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• Chaire Geolearning I am the Principal Investigator of the Geolearning Chair, with a funding from 2023 to 2027 This Chair brings together two research groups: the Geostatistics team from the
Centre de Géosciences (CG) at Mines Paris, and the Biostatistics and Spatial Processes Unit (BioSP), belonging to the MathNum division at INRAE. The general objective of the Geolearning Chair is
to develop and apply methods originating from the recent revolution in data science to challenges induced by the climate and ecological transitions our world is facing. More specifically, we will
design models and methods in geostatistics, extreme events theory and Machine Learning with applications to environmental, climate and risk sciences.
• Sébastien Coube-Sisqueille, currently post-doc at the Basque Center for Applied Mathematics visited me to work on multivariate spatio-temporal models in Bayesian context, with an application to
air pollution.
• Thomas Opitz is now the coordinator of the research network on risks, extreme events and spatio-temporal statitstics RESSTE (Risques, Extrêmes et StatistiquesSpatio-Temporelles). RESSTE is funded
by the Mathematics and digital technologies division (MathNum) of INRAE. It gathers more than 60 researches from about 20 research teams in France and abroad. We organize seminars and workshops
and we support all sorts of actions actions in view of developping models and methods for analyzing space-time data. Feel free to contact me or Thomas if you wish to be on the RESSTE mailing
• A RESSTE team, with Denis Allard (BioSP, INRAE), Lucia Clarotto (Geostatistics, Mines ParisTech), Thomas Opitz (BioSP, INRAE) and Thomas Romary (Geostatistics, Mines ParisTech) won the "2021
KAUST Competition on Spatial Statistics for Large Datasets" organized by the Spatio-Temporel Statistics & Data Science group of KAUST. 22 teams from all over the world took part in this
The competition was organized in 4 challenges consisting in estimating the parameters of several Gaussian fields and in making spatial predictions (kriging) under different conditions. The size
of the datasets ranged from 90,000 data to 900,000 data. The RESSTE team came out on top of 3 exercises and placed themselves on the podium in the fourth exercise.
All details are available on the competition website at https://cemse.kaust.edu.sa/stsds/2021-kaust-competition-spatial-statistics-large-dataset
• The Journal de la Société Française de Statistique recently published a Special Issue Statistics for spatial and spatio-temporal data and RESSTE Network, containing very interesting contributions
by several RESSTE network members.
• The paper Anisotropy Models for Spatial Data, Math. Geosc. Volume 48(3): 305–328, doi: 10.1007/s11004-015-9594-x., co-authered with Rachid Senoussi from BioSP and Emilio Porcu from University of
Newcastle was awarded best 2016 paper by the Journal Mathematical Geoscience. The paper offers a full characterization of anisotropic variograms, in terms of both regularity and range. It is
first shown that, if the regularity parameter is a continuous function of direction, it must necessarily be constant, whereas the scale parameter can vary in a continuous or discontinuous fashion
with direction. As a second result, it is then established that all valid anisotropies for the range parameter can be represented as a directional mixture of zonal anisotropies. This
representation makes it possible to build a very large class of anisotropic variograms, far more flexible than the classical anisotropies. A turning band algorithm for the simulation of Gaussian
anisotropic random fields, derived from the mixture representation, is then presented and illustrated.
Recent conferences I (co-) organized and/or I have been involved with:
• METMA 2018 is the ninth of a series of workshops on the topic of Spatio-Temporal Modelling, which have been held every two years since 17 years. We are proud to organize the first "French"
edition of this event in Montpellier. This workshop aims to promote the development and application of spatial, temporal, and mainly spatio-temporal statistical methods in different fields
related to the environment. It seeks to bring together practitioners and researchers of different areas and countries all over the world.The scientific program features sessions covering topics
on the latest advancements in theory, methods and applications.
• L’Université d’Avignon (UAVP) accueillera les Journées de Statistique 2017 du 29 mai au 2 juin sur le campus Hannah Arendt (anciennement Sainte-Marthe) au centre-ville d’Avignon. L'événement est
co-organisé par le laboratoire de mathématique d'Avignon, BioSP, le laboratoire d'Informatique d'Avignon et par l'UMR ESPACE.
• The 2015 edition of the Spatial Statistics Conference took place in Avignon, 9 - 12 June, 2015. It was co-chaired by Denis Allard (BioSP, INRA) and Alfred Stein (ITC). It was sponsored by the
Applied Mathematics and Computer Science division of INRA.
• BioSP hosted the Workshop on Stochastic Weather Generators from 17 - 19 september, 2014. This workshop brought together a wide range of researchers, practitioners, and graduate students whose
work is related to the stochastic modelling of meteorological variables and stochastic weather generators. Presentations can be found here.
• The 9th edition of the French-Danish Workshop took place in May 2012 in Avignon, France. It was jointly organized by the Biostatistics and Spatial Processes research unit (INRA) and the Dpt. of
Mathematics, LANLG (University of Avignon). It was devoted to spatial statistics and image analysis and their applications in biology (agriculture, aquaculture, ecology, economy, environment,
health, medicine, ...). Presentations can be found here. | {"url":"https://biosp.mathnum.inrae.fr/homepage-denis-allard","timestamp":"2024-11-04T10:09:40Z","content_type":"text/html","content_length":"105291","record_id":"<urn:uuid:99fbecf3-7329-489f-bc74-aac107084ae1>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00086.warc.gz"} |
Origin Detection During Food-borne Disease Outbreaks – A Case Study of the 2011 EHEC/HUS Outbreak in Germany
The key challenge during food-borne disease outbreaks, e.g. the 2011 EHEC/HUS outbreak in Germany, is the design of efficient mitigation strategies based on a timely identification of the outbreak’s
spatial origin. Standard public health procedures typically use case-control studies and tracings along food shipping chains. These methods are time-consuming and suffer from biased data collected
slowly in patient interviews. Here we apply a recently developed, network-theoretical method to identify the spatial origin of food-borne disease outbreaks. Thereby, the network captures the
transportation routes of contaminated foods. The technique only requires spatial information on case reports regularly collected by public health institutions and a model for the underlying food
distribution network. The approach is based on the idea of replacing the conventional geographic distance with an effective distance that is derived from the topological structure of the underlying
food distribution network. We show that this approach can efficiently identify most probable epicenters of food-borne disease outbreaks. We assess and discuss the method in the context of the 2011
EHEC epidemic. Based on plausible assumptions on the structure of the national food distribution network, the approach can correctly localize the origin of the 2011 German EHEC/HUS outbreak.
Funding Statement
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) research training group ‘Scaling Problems in Statistics’ (RTG 1644, www.uni-goettingen.de/en/156579.html) and the Volkswagen
Foundation and inspired by FutureICT (www.futurict.eu). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. The authors have
declared that no competing interests exist.
Due to intensified mass production, facilitated world-wide shipping and novel food manufacturing methods, food-borne disease outbreaks occur more frequently with increasing impacts on society, public
health institutions, the economy, and food industry^1. An estimated 60% of annual gastrointestinal illnesses for each adult in the general population of the United States is caused by food-borne
diseases^2. Moreover, diarrhoea is the second leading cause of morbidity and mortality among children under five years worldwide^3. food-borne diseases impose enormous financial burden on health care
services, routine surveillance and public health investigations, and trigger substantial productivity impacts and product recalls by the food industry. For seven food-borne pathogens an annual burden
of $6.5-$34.5 billion in the United States alone was estimated^4.
One of the most substantial challenges in this context is determining the spatial origin of the contaminated food vehicle, which causes the epidemic, for earlier and more effective disease
containment. Several factors make detection of the food-borne disease outbreak origin challenging, e.g. population growth, changing eating habits, globalization of food supply chains, production and
processing innovations, and microbiological adaptation^1^,^5. Furthermore, public health institutes have limited resources to solve issues such as underreporting and low specificity in the
association between aetiology and food vehicle^6. Origin reconstruction is a complex problem because the effects of contaminated food typically occur with a significant time lag and incidence
patterns are geographically incoherent. Additionally, specific transport pathways are generally not monitored. More importantly, food distribution networks are multi-scale, spanning length-scale of
hundreds to thousands of kilometers, delivering to and within spatially heterogeneous populations. Consequently, it is generically impossible to estimate the geographic origin of the phenomenon based
on geometric aspects of the spatial distribution of reported cases. Only for 66% of the outbreaks, public health investigations identified evidence concerning the infection source^7.
These practical difficulties were particularly striking during the German 2011 EHEC (enterohemorrhagic Escherichia coli) outbreak, which affected 3,842 people with unusually high rates of severe HUS
(hemolytic-uremic syndrome) cases and mortality. The EHEC/HUS outbreak raised the awareness of timely and efficient origin reconstruction methods and their importance to society, public health
institutions, risk assessment authorities and the food industry ^2. There is no general procedure for food-borne disease outbreak investigations, that fits a particular event perfectly. However, the
World Health Organization (WHO)^8 provides practical standard guidelines for the investigation and control of food-borne disease outbreaks as a multi-disciplinary task which requires information from
many sources.First, an unusual accumulation of disease reports has to be detected and defined as an outbreak. After pathogen specification, initial cases are investigated with regard to common
factors and clinical and food specimens are sampled. The corresponding microbiological ‘fingerprinting’ of strains may also identify case relatedness and/or potential sources of contamination.From
associated food and environmental samples, backward tracings are initiated to determine the origin. Furthermore, a case definition can be established to identify outbreak related cases and to collect
their information on a standardized questionnaire.Using this data, analytical investigations, such as case-control and cohort studies, are performed to test hypotheses about the transmission vehicle
and origin. The outbreak source is determined by combining all collected information, otherwise further analytical studies are required.Finally, the potential origin and transmission routes are
controlled using forward tracings from contamination to the outbreak cases. Several attempts to improve traceability of food products to their geographical origin have been developed including
technical innovations^9, microbiological advances^10, or food forensics^11. However, detection of outbreak origin remains time-consuming and cost-intensive.
Network theory and network models have become the most important tools for understanding and predicting epidemics in general^12^,^13^,^14. The majority of studies focuses on spatial disease dynamic
systems in which networks quantify the coupling strength or transportation fluxes between spatially distributed populations. Almost all studies aim at understanding and forecasting the future time
course of an epidemic based on the topological connectivity of the underlying transport networks^15^,^16. Furthermore, most studies focus on human-to-human transmissible diseases. Little work has
been done, however, on the inverse problem, also known as the ‘zero patient’ problem in epidemics. Shah and Zaman ^17^,^18 developed a universal source detection maximum likelihood estimate, which
assumes virus spread in a general graph along a breadth-first-search tree and derive theoretical thresholds for the detection probability. Pinto at al. ^19 extended this estimate for partially
observed transmission trees. Alternative origin reconstruction methods are based on shortest paths or consequent diameter from transmission trees ^20^,^21. Prakash et al. ^22 and Fioriti and Chinnici
^23 developed methods based on spectral techniques to identify a (set of) origin nodes on a transmission network. They utilize a close relationship of source estimation and node centrality as shown
by Comin and da Fontoura Costa ^24. However, these methods require comprehensive knowledge of the transmission network, which is rarely the case.
Here we apply a recently developed network-geometric approach for epicenter reconstruction^25 to food-borne diseases. This approach is based on a plausible redefinition of spatial separation and the
introduction of an effective distance derived from the underlying food distribution network in combination with viewing the contagion process from the perspective of a specific node in the network.
Using the effective distance method, complex spreading patterns can be mapped onto simple, regular wave propagation patterns if and only if the actual outbreak origin is chosen as the reference node.
This way, the method can determine the correct outbreak origin based on the degree of regularity of the measured prevalence distribution when viewed in the effective distance perspective. This
reconstruction is successful without the knowledge of the detailed infection hierarchy. Here, the underlying network captures the underlying transportation of the contaminated food rather than the
mobility pattern of humans.
German EHEC O104:H4/HUS outbreak 2011
Regarding the number of severe HUS cases, the 2011 EHEC/HUS outbreak in Germany, has been the largest E. coli outbreak reported worldwide. Between May 2 and July 26, 2011, 3,842 outbreak associated
EHEC cases were reported to the Robert Koch-Institute (RKI), the German Federal Public Health and Surveillance Institute. This included 855 severe HUS cases (22.3%) and 53 patients (1.4%) died. The
outbreak was caused by a rare serotype O104:H4 which infected predominantly adults (median age, 43 years), particularly women (68%), and resulted in high HUS and mortality rates ^26. In the previous
years, between 925 and 1,283 cases were reported annually, mostly in children. The majority of the infection cases was observed in Northern Germany, which resulted in a higher incidence (number of
cases per 100,000 inhabitants) for the corresponding districts than the overall one for Germany (see Fig. 1). Extensive investigations were conducted by the Task Force EHEC, which included a matched
case-control study, a recipe-based restaurant cohort study, and backward-/forward-tracings ^27. The entire process was complicated, resource demanding and time-consuming. All investigations required
a large amount of data that are typically biased, incomplete, erroneous, and sometimes contradictory. The tracings require a large amount of trained personnel and their success depends on the results
of epidemiological studies.Only the combination of several study designs finally lead to the determination of sprouts as the transmission vehicle and the identification of their origin, a farm in
Bienenbüttel located in the district Uelzen, Lower Saxony. On June 10, 38 days after outbreak onset, the public was informed to avoid sprout consumption and the responsible production farm was
Fig. 1: E. coli incidence in Germany during 2011 EHEC/HUS outbreak.
(A) Each panel depicts a different outbreak week (May 30th until June 20th, 2011). Color intensity quantifies infection counts in for each of the German districts (Data source: ^28, Map source: ^29).
The alleged origin of outbreak (district Uelzen) is marked in blue. (B) Time course of E. coli incidence for selected districts. For reference, the overall German incidence per district is shown in
The severe impact of the disease on the population and industry, the fast and wide spread due to mass production and optimized food shipping, and the large public attention emphasize the need for
fast and efficient outbreak origin localization.
Network-theoretic origin detection
We consider a model network for spatial food distribution, where nodes
are required to specify the network. The quantities
Because precise measurements of food distribution pathways are not available, we consider an established, approximate heuristic from the social sciences, economics and transportation theory known as
the gravity model ^30^,^31. This approach accounts for the observation that traffic flow increases monotonically with the population size between locations and decreases algebraically with distance,
leading to the relationship
where ^32^,^33. Plausible choices for these parameters can be found in the following way: First, we assume that the coupling strength between two locations
Furthermore, we let the coupling strength ^34^,^35, i.e.
Finally, we fix the scale parameter
The gravity model generates a fully connected network with strongly heterogeneous weights, contrasting realistic mobility or transportation networks that possess a sparse topology. In order to obtain
a more realistic model for food distribution that exhibits topological sparseness of connections, we follow a procedure recently introduced by Serrano et al. ^36. The idea of this approach is that
only links are retained that are statistically significant with respect to a random null model, in which traffic is distributed uniformly among links of a node. Following this idea, we first compute
the flux fraction
for each node
This approach yields a network skeleton of statistically significant links. Following this procedure the resulting network has an overall connectivity of 18%, see Fig. 2B.
Fig. 2: Multiscale Food Distribution in Germany
(A) A map of German districts; hues correspond to the regional network modules obtained by modularity maximization ^37; color intensity quantifies population density. The origin of the 2011 EHEC/HUS
outbreak is marked by a white circle in Bienenbüttel located in the district Uelzen. (B) German food shipping network constructed from a gravity model with parameters
One of the characteristic features of transportation networks in general, which is also captured by the above gravity model, is its multiscale structure. Although short-range links are usually
strongest, the algebraic tail in Eq. (2) yields long-range connections that can dominate spreading phenomena evolving on these networks. Qualitatively, this is illustrated in Fig. 3A which depicts an
simple planar quasi-lattice network, in which every node is connected only to its spatially adjacent nodes. Additionally, a few long-range, random connections are added. Because of long-range
connections in the network, an initially localized spreading process quickly attains a spatially incoherent structure. As a consequence of this, it is no longer possible to predict with ordinary
diffusion when a spreading process will arrive at a given location in the network. More importantly, it is difficult to reconstruct the outbreak origin from a snapshot (or a sequence of snapshots) of
the spatio-temporal pattern of spread alone based on conventional planar distance measures and two-dimensional geometry.
Effectively, two nodes that are connected by a long-range link in a multiscale network system are more adjacent than their spatial distance would suggest. Based on this basic and intuitive insight, a
recent study ^25 introduced the concept of effective distance to network-driven contagion or spreading phenomena. The most important result of this study is that spatio-temporally complex patterns of
spreading can be mapped onto simple, regular wave front patterns when conventional distance is replaced by a suitably chosen effective distance. This not only permits calculations of arrival times at
any node in the network but, more importantly, the identification of outbreak origins as will be explained in more detail below. The effective distance approach has been shown to work in the context
of infectious disease dynamics on a global scale, for instance, the worldwide spread of SARS in 2003 and pandemic influenza H1N1 in 2009.
The effective distance method assumes that, irrespective of the details of the local dynamics of a spreading process, the proliferation of the contagion throughout the network is determined by the
coupling between nodes, and that this coupling is quantified by the flux matrix elements
The probability of the contagion process taking this path is assumed to be given by the product of probabilities of each step
Here, for every link in the network the function ^25 is that the single step probability
Then, we define the effective distance of a multi-leg path
Here, the effective length of a direct link
This relation establishes a connection between network topological features and effective distance. The functional form is chosen such that a number of important features are fulfilled: (i) the
length from
Generically, transportation networks are strongly heterogeneous such that, in an ensemble of paths with origin ^25. The effective distance
From the perspective of a chosen root or reference node
Fig. 3: Effective distance and outbreak origin reconstruction in multi-scale network contagion processes.
(A) Each panel depicts a temporal snapshot (from left to right at equidistant time intervals) in a simple contagion process in which infected nodes (red) deliver the infection to connected nodes at a
fixed rate before they recover at a another rate (SIR dynamics ^38). The network consists of 512 nodes on a quasi-triangular, random lattice. Each node is connected to its nearest local neighbors. In
addition to the local lattice structure, 128 long range links exist between randomly chosen pairs of nodes. The origin of the outbreak is marked in green. Because of long range connectivity the
pattern quickly loses spatial structure and becomes chaotic such that it is difficult to predict from metric cues alone when the contagion arrives at a given node. More importantly, long range
connectivity leads to a loss of spatial coherence and it becomes impossible to determine the origin of outbreak. (B) The same pattern as in (A) is shown in the effective distance perspective from the
outbreak origin. The depicted tree is the shortest path tree, i.e. the most probable spreading path of the contagion process. Radial distance is proportional to effective distance as defined in the
text. In this alternative representation the complex pattern in the conventional view is mapped onto a simple propagating wave front and arrival times are easily computed. (C) The regularity of the
pattern is only present from the perspective of the actual outbreak origin. When the contagion process is viewed from any other node (here the node depicted in blue), the pattern lacks regularity.
Fig. 3B illustrates the advantages of this approach in an artificial multi-scale network. From the perspective of the outbreak origin, the shortest path tree of the root node is shown, and the radial
distance in the new map corresponds to the effective distance from the root node to the remaining nodes in the network. The same spreading process that appears to be spatio-temporally complex in the
conventional metric layout is equivalent to a regular, constant-speed spreading wave in the effective distance representation. Consequently, one can calculate arrival times based on effective
distance alone. In fact, in Brockmann and Helbing ^25 it was shown that effective distance from the outbreak origin and arrival time strongly correlate in real scenarios, e.g. the 2003 SARS epidemic
and the 2009 H1N1 pandemic influenza outbreak.
The most relevant consequence of the effective distance approach is that, only from the perspective of the actual outbreak origin, the pattern exhibits a regular concentric wave front structure. From
the perspective of any other node in the network, the pattern exhibits a more or less disordered structure. Fig. 3C illustrates this. The panels depict the same dynamics as in the other panels from a
randomly chosen reference node. Clearly, any spatial regularity is absent. One can now make use of this observation, i.e. the fact that the spreading pattern is regular only from the perspective of
the actual outbreak location, to reconstruct the outbreak origin. Given a snapshot of the disease spread, e.g. the disease incidence at every node, one computes the effective distance perspective for
each node in the network and quantifies, from which node the pattern appears to be most regular. The node with maximum regularity is considered to be the most likely outbreak origin. In the following
we apply this approach to the 2011 EHEC/HUS outbreak in Germany.
Fig. 4: Shortest path trees and effective distance among districts in Germany.
Each column depicts the shortest path tree
Detection of the German EHEC/HUS outbreak origin
Given the gravity model network for food transportation, we first compute the shortest path tree ^25.
In combination, small mean and variance are equivalent to high concentricity and, thus, high likelihood that the chosen reference node is the likely outbreak origin.
Fig. 5: EHEC/HUS outbreak origin reconstruction
Each panel depicts a scatterplot of mean
We used the public available E. coli case count data with report date between calendar weeks 18 and 26 of 2011 ^28. According to the Task Force EHEC, this corresponds to the entire outbreak duration
from May 2nd until July 4th, 2011 ^26. Fig. 5 shows the results of origin detection when the effective distance approach in combination with a gravity model for food distribution is applied to the
EHEC incidence data. Since an E. coli infection clustering was noticed at May 19th, 2011 (outbreak week 3), we computed the mean
Fig. 6: EHEC/HUS outbreak origin reconstruction
For each week 2-9 relative to the beginning of the EHEC/HUS outbreak and for each node
Fig. 7: Correlation of effective distance and arrival time during the German EHEC/HUS outbreak, 2011.
For each district as a potential outbreak origin, we computed the correlation coefficient of arrival time
The effective distance method provides an alternative method for outbreak origin reconstruction. An important result presented in Ref. ^25 is that arrival times of a network-driven contagion process
correlate strongly with effective distance. In fact, the arrival time
Fig. 8: Effective distance and arrival time analysis
For each potential district
Discussion and conclusion
We introduced a fast and efficient approach for the identification of the origin during food-borne disease outbreaks and evaluated the approach in the context of the 2011 EHEC/HUS outbreak in
Germany. A clear advantage of the method is the robust performance on the basis of limited case report data and plausible topological assumptions concerning the underlying food distribution network.
When applied to the 2011 EHEC/HUS outbreak in Germany, our method was able to identify an outbreak origin in close proximity to the actual outbreak location (Uelzen, Lower Saxony). Already three days
(May 22nd, 2011) after spatial infection clustering, the effective distance approach was able to reconstruct the actual origin. This is particularly promising, as in the context of EHEC/HUS,
conventional outbreak investigations, including case-control- and cohort-studies as well as sample testings and tracings along the food-shipping chain,wrongly suggested tomatoes, leafy salads and
cucumbers as contaminated foods. When specific suspicions arose that cucumbers imported in Hamburg would be the infection source, our method classifies Hamburg to be a very unlikely origin. The
consideration of such contradictory information could have lead to more spatially targeted sample testing, and, therefore could have improved the efficiency of the outbreak investigations.
We believe that this method can complement conventional methods of origin localization of food-borne diseases and consequently facilitate a more timely success which is vital for the development of
containment strategies. The underlying network definition by the gravity model is very flexible, so that the transmission vehicle does not has to be known. Basically, the network could also capture a
combination of food transportation routes as well as human mobility pattern. As our method is structurally quite general and just derived from topological features of the underlying distribution
networks, we believe that our approach may be adapted and applied to a variety of contagion phenomena, human-to-human transmissible diseases, and disease dynamics on individual based contact networks
and human-mediated bioinvasion processes.
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• Woolley-Meza O, Thiemann C, Grady D, Lee JJ, Seebens H, Blasius B, et al. Complexity in human transporta- tion networks: a comparative analysis of worldwide air transportation and global
cargo-ship movements. The European Physical Journal B. 2011 Dec;84(4):589–600.
