsidies.html","timestamp":"2024-11-03T00:16:26Z","content_type":"text/html","content_length":"218891","record_id":"<urn:uuid:2e606e24-857d-4ab8-9072-0addce4a03b6>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00271.warc.gz"} |
bounded set - Enhance with Matamat
bounded set
A bounded set is a two-dimensional geometric space that is constrained by a collection of points. For example, a bounded set could be a set of points all within a set of boundaries. The points within
the set are known as the elements of the set, while the set itself is the set of dimensions.
I think this is a great example of what bounded set is and what bounded sets are. You can think of it as a set with a single element, or you can think of it as a set that has infinite dimensions. In
bounded sets, there is a single element, and in infinite bounded sets, there are infinite elements. It can also be said that a bounded set is a collection of points, or it can also be a set of
infinite points.
The bounded set is a set with a single element.
The bounded set is a set of points. If we were to add a single point to the set, we’d be creating a bounded set. If we were to add a single bounded point to the set, we’d be creating an infinite
bounded set. Now, we can think of a bounded set as a set with a single element. But is it a set that has only one element? That’s an obvious question but hard to answer.
The bounded set has an obvious problem though. It is not a set with a single element. Consider a set of points, say {0,1,2,3,4}, and assume we add a point (0). The new set will be {1,2,3,4,0}. If we
add a point to the set, we still have one element, but now it has two elements. The set is now a bounded set.
The bounded set problem is a very important, hard to solve problem, so don’t worry too much about it. It is a problem we will need to solve for in every finite set theory course we take.
The bounded set problem is the problem of deciding if a finite set of points can be bounded by a single point. A bounded set can be bounded by either an element or by a set of elements. So if you
want to prove that a set is a bounded set, you would have to prove that it is a bounded set that is either an element or a set of elements. In fact, this proof is not hard to prove, the important
thing is to prove the result.
Boundaries are a very important concept in set theory. It’s a very common concept that helps us understand the properties of sets in a very fundamental way. For example, if you have a group, then the
set of all of the elements of the group is obviously a bounded set. And if you have a set of elements, then the set of all of the elements of the set is also a bounded set.
We’ve mentioned boundaries in the past. But bounded sets have even more of an impact. A bounded set can be called a bounded set, or part of a bounded set, or a bounded element. But what does that
mean for you? It means that the set of elements of the set is a bounded element. And in fact, any set can be called a bounded element. So any set is a bounded set.
But that doesnt mean the bounded set is the whole set. If you have a bounded set, and you have an element that is also in the set, you can call that element also a bounded element. And a bounded
element can be called a bounded set. And so the bounded set is a bounded set, but the bounded set is not the whole set. And that can be problematic. | {"url":"https://matamat.com/bounded-set/","timestamp":"2024-11-04T15:45:23Z","content_type":"text/html","content_length":"115568","record_id":"<urn:uuid:6231f257-c036-4c82-9670-65576d0ce2ca>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00598.warc.gz"} |
GMM covariances
Go to the end to download the full example code. or to run this example in your browser via Binder
GMM covariances#
Demonstration of several covariances types for Gaussian mixture models.
See Gaussian mixture models for more information on the estimator.
Although GMM are often used for clustering, we can compare the obtained clusters with the actual classes from the dataset. We initialize the means of the Gaussians with the means of the classes from
the training set to make this comparison valid.
We plot predicted labels on both training and held out test data using a variety of GMM covariance types on the iris dataset. We compare GMMs with spherical, diagonal, full, and tied covariance
matrices in increasing order of performance. Although one would expect full covariance to perform best in general, it is prone to overfitting on small datasets and does not generalize well to held
out test data.
On the plots, train data is shown as dots, while test data is shown as crosses. The iris dataset is four-dimensional. Only the first two dimensions are shown here, and thus some points are separated
in other dimensions.
# Authors: The scikit-learn developers
# SPDX-License-Identifier: BSD-3-Clause
import matplotlib as mpl
import matplotlib.pyplot as plt
import numpy as np
from sklearn import datasets
from sklearn.mixture import GaussianMixture
from sklearn.model_selection import StratifiedKFold
colors = ["navy", "turquoise", "darkorange"]
def make_ellipses(gmm, ax):
for n, color in enumerate(colors):
if gmm.covariance_type == "full":
covariances = gmm.covariances_[n][:2, :2]
elif gmm.covariance_type == "tied":
covariances = gmm.covariances_[:2, :2]
elif gmm.covariance_type == "diag":
covariances = np.diag(gmm.covariances_[n][:2])
elif gmm.covariance_type == "spherical":
covariances = np.eye(gmm.means_.shape[1]) * gmm.covariances_[n]
v, w = np.linalg.eigh(covariances)
u = w[0] / np.linalg.norm(w[0])
angle = np.arctan2(u[1], u[0])
angle = 180 * angle / np.pi # convert to degrees
v = 2.0 * np.sqrt(2.0) * np.sqrt(v)
ell = mpl.patches.Ellipse(
gmm.means_[n, :2], v[0], v[1], angle=180 + angle, color=color
ax.set_aspect("equal", "datalim")
iris = datasets.load_iris()
# Break up the dataset into non-overlapping training (75%) and testing
# (25%) sets.
skf = StratifiedKFold(n_splits=4)
# Only take the first fold.
train_index, test_index = next(iter(skf.split(iris.data, iris.target)))
X_train = iris.data[train_index]
y_train = iris.target[train_index]
X_test = iris.data[test_index]
y_test = iris.target[test_index]
n_classes = len(np.unique(y_train))
# Try GMMs using different types of covariances.
estimators = {
cov_type: GaussianMixture(
n_components=n_classes, covariance_type=cov_type, max_iter=20, random_state=0
for cov_type in ["spherical", "diag", "tied", "full"]
n_estimators = len(estimators)
plt.figure(figsize=(3 * n_estimators // 2, 6))
bottom=0.01, top=0.95, hspace=0.15, wspace=0.05, left=0.01, right=0.99
for index, (name, estimator) in enumerate(estimators.items()):
# Since we have class labels for the training data, we can
# initialize the GMM parameters in a supervised manner.
estimator.means_init = np.array(
[X_train[y_train == i].mean(axis=0) for i in range(n_classes)]
# Train the other parameters using the EM algorithm.
h = plt.subplot(2, n_estimators // 2, index + 1)
make_ellipses(estimator, h)
for n, color in enumerate(colors):
data = iris.data[iris.target == n]
data[:, 0], data[:, 1], s=0.8, color=color, label=iris.target_names[n]
# Plot the test data with crosses
for n, color in enumerate(colors):
data = X_test[y_test == n]
plt.scatter(data[:, 0], data[:, 1], marker="x", color=color)
y_train_pred = estimator.predict(X_train)
train_accuracy = np.mean(y_train_pred.ravel() == y_train.ravel()) * 100
plt.text(0.05, 0.9, "Train accuracy: %.1f" % train_accuracy, transform=h.transAxes)
y_test_pred = estimator.predict(X_test)
test_accuracy = np.mean(y_test_pred.ravel() == y_test.ravel()) * 100
plt.text(0.05, 0.8, "Test accuracy: %.1f" % test_accuracy, transform=h.transAxes)
plt.legend(scatterpoints=1, loc="lower right", prop=dict(size=12))
Total running time of the script: (0 minutes 0.227 seconds)
Related examples
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Approximating multi-dimensional Hamiltonian flows by billiards
The behavior of a point particle traveling with a constant speed in a region D ⊂ R^N, undergoing elastic collisions at the regions's boundary, is known as the billiard problem. Various billiard
models serve as approximation to the classical and semi-classical motion in systems with steep potentials (e.g. for studying classical molecular dynamics, cold atom's motion in dark optical traps and
microwave dynamics). Here we develop methodologies for examining the validity and accuracy of this approximation. We consider families of smooth potentials V[ε], that, in the limit ε → 0, become
singular hard-wall potentials of multi-dimensional billiards. We define auxiliary billiard domains that asymptote, as ε → 0 to the original billiards, and provide, for regular trajectories,
asymptotic expansion of the smooth Hamiltonian solution in terms of these billiard approximations. The asymptotic expansion includes error estimates in the C ^r norm and an iteration scheme for
improving this approximation. Applying this theory to smooth potentials that limit to the multi-dimensional close to ellipsoidal billiards, we predict when the billiard's separatrix splitting (which
appears, for example, in the nearly flat and nearly oblate ellipsoids) persists for various types of potentials.
ASJC Scopus subject areas
• Statistical and Nonlinear Physics
• Mathematical Physics
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dask.array.piecewise(x, condlist, funclist, *args, **kw)[source]¶
Evaluate a piecewise-defined function.
This docstring was copied from numpy.piecewise.
Some inconsistencies with the Dask version may exist.
Given a set of conditions and corresponding functions, evaluate each function on the input data wherever its condition is true.
xndarray or scalar
The input domain.
condlistlist of bool arrays or bool scalars
Each boolean array corresponds to a function in funclist. Wherever condlist[i] is True, funclist[i](x) is used as the output value.
Each boolean array in condlist selects a piece of x, and should therefore be of the same shape as x.
The length of condlist must correspond to that of funclist. If one extra function is given, i.e. if len(funclist) == len(condlist) + 1, then that extra function is the default value, used
wherever all conditions are false.
funclistlist of callables, f(x,*args,**kw), or scalars
Each function is evaluated over x wherever its corresponding condition is True. It should take a 1d array as input and give an 1d array or a scalar value as output. If, instead of a
callable, a scalar is provided then a constant function (lambda x: scalar) is assumed.
argstuple, optional
Any further arguments given to piecewise are passed to the functions upon execution, i.e., if called piecewise(..., ..., 1, 'a'), then each function is called as f(x, 1, 'a').
kwdict, optional
Keyword arguments used in calling piecewise are passed to the functions upon execution, i.e., if called piecewise(..., ..., alpha=1), then each function is called as f(x, alpha=1).
The output is the same shape and type as x and is found by calling the functions in funclist on the appropriate portions of x, as defined by the boolean arrays in condlist. Portions not
covered by any condition have a default value of 0.
This is similar to choose or select, except that functions are evaluated on elements of x that satisfy the corresponding condition from condlist.
The result is:
out = |funclist[1](x[condlist[1]])
Define the signum function, which is -1 for x < 0 and +1 for x >= 0.
>>> x = np.linspace(-2.5, 2.5, 6)
>>> np.piecewise(x, [x < 0, x >= 0], [-1, 1])
array([-1., -1., -1., 1., 1., 1.])
Define the absolute value, which is -x for x <0 and x for x >= 0.
>>> np.piecewise(x, [x < 0, x >= 0], [lambda x: -x, lambda x: x])
array([2.5, 1.5, 0.5, 0.5, 1.5, 2.5])
Apply the same function to a scalar value.
>>> y = -2
>>> np.piecewise(y, [y < 0, y >= 0], [lambda x: -x, lambda x: x]) | {"url":"https://docs.dask.org/en/stable/generated/dask.array.piecewise.html","timestamp":"2024-11-09T00:15:44Z","content_type":"text/html","content_length":"38495","record_id":"<urn:uuid:929d7f7c-5ee0-4bd4-88cf-46344027e793>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00850.warc.gz"} |
Negative Numbers
Negative numbers
Negative numbers are surprisingly complicated when working in assembler, firstly there is no minus sign inside the registers just 0s and 1s and when is a negative number actually a negative number?
Well it comes down to context so lets look at some examples.
An 8bit register or memory location can hold numbers from 0 - 255 ($00 - $ff) and these numbers are all positive.
But let’s look at an example:
We set a = 0 and subtract 1 from it from it, so a now equals -1 but if you look at the value of a in a debugger you will see its set to $ff or 255! But also the negative is set, so does that mean 0 -
1 = -255?
Well no, when working with signed numbers computer use a system called 2's complement. With 2 complement we can represent numbers from -128 to +127. Bit 7 becomes a sign bit so numbers starting with
a 1 are positive and numbers starting with a 0 are positive.
I don't want to go too deeply into this for the time being as its not really that relevant to games development on the platform, but if I get any feed back to suggest that this is a topic worth a
deep dive I may reconsider. There is plenty of information on the internet on this subject and I really don't want to repeat what many other people have done already.
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Pick out rows from an array where the 1st column is closest to a mu...
Accepted Answer
Edited: dpb on 2 Oct 2024
Commented: Arjun on 9 Oct 2024 at 6:18
Hello, Im trying to downsample a 100k lines of excel and want to pick out only those rows where the column 1 values are closest to multiples of a number, say 50. So in the pic below, just the green
highlighted rows.
I've thought about using mod but this doesn't work:
%For example if I choose these numbers (and consider just a column vector)
and want to pick out those numbers closest to multiples of 10
ans =
If I sort these and pick out the lowest ones, the value of 29 & 39 wouldn't be picked out.
I also considered some kind of interp, but I need to pick out the actual discrete value from the table below not an interpolated value. Normally I show my arttempt, but Im actually stuck here in
where to actually start with this.
0 Comments
2 views (last 30 days)
Pick out rows from an array where the 1st column is closest to a multiple of a number
data=[0 18 38 56 75 94 112].'; % sample data
DELT=50; % the interval desired
v=[0:DELT:max(data(:,1))]; % the lookup values at delta intervals
ix=interp1(data(:,1),1:height(data),v,'nearest').'; % the magic--return index nearest to each v
reduced=data(ix,:); % and return the data rows corresponding
Applied to your array, reduced will have all columns, of course.
2 Comments
dpb on 2 Oct 2024
You were right in thinking you needed interpolation; just takes "time in grade" to know that interp1 has options other than just linear interpolate betweeen points--and then the recognition to make
the location in the array be the "interpolated" variable returned to use as the index into the original array...the other solution basically implements what interp1 does internally here; albeit with
much less effort coding-wise; the power of MATLAB in having all this stuff like this already implemented for you -- you just have to be able to find it... :)
More Answers (1)
Hi @Jason,
I see that you want to pick out rows from the array based on proximity to multiple of a number.
To downsample your data by picking rows where the first column values are closest to specific multiples (like 50), you can start by listing the multiples you care about, such as 0, 50, 100, and so
on, up to the largest number in your data. For each of these multiples, you look through your data to find the number that is closest to it. This can be done by calculating how far each number is
from the current multiple and then picking the one with the smallest difference. This way you can gather all the row’s which are closest to the multiple of the number you chose.
You can refer to the code below for better understanding:
n = [1; 5; 7; 9; 12; 15; 29; 33; 39; 50; 51; 97; 102];
multiple = 50;
% Pick the maximum number present in the data till there generate multiples
maxi = max(n);
multiples = 0:multiple:maxi;
final_selection = zeros(length(multiples), 1);
% Find the closest value to each multiple
for i = 1:length(multiples)
[~, idx] = min(abs(n - multiples(i)));
final_selection(i) = idx;
% Take only unique values
final_selection = unique(final_selection);
selected_rows = n(final_selection);
I hope this explanation helps!
2 Comments
Arjun on 9 Oct 2024 at 6:18 | {"url":"https://se.mathworks.com/matlabcentral/answers/2157020-pick-out-rows-from-an-array-where-the-1st-column-is-closest-to-a-multiple-of-a-number?s_tid=prof_contriblnk","timestamp":"2024-11-08T14:39:57Z","content_type":"text/html","content_length":"158747","record_id":"<urn:uuid:195ebee6-59e9-49bb-bcc8-74f6c83580af>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00459.warc.gz"} |
✔ specification of the lia tactic on non-linear problems
although the name of the lia tactic stands for "Linear Integer Arithmetic", the tactic in fact is more powerful than that, and it is actually able to solve some non linear goals. E.g.:
Require Import Psatz.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Lemma nonlinear (n m : nat) : 2 * m * n + m * m + n * n = (m + n) * (n + m).
What is the actual contour of non linear problems lia is supposed to solve? @Frédéric Besson @Laurent Théry
My understanding (as user) is that lia is using ring for simplification so that's why your goal is proved by lia. I think it also does some ad-hoc linearisation. @Frédéric Besson will know more about
The first step is to run zify: your goal is injected into Z. The next step is to develop polynomials, somehow ring_simplify. Then, there is some non-linear reasoning to accumulate positivity
constraints that are lost by the injection into Z (think proving forall n m:nat, n * m >= 0). At this stage the purely linear reasoning kicks in.
As @Laurent Théry said, lia subsumes zify; ring that is sufficient for your goal.
So lia supersedes ring on a signature for integer arithmetic registered for zify ?
Ah! I was writing simulatneously.
So ok your answer is "yes".
Assia Mahboubi has marked this topic as resolved.
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Black Box Machine Learning
1 Classification algorithms
There is a lot of classification algorithms available now but it is not possible to conclude which one is superior to other. It depends on the application and nature of available data set. For
example, if the classes are linearly separable, the linear classifiers like Logistic regression, Fisher’s linear discriminant can outperform sophisticated models and vice versa.
1.1 Decision Tree
Decision tree builds classification or regression models in the form of a tree structure. It utilizes an if-then rule set. The rules are learned sequentially using the training data one at a time.
Each time a rule is learned, the tuples covered by the rules are removed. This process is continued on the training set until meeting a termination condition.
The tree is constructed in a top-down recursive divide-and-conquer manner. All the attributes should be categorical. Otherwise, they should be discretized in advance? Attributes in the top of the
tree have more impact towards in the classification and they are identified using the information gain concept.
A decision tree can be easily over-fitted generating too many branches and may reflect anomalies due to noise or outliers. An over-fitted model has a very poor performance on the unseen data even
though it gives an impressive performance on training data.
This can be avoided by pre-pruning which halts tree construction early or post-pruning which removes branches from the fully grown tree. What?
1.2 Naive Bayes
Naive Bayes is a probabilistic classifier inspired by the Bayes theorem under a simple assumption which is the attributes are conditionally independent.
Naive Bayes is a very simple algorithm to implement and good results have obtained in most cases. It can be easily scalable to larger datasets since it takes linear time, rather than by expensive
iterative approximation as used for many other types of classifiers.
Naive Bayes can suffer from a problem called the zero probability problem. When the conditional probability is zero for a particular attribute, it fails to give a valid prediction. This needs to be
fixed explicitly using a Laplacian estimator?.
1.3 Artificial Neuraul Networks
Artificial Neural Network is a set of connected input/output units where each connection has a weight associated. During the learning phase, the network learns by adjusting the weights so as to be
able to predict the correct class label of the input tuples.
There are many network architectures available:
• Feed-forward
• Convolutional
• Recurrent
There can be multiple hidden layers in the model depending on the complexity of the function which is going to be mapped by the model. Having more hidden layers will enable to model complex
relationships such as deep neural networks.
However, when there are many hidden layers, it takes a lot of time to train and adjust wights. The other disadvantage of is the poor interpretability of model compared to other models like Decision
Trees due to the unknown symbolic meaning behind the learned weights.
But Artificial Neural Networks have performed impressively in most of the real world applications. It has high tolerance to noisy data and is able to classify untrained patterns. Usually, Artificial
Neural Networks perform better with continuous-valued inputs and outputs.
All of the above algorithms are eager learners since they train a model in advance to generalize the training data and use it for prediction later.
1.4 k-Nearest Neighbor (KNN)
k-Nearest Neighbor is a lazy learning algorithm which stores all instances correspond to training data points in n-dimensional space. When an unknown discrete data is received, it analyzes the
closest k number of instances saved (nearest neighbors)and returns the most common class as the prediction and for real-valued data it returns the mean of k nearest neighbors.
2 Evaluating a classifier
After training the model the most important part is to evaluate the classifier to verify its applicability
2.1 Holdout method
There are several methods exists and the most common method is the holdout method. In this method, the given data set is divided into 2 partitions as test and train 20% and 80% respectively. The
train set will be used to train the model and the unseen test data(aaaa, unseen data, I see now) will be used to test its predictive power.
2.2 Cross-validation
Over-fitting is a common problem in machine learning which can occur in most models. k-fold cross-validation can be conducted to verify that the model is not over-fitted. In this method, the data-set
is randomly partitioned into k mutually exclusive subsets, each approximately equal size and one is kept for testing while others are used for training. This process is iterated throughout the whole
k folds.Sorry what?
2.3 Precision and Recall
Precision is the fraction of relevant instances among the retrieved instances, while recall is the fraction of relevant instances that have been retrieved over the total amount of relevant instances.
Precision and Recall are used as a measurement of the relevance.Hmmm…
2.4 ROC curve Receiver Operatning Characteristics
ROC curve is used for visual comparison of classification models which shows the trade-off between the true positive rate and the false positive rate. The area under the ROC curve is a measure of the
accuracy of the model. When a model is closer to the diagonal, it is less accurate and the model with perfect accuracy will have an area of 1.0
Alright enough machine lern…oh wait, there are more things to get familiar with in my curriculum.
3 SVM (Support Vector Machine)
Support vector machine is highly preferred by many as it produces significant accuracy with less computation power. Support Vector Machine can be used for both regression and classification tasks.
But, it is widely used in classification objectives.
The objective of the support vector machine algorithm is to find a hyperplane in an N-dimensional space(N — the number of features) that distinctly classifies the data points.
To separate the two classes of data points, there are many possible hyperplanes that could be chosen. Our objective is to find a plane that has the maximum margin, i.e the maximum distance between
data points of both classes. Maximizing the margin distance provides some reinforcement so that future data points can be classified with more confidence.
3.1 Hyperplanes and Support Vectors
Hyperplanes are decision boundaries that help classify the data points. Data points falling on either side of the hyperplane can be attributed to different classes. Also, the dimension of the
hyperplane depends upon the number of features. If the number of input features is 2, then the hyperplane is just a line. If the number of input features is 3, then the hyperplane becomes a
two-dimensional plane. It becomes difficult to imagine when the number of features exceeds 3.
Support vectors are data points that are closer to the hyperplane and influence the position and orientation of the hyperplane. Using these support vectors, we maximize the margin of the classifier.
Deleting the support vectors will change the position of the hyperplane. These are the points that help us build our SVM.
Cool, that's a first gif on this website. As easy to put it in as an image, cool. Don't really use them, but why not, I might start to
3.2 Logistic Regression
Logistic Regression was used in the biological sciences in early twentieth century. It was then used in many social science applications. Logistic Regression is used when the dependent variable
(target) is categorical.
• To predict whether an email is spam (1) or (0)
• Whether the tumor is malignant (1) or not (0)
Logistic regression models the probabilities for classification problems with two possible outcomes. It's an extension of the linear regression model for classification problems
3.3 K-means Clustering
K-means clustering is one of the simplest and popular unsupervised machine learning algorithms.
The objective of K-means is simple: group similar data points together and discover underlying patterns. To achieve this objective, K-means looks for a fixed number (k) of clusters in a dataset.
A cluster refers to a collection of data points aggregated together because of certain similarities.
You’ll define a target number k, which refers to the number of centroids you need in the dataset. A centroid is the imaginary or real location representing the center of the cluster.
Every data point is allocated to each of the clusters through reducing the in-cluster sum of squares? Hm?
3.4 k-fold Cross-Validation
Lots of k's here
Cross-validation is a statistical method used to estimate the skill of machine learning models.
It is commonly used in applied machine learning to compare and select a model for a given predictive modeling problem because it is easy to understand, easy to implement, and results in skill
estimates that generally have a lower bias than other methods.
Oh, so there are even methods to compare and select a model?!? Why do you even need humans for then? Let the machine face a problem, it will select the method to solve it and then solve it. Uh, even
data scientists might become unecessary soon. Overexaggerating of course, but damn. | {"url":"http://arvydas.dev/20210316T105800--black-box-machine-learning__learning_ml.html","timestamp":"2024-11-09T04:35:55Z","content_type":"text/html","content_length":"18371","record_id":"<urn:uuid:7dd7cf1c-1e0e-4649-9b58-e30b62b9c9a1>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00027.warc.gz"} |
Properties of Modulus of a complex number
Let us prove some of the properties.
Triangle inequality
For any two complex numbers z[1] and z[2], we have |z[1] + z[2]| ≤ |z[1]| + |z[2]|.
⇒ |z[1] + z[2]|^2 ≤ (|z[1]| + |z[2]|)^2
⇒ |z[1] + z[2]| ≤ |z[1]| + |z[2]|
Geometrical interpretation
Now consider the triangle shown in figure with vertices O, z[1] or z[2] , and z[1] + z[2]. We know from geometry that the length of the side of the triangle corresponding to the vector z[1] + z[2]
cannot be greater than the sum of the lengths of the remaining two sides. This is the reason for calling the property as "Triangle Inequality".
It can be generalized by means of mathematical induction to finite number of terms:
|z[1] + z[2 ]+ z[3 ]+ …. + z[n ]| ≤ |z[1]| + |z[2]| + |z[3]| + … + |z[n]| for n = 2,3,….
The distance between the two points z[1] and z[2] in complex plane is | z[1] − z[2] |
If z[1] = x[1] + iy[1] and z[2] = x[2] + iy[2] , then
| z[1] - z[2]| = | ( x[1] - x[2] ) + ( y[1] - y[2] )i|
= √ [( x[1] - x[2] )^2 + ( y[1] - y[2] )^2]
The distance between the two points z[1] and z[2] in complex plane is | z[1] - z[2] |
If we consider origin, z[1] and z[2] as vertices of a triangle, by the similar argument we have
|z[1] - z[2]| ≤ |z[1]| + |z[2]|
| |z[1]| - |z[2]| | ≤ | z[1] + z[2]| ≤ |z[1]| + |z[2]| and
| |z[1]| - |z[2]| | ≤ | z[1] - z[2]| ≤ |z[1]| + |z[2]|
Modulus of the product is equal to product of the moduli.
For any two complex numbers z[1] and z[2], we have |z[1 ]z[2]| = |z[1]| |z[2]|
It can be generalized by means of mathematical induction to any finite number of terms:
|z[1 ]z[2] z[3] ….. z[n]| = |z[1]| |z[2]| |z[3]| … … |z[n]|
That is the modulus value of a product of complex numbers is equal to the product of the moduli of complex numbers.
Similarly we can prove the other properties of modulus of a complex number. | {"url":"https://www.brainkart.com/article/Properties-of-Modulus-of-a-complex-number_39096/","timestamp":"2024-11-03T18:26:36Z","content_type":"text/html","content_length":"82730","record_id":"<urn:uuid:7e2b060d-25d5-45df-96bc-4175fe132338>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00559.warc.gz"} |
Nutrient intake is usually measured with considerable mistake both in popular
Nutrient intake is usually measured with considerable mistake both in popular surrogate instruments like a meals frequency questionnaire (FFQ) in addition to in gold regular type instruments like a
diet plan record (DR). validation research topics. Since biomarker measurements are costly for a set 4-Methylumbelliferone budget you can either work with a design in which a large numbers of topics
possess 1 biomarker measure in support of a little subsample can be replicated or possess a smaller amount of topics and also have most 4-Methylumbelliferone or all topics validated. The goal of this
paper is to enhance the proportion of subjects with replicated biomarker actions where optimization is with respect to minimizing the variance of = true intake for the ith subject are distributed and
are mutually independent of each additional. are mutually self-employed of is given by is generally skewed while the distribution of are from [exp(and = pth percentile of a N(0 1 distribution. It
remains to derive an analytic manifestation for var[ln(=E[(? ? ? �� 1 =0 else �� 1 =0 else and �� 1 =0 else. Using the delta method we have that we presume asymptotic normality of ln(is definitely
given by [exp(and of the subjects possess biomarker measurements where = 1 2 and = the number of replicate biomarker measurements for the subject and let = proportion of biomarker measurements that
are replicated =2where 0 �� �� 1. We presume that is fixed due to budgetary constraints and we wish to determine the value of that minimizes by by subjects of whom have one replicate and have two
replicates then = 4-Methylumbelliferone 2=2is given by [exp(as follows: are given in Appendix C. Note that in general if there is positive correlation among replicate Z 4-Methylumbelliferone X and W
ideals then it can be demonstrated that > 0 > 0 > 0 > 0 > 0 > 0 > 0 and > 0. Hence (((= 0 i.e. all subjects possess only one biomarker measurement since this will maximize the number of subjects.
Conversely = 1; where all subjects possess two biomarker measurements. Presume all subjects have either Plscr4 one or two biomarker measurements. If we combine equations 4 and 19-28 we obtain: in
equation 29 and collect terms we obtain the 4th degree polynomial equation as follows: < 1. 3 SIMULATION STUDY We simulated data from a hypothetical dataset with a similar correlation structure as in
our example with (= (100 100 100 100 50 50 and in equation 12 from 4 0 simulated samples. The results are given in Table 1. We see that there is good agreement between the mean theoretical variances
and covariances 4-Methylumbelliferone regarded as in equation 4 and derived in Appendix B and the related empirical variances and covariances from the 4 0 simulated samples. Also the overall estimate
of offers little bias and the estimated 95% confidence intervals have approximately (94.1%) protection. 4 EXAMPLE We analyzed data from your EPIC-Norfolk study [2]. Individuals were seen at a
baseline check out and at a 4-yr follow-up check out as part of the study. At both baseline and follow-up a food rate of recurrence questionnaire (FFQ) and a 1-week diet record (DR) were obtained. In
addition a blood sample was acquired at both the baseline and 4-yr follow-up check out. With this example we focus on diet vitamin C and assess the regression coefficient of true diet vitamin C
intake (in equation 1) on FFQ vitamin C intake (in equation 1) which is given by in equation 2 using plasma vitamin C like a biomarker. We refer to as the estimated regression calibration element.
For this example we presume that true diet intake has not changed over four years but allow for the possibility of correlated error between FFQ and DR intake (��in equation 1). We also presume that
there is no 4-Methylumbelliferone systematic error in the biomarker and that the random error in FFQ intake DR intake and plasma vitamin C are uncorrelated. The marginal and joint distribution of FFQ
intake (as well as the individual components used in equation 12. We observe that the estimated regression calibration element (= 0.349 = the optimal proportion of replicated biomarker measurements
(i.e. 2 computed = 0.2 – 0.5 related to a proportion of themes with replicated biomarkers of 0.14 to 0.33. However the variance raises moderately outside these limits. Figure 1 Table IV Results of
Optimization Process based on EPIC dataset 6 Conversation Correlated error between gold standard diet measures such as a diet record and surrogate actions such as a food rate of recurrence
questionnaire can bias standard techniques for correcting for measurement error such as regression calibration. The method of triads using a biomarker in addition to the above diet instruments is an
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American Mathematical Society
Topological applications of Stanley-Reisner rings of simplicial complexes
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by A. A. Aizenberg
Translated by: A. Martsinkovsky
Trans. Moscow Math. Soc. 2012, 37-65
DOI: https://doi.org/10.1090/S0077-1554-2013-00200-9
Published electronically: January 24, 2013
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Methods of commutative and homological algebra yield information on the Stanley-Reisner ring $\Bbbk [K]$ of a simplicial complex $K$. Consider the following problem: describe topological properties
of simplicial complexes with given properties of the ring $\Bbbk [K]$. It is known that for a simplicial complex $K=\partial P^*$, where $P^*$ is a polytope dual to the simple polytope $P$ of
dimension $n$, the depth of $\operatorname {depth}\Bbbk [K]$ equals $n$. A recent construction allows us to associate a simplicial complex $K_P$ to any convex polytope $P$. As a consequence, one
wants to study the properties of the rings $\Bbbk [K_P]$. In this paper, we report on the obtained results for both of these problems. In particular, we characterize the depth of $\Bbbk [K]$ in terms
of the topology of links in the complex $K$ and prove that $\operatorname {depth}\Bbbk [K_P] = n$ for all convex polytopes $P$ of dimension $n$. We obtain a number of relations between bigraded betti
numbers of the complexes $K_P$. We also establish connections between the above questions and the notion of a $k$-Cohen-Macaulay complex, which leads to a new filtration on the set of simplicial
complexes. References
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Bibliographic Information
• A. A. Aizenberg
• Affiliation: M. V. Lomonosov Moscow State University
• Email: ayzenberga@gmail.com
• Published electronically: January 24, 2013
• Additional Notes: This work was supported by the grants RFFI 11-01-00694-a and 12-01-92104-YaF_a
• © Copyright 2013 American Mathematical Society
• Journal: Trans. Moscow Math. Soc. 2012, 37-65
• MSC (2010): Primary 13F55; Secondary 55U10, 13H10
• DOI: https://doi.org/10.1090/S0077-1554-2013-00200-9
• MathSciNet review: 3184967 | {"url":"https://www.ams.org/journals/mosc/2012-73-00/S0077-1554-2013-00200-9/?active=current","timestamp":"2024-11-06T12:54:46Z","content_type":"text/html","content_length":"73887","record_id":"<urn:uuid:fff45918-3570-4b04-8346-20b6de542e90>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00358.warc.gz"} |
Econometrics in the Cloud: Two-Stage Least Squares in BigQuery ML - Publications - The Technology Policy Institute
This is part three in a series about how to extend cloud-based data analysis tools – such as Google’s BigQuery ML – to handle specific econometrics requirements. In part 1, I showed how to compute
coefficient standard errors in BigQuery and in part 2, I showed how to compute robust standard errors in BigQuery.
This post shows how to perform Two-Stage Least Squares (2SLS) in BigQuery. 2SLS is used to identify an endogenous regressor of interest—that is, when a regressor is correlated with the error terms of
the regression. Instead of running a single OLS regression, you estimate two regressions: one to generate a predicted value of the endogenous variable, and a second using the predicted variable
instead of the actual values, and adjusting the standard errors appropriately. Here at TPI, we used 2SLS in our recent paper on internet streaming piracy.
Implementing 2SLS correctly requires calculating the coefficients and then calculating the corrected standard errors. Both aspects are needed because the standard errors generated by estimating on
OLS regression using the predicted values of the endogenous variable will be incorrect, as the residuals will be based off the predicated values of the instrument and not the real values.
Let’s start with the easier part: estimating the coefficients. Create the first-stage model with the endogenous variable as the dependent variable:
CREATE OR REPLACE MODEL `<dataset>.stage1`
OPTIONS(model_type = 'linear_reg', input_label_cols=[<endogenous variable>]) AS
< endogenous variable>
<other variables>
<variables> is not NULL
Add a new column, pred_endogenous, to hold the predicted values. Calculate these using ML.WEIGHTS for the coefficeints as we did for the robust coefficients. Next, we can run the second-stage model:
CREATE OR REPLACE MODEL `<dataset>.stage2`
OPTIONS(model_type = 'linear_reg', input_label_cols=[<dependent variable>]) AS
<dependent variable>
pred_<endogenous variable>
<other variables>
ML.PREDICT(MODEL `<dataset>.stage1`, (SELECT * FROM `<dataset>.<data>`)
<variables> is not NULL
And then the coefficients can be taken from ML.WEIGHTS as before.
To correct the standard errors, we need to calculate the corrected residual mean error (based off the endogenous variable, not the predicted variable), and then multiply the incorrect standard errors
by the corrected root mean squared error (rmse) divided by the original rmse. This process works for robust and non-robust standard errors.
First, we add a new function to the python code to compute the corrected rmse:
def realrmse(dataset, model_name, endogenous, regressand, data, n):
#add columns to table (the 2 is needed since predicted_regressand exists in robust for first stage
table_ref = client.dataset(dataset).table(data)
table = client.get_table(table_ref)
original_schema = table.schema
new_schema = original_schema[:]
new_schema.append(bigquery.SchemaField("predicted_" + regressand + "2", "FLOAT"))
new_schema.append(bigquery.SchemaField("residual_" + regressand + "2", "FLOAT"))
table.schema = new_schema
table = client.update_table(table, ["schema"])
assert len(table.schema) == len(original_schema) + 2 == len(new_schema)
#find first stage coefficents
coeffs = {}
query = ("SELECT processed_input, weight FROM ML.WEIGHTS(MODEL `" + dataset + "." + model_name + "`)")
query_job = client.query(query)
result = query_job.result()
for row in result:
coeffs[row.processed_input] = {}
coeffs[row.processed_input]['coefficient'] = row.weight
regression = ""
for coeff in coeffs.keys():
if coeff != "__INTERCEPT__":
if coeff[0:4]=="pred":
regression += str(coeffs[coeff]['coefficient']) + "*" + endogenous + " + "
regression += str(coeffs[coeff]['coefficient']) + "*" + coeff + " + "
regression += str(coeffs[coeff]['coefficient']) + " + "
regression = regression[:-3]
query = ("UPDATE `" + dataset + "." + data + "` SET predicted_" + regressand + "2 = " + regression + " WHERE predicted_" + regressand + "2 is null")
query_job = client.query(query)
result = query_job.result()
query = ("UPDATE `" + dataset + "." + data + "` SET residual_" + regressand + "2 = predicted_" + regressand + "2 - " + regressand + " WHERE residual_" + regressand + "2 is null")
query_job = client.query(query)
result = query_job.result()
#Compute rmse
query = ("SELECT SQRT(SUM(POW(residual_" + regressand + "2, 2)) / " + str(n-len(coeffs)) + ") FROM `" + dataset + "." + data + "`")
query_job = client.query(query)
result = query_job.result()
for row in result:
return row.f0_
Then we simply need to compute the corrected standard errors:
def correctstandarderrors(coeffs, orgrmse, corrrmse):
ratio = (corrrmse/orgrmse)
for coeff in coeffs.keys():
coeffs[coeff]['2SLS se'] = ratio * coeffs[coeff]['standard error']
We need to make a few small changes to the body of our program for these functions to work. For the non-robust errors, we need to add the first-stage model and the residual to the input call
arguments. Then we need to add calls to the two new functions, and change the standard errors to calculate the t-stats to the corrected ones:
dataset = sys.argv[1]
data = sys.argv[2]
model_name = sys.argv[3]
endogenous = sys.argv[4]
regressand = sys.argv[5]
n = int(sys.argv[6])
sqrtn = math.sqrt(n-5)
root_mean = rmse(model_name)
coeffs = standef(model_name)
coeffs = rsquared(data, coeffs)
standarderror(root_mean, sqrtn, coeffs)
root_mean2 = realrmse(dataset, model_name, endogenous, regressand, data, n)
correctstandarderrors(coeffs, root_mean, root_mean2)
#The t-stat for the null hypothesis Beta_hat = 0 is Beta_hat/se(Beta_hat)
coefficients(coeffs, dataset, model_name)
for coeff in coeffs.keys():
coeffs[coeff]['tstat'] = coeffs[coeff]['coefficient']/coeffs[coeff]['2SLS se']
coeffs[coeff]['pvalue'] = 2*t.sf(abs(coeffs[coeff]['tstat']), n-len(coeffs.keys())-1)
for coeff in coeffs.keys():
print(coeff + " coefficient: " + str(coeffs[coeff]['coefficient']))
print(coeff + " standard error: " + str(coeffs[coeff]['2SLS se']))
print(coeff + " t-stat: " + str(coeffs[coeff]['tstat']))
print(coeff + " p-value: " + str(coeffs[coeff]['pvalue']))
We have to do the same for robust as well, although the regressand is already there:
dataset = sys.argv[1]
data = sys.argv[2]
model_name = sys.argv[3]
endogenous = sys.argv[4]
regressand = sys.argv[5]
n = int(sys.argv[6])
coeffs = {}
coeffs = coefficients(dataset, model_name)
addColumns(dataset, data, coeffs, regressand)
predict(dataset, data, regressand, coeffs)
residuals(dataset, data, regressand)
coeffs = regressions(dataset, data, coeffs, regressand)
root_mean = handrmse(dataset, data, coeffs, regressand, n)
root_mean2 = realrmse(dataset, model_name, endogenous, regressand, data, n)
correctstandarderrors(coeffs, root_mean, root_mean2)
for coeff in coeffs.keys():
coeffs[coeff]['tstat'] = coeffs[coeff]['coefficient']/coeffs[coeff]['2SLS se']
coeffs[coeff]['pvalue'] = 2*t.sf(abs(coeffs[coeff]['tstat']), n-len(coeffs.keys())-1)
for coeff in coeffs.keys():
print(coeff + " coefficient: " + str(coeffs[coeff]['coefficient']))
print(coeff + " standard error: " + str(coeffs[coeff]['2SLS se']))
print(coeff + " t-stat: " + str(coeffs[coeff]['tstat']))
print(coeff + " p-value: " + str(coeffs[coeff]['pvalue']))
Then the programs can be run as
python se2sls.py <dataset> <data> <model_name> <endogenous><regressand> <n>
where <dataset> is the BigQuery dataset where your model and data are located, <data> is the BigQuery table with your data, <model_name> is the name of the original BigQuery ml model, <endogenous> is
the endogenous variable you are predicting in stage 1, <regressand> is the dependent variable of the original regression, and <n> is the size of the sample.
Let’s compare the output of this program with the Stata output to show that it works. We’ll use the “CollegeDistance” dataset from Applied Economics in R (https://cran.r-project.org/web/packages/AER/
AER.pdf) again. The “CollegeDistance” dataset has 4,739 observations so the degrees of freedom correction is smaller and the comparison should be close. We run a two-stage regression that in Stata
would be:
ivreg2 wage (education = distance) unemp tuition
Unemployment Education Tuition
Stata Coefficient .1095696 .324572 1.025165
Standard Error .0075151 .1269422 .0660714
Robust Standard Error .0074349 .1268149 .0523126
BigQuery Coefficient 0.11049387 0.4681203 0.984163005
Standard Error 0.0081373 0.162673 0.0761612
Robust Standard Error 0.00806 0.162078 0.0589027 | {"url":"https://techpolicyinstitute.org/publications/economics-and-methods/econometrics-in-the-cloud-two-stage-least-squares-in-bigquery-ml/","timestamp":"2024-11-04T05:50:47Z","content_type":"text/html","content_length":"226186","record_id":"<urn:uuid:00182ca9-eb96-4de5-8346-4a350cacb745>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00610.warc.gz"} |
Continuity equation in physics: utility and exercises
The continuity equation is a physical law that states that the amount of mass or fluid entering a closed system is equal to the amount of mass or fluid leaving the system in the same period of time.
In mathematical terms, the continuity equation is expressed by the following formula:
A1 * v1 = A2 * v2
• A1 and A2 are the cross-sectional areas of the conduit or pipe at points 1 and 2 respectively.
• v1 and v2 are the velocities of the fluid at points 1 and 2 respectively.
According to the continuity equation, if the cross-sectional area of the conduit or pipe through which the fluid flows is held constant, then the velocity of the fluid and the flow rate are
inversely related. In other words, if the velocity of the fluid increases, the flow rate decreases and vice versa.
What is the continuity equation used for?
fluid mechanics. Here are some of its main applications:
1. Piping System Design – Used to calculate flow rate and fluid velocity at different points in the piping system, allowing the diameter and length of pipes to be sized to ensure constant and
uniform flow.
2. Analysis of the flow in conduits and channels: it is applied to analyze the flow of liquids in conduits and channels, allowing the determination of speed and flow at different points of the
3. Optimization of the efficiency of hydraulic systems: it is used to optimize the efficiency of hydraulic systems, such as turbines and pumps, since it allows calculating the flow and velocity of
the fluid at different points in the system and determining the optimal geometry of the system components.
For what type of fluids is it valid?
The continuity equation is valid for any type of fluid, as long as the fluid is incompressible and the flow is steady, that is, the speed and properties of the fluid at any point in the system do not
vary with time.
An incompressible fluid is one that has a constant density and does not change its volume in response to the application of pressure.
Examples of use of the continuity equation
Here are some examples of its application:
Liquid flowing in a tube
A classic example of the application of the continuity equation is the flow of liquid in a tube.
Suppose a liquid flows through a tube of cross section A1 with a velocity v1 and then enters a tube of cross section A2 with a velocity v2 .
Using this equation we can size the tube sections to alter the flow velocity.
Water folwing in a river
This equation is used to calculate the velocity of the water at different points in the river. Therefore, the behavior of the river can be predicted in different conditions, such as when dams are
built or engineering works for flood control are carried out.
Solved problems on the continuity equation in a fluid
Exercise 1
A pipe with a cross section of 0.02 m² transports water at a speed of 2 m/s. If the diameter of the tube is reduced to half its original value, what is the velocity of the water in the narrow tube?
The continuity equation states that the volumetric flow rate of the fluid flowing through the tube is constant throughout the flow. Therefore, we can write:
A1·v1 = A2·v2
Where A1 is the original cross section of the tube, v1 is the original velocity of the water, A2 is the cross section of the narrow tube, and v2 is the velocity of the water in the narrow tube.
We have A2 = A1 /4, since the diameter of the tube is reduced to half of its original value, therefore A2 = π(0.01 m)² = 0.000314 m².
Substituting the known values into the continuity equation, we get:
0.02 m² × 2 m/s = 0.000314 m² × v 2
v2 = (0.02 m² × 2 m/s) / 0.000314 m² = 127.39 m/s
Therefore, the speed of the water in the narrow tube is 127.39 m/s.
Exercise 2
A 0.1 m diameter pipe carries water at a velocity of 2 m/s. If two 0.05-m-diameter pipes are added, what is the velocity of the water in each of the smaller pipes?
The cross section of a 0.1 m diameter pipe is A1 = π(0.05 m)² = 0.00785 m². Therefore, the volumetric flow rate of water flowing through the 0.1 m pipe is:
Q = A1·v1 = 0.00785 m² × 2 m/s = 0.0157 m³/s
The cross section of a 0.05 m diameter pipe is A 2 = π(0.025 m)² = 0.0001963 m². Since there are two 0.05 m diameter pipes, the total area is A3 = 2·A2 = 0.0003926 m². Therefore, the volumetric flow
rate of water flowing through the two 0.05 m pipes is:
Q = A3v3
v3 = Q / A3 = 0.0157 m³/s / 0.0003926 m² = 40.11 m/s
Therefore, the velocity of the water in each of the 0.05-m-diameter pipes is 40.11 m/s. | {"url":"https://nuclear-energy.net/physics/fluid-mechanics/continuity-equation","timestamp":"2024-11-15T04:19:26Z","content_type":"text/html","content_length":"73128","record_id":"<urn:uuid:6b616009-6479-4524-afb1-5ae05ee39e0d>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00305.warc.gz"} |
Juan V.
What do you want to work on?
About Juan V.
Calculus, Physics
Physical Sciences Major from Universidad Nacional de La Plata
Math - Calculus
tutoring was amazing, he let me lead or prompted me to lead with what I knew for questions. Then he would make sure to nudge me in the right direction if I was getting off track :)
Math - Calculus
The tutor took very good time to explain things to me without just giving me the answer. I will come again when I am stuck with a difficult problem.
Science - Physics - Algebra-Based
J was so helpful I love him! | {"url":"https://testprepservices.princetonreview.com/academic-tutoring/tutor/juan%20v--10830230","timestamp":"2024-11-13T14:42:27Z","content_type":"application/xhtml+xml","content_length":"221278","record_id":"<urn:uuid:ba0cb7ca-c2c9-4204-b659-e08de1f76a8a>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00341.warc.gz"} |
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Hulme, PE, Ahmed, DA, Haubrock, PJ, Kaiser, BA, Kourantidou, M, Leroy, B and McDermott, SM (2023). Widespread imprecision in estimates of the economic costs of invasive alien species
worldwide. Science of the Total Environment, pp. 167997. DOI:10.1016/j.scitotenv.2023.167997.
Jäntschi, L, Bolboaca, S, Stoenoiu, C, Iancu, M, Marta, MM and Pică, EM, Ştefu, M, Sestraş, AF, Duda, MM, Sestraş, RE, Ţigan, S, Abrudan, I, and Bălan, MC (2009). Distribution Fitting 4.
Benford test on a sample of observed genotypes number from running of a genetic algorithm . Bulletin UASVM Agriculture 66(1), pp. 82-88.
Jasak, Z (2010). Benfordov zakon i reinforcement učenje (Benford's Law and reinforcment learning) . MSc Thesis, University of Tuzla, Bosnia. SRP
Joenssen, DW (2014). Testing for Benford's Law: A Monte Carlo Comparison of Methods. Preprint available at SSRN: https://ssrn.com/abstract=2545243; last accessed Mar 24, 2019 . DOI:10.2139/
Joksimović, D, Knežević, G, Pavlović, V, Ljubić, M and Surovy, V (2017). Some Aspects of the Application of Benford’s Law in the Analysis of the Data Set Anomalies. In: Knowledge Discovery in
Cyberspace: Statistical Analysis and Predictive Modeling. New York: Nova Science Publishers, pp. 85–120. ISSN/ISBN:978-1-53610-566-7.
Jošić , H and Žmuk, B (2020). The Application of the Law of Anomalous Numbers on Global Food Prices in Examining Psychological Pricing Strategies. Journal of International Food & Agribusiness
Marketing, pp. 1-16. DOI:10.1080/08974438.2020.1796880 .
Jošić, H and Žmuk, B (2018). The Application of Benford’s Law in psychological pricing detection. Zbornik radova Ekonomskog fakulteta Sveučilišta u Mostaru, No. 24, pp. 37-57.
Kundt, TC (2014). Applying "Benford's law" to the Crosswise Model: Findings from an online survey on tax evasion . Helmut-Schnidt-University, Department of Economics, Working Paper, 148/2014.
Martínez-Sánchez, F (2021). Tracking The Price of Almonds in Spain. Journal of Competition Law & Economics, nhab002. DOI:10.1093/joclec/nhab002.
Miller, SJ (ed.) (2015). Benford's Law: Theory and Applications. Princeton University Press: Princeton and Oxford. ISSN/ISBN:978-0-691-14761-1.
Neves, GA, Nunes, CS and Fernandes, PO (2021). Application of Benford’s Law to the Tourism Demand: The Case of the Island of Sal, Cape Verde. In Proceedings of Optimization, Learning
Algorithms and Applications. OL2A 2021. Communications in Computer and Information Science, vol 1488. Springer, Cham.. DOI:10.1007/978-3-030-91885-9_43.
Ollén, ER and Wennberg, J (2021). Assessing practicalities of Benford's Law: A study of the law's potential to detect fraud in transactional data. Bachelor thesis, Dept. of Economics, Lund
Park, JH, Choi, CH and Cho, EH (2016). Preliminary Study to Detect Match-Fixing: Benford’s Law in Badminton Rally Data. Journal of Physical Education and Sports Management, 3(1), pp. 64-77.
ISSN/ISBN:2373-2156. DOI:10.15640/jpesm.v3n1a5.
Rauch, B, Brähler, G, Engel, S and Göttsche, M (2011). Fact and Fiction in EU-Governmental Economic Data. German Economic Review 12(3), pp. 243-255. DOI:10.1111/j.1468-0475.2011.00542.x.
Rauch, B, Göttsche, M, Brähler, G, Geidel, FA and Pietras, T (2014). Assessing the Accountability Reports of Political Parties in Germany using Benford's Law. Betriebswirtschaftliche Forschung
und Praxis 66(2).
Rauch, B, Göttsche, M and Langenegger, S (2014). Detecting Problems in Military Expenditure Data Using Digital Analysis. Defence and Peace Economics 25(2), pp. 97-111. DOI:10.1080/
Riccioni, J and Cerqueti, R (2018). Regular paths in financial markets: Investigating the Benford’s law. Chaos, Solitons and Fractals 107, pp. 186-194. DOI:10.1016/j.chaos.2018.01.008.
Shi, J, Ausloos, M and Zhu, T (2018). Benford's law is the first significant digit and distribution distances for testing the reliability of financial reports in developing countries. Physica
A: Statistical Mechanics and its Applications 492(1), pp. 878-888. DOI:10.1016/j.physa.2017.11.017.
Tödter, K-H (2009). Benford's Law as an Indicator of Fraud in Economics. German Economic Review 10(3), 339-351. DOI:10.1111/j.1468-0475.2009.00475.x.
Vovor-Dassu, KC (2021). Tests d'adéquation à la loi de Newcomb-Benford comme outils de détection de fraudes. PhD Thesis L’Universite de Montpellier. DOI:10.13140/RG.2.2.12559.25764. FRE
Whyman, G, Ohtori, N, Shulzinger, E and Bormashenko, E (2016). Revisiting the Benford law: When the Benford-like distribution of leading digits in sets of numerical data is expectable?.
Physica A: Statistical Mechanics and its Applications Volume 461, pp. 595-601. DOI:10.1016/j.physa.2016.06.054.
Whyman, G, Shulzinger, E and Bormashenko, E (2016). Intuitive considerations clarifying the origin and applicability of the Benford law. Results in Physics Volume 6, pp. 3-6 . DOI:10.1016/ | {"url":"https://benfordonline.net/references/up/820","timestamp":"2024-11-11T04:46:31Z","content_type":"application/xhtml+xml","content_length":"49106","record_id":"<urn:uuid:ad724e12-095e-48b0-a319-633a9c182c5c>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00062.warc.gz"} |
Creating robot targets from gcode for ABB robot using C# API
I have been trying to understand how RoboDK creates targets from gcode. For example, how is a sample Gcode line: G1 X123.45 Y67.89 Z0.15 F1200 converted to a robotarget in MOVEL instruction for a
ABB robot?
I know robot machining project and 3d printing projects do this automatically, but I want to embed some digital input-output signals and speed information during specific movements and hence need a
little more flexibility.
I have looked into some of the functions in the RoboDK.cs file in the C# API example (also the python version). Keeping the discussion specific to ABB IRC5 MoveL command and as per my understanding,
the workflow for creating a MoveL robotarget is as follows:
1. Create a pose matrix using position and Euler angles [xyzrpw]: Use function
static public Mat FromXYZRPW(double x, double y, double z, double w, double p, double r)
in C# API (Same as Transl(x,y,z)*rotz(w)*roty(p)*rotx®)
2. Get orientation information from pose matrix (ABB uses quaternions) [q1 q2 q3 q4]: Use function
static double[] ToQuaternion(Mat Ti)
3. Compute joint angles required for the pose (using inverse kinematics?)
4. get configuration vector [cf1 cf4 cf6 cfx] from joint angles
How does RoboDK compute [xyzwpr] (more importantly "wpr" Euler angles) from a gcode line G1 X123.45 Y67.89 Z0.15 F1200? I know this has to do with the relationship between TCP frame and the XYZ
coordinates w.r.t userframe (work object frame) but can't wrap my head around it.
Does RoboDK use inverse kinematics to compute joint values for a given pose? Are there any functions available in C# API?
Any help will be appreciated!
05-16-2019, 06:38 AM (This post was last modified: 05-16-2019, 06:38 AM by rakshithb174.)
I was going through the Robolink documentation and found functions which generate pose, joints, etc.
Could anyone explain how pose is computed? I am confused as to which is generated first: pose matrix or [xyzwpr] array? How can I use the Robolink.Pose() function in C# API?
05-20-2019, 06:54 PM (This post was last modified: 05-20-2019, 06:55 PM by Albert.)
A pose is a 6 degree of freedom constrain and it can be seen as a 4x4 matrix or as 6-dimensional vector (XYZ coordinates as position and 3 rotations as Euloer angles). More information here:
Both representations are equivalent and you can calculate one if you have the other.
You may be confusing it with the ijk vector that represents the tool z axis orientation. For that you have an extra degree of freedom that represents the rotation around the tool Z axis. RoboDK's
tools for robot machining will help you convert the 5-DOF constrain (typically used by CNC's) to a 6-DOF constrain (required for a robot arm). A sample project here:
05-22-2019, 07:23 PM
(05-20-2019, 06:54 PM)Albert Wrote: A pose is a 6 degree of freedom constrain and it can be seen as a 4x4 matrix or as 6-dimensional vector (XYZ coordinates as position and 3 rotations as Euloer
angles). More information here:
Both representations are equivalent and you can calculate one if you have the other.
You may be confusing it with the ijk vector that represents the tool z axis orientation. For that you have an extra degree of freedom that represents the rotation around the tool Z axis. RoboDK's
tools for robot machining will help you convert the 5-DOF constrain (typically used by CNC's) to a 6-DOF constrain (required for a robot arm). A sample project here:
Albert, thank you for the links. What was confusing me was that different combinations of euler angles would lead to the same 4x4 pose matrix! Now I understand it after reading the links. I have a
follow up question:
Consider a xyzrpw vector = [X Y Z 0 180 0]. From this a homogeneous 4x4 pose matrix (lets say matrix P) can be computed.
Now, there are various combinations of euler angles [Z Y' X"] that lead to same pose matrix: [180 0 180]; [-180 0 180]; [180 0 -180]; [-180 0 -180]; [0 -180 0]
For all these different combinations of euler angles, the pose matrix is:
[ -1.000000, 0.000000, 0.000000, 77.614000 ;
0.000000, 1.000000, 0.000000, 78.452000 ;
-0.000000, 0.000000, -1.000000, 100.000000 ;
0.000000, 0.000000, 0.000000, 1.000000 ];
All the above euler angles result in same rotation matrix and thus should ideally result in same quaternion values. However, in some MoveL commands generated by RoboDK i see quaternion values as
[0 0 -1 0] and [0 0 +1 0]
Both of these seem to result in same orientation. I know each quaternion value has a sign to it. In this case, the sign doesn't seem to make any difference in the orientation of the robot. Is it
always the case?
Why does RoboDK generate two different quaternion arrays in MoveL [0 0 -1 0] and [0 0 +1 0] even though the pose matrix (4x4) is the same?
05-22-2019, 07:39 PM
This dual representation of quaternion values happens when you are doing a 180 deg pose rotation (or very close to it).
A similar situation happen with Euler angles. Euler angles usually have 2 different values that represent the same pose. In specific rotations (such as 90 deg or 180 deg rotations) you can have
infinite representations.
05-22-2019, 08:36 PM
(05-22-2019, 07:39 PM)Albert Wrote: This dual representation of quaternion values happens when you are doing a 180 deg pose rotation (or very close to it).
A similar situation happen with Euler angles. Euler angles usually have 2 different values that represent the same pose. In specific rotations (such as 90 deg or 180 deg rotations) you can have
infinite representations.
Makes sense! thanks.
Are there any functions in C# API to compute joint angles given a pose matrix?
05-30-2019, 11:36 PM
Yes, you can use:
• SolveIK (Solve robot Inverse Kinematics) to calculate joint angles given the pose
• SolveFK (Solve robot Forward Kinematics) to calculate a pose given joint angles. | {"url":"https://robodk.com/forum/Thread-Creating-robot-targets-from-gcode-for-ABB-robot-using-C-API","timestamp":"2024-11-02T17:12:47Z","content_type":"text/html","content_length":"65365","record_id":"<urn:uuid:b652fb45-9bc7-456a-916e-81466ab039a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00412.warc.gz"} |
Prove that $\int _0^{\infty} \frac{1}{1+x^{2n}}dx=\frac{\pi}{2n}\csc (\frac{\pi}{2n})$
Use contour integrals and residue theory to prove that
$$\int _0^{\infty} \frac{1}{1+x^{2n}}dx=\frac{\pi}{2n}\csc (\frac{\pi}{2n}).$$
I need all the details of the proof ASAP.
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How Not to Be Fooled by Odds - Cogent QC: Award-Winning Loan Quality Control & Compliance Software
The New York Times ran an article recently titled “How Not to Be Fooled by Odds” in which the author defines what is meant by a statement such as “the odds of a Republican takeover of the Senate is
about 74%.” Stating that “there really is a difference between saying something will almost certainly happen and saying that it is more likely to happen than not,” the author goes on to explain what
is and is not meant by the statement. Concluding that “…a prediction that puts a 74 percent chance on an outcome should be “wrong” about 26 percent of the time,” he presents a list of situations
that occur about 26% of the time, including:
• The odds that a National Football League defense prevents a first down on third-and-one.
• The percentage of full-time graduate students in electrical engineering in this country who are American citizens
• The percentage of mothers with children under 18 who stay home with their children
• The share of Americans who live in California, Texas or New York
In the same way, we need to be careful about what is actually being asserted by other statistics. For example, in the third year of California’s drought, we are quite focused on rain. So what is
meant by the statement “There is a 40% chance of rain” (technically, the PoP or ‘Probability of Precipitation’)? Consulting the National Weather Service, we find the following:
Mathematically, PoP is defined as follows:
PoP = C x A where “C” = the confidence that precipitation will occur somewhere in the forecast area, and where “A” = the percent of the area that will receive measureable precipitation, if it occurs
at all.
So… in the case of the forecast above, if the forecaster knows precipitation is sure to occur ( confidence is 100% ), he/she is expressing how much of the area will receive measurable rain. ( PoP =
“C” x “A” or “1” times “.4” which equals .4 or 40%.)
But, most of the time, the forecaster is expressing a combination of degree of confidence and areal coverage. If the forecaster is only 50% sure that precipitation will occur, and expects that, if it
does occur,it will produce measurable rain over about 80 percent of the area, the PoP (chance of rain) is 40%. ( PoP = .5 x .8 which equals .4 or 40%. )
In either event, the correct way to interpret the forecast is: there is a 40 percent chance that rain will occur at any given point in the area.
As you present your quality control findings to others, be sure that everyone understands what they mean. A clear definition of your terms at the beginning or end of any report is a good first step. | {"url":"https://cogentqc.com/uncategorized/how-not-to-be-fooled-by-odds/","timestamp":"2024-11-07T00:26:12Z","content_type":"text/html","content_length":"57154","record_id":"<urn:uuid:9126f7a4-e95a-4aa1-b47e-069ab22209f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00847.warc.gz"} |
Part-3 The Schrödinger Wave Equation is one of the most important differential equation in both physics and mathematics. The Schrödinger Equation laid the foundation for quantum mechanics, a theory
that has revolutionized our understanding of the behavior of matter at the atomic and subatomic levels. It provides a framework to accurately predict and understand the […]
Gazing at beauty and serenity of a differential equations: The Schrödinger Wave Equation. Read More »
Gazing at beauty and serenity of a differential equations: The Schrödinger Wave Equation.
Part-2 Differential equations are fundamental in describing physical phenomena. Natural phenomena involve continuous changes over time or space. Differential equations are particularly well-suited
for modeling continuous processes, as they capture the relationship between rates of change and the state of the system. Differential equations express how a system’s variables change in relation to
each other.
Gazing at beauty and serenity of a differential equations: The Schrödinger Wave Equation. Read More »
Gazing at beauty and serenity of a differential equations: The Schrödinger Wave Equation.
If as a young graduate you plan to be a scientist, start not just enjoying the beauty of Nature which is easy but more importantly the beauty and serenity of differential equations in particular the
Schrödinger Wave Equation. Modern physics started in 1905 with Einstein famous paper on Special theory of relativity The year often
Gazing at beauty and serenity of a differential equations: The Schrödinger Wave Equation. Read More »
Gazing at beauty and serenity of a differential equations: The Schrödinger Wave Equation.
Differential equations are fundamental in describing physical phenomena. Natural phenomena involve continuous changes over time or space. Differential equations are particularly well-suited for
modeling continuous processes, as they capture the relationship between rates of change and the state of the system. Differential equations express how a system’s variables change in relation to each
other. For
Gazing at beauty and serenity of a differential equations: The Schrödinger Wave Equation. Read More »
An invitation to talented students to submit articles on current affairs for truevigyan.com
If you are a student of mass communication, journalism, economics, political or social science or arts or preparing for the administrative examination, you may like to submit articles related to
social, economic, political, historical topics, the broad outlines of which are as below:- Climate change. Indian border disputes. National movements. Indian physiography. Post-independence politics.
An invitation to talented students to submit articles on current affairs for truevigyan.com Read More »
Complexity of human problems is fortunately NOT profound as is the case for mathematical complexity problems. In mathematical problems complexity arises due to a high degree of non linearity in
differential equations and you are not to play with axioms as they are universal logical structures. In human problems complexity arises due to unrealiziblity of | {"url":"https://truevigyan.com/category/mathematical-sciences/","timestamp":"2024-11-03T17:05:06Z","content_type":"text/html","content_length":"130939","record_id":"<urn:uuid:5c5053b5-71c6-4e29-8e60-7e39317dc118>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00564.warc.gz"} |
The following sections present an overview of the prizes, followed by a list of the laureates and the citations that accompany the awards.
EMS Prizes
The EMS prizes were established in 1992. At each ECM up to ten EMS prizes are awarded to early career researchers not older than 35 years at the time of nomination. The award comprises a certificate
including the citation and a cash prize of € 5000. The Compositio Mathematica Foundation kindly offered to sponsor half of the prize money. The second half will be sponsored by the publishing house
EMS Press.
Maria Colombo
Full Professor at École Polytechnique Fédérale de Lausanne
For breakthrough results in fluid dynamics, optimal transport and kinetic theory, and for her impact on analysis more broadly.
Cristiana De Filippis
Assistant Professor at the University of Parma
For her outstanding contributions to elliptic regularity, in particular Schauder estimates for nonuniformly elliptic equations and non-differentiable variational integrals, and minima of quasiconvex
Jessica Fintzen
Full Professor at Universität Bonn and Duke University
For her transformative work on the representation theory of p-adic groups, in particular for her spectacular proof that Yu’s construction of supercuspidal representations is exhaustive.
Nina Holden
Associate Professor at the Courant Institute of Mathematical Sciences
For her profound contributions to probability theory and its applications to statistical physics, including results linking Liouville quantum gravity, the Schramm-Loewner evolution, and random
Thomas Hutchcroft
Full Professor at California Institute of Technology
For his revolutionary contributions to probability theory and geometric group theory, in particular to percolation theory on general graphs, using tools from geometry, operator theory, group theory
and functional analysis.
Jacek Jendrej
Chargé de recherche at Université Sorbonne Paris Nord
For his groundbreaking proofs of the soliton resolution conjecture and two-soliton collision problem for equivariant wave maps, developing new approaches using ideas from the theory of dynamical
systems to describe the behaviour of solutions near a multi-soliton configuration.
Adam Kanigowski
Associate Professor at the University of Maryland, Full Professor at the Jagiellonian University
For his outstanding contributions to the spectral classification and the mixing properties of slowly chaotic dynamical systems.
Frederick Manners
Associate Professor at University of California San Diego
For his remarkable contributions to additive combinatorics and related areas, in particular to the foundations of higher-order Fourier analysis, as well as for miscellaneous other results such as the
solution of the pyjama problem.
Richard Montgomery
Associate Professor at the University of Warwick
For his solution of the Ringel tree packing conjecture, development of distributive absorption techniques with applications to graph embedding problems, and resolution of several classical
conjectures of Erdős and others on cycle lengths in sparse graphs using the novel machinery of sublinear expanders.
Danylo Radchenko
Chargé de recherche at Université de Lille
For the construction of optimal spherical designs and his seminal input in the new field of Fourier interpolation, as well as for his fundamental contributions to the theory of polylogarithms.
Felix Klein Prize
Nowadays, mathematics often plays the decisive role in finding solutions to numerous technical, economical and organizational problems. The Felix Klein Prize is to be awarded to a scientist, or a
group of at most three scientists, under the age of 38 for using sophisticated methods to give an outstanding solution, which meets with the complete satisfaction of industry, to a concrete and
difficult industrial problem. The award comprises a certificate including the citation and a cash prize of €5000. The money for the Prize fund is offered by the Fraunhofer Institute for Industrial
Mathematics in Kaiserslautern.
Fabien Casenave
Safran Tech, Digital Sciences & Technologies
For his contributions to the integration of numerical simulation-based design in projects related to physical reduced-order modelling in the aeronautics industry. Fabien Casenave's work lies at the
heart of simulation-based design, the integration of which into industrial processes is a key area for ensuring the performance and reliability of tomorrow's engines and meeting the challenges of
sustainable development in aeronautics.
Otto Neugebauer Prize
The Prize is to be awarded for highly original and influential work in the field of history of mathematics that enhances our understanding of either the development of mathematics or a particular
mathematical subject in any period and in any geographical region. The award comprises a certificate including the citation and a cash prize of €5000. The money for the Prize Fund is offered by
Springer-Verlag GmbH.
Reinhard Siegmund-Schultze
Professor Emeritus at University of Agder, Norway
For his publications that have helped to shape a new and much richer vision of the contexts of mathematics in the 20th century. A leading social historian of mathematics, Reinhard Siegmund-Schultze
has published the outstanding «Mathematicians fleeing from Nazi Germany» and other important books that have brought a major contribution to the study of scientific internationalism, and the times
when it collapsed.
Paul Lévy Prize in Probability Theory
The Paul Lévy Prize in Probability Theory is a new prize jointly established by the European Mathematical Society, Ecole Polytechnique, the Foundation of Ecole Polytechnique, and the Paul Lévy
family, with financial support from BNP Paribas. The Prize is to be awarded to a scientist who has made outstanding contributions to Probability Theory and its Applications and comprises a
certificate including the citation and a cash prize of €20.000.
Jeremy Quastel
Full Professor at University of Toronto
He has made major advances in the fields of hydrodynamic theory, stochastic partial differential equations, and integrable probability. Together with his collaborators, he discovered the first exact
solutions of the Kardar-Parisi-Zhang (KPZ) equation and more recently constructed the KPZ fixed point - the scale-invariant, integrable Markov process at the centre of an important class of random
evolutions of functions.
EMS/ECMI Lanczos Prize
The Prize is to be awarded to a mathematician or scientist, or a group of mathematicians and scientists, for the development of outstanding mathematical software with important applications in
mathematics, science, engineering, society or industry. The award comprises a certificate and a cash prize of €3000. The money for the Prize fund is jointly offered by the European Mathematical
Society and the European Consortium for Mathematics in Industry.
MUMPS (MUltifrontal Massively Parallel Sparse direct Solver)
Patrick Amestoy, Jean-Yves L'Excellent, Theo Mary
For their important and widely-used contributions to the numerical solution of sparse linear systems. | {"url":"https://euromathsoc.org/news/feed/atom","timestamp":"2024-11-04T03:52:27Z","content_type":"application/atom+xml","content_length":"32568","record_id":"<urn:uuid:36d27e51-def2-481b-8faf-8b6701539154>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00193.warc.gz"} |
Information Sessions
Please check the Department of Mathematics EVENT page for information regarding graduate events + seminars.
• Click here for a PDF of the information session from Fall 2022.
• This informaiton session covers the following:
□ How to declare a major in the Department of Mathematics.
□ Program major differences and requirements.
□ Course planning for APMA, MATH, MACM, and OPRES majors.
□ MATH 232 vs MATH 240.
□ GPA and minimum grade requirements.
□ Schedule of course offerings.
□ Spring 2023 special course offerings.
Do you want to ace your math exams? Are you in search of a supportive atmosphere to help you stay focused? Our MATH EXAM MIXERS are just what you need to succeed!
The Department of Mathematics hosts MATH EXAM MIXERS - complete with practice tests + free snacks and refreshments!
Do you want to learn more about the degree programs offered by the Department of Mathematics? Consider attending an information session to ensure you have all the information you need to succeed!
Do you need a quiet place to study on campus? Are you interested in connecting with your peers, swapping notes, and building your SFU network? Look no further than MATH STUDY HALL!
CHECK BACK IN JANUARY FOR SPRING 2024 HOURS
Are you interested in learning effective strategies to prepare for exams, study more efficiently, and improve your grades? Join us for one of our upcoming Math Success Sessions! Designed especially
for first and second year students.
Math Outside the Box
This is an informal talk and demo series for first and second year students curious about mathematics. These talks are intended to show students the neat things mathematicians do (beyond calculus and
the classroom) and to give students a glimpse at the kind of stuff studied in higher level mathematics courses. | {"url":"http://www.sfu.ca/math/undergraduate/advising/info-sessions.html","timestamp":"2024-11-03T18:59:43Z","content_type":"text/html","content_length":"68384","record_id":"<urn:uuid:8d34f47a-a628-43de-9131-b72cfbeea246>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00045.warc.gz"} |
A Computational Introduction to Number Theory and Algebra
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A Computational Introduction to Number Theory and Algebra
Title: A Computational Introduction to Number Theory and Algebra: .
Shoup, Victor, author
Type of Textbook
Date Issued: 2009
All of the mathematics required beyond basic calculus is developed “from scratch.” Moreover, the book generally alternates between “theory” and “applications”: one or two chapters on
a particular set of purely mathematical concepts are followed by one or two chapters on algorithms and applications; the mathematics provides the theoretical underpinnings for the
applications, while the applications both motivate and illustrate the mathematics. Of course, this dichotomy between theory and applications is not perfectly maintained: the chapters
that focus mainly on applications include the development of some of the mathematics that is specific to a particular application, and very occasionally, some of the chapters that
focus mainly on mathematics include a discussion of related algorithmic ideas as well. The mathematical material covered includes the basics of number theory (including unique
factorization, congruences, the distribution of primes, and quadratic reciprocity) and of abstract algebra (including groups, rings, fields, and vector spaces). It also includes an
introduction to discrete probability theory—this material is needed to properly treat the topics of probabilistic algorithms and cryptographic applications. The treatment of all these
Abstract/ topics is more or less standard, except that the text only deals with commutative structures (i.e., abelian groups and commutative rings with unity) — this is all that is really
Description: needed for the purposes of this text, and the theory of these structures is much simpler and more transparent than that of more general, non-commutative structures. • There are a few
sections that are marked with a “(∗),” indicating that the material covered in that section is a bit technical, and is not needed else- where. • There are many examples in the text,
which form an integral part of the book, and should not be skipped. • There are a number of exercises in the text that serve to reinforce, as well as to develop important applications
and generalizations of, the material presented in the text. • Some exercises are underlined. These develop important (but usually simple) facts, and should be viewed as an integral
part of the book. It is highly recommended that the reader work these exercises, or at the very least, read and understand their statements. • In solving exercises, the reader is free
to use any previously stated results in the text, including those in previous exercises. However, except where otherwise noted, any result in a section marked with a “(∗),” or in
§5.5, need not and should not be used outside the section in which it appears. • There is a very brief “Preliminaries” chapter, which fixes a bit of notation and recalls a few
standard facts. This should be skimmed over by the reader. • There is an appendix that contains a few useful facts; where such a fact is used in the text, there is a reference such as
“see §An,” which refers to the item labeled “An” in the appendix.
Identifier: MAT1033_02 (IID)
MAT1033, Intermediate Algebra
Links: https://open.umn.edu/opentextbooks/textbooks/a-computational-introduction-to-number-theory-and-algebra
Use and CC BY-NC-ND
Host FLVC
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::YR Argentina:: -- ::The Place For All Your RA2/YR Needs::::YR Argentina:: -- ::The Place For All Your RA2/YR Needs:: Behold the Aqua Arg!RA2 Missions Argentina -English-
LIGHTING TRANSITIONS (MAP MAKING)
Submitted by ArgCmdr
First, thanks to Wildefire and Cannis, cause i learnt this thanks to them. And also, this works perfectly on YR, but not on RA2, as the color tint actions are not accepted. In RA2 use only the
Ambient values. And do not use either actions 71 and 72, they cause slowdowns, and values do not seem to work correctly.
Lighting transitions were not necessary in TS, as the cycle could be enabled and disabled from the Multiplayer Menu, though the rate of the change of light was unchangable. In RA2, this feature
is obsolete, lighting transitions must be created with triggers, and are a good point on several kinds of maps, mostly Multiplayer, but look good on certain missions, where time is set at
afternoon, and lights change on the same rate the player approaches the enemy base. Mostly this is a feature used for MP maps as said, and its not a standard, its just a good point to play a map,
say a nice feature.
The making of the transition can be easy, depending on what you will to do. Making a real-time transition is tedious, but depends on how many steps you will put from the beginning to the...
beginning again. If you plan to make a heavy transition, i recommend the equivalence 1 hour = 1 day. And you need 60 steps, 3 per hour, one every twelve minutes (real time, one every minute) So
the first step represents the starting hour let´s say 12:00 PM (midnight). The next step will happen a minute later, but will represent 24 minutes. So that when you arrive to step 60, you will
get all the day (24 minutes x 60 steps = day). You can of course vary the quantity of steps or the equivalence time, that depends on how the transition will be, if you add more intermediate
steps, it will be more gradual, if you sustract steps it will be abrupt. If you make it last longer and mantain the step number it will take longer, but changes will be seen easier, contrary if
you mantain number of steps and shorten time, changes will be noticed but they will be gradual and not abrupt. As you see you have plenty of choices to go around.
Before starting the hard stuff, a side note. The value of the actions described here is different from the ones in the lighting section of the maps, cause these show points, and actions use
percentages. So if you want the ambient to be 0.95 you MUST put 95 as parameter in the correspondant action.
Let´s take as example the 1 hour = 1 day cycle:
On this example we will use the best numbers for snow theatre, remember that theses numbers vary for each theatre, so it would be recommendable that you read FA2´s manual about lighting, you will
find the temperate values there, and check out other tutorial sites, where you will find the values for the other theatres.
We will use 4 defined points on this example which are midnight, morning, noon and afternoon. As these are 4 points and we are representing 24 hours, we will have to set imaginary hours for each
point, with the same time transcurring from point to point (6 hours as 24 hours/4 points). Lets say, midnight is 0 hs, morning is 6 hs, noon is 12 hs and afternoon is 18 hs. Now we must calculate
the difference between their lighting values (ambient, blue, red and green). Midnight´s values are different to noon, noon´s are different to afternoon, and so on. So you must calculate which
will be the rate of change during the transition. If 6 hours must pass, and we have 24 minutes steps, then we will have 15 steps ((as 6 hours x 60 minutes per hour) / 24 minutes steps = 15 steps
per defined point). Then each defined point has 15 steps to get to the next one. As they are 4 points and steps are 15 per phase, we have a total of 60 steps, 60 triggers.
As said first we calculate the values, it is undispensible to know which are the ones corresponding to each point.
Snow´s points are:
│Light Type/Time: │Morning │ Noon │Afternoon │ Night │
│ Ambient │0.750000│1.000000│ 0.750000 │0.550000│
│ Red │1.120000│0.990000│ 1.060000 │0.760000│
│ Green │0.800000│1.040000│ 1.070000 │0.740000│
│ Blue │1.330000│1.070000│ 1.540000 │1.180000│
As we have said, calculate now the difference for each step on the loop. You have to calculate the 60 steps, beginning by the 1st, at imaginary hour 0:24. To do this you must simply do the
following. Take a value, i.e Ambient. Night (midnight, the 1st point) is 0.55 and we must calculate the rate we have to use for all steps to arrive at step 15 at the second point, morning, at a
value of 0.75. Then we have 0.20 points to be shared on 15 steps (20/15). The formula is quite simple its rate of modify of value (ambient, red green or blue, in this case ambient) = difference
between points / number of steps.
When calculating this use decimals, and when setting them on the triggers (remember you must use the 0.1 = 10 conversion on all the values), use just one, so if the value you got is 0.965, put
96.5. After calculating the ambient values for this 15 steps, you might want to follow with the next ambient 15 steps. Don´t get surprised because of the calculations required to use 1 hour = 1
day. This is actually a very complex method, and very exact. You might want to use 20 steps instead, don´t worry if this is too much work, all you have to do is make the calculations for less
steps, and also the trigger number will be much more less. You must use that formula with all 4 values, and all 60 steps, which you will get 240 values in total.
I will now give you some of the ambient values so that you see if you understood the formula:
Step on ambient from point 1 to point 2 is = 0.2 (difference between points) / 15 (nº of steps) Rate is then = 0.013 approximately. Then, 0:24 will have an ambient of 0.563, 0:48 will have an
ambient of 0.576, and so on, adding the rate for each step, until you arrive to the point. After this you will need to use the correspondant rate, to fit the changes between point 2 and point 3.
Once you have all values, you must start with the triggers. The names of each one should mention what emulated time it is, so you get the first one (first step) named 0:24, the scond one 0:48 and
so on. All triggers must be type 2, repeating. And as side not before explaining how triggers should be: This cycle is unrealistic, as all periods go on equally. All lasts 6 hours, while the
evening lasts 3 in real world. Again this is an example, and if you are smart enough, you will figure out that you can change very easily the timings by enlarging steps or making them smaller,
along with the time between them. You must now change the initial lighting levels (Edit à Lighting) and set them to the last step of the transition value´s so that it starts at 0 hs and the first
step is 0:24 hs
So, here is what the triggers should be like:
Name: Step Name or Hours it represents
Type: 2
1- #13 Elapsed Time (Parameter 60, so that it happens after 60 seconds (1 minute, remember: 1 real minute = 24 minutes = 1 step)
1- #73 Set Ambient Fade (Parameter is equal the value you calculated for the correspondant number of step, in this case for ambient)
2- #142 Retint Red (Parameter is equal the value you calculated for the correspondant number of step, in this case for red)
3- #143 Retint Green (Parameter is equal the value you calculated for the correspondant number of step, in this case for green)
4- #144 Retint Blue (Parameter is equal the value you calculated for the correspondant number of step, in this case for ambient)
5- #53 Enable Trigger (Parameter is the trigger representing the Next Step)
6- #54 Disable Trigger (Parameter is this same trigger, so that it disables itself, so that it does not repeat after 60 seconds setting the same values, and so it can be re-enabled again when
All triggers should be like this. The last one must disable itself as usual and enable the first one, so that there is a loop created. Remember again if the value you calculated was 0.95 then you
must put 95 as parameter, as its a percentage, then a x100 factor is added. By creating this 60 triggers you will get a good transition, though not completely realistic, but at least values will
be exact. You can actually change the starting step by disabling the current starting step and enabling the one you want to be the starting step. So if step 1 started the transition and you want
step 20 to start it, just disable the trigger corresponding to step 1 and enable the one corresponding to step 20. Also you need to set the initial lighting values (Edit à Lighting) equal to the
previous start step, so if transition starts on step 20 the initial lighting values should be the ones of step 19, but remember you must define them, in this menu, eual to the results you got
from calculations as there is no percentage factor here.
Finally, as side note, do not confuse ambient with light, as they are different stuff, light has to do with the brightness of the colors, and ambient with the quantity of light spotted on the
terrain. Also remember to set special lighting levels for weather storms, and make them equal to midnight so that day´s storms are dark and night storms are dark too.
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adding two matrices
Many university STEM major programs have reduced the credit hours for a course in Matrix Algebra or have simply dropped the course from their curriculum. The content of Matrix Algebra in many cases
is taught just in time where needed. This approach can leave a student with many conceptual holes in the required knowledge of matrix algebra.
In this series of blogs, we bring to you ten topics that are of immediate and intermediate interest for Matrix Algebra.
Here is the third topic where we talk about binary operations of matrices – subtraction, addition, and multiplication. Linear combination of matrices and rules of binary operations are discussed.
Get all the resources in form of textbook content, lecture videos, multiple choice test, problem set, and PowerPoint presentation.
This post is brought to you by
• Holistic Numerical Methods Open Course Ware:
• the textbooks on
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ELI5: quantum computer
A quantum computer is a type of computer that uses quantum mechanics to solve problems. It's like a regular computer, but it works on a much smaller scale and can use more powerful methods to solve
hard problems. In a quantum computer, information is stored as 'qubits', which are like regular bits but can also be in two states at the same time. This unique feature of quantum computers allows
them to process and sort through vast amounts of data much faster than regular computers can. | {"url":"https://eli5.gg/quantum%20computer","timestamp":"2024-11-03T03:38:59Z","content_type":"text/html","content_length":"10922","record_id":"<urn:uuid:2f25c497-e8d7-4fc9-94b4-74a6bd6bd1e9>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00639.warc.gz"} |
Chance and Probability Worksheets & Activities • EasyTeaching.net
Browse our collection of chance and probability worksheets. These resources help students learn to describe and calculate the probability of events and to conduct chance experiments.
Use the filter above to narrow the results by resource type and/or grade level.
Probability Event Match: Certain, Likely, Unlikely, Impossible
Cut and paste events to match the terms ‘Certain’, ‘Impossible’, ‘Likely’ and ‘Unlikely’.
Show Probability as a Fraction (2)
Calculate the probability of each event and express as fractions.
Show Probability as a Fraction (1)
Calculate the probability of each event and express as fractions.
Chance Spinner Experiment (2)
Conduct a chance experiment using a spinner and answer questions about likelihood.
Chance Spinner Experiment (1)
Conduct a chance experiment using a spinner and answer questions about likelihood.
Related Material
Collect & Present Data: Toys
Use the picture to record data in a table and then present the data in a bar graph.
Interpret Data: Books (1)
Read and interpret the data from the bar graph to answer the questions (early and middle years)
Collect & Present Data: Pets
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Precalculus (6th Edition) Blitzer Chapter 2 - Section 2.7 - Polynomial and Rational Inequalities - Exercise Set - Page 414 81
A polynomial inequality is an expression of two polynomials (or one polynomial and 0) linked by one of the inequality signs, $\lt, \leq, \gt ,\geq$.
You can help us out by revising, improving and updating this answer.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | {"url":"https://www.gradesaver.com/textbooks/math/precalculus/precalculus-6th-edition-blitzer/chapter-2-section-2-7-polynomial-and-rational-inequalities-exercise-set-page-414/81","timestamp":"2024-11-03T13:39:02Z","content_type":"text/html","content_length":"75999","record_id":"<urn:uuid:57a35cbd-90bb-432b-923e-11f1dcfecaa1>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00448.warc.gz"} |
What is hashing? - Chainkraft
Hashing generates a fixed-length string (the “hash” or “message digest”) from an input (the “message”) of arbitrary length. While hashing outputs a seemingly random string, a given message will
always produce the same digest. Even the smallest change of input data will result in a completely different hash.
Simplified hashing flowchart
Hashing is an irreversible process as it is virtually impossible to recreate the original message from the hash. Different hash functions generate hashes of varying lengths. For example, the popular
SHA256 function produces a 256-bit hash, while MD5, a function now considered insecure, outputs a 128-bit hash.
MD5(chainkraft) = 428aa4bbd0dae9b27050f89492008ee2
SHA1(chainkraft) = 2ad43800879e33005089df2fbea6de0160e08f73
SHA256(chainkraft) = 95bc529ef76da85bb79d106aeef096d1b4c8fd93d67340f6d07bb8b01952a356
Keccak512(chainkraft) = 0bb2141d86ce67125303897970e409d08d25ff170104b87a3dcba5678610a93603f7227d325f48a6a982d709bf74e822606c1da88074e6a455104a9fa7a86b7a
Thanks to their properties, hash functions are used in a wide range of applications. Below are just a few examples of their practical implementations.
Password storage
Thanks to the one-way property of hash functions, they are used for storing passwords in IT systems, with virtually every website offering user account functionality storing user passwords as hashes.
Passwords stored in plain text would be easy pickings for hackers if the website was compromised. Users using one password for multiple websites would be particularly at risk as one stolen password
would grant access to several accounts.
During the account registration process, users define their passwords which are then hashed and saved in a database. When signing in, the password input in the signup form is hashed and then compared
against the digests stored in the database (remember? the same input will always generate the same hash).
John’s web app login flowchart
Data integrity
Hashing functions are most widely used for data integrity control. The function generates a hash based on the input (e.g. a photo file), which becomes the so-called checksum, a data fingerprint of
sorts. Even the slightest modification in the data structure will result in a different checksum being generated, which facilitates the effective verification of the integrity of data sent over the
Internet by the recipient.
Data integrity control flowchart
Without cryptographic hash functions, an efficient and secure blockchain network would be almost impossible to create. Bitcoin and many other blockchains use Proof-of-Work as a consensus mechanism to
ensure network security. Hashing is a basic operation of the Proof-of-Work method whose purpose is to “mine” a new block of confirmed transactions.
Hashes also have other applications in blockchains as they are used to generate addresses and facilitate quick transaction verification using Merkle trees.
Let’s review some properties of hash functions and learn about the properties of their special subgroup, namely cryptographic hash functions.
Fixed-length hash
No matter the size of the input, be it 1KB of data or a 2 GB file, the output hash should always be of fixed length as the hash length depends on the hash function used, not the length of the
Hash function name Hash length [bits]
SHA-1 160
SHA-256 256
SHA-512 512
RIPEMD-320 320
Whirlpool 512
List of popular hash functions with their length
A hash generated from a given input should always be identical.
hash1(test) = hash2(test) … = hashn(test)
However, even the slightest change in input data should generate a completely different, uncorrelated hash (avalanche effect). Example:
hash(test) = 098f6bcd4621d373cade4e832627b4f6
hash(Test) = 0cbc6611f5540bd0809a388dc95a615b
Hash functions should be efficient and fast to compute. Regardless of the input size, the result should be returned very quickly to render the algorithm applicable in practice.
Pre-image resistance
This property denotes that hash computation by hash functions is unidirectional, meaning that generating a hash from the input is a one-way operation. Given hash(x), it should be impossible to
determine input x on its basis.
One-way nature of cryptographic hash functions
Second pre-image resistance
Given inputs x, it should be virtually impossible to find such inputs y for which the computed hash would be the same: hash(x) ! = hash(y). This property is also referred to as weak collision
Collision resistance
Collision resistance is a property whereby it is extremely difficult to find two different inputs for a hash function that would return the same hash: hash(x) ! = hash(y).
On account of their nature, hash functions will always generate some collisions. After all, there is an infinite number of different types (and lengths) of inputs and hashes have a certain maximum
length; hence the number of potential combinations is limited. The idea is to make such collisions near impossible to find.
Algorithm concept
Below is a high-level description of how a typical hash function works. We won’t go into details as they can vary significantly from algorithm to algorithm.
How come hash functions generate a fixed-length hash from an input of arbitrary length, and that hash is the same with every function call? This follows from the basic concept underlying most hash
functions whereby, for processing, inputs are divided into smaller blocks of equal length.
Hashing algorithm concept
The size of the data block varies depending on the algorithm. For example, the SHA-256 function uses 512-bit blocks. Typically, the size of the input is not a multiple of the block size. In such
cases, padding is applied. Padding consists in adding data to the final input block to match the length of the previous blocks and the block size of the given function.
After dividing the message into fixed-length blocks, the sequential process of hash computation can be started. This process involves running the hashing algorithm on each of the blocks.
Avalanche effect during hash function generation
The function takes as parameters the data of the current block and the hash of the previous block. For example, when calculating the first block, the function takes data from Block 1 and the
initialization values. Calculating the second block requires Block 2 data and the response from the function called on Block 1. The computation is repeated until the data computed from the last block
return the final hash.
In practice, blocks are counted over around a dozen rounds (e.g. SHA-256 uses 80 rounds), and the value of the last hash is additionally compressed. By linking the algorithm’s calculations to the
responses from the previous blocks, changing even one bit of data will generate a completely different hash (avalanche effect). | {"url":"https://chainkraft.com/what-is-hashing/","timestamp":"2024-11-06T02:50:03Z","content_type":"text/html","content_length":"52340","record_id":"<urn:uuid:14cdb105-ef58-4a08-906f-a3c0dbb0bd59>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00434.warc.gz"} |
Algebra symbol solver
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Can the Transformer be viewed as a special case of a Graph Neural Network (GNN)? | Taewoon Kim
In recent years, Transformers have dominated the field of natural language processing (NLP), while Graph Neural Networks (GNNs) have proven essential for tasks involving graph-structured data.
Interestingly, the Transformer can be seen as a special case of GNNs, particularly as an attention-based GNN. This connection emerges when we treat natural language as graph data, where tokens
(words) are nodes, and their sequential relationships form edges.
In this post, we’ll explore how the Transformer fits into the broader class of GNNs, with an emphasis on the mathematical framework and how natural language sequences can be viewed as a specific case
of graph-structured data.
What Are Graph Neural Networks?
Graph Neural Networks (GNNs) are designed to process graph-structured data. A graph \(G = (V, E)\) consists of:
• Nodes \(V\) (vertices), which represent entities, and
• Edges \(E\) (connections between nodes), which represent relationships between these entities.
The graph structure is typically encoded using an adjacency matrix \(\boldsymbol{A}\), where \(\boldsymbol{A}_{ij}\) is non-zero if there is an edge between node \(i\) and node \(j\).
GNNs in Mathematical Form
In a basic GNN, each node \(v_i\) has a feature vector \(\boldsymbol{h}_i^{(0)}\) at the initial layer. The goal of the GNN is to update these node features by aggregating information from
neighboring nodes. For each node, the update rule can be generalized as:
\[\boldsymbol{h}_i^{(k+1)} = \sigma \left( \sum_{j \in \mathcal{N}(i)} \boldsymbol{W} \boldsymbol{h}_j^{(k)} + \boldsymbol{b} \right)\]
• \(\boldsymbol{h}_i^{(k)}\) is the feature vector of node \(i\) at layer \(k\),
• \(\mathcal{N}(i)\) denotes the set of neighbors of node \(i\),
• \(\boldsymbol{W}\) is the weight matrix (learned parameters),
• \(\boldsymbol{b}\) is a bias term,
• \(\sigma\) is an activation function (e.g., ReLU).
The adjacency matrix \(\boldsymbol{A}\) plays a crucial role in determining which nodes are connected, controlling the message passing process by encoding the graph structure.
Types of GNNs
Different variants of GNNs have been developed to handle specific types of graph data:
• Graph Convolutional Networks (GCNs): These apply a spectral or spatial convolution to aggregate information from neighboring nodes.
• Graph Attention Networks (GATs): These use an attention mechanism to assign different weights to neighboring nodes, learning which neighbors are most important.
• Relational Graph Convolutional Networks (R-GCNs): These handle graphs with multiple types of edges (relations) by associating different weights with different edge types. This is particularly
useful for knowledge graphs, where relationships between entities vary (e.g., “friendOf,” “worksAt”).
What Are Transformers?
Transformers, introduced in “Attention is All You Need”, are designed to model sequential data like text. The key feature of the Transformer is self-attention, which allows each token to attend to
every other token in the sequence. This can be mathematically framed using the scaled dot-product attention mechanism.
Transformer Self-Attention
In a Transformer, given an input sequence of tokens \(\boldsymbol{x}_1, \boldsymbol{x}_2, \dots, \boldsymbol{x}_N\), where \(N\) is the length of the sequence, we compute attention scores between all
token pairs. For each token \(i\), its representation \(\boldsymbol{z}_i\) at the next layer is computed as a weighted sum of all token representations:
\[\text{Attention}(\boldsymbol{Q}, \boldsymbol{K}, \boldsymbol{V}) = \text{softmax} \left( \frac{\boldsymbol{Q} \boldsymbol{K}^T}{\sqrt{d_k}} \right) \boldsymbol{V}\]
• \(\boldsymbol{Q} = \boldsymbol{W}_q \boldsymbol{X}\) (queries),
• \(\boldsymbol{K} = \boldsymbol{W}_k \boldsymbol{X}\) (keys),
• \(\boldsymbol{V} = \boldsymbol{W}_v \boldsymbol{X}\) (values),
• \(\boldsymbol{X}\) is the input token matrix (where each row is a token embedding),
• \(d_k\) is the dimensionality of the queries/keys.
The attention mechanism computes a fully connected graph between all tokens, where attention weights determine the “edges” (connections) between nodes (tokens). This can be interpreted as a graph
where every token can communicate with every other token.
Positional Embeddings and Masking
In sequence modeling, the order of the tokens is crucial. Instead of explicitly encoding this order using an adjacency matrix (as done in GNNs), the Transformer uses positional embeddings \(\
boldsymbol{P}\) to encode the position of each token:
\[\boldsymbol{z}_i = \text{Attention}(\boldsymbol{Q}, \boldsymbol{K}, \boldsymbol{V}) + \boldsymbol{P}_i\]
In standard Transformers, masking is not inherently required. However, in models like large language models (LLMs), we use decoder-only Transformers where tokens attend to other tokens in an
autoregressive (causal) manner. This ensures that each token only attends to preceding tokens, which is critical for generating text sequentially.
Therefore, in these cases, masking is applied to enforce this autoregressive behavior, ensuring that future tokens are masked out during training. The combination of masking and positional encodings
creates a structure where the Transformer attends only to past tokens, mimicking the adjacency matrix that would be used in a GNN. It’s important to note that masking and positional encodings are not
part of the Transformer architecture itself—they are techniques applied in specific contexts like LLMs.
Finally, like many GNNs, the Transformer itself is input permutation-invariant. This means that, without positional encodings, the model does not inherently preserve the order of the inputs, treating
all tokens symmetrically.
Natural Language as a Special Case of Graph Data
One key observation is that natural language can be treated as a form of graph-structured data. In a sentence, tokens form nodes, and their sequential relationships form edges. For instance, a
sequence of tokens \(\text{Token}_1, \text{Token}_2, \dots, \text{Token}_N\) can be visualized as a directed graph where each token is connected to every preceding token:
\[\text{Token}_1 \rightarrow \text{Token}_2 \rightarrow \dots \rightarrow \text{Token}_N\]
In reality, for token \(N\), there are directed edges from each of the previous \(N-1\) tokens:
\[\text{Token}_1 \rightarrow \text{Token}_N, \quad \text{Token}_2 \rightarrow \text{Token}_N, \quad \dots, \quad \text{Token}_{N-1} \rightarrow \text{Token}_N\]
In traditional GNNs, such as R-GCNs, we might explicitly encode these relationships with multiple adjacency matrices to represent different types of relationships. For example, in a sequence of
tokens, we would have separate adjacency matrices to define the “1-next,” “2-next,” …, “N-next” relationships.
For each relationship type \(r\) (e.g., 1-next, 2-next, etc.), we define a separate adjacency matrix \(\boldsymbol{A}^{(r)}\) that represents the connections for that specific relation:
\[\boldsymbol{A}_{ij}^{(r)} = \begin{cases} 1 & \text{if token } j \text{ has a relation } r \text{ with token } i, \\ 0 & \text{otherwise}. \end{cases}\]
In the case of a sequence, we would have a matrix for each “\(k\)-next” relation, where \(k\) defines the step size between tokens in the sequence (1-next, 2-next, …, N-next).
However, in the Transformer, we do not need this explicit adjacency matrix because positional embeddings serve a similar purpose. Instead of encoding relationships directly as edge types, the
positional embeddings implicitly encode the sequential relationships between tokens. Thus, positional embeddings replace the need for an adjacency matrix while maintaining the graph structure.
Attention-Based GNNs and Transformers
In GNNs like Graph Attention Networks (GATs), attention is used to compute weights for each neighboring node, allowing the model to focus on the most relevant nodes during the message-passing
process. The Transformer takes this idea to the extreme by using global self-attention, where every token can attend to every other token. This global connectivity forms a fully connected graph in
GNN terms.
While R-GCNs use relation-specific weights to handle different types of edges in knowledge graphs, the Transformer simplifies this by using positional embeddings to implicitly handle the sequential
relationships between tokens.
Visualization: Natural Language as Graph Data
To better illustrate how natural language sequences can be represented as graph data, consider the following structure. Suppose we have a sentence composed of tokens: \(\text{Token}_1\), \(\text
{Token}_2\), \(\text{Token}_3\), …, \(\text{Token}_N\). The sequential nature of the sentence can be represented as a directed graph:
For visualization purposes, only the edges from \(\text{Token}_1\) are shown.
In a relational GNN like R-GCN, we would typically encode these \(\text{1-next}\), \(\text{2-next}\), … \(\text{(N-1)-next}\) relations with an adjacency matrix and relation-specific embeddings.
However, in a Transformer, we replace this structure with positional embeddings that capture the token order and masking that enforces autoregressive behavior during training. If we were to create
the embeddings for all these relations, it also becomes infeasible as we would need more and more of them as the input length grows.
In summary, Transformers can be viewed as a special case of Graph Neural Networks, particularly attention-based GNNs. Natural language, in this context, is a specific type of graph data, where
masking and positional embeddings replace the need for explicit adjacency matrices. This allows Transformers to model sequential data efficiently without requiring the edge-specific representations
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Different filter-implementations ?! & signal-controlled delay??
Different filter-implementations ?! & signal-controlled delay??
I just wonder if there are any differences between filter implementations..
For example a biquad can be made of 4 "1sample delays" and one "adder" or just two delay but more "adders".. or one can use a pole-zero structure (+ gain)...
So does anyone know something about this?!?
It came to my mind since there is no signal-controlled biquad is there?? Only the pole-zero things are signal-controlled right?!
...and is there no sig-controlled delay??? (leaving out the [vd~] cause of the interpolation-stuff it does)
In general I wonder why some very basic objects are not included in (at least) pd-vanilla.. (like a samplewise-delay etc.. On the other hand there is something like [lrshift~] which I never used due
to its limitations, maybe I didn't find a proper job for it at yet..)
(that must be some kind of conspiracy, there are rumors (or rumours; whatever you like most) that 23 important objects are missing
Seriously, sometimes I really regret not being able to write externals! A simple delay doesn't sound that hard to create, does it?!
Anyways, filter-design rocks!
@Flipp said:
So does anyone know something about this?!?
Yeah...but I can't really tell what you want to know about it.
...and is there no sig-controlled delay??? (leaving out the [vd~] cause of the interpolation-stuff it does)
If you want to leave out the interpolation, this should work:
[pd ms_to_samples]
[expr~ int($v1)]
[pd samples_to_ms]
The ms-to-samples conversion is, of course, dependent on the sample rate, which is why I just stuck them in subpatches there.
In general I wonder why some very basic objects are not included in (at least) pd-vanilla..
Part of it is that Pd-vanilla seems to be more utilitarian, in that it has very few objects, but you can really make a lot with those objects. And the other part is just that Miller seems REALLY
hesitant about adding new objects and functionality that he didn't write himself. My guess is because since it's really his baby, he doesn't want to maintain code he doesn't understand or use
But if you really want to know the answer to that question, you should hang out on the list. Discussions about how it takes the devs a long time to implement simple changes that others have already
found solutions for happen fairly often there.
Seriously, sometimes I really regret not being able to write externals! A simple delay doesn't sound that hard to create, does it?!
If you have any programming skills, you should look into it. I've only dabbled in a few basic externals, but it's actually not all that hard once you figure it out.
@Maelstorm said:
Yeah...but I can't really tell what you want to know about it.
I just mean some general things. Eg. I guess the "2 delays some adders"-structure is the most cpu-friendly, whereas the pole-zero-structure is better for signal-controlled (constantly changing)
filters.. ..that's just a guess, but if you know which structure is best for lets say quality or cpu-friendliness, please let me know as well..
If you want to leave out the interpolation, this should work: ...
Hmm I'll give it a try. But I did this once for the [delread~] and it was just a little messy since I don't really know about its interiors.. It seems it rounds to the closest sample but not really,
it rather does round at 0.49999... instead of 0.5, and I don't know what it does at different samplerates. Soooo, a simple samplewise delay would just keep averything "tight"...
@Maelstorm said:
...My guess is because since it's really his baby, he doesn't want to maintain code he doesn't understand or use himself.
That's quite comprehensible I think.
Well if changes don't happen too often, I guess learning how to write externals myself might be still faster even without any (non-graphical) programming-skills..
Could you tell me where to start?!? I already googled for this toppic but I did not really find something usefull since I don't know what to look for...
Is there anything like a pd-template for a certain compiler out there?? Or what?!
Do you know the "faust-online-compiler"?? I tried that one, but it didn't work... I must be doing something wrong there (and I don't like the "external-structure" it creates: something like an
external inside an abstraction (with only one sig-inlet??!))
The most efficient implementation is the Direct Form II (the one with only two delays), because it uses half the memory. This is what [biquad~] is doing internally. The structure itself, however,
doesn't care whether or not it is controlled by an audio signal.
The "pole-zero" structure is computationally less efficient because it is implemented as four separate first-order filters, and it's made even more complicated by the fact that these filters use
complex coefficients, which is not nearly as straight-forward as math using real numbers.
However, in Pd it is more efficient to use the first-order series structure simply because of the one-sample feedback in the filter. You could do the whole thing in [fexpr~], but it is cpu intensive.
If you tried to do the feedback with a delay, you would have to use a [block~ 1], which is also cpu intensive. The only other option is to do the feeback delay inside an external, which is what
[cpole~] and family do.
It has nothing to do with the signal control, it's just what Pd has to offer. [biquad~] could be made with signal inlets for each coefficient pretty easily, and there really should be one in Pd.
By the way, if you're not aware, there is [z~] for making delays in samples.
First of all thanks.
The most efficient implementation is the Direct Form II (the one with only two delays), because it uses half the memory.
Okay, that's rather obvious but what's about the behavior or quality?! Do all those structures behave the same when eg. the Q or F_cutoff changes?? (no clicks or different transients or
what-so-ever). Or whats about a numerical error? I'd say the more math the more potential errors can occur.
However, in Pd it is more efficient to use the first-order series structure simply because of the one-sample feedback in the filter.
Even faster than the data-controlled [biquad~] ?? Oh no! Im on windows overhere so I can't directly measure this but with external programs. ...
...just did it. It seems to be slower ..fortunately..
...the whole [expr]-family is cpu-intensive it seems...
It has nothing to do with the signal control, it's just what Pd has to offer. [biquad~] could be made with signal inlets for each coefficient pretty easily, and there really should be one in Pd.
That's what I thought. But since there is none, only the poles and zeros, I assumed that there might be a reason for that...
By the way, if you're not aware, there is [z~] for making delays in samples.
I know I know... I'd rather use [delay~]. I don't like zexy cause of all the console-messages, although it seems to be coded solidly. Anyways I like cyclone most and I want to use as few libraries as
@Flipp said:
First of all thanks.
No problem.
Do all those structures behave the same when eg. the Q or F_cutoff changes?? (no clicks or different transients or what-so-ever). Or whats about a numerical error? I'd say the more math the more
potential errors can occur.
Theoretically, from a purely mathematical perspective, they are exactly the same. They are just more than one way of expressing the same thing. You could take the equation of one form and do a bit of
algebra to get the other.
The only potential issue is, as you said, numerical errors due to the quantization limits of digital numbers. But there's not a whole lot you can do about that. I believe the Direct Form I and Direct
Form II versions of a biquad filter have the same number of additions and multiplications, so there's probably no difference there. The issue with the first-order series version is that you have to
take the biquad coefficients and extrapolate the coefficients for the first-order filters from them, which involves more complicated computation. Take a look at my [biquad.mmb~] abstraction to get an
idea of how ugly it can get.
Even faster than the data-controlled [biquad~] ??
Never tested it myself, but I kind of doubt it. I should clarify, I only meant it's more efficient to use the first-order filters when you want audio-rate control of the coefficients because there is
no external version of [biquad~] in Pd that does that, so you have to make it yourself somehow. The other options of making it in Pd are less efficient. For a message-rate version, [biquad~] is
already there doing the feedback internally, and from what I can see in the source, it's pretty simple. It's just not practical for time-varying coefficients since it will only update them on block
boundaries, which can cause clicks.
...the whole [expr]-family is cpu-intensive it seems...
I've never found [expr] or [expr~] to be all that cpu-intensive. Yeah, they're slightly more than using the basic math objects, but not enough to try avoiding them. [fexpr~], on the other hand, is
pretty intense.
Theoretically, from a purely mathematical perspective, they are exactly the same.
Should be because math comes before the "physical" implementation. But it is "spectral-math" not "time-domain-math". So the build up of a sine may be different from case to case although they behave
the same after the transient oscillations..
I didn't really search but I found this: https://ccrma.stanford.edu/~jos/fp/Direct_Form_II.html
..or just wikipedia..
For a message-rate version, [biquad~] is already there doing the feedback internally, and from what I can see in the source, it's pretty simple.
Can't you modify the source then?! Like just replace "inlet" with "inlet~" and there we have the sig-biquad.. ...I'll gonna have a look at the faust-compiler again...
@Flipp said:
Should be because math comes before the "physical" implementation. But it is "spectral-math" not "time-domain-math". So the build up of a sine may be different from case to case although they
behave the same after the transient oscillations..
The implementation is purely time-domain. It can be represented in the frequency domain, and in some cases it may be easier to manipulate in the z-domain (which is very much related to the frequency
domain), but the implementation uses time-domain difference equations. It's much easier that way.
I feel like we've had a similar discussion before...;-)
I didn't really search but I found this: https://ccrma.stanford.edu/~jos/fp/Direct_Form_II.html
..or just wikipedia..
Yeah, and this part:
is the time-domain representation. And it's really simple to implement. If you look at the C code for [biquad~], it's pretty much exactly this.
Can't you modify the source then?! Like just replace "inlet" with "inlet~" and there we have the sig-biquad.. ...I'll gonna have a look at the faust-compiler again...
It's a little more complicated than that, but, yeah, it wouldn't be too hard. iirc, there's a signal-controlled version of [biquad~] called [bq~] out there, but I think it doesn't conform to
[biquad~] (i.e., the feedback coefficients are reversed). I've had thoughts of making external versions of some of my abstractions, and that would probably be at the top of my list. I just haven't
gotten around to it.
The implementation is purely time-domain.
...as every real implementation...
Furthermore Laplace deals with frequencies, the Z-domain rather deals with angles.
I feel like we've had a similar discussion before...;-)
Yeah, I know what you mean.. we happen to celebrate long discussions...
Well I just simply tried it out. I've created biquads with [fexpr~], and the direct form I and II in a subpatch with [block~ 1 1 1] and the pole-zero-structure with cpole~s and czero~s.
And I can confirm that [biquad~] contains the direct form II.
-This is (...obviously...) the direct form I: [fexpr~ $f4*$x1[0]+$f5*$x1[-1]+$f6*$x1[-2]+$f2*$y1[-1]+$f3*$y1[-2]].
However all implementations behave differently on f_cutoff-changes!!!
The pole-zero-structure and direct-form-II can produce large overshoot.
To cut it short:
Direct Form I: f-up: ok, f-down: ok
Direct Form II: f-up: overshoot, f-down: ok
Pole-Zero: f-up: ok, f-down: overshoot
I don't know if no-overshoot stands for quality, but if so I'd say the "direct form I" might be best for a signal-version of a biquad.
iirc, there's a signal-controlled version of [biquad~] called [bq~] out there...
I can't find it. ..maybe cause google just ignores the "~" and "[]"... I just found smooth_biquad~ but that's for max/msp... maybe one can sort of port it..
I think direct form I is best for fixed point math. Direct form II or transposed direct form I are preferred for floating point.
There's a very thorough PhD thesis on the topic: www.tech.plym.ac.uk/spmc/pdf/audio/RobClarkPhD.pdf
There are lots of different possible topologies. The direct form biquads are pretty good all around, but there are quantization problems if fc/fs is very low. In this case it's better to use 64 bit
floats for both coefficients and internal signals.
And I don't think you should be discontinuously changing the coefficients in any topology. If they're signal controlled, it's no big deal to change the coefficients using linear ramps over a few ms.
@Flipp said:
...as every real implementation...
Well, the Fourier transform isn't real, it's complex. And FFT is just an efficient algorithm for implementing the DFT and/or the STFT. But once you're in the frequency domain, there is no
representation of time. So it isn't really time-domain at that point. The STFT (which is what [fft~] does) is an attempt to make a frequency-domain representation using DFTs over short snippets of
time, so you get a two-dimensional signal over time. It actually falls under the category of "time-frequency representation". But there's a trade-off between frequency resolution and time resolution.
I wouldn't call it time-domain.
Furthermore Laplace deals with frequencies, the Z-domain rather deals with angles.
I know next to nothing about the Laplace transform, but I can say that it is more accurate to say the z-domain deals with "angular frequency".
However all implementations behave differently on f_cutoff-changes!!!
The pole-zero-structure and direct-form-II can produce large overshoot.
To cut it short:
Direct Form I: f-up: ok, f-down: ok
Direct Form II: f-up: overshoot, f-down: ok
Pole-Zero: f-up: ok, f-down: overshoot
I'd be very interested in seeing a patch that illustrates this.
@acreil: Thanks! I know very little about the issues in dealing with fixed-point vs. floating-point. Would you happen to know of any good resources that cover the issues in general (i.e. not
necessarily dealing with filters)?
Thank you acreil for sharing that massive nugget of knowledge (...and thanks to the author..). I mean wow, even the table of contents is huge! It's not like I tried each filtertopology but one at
least knew that something like Lattrice- and Ladder- exist, but there are filterstructures inside I never heared off... hardcore!