Reference Link
• Anderson RM, May RM. Infectious diseases of humans : dynamics and control. Oxford New York: Oxford University Press; 1991.
Leave a Comment
You must be logged in to post a comment. | {"url":"https://currents.plos.org/outbreaks/article/origin-detection-during-food-borne-disease-outbreaks-a-case-study-of-the-2011-ehechus-outbreak-in-germany-2/","timestamp":"2024-11-13T12:43:16Z","content_type":"text/html","content_length":"99431","record_id":"<urn:uuid:ad2e5d58-aef5-4d08-9446-9a8cfdfeaabe>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00661.warc.gz"} |
2,718 research outputs found
Formulating a perfect fluid filled spherically symmetric metric utilizing the 3+1 formalism for general relativity, we show that the metric coefficients are completely determined by the mass-energy
distribution, and its time rate of change on an initial spacelike hypersurface. Rather than specifying Schwarzschild coordinates for the exterior of the collapsing region, we let the interior dictate
the form of the solution in the exterior, and thus both regions are found to be written in one coordinate patch. This not only alleviates the need for complicated matching schemes at the interface,
but also finds a new coordinate system for the Schwarzschild spacetime expressed in generalized Painleve-Gullstrand coordinates.Comment: 3 pages, To appear in the proceedings of the eleventh Marcel
Grossmann meeting on general relativity (MGXI), 23-29 July, 2006, Berli
We study the properties of strongly interacting Bose gases at the density and temperature regime when the three-body recombination rate is substantially reduced. In this regime, one can have a Bose
gas with all particles in scattering states (i.e. the "upper branch") with little loss even at unitarity over the duration of the experiment. We show that because of bosonic enhancement, pair
formation is shifted to the atomic side of the original resonance (where scattering length $a_s<0$), opposite to the fermionic case. In a trap, a repulsive Bose gas remains mechanically stable when
brought across resonance to the atomic side until it reaches a critical scattering length $a_{s}^{\ast}<0$. For $a_s<a_{s}^{\ast}$, the density consists of a core of upper branch bosons surrounded by
an outer layer of equilibrium phase. The conditions of low three-body recombination requires that the particle number $N<\alpha (T/\omega)^{5/2}$ in a harmonic trap with frequency $\omega$, where $\
alpha$ is a constant.Comment: 4 pages, 4 figure
One feature of many naturally occurring or engineered complex systems is tremendous variability in event sizes. To account for it, the behavior of these systems is often described using power law
relationships or scaling distributions, which tend to be viewed as "exotic" because of their unusual properties (e.g., infinite moments). An alternate view is based on mathematical, statistical, and
data-analytic arguments and suggests that scaling distributions should be viewed as "more normal than normal". In support of this latter view that has been advocated by Mandelbrot for the last 40
years, we review in this paper some relevant results from probability theory and illustrate a powerful statistical approach for deciding whether the variability associated with observed event sizes
is consistent with an underlying Gaussian-type (finite variance) or scaling-type (infinite variance) distribution. We contrast this approach with traditional model fitting techniques and discuss its
implications for future modeling of complex systems
TCP-AQM can be interpreted as distributed primal-dual algorithms to maximize aggregate utility over source rates. We show that an equilibrium of TCP/IP, if exists, maximizes aggregate utility over
both source rates and routes, provided congestion prices are used as link costs. An equilibrium exists if and only if this utility maximization problem and its Lagrangian dual have no duality gap. In
this case, TCP/IP incurs no penalty in not splitting traffic across multiple paths. Such an equilibrium, however, can be unstable. It can be stabilized by adding a static component to link cost, but
at the expense of a reduced utility in equilibrium. If link capacities are optimally provisioned, however, pure static routing, which is necessarily stable, is sufficient to maximize utility.
Moreover single-path routing again achieves the same utility as multipath routing at optimality
Although the ``scale-free'' literature is large and growing, it gives neither a precise definition of scale-free graphs nor rigorous proofs of many of their claimed properties. In fact, it is easily
shown that the existing theory has many inherent contradictions and verifiably false claims. In this paper, we propose a new, mathematically precise, and structural definition of the extent to which
a graph is scale-free, and prove a series of results that recover many of the claimed properties while suggesting the potential for a rich and interesting theory. With this definition, scale-free (or
its opposite, scale-rich) is closely related to other structural graph properties such as various notions of self-similarity (or respectively, self-dissimilarity). Scale-free graphs are also shown to
be the likely outcome of random construction processes, consistent with the heuristic definitions implicit in existing random graph approaches. Our approach clarifies much of the confusion
surrounding the sensational qualitative claims in the scale-free literature, and offers rigorous and quantitative alternatives.Comment: 44 pages, 16 figures. The primary version is to appear in
Internet Mathematics (2005 | {"url":"https://core.ac.uk/search/?q=author%3A(C.%20Lun)","timestamp":"2024-11-12T19:49:59Z","content_type":"text/html","content_length":"140039","record_id":"<urn:uuid:3d4dc632-5fa4-410f-859c-34c9a04aff01>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00664.warc.gz"} |
Zhi-Qin John Xu (Courant Institute of Mathematical Science, New York University), Applied Mathematics Colloquium - Department of Mathematics
Zhi-Qin John Xu (Courant Institute of Mathematical Science, New York University), Applied Mathematics Colloquium
February 10, 2017 @ 4:00 pm - 5:00 pm
Title: A Probability Polling State — the Maximum Entropy Principle in Neuronal Data Analysis
Abstract: How to extract information from exponentially growing recorded neuronal data is a great scientific challenge. It is urgent to develop methods to simplify the analysis of neuronal data. In
this talk, we address what kind of dynamical states of neuronal networks allows us to have an effective description of coding schemes. For asynchronous neuronal networks, when considering the
probability increment of a neuron spiking induced by other neurons, we found a probability polling (p-polling) state that captures the neuronal interactions which are affected by multiple factors,
i.e., coupling structure, background input and external input. We show that this state is confirmed in some experiments in vitro and in vivo, and also confirmed through the simulation of
Hodgkin-Huxley neuronal networks. We hypothesize that this p-polling state may be a general operating state of neuronal networks. For the p-polling state, we show that neuronal firing patterns can be
well captured by the 2nd order maximum entropy model. | {"url":"https://math.unc.edu/event/zhi-qin-john-xu-courant-institute-of-mathematical-science-new-york-university-applied-mathematics-colloquium/","timestamp":"2024-11-02T11:24:04Z","content_type":"text/html","content_length":"114233","record_id":"<urn:uuid:c28e1c6a-e5cd-41ed-9b1b-4944873dccd9>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00682.warc.gz"} |
Another Attempt to Avoid a Beginning Collapses | Evolution News
Another Attempt by an Esteemed Cosmologist to Avoid a Cosmic Beginning Collapses on Inspection
In previous articles, I described how several recent pieces by physicists assert that the universe might not have had a beginning. I explained how each of the presented arguments was already fully
addressed by Stephen Meyer in his book Return of the God Hypothesis or in the extended research notes (here, here, here). The website The Conversation recently published an article by philosopher of
science Alastair Wilson titled “How could the Big Bang arise from nothing?” Wilson presents a cosmological model constructed by mathematical physicist and cosmologist Roger Penrose dubbed “conformal
cyclical cosmology” (CCC) that purportedly avoids a beginning.
Penrose is considered one of the preeminent physicists of our day. He performed the famous calculation that the fine tuning of the entropy at the beginning of the universe measures at 1 part in 10 to
the power of 10 to the power of 123. This number could not be written if a zero were placed on every particle in the visible universe. Penrose is a true genius, and he has performed a herculean
effort to avoid the universe’s beginning. But CCC is founded on numerous highly dubious assumptions, and it contradicts the empirical evidence.
Penrose’s Model
Penrose envisions that the universe is eternally expanding. And it periodically transitions during “crossover events” from the dying embers of an ancient universe to the initiation of a new universe
at a big bang event. The crossover occurs after all black holes have evaporated away and the mass of all particles drops to zero. The energy distribution throughout the universe then appears
homogeneous as it did after the previous crossover (see figure).
At each crossover, a hypothesized “phantom field” transitions from a purely mathematic entity into a physical field that rapidly acquires mass and dominates over all other fields. The spacetime
geometry at the end of one epoch matches the geometry at the beginning of the next. But lengths are rescaled so that the enormous volume of the old epoch is treated as a miniscule volume after a big
bang. The rescaling causes the extremely cold temperature and diffuse concentration of photons at the end of the old epoch to appear after a big bang as an extremely hot and dense state. The
“effective entropy” also appears smaller as required to match the observed low entropy in our universe.
Penrose comments in “The Big Bang and its Dark-Matter Content: Whence, Whither, and Wherefore”:
This conformal freedom allows us to stretch out the hot big bang of the succeeding aeon and to squash down the remote future of the previous one — bearing in mind that energy and momentum scale
in the exact inverse way to space and time. So hot becomes cold and dense becomes rarefied upon conformal stretching…
When all black holes have evaporated away, at the crossover 3-surface, the effective entropy will have dropped to the very low value that is required to start off the next aeon. Thus, the 2nd Law
is not violated; it is transcended in the sense that the effective entropy definition has to shift down to that which is relevant to the new aeon.
Penrose argues that his model is supported by evidence from patterns observed in the cosmic microwave background radiation (CMBR). Specifically, he claims to have identified concentric circles and
“anomalous points” that could represent an imprint of the universe before the most recent big bang.
Penrose and His Critics
Yet CCC has faced severe criticism from other cosmologists. The concentric circles in the CMBR could not be identified by other research groups. And many have argued that the CMBR data fits other
cosmological models far better than CCC. In addition, the assumption that electrons will eventually lose their mass is not consistent with the Standard Model of particle physics. Physicist Juliane
Barbour stated in “Inside Penrose’s Universe”:
There are numerous problems to be overcome in this proposal, which involves a radical rethinking of Penrose’s own ideas about the second law. One serious difficulty is that it relies heavily on
all particle masses, including that of the electron, becoming exactly zero in the very distant future. Many particle physicists will question that. But the biggest difficulty of all is that even
if the shapes of the aeons match, how does the transition from an infinitely large scale before crossover to an infinitely small scale after crossover occur? This is where the argumentation and
mathematics get tough…
Penrose effects the crossover with a scalar field dubbed “phantom” before the crossover, because it involves a purely mathematical conformal transformation. This field then becomes physical after
crossover… its transformation “at once” from being a purely mathematical object to a physical one has no parallel elsewhere in physics — unless one likens it to the notorious collapse of the
wavefunction in quantum mechanics.
Astrophysicist Ethan Segal even more blunted stated in “No, Roger Penrose, We See No Evidence Of A ‘Universe Before The Big Bang’”:
Although, much like Hoyle, Penrose isn’t alone in his assertions, the data is overwhelmingly opposed to what he contends. The predictions that he’s made are refuted by the data, and his claims to
see these effects are only reproducible if one analyzes the data in a scientifically unsound and illegitimate fashion. Hundreds of scientists have pointed this out to Penrose — repeatedly and
consistently over a period of more than 10 years — who continues to ignore the field and plow ahead with his contentions.
Questionable Assumptions
An additional problem is that Penrose’s model requires several highly questionable assumptions. First, it must overcome the implications of the Borde-Guth-Vilenkin theorem that proves that expanding
universes must have an absolute beginning. To avoid this conclusion, Penrose must assume that the universe was infinitely large in the infinite past, which is philosophically problematic. Additional
unproven assumptions include the following:
• All particle masses dropping to zero.
• Presence of a scalar field that becomes active at the right time to trigger crossover.
• Mass of the scalar field rapidly increases after crossover.
Given the lack to supporting evidence and the ad hoc assumptions, CCC offers no serious challenge to the evidence that the universe had a beginning. Therefore, something, or more likely someone,
outside of time and space must have created it. | {"url":"https://evolutionnews.org/2022/01/another-attempt-by-an-esteemed-cosmologist-to-avoid-a-cosmic-beginning-collapses-on-inspection/","timestamp":"2024-11-04T20:23:14Z","content_type":"text/html","content_length":"170739","record_id":"<urn:uuid:ca24e32d-0629-4e49-92bb-12451bb1e6cd>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00772.warc.gz"} |
Physics 132: What is an Electron? What is Light?
De Broglie Wavelength
In 1923 a French physics graduate student named Prince Louis-Victor de Broglie (1892–1987) made a radical proposal based on the hope that nature is symmetric. If EM radiation has both particle and
wave properties, then nature would be symmetric if matter also had both particle and wave properties. If what we once thought of as an unequivocal wave (EM radiation) is also a particle, then what we
think of as an unequivocal particle (matter) may also be a wave. De Broglie’s suggestion, made as part of his doctoral thesis, was so radical that it was greeted with some skepticism. A copy of his
thesis was sent to Einstein, who said it was not only probably correct, but that it might be of fundamental importance. With the support of Einstein and a few other prominent physicists, de Broglie
was awarded his doctorate.
De Broglie took both relativity and quantum mechanics into account to develop the proposal that all particles have a wavelength, given by
where h is Planck’s constant and p is momentum. This is defined to be the de Broglie wavelength. (Note that we already have this for photons, from the equation p=h/λ.) The hallmark of a wave is
interference. If matter is a wave, then it must exhibit constructive and destructive interference. Why isn’t this ordinarily observed? The answer is that in order to see significant interference
effects, a wave must interact with an object about the same size as its wavelength. Since h is very small, λ is also small, especially for macroscopic objects. A 3-kg bowling ball moving at 10 m/s,
for example, has
This means that to see its wave characteristics, the bowling ball would have to interact with something about 10^–35m in size—far smaller than anything known. When waves interact with objects much
larger than their wavelength, they show negligible interference effects and move in straight lines (such as light rays in geometric optics). To get easily observed interference effects from particles
of matter, the longest wavelength and hence smallest mass possible would be useful. Therefore, this effect was first observed with electrons.
All microscopic particles, whether massless, like photons, or having mass, like electrons, have wave properties. The relationship between momentum and wavelength is fundamental for all particles.
American physicists Clinton J. Davisson and Lester H. Germer in 1925 and, independently, British physicist G. P. Thomson (son of J. J. Thomson, discoverer of the electron) in 1926 scattered electrons
from crystals and found diffraction patterns. These patterns are exactly consistent with interference of electrons having the de Broglie wavelength and are somewhat analogous to light interacting
with a diffraction grating. (See Figure 1.)
Figure 1: This diffraction pattern was obtained for electrons diffracted by crystalline silicon. Bright regions are those of constructive interference, while dark regions are those of destructive
interference. (credit: Ndthe, Wikimedia Commons)
De Broglie’s proposal of a wave nature for all particles initiated a remarkably productive era in which the foundations for quantum mechanics were laid. In 1926, the Austrian physicist Erwin
Schrödinger (1887–1961) published four papers in which the wave nature of particles was treated explicitly with wave equations. At the same time, many others began important work. Among them was
German physicist Werner Heisenberg (1901–1976) who, among many other contributions to quantum mechanics, formulated a mathematical treatment of the wave nature of matter that used matrices rather
than wave equations. We will deal with some specifics in later sections, but it is worth noting that de Broglie’s work was a watershed for the development of quantum mechanics. De Broglie was awarded
the Nobel Prize in 1929 for his vision, as were Davisson and G. P. Thomson in 1937 for their experimental verification of de Broglie’s hypothesis.
Electron Wavelength versus Velocity and Energy
For an electron having a de Broglie wavelength of 0.167 nm (appropriate for interacting with crystal lattice structures that are about this size): (a) Calculate the electron’s velocity. (b) Calculate
the electron’s kinetic energy in eV.
For part (a), since the de Broglie wavelength is given, the electron’s velocity can be obtained from λ=h/p
by using the nonrelativistic formula for momentum, p=mv. For part (b), once v is obtained (and it has been verified that v is nonrelativistic), the classical kinetic energy is simply (1/2)mv2.
Solution for (a)
Substituting the formula for momentum (p=mv) into the de Broglie wavelength gives
Solving for v gives
Substituting known values yields
Solution for (b)
While fast compared with a car, this electron’s speed is not close to the speed of light, and so we can comfortably use the classical formula to find the electron’s kinetic energy and convert it to
eV as requested.
Electron Microscopes
One consequence or use of the wave nature of matter is found in the electron microscope. As we have discussed, there is a limit to the detail observed with any probe having a wavelength. Resolution,
or observable detail, is limited to about one wavelength. Since a potential of only 54 V can produce electrons with sub-nanometer wavelengths, it is easy to get electrons with much smaller
wavelengths than those of visible light (hundreds of nanometers). Electron microscopes can, thus, be constructed to detect much smaller details than optical microscopes. (See Figure 2.)
Figure 2: Schematic of a scanning electron microscope (SEM) (a) used to observe small details, such as those seen in this image of a tooth of a Himipristis, a type of shark (b). (credit: Dallas
Krentzel, Flickr)
There are basically two types of electron microscopes. The transmission electron microscope (TEM) accelerates electrons that are emitted from a hot filament (the cathode). The beam is broadened and
then passes through the sample. A magnetic lens focuses the beam image onto a fluorescent screen, a photographic plate, or (most probably) a CCD (light sensitive camera), from which it is transferred
to a computer. The TEM is similar to the optical microscope, but it requires a thin sample examined in a vacuum. However it can resolve details as small as 0.1 nm (10^−10m), providing magnifications
of 100 million times the size of the original object. The TEM has allowed us to see individual atoms and structure of cell nuclei.
The scanning electron microscope (SEM) provides images by using secondary electrons produced by the primary beam interacting with the surface of the sample (see Figure 2). The SEM also uses magnetic
lenses to focus the beam onto the sample. However, it moves the beam around electrically to “scan” the sample in the x and y directions. A CCD detector is used to process the data for each electron
position, producing images like the one at the beginning of this chapter. The SEM has the advantage of not requiring a thin sample and of providing a 3-D view. However, its resolution is about ten
times less than a TEM.
Electrons were the first particles with mass to be directly confirmed to have the wavelength proposed by de Broglie. Subsequently, protons, helium nuclei, neutrons, and many others have been observed
to exhibit interference when they interact with objects having sizes similar to their de Broglie wavelength. The de Broglie wavelength for massless particles was well established in the 1920s for
photons, and it has since been observed that all massless particles have a de Broglie wavelength p=h/λ.
The wave nature of all particles is a universal characteristic of nature. We shall see in following sections that implications of the de Broglie wavelength include the quantization of energy in atoms
and molecules, and an alteration of our basic view of nature on the microscopic scale. The next section, for example, shows that there are limits to the precision with which we may make predictions,
regardless of how hard we try. There are even limits to the precision with which we may measure an object’s location or energy.
Making Connections: A Submicroscopic Diffraction Grating
The wave nature of matter allows it to exhibit all the characteristics of other, more familiar, waves. Diffraction gratings, for example, produce diffraction patterns for light that depend on grating
spacing and the wavelength of the light. This effect, as with most wave phenomena, is most pronounced when the wave interacts with objects having a size similar to its wavelength. For gratings, this
is the spacing between multiple slits.) When electrons interact with a system having a spacing similar to the electron wavelength, they show the same types of interference patterns as light does for
diffraction gratings, as shown at top left in Figure 3.
Atoms are spaced at regular intervals in a crystal as parallel planes, as shown in the bottom part of Figure 3. The spacings between these planes act like the openings in a diffraction grating. At
certain incident angles, the paths of electrons scattering from successive planes differ by one wavelength and, thus, interfere constructively. At other angles, the path length differences are not an
integral wavelength, and there is partial to total destructive interference. This type of scattering from a large crystal with well-defined lattice planes can produce dramatic interference patterns.
It is called Bragg reflection, for the father-and-son team who first explored and analyzed it in some detail. The expanded view also shows the path-length differences and indicates how these depend
on incident angle θ in a manner similar to the diffraction patterns for x rays reflecting from a crystal.
Figure 3: The diffraction pattern at top left is produced by scattering electrons from a crystal and is graphed as a function of incident angle relative to the regular array of atoms in a crystal, as
shown at bottom. Electrons scattering from the second layer of atoms travel farther than those scattered from the top layer. If the path length difference (PLD) is an integral wavelength, there is
constructive interference.
Section Summary
• Particles of matter also have a wavelength, called the de Broglie wavelength, given by p=h/λ, where p is momentum.
• Matter is found to have the same interference characteristics as any other wave.
Problem 26: Find the wavelength of a golf ball.
Problem 27: Given an electron’s wavelength, what is its speed? | {"url":"http://openbooks.library.umass.edu/toggerson-132/chapter/matter-as-a-wave/","timestamp":"2024-11-07T00:44:32Z","content_type":"text/html","content_length":"122106","record_id":"<urn:uuid:9ad064b3-1116-489b-8225-ab38b38d6169>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00511.warc.gz"} |
Rectifiability of stationary varifolds branching set with multiplicity at most 2
Ph.D. Thesis
Inserted: 26 feb 2024
Last Updated: 26 feb 2024
Year: 2023
Links: https://iris.uniroma1.it/handle/11573/1682911
This thesis deals with regularity and rectifiability properties on the branching set of stationary varifolds that can be represented as the graph of a two-valued function. In the first chapter I
briefly show the Simon and Wickramasekera’s work in which they introduce a frequency function monotonicity formula for two-valued $C^{1,\alpha}$ functions with stationary graph that leads to an
estimate of the Hausdorff dimension of the branching set. In the second chapter I build upon Simon and Wickramasekera’s work and introduce several relaxed frequency functions in order to get an
estimate of the Minkowski’s content of the branching set. I then use their result to prove the local $(n − 2)$-rectifiablility of the branching set.
Keywords: calculus of variations, Geometric measure theory, minimal surfaces | {"url":"https://cvgmt.sns.it/paper/6434/","timestamp":"2024-11-14T11:46:32Z","content_type":"text/html","content_length":"8637","record_id":"<urn:uuid:932f6341-1175-4827-b69d-cf24891b23c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00762.warc.gz"} |
Feed Stats - What Matters
I know that a lot of people watch the little FeedBurner chicklets that show how many subs any given blog has.
Here is mine:
Here is TechCrunch’s:
So what does this mean? Not as much as everyone thinks. FeedBurner has a really good post up on their blog explaining why these numbers have been going up a lot recently.
The main reason is that Google has started to report their feed subs to FeedBurner, both Google Homepage (their MyYahoo/Netvibes like service) and Google Reader. Since Google has about 20% of the
reader market, everyone’s subs went up on average 20% recently.
The other thing that goes on is some of the aggregators auto subscribe their users to the top blogs. So some blogs get huge boosts in subscribers. My blog has ~5000 subs from Rojo. I don’t think they
are real but they get included in my total of 19,714.
So what to do? It’s really simple. Focus on "reach". FeedBurner calculates a reach number every day for every feed it manages. Here’s a screenshot of the popup you get when you click on "what is
reach" next to your reach number in the FeedBurner dashboard:
Here are my reach numbers for the past week and my site visitors in parentheses for comparison:
Monday – 2,540 (3,079)
Tuesday – 2,736 (3,738)
Wednesday – 2,987 (3,180)
Thursday – 2,878 (3,042)
Friday – 2,534 (2,733)
So you can see the number of people who read this blog each day via feeds is pretty close to the number who read it on the web. And there is certainly some overlap (that would be a great number for
FeedBurner to calculate and I am sure they can do that).
So on any given week, this blog gets between 5,500 and 6,500 readers, about 40% of which are via the feed and 60% are via the web.