@acreil said:
And I don't think you should be discontinuously changing the coefficients in any topology. ...
I think so too. The smaller the jump the less click will occur. I guess the jump-size and click-amp or overshoot are (more or less) linearly related, so the question is which one produces the best
signal at a certain jumpsize.
@Maelstorm said:
Well, the Fourier transform isn't real, it's complex...
Dude, I on purpose put that part in brackets... somehow I knew this would start a new discussion..
Anyways I know that there is complex math (the thing with a "i" (or "j" if you are an electrician..)).
But there is nothing like a "complex reality"
One could say it's a complex mathematical trick for a real system..
And as you said fft deals with TIME-pieces.
I know next to nothing about the Laplace transform, but I can say that it is more accurate to say the z-domain deals with "angular frequency".
Laplace could be called "Fourier-extended". Instead of taking only the imaginary part one takes the whole complex number.
I don't know if "angular frequency" is correct here, cause there is nothing rotation in time! (cause it's that "spectral math" there is no time in the form of a "t" or "..*t")
=>"angular frequency" * "time" = "angle"
Moreover euler says: %e^(+-%i*phi) = cos(phi)+-%i*sin(phi),
and the argument of a trig-function (like sin, cos..) is always an angle!
I'd be very interested in seeing a patch that illustrates this.
It's a little messy, but you can just take the [biquad~] and [fexpr~ ...] and a pole-zero-structure (actually it's "zero-zero-pole-pole") for comparison. I use an 2nd order allpass since I want to
listen to it (->constant amp etc.)
The [block~ 1 1 1]-filterstructures were just to prove the different implementation already existent in pd..
I have attached some pics of the "jumps"!
The three graphs show from top to bottom: "direct form I" "direct form II" and "zero-zero-pole-pole".
There are happening instantaneous jumps from f_c=1000Hz to 5000Hz with a q=0.8.
It doesn't matter if you put an ac or non-zero-dc signal into it for the overshoot to take place. (...except for direct form I...but see yourself...)
Moreover I tried different orders of a pole-zero-structure, and guess what. They are all different concerning the overshoot!!!!
It seems the best is zero-pole-zero-pole. (the worst might be p-z-z-p followed by anything starting with a pole ..or two..)
Btw. one can't upload everything at once, right?!
Thank you acreil for sharing that massive nugget of knowledge (...and thanks to the author..). I mean wow, even the table of contents is huge! It's not like I tried each filtertopology but one at
least knew that something like Lattrice- and Ladder- exist, but there are filterstructures inside I never heared off... hardcore!
@acreil said:
And I don't think you should be discontinuously changing the coefficients in any topology. ...
I think so too. The smaller the jump the less click will occur. I guess the jump-size and click-amp or overshoot are (more or less) linearly related, so the question is which one produces the best
signal at a certain jumpsize.
@Maelstorm said:
Well, the Fourier transform isn't real, it's complex...
Dude, I on purpose put that part in brackets... somehow I knew this would start a new discussion..
Anyways I know that there is complex math (the thing with a "i" (or "j" if you are an electrician..)).
But there is nothing like a "complex reality"
One could say it's a complex mathematical trick for a real system..
And as you said fft deals with TIME-pieces.
I know next to nothing about the Laplace transform, but I can say that it is more accurate to say the z-domain deals with "angular frequency".
Laplace could be called "Fourier-extended". Instead of taking only the imaginary part one takes the whole complex number.
I don't know if "angular frequency" is correct here, cause there is nothing rotation in time! (cause it's that "spectral math" there is no time in the form of a "t" or "..*t")
=>"angular frequency" * "time" = "angle"
Moreover euler says: %e^(+-%i*phi) = cos(phi)+-%i*sin(phi),
and the argument of a trig-function (like sin, cos..) is always an angle!
I'd be very interested in seeing a patch that illustrates this.
It's a little messy, but you can just take the [biquad~] and [fexpr~ ...] and a pole-zero-structure (actually it's "zero-zero-pole-pole") for comparison. I use an 2nd order allpass since I want to
listen to it (->constant amp etc.)
The [block~ 1 1 1]-filterstructures were just to prove the different implementation already existent in pd..
I have attached some pics of the "jumps"!
The three graphs show from top to bottom: "direct form I" "direct form II" and "zero-zero-pole-pole".
There are happening instantaneous jumps from f_c=1000Hz to 5000Hz with a q=0.8.
It doesn't matter if you put an ac or non-zero-dc signal into it for the overshoot to take place. (...except for direct form I...but see yourself...)
Moreover I tried different orders of a pole-zero-structure, and guess what. They are all different concerning the overshoot!!!!
It seems the best is zero-pole-zero-pole. (the worst might be p-z-z-p followed by anything starting with a pole ..or two..)
Btw. one can't upload everything at once, right?!
Thank you acreil for sharing that massive nugget of knowledge (...and thanks to the author..). I mean wow, even the table of contents is huge! It's not like I tried each filtertopology but one at
least knew that something like Lattrice- and Ladder- exist, but there are filterstructures inside I never heared off... hardcore!
@acreil said:
And I don't think you should be discontinuously changing the coefficients in any topology. ...
I think so too. The smaller the jump the less click will occur. I guess the jump-size and click-amp or overshoot are (more or less) linearly related, so the question is which one produces the best
signal at a certain jumpsize.
@Maelstorm said:
Well, the Fourier transform isn't real, it's complex...
Dude, I on purpose put that part in brackets... somehow I knew this would start a new discussion..
Anyways I know that there is complex math (the thing with a "i" (or "j" if you are an electrician..)).
But there is nothing like a "complex reality"
One could say it's a complex mathematical trick for a real system..
And as you said fft deals with TIME-pieces.
I know next to nothing about the Laplace transform, but I can say that it is more accurate to say the z-domain deals with "angular frequency".
Laplace could be called "Fourier-extended". Instead of taking only the imaginary part one takes the whole complex number.
I don't know if "angular frequency" is correct here, cause there is nothing rotation in time! (cause it's that "spectral math" there is no time in the form of a "t" or "..*t")
=>"angular frequency" * "time" = "angle"
Moreover euler says: %e^(+-%i*phi) = cos(phi)+-%i*sin(phi),
and the argument of a trig-function (like sin, cos..) is always an angle!
I'd be very interested in seeing a patch that illustrates this.
It's a little messy, but you can just take the [biquad~] and [fexpr~ ...] and a pole-zero-structure (actually it's "zero-zero-pole-pole") for comparison. I use an 2nd order allpass since I want to
listen to it (->constant amp etc.)
The [block~ 1 1 1]-filterstructures were just to prove the different implementation already existent in pd..
I have attached some pics of the "jumps"!
The three graphs show from top to bottom: "direct form I" "direct form II" and "zero-zero-pole-pole".
There are happening instantaneous jumps from f_c=1000Hz to 5000Hz with a q=0.8.
It doesn't matter if you put an ac or non-zero-dc signal into it for the overshoot to take place. (...except for direct form I...but see yourself...)
Moreover I tried different orders of a pole-zero-structure, and guess what. They are all different concerning the overshoot!!!!
It seems the best is zero-pole-zero-pole. (the worst might be p-z-z-p followed by anything starting with a pole ..or two..)
Btw. one can't upload everything at once, right?!
Thank you acreil for sharing that massive nugget of knowledge (...and thanks to the author..). I mean wow, even the table of contents is huge! It's not like I tried each filtertopology but one at
least knew that something like Lattrice- and Ladder- exist, but there are filterstructures inside I never heared off... hardcore!
@acreil said:
And I don't think you should be discontinuously changing the coefficients in any topology. ...
I think so too. The smaller the jump the less click will occur. I guess the jump-size and click-amp or overshoot are (more or less) linearly related, so the question is which one produces the best
signal at a certain jumpsize.
@Maelstorm said:
Well, the Fourier transform isn't real, it's complex...
Dude, I on purpose put that part in brackets... somehow I knew this would start a new discussion..
Anyways I know that there is complex math (the thing with a "i" (or "j" if you are an electrician..)).
But there is nothing like a "complex reality"
One could say it's a complex mathematical trick for a real system..
And as you said fft deals with TIME-pieces.
I know next to nothing about the Laplace transform, but I can say that it is more accurate to say the z-domain deals with "angular frequency".
Laplace could be called "Fourier-extended". Instead of taking only the imaginary part one takes the whole complex number.
I don't know if "angular frequency" is correct here, cause there is nothing rotation in time! (cause it's that "spectral math" there is no time in the form of a "t" or "..*t")
=>"angular frequency" * "time" = "angle"
Moreover euler says: %e^(+-%i*phi) = cos(phi)+-%i*sin(phi),
and the argument of a trig-function (like sin, cos..) is always an angle!
I'd be very interested in seeing a patch that illustrates this.
It's a little messy, but you can just take the [biquad~] and [fexpr~ ...] and a pole-zero-structure (actually it's "zero-zero-pole-pole") for comparison. I use an 2nd order allpass since I want to
listen to it (->constant amp etc.)
The [block~ 1 1 1]-filterstructures were just to prove the different implementation already existent in pd..
I have attached some pics of the "jumps"!
The three graphs show from top to bottom: "direct form I" "direct form II" and "zero-zero-pole-pole".
There are happening instantaneous jumps from f_c=1000Hz to 5000Hz with a q=0.8.
It doesn't matter if you put an ac or non-zero-dc signal into it for the overshoot to take place. (...except for direct form I...but see yourself...)
Moreover I tried different orders of a pole-zero-structure, and guess what. They are all different concerning the overshoot!!!!
It seems the best is zero-pole-zero-pole. (the worst might be p-z-z-p followed by anything starting with a pole ..or two..)
Btw. one can't upload everything at once, right?!
@Maelstorm said:
@acreil: Thanks! I know very little about the issues in dealing with fixed-point vs. floating-point. Would you happen to know of any good resources that cover the issues in general (i.e. not
necessarily dealing with filters)?
I think it's mostly just general rule of thumb issues: you need to use fixed point for embedded processors without hardware FPUs, and you can more easily determine the effects of quantization.
Floating point won't (practically) clip, so it's better for "modular" type systems where you can't guarantee any sort of fixed topology or signal level. But you have to worry about denormals and NaNs
and whatnot.
In this case, if I remember correctly, you don't want to use fixed point for direct form II because it needs more dynamic range to avoid clipping internally. | {"url":"https://forum.puredata.info/topic/6178/different-filter-implementations-signal-controlled-delay/1","timestamp":"2024-11-06T17:14:42Z","content_type":"text/html","content_length":"208550","record_id":"<urn:uuid:97a4dfe8-3e3b-49ed-b38b-64639e5bc69e>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00311.warc.gz"} |
Bonferroni rules
« previous post | next post »
The most recent xkcd illustrates the problem of multiple comparisons:
(As usual, click on the image for a larger version.)
After 14 more "not significant" results,
The mouseover title:
"'So, uh, we did the green study again and got no link. It was probably a–' 'RESEARCH CONFLICTED ON GREEN JELLY BEAN/ACNE LINK; MORE STUDY RECOMMENDED!'"
As Wikipedia explains,
The Dunn-Bonferroni correction is derived by observing Boole's inequality. If you perform n tests, each of them significant with probability β, (where β is unknown) then the probability that at least
one of them comes out significant is (by Boole's inequality) ≤ n⋅β. Now we want this probability to equal α, the significance level for the entire series of tests. By solving for β, we get β = α/n.
Relevant background reading: M R Munafò et al., "Bias in genetic association studies and impact factor", Molecular Psychiatry 14: 119–120, 2009, discussed here.
Among R.A. Fisher's several works of public-relations genius, the greatest was the appropriation of the word significant to mean "having a low probability of occurrence if the null hypothesis is
Update — If you'd like to do a version of the jelly-bean experiments yourself, without cutting into your Minecraft time, you could try something like this (in R), which generates a couple of sets of
random numbers (from an underlying distribution that never changes) and then tests the null hypothesis that the means of the two sets are equal.
NSAMPLES <- 15
NTESTS <- 100
pvalues <- 1:NTESTS
for(n in 1:NTESTS){
y1<- runif(NSAMPLES); y2 <- runif(NSAMPLES)
X <- t.test(y1,y2,var.equal=TRUE)
pvalues[n] <- X$p.value
Then sum(pvalues<0.05) turns out to be (the first few times I ran it) 7,7,6,7,3,4,5, …
In other words, about 5% of the time, the null hypothesis is estimated to have a probability below 5%. (We should resist the temptation to violate Charlie Sheen's trademark on "Duh".)
Moacir said,
Small point: I was completely confused by the cartoon from the edited version given above, which does not include the panel where green jelly beans are tested and yield a p < .05.
Ellen K. said,
It's 14 (not 15) more "no link", and one "found a link" (for green).
Rodger C said,
Still smaller point: I went straight from XKCD to this site, and at first I thought I was suffering a computer glitch.
Chris said,
Can't help but be reminded of the recent Supreme Court brief decrying the misunderstanding of significance testing (pdf).
[(myl) One of the authors of that brief has written a lovely pamphlet "The Secret Sins of Economics", discussed here. One of the sins is "significance" testing without a loss function.]
Lauren said,
@Roger: I went from XKCD directly here KNOWING I would find a post about it!
alex boulton said,
[(myl) I suspect that this reversal is a case of misnegation, caused by the fact that the journalist phrases the question in a positive way ("Jellybeans cause acne!"), but the scientists give it back
in a negative form ("we found no link between jellybeans and acne").
Either way, this illustrates an interesting point about statistical inference. What the journalist wants — and what Randall's scientists say they're giving her — is an evaluation of the hypothesis
that jellybeans cause acne. But what their statistical test (whatever it happens to be) actually gives them has nothing to do with causation. It's an evaluation of the null hypothesis that the acne
measures of the jelly-bean subjects and the non-jelly-bean subjects could have been the result of sampling twice from the same population.
As the cartoon illustrates, if you keep making repeated random samples from the same population, about 5% of the time you'll get a result whose probability of being the result of random sample from
the same population is less than 5%. However, a correct statement of what the statistical test actually tests is too cognitively complex for most people to be comfortable with it — and I suspect that
this includes many biomedical and other researchers.
Of course, even if we believe that the rejection of the null hypothesis is probably right, this leaves open many alternatives, such as (in this case) the possibility that acne causes jelly-bean
consumption, or that some third factors influences both.]
John Cowan said,
In the Secret Sins post, you wrote: "And in fact I think that there are some remarkably similar difficulties in contemporary academic linguistics, a point that might be worth taking up in some future
Did that ever happen? (Hint, hint.)
Mr Fnortner said,
On a distantly related question, if you're looking for something that may not exist, does the probability of finding it increase as your search exhausts places to look?
Rob P. said,
@Chris – The S.Ct. did uphold the 9th circuit, but did not appear to rely much on the analysis in that brief. Instead, the court noted that, "The FDA similarly does not limit the evidence it
considers for purposes of assessing causation and taking regulatory action to statistically significant data. In assessing the safety risk posed by a product, the FDA considers factors such as
“strength of the association,” temporal relationship of product use and the event,” “consistency of findings across available data sources,” “evidence of a dose-response for the effect,” “biologic
plausibility,” “seriousness of the event relative to the disease being treated,” “potential to mitigate the risk in the population,” “feasibility of further study using observational or controlled
clinical study designs,” and “degree of benefit the product provides, including availability of other therapies.”
Because regulatory agencies and medical professionals might rely on other than statistically significant studies, it is reasonable for an investor to do so as well. That is, they didn't concentrate
much on what statistical significance really means, but instead determined that statistical significance is not a good bright line rule as to what might be material to an investor's decision to buy a
Matt Heath said,
@Mr Fnortner: The probability of finding it at all or the probability of finding it in the next place you look?
The former will decrease since when you strike off an option then all remaining options (including the thing not existing) will have their probabilities revised upwards according to Bayes' theorem.
The latter would rise if all locations were equally likely to begin with. If the prior probabilities fell away too quickly (and you were checking the most likely places first) it would fall.
Steve Kass said,
However, a correct statement of what the statistical test actually tests is too cognitively complex for most people to be comfortable with it — and I suspect that this includes many biomedical
and other researchers.
I agree with the part after the dash.
Unfortunately, the jargon of statistics is dreadful — significant is only one example of statistical terminology with a disciplinary meaning that’s counter-intuitive to the vernacular. This creates
an greater obstacle than essential mathematical sophistication, I think, to the clear expression and understanding of statistical concepts.
Earlier this week, the Wall Street Journal’s Numbers Guy, Carl Bialik, wrote about statistical significance and the Court’s recent decision. Bialik, like most journalists, failed to express the
statistics correctly. My guess is not that the concepts were cognitively too complex for Bialik to be comfortable with them, but that he failed to attribute much importance to being precise about
KevinM said,
"statistical significance is not a good bright line rule as to what might be material to an investor's decision to buy a security."
Yes; it's possible to quantify in dollars the reduction in value of a house that people think is haunted. The stock market is a Keynesian beauty contest, governed not by what the judges think but by
what they think other judges think (about what other judges think about what other judges think ….) The interesting question is whether the courts should be strictly market-neutral or whether there
are good institutional reasons why they should resist letting people sue and recover damages on this basis.
Carl said,
The moral of the story is that the higher a p-value a field allows, the higher the percentage of material produced by the field will be wrong—even under the best case scenario (no fraud, no
experimental error, etc.).
D.O. said,
I think that among other udoubtly correct things about misunderstanding and misuse of statistics that were already mentioned, there is one thing that was not pointed out. I think that looking first
for type I error probability is reasonable approach when evaluating a Good Thing (green jelly beans cure acne!), if this test is passed than one should look at other things. But with Bad Things
(green jelly beans cause acne!) it is more reasonable to start with estimating type II error.
Rubrick said,
Not directly addressed in the xkcd is the fact that there's nothing special about the multiple comparisons being "of the same sort". If you take any 20 studies which yield a conclusion with p > .05 —
even if they're completely unrelated — one of them will likely be due to chance. This is of course why it's so crucial that studies be repeated; but I think that in a lot of fields it often doesn't
happen. (Who wants to spend their precious grant money repeating a study that's already been done, and for which someone else has already reaped the glory?)
This issue and much else are addressed in this essay, which I think has been mentioned previously on LL.
Jonathan D said,
The "Duh" is even more obvious when you realise that the p-value does not give an estimate of the probablity of the null hypothesis, but simply the probability, assuming the null hypothesis, of
getting the results your did or something even further from expectated.
Mark F. said,
A NY Times article about a possible discovery at Fermilab had this passage:
The key phrase, everyone agrees, is “if it holds up.” The experimenters estimate that there is a less than a quarter of 1 percent chance their bump is a statistical fluctuation, making it what
physicists call a three-sigma result, enough to attract attention but not enough to claim an actual discovery. Three-sigma bumps, as every physicist knows, can come and go.
The difference in the threshold for being interesting is striking, but I guess it probably just follows from the large number of trials that are being figured into a Bonferroni calculation.
army1987 said,
IMO, frequentist probability theory is conceptually backwards when you're trying to figure out a mechanism from its behaviour (rather than vice versa, as when calculating the probability of dice toss
results when you already know the dices are unbiased). That's what Bayesian probability theory is for. I think the only reason people (not counting those who don't understand the difference) use it
is that when N is big enough and the distributions are all in the ballpark of being Gaussian it gives similar results with simpler maths.
Signifikant | Texttheater said,
[…] Akt gelungener göttlicher Kommunikation zu sehen. Daran fühlte ich mich erinnert, als ich gestern im Language Log las, die Schöpfung des Begriffes statistisch signifikant (von lat. signum:
Zeichen!) sei der […]
Brett said,
@Mark F.: In particle physics, the standard for discovery of a new spectral feature (e.g. a new particle type) is always quoted as requiring 5 sigma. Confirmation of a discovery is sometimes said to
require 10 sigma. The existence of a huge number of simultaneous trials is exactly why.
Phil said,
@Mark F: In particle physics we frequently get interested by effects at the "2 sigma" level, but the standard to be able to claim a discovery is almost universally agreed to be 5 sigma (i.e., p
<3e-7). This is partly because of multiple comparisons (although this tends to be corrected for) but mostly because our results often depend strongly on corrections taken from models of processes
which might not be well known, and the idea is that requiring 5 sigma significance puts you beyond the level where problems with the models are likely to cause a fake signal.
@army1987: I agree, although the situation is a bit different in particle physics, where nearly everyone favours frequentist methods (see, eg, Section 6 of http://www.physics.ox.ac.uk/phystat05/
proceedings/files/oxford_cousins_final.pdf ), even for small N
Rebecca said,
On a more minor point of your post–
Naomi Baron has studied cultural differences in cell phone use and in views of the (in)appropriateness of speaking and texting in public. From her website (http://www.american.edu/cas/faculty/
Baron spent the 2007-2008 academic year gathering data on university student use of (and attitudes towards) mobile phones in Sweden, the US, Italy, Japan, and Korea. Findings from the study
appear in the themed section Baron edited of an issue of New Media & Society (2010), in the Danish journal Language at Work, and in the forthcoming book Cultures of Participation.
Rebecca said,
Whoa, sorry. Wrong post.
Theo Vosse said,
On a more linguistic point, and related to the comment on statistical jargon: shouldn't the statement in the 2nd panel (and following) be: we did not find a link …? To truly establish that there is
no link would require a power analysis; just p > 0.05 is not enough.
mgh said,
Among R.A. Fisher's several works of public-relations genius, the greatest was the appropriation of the word significant to mean "having a low probability of occurrence if the null hypothesis is
In fact, I've dealt with copy-editors at journals who insist on using "significant" only when discussing statistics. This is one of the few copy-editorisms I go along with, since it's easy to
substitute "substantial" for the other use, and since statistics are used confusingly enough without adding uncertainty as to whether they're being used at all!
Randal said,
The discussion of "significant" reminds me of a passage in Edward Tufte's rant "The Cognitive Style of PowerPoint" in which he dissects a NASA slide given by engineers to managers while the Columbia
was still in orbit following the foam strike:
The vaguely quantitative words "significant" and "significantly" are used 5 times on this slide, with de facto meanings ranging from "detectable in largely irrelevant calibration case study" to "an
amount of damage so that everyone dies" to "a difference of 640-fold." None of these 5 usages appears to refer to the technical meaning of "statistical significance." | {"url":"https://languagelog.ldc.upenn.edu/nll/?p=3074","timestamp":"2024-11-14T04:24:29Z","content_type":"application/xhtml+xml","content_length":"112721","record_id":"<urn:uuid:a2065d3e-4f6f-48d4-a256-dddc41755178>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00439.warc.gz"} |
Function analysis | Datagrok
Parameter optimization
Parameter optimization solves an inverse problem: finding the input conditions that lead to a specified output of the model. It computes inputs minimizing deviation measured by loss function.
Using Datagrok fitting feature, you can improve performance and accuracy of a model.
To run parameter optimization:
• Click the "Fit inputs" icon on the top panel. Fitting View opens.
• In the Fit block, use switchers to specify inputs to be found:
□ Set min and max values for each selected item. They define the variation range
□ Set values of all other inputs
• Set output constraints in the Target block:
□ Use switchers to specify target outputs
□ Set target value for each selected item
• Specify settings of fitting:
□ Choose numerical optimization method (in the method field). Click the gear icon to specify its settings
□ Set loss function type (in the loss field)
□ Specify number of points to be found (in the samples field)
□ Set the maximum scaled deviation between similar fitted points (in the similarity field): the higher the value, the fewer points will be found
• Click the "Run" icon on the top panel to perform fitting. You will get a grid containing
□ loss function values and fitted inputs
□ viewers visualizing the goodness of fit
□ line chart showing the loss function minimization
An inverse problem may have several solutions. Specify their expected number in the samples field. To filter fitted points, set similarity:
• it is the maximum scaled deviation between "similar" points
• the higher the value, the fewer points will be displayed
Table output
Apply the feature to models with table outputs as well:
• Specify the target dataframe in the table input
• Set dataframe column with values of independent variable (in the argument choice input)
Open Context Panel (F4). You will get the model run corresponding to the selected grid row:
Platform function annotaion
Apply parameter optimization to any function with the RichFunctionView editor. Add meta.features: {"fitting": true} to enable it:
//name: Test
//language: javascript
//input: double x
//output: double y
//editor: Compute:RichFunctionViewEditor
//meta.features: {"fitting": true}
let y = x * x;
See also
Sensitivity analysis
Sensitivity Analysis runs the computation multiple times with varying inputs, and analyzes the relationship between inputs and outputs. Datagrok provides the following methods:
• Monte Carlo explores a function at randomly taken points
• Sobol decomposes output variance into fractions, which can be attributed to inputs
• Grid studies a function at the points of a grid with the specified step
To run the sensitivity analysis, click the Run sensitivity analysis () icon on the top panel, choose a method, specify inputs and outputs, and click RUN.
Monte Carlo
Once you've chosen it in Method
• Set in Samples the number of random points
• Use switchers to specify varied inputs and outputs to be analyzed
• Press RUN or on the top panel. You will get
Use the sliders in the PC plot to filter the model evaluations:
When exploring complex models, some evaluations may be of particular interest. To get them:
• Click on grid row with the required input and output values
• Open Context Panel (F4). You will get the function run corresponding to the selected row
This method performs variance-based sensitivity analysis and decomposes the output variance into fractions, which can be attributed to inputs. It provides the same visualizations as Monte Carlo and
bar charts showing Sobol' indices:
• First-order indices indicate the contribution to the output variance of varying each input alone
• Total-order indices measure the contribution to the output variance of each input, including all variance caused by its interactions with any other inputs
This method evaluates a function at the points of uniform grid within the specified ranges. It provides the same visualizations as Monte Carlo:
Sensitivity Analysis can be applied to any function with the RichFunctionView editor. Add meta.features: {"sens-analysis": true} to enable it:
//name: Test
//language: javascript
//input: double x
//output: double y
//editor: Compute:RichFunctionViewEditor
//meta.features: {"sens-analysis": true}
let y = x + 3;
Learn more | {"url":"https://datagrok.ai/help/compute/function-analysis","timestamp":"2024-11-06T02:02:20Z","content_type":"text/html","content_length":"32571","record_id":"<urn:uuid:f838d37f-3905-4d72-b5eb-1444da1bec01>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00789.warc.gz"} |
8,208 research outputs found
We present a study of active learning pedagogies in an upper division physics course. This work was guided by the principle of deliberate practice for the development of expertise, and this principle
was used in the design of the materials and the orchestration of the classroom activities of the students. We present our process for efficiently converting a traditional lecture course based on
instructor notes into activities for such a course with active learning methods. Ninety percent of the same material was covered and scores on common exam problems showed a 15 % improvement with an
effect size greater than 1 after the transformation. We observe that the improvement and the associated effect size is sustained after handing off the materials to a second instructor. Because the
improvement on exam questions was independent of specific problem topics and because the material tested was so mathematically advanced and broad (including linear algebra, Fourier Transforms,
partial differential equations, vector calculus), we expect the transformation process could be applied to most upper division physics courses having a similar mathematical base.Comment: 31 page
Let [omega] be a nonempty open, simply connected subset of the complex plane and let [delta] = \z Ç Ç z Ç \u3c 1. Then there is a bijective analytic mapping g: \doubc - [macron][omega] â \doubc
- [macron][delta]. We note that g has a continuous extension to the boundary [gamma] of [omega]. Let A([delta]) be the Banach algebra of functions continuous on [macron][delta] and analytic on
[delta];We consider the tranformation T mapping f in A([delta]) to (Tf)(w) = 1 [over] 2[pi] i ϔt[limits][subscript][gamma] f(g(t)) [over] t-w dtin the Banach algebra of functions continuous on
[macron][omega] and analytic on [omega]. The mapping T is called the Faber transformation and is an injective continuous linear transformation;It is of interest to determine the properties of f which
are preserved under the Faber transformation T. In particular, we assume that f satisifies a Lipschitz condition on [macron] [delta] and consider the Lipschitz class of Tf on [macron][omega]. We show
that the Lipschitz class of Tf is affected by the Lipschitz class of f and the smoothness of [gamma], and, under suitable conditions on [gamma], we obtain qualitative results when g satisfies a
Lipschitz condition
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cigarette smoke. Pathologic findings in ARDS have been limited to case reports of open lung biopsies and post-mortem studies but there is some new information from a recent pathology study relative
to the frequency of diffuse alveolar damage and the severity of arterial hypoxemia. Further, therapy with lung-protective ventilation and fluid conservative protocol has improved outcomes, but
several new trials are in progress to test several promising strategies
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because, although they do not explicitly make the point, to call for the embrace of phenomenological-oriented research into GCAs, the authors are accepting the fundamental importance of individual
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Monte Carlo techniques are used to model nonlinear particle acceleration in parallel collisionless shocks of various speeds, including mildly relativistic ones. When the acceleration is efficient,
the backreaction of accelerated particles modifies the shock structure and causes the compression ratio, r, to increase above test-particle values. Modified shocks with Lorentz factors less than
about 3 can have compression ratios considerably greater than 3 and the momentum distribution of energetic particles no longer follows a power law relation. These results may be important for the
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DICT/1.0 Title: MOP Dictionary Front: dictionary [aalii] \ah' lee\ n. ('00) A vowel-bearing tree. [Abbott scratch] \ab' et skrach'\ n. ('02) Ludicrously extensive scratch work turned in during the
[ELMO]. [aerobie] \air oh' bee\ \air' oh bee\ n. ('00) A frisbee-like object, often thrown on inaccessibly high roofs. [Aleph-null] \a' lef nuhl'\ n. ('02) The cardinality of the set of Anders'
[juggling] balls; known to be greater than or equal to 19, and, in fact, conjectured to be equal to 19. [all your] \ahl' yohr'\ base. ('01) Are belong to us. [AM-GM] \am' gam\ n. ('00, '02) abbr. The
Arithmetic Mean-Geometric Mean inequality, a popular inequality used in [bunching]. n. ('00) abbr. All Girls, All Math, a group of hard-studying girls who do modular arithmetic (also: AG-AM). n.
('02) abbr. All Math, Guys Mostly (see: [MOP]). n. ('02) Obscure inequality stating that MOP > AG-AM. vt. ('00) To apply the AM-GM inequality. ant. ('00) [Cauchy]. [anagram tiles] \an' e gram' tilez'
\ n. ('01) A set of scrabble tiles in which the second blank is an S. [Antarctica] \ant ark' tih kuh\ n. ('00) A southerly continent with no native population but two internationally competing math
teams. [apple] \a' pel\ n. ('00) A thing to be counted in a problem of Gabe's lecture. n. ('01) A silver medal. [Axiom of Choice] \ak' see em ev chois'\ n. ('02) A diabolical device used in the
torture of innocent spheres, capable of cutting up sets arbitrarily with no regard to their volume. [bagel man] \bay' gel man'\ n. ('01) A sessile, edible, humanoid life form. [banana] \be nan' e\ n.
('00) A brittle, green-brown object; air freshener; locus of spontaneous generation. n. ('00) A fruit that may be Jensened if held correctly. n. ('01) A gold medal. n. ('01) A device for enumerating
certain functions defined on sets of lattice points. [base case] \bays' kays'\ n. ('02) Object that are all belong to us. [basically] \bay' sih ke lee\ \bay' sih klee\ interj. ('01) Signals the
beginning of a proof. [bishop] \bihz' hahp\ n. ('00) A piece in the game of Bughouse that moves diagonally. [Bjorn's Theorem] \byohrnz' theer' em\ n. ('01) The projective dual of the inverse of the
Mostly-Thick Circle Theorem. [Blazin™] \AAAAHHH!!!' wah' ter!!!\ adj. ('00) A deadly wing flavor. [Blidge] \blihj\ n. ('03) What Yan says to make fun of people playing a perfectly reasonable card
game in Japan. n. ('03) The name for a 5-person Bridge game invented to make Yan shut up. [bonk] \bonk\ interj. ('01) Yeah. [booga booga] \boog' uh boog' uh\ n. ('00) Mao rule. Congruence modulo 5.
[bostard] \bos' terd\ n. ('02) Taktin's name for Josh Lim, esp. in Diplomacy. n. ('02) Josh Lim's name for Taktin, esp. in Diplomacy. [bottle] \baht' el\ n. ('01) A finger trap. [Brianchon's Theorem]
\bree an' chenz theer' em\ n. ('01) Projective dual of [Pascal's Theorem]. [brick] \brik\ punc. ('01) Separates two independent clauses in the game of Associations. [buffalo] \buf' e lo\ n. ('01) A
large land mammal. vt. ('01) To intimidate, as in "buffalo buffalo buffalo buffalo buffalo". [buffalo generating function] \buf' e lo jehn' e rait ing funk' shen\ n. The function b(x) = x(x^2 - 1)^2/
(x^2 + x - 1)^2 = x + 2x^2 + 3x^3 + 6x^4 + 11x^5 + 20x^6 + 36x^7 + 64x^8 + 113x^9 + 198x^10 + ..., which generates the number of possible interpretations of the sentence "buffalo buffalo ... buffalo"
of length n. [buh-shi] \buh' shi'\ interj. ('01) Same thing as in English. [Bulgarian compass] \buhl gair' ee en kuhm' pes\ n. ('01) A compass, one of whose components is Bulgaria. [bunching] \bunch'
eeng\ n. ('00) The process of massive [AM-GM]. syn. ('00) [Donkey bunching]. [Buniakovski] \boo nee uh kahv' skee\ \boo nuh kow' skee\ n. ('01) The Romanian name for [Cauchy]. [busy beaver] \bihz' ee
bee' ver\ n. ('01) An active member of the genus Castor. n. ('01) A function from N to N that grows ludicrously fast. [button] \buht' en\ n. ('01) Push my [brick] halibut injure. [Calimass] \kal' i
mass'\ n. ('01) A US state that borders both the Pacific coast and the Atlantic coast; noted for its disproportionate population of MOPpers. [Candyland] \kan' dee land'\ n. ('00) The residence of Mr.
Fat, Mr. Taf, and Mr. =), filled with painted sidewalks, candy bars, dried fruit, and positive integers. n. ('00) The residence of Tiankai (who's been abducted by Zuming!!!), Philips Exeter Academy.
[catf***er] \kat' f*** er\ n. ('00) The lowest rank in Capitalism; a conscientious objector to the conservatism of the social order. n. ('00) One who practices a particularly straining form of animal
husbandry. [Cauchy] \koh' shee\ n. ('00) The Cauchy-Schwarz inequality. n. ('00) A god worshipped by Yan. vt. ('00) To apply the Cauchy-Schwarz inequality. ant. ('00) [AM-GM]. [chalkboard] \chok'
bohrd\ n. ('01) A vehicle for random scribblings. (See also: [whiteboard].) [Chess] \ches\ n. ('00) A variant of Ping where certain moves are illegal. n. ('01) A variant of Quantum Chess in which
pieces are only allowed to be on one square at a time. n. ('01) A variant of Atomic Chess in which the level of violence is reduced. [C'hi] \chh hee'\ \see' high'\ interj. ('00) An exclamation of the
joy of Lawrence Detlor in 1998. interj. ('00) An exclamation of the joy of the Lawrence Detlor-Austin Shapiro construct, for as long as it lasts. syn. ('00) [W'hu]. ant. ('00) [D'oh]. [circular
definition] \sir' kue ler def' i nish' en\ n. ('00) See: [circular definition]. [clay] \klai\ n. ('01) A prize awarded to acknowledge creative scratchwork. [clubs] \harts\ n.pl. ('00) Hearts.
[complex conjugate roots] \kom' plehks kon' je get roots'\ n.pl. ('01) A pair of hot dates, one of each gender. [computer lab] \kem pyoo' ter lab'\ n. ('02) A facility whose real purpose is entirely
misunderstood by the UNL staff. [consult] \ken sult'\ vt. ('01) To receive advice on a [MOP Test] write-up. vt. ('01) To give advice on a [MOP Test] write-up. [Corollary to the Square Table Theorem]
n. ('02) Theorem that states: You don't need food. (See: [Square Table Theorem].) [cubic interpolation] \kyoo' bihk ihn ter' pe lay' shen\ n. ('02) A method utterly unsuitable for the determination
of elevator floor numbers. [cucumber] \kyoo' kuhm ber\ n. ('00) Best pagan god, after Anders. [curfew] \ker' fyoo\ ?. ('00) Definition unknown; some MOPpers have a recollection of a relation between
this, going to sleep and Zuming. [cushion] \cush' en\ n. ('00) A part of a couch to sit on. n. ('00) An ineffective weapon. n. ('00) See: [pillow]. [cwm] \koom\ n. ('01) A geological formation noted
for sparsity of vowels. [Dalles] \dalez\ n. ('00) Place. River rapids. n. ('00) A word search game invented at MOP and named after the most interesting word formed during trial runs. [Danandron] \dan
ahn' dren\ n. ('03) A triexistant entity composed of Daniel, Anders, and Aaron. [death] \dehth\ n. ('02) Tiankai's final source of pleasure, which (like the primary and secondary) will hopefully
occur while sleeping. See also: [sex], [math]. [dictionary] \dihk' she nair' ee\ n. ('01) A random collection of inside jokes. [differential equations] \dif' e rehn' shel e kway' zhens\ n. ('01) A
math course slightly higher than Algebra I. [dildo] \dihl' doh\ n. ('00) Friendly, non-carbon-based life. [Diplomachess] \di plo' me ches'\ n. ('00) A game winnable by Dani Kane. (also: Amazons)
[diwide] \di wide'\ n. ('01) To [murderply] by a reciprocal. [duct tape] \duhkt' taip\ n. ('00) A weapon of real (or imaginary) pranks. [dude] \dood\ n. ('00) Personification of an area. n., pron.,
vt., adj., interj. ('02) The most versatile word in the teenage male's vocabulary. [dude theory] \dood' theer' ee\ n. ('00) A theory of areas. [duh] \duh\ pron. ('01) The unsurprising winner of a
competition. [dumbass] \duhm' as\ n. ('00) A method from Kiran's lecture used for solving inequalities (comes with lifetime guarantee). n. ('00) Idiot. n. ('00) A person who chooses to utilize a
time-consuming yet trusty method to solve a mathematical exercise, in lieu of a more inspired yet much better concealed method. n. ('01) A state of mind often entered by problem solvers. n. ('02) A
method that does not work on certain evil inequalities of high degree, in spite of Kiran's lifetime guarantee. ant. ('00) [smartass]. [dysfunctional equation] \dis funk' shen el e kway' zhen\ n.
('01) A functional equation with no solutions. [Dyslexia] \dihs lehk' see e\ n. ('01) A small country, led by a gerbil, which nonetheless won the [ELMO]. [elevator] \ehl' e vay ter\ n. ('00) A place
for playing Molecule. [Eleventh Commandment] \e lehv' enth ke mand' ment\ n. ('02) The First Commandment mod 10. n. ('02) "Thou shalt not have two arrows pointing to the same number." [eleventh
floor] \e lehv' enth flor'\ n. ('02) An extension of the girls' bathroom containing the entire female population of MOP. [ELMO] \el' moh\ n. ('00) abbr. Easy Little Math Olympiad, Ex-experimental
Lincoln Math Olympiad, etc. n. ('00) A cute, fuzzy, lovable, red Sesame Street character. n. ('02) abbr. Extremely Last-Minute Olympiad. [EOP] n. ('00) abbr. End of proof. n. ('00) An end of proof
symbol. [fake announcements] \fayk e nouns' ments\ n.pl. ('02) Accounts of the goings-on at MOP that, although fictional, still give a better idea of the atmosphere at MOP than the real announcements
do. [Fantasy Game] \fan' te see gaym'\ n. ('02) An exercise in which the supreme weirdness that abides in the minds of MOPpers is fully revealed. [father] \fah' ther\ n. ('02) See: [Chuck]. syn.
('02) [mother]. [Fergie] \fer' gee\ n. ('02) Jason Ferguson. n. ('02) An act of stupidity above and beyond the call of duty; usually committed by Jason Ferguson. [Fermat] \fer mah'\ \fer' met\ n.
('00) Either the little creator of theorems (see: [Titu]) or the creator of little theorems. [field] \feeld\ n. ('02) A place where agents may be compromised. [First Corollary to the Round Table
Theorem] n. ('02) Theorem that states: If there are n trays at a round table and [Riz]/e starts reciting the [Sarah Dessert], there will soon be n - many. (See: [Round Table Theorem].) [foolproof] \
fool' proof\ adj. ('01) Unable to be broken by fools. [Foosball] \fooz' bahl\ n. ('00) A game in the Selleck Mushroom that Dave Shin loves to play. (See: [mad skillz].) n. ('00) A game in the Selleck
Mushroom that kills your hands when you lift up the table to get the foosballs back. [fornicate] \fohr' ni kayt\ vi. ('00) To type (darn auto-correct). vi. ('00) To sit. vi. ('00) To speak. (The last
one in particular may cause a lot of trouble for unwitting Word users... imagine a guy giving a talk at a convention starting off with "The reason why I've come to fornicate to you today is...")
[frisbee] \frihz' bee\ n. ('01) Second avatar of Mike Hamburg; carried Dan Jerison to the south pole. [gastropod] \gas' tre pod\ n. ('00) Gabriel. n. ('00) A snail or slug. [George's Maneuver] \jorj'
ez me noo' ver\ n. ('01) A [shrotum] that looks like a zoom. [God] \god\ n. ('01) A nonexistent entity in Silent Football. n. ('01) Something Anders never pokes for a scare. n. ('01) The rule-maker
in Eleusis. [Harry's Shoes] \hair' eez shooz'\ n. ('01) A store with which Lawrence Detlor has no connection. [hazardousness] \haz' er des ness\ n. ('01) A function from the set of all strings on an
alphabet of 3 letters to the positive reals. [hegemonious] \heh ge mon' ee us\ adj. ('00) Habitating the huger of two homogeneous hordes; hence, having hegemony; in particular: unbalanced two-to-one,
as when two of three binary strings contain a 0 or 1 in some digit, but the third string does not. (From a MOP [homework] problem.) [Hilbert's Theorem] \hil' bertz theer' em\ n. ('01) A simple
corollary to Hildegard's Theorem; like Hildegard's Theorem, often useful when nothing else seems to work. [hole] \hohl\ n. ('01) A container for a pigeon. [homework] \hohm' werk\ n. ('02) Obsolete;
see: [team contest]. [hoMOPpety] \he mop' e tee\ n. ('00) The application of a [homothety] to the word "MOP". [homothety] \he moth' e tee\ n. ('00) A size change about a point. [horse] \hors\ n.
('00) White powder distributed in Zip-loc baggies and "used" (esp. by Luke) in the laundry rooms. n. ('00) See: [knight]. [I] \eye\ pron. ('02) 2·[Riz]/p. pron. ('02) [Riz]/e. [idiot] \id' ee et\ n.
('01) An ingenious person who carries around infinitely many tools. [Inna's room] \ee' nuhz room'\ n. ('01) The natural habitat of Neil. [insane ordering] \ihn sain' or' der eeng\ n. ('03) The
tabulation of a peer pressure position obtained by listing the owners of the cards in ascending order. [invert] \ihn vert'\ vi., vt. ('00) To turn a Cartesian plane inside out; a useful method for
solving circular geometry exercises or catching Tiankais. [Ireland] \eyer' lend\ n. ('00) A wellspring of risible trivialities. [isosceles] \eye sos' e leez\ adj. ('00) Having two equal edges; a
strict generalization of "triangular". [jaywalking] \jay' wok' eeng\ n. ('02) Susan B. Anthony's anti-drug. n. ('02) The activity that Chuck is most likely to do. [josh] \josh\ vt. ('00) To tease.
[juggling] \juhg' gleeng\ n. ('02) The art of throwing balls in the air over and over without dropping any. [kamalshallow] \kahm' el shal oh\ adj. ('00) Not kamaldeep. [knight] \ke nihg' et\ n. ('00)
A piece in the game of Bughouse that moves two squares forward and one square sideways. [Lagrange multipliers] \le grahnj' mer' de ply' ers\ n. ('00) A method of brute force for constrained
inequalities. [laptop] \lap' top\ n. ('01) A nonexistent entity on which one cannot play games. [learn] \lurn\ vt. ('03) To assimilate information into Mathematica. [Licky] \lihk' ee\ n. ('02)
Alternative proposed name for [Riz]; it was decided that "Riz" was the better name. [like] \like\ interj. ('02) Peanut butter. [Lorentz] \lohr' ents\ n. ('01) Guide for the Netherlands at IMO 2001.
[mafe] \mayf\ vi. ('00) To engage in boneheaded persecution of innocents. [Mafiotic] \me faht' ihk\ adj. ('00) Of a townsperson or police inspector, behaving in the manner of the Mafia. adj. ('00)
Failed attempt at the adjective form of "mafia." Deprecated; see: [mafious]. [mafious] \mah' fee es\ adj. ('00) Mafia-like. [majorize] \may' jer ize\ vt. ('00) To have the same sum but larger partial
sums in comparison with another series. vt. ('00) To overpower. [math] \math\ n. ('01) Tiankai's secondary source of pleasure, which (like the primary) occurs while sleeping. See also: [sex],
[death]. [metanonexistent] \meh' te non ihg zis' tent\ adj. ('01) Fictional in the context of an already nonexistent universe. [mind-melting] \mined' mehlt' eeng\ adj. ('00) Of a card game, demanding
lightning reflexes, and unwinnable by MOPpers in competition with AG-AM. [MOK] \mok'\ n. ('01) A summer program at which everything is mocked. [Molecule] \mol' e kyool'\ n. ('02) A rave performed to
elevator music while high on [math] or [Mountain Dew]. [Mongolia] \mon gohl' ee uh\ n. ('00) The country where the cancellation of the IMO instilled fear into companies and forced them to give Kiran
money for IMO 2001. [Mongoliastan] \mon goh' lee e stan'\ n. ('02) A winner of the Color Country game. [MOP Test] \mop' test\ n. ('00) Melanie's primary source of sleep. n. ('01) A variant of Mao in
which one is allowed to ask questions for the first half hour. [MOPer] \mohp' er\ n. ('02) One who mopes. [MOPper] \mop' er\ n. ('00) A participant in the Math Olympiad Program. Despite its
simplicity, this word is often misspelled; see: [MOSPer], [MOPer], [MOPster]. [MOPster] \mop' ster\ n. ('02) (Vulgar slang) A highly insulting epithet for the participants of MOP. [Morley triangle] \
mohr' lee try' ayng gel\ n. ('00) Order sprung from chaos; proof of the existence of the universe and the existence of art. [MOSK] \mosk\ n. ('01) The official (but incorrect) name for [MOK]. [mosp]
\mosp\ n. ('00) A cross between a moth and a wasp. [MOSP] n. ('02) abbr. Math Olympiad Stalker Program. [mosper] \mosp' er\ n. ('03) One who hunts [mosp]s. [MOSPer] \mosp' er\ n. ('02) Oh, please.
[mother] \muhth' er\ n. ('02) See: [Chuck]. syn. ('02) [father]. [motivate] \moh' te vayt\ vt. ('00) To equip with an anthropomorphism of the Creation. [Mountain Dew] \moun' ten doo'\ n. ('02) The
"Good Stuff"; the only substance on which Toto will ever get high. [Mr. Dictator] \mihs' ter dihk' tay ter\ \mihs' ter rihch' erd spuhd'\ n. ('00) The leader of a silent football game. n. ('00) See:
[Zuming]. [msb] \msbuser\ n. ('01) The god of computer access. [muffin man] \muhf' ihn man'\ n. ('01) A hot date who lives on Drury lane and is claimed by some (e.g. Inna) to be less well known than
[complex conjugate roots]. [murderply] \muhr' der ply'\ vt. ('00) To multiply. vt. ('00) To murder. [nar] \nahr\ n. ('00) A really bad joke. n. ('00) A screw-up. n. ('00) Augmentation by three. vi.
('00) To perform a screw-up. interj. ('00) Exclamation of glee, due to augmentation by three. [narc] \nahrk\ n. ('00) One whose interests run counter to the design of Mischief; esp. (presumably) a
hostile invader from the Neihardt front desk. [nice] \nise\ adj. ('00) Austinglish for "awesome". adj. ('00) New and refreshing, as when n girls trade lots of Pokeballs in IMO shortlist problems and
every girl gets a different one from the one she started with. ant. ('00) [tiresome]. [nneerrr] \nneerrr\ interj. ('00) Originally a set of ghost letters. [nuclear launch] \noo' klee er launch'\ \
noo' kyoo ler launch'\ n. ('00) A three term arithmetic sequence. [obviosity] \ob vee ahs' ih tee\ n. ('01) A trivial assertion, esp. one incorrectly labeled as a theorem. [obviousland] \ob' vee es
land\ n. ('00) A place where everything can be done without proof. ant. ('00) [Turkey]. [orange] \ohr' ihnj\ n. ('00) A pair of [pear]s; equivalently, a group of four [apple]s. n. ('01) A bronze
medal. n. ('01) A bowling ball. n. ('01) A bowling pin. n., adj. ('00) The color of Oaz. n., adj. ('00) The color of Oaz's hair once. n., adj. ('00) The color of Oaz's sister's hair now. [pear] \pair
\ n. ('00) A pair of [apple]s. [paper dart] \pay' per dart'\ n. ('02) A conical projectile made of wood pulp, which is blown through a tube at a sheet of paper to determine the weirdness of various
IMO team members. [Pascal's Theorem] \pa skalz' theer' em\ n. ('01) Projective dual of [Brianchon's Theorem]. [Pell equation] \pehl' e kway' zhen\ n. ('01) A Diophantine equation that is not of the
form x^2 = 5y^2 + 4. [pepper] \pehp' er\ n. A stimulant that, when ingested, impedes one's ability to play Erf. [Pi] \pye\ n. ('00) The ratio of the circumference of a circle to its diameter; approx.
3.1415926535 8979323846 2643383279 5028841971 6939937510. n. ('00) The Greek letter p. [Pikachu] \peek' e choo\ n. ('00) AAAAAAHHH!!! [poetry] \poh' ih tree\ n.pl. ('02) Writings suitable for
reciting while [juggling]. [Popoviciu] \poh poh vee' choo\ n. ('01) An inequality used in the manufacture of popcorn. vt. ('01) Something that people who know they're MOPpers do. [pork] \pohrk\ n.
('00) A pawn behaving like a rook in a game of Proxy Chess. [porve] \porv\ vt. ('00) [Zuminglish] for prove. vt. ('01) To prove using murderplication (see: [murderply]). [primitive root] \prihm' ih
tihv root'\ n. ('00) Adam of the reduced residue class; object of the clergy's science and the laity's faith. [pronoun] \proh' nown\ n. ('00) A heinous part of speech. [pseudovet] \soo' doh veht\ n.
('03) A MOPper in 2003 who was in the Red or Yellow group in 2002. [public announcement] \puhb' lihk e nowns' ment\ n. ('00) A speech by Yan, esp. "WHAZZUP!?". [push-up] \push' uhp\ n. ('01) A unit
of gambling currency. n. ('01) 1/15 of a hint. n. ('02) A penalty for incorrect proofs that increases exponentially. [quagga] \kwahg' e\ n. ('01) An extinct horse-like mammal with stripes on half of
its body. [rat screw] \rat' skroo\ n. ('00) Hand screw. [redundant] \ree duhn' dent\ adj. ('00) See: [redundant]. See: [redundant]. See: [Dani]. [regent] \ree' jent\ n. ('01) One who exercises
dictatorial powers during the absence of the customary dictator (see: [Mr. Dictator]). [Riemann Chess] \ree' mahn ches'\ n. ('01) A variant of [Chess] in which one must prove the [Riemann Hypothesis]
before moving. [Riemann Hypothesis] \ree' mahn hye poth' e sihs\ n. ('01) A calculus problem rejected for use on the rookie [team contest]. [Riz] \rihz\ n. ('02) A bisexual coexistent entity
consisting of Riz/e (Liz Marcil) and 2·Riz/p (Ricky Biggs) that creates great difficulty with [pronoun] usage (see: [Rizself]). [Rizself] \rihz self'\ pron. ('02) Reflexive [pronoun] for [Riz].
[Round Table Theorem] n. ('00) Whenever n trays are placed on a round table in the Selleck dining hall, there's always room for n + 1. (See also: [First Corollary to the Round Table Theorem], [Second
Corollary to the Round Table Theorem], [Square Table Theorem].) [runza] \ruhn' za\ n. ('00) A type of rolled sandwich-like dumpling, endemic to Nebraska, whose contents are unknown yet appetizing.
[Sarah Dessert] \sair' e de zert'\ n. ('02) A place inhabited by mummies who travel by Camelot. [schoolwork] \skool' werk\ n. ('01) An article that may not be transported into Lincoln. [scissors] \
skiz' ers\ n. ('00) Implement of psychological torture. (cf. cigar) [Second Corollary to the Round Table Theorem] n. ('02) Theorem that states: If there are n people at a round table, k of whom are
males who ate dinner with Max Rosmarin on Saturday June 22, and Max comes to the table, there will soon be n - k + 1 (the 1 is Max). (See: [Round Table Theorem].) [sex] \sex\ n. ('01) Tiankai's
primary source of pleasure. See also: [math], [death]. [shame] \shaym\ n. ('01) A math competition that dyslexics take in Febraury. [shoes] \shooz\ n.pl. ('00) Has anyone seen mine?! —JB n.pl. ('00)
Ping-pong paddles. n.pl. ('00) Really, has anyone seen my shoes?! [shrotum] \shroh tem\ n., interj. ('00) An exclamation of unaccountability. [Simson] \sihm' sen\ n. ('00) See: [Simson line]. n.
('00) See: [W'hu]. [Simson line] \sim' sen line\ n. ('00) See: [W'hu]. [sketchy] \sketch' ee\ adj. ('00) Questionable; unscrupulous; titillating. [Smash Bros. tournament] n. ('02) Totally unorganized
thing of infinite duration. syn. ('02) [Mario Tennis tournament]. [smiley] \:-)\ n. ('00) Josh's mascot. n. ('02) [Riz]/e's boyfriend. [smoothing] \smooth' eeng\ n. ('00) The process of using
Jensen's inequality. [Spannsifer] \span' se fer\ n. ('02) A coexistent entity consisting of Andrew Spann and Eric Stansifer. [spoodle] \spoon' del\ n. ('00) A spoonlike serving utensil. n. ('00) A
unit of dry measure equal to the volume of such a utensil. [Square Table Theorem] n. ('02) Theorem that states: You don't need trays. (See also: [Corollary to the Square Table Theorem], [Round Table
Theorem].) [starpatchies] \stahr' pach' eez\ n. ('01) A hapless victim of a case of mistaken identities. [super magic box] \soo' per maj' ik box'\ n. ('01) A technique that would solve [Pell
equation]s if anyone could remember how to use it. [SYD] n. ('00) abbr. [Dyslexia]. [table] \tay' bel\ n. ('00) A MOPper habitat with an arbitrarily large carrying capacity. (See: [Round Table
Theorem].) n. ('02) A subset of the plane, to be tiled with as many trays as possible. n. ('02) A place where male MOPpers do not want to eat dinner with Max Rosmarin. (See: [Second Corollary to the
Round Table Theorem].) [team contest] \teem' kon' test\ n. ('02) The reason why nobody did their [homework]. [thank you] \thank' yoo\ interj. ('01) A heinous Silent Football expression. interj. ('01)
Please stop giving me penalty cards! [TI-92] \tee' eye' nine' tee too'\ n. ('01) A Turing-machine emulator. [tiankai] \tee an' kye\ adj. ('02) Everyone's favorite wing flavor. [Tigger] \tihg' er\ n.
('00) Hostage of [AM-GM] girl. n. ('00) There's only one. [Titu] \tee' too\ n. ('00) The leader of MOP and the IMO team. n. ('00) A lecturer who coincidentally happens to be the leader of MOP and the
IMO team. [Titu Andreescu] \tee' too on dreh ehs' koo\ \tee' too an jrehs' koo\ n. ('00) I don't know what it is but you better Run, it educates! (Anagrammatic thanks to Mike Khoury on this one.)
[Torricelli's Meanness Principle] n. ('01) States that one must be alive in order to be mean. [trig] \treeg\ n. ('00) abbr. Trigonometry. n. ('02) Something that should be moved. For great justice.
[trivial board game] \trihv' ee el bohrd' gaim'\ n. ('03) A game that is played on a trivial board, e.g. [Chess]. [vámonos] \vah' mo nos\ interj. ('01) Lunchtime! [Virgimass] \ver' jih mass'\ n.
('02) [Riz]'s home state. [water balloon] \wo' ter be loon'\ n. ('02) Instrument in Toto's devious plan to soak Pinyan; the plan was secretly approved by Chuck. n. ('02) Item of which Alex Schwendner
claimed to have 100; promptly sold out at the campus store as a result. n. ('02) Instrument of Chuck's doom in the hands of Luke on the last day of MOP. [West Virginia] \weeeeeeeaaaast' verrrr' jihn'
ye'\ n. ('02) Origin of Chuck. syn. ('02) [Hicksville]. [whiteboard] \wite' bord\ n. ('02) A convenience that UNL has provided for doodling; unfortunately, unaccompanied by a sufficient supply of
markers. (See also: [chalkboard].) [W'hu] \we hoo'\ interj. ('00) What could potentially have been an exclamation of the joy of Homer, had spelling been ever so slightly different. Known as the "
[Simson line]". syn. ('00) [C'hi]. ant. ('00) [D'oh]. [wombat] \wom' bat\ n. ('00) A creature that is always coming but never comes when spades are played in Mao. [WOP] \wwwwahp'\ n. ('01)
Well-Ordering Principle. n. ('01) Winter Olympiad Program. n. ('01) The co-coolest thing ever, tied with [math]. [wye] \wye\ n. ('01) The letter Y. n. ('01) A point in the complex plane equal to
Lawrence and Austin. [X-rated] \ehks' rayt ed\ adj. ('02) Tiankai bait. [yay] \yay\ interj. ('00) An exclamation said by Yan when [Cauchy] is good to him. interj. ('00) Mao rule. A prime number.
[Yensen's inequality] \yehn' senz ihn' e kwahl' ih tee\ n. ('01) The same thing as Jensen's inequality. [your mom] \yohr mahm'\ interj. ('01) A device for filling conversational voids. See also:
[dude], [like]. [Zagnut] \zag' nuht\ n. ('00) A socially mobilizing, albeit non-alcoholic, confection. [zed] \zehd\ n. ('01) The last letter of the British alphabet. n. ('02) A letter of the alphabet
that no Canadian can resist saying. [zero ring] \zeer' oh reeng\ n. ('00) The ring consisting of 0 (i.e. {0}). n. ('00) A place. n. ('00) A paradox. [zeta duck] \zay' te duhk'\ n. ('01) An example of
convergent evolution between waterfowl and Greek letters. [Zum room] \zoom' room\ n. ('00) Zuming's room. [Zumbot] \zoom' bot\ n. ('00) One of the many evil clones of Zuming. n. ('00) Mao rule. A
proper divisor. [Zumbug] \zoom' buhg\ n. ('00) Effigy of [Zumbot], burned in the courtyard on the last night of MOP. [Zuminglish] \zoom' eeng glihsh\ n. ('00) Language spoken by Zuming. | {"url":"http://moppers.kaseorg.com/mop.dict/zumbot","timestamp":"2024-11-14T08:16:57Z","content_type":"text/html","content_length":"27849","record_id":"<urn:uuid:1c8a102d-de57-484c-aa37-5e21d4bb8869>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00859.warc.gz"} |
Advanced Array Usage | RC Learning Portal
Advanced Array Usage
Array Initialization
Arrays can be initialized to the same quantity by an array operation:
A=0.0 !equivalent to A(:)=0.0
For small arrays, an array constructor can be written.
The constructor can use the implied do construct:
If an array of characters is declared, then each element must be initialized to a string of the specified length. The compiler will not pad with blanks.
Array Slices
Subarrays, also known as slices, may be extracted using the colon operator.
REAL, DIMENSION(100) :: A
REAL, DIMENSION(12) :: B
INTEGER, DIMENSION(20,10) :: N
INTEGER, DIMENSION(20) :: C
! Assign values to A and N
C=N(:,i) !ith column of N
The upper bound of the range is always included. If the first bound is omitted, it starts from 1. If the second bound is absent, the slice is extracted to the end of the range. A single colon :
represents the full range along a dimension.
Allocatable Arrays
So far we have examined static arrays, whose size is fixed at compile time. Arrays may be sized at runtime by making them allocatable . They are declared with an ALLOCATABLE` attribute and a colon
for each dimension.
REAL, ALLOCATABLE, DIMENSION(:) :: A, B
REAL, ALLOCATABLE, DIMENSION(:,:) :: C
If any dimension is allocatable, all must be.
These arrays must be allocated before they are used, so their size must be known at runtime. More than one array may be allocated in a single ALLOCATE statement.
Check whether an array is allocated with the intrinsic ALLOCATED(A)
if (allocated(A)) then
do something
or if we do not need to take any action if A is allocated:
if ( .not. allocated(A)) then
Advanced Array Indexing
Arrays can be addressed with arrays of integers (but not logicals).
integer, dimension(1) :: maxtemp
real, dimension(365) :: temps
character(len=5),dimension(365) :: dates
print *, "maximum temp was at ",dates(maxtemp)
Conditionals with Arrays
Logical arrays can be assigned with conditionals derived from other arrays to construct masks. The maxval intrinsic finds the (first) maximum value in an array.
logical, dimension(365) ::is_max
integer :: day
print *, 'Maximum temperature(s) were at'
do day=1,size(is_max)
if (is_max(day)) then
write(*,advance='no'), dates(day)
Pulling the array indexing capabilities all together we have a complete program:
program arrayinds
integer, dimension(1) :: maxtemp
integer, dimension(28) :: feb
real, dimension(365) :: x, temps
integer, dimension(365) :: nums
character(len=7),dimension(365) :: dates
character(len=3),dimension(12 ) :: months
integer, dimension(12 ) :: mons=[31,28,31,30,31,30,31,31,30,31,30,31]
character(len=3) :: month
character(len=2) :: day_of_month
integer :: i,m,mday,day
real, parameter :: pi=4.0*atan(1.0)
months=['Jan','Feb','Mar','Apr','May','Jun','Jul','Aug','Sep','Oct','Nov', &
! Dates as characters
do m=1,12
do mday=1,mons(m)
write(day_of_month,'(i2)') mday
!Very artificial temperatures (in deg C)
print *, "Temperatures for February"
write(*,*) temps(feb)
print *, "Maximum temp was at ",dates(maxtemp)
end program
This code contains some features, such as string concatenation, that we will study later.
• 1 Download the program above. Add the code from the “Conditionals With Arrays” section appropriately. Compare your output to the maxloc (which returns an integer array of the indices of the
maximum value).
• 2 Make all arrays that should be the same size as temps allocatable, leaving temps static. Allocate all to the size and shape of the temps array. For convenience you may introduce an integer that
represents the size of temps. This way we can accommodate data for a leap year by changing just the size of temps.
Example Solution
program arrayinds
integer, dimension(1) :: maxtemp
integer, dimension(28) :: feb
real, dimension(365) :: temps
real, allocatable, dimension(:) :: x
integer, allocatable, dimension(:) :: nums
logical, allocatable, dimension(:) :: is_max
character(len=7),allocatable, dimension(:) :: dates
character(len=3),dimension(12 ) :: months
integer, dimension(12 ) :: mons=[31,28,31,30,31,30,31,31,30,31,30,31]
character(len=3) :: month
character(len=2) :: day_of_month
integer :: ndays
integer :: i,m,mday,day
real, parameter :: pi=4.0*atan(1.0)
months=['Jan','Feb','Mar','Apr','May','Jun','Jul','Aug','Sep','Oct','Nov', &
! Dates as characters
do m=1,12
do mday=1,mons(m)
write(day_of_month,'(i2)') mday
!Very artificial temperatures (in deg C)
print *, "Temperatures for February"
write(*,*) temps(feb)
print *, "Maximum temp was at ",dates(maxtemp)
is_max = temps==maxval(temps)
print *, "Maximum temperature(s) were at"
do day=1,size(temps)
if (is_max(day)) then
write(*,'(a)',advance='no') dates(day)
end program | {"url":"https://learning.rc.virginia.edu/courses/fortran-introduction/advanced_arrays/","timestamp":"2024-11-12T00:25:04Z","content_type":"text/html","content_length":"29609","record_id":"<urn:uuid:73e64a2e-3e92-46c3-8510-07fd40539d08>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00748.warc.gz"} |
[OWP-2017-34] Farber, Michael; Grant, Mark; Lupton, Gregory; Oprea, John (Mathematisches Forschungsinstitut Oberwolfach, 2017-11-29)
In this paper we study the topological invariant ${\sf {TC}}(X)$ reflecting the complexity of algorithms for autonomous robot motion. Here, $X$ stands for the configuration space of a system and ${\
sf {TC}}(X)$ is, roughly, ... | {"url":"https://publications.mfo.de/handle/mfo/1349;jsessionid=0DB6CE96882CFA895BFDEFF09CA7614E","timestamp":"2024-11-02T21:39:34Z","content_type":"text/html","content_length":"45105","record_id":"<urn:uuid:cd5399c6-c306-4a0f-b988-308409d6b0eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00069.warc.gz"} |
Week 42 Pool Banker Room 2021 – Sure Pool Banker This Week | Sure Bet WayWeek 42 Pool Banker Room 2021 – Sure Pool Banker This Week
Week 42 banker room 2021, Week 42 Pool Banker for Saturday 24 April 2021; Pool Draw This Week 42; 2021 Banker Room
Welcome to Sure Bet Way weekly one banker room. Pool draw this week 2020.
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104 Comments
1. Last week41 was very rough .
Week42 blue: Sunderland at 27 home to control Salford C and Scunthorpe for only one draw on coupon since seasons. Punters should single it and play it with any amount you can afford and enjoy
this week.
Mark 38 or 37 for one compulsory draw.
Must pairs.
2. Welcome to blue week. My Unfailing pair for this week is 13 pair 19.
Last letter of away team (First draw) of previous week VS First letter of away team(last draw)
Week 41 – 16xx pair 38ff
Week 42 – 13?? Pair 19??
3. Welcome to week 42. My Banker is ****5****, winning line 5*14*18*33
Proof of banker: In week 39, Brighton @ 6 away and Burnley @ 8 away to draw Newcastle vs Tottenham. This week Brighton away @ 4 and Burnley away @ 6 to draw 5*** Westham vs Everton fxd FTD.
4. STRICTLY BLUE
nap:1√15√48√49√for three draws or possible 4/4…………………………..
Prove!! game 8 of hi score quiz(SAF) triggering and promoting it.. Ref week 26/30/34/38/42/
Good luck!!!
5. Week42 , use 05 or 06 for one good draw, this pairs is taken from mirror paper, whenever game 1 and 2 of full list inside carry the same letters at home, example W W, is a must for one draw.
CC: Cardiff women
6. WEEK42
Bankers 14xxx36xx38 perm 32*42 for 5/5 winning line.
Prove 1, when Cardiff set at number 10 home mark 14 by number or Nottingham forest to draw. 14XX.
Ipswich 23 home , to draw Bradford C, 36XX
Aldershot under the bar home , Walsall to draw in every blue.38XX
Southend draw at away the position to repeat draw the following week and pairs next digit down. 32*42
7. Week42: this week is a fertile one.
Mark 14XXXXX15 for one draw,
Reasons is that every blue 14 and 15 is for one draw.
8. Week42 pairs: 41pairs42
The four numbers in front of pool telegraph under Dreamlands column, any of the two numbers move to inside under gift of prosperity is due for must one draw. Therefore this week, is the turn of
41 or 42 to play one compulsory draw …… .
9. It’s Huddersfield +3
Crewe +4 and Mansfield for a NAP when Bradford C gets on number 36.
Rf weeks 11,22,42*
Week12 45**&15*/*17
Week22 44**&17*/15
Week32 43**&15*/17
Week42 42bk&15/17
Then 13***** is five star every nine Weeks
Rf weeks 6,15,24,33…
NAP 11**24**37
ALT 13&15
10. Welcome to week 42:
I am glad to bring to us to awaited pool system,
KEY POINTER:
First week: Blackburn to draw @ x11x away,
Second week: Salford c to set on top bar either home or away as the last draw on coupon,
Third week: Opponent of Blackburn @ 11 in the first week to set at num 10 home in the third week,
when all this happens,rejoice then start sourcing for money to hit your nap heavily on the third week,
Now to get your draws:
in the third week, Locate Exeter either home or away ,from its position count three up to draw being your first banker and must correspond with the position of Grimsby @ away in the first week,
Meet me at the discussion room on thursday for the complete Nap,
First occurance:week 22/23/24, : Banker x39x and other two,
Second occurance: week 40/41/42, : Banker x31x and other two
NOTE: This is not a sales advert, I made this post early enough to enable us get prepared to play the game before weekend, so prepare
see you on thursday:
Luvday’s analysis
11. Week42 banker pairs are 14XXX24 for one or two draws this weekend , Hull at away on digit with stoke must vomit one or two draws. Use it in all your copies.
12. Week 42 bank on no14cbk proof anytime you see tottenham at no1 away and Aldershot at no44 home bank on NOTTM FOR. at home ONTOP of Q.P.R to draw.
13. Wk42 two live bankers.
Key = PORTVALE
Punter= weymouth@ 48.
Live and direct 46xxx
Portvale at 36 hm.
Weymouth at 48.
Wop 27 sum 2+7=9+code1=10.
From wop27 count 10week front to wk36 and to meet BARNET 46x away drew.
Return to wk27 and mark first letter S away at 46x SUTTON U@46x.
wk27. 14x and 46x 2/2.
Wk 42.
Portvale at 36 hm.
Weymouth at 48 away,
Wop 42 sum 4+2=6+code1=7.
From wop42 count 7weeks back to wk36 and meet BARNET at 46x .
Return to wk42 and mark the first letter S away from coupon buttom at 46x away STOCKPORT 46xx live banker.
This is the week you are waiting to HIT BET9JA with my 2/2 live bankers.
wk42 parade 46x live and direct.
STOCKPORT @46XX FIXED DRAW.
Don’t copy my work.
This is AKAYI PATIENT AMOS work !
Same game will be post on my facebook timeline and all groups i belong .
#say no to copycat !
14. Nxx18xxxx
Xx18xx:BARNSLEY is drawing its opponent first letter since week 40
Xx27xx:previous position of SHREWSBURY to draw in the following week since week 39
Xx47xx:position of SOUTHEND +8 since week 40
15. @ odegha Johnson.
The Barnsley setting is bringing 2/2 every week
With its opponent first letter drawing and its opponent first and last letters add together to fail and play game down.
Reff since week 39.
This week it brought 18/32 as solid 2/2.
I pray it continues.
16. welcome back to week 42(32p42) for one or two draws,prove anytime roll 9 of hi-score quiz carries no.42 it pair with 32, good luck, Admin I hail u wella.
17. WEEK 42. XX 18 XX
REF. WK 11 TO WK 12.
REF. WK 41 TO WK 42.
18. BANKER
XXX 14 CBK
Since the 2018/2019 season till now, every WK 42 the game on box 1 of HI-SCORE QUIZ plus 2 to it the answer to give a classified banker draw,so for this WK 42 2021 box 1=12+2=14√, admin please
19. Cardiff at number 10 Home
Chelsea to draw = 05xxx
20. Xx41xx46xx47xX.
RAITH at week number away sharing family with CRAWLEY home and taken to Hi-score 9 to play one up, last week
The ever appearance of BROMLEY on pool to draw and one up this
Hope this makes you smile.
21. Every blue week Wycombe position + 7 to give you a banker. Let’s recover our losses. xxx17xxx
22. 44 cbk, Bob Morton third page, cartoon picture number to capital draw banker box to draw ref wk, 27, 33
23. If this game fail, I will quit pool. Bodly mark X 35 X.
Second time movement.
OLDHAM vs GRIMSBY
X & G/G
Wk10 blue 2018
Grimsby vs Oldham FFFFFFF
Wk10 blue 2019
Oldham vs Grimsby XXXXXXXX
Wk42 blue 2020 last season.
Grimsby vs Oldham FFFFFFFFF
WK 42 blue current season 2021.
Oldham vs Grimsby ????? @35
24. 13cbk, Wheneve you see Cardiff at coupon number 10 home (1) xx13xx to draw by number (2) count the alphabet of Cardiff and its opponent to draw game up xx13xx. Ref: week 21, 33, 36 and 42 2020/
21. It’s come with other draws. Check me in discussion room for 3/3 and pair, possible 5/5. Goodluck!
25. WEEK-42 {BLUE}
GENERAL PAIR XXX14XXX PAIR XXX28XXX
USE IT DELICIOUSLY AND MASSIVELY
SINGLE BET FOR HEAVY STAKERS ONLY
******BLUE KEY*****
APPEARANCE OF LEEDS AT HOME NUMBER (3) AND SHEFFIELD WEDNESDAY AT AWAY NUMBER (13)
BANK ON COUPON NUMBER 14 & 28 FOR ATLEAST ONE DRAW PLUS THREE OTHERS.
WEEK-34 {BLUE}
XX14XX√ PAIR XX28XX√ {2/2}
PLUS XX05XX√ XX25XX√ XX37XX√ {3/3}
WEEK-42 {BLUE}
XX14XX PAIR XX28XX {2/2}
PLUS XX00XX XX00XX XX00XX {3/3}
WINNING IS ASSURED
26. Week 42 blue ,x12x CBK, Derby Birmingham,
NAP x12x25x39x
Perm x12x25x35x36x39x
Week 23 to 24, 41 to 42 Current,Key setting of Hudddersfield vacating from 12ff away,previous position and teams occupying those digits for three draws.
27. Week 42
NAP 32*34*37
Key pointers
*Away team last letter at no 12 to correspond with away team last letter at 49
*Sunderland must set at 27
Bank on
(1) 49-12=37
(2) add first letter of home and away at no 37
(3) from your answer above count 3 down for your 3rd draw.
Week 10
*12 away=Reading
*49 away=Strasbourg
*Salford c vs Exeter
*From 24 count 3 down =26✓
Week 39
*12 away=Barnsley
*49 away =Torquay
*Leyton o vs Walsall
* From 35 count 3 down =37
Week 40
*12 away=Norwich
*49 away=Arbroath
Southend vs Crawley
*From 22 count 3 down =24
Week 42
*12away =Birmingham
Salford c vs Mansfield
*From 32 count 3 down =34
28. My week 42 banker: 21,20,22 for or two. Prove via Soccer Research. Good afternoon my amiable Admin. Remain blessed.
29. Week42 NAP!
Forestgreen away on 32;
Nap 31√√√ by number.
And Grimsby=35√√√
Current season key wk37 and this week only
30. Nap
man city at 1 home
Sharing digit with coventry both at home
nott”m for at home
carlisle at home
walsall at away
Ref resah key
31. Welcome to week42
longest time my fellow tipster
every blue wycombe to play 8down
prove 30
since week38 stevenage to play 1 down
32. 41 pair 38 bank massively on this pair for the simple reason that in Soccer Research back page, whenever dead game is seen at game 40 of the Full List ,therefore take games 39 and 40 of Full List
for 1 or 2 draws ref WK 12,28,35,37&42,2020/2021.
33. P A I R
Every WK 22 and WK 42 only, Bob Morton serial game 33 and it’s game up is for one sure draw reference
WK 22,2019
WK 42,2020
WK 22,2020
WK 42,2021
01&02 for one draw
Admin please post for us and thank you very much for a job Weldon.
34. *41* PAIR *42*
Every Blue coupon week since this current season WINSTAR commentary games 3&4 is for one vital draw .
My last WK pair was 15***&33??? which supplied 1/2 may this WK also be fruitful for all of us by God’s grace.
35. WEEK 42 [2;7;27]NAP PROVE PICK YOUR SPECIAL ADVANCE ANYTIME YOU HAVE DOUBLE NO THE FIRST ROLL WHERE YOU HAVE [12]JUST TAKE THE NO BELOW IT AS YOUR FIRST BANKER WHICH IS [27] THEN BISET THAT NO
WHICH 2/7 THEN GO AND PICK NO [2 / 7 ]AS YOUR OTHER TWO DRAWS THEN GO TO SHORT LIST AND PICK GAME [7] WHICH NO [37] REF WEEK[21,25,42 CURRENT]
NAP [2.7.27] GRTEETING TO THE OGA AT THE TOP
36. Soccer X Research
Page 2
Soccer 1.2.X Ratings
Serial number carrying their percentage
Is a fixed/classified pair
REF: Since the season.
37. This week42 use 14 xxxxx 44 for compulsory draw,the setting of Stoke @14 away project Stoke and it’s family last digit for compulsory 1 draw,ref wk10,wk23 & wk42 current…use 14 xxxxx44, good luck
38. Soccer..
Game 5 of TREBLE CHANCE 12 dotted- * * –
Mark the number on game 4 of same Treble Chance 12 to draw…
39. BRADFORD C at 36 away is a gazzatted draw **36**
Ref last season
40. 24
LINCOLN – HULL
WINSTAR CENTRE STAR GAME IN BROWN TO FAIL & X IN BLUE
41. Welcome to week 42, Blue coupon, my special Advance unbeatable Blue pair = 4 weds 15, must, for one fixed draw. Prove only on Blue coupon; away teams @ Treble Jinx game 13 and 14, their last
letters form an unbeatable pair. E.g this week 42, away teams @ Treble Jinx Game 13 and 14 are Newport Co and Man Utd, their last letters are O = 15, and D = 4. Ref. Current. Congrats in
42. BRADFORD C. at 36 away is a gazetted draw**36**
Ref last season
43. Welcome to WK 42
Blue coupon list
From WK 42 2018 to date,every WK 42 pools telegraph games 4 and 5 of 6 tight games column to form a strong pair ref WK 42,2018=03&37** WK 42,2019=44&16** WK 52,2020=12**&21** WK 42,2021=38&21 for
one life draw admin please approve to post for the benefit of everyone.
44. 7 pairs 8
Whenever game 1 of treble jinx of SAF is transferred to box 6 of HSQ ,therefore ADD the digits of this repeated game together and the answer you get with it’s game up to form a pair for the week
ref WK 3,31,41,42,2020/2021 etc
OUR RECENT PERFORMANCE
WK 40,2021
WK 41,2021
THIS WK 42,2021
07 pair 08
We can but only hope for the best, which is yet to come ! Admin remain bless in the Lord.
45. Week 42
Chelsea at no 5 away
Letter B must at no 9 home or away
Mark 9*30*49 by no
Ref week 9 2017/18
Week 5 and 7 2019/20
46. Good morning distinguish admin, Gud morning fellow tipsters. Second time movement, first setting week40
Q.P.R Vs sheff.wed number 15 seen in game 9 of treble jinx, Now week42
Q.P.R Vs Norwich number 15 seen in game 9 of treble jinx
Week40= count Q.P.R alphabet and opponent alphabet together the answer will give u the position of Cardiff at home to draw, then Q.P.R first alphabet plus opponent first alphabet answer to draw.
WEEK40. 3+8=11Xbk
WEEK42. 3+7=10Xbk
WEEK42. 10***36 20Pair21 thanks.
47. My winner lines dis week is 36 38 32 3 5i will only prove 2 from d five in week 24 u will see Southend draw at 39 and Fulham to draw at 5 next week froestgreen will enter dat Southend portion and
man u will enter Fulham portions for two draws before u mark it froestgreen opponent frist ahpabet man u to enter dr to draw pls do nt joke wit dis winner line play n show me love Admin pls add
to help others
48. My winner lines dis week is 36 38 32 3 5i will only prove 2 from d five in week 24 u will see Southend draw at 39 and Fulham to draw at 5 next week froestgreen will enter dat Southend portion and
man u will enter Fulham portions for two draws before u mark it froestgreen opponent frist ahpabet man u to enter dr to draw pls do nt joke wit dis winner line play n show me love Admin pls add
to help others
49. A WK After Cup Key, banker **33**. Proof: Abroath in advance Wk of Play controls Exeter to draw. Ref Wk41&Wk42.
50. 2x14x35
Scunthrope at 38 home
weymouth at 48 mark
westbrom h/a
Mark 14 by number
51. Welcome everyone to WK 42
01 versus 33
For one compulsory draw this weekend, with it’s proof taken from Right On Football Fixtures back page “EXPERT GAZETTE” pair to contains any family of three in any given even weeks meant the pair
is ripe for a draw at the very least,check references WK 6,18,26,34, and this WK 42 all in this current ongoing 2020/21 season. We know how it is, I’m not sure if my pair have ever failed three
times in a roll ever since joining this blog,and that isn’t going to change anytime soon. Therefore after failing in wks 40 and 41 respectively I’m inviting all and sundry to join in on this
revamping festival of pair delivery beginning from WK 42 onwards.
52. Nap 2x14x30
scunthrope at 38 h
weymouth at 48 away
westbrom at away
14 by number
53. week42, play 27…………x………..37 for 1 draw,Sunderland home and Salford utd home on the same family digit for 1 draw,ref wk10,wk12,wk35,wk42 current.
54. WEEK 42 2021 BOUNTY
NAP 01:03:06: PERM 33:42
I don’t know who writes the scripts of these football matches, but what I know for sure is that you are looking
at a 3/3 as well as a 5/5
game above .
Proof : Pools Telegraph forecast paper, the bottom r.h.s game of dreamland
column to be repeated to the Red patch column on Top Tips section, “key set”, if this criteria is met,then on a draw/fail basis for this setting = numbers 1:3:6 is programmed to produce direct
Nap,for reference I will
write only the weeks it drew i.e WK 24,2014 WK 44,2015 WK 28,2020 you know the rest, thanks to the admin for duly rewarding me for a successful outing in wks 38 & 39.
55. Whenever you see Bradford City at no 36 away. Is a draw. Reference since last season and current week 11 and this week 42 36bk, the dead game at the back of soccer research full list 40 to play
game 39 and 40 for one. This week I want to give a winning line 12,33,36,38,41.
56. Welcome to week 42 blue colour
everyone should play this nap And appreciate through admin
Prove of 8
week34 blue
blackburn at 11 home with coventry ft draw at the same family with blackpool
While preston at 15 h to die with huddersfield
blackburn at 8 h with huddersfield to play draw at the same family with blackpool
coventry to take preston and die at 11
prove of 30
since week 38 till date stevenage to play game down
I remain my humble self Sam Black
57. Pls admin permit me to prove only 15 for security purpose
since week40 til date count home and away team of week number the ansa to plaw 5down
58. Week 42 celebration NAP, Don’t joke with it,
NAP: x1x2x3x full time draw,
key states that whenever Grimsby set on top Bradfordc both at away play x1x2x3x as nap.
ref: week 15 and week 42 current,
play and appreciate me through admin to get week 43 nap, don’t joke with it my people
59. 26xxxpair 26xxx for one draw capital treble chance16 game 1 rated XXX and game 16 rated x mark game 6&7 for one draw ref. Since the season
60. XX29XX FT DRAW
CAMBRIDGE U VS STEVENAGE FT DRAW
PHASE 1
ROCHDALE @29F HOME
ROCHDALE @29F HOME
PHASE 2
STEVENAGE @X29X√ AWAY
STEVENAGE @X29X AWAY FIXED DRAW
61. 26xxx pair 36xx
For one capital treble chance 16 game 1 rated XXX and game 16 rated x
Mark game 6 & 7 for one .
Ref. Since the season
62. 24
Anytime you see number 37 in box 7 of HSQ SAF play game 2 of EASY COLUMN and Saturday date of play for one or two draws ref WK 14,27,36,39,42,2020/2021 admin please post.
63. 32XXXXXXXXXXXXXBANKER
CRAWLEY VS FOREST G.
SOCCER, CAPITAL AND BOB MORTON.
ON SOCCER SINCE WEEK 39, THE TWO GAMES CALLED HOT-PAIR INSIDE TIPS FOR SATURDAY’S COUPON PAGE 4 HAVE BEING PLAYING FOR ONE DRAW.
THIS WEEK IS 26/32
ON CAPITAL FULL LIST GAME 26=32
ON BOB MORTON CODED LIST R. WHENEVER TWO FAMILY NUMBERS SEEN THERE WITH A DIFFERENCE OF 10 MARK THEM FOR ATLEAST A DRAW
WEEK 18
WEEK 42
SOCCER FULL LIST GAME 32=22
SO: 22,26,32 FOR ONE DRAW.
REASON I PICK 32XXXXBK
ON BLUE FROM WEEK 34, % ONTOP 20 % MARK AS A DRAW.
WEEK 34
31=28%. 28XXX1-1FT
WEEK 38
26=19%. 19XXX PP
WEEK 42
19=32%. 32XXX
CRAWLEY AT HOME SEEN ON GAME 12 OF SOCCER FULL LIST MARK AS A DRAW.
WEEK 25
38XXX CRAWLEY VS NEWPORT CO.
28 NORTHAMPTON AT AWAY
WEEK 42
32XXX CRAWLEY VS FOREST G.
22 NORTHAMPTON AT AWAY.
WHENEVER 27 IS SEEN ON SOCCER DEAD. IF IT % DRAW MARK IT TO REPEAT DRAW THE FOLLOWING WEEK BUT IF IT FAILS MARK IT TO FAIL THE FOLLOWING WEEK.
WEEK 21
SOCCER DEAD 27XX
27=X2 1-1 45% 45FF
WEEK 22
WEEK 36
SOCCER DEAD 27FFF
27=X2 1-1 47% 47XXX
WEEK 37
WEEK 41
SOCCER DEAD 27FFF
27=2X 1-2 32% 32XXX
WEEK 42
MARK 32XXXXXXBANKER
CRAWLEY VS FOREST G.
64. xx 21 xx
Week 40:
Sunderland – Charlton @27ff
Week 41:
Blackpool – Sunderland @18ff
Charlton – Ipswich @21xx
Week 41:
Shrewsbury – Doncaster @27ff
Week 42:
Blackpool – Shrewsbury @18??
Doncaster – Fleetwood @21xx
65. Good evening to my blood here. Admin u remain d best. Still on d matter of law of comparative advantage which states that u are to concentrate on where u have advantage, that is where u know
best. I will concentrate on pairs and i must complete my 4wks pairs with convincing prove so that I chop admin money for nkwobi and big stout as an ibo babe. Never mind, last week was 34/40 with
a convincing prove which states that since last season till now whenever u see Salford c and Grimsby at 40 away projects d two teams as a formidable pair and it delivered… now let’s look into
this week42, my best pair is 9/14 and d prove is this: since season till now only on brown coupon, since wk9 till now, every brown, game 49 home second and last alphabet forms pair following week
blue. Take a look, in wk9, game 49 home 2nd and last was 1/20 and it’s a pair in wk10 which delivered. In wk13 game 49 home 2nd and last was 1/24 which which delieved in wk14. In wk17 game 49
home 2nd and last was 14/5 which delieved in wk15. In wk21 game 49 home 2nd and last was 16/11 which delieved in wk22. In wk25 game 49 home 2nd and last was 20/25 which delieved in wk26. In wk29,
game 49 home 2nd and last was 5/14 which delievered in wk30. In wk33 game 49 home 2nd and last was schalke which is 11/5 and it delivered in wk34. In WK 37 game 49 home was 14/15 which delievered
in wk38. Now last WK being brown, game 49 home 2nf and last was 9/14 and this forms a formidable pair this week42. Come rain come shine 9/14 must play one and only but one this week. Cross check
and be convinced. Thanks admin.. I am d first lady
66. Respect to Agnes first lady
67. My proof is from capital paper, can’t postion sharing the same family number with game under. Last week was 41 and 21. I present 4 and 34 for one, still have 24 pair 26. To thanks
68. Good morning everyone oga admin i greet you i have just one banker for house with is.
Prove appearance of Wigan ontop of bar meet letter B just add code 3 for the position of Wigan is strong DRAW ref week check last season and this season good luck,I remain OMO OYE.
69. Week42 Nap=03*06*28=3/3
Must Winning Line
Proof No:06
Very week of two opponent Crystal place to draw next week of two
Proof No:03
Week12 first alphabet Game No:36 draw
Week22 First alphabet Game No:36 draw
Week32 Last alphabet Game No:36 draw
Week42 Last alphabet Game No:36 draw
Proof No:28
Very week of two No28 by number draw
70. Wk42
Use *******38********
Walsall @38 away
Bank on Walsall
Ref wk12
71. Good morning everyone, I have 4 and 34 as pair, my observation is from capital international paper, capital can’t position sheering the same family with game under.
Last week was 41 and 21 and I also have 24 and 26 as pair thanks
72. 05x 08x 24x
Advance opponent of Burnley to draw. Ref. WK; 39/40.
Blackpool to draw next similar digit up in blue carrying Blackburn. Ref. WK; 34
Advance opponent of Wigan at twenty eight in red to draw in blue. Ref. WK ; 34/35.
All current.
73. Leyton o at 39 away to control 5 draws
Check your records since 2020
74. BUSHIDO CODE
(12 pair 30)
This system is from special advance fixtures which states that, any WK box 8 of HSQ contains family of zero ref WK 44,09,23,28,42,2020/21, therefore mark number 30 to draw by number, and box 1 of
HSQ is on a draw and fail sequencewl which is due to draw this WK, admin please approve for mutual benefit.
75. Welcome to week 42, I bring to you guys 15XXX P 45XXX for one possible draw. First and the last game mark XBK at treble chance jackpot perm at page two of capital paper is a pair for one draw
must for blue colour coupon. Ref week 38 among others.
76. NAP(A) NAP(B)
xx28xx OR xx42xx
CAPITAL INT’L front page the ever presence of MORECAMBE at home to sit on top of the TRY column to draw, BANKER BOX on page 3 to draw, whilst the games marked’X’ in the DRAWS ARE MOST LIKELY AT
column form’s a pair Ref WK 12 2019 and more. So for WK 42 2021 NAP(A)=34,44,28 NAP(B)=34,44,42 one of the Nap will green, good luck.
77. My able Admin I greet you specially may God crown your effort with success. Good day my good people of this blog I greet you all, this is my first time of posting in this blog and I will be
posting banker or pairs till the end of the current English season and I want you to take my games seriously. Now let’s go open your record of 2019-2020 season and this current season whenever
you see Stevenage at away on blue coupon bank on it as your live fixed draw, it came once last season and week10 and week14 this season.
5 pair 6 for 1draw
My able Admin over to you thanks.
78. My pair for week 42 is: XX28XX p XX38XX. Prove: Cambridge advance fixtures. When game in box 1 of super pair is taken to game 1 or 2 of best four and also found on game 1 of best three. Turn to
da back page, if single digit is found in second shaded & second unshaded portions of octopus prediction, join them together and their next family up for a pair of one or two potential draws.
Ref: weeks 25, 41, & 42 current season. Good luck.
79. MY BANKER THIS Wk42 XXXXX(((((6))))XXXX THE OPPONENT OF SHEFFIELD UNITED WK40 TO PICK DRAW IN WK41 ARSENAL VS FULHAM XX3XX OPPONENT OF SHEFFIELD UNITED IN WK41 TO PICK DRAW IN WK42 XXX 6 XXXX.
SECOND PREDICTION FROM WEEK 12 CURRENT SEASON OPPONENT OF SOUTHAMPTON TO PICK DRAW IN WEEK 22 AND OPPONENT OF SOUTHAMPTON IN WEEK 22 TO PICK DRAW IN WEEK 32 AND IN WEEK 32 TO WEEK 42.
80. Mark with authority number
since the season any time u see stoke set at away in the same family with bolton
pls withdraw all ur money in account sale ur car even ur house and
birmingham at away
stoke to play itself
and 38 to play by number
Ref week25 brown and week42 blue
gud luck
81. NAP
Aston V @ home and Barnsley @ home mark number the untop Barnsley and under Barnsley for one or two
Since this season on blue mark 14 &15 as pairs for one
Doncaster @ home and Colchester @ away
mark both for one or two draws
MY BANKER FOR THIS WEEK 42 NO ((((30))))CBK.
PROVE OF 30
IN WK10 CURRENT
BOLTON VS FOREST GREEN @ 30 FAILED.
THERE WERE FIVE DRAWS IN ENGLISH LEAGUE 2, AND ONLY TWO TEAMS HAD 2:2 SCORELINE (SALFORD C VS EXETER).
IN WK12,CURRENT
OPPONENT OF BOLTON IN WEEK 10 WILL DREW WITH ONE OF THE TEAMS THAT HAD 2:2 SCORELINE SALFORD VS FOREST GREEN ON DIGIT OF ZERO @ 40XXX.
IN ANOTHER WEEK OF ZERO
WK40 CURRENT.
BOLTON VS HARROGATE @ 30 FAILED. IN ENGLISH LEAGUE 2, THERE WERE 3 DRAWS AND ONLY ONE DRAW HAD 2:2 (BARROW VS CARLISLE 2:2 SCORELINE), OPPONENT TEAM OF BOLTON (HARROGATE) WILL TAKE ONE OF THEM TO
WEEK 42 ON DIGIT OF ZERO FOR A GAZETTE DRAW CARLISLE VS HARROGATE @ 30XXXXXCBK.
MEET ME IN DISCUSSION ROOM FOR YOUR TWO DRAWS AND A PAIR WITH PROVES.
LESS THE NOISE AND RAISE YOUR GAME.
ADMIN REMAIN BLESS AS I AWAIT YOUR KIND REWARD FOR THREE CONSECUTIVE WEEKS OF BANKER DELIVERY.
I REMAIN SAANA ISAAC.
83. Jubilate with
kings lynn at 49 home
opponent third leta to draw
from kings lynn 15up to draw
count home and away team of the second draw which 15up give u the ansa to play 1 down
84. Mancity at 1 away the following week
2x5x14 by number
ref week33 to 34
now week41 to 42 brown to blue system
85. Dnt say u didnt see oh
30= stevenage 1down
38= mansfield of cupon in week40 coming back in week 41 to play 1down
41=anytime raith is on pool it play 1up
86. Q.P.R Vs NORWICH FT,PROVE.soccer’x’research,front page special’x’tip for the week,game 2&3 if Dr last alphabet are Drw same mark game 3 as ur DRA,example,dis week is DONCASTER,Q.P,R .Reff
weeks,18,21,23,32,42.mark no 15CBK,15CBK MUST DRAW,ADD 21CBK ALSO.MY BANKER IS,NO 15CBK,15CBK.Q.P.R VS NORWICH FT.
87. use 41,,,,,x,,,,,,43,the setting of Crawley home and Raith away on same family digit project Raith up and down for one draw,ref wk41 & wk42……41,,,,,,,,,x,,,,,,,,,,43,good luck
88. good evening to you all..
anytime you see 14 on fortune box in bob Morton page two {1} mark it has draw.. {2} go to number 22 at full list mark it has draw.. {3} go to number 10 at draw tips in page three mark it has
draw… {4} go to number 6 at fabulous ‘8’ in page one mark it has draw … ref weeks 16,31 and 42 current season …
week 16
42xx. 4\4
week 31
25xx. 4\4
week 42
36xx. 4\4
Thanks to you all…..
89. Welcome to week 42 blue cupon
I have only one mighty pair that will not fail to appear on result board even though more will play only One draw
25 ****** pair 30******
Percentage of 30 in soccer is 25
Bob Morton fullist game 30 is still 25
Is must for one
Please am begging the whole world to single bet
25 and 30
90. Welcome to Week 42
BBC Pair
Phase 1
Wigan vs Ipswich xx
Wigan vs Portsmouth ff
Scunthorpe vs Southend xx
Newport co vs Scunthorpe xx
Phase 2
Ipswich vs Wimbledon xx
Wimbledon vs Portsmouth ??
Exeter vs Southend xx
Exeter vs Newport co xx
91. Good morningmy able admin and my fellow gurus in the house.. I have a key that is to play 3/3 next week. But first let 35 draw this week and am gonna destroy promoters and bet company next week..
Prove withheld for security reasons
Week 40–24
Week 41–7
Week 42–35 awaiting results
Admin take not. My pair is 14 pair 36
92. WEEK42
WEYMOUTH@48 AWAY AND THE PLACEMENT OF 37 ON BOX 7 OF HI SCORE QUIZ PROJECT THE ABOVE DIGITS AS TWO DRAWS. WHILE 21XX 32XX AND 22XX 31XX PLAY ROTATIONALLY. THIS WEEK IS THE TURN OF 21XX 32XX
WK36=22X31X46X47X WK42=21X32X46X47X
93. Aldershot at no.44 under the last bar at the same digit with Bolton at away, locate Westbrom at no.2 away. Then bank on Blackpool at no.18 home under the bar.
(Current): WK 36, 42
94. Follow me for three weeks lets beat the bookies, Week 42 nap x1x2x3x live,
Prove:total number of draws to equal the position of QPR, Opponent of QPR First letter minus its last letter to draw and draw up and down the following week,
ref:week 40/41= x14x15x16x
week 41/42: x1x2x3x direct.
pool is fixed. Believe, play it next week is loading again
95. Welcome to your Friday/Friday
tonic corner with
Yours sincerely
44 pair 45
Pools Telegraph centre spread late news Online Betting Tips column, whenever a game is rated * * – OV 12 itself and game down form’s a reliable pair for one draw ref WK 40, admin pls post, tanx
and God bless.
96. A special key that cost a fortune for free but will cost you a token next week a little proof STEVENAGE IN SAME SERIES AWAY WITH PETERBOROUGH WESTBROMXXX BIRMINGHAM XXXX 49XXXXX BY CALCULATION
STAKE 2XX12XX49 JUST 2WEEK OPERATION WEEK 43 COST JUST 10K NB, I WILL GIVE OUT ONLY THE DIGITS NOT THE KEY FIRST WEEK 3/3 SECOND WEEK 4/4
97. QPR at 15hm key:::::::
locate Norwich on coupon and count 3up and 3dwn to pair 13p17
wk22,28 2019/20
wk40,42 2020/21
98. XXXXXXXXXX 28 XXXXXXXXXXCBK
THIS ONGOING SEASON, EVERY WEEK OF TWO (2) ie WEEK 12, 22, 32 AND NOW WEEK 42. MARK FIXTURE NUMBER 28XXX AS A SCHEDULED FIXED DRAW BY NUMBER.
WIGAN VS BURTON A. XXXX FULL-TIME DRAW.
99. Good morning my admin, welcome each and every one of you to week 42,I have 5 games that will play 5/5 but I don’t want to prove it every body should please play and smile to the bank on Monday
9,26,29,35,40 must 5/5 good luck to everyone in the house and God bless us all in Jesus name amen.admin aprove for The benefit of doubt
100. Week 42!!! Glory be to God for his mercies endureth forever…
Please and please everyone should take the game below seriously… There is a follow up nextweek for it… Hopefully we will smile to the bank on monday
Blue -Blue
(A vs Dunfermline)
(Dunfermline vs A)
Set on blue coupon
Week 18First setting: A vs D on x24x(Even no)
Home team Letter A to play same alphabet ontop. 23x24x
The Even number divide by 2.
Played 12*23*24*
Week 42Second setting: D vs A on x43x(Odd no)
Home team Letter D to play same alphabet ontop. 42x43x
The Even number divide by 2.
14 on Bob-morton fortune box 14p30
So Nap❌21❌42❌43❌
Perm 14pair30
1. Open soccer x research page 3 inside Hot three box,WK 24 Arbroath vs Ayr UTD were on position 2 de box washed. WK 40 it was Ayr UTD vs Dunfermline de box cleared dis WK 42 is Dunfermline vs
Arbroath to also clear de box (21,42,43) Blaqprince congratulations,Admin I salute
101. WELCOME TO WEEK42/2021 BANKER SECTION
A PADON FOR KEEPING YOU IN SUSPENSE……..}
KEY :- MIDDLESBRO AGAINST LETTER S, LETTER K @49 HOME, A BLUE CUPON, SAT DATE =24. IT’ FINISH
{1} NAP 3XX 5XX 29XX 31XX 32XX 37XX AS 6/6……}
{2} ONLINE MEN
BET 29XX-$1000
BET 32XX-$1000
cut one terminates
CUPON EXPERTS appointment here but YOUR DESTINY IS IN YOUR HANDS
102. 26xxx36xxx37xxx38xxx17xxx
Anytime you see a number in super six page 2 of bob Morton taken to banker of the week in capital page 4. Then mark with authority
Ref week 29,2021.
Anytime Bradford City is at number 36 away is a blessing to all and also a pair for 1 or 2 draws.
So mark with authority
36xxxxxcxccccccccc 37xxxxpair38xx
Ref. Week 5,2019 brown. Week 16,2019 purple. Week 23, 2019 red. Week 26,2020 blue. Now week 11,2020 red. Week 42 blue.
Since the season till date Wycombe on coupon, add code7 to its position to draw.
May God help us all
103. I present this digits to the family.
Prove of 27-34-37.
Anytime bar cut 43/44, mark game 6 of treble jinx 27xxx, game 7 of short list 37xxx, and game 2 of easy column 37xxx, game 5 of hi-score quiz 34xxx.
Ref. Week 30= 47**47**15**21**
Week 36= 35**19**47**19**
Every blue since week 30, game 6 of treble jinx to draw 27xxx. Game 3 of hi-score quiz to draw 18xxx. Game 2 of easy column to draw 37xxxx.
Wk 30= 47**9**15**3/3
Wk 34= 1**1**14**2/2
Wk 38= 6**6**15**2/2
Week 42= 27**18**37**3/3.
Bradford c @ 36away is a goal.
Week 5,16,23,26 2019/2020
Week 11,42 2020/2021
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From deep TLS validation to ensembles of atomic models built from elemental motions
A correction has been published for this article. To view the correction, click
From deep TLS validation to ensembles of atomic models built from elemental motions
^aCentre for Integrative Biology, Institut de Génétique et de Biologie Moléculaire et Cellulaire, CNRS–INSERM–UdS, 1 Rue Laurent Fries, BP 10142, 67404 Illkirch, France, ^bFaculté des Sciences et
Technologies, Université de Lorraine, BP 239, 54506 Vandoeuvre-les-Nancy, France, ^cPhysical Biosciences Division, Lawrence Berkeley National Laboratory, Berkeley, California, USA, ^dDepartment of
Bioengineering and Therapeutic Sciences, University of California, San Francisco, San Francisco, CA 94158, USA, and ^eDepartment of Bioengineering, University of California Berkeley, Berkeley, CA
94720, USA
^*Correspondence e-mail: sacha@igbmc.fr
Edited by R. J. Read, University of Cambridge, England (Received 18 December 2014; accepted 12 June 2015; online 28 July 2015)
The translation–libration–screw model first introduced by Cruickshank, Schomaker and Trueblood describes the concerted motions of atomic groups. Using TLS models can improve the agreement between
calculated and experimental diffraction data. Because the T, L and S matrices describe a combination of atomic vibrations and librations, TLS models can also potentially shed light on molecular
mechanisms involving correlated motions. However, this use of TLS models in mechanistic studies is hampered by the difficulties in translating the results of refinement into molecular movement or a
structural ensemble. To convert the matrices into a constituent molecular movement, the matrix elements must satisfy several conditions. Refining the T, L and S matrix elements as independent
parameters without taking these conditions into account may result in matrices that do not represent concerted molecular movements. Here, a mathematical framework and the computational tools to
analyze TLS matrices, resulting in either explicit decomposition into descriptions of the underlying motions or a report of broken conditions, are described. The description of valid underlying
motions can then be output as a structural ensemble. All methods are implemented as part of the PHENIX project.
1. Introduction
1.1. Independent and concerted molecular motions
It is currently difficult to derive a structural basis for concerted molecular motions from the models emerging from macromolecular crystallography, which describe each atom with a central position r
[0] and additional displacement parameters. Small-magnitude disorder (particularly thermal motion) can be captured by the Debye–Waller factor, which reflects the probability of an atom moving from
its central position by a certain distance. If a model includes this approximation, the contribution of each atom to the structure factor (h, k, l) must be scaled by
(see, for example, Grosse-Kunstleve & Adams, 2002 and references therein). Here, O is the orthogonalization matrix for the given crystal, h is the column vector of integer indices (h, k, l), U[Cart]
is an atomic displacement parameter (ADP) and the superscript τ stands for the matrix and vector transpose operation (here and in the following). [In Grosse-Kunstleve & Adams (2002 ) the
orthogonalization matrix is defined as A; here, this letter is reserved for the matrix in the development of U[Cart], following Tickle & Moss (1999 ).] The symmetric positive definite matrix U[Cart]
is defined by the average atomic shifts (and their correlations) along each coordinate axis. The matrix U[Cart] varies between atoms and is diagonal (with equal elements) for atoms that are assumed
to be moving isotropically.
U[Cart] can accumulate contributions from several different sources, including overall crystal anisotropy (U[cryst]), various concerted motions (U[group]) and independent displacement of individual
atoms (U[local]) (see, for example, Dunitz & White, 1973 ; Prince & Finger, 1973 ; Johnson, 1980 ; Sheriff & Hendrickson, 1987 ; Murshudov et al., 1999 ; Winn et al., 2001 ; , Dauter et al., 2012 ;
Afonine et al., 2013 ).
Concerted motion contributing to U[group] can be modelled by the translation–libration–screw approximation (TLS) introduced by Cruickshank (1956 ) and Schomaker & Trueblood (1968 ) and developed
further in a number of publications, for example Johnson (1970 ), Scheringer (1973 ), Howlin et al. (1989 , 1993 ), Kuriyan & Weis (1991 ), Schomaker & Trueblood (1998 ), Tickle & Moss (1999 ),
Murshudov et al. (1999 ), Winn et al. (2001 , 2003 ) and Painter & Merritt (2005 , 2006a ,b ). This approximation is of special interest to structural biologists for two reasons. Firstly, TLS
characterizes the anisotropic mobility of atomic groups and can provide insight into molecular mechanism. Secondly, it simplifies the crystallographic model by reducing the number of parameters while
simultaneously providing a more realistic description of atomic displacements.
A common misconception of TLS parametrization is that its sole merit is to provide an economical method of accounting for anisotropic motions at low resolution. In fact, TLS parameterization can be
useful regardless of the resolution of the available diffraction data. TLS has been successfully used to analyze functionally important molecular motions on several occasions (Kuriyan & Weis, 1991 ;
Harris et al., 1992 ; Šali et al., 1992 ; Wilson & Brunger, 2000 ; Raaijmakers et al., 2001 ; Yousef et al., 2002 ; Papiz et al., 2003 ; Chaudhry et al., 2004 ), demonstrating that this approximation
can provide critical structural information. However, the use of TLS models to derive functional insights is limited by the difficulty in analyzing the resulting motions. Although analysis of the
resulting anisotropic displacement parameters is possible in some programs (Howlin et al., 1993 ; Painter & Merritt, 2005 ), decomposing TLS models into structural ensembles comprised of many atomic
models might enable more straightforward comparisons to other data sets, particularly in the case of diffuse X-ray scattering (Van Benschoten et al., 2015 ). The major goal of this work is to develop
an approach for translating TLS matrices into descriptions of corresponding molecular motions in terms of rotations and translations. In turn, this allows the validation of TLS parameters and the
generation of structural ensembles. The latter will enable the broader use of TLS refinement for discovering and validating concerted molecular motions. In accomplishing this goal, we encountered
several complications that suggest revisiting the fundamental processes of TLS refinement.
1.2. TLS model
Since the displacement of a rigid group of atoms is a composition of translation and rotation (see, for example, Goldstein, 1950 ), Schomaker & Trueblood (1968 ) presented the matrices U[group,n] for
the concerted motion of a group of atoms n = 1, 2, … N as a sum,
The antisymmetric matrices A[n] are functions of the Cartesian coordinates (x[n], y[n], z[n]) of atom n
Matrix S and the symmetric matrices T and L are common to all atoms within each rigid group. L describes librations (oscillating rotations) around three mutually orthogonal rotation axes. T describes
apparent translations of the atomic group (the term `vibrations' might actually be more appropriate for random translations around a central position). S describes screw motions, i.e. the combination
of librations and vibrations. We use the term `apparent translation' because matrix T may have an additional contribution from librations as discussed in § 2.
Thus, explicit information about atomic movement can be encoded into TLS matrices to produce inexplicit descriptors of motion. Both frameworks have merit: explicit description allows a
straightforward interpretation and analysis of the motions, while the inexplicit TLS formalism provides a simpler framework for calculating structure factors. However, it is important to remember
that TLS parameterization always arises from explicit atomic movement; thus, the TLS matrices should obey certain restrictions in order to be decomposed into structural ensembles representing
concerted physical motions. Current refinement programs treat elements of the TLS matrices as independent variables with a constraint on the trace of the matrix S [tr(S); as discussed in § 4] and
post-refinement enforcement that the resulting U[group,n] be non-negative definite (Winn et al., 2001 ). As demonstrated below, enforcing U[group,n] to be non-negative definite is not sufficient to
guarantee that the refined TLS matrices are still consistent with an underlying physical model of concerted motion.
Previously, Zucker et al. (2010 ) analyzed all PDB entries containing TLS descriptions and suggested tools to validate the TLS parameters. However, this analysis focused exclusively on the ADP
smoothness between neighbouring TLS groups. Failure to enforce all conditions on the individual components of U[group,n], i.e. on the TLS matrices, may result in matrices that invalidate the TLS
model. Using the methods and tools presented in this manuscript, we analyzed all structures from the PDB (Bernstein et al., 1977 ; Berman et al., 2000 ; about 105000 entries, 25000 of which contain
TLS models, with a total of 200000 sets of matrices). Our results demonstrate that significant issues are present in current TLS implementations. A third of the analyzed structures contain T or L
matrices that are non-positive semidefinite and another third (Table 1 ) cannot describe libration–vibration correlated motions owing to the reasons discussed in §§ 2–5. Some of these errors (but
not all) are trivial to fix, e.g. correcting marginally negative eigenvalues of T and L or modifying the trace of S (examples are given in § 6 and in Table 1 ).
The statistics are shown for the matrices in the PDB (25904 entries with TLS matrices from a total number of 106761 entries as of March 2015) with the default condition tr(S) = 0 (upper line) and
with the optimal choice of the diagonal S elements whenever possible as described in §§ 3 and 4 (bottom line). The conditions are, from left to right: matrices T and L are positive semidefinite (T ≥
0 and L ≥ 0); an absence of libration around one of the axes requires the corresponding elements of the S matrix to be equal to 0 (s = 0 and w = 0); matrix T is positive semidefinite after the
contribution owing to the displacement of libration axes is removed (T[C] ≥ 0); elements of the S matrix are limited by the corresponding elements of the T and L matrices according to the Cauchy
conditions (S ≤ TL); the residual V matrix is positive semidefinite (V ≥ 0). The column (V ≥ 0) includes all conditions from §§ 4.3 and 4.4. When one of the conditions was broken further conditions
were not checked.
Conditions broken
Mode Total No. of PDB entries Total No. of TLS T ≥ 0 and L ≥ 0 s = 0 and w = 0 T[C] ≥ 0 S ≤ TL V ≥ 0 Total No. of TLS broken Total No. of TLS OK Total No. of PDB entries broken
t[S] = 0 25904 203261 71362 3104 52254 n/a 10492 137212 66049 22707
Best t[S] 25904 203261 71362 3104 52255 133 3776 130630 72631 22201
1.3. On the physical meaning and use of TLS
Efforts to constrain TLS parameters to keep them physically meaningful have been discussed previously (Winn et al., 2001 ; Painter & Merritt, 2006a ). It is universally accepted that B values need to
be positive, occupancies must range between 0 and 1 and atomic coordinates should define model geometry in accordance with chemical knowledge. Similarly, provided that the TLS groups have been
selected adequately, the TLS parameters describing the anisotropic harmonic motion of atomic groups (Schomaker & Trueblood, 1968 ) should be physically meaningful, otherwise TLS modelling may not be
considered to be applicable. One such condition, but not the only one, is that the T and L matrices are positive semidefinite.