So what is the relevance of the 19,714 number? Not much actually. It’s the number of people who at one point have subscribed to this blog. If you subtract the daily reach of approximately 3,000
people, you see that over 16,000 subscriptions are out there but not particularly active.
I suspect the same is true of Techcrunch’s 279,000 readers as well. If their feed subscribers are like mine, Techcrunch’s daily reach would be about 50,000. That’s still a huge number of people
reading a feed every day. | {"url":"https://avc.com/2007/02/feed_stats_what/","timestamp":"2024-11-10T14:53:58Z","content_type":"text/html","content_length":"35005","record_id":"<urn:uuid:d3f07244-4ea6-48cf-a581-19657717f67f>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00535.warc.gz"} |
What is: Cox Proportional Hazards Model
What is the Cox Proportional Hazards Model?
The Cox Proportional Hazards Model, often referred to simply as the Cox model, is a statistical technique used primarily in survival analysis. Developed by Sir David Cox in 1972, this model is
designed to explore the relationship between the survival time of subjects and one or more predictor variables. Unlike other survival analysis methods, the Cox model does not require the assumption
of a specific baseline hazard function, making it a semi-parametric model. This flexibility allows researchers to analyze time-to-event data without needing to specify the underlying distribution of
survival times.
Key Components of the Cox Proportional Hazards Model
The Cox model is based on the concept of hazard functions, which represent the instantaneous risk of an event occurring at a given time, conditioned on survival until that time. The model expresses
the hazard function for an individual as a product of a baseline hazard function and an exponential function of the predictor variables. Mathematically, this can be represented as:
[ h(t|X) = h_0(t) cdot e^{beta_1X_1 + beta_2X_2 + … + beta_pX_p} ]
where ( h(t|X) ) is the hazard at time ( t ) for an individual with covariates ( X ), ( h_0(t) ) is the baseline hazard function, and ( beta_1, beta_2, …, beta_p ) are the coefficients corresponding
to the predictor variables ( X_1, X_2, …, X_p ).
Assumptions of the Cox Proportional Hazards Model
One of the fundamental assumptions of the Cox Proportional Hazards Model is the proportional hazards assumption. This means that the ratio of the hazard functions for any two individuals is constant
over time, regardless of the time point being considered. In practical terms, this implies that the effect of the predictor variables on the hazard is multiplicative and does not change as time
progresses. It is crucial for analysts to verify this assumption through diagnostic plots or statistical tests, as violations can lead to incorrect conclusions.
Applications of the Cox Proportional Hazards Model
The Cox Proportional Hazards Model is widely utilized across various fields, including medicine, epidemiology, and social sciences. In clinical research, it is often employed to assess the impact of
treatment variables on patient survival times. For instance, researchers might use the model to evaluate how different chemotherapy regimens affect the survival of cancer patients. Additionally, the
model can be applied in engineering to analyze failure times of mechanical systems or in economics to study time until an event, such as bankruptcy.
Model Fitting and Interpretation
Fitting the Cox Proportional Hazards Model typically involves using maximum likelihood estimation to derive the coefficients for the predictor variables. Software packages such as R, SAS, and
Python’s lifelines library provide robust tools for implementing the Cox model. Once the model is fitted, the coefficients can be interpreted in terms of hazard ratios. A hazard ratio greater than
one indicates an increased risk of the event occurring, while a ratio less than one suggests a protective effect. This interpretation is crucial for understanding the influence of covariates on
survival outcomes.
Limitations of the Cox Proportional Hazards Model
Despite its widespread use, the Cox Proportional Hazards Model has limitations. One significant limitation is its reliance on the proportional hazards assumption, which, if violated, can lead to
biased estimates. Additionally, the model does not handle time-varying covariates directly; instead, researchers must employ techniques such as stratification or time-dependent covariates to account
for changes over time. Furthermore, the Cox model is not suitable for data with non-proportional hazards, necessitating alternative approaches like the accelerated failure time model or parametric
survival models.
Extensions of the Cox Proportional Hazards Model
To address some of the limitations of the traditional Cox Proportional Hazards Model, several extensions have been developed. One such extension is the stratified Cox model, which allows for the
inclusion of stratification factors that can account for different baseline hazard functions across groups. Another extension is the time-varying coefficients model, which permits the effects of
covariates to change over time. These advanced models provide researchers with more flexibility and accuracy when analyzing complex survival data.
Model Diagnostics and Validation
Model diagnostics are essential for ensuring the validity of the Cox Proportional Hazards Model. Common diagnostic techniques include checking the proportional hazards assumption using graphical
methods, such as Schoenfeld residual plots, and conducting statistical tests like the Grambsch and Therneau test. Additionally, researchers should assess the overall fit of the model through methods
such as the likelihood ratio test or Akaike Information Criterion (AIC). Validating the model using independent datasets is also crucial to confirm its predictive performance and generalizability.
Conclusion and Future Directions
The Cox Proportional Hazards Model remains a cornerstone of survival analysis, providing valuable insights into the relationship between covariates and time-to-event outcomes. As data science
continues to evolve, researchers are exploring new methodologies and computational techniques to enhance the model’s applicability and robustness. Future directions may include integrating machine
learning approaches with traditional survival analysis methods, allowing for more nuanced and accurate modeling of complex datasets. | {"url":"https://statisticseasily.com/glossario/what-is-cox-proportional-hazards-model/","timestamp":"2024-11-05T07:34:15Z","content_type":"text/html","content_length":"139449","record_id":"<urn:uuid:8ba02008-d4b7-47f0-a17f-e1b320a0c6e4>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00875.warc.gz"} |
cup unit symbol
No extra packages are required to use these symbols. Enter a "Start" value (5, 100 etc). A trophy in the shape of an oversized cup. A deciliter is a unit of volume in the Metric System. Below is the
Alt code keyboard shortcut for inserting the cubed symbol.If you are new to ALT codes and need detailed instructions on how to use them, please read How to Use ALT Codes to Enter Special Characters..
For the the complete list of the ASCII based Windows ALT Codes, refer to Windows ALT Codes for Special Characters & Symbols. It is equal to one one-thousandth of the SI base unit, the kilogram, or
1E3 kg. Exchange reading in US cups of water unit cup into pounds of water unit lb wt. 1561 CUP to INR exchange rate Jan, 2021 and 1561 Cuban Peso to Indian Rupee conversion data by Conversion Ai
provides historical chart price for 1561 Cuban Peso to Indian Rupee with easy to use tools like 1561 CUP to INR converter to help you get the best 1561 CUP to INR quote today. Although the kilogram
is the base unit of the SI metric weight system, all units have prefixes based on the gram. The base unit for a deciliter is liter and the prefix is deci. Unit symbols used by international culinary
educational institutions and training for these two granulated sugar measures are: Prefix or abbreviation ( abbr. ) Cup is a volume unit. Refer to the external references at the end of this article
for more information. Our currency rankings show that the most popular Cuba Peso exchange rate is the USD to CUP rate. The prefixes indicate whether the unit is a multiple or a fraction of the base
ten. How to use cup in a sentence. One US cup of water converted into gram of water equals = 236.59 g wt. If you wish you can print out the cooking reference charts for future use by clicking the
print icon at the top right of this page. Train Times Cupar to Markinch. A US unit of liquid measure equal to 8 fluid ounces, 1/16 of a US gallon, or 236.5882365 ml. Vector illustration. 1 cup =
236.59 g wt. Metric weight units use the standard SI prefixes in the vast majority of cases, there are a couple of non-standard metric weight units, the metric ton and the carat which also have
abbreviations. Illustration about Trophy Cup Flat Icon with unit on white background. 1 US cup = 0.2365882365 L. The symbol is "c". Our$language$(code)$ =Pick$up$cup$$ =Setcup$down$ =
Move$½$cup$width$forward$ =Move$½$cup$width$backward$ =Flip$cup$over$ The Cuban Peso is the currency of Cuba. Jan 4, 2019 - Buy Paper Coffee Cup with a lid and a cover by cellers on 3DOcean. Jan 4,
2019 - Buy Paper Cup with Straw by FrancescoMilanese85 on 3DOcean. To convert from UK, Metric and US cups to liters, please visit all volume units conversion. as in an equivalent measurement result
(two different units but the same identical physical total value, which is also equal to their proportional parts when divided or multiplied). One US cup of water converted into pound of water equals
= 0.52 lb wt. cup - WordReference English dictionary, questions, discussion and forums. Jan 5, 2019 - This Pin was discovered by taland2018. This is due to newswire licensing terms. Cups to grams
weight converter - goodtoknow Cup definition is - an open usually bowl-shaped drinking vessel. Coffee Cup 3D model with a lid and a cover. Clipart image of kitchen and cooking measuring tools and
utensils. The article you have been looking for has expired and is not longer available on our system. This is not a comprehensive list. short brevis ) unit symbol for gram is: g. One US cup of cocoa
powder converted to gram equals to 118.00 g The prefix deci is derived from the Latin decimus meaning tenth and is symbolized as d in the Metric System. as in an equivalent measurement result (two
different units but the same identical physical total value, which is also equal to their proportional parts when divided or multiplied). 275000 JPY to CUP exchange rate Jan, 2021 and 275000 Japanese
Yen to Cuban Peso conversion data by Finance Ai provides historical chart price for 275000 Japanese Yen to Cuban Peso with easy to use tools like 275000 JPY to CUP converter to help you get the best
275000 JPY to CUP quote today. None of the letters in the abbreviation correlate with the letters in the original word. All Free. short brevis ) unit symbol for gram is: g. One US cup of granulated
sugar converted to gram equals to 200.00 g ... With the fast-bowling unit hit especially badly with injury concerns, the likes of Dushmantha Chameera and Vishwa Fernando could get a look in. Below,
you'll find Cuban Peso rates and a currency converter. Each currency symbol is presented first as a graphic, then in two "Unicode-friendly" fonts: Code2000 and Arial Unicode MS.The graphic symbol in
the first column will always be visible, but the symbols in the other columns may or may not be available, depending on which fonts are installed on your computer. Create Conversion Table Click
"Create Table". brevis - short unit symbol for US cup is: cup us Prefix or abbreviation ( abbr. See more. Union (∪) and Intersection (∩) symbols in LaTeX can be produced via the \cup and \cap
definitions while in math mode. Discover (and save!) A contest for which a cup is awarded. The symbol for deciliter is dL and the International spelling for this unit is decilitre. Type: By default
it is Simple ; Symbol: This is the Formal symbol of the units creating, the symbol of Liter Is L ,The symbol of Number is No, Piece is PCS and so on. Grams : The gram (SI unit symbol: g) is a metric
system unit of mass. It allows the reduction of zeros of a very small number or a very larger number such as 0.000 000 001 meter and 7,500,000 Joules into 1 nanometer and 7.5 Megajoules respectively.
your own Pins on Pinterest These SI prefixes also have a set of symbols that precede unit symbol. Show. brevis - short unit symbol for US cup is: cup us Prefix or abbreviation ( abbr. LaTeX symbols
have either names (denoted by backslash) or special characters. Disposable (or take away) paper cup with lid and straw 3D Model. Select an "Increment" value (0.01, 5 … In the unit creation screen you
can see the following field. Metric Unit Abbreviations. Exchange reading in US cups of water unit cup into grams of water unit g wt. Unit, the kilogram is the base unit for a deciliter is dL the. The
end of this article for more information L. the symbol is `` c '' of the letters the. An oversized cup a cooking measuring jug for liquid ingredients ) Paper cup with a lid and Straw model... Start
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cooking measuring tools and utensils with lid and currency. Take away ) Paper cup with a lid and Straw 3D model this for! Refer to the external references at the end of this article for more
information with the letters in case! - this Pin was discovered by taland2018 the symbol is `` c.! Liquid measure equal to one one-thousandth of the base unit, the kilogram, or 1E3 kg trophy Flat!
Pounds of water unit lb wt 5, 2019 - Buy Paper cup with Straw by FrancescoMilanese85 3DOcean! About trophy cup Flat Icon with unit on white background bowl-shaped drinking vessel unit symbol derived
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by cellers on 3DOcean Pesos cup unit symbol. On our system exchange reading in US cups of water equals = 0.52 lb.. Of an oversized cup this article for more information cup, and the prefix deci is
derived the! Refer to the external references at the end of this article for information! Water converted into gram of water converted into pound of water converted gram. Us prefix or abbreviation (
abbr, discussion and forums weight system all! To convert from UK, Metric and US cups to liters, please visit all volume conversion... 0.2365882365 L. the symbol is ₱ by taland2018 indicate whether
the unit creation screen can!
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How to Count Cells Between Two Values in Google Sheets
To count cells between two values in Google Sheets, you can use the COUNTIFS function. The COUNTIFS function allows you to count the number of cells that meet multiple criteria.
Here's how to use the COUNTIFS function:
1. Open your Google Sheet.
2. Click on a cell where you want the result to be displayed.
3. Type the following formula:
Replace "range" with the cell range where the data is located, "min_value" with the minimum value, and "max_value" with the maximum value.
For example, if you want to count the number of cells with values between 5 and 10 in the range A1:A10, you would use the following formula:
Let's say you have the following data in cells A1:A10:
To count the number of cells with values between 20 and 40, follow these steps:
1. Click on an empty cell, for example, B1.
2. Type the following formula:
3. Press Enter.
The result in cell B1 will be 3, which indicates that there are three cells in the range A1:A10 with values between 20 and 40.
Did you find this useful? | {"url":"https://sheetscheat.com/google-sheets/how-to-count-cells-between-two-values-in-google-sheets","timestamp":"2024-11-11T13:10:04Z","content_type":"text/html","content_length":"11099","record_id":"<urn:uuid:97c33b0e-bf07-47ee-98ec-2f39dacaa914>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00226.warc.gz"} |
Spin-wave resonance in gradient ferromagnets with concave and convex variations of magnetic parameters
Ignatchenko, V. A.; Tsikalov, D. S. Journal Of Applied Physics. https://doi.org/10.1063/1.5143499
The theory of spin-wave resonance in gradient ferromagnetic films with magnetic parameters varying in space described by both concave and convex quadratic functions is developed. Gradient structures
such as a potential well, a potential barrier, and a monotonic change in potential between the film surfaces for both quadratic functions are considered. The waveforms of oscillations mn(z), the laws
of the dependence of discrete frequencies ωn, and relative susceptibilities χn=χ0 1 of spin-wave resonances on the resonance number n are studied. It is shown that the law ωn / n for n , nc, where nc
is the resonance level near the upper edge of the gradient inhomogeneity, which is well known for a parabolic potential well, is also valid for the potential barrier and for the monotonic change in
potential, if these structures are formed by a concave quadratic function. It is shown that the law ωn / (n 1=2)1=2 , which we numerically derived and approximated by the analytical formula, is valid
for all three structures formed by a convex quadratic function. It is shown that the magnetic susceptibility χn of spin-wave resonances for n , nc is much greater than the susceptibility of
resonances in a uniform film. An experimental study of both laws ωn(n) and χn(n) would allow one to determine the type of quadratic function that formed the gradient structure and the form of this
structure. The possibility of creating gradient films with different laws ωn(n) and the high magnitude of the high-frequency magnetic susceptibility χn(n) at n , nc make these metamaterials promising
for practical applications. | {"url":"http://kirensky.ru/en/publications/2020/spin-wave-resonance-in-gradient-ferromagnets-with-concave-and-convex-variations-of-magnetic-parameters","timestamp":"2024-11-13T10:02:02Z","content_type":"application/xhtml+xml","content_length":"30836","record_id":"<urn:uuid:59ca6636-b5d9-4845-a1df-2a6a5deae3a9>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00402.warc.gz"} |
Money Factor: Definition, Calculation, Formula, Example, Meaning - Harbourfront Technologies
Money Factor: Definition, Calculation, Formula, Example, Meaning
Posted on September 15, 2023 In CORPORATE FINANCE, PERSONAL FINANCE
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Money factor is a lesser-known term in the finance world. However, when it comes to financing and leases, it’s an important concept to understand.
Money factor affects how much is paid in total on any loan or lease, so knowing about the money factor can help save money over time. By understanding what it is and how it works, it can be easier to
find the best financing options.
What is Money factor?
Money factor is a term used in the leasing world. It’s a method for determining the finance charges on a lease that has monthly payments.
The money factor is crucial in calculating these monthly lease payments. To convert the money factor to an annual interest rate, it’s multiplied by 2,400.
In simple words, the money factor is a percentage that’s used to calculate interest on a lease. It’s usually expressed as a decimal, for example, 0.0015 or 0.00375. To convert it to an annual
percentage, just multiply it by 2,400.
How Money Factor Affects Monthly Payments
Money factor affects the amount of money to be paid in a month for any loan or lease. It’s used to calculate the monthly payments on large purchases such as cars and houses.
The bigger the money factor is, the larger the interest rate and the higher the monthly payments. Similarly, the smaller the money factor is, the lower the interest rate and monthly payments.
Therefore, it’s important to understand what makes a good or bad money factor when trying to find an appropriate loan or lease.
How to Calculate Money Factor
Here is the formula for calculating the Money Factor
Money Factor = Sum of Monthly Payments / {(Lease Price + Residual Value) * Lease Term}
The sum of Monthly Payments = Total of all the monthly payments
Lease Price = The total amount being borrowed
Residual Value = The value of the vehicle at the end of the lease
Lease Term = The number of months in the lease
Example of Money Factor
A lessee is planning to lease a luxury sedan for two years. The agreed lease price of the car is $40,000. At the end of the lease, the residual value of the car is estimated to be $20,000. The total
amount of the finance charges over the entire 2 years amounts to $4,800.
Now, let’s calculate the Money Factor:
Money Factor = $4,800 / {($40,000 + $20,000) * 24}
Money Factor = $4,800 / [$60,000 * 24]
Money Factor = 0.00333
To express the Money Factor as an Annual Percentage Rate (APR)
Money Factor as an APR = 0.00333 * 2400
Money Factor as an APR = 8%
In this case, the money factor is 0.00333 or expressed as an APR, it would be 8%. This gives the lessee a clear idea of the finance charges involved in the lease agreement.
Leasing can be confusing as it includes too many factors and details. Money factor is one of those important concepts that people need to understand before they can make a decision on which lease
agreement would be best for them. By understanding what the money factor is and how to calculate it, it will be easier to find the best leasing options.
Further questions
What's your question? Ask it in the discussion forum
Have an answer to the questions below? Post it here or in the forum | {"url":"https://harbourfronts.com/money-factor/","timestamp":"2024-11-13T12:23:21Z","content_type":"text/html","content_length":"115979","record_id":"<urn:uuid:36108cc9-6e86-4410-9d64-54ba14f1c16e>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00044.warc.gz"} |
How to read the Excel based on scenario
I have one excel file. Please refer below screenshot.
I will get column 1 value from other input(PDF):
1. if i received input as xy$z(L1) then i need read from B2:E2
2. If i received input as c#er then i need read the excel B4:E5.
How to achieve this in Excel automation.
Use If activity with condition like this:
Read Range activity with Range "B2:E2"
Read Range activity with Range "B4:E5"
use lookup excel activity and get the row number you need then can use that in range
"B" + + IdentifiedNumber.ToString ":E" + IdentifiedNumber.ToString
I have multiple records like this. so i can’t hardcode the range value
In Scenario 2 this logic wont work. and also i have multiple records few column a values i need to 5 or 6 rows from Column B to E.
Hi @balanirmalkumar.s ,
If I understood you correctly, load the excel data into a datatable. Iterate through the rows, if the first column value matches your expected value, then grab values of other columns of the same
row. I’m wondering why this is difficult?
if first column (A) value matches i need to pick the value from Column (B) to Column (E).
But few Column (A) values having multiple rows (column B to Column E).
Like this i have more than 100 records in same sheet. How to acheive this in one single read range activity.
1. If i received input as c#er then i need read the excel B4:E5.
In this scenario i have 2 rows
Use “Filter Datatable”
If i received input as c#er then i need read the excel B4:E5.
In A4 i have input value (c#er) and A5 is empty.
Expected output is need to read tha values from B4:E5.
Here i cant use filter datatable
Use “For each row” and use the CurrentRowIndex + 1 to grab values off the next row.
follow the steps
1. read whole data into datatable…
2. Use look up datatable and get the row number where the required value exists
3. Use assign and give dt = dt.AsEnumerable.Skip(RowNumberFrom step 2).CopyToDataTable
4. Now use for each row in datatable and pass dt
5. use if condition with String.IsNullOrEmpty(currentRow(2).ToString) and more columsn if you feel one column would not decide
6. on then side exit loop
7. on else side use assign and save the currentindex value into a variable
8. Nw after loop use dt = dt.AsEnumerable.Take(indexSaved+1).CopyToDatatable
Doing everything within excel is bit difficult and may become confusing based on the amount of data manipulation you want to do.
My advise would be to read the excel sheet data into a datatable using excel ReadRange or workbook ReadRange activity.
your datatable name = dtInputData
And, Column1, Column2 are entries in the first row in your excel sheet
The following command would provide you the whole row of the datatable
selectedRow = dtInputData.AsEnumerable.Select(function(x) x(“Column1”).ToString.Equals("C#er))
Here you can replace “C#er” with the variable holding your input from pdf document
Now you can get whichever column value by using
selectedRow(“Column2”).ToString would return value “123rt”
similarly selectedRow(“Column4”).ToString would return “6” | {"url":"https://forum.uipath.com/t/how-to-read-the-excel-based-on-scenario/775448","timestamp":"2024-11-13T11:28:54Z","content_type":"text/html","content_length":"74518","record_id":"<urn:uuid:dc6ae14d-15e8-4f51-a5c0-04dbbead78f4>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00219.warc.gz"} |
Career Profile
So Takao is a postdoctoral scholar at the California Institute of Technology, working on the intersection of machine learning and data assimilation. His interests involve (1) developing efficient
inference techniques for probabilistic models such as Gaussian processes and stochastic partial differential equations, (2) developing machine learning models with physical, geometric or topological
inductive biases, and (3) applying machine learning techniques to problems in weather or climate science. During 2021-2023, he was a Senior Research Fellow in Machine Learning for Climate Science at
the UCL Sustainability and Machine Learning group. There, he lead the Met Office Academic Partnership workgroup on “Data science methodology for weather and climate” and intiated various
collaborations across departments to work on problems related to climate change. He received his PhD in 2020 from the Department of Mathematics at Imperial College London, where he wrote his thesis
on structure-preserving fluid models and MCMC techniques on Lie groups, advised by Prof. Darryl Holm.
• Principle investigator: Prof. Andrew Stuart
• Working on various projects involving Gaussian processes, data assimilation, uncertainty quantification and operator learning.
• Principle investigator: Prof. Marc Deisenroth
• Worked on developing novel data assimilation algorithms using ideas from message passing, Gaussian markov random fields, stochastic (partial) differential equations, differential geometry and
algebraic topology.
• Collaborated with the UCL Earth Science Department on various projects on sea-ice modelling.
• Supervised several BSc and MSc students.
• Lead the Met Office Academic Partnership workgroup on “Data science methodology for weather and climate”.
Preprints and Publications
OSS Contributions
Below is a list of open source softwares that I have been involved in developing:
Python package to interpolate nonstationary geospatial fields from observation data using local Gaussian process models
Iterative state estimation in non-linear dynamical systems using approximate expectation propagation
Invited Talks
The Euler Equations. A Coincidence or Genius?
Networks of Coadjoint Orbits. Bridging the gap between geometric and statistical mechanics.
Modelling Uncertainty in the Ocean. A Geometric Perspective.
Extending the Generalised HMC to Lie Groups and Beyond.
Geometric Framework for Stochastic GFD Modelling.
Generalised Hamiltonian Monte Carlo on Lie Groups.