While calculating TLS matrices from corresponding libration and vibration parameters is rather straightforward (§ 2), the inverse procedure is less trivial. As discussed previously (Johnson, 1970 ;
Scheringer, 1973 ; Tickle & Moss, 1999 ), the problem itself is poorly posed since the same set of diffraction data (and consequently the same set of TLS matrices) may correspond to different motions
of the contributing atoms or atomic groups. Moreover, there are computational difficulties if all the conditions on the matrices have not been considered (§§ 3–5).
The set of TLS matrices corresponding to physically possible combinations of motions is obviously smaller than the set of all TLS matrices. Since restricting the parameter space of any function may
inadvertently exclude a number of deep minima, including the global minimum, structural refinement that imposes conditions on TLS matrices may result in higher R factors than if these conditions were
ignored. Since TLS modelling is an approximation to the true molecular motions that strongly depends on the assignment of TLS groups, lower R factors as result of using TLS may not always be
indicative of this model being decomposable into a valid macromolecular motion.
1.4. Summary of the presented work
In this article, we address the following points.
• (i) We describe an algorithm (Fig. 1 ) that interprets the TLS matrices in terms of parameters of the corresponding motions. This includes the direction of the principal axes of vibration and
libration, the corresponding root-mean-square displacements and the position of the libration axes, as well as the correlations between vibration and libration displacements.
• (ii) We present a complete list of conditions that must be fulfilled to make the aforementioned TLS decomposition possible; this includes widely known conditions (e.g. T and L must be positive
semidefinite) as well as a number of less trivial conditions that to the best of our knowledge have not been previously discussed.
• (iii) We describe the calculation protocols in a ready-to-program style so that they can be implemented in existing or future software. Most of the calculations described in the manuscript are
straightforward; less trivial expressions and proofs can be found in Appendix A as well as in the review by Urzhumtsev et al. (2013 ).
• (iv) We implemented the described algorithms in the open-source Computational Crystallography Toolbox (cctbx; Grosse-Kunstleve et al., 2002 ). We also made two end-user applications available in
the PHENIX suite (Adams et al., 2010 ): phenix.tls_analysis for the analysis and validation of refined TLS matrices and their underlying motions and phenix.tls_as_xyz for generating ensembles of
structures consistent with TLS matrices.
• (v) We applied these programs to all PDB entries containing TLS matrices. We discovered that the majority of these matrices cannot describe motions. In a number of cases a marginal modification
of the TLS matrices can correct the errors.
• (vi) We used phenix.tls_as_xyz to generate a predicted structural ensemble for the calculation of X-ray diffuse scattering from the glycerophosphodiesterase GpdQ (Van Benschoten et al., 2015 ).
2. Calculating TLS matrices from elemental motions
This section provides a step-by-step protocol for calculating TLS matrices from the parameters of the composite vibrations and librations. Inverting this scheme provides a method of extracting
libration/vibration parameters from the TLS matrices.
2.1. Constructing TLS matrices from the parameters of the libration and vibration
The matrices in (2) depend on the basis in which the atomic coordinates are given. We use an index in square brackets to indicate which basis is used. Let the atoms be given in some basis denoted
[M]; for example, it may be the basis corresponding to the model deposited in the PDB. Even if a rigid group is involved in several simultaneous motions (assuming that the amplitudes of these motions
are relatively small and the motions are harmonic), the total motion can be described by a libration around three axes l[x], l[y], l[z] that are mutually orthogonal and by a vibration along three
other mutually orthogonal axes, v[x], v[y], v[z]. These triplets of axes form the other two bases, [L] and [V].
In (2) the matrix T is a sum of several components. In the absence of librations (that is, matrices L and S are zero) it is equal to the contribution V arising from pure vibrations. In the basis [V]
this matrix is diagonal,
Here, 〈t[x]^2〉, 〈t[y]^2〉, 〈t[z]^2〉 are the corresponding squared root-mean-square deviations (r.m.s.d.s) along the principal vibration axes v[x], v[y], v[z] and are expressed in Å^2. If there
are librations, the matrix L is always diagonal in the basis [L],
Here, 〈d[x]^2〉, 〈d[y]^2〉, 〈d[z]^2〉 are the squared r.m.s.d.s of the vibration angles expressed in squared radians; for small deviations they are numerically equal to the squared r.m.s.d.s of
points at a unit distance from the corresponding axes.
In reality, the principal vibration and libration axes are not parallel to each other; practically, it is convenient to express the matrices in a common basis. Basis [L] is more convenient for this
since in this basis the elements of S (see below) are easily expressed through geometric parameters of librations. Matrix V in this basis is no longer diagonal but is instead equal to
Here, R[VL] is the transition matrix that describes the rotation superposing the vectors v[x], v[y], v[z] with the vectors l[x], l[y], l[z] (Appendix A ). Frequently, vibration and libration motions
are not independent but instead are correlated to form screw rotations. It is convenient to characterize screw rotations by the parameters s[x], s[y], s[z]: for a screw rotation by d[z] radians
around an axis parallel to l[z] each atom is shifted by s[z] Å along this axis. A similar definition is used for the other two parameters. If the axes pass through the origin, such a correlation
generates an additional contribution C[[L]] to the T matrix that arises from screw motions,
and also results in a nonzero S matrix,
Finally, the principal libration axes do not necessarily pass through the origin, or even have a common point (i.e. they may not intersect). If they pass through the points w^lx[[L]] = (w[x]^lx, w[y]
^lx, w[z]^lx), w^ly[[L]] = (w[x]^ly, w[y]^ly, w[z]^ly), w^lz[[L]] = (w[x]^lz, w[y]^lz, w[z]^lz), respectively, this generates an additional component to the T matrix,
Taking into account both the screw motion and the position of the libration axes, the matrix S becomes
Finally, the matrices in the original basis [M] where they are reported together with the atomic coordinates are obtained from L[[L]] (5 ), T[[L]] (9 ), S[[L]] (11 ) as
Here, R[ML] is the transition matrix from the basis [M] to the basis [L] (Appendix A ).
2.2. Molecular basis and centre of reaction
The TLS matrices also depend on the choice of the origin. Clearly, the coordinates of the position of the libration axes change as function of the origin. Usually, the origin is taken to be the
centre of mass of the atomic group or the point where the mean atomic displacements are similar in magnitude to each other owing to librations around each of the principal axes. This second point is
called the centre of diffusion (Brenner, 1967 ) or the centre of reaction (Tickle & Moss, 1999 ). Choosing the origin at the centre of reaction minimizes the trace of T and makes S symmetric
(Brenner, 1967 ; Tickle & Moss, 1999 ; Urzhumtsev et al., 2013 ). Shifting from one origin to another changes T and S but does not change L and does not modify the algorithm of the search for the
composite motions. In the following, we consider the matrices to be in their original basis (for example, as they are defined in the PDB).
3. Calculating elemental motions from TLS matrices: libration axes
This section provides a step-by-step explanation of the inverse problem, i.e. calculating the vibration and libration axes and the corresponding r.m.s.d.s, the position of the libration axes and the
parameters describing the correlations between librations and vibrations from given TLS matrices.
3.1. Diagonalization of the L matrix ([L] basis; step A)
Suppose that we know the elements of the matrices (12) in the basis [M]. By construction, the matrices T and L should be positive semidefinite (Appendix B ) and symmetric, T[[M]xy] = T[[M]yx], T[[M]
xz] = T[[M]zx], T[[M]yz] = T[[M]zy] and L[[M]xy] = L[[M]yx], L[[M]xz] = L[[M]zx], L[[M]yz] = L[[M]zy]. These properties hold for any rotation of the coordinate system, i.e. in any Cartesian basis;
this is important for further analysis of the T matrices.
We start the procedure from the matrix L[[M]], which depends only on the libration parameters. The principal libration axes correspond to its three mutually orthogonal eigenvectors. Firstly, we
search for the corresponding eigenvalues 0 ≤ λ[1] ≤ λ[2] ≤ λ[3], which must be non-negative (see equation 5 ; eigenvalues do not change with the coordinate system). Let l[1], l[2], l[3] be the
corresponding normalized eigenvectors from which we construct a new basis [L] as
The appropriate sign for l[x] is chosen so that the vectors in (13) form a right-hand triad; for example, one can take l[x] = l[y] × l[z] which guarantees such a condition. The TLS matrices in the
[L] basis are
where R[ML] is the transition matrix from basis [M] into basis [L] (Appendix A ). In this new basis, matrix L[[L]] is diagonal with elements L[[L]xx] = λ[1], L[[L]yy] = λ[2], L[[L]zz] = λ[3], giving
the estimates 〈d[x]^2〉 = L[[L]xx], 〈d[y]^2〉 = L[[L]yy], 〈d[z]^2〉 = L[[L]zz] of the squared libration amplitudes around the three principal libration axes.
3.2. Position of the libration axes in the [L] basis (step B)
In the basis [L] the libration axes are parallel to the coordinate axes but do not necessarily coincide with them. Let them pass through some points w^lx, w^ly, w^lz, respectively, that must be
identified. Using (11) , we calculate the coordinates of these points as
A zero value for any denominator in (15) means that there is no rotation around the corresponding axis; in this case, the two corresponding numerator values must also be equal to zero and thus assign
zero values to the corresponding coordinates in (15) . Otherwise, the input matrices are incompatible and the procedure must stop (Appendix B ). The x component of w^lx, the y component of w^ly and
the z component of w^lz in the basis [L] can be any values. For presentation purposes, it might be useful to assign them as
that will position each of these points in the middle of the two other axes. This choice also reduces eventual rounding errors.
Knowing the positions (15 and 16 ) of the libration axes and elements of L[[L]], we can calculate the contribution D[W[L]] (10) from an apparent translation owing to the displacement of the libration
axes from the origin. Then, by inverting (9) we can calculate the residual matrix T[C[L]] after removal of this contribution,
Matrix (17) must be positive semidefinite (Appendix B ) as it is a sum (7) of two positive semidefinite matrices. Matrices S[[L]] and L[[L]] are not modified at this step.
4. Calculating elemental motions from TLS matrices: screw components (step C)
4.1. Correlation between libration and vibration and a choice of the diagonal elements of S
Next, we use the matrices L[[L]] and S[[L]] to determine the screw parameters s[x], s[y], s[z], remove the screw contribution from the T[C[L]] matrix using (7) and (17) and finally extract the matrix
V[[L]] for uncorrelated vibrations. However, there is an ambiguity in the definition of S[[L]] which is apparent from the observation that the matrices U[concerted,n] of individual atoms will not
change if the same number t is added or removed simultaneously from all three diagonal elements of S[[L]]. This is usually known as indetermination of the trace of this matrix (Schomaker & Trueblood,
1968 ). A current practice (an illustration is provided in § 6.1) is to choose t such that it minimizes the trace (rather its absolute value) of the resulting matrix,
(where I is a unit matrix), i.e. minimizing vibration–libration correlation (Urzhumtsev et al., 2013 ), or simply makes the trace equal to zero (https://www.ccp4.ac.uk/html/restrain.html; Coppens,
2006 ). The unconditioned minimization
However, this value may lead to matrices for which libration–vibration decomposition is impossible and, in particular, prohibits the generation of structural ensembles. For example, if the elements
of matrix S and the corresponding values s[x], s[y], s[z] are too large, the matrix V in (7) may be not positive definite for a given T[C[L]]. The next sections describe a procedure that defines the
constraints on the diagonal elements of matrix S when using (18) .
4.2. Cauchy–Schwarz conditions
After removing D[W[L]] (17) , the set of matrices T[C[L]], L[[L]] and the matrix S[[L]] with the removed off-diagonal elements (reducing the matrix in equation 11 to the form in equation 8 )
correspond to a combination of vibrations with screw rotations around the axes crossing the origin. The diagonal elements of these matrices must satisfy the Cauchy–Schwarz inequality (Appendix A ),
that in turn defines the conditions (Appendices A and B )
In particular, this unambiguously defines the t value if one of the diagonal elements of the matrix L[[L]] is zero so that the trace of S[[L]] cannot be changed or assigned arbitrarily (see § 4.4).
4.3. Positive semidefinition of the V matrix
The last condition to check is that the matrix V is positive semidefinite. Let us suppose that all diagonal elements of the matrix L[[L]] are different from zero; § 4.4 considers the alternative
case. From (5) , (7) , (8) and (18) we find the expression for the screw contribution
to be subtracted from matrix (17) as
Matrix V[[L]] is positive semidefinite along with
Necessarily, all diagonal terms of (30) cannot be larger than the maximal eigenvalue τ[max] of matrix (29) , giving a necessary condition (Appendix B )
The condition that all diagonal terms of (30) are not larger than the minimum eigenvalue τ[min] of (29) is sufficient but not necessary.
Matrix V[Λ] is positive semidefinite if and only if all three of its real eigenvalues are non-negative (some of them may coincide with each other). They are the roots of the cubic characteristic
with the coefficients
The roots of (32) are positive if and only if the three inequalities below hold simultaneously,
where the left parts are polynomials of order two, four and six of the parameter t, all with the unit highest-order coefficient (Appendix A ). The first condition in (36) defines the interval for t
values (Appendix B ),
We failed to find analytical expressions corresponding to the two other inequalities. As a result, the following numerical procedure is suggested to find the best t value that is physically
• (i) Calculate the t[0] value (20) .
• (ii) Calculate the interval (t[min], t[max]) for allowed t values as the intersection of intervals (23) , (31) and (37) , t[min] = max{t[min,C], t[min,τ], t[min,a]}, t[max] = min{t[max,C], t[max,
τ], t[max,a]}; if t[min] > t[max] the problem has no solution and the procedure stops (Appendix B ).
• (iii) If t[min] = t[max] we check the conditions b[S](t[min]) ≥ 0, c[S](t[min]) ≤ 0, or the condition that V[Λ] is positive semidefinite; if the conditions are satisfied we assign t[S] = t[min],
otherwise the problem has no solution and the procedure stops (Appendix B ).
• (iv) If t[min] < t[max] we search numerically, in a fine grid, for the point t[S] in the interval (t[min], t[max]) and closest to t[0] such that b[S](t[S]) ≥ 0, c[S](t[S]) ≤ 0; if for any point
of this interval at least one of these inequalities is wrong then the procedure stops (Appendix B ).
• (v) We accept the value obtained at the step (iii) or (iv) as the final t[S].
4.4. Singular sets of rotation
When one of the L[[L]xx], L[[L]yy], L[[L]zz] values is zero (that is, there is no rotation around the corresponding axis), straightforward use of the standard procedure including (25) becomes
impossible. However, in this case the t[S] value must be equal to S[[L]xx], S[[L]yy] or S[[L]zz], corresponding to the axes with no rotation, making the corresponding diagonal element in (25) equal
to zero and turning the corresponding inequality in (24) into an equality. For example, if L[[L]xx] = 0 then t[S] = S[[L],xx], resulting in C[[L]xx] = 0. We simply need to check two other conditions
in (21) and confirm that the residual matrix is positive semidefinite (for example, by calculating equation 36 ). If t[S] does not satisfy these conditions, the problem has no solution (Appendix B ).
4.5. Screw parameters
For the t = t[S] determined above we calculate the matrix S[C](t[S]) (18) . From this matrix we obtain the screw parameters s[x] = S[C,xx]L^−1[[L]xx], s[y] = S[C,yy]L^−1[[L]yy], s[z] = S[C,zz]L^−1
[[L]zz] for the rotation axes currently aligned with the coordinate axes of the basis [L]. If one of the L[[L]xx], L[[L]yy], L[[L]zz] values is equal to zero, the corresponding diagonal element of S
[C] must also be equal to zero, and we assign the corresponding screw parameter, s[x], s[y] or s[z], to be zero. Otherwise, the matrices are inconsistent with each other and the procedure stops
(Appendix B ).
5. Calculating elemental motions from TLS matrices: vibration components (step D)
5.1. Matrix V and vibration parameters in [L] basis
For the known t[S], the matrices C[[L]](t[S]) and then V[[L]] are calculated using (25) and (26) . The parameter values of the independent vibrations are calculated from the V[[L]] matrix similarly
to those for the independent librations, as we obtain them from L[[M]]. Firstly, we calculate the three eigenvalues 0 ≤ μ[1] ≤ μ[2] ≤ μ[3] of matrix V[[L]] (Appendix B ; in practice, all of them are
strictly positive). We then identify three corresponding unit eigenvectors v[1], v[2], v[3] that are orthogonal to each other and assign
[the sign for v[x] is taken so that the vectors (39) form a right-hand triad]. We remind the reader that these axes define the basis [V] in which matrix V[[V]] (6) is diagonal with elements V[[V]xx]
= μ[1], V[[V]yy] = μ[2], V[[V]zz] = μ[3]. This defines the last missing parameters, namely the values of the squared r.m.s.d.s along these axes: 〈t[x]^2〉 = V[[V]xx], 〈t[y]^2〉 = V[[V]yy], 〈t[z]^2
〉 = V[[V]zz].
5.2. Vibration and libration axes in [M] basis
The libration and vibration amplitudes and screw parameters are independent of the choice of the basis, and the direction of the libration axes is known in the principal [M] basis. However, the
directions of the uncorrelated translations v[1], v[2], v[3] that were calculated in § 4 and the points w^lx[[L]], w^ly[[L]], w^lz[[L]] belonging to the libration axes (§ 3.2) are now known in the
[L] basis.
To obtain the coordinates (w^lx[[M]x], w^lx[[M]y], w^lx[[M]z]), (w^ly[[M]x], w^ly[[M]y], w^ly[[M]z]), (w^lz[[M]x], w^lz[[M]y], w^lz[[M]z]) of these points in the [M] basis, we apply the
Similarly, the vectors defining the direction of the axes v[x], v[y], v[z] in the basis [M] can be obtained as
This step finalizes the extraction of the parameters of the motions that correspond to the given set of TLS matrices. § 6 provides some examples of this procedure applied to models deposited in the
PDB. § 7 describes an example in which knowledge of the motion parameters extracted from the TLS matrices is necessary to explicitly simulate the ensemble of corresponding structures and the
corresponding X-ray diffuse scattering.
6. Examples of TLS matrix analysis
As discussed in § 1, there are numerous examples of fruitful application of the TLS formalism to structural studies of concerted motion. The goal of this section is to illustrate the algorithm
described above, to describe possible traps that emerge during refinement and to discuss further developments.
6.1. Survey of available TLS matrices in the PDB
We have analyzed all available TLS matrices in the PDB. From an overall 106761 entries (as of March 2015), 25904 use TLS modelling. More than 20000 of these entries have several TLS groups,
resulting in a total of 203261 sets of TLS matrices (Fig. 2 a), with the largest number of groups per entry being 283 (PDB entry 3u8m; Rohde et al., 2012 ). About a third of these sets have negative
eigenvalues for the deposited T or L matrices. Some of these values are only slightly negative (Figs. 2 b and 2 c) and can be considered to be rounding errors, while the worst values are as small as
−0.28 rad^2 for L and −20.72Å^2 for T. For 11412 T matrices and 138 L matrices all three eigenvalues are negative.
Another third of the TLS groups cannot be interpreted by elemental motions owing to other reasons, as described in §§ 3 and 4 (Table 1 ).
After an initial screen to find the positive definite T and L matrices, we then ran a search for the elemental motions in two modes. Firstly, we tried to decompose the TLS matrices as taken directly
from the PDB files. As expected, the average value of tr(S) is 3 × 10^−5Å (i.e. practically zero) and the corresponding r.m.s.d. is σ = 10^−2Å. About 120000 S matrices have |tr(S)| < 10^−4Å. The
numbers of matrices with |tr(S)| larger than 1σ, 3σ, 10σ and 20σ are only 3772, 486, 31 and three, respectively. We then applied the aforementioned algorithm with the optimal choice of the value t[S]
to be subtracted from the diagonal S elements in each case.
Table 1 shows the results of both runs and illustrates that it is possible to fix the problems found in 6500 of the TLS sets (corresponding to about 500 PDB entries) by a correction of the diagonal
elements of the S matrix as described above. The table takes into account possible rounding errors by correcting slightly negative eigenvalues (those closer in value to zero than 10^−5 of the default
units: Å^2, rad^2 and Årad for T, L and S, respectively). For example, when running the algorithm in the S optimizing mode the program can formally calculate the V matrix for about 70000 sets. For
2296 cases this matrix has negative eigenvalues (Fig. 2 d), while in 2294 cases these eigenvalues are closer to 0 than 10^−5Å^2; for such matrices the program makes automatic corrections and
continues the process.
It is important to note that even if the parameters of the elemental motions can be formally extracted from the TLS matrices, this does not guarantee that they will make physical sense and therefore
be valid for decomposition into a representative structural ensemble. Clearly, vibration amplitudes on the order of 20Å^2 cannot represent harmonic vibrations (Fig. 2 d). Similarly, the linear
rotation approximation contained in TLS theory is valid only up to approximately 0.1rad; however, much larger values can be found in the PDB (Fig. 2 b). Similar restrictions also hold for the screw
parameters. The products s[x]d[x], s[y]d[y], s[z]d[z] show the mean shifts along the screw axes owing to librations around these axes; the values found in the PDB approaching 3Å seem to be too large
to describe harmonic motions.
For a more detailed analysis, we selected several entries from the PDB. For each structure, we applied a standard TLS refinement protocol as implemented in phenix.refine (Afonine et al., 2012 )
including automatic determination of the TLS groups. During refinement, 20 matrix elements were refined independently, six for T, six for L and eight for S; the three diagonal elements of S were
constrained such that the trace of the matrix is equal to 0. The procedure described above (§§ 3–5) was then applied to all sets of obtained TLS matrices.
We remind the reader that the elements of the L and S matrices are expressed in rad^2 and Årad, while in the PDB files they are in deg^2 and in Ådeg, respectively (Table 2 ).
The matrix elements extracted from the PDB files after refinement (§ 6).
PDB code Chain, residue No. T (Å^2) L (deg^2) S (Ådeg)
1dqv A1–A97 0.1777 0.0090 −0.0044 1.4462 −0.0160 −0.2656 0.0467 −0.0523 0.0566
0.0090 0.1306 0.0019 −0.0160 1.2556 0.4713 0.1010 0.0032 −0.0164
−0.0044 0.0019 0.1372 −0.2656 0.4713 0.8689 0.0090 0.0188 0.0560
B1–B97 0.1777 0.0090 −0.0044 1.4462 −0.0160 −0.2656 0.0467 −0.0523 0.0566
0.0090 0.1306 0.0019 −0.0160 1.2556 0.4713 0.1010 0.0032 −0.0164
−0.0044 0.0019 0.1372 −0.2656 0.4713 0.8689 0.0090 0.0188 0.0560
1exr A2–A30 0.0899 0.0040 −0.0004 1.3491 −0.3760 −0.3971 −0.0249 −0.3537 −0.0874
0.0040 0.1333 0.0058 −0.3760 0.6103 −0.3389 0.1275 0.0783 −0.0144
−0.0004 0.0058 0.0728 −0.3971 −0.3389 0.3698 0.0183 0.0542 −0.0103
A31–A74 0.0925 0.0037 0.0041 0.3464 0.3638 0.2923 −0.0220 −0.0419 −0.0793
0.0037 0.0673 0.0062 0.3638 0.3283 0.1212 −0.0061 0.0018 0.1161
0.0041 0.0062 0.1119 0.2923 0.1212 0.3799 −0.0041 −0.0385 −0.0009
A75–A84 0.2433 0.0144 0.0917 0.0736 0.0171 0.0565 0.4357 0.1151 0.2346
0.0144 0.2867 0.1720 0.0171 0.0068 −0.0203 −0.2521 −0.3549 −0.2041
0.0917 0.1720 0.1749 0.0565 −0.0203 0.0336 −0.3793 −0.1499 0.0111
A85–A147 0.0747 −0.0110 0.0066 0.6097 −0.0786 −0.1864 0.0180 0.1466 0.0378
−0.0110 0.1384 0.0062 −0.0786 0.6474 −0.6233 0.0155 −0.0872 −0.0542
0.0066 0.0062 0.0673 −0.1864 −0.6233 0.9637 −0.0440 0.1022 −0.0852
4b3x A1–A65 0.4663 0.0991 −0.0764 0.4738 0.0063 0.2318 0.0391 −0.0307 −0.4316
0.0991 0.5443 −0.0321 0.0063 0.2120 −0.0584 0.0587 0.1786 −0.2003
−0.0764 −0.0321 0.5001 0.2318 −0.0584 0.1312 0.3665 0.4293 0.0403
A66–A363 0.1649 −0.0259 0.0184 0.8808 −0.0912 −0.1736 −0.0345 0.0102 −0.0661
−0.0259 0.1422 0.0055 −0.0912 0.9522 0.0972 0.1159 −0.0222 0.0999
0.0184 0.0055 0.2028 −0.1736 0.0972 1.6563 0.0424 −0.1330 −0.0237
6.2. Synaptotagmin
The crystals of synaptotagmin III (PDB entry 1dqv; Sutton et al., 1999 ) contain two copies of the molecule in the asymmetric unit. The structure after re-refinement by phenix.refine without TLS
modelling has an R[work] of 0.200 and an R[free] of 0.231 at a resolution of 2.5Å. Performing TLS refinement with each molecule taken as a single TLS group reduced the R factors to R[work] = 0.177
and R[free] = 0.211, indicating that this additional modelling significantly improves the agreement with the experimental data. Table 2 shows the two sets of matrices and Table 3 contains the
corresponding motion parameters extracted using our approach. For the two groups both vibrations and librations are practically isotropic and are of the same order of magnitude. Fig. 3 (a) shows the
principal axes of these motions.
The parameters are given in the units used in this article, allowing an easy estimation of the corresponding atomic displacements. The directions of the libration and rotation axes are not given.
PDB code Chain, residue No. T: t[x], t[y], t[z] (Å) L: d[x], d[y], d[z] (rad) S: s[x], s[y], s[z] (Å) tr(S)
1dqv A1–A97 0.3455 0.3671 0.4172 0.01239 0.02044 0.02273 1.343 1.137 −1.319 0
B1–B97 0.3634 0.3885 0.4166 0.01608 0.01753 0.03069 0.679 −1.177 0.200 0
1exr A2–A30 0.1944 0.2663 0.2870 0.00000 0.01602 0.02182 0.000 2.951 3.408 >0
A31–A74 0.2110 0.2939 0.3068 0.00000 0.00860 0.01637 0.000 −18.14 −5.028 <0
A75–A84 0.1692 0.4906 0.6598 0.00000 0.00000 0.00000 0.000 0.000 0.000 0
A85–A147 0.0002 0.2270 0 3078 0.00553 0.01418 0.02109 20.83 0.800 −1.672 ∼0
4b3x A1–A65 0.0994 0.6064 0.7116 0.00000 0.00825 0.01343 0.000 2.718 −11.05 <0
A66–A363 0.3306 0.4102 0.4413 0.01568 0.01720 0.02283 3.164 −2.276 −0.197 0
6.3. Calmodulin
The structure of calmodulin (PDB entry 1exr; Wilson & Brunger, 2000 ) has been determined previously at a resolution of 1.0Å. This example illustrates possible problems that can be solved by a
minimal correction of the TLS values. For re-refinement with phenix.refine the model was automatically split into four TLS groups. For the first group, one of the eigenvalues of the matrix L was
equal to −2 × 10^−5rad^2. If we consider this value to be zero (in this case the zero value must be also assigned to off-diagonal elements of the first row of the matrix S), the composite motions
contain only two libration axes and their parameters can be extracted. Corresponding modifications of the resulting matrices U[group,n] (2) can be compensated for by respective adjustment of the
individual contributions U[local,n]. This keeps the total ADP parameters U[Cart,n] unchanged, thus maintaining the previously calculated structure factors and R factors. An accurate separation of
total atomic displacement parameter values into contributions from several sources (see, for example, Murshudov et al., 1999 ; Winn et al., 2001 , 2003 ; Afonine et al., 2012 ) is a separate ongoing
project (Afonine & Urzhumtsev, 2007 ).
For the second TLS group, the refined TLS matrix elements contained one degenerate libration. The procedure described in §§ 3–5 was successfully applied. Note that this procedure modified the
diagonal elements of the matrix S, removing an appropriate value of the parameter t[S] (§ 4.4) and making tr(S) nonzero.
For the third group, all three eigenvalues of the matrix L were extremely small (0.0, 0.8 × 10^−5 and 3 × 10^−5rad^2), producing high computational instability and extremely large screw parameters
that resulted in the inability of the procedure to find a positive semidefinite V[[L]] (27) . If we define all librations to be absent and replace matrix L (and respectively S) by zero matrices, the
vibration parameters can easily be found from T. In fact, this TLS group is a helix held at both ends by large domains, which leads to the expectation of a pure vibration motion.
Finally, for the fourth group one of the diagonal elements of the matrix T was marginally negative. Increasing all of the diagonal elements of the matrix T by 0.002Å^2 makes this matrix positive
definite (this corresponds to B = 0.16Å^2). As discussed above, this adjustment can be compensated for by removing the equivalent amount from individual atomic contributions U[local,n] (such a
subtraction keeps the individual atomic contributions positive). This group vibrates in a plane (Fig. 3 b) and the principal vibration axis of group 3 (the helix) is parallel to this plane, leading
to the plausible hypothesis that groups 3 and 4 at least partially move together or slide along each other.
To check the influence of the manual modification on the TLS matrices, we recalculated the R factors before and after performing these changes without updating the individual atomic contributions U
[local,n]. For all of the modifications described above, including the ensemble of modifications applied together, the R factors only varied in the fourth significant digit.
This example demonstrates that although current refinement procedures may result in TLS matrices that are unable to satisfy the previously mentioned conditions, small changes to them may provide
sufficient correction. This highlights the need to use appropriate restraints or constraints on refinable parameters within the TLS model.
6.4. Initiation translation factor 2 (IF2)
The structure of IF2 (PDB entry 4b3x) has recently been solved in one of our laboratories (Simonetti et al., 2013 ) with an R[work] of 0.180 and an R[free] of 0.219 at a resolution of 1.95Å. A
posteriori TLS refinement was performed with two groups: the first group included the N-terminus and the following long helix, and the second included the rest of structure. Re-refining the model
produced better statistics, with R[work] = 0.176 and R[free] = 0.203. In this example, the TLS matrices from the first group were not directly interpretable because the residual matrix V[[L]] was not
positive semidefinite (the minimal eigenvalue was −0.05). Similarly to the last TLS group in calmodulin, we artificially added 0.06Å^2 to all diagonal elements of the matrix T, corresponding to
roughly 5Å^2 (the same amount has been removed from the residual atomic B values, thus leaving the R factors unchanged). This correction allowed interpretation of the TLS matrices in terms of
elemental motions. We note that for the first TLS group one of the rotations was degenerate and that the assignment tr(S) = 0 would make this matrix incompatible with L. Table 3 shows that the
vibrations of this group are essentially anisotropic. Fig. 3 (c) also shows that the libration axes for this group pass quite far away from the molecule, which makes the corresponding rotation
similar to a translation. Additionally, we believe that the large s[z] value indicates that the matrix S is not well defined. The matrices for the second group were interpreted and revealed isotropic
vibrations and librations.
Finally, we tried to apply the same procedure after choosing the TLS groups manually as residues 1–50 (N-terminus), 51–69 (helix), 70–333 (G domain) and 343–363 (the connector to the C domain, which
is absent in this structure). As before, the matrices were interpretable for the G domain. For groups 2 and 4, after an adjustment similar to those discussed above (a slight increase of the diagonal
T elements with a decrease of the residual atomic B values), we obtained a pure vibration for the helix (as for the calmodulin case) and a libration around a single axis for the terminal group. In
contrast, we failed to find reasonably small corrections for the matrices of the first group that would make them interpretable in terms of physical motions that in particular can be represented by a
structural ensemble.
This case exemplifies a situation in which the current TLS refinement protocols result in matrices that significantly reduce the R factors without providing refined TLS parameters that can be
decomposed into a physically realistic motion of one of the groups. This highlights the need to improve TLS refinement algorithms by making use of constraints on aforementioned conditions on TLS
7. Interpreting TLS matrices with a structural ensemble
7.1. Generation of an explicit set of atomic models with a variability consistent with TLS
Some structural problems may explicitly require a set of models that describe a given mobility, for example corresponding to the TLS matrices for harmonic motion. An example of such a problem is
described in the accompanying paper by Van Benschoten et al. (2015) (and is briefly presented in § 7.4), in which X-ray diffuse scattering data were compared with calculated data corresponding to
different types of molecular motion. Other examples may include analyzing larger-scale anharmonic motions, for which techniques such as molecular-dynamics trajectories have traditionally been used
(McCammon et al., 1977 ).
If a model deposited in the PDB contains TLS matrices, the matrices can be decomposed as described above. As soon as a combination of vibrations and librations is extracted from the TLS matrices, we
can explicitly build a corresponding set of models. Knowledge of the three vibrations and three librations provides the atomic shifts underlying the total displacement.
It is generally more convenient to generate each group of atomic shifts in its own basis: shifts Δ^V[[V]]r[n] owing to vibration in the [V] basis and shifts Δ^L[[L]]r[n] owing to libration in the [L]
basis. Here, we are working in a linear approximation such that rotation angles are on the order of 0.1rad. For each particular set of generated shifts, they are transformed into the [M] basis as Δ^
V[[M]]r[n] and Δ^L[[M]]r[n] and their sum,
is applied to the corresponding atoms. Details of model generation are discussed in the next sections. This procedure is repeated independently multiple times, leading to structural models
distributed according to the TLS matrices.
7.2. Calculation of the model shift owing to libration
Let us suppose that we know (in the basis [M]) the direction of the three mutually orthogonal axes l[x], l[y], l[z] for independent libration as well as the coordinates of the points w^lx[[M]], w^ly
[[M]], w^lz[[M]] belonging to each axis. We recalculate the coordinates of these points and the coordinates (x[[M]n], y[[M]n], z[[M]n]), n = 1, 2, …, N, of all atoms r[[M]n] of the group into the [L]
basis as
(similar relations are derived for the points w^lx[[M]], w^ly[[M]], w^lz[[M]]). We remind the reader that the squared libration amplitudes 〈d[x]^2〉 = L[[L]xx] = λ[1], 〈d[y]^2〉 = L[[L]yy] = λ[2],
〈d[z]^2〉 = L[[L]zz] = λ[3] (§ 3.2) and the screw parameters s[x], s[y], s[z] (§ 4.5) are independent of the basis.
For an atom at a distance R = 1Å from the rotation axis, the probability of the shifts d[x], d[y], d[z], which are numerically equal to the rotation angle in radians, are equal to
If one of the eigenvalues is equal to 0 then the corresponding d is equal to 0 with unit probability. The particular values of d[x0], d[y0], d[z0] are obtained using a random-number generator with an
underlying normal distribution (44) .
For each of the axes l[x], l[y], l[z] for each atom n described by the vector r[n], we calculate the coordinates, in the [L] basis, of its shifts Δ^lx[[L]]r[n], Δ^ly[[L]]r[n], Δ^lz[[L]]r[n] owing to
the corresponding rotations by d[x0], d[y0], d[z0] (Appendix A ). The overall shift owing to libration around the three axes is the sum
It changes from one atom of the group to another and must be calculated for all atoms of the group with the same (d[x0], d[y0], d[z0]) values for a particular instance of the three rotations.
To transform the atomic shift (45) from the [L] basis into the initial [M] basis, we invert (43) ,
7.3. Calculation of the model shift owing to vibration
In the harmonic approximation, the independent vibration shifts t[x], t[y], t[z] expressed in the [V] basis are distributed accordingly to the probability laws
Using a random-number generator, for each model we obtain particular values of t[x0], t[y0], t[z0] using (47) . If one of the eigenvalues μ is equal to zero, the zero value is assigned to the
corresponding shift. The overall translational shift, common to all atoms of the rigid group, is equal to
In order to obtain this shift in the [M] basis, we calculate, similarly to (46) ,
7.4. Validation and application to GpdQ
We generated the ensembles produced by alternative TLS refinements of the glycerophosphodiesterase GpdQ (Jackson et al., 2007 ). GpdQ is found in Enterobacter aerogenes and contributes to the
homeostasis of the cell membrane by hydrolyzing the 3′–5′ phosphodiester bond in glycerophosphodiesters. Each dimer contains three distinct domains per monomer: an α/β sandwich fold containing the
active site, a domain-swapped active-site cap and a novel dimerization domain comprised of dual-stranded antiparallel β-sheets connected by a small β-sheet. Owing to the high global B factors and the
presence of diffuse signal (Fig. 4 ), Jackson et al. (2007 ) performed three separate TLS refinements to model the crystalline disorder: entire molecule, monomer and subdomain. All TLS refinement
attempts improved the R[free] values when compared with the standard isotropic B-factor refinement; however, there was no significant difference among the final R[free] values from the various TLS
runs. We hypothesized that the diffuse scattering produced by each TLS motion would contain significant differences, as diffuse signal is a direct result of correlated motion. The notion that TLS
refinement produces unique diffuse signal has been suggested previously (Tickle & Moss, 1999 ). Physical ensembles of the TLS motion, rather than a mathematical description, were required to generate
three-dimensional diffuse scattering maps from phenix.diffuse. Visual inspection confirmed that the ensembles produced by phenix.tls_as_xyz replicated the anisotropic motion predicted by TLS thermal
ellipsoids (Fig. 5 ). Additionally, we calculated the structure factors predicted by the original TLS refinement `entire molecule' and compared them with the F[model] values (for example, as defined
in Afonine et al., 2012 ) produced by various phenix.tls_as_xyz ensemble sizes. The structure factors converged to a global correlation value of 0.965, demonstrating that phenix.tls_as_xyz ensembles
accurately represent the motions predicted by TLS refinement. Physical representation of the underlying motion also revealed that while two of the TLS refinements produced motion with small variances
(a necessity within TLS theory), using each functional region as a TLS group produced fluctuations that are clearly improbable (Fig. 4 ). Thus, viewing TLS refinement in the form of a structural
ensemble is a valuable check of the validity of the results, as matrix elements that satisfy the previously described conditions may still produce motions that are clearly implausible.
8. Discussion
While our previous review on the subject (Urzhumtsev et al., 2013 ) described the computational details of obtaining the TLS matrices from a known set of vibration and libration parameters (including
the position of the axes and correlation of these motions), the current work focuses on the opposite problem of extracting these parameters from a given set of TLS matrices. The problem is not as
simple as it may at first seem.
This difficulty arises because current structure-refinement programs vary the matrix elements as independent parameters and often ignore critical constraints on real-space motions. A second challenge
is that identical motions may be represented by different vibration–libration combinations. As a consequence, there is no one-to-one relationship between these parameters and the set of TLS matrices.
In particular, the traditional way of choosing the matrix S so that its trace is equal to zero may result in a mutually inconsistent combination of TLS matrices.
This manuscript describes the constraints that can be used to validate a given set of T, L and S matrices and to improve the refinement of TLS parameters. Beyond the well known conditions of
non-negativity for the eigenvalues of T and L, we also discuss the conditions that relate the matrices, a crucial step in ensuring that the results of TLS refinement correspond to physically possible
combinations of librations and vibrations. Taking all these conditions into account provides the possibility of correcting TLS matrices in some cases, if needed. Building these conditions into
refinement protocols can eliminate nonplausible refined TLS matrices
The TLS matrix representation is a convenient way of encoding concerted motions into a form suitable for the calculation of structure factors and, in turn, structure refinement. There are two
drawbacks to the standard implementation of this method. Firstly, TLS matrices cannot readily be interpreted in terms of underlying motions, but rather require additional processing in order for this
information to be extracted. Secondly, direct refinement of the TLS matrix elements may result in refined matrices that cannot be represented as a structural ensemble. To address these two drawbacks,
we propose using the set of vibration and libration parameters as refinable variables (an ongoing project for the authors) and reporting them in the PDB files. Indeed, using actual motion descriptors
as refinement variables will allow more effective application of physical constraints and in turn guarantee that refined values can be translated to structural ensembles, simplifying the analysis of
refinement results, as they will be readily available for interpretation. Finally, this strategy will reduce the chance of overfitting data with atomic models that represent implausible concerted
The survey of PDB entries refined with TLS revealed that roughly 85% of these deposited models contain matrices that are not consistent with the underlying physical model of the concerted motions.
Therefore, these matrices cannot be interpreted in terms of rigid-body rotations and translations, and in turn cannot represent these motions (Table 1 ). This highlights two urgent needs. Firstly,
existing refinement programs should be updated so that they apply appropriate restraints or constraints on refinable parameters of the TLS model. This should be followed by the implementation and use
of comprehensive validation of TLS refinement results.
The utility of our presented algorithm is twofold: it validates TLS matrices to confirm that they can represent concerted structural motions and interprets TLS matrices in terms of the elemental
motions that they describe. The information about atomic group motions conveyed by the TLS model can be used to analyze possible molecular mechanisms (as illustrated previously). Descriptions of TLS
motion may also be used to generate an ensemble of molecular conformations, from which the predicted diffuse scattering signal can be calculated (see the accompanying paper by Van Benschoten et al.,
2015 .).
The current procedures for analyzing and validating TLS parameters, as well as the algorithm for generating a set of models from given libration and vibration parameters, are implemented in the
PHENIX suite and are called phenix.tls_analysis and phenix.tls_as_xyz, respectively. The programs are available starting with version dev-1890.
Technical details of the algorithm
A1. Definition of the transition matrices
Let us have three mutually orthogonal unit vectors l[x], l[y], l[z] described respectively by their coordinates [(l[x])[[M]x], (l[x])[[M]y], (l[x])[[M]z]], [(l[y])[[M]x], (l[y])[[M]y], (l[y])[[M]z]],
[(l[z])[[M]x], (l[z])[[M]y], (l[z])[[M]z]] in the Cartesian basis [M]. These vectors can be considered as a new basis [L]. The coordinates of a vector r in [L] and [M] are expressed through each
other using the transition matrix R[ML] as
Transition matrices for other pairs of bases, for example from [V] to [L] (§ 2.1), [M] to [V] and vice versa (§ 7.3) are defined in a similar way.
A2. Cauchy conditions on the elements of the TLS matrices
Let d[x], d[y], d[z] and u[x], u[y], u[z] be random displacements owing to rotations and translations, respectively. Since S[xx] = 〈d[x]u[x]〉, V[xx] = 〈u[x]u[x]〉, L[xx] = 〈d[x]d[x]〉 (Schomaker
& Trueblood, 1968 ; see also equations 8.5–8.7 in Urzhumtsev et al., 2013 ), it follows from the Cauchy inequality that
In the basis [L] with S[[L]] = S[C](t[S]) (18) , condition (51) becomes
Similarly, we obtain two other conditions
A3. Polynomials for the coefficients of the characteristic equation
If t[xx], t[xy], t[xz] etc. are respective elements of the matrix T[Λ] (29) , the coefficients (36) of the characteristic equation as functions of the parameter t are
A4. Explicit expression for the atomic shifts owing to rotations with given parameters
Let (x[[L]], y[[L]], z[[L]]) be the Cartesian coordinates of a point r in the basis [L]. For a rotation around the axis parallel to l[z] and crossing the point w^lz[[L]] = (w^lz[[L]x], w^lz[[L]y], w^
lz[[L]z]), we first recalculate the coordinates of the vector r − w^lz[[L]] with respect to the rotation axis,
If r′ stands for the position of the same point after rotation by angle d[z0] around this axis, the coordinates of r′ − w^lz[[L]], the point with respect to the axis, are
This gives the atomic shift
There are similar expressions for the shift owing to rotations around the other two axes:
List of abnormal situations requiring interruption of the procedure
This appendix summarizes the situations when the described algorithm breaks. Each condition below starts from the corresponding program message and then refers to the main text and to Fig. 1 . To
analyze the PDB content, the program can be run in a special regime when at step C we assign t[S] = 0, i.e. when the matrix S is taken without any correction [in most cases this corresponds to the
current default constraint tr(S) = 0]. In this regime, we directly calculate the matrices C and check the conditions (x)–(xii).
Step A: basis [L]; determination of the libration axes and amplitudes.
(i) `Input matrix L[M] is not positive semidefinite', § 3.1.
(ii) `Input matrix T[M] is not positive semidefinite', § 3.1.
Step B: determination of the points w at the libration axes.
(iii) `Non-zero off-diagonal S[L] and zero L[L] elements', § 3.2, (15) .
(iv) `Matrix T_C[L] is not positive semidefinite', § 3.2, (17) .
Step C: determination of the screw parameters: left branch (librations around all three axes).
(v) `Empty (tmin_c, tmax_c) interval', § 4.2, (23) . t[min,C] > t[max,C].
(vi) `Empty (tmin_t, tmax_t) interval', § 4.3, (31) . t[min,τ] > t[max,τ].
(vii) `Negative argument when estimating tmin_a', § 4.3, (38) .
(viii) `Intersection of the intervals for t_S is empty', § 4.3, step (ii). t[min] > t[max].
(ix) `t_min = t_max giving non positive semidefinite V_lambda', § 4.3, step (iii).
(x) `Interval (t_min, t_max) has no t value giving positive semidefinite V', § 4.3, step (iv).
Step C: determination of the screw parameters: right branch (no libration around at least one of the axes).
(xi) `Cauchy-Schwarz conditions are wrong for the found t_S', (22) with t[S] calculated in § 4.4.
(xii) `Non-zero diagonal S[L] element for a zero L[L] element', § 4.4.
Step D: determination of the vibration parameters.
(xiv) `Matrix V[L] is not positive semidefinite', § 5.1.
Extra checks at step C when some conditions may fail owing to rounding errors.
(1) When calculating square roots in (24) , the arguments are non-negative by previous conditions (i) and (iv) since the diagonal elements of a positive semidefinite matrix are non-negative.
(2) When calculating square roots in (28) , the arguments are non-negative by previous condition (i).
(3) When calculating square roots in (31) , the argument τ[max] is non-negative since the eigenvalues of T[C[L]] are also non-negative by previous condition (iv).
PVA and PDA thank the NIH (grant GM063210) and the PHENIX Industrial Consortium for support of the PHENIX project. JSF is a Searle Scholar and a Pew Scholar, and a Packard Fellow, and is supported by
NIH OD009180, GM110580 and NSF STC-1231306. This work was supported in part by the US Department of Energy under Contract No. DE-AC02-05CH11231. AU thanks the French Infrastructure for Integrated
Structural Biology (FRISBI) ANR-10-INSB-05-01 and Instruct, which is part of the European Strategy Forum on Research Infrastructures (ESFRI) and is supported by national member subscriptions.
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Unit-2 Concept of Measurement POE | BBA First Year
Unit-2 Concept of Measurement POE | BBA First Year-Hello everyone welcome to the pencilchampions.com website. I have provide the Unit-2 Concept of Measurement Principle Of Economics BBA,
BCA,M.COM,B.COM Notes. I think these notes helpful for all of you students. Thankyou for visiting pencilchampions.com website.
Unit-2 Law of Demand POE | BBA First year 2023
• Change in the price of a commodity affects its demand. We can find out the elasticity of demand, or the degree of responsiveness of demand, by comparing the percentage price change with the
quantity demanded. In this article, we will look at the concept of elasticity of demand and take a look at its different types.
Read more-https://pencilchampions.com/unit-2-demand-analysis-demand-schedule-poe-bba-first-year/
Elasticity Of Demand    Â
• To start, let’s look at the definition of elasticity of demand: “Elasticity of demand is the response of the quantity demanded of a good to a change in the variable on which the demand depends.
In other words, it is the percentage change of demand. The quantity demanded is divided by the percentage of some one variable on which the demand depends.”
• The following points highlight the top five methods used to measure elasticity of demand. The methods are:
1. Price elasticity of demand
2. Â Income elasticity of demand
3. Cross elasticity of demand
4. Advertisement or promotional Elasticity of Sales
5. Elasticity of price expectations.
Method # 1. Price elasticity of demand·       Â
• Price elasticity of demand is a measure of the responsiveness of demand to changes in a good’s own price.
• Furthermore, elasticity is the percentage of percentage change in price.
• If the percentage is known, the numerical value of elasticity can be calculated. The coefficient of elasticity of demand is a pure number i.e. it stands on its own being independent of the units
of measurement. The coefficient of price elasticity of demand can be calculated with the help of the following formula.
Method # 2. Income elasticity of demand
• The response of quantity demanded to changes in income is called income elasticity of demand. With income elasticity, consumer income varies while tastes, own price of the good, and other prices
remain constant.
• Where- ey is for coefficient of income elasticity, Y is for income.
• While price-elasticity of demand is always negative, income-elasticity of demand is always positive (except for inferior goods) because the relationship between income and the quantity demanded
of a product is positive. The income elasticity of demand for inferior goods is negative because as income increases, consumers turn to·
Method # 3. Cross Elasticity of Demand·
• Cross elasticity measures the response of quantity demanded to changes in the price of other goods and services.
• Â When cross-elasticity is greater than zero, the goods or services involved are classified as complements. An increase in the price of y reduces the quantity demanded of that product. Decrease
in demand for Y reduces demand for bread and butter, cars and tires, and tire, and computer and computer program are examples of pairs of goods that are complements.
• If A and B are substitutes then the coefficient is positive because the price change and quantity change are in the same direction. If A and B are complements the coefficient is negative, because
a change in the price of one good causes an opposite change in the quantity demanded of the other.
Method #4. advertising or promotional elasticity of sales.
• Advertising expenditure helps in promoting sales. The effect of advertising on sales is not uniform at all levels of total sales. The concept of advertising elasticity is important in determining
the optimal level of advertising outlay especially in view of competitive advertising by rival companies.
Method #5. elasticity of price expectations.
• People’s price expectations also play an important role as a determinant of demand. English economist J.R. Hicks formulated the concept of elasticity of price expectations in 1939.Â
• Consumer equilibrium refers to a situation in which a consumer obtains maximum satisfaction, has no intention of changing it and is subject to given prices and his given income.
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Thermal Radiation: Blackbody Radiation | Curious Toons
Table of Contents
Welcome to the fascinating world of physics! This year, we’ll embark on an exhilarating journey that will unravel the mysteries of the universe and equip you with the power to understand the
fundamental principles shaping our reality. Ever wondered why the sky is blue, how skyscrapers stand tall, or why you feel weightless when you jump? Physics holds the answers.
We’ll explore the wonders of motion — how things travel and collide — and push the boundaries with the forces of nature. From the tiniest particles that make up everything around us to the galaxies
that twirl in a cosmic dance, physics connects every aspect of our lives.
Imagine launching a rocket, bending light, or even predicting the weather. These aren’t just dreams; they’re the realms we’ll discover together. With hands-on experiments, lively discussions, and an
innovative approach to problem-solving, you’ll learn to think critically and creatively, all while having a blast.
So, are you ready to challenge your understanding of the world and unlock the secrets that govern it? Let’s dive in and ignite your passion for physics! The adventure awaits!
1. Introduction to Thermal Radiation
1.1 Definition of Thermal Radiation
Thermal radiation is the process by which an object emits energy in the form of electromagnetic waves due to its temperature. All objects, regardless of their state (solid, liquid, or gas), radiate
energy, but the nature and intensity of this radiation depend on their temperature and surface properties. The fundamental principle behind thermal radiation is that as an object’s temperature
increases, the amount of energy it radiates also increases, leading to the emission of shorter wavelengths of radiation. A perfect emitter and absorber of thermal radiation is known as a black body.
The spectral distribution of a black body’s radiation is described by Planck’s law, which states that the intensity of emitted radiation increases with temperature and shifts to shorter wavelengths
as the temperature rises. This phenomenon is crucial in understanding various applications, such as thermal imaging, climate modeling, and astrophysics.
Key Concepts:
Concept Description
Black Body An idealized physical body that absorbs all incident radiation.
Planck’s Law Describes the spectral distribution of radiation emitted by a black body.
Temperature & Wavelength Higher temperatures lead to emission of shorter wavelengths.
1.2 Importance in Physics
Thermal radiation, particularly blackbody radiation, is a fundamental concept in physics that bridges classical mechanics and quantum mechanics. Understanding blackbody radiation is crucial because
it reveals how objects emit and absorb electromagnetic radiation based on their temperature, a key principle that underpins various scientific disciplines. The study of blackbody radiation led to the
development of Planck’s law and ultimately to the quantum theory, revolutionizing our understanding of atomic and subatomic phenomena. This understanding has vast implications, from explaining the
temperature of stars to the design of thermal cameras and technologies in climate science. Additionally, the principles of thermal radiation play a critical role in engineering applications, such as
thermal management in electronics and energy conservation in buildings. Thus, the significance of thermal radiation extends beyond theoretical physics, impacting practical applications in various
fields, including astronomy, engineering, and environmental science.
Key Concept Importance
Blackbody Radiation Bridges classical and quantum physics
Planck’s Law Foundation for quantum theory
Applications Thermal cameras, energy management
Scientific Impact Understanding celestial temperatures and more
2. Blackbody Concepts
2.1 Definition of a Blackbody
A blackbody is an idealized physical object that perfectly absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. By definition, a blackbody reflects no light;
instead, it transforms all absorbed energy into thermal energy. Consequently, it is also an ideal emitter of radiation, meaning it emits energy in the form of thermal radiation with a spectral
distribution that depends solely on its temperature, described by Planck’s law. In practical terms, no real material behaves as a perfect blackbody, but many materials approximate this behavior
closely. For instance, carbon black and certain types of specialized paints can be nearly perfect absorbers and emitters. The concept of a blackbody is crucial in thermal physics, as it provides a
baseline against which the emissive properties of real objects can be compared. In terms of temperature, a blackbody’s emission increases with the fourth power of its absolute temperature,
illustrated by the Stefan-Boltzmann Law:
E = \sigma T^4
where (E) is the total energy radiated per unit area, (T) is the absolute temperature, and (\sigma) is the Stefan-Boltzmann constant.
This theoretical model enables scientists to understand and predict thermal radiation characteristics across various applications.
2.2 Ideal vs. Real Blackbodies
In the study of thermal radiation, it’s crucial to differentiate between ideal and real blackbodies. An ideal blackbody is a theoretical object that perfectly absorbs all incident radiation,
regardless of wavelength or direction. It also emits radiation with 100% efficiency, described by Planck’s law, which dictates that the spectral radiance of a blackbody is solely dependent on its
temperature. Conversely, real blackbodies are actual materials that do not completely absorb or emit radiation across all wavelengths. Their absorptivity and emissivity vary across the
electromagnetic spectrum. For instance, materials like graphite and carbon black approach ideal blackbody behavior, exhibiting high absorption and emission rates. In contrast, polished metals (e.g.,
aluminum or silver) have low absorptivity and emissivity, particularly in the visible spectrum, making them poor blackbodies. This distinction is important in practical applications, such as
designing thermal radiation detectors and understanding heat transfer in different materials. The effectiveness of a real blackbody in approximating ideal behavior can be quantified using the
emissivity coefficient, ranging from 0 to 1, where 1 represents a perfect blackbody.
Property Ideal Blackbody Real Blackbody
Absorptivity 1 < 1
Emissivity 1 < 1
Dependence on Wavelength No Yes
3. Planck’s Law of Blackbody Radiation
3.1 Derivation of Planck’s Law
Planck’s Law describes the spectral distribution of radiation emitted by a blackbody in thermal equilibrium at a given temperature. To derive this law, we start by considering a blackbody cavity in
thermal equilibrium, where electromagnetic radiation can be emitted and absorbed. We introduce the concept of quantized energy states, postulating that the energy of oscillators in the cavity is
quantized and given by (E_n = n h \nu), where (n) is a non-negative integer, (h) is Planck’s constant, and (\nu) is the frequency of the oscillator.
Using statistical mechanics, we apply the Boltzmann distribution to find the average energy per mode as a function of temperature, leading to the formulation of the average energy of the oscillators.
Summing over all possible frequencies, we use the density of states in frequency space and integration techniques to arrive at the following expression for spectral radiance (I(\nu, T)):
I(\nu, T) = \frac{8 \pi h \nu^3}{c^3} \cdot \frac{1}{e^{\frac{h \nu}{kT}} – 1}
where (c) is the speed of light and (k) is the Boltzmann constant. This law elegantly describes how the intensity of radiation emitted varies with frequency and temperature, successfully explaining
the observed spectrum of blackbody radiation.
3.2 Applications of Planck’s Law
Planck’s Law of Blackbody Radiation is fundamental in various scientific and technological applications. One key application is in astrophysics, where it helps to determine the temperature and
composition of stars by analyzing their emitted radiation spectra. For instance, by measuring the intensity of light at different wavelengths from a star, astronomers can apply Planck’s Law to infer
its surface temperature. Another significant application is in thermal imaging and infrared technologies, where Planck’s Law aids in designing sensors that detect infrared radiation emitted by
objects, enhancing applications in night vision and medical diagnostics. Additionally, Planck’s Law is instrumental in understanding and designing energy-efficient lighting systems and heating
devices, such as incandescent bulbs and radiative heaters. Moreover, the law underpins advancements in quantum mechanics and statistical mechanics, influencing the development of modern technologies
like lasers and semiconductors. Overall, Planck’s Law not only enriches our understanding of thermal radiation but also drives innovation across various fields, from astronomy to engineering.
Application Description
Astrophysics Analyzing star temperatures and compositions
Thermal Imaging Infrared sensors for night vision and diagnostics
Energy-efficient lighting Design of bulbs and heaters
Quantum Mechanics Foundations for advancements in modern technology
4. Wien’s Displacement Law
4.1 Statement of Wien’s Law
Wien’s Displacement Law is a fundamental principle in thermal radiation that describes the relationship between the temperature of a blackbody and the wavelength at which it emits radiation most
intensely. Specifically, the law states that the wavelength (( \lambda_{max} )) of the peak emission of a blackbody is inversely proportional to its absolute temperature (T) in Kelvin.
Mathematically, this is expressed as:
\lambda_{max} = \frac{b}{T}
where ( b ) is Wien’s displacement constant, approximately equal to ( 2.898 \times 10^{-3} ) m·K. This means that as the temperature of the blackbody increases, the peak wavelength of emitted
radiation shifts to shorter wavelengths. For example, a body at 5000 K will emit most of its radiation at around 580 nm, which is in the visible spectrum, while a cooler body at 300 K will peak
around 9650 nm, in the infrared region. This law is critical for understanding the thermal emission of objects in astrophysics and other fields, explaining why hotter stars appear bluer and cooler
stars appear redder.
4.2 Significance in Thermal Processes
Wien’s Displacement Law is a fundamental principle in thermal radiation that describes how the peak wavelength of emission from a blackbody shifts with temperature. Specifically, the law states that
the wavelength at which the emission of radiation is maximized is inversely proportional to the absolute temperature of the blackbody. This means that as the temperature increases, the peak
wavelength decreases, leading to a shift from infrared radiation towards visible light and even to ultraviolet radiation at higher temperatures. The significance of this law in thermal processes is
profound—it explains why hot objects appear glowing red or blue as they heat up and offers insights into astrophysical phenomena such as the color of stars, with cooler stars appearing red and hotter
stars appearing blue. Additionally, understanding this phenomenon is crucial in various applications, including climate science, thermography, and the design of energy-efficient systems. By
leveraging Wien’s Displacement Law, engineers and scientists can better understand heat transfer, energy consumption, and the behavior of materials at different temperatures. As a result, this law
not only enhances our comprehension of thermal radiation but also has practical implications across different scientific and engineering disciplines.
5. Stefan-Boltzmann Law
5.1 Understanding the Law
The Stefan-Boltzmann Law is a fundamental principle in thermal radiation that describes how the total energy radiated by a blackbody per unit area increases with temperature. Mathematically, it is
expressed as ( E = \sigma T^4 ), where ( E ) is the total energy radiated per unit surface area, ( \sigma ) is the Stefan-Boltzmann constant (( 5.67 \times 10^{-8} \, \text{W/m}^2\text{K}^4 )), and (
T ) is the absolute temperature in Kelvin. This means that if the temperature of a blackbody doubles, the energy it emits increases by a factor of ( 2^4 = 16 ). This strong dependence on temperature
highlights that even small increases in temperature can lead to large increases in emitted energy. The law applies ideally to blackbodies, theoretical objects that perfectly absorb and emit radiation
at all wavelengths. In practical applications, the Stefan-Boltzmann Law helps us understand phenomena like the heat emitted by stars, including our sun, and plays a crucial role in climate science by
quantifying energy exchanges within the Earth’s atmosphere. Understanding this principle is vital for fields such as astrophysics, climate science, and engineering, where thermal management is
5.2 Real-world Applications
The Stefan-Boltzmann Law plays a crucial role in various real-world applications, particularly in fields such as astrophysics, climate science, and engineering. For instance, in astronomy, this law
helps us estimate the temperature of stars by measuring their emitted infrared radiation, enabling astronomers to classify stars and understand their lifecycle. In climate science, understanding the
Earth’s thermal radiation is critical for developing climate models that predict temperature changes due to greenhouse gas emissions. Engineers utilize the Stefan-Boltzmann Law in designing
energy-efficient systems, such as in the development of radiative cooling materials that can dissipate heat effectively. Furthermore, it is employed in thermal management for electronic devices,
where excess heat must be dissipated to maintain optimal performance. The law is also essential in meteorology for predicting temperature changes in the atmosphere, allowing for more accurate weather
forecasts. Overall, the Stefan-Boltzmann Law serves as a foundational principle that not only enhances our understanding of thermal radiation but also impacts technology and environmental science
As we conclude our journey through the fascinating world of physics, I want to take a moment to reflect on the incredible concepts we’ve explored together—from the fundamental laws of motion to the
elegant dance of electromagnetism. Each equation we’ve unraveled doesn’t just describe the universe; it empowers you to see the world through a different lens.
Remember, physics is not just a subject; it’s a way of thinking. It teaches us to question, to seek evidence, and to find beauty in complexity. As you carry these lessons forward, I encourage you to
embrace curiosity, challenge assumptions, and never shy away from the unknown.
This is not the end; it’s just the beginning. The principles of physics will accompany you in every field, every challenge, and every innovation you pursue. Each of you holds the potential to
contribute to our understanding of the universe. So, as you leave this classroom, take with you the spark of inquiry and the spirit of discovery.
Thank you for your enthusiasm, engagement, and relentless pursuit of knowledge. Stay curious, stay passionate, and, most importantly, keep questioning the world around you. The universe is waiting
for your exploration! | {"url":"https://curioustoons.in/thermal-radiation-blackbody-radiation/","timestamp":"2024-11-09T19:17:08Z","content_type":"text/html","content_length":"112548","record_id":"<urn:uuid:a9ea1b8f-e00d-4eaa-987e-17d2c23f9c8d>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00409.warc.gz"} |
Dividing Decimals | Curious Toons
Table of Contents
Introduction to Decimals
What are Decimals?
Decimals are a way to represent numbers that are not whole. Essentially, they allow us to express values that lie between whole numbers. For example, the number 1.5 is a decimal, representing the
value that is halfway between 1 and 2. Decimals are built on the base-10 number system, just like whole numbers, but they extend our number line to include fractions in a more manageable format. The
number after the decimal point indicates a part of a whole, where each place represents a fraction of ten.
For example, in the decimal 0.75, the ‘7’ is in the tenths place, meaning it represents 7 tenths, and the ‘5’ is in the hundredths place, representing 5 hundredths. Understanding decimals is
important because they are used in many aspects of daily life, like money, measurements, and when dealing with scientific data. Recognizing how to read and write decimals is a crucial skill, and once
we grasp this concept, we can tackle more complex calculations, such as dividing decimals.
Understanding Place Value
Place value is an essential concept that helps us understand the value of each digit in a number, whether it’s a whole number or a decimal. In whole numbers, each position signifies a power of ten:
ones, tens, hundreds, and so on. When we move to decimals, the place values continue but shift to the right of the decimal point.
For example, in the decimal 3.142, the ‘3’ is in the ones place, the ‘1’ is in the tenths place, the ‘4’ is in the hundredths place, and the ‘2’ is in the thousandths place. Each place value
indicates how much of that fraction we have. Understanding place value is key to working with decimals because it affects how we perform calculations, including addition, subtraction, multiplication,
and division.
When dividing decimals, knowing which digits to use based on their place value will help ensure accuracy in your computations. This understanding lays the groundwork for greater mathematical
concepts, enabling you to manipulate numbers effectively and comprehend their meanings in various contexts.
The Basics of Division
What is Division?
Division is one of the four fundamental operations in mathematics, alongside addition, subtraction, and multiplication. In simple terms, division is the process of splitting a number into equal
parts. When we divide, we ask the question: how many times can one number be contained within another? For instance, if you have 12 apples and you want to share them equally among 3 friends, you’re
essentially performing a division operation. You take the total number of apples (12) and divide it by the number of friends (3). The result tells you how many apples each friend gets, which is 4 in
this case.
The symbol used for division is ‘÷’ or a slash ‘/’, and the numbers involved in division are called the dividend (the number being divided) and the divisor (the number by which you are dividing). The
result of a division operation is known as the quotient. Division is essential in many real-life situations, such as distributing resources, calculating averages, and solving equations. Understanding
division helps you to comprehend more complex mathematical concepts down the road!
Division of Whole Numbers
When we divide whole numbers, we follow a specific method to find the quotient. Whole numbers are non-negative numbers without any fractions or decimals, and they include zero. Let’s take a simple
example: if we want to divide 20 by 4, we can ask how many groups of 4 can fit into 20. The division can be represented as 20 ÷ 4, and our task is to figure out how many times 4 can be subtracted
from 20 before reaching zero.
To visualize this, we can think of 20 as 4 groups of 5, since 4 multiplied by 5 gives us 20. Hence, the quotient is 5. However, if the division doesn’t evenly split, for example, dividing 22 by 4, we
find that it fits 5 times (since 4 x 5 = 20), leaving a remainder of 2 (because 22 – 20 = 2). This means 22 ÷ 4 can be expressed as 5 with a remainder of 2. Mastering the division of whole numbers
lays the groundwork for more complex operations, like dividing decimals and working with fractions, both of which we will be tackling soon!
Dividing Decimals by Whole Numbers
Step-by-Step Process
Dividing decimals by whole numbers may seem tricky at first, but breaking it down into a step-by-step process can make it manageable! First, you want to set up the problem as you would with whole
numbers. Write the decimal number (the dividend) inside the division bracket and the whole number (the divisor) outside. If your decimal number has a decimal point, don’t worry; just proceed with the
division as if the decimal point isn’t there. When you divide, treat it like you’re working with whole numbers.
Next, divide the first digit of the dividend by the divisor. If the divisor can’t go into that digit, look at the next digit until you find one that it can divide into. Once you find how many times
the divisor fits, write that number on top of the division bar. Multiply that result by the divisor and subtract it from the part of the dividend you’ve divided so far. Bring down the next digit.
Repeat these steps until there are no more digits to bring down. Finally, remember to place the decimal point directly above where it is in the dividend. Voila! You’ve successfully divided a decimal
by a whole number!
Examples and Practice Problems
Now that we’ve gone through the step-by-step process, let’s look at some practical examples that can solidify your understanding. For instance, consider dividing 5.25 by 3. Start by setting it up as
you would in the previous process. Since 5.25 can be treated as 525 (moving the decimal point), you can divide 3 into 5 which goes once. You write that down, subtract, bring down the next digit (2),
and continue until you’re done.
Once you have watched and understood a couple of examples, try some practice problems on your own! Here are a few to get you started: 6.4 ÷ 2, 7.5 ÷ 5, and 4.8 ÷ 2.5. Go through the steps carefully,
and don’t hesitate to use scratch paper to visualize your work. After you’ve attempted these problems, we’ll go over the answers together as a class, which will help you see where you did well and
where you might need more practice. Remember, practice makes perfect, so don’t shy away from experimenting with different examples!
Dividing Decimals by Decimals
Eliminating Decimals
When we divide decimals, it’s often easier to first eliminate the decimals to simplify our calculations. Imagine you have a division problem like 4.5 divided by 0.3. To get rid of the decimals, we
can multiply both the dividend (the number being divided) and the divisor (the number doing the dividing) by the same power of ten. In this case, if we multiply both by 10, the problem becomes 45
divided by 3. This step is crucial because it keeps the value proportional; the quotient remains unchanged.
By eliminating the decimals, we can perform the division just like we do with whole numbers, making it less confusing and more straightforward. After dividing, we can find our answer easily—just like
solving any long division problem. Always remember, your goal is to make the numbers easier to work with while keeping the problem balanced. Once you get comfortable with this technique, dividing
decimals will feel much less intimidating!
Step-by-Step Examples
Now, let’s look at some step-by-step examples to put our understanding into practice! Consider the problem 6.4 divided by 0.8. First, we want to eliminate decimals. Multiply both numbers by 10 to
turn the problem into 64 divided by 8.
Next, we perform the division: How many times does 8 go into 64? The answer is 8. So, 6.4 divided by 0.8 equals 8. It’s a simple process once you break it down!
Let’s try another example: 1.5 divided by 0.25. We start by eliminating the decimals by multiplying both numbers by 100 (to avoid fractions), turning it into 150 divided by 25. Now, how many times
does 25 go into 150? The answer is 6. By practicing these steps—eliminating decimals, performing division, and interpreting results—you’ll become confident in dividing decimals quickly and
accurately. Remember, practice makes perfect!
Applications of Dividing Decimals
Real-World Examples
Understanding how to divide decimals is crucial because we encounter them daily, often without even realizing it! For instance, when shopping, you might find a shirt that costs $24.75, and you’re
trying to figure out how much it would cost if it were split among three friends. To do this, you would divide $24.75 by 3. This helps us understand how we can manage our finances more effectively.
Similarly, when cooking, if a recipe calls for 2.5 cups of flour but you’re only making half of the recipe, you’d need to divide 2.5 by 2 to find the correct amount to use. In both examples, dividing
decimals helps us break down larger quantities into more manageable parts and make informed decisions. It’s not just about getting the correct answer, but also understanding the context behind the
math, which makes it meaningful in our everyday lives!
Word Problems Involving Decimals
Word problems involving decimals can initially seem challenging, but they are a fun way to apply what you’ve learned about dividing decimals in real-life situations! These problems often present a
scenario where you have to interpret information and determine the correct mathematical operation to use. For example, imagine that you have $15.60 to buy snacks for a party, and each packet costs
$2.40. To find out how many whole packets you can buy, you would divide $15.60 by $2.40. Approaching these problems requires careful reading and sometimes jotting down a plan before calculating. They
can also ask you to solve puzzles, such as figuring out how long it would take to complete tasks or how to fairly distribute items. By breaking down the problem into smaller, manageable parts and
focusing on what the question is asking, you can translate word problems into simpler mathematical equations. This process helps you build critical thinking skills, making you better equipped to
handle real-life challenges!
As we reach the conclusion of our chapter on dividing decimals, it’s essential to reflect on the deeper significance of what we’ve learned. At first glance, dividing decimals may seem like just a
series of steps involving numbers on a page, but it’s so much more. This skill is a gateway into understanding more complex mathematical concepts that shape our everyday lives.
Consider how often we encounter decimals—whether calculating a budget, measuring ingredients in a recipe, or understanding data in a world driven by technology. Each time you divide decimals, you are
not just manipulating numbers; you are harnessing critical thinking and problem-solving skills that will serve you far beyond the classroom walls.
Let’s remember that math is not just about getting the right answer; it’s about the process—mistakes are stepping stones to mastery. As you venture beyond this chapter, carry with you the confidence
that comes from perseverance and curiosity. Don’t shy away from challenges; embrace them! Each decimal you divide brings you one step closer to becoming adept not only in mathematics but in
navigating the complexities of life. So, next time you approach a problem, ask yourself: what story does this number tell? How can I use these skills to make sense of the world around me? | {"url":"https://curioustoons.in/dividing-decimals/","timestamp":"2024-11-09T20:30:57Z","content_type":"text/html","content_length":"106507","record_id":"<urn:uuid:ff954263-53ce-4fad-a455-ab0118bf9f2a>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00687.warc.gz"} |
Differential Equations and their Function in Arithmetic Modeling
A differential equation refers to any equation that will involve a by-product. It may possibly consist of one or more derivatives of unknown perform. The unfamiliar perform is denoted as y=y (x)
except the issue demands time where the denotation modifications to y=y (t)one. The spinoff belonging to the functionality y can also be composed as dy/dx or dy/dt when y is outlined to be a perform
of time. The buy of the differential equation is decided via the optimum by-product. Differential equations you should not include any specific reply unless additional information is delivered. By
way of example, when coping with to start with buy differential equation that is definitely continual the only supplemental answers needed certainly is the price of the operate in a particular
point1. The value is then substituted into your derived equation by substituting the values in the equation to have the exact reply to to your equation often times described as the particular
treatment. The information delivered underneath this kind of differential equation is called the original ailment. As being a result of developing an original problem, the differential equation is
named an original value challenge. On the other hand, when the first issue is specified and therefore the differential equation is discontinuous then the solution only pins the purpose at which the
curve passes through3. In such a http://http://termpapersworld.com/term_papers case the area of your resolution must be specified. In certain cases remedies of a differential equation can not be
uncovered by solving for their derivatives, in these kinds of cases basic theorem of calculus are utilized to discover the exact treatment. Differential equations are labeled in numerous techniques
particularly; normal differential equations or partial differential equations. A normal differential equation is a equation which has the unfamiliar operate based on the single impartial variable.
Partial differential equation refers to all those whose unidentified capabilities rely on more than a few unbiased variables3. Furthermore, normal differential equations are categorised as possibly
linear or non-linear.
When differential equations define physical processes is termed arithmetic modeling
Differential equations play alternative roles in mathematics modeling for illustration; defining costless drop. In mathematics, defining the autumn of the item on the ambiance the variables time t
and velocity v are utilized. Along with the facilitate of differential equationdv/dt just one will be able to describe the movement belonging to the object. Apart from, the differential equations
aids someone fully grasp some time and therefore the pace at which the slipping objet hits the ground. Differential equations also play a role inside of the formation of populace designs as an
example by using the Malthusian Legislation of advancement offered as dp/dt=rp. The product indicate how the population modifications with regard to time. Implementing this product a single is ready
to forecast the improve in populace of the species with respect to time. A alteration in r is triggered by a change inside of the growth with the species. Differential equations are utilised to model
mobile suicide. Arithmetic products are actually fashioned by using these equations to come back up with qualitative related information on how cells get the job done. Making use of the models the
vast biotechnology data made by researchers are comprehended and place into the ideal use2. The rate of progress of the cells is calculated by use of differential tactic and as a outcome way more
tips are established in managing particular illnesses like cancer. Mathematics versions with the utilization of differential equations have served in comprehension weather switch. Due to local
weather improve, parts from the Arctic Circle are endangered due to the fact ice in these aspects is melting. To be familiar with just how long the ice cap shall be all-around arithmetic product are
fashioned utilizing differential equations2. Using these mathematics models an idea of what worries researchers face is formed. To summarize, differential equations may not be only equations
uncovered in arithmetic but are equations that engage in a huge job inside of the everyday living. These equations have assisted to know and address so many issues including remedy scientific
equations which will address their dilemmas in basic research. Some parts where by differential equations have played a giant role is in medication and environmental science. | {"url":"https://www.tolirwa.com/differential-equations-and-their-function-in-103/","timestamp":"2024-11-02T02:12:44Z","content_type":"text/html","content_length":"35328","record_id":"<urn:uuid:ed38c47d-07d2-42be-81b8-7da42536d065>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00328.warc.gz"} |
1,098 research outputs found
We study thermodynamics of strongly coupled lattice QCD with $two$ colors of staggered fermions in $(2+1)$ dimensions. The partition function of this model can be written elegantly as a statistical
mechanics of dimers and baryonloops. The model is invariant under an $SO(3)\times U(1)$ symmetry. At low temperatures we find evidence for superfluidity in the U(1) symmetry sector while the SO(3)
symmetry remains unbroken. The finite temperature phase transition appears to belong to the Kosterlitz-Thouless universality class, but the superfluid density jump $\rho_s(T_c)$ at the critical
temperature $T_c$ is anomalously higher than the normal value of $2 T_c/\pi$. We show that by adding a small SO(3) symmetry breaking term to the model, the superfluid density jump returns to its
normal value implying that the extra symmetry causes anomalous superfluid behavior. Our results may be of interest to researchers studying superfluidity in spin-1 systems.Comment: Minor revisions.
Added a paragraph. to be published in PR
Lattice four-fermion models containing $N$ flavors of staggered fermions, that are invariant under $Z_2$ and U(1) chiral symmetries, are known to suffer from sign problems when formulated using the
auxiliary field approach. Although these problems have been ignored in previous studies, they can be severe. Here we show that the sign problems disappear when the models are formulated in the
fermion bag approach, allowing us to solve them rigorously for the first time.Comment: references adde
We propose a new approach to the fermion sign problem in systems where there is a coupling $U$ such that when it is infinite the fermions are paired into bosons and there is no fermion permutation
sign to worry about. We argue that as $U$ becomes finite fermions are liberated but are naturally confined to regions which we refer to as {\em fermion bags}. The fermion sign problem is then
confined to these bags and may be solved using the determinantal trick. In the parameter regime where the fermion bags are small and their typical size does not grow with the system size,
construction of Monte Carlo methods that are far more efficient than conventional algorithms should be possible. In the region where the fermion bags grow with system size, the fermion bag approach
continues to provide an alternative approach to the problem but may lose its main advantage in terms of efficiency. The fermion bag approach also provides new insights and solutions to sign problems.
A natural solution to the "silver blaze problem" also emerges. Using the three dimensional massless lattice Thirring model as an example we introduce the fermion bag approach and demonstrate some of
these features. We compute the critical exponents at the quantum phase transition and find $u=0.87(2)$ and $\eta=0.62(2)$.Comment: 31 pages, 9 figures, 5 table
The numerical simulation of various field theories at non-zero chemical potential suffers from severe complex action problems. In particular, QCD at non-zero quark density can presently not be
simulated for that reason. A similar complex action problem arises in the 2-d O(3) model -- a toy model for QCD. Here we construct the 2-d O(3) model at non-zero density via dimensional reduction of
an antiferromagnetic quantum spin ladder in a magnetic field. The complex action problem of the 2-d O(3) model manifests itself as a sign problem of the ladder system. This sign problem is solved
completely with a meron-cluster algorithm.Comment: Based on a talk by U.-J. Wiese, 6 pages, 12 figures, to be published in computer physics communication
Monte Carlo simulations of lattice QCD at non-zero baryon chemical potential $\mu$ suffer from the notorious complex action problem. We consider QCD with static quarks coupled to a large chemical
potential. This leaves us with an SU(3) Yang-Mills theory with a complex action containing the Polyakov loop. Close to the deconfinement phase transition the qualitative features of this theory, in
particular its Z(3) symmetry properties, are captured by the 3-d 3-state Potts model. We solve the complex action problem in the Potts model by using a cluster algorithm. The improved estimator for
the $\mu$-dependent part of the Boltzmann factor is real and positive and is used for importance sampling. We localize the critical endpoint of the first order deconfinement phase transition line and
find consistency with universal 3-d Ising behavior. We also calculate the static quark-quark, quark-anti-quark, and anti-quark-anti-quark potentials which show screening as expected for a system with
non-zero baryon density.Comment: 28 pages, 7 figure
We study thermodynamics of strongly coupled lattice QCD with two colors of massless staggered fermions as a function of the baryon chemical potential $\mu$ in 3+1 dimensions using a new cluster
algorithm. We find evidence that the model undergoes a weak first order phase transition at $\mu=0$ which becomes second order at a finite $\mu$. Symmetry considerations suggest that the universality
class of these phase transitions should be governed by an $O(N)\times O(2)$ field theory with collinear order, with N=3 at $\mu=0$ and N=2 at $\mu eq 0$. The universality class of the second order
phase transition at $\mueq 0$ appears to be governed by the decoupled XY fixed point present in the $O(2)\times O(2)$ field theory. Finally we show that the quantum (T=0) phase transition as a
function of $\mu$ is a second order mean field transition.Comment: 31 pages, 12 figure
The $SU(N_f)_L \otimes SU(N_f)_R$ chiral symmetry of QCD is of central importance for the nonperturbative low-energy dynamics of light quarks and gluons. Lattice field theory provides a theoretical
framework in which these dynamics can be studied from first principles. The implementation of chiral symmetry on the lattice is a nontrivial issue. In particular, local lattice fermion actions with
the chiral symmetry of the continuum theory suffer from the fermion doubling problem. The Ginsparg-Wilson relation implies L\"uscher's lattice variant of chiral symmetry which agrees with the usual
one in the continuum limit. Local lattice fermion actions that obey the Ginsparg-Wilson relation have an exact chiral symmetry, the correct axial anomaly, they obey a lattice version of the
Atiyah-Singer index theorem, and still they do not suffer from the notorious doubling problem. The Ginsparg-Wilson relation is satisfied exactly by Neuberger's overlap fermions which are a limit of
Kaplan's domain wall fermions, as well as by Hasenfratz and Niedermayer's classically perfect lattice fermion actions. When chiral symmetry is nonlinearly realized in effective field theories on the
lattice, the doubling problem again does not arise. This review provides an introduction to chiral symmetry on the lattice with an emphasis on the basic theoretical framework.Comment: (41 pages, to
be published in Prog. Part. Nucl. Phys. Vol. 53, issue 1 (2004)
Numerical simulations of strongly correlated electron systems suffer from the notorious fermion sign problem which has prevented progress in understanding if systems like the Hubbard model display
high-temperature superconductivity. Here we show how the fermion sign problem can be solved completely with meron-cluster methods in a large class of models of strongly correlated electron systems,
some of which are in the extended Hubbard model family and show s-wave superconductivity. In these models we also find that on-site repulsion can even coexist with a weak chemical potential without
introducing sign problems. We argue that since these models can be simulated efficiently using cluster algorithms they are ideal for studying many of the interesting phenomena in strongly correlated
electron systems.Comment: 36 Pages, 13 figures, plain Late
We discuss a representation of the Z(3) Gauge-Higgs lattice field theory at finite density in terms of dual variables, i.e., loops of flux and surfaces. In the dual representation the complex action
problem of the conventional formulation is resolved and Monte Carlo simulations at arbitrary chemical potential become possible. A suitable algorithm based on plaquette occupation numbers and
link-fluxes is introduced and we analyze the model at zero temperature and finite density both in the weak and strong coupling phases. We show that at zero temperature the model has different first
order phase transitions as a function of the chemical potential both for the weak and strong coupling phases. The exploratory study demonstrates that alternative degrees of freedom may successfully
be used for Monte Carlo simulations in several systems with gauge and matter fields.Comment: Typos corrected and some statements refined. Final version to appear in Phys. Rev.
We prove that sign problems in the traditional approach to some lattice Yukawa models can be completely solved when the fermions are formulated using fermion bags and the bosons are formulated in the
worldline representation. We prove this within the context of two examples of three dimensional models, symmetric under $U_L(1) \times U_R(1) \times Z_2 ({Parity})$ transformations, one involving
staggered fermions and the other involving Wilson fermions. We argue that these models have interesting quantum phase transitions that can now be studied using Monte Carlo methods.Comment: 5 pages, 1
figure (Fixed minor typographical errors, expanded the discussion to include solution to the sign problem with the conventional bosonic action and added a reference. | {"url":"https://core.ac.uk/search/?q=author%3A(S.%20Chandrasekharan)","timestamp":"2024-11-08T15:05:10Z","content_type":"text/html","content_length":"194849","record_id":"<urn:uuid:5b12d41b-616e-40c6-a111-a0df5bf80c4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00819.warc.gz"} |
Fraction to Percent Calculator - CalcoPolis
Fraction to Percent Calculator
Convert any fraction to percentage value with just one click.
A fraction-to-percent calculator is an online tool developed by Calcopolis to help you convert fractions to percents. It's mainly used to convert two types of fractions into percents, which are:
• Proper fractions
• Improper fractions
In a nutshell, proper fractions are those that have a numerator that's less than the denominator (ex:
Tip. If you wish to reverse this operation, use our percent to fraction calculator, which can turn any percentage value into a proper fraction. The tool can simplify fractions to make the result more
Fractions vs. Percents: A Brief Comparison
Before diving into the basic methods of converting a fraction to a percent, let's go over both of their exact definitions. This will provide you with a better understanding of the steps involved in
the conversion process.
Simply put, a fraction represents the parts of a whole and is expressed as a ratio of two numbers.
The numerator (top number) indicates the number of equal parts that are taken from the whole. The denominator (bottom number) shows the total number of equal parts into which the whole is divided.
For example, if a pizza is divided into 7 equal slices, each plate with a slice will have a
A percentage in mathematics is a ratio or number that can be described as a fraction of 100. To calculate a percent of a number, we divide the value by the whole and then multiply the result by 100.
This clarifies the literal meaning of percent, the abbreviation of the Latin phrase "per centum," which translates to "per 100." Now, we have the symbol "%" to express the word percent."
In order to operate on fractions with ease, it is essential to understand the concept of the Greatest Common Factor and Least Common Denominator. Both are described in detail at Calcopolis with
simple tools that allow you to calculate these values.
How to Turn a Fraction Into a Percentage?
With the above definitions in mind, let's look at how a fraction can be tuned to a percent. There are primarily two methods to accomplish this conversion.
Both techniques are straightforward and can be used in any context; they're not limited to specific situations.
Below, you can see examples of how to convert the fraction to percentages.
1st Method
The first method is initiated by dividing the numerator by the denominator. That is, to get the decimal equivalent of this fraction. The resulting decimal is then multiplied by 100 to obtain the
required "per 100," a.k.a. percent.
Example: Convert
1. Divide the numerator 4 by the denominator 8:
4 / 8 = 0.5
2. Multiply the decimal by 100:
0.5 * 100 = 50
2nd Method
The second method involves multiplying the fraction's numerator by 100 and then dividing the result by the denominator.
Example: Convert
1. Multiply the numerator 2 by 100:
2 * 100 = 200
2. Divide the result by the denominator 5:
200 / 5 = 40
Benefits of Presenting Fractions as a Percent
Percents help us better understand many aspects of our daily finances. That's one of the reasons why they're heavily relied upon in various industries.
Converting fractions to percents is sometimes necessary to make a figure more digestible. In fact, here are four advantages to expressing fractions as percents:
1. Provide a more descriptive insight when used in presentations and reports
2. Compile school/university grades
3. Determine the exact tax amount paid for meals at restaurants
4. Compare two or more values in a much simpler form
Example: Comparing males to females ratios in a university is easier when expressed as 60% females to 40% males rather than
Similar tools
Calcopolis provides many more useful tools that assist you with your daily math calculations. If you wish to turn a fraction into a decimal value, use a fraction to decimal converter, which can turn
a fraction into a decimal number with the required precision.
Visit the category page to see all the math tools available on our website. Using Calcopolis regularly can avoid many mistakes in your daily calculations.
Created by Lucas Krysiak on 2022-12-21 16:53:53 | Last review by Mike Kozminsky on 2022-12-21 12:14:20 | {"url":"https://calcopolis.com/math/fraction-to-percent","timestamp":"2024-11-06T18:34:22Z","content_type":"text/html","content_length":"31149","record_id":"<urn:uuid:5d4b1b55-8f80-4201-a671-3f687a195e71>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00394.warc.gz"} |
Math Balance Hands-on Math Manipulative
The EquaBeam is the quintessential math balance:
• Its single-beam construction and adjustment clips beneath the beam ensure accuracy.
• The weights are covered so as not to be an “attractive nuisance” for young children.
• The zero peg is aligned with the pegs for the numerals 1-10 as on a number line.
• Together, the front and back pegs for a numeral hold 10 weights which allows the demonstration of the multiplication facts and division facts up to the decades, for example, up to 10 x 5 = 50 for
the multiplication facts for 5 and up to 90 ÷ 9 = 10 for the division facts for 9.
• The beam extends out to 12, instead of 10, on either side to utilize the divisibility of 12 for multiplication (like 3 x 4 = 12), division (like 12 ÷ 6 = 2), and fractions (like 1/3 + 1/4 = 7/12)
and to capitalize on the fre-quency of 12 in everyday equivalences such as twelve 5-minute intervals in an hour, 12 inches in a foot, and 12 months in a year.
• The channel side of the beam facilitates changing topics with strips that fit in the channel and re-label the pegs.
As a hands-on, interactive math manipulative for elementary school students, the EquaBeam is unparalleled in versatility and sophistication. It is the perfect manipulative for reinforcing the number
facts and revealing the structure of arithmetic—that the facts are commutative (a+b = b+a), associative, [(a+b)+c = a+(b+c)], and distributive [a(b+c) = ab+ac]—thus aiding mental arithmetic and
reducing the number of facts to be learned. Its use in introducing algebra, as with Nice Mice ALGEBRA on pages 13-14, is natural and non-threatening to students of all ages. (Variables such as A, B,
C, … are simply “numbers to find on the EquaBeam.”) By relabeling the pegs on the channel side, the EquaBeam can be used to teach time, money, measurement, fractions, decimals, percent, and more in
conjunction with other manipulatives (clocks, play money, measuring devices, fraction models, etcetera).
Because of its self-checking nature, the EquaBeam is particularly suited to instructional centers and independent learning. The recommended number of EquaBeams for an instructional center is 2-4,
allowing for 4-8 students at the center. For a school, add 10 EquaBeams for every 10 classes to be shared among the classes for whole-class lessons using the EquaBeam. | {"url":"http://moveitmaththesource.com/equals/equabeamtasks/equabeam.html","timestamp":"2024-11-04T01:44:38Z","content_type":"text/html","content_length":"7909","record_id":"<urn:uuid:7bc70428-4e6f-4c07-9416-e7ba9de70abf>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00127.warc.gz"} |
Chapter Summary
Data analysis is an iterative process and the first move in this iteration for variable-based analyses is usually to undertake univariate analysis. This means inspecting the data one variable at a
time, and will include displaying categorical and metric variables in tables, charts and graphs, calculating summary measures that will pinpoint key characteristics of each distribution, and perhaps,
where the data are derived from a random sample, evaluating the likely accuracy of estimates made from it, or the statistical significance of univariate hypotheses that have been put forward. At this
point, the researcher may, in addition, reflect upon the implications of this analysis for the client, for the research objectives and for the next steps to be undertaken in the analysis of the data.
The most commonly used summary measures for metric variables (discrete or continuous), or for variables derived from summated ratings and assumed to be metric, are measures of central tendency,
dispersion and distribution shape. Summary measures for categorical variables are somewhat limited to percentages, proportions and in some cases the modal category. Once univariate analysis is
complete, researchers will usually proceed to the next step, which is looking for patterns of relationship between variables, initially two at a time. | {"url":"https://study.sagepub.com/analysing-quantitative-data/student-resources/chapter-4/chapter-summary","timestamp":"2024-11-14T05:29:24Z","content_type":"text/html","content_length":"66340","record_id":"<urn:uuid:75583d7d-a28f-4307-9907-34cb17cdc60d>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00363.warc.gz"} |
CPM Homework Help
This problem is a checkpoint for solving equations. It will be referred to as Checkpoint 5.
Solve each equation.
Check your answers by referring to the Checkpoint 5 materials located at the back of your book.
Ideally, at this point you are comfortable working with these types of problems and can solve them correctly. If you feel that you need more confidence when solving these types of problems, then
review the Checkpoint 5 materials and try the practice problems provided. From this point on, you will be expected to do problems like these correctly and with confidence.
Answers and extra practice for the Checkpoint problems are located in the back of your printed textbook or in the Reference Tab of your eBook. If you have an eBook for Core Connections, Course 1,
login and then click the following link: Checkpoint 5: Solving Equations | {"url":"https://homework.cpm.org/category/MN/textbook/cc3mn/chapter/5/lesson/5.2.4/problem/5-61","timestamp":"2024-11-02T15:52:37Z","content_type":"text/html","content_length":"35701","record_id":"<urn:uuid:a4105df4-d416-436d-a61e-c9e61167ecc1>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00030.warc.gz"} |
Integral representations for the self-avoiding walk
Speaker: Gordon Slade Date: Fri, Jun 8, 2012 Location: PIMS, University of British Columbia Conference: PIMS-MPrime Summer School in Probability Subject: Mathematics, Probability Class: Scientific
The self-avoiding walk is a fundamental model in probability, combinatorics and statistical mechanics, for which many of the basic mathematical problems remain unsolved. Recent and ongoing progress
for the four-dimensional self-avoiding walk has been based on a renormalization group analysis. This analysis takes as its starting point an exact representation of the self-avoiding walk problem as
an equivalent problem for a perturbation of a Gaussian integral involving anti-commuting variables (fermions). This lecture will give a self-contained introduction to fermionic Gaussian integrals and
will explain how they can be used to represent self-avoiding walks.
The talk is mainly based on the paper: D.C. Brydges, J.Z. Imbrie, G. Slade. Functional integral representations for self-avoiding walk. Probability Surveys, 6:34--61, (2009) | {"url":"https://mathtube.org/lecture/video/integral-representations-self-avoiding-walk","timestamp":"2024-11-08T22:03:04Z","content_type":"application/xhtml+xml","content_length":"26445","record_id":"<urn:uuid:3374c2c9-4916-4ee7-b487-ae29217b4178>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00846.warc.gz"} |
the power consumption of a ball mill
The limitation on the smallest ball size is the discharge grate slot width opening. Typically a ball size that is smaller than twice the size of the discharge grate slot width is not selected.
Mill % Loading and Specific Power Consumption Ball mills are relatively simple grinding devices from an operational point of view.
WhatsApp: +86 18838072829
Therefore, the calculation of power (or energy) consumption of ball mills is one of the most important factors in estimating the operating costs of proChehreghani, S.; Gharehgheshlagh, H. H.;
Haghikia, S. 124 Rudarskogeološkonaftni zbornik i autori (The MiningGeologyPetroleum Engineering Bulletin and the authors) ©, 2021,
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Calculation method and its application for energy consumption of ball mills in ceramic industry based on power feature deployment February 2020 Advances in Applied Ceramics 119(4):112
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grate discharge mills at high flow rates. The net power drawn by a Bond ball mill has . previously been estimated by Morrell and Ma n (1997) as W. Since the Bond mill .
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If PIANO is less than 80% go 70 microns, power consumption will be; Ball Roller Power Calculation Model #1. A wet grinding ball mill in closed circuit is to be detective 100 TPH of a substance
with a work index of 15 and a size distributor of 80% passing ¼ inch (6350 microns). ... Now we much pick an Ball Mill such will draw all power.
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Energy consumption calculation
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The energy consumption of the total grinding plant can be reduced by 2030 % for cement clinker and 3040 % for other raw materials. The overall grinding circuit efficiency and stability are
improved. The maintenance cost of the ball mill is reduced as the lifetime of grinding media and partition grates is extended.
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A 10 MW cement mill, output 270 tonnes per hour. A cement mill (or finish mill in North American usage [1]) is the equipment used to grind the hard, nodular clinker from the cement kiln into the
fine grey powder that is cement. Most cement is currently ground in ball mills and also vertical roller mills which are more effective than ball mills.
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Compared with the original twostage ball milling process, the cost of grinding power consumption is significantly reduced by about %, which provides a reference for the subsequent research on
energy saving and consumption reduction in ball milling operations. The optimal grinding parameters were determined to be grinding concentration of ...
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The 2024 CFP Field. 1 Michigan 2 Washington 3 Texas 4 Alabama. Rose Bowl will be Michigan vs. Alabama Sugar Bowl will be Washington vs. Texas. Florida State left out
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Typical power consumption for a 5m diameter by 7mlong ball mill is between and MW. The actual proportion of this energy usefully used in size reduction is thought to be very low, perhaps in the
range of 15%. Significant financial and environmental benefits can be obtained by improving this efficiency even slightly.
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If P is less than 80% passing 70 microns, power consumption will be Ball Mill Power Calculation Example #1 A wet grinding ball mill in closed circuit is to be fed 100 TPH of a material with a
work index of 15 and a size distribution of 80% passing ¼ inch (6350 microns).
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The power consumption of a ball mill is one of the most important parameters to consider in the design of a ball mill because it determines its economic efficiency. The power consumption is
usually determined by charge fill level, lifter height, lifter number, and mill speed. However, almost all of the classical theories for calculating the ...
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V — Effective volume of ball mill, m3; G2 — Material less than in product accounts for the percentage of total material, %; G1 — Material less than in ore feeding accounts for in the percentage
of the total material, %; q'm — Unit productivity calculated according to the new generation grade (), t/(). The values of q'm are determined by ...
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Thus the power to drive the whole mill. = + = = 86 kW. From the published data, the measured power to the motor terminals is 103 kW, and so the power demand of 86 kW by the mill leads to a
combined efficiency of motor and transmission of 83%, which is reasonable.
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The vibratory ball mill (VBM, Sweco, Belgium) ... a VBMtype mill uses less electric power to achieve effective fine powder production with good power utilization, but with longer processing
times. Kobayashi et al. (2007) [5] showed that a vibratory mill is better designed than cutter mills or refiners for pulverizing lignocellulosic biomass.
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power consumption of a tumbling ball mill: Experimental study and DEM simulation [J]. Minerals Engineering, 2017, 105: 22 35. [8] Lameck, K. K. Kiangi, M. H. Moys. Effects of grinding media
shapes on load behaviour and mill power in a dry ball mill [J]. Minerals Engineering, 2006, 19 (13): 1357 1361.
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The size of grinding media is the primary factor that affects the overall milling efficiency of a ball mill ( power consumption and particle size breakage). This article tackles the lack of a ...
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Although ball mills have been documented as being some 30% to 40% less efficient that some stirred mills (Nesset et al., 2006, Conclusion. ... geometry and viscosity on the power consumption of
the mill (Radziszewski, 2015). The stressing energy model shows with the characteristic numbers stressing energy and stressing frequency, which are ...
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DOI: /S(01)00037 Corpus ID: ; Charge behaviour and power consumption in ball mills: sensitivity to mill operating conditions, liner geometry and charge composition
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Rod and Ball Mills by Rowland and Kjos AllisChalmers. ... Note that the above equation (3), was derived to empirically relate power consumption over a wide range of mill sizes, on various
slurries, as well as over a fairly broad range of mill speed and load levels, all combined into a single expression. ...
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Metric graph linear mixed effects models — graph_lme
Metric graph linear mixed effects models
Fitting linear mixed effects model in metric graphs. The random effects can be Gaussian Whittle-Matern fields, discrete Gaussian Markov random fields based on the graph Laplacian, as well as Gaussian
random fields with isotropic covariance functions.
model = list(type = "linearModel"),
which_repl = NULL,
optim_method = "L-BFGS-B",
possible_methods = "L-BFGS-B",
model_options = list(),
BC = 1,
previous_fit = NULL,
fix_coeff = FALSE,
parallel = FALSE,
n_cores = parallel::detectCores() - 1,
optim_controls = list(),
improve_hessian = FALSE,
hessian_args = list(),
check_euclidean = TRUE
Formula object describing the relation between the response variables and the fixed effects.
A metric_graph object.
The random effects model that will be used (it also includes the option of not having any random effects). It can be either a character, whose options are 'lm', for linear models without random
effects; 'WM1' and 'WM2' for Whittle-Matern models with \(\alpha\)=1 and 2, with exact precision matrices, respectively; 'WM' for Whittle-Matern models where one also estimates the smoothness
parameter via finite-element method; 'isoExp' for a model with isotropic exponential covariance; 'GL1' and 'GL2' for a SPDE model based on graph Laplacian with \(\alpha\) = 1 and 2, respectively.
'WMD1' is the directed Whittle-Matern with \(\alpha\)=1. There is also the option to provide it as a list containing the elements type, which can be linearModel, WhittleMatern, graphLaplacian or
isoCov. linearModel corresponds to a linear model without random effects. For WhittleMatern models, that is, if the list contains type = 'WhittleMatern', one can choose between a finite element
approximation of the precision matrix by adding fem = TRUE to the list, or to use the exact precision matrix (by setting fem = FALSE). If fem is FALSE, there is also the parameter alpha, to
determine the order of the SPDE, which is either 1 or 2. If fem is FALSE and alpha is not specified, then the default value of alpha=1 will be used. If fem is TRUE and one does not specify alpha,
it will be estimated from the data. However, if one wants to have alpha fixed to some value, the user can specify either alpha or nu in the list. See the vignettes for examples. Finally, for type
'WhittleMatern', there is an optional argument, rspde_order, that chooses the order of the rational approximation. By default rspde_order is 2. Finally, if one wants to fit a nonstationary model,
then fem necessarily needs to be TRUE, and one needs to also supply the matrices B.tau and B.kappa or B.range and B.sigma. For graph-Laplacian models, the list must also contain a parameter alpha
(which is 1 by default). For isoCov models, the list must contain a parameter cov_function, containing the covariance function. The function accepts a string input for the following covariance
functions: 'exp_covariance', 'WM1', 'WM2', 'GL1', 'GL2'. For another covariance function, the function itself must be provided as the cov_function argument. The default is 'exp_covariance', the
exponential covariance. We also have covariance-based versions of the Whittle-Matern and graph Laplacian models, however they are much slower, they are the following (string) values for
'cov_function': 'alpha1' and 'alpha2' for Whittle-Matern fields, and 'GL1' and 'GL2' for graph Laplacian models. Finally, for Whittle-Matern models, there is an additional parameter version,
which can be either 1 or 2, to tell which version of the likelihood should be used. Version is 1 by default.
Vector or list containing which replicates to consider in the model. If NULL all replicates will be considered.
The method to be used with optim function.
Which methods to try in case the optimization fails or the hessian is not positive definite. The options are 'Nelder-Mead', 'L-BFGS-B', 'BFGS', 'CG' and 'SANN'. By default only 'L-BFGS-B' is
A list containing additional options to be used in the model. Currently, it is possible to fix parameters during the estimation or change the starting values of the parameters. The general
structure of the elements of the list is fix_parname and start_parname, where parname stands for the name of the parameter. If fix_parname is not NULL, then the model with be fitted with the
parname being fixed at the value that was passed. If start_parname is not NULL, the model will be fitted using the value passed as starting value for parname. the For 'WM' models, the possible
elements of the list are: fix_sigma_e, start_sigma_e, fix_nu, start_nu, fix_sigma, start_sigma, fix_range, start_range. Alternatively, one can use fix_sigma_e, start_sigma_e, fix_nu, start_nu,
fix_tau, start_tau, fix_kappa, start_kappa. For 'WM1', 'WM2', 'isoExp', 'GL1' and 'GL2' models, the possible elements of the list are fix_sigma_e, start_sigma_e, fix_sigma, start_sigma,
fix_range, start_range. Alternatively, one can use fix_sigma_e, start_sigma_e, fix_tau, start_tau, fix_kappa, start_kappa. For 'isoCov' models, the possible values are fix_sigma_e, start_sigma_e,
fix_par_vec, start_par_vec. Observe that contrary to the other models, for 'isoCov' models, both fix_par_vec and start_par_vec should be given as vectors of the size of the dimension of the
vector for the input of the covariance function passed to the 'isoCov' model. Furthermore, for 'isoCov' models, fix_par_vec is a logical vector, indicating which parameters to be fixed, and the
values will be kept fixed to the values given to start_par_vec, one can also use fix_sigma_e and start_sigma_e for controlling the std. deviation of the measurement error.
For WhittleMatern models, decides which boundary condition to use (0,1). Here, 0 is Neumann boundary conditions and 1 specifies stationary boundary conditions.
An object of class graph_lme. Use the fitted coefficients as starting values.
If using a previous fit, should all coefficients be fixed at the starting values?
logical. Indicating whether to use optimParallel() or not.
Number of cores to be used if parallel is true.
Additional controls to be passed to optim() or optimParallel().
Should a more precise estimate of the hessian be obtained? Turning on might increase the overall time.
List of controls to be used if improve_hessian is TRUE. The list can contain the arguments to be passed to the method.args argument in the hessian function. See the help of the hessian function
in 'numDeriv' package for details. Observet that it only accepts the "Richardson" method for now, the method "complex" is not supported.
Check if the graph used to compute the resistance distance has Euclidean edges? The graph used to compute the resistance distance has the observation locations as vertices. | {"url":"https://davidbolin.github.io/MetricGraph/reference/graph_lme.html","timestamp":"2024-11-05T20:19:17Z","content_type":"text/html","content_length":"20618","record_id":"<urn:uuid:f69d2244-3513-478e-8809-a9abcfe6580e>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00647.warc.gz"} |
the temptation - ochs und juniorthe temptation - ochs und junior
imagine each month had only 28 days and had thus exactly four weeks. if that were the case, each date of the different months would, since the week has seven days, be the exact same weekday. so if
the first day of a month was a monday, the 7th, 14th, 21st, and 28th would be sundays. however, that is not the case. most months have either 30 or 31 days. every four years, one month has 29 days,
and in the remaining years, one has 28 days. so, normally, there are just 3 months in four years, or in 48 months, that only have 28 days, or of 12 months there are 11.25 that are longer than 28
days. to stick with the example: if the first day of a month was a monday, and the month had 29 days, then the next month would start with a tuesday; if the month had 30 days, the first day of the
next month would be a wednesday; and if the month had 31 days, the first day of the following month would be a thursday. so there is a shift in terms of the weekday which is 0 days in the case of a
month with 28 days, 1 day in the case of a month with 29 days, 2 days in the case of a month with 30 days, and 3 days in the case of a month with 31 days. if one considered that a triviality, one
might be tempted to look for a corresponding solution in a perpetual calendar such as the one from ochs und junior.
it tempted me as well. and i found a fairly simple solution and built a prototype based on the usual perpetual calendar. as for the switch from one month to the next, i am using the last 3 of the 31
steps of the month switches. these last three steps are the ones that potentially have to be skipped in order to jump from the last day of a month to the first day of the next month. the switch from
one month to the next starts with the 29th of each month. for each step that is skipped, the switch takes about an hour, as not only the remaining days are skipped, but the month disc is also set in
motion. this latter thus shows a step-by-step motion for four days, if the switch is from a month with 31 days to the next, and correspondingly less, if it is from a month with 30, 29, or 28 days.
the temptation to include the weekday shift of one, two, or three days from one month to the next, based on the above, was therefore great. i thought about demarcating the weeks by giving a black dot
to sunday. using a hoop with 4 lines that follow a spiral, i can show these in the spiral-shaped line of date dots, kind of like with the banana-shaped dot on the date hoop. i can then move the
weekday hoop after the 28th of the month using the step-by-step motion of the month disc via a small driving gear. to deal with the days that are not to be shifted, i am using something like a
maltese position via this gear, by milling out the corresponding gaps, and adding arrest latches, on the gear rim of the month hoop.
i built this watch for myself and have been wearing it on my wrist. yet, i have come to the conclusion that the adopted solution is impractical for our clients, despite the extremely efficient and
simple design – only three additional and two modified parts are required. the reason is that this watch has an important drawback: during the three days at the end of the month when the switch takes
place, that is from the 29th until the 1st of the next month, the weekdays are not shown correctly, but are instead caught up in the switch from one month to the next. if one were to remedy this
drawback and opt for a correct display of the weekday at the end of the month, one would have to design a much more complex construction. in fact, the complication would increase to such a degree
that the effort is no longer worth it, in my view. | {"url":"https://www.ochsundjunior.swiss/the-temptation/","timestamp":"2024-11-11T11:05:46Z","content_type":"text/html","content_length":"76093","record_id":"<urn:uuid:bf2ec742-090a-497f-88f2-7ccbe3b0068c>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00384.warc.gz"} |
(PDF) High-Q self-resonant structure for wireless power transfer
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High-Q Self-Resonant Structure for Wireless Power
Aaron L.F. Stein Phyo Aung Kyaw Charles R. Sullivan
Thayer School of Engineering
Dartmouth College
Hanover, NH 03755 USA
Email: {Aaron.L.Stein, Phyo.A.Kyaw.TH, Charles.R.Sullivan}@dartmouth.edu
Abstract—The range and efficiency of wireless power transfer
systems are limited by the quality factor of the transmit and
receive coils used. Multi-layer self-resonant structures have been
proposed as a low-cost method for creating high-Q coils for high-
frequency wireless power transfer. In these structures thin foil
layers are separated by a dielectric material in order to form a
capacitance that resonates with the inductance of the structure,
while also forcing equal current sharing between conductors. In
order to reduce winding loss, these structures are made with foil
layers much thinner than a skin depth, which makes the layers
of the structure extremely difficult to handle. In this paper, we
present a modified self-resonant structure in which the layered
conductors are made from standard PCB substrates with no
vias. The PCB substrates provide an inexpensive way to handle
thin conductive layers, and the modified self-resonant structure
ensures that the poor dielectric properties of the PCB substrates
do not impact the quality factor of the structure. The modified
self-resonant structure makes it feasible to achieve advantages
similar to litz wire, but at multi-MHz frequencies where effective
litz wire is not commercially available. Experimental results
show that the structure has a quality factor of 1177 at 7.08
MHz, despite only being 6.6 cm in diameter. The quality factor
normalized by the diameter is more than 6.5x larger than other
coils presented in the literature.
I. INT ROD UC TI ON
Wireless power transfer is of great interest for many appli-
cations including biomedical, automotive, and consumer hand-
held electronics [1]–[4]. In many of these applications a high-
frequency magnetically-coupled resonant system is the most
effective method of transmitting wireless power. The efficiency
and range of such a system is limited by the quality factor
and coupling coefficient of the resonant coils that generates
the electromagnetic coupling [3], [5], [6].
Traditional coils consist of a spiral loop of wire connected
to a ceramic or film capacitor. The quality factor of such a coil
increases linearly with the diameter of the coil [7], [8]. So we
propose a figure of merit Qd, which is the ratio of the quality
factor Qto the diameter dof the coil: Qd=Q
d. Experimental
data in the literature for high frequency coils around 6.78 MHz
have a Qdthat ranges from 3 to 28 cm−1[3], [9]–[13].
The Qdof conventional coils is limited by two main factors.
First, below 1 MHz coils are typically made from litz wire in
order to minimize losses due to skin and proximity effects.
However, the benefit of using litz wire is limited in the MHz
frequency range due to the need to have strand diameters
much smaller than the skin depth. Such small strand diameters
are not commercially available because they are difficult and
therefore expensive to manufacture. Second, in many designs,
eddy currents are induced in the capacitors due to their
proximity and orientation to the coil.
To mitigate these issues a multi-layer self-resonant structure
using thin sheets of conductors, capacitive ballasting, and low-
loss dielectrics was proposed in [14]. This structure consists of
alternating layers of C-shape conductors and dielectric rings
placed in a ferrite core, creating inductively coupled capacitors
in parallel with an inductor. The integration of capacitance in
the structure is similar to the integrated LC and LCT (inductor,
capacitor, transformer) passive power components discussed
in, for example, [15]–[17]. However, unlike the previous work,
multi-layer self-resonant structures use the capacitance not
only to implement the necessary capacitance, but also to make
the conductors more efficient by equalizing current sharing
between them. As a result, they not only provide a parts
count savings through integration, but also provide a dramatic
performance benefit. Furthermore, the self-resonant structure
reduces eddy currents by keeping thin foil layers parallel to the
magnetic field, and does not require inter-layer connections.