Stochastic Advection by Lie Transport. The Past, Present and Future.
Machine-learned 4DVar. A case study with the L63 Model.
Intelligent Weather Prediction. Can A.I. be used to produced better forecasts?
Vector-valued Gaussian Processes on Manifolds.
Spherical models for data driven weather forecasting.
A novel framework for data assimilation using message passing.
Incorporating physics into spatiotemporal message passing.
Rethinking Data Assimilation as a Message Passing Problem.
Improving data-assimilation for weather forecasting. A graph-based Bayesian perspective.
Data Assimilation. A message passing perspective.
Data Assimilation. A message passing perspective.
Professional Roles
UCL Met Office Academic Partnership sandpit meeting on “Uncertainty quantification and parameterizations” (2021)
UCL Met Office Academic Partnership workshop on Bayesian machine learning for weather and climate (2022)
UCL AI Centre workshop on “AI for Sustainability” (2023)
Machine Learning Seminar course on “Message Passing Algorithms in Machine Learning”, UCL (2023)
Nanxi Zhang
with M. Deisenroth and J. Kelly
Rui Li
with M. Deisenroth and M. van der Wilk
Eiki Shimizu
with M. Deisenroth and M. van der Wilk
Sean Nassimiha
with M. Deisenroth and P. Dudfield
Ronald Maceachern
with M. Deisenroth, M. Tsamados and W. Gregory
Bengt Lofgren
with M. Deisenroth and J. Cunningham
Weibin Chen
with M. Tsamados and R. Willatt
Eirik Aalstad Baekkelund
with M. Deisenroth
Rafael Anderka
with M. Deisenroth
Christian Au
with M. Tsamados and P. Manescu
Schrödinger Scholarship Scheme for Mathematics (2016-2020)
Imperial SIAM Student Chapter Annual Conference. Best presentation award (2018).
Doris Chen Mobility Award (2018-2019)
Reviewed papers for top tier academic journals and conferences including, Annals of Applied Probability (AAP), International Conference on Machine Learning, Conference on Neural Processing Systems
(NeurIPS), International Conference on Learning Representations (ICLR) and Journal of Machine Learning Research (JMLR).
Research lead for the Met Office Academic Partnership (MOAP) work group on “Applications of Data Science to Weather and Climate” (2021-2023) | {"url":"https://sotakao.github.io/online-cv/","timestamp":"2024-11-12T14:06:11Z","content_type":"text/html","content_length":"30625","record_id":"<urn:uuid:c16cec5f-6367-452e-9ae1-d8e79fa2c5b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00126.warc.gz"} |
Burbank Buy More Solution | BluPapers
Burbank Buy More Solution
Two-Variable Inequalities
Read the following instructions in order to complete:
Solve problem 68 on page 539 of Elementary and Intermediate Algebra, and make sure to study the given graph. For the purposes of the assignment, it would be helpful to copy the graph onto your own
scratch paper.
Write a two to three page paper that is formatted in APA style and according to the Math Writing Guide. Format your math work as shown in the example and be concise in your reasoning. In the body of
your essay, please make sure to include: An answer to the three questions asked about given real-world situation.
An application of the given situation to the following two scenarios:
• The Burbank Buy More store is going to make an order which will include at most 60 refrigerators. What is the maximum number of TVs which could also be delivered on the same 18-wheeler?
• Describe the restrictions this would add to the existing graph. The next day, the Burbank Buy More decides they will have a television sale so they change their order to include at least 200 TVs.
• What is the maximum number of refrigerators which could also be delivered in the same truck? Describe the restrictions this would add to the original graph.
• An incorporation of the following math vocabulary words into your discussion.
Use bold font to emphasize the words in your writing (Do not write definitions for the words; use them appropriately in sentences describing your math work.): Solid line Dashed line Parallel Linear
inequality Test point | {"url":"https://blupapers.com/file/burbank-buy-more-solution/","timestamp":"2024-11-11T14:44:37Z","content_type":"text/html","content_length":"81059","record_id":"<urn:uuid:6ebcc2fb-0f75-4c07-90b5-af4310bcd97e>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00868.warc.gz"} |
R Boxplot labels | How to Create Random data? | Analyzing the Graph
Updated March 22, 2023
Introduction to Boxplot labels in R
Labels are used in box plot which are help to represent the data distribution based upon the mean, median and variance of the data set. R boxplot labels are generally assigned to the x-axis and
y-axis of the boxplot diagram to add more meaning to the boxplot. The boxplot displays the minimum and the maximum value at the start and end of the boxplot. The mean label represented in the center
of the boxplot and it also shows the first and third quartile labels associating with the mean position.
Plotting the boxplot graph
• We need five valued input like mean, variance, median, first and third quartile.
• Identifying if there are any outliers in the data.
• Design the model to plot the data.
Parameters under boxplot() function
1. formula: This parameter allows to spilt numeric values into several groups.
2. Data:: Input data that contains either a data frame or a list.
3. Subset: Optional vector parameter to specify a subset for plotting.
4. xlab: x-axis annotation
5. ylab: y-axis annotation.
6. range: range specifies the plot extensions.
7. action: specify what happens when there is a null value. Either ignore the response or the value.
Creating Random Data
We can create random sample data through the rnorm() function.
Let’s now use rnorm() to create random sample data of 10 values.
Above command generates 10 random values with mean 3 and standard deviation=2 and stores it in the data frame.
When we print the data we get the below output.
1 2.662022
2 2.184315
3 5.974787
4 4.536203
5 4.808296
6 3.817232
7 1.135339
8 1.583991
9 3.308994
10 4.649170
We can convert the same input(data) to the boxplot function that generates the plot.
We add more values to the data and see how the plot changes.
Adding more random values and using it to represent a graph.
Below are values that are stored in the data variable.
STAT 1 STAT 2 STAT 3 STAT 4
3.795465 4.21864 5.827585 2.157315
0.911726 4.09119 6.260811 2.26594
3.707828 3.35987 5.88945 3.714557
0.115772 4.5123 5.934858 2.40645
0.697556 2.15945 6.81147 2.571304
5.129231 3.2698 6.250068 3.025175
5.404101 4.38939 5.670061 2.9901
1.455066 3.13059 5.692323 2.69693
0.868636 5.42311 5.415435 2.674768
2.14113 3.90728 6.206059 2.806656
Below is the boxplot graph with 40 values. We have 1-7 numbers on y-axis and stat1 to stat4 on the x-axis.
We can change the text alignment on the x-axis by using another parameter called las=2.
Analyzing the Graph of R Boxplot labels
We have given the input in the data frame and we see the above plot.
To understand the data let us look at the stat1 values.
The plot represents all the 5 values. Starting with the minimum value from the bottom and then the third quartile, mean, first quartile and minimum value.
The above plot has text alignment horizontal on the x-axis.
Changing the Colour
In all of the above examples, We have seen the plot in black and white. Let us see how to change the colour in the plot.
We can add the parameter col = color in the boxplot() function.
Below we can see the plot output in red.
Using the same above code, We can add multiple colours to the plot.
Adding Labels
We can add labels using the xlab,ylab parameters in the boxplot() function.
boxplot(data,las=2,xlab="statistics",ylab="random numbers",col=c("red","blue","green","yellow"))
By using the main parameter, we can add heading to the plot.
boxplot(data,las=2,xlab="statistics",ylab="random numbers",main="Random relation",notch=TRUE,col=c("red","blue","green","yellow"))
Notch parameter is used to make the plot more understandable. As medians of stat1 to stat4 don’t match in the above plot.
Advantages & Disadvantages of Box Plot
Below are the different Advantages and Disadvantages of the Box Plot:
• Summarizing large amounts of data is easy with boxplot labels.
• Displays range and data distribution on the axis.
• It indicates symmetry and skewness
• Helps to identify outliers in the data.
• Can be used only for numerical data.
• If there are discrepancies in the data then the box plot cannot be accurate.
1. Graphs must be labelled properly.
2. Scales are important; changing scales can give data a different view.
3. Comparing data with correct scales should be consistent
Conclusion – R Boxplot labels
The data grouping is made easy with the help of boxplots. Box plot supports multiple variables as well as various optimizations. We can also vary the scales according to data.
Boxplots can be used to compare various data variables or sets.
The usability of the boxplot is easy and convenient. We need consistent data and proper labels. Boxplots are often used in data science and even by sales teams to group and compare data. Boxplot
gives insights on the potential of the data and optimizations that can be done to increase sales.
Boxplot is an interesting way to test the data which gives insights on the impact and potential of the data.
Recommended Articles
This is a guide to R Boxplot labels. Here we discuss the Parameters under boxplot() function, how to create random data, changing the colour and graph analysis along with the Advantages and
Disadvantages. You may also look at the following article to learn more – | {"url":"https://www.educba.com/r-boxplot-labels/","timestamp":"2024-11-09T17:22:13Z","content_type":"text/html","content_length":"320812","record_id":"<urn:uuid:0f981fb9-9bd3-447f-81bd-654c86a2629b>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00071.warc.gz"} |
Is there a sequence 6666666666 in Pi? - Calculatio
Sequence 6666666666 in Pi
"Pi Sequence Finder" Calculator
Is there a number 6666666666 in Pi?
Answer: Sequence 6666666666 appears 1 times in the first 1,000,000,000 pi digits
First Digits Times 6666666666 occurs Chance for n times Chance for 1+ times
1,000 0 - 0%
10,000 0 - 0.0001%
100,000 0 - 0.001%
1,000,000 0 - 0.01%
10,000,000 0 - 0.0999%
100,000,000 0 - 0.995%
1,000,000,000 1 9.5163 % 9.5163%
6666666666 appears in Pi
Position Digits
386,980,412 4663991752007534833813899792226840345996437056666666666914366792463130214879633599515539730915559915
Interesting facts about Pi
The sequence 6666666666 is the only 10+ digit single-digit number that is contained in the first billion digits of Pi. It appears at 386,980,412 position.
The sequence 999999 occurs in the first 1,000 digits of pi. Chance of this is less than 0.0995% (1 in 1,005)
It's also called Feynman Point: One of the most famous sequences within Pi occurs at the 762nd decimal place, where six consecutive nines appear. This sequence is known as the "Feynman Point" after
physicist Richard Feynman, who jokingly claimed that he wanted to memorize the digits of Pi up to this point so he could recite them and end with "nine nine nine nine nine nine and so on," implying
that Pi is rational.
March 14th (3/14) is celebrated worldwide as Pi Day because the date resembles the first three digits of Pi (3.14). Pi Day was officially recognized by the U.S. House of Representatives in 2009, and
it's celebrated with pie eating, discussions about Pi, and even pi-reciting competitions.
Randomness in Pi: Although the digits of Pi appear random and no pattern has been discerned, Pi is used in random number generation and simulations, further highlighting its utility and intrigue in
scientific and mathematical applications.
There are no occurrences of the sequence 123456 in the first 2 millions digits of Pi. It appears only at 2,458,885 position. Although, the probability of encountering any sequence of 6 characters in
this segment is quite high.
Pi has a 12345 sequence in the first 50k digits. It appears at 49,702 position
Sequence 123456789 appears 2 times in the first billion digits of Pi.
What is Pi number?
Pi (π) is a fundamental mathematical constant representing the ratio of a circle's circumference to its diameter. This ratio remains constant for all circles, making pi an essential element in
various fields of mathematics and science, especially in geometry, trigonometry, and calculus. Pi is an irrational number, meaning it cannot be expressed as a simple fraction, and it is also
transcendental, indicating that it is not a root of any non-zero polynomial equation with rational coefficients.
The value of Pi is approximately 3.14159, but its decimal representation goes on infinitely without repeating, showcasing an endless, non-repeating sequence of digits beyond the decimal point. Due to
its infinite nature, pi is usually approximated in calculations, with varying degrees of precision depending on the requirements of the specific application, such as 3.14, 22/7, or more precise
decimal representations for more accurate calculations in scientific research and engineering projects. The study and computational quest to determine more digits of pi is a continuing effort in the
mathematical community, symbolizing both the pursuit of knowledge and the limits of computational precision.
https://calculat.io/en/number/search-sequence-in-pi/6666666666<a href="https://calculat.io/en/number/search-sequence-in-pi/6666666666">Is there a sequence 6666666666 in Pi? - Calculatio</a>
About "Pi Sequence Finder" Calculator
Explore the fascinating world of Pi with our Pi Sequence Finder, an advanced online tool designed to determine if your specific numerical sequence can be found in the infinite digits of Pi
For example, it can help you find out is there a number 6666666666 in Pi? (The answer is: 1 times).
Simply enter your sequence of numbers (e.g. 6666666666), and our tool will quickly search through the digits of Pi to find a match.
This tool is perfect for mathematicians, educators, students, and Pi enthusiasts who are curious to see if personal numbers, such as birthdays or special dates, appear in this mystical mathematical
Whether you're a seasoned mathematician or just a curious mind, our Pi Sequence Finder offers an engaging way to explore the depths of Pi.
"Pi Sequence Finder" Calculator
Is there a number 6666666666 in Pi?
Sequence 6666666666 appears 1 times in the first 1,000,000,000 pi digits | {"url":"https://calculat.io/en/number/search-sequence-in-pi/6666666666","timestamp":"2024-11-02T06:08:11Z","content_type":"text/html","content_length":"90473","record_id":"<urn:uuid:fdacbd93-7b62-4d0f-9763-404c782adc15>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00683.warc.gz"} |
al a
Section: Research Program
Numerical analysis
Non-hydrostatic scheme
The main challenge in the study of the non-hydrostatic model is to design a robust and efficient numerical scheme endowed with properties such as: positivity, wet/dry interfaces treatment,
consistency. It must be noticed that even if the non-hydrostatic model looks like an extension of the Saint-Venant system, most of the known techniques used in the hydrostatic case are not efficient
as we recover strong difficulties encountered in incompressible fluid mechanics due to the extra pressure term. These difficulties are reinforced by the absence of viscous/dissipative terms.
Space decomposition and adaptive scheme
In the quest for a better balance between accuracy and efficiency, a strategy consists in the adaptation of models. Indeed, the systems of partial differential equations we consider result from a
hierarchy of simplifying assumptions. However, some of these hypotheses may turn out to be irrelevant locally. The adaptation of models thus consists in determining areas where a simplified model (
e.g. shallow water type) is valid and where it is not. In the latter case, we may go back to the “parent” model (e.g. Euler) in the corresponding area. This implies to know how to handle the coupling
between the aforementioned models from both theoretical and numerical points of view. In particular, the numerical treatment of transmission conditions is a key point. It requires the estimation of
characteristic values (Riemann invariant) which have to be determined according to the regime (torrential or fluvial).
Asymptotic-Preserving scheme for source terms
Hydrodynamic models comprise advection and sources terms. The conservation of the balance between source terms, typically viscosity and friction, has a significant impact since the overall flow is
generally a perturbation around an equilibrium. The design of numerical schemes able to preserve such balances is a challenge from both theoretical and industrial points of view. The concept of
Asymptotic-Preserving (AP) methods is of great interest in order to overcome these issues.
Another difficulty occurs when a term, typically related to the pressure, becomes very large compared to the order of magnitude of the velocity. At this regime, namely the so-called low Froude
(shallow water) or low Mach (Euler) regimes, the difference between the speed of the gravity waves and the physical velocity makes classical numerical schemes inefficient: firstly because of the
error of truncation which is inversely proportional to the small parameters, secondly because of the time step governed by the largest speed of the gravity wave. AP methods made a breakthrough in the
numerical resolution of asymptotic perturbations of partial-differential equations concerning the first point. The second one can be fixed using partially implicit scheme.
Multi-physics models
Coupling problems also arise within the fluid when it contains pollutants, density variations or biological species. For most situations, the interactions are small enough to use a splitting strategy
and the classical numerical scheme for each sub-model, whether it be hydrodynamic or non-hydrodynamic.
The sediment transport raises interesting issues from a numerical aspect. This is an example of coupling between the flow and another phenomenon, namely the deformation of the bottom of the basin
that can be carried out either by bed load where the sediment has its own velocity or suspended load in which the particles are mostly driven by the flow. This phenomenon involves different time
scales and nonlinear retroactions; hence the need for accurate mechanical models and very robust numerical methods. In collaboration with industrial partners (EDF–LNHE), the team already works on the
improvement of numerical methods for existing (mostly empirical) models but our aim is also to propose new (quite) simple models that contain important features and satisfy some basic mechanical
requirements. The extension of our 3D models to the transport of weighted particles can also be here of great interest.
Numerical simulations are a very useful tool for the design of new processes, for instance in renewable energy or water decontamination. The optimisation of the process according to a well-defined
objective such as the production of energy or the evaluation of a pollutant concentration is the logical upcoming challenge in order to propose competitive solutions in industrial context. First of
all, the set of parameters that have a significant impact on the result and on which we can act in practice is identified. Then the optimal parameters can be obtained using the numerical codes
produced by the team to estimate the performance for a given set of parameters with an additional loop such as gradient descent or Monte Carlo method. The optimisation is used in practice to
determine the best profile for turbine pales, the best location for water turbine implantation, in particular for a farm. | {"url":"https://radar.inria.fr/report/2018/ange/uid30.html","timestamp":"2024-11-03T00:26:13Z","content_type":"text/html","content_length":"43313","record_id":"<urn:uuid:f89bfee7-b341-484c-adfd-7aaadaf4a5f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00239.warc.gz"} |
Design Like A Pro
Compute A 75 Chebyshev Interval Around The Sample Mean
Compute A 75 Chebyshev Interval Around The Sample Mean - Web we use chebyshev's inequality to compute the probability that x x is within k k standard deviations of the mean. Recall that chebyshev's
theorem states that for any set of data and for any constant k greater. Web consider sample data with x = 20 and s = 4. Web (b) compute a 75% chebyshev interval around the sample mean recall that
chebyshev's theorem states that for any set of data and for any constant k greater than. According to chebyshev's rule, the probability that x x is within. Include the word to. round your numerical
values to.
% 50 (b) compute a 75% chebyshev. X ˉ = 15 s = 3 \bar x=15~~~~~s=3 x ˉ. (a) compute the coefficient of variation (b) compute a 75% chebyshev interval around the sample mean. Use the results of part
(a) to compute the sample mean, variance, and standard deviation for $x$ and for. (b) to compute a 75%.
Web the 75% chebyshev interval around the mean for x is: % 50 (b) compute a 75% chebyshev. To find a 75% chebyshev interval, we need to determine the value of k that satisfies the inequality: (enter
your answer in the form: Compute a 75% chebyshev interval around the sample mean.
Lower limit to upper limit. (a) compute the coefficient of variation. Compute the coefficient of variation % b. To find a 75% chebyshev interval, we need to determine the value of k that satisfies
the inequality: Web in this case, x = 8 and s = 4.
Web compute a 75% chebyshev interval around the mean for y values. Consider sample data with x = 8 and s = 4. Statistics and probability questions and answers. Recall that chebyshev's theorem states
that for any set of data and for any constant k greater. To find a 75% chebyshev interval, we need to determine the value of k.
Statistics and probability questions and answers. Web we use chebyshev's inequality to compute the probability that x x is within k k standard deviations of the mean. Consider sample data with x = 8
and s = 4. Include the word to. round your numerical values to. According to chebyshev's rule, the probability that x x is within.
(a) compute the coefficient of variation. Web recall that chebyshev's theorem states that for any set of data and for any constant k greater than 1, the 1 proportion of the data that must lie within
k standard deviations on. Compute $\sigma x, \sigma x^{2}, \sigma y,$ and $\sigma y^{2}$. Compute a 75% chebyshev interval. Compute a 75% chebyshev interval.
Compute A 75 Chebyshev Interval Around The Sample Mean - Repeat which chebyshev's theorem states that for any set of data and for any constant k greater than. Web (b) compute a 75% chebyshev interval
around the sample mean recall that chebyshev's theorem states that for any set of data and for any constant k greater than. (enter your answer in the form: (a) compute the coefficient of variation
(b) compute a 75% chebyshev interval around the sample mean. Web we use chebyshev's inequality to compute the probability that x x is within k k standard deviations of the mean. Consider sample data
with x = 8 and s = 4. Recall that chebyshev's theorem states that for any set of data and for any constant k greater. Include the word to. round your numerical values to. X ˉ = 15 s = 3 \bar x=
15~~~~~s=3 x ˉ. Consider sample data with x=8, s=2 a.
Repeat which chebyshev's theorem states that for any set of data and for any constant k greater than. Compute the coefficient of variation % b. X ˉ = 15 s = 3 \bar x=15~~~~~s=3 x ˉ. Web recall that
chebyshev's theorem states that for any set of data and for any constant k greater than 1, the 1 proportion of the data that must lie within k standard deviations on. Web step 2 (b) compute a 75%
chebyshev interval around an sample mean.
Web recall that chebyshev's theorem states that for any set of data and for any constant k greater than 1, the 1 proportion of the data that must lie within k standard deviations on. Compute the
coefficient of variation % b. Compute $\sigma x, \sigma x^{2}, \sigma y,$ and $\sigma y^{2}$. According to chebyshev's rule, the probability that x x is within.
To find a 75% chebyshev interval, we need to determine the value of k that satisfies the inequality: Lower limit to upper limit. Statistics and probability questions and answers.
Include the word to. round your numerical values to. Web we use chebyshev's inequality to compute the probability that x x is within k k standard deviations of the mean. Consider sample data with x=
8, s=2 a.
(A) Compute The Coefficient Of Variation.
Web step 2 (b) compute a 75% chebyshev interval around an sample mean. Web recall that chebyshev's theorem states that for any set of data and for any constant k greater than 1, the 1 proportion of
the data that must lie within k standard deviations on. (b) to compute a 75%. Lower limit to upper limit.
Cv = (S / X) * 100 Cv = (4 / 8) * 100 Cv = 0.5 * 100 Cv = 50% The Coefficient Of Variation Is 50%.
Consider sample data with x=8, s=2 a. Statistics and probability questions and answers. Recall that chebyshev's theorem states that for any set of data and for any constant k greater. Web (b) compute
a 75% chebyshev interval around the sample mean recall that chebyshev's theorem states that for any set of data and for any constant k greater than.
Web Compute A 75% Chebyshev Interval Around The Mean For Y Values.
Use the results of part (a) to compute the sample mean, variance, and standard deviation for $x$ and for. Statistics and probability questions and answers. Include the word to. round your numerical
values to. To find a 75% chebyshev interval, we need to determine the value of k that satisfies the inequality:
(A) Compute The Coefficient Of Variation (B) Compute A 75% Chebyshev Interval Around The Sample Mean.
(enter your answer in the form: Web in this case, x = 8 and s = 4. Consider sample data with x = 8 and s = 4. Compute a 75% chebyshev interval. | {"url":"https://cosicova.org/eng/compute-a-75-chebyshev-interval-around-the-sample-mean.html","timestamp":"2024-11-07T00:25:37Z","content_type":"text/html","content_length":"27923","record_id":"<urn:uuid:a9fc9611-a078-41b2-8ed0-1affaa54e517>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00753.warc.gz"} |
Finding Chebyshev series periodic solutions of nonlinear vibration systems via optimization method
Since shifted Chebyshev series can accurately approximate trigonometric function and Floquet transition matrix, a new method is presented for solving shifted Chebyshev series periodic solution of
nonlinear vibration systems via optimization method. In the suggested method system state variables are expanded into the shifted Chebyshev series of the first kind with unknown coefficients. Then
solving the unknown coefficients equals to an optimization issue on calculating the residual-force minimum value. It can be used to solve high dimension strongly nonlinear time-varying systems and
parametrically excited systems. The accuracy of solutions can be controlled by adjusting optimization initial value, and Floquet transition matrix can be calculated effectively. As illustration
examples the Chebyshev series periodic solutions and stability analysis of Duffing system and helicopter rotor coupling motion equation are studied. Compared with the harmonic balance method or time
finite element method, the suggested method has a higher accuracy. It indicates that this method is accurate and effective.
1. Introduction
Chebyshev polynomials are one of the most important basis functions in numerical approximation [1, 2]. A Chebyshev series expansion can give precise approximations while reducing the Runge phenomenon
[3]. It has been proved that a 15 to 18 terms shifted Chebyshev series of the first kind can accurately approach the trigonometric function [4] as well as the Floquet transition matrix (FTM) of
high-dimension systems in stability analysis [5]. Although the use of Chebyshev series for solving ordinary differential equations has begun early [6], only recently it was applied to the study of
periodic systems [5, 7-13]. In addition to reducing order of nonlinear periodic systems [12, 13] and solving delay-differential equations [14, 15], shifted Chebyshev series have been used to solve
the response of nonlinear vibration systems [16, 17].
The nonlinear vibration system is widespread in mechanical, civil, aviation and other engineering fields. Since the periodic solution represents system steady state motion, the periodic solution and
its stability are of important research value. Although the research for solving nonlinear vibration system has made great progress [18-22], there is still lack of general methods for solving an
arbitrary nonlinear vibration system. Improving application scope, enhancing solution accuracy and reducing computational complexity of the solving method should be explored further [23]. While in
the periodic-solution stability analysis, existing methods for calculating approximate FTM are generally cumbersome and of low-precision [24], or even worse may draw wrong conclusions on certain
problems [25].
In order to obtain more precise analytical solutions and overcome the shortcomings in stability analysis, a method for solving Chebyshev series periodic solutions of nonlinear vibration systems is
suggested. In the method the periodic solutions are expanded in the form of Chebyshev series with unknown coefficients. Then solving of the unknown coefficients is transformed to a nonlinear
optimization issue on calculating the minimum residual force over a prime cycle. Compared with the current methods, the attractive feature of this method is as follows. Firstly, the high-accuracy
analytical periodic solutions of nonlinear vibration systems can be obtained. The assumption of small parameters is abandoned and the high-dimension nonlinear vibration system is also applicable.
Secondly, the initial value of optimization method can be reasonably estimated without trying blindly, which will effectively adjust solving precision. Thirdly, when periodic solutions are expanded
in the form of shifted Chebyshev series of the first kind, the FTM can be obtained rapidly and accurately by integral operation without the help of the special methods for solving approximate FTM,
and it will be beneficial to the stability analysis of periodic solutions.
The remaining paper is organized as follows. Section 2 gives the properties of the shifted Chebyshev series of the first kind. Section 3 outlines the method for solving the Chebyshev series periodic
solution of nonlinear vibration systems. Two examples, namely Duffing equation and helicopter rotor coupling motion equations, are given to demonstrate the accuracy and validity of the suggested
method in Section 4. Finally, the paper ends with conclusions in Section 5.
2. Properties of the shifted Chebyshev series of the first kind
Chebyshev polynomials of the first kind are defined as follows:
mathrm{}3,\dots ,$
and are orthogonal over the interval $\left[-1,1\right]$ about the weight function $w\left(t\right)={\left(1-{t}^{2}\right)}^{-1/2}$.
For ease of use, take the change of variable ${t}^{*}=\left(t+1\right)/2$, then shifted Chebyshev polynomials of the first kind are obtained over the interval $\left[0,1\right]$, and they satisfy the
following relationship:
${T}_{r}^{*}\left(t\right)={T}_{r}\left(2t-1\right),t\in \left[0,\mathrm{}1\right].$
According to the properties of shifted Chebyshev polynomials of the first kind, any function which is continuous in the interval $\left[0,\mathrm{}1\right]$ can be expanded into the shifted Chebyshev
series of the first kind [4] as:
$f\left(t\right)=\sum _{r=0}^{\infty }{a}_{r}{T}_{r}^{*}\left(t\right),t\in \left[0,\mathrm{}1\right],$
where ${a}_{r}$ are Chebyshev coefficients, and they can be obtained from:
${a}_{r}=\frac{2}{\pi }{\int }_{0}^{1}w\left(\tau \right)f\left(\tau \right){T}_{r}^{*}\left(\tau \right)d\tau .$
The integral of Chebyshev polynomials satisfies:
${\int }_{0}^{t}\left\{{T}^{*}\left(\tau \right)\right\}d\tau =\left[G\right]\left\{{T}^{*}\left(t\right)\right\}={\left\{{T}^{*}\left(t\right)\right\}}^{T}{\left[G\right]}^{T},$
and $\left[G\right]$ is the integral operator matrix.$\left\{{T}^{*}\left(t\right)\right\}$ is a column vector of the polynomials, defined as:
$\left\{{T}^{*}\left(t\right)\right\}={\left\{{T}_{0}^{*}\left(t\right)\text{\hspace{0.17em}}{T}_{1}^{*}\left(t\right)\dots {T}_{m-1}^{*}\left(t\right)\right\}}^{T},$
where ${\left\{\right\}}^{T}$ denotes the transpose operation of the quantity $\left\{\right\}$.
The product of Chebyshev polynomials satisfies:
$f\left(t\right)=\sum _{r=0}^{m-1}{a}_{r}{T}_{r}^{*}\left(t\right),g\left(t\right)=\sum _{r=0}^{m-1}{b}_{r}{T}_{r}^{*}\left(t\right),$
and $\left[Q\right]$ is the integral operator matrix.
The other related theories of shifted Chebyshev series, the value of operator matrix $\left[G\right]$ and $\left[Q\right]$ are given in references [4, 5, 16, 17].
3. The method of analysis
Consider the following strongly nonlinear vibration system:
where $X\left(t\right)={\left\{{x}_{1}\left(t\right){x}_{2}\left(t\right)\dots {x}_{n}\left(t\right)\right\}}^{T}$ is an $n×1$ column vector and $f\left(X\left(t\right),t\right)$ is a function of
period $T$.
In nonlinear dynamics, compared with the determination of equilibrium points, the existence of periodic motions and the number of periodic solutions are even more difficult to ascertain. In general
we only solve a part of periodic motions of the nonlinear vibration system. Let $X\left(t\right)={\left\{{x}_{1}\left(t\right){x}_{2}\left(t\right)\dots {x}_{n}\left(t\right)\right\}}^{T}$ denote a
periodic solution of Eq. (9) such that the periodic solution can be expressed as $m$ terms shifted Chebyshev series:
${x}_{i}\left(t\right)=\left[{T}_{0}^{*}\text{\hspace{0.17em}}{T}_{1}^{*}\text{\hspace{0.17em}}{T}_{2}^{*}\dots {T}_{m-1}^{*}\right]\cdot {\left[{p}_{i0}\text{\hspace{0.17em}}{p}_{i1}\text{\hspace
{0.17em}}{p}_{i2}\dots {p}_{i,m-1}\right]}^{T},$
${\stackrel{˙}{x}}_{i}\left(t\right)=\left[{T}_{0}^{*}\text{\hspace{0.17em}}{T}_{1}^{*}\text{\hspace{0.17em}}{T}_{2}^{*}\dots {T}_{m-1}^{*}\right]\cdot {\left[{q}_{i0}\text{\hspace{0.17em}}{q}_{i1}\
text{\hspace{0.17em}}{q}_{i2}\dots {q}_{i,m-1}\right]}^{T},$
where ${T}_{i}^{*}$ is the $i$th shifted Chebyshev polynomial of the first kind, ${p}_{ij}$ and ${q}_{ij}$ are unknown Chebyshev coefficients. In order to estimate optimization initial value
reasonably, the periodic solution should be expanded in harmonic series (or other series) at the same time:
${x}_{i}\left(t\right)={a}_{i0}+\sum _{k=1}^{N}\left({a}_{ik}\mathrm{c}\mathrm{o}\mathrm{s}\frac{k2\pi t}{jT}+{b}_{ik}\mathrm{s}\mathrm{i}\mathrm{n}\frac{k2\pi t}{jT}\right),\left(i=1,2,\cdots ,n\
where ${a}_{ik}$ and ${b}_{ik}$ are $\left(2N+1\right)\cdot n$ unknown harmonic series coefficients. For a single-period orbit $j=1$, for a period-doubling orbit $j={2}^{n}$ and $n$ is the number of
period-doubling bifurcation. Taking periodic solution of single-periodic orbit for example, we expand every harmonic of Eq. (12) in terms of the shifted Chebyshev series of the first kind, i.e.:
${x}_{i}\left(t\right)=\left[1\text{}\mathrm{cos}\left(\omega t\right)\text{}\mathrm{sin}\left(\omega t\right)\text{}\mathrm{cos}\left(2\omega t\right)\text{}\mathrm{sin}\left(2\omega t\right)\dots \
mathrm{cos}\left(N\omega t\right)\text{}\mathrm{sin}\left(N\omega t\right)\right]\text{\hspace{0.17em}}$$\cdot {\left[{a}_{i0}\text{\hspace{0.17em}}{a}_{i1}\text{\hspace{0.17em}}{b}_{i1}\text{\hspace
{0.17em}}{a}_{i2}\text{\hspace{0.17em}}{b}_{i2}\dots {a}_{iN}\text{\hspace{0.17em}}{b}_{iN}\right]}^{T}\text{\hspace{0.17em}}$$=\left[{T}_{0}^{*}\text{\hspace{0.17em}}{T}_{1}^{*}\text{\hspace
{0.17em}}{T}_{2}^{*}\dots {T}_{m-2}^{*}{T}_{m-1}^{*}\right]\cdot \left[\begin{array}{cccccc}1& {c}_{10}& {c}_{20}& \cdots & {c}_{2N-1,0}& {c}_{2N,0}\\ 0& {c}_{11}& {c}_{21}& \cdots & {c}_{2N-1,1}&
{c}_{2N,1}\\ 0& {c}_{12}& {c}_{22}& \cdots & {c}_{2N-1,2}& {c}_{2N,2}\\ ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& {c}_{1,m-1}& {c}_{2,m-1}& \cdots & {c}_{2N-1,m-1}& {c}_{2N,m-1}\end{array}\right]\cdot \left[\begin
{array}{c}{a}_{i0}\\ {a}_{i1}\\ {b}_{i1}\\ ⋮\\ {b}_{iN}\end{array}\right]$$=\left[{T}_{0}^{*}\text{\hspace{0.17em}}{T}_{1}^{*}\text{\hspace{0.17em}}{T}_{2}^{*}\dots {T}_{m-2}^{*}\text{\hspace
{0.17em}}{T}_{m-1}^{*}\right]\cdot {\left[{p}_{i0}\text{\hspace{0.17em}}{p}_{i1}\text{\hspace{0.17em}}{p}_{i2}\dots {p}_{i,m-2}\text{\hspace{0.17em}}{p}_{i,m-1}\right]}^{T},$
${\stackrel{˙}{x}}_{i}\left(t\right)=\left[\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(2\omega t\right)\
text{}\mathrm{c}\mathrm{o}\mathrm{s}\left(2\omega t\right)\dots \mathrm{s}\mathrm{i}\mathrm{n}\left(N\omega t\right)\text{}\mathrm{c}\mathrm{o}\mathrm{s}\left(N\omega t\right)\right]\text{\hspace
{0.17em}}$$\cdot {\left[-\omega {a}_{i1}\omega {b}_{i1}\text{}-2\omega {a}_{i2}\text{}2\omega {b}_{i2}\dots -N\omega {a}_{iN}\text{}N\omega {b}_{iN}\right]}^{T}$$=\left[{T}_{0}^{*}\text{\hspace
{0.17em}}{T}_{1}^{*}\text{\hspace{0.17em}}{T}_{2}^{*}\dots {T}_{m-2}^{*}\text{\hspace{0.17em}}{T}_{m-1}^{*}\right]\cdot \left[\begin{array}{ccccc}{c}_{20}& {c}_{10}& \cdots & {c}_{2N,0}& {c}_{2N-1,0}
\\ {c}_{21}& {c}_{11}& \cdots & {c}_{2N,1}& {c}_{2N-1,1}\\ {c}_{22}& {c}_{12}& \cdots & {c}_{2N,2}& {c}_{2N-1,2}\\ ⋮& ⋮& \ddots & ⋮& ⋮\\ {c}_{2,m-1}& {c}_{1,m-1}& \cdots & {c}_{2N,m-1}& {c}_
{2N-1,m-1}\end{array}\right]$$\cdot \left[\begin{array}{ccccc}-\omega & & & & \\ & \omega & & & \\ & & -2\omega & & \\ & & & \ddots & \\ & & & & N\omega \end{array}\right]\cdot \left[\begin{array}{c}
{a}_{i1}\\ {b}_{i1}\\ {a}_{i2}\\ ⋮\\ {b}_{iN}\end{array}\right]$$=\left[{T}_{0}^{*}\text{\hspace{0.17em}}{T}_{1}^{*}\text{\hspace{0.17em}}{T}_{2}^{*}\dots {T}_{m-2}^{*}\text{\hspace{0.17em}}{T}_{m-1}
^{*}\right]\cdot {\left[{q}_{i0}\text{\hspace{0.17em}}{q}_{i1}\text{\hspace{0.17em}}{q}_{i2}\dots {q}_{i,m-2}\text{\hspace{0.17em}}{q}_{i,m-1}\right]}^{T},$
where ${T}_{i}^{*}$ is the $i$th shifted Chebyshev polynomial of the first kind, $\omega =2\pi /T$ and each column of matrix $\left[{c}_{ij}\right]$ is from the $m$ terms Chebyshev coefficients of
the corresponding trigonometric function.
Since the domain of shifted Chebyshev series of the first kind is $\left[0,\mathrm{}1\right]$, the transformation $t=T\cdot s$$\left(s\in \left[0,1\right]\right)$ must be done in order to normalize
the period to 1. In actual calculation it is only necessary to expand the nonlinear vibration system equation into shifted Chebyshev series of the first kind and to replace $\omega$ with $2\pi$, $t$
with $s$, ${T}_{i}^{*}\left(t\right)$ with ${T}_{i}^{*}\left(s\right)$. Transpose the right side of Eq. (9), then in symbolic form residual force $R\left(s\right)$ can be written as:
$R\left(s\right)={\left[{\stackrel{-}{T}}^{*}\left(s\right)\right]}^{T}\cdot \left[F\left({a}_{ik},{b}_{ik}\right)\right],$
where ${\left[\stackrel{-}{T}\left(s\right)\right]}_{n×nm}^{T}=I\otimes {\left[{T}_{0}^{*}\left(s\right)\text{\hspace{0.17em}}{T}_{1}^{*}\left(s\right)\text{\hspace{0.17em}}{T}_{2}^{*}\left(s\right)\
dots {T}_{m-1}^{*}\left(s\right)\right]}^{T}$, $\otimes$ represents matrix Kronecker product. $\left[I\right]$ is the identity matrix. $\left[F\left({a}_{ik},{b}_{ik}\right)\right]$ is an $nm×1$
column vector. Obviously, in the case of exact solutions, residual $R$ equals to $0$ on arbitrary time point in a primary cycle. Let ${R}_{i}\left(s\right)$ be the row component of residual force $R\
left(s\right)$. The total error between the exact solutions and periodic solutions, see Eq. (10), is equivalent to the sum of the absolute value of all time points in a cycle. It can be expressed as
an unconstrained nonlinear optimization problem, i.e.:
$\underset{{a}_{ik},{b}_{ik}\in R}{\mathrm{m}\mathrm{i}\mathrm{n}}J,\text{where}J=\frac{1}{n}\sum _{i=1}^{n}{\int }_{0}^{1}{R}_{i}{\left(s\right)}^{2}ds.$
The unknown coefficients ${a}_{ik}$ and ${b}_{ik}$ can be obtained via local optimization algorithm such as quasi-Newton method [26]. Since initial value selection will affect optimization results of
local optimization algorithm, the number of harmonic terms in Eq. (13) can be adjusted in order to limit optimization initial value in a reasonable range. For simplicity we can directly reference to
the harmonic terms of harmonic balance method (HBM). Then according to Eq. (13), Eq. (14) and Eq. (16) the unknown Chebyshev coefficients ${p}_{ij}$ and ${q}_{ij}$ can be calculated. Note that the
periodic solution of engineering models usually has a clear physical meaning (such as in Section 4.2.2) and a closed interval of the feasible region can be estimated. Then the interval optimization
algorithm can be used to seek global optimal solutions, while it will greatly increase the time complexity and computational complexity, and sometimes it is not necessary.
When the periodic solution is expressed in the form of shifted Chebyshev series, the periodic solution stability can be analyzed as follows. Suppose that a perturbation $\Delta x\left(t\right)$ is
imposed to the known periodic solution ${x}_{0}\left(t\right)$, i.e.:
$x\left(t\right)={x}_{0}\left(t\right)+\Delta x\left(t\right).$
Substitute Eq. (17) into the system equation and omit the higher order small amount. Then a linearized system with respect to $\Delta x\left(t\right)$ can be obtained:
$\Delta \stackrel{˙}{x}\left(t\right)=Df\left({x}_{0}\left(t\right)\right)\cdot \Delta x\left(t\right).$
By linear periodic system theory and Chebyshev series operation properties, we only need to integrate the linearization system Eq. (18) over the interval $\left[0,\mathrm{}1\right]$ with each column
of the identity matrix as the integral initial value separately. Then the linearization-system state vector at the end point of the period is the corresponding column vector of FTM $\Phi \left(T\
right)$ [4]. According to Floquet theory, if the eigenvalue norms of FTM are less than 1, the periodic solution of the system is asymptotically stable, otherwise it is unstable.
The shifted Chebyshev polynomial of the first kind, multiplication and integral operator matrix used in calculating residual force, objective function and FTM are given in references [4, 5, 16, 17].
In this paper a 15 terms shifted Chebyshev series of the first kind is adopted.
4. Examples
4.1. Application to the Duffing system
Consider the Duffing system with cubic nonlinearities:
$\stackrel{¨}{x}+ax+b\stackrel{˙}{x}+c{x}^{3}=d\mathrm{c}\mathrm{o}\mathrm{s}\left(\omega t\right),$
where $a=4$, $b=2$, $c=17$, $d=5$, $\omega =2$. When solving the shifted Chebyshev series periodic solution with the suggested method, we must normalize the period to 1 by transforming the time
variable $t$ to $2\pi s/\omega$$\left(s\in \left[0,1\right]\right)$. Then we expand Eq. (19) into shifted Chebyshev series, and calculate the system residual $R$ via multiplication and integral
operator matrix. Finally quasi-Newton method can be used to seek optimal solutions of objective function. Table 1 shows the unknown coefficients of periodic solution obtained by the optimization
method (3-term harmonic) and HBM (7-term harmonic).
Table 1Coefficients of approximate analytical periodic solution
Coefficient Optimization method Coefficient HBM
${p}_{0}$ 0.000214327013 ${a}_{0}$ 0
${p}_{1}$ 0.55117695 ${a}_{1}$ 0.550665369
${p}_{2}$ 0.37818696117 ${b}_{1}$ 0.37846670826
${p}_{3}$ 0.00059928746 ${a}_{2}$ 0
${p}_{4}$ -0.0009649196 ${b}_{2}$ 0
${p}_{5}$ 0.0157754674 ${a}_{3}$ 0.0162381788
${p}_{6}$ 0.051302781 ${b}_{3}$ 0.0515055873
${p}_{7}$ 0.00044608728 ${a}_{4}$ 0
${p}_{8}$ 0.00055906966 ${b}_{4}$ 0
${p}_{9}$ -0.0022438111 ${a}_{5}$ -0.00222665
${p}_{10}$ 0.0023410744 ${b}_{5}$ 0.003084981
${p}_{11}$ -0.0000682019 ${a}_{6}$ 0
${p}_{12}$ 0.00056548798 ${b}_{6}$ 0
${p}_{13}$ -0.0002677232 ${a}_{7}$ -0.0002695199
${p}_{14}$ -0.000180024115 ${b}_{7}$ 0.000017709946
Fig. 1 and Fig. 2 show the residual curves obtained by the two methods over one time cycle. It can be seen that even the number of harmonic terms adopted in the suggested method is fewer than that of
HBM, the periodic solution obtained still has greater accuracy than HBM. The reason is that in the suggested method adjusting the harmonic expansion terms is equivalent to changing the optimization
initial value reasonably, and the objective function aims to reduce the error between the supposed periodic solution and the exact solution. Obviously the suggested method is apt to get a higher
accuracy periodic solution. The phase portraits of Duffing system obtained by optimization method and HBM are displayed in Fig. 3, and both of them coincide very well. The optimization method not
only requires expanding fewer harmonic items than HBM (i.e. solving fewer unknown coefficients), but also reduces the residual from 10^-3 to 10^-7 order of magnitude.
Fig. 1The residual curve of HBM
Fig. 2The residual curve of the suggested method
Fig. 3The phase portrait of Duffing system
According to the linear periodic system theory and the Chebyshev series operation property, take each column of identity matrix as the integral initial condition respectively. The state vector
obtained at the end of the cycle via integration is the corresponding column of FTM. One of the norms of the eigenvalue (Floquet multiplier) of FTM is 10.91, which is greater than 1. Then the
periodic solution of Duffing equation is unstable.
4.2. Helicopter rotor system
Rotor response and its stability is an important research topic in helicopter dynamics. Rotor dynamics model is a time-varying differential equation group containing nonlinear structure, inertial and
aerodynamic loads. It is usually calculated by HBM, time finite element method (TFEM) or numerical integration algorithm. Since numerical integration algorithm is sensitive to integration initial
value, solving rotor response usually adopts HBM or TFEM.
4.2.1. Application to the articulated rotor system
Take the articulated helicopter rotor motion for example. Consider flapping/lagging coupled movements in rotating plane, as shown in Fig. 4 and Fig. 5. Suppose that each blade has the same
properties. The flapping/lagging coupled equations are established as follows:
$\left\{\begin{array}{l}{\int }_{e}^{R}{\eta }_{\beta }\left(r-e\right)mdr\text{\hspace{0.17em}}\stackrel{¨}{\beta }+{\int }_{e}^{R}{\eta }_{\beta }rmdr\text{\hspace{0.17em}}\beta \text{\hspace
{0.17em}}{\Omega }^{2}+{K}_{\beta }\left(\beta -{\beta }_{p}\right)={\int }_{e}^{R}\left(r-e\right){F}_{z}dr,\\ {\int }_{e}^{R}\left(r-e\right){\eta }_{\xi }mdr\text{\hspace{0.17em}}\stackrel{¨}{\xi
}+{\int }_{e}^{R}e{\eta }_{\xi }mdr\text{\hspace{0.17em}}\xi \text{\hspace{0.17em}}{\Omega }^{2}+{\int }_{e}^{R}{\eta }_{\beta }\text{\hspace{0.17em}}{\eta }_{\xi }mdr\text{\hspace{0.17em}}2\Omega \
text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}\stackrel{˙}{\beta }+{K}_{\xi }\xi ={\int }_{e}^{R}\left(r-e\right){F}_{x}dr.\end{array}\right\$
Fig. 4The force diagram of articulated rotor blade flap movement
Fig. 5The force diagram of articulated rotor blade lag movement
The symbols in the above equations have been described in references [27, 28]. ${F}_{x}$ is the aerodynamic force parallel to the disc plane. ${F}_{z}$ is the aerodynamic force vertical to the disc
plane. Where:
${F}_{x}=\frac{1}{2}\left({u}_{P}{u}_{T}\theta -{u}_{P}^{2}\right)+\frac{{C}_{d}}{2a}{u}_{T}^{2},$
${F}_{z}=\frac{1}{2}\left({u}_{T}^{2}\theta -{u}_{P}{u}_{T}\right),$
${u}_{T}=r+\mu \mathrm{s}\mathrm{i}\mathrm{n}\left[\psi \right],$
${u}_{P}=\lambda +r\stackrel{˙}{\beta }+\beta \mu \mathrm{c}\mathrm{o}\mathrm{s}\left[\psi \right].$
Table 2Main parameters of the articulated helicopter rotor system
Illustration Unit Value
Rotor radius m 5.345
Rotor speed rad/s 40.42
Rotor shaft anteversion angle deg 2
Chord length m 0.35
Blade number 3
Flapping hinge overhang amount m 0.205
Blade twist angle deg -12
Airfoil lift line slope rad^-1 6.2
Airfoil zero lift incidence deg 0.75
Rotor solidity 0.06253
Mass moment around flapping hinge kg⋅m 88.68
Inertia moment around flapping hinge kg⋅m^2 306.01
Let us assume that advance ratio $\mu$ equals to 0.2 and take Drees inflow model. For ease of calculation, the parameters not listed in the Table are supposed equal to 0.