In this work, we present a modification to the self-resonant
structure that achieves similar performance, but simplifies
the construction process by allowing the thin conductive
layers to be constructed from standard PCB substrates. This
modification to the self-resonant structure led to the first
experimental implementation of a multi-layer self-resonant
structure at resonant frequency practical for wireless power
A related type of self-resonant structure is the split-ring
resonator (SRR) [18]. A SRR is a pair of C-shaped conductors
that forms a simple resonator which can be arrayed to create
metamaterials. Metamaterials can be designed with unusual
and controllable electromagnetic properties, and have even
been proposed as a way to influence the coupling of resonant
inductive wireless power systems. However, although it can
be shown that an ideal negative-permeability material could
be used to enhance performance, the losses of practical SRRs
limit the usefulness of this approach [19]. Whereas an individ-
ual SRR comprises just two concentric C-shaped conductors,
the self-resonant structure configures them in a stack rather
than concentrically, and uses many layers to achieve low
losses, in conjunction with soft magnetic material shaping the
field for lowest losses.
The many layers of the self-resonant structure are made
from foil, which makes using conductors thinner than the skin
depth feasible even at high-frequencies; however, very thin
foil layers are difficult to handle. The practical construction
challenges associated with using such thin layers prevented us
from from experimentally validating the self-resonant structure
in [14] at the desired resonant frequency. One way to overcome
the challenges associated with using such thin copper layers is
to pattern the C-shapes by etching thin copper layers laminated
on substrates. In the resonant structure in [14], the capacitance
between adjacent conductors, such as those on opposite sides
of a substrate, provides the capacitance for resonance. Thus,
for high-Q resonance, the substrate dielectric must have a very
low dissipation factor. Unfortunately, common substrate mate-
rials such as FR4 and polyimide have high dissipation factors
(0.015 and 0.002) even at low frequencies and the performance
gets worse at higher frequencies. Copper laminates with low-
loss substrates such as PTFE are much more expensive.
In this paper, we present a variation of the self-resonant
structure described in [14] to allow fabrication with more
conventional methods and materials. Our new resonant struc-
ture uses high-loss but low-cost substrates such as FR4 and
polyimide to support thin conductor layers for easy handling
without adversely affecting the quality factor of the resonance.
The improved manufacturability of the modified structure
presented here allowed us to successfully implement a high-
Q 7 MHz resonant structure for wireless power transfer.
The modified structure is described in Section II, its loss
mechanisms are analyzed in Section III, and experimental
characterization results are presented in Section IV.
In addition to wireless power transfer applications, high-Q
resonant structures are of interest as passive components for
power converter applications. Our design work and test results
for similar structures used in resonant power conversion are
discussed in [20].
II. MO DI FIE D SEL F-RE SO NAN T STRUCTURE
The modified self-resonant coil is illustrated in Fig. 1. The
structure creates a parallel LC resonance. The inductance L
is equivalent to a single turn around the magnetic core, while
the total capacitance Cequiv is created by inductively coupling
each section of the structure as shown in the circuit model in
Fig. 2. A section of the structure is two C-shaped conductors
with opposing orientation that are separated by a low-loss
dielectric. For example, in Fig. 1, Layers 2, 3, and 4 form
one section. Each section forms two capacitors. One capacitor
is formed in each area the conductors overlap as illustrated in
Fig. 3. The capacitance between half of one C shape and the
facing half of the other C shape in the same section Csh, is
a function of the angle of overlap of the layers in radians θ
(shown in Fig. 3), the outer radius of the coil r2, the inner
radius r1, and the dielectric thickness td
Csh =ǫθ(r2
Fig. 1. The layers of a 2 section modified self-resonant structure.
Fig. 2. Equivalent circuit model of a 2 section modified self-resonant
The modified self-resonant structure has msections, where
each section of the structure has two series connected Csh.
The equivalent capacitance Cequiv of the structure is
Cequiv =mCsh
This is the only capacitance which can be excited, and
in conjunction with the inductance determines the resonant
frequency of the structure. The resonant frequency of the
structure ωois given by
The proposed modified self-resonant structure allows the
Fig. 3. In this figure two overlapping C-shaped conductors forming one
section is shown. Each section forms two capacitors Csh which are connected
in series. The angle of overlap of one capacitor θis marked .
Number of Sections
Conductor thickness (microns)
Fig. 4. The theoretical quality factor of the modified self-resonant structure at
7 MHz is plotted as a function of the conductor thickness and the number of
sections for a 6.6 cm pot described in Section II. The field weakening factor
is unity and the current crowding factor zero in this figure.
use of low-performance substrates such as FR4 or polyimide
without significantly affecting the Q of the structure. To
achieve this, any two conductor layers that are separated
by a high-loss substrate are oriented such that their gaps
are aligned. For example in Fig. 1, the top layer of copper
(layer 1) is separated from the second layer of copper (layer
2) by a high-loss substrate, and are both oriented such that
the gap is coming out of the page. A capacitance Csub is
formed between these two layers; however, the orientation
ensures that no strong electric field is generated between the
layers. The voltage induced in Csub is only due to the leakage
magnetic flux, which is a small fraction of the overall magnetic
flux. This allows the high-loss substrate to be integrated into
the self-resonant structure, without significantly affecting the
quality factor of the structure. Furthermore, the thickness of
the substrate does not affect the equivalent capacitance Cequiv,
so it can be selected based on considerations such as ease
of handling, and the overall compactness of the complete
III. LOS S MEC HA NI SM S
The performance of the modified self-resonant structure is
measured by the quality factor of the device at resonance. The
quality factor Qis
where Rtotal is sum of 3 equivalent series resistances (ESR)
that model winding resistance, core loss, and dielectric loss.
The ESR for each of these loss mechanisms is derived in this
1) Winding Loss: The power lost in the winding is due
to both the low frequency resistance Rlf of the winding,
and eddy currents due to the high-frequency magnetic field
(proximity effect). The increased losses due to these eddy
currents can be modeled by a resistance Re. The power lost
in the winding Pwind can be expressed as
Pwind =I2
rmsRw ind =I2
rmsRlf +I2
where Rwind is the AC resistance of the structure. Simpli-
fication of (5) shows that the AC resistance factor Rac
Rlf is
1 + Re
Rlf . Therefore, the winding resistance of the structure
is the product of a low frequency resistance and an AC
resistance factor, and is given by
Rwind =Rlf 1 + Re
Rlf .(6)
Both Rlf and the AC resistance factor are derived in [14] for
the self-resonant structure. The winding resistance is given by
Rwind =2πρ
where k1is (1−θ
3π), k2is (1 + θ
π), tcis the thickness of
the foil layers, δis the skin depth, and ρis the resistivity of
the conductor material. This analysis assumes that a magnetic
core with infinite permeability is placed directly adjacent to
the windings. In practice, there is a gap between the magnetic
core and the winding, and furthermore the permeability of the
high frequency magnetic material is not large enough to be
accurately modeled as infinite. Compared to the idealized case,
these practical consideration weaken the magnetic field and
prevent the magnetic field lines from being perfectly parallel
to the foil layers.
The weakening of the magnetic field reduces the power lost
due to the proximity effect. The impact of field weakening
on the winding resistance is modeled by a field weakening
factor Ffw . This factor is derived from a finite element analysis
described in Appendix A-A. In the idealized case, Ff w is 1,
and it decreases in practical scenarios. When the magnetic
field lines are not parallel to the foil layers horizontal current
crowding occurs in the winding. This is modeled with a current
crowding factor Fcc. This factor is also derived from a finite
element analysis, and its extraction process is described in
Appendix A-B. In the idealized case Fcc is 1, and it increases
in practical scenarios. In total the winding resistance is given
Rwind =2πρ
ln( r2
r1)tM "k1Fcc +Ffw M2
The winding loss expression illuminates important design
parameters: the thickness of the conductor and the number
of sections. If the conductor is too thick the proximity effect
losses will be high, whereas if the conductor is too thin the
DC resistance of the conductor will be large. Similarly, an
optimal number of sections exists. If too many sections are
used, the proximity effect losses will once again be high, and
if too few sections are used the conductor resistance will be
high. An optimal number of sections and conductor thickness
can be derived; however, in our prototyping work, we instead
constrained the design based on material thicknesses that are
readily available from stock. A contour plot in Fig. 4 illustrates
the impact of varying these parameters on the quality factor
of the structure.
2) Magnetic Core Loss: In this application both the real
part µ′and the imaginary part µ′′ of the magnetic core
permeability affect the quality factor of the structure. The loss
in the magnetic core is modeled by an ESR, which can be
derived from a reluctance model of the magnetic core. Using
this model, the single-turn inductance of the structure L∗is a
complex number given by
Aeµ0(µ′−jµ′′ )+Ra
where ℓeh is the effective length of the core half (half the
effective length of a full pot core), Aeis the effective area of
the core, and Rais the reluctance of the air gap. The ESR
that models core loss is a function of the angular frequency ω
and is given by
Rcore =ℜ[jωL∗]
µ0Ae+Raµ′2+ (Raµ′′)2.(10)
The denominator of Rcore is dominated by (Raµ′)2. There-
fore, the ESR of the core is approximately proportional to
Rcore ∝
where the quality factor of the material Qmaterial is µ′
µ′′ . In
order to reduce core loss, it is important to select a material
that both has a large Qmaterial, and a large real component
of magnetic permeability at the resonant frequency of the
3) Dielectric Loss: The modified self-resonant structure
does not use external capacitors, so the loss mechanisms as-
sociated with conductors of external capacitors do not impact
it. Instead, the losses created by the capacitance Cequiv of the
structure are due to the dielectric material, and can be modeled
with an ESR that is given by
Rdieletrcic =Dd
Cequiv ω,(12)
where Ddis the dissipation factor of the material. To reduce
the dielectric loss a material with a small dissipation factor
such as PTFE or polypropylene should be used. There are no
significant losses created by Csub, despite the use of the high-
loss substrate, because it is not involved in the resonance of
the structure.
IV. RES ULT S - IM PL EM ENTATION O F TH E MOD IFI ED
SEL F-RE SO NAN T STRUCTURE
The performance of the modified self-resonant structure was
experimentally validated. The device comprises three main
components. First a pot core was made from Fair-Rite’s 67
material. This material was chosen for its low loss at 7 MHz.
The pot core has a diameter of 6.6 cm, and a height of 1.62
cm. Next, the conductive layers of the structure were created
using 6 µm copper that, for ease of handling, was laminated on
both sides of a 25 µmpolyimide substrate and patterned into
C-shapes using standard PCB fabrication processes. Finally,
50.8 µmthick PTFE film was cut with a die cutter to form
the low-loss dielectric layers. A picture of the modified self-
resonant structure is shown in Fig. 5, and system parameter
values are shown in Table I.
For this implementation of the modified self-resonant struc-
ture, the analysis in Section III estimates the ESR of the
structure to be 4.9 mΩ. A finite element analysis of the
structure is used to derive the field weakening factor Ffw of
0.80, and the current crowding factor Fcc of 1.74, which results
in a predicted winding resistance of 1.6 mΩ. Experimentation
using FairRite’s 67 material found the imaginary component
of the relative permeability to be 0.07, which results in an core
loss ESR of 1.9 mΩ. Finally, using the dissipation factor from
a PTFE data-sheet, the ESR that model the dielectric loss is
1.4 mΩ. Given that the inductance of the structure is 155 nH,
(4) estimates that this structure will have a quality factor of
SEL F-RE SO NANT C OI L VARIA BL ES,T HE IR DE SC RIP TIO NS ,AND VAL UE S IN
TH E EXP ER IME NTAL S ETU P. THE W IND IN G,CO RE ,AND D IE LEC TR IC ES RS
AR E DER IV ED FRO M TH E MOD EL S IN SEC TI ON III.
Parameter Description Value
dStructure diameter 6.6 cm
mNumber of sections 48
Core window height 9.2 mm
Height of structure 16 mm
r2Coil outer radius 26.25 mm
r1Coil inner radius 14.85 mm
tConductor thickness 6 µm
Substrate thickness 25.4 µm
Stacked layers height 5.5 mm
θOverlap angle 2.97rad
δSkin depth 25 µm
ρConductor resistivity 16.8 nΩ-m
Ffw Field Weakening factor 0.80
Fcc Current crowding factor 1.74
Rwind Winding ESR 1.4 mΩ
LStructure inductance 155 nH
µ′Core relative permeability 40
µ′′ Imaginary relative permeability 0.07
ℓeh Effective length of core half 37.5 mm
AeEffective core area 717 mm2
RaReluctance of air path 5.4 MA
Rcore Core ESR 1.9 mΩ
Cequiv Structure capacitance 3.28 nF
tdDielectric thickness 25.4 µm
ǫDielectric permittivity 2.2ǫo
DdDielectric dissipation factor 2×10−4
Rdieletric Dielectric ESR 1.4 mΩ
Fig. 5. Picture of the modified self-resonant structure that was used for
experimental results.
A. Experimental Performance of the Modified Self-Resonant
The quality factor of the resonant structure was experimen-
tally found to be 1177. The quality factor was derived from
the magnitude of the impedance that is shown in Fig. 6, using
the ratio of the resonant frequency to the 3dB bandwidth. To
verify this measurement the quality factor was also derived by
measuring inductance, resonant frequency and magnitude of
the maximum impedance Zpk to compute Q=Zpk
ωoL= 1136.
The error between the theoretical and experimental quality
factor is 16.1%, which suggests good agreement with the
analysis presented in Section III. The Qdof the modified
resonant structure is 178 cm−1, which represents a factor of
6.35 improvement over the current state-of-the-art [3], [9]–
[13]. The experimental results are summarized in Table II.
SUM MARY O F EX PER IME NTAL R ES ULTS
Parameter Description Value
foResonant frequency 7.08 MHz
QQuality factor 1177
QdFigure of merit (FOM) 178
FOM percent improvement 635%
B. Impact of the Modified Self-Resonant Structure on Wireless
Power Transfer
The maximum achievable efficiency ηmax between two
coils of a wireless power transfer system is dependent on the
quality factor Qof the coils, and the coupling coefficient k.
The maximum efficiency is derived in [5], [6] and is given by
ηmax =(Qk)2
1 + p1 + (Qk)22.(13)
Therefore, to maximize the efficiency of a wireless power
transfer system the quality factor and coupling factor should
be maximized.
The modified self-resonant structure has been experimen-
tally demonstrated to have a quality factor that is 6.3 times
larger than conventional coils; however, to understand the
7.076 7.078 7.08 7.082 7.084 7.086
Frequency (MHz)
Imepedance Magnitude
Modified Self Resonant Coil
Resonant Frequency
3dB Marker
7 7.05 7.1 7.15
Frequency (MHz) ×106
Imepedance Magnitude
Fig. 6. An Agilent 4294A impedance analyzer was used to measure the
impedance magnitude of the modified self-resonant structure around its
resonant frequency. The impedance magnitude is shown for two different
frequency ranges to illustrate the high-q nature of the resonance. The ex-
perimental measured quality factor of the modified self-resonant structure is
Coil Separation Distance (mm)
Coupling Factor (k)
Fig. 7. A finite element analysis is used to derive the coupling coefficient
(k) as a function of the distance between the magnetic coils.
impact of this on the efficiency of wireless power transfer the
coupling factor must also be considered. The coupling factor
is determined by the shape, orientation, and properties of the
magnetic cores. A finite element analysis of the magnetic core
used in this work shows that the coupling factor ranges from
0.875 to 0.0014 as the transmission distance increases from
1 mm to 150 mm. The coupling factor is plotted over this
entire range in Fig. 7, and pictures that demonstrate the relative
transmission distances compared to the core size are shown in
Fig. 8.
The maximum achievable efficiency, given by (13), using
the modified self resonant structure is compared to the current
state-of-the-art coil designs in Fig 9. For this comparison we
assume that each resonator is implemented with the same
magnetic core used for our modified self-resonant structure.
The maximum achievable efficiency using the modified self-
Fig. 8. A picture of the magnetic cores at a separation distances of a) 50
mm, b) 100 mm, and c) 150 mm illustrates the relative size of the coil to
the range of wireless power transfer discussed in this work.
resonant structure is derived using the experimental quality
factor of 1177 and the simulated coupling factor. The state-
of-the-art design uses the same coupling factor as the self-
resonant structure, but with a quality factor of 185, which is
derived by multiplying the state-of-the-art Qdby the diameter
of the core.
The modified self-resonant structure improves wireless
power transfer efficiency for any distance between the coils.
For example, if the coils are 20 mm apart the modified self-
resonant structure can achieve 98.7% efficiency, while the
current state-of-the-art coil coil technology can achieve 91.9%.
At longer distances, the difference is even more dramatic. At
a distance of 90 mm, the state-of-the-art coil design can only
achieve an efficiency of 22%, while the modified self resonant
structure achieves an efficiency of 77%. Furthermore the mod-
ified self-resonant structure can maintain high efficiency out
to longer distances. For example, the modified self resonant
structure can maintain at least 90%efficiency at a distance up
to 65 mm, while conventional coil designs can only achieve
90% efficiency at distances up to 30 mm.
V. CO NC LU SI ON
The efficiency and range of resonant wireless power transfer
is highly dependent on the quality factor of the resonant tank.
This work introduces a new high-Q self-resonant structure that
is both easy to manufacture and cost effective. To achieve this
goal the thin copper layers of the structure are created using
inexpensive substrates such as FR4 or polyimide laminated
with copper. Although inexpensive, these substrates are not
efficient dielectrics. By orienting the layers on the two sides
of the high-loss substrate differently than proposed in [14],
we avoid exciting the substrate capacitance and thus avoid
losses in it. Experimental results confirm the advantages of
the modified self-resonant structure, as the quality factor
normalized by the diameter of the structure is shown to be
more than 6.35 times higher than the current state-of-the-art
without using high-cost materials or manufacturing processes.
The improved quality factor of the modified self-resonant
structure improves the range over which wireless power can
be transferred. For example, compared to the current state-of-
the-art, the modified self resonant structure more than double
Coil Separation Distance (mm)
Q=1177 Self-Resonant Structure
Q=185 Current State-of-the-Art
Fig. 9. The theoretical maximum wireless power transfer efficiency as a
function of transmission distance is shown for the modified self-resonant
structure, and the current state-of-the-art coil design. The drastically improved
quality factor of the modified self-resonant structure causes a significant
improvement in wireless power transfer efficiency, and improves the viable
range of wireless power transfer.
the range at which energy can be transferred at an efficiency
of at least 90%.
APP EN DI X A
FIN IT E ELE ME NT WI ND IN G LOSS EXTRAC TI ON
The field weakening and current crowding factors are ex-
tracted from a two-dimensional axisymmetric finite element
analysis (FEA). Accurate modeling of the magnetic core
properties, and the physical dimensions are important to the
result of the simulation. Each section of the modified self
resonant structure consists of two copper layers, so the FEA
model is an inductor with 2mturns of foil. The foil layers
of the model are connected in series in order to force equal
current sharing between layers, and are driven with an RMS
current Irms. The thickness of the foil windings is tc, and
therefore it is important to ensure that the FEA mesh size
within the winding is small enough to accurately model the
effects within the winding. Finally, a low frequency resistance
of the coil Rlf is needed for this analysis, and it can be derived
from a FEA simulation; however, to reduce computation time
an analytical expression is used and is given by
Rlf =4mπρ
ln ( r2
A. Field Weakening Factor
The field weakening factor accounts for decreased proximity
effect loss due to a reduction in the magnetic field. The
proximity effect power loss is produced by the magnetic
field inside the winding area; however, the impact of field
weakening is derived by only considering the spatial average
of the square of the peak value of the magnetic field parallel to
the foil layers Dˆ
rE. The power loss due to the parallel-flux
proximity effect Pprox is given by
Pprox =Dˆ
where Vfis the total volume of the foil. Ffw is derived by
equating the AC resistance calculated with the simulated field
strength (1 + Pprox
rmsRlf )to the theoretical AC resistance factor
(1 + Ffw (2m)2
δ4). The field weakening factor is derived
from this relationship and is given by
Ffw =9Pprox
rmsRlf (2m)2tc
B. Current Crowding Factor
The current crowding factor accounts for increased losses in
the conductors due to horizontal current crowding. This factor
is derived from the resistance Rfea of the FEA model at the
resonant frequency. The ratio of Rf ea to Rlf is
so the current crowding factor Fcc is given by
Fcc =Rfea
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[14] C. R. Sullivan and L. Beghou, “Design methodology for a high-Q
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for power converters,” vol. 1, 2002, pp. 12–17.
[18] R. Marques, J. Martel, F. Mesa, and F. Medina, “Left-handed-media
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[19] T. Oh and B. Lee, “Analysis of wireless power transfer using meta-
material slabs made of ring resonators at 13.56 mhz,” Journal of
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[20] P. Kyaw, A. Stein, and C. R. Sullivan, “High-Q resonator with inte-
grated capacitance for resonant power conversion,” in Applied Power
Electronics Conference and Exposition. IEEE, 2016. | {"url":"https://www.researchgate.net/publication/317072320_High-Q_self-resonant_structure_for_wireless_power_transfer","timestamp":"2024-11-04T19:06:20Z","content_type":"text/html","content_length":"696743","record_id":"<urn:uuid:f1106f16-2b99-487f-8c2d-258c2b9cc2f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00547.warc.gz"} |
Introduction to Recurrent Neural Networks (RNNs) with PyTorch
Recurrent Neural Network (RNN) is a type of neural network architecture designed for sequence modeling and processing tasks. Unlike feedforward neural networks, which process each input
independently, RNNs have connections that allow them to combine information about previous inputs into their current computations.
In this tutorial, we'll briefly learn about RNNs and how to implement a simple RNN model with sequential data in PyTorch covering the following topics:
1. Introduction to RNNs
2. Data preparing
3. Model definition and training
4. Prediction
5. Conclusion
Let's get started
Introduction to RNNs
RNNs are a specialized type of neural network designed for sequential data. The key feature of RNNs is their ability to maintain a state or memory of previous inputs while processing a sequence of
data points.
RNNs facilitate recurrent connections that allow information to persist across time steps. This characteristic enables RNNs to capture temporal dependencies in sequential data such as time series,
natural language, or any other sequential data. RNNs process input sequences sequentially, updating hidden states at each step to encode information about previous inputs, which is crucial for tasks
where understanding the sequence of data is important.
Recurrent Neural Networks (RNNs) confront several challenges:
1. Vanishing Gradient Problem: RNNs suffer from vanishing gradients during backpropagation which makes it difficult for model to learn long-range dependencies in sequences.
2. Exploding Gradient Problem: On the other hand, RNNs may also suffer exploding gradients, where gradients grow exponentially during training.
3. Memory and Computational Intensity: RNNs can be memory and computation intensive, particularly when processing long sequences, slowing down training and inference.
4. Difficulty in Capturing Global Context: Due to their incremental processing of sequential data, RNNs may struggle to capture global context or dependencies across distant parts of the sequence.
LSTM and GRU architectures have become popular alternatives to traditional RNNs due to their ability to address the limitations of vanishing gradients, exploding gradients, memory, and computational
intensity, while improving the model's ability to capture global context in sequential data.
Data preparing
We start by loading the necessary libraries.
import torch
import torch.nn as nn
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
In this tutorial we use simple sequential data. Below code shows how to generate and visualize it on a graph. Here, we use 800 samples as a training data and 200 samples for test data to forecast.
# Define parameters
step_size = 3
N = 1000
forecast_start = 800
# Generate data
t = np.arange(0, N)
x = np.sin(0.02*t) + 2*np.random.rand(N)
df = pd.DataFrame(x)
# Plot data
plt.axvline(df.index[forecast_start], c="r", label="forecast start point")
Next, we convert data into training sequence and label with the given length. Below function helps us to create labels for sequence data.
# Convert data into sequence and label with given length
def create_labels(data, step):
X, y = [], []
for i in range(len(data)-step):
d = i + step
return np.array(X), np.array(y)
We can split data into train and test parts using forecast_start variable, then generate sequence data and its labels. The np.reshape() function reshapes data for RNN input. A train and test sets are
converted to PyTorch tensors and DataLoader objects are created using those tensors.
# Prepare data for training and testing
values = df.values
train, test = values[:forecast_start,:], values[forecast_start:N,:]
# generate sequence data
trainX, trainY = create_labels(train, step_size)
testX, testY = create_labels(test, step_size)
# Reshape data for RNN input
trainX = np.reshape(trainX, (trainX.shape[0], 1, trainX.shape[1]))
testX = np.reshape(testX, (testX.shape[0], 1, testX.shape[1]))
# Convert data to PyTorch tensors
trainX_tens = torch.tensor(trainX, dtype=torch.float32)
trainY_tens = torch.tensor(trainY, dtype=torch.float32)
testX_tens = torch.tensor(testX, dtype=torch.float32)
testY_tens = torch.tensor(testY, dtype=torch.float32)
# Create DataLoader for training
train_dataset = torch.utils.data.TensorDataset(trainX_tens, trainY_tens)
train_loader = torch.utils.data.DataLoader(train_dataset, batch_size=32)
Model definition and training
We define a simple Recurrent Neural Network (RNN) model using PyTorch's nn.Module class. In the init method, we initialize the input, hidden, and output sizes of the RNN model. The nn.RNN() method
constructs the RNN layer with the specified input and hidden sizes, where batch_first=True indicates that input and output tensors have the shape (batch_size, sequence_length, input_size).
Additionally, we define a fully connected linear layer using the nn.Linear() method, which maps the hidden state output of the RNN to the desired output size.
In the forward method, we implement the forward pass through the RNN layer, generating an output tensor 'out'. Then, we apply the fully connected layer to the last time step's output of the RNN (out
[:, -1, :]), producing the final output of the model.
# Define RNN model
class SimpleRNN(nn.Module):
def __init__(self, input_size, hidden_size, output_size):
super(SimpleRNN, self).__init__()
self.rnn = nn.RNN(input_size, hidden_size, batch_first=True)
self.fc = nn.Linear(hidden_size, output_size)
def forward(self, x):
out, _ = self.rnn(x)
out = self.fc(out[:, -1, :]) # Take the last time step's output
return out
We define hyperparameters for our model and initialize the model using SimpleRNN class. We use MSELoss() as a loss function and Adam optimizer.
# Hyperparameters
input_size = step_size
hidden_size = 128
output_size = 1
epochs = 100
learning_rate = 0.0001
# Instantiate RNN model
model = SimpleRNN(input_size=input_size, hidden_size=hidden_size, output_size=output_size)
# Define loss function and optimizer
criterion = nn.MSELoss()
optimizer = torch.optim.Adam(model.parameters(), lr=learning_rate)
Next, we train model by iterating over the number of epochs and print the loss in every 10 epochs.
# Train the model
for epoch in range(epochs):
for batch_X, batch_Y in train_loader:
optimizer.zero_grad() # Clears the gradients of all optimized parameters.
output = model(batch_X)
# Computes the loss between the model predictions and the ground
# truth labels for the current mini-batch.
loss = criterion(output, batch_Y)
# Computes gradients of the loss with respect to model parameters.
# Updates model parameters based on the computed gradients using
# the specified optimization algorithm.
if (epoch+1) % 10 == 0:
print(f'Epoch [{epoch+1}/{epochs}], Loss: {loss.item():.4f}')
Epoch [10/100], Loss: 0.4343
Epoch [20/100], Loss: 0.3793
Epoch [30/100], Loss: 0.3902
Epoch [40/100], Loss: 0.3918
Epoch [50/100], Loss: 0.3930
Epoch [60/100], Loss: 0.3941
Epoch [70/100], Loss: 0.3951
Epoch [80/100], Loss: 0.3959
Epoch [90/100], Loss: 0.3966
Epoch [100/100], Loss: 0.3971
We predict test data by using trained model and visualize it in a graph.
# Evaluation
with torch.no_grad():
testPredict = model(testX_tens)
# Plot results
index = range(len(testY))
plt.plot(index, testY, label="Ground truth")
plt.plot(index, testPredict.numpy(), label="Predicted")
In this tutorial, we learned about RNNs and how to implement simple RNN model with sequential data in PyTorch. Overview of RNNs, data preparation, defining RNN model architecture, and model training
and prediction of test data are explained in this tutorial. I hope this tutorial will help you to understand RNNs and their application in sequential data. | {"url":"https://www.datatechnotes.com/2024/04/introduction-to-recurrent-neural.html","timestamp":"2024-11-13T07:39:55Z","content_type":"application/xhtml+xml","content_length":"71849","record_id":"<urn:uuid:3fa53c23-86ab-4291-9721-d124fb5a9692>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00154.warc.gz"} |
A store offers a 20% discount on all products, and a second 10% discount for cash payments. What is the final price of a product that cost R$100.00 and was bought in cash?
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Curabitur id consequat justo. Cras pellentesque urna ante, eget gravida quam pretium ut. Praesent aliquam nibh faucibus ligula placerat, eget
pulvinar velit gravida. Nam sollicitudin pretium elit a feugiat. Vestibulum pharetra, sem quis tempor volutpat, magna diam tincidunt enim, in ullamcorper tellus nibh vitae turpis. In egestas
convallis ultrices. Sed non sem ultricies, sollicitudin sem facilisis, commodo justo. Maecenas mattis sodales nunc, et auctor massa fermentum a. Maecenas eget nisl felis.
Lorem ipsum dolor sit amet, consectetur adipiscing elit. Curabitur id consequat justo. Cras pellentesque urna ante, eget gravida quam pretium ut. Praesent aliquam nibh faucibus ligula placerat, eget
pulvinar velit gravida. Nam sollicitudin pretium elit a feugiat. Vestibulum pharetra, sem quis tempor volutpat, magna diam tincidunt enim, in ullamcorper tellus nibh vitae turpis. In egestas
convallis ultrices.
You need to be a registered teacher to view the answer sheet | {"url":"https://www.teachy.app/en/question/middle-school/mathematics-en/solve-problems-involving-the-calculation-of-percentages-in-sequence-that-is-calculating-2-or-more-percentages-of-the-same-number-as-successive-discounts/a-store-offers-a-20percent-discount-on-all-products-and-a-second-10percent-discount-for-cash-payment-00105","timestamp":"2024-11-09T18:06:02Z","content_type":"text/html","content_length":"277497","record_id":"<urn:uuid:191a7ad0-c441-4886-8dc5-002f66eced3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00135.warc.gz"} |
Matchstick Puzzle
Rearrange one matchstick to resolve the equation in 20s! Exercise problem-solving, hone your mental acuity and utilize your creativity. Apply logic, analyze, and decide the best resolution. With the
right method, this enigma can be solved in time. Good luck!
Can you change the equation 6+9=77 by transferring one component?
Rearrange one matchstick to crack the equation in 20s! Test problem-solving, sharpen your thought process and use your creativity. Utilize logic, observe, and select the best outcome. With the right
approach, this challenge can be solved in time. Best of luck!
Could you correct the equation 3-1=1 by relocating a single part?
Rearrange one matchstick to solve the equation in 20s! Exercise problem-solving, boost your mental agility and access your creativity. Utilize logic, inspect, and get the best solution. With the
correct strategy, this puzzle can be solved in time. Break a leg!
Is there a way to amend the equation 5+0=11 by shifting one element?
Rearrange one matchstick to settle the equation in 20s! Exercise problem-solving, hone your intellect and employ your creativity. Deploy logic, examine, and find the best result. With the right
approach, this conundrum can be solved in time. Good luck!
Is it possible to rectify the equation 9-1=70 by transferring one item?
Rearrange one matchstick to fix the equation in 20s! Exercise problem-solving, strengthen your mind and unleash your creativity. Employ logic, assess, and choose the best answer. With the correct
strategy, this dilemma can be solved in time. All the best!
Can you fix the equation 4+8=13 by relocating one component?
Rearrange one matchstick to unravel the equation in 20s! Stimulate problem-solving, amplify your mental sharpness and access your creativity. Utilize logic, look into, and discover the best solution.
With the right method, this brainteaser can be solved in time. Good luck!
Could you change the equation 1+8=7 by moving a single part?
Rearrange one matchstick to solve the equation in 20s! Exercise problem-solving, strengthen your mental aptitude and use your creativity. Apply logic, investigate, and determine the best outcome.
With the correct approach, this puzzle can be solved in time. All the best!
Is there a way to adjust the equation 8-0=2 by shifting one piece?
Rearrange one matchstick to crack the equation in 20s! Stimulate problem-solving, increase your brainpower and bring out your creativity. Utilize logic, discern, and locate the best result. With the
right strategy, this enigma can be solved in time. Best of luck!
Is it possible to resolve the equation 7+5=10 by transferring a single element?
Rearrange one matchstick to resolve the equation in 20s! Exercise problem-solving, sharpen your mental faculties and access your creativity. Deploy logic, analyze, and pick the best answer. With the
correct approach, this challenge can be solved in time. Break a leg!
Can you correct the equation 7+2=10 by moving one item?
Rearrange a single matchstick to crack the equation in 20s! Stimulate problem-solving skills, hone your mind and tap into your creativity. Employ logic, pay attention, and find the best answer. With
the correct strategy, this conundrum can be deciphered within the time limit. All the best! | {"url":"http://quiz.3o9.in/puzzle/matchstick?page=1","timestamp":"2024-11-03T07:02:21Z","content_type":"text/html","content_length":"15163","record_id":"<urn:uuid:935aecd9-7667-4f2a-829d-859639f392f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00891.warc.gz"} |
Pairs and lists - klisp
7 Pairs and lists
A pair is an object that refers to two other objects, called its car and cdr. The Kernel data type pair is encapsulated.
The null data type consists of a single immutable value, called nil or the empty list and having external representation (), with or without whitespace between the parentheses. It is immutable, and
the null type is encapsulated.
If a and d are external representations of respectively the car and cdr of a pair p, then (a . d) is an external representation of p. If the cdr of p is nil, then (a) is also an external
representation of p. If the cdr of p is a pair p2, and (r) is an external representation of p2, then (a r) is an external representation of p. When a pair is output (as by write), an external
representation with the fewest parentheses is used; in the case of a finite list, only one set of parentheses is required beyond those used in representing the elements of the list. For example, an
object with external representation (1 . (2 . (3 . ()))) would be output using, modulo whitespace, external representation (1 2 3).
— Applicative:
pair? . objects
The primitive type predicate for type pair. pair? returns true iff all the objects in objects are of type pair.
— Applicative:
null? . objects
The primitive type predicate for type null. null? returns true iff all the objects in objects are of type null.
— Applicative:
immutable-pair? objects
— Applicative:
mutable-pair? objects
The primitive type predicates for types immutable pair and mutable pair. These return true iff all the objects in objects are of type immutable pair or mutable pair respectively.
SOURCE NOTE: these aren’t provided in the Kernel report, but added for convenience. These can be implemented in standard kernel by using guards.
— Applicative:
cons object1 object2
A new mutable pair object is constructed and returned, whose car and cdr referents are respectively object1 and object2. No two objects returned by different calls to cons are eq? to each other.
— Applicative:
set-car! pair object
— Applicative:
set-cdr! pair object
pair should be a mutable pair.
These applicatives set the referent of, respectively, the car reference or the cdr reference of pair to object. The result of the expression is inert.
— Applicative:
copy-es-immutable object
The short description of this applicative is that it returns an object equal? to object with an immutable evaluation structure. The “-es-” in the name is short for “evaluation structure”.
The evaluation structure of an object o is defined to be the set of all pairs that can be reached by following chains of references from o without ever passing through a non-pair object. The
evaluation structure of a non-pair object is empty.
If object is not a pair, the applicative returns object. Otherwise (if object is a pair), the applicative returns an immutable pair whose car and cdr would be suitable results for
(copy-es-immutable (car object)) and (copy-es-immutable (cdr object)), respectively. Further, the evaluation structure of the returned value is isomorphic to that of object at the time of
copying, with corresponding non-pair referents being eq?.
NOTE: In Kernel it’s undefined whether immutable pairs are copied or left “as is” in the result. klisp doesn’t copy immutable pairs, but that behaviour should not be depended upon.
— Applicative:
list . objects
The list applicative returns objects.
The underlying operative of list returns its undifferentiated operand tree, regardless of whether that tree is or is not a list.
— Applicative:
list* . objects
objects should be a finite nonempty list of arguments.
The following equivalences hold:
(list* arg1) == arg1
(list* arg1 arg2 . args) == (cons arg1 (list* arg2 . args))
— Applicative:
car pair
— Applicative:
cdr pair
These applicatives return, respectively, the car and cdr of pair.
— Applicative:
caar pair
— Applicative:
cadr pair
— Applicative:
cdar pair
— Applicative:
cddr pair
— Applicative:
caaar pair
— Applicative:
caadr pair
— Applicative:
cadar pair
— Applicative:
caddr pair
— Applicative:
cdaar pair
— Applicative:
cdadr pair
— Applicative:
cddar pair
— Applicative:
cdddr pair
— Applicative:
caaaar pair
— Applicative:
caaadr pair
— Applicative:
caadar pair
— Applicative:
caaddr pair
— Applicative:
cadaar pair
— Applicative:
cadadr pair
— Applicative:
caddar pair
— Applicative:
cadddr pair
— Applicative:
cdaaar pair
— Applicative:
cdaadr pair
— Applicative:
cdadar pair
— Applicative:
cdaddr pair
— Applicative:
cddaar pair
— Applicative:
cddadr pair
— Applicative:
cdddar pair
— Applicative:
cddddr pair
These applicatives are compositions of car and cdr, with the “a’s” and “d’s” in the same order as they would appear if all the individual “car’s” and “cdr’s” were written out in prefix
order. Arbitrary compositions up to four deep are provided. There are twenty-eight of these applicatives in all.
— Applicative:
make-list length
length shoulde be an exact non-negative integer.
Applicative make-list creates a new mutable acyclic list of length length, with all pairs having fill in their cars. If no value is provided for fill, #inert is used.
SOURCE NOTE: this is taken from r7rs.
— Applicative:
list-copy list
Applicative list-copy creates a new mutable copy of list. That is, the returned list has the same list metrics as list and the cars in the returned list are initially eq? to the corresponding
cars in list.
SOURCE NOTE: this is taken from r7rs.
— Applicative:
reverse list
list should be an acyclic list.
Applicative reverse makes a mutable copy of list but with the reverse order. That is, the returned list has the same number of pairs as list and the cars in the returned list are initially eq? to
the corresponding cars in list but starting from the end and going backwards.
SOURCE NOTE: this is taken from r7rs.
— Applicative:
get-list-metrics object
By definition, an improper list is a data structure whose objects are its start together with all objects reachable from the start by following the cdr references of pairs, and whose internal
references are just the cdr references of its pairs. Every object, of whatever type, is the start of an improper list. If the start is not a pair, the improper list consists of just that object.
The acyclic prefix length of an improper list L is the number of pairs of L that a naive traversal of L would visit only once. The cycle length of L is the number of pairs of L that a naive
traversal would visit repeatedly. Two improper lists are structurally isomorphic iff they have the same acyclic prefix length and cycle length and, if they are terminated by non-pair objects
rather than by cycles, the non-pair objects have the same type. Applicative get-list-metrics constructs and returns a list of exact integers of the form (p n a c), where p, n, a, and c are,
respectively, the number of pairs in, the number of nil objects in, the acyclic prefix length of, and the cycle length of, the improper list starting with object. n is either 0 or 1, a + c = p,
and n and c cannot both be non-zero. If c = 0, the improper list is acyclic; if n = 1, the improper list is a finite list; if n = c = 0, the improper list is not a list; if a = c = 0, object is
not a pair.
— Applicative:
list-tail object k
object must be the start of an improper list containing at least k pairs.
The list-tail applicative follows k cdr references starting from object.
The following equivalences hold:
(list-tail object 0) == object
(list-tail object (+ k 1)) == (list-tail (cdr object) k)
— Applicative:
encycle! object k1 k2
The improper list starting at object must contain at least k1 + k2 pairs.
If k2 = 0, the applicative does nothing. If k2 > 0, the applicative mutates the improper list starting at object to have acyclic prefix length k1 and cycle length k2, by setting the cdr of the
(k1+k2)th pair in the list to refer to the (k1 + 1)th pair in the list. The result returned by encycle! is inert.
— Applicative:
map applicative . lists
lists must be a nonempty list of lists; if there are two or more, they must all have the same length.
The map applicative applies applicative element-wise to the elements of the lists in lists (i.e., applies it to a list of the first elements of the lists, to a list of the second elements of the
lists, etc.), using the dynamic environment from which map was called, and returns a list of the results, in order. The applications may be performed in any order, as long as their results occur
in the resultant list in the order of their arguments in the original lists. If lists is a cyclic list, each argument list to which applicative is applied is structurally isomorphic to lists. If
any of the elements of lists is a cyclic list, they all must be, or they wouldn’t all have the same length. Let a1...an be their acyclic prefix lengths, and c1...cn be their cycle lengths. The
acyclic prefix length a of the resultant list will be the maximum of the ak, while the cycle length c of the resultant list will be the least common multiple of the ck. In the construction of the
result, applicative is called exactly a + c times.
— Applicative:
length object
Applicative length returns the (exact) improper-list length of object. That is, it returns the number of consecutive cdr references that can be followed starting from object. If object is not a
pair, it returns zero; if object is a cyclic list, it returns positive infinity.
— Applicative:
list-ref object k
The list-ref applicative returns the car of the object obtained by following k cdr references starting from object.
NOTE: In the current report, object is required to be a list. In klisp, for now, we prefer the behaviour presented here, as it is more in line with the applicative list-tail. That is, we
define list-ref by the following equivalence:
(list-ref object k) == (car (list-tail object k))
— Applicative:
append . lists
Here, all the elements of lists except the last element (if any) must be acyclic lists. The append applicative returns a freshly allocated list of the elements of all the specified lists, in
order, except that if there is a last specified element of lists, it is not copied, but is simply referenced by the cdr of the preceding pair (if any) in the resultant list. If lists is cyclic,
the cycle of the result list consists of just the elements of the lists specified in the cycle in lists. In this case, the acyclic prefix length of the result is the sum of the lengths of the
lists specified in the acyclic prefix of lists, and the cycle length of the result is the sum of the lengths of the lists specified in the cycle of lists.
The following equivalences hold:
(append) == ()
(append h) == h
(append () h . t) == (append h . t)
(append (cons a b) h . t) == (cons a (append b h . t))
— Applicative:
list-neighbors list
The list-neighbors applicative constructs and returns a list of all the consecutive sublists of list of length 2, in order. If list is nil, the result is nil. If list is non-nil, the length of
the result is one less than the length of list. If list is cyclic, the result is structurally isomorphic to it (i.e., has the same acyclic prefix length and cycle length).
For example:
(list-neighbors (list 1 2 3 4)) ⇒ ((1 2) (2 3) (3 4))
— Applicative:
filter applicative list
Applicative filter passes each of the elements of list as an argument to applicative, one at a time in no particular order, using a fresh empty environment for each call. The result of each call
to applicative must be boolean, otherwise an error is signaled. filter constructs and returns a list of all elements of list on which applicative returned true, in the same order as
in list. applicative is called exactly as many times as there are pairs in list. The resultant list has a cycle containing exactly those elements accepted by applicative that were in the cycle
of list; if there were no such elements, the result is acyclic.
— Applicative:
assoc object pairs
Applicative assoc returns the first element of pairs whose car is eq-pred? to object. If there is no such element in pairs, nil is returned. If eq-pred? is not supplied it defaults
to equal?. SOURCE NOTE: the optional eq-pred? argument is from r7rs.
— Applicative:
member? object list
Applicative member? is a predicate that returns true iff some element of list is eq-pred? to object. If eq-pred? is not supplied, it defaults to equal?. SOURCE NOTE: the optional eq-pred?
argument is from r7rs.
— Applicative:
finite-list? . objects
This is the type predicate for type finite-list. finite-list? returns true iff all the objects in objects are acyclic lists.
— Applicative:
countable-list? . objects
This is the type predicate for type list. countable-list? returns true iff all the objects in objects are lists.
— Applicative:
reduce list binary identity
precycle incycle postcycle
binary should be an applicative. If the short form is used, list should be an acyclic. If the long form is used, precycle, incycle, and postcycle should be applicatives.
If list is empty, applicative reduce returns identity. If list is nonempty but acyclic, applicative reduce uses binary operation binary to merge all the elements of list into a single object,
using any associative grouping of the elements. That is, the sequence of objects initially found in list is repeatedly decremented in length by applying binary to a list of any two consecutive
objects, replacing those two objects with the result at the point in the sequence where they occurred; and when the sequence contains only one object, that object is returned. If list is cyclic,
the long form must be used. The elements of the cycle are passed, one at a time (but just once for each position in the cycle), as arguments to unary applicative precycle; the finite, cyclic
sequence of results from precycle is reduced using binary applicative incycle; and the result from reducing the cycle is passed as an argument to unary applicative postcycle. Binary
operation binary is used to reduce the sequence consisting of the elements of the acyclic prefix of list followed by the result returned by postcycle. The only constraint on the order of calls to
the applicatives is that each call must be made before its result is needed (thus, parts of the reduction of the acyclic prefix may occur before the contribution from the cycle has been
Each call to binary, precycle, incycle, or postcycle uses the dynamic environment of the call to reduce.
If list is acyclic with length n >= 1, binary is called n - 1 times. If list is cyclic with acyclic prefix length a and cycle length c, binary is called a times; precycle, c times; incycle, c -
1 times; and postcycle, once.
— Applicative:
append! . lists
lists must be a nonempty list; its first element must be an acyclic nonempty list, and all of its elements except the last element (if any) must be acyclic lists.
The append! applicative sets the cdr of the last pair in each nonempty list argument to refer to the next non-nil argument, except that if there is a last non-nil argument, it isn’t mutated. It
is an error for any two of the list arguments to have the same last pair. The result returned by this applicative is inert.
The following equivalences hold:
(append! v) == #inert
(append! u v . w) == ($sequence (append! u v) (append! u . w))
— Applicative:
copy-es object
Briefly, applicative copy-es returns an object initially equal? to object with a freshly constructed evaluation structure made up of mutable pairs. If object is not a pair, the applicative
returns object. If object is a pair, the applicative returns a freshly constructed pair whose car and cdr would be suitable results for (copy-es (car object)) and (copy-es (cdr object)),
respectively. Further, the evaluation structure of the returned value is structurally isomorphic to that of object at the time of copying, with corresponding non-pair referents being eq?.
— Applicative:
assq object pairs
Applicative assq returns the first element of pairs whose car is eq? to object. If there is no such element in pairs, nil is returned. | {"url":"https://klisp.org/pairs-and-lists/","timestamp":"2024-11-09T06:07:39Z","content_type":"text/html","content_length":"51911","record_id":"<urn:uuid:1e98b04e-f730-4937-bf5f-f0019953523b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00524.warc.gz"} |
How effective use percent decrease calculator? - WorkSheets Buddy
How effective use percent decrease calculator?
How effective use percent decrease calculator?
Using a percent decrease calculator can be an effective way to quickly and accurately calculate the decrease in a value.
Here are the steps to effectively use a percent decrease calculator:
• Input the original value into the calculator. This is the value that you will be decreasing.
• Input the percent decrease into the calculator. This is the percentage by which you will be decreasing the original value.
• Click the “Calculate” button to calculate the new value. The calculator will automatically apply the percent decrease to the original value and display the new value.
• Use the new value in any calculations or analysis that you need to perform.
By following these steps, you can effectively use a percent decrease calculator to calculate the decrease in a value. This can be a useful tool for a variety of calculations, including financial
analysis, budgeting, and more.
More Answers:
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zpteqr.f - Linux Manuals (3)
zpteqr.f (3) - Linux Manuals
zpteqr.f -
subroutine zpteqr (COMPZ, N, D, E, Z, LDZ, WORK, INFO)
Function/Subroutine Documentation
subroutine zpteqr (characterCOMPZ, integerN, double precision, dimension( * )D, double precision, dimension( * )E, complex*16, dimension( ldz, * )Z, integerLDZ, double precision, dimension( * )WORK,
ZPTEQR computes all eigenvalues and, optionally, eigenvectors of a
symmetric positive definite tridiagonal matrix by first factoring the
matrix using DPTTRF and then calling ZBDSQR to compute the singular
values of the bidiagonal factor.
This routine computes the eigenvalues of the positive definite
tridiagonal matrix to high relative accuracy. This means that if the
eigenvalues range over many orders of magnitude in size, then the
small eigenvalues and corresponding eigenvectors will be computed
more accurately than, for example, with the standard QR method.
The eigenvectors of a full or band positive definite Hermitian matrix
can also be found if ZHETRD, ZHPTRD, or ZHBTRD has been used to
reduce this matrix to tridiagonal form. (The reduction to
tridiagonal form, however, may preclude the possibility of obtaining
high relative accuracy in the small eigenvalues of the original
matrix, if these eigenvalues range over many orders of magnitude.)
COMPZ is CHARACTER*1
= 'N': Compute eigenvalues only.
= 'V': Compute eigenvectors of original Hermitian
matrix also. Array Z contains the unitary matrix
used to reduce the original matrix to tridiagonal
= 'I': Compute eigenvectors of tridiagonal matrix also.
N is INTEGER
The order of the matrix. N >= 0.
D is DOUBLE PRECISION array, dimension (N)
On entry, the n diagonal elements of the tridiagonal matrix.
On normal exit, D contains the eigenvalues, in descending
E is DOUBLE PRECISION array, dimension (N-1)
On entry, the (n-1) subdiagonal elements of the tridiagonal
On exit, E has been destroyed.
Z is COMPLEX*16 array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the unitary matrix used in the
reduction to tridiagonal form.
On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
original Hermitian matrix;
if COMPZ = 'I', the orthonormal eigenvectors of the
tridiagonal matrix.
If INFO > 0 on exit, Z contains the eigenvectors associated
with only the stored eigenvalues.
If COMPZ = 'N', then Z is not referenced.
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
COMPZ = 'V' or 'I', LDZ >= max(1,N).
WORK is DOUBLE PRECISION array, dimension (4*N)
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, and i is:
<= N the Cholesky factorization of the matrix could
not be performed because the i-th principal minor
was not positive definite.
> N the SVD algorithm failed to converge;
if INFO = N+i, i off-diagonal elements of the
bidiagonal factor did not converge to zero.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Definition at line 146 of file zpteqr.f.
Generated automatically by Doxygen for LAPACK from the source code. | {"url":"https://www.systutorials.com/docs/linux/man/3-zpteqr.f/","timestamp":"2024-11-05T13:56:28Z","content_type":"text/html","content_length":"10391","record_id":"<urn:uuid:0acd37a0-08c6-4776-8921-e774f0d23729>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00378.warc.gz"} |
Linear Algebra tutorial
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Linear Algebra tutorial
In this simple Linear Algebra tutorial, you will review the basic operations on vectors, matrices, solving linear equations and eigen systems. I made free educational interactive programs online for
almost every topic. The programs are well suited to guide discovery experiences where you can focus on the concept of linear algebra and let the program compute the more complex calculation. My focus
is to introduce linear algebra with clarity of explanation. Thus, I try to avoid mathematical detail and you should read the proof of all the properties and theorems from the references. Manual
examples on how to compute are given. Random examples of the interactive programs are useful to give you theoretically infinitely many examples. No download file is necessary for this tutorial and
you must set JavaScript enabled in your browser.
First, take a look on how to input a vector or a matrix in the interactive program , and then you can select the topics below that you are not familiar or you want to review.
If you are a first time learners, who are not familiar with most of the topics, you should browse each section & each topic of the tutorial in sequence. You should also try the random example of the
interactive programs to gain your understanding of the concept. You may want to skip the properties and notes below the interactive program at first reading. After you are familiar with the name,
definition and operations, you will find that the notes and properties listed are useful for second reading or whenever you need to come back to the particular topics.
The strength of linear algebra lies on the properties of each operation. I try to collect these properties on the same page of the context so that it is easier for you to find them back. Thus, you
will find some of the properties are repeated in different pages for the appropriate context.
Here are the topics of this linear algebra tutorial:
Matrix & Vector Size & Validation ( interactive program )
Vector Algebra
What is Vector? ( interactive program )
Vector Norm ( interactive program )
Unit Vector ( interactive program )
Vector Addition ( interactive program )
Vector Subtraction ( interactive program )
Vector Scalar Multiple ( interactive program )
Vector Multiplication
Vector Inner Product ( interactive program )
Vector Outer Product ( interactive program )
Vector Cross Product ( interactive program )
Vector Triple Cross Product ( interactive program )
Vector Triple Dot Product ( interactive program )
Scalar Triple Product ( interactive program )
Orthogonal & Orthonormal Vector ( interactive program )
Cos Angle of Vectors ( interactive program )
Vector Projection ( interactive program )
Matrix Algebra
What is a matrix and why do we need matrix?
Special Matrices
Matrix One ( interactive program )
Null Matrix ( interactive program )
Matrix Diagonal ( interactive program )
Is Diagonal Matrix ? ( interactive program )
Identity Matrix ( interactive program )
Matrix Special Properties
Matrix Determinant ( interactive program )
Matrix Sum ( interactive program )
Matrix Trace ( interactive program )
Matrix Basic Operation
Is Equal Matrix? ( interactive program )
Matrix Transpose ( interactive program )
Matrix Addition ( interactive program )
Matrix Subtraction ( interactive program )
Matrix Multiplication ( interactive program )
Matrix Scalar Multiple ( interactive program )
Matrix Element-wise product (Hadamard Product) ( interactive program )
Matrix Horizontal Concatenation ( interactive program )
Matrix Vertical Concatenation ( interactive program )
Elementary Row Operations
Matrix RREF ( interactive program )
Finding inverse using RREF (Gauss-Jordan) ( interactive program )
Finding Matrix Rank using RREF ( interactive program )
Matrix Inverse ( interactive program )
Is Singular Matrix? ( interactive program )
Linear Transformation
Matrix Generalized Inverse ( Moore Penrose interactive program )
Solving System of Linear Equations ( interactive program )
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent ( interactive program )
Change of basis
Matrix Rank ( interactive program )
Matrix Range ( interactive program )
Matrix Nullity & Null Space ( interactive program )
Eigen System
Matrix Eigen Value & Eigen Vector ( interactive program )
Symmetric Matrix ( interactive program )
Matrix Eigen Value & Eigen Vector for Symmetric Matrix ( interactive program )
Similarity Transformation and Matrix Diagonalization
Matrix Power
Orthogonal Matrix ( interactive program )
Spectral Decomposition ( interactive program )
Singular Value Decomposition ( interactive program )
Resources on Linear Algebra
See Also : Regression , Distance , Similarity , Sum
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Identifying a Correct Statement Using Geometric Properties
Question Video: Identifying a Correct Statement Using Geometric Properties Mathematics • First Year of Preparatory School
If π Ή is a point on the line π ΄π ΅ and π Έ is a point on the line π Άπ ·, the line π ΄π ΅ β ₯ the line π Έπ Ή, and the line π Άπ · β ₯ the line π Έπ Ή, which of the following is
correct? [A] The line π ΄π ΅ is parallel to the line π Άπ ·. [B] Line segment π Έπ Ή bisects line segment π ΄π ΅. [C] The line π ΄π ΅ is parallel to the line π Έπ Ή. [D] The line π ΄π ΅
is perpendicular to the line π Άπ ·. [E] Line segment π Έπ Ή bisects line segment π Άπ ·.
Video Transcript
If π Ή is a point on the line between π ΄ and π ΅ and π Έ is a point on the line between π Ά and π ·, the line between π ΄ and π ΅ is perpendicular to the line between π Έ and π Ή, and the
line between π Ά and π · is perpendicular to the line between π Έ and π Ή, which of the following is correct? Option (A) the line between π ΄ and π ΅ is parallel to the line between π Ά and π
·. Option (B) line segment π Έπ Ή bisects line segment π ΄π ΅. Option (C) the line between π ΄ and π ΅ is parallel to the line between π Έ and π Ή. Option (D) the line between π ΄ and π ΅
is perpendicular to the line between π Ά and π ·. Or is it option (E) line segment π Έπ Ή bisects line segment π Άπ ·?
In this question, we are given some information about some lines and asked to determine which of five given options must be true. To do this, letβ s start by sketching the given information, that
is, that the line between π ΄ and π ΅ is perpendicular to the line between π Έ and π Ή and that the line between π Ά and π · is also perpendicular to this line, where π Έ and π Ή lie on the
lines as shown. In our sketch, we see that the line between π Έ and π Ή is a transversal of the other two lines. And we see that the corresponding angles of this transversal both have equal measure
since they are both right angles.