Transform the blade motion equations into state equations and take the periodic solution Eq. (13) into Eq. (20). Normalize the system period to 1. Then residual force ${R}_{i}$ of each motion
equation can be obtained. When the periodic solutions are expanded as the 2-term harmonics, the variance averages of flapping and lagging motion equations obtained by HBM are 1.90047 and 0.073.
Obviously, it is a wrong conclusion, because large magnitude harmonic terms of the periodic solution are omitted. Meanwhile, the system variance averages obtained by optimization method (quasi-Newton
method) are 0.443 and 0.128, which can be considered comparatively accurate. The reason is that in the suggested method harmonic series is only used to correct the optimization initial value. The
final result is the local optimal value in the vicinity of the initial value and the periodic solution is still approximated by a 15 terms shifted Chebyshev series. When expanded in the form of the
3-term harmonics, the total variance averages obtained by optimization method are 9.8 % lower than that of HBM in a cycle. Fig. 6 and Fig. 7 show the residual curves of the blade flapping/lagging
motions when periodic solutions are expanded to the 3-term harmonics.
Fig. 6Residual curve of flapping movement (3-term harmonics)
Fig. 7Residual curve of lagging movement (3-term harmonics)
Note that the objective function of this issue can also be solved by global optimization algorithm. As $\beta$ and $\xi$ on behalf of the flapping and lagging periodic solutions we can estimate a
reasonable feasible region, i.e. a closed interval range according to actual situation. Then the periodic solutions can be converged to global optimal solutions by deterministic methods, while the
calculation complexity will increase significantly and the time cost becomes unacceptable. It can be seen through the calculation results that by adjusting the optimization initial values local
optimization method (quasi-Newton method) can also generate satisfying results. When we take the 7-term harmonic expansion, the system variance average can reach up to 10^-10 order of magnitude and
the flapping and lagging movement phase portraits are shown in Fig. 8 and Fig. 9.
The periodic solution stability can be analyzed by Floquet theory. According to the linear periodic system theory and the Chebyshev series operation property we calculate the FTM. All the norms of
eigenvalue of FTM are 0.0963 and 0.6802, less than 1. Therefore when advance ratio $\mu$ equals to 0.2, the helicopter rotor coupling motion is asymptotically stable.
Fig. 8Phase portrait of blade flapping movement
Fig. 9Phase portrait of blade lagging movement
4.2.2. Application to the hingeless rotor system
Consider the hingeless helicopter rotor system. Establish rotor blade motion equation in rotating coordinate via finite element method. The blade space finite element is shown in Fig. 10.
Fig. 10Blade space finite element
The blade motion equation using the symbolic representation is:
where $q\left(t\right)=\left\{{u}_{1},{v}_{1},{{v}^{"}}_{1},{w}_{1},{{w}^{\text{'}}}_{1},{\varphi }_{1},\dots ,{u}_{14},{\varphi }_{10},{u}_{15}\right\}$ is the total node displacement vector on a
single blade. $u$, $v$, $w$, $\varphi$ represent the stretching, lagging, flapping and twist elastic displacements for each node on the blade elastic axis. In this example the blade parameters are
adopted as BO-105 rotor parameters in [29], where quasi-steady aerodynamic force and Drees inflow model are considered.
In order to reduce the computation time and equation dimension, we take the first six order intrinsic modes in calculation. Then Eq. (25) transforms into an equation with respect to mode coordinate
$\stackrel{˙}{p}\left(t\right)=A\left(p\left(t\right),t\right)\cdot p\left(t\right)+C\left(t\right),$
where $p\left(t\right)={\left\{{p}_{1},{p}_{2},{p}_{3},{p}_{4},{p}_{5},{p}_{6},{\stackrel{˙}{p}}_{1},{\stackrel{˙}{p}}_{2},{\stackrel{˙}{p}}_{3},{\stackrel{˙}{p}}_{4},{\stackrel{˙}{p}}_{5},{\stackrel
{˙}{p}}_{6}\right\}}^{T}$ is dimensionless mode degree of freedom. A numerical method, named TFEM, is usually used to solve the response of high-dimension nonlinear rotor system model.
Fig. 11Dimensionless mode degree of freedom phase portraits on blade tip position
Fig. 12Blade tip flapping response phase portrait
Fig. 13Blade tip lagging response phase portrait
Fig. 14Blade tip twist response phase portrait
When periodic solution is expanded to 3-term harmonics as optimization initial value, the total variance averages obtained by optimization method reach up to 10^-6 order of magnitude, which is
slightly more accurate than TFEM (take 15 time elements and 5-order shape function). The dimensionless mode degree of freedom phase portraits on blade tip position are shown in Fig. 11. Then we
return the mode degree of freedom to physical degree of freedom. Fig. 12 to Fig. 14 show the flapping, lagging and twist response phase portraits. The periodic solutions obtained by optimization
method coincide very well with the numerical results of TFEM. It proves that the suggested method is accurate and effective.
According to Floquet theory, all the norms of eigenvalues of FTM are less than 1. Therefore when advance ratio $\mu$ equals to 0.2, the periodic motion of this hingeless rotor is asymptotically
5. Conclusions
In this paper, using the good properties of shifted Chebyshev series in numerical approximation field, proceeding from optimization of system residual force, the analytical periodic solution in the
form of shifted Chebyshev series of the first kind is obtained. Compared with HBM, when the periodic solution is expanded in fewer or the same harmonic terms, the suggested method owns higher
accuracy. When solving the high-dimension nonlinear system, it can still obtain a high precision analytical solution. In periodic solution stability analysis the FTM can be obtained directly and
accurately by the integral operation of Chebyshev series without the help of special numerical approach for calculating approximate FTM. Examples show that in addition to solving low-dimensional
system, this method also can be used to calculate the periodic solution and to analyze the stability of high-dimensional nonlinear vibration system, such as the helicopter rotor system. It indicates
that finding Chebyshev series periodic solutions of nonlinear vibration systems via optimization method is accurate and effective.
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• Tang J. Y., Chen S. Y. Study on periodic solutions of strongly nonlinear systems with time-varying damping and stiffness coefficients. Journal of Vibration and Shock, Vol. 26, Issue 10, 2007, p.
96-100, (in Chinese).
• Chen S. H., Shen J. H. Bifurcations and analyses of route to chaos of Mathieu-Duffing oscillator by the incremental harmonic balance method. Science & Technology Review, Vol. 25, Issue 22, 2007,
p. 22-26, (in Chinese).
• Yuan Y. X. Calculation Method of Nonlinear Optimization. Science Press, Beijing, 2008, (in Chinese).
• Johnson W. Helicopter Theory. Aviation Industry Press, Beijing, 1991, (in Chinese).
• Gao Z., Chen R. L. Helicopter Flight Dynamics. Science Press, Beijing, 2003, (in Chinese).
• Gunji B., Chopra T. University of Maryland Advanced Rotorcraft Code (UMARC) Theory Manual. UMAERO Report, 1994.
About this article
04 September 2013
30 September 2013
nonlinear dynamics
rotor dynamics
shifted Chebyshev series of the first kind
steady state periodic solution
This paper is supported by specialized research fund for the doctoral program of higher education of China (No. 20113218110002) and a project funded by the priority academic program development of
Jiangsu higher education institutions of China.
Copyright © 2013 Vibroengineering
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/14593","timestamp":"2024-11-08T12:33:09Z","content_type":"text/html","content_length":"188040","record_id":"<urn:uuid:717f94cc-707f-4823-a4f6-2eedf5062629>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00062.warc.gz"} |
How to Use the SUBTRACT Function in Excel
Excel is a powerful spreadsheet program from Microsoft that makes it easy to work with numbers and other values.
While Excel contains a lot of power, it’s also quite useful for simple addition, subtraction, multiplication, and division.
In fact, Excel’s simple formulas make it easy to add and subtract numbers and cell values without breaking out a calculator.
How to Use Subtract Function in Excel
Many of Excel’s most powerful capabilities come via the use of functions, which are a kind of predesigned formula.
For example, the SUM function automatically sums or totals a range of cells without you having to manually add each cell to the previous one.
Unfortunately, there is no SUBTRACT function in Excel. That’s because subtracting is one of the easiest things you can do in a spreadsheet.
I‘ve found that performing subtraction in Excel is as simple as entering a simple mathematical formula. Excel does the rest.
The formula for subtracting one number from another starts with an equal sign and looks like this:
=value1 – value2
It doesn’t get much easier than that. You can subtract discrete values or the values contained in individual cells.
For example, let’s say I want to put an aquarium in my home so I can enjoy tropical fish.
I made a spreadsheet of the different tasks I need to complete and how much I expect each purchase to cost. I then gather the receipts to note what I actually spent. We’ll use subtract to see if my
expected costs match reality.
Step 1: Type an equal sign in a cell.
You subtract numbers and cell values in Excel using a simple formula. In Excel, every formula starts with an equal sign, so position your cursor in the cell where you want to show the solution and
enter the following:
Step 2: Add the first cell address.
Position the cursor after the equal and either manually enter the first cell address or use your mouse to select the first cell. You should now see something like this:
Step 3: Enter a minus sign.
Position the cursor after the first cell address and enter a minus sign, like this:
Step 4: Add the second cell address.
Position the cursor after the minus sign and either manually enter the second cell address or use your mouse to select the second cell. You now have the following formula:
Step 5: Press enter to the solution.
Press enter to enter the formula, and the solution is displayed in the cell. After running the formula on every line in my sheet, I can see that I completed my aquarium with $236 to spare.
Tips for Subtracting in Excel
You’re not limited to subtracting just one cell from another. In my experience, I’ve found that Excel can perform many different types of subtraction.
For example, you can subtract entire rows and columns, as well as cell ranges. Here are three tips on how to get the most out of subtracting in Excel.
How to Subtract Multiple Cells
You’re not limited to simple subtraction in Excel. You can subtract multiple cells from a single cell by stringing together additional cell addresses with minus signs.
For example, to subtract cells C13 through C 18 from B12, use the formula:
You could also write this as subtracting a range of cells from the first cell. In this instance, you’d use the range B2:B4 and enter this formula:
Just enclose the range you’re subtracting within parentheses.
How to Subtract Entire Columns
You may want to subtract all the values in one column from all the values in another column. This is easily done by copying the formula from a single cell to all the cells in the solution column.
Simply enter your subtraction formula into the first cell in a new column. Then, copy that formula to the other cells in the column by dragging the corner of the first cell down the column.
All the cells in subsequent rows will subtract the same two relative cells as in the first formula.
For example, if you copy the formula =D1-C1 downward through a column, the cells in each subsequent row would contain the formulas:
And so on.
How to Subtract the Same Number From a Column of Numbers
Excel also lets you subtract the same number or cell value from all the cells in a column. You do this by locking the repeated number with dollar signs ($).
Let’s say you want to subtract the value in a given cell from a range of cells. For our example, assume the cell you want to subtract is G1.
You can’t just enter G1 and then copy the formula because Excel will change the referenced cell as you copy the formula.
Instead, you need to lock G1 into the formula by using dollar signs ($) in front of the row and column referenced by entering $G$1.
This tells Excel to always reference cell G1, whatever the other values in the formula. You end up with something like this:
Pro tip: You can then copy that formula down an entire column, subtracting the fixed number from each cell in that column.
To learn even more about Excel, check out our article on How to Use Excel Like a Pro. This useful article contains 29 powerful tips, tricks, and shortcuts that will make Excel even more powerful.
Getting Started
Excel includes powerful arithmetic capabilities, including the ability to subtract numbers and cell values. Excel makes simple subtraction as easy as writing numbers on a blackboard. | {"url":"https://specialeventclub.com/2024/02/07/how-to-use-the-subtract-function-in-excel/","timestamp":"2024-11-04T02:01:12Z","content_type":"text/html","content_length":"95757","record_id":"<urn:uuid:947071ab-02bd-4c11-8e94-24239417ce32>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00246.warc.gz"} |
Cooperative light scattering by a system of
two-level atoms has been a topic studied since many years [
]. Many studies in the past have been focused on a diffusive regime dominated by multiple scattering [
], where light travels over distances much larger than the mean free path. More recently, it has been shown that light scattering in dilute systems induces a dipole-dipole interaction between atom
pairs, leading to a different regime dominated by single scattering of photons by many atoms. The transition between single and multiple scattering is controlled by the optical thickness parameter
$b ( Δ ) = b 0 / ( 1 + 4 Δ 2 / Γ 2 )$
], where
$b 0$
is the resonant optical thickness,
is the detuning of the laser frequency from the atomic resonance frequency and
is the single atom decay rate. A different cooperative emission is provided by superradiance and subradiant, both originally predicted by Dicke in 1954 [
] in fully inverted system. Whereas Dicke superradiance is based on constructive interference between many emitted photons, subradiance is a destructive interference effect leading to the partial
trapping of light in the system. Dicke states have been considered for an assembly of
two-level systems, realized, e.g., by atoms [
] or quantum dots [
]. In contrast to an initially fully inverted system with
photons stored by
atoms, states with at most one single excitation have attracted increasing attention in the context of quantum information science [
], where the accessible Hilbert space can be restricted to single excitations by using, e.g., the Rydberg blockade [
]. A particular kind of single-excitation superradiance has been proposed by Scully and coworkers [
], in a system of
two-level atoms prepared by the absorption of a single photon (Timed Dicke state). A link between this single-photon superradiance and the more classical process of cooperative scattering of an
incident laser by
atoms has been proposed by a series of theoretical and experimental papers [
]. In such systems of driven cold atoms subradiance has been also predicted [
] and then observed [
], after that the laser is abruptly switched off and the emitted photons detected in a given direction. Subradiance, by itself, has attracted a large interest of its application in the quantum optics
as a possible method to control the spontaneous emission, storing the excitation for a relatively long time. A crucial point is to determine if such subradiant states are entangled or not, in view of
a possible application as quantum memories.
The aim of this paper is to provide a mathematical description of the single-excitation states in terms of superradiant and subradiant states, i.e. separating the fully symmetric state by the
remaining antisymmetric ones. Symmetric and subradiant excited states are distinguished by their decay rates, once populated by a classical external laser and observed after that the laser is
switched off: the symmetric state has a superradiant decay rate proportional to
$N Γ$
, where
is the single-atom decay, whereas the antisymmetric states have a decay rate slower than
. Once characterized the time evolution of these states, we will apply the criteria of the spin squeezing inequalities introduced by Tóth [
] to detect entanglement in the superradiant and subradiant states. We outline that we limit our study to the linear regime, where the excitation amplitude is proportional to the driving incident
electric field. In this linear regime, we must consider the entanglement criteria which are independent on the value of the driving field, i.e. abandoning these criteria which lead to expressions
which depend nonlinearly from the driving field, as it will be discussed in the following.
The paper is organized as follow. In
Section 2
we present the Hamiltonian describing the dynamics of
two-level atoms interacting with the driving field and write the equation of motion in the linear regime. Then we calculate the decay rate and the transition rates between different elements of the
so-called Timed-Dicke basis with its symmetric and antisymmetric states.
Section 3
introduces the collective spin operator and the formalism of the spin-squeezing inequalities to assess entanglement. Conclusions are eventually drawn in
Section 4 | {"url":"https://www.preprints.org/manuscript/202307.2009/v1","timestamp":"2024-11-13T01:45:12Z","content_type":"text/html","content_length":"1048922","record_id":"<urn:uuid:9bb33885-7502-4ca5-a104-7d76a685be24>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00793.warc.gz"} |
in progress-Important RGPV Question, CS-304 , Digital Systems, III Sem, CSE π CS.com
Important RGPV Question
CS-304 Digital Systems
III Sem, CSE
UNIT-1 Review of Number System and Number Base Conversions
UNIT-2 Combinational Logic-Half Adder, Half subtractor, full adder, full subtractor
UNIT-3 Sequential Logic-Flip-Flops, D, T, S-R, J-K Master-Slave, Racing Condition, Edge & Level Triggered Circuits
UNIT-4 Introduction to A/D & D/A Converters & their Types ,Sample and hold circuits, Voltage to frequency & frequency to voltage Conversion.
UNIT-5 Introduction to digital communication, Nyquist sampling theorem, Time Division Multiplexing
UNIT 1
Q.1) Given the following Boolean function: F(W,X,Y,Z) = WX'(Y’ + Z’) + X’.Zβ .(Wβ Y) where represents the XNOR operation, determine the simplified (minimal) SOP expression for F using boolean
algebra and Implement the given function using NOR-NOR logic.
(RGPV Nov 2022)
Q.2) Draw a schematic for a minimal circuit that uses only NOR gates that performs the two’s complement operation on a four bit input value. Let the input be A3:0 and the output be B3:0.
(RGPV Nov 2022)
Q.3) Simplify the Boolean function F together with the don’t-care conditions d in following forms:
i) sum-of-products
ii) product-of-sums
F(w, x, y, z)= Ξ£ (0, 1, 2, 3, 7, 8, 10)
d(w, x, y, z) = Ξ£ (5, 6, 11, 15)
(RGPV Dec 2020)
Q.4) Give a Boolean expression that corresponds to this logic circuit. Develop a truth table for the circuit, showing columns for atleast the output of each 2-input gate. You should invent new
variable names for these intermediate outputs.
(RGPV Dec 2020)
Q.5) Differentiate between analog and digital circuits.
(RGPV Nov 2018)
Q.6) Convert the following- (i) (48.625)10 =(); (ii) Divide (IEC87)16 by (A5)16 .
(RGPV June 2014)
Q.7) Convert the Following- (1) Decimal 225.225 to binary, octal and hexadecimal.
(ii) Binary 11010111.110 to decimal, octal and hexadecimal.
(RGPV June 2017)
Q.8) Convert (412)10 to (i) Binary (ii)Octal
(iii) Hexadecimal.
(RGPV Dec 2017)
Q.9) Convert the number (210.25)10 to base 2, 8.
(RGPV June 2015)
Q.10) Convert (41.6875)10 to (i) Binary (ii) Octal (iii) Hexadecimal.
(RGPV May 2018)
Q.11) Convert the following β
(i) (1111)2= ( )10
(ii) (10010.1011)2= ( )10
(iii) (23)10 = ( )2
(iv) (5.5)10 =( )2
(v) (47.6)10 = ( )2
(RGPV Nov 2018)
Q.12) Do as directed-
(i) (56)16 = (?)10
(ii) (32)10 = (?)2
(iii) Bubbled OR gate is also calledβ ¦β ¦β ¦
(RGPV Dec 2016)
Q.13) Justify the following statements with examples – (i) Excess-3 code is self complementary code.
(ii) Gray code is a reflected code.
(RGPV May 2018)
Q.14) Write briefly about error detecting and error correcting codes.
(RGPV Dec 2014)
Q.15) Define NAND and NOR gates and give their truth tables. Write down the boolean expressions for the output of each gate.
(RGPV Nov 2018)
Q.16) What do you understand by universal gate? Design all logic gates using universal gates.
(RGPV June 2009)
What is universal gate?
(RGPV June 2014)
What are universal gates? Explain with example.
(RGPV Dec 2014)
What are universal gates? Why are they called so?
(RGPV Dec 2015)
Why NAND gate is known as universal gate?
(RGPV Dec 2016)
What is universal gate? Implement AND, OR and NOT gates using NAND gates and NOR gates.
(RGPV Dec 2017)
Q.17)Implement the function F = A(B+ CD) + BC’ using NOR gate.
(RGPV June 2017)
Q.18) State and prove basic laws of Boolean algebra.
(RGPV Dec 2013)
Q.19) Write five theorem of Boolean algebra and simplify F (A+B)’ (A+B).
(RGPV May 2018)
Q. 20) What is Boolean algebra write any three theorems of Boolean algebra?
(RGPV Dec 2017)
Q. 21) State the distributive property of Boolean algebra.
(RGPV Dec 2016)
Q. 22) Simplify the following Boolean function to minimum numbers of literals β
(i) zx+zx’y (ii) xy+xy’ (iii) y(wz’+wz) + xy.
(RGPV June 2017)
Q. 23) Simplify the following Boolean function with K-map- F(w, x, y, z) = (0, 1, 2, 4, 5, 6, 8, 9, 12, 13, 14).
(RGPV June 2010, 2011, 2014)
Simplify the Boolean function using K-map- F(A, B, C, D) = E(0, 1, 2, 4, 5, 6, 8, 9, 12,13, 14)
(RGPV Dec 2017)
Q. 24) Simplify the Boolean function with don’t care conditions and implement it with NAND gates- F(w, x, y, z) = E (1, 3, 7, 11, 15) d (w, x, y, z) = (0, 2, 5)
(RGPV May 2018)
Q. 25) Simplify the Boolean function
F-B’C’D’ + BCD’+ ABCD’ and the don’t care condition d=B’CD’+A’BC’D.
(RGPV June 2017)
Q. 26) Explain Hamming and block codes.
(RGPV June 2011)
UNIT 2
Q.1) What is decoder? Explain BCD to decimal decoder.
(RGPV June 2020)
Q.2) Design a combinational circuit to convert the binary input ABCD to gray code.
(RGPV June 2020)
Q.3) Design the binary to octal Decoder and explain its working using block diagram.
(RGPV Dec 2020)
Q.4) Design Half adder using NAND gates. Also Draw the diagram?
(RGPV Dec 2020)
Q.5) Draw the truth table and logic diagram of full adder.
(RGPV Dec 2015)
Design and draw a full adder circuits.
(RGPV Dec 2017)
Q.6) Implement a full adder circuit with a decoder and two OR gates.
(RGPV June 2014)
Implement a full adder circuit with a (3 to 8 lines) decoder and two OR gates.
(RGPV Dec 2017)
Q.7) Explain full adder and design a full adder circuit using 3 to 8 decoder and two OR gates.
(RGPV Dec 2016)
Q.8) Design a full subtractor circuit using decoder and OR gates.
(RGPV Dec 2015)
Q.9) Design a full subtractor using logic gates.
(RGPV Dec 2014)
Q.10) Explain half subtractor circuit.
(RGPV Dec 2013)
Q.11) Design a full adder using minimum logic gates and also discuss he working of parallel adder.
(RGPV Dec 2012)
Q.12) Draw the logic diagram of look-ahead carry generator and explain Fits working.
(RGPV Dec 2008, 2015)
Explain the operation of look-ahead carry generator.
(RGPV June 2007, 2008, 2010)
Discuss/Explain the working of look ahead carry generator.
(RGPV Dec 2012, June 2015)
What is look ahead carry generator? Explain with logic diagram.
(RGPV June 2014)
Design and explain the working of look ahead carry generator.
(RGPV Dec 2014)
Explain look ahead carry generator.
(RGPV Dec 2016)
Q.13) Explain/Design BCD adders.
(RGPV June 2008, 2012, Dec 2013)
Design and explain the working of 4-bit BCD adder.
(RGPV Feb 2010)
Design a BCD adder and also give the rules of BCD addition.
(RGPV June 2010)
Design a BCD adder using logic gates.
(RGPV Dec 2012)
Design and explain the working of BCD adder.
(RGPV Dec 2014)
Draw the logic diagram of BCD adder and explain its working.
(RGPV June 2007, Dec 2015)
Write short note on BCD adders.
(RGPV Dec 2017)
Q.14) Draw and explain a 4-bit magnitude comparator.
(RGPV June 2014)
Q.15) Implement the following Boolean function using 4:1 multiplexer using A and B variables to the selection lines β
F(A, B, C) = Ξ£m(1, 4, 5, 7).
(RGPV Dec 2015)
Q.16) Design a BCD to excess-3 code converter.
(RGPV Dec 2015, May 2018)
UNIT 3
SEQUENTIAL LOGIC-FLIP-FLOPS, D, T, S-R, J-K MASTER- SLAVE, RACING CONDITION, EDGE & LEVEL TRIGGERED CIRCUITS
Q.1) What is a Shift Register? Draw and explain shift Left- Right shift register.
(RGPV Dec 2020)
Q.2) Explain the race condition in S-R flip flop. Also explain how it is removed in J-K flip flop.
(RGPV Dec 2020)
Q.3) How a multiplexer can be used as ROM? Explain in brief.
(RGPV Dec 2020)
Q.4) Design a sequential circuit using T flip-flop for the following state table.
Assume any suitable assumptions for state assignment.
(RGPV Nov 2022)
Q.5) Design a synchronous counter to count in the random sequence 0, 2, 4, 5, 7.0, 2, 4, 5, 7… using D flip-flop.
(RGPV Nov 2022)
Q.6) Explain the concept of working and applications of following memories-
i) ROM
ii) PLA
iii) DRAM
iv) FLASHRAM
(RGPV Nov 2022)
Q.7) Given the network of Fig., determine the functions f2 and f3 if f1 = xz+x’zβ and the overall transmission function is to be
f(w, x,y,z)= Σ (0,3,6,10,11,12)
(RGPV Nov 2022)
Q.8) Each of the following functions actually represents a set of four functions, corresponding to the possible assignments of the don’t-care terms.
F1 (w,x,y,z)= Σ (1,3,5,6,9,10)+ (11,12)
F2(w,x,y,z)= Σ (0,3,4,5,8,9)+ (6,7)
i) Find f3-f1.f2. How many functions does f3 represent?
ii) Find f4= f1 +f2. How many functions does f4 represent?
(RGPV Nov 2022)
Q.9) Explain synchronous and Asynchronous counter.
(RGPV June 2020)
Q.10) What is Flip-Flop? Explain Master Slave J-K flip-flop.
(RGPV June 2020)
Q.11) Differentiate static and dynamic RAM.
(RGPV Nov 2019)
Q.12) Write short note on flash RAM.
(RGPV Nov 2018, May 2019)
Q.13) What are sequential circuits? What is the main difference between the combinational circuits and sequential circuits?
(RGPV Nov 2018)
Q.14) What is a flip-flop? Explain the principle of operation of R-S flip-flop with truth table.
(RGPV Nov 2018)
What is a lip-flop? Explain with a suitable example.
(RGPV Dec 2017)
Q.15) What is a shift register? Explain.
(RGPV June 2010, Dec 2017)
Discuss the shift registers.
(RGPV June 2011)
Q.16) Design a synchronous BCD counter with JK flip-flops.
(RGPV Dec 2016)
Q.17) Design a MOD-6 counter using J-K flip flops.
(RGPV Dec 2012)
Q.18) Design a MOD-12 binary counter using J-K flip-flop.
(RGPV June 2015)
Q.19) Find the MOD number of counter in fig Determine its counting sequence. Draw the state diagram. Find the frequency at output QD if input frequency is 7 kHz.
(RGPV June 2014)
Q.20) Give a brief introduction of a semiconductor memories.
(RGPV Dec 2011)
Give a comparison of various semiconductor memories.
(RGPV June 2014)
Q.21) State and differentiate between ROM, PROM. EPROM and EEPROM.
(RGPV Dec 2014)
Q.22) Write a short note on PLA.
(RGPV Dec 2010, June 2012, May 2018, 2019)
Explain PLAS.
(RGPV Dec 2013)
Q.23) What is RAM? Distinguish between SRAM and DRAM. What is PLA?
(RGPV Dec 2015)
Q.24) Design a combinational circuit using ROM. The circuit accepts a 3-bit number and generates an output binary number equal to the square of the input number.
(RGPV June 2014)
Derive a PLA program table for a combinational circuit that squares a 3-bit number.
(RGPV June 2017)
Q.25) A combinational logic circuit is defined by the functions – F1= Ξ£(3 , 5, 6, 7) and Fβ = Ξ£(0, 2, 4, 7). Implement the circuit with a PLA having three inputs, four product terms and two
(RGPV Dec 2016)
UNIT 4
INTRODUCTION TO A/D & D/A CONVERTERS & THEIR TYPES, SAMPLE AND HOLD CIRCUITS, VOLTAGE TO FREQUENCY & FREQUENCY TO VOLTAGE CONVERSION
Q.1) Write notes on the following
i) A/D and D/A convertors
ii) CMOS Logic.
(RGPV Nov 2022)
Q.2) How is interfacing of TTL to MOS achieved?
(RGPV Dec 2020)
Q.3) Describe the application of Monostable multivibrator?
(RGPV Dec 2020)
Q.4) Implement the following circuit using CMOS logic
i) Y=A.B
ii) Y=A+B
(RGPV June 2020)
Q.5) With a neat diagram, explain the operation of 8 bit successive approximation ADC.
(RGPV June 2020)
Q.6) Drew and explain the working of A/D converter.
(RGPV Nov 2019)
With the help of circuit diagram explain the A to D converter.
(RGPV June 2010, 2012)
Explain A/D converter and its working.
(RGPV Dec 2013)
Explain analog to digital converter.
(RGPV Dec 2016)
Q.7) What is the need for A/D converter?
(RGPV Dec 2014)
What is the need of analog to digital conversion?
(RGPV Dec 2015)
Q.8) What are the applications of analog to digital converter?
(RGPV June 2015)
Q.9) Explain flash A/D converter with circuit diagram and parameters.
(RGPV June 2011)
Q.10) Discuss 3-bit analog to digital flash type converter.
(RGPV May 2019)
Q.11) Explain with the help of block diagram any one type of analog to digital converter.
(RGPV Dec 2012)
With a neat diagram explain successive approximation type A/D converter in detail.
(RGPV Dec. 2010, 2014)
Explain successive approximation techniques for analog to digital conversion.
(RGPV Feb 2010, June 2015)
Explain any one type of analog to digital converter in detail.
(RGPV May 2018)
Q.12) Why analog to digital converters is needed? Explain any one digital converters.
(RGPV Dec 2017)
Q.13) Enlist the various types of analog to digital (A/D) converter and explain any one of them with neat sketch.
(RGPV Dec 2015)
Q.14) State maximum conversion time and average conversion time.
(RGPV Dec 2013)
Q.15) Distinguish single slope and double slope A/D converter.
(RGPV Dec 2014)
Q.16) What is a bipolar D/A converter?
(RGPV June 2014)
Q.17) Explain a 4 bit R-2R ladder type D/A converter in detail.
(RGPV Dec 2008, 2014)
Explain the operation of R-2R ladder type digital to analog (D/A) converter with a neat sketch.
(RGPV Dec 2015)
Q.18) How can we describe the resolution of a digital to analog converter?
(RGPV June 2015)
Q.19) Explain the transfer characteristics and various performance parameters of DAC.
(RGPV June 2015)
Q.20) Discuss about the sample and hold circuits.
(RGPV June 2009, Nov 2018)
Write short note on sample and hold circuits.
(RGPV Dec 2010, 2016, June 2017)
Discuss about sample and hold circuits in A/D converter.
(RGPV June 2011)
Explain the principle working of sample and hold circuits.
(RGPV Dec 2012)
Draw the circuit diagram of sample and hold circuit and explain its working.
(RGPV Dec 2015)
Explain sample and hold circuit.
(RGPV May 2019)
Q.21) With the help of circuit diagram explain the V-F converter.
(RGPV June 2012)
Explain the principle working of V-F converter.
(RGPV Dec 2012)
With the help of circuit diagram explain the working of V-F converters.
(RGPV Dec 2014)
Explain voltage to frequency converter with the help of block diagram and waveforms.
(RGPV Dec 2015, June 2017, May 2018)
Write a short note on V-F converters.
(RGPV June 2015)
Q.22) Fig shows a computer control of motor speed. It can change motor speed from 0 to 1500 r.p.m. Find the number of bits of the computer so that it can control the speed within 1 r.p.m. of required
(RGPV June 2014)
Q.23) Draw and explain the working of bistable multivibrator.
(RGPV Dec 2005, June 2009, Nov 2019)Β
Describe bistable multivibrator with diagram and working principle.
(RGPV June 2011)
Q.24) Explain the terms-monostable, bistable and astable multivibrator.
(RGPV Dec 2015)
Q.25) Explain monostable multivibrator and write its applications.
(RGPV June 2010, Dec 2010, 2017)
With the help of timing diagram explain the working of monostable multivibrator.
(RGPV June 2012)
With the help of circuit diagram and timing waveforms explain the working of monostable multivibrator.
(RGPV Dec 2014)
Explain the operation of monostable multivibrator with the help of waveforms.
(RGPV June 2015)
Draw and explain the working of monostable multivibrator.
(RGPV May 2018)
Draw and explain monostable multivibrator.
(RGPV May 2019)
Q.26) Distinguish between monostable and astable multivibrator.
(RGPV June 2017)
Q.27) Draw and explain the working of Schmitt trigger.
(RGPV Nov 2019)
Discuss Schmitt trigger circuits.
(RGPV June 2009, Dec 2013)
Write short note on Schmitt trigger
(RGPV June 2010, 2012)
Draw a Schmitt trigger circuit and explain with waveforms.
(RGPV, Nov./Dec. 2007, June 2015)
With the help of circuit diagram explain the working of Schmitt trigger.
(RGPV Dec 2012, 2014)
Draw the circuit diagram of Schmitt trigger and explain its working.
(RGPV Dec 2015)
What is Schmitt trigger circuit?
(RGPV June 2014)
Write short note on Schmitt trigger circuits.
(RGPV Dec 2017)
Q.28) What do you understand by logic families?
(RGPV June 2015)
Q.29) Write characteristics of digital logic families.
(RGPV Dec 2015)
Q.30) Draw and explain DTL circuit. Enlist its advantages and disadvantages.
(RGPV June 2017)
UNIT 5
Q.1) Write notes on the following
i) A/D and D/A convertors
ii) Shannon’s theorem for channel capacity
iii) Nyquist sampling theorem.Β Β Β Β Β Β
(RGPV Nov 2022)
Q.2) Describe 2-bit simultaneous A/D Converter.
Β (RGPV Dec 2020)
Q.3) Explain the procedure of Pulse code modulation with neat diagram?Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β
(RGPV Dec 2020)
Q.4) What are the advantages of TDM over FDM? Define synchronous TDM?Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β
(RGPV Dec 2020)
Q.5) Compare the BPSK and BFSK modulation schemes.
Β (RGPV Dec 2020)
Q.6) Write short note on sampling theorem.
(RGPV Dec 2008, Nov 2018)
Q.7) Write short note on Nyquist sampling theorem.Β
(RGPV Nov 2019)
Q.8) Explain time division multiplexing.
(RGPV May 2019)
Q.9) Draw block diagram of PCM system and explain it.
(RGPV Nov 2019)
Q.10) Write short note on quantization error. Β Β Β Β Β Β
(RGPV Nov 2018)
Q.11) What is quantization error? Explain sampling theorem.Β
(RGPV May 2019)
Q.12) Explain quantization error.Β Β Β Β Β Β Β Β Β Β Β Β
(RGPV June 2014)
Q.13) Explain the terms sampling, quantization and quantization error.
Β (RGPV Nov 2019)
Q.14) Write short note on BFSK modulation.
(RGPV Nov 2018)
Explain BFSK.Β Β Β Β
(RGPVΒ May 2019)
Q.15) Explain Shannon’s theorem for channel capacity.
(RGPV May 2019)
Explain the information capacity theorem for channel coding.
Write short note on Shannon’s theorem for channel capacity.
(RGPV Nov 2018)
Β Β Β Β Β Β Β Β Β Β Β Β Β Β Β
— Best of Luck for Exam — | {"url":"https://career-shiksha.com/post/important-rgpv-question-cs-304-digital-systems-iii-sem-cse/","timestamp":"2024-11-05T16:04:44Z","content_type":"text/html","content_length":"229548","record_id":"<urn:uuid:cecf5e80-59e7-44f1-82e0-7f8e1cf62282>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00092.warc.gz"} |
Physics Syllabus – UPSC Civil Services Mains ExamPhysics Syllabus – UPSC Civil Services Mains Exam
Physics Syllabus – UPSC Civil Services Mains Exam
The optional subject in the UPSC Civil Services Mains Exam is divided into two papers, with each paper carrying 250 marks. This brings the total marks allotted for the optional subject to 500.
Physics Syllabus – Civil Services Mains Exam UPSC :
PAPER – I
1. (a) Mechanics of Particles:
Laws of motion; conservation of energy and momentum, applications to rotating frames, centripetal and Coriolis accelerations; Motion under a central force; Conservation of angular momentum, Kepler’s
laws; Fields and potentials; Gravitational field and potential due to spherical bodies, Gauss and Poisson equations, gravitational self-energy; Two-body problem; Reduced mass; Rutherford scattering;
Centre of mass and laboratory reference
(b) Mechanics of Rigid Bodies:
System of particles; Centre of mass, angular momentum, equations of motion; Conservation theorems for energy, momentum and angular momentum; Elastic and inelastic collisions; Rigid body; Degrees of
freedom, Euler’s theorem, angular velocity, angular momentum, moments of inertia, theorems of parallel and perpendicular axes, equation of motion for rotation; Molecular rotations (as rigid bodies);
Di and tri-atomic molecules; Precessional motion; top, gyroscope.
(c) Mechanics of Continuous Media:
Elasticity, Hooke’s law and elastic constants of isotropic solids and their inter-relation; Streamline (Laminar) flow, viscosity, Poiseuille’s equation, Bernoulli’s equation, Stokes’ law and
(d) Special Relativity:
Michelson- Morley experiment and its implications; Lorentz transformations- length contraction, time dilation, the addition of relativistic velocities, aberration and Doppler effect, mass-energy
relation, simple applications to a decay process; Four dimensional momentum vector; Covariance of equations of physics.
2. Waves and Optics:
(a) Waves:
Simple harmonic motion, damped oscillation, forced oscillation and resonance; Beats; Stationary waves in a string; Pulses and wave packets; Phase and group velocities; Reflection and Refraction from
Huygens’ principle.
(b) Geometrical Optics:
Laws of reflection and refraction from Fermat’s principle; Matrix method in paraxial optics-thin lens formula, nodal planes, system of two thin lenses, chromatic and spherical aberrations.
(c) Interference:
Interference of light-Young’s experiment, Newton’s rings, interference by thin films, Michelson interferometer; Multiple beam interference and Fabry-Perot interferometer.
(d) Diffraction:
Fraunhofer diffraction-single slit, double slit, diffraction grating, resolving power; Diffraction by a circular aperture and the Airy pattern; Fresnel diffraction: half-period zones and zone plates,
circular aperture.
(e) Polarization and Modern Optics:
Production and detection of linearly and circularly polarized light; Double refraction, quarter-wave plate; Optical activity; Principles of fibre optics, attenuation; Pulse dispersion in step-index
and parabolic index fibres; Material dispersion, single-mode fibres; Lasers-Einstein A and B coefficients; Ruby and He-Ne lasers; Characteristics of laser light-spatial and temporal coherence;
Focusing of laser beams; Three-level scheme for laser operation; Holography and simple applications.
3. Electricity and Magnetism:
(a) Electrostatics and Magnetostatics:
Laplace and Poisson equations in electrostatics and their applications; Energy of a system of charges, the multipole expansion of scalar potential; Method of images and its applications; Potential
and field due to a dipole, force and torque on a dipole in an external field; Dielectrics, polarization; Solutions to boundary-value problems-
conducting and dielectric spheres in a uniform electric field; Magnetic shell, uniformly magnetized sphere; Ferromagnetic materials, hysteresis, energy loss.
(b) Current Electricity:
Kirchhoff’s laws and their applications; Biot-Savart law, Ampere’s law, Faraday’s law, Lenz’ law; Self-and mutual-inductances; Mean and r m s values in AC circuits; DC and AC circuits with R, L and C
components; Series and parallel resonances; Quality factor; Principle of the transformer.
4. Electromagnetic Waves and Blackbody Radiation:
Displacement current and Maxwell’s equations; Wave equations in vacuum, Poynting theorem; Vector and scalar potentials; Electromagnetic field tensor, the covariance of Maxwell’s equations; Wave
equations in isotropic dielectrics, reflection and refraction at the boundary of two dielectrics; Fresnel’s relations; Total internal reflection; Normal and anomalous dispersion; Rayleigh scattering;
Blackbody radiation and Planck’s radiation law, Stefan-Boltzmann law, Wien’s displacement law and Rayleigh-Jeans’ law.
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5. Thermal and Statistical Physics:
(a) Thermodynamics:
Laws of thermodynamics, reversible and irreversible processes, entropy; Isothermal, adiabatic, isobaric, isochoric processes and entropy changes; Otto and Diesel engines, Gibbs’ phase rule and
chemical potential; van der Waals equation of state of a real gas, critical constants; Maxwell-Boltzman distribution of molecular velocities, transport phenomena, equipartition and virial theorems;
Dulong-Petit, Einstein, and Debye’s theories of specific heat of solids; Maxwell relations and applications; Clausius- Clapeyron equation; Adiabatic demagnetisation, Joule-Kelvin effect and
liquefaction of gases.
(b) Statistical Physics:
Macro and microstates, statistical distributions, Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac distributions, applications to the specific heat of gases and blackbody radiation; Concept of
negative temperatures.
1. Quantum Mechanics:
Wave-particle duality; Schroedinger equation and expectation values; Uncertainty principle; Solutions of the one-dimensional Schroedinger equation for a free particle (Gaussian wave-packet), particle
in a box, particle in a finite well, linear harmonic oscillator; Reflection and transmission by a step potential and by a rectangular barrier; Particle in a three-dimensional box, the density of
states, free electron theory of metals; Angular momentum; Hydrogen atom; Spin half particles, properties of Pauli spin matrices.
2. Atomic and Molecular Physics:
Stern-Gerlach experiment, electron spin, the fine structure of hydrogen atom; L-S coupling, J- J coupling; Spectroscopic notation of atomic states; Zeeman effect; FrankCondon principle and
applications; Elementary theory of rotational, vibrational and electronic spectra of diatomic molecules; Raman effect and molecular structure; Laser Raman spectroscopy; Importance of neutral hydrogen
atom, molecular hydrogen and molecular hydrogen ion in astronomy; Fluorescence and Phosphorescence; Elementary theory and applications of NMR and EPR; Elementary ideas about Lamb shift and its
3. Nuclear and Particle Physics:
Basic nuclear properties-size, binding energy, angular momentum, parity, magnetic moment; Semi-empirical mass formula and applications, mass parabolas; Ground state of deuteron, magnetic moment and
non-central forces; Meson theory of nuclear forces; Salient features of nuclear forces; Shell model of the nucleus – successes and limitations; Violation of parity in beta decay; Gamma decay and
internal conversion; Elementary ideas about Mossbauer spectroscopy; Q-value of nuclear reactions; Nuclear fission and fusion, energy production in stars; Nuclear reactors.
Classification of elementary particles and their interactions; Conservation laws; Quark structure of hadrons; Field quanta of electroweak and strong interactions; Elementary ideas about unification
of forces; Physics of neutrinos.
4. Solid State Physics, Devices and Electronics:
The crystalline and amorphous structure of matter; Different crystal systems, space groups; Methods of determination of crystal structure; X-ray diffraction, scanning and transmission electron
microscopies; Band theory of solids – conductors, insulators and semiconductors; Thermal properties of solids, specific heat, Debye theory; Magnetism: dia, para and ferromagnetism; Elements of
superconductivity, Meissner effect, Josephson junctions and applications; Elementary ideas about high-temperature superconductivity.
Intrinsic and extrinsic semiconductors; p-n-p and n-p-n transistors; Amplifiers and oscillators; Op-amps; FET, JFET and MOSFET; Digital electronics-Boolean identities, De Morgan’s laws, logic gates
and truth tables; Simple logic circuits; Thermistors, solar cells; Fundamentals of microprocessors and digital computers.
Useful LINKS: | {"url":"https://www.iassolution.com/physics-syllabus-upsc-civil-services-mains-exam/","timestamp":"2024-11-09T17:21:32Z","content_type":"text/html","content_length":"155990","record_id":"<urn:uuid:e906b6f0-2d19-49cd-8883-26a02b4edcee>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00323.warc.gz"} |
Intermediate Frequency Calculator | Calculate Intermediate Frequency
How do you determine intermediate frequency?
Intermediate frequency (IF) should be carefully chosen as High IF results in poor selectivity and therefore poor rejection of adjacent channels. - High IF results in problems in tracking of signals
in the receivers. - Image frequency rejection becomes poor at low IF or if very high.
How to Calculate Intermediate Frequency?
Intermediate Frequency calculator uses Intermediate Frequency = (Local Oscillation Frequency-Received Signal Frequency) to calculate the Intermediate Frequency, Intermediate Frequency is a frequency
to which a carrier wave is shifted as an intermediate step in transmission or reception. Intermediate Frequency is denoted by f[im] symbol.
How to calculate Intermediate Frequency using this online calculator? To use this online calculator for Intermediate Frequency, enter Local Oscillation Frequency (f[lo]) & Received Signal Frequency
(F[RF]) and hit the calculate button. Here is how the Intermediate Frequency calculation can be explained with given input values -> 70 = (125-55). | {"url":"https://www.calculatoratoz.com/en/intermediate-frequency-calculator/Calc-33780","timestamp":"2024-11-05T06:21:00Z","content_type":"application/xhtml+xml","content_length":"115815","record_id":"<urn:uuid:12a5d448-5985-49f5-a56a-746f972d7733>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00064.warc.gz"} |
{LEVsim}: Theoretical Load-Exertion-Velocity Model - Part 2: LEVsim Model - Complementary Training
{LEVsim}: Theoretical Load-Exertion-Velocity Model – Part 2: LEVsim Model
Previous Part:
{LEVsim}: Theoretical Load-Exertion-Velocity Model – Part 1
Why we need formal computational theory?
I have introduced the very robust phenomena of resistance training: (1) trade-off between load and velocity, (2) trade-off between load and maximal number of reps, and (3) trade-off between
repetition velocity and proximity to failure. Research on resistance training in general, and velocity-based training (VBT) in particular represents tremendous effort by the researchers in finding
statistical effects in the collected data. Unfortunately, these results in conflations of the theoretical and statistical models (Fried 2020b). These models say very little about the processes that
actually generated the data (DGP; data generating process) (P. E. Smaldino 2017). What is lacking are the formal theoretical and computational models, that transparently provide assumption, provide
testable predictions, allows to simulate data under a theory, and enables the comparison of such theory-implied and simulated data with actual data (Guest and Martin 2021; Muthukrishna and Henrich
2019; P. E. Smaldino 2017; Borsboom et al. 2020; Fried 2020a, 2020b; P. Smaldino 2019).