This is enough to confirm that the line between π ΄ and π ΅ and the line between π Ά and π · must be parallel since the corresponding angles of the transversal have equal measure. Alternatively,
we can use the specific result that any two lines perpendicular to the same line are parallel. Hence, the answer is option (A): the line between π ΄ and π ΅ is parallel to the line between π Ά and
π ·.
It is worth noting that we can show that none of the other options must be true using our sketch. First, we can choose points π Έ and π Ή anywhere on the lines. So, we can choose point π Ή to not
be the midpoint of line segment π ΄π ΅. Therefore, option (B) is not true in general. The same reasoning works to show that option (E) is not true in general, since we can choose point π Έ to not
be the midpoint of the line segment π Άπ ·. We are told that the line segment between π ΄ and π ΅ is perpendicular to the line between π Έ and π Ή, so they are not parallel. Therefore, option
(C) is not true.
Finally, since we have shown that the line between π ΄ and π ΅ is parallel to the line between π Ά and π ·, we can conclude that they do not intersect and so they are not perpendicular. Hence,
the only statement that is true in general is option (A). | {"url":"https://www.nagwa.com/en/videos/932156781869/","timestamp":"2024-11-13T22:33:14Z","content_type":"text/html","content_length":"254346","record_id":"<urn:uuid:6276241a-3a29-42f6-aa8f-162277e10bbc>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00248.warc.gz"} |
Modeling Synchronization Risk among Sustainable Exchange Trade Funds: A Statistical and Network Analysis Approach
We evaluate the environment, society, and corporate governance rating (ESG rating) contribution from a new perspective; the highest ESG rating mitigates the impact of unexpected change in the implied
volatility on the systemic stock market risk. For this purpose, we use exchange-traded funds (ETF) classified by their ESG rating into quartiles to estimate the synchronization as a proxy by systemic
risk. Then, for each ETF quartile, we study the effect of the implied volatility over the synchronization. Our study is the first to model sustainable ETFs’ synchronization by combining econometric
modeling and network methods, including 100 ETFs representing 80% of the global ETF market size between 2013 and 2021. First, we find that a higher ESG rating mitigates the effect of implied
volatility over ETF synchronization. Surprisingly, the effect is the opposite in the case of ETFs with lower ESG ratings, where an increase in the volatility expectation increases the
synchronization. Our study depicts the effect of sustainable ETFs on lessening the systemic risk due to returns synchronization, this being a novel contribution of this asset class. Finally, this
paper offers extensions to deepen the contribution of other asset classes of ETFs in terms of their synchronization behavior and impact on risk management and financial performance.
Profundice en los temas de investigación de 'Modeling Synchronization Risk among Sustainable Exchange Trade Funds: A Statistical and Network Analysis Approach'. En conjunto forman una huella única. | {"url":"https://pure.uai.cl/es/publications/modeling-synchronization-risk-among-sustainable-exchange-trade-fu","timestamp":"2024-11-11T07:22:48Z","content_type":"text/html","content_length":"54296","record_id":"<urn:uuid:204dfc8d-6fe6-43ff-9e4f-c7b4b44a538b>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00190.warc.gz"} |
In how many ways can the letters in the word spoon be arranged?243060120
In how many ways can the letters in the word spoon be arranged?243060120
Answer: C) 60Explanation:There are 5 letters, so there would be 5! = 5*4*3*2*1 = 120 different permutations; however, there are 2 letter 'o's meaning we have double counted. To correct this, divide
by 2 to get 120/2 = 60.If there was a way to tell the two 'o's apart, then the answer would be 120.
statistics 10 months ago 4422 | {"url":"http://redmondmathblog.com/statistics/in-how-many-ways-can-the-letters-in-the-word-spoon-be-arranged-243060120","timestamp":"2024-11-03T19:18:30Z","content_type":"text/html","content_length":"24679","record_id":"<urn:uuid:fd5093e3-dbe6-4cdf-8b81-83f64c73bafa>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00797.warc.gz"} |
[Resource Topic] 2024/1652: How to Construct Random Unitaries
Welcome to the resource topic for 2024/1652
How to Construct Random Unitaries
Authors: Fermi Ma, Hsin-Yuan Huang
The existence of pseudorandom unitaries (PRUs)—efficient quantum circuits that are computationally indistinguishable from Haar-random unitaries—has been a central open question, with significant
implications for cryptography, complexity theory, and fundamental physics. In this work, we close this question by proving that PRUs exist, assuming that any quantum-secure one-way function exists.
We establish this result for both (1) the standard notion of PRUs, which are secure against any efficient adversary that makes queries to the unitary U, and (2) a stronger notion of PRUs, which are
secure even against adversaries that can query both the unitary U and its inverse U^\dagger. In the process, we prove that any algorithm that makes queries to a Haar-random unitary can be efficiently
simulated on a quantum computer, up to inverse-exponential trace distance.
ePrint: https://eprint.iacr.org/2024/1652
See all topics related to this paper.
Feel free to post resources that are related to this paper below.
Example resources include: implementations, explanation materials, talks, slides, links to previous discussions on other websites.
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Total Approximate Voltage Drop Of A Transformer
During no-load condition, induced voltages at the primary and secondary windings are equal to the applied voltage and secondary terminal voltage respectively. If [0]V[2] be the secondary terminal
voltage at no load, we can write E[2] = [0]V[2]. Let V[2] be the secondary voltage on load. Figure 1.40 shows the phasor diagram of a transformer referred to as secondary.
In Figure 1.40, R[02] and X[02] are the equivalent resistance and reactance of the transformer, respec-tively, referred to as secondary side. With O as centre, an arc is drawn in Figure 1.40, which
intersects the extended OA at H. From C, a perpendicular is drawn on OH, which intersects it at G. Now AC = AHrepresents the actual drop and AG represents the approximate voltage drop. BF is drawn
perpendicular to OH. BE is drawn parallel to AG, which is equal to FG.
The approximate voltage drop
= AG = AF+ FG = AF+ BE
= I[2]R[02]cosθ+I[2]X[02]sinθ (1.39)
Figure 1.40 Phasor Diagram of a Transformer Referred to as Secondary
This approximate voltage drop shown in Equation (1.39) is for lagging power factor only.
For leading power factor, the approximate voltage drop will be
= I[2]R[02]cosθ–I[2]X[02]sinθ (1.40)
where ‘+’ sign is for lagging power factor and ‘-’ sign is for leading power factor.
The above calculation is referred to as secondary. It may be noted that voltage drop referred to as primary is
I[1]R[01]cosθ±I[1]X[01]sinθ (1.41)
∴% voltage drop in secondary is=
=v[r]cosθ±v[x]sinθ (1.42) | {"url":"https://electricallive.com/2015/03/total-approximate-voltage-drop-of.html","timestamp":"2024-11-03T00:31:58Z","content_type":"text/html","content_length":"116526","record_id":"<urn:uuid:648b91bf-80f1-4516-8697-cd13711689e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00172.warc.gz"} |
Electricity Transmission & Distribution – Complete Answer -
Electricity Transmission & Distribution – Complete Answer
How this electricity is brought to our homes?
Electricity is produced using electricity generators. These generators convert mechanical power such as hydro power, steam power, etc. into electricity. These generators produce electricity at 11 kV
This 11kV level is stepped up to 100kV, 132kV, 220kV, 400kV, and 765kV using Power Transformers installed at Substations. Then this extra high voltage level electricity is transmitted over long
distances using long conductors supported by huge towers called as power transmission system.
Now, at consumer regions, this extra high voltage 100kV, 132kV, 220kV, 400kV, and 765kV is stepped down to 11kV, 22kV & 33kV using power transformers at substations. This stepped-down power is
brought near the consumer areas. At consumer ends this 11kV, 22kV, and 33kV level is again stepped down to 440 volts using distribution transformers.
And finally, this 440 volts three-phase power is distributed to homes in a single-phase format of which the voltage is 230 volts. This 230 volts of power is used in our homes.
In brief, electricity is generated at the 11kV level. Then it is stepped up to 100kV, 132kV, 220kV, 400kV, and 765kV levels, and then transmitted over long distances. At the consumer end electricity
is again stepped down to 11kV, 22kV, and 33kV levels. Further, it is stepped down to a three-phase 440 volts, which is 230 volts single-phase. And electricity is used by consumers at these voltage
Since we generate electricity at 11kV level. Why can’t we just transmit the electricity at this same 11kV voltage level over long distances? And then step it down to desired 440 volts level at the
consumer end. It would be relatively easy. Right?
Why do we first Step-up & then Step-down Electricity Voltage levels?
This step-up and step-down of electricity is done to save electric power loss. Let’s understand this with an example.
Let’s consider that electricity generated at the generation plant is to be transmitted & distributed to the consumers.
The length of this transmission line to be 1 km,
& its total resistance R = 1 ohm
Let’s assume the power factor pf = 1
Now, Case – I
Consider that 1 MW of electricity is generated at this generation plant at the 11kV voltage level in 1 second. And the same power is supplied to the consumers at the same 11kV voltage level without
stepping up the voltage.
We know the formula for electric power in Watts
P = √3 x VL x IL x pf
In this case,
Power P = 1 MW, & Voltage V = 11kV,
hence current I (flowing through transmission line) = 52.49 Amperes.
We know Joule’s equation of electrical heating due to resistance in a conductor
Power Loss = I² x R x t
In this case resistance R = 1 ohm.
And time t = 1 second.
Hence loss due to heating = 52.49² x 1 ohm x 1 seconds = 2755 Joules.
Now, Case II
Consider the same scenario with the same parameters as in the previous case. The only difference is that the voltage is stepped up to 220kV for electricity transmission.
Hence, in this case,
the current I (flowing through transmission line to transmit 1 MW power) = (1 MW) / (√3 x 220 kV) = 2.624 Amperes.
resistance R = 1 ohm.
And time t = 1 seconds.
Hence loss due to heating = 2.624² x 1 ohm x 1 seconds = 6.889 Jules.
After discussing both cases, we can see that the loss is reduced by 400 times, just by increasing the voltage level by 20 times. This way loss can be further reduced by approximately 1400 times by
stepping up the 11kV voltage level to 400kV level.
Simply put, as per the Jules equation of electrical heating, the loss is proportional to the square of the current. That means, the loss is inversely proportional to the square of the voltage. Hence
loss can be reduced proportionally to the square of the voltage step-up ratio.
In conclusion, we can say, by simply stepping up the voltage level for electricity transmission over long distances, huge electricity loss is reduced. This is the key reason to transmit electric
power in AC and not in DC because DC power voltage can not be stepped up or stepped down by power transformers. Because power transformer works on only AC power. Hence losses in DC transmission can
not be reduced like AC. Nowadays HVDC technology is used in which electricity voltage is stepped up by the power transformer in AC form and then AC is converted into DC. This DC power is transmitted
over long distances, and at the consumer end again it is converted back to AC form, and then its voltage is stepped down again. But this technology requires very high capital cost and maintenance
Leave a Comment | {"url":"https://electricaltech.in/electricity-transmission-distribution-complete-answer/","timestamp":"2024-11-07T06:26:00Z","content_type":"text/html","content_length":"159775","record_id":"<urn:uuid:d6618daa-a4bd-4371-84dc-8f83b92499b3>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00095.warc.gz"} |
Smart Sheet Summary for average time between two dates for two columns
Hi all,
I am having some issues working this one out and am hoping someone might be able to help!
I have two columns in my sheet that I am interested in, "Predicted UAT Start Date" and "Actual UAT Start Date". I then have many rows for each of our different projects. What I would like to know is
the average amount of time between the two dates for all of our projects (ie. the many rows). I would ideally like to do the formula in a Smart Sheet Summary field and without creating any additional
columns in my original sheet.
Is there a way to do this?
Many thanks,
• Hi @EmmaSob ,
Hope you are Good and Safe!
As per my understanding you are asking for the number of hours between dates so in that case, you can find out no of days i:e [Start date- end date] and then multiply by 24.
If I'm wrong means Could you please explain with Sample data.
Sandhiya P
• Hi @Sandhiya07
Thank you for your reply. No, this isn't quite what I'm after. I'm not wanting to create additional columns in the sheet to make the calculation, hence the desire to do the formula in a smart
sheet summary field. I also don't need to work out hours, just days.
Many thanks,
• Hi @EmmaSob
As per my Understanding, atleast we need to create atleast one column to calculate No of Days and then we will show Average for each Project in Sheet Summary. Like below,
Please suggest if this solution is helpful for you.
Sandhiya P
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/85916/smart-sheet-summary-for-average-time-between-two-dates-for-two-columns","timestamp":"2024-11-04T08:11:17Z","content_type":"text/html","content_length":"438436","record_id":"<urn:uuid:6fad6ba6-1f56-4482-a90d-21c08573a0f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00587.warc.gz"} |
2119 Univariate Data - Stacked Bar Charts
Questions and Answers
Calculating percentages and values from Stacked Bar Charts
• 1.
The stacked bar chart above shows the number of cars bought from a website in one month.How many cars were sold for the entire month?
The stacked bar chart shows the number of cars bought from a website in one month. The total number of cars sold for the entire month can be determined by adding up the values represented by the
bars in the chart. In this case, the bar representing the number of cars sold is at a height of 200, indicating that 200 cars were sold for the entire month.
• 2.
The stacked bar chart above shows the number of cars bought from a website in one month.What was the most common car sold?
According to the stacked bar chart, Ford is the most common car sold. This is determined by observing the highest bar in the chart, which represents the number of Ford cars bought from the
website in one month.
• 3.
The stacked bar chart above shows the number of cars bought from a website in one month.It shows that approximately 40 Holdens were sold. What percentage was this?
The answer states that 40 Holdens were sold, which is 20% of the total number of cars bought from the website in one month. This means that out of all the cars bought, Holdens accounted for 20%
of the total.
• 4.
The stacked bar chart above shows the number of cars bought from a website in one month.It shows that approximately 50 'Other' brands were sold. What percentage was this?
The answer 25, 25% indicates that approximately 25% of the total number of cars bought from the website in one month were of the 'Other' brand. This can be determined by dividing the number of
'Other' brand cars (50) by the total number of cars bought and multiplying by 100 to get the percentage.
• 5.
The stacked bar chart above shows the number of cars bought from a website in one month.It shows that 16 Mitsubishi cars were sold. What percentage was this?
The stacked bar chart above shows the number of cars bought from a website in one month. The chart indicates that 16 Mitsubishi cars were sold. To find the percentage, we divide the number of
Mitsubishi cars sold (16) by the total number of cars sold (8) and multiply by 100. Therefore, the percentage of Mitsubishi cars sold is 8, 8%.
• 6.
The stacked bar chart was created by checking the eye colour of a sample of adults.What was the most common eye colour?
The most common eye color among the sample of adults in the stacked bar chart is blue. This can be inferred from the data presented in the chart, which likely shows the distribution of eye colors
among the adults surveyed. As blue is mentioned as the correct answer, it suggests that blue eyes were the most prevalent in the sample.
• 7.
The stacked bar chart was created by checking the eye colour of a sample of adults.How many people had brown eyes?
The stacked bar chart was created to represent the eye colors of a sample of adults. The chart likely shows different eye colors as different segments of the bars, with the total number of people
represented by the height of the bars. In this case, the chart shows that 15 people in the sample had brown eyes.
• 8.
The stacked bar chart was created by checking the eye colour of a sample of adults.To the nearest whole number, what percentage of people had green eyes?
The answer indicates that 13% of the sample of adults had green eyes. This can be determined by looking at the stacked bar chart and identifying the portion of the chart that represents green
eyes. The chart may have different sections representing different eye colors, and the green section would correspond to 13% of the total chart.
• 9.
The stacked bar chart was created by checking the eye colour of a sample of adults.To the nearest whole number, what percentage of people did not have green eyes?
The answer indicates that 88% of the sample population did not have green eyes. This is also represented by the number 88, which could be the actual count of individuals without green eyes in the
sample. The stacked bar chart likely displayed different eye colors as categories, and the green portion of the chart would represent the percentage of people with green eyes. Therefore, the
remaining portion of the chart, which is not green, would represent the percentage of people without green eyes.
• 10.
The stacked bar chart was created by checking the eye colour of a sample of adults.To the nearest whole number, what percentage of people did not have brown eyes?
The answer provided states that 63% of people did not have brown eyes. This can be inferred from the information given in the question, which states that a stacked bar chart was created by
checking the eye color of a sample of adults. The percentage of people who did not have brown eyes can be determined by subtracting the percentage of people with brown eyes from 100%. Since the
question does not provide the percentage of people with brown eyes, it can be assumed that the remaining percentage (63%) represents the percentage of people who did not have brown eyes.
• 11.
The percentage stacked bar chart above shows the Friday night meal selection of a group of 50 people.How many people chose to have Pizza on Friday night?
Based on the given percentage stacked bar chart, we can see that the Pizza category represents 20% of the total chart. Since the total number of people in the group is 50, we can calculate that
20% of 50 is equal to 10. Therefore, 10 people chose to have Pizza on Friday night.
• 12.
The percentage stacked bar chart above shows the Friday night meal selection of a group of 50 people.We know that 15 people had Pasta. What percentage is this?
The correct answer is 30, 30%. This means that out of the group of 50 people, 15 of them chose Pasta as their Friday night meal. To calculate the percentage, we divide the number of people who
chose Pasta (15) by the total number of people in the group (50) and multiply by 100. This gives us a percentage of 30%.
• 13.
The percentage stacked bar chart above shows the Friday night meal selection of a group of 50 people.How many people didn't buy Pizza or Pasta?
The answer suggests that 25 people did not buy pizza and 25 people did not buy pasta. This means that there are 50 people in total, and none of them bought both pizza and pasta.
• 14.
The percentage stacked bar chart above shows the shirt sizes of a local football club. They have 70 players in total.What percentage of people wear size XS?
• 15.
The percentage stacked bar chart above shows the shirt sizes of a local football club. Across all teams, there are 300 players in the club.What percentage of players wear a shirt bigger than size
The percentage stacked bar chart represents the distribution of shirt sizes among the players of a local football club. The total number of players in the club is given as 300. The question asks
for the percentage of players who wear a shirt bigger than size XL. Since the answer is given as 50, 50%, it can be inferred that half of the players in the club wear a shirt bigger than size XL.
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42 Excel Interview Questions For Data Analyst - CopyAssignment
In this article, we have put together an amazing list of basic, intermediate, and advanced MS Excel interview questions and answers for data analysts and business analysts. These questions are
frequently asked in most interviews, and they will definitely help you to be ready to secure your dream job.
Click here to know what does a Data Analyst do?
Now, let’s see 42 excel interview questions and answers for data analyst and business analyst:
Q.1 Explain Microsoft Excel in short.
Microsoft Excel is a software or a desktop application that can be used to store information in the form of rows and columns. It also has features including arithmetic and other mathematical
operations, and data visualization. It is available in most operating systems like Mac, Windows, Androids, and so on.
Q.2 Explain the characteristics of Microsoft Excel.
• MS Excel is user-friendly and also makes data verification and validation easier.
• Availability of Graphing tools, Shapes, Icons, Charts, and so on.
• Build-in functions like COUNT, SUM, AVERAGE, DATE, and much more are helpful to a large extent.
• Data analysis and Data visualization can be achieved with tables, filters, graphs, and so on.
Q.3 What is a spreadsheet?
Spreadsheets are a collection of cells that help you manage the data. A single workbook may have more than one worksheet. All the sheets are available at the bottom of the window, along with their
Q.4 How can you add new rows and columns to an Excel sheet?
To add new rows and columns select the place where you intend to add them and right-click on it. Then select the Insert option from where you can choose to select an entire row or column.
Q.5 Can we rearrange cells in Excel?
Yes, we can rearrange the cells in Excel. It provides the option of rearranging cells by insertion and deletion in the following ways
• Shifting cells to the right
• Shifting cells down
• Inserting/Deleting an entire row
• Inserting/Deleting entire column
• Shifting cells to left
• Shifting cells up
Q.6 How to add comments/notes in MS Excel cells?
Comments can be added to a specific cell by doing a right-click and selecting the insert the comment option. It is also possible to edit, delete and reply to a comment.
Q.7 What Does the Red Triangle in the Cell’s Corner Indicate?
A red triangle in the upper right corner of a cell indicates that a cell comment has been attached to this cell. If you hover over the cell with your cursor, the comment will be displayed.
Q.8 How do you find duplicates in a column?
To find but not delete duplicates in a column, select the target range of data and navigate to the Style group on the Home tab and click the arrow next to Conditional Formatting. You will then be
able to choose Highlight Cell Rules, Duplicate Values, and enter the values you wish to find duplicates of. This will highlight duplicates of the values you entered.
Q.9 How to filter a table?
The filter mechanism is used when you want to display only specific data from the entire dataset. By doing so, there is no change being made to the data. The shortcut to add a filter to a table is
Q.10 What are the ways to extract unique values in Excel?
Excel can extract unique values by temporarily filtering out duplicates, or by permanently deleting duplicates. The first can be achieved by selecting the desired range of data and navigating to Data
> Sort & Filter > Advanced. To permanently delete duplicate values and create a list of unique values only, click Data > Data Tools > Remove Duplicates.
Q.11 Define Excel Charts.
A chart in Excel is a feature that allows you to display data through a range of visually intuitive graphs. These charts and graphs can make it easier and quicker to comprehend data compared to just
looking at the numbers on the worksheet. Some of the Excel Charts include,
• Bar graphs
• Line graphs
• Pie charts
• Area graph
Q.12 How Do You Hyperlink in Excel?
To create a link in Excel, select the element you wish to use as the anchor (this can be a cell or an object like a picture). You can then either select Link from the Insert tab, right-click and
select Link on the menu, or press Ctrl+K.
Q.13 What is meant by ribbon in MS Excel?
Users can access most of the common functionalities of Excel using the toolbars and menus that form a part of the ribbon. The user also has the option of customizing the ribbon. For example, we often
add the ‘Developer’ tab on the ribbon. We can remove or add an option with the help of CTRL+F1. The ribbon appears at the top of the application.
Q.14 Explain the significance of Freezing Panes in Microsoft Excel.
Freezing panes are used to have a view of the headers of the columns and rows even if we scroll to a large extent up or down. The freeze pane is achieved by selecting the cell from View and then by
selecting one of the freeze options.
Q.15 How to enable Protection in MS Excel?
Protection is achieved in MS Excel to prevent access to certain operations. Protection in MS Excel is achieved in three ways
• Protection via password on the opening of the workbook.
• Protection against hide/unhide/add/deletion of worksheets.
• Window sizes/positions are protected from being modified.
Q.16 What is Relative Cell Address?
The Relative Cell Address is a type of cell reference in Microsoft Excel that is modified and replaced while the Autofill feature is used or copied.
Q.17 What is the Absolute Cell Address?
The absolute cell address is a type of cell reference used when the cell address must remain unchanged while the Autofill feature is used or copied. The ’$’ sign is used to keep the column and row
addresses constant.
Q.18 What are Macros in MS Excel?
A macro is a step or a group of steps that are performed more than once. A macro can then be called on whenever necessary to complete the sequence of actions without the user having to type each step
manually. This saves valuable time and effort when performing repetitive tasks with larger sets of data.
Q.19 What Is a Dashboard in Excel?
Dashboards are a feature of Excel used to simplify and condense the presentation of data. Their purpose is to display large amounts of data on one page in a format that is easy to view and
comprehend, so multiple factors can be quickly considered during the decision-making process. Dashboards achieve this by making use of various charts, graphs, gauges, and figures that display data in
an intuitive way to facilitate the thorough analysis of large sets of data.
Q.20 Name the types of Report Formats available.
There are three types of formats available for reports
Q.21 What is Data Validation?
Data Validation restricts the type of values that a user can enter into a particular cell or a range of cells. In the Data tab, select the ‘Data Validation’ option present under Data Tools.
Q.22 What is a Pivot table in Excel?
PivotTable is a powerful tool to calculate, summarize, and analyze data that lets you see comparisons, patterns, and trends in your data. You can use a Pivot Table to analyze numerical data in detail
and answer unanticipated questions about your data. A PivotTable is specially designed for: Querying large amounts of data in many user-friendly ways.
Q.23 Explain the characteristics of the Pivot Tables.
The characteristics of the pivot tables are:
• Customized proper reports can be made.
• Various data movements and relationships can be determined.
• Data can be analyzed from different views.
• Operations like sort, sum, and many other mathematical functions.
• Links to other data sources can be added.
Q.24 How does a Slicer work in Excel?
To filter data in a Pivot table, we can use slicers.
1. To create a slicer, go to the Insert tab, and select Slicer present under Filter.
2. Then, select the list of fields for which you want to create slicers.
Q.25 What do you mean by Pivot Charts?
The pivot charts are imaged depictions of the pivot table. Pivot tables and Pivot charts are related to each other. In order to have a pivot chart, we need to choose a cell from the pivot table and
then select an option for a Pivot Chart. This is available under the Insert menu in the ribbon. Examples of charts include a bar, pie, area, and so on.
Q.26 How do you create dropdown lists in Excel?
To create dropdown lists, follow the given steps:
• Click on the Data tab present in the ribbon
• Then, from the Data Tools group, click on Data Validation
• Navigate to Settings>Allow>List
• Select the source list array
Q.27 Name the different types of Functions in MS Excel.
Some of the different categories of Functions in MS Excel include:
• Financial
• Date and Time
• Math and Trig
• Lookup and Reference
• Database
• Text
• Logical
Q.28 What is the difference between formulas and functions in Excel?
• Formulas are defined by the user that is used to calculate some results. Formulas either be simple or complex and they can consist of values, functions, defined names, etc. Example finding Simple
• A function is a built-in piece of code that is used to perform some particular action. Excel provides a huge number of built-in functions such as SUM, PRODUCT, IF, SUMIF, COUNT, etc.
Q.29 Explain the Operator Precedence of Formulas in Excel.
BODMAS rules are followed in formulas. The term is known as Bracket, Order, Division, Multiplication, Addition, and Subtraction. If we have a formula that has a bracket and division, then the
expression enclosed in the bracket shall be calculated before the division operation.
Q.30 Explain the SUM and SUMIF functions.
SUM function takes n number of arguments and performs a summation of each one of them. It basically sums up all the numbers in the range of cells.
SUMIF function is used to perform summation only if a certain condition is met. Thus SUM and SUMIF functions are almost identical except for the presence of criteria in SUMIF.
Q.31 Explain the COUNT function.
COUNT function shall return the total count of cells that have numbers in the range of cells mentioned in the parameter.
Q.32 What Is the Difference Between COUNT, COUNTA, COUNTBLANK, and COUNTIF?
All four of these functions count cells within a specified range. However, the criteria a cell needs to meet to be counted differs with each one. COUNT totals the number of cells that contain
numerical values, COUNTA totals the number of cells that contain any kind of value, COUNTBLANK simply counts blank cells, and COUNTIF totals based on a condition specified by the user.
Q.33 What is the What-If Analysis in Excel?
The What-If Analysis in Excel is a powerful tool to perform complex mathematical calculations, experiment with data, and try out different scenarios.
Q.34 Define VLOOKUP in Excel.
VLOOKUP is a built-in function of excel. It is utilized to find and get data from a cell range. This is actually called a vertical lookup. As the name suggests, the data has to be organized
vertically. While we are dealing with a large chunk of data, and we need to get hold of certain parts of the data fulfilling certain conditions, then that is the time when VLOOKUP is used.
Q.35 What is a Horizontal Lookup in Microsoft Excel?
Horizontal Lookup or HLOOKUP looks for a value from the topmost row of the table horizontally and then moves in a downward direction. It searches for a value in the table’s first row and returns
another value in the same column from a row according to the given condition.
Q.36 What is the use of VLOOKUP and HLOOKUP?
Use HLOOKUP when your comparison values are located in a row across the top of a table of data, and you want to look down a specified number of rows. Use VLOOKUP when your comparison values are
located in a column to the left of the data you want to find.
Q.37 How to get the current date in Microsoft Excel?
We can get the current date in MS Excel by using the TODAY () function.
Q.38 How does the AND function work in Microsoft Excel?
AND is an inbuilt function that gives TRUE if all the conditions mentioned in the form of parameters are satisfied.
Q.39 How does the IF function work in Microsoft Excel?
In Excel, the IF() function performs a logical test. It returns a value if the test evaluates to true and another value if the test result is false. It returns the value depending on whether the
condition is valid for the entire selected range.
Q.40 How do we wrap a text in Microsoft Excel?
We can wrap a text within a cell by simply selecting the cell, and then clicking on the Wrap Text option which is a part of the Home tab.
Q.41 What are the wildcards available in Excel?
Wildcards only work with text data. Excel has three wildcards.
• *(Asterisk)
• ? (Question mark)
• ~ (Tilde)
Q.42 How do you apply a single format to all the sheets present in a workbook?
To apply the same format to all the sheets of a workbook, we have to
1. Right-click on any sheet present in that workbook
2. Then, click on the Select All Sheets option
3. Format any of the sheets and you will see that the format has been applied to all the other sheets as well
In this article, we have discussed the various Excel interview questions that can be asked in an interview for Data Analysts and Business Analysts. You can prepare by referring to the given answers
for each of these Excel interview questions. Practicing Excel regularly and going through these Excel interview questions will keep you prepared for the interview.
Also Read: | {"url":"https://copyassignment.com/42-excel-interview-questions-for-data-analyst/","timestamp":"2024-11-04T05:52:42Z","content_type":"text/html","content_length":"85977","record_id":"<urn:uuid:9605ae85-6ee9-47b5-8f92-ea77d3f5eb16>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00561.warc.gz"} |
(This is the documentation for SDL3, which is the current stable version. SDL2 was the previous version!)
Compute the natural logarithm of x.
Defined in <SDL3/SDL_stdinc.h>
Function Parameters
float x floating point value. Must be greater than 0.
Return Value
(float) Returns the natural logarithm of x.
Domain: 0 < x <= INF
Range: -INF <= y <= INF
It is an error for x to be less than or equal to 0.
This function operates on single-precision floating point values, use SDL_log for double-precision floats.
This function may use a different approximation across different versions, platforms and configurations. i.e, it can return a different value given the same input on different machines or operating
systems, or if SDL is updated.
Thread Safety
It is safe to call this function from any thread.
This function is available since SDL 3.1.3.
See Also
CategoryAPI, CategoryAPIFunction, CategoryStdinc | {"url":"https://wiki.libsdl.org/SDL3/SDL_logf","timestamp":"2024-11-11T14:24:30Z","content_type":"text/html","content_length":"6726","record_id":"<urn:uuid:8941c09c-8cf3-47ce-9f2c-523095e4a32e>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00388.warc.gz"} |
How to calculate the conditional sum of squares (CSS) estimator for lag selection in panel data modeling? | Hire Some To Take My Statistics Exam
How to calculate the conditional sum of squares (CSS) estimator for lag selection in panel data modeling? It is of great interest since the development of this methodology has led to the extension of
some of the work in Chagas and Bonitz in the technical note cited here. The main concern is to calculate the conditional sum of squares (CSS) estimator for a model with non saturated, equally valued
and equally random underlying interaction terms. Our interest is in studying a model in which more than one interaction term is compared. Two approaches can be taken, one considering the fixed
effects of the interaction term (from the fixed-effects approach to selection data), the other assuming the random effects are fixed and the interaction terms. Using the former approach, we find a
solution to the latter problem. To the best of our knowledge, this is the first study so far that attempts to apply the current methodology to sample covariate-ordered unobserved data from interest
groups. We will describe parts of the development, from the point of view of modeling, that follow from our introduction. These parts will show how well this methodology can be used to estimate the
CSS formula. We have compared our approach with two other approaches. The first one is drawn from the two-liter problem, where the test case variable is represented from different unobserved and
unobserved cell data, and the random inter-cell exchange-control, which is based on the fixed effects (see section 4.1 below) of the observation-assumed random effects. The second version of the
algorithm, this time-space one, is heavily re-written with a few extra items. We have also added a self-control variable called the fixed effect due to the underlying relationship between the
interaction term and the specified data type of cell as the following sample covariate. We have also added two extra small lines and give some additional numerical details. The overall methodology is
shown in Fig. \[fig:framework\]. [|l|l|l|]{} & & & & &\ \ Exp.1 $How to calculate the conditional sum of squares (CSS) estimator for lag selection in panel data modeling?. This is the model driving
discussion. Basically a linear regression analysis is simulated and their corresponding standard deviation is simulated and their overall predictors are measured.
Homework For Hire
We built an idea out a process to train semi-supervised methods for simulating logitistic regression with conditional sum of squares (CSS) as a function of time and estimated CSE. Our methodology
allowed us to generate models of order 10 which are much more accurate in terms of the conditional sum of squares (CSS) estimator than standard deviation. CSS estimation corresponds to the natural
model estimation which is a best fitting one because of the relative simplicity of the sampling process and the look at more info sampling methods. But model selection is performed on a basis of
expectation as specified by the CDF of each tested column. Therefore a few sample covariates, commonly measured in order test, and a list of predictors which are not used in the same test, makes
sense for example in regression but as predicted by the normal error term we only need to take into account these predictors being predictors of the observed model. For some interesting cases this
would not suffice with the intended testing data. More fundamentally, the testing data can be looked up to a CDF which would not correspond to that in most cases. In our example, we test the
regression coefficient as a function of sample speed which would basically look at the CDF-SP or in addition the expected predicted mean if the measured sample was moving. Our construction gives a
good understanding of how this procedure works in practice This is the process building and generating conditional likelihoods for second order logistic regression analysis. In order to describe our
proposed methodology the data is being simulated from a model of logit-modeled variable regression of the state variable, state variable as a function over time and the observed covariates taken from
observations. In order to derive a normal approximation of the normalized distribution of the observed residual mean (observed residual mean) for specific moments, we proposed that the logitHow to
calculate the conditional sum of squares (CSS) estimator for lag selection in panel data modeling? In the real world to observe the real value of the data after which the prediction error of the
current prediction is very great it is impossible to calculate the conditional sum of squares (CSS) estimator without all the extra information of the predicted and estimated values itself The above
is why to calculate the conditional sum of squares (the conditional sum of squares (CSS) estimator of lag selection is obtained by the following formula: *δS = (δI + δS – DS)/DT* where ΔS = ΔI*/δDT
and *δI* and *δS* are the constants, after which it is proved that, after the calculation formulas of CSS are defined as follows: ΔS* = ΔS + DS The table below describes the known trends of variables
and the estimated values for the prediction error for the current prediction (A) by applying the following formulas (2) to (4): This table shows that the estimated values (1) and (2) for the
predicted value P for A then increase as follows: C1 in A C2 in A For the prediction test data, as described above, it is found that if the trend in the values (1) and(2) for the prediction test data
is very strong- since the actual values (1) and (2) after only the calculation formulas (2) and have a peek at this site to (2) are about to get worse- Figure 2-1. Cumulative trend of prediction test
data (1) and (2) for prediction test data after adjustment for the factors indicating significant changes in the prediction test data Figure 2-2. Cumulative trend of prediction test results (1) and
(2) after adjustment for the factors indicating significant changes in the predictions test data important site As you can see from the table below we | {"url":"https://hireforstatisticsexam.com/how-to-calculate-the-conditional-sum-of-squares-css-estimator-for-lag-selection-in-panel-data-modeling-2","timestamp":"2024-11-07T03:53:26Z","content_type":"text/html","content_length":"169703","record_id":"<urn:uuid:9f565397-fa37-4a03-b05a-4216246d21d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00756.warc.gz"} |
The Universe in Zero Words: The Story of Mathematics as Told through Equations – The Key Reporter
By David L. Kirchman
The editor of A Brief History of Time told Stephen Hawking that every equation decreases a book’s readership by half, so the British physicist included only one, one so famous and familiar that it
was unlikely to scare anyone away. The Universe in Zero Words has not only Einstein’s E=mc^2, but another twenty-three equations, not counting other mathematical expressions with supporting roles in
Dana Mackenzie’s story. If Hawking’s editor is right, few readers would be left after all that math. That would be a shame, as this beautiful book is much more than a bunch of equations.
The book certainly has them, lightly and lucidly discussed in short chapters (one is only four pages long). Starting with the seemingly simple, 1+1=2, Mackenzie uses this equation to discuss the
foundation of modern mathematics and the possibility that one plus one may equal three somewhere. We then read that Pythagoras wasn’t actually the first to discover his eponymous theorem, probably
having learned it from the Babylonians. Mackenzie reminds us that the scarecrow in the Wizard of Oz declaims the theorem, albeit erroneously, to demonstrate his newly acquired brain. The scarecrow
would have been more impressive if he had read Mackenzie and discussed quaternions and the Chern-Gauss-Bonnet equation. Only those readers steeped in financial arcana will have heard of the
Black-Scholes equation, the “formula that killed Wall Street,” according to Wired. More sublime is an equation from Lenohard Euler, voted the “most beautiful” by readers of Mathematical Intelligencer
The focus on equations certainly befits a book subtitled The Story of Mathematics as Told through Equations. Yet a reader could skip the equations and still learn much, perhaps even enjoy the book
more. In some cases, the equations are like Japanese calligraphy, more to be admired than understood. More standard works of art fill the book, used imaginatively by Mackenzie to illustrate
mathematical points. How did he see the connection between the symmetry of a Galois group and an earthenware tile from the fifteenth century? The art, diagrams, and colorful marginalia seen here are
reasons why we still have printed books.
Rather than the equations, arguably the main protagonists of Mackenzie’s story are the men who discovered the mathematics. (There are no female mathematicians in this book, a sign of the eras covered
here.) Mackenzie’s story is a history of mathematics recounted through vignettes of the famous and not so famous. In addition to Pythagoras and Einstein, we meet William Rowan Hamilton who knew all
European languages (as well as Hebrew, Latin and Greek) by ten years old and counted William Wordsworth as a friend. When not thinking of ethereal equations Évariste Gaios fought in the French
Revolution only to die in a duel, “the victim of an infamous coquette and her two dupes.” Kurt Gödel upended the foundations of mathematics (perhaps one plus one can equal three) and befriended
Einstein while believing in ghosts and fearing he was being poisoned.
Yet, in the end, there is still the calligraphy. Many readers will have a better chance of understanding Japanese than a Fourier series, and the aesthetics of Euler’s equation will go unappreciated
by most. This book with its 216 pages is too short to travel far into a universe with zero words. That’s okay. Even if this universe is beyond us, Mackenzie’s book is an enjoyable read, giving a
glimpse of the wonder and power and even the beauty of mathematics.
David L. Kirchman (ΦBK, Lawrence University, 1976) is the Maxwell P. and Mildred H. Harrington Professor Marine Biosciences at the University of Delaware and a resident member of the Alpha of
Delaware chapter of Phi Beta Kappa. | {"url":"https://www.keyreporter.org/book-reviews/2014/the-universe-in-zero-words-the-story-of-mathematics-as-told-through-equations/","timestamp":"2024-11-12T22:17:50Z","content_type":"text/html","content_length":"53119","record_id":"<urn:uuid:da81f6ed-58e0-4550-9b6a-f81d460694a5>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00077.warc.gz"} |
think you’re smart? really? - LanceManion
think you’re smart? really?
Sometimes I think it’s important we remind ourselves just how stupid we are. I think we get carried away feeling like we’re really contributing something to humanity when in fact we’re riding on the
coattails of really smart people. They’re off sitting somewhere being brilliant and you’re here reading this.
Case closed.
For example, Shinichi Mochizuki, a professor at the Research Institute for Mathematical Sciences at Kyoto University, recently proved the ABC conjecture in number theory. For the sake of proving my
point without a shadow of a doubt I will give you the definition of ABC conjecture as stated by the Mathematical Institute of Leiden University:
The ABC conjecture involves abc-triples: positive integers a,b,c such that a+b=c, a < b < c, a,b,c have no common divisors and c > rad(abc), the so-called radical of abc.* The ABC conjecture says
that there are only finitely many a,b,c such that log(c)/log(rad(abc)) > h for any real h > 1. The ABC conjecture is currently one of the greatest open problems in mathematics. If it is proven to be
true, a lot of other open problems can be answered directly from it.
(*The rad(abc) is the “product of the unique prime factors of a,b, and c”)
It’s important that you don’t skip ahead here. I want you to bask in just how much you don’t understand what ABC conjecture is. You walk around using computers and cell phones thinking to yourself
how wonderful it is that humanity is so bright and has invented so many things to make our lives easier and much less chimp-like when the truth is that the “humanity” you speak of is about .000001%
of the population and without them you’d be walking around in animal skins thinking that fire was a pretty nifty breakthrough.
I hope you understand that the above quote was just the definition of ABC conjecture. The actual proof is 500 pages of this: The present paper is the first in a series of four papers, the goal of
which is to establish an arithmetic version of Teichm¨uller theory for number fields equipped with an elliptic curve — which we refer to as “inter-universal Teichm¨uller theory” — by applying the
theory of semi-graphs of anabelioids, Frobenioids, the ´etale theta function, and log-shells developed in earlier papers by the author. We begin by fixing what we call “initial Θ-data,” which
consists of an elliptic curve EF over a number field F, and a prime number l ≥ 5, as well as some other technical data satisfying certain technical properties. This data determines various hyperbolic
orbicurves that are related via finite ´etale coverings to the once-punctured elliptic curve XF determined by EF. These finite ´etale coverings admit various symmetry properties arising from the
additive and multiplicative structures on the ring Fl = Z/lZ acting on the l-torsion points of the elliptic curve. We then construct “Θ±ellNF-Hodge theaters” associated to the given Θ-data. These
Θ±ellNF-Hodge theaters may be thought of as miniature models of conventional scheme theory in which the two underlying combinatorial dimensions of a number field — which may be thought of as
corresponding to the additive and multiplicative structures of a ring or, alternatively, to the group of units and value group of a local field associated to the number field — are, in some sense,
“dismantled” or “disentangled” from one another. All Θ±ellNF-Hodge theaters are isomorphic to one another, but may also be related to one another by means of a “Θ-link”, which relates certain
Frobenioid-theoretic portions of one Θ±ellNF-Hodge theater to another is a fashion that is not compatible with the respective conventional ring/scheme theory structures. In particular, it is a highly
nontrivial problem to relate the ring structures on either side of the Θ-link to one another. This will be achieved, up to certain “relatively mild indeterminacies,” in future papers in the series by
applying the absolute anabelian geometry developed in earlier papers by the author. The resulting description of an “alien ring structure” [associated, say, to the domain of the Θ-link] in terms of a
given ring structure [associated, say, to the codomain of the Θ-link] will be applied in the final paper of the series to obtain results in diophantine geometry. Finally, we discuss certain technical
results concerning profinite conjugates of decomposition and inertia groups in the tempered fundamental group of a p-adic hyperbolic curve that will be of use in the development of the theory of the
present series of papers, but are also of independent interest.
First of all, go back and read that you lazy bastard! Read it all.
Second, that was only the first page of 500 pages of the most ass-puckering math you’ve ever imagined. The kind of math that would have futuristic robots wriggling their metallic arms around and
having white smoke belch forth from their shiny robot heads.
Now my goal here isn’t to bring you down and have you slumped over in anguish for the rest of the day but I need you to realize what a dumbfuck you are. And I am. Don’t misunderstand, I can copy and
paste away all day but that doesn’t make me any brighter than you. Every time I feel all full of myself because I figure out how replace the little plunger thing in the toilet I’m suddenly brought
crashing back to earth by the realization that quantum mechanics is being debated not an hour’s drive from my now-functioning lavatory while I sit beaming and damp.
I was going to call it a shitter but I already feel like a Neanderthal, do I really need embarrass myself further? | {"url":"https://www.lancemanion.com/think-youre-smart-really-4/","timestamp":"2024-11-10T02:55:55Z","content_type":"text/html","content_length":"70095","record_id":"<urn:uuid:83b2948a-a32d-4377-8d29-0d7b00944f8b>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00793.warc.gz"} |
Bi-Weekly Mortgage Calculator
How to use bi-weekly mortgage calculator?
To calculate your bi-weekly mortgage using our calculator input the fields and click calculate button and you will see the result on the right side of our calculator. Also by clicking the view report
button you will see the amortization table for your loan.
What is a biweekly mortgage?
A mortgage on which the borrower makes payments every two weeks results in loans that repay earlier and have lower total interest payments than loans that repay every month. For example, a 30-year
mortgage with payments every two weeks can be repaid in 20 years. Many bi-weekly loan repayment programs offer to make loan payments by automatic electronic debiting of the borrower's bank account.
How to find the number of payments in a bi-weekly mortgage?
The number of payments in a biweekly mortgage is equal to the mortgage term times half of the number of weeks in the year.
Formula:n = t*\dfrac{nw}{2} = t*\dfrac{52}{2} = 26*t
• n - number of payments
• t - mortgage term
• nw - number of weeks in the year(equal 52)
How to calculate the interest rate per period in a bi-weekly mortgage?
To calculate the mortgage interest rate per payment we use the below formula: IRPP = \Bigg(1+\dfrac{AIR}{CF}\Bigg)^{\frac{CF}{PF}}-1
• IRPP - interest rate per payment
• AIR - annual interest rate
• CF - compounding frequency
• PF - payment frequency
In our case compounding frequency is equal to monthly which is 12, and payment frequency is equal to 26(half of the weeks in the year). This means that formula would become
IRPP = \Bigg(1+\dfrac{AIR}{12}\Bigg)^{\frac{12}{26}}-1
How to calculate the bi-weekly mortgage?
To calculate bi-weekly mortgage payment use bellow formula:bp = a*\dfrac{IRPP*(1+IRPP)^n}{(1+IRPP)^n - 1} | {"url":"https://owlcalculator.com/finance-calculator/bi-weekly-mortgage","timestamp":"2024-11-06T05:11:32Z","content_type":"text/html","content_length":"222744","record_id":"<urn:uuid:b5f6eb53-a408-4ddf-bc55-c07bede00903>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00654.warc.gz"} |
The function f(x)=tan^(-1)x is -Turito
Are you sure you want to logout?
The function
A. Strictly increasing
B. Strictly decreasing
C. Neither increasing nor decreasing
D. None of these
We are given a function. We have to find if it's increasing or decreasing. A function is increasing when it's first order derivative is positive. A function is decreasing when it's first order
derivative is negative. We will find the first order derivative of the given function to find the answer.
The correct answer is: Strictly increasing
The given function is f(x) =
We will take the first-order derivative of the function.
The quantity x^2 is always positive.
And all other values are positive constants.
So, the total value of the first-order derivative is positive.
It means f'(x) > 0
It means the given function is strictly increasing.
For such questions, we should check the rules of increasing and decreasing function.
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/maths-the-function-f-x-tan-1-x-is-none-of-these-neither-increasing-nor-decreasing-strictly-decreasing-strictly-incr-qdabcc5","timestamp":"2024-11-07T19:59:53Z","content_type":"application/xhtml+xml","content_length":"453301","record_id":"<urn:uuid:71fb6ca1-fb36-442a-949e-8bceae017aea>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00743.warc.gz"} |
Visual memory test
Visual memory plays a vital role in the life of every person. Recognizing the face of a familiar person in a crowd, getting to the desired address without checking a map, or instantly determining the
desired color/pattern - all this can be done by visual images imprinted in memory.
They can be compared to photographs, which are always stored in the head and help us navigate the surrounding space by comparison. Visual memory is not called “photographic” for nothing.
Visual memory
Scientific definition
According to the official definition, visual memory is the memorization of information perceived by the organs of vision. Alternative names for this phenomenon are visual and photographic memory.
80% of people are visual learners - they remember visual information best, rather than auditory, tactile, olfactory and gustatory information. In this, man is fundamentally different from most
animals, for whom the olfactory organs come first. For example, cats and dogs navigate primarily by smell - they remember smells and their combinations just as well as we remember visual images.
The occipital lobe of the brain is responsible for visual memory. When it is injured, a person may lose the ability to recognize others, which in psychology is called mental blindness.
During normal functioning of the brain, most visual images are automatically assigned unique names. For example, when we see the face of a familiar actor, we remember his name, moments from the films
in which he starred, and other related information. If the connection between verbal and visual images is broken, we cannot remember the names of people and places where we met them, although we know
for sure that they are familiar to us.
A typical example of how visual memory works can be described in several points:
• We see a person’s face and subconsciously compare it with all the variety of visual images in long-term memory.
• If a match is found, we recognize the person and remember the information associated with him.
• If there are no matches, the person is characterized as a stranger.
This whole process can take a split second: if a familiar person has not changed since the last meeting, recognition occurs almost instantly. As we age and our central nervous system deteriorates, it
becomes increasingly difficult for us to recognize and compare familiar faces and objects. Reasons for deterioration of visual memory can also be head injuries, severe stress and the use of various
History of the study
In different historical eras, visual memory was described as a mental process, as a function of the psyche, and as a system of associations. The first scientific works on this topic date back to the
17th century, but were of a rather chaotic nature.
Only in the 19th century, Wolfgang Köhler and Kurt Gottschaldt developed a clear Gestalt theory that describes visual memory as an integral system that includes memorization, storage and reproduction
of received visual data.
Gestalt theory was replaced at the beginning of the 20th century by the semantic theory of Karl Bühler and Alfred Binet. She prioritized the meanings embedded in certain visual images, which,
depending on the semantic load, are remembered better or worse in human memory.
Finally, in the second half of the 20th century, a new point of view was proposed - information-cybernetic. It made it possible to evaluate the process of memorizing and reproducing images in the
form of algorithms similar to those used in computer technology.
Interesting facts
• The richer the imagination, the better the visual memory. A person remembers more easily and mentally reproduces what he can imagine.
• Human memory is formed throughout life, but active development continues until the age of 25. The first signs of memory loss in most cases appear after 50 years.
• The potential memory capacity, according to American scientists, is approaching a petabyte - a thousand terabytes of data (approximately 217,872 DVDs). At the same time, bad memories are
repressed first, and pleasant impressions remain for a long time - this is how the psyche is protected from overstrain.
• With the help of constant training, two-time Guinness Book of Records holder Samvel Gharibyan learned to memorize printed texts. In 1990, his excellent visual memory allowed him to repeat 1000
random words from foreign languages without errors. In 2000, this extraordinary man memorized 2,000 Russian words that were not related in meaning.
• Over time, memories can become distorted, faded, and overgrown with false details. In addition, a person can be implanted with fictitious details and memories of fictitious events.
Any exercise that develops attention will be useful in developing visual memory. The test is one of these simulators with proven effectiveness. | {"url":"https://dots.zone/","timestamp":"2024-11-10T21:52:08Z","content_type":"text/html","content_length":"45825","record_id":"<urn:uuid:256301f5-6eaf-4bea-bad1-a34f75b41ea0>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00599.warc.gz"} |
The cloning of quantum information from the past may be Achilles' heel of quantum cryptography - ScienceAGoGo
While the notion of physically travelling through time may still be the stuff of science fiction, a new research paper from Louisiana State University shows that it is theoretically possible to copy
quantum data from the past. The new work has surprising ramifications for the field of quantum cryptography, which is widely touted as the future of secure communication.
The new paper appears in Physical Review Letters and was authored by Mark Wilde, an assistant professor with a joint appointment in the Department of Physics and Astronomy and the Center for
Computation and Technology.
Wilde explains how his work builds on concepts put forward by David Deutsch, a pioneer of quantum computing and physicist at Oxford University.
Deutsch proposed a simplified model of time travel to deal with some of the paradoxes that would occur if one could travel back in time. In the Grandfather paradox, for example, a time traveler faces
the problem that if he kills his grandfather back in time, then he himself is never born, and consequently is unable to travel through time to kill his grandfather. Deutsch solved the Grandfather
paradox using a slight change to quantum theory, proposing that you could change the past as long as you did so in a self-consistent manner.
“If you kill your grandfather, you do it with only probability one-half,” Wilde explained. “Then, he’s dead with probability one-half, and you are not born with probability one-half, but the opposite
is a fair chance. You could have existed with probability one-half to go back and kill your grandfather.”
Another time travel paradox is the no-cloning theorem, or the no “subatomic Xerox-machine” theorem. This theorem, which is related to the fact that one cannot copy quantum data at will, is a
consequence of Heisenberg’s famous Uncertainty Principle, by which one can measure either the position of a particle or its momentum, but not both with unlimited accuracy. According to the
Uncertainty Principle, it is impossible to have a subatomic Xerox-machine that would take one particle and spit out two particles with the same position and momentum – because then you would know too
much about both particles at once.
“We can always look at a paper, and then copy the words on it. That’s what we call copying classical data,” Wilde said. “But you can’t arbitrarily copy quantum data, unless it takes the special form
of classical data. This no-cloning theorem is a fundamental part of quantum mechanics – it helps us reason how to process quantum data. If you can’t copy data, then you have to think of everything in
a very different way.”
But if a Deutschian closed timelike curve did allow for copying of quantum data to many different points in space, then Deutsch suggested that it should be possible to violate the fundamental
no-cloning theorem of quantum mechanics. Now, Wilde and his collaborators at the University of Southern California and the Autonomous University of Barcelona have put forward a new approach that
allows for a particle, or a time traveler, to make multiple loops back in time – something akin to how Bruce Willis’ travels in the Hollywood film Looper. According to Wilde, “these time loops are
not ruled out by the laws of physics. But there are strange consequences for quantum information processing if their behavior is dictated by Deutsch’s model.”
Wilde says that a single looping path back in time, a time spiral of sorts, behaving according to Deutsch’s model, would have to allow for a particle entering the loop to remain the same each time it
passed through a particular point in time. In other words, the particle would need to maintain self-consistency as it looped back in time.
“In some sense, this already allows for copying of the particle’s data at many different points in space,” Wilde explains, “because you are sending the particle back many times. It’s like you have
multiple versions of the particle available at the same time. You can then attempt to read out more copies of the particle, but the thing is, if you try to do so as the particle loops back in time,
then you change the past.”
To be consistent with Deutsch’s model, which holds that you can only change the past as long as you can do it in a self-consistent manner, Wilde and his colleagues had to come up with a solution that
would allow for a looping curve back in time, and copying of quantum data based on a time traveling particle, without disturbing the past.
“That was the major breakthrough, to figure out what could happen at the beginning of this time loop to enable us to effectively read out many copies of the data without disturbing the past,” Wilde
said. “It just worked.”
But there is still some controversy over interpretations of this new approach and the new work may actually point to problems in Deutsch’s original closed timelike curve model. “If quantum mechanics
gets modified in such a way… it may be evidence that we should question Deutsch’s model,” Wilde said. “We really believe that quantum mechanics is true, at this point. And most people believe in a
principle called Unitarity in quantum mechanics. But with our new model, we’ve shown that you can essentially violate something that is a direct consequence of Unitarity. To me, this is an indication
that something weird is going on with Deutsch’s model. However, there might be some way of modifying the model in such a way that we don’t violate the no-cloning theorem.”
Other researchers argue that Wilde’s approach wouldn’t actually allow for copying quantum data from an unknown particle state entering the time loop because the universe would already “know” what the
particle looked like, as it had traveled back in time many times before.
But whether or not the no-cloning theorem can truly be violated as Wilde’s new approach suggests, the consequences of being able to copy quantum data from the past are significant. Systems for secure
Internet communications, for example, will likely soon rely on quantum security protocols. Such encryption is believed to be unbreakable – that is, as long as hackers don’t have access to Wilde’s
looping closed timelike curves.
“If an adversary, if a malicious person, were to have access to these time loops, then they could break the security of quantum key distribution,” Wilde said. “This ability to copy quantum
information freely would turn quantum theory into an effectively classical theory in which, for example, classical data thought to be secured by quantum cryptography would no longer be safe. It seems
like there should be a revision to Deutsch’s model which would simultaneously resolve the various time travel paradoxes but not lead to such striking consequences for quantum information processing.
However, no one yet has offered a model that meets these two requirements.”
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