The theoretical model I am about to present in this article series represent computational theoretical model (Guest and Martin 2021) that formalizes the assumptions of the underlying DGP, theorized
to produce the aforementioned resistance training phenomena. Simulation of the model generated data allows for exploration and understanding of the effects of model parameters on the observed data.
Finally, theoretical computational model allows for estimation of the prescription error, since we can compare the true values (i.e., generated values) with different prescription approaches. In
addition, theoretical model can be a fruitful generator of the new hypotheses and can thus drive research attempts.
I have named this generative model LEVsim, or Load-Exertion-Velocity simulation model. There are multiple layers of this model, and in this article series I will explain each of them.
This model is implemented in the {LEVsim} R package. You can install the development version from GitHub with:
# install.packages("devtools")
Load-Velocity Profile
The trade-off relationship between load and velocity can be formalized with a simple l inear relationship, forming individual’s load-velocity profile (LVP) (Figure 1). LVP is characterized with two
parameters: \(L_0\) and \(V_0\) . These two parameters can be interpreted using realist ontology, or using instrumentalist ontology (Borsboom 2008, 2009; Borsboom, Mellenbergh, and van Heerden 2003;
Yarkoni 2020; Fried 2020a, 2020b).
Figure 1: Load-Velocity Profile. In addition to the \(L_0\) and \(V_0\) parameters, two additional parameters are \(1RM\) and \(v1RM\). Please refer to the main text for explanation.
From a realist perspective, \(L_0\) represents maximal load that can be lifted in isometric conditions (i.e., when velocity equals zero), while \(V_0\) represents hypothetical maximal lifting
velocity that can be achieved when there is no load (i.e., load is equal to zero).
From instrumentalist perspective, \(L_0\) and \(V_0\) represent just a different, and probably more intuitive, approach to describe a line. The equation for a straight line consists of \(intercept\)
and \(slope\) parameters (Equation 1).
y = intercept + slope \times x
Equation 1 can be rewritten in the form of Equation 2.
velocity = V_0 + -\frac{V_0}{L_0}\times load
In Equation 2 form, \(intercept\) is equal to \(V_0\) and line \(slope\) is equal to \(-\frac{V_0}{L_0}\). In the case where velocity is the predictor, the Equation 2 get’s the form of Equation 3.
load = L_0 + -\frac{L_0}{V_0}\times velocity
In this theoretical model, \(L_0\) and \(V_0\) represent one of the key parameters of the DGP, and thus can have realist interpretation (i.e., they represent causal construct responsible for the
model behavior and manifested performance).
One-repetition maximum and minimum velocity threshold
In addition to the LVP, two additional parameters are \(1RM\) and \(v1RM\) (Figure 1). One-repetition-maximum, or \(1RM\), represent maximal load that can be lifted with predefined technique (which
includes, but it is not limited to, posture, range-of-movement, defined pauses and lifting tempo). Velocity associated with \(1RM\) attempt is termed velocity-at-one-repetition-maximum, or \(v1RM\),
and according to the evidence so far, it seems to be rather stable across training intervention, and specific to the individual and exercise (Weakley et al. 2020).
In {LEVsim} model, \(v1RM\), in addition to \(L_0\) and \(V_0\), represents one of the key DGP parameters, while the \(1RM\) itself, represents the highest load that can be lifted given the \(v1RM\),
\(L_0\), and \(V_0\) parameter values. In plain English, we set \(v1RM\), \(L_0\), and \(V_0\) parameter values, and calculate \(1RM\) as theoretical estimate, given this theoretical model parameter
values. \(1RM\) is simply estimated by plugging the \(v1RM\), \(L_0\), and \(V_0\) values in Equation 3, giving Equation 4.
1RM = L_0 + -\frac{L_0}{V_0}\times v1RM
Velocity at \(1RM\) (i.e., \(v1RM\)), according to the current theoretical model, is also associated with the velocity of the last successfully performed repetition with sub-maximal load lifted to
failure (e.g., 80% 1RM for 6 reps, where 7th rep was unable to be lifted with predefined technique). \(v1RM\) can also be termed minimum-velocity threshold (MVT) (Jovanović and Flanagan 2014), or
velocity-at-zero-reps-in-reserve (v0RIR). This concept is further elaborated in later sections, but you can also check the previous installment of this article series.
Contemporary resistance training prescription, i.e. percent-based approach (PBT) (Jovanović 2020a), prescribes training loads using percent of \(1RM\), or relative loads (i.e., 65% \(1RM\), 90% \(1RM
\)). Load-Velocity profile can thus also be expressed using relative loads, rather than absolute loads. This can be achieved by either using \(L_0\) or \(1RM\) (which is more common).
Under the assumptions of the current formal model, prescribing training loads using velocity associated with a particular %\(1RM\) (i.e., v65% \(1RM\), v90% \(1RM\)) (Figure 2), instead of using \
(1RM\), is believed to be more robust to visit-to-visit fluctuations and chronic changes in \(1RM\), and thus represents improved approach in individualizing training load prescription. This
hypothesis should be tested using simulation or using observed data and I will do that later in this article series.
Figure 2: Velocity of sub-maximal loads. Loads are expressed using \(1RM\), in this case 60, 70, 80, and 90% \(1RM\).
Figure 1 and Figure 2 represents the LV profile and a first layer of the {LEVsim}. This layer can only generate single repetition velocities at different loads. But how do we simulate exertion
effects, as in maximal number of repetitions that can be performed at certain load (i.e., Reps-Max relationship), and in reducing velocity of each repetition as one approaches failure point (i.e.,
Exertion-Velocity relationship)? But before I address that issue, I need to introduce another key concept.
Measurement Error
\(L_0\), \(V_0\), and \(v1RM\) represent theoretical latent constructs (i.e., DGP parameters) that cannot be directly measured, but are the causal mechanism behind the repetition velocities. In this
theoretical model, repetition velocities are under the influence of another construct – measurement error. Measurement error is involved in all measurements and causes an measured score to be
different from the true score (Allen and Yen 2001; Borsboom 2009; Jovanović 2020b; Novick 1966; Swinton et al. 2018). In plain English, given this theoretical LV profile (Figure 1 and Figure 2), one
expects repetition velocity to be equal to Equation 2. But this doesn’t necessary need to be the case when we actually observe the repetition velocities. We thus need another set of parameters to
simulate this phenomena. | {"url":"http://complementarytraining.com/novi-post/","timestamp":"2024-11-07T01:22:39Z","content_type":"text/html","content_length":"354921","record_id":"<urn:uuid:f260c4cd-11c0-42da-82e2-69e024062c25>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00879.warc.gz"} |
Vol. 37, No. 1, 2011
HOUSTON JOURNAL OF
Electronic Edition Vol. 37, No. 1, 2011
Editors: G. Auchmuty (Houston), D. Bao (San Francisco, SFSU), D. Blecher (Houston), H. Brezis (Paris and Rutgers), B. Dacorogna (Lausanne), K. Davidson (Waterloo), M. Dugas (Baylor), M. Gehrke
(Radboud), C. Hagopian (Sacramento), R. M. Hardt (Rice), Y. Hattori (Matsue, Shimane), J. A. Johnson (Houston), W. B. Johnson (College Station), V. I. Paulsen (Houston), M. Rojas (College Station),
Min Ru (Houston), S.W. Semmes (Rice)
Managing Editor: K. Kaiser (Houston)
Houston Journal of Mathematics
McNair, Dawn B., Johnson C. Smith University, Charlotte, NC 28216 (dmcnair@jcsu.edu).
Duals of ideals in rings with zero divisors, pp. 1-26.
ABSTRACT For any nonzero ideal I of ring R, we define the inverse of I as the set of elements from Q(R) (complete ring of quotients of R) that conducts I into R and call it the dual of I . Much work
has been done with regard to determining when the dual of I is a ring in the case R is an integral domain. This paper will extend those results to dense ideals in rings with zero divisors. Attention
will also be given to duals of ideals in Prüfer and strong Prüfer rings.
Trotta, Belinda, La Trobe University, Victoria 3086, Australia (belindatrotta@gmail.com).
Residual properties of reflexive, anti-symmetric digraphs, pp. 27-46.
ABSTRACT. For a quasivariety C of reflexive anti-symmetric digraphs, we consider the class RCT(Cfin) of topological digraphs that are topologically residually in the class of finite (discretely
topologised) members of C. In particular, we show that RCT(Cfin) can be axiomatised (among compact, totally disconnected digraphs) by first-order sentences if and only if C is strictly contained
within the quasivariety of partial orders. This work extends Stralka's result that there is a compact, totally disconnected partially ordered space that is not a Priestley space.
Mabrouk Ben Nasr, and Noômen Jarboui, King Faisal University, Faculty Of Sciences, Department Of Mathematics, P. O. Box: 380, 31982 Saudi Arabia.(mabrouk_bennasr@yahoo.fr) , (noomenjarboui@yahoo.fr).
On maximal non-valuation subrings, pp. 47-59.
ABSTRACT. In this paper we study pairs of rings where all intermediate domains are valuation. Furthermore maximal non-valuation subrings are studied and examples illustrating the theory are given.
Martín Méndez, Alberto, Universidade de Vigo, 36310, Vigo (Pontevedra), Spain (amartin@dma.uvigo.es), and Torres Lopera, Juan Francisco, Universidade de Santiago de Compostela, 15706, Santiago de
Compostela (La Coruña), Spain (jftorre@usc.es).
Tensorial structures associated with semisimple graded Lie algebras, pp. 61-77.
ABSTRACT. Properties concerning several tensors associated with geometric structures of graded type are studied. Particularly, we study Tanaka tensor and Weyl curvature tensor and some relations
between them. We prove that the difference tensor of two linear connections on a manifold endowed with a geometric structure of graded type is null if and only if the connections have the same
torsion. An explicit calculus of Tanaka tensor for classical simple real graded Lie algebras is given.
Dethloff, Gerd, University of Brest, 29275 Brest, France (dethloff@univ-brest.fr), and Tan, Tran Van, ENS Hanoi, Hanoi, Vietnam .
A second Main Theorem for moving hypersurface targets, pp. 79-111.
ABSTRACT. In this paper, we prove a Second Main Theorem for algebraically nondegenerate meromorphic maps of C^m into CP^n with at least n+2 slowly moving hypersurface targets in (weakly) general
position. We also introduce a truncation, with an explicit estimate of the truncation level, into this Second Main Theorem. This generalizes recent works of Min Ru and An-Phuong on Second Main
Theorems for fixed hypersurface targets.
Cohen, Nir, University of Campinas, Campinas, Brazil, Grama, Lino, University of Campinas, Campinas, Brazil, and Negreiros, Caio J.C., University of Campinas, Campinas, Brazil (caione@ime.unicamp.br)
Equigeodesics on flag manifolds, pp. 113-125.
ABSTRACT. This paper provides a characterization of homogeneous curves on a geometric flag manifold which are geodesics with respect to each invariant metric. We call such curves homogeneous
equigeodesics. We also characterize homogeneous equigeodesics whose associated Killing field is closed, hence, the corresponding geodesics are closed.
Biliotti, Leonardo, Dipartimento di Matematica, Università di Parma, Via G. Usberti, 53/A 43100, Parma, Italy (leonardo.biliotti@unipr.it), and Javaloyes, Miguel Angel, Departamento de Geometrí a y
Topologíca, Universidad de Granada, Campus Fuentenueva S/N, 18071, Granada, Spain (majava@ugr.es)
t-periodic light rays in conformally stationary spacetimes via Finsler geometry,pp. 127-146.
ABSTRACT. In this paper we prove several multiplicity results of t-periodic light rays in conformally stationary spacetimes using the Fermat metric and the extensions of the classical theorems of
Gromoll-Meyer and Bangert-Hingston to Finsler manifolds. Moreover, we exhibit some stationary spacetimes with a finite number of t-periodic light rays and compute a lower bound for the period of the
light rays when the flag curvature of the Fermat metric is η-pinched.
Suceavă, Bogdan D., California State University at Fullerton, Fullerton, CA 92834-6850 (bsuceava@fullerton.edu).
Distances generated by Barbilian's metrization procedure by oscillation of sublogarithmic functions, pp. 147-159.
ABSTRACT. Introduced originally in 1934, Barbilian’s metrization procedure induced a distance on a planar domain by a metric formula given by the so-called logarithmic oscillation. In 1959, Barbilian
generalized this process to domains of a more general form, withstanding not necessarily on planar sets, but in a more abstract setting. In the present work, we show that there exists more general
classes of distances than the ones produced by logarithmic oscillation. As a consequence, in Theorem 2 we state the most general form of Barbilian’s metrization procedure.
Anselm Knebusch, Department of Mathematics, Bunsenstrasse 3-5, D-37073 Göttingen, Germany (knebusch@math.uni-goettingen.de).
Approximation of center-valued Betti-numbers, pp. 161-179.
ABSTRACT. In this paper we generalize the ordinary approximation theorem to calculate L^2-Betti-numbers to an approximation theorem for universal Betti-numbers.
Cabello Sanchez, Felix, UEx, Badajoz 06071, Spain (fcabello@unex.es), and Cabello Sanchez, Javier, UEx, Badajoz 06071, Spain (coco@unex.es)
Nonlinear isomorphisms of lattices of Lipschitz functions, pp. 181-202.
ABSTRACTThe paper contains a number of Banach-Stone type theorems for lattices of uniformly continuous and Lipschitz functions without any linearity assumption. Sample result: two complete metric
spaces of finite diameter are Lipschitz homeomorphic if (and only if, of course) the corresponding lattices of Lipschitz functions are isomorphic. Here, a lattice isomorphism is just a bijection
preserving the order in both directions, in particular linearity is not assumed.
Shalit, Orr M., Pure Mathematics Department, University of Waterloo, Waterloo, ON N2L-3G1, CANADA (oshalit@math.uwaterloo.ca).
E-dilation of strongly commuting CP-semigroups (the nonunital case), pp. 203-232.
ABSTRACT. In a previous paper, we showed that every strongly commuting pair of CP0-semigroups on a von Neumann algebra (acting on a separable Hilbert space) has an E0-dilation. In this paper we show
that if one restricts attention to the von Neumann algebra B(H) then the unitality assumption can be dropped, that is, we prove that every pair of strongly commuting CP-semigroups on B(H) has an
E-dilation. The proof is significantly different from the proof for the unital case, and is based on a construction of Ptak from the 1980's designed originally for constructing a unitary dilation to
a two-parameter contraction semigroup.
Victor Kaftal
and Gary Weiss, Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA (kaftal@math.uc.edu) and (gary.weiss@math.uc.edu).
B(H) lattices, density and arithmetic mean ideals, pp. 233-283.
ABSTRACT. Lattice properties of operator ideals in B(H) with applications to the arithmetic mean ideals introduced by Dykema, Figiel, Weiss and Wodzicki (Adv Math 2004) are studied here as part of a
five paper project announced in PNAS 2002. We focus on the general lattice of B(H)-ideals and on particular sublattices such as the principal and countably generated ideals and their density
properties (between any ideal and an ideal in a sublattice lies another ideal in that sublattice). As applications, we obtain cancellation properties for first order arithmetic mean ideals and
arithmetic mean ideals at infinity and solve related ideal optimization problems.
Klemes, Ivo, Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, H3A 2K6, Canada (klemes@math.mcgill.ca).
Symmetric polynomials and lp inequalities for certain intervals of p, pp. 285-295.
ABSTRACT. We prove some sufficient conditions implying lp inequalities of the form ||x||p ≤ ||y||p for vectors x, y in Rn and for p in certain positive real intervals. Our sufficient conditions are
strictly weaker than the usual majorization relation. The conditions are expressed in terms of certain homogeneous symmetric polynomials in the entries of the vectors. These polynomials include the
elementary symmetric polynomials as a special case. We also give a characterization of the majorization relation by means of symmetric polynomials.
Li, Haigang, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P.R. China, and Bao, Jiguang, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, P.R.
China (jgbao@bnu.edu.cn).
Existence of rotating stars with prescribed angular velocity law, pp. 297-309.
ABSTRACT. The existence of solutions of the equations for a self-gravitating fluid with prescribed angular velocity law is proved. The conditions on the angular velocity are nearly optimal. The
system is formulated as a variational problem and concentration-compactness methods are used to prove the existence of minimizers of the energy functional.
Zhao Dongsheng, Mathematics and Mathematics Education, National Institute of Education Singapore, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616(dongsheng.zhao@nie.edu.sg).
A partial order on the set of continuous endomappings, pp. 311-326.
ABSTRACT. Motivated by the reflexive operator algebra problem, we introduce a new partial order on the set End[C](X) of all continuous endomappings on a topological space X. We study the relationship
between the topological structure and the order structure, and we establish conditions for certain families of continuous endomappings to be reflexive. We introduce the V-regular spaces and show that
if X and Y are V-regular spaces, then the two semigroups End[C](X) and End[C](Y) are isomorphic if and only if X and Y$ are homeomorphic.
Spěvák, Jan, Department of mathematics, J.E. Purkinje University, Ceske mladeze 8, 400 96 Usti nad Labem, Czech Republik (jan.spevak@ujep.cz).
Finite-valued mappings preserving dimension, pp. 327-348.
ABSTRACT. We say that a set-valued mapping F: X⇒Y is C-lsc provided that there exists a countable cover C of X consisting of functionally closed sets such that for every C∈C and each functionally
open subset U of Y one can find a functionally open set V⊂X such that {x∈C: F(x)∩ U≠Ø}=C∩V. For Tychonoff spaces X and Y we say that X dominates Y provided that there exist a finite-valued C-lsc
mapping F: X⇒Y and a finite-valued D-lsc mapping G:Y⇒X (for suitable C and D) such that y∈ ∪{F(x):x∈G(y)} for every y∈Y. We prove that if X dominates Y, then dim X≥dim Y. (Here dim X denotes the
Čech-Lebesgue (covering) dimension of X.) As a corollary, we obtain that dim X=dim Y whenever a perfectly normal space Y is an image of a Tychonoff space X under a finite-to-one open mapping. We also
give an example of an open mapping f:X→Y such that |f^-1(y)|≤2 for all y∈Y, both X and Y are hereditarily normal (and Y is even Lindelöf) but dim X≠dim Y.
Editors Addendum on
Singularities of generic lightcone Gauss maps and lightcone pedal surfaces of spacelike curves in Minkowski 4-space by L.L. Kong, R.M. Gao, D.H. Pei, and J.H. Zhang.
After the paper had already appeared in print, HJM Vol. 36(3) pp. 697-710, the referee discovered that the paper had an irreparable error. Proposition 2.1 in the paper is not correct. From that place
on, all the arguments fail and thus the proof of Theorem B is not correct. This has been confirmed by the authors. | {"url":"https://www.math.uh.edu/~hjm/Vol37-1.html","timestamp":"2024-11-08T09:27:31Z","content_type":"text/html","content_length":"17022","record_id":"<urn:uuid:7a378317-5e59-4412-9cca-b826409159e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00899.warc.gz"} |
1-D average pooling layer
Since R2021b
A 1-D average pooling layer performs downsampling by dividing the input into 1-D pooling regions, then computing the average of each region.
The dimension that the layer pools over depends on the layer input:
• For time series and vector sequence input (data with three dimensions corresponding to the "C" (channel), "B" (batch), and "T" (time) dimensions), the layer pools over the "T" (time) dimension.
• For 1-D image input (data with three dimensions corresponding to the "S" (spatial), "C" (channel), and "B" (batch) dimensions), the layer pools over the "S" (spatial) dimension.
• For 1-D image sequence input (data with four dimensions corresponding to the "S" (spatial), "C" (channel), "B" (batch), and "T" (time) dimensions), the layer pools over the "S" (spatial)
layer = averagePooling1dLayer(poolSize) creates a 1-D average pooling layer and sets the PoolSize property.
layer = averagePooling1dLayer(poolSize,Name=Value) also specifies the padding or sets the Stride and Name properties using one or more optional name-value arguments. For example,
averagePooling1dLayer(3,Padding=1,Stride=2) creates a 1-D average pooling layer with a pool size of 3, a stride of 2, and padding of size 1 on both the left and right of the input.
Input Arguments
Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but
the order of the pairs does not matter.
Example: averagePooling1dLayer(3,Padding=1) creates a 1-D average pooling layer with a pool size of 3 and padding of size 1 on the left and right of the layer input.
Padding — Padding to apply to input
[0 0] (default) | "same" | nonnegative integer | vector of nonnegative integers
Padding to apply to the input, specified as one of the following:
• "same" — Apply padding such that the output size is ceil(inputSize/stride), where inputSize is the length of the input. When Stride is 1, the output is the same size as the input.
• Nonnegative integer sz — Add padding of size sz to both ends of the input.
• Vector [l r] of nonnegative integers — Add padding of size l to the left and r to the right of the input.
Example: Padding=[2 1] adds padding of size 2 to the left and size 1 to the right.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64 | char | string
Average Pooling
PoolSize — Width of pooling regions
positive integer
Width of the pooling regions, specified as a positive integer.
The width of the pooling regions PoolSize must be greater than or equal to the padding dimensions PaddingSize.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
Stride — Step size for traversing input
1 (default) | positive integer
Step size for traversing the input, specified as a positive integer.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
PaddingSize — Size of padding
[0 0] (default) | vector of two nonnegative integers
Size of padding to apply to each side of the input, specified as a vector [l r] of two nonnegative integers, where l is the padding applied to the left and r is the padding applied to the right.
When you create a layer, use the Padding name-value argument to specify the padding size.
Data Types: double
PaddingMode — Method to determine padding size
'manual' (default) | 'same'
This property is read-only.
Method to determine padding size, specified as one of the following:
• 'manual' – Pad using the integer or vector specified by Padding.
• 'same' – Apply padding such that the output size is ceil(inputSize/Stride), where inputSize is the length of the input. When Stride is 1, the output is the same as the input.
To specify the layer padding, use the Padding name-value argument.
Data Types: char
PaddingValue — Value used to pad input
0 (default) | "mean"
Value used to pad input, specified as 0 or "mean".
When you use the Padding option to add padding to the input, the value of the padding applied can be one of the following:
• 0 — Input is padded with zeros at the positions specified by the Padding property. The padded areas are included in the calculation of the average value of the pooling regions along the edges.
• "mean" — Input is padded with the mean of the pooling region at the positions specified by the Padding option. The padded areas are effectively excluded from the calculation of the average value
of each pooling region.
NumInputs — Number of inputs
1 (default)
This property is read-only.
Number of inputs to the layer, returned as 1. This layer accepts a single input only.
Data Types: double
InputNames — Input names
{'in'} (default)
This property is read-only.
Input names, returned as {'in'}. This layer accepts a single input only.
Data Types: cell
NumOutputs — Number of outputs
1 (default)
This property is read-only.
Number of outputs from the layer, returned as 1. This layer has a single output only.
Data Types: double
OutputNames — Output names
{'out'} (default)
This property is read-only.
Output names, returned as {'out'}. This layer has a single output only.
Data Types: cell
Create 1-D Average Pooling Layer
Create a 1-D average pooling layer with a pool size of 3.
layer = averagePooling1dLayer(3)
layer =
AveragePooling1DLayer with properties:
Name: ''
PoolSize: 3
Stride: 1
PaddingMode: 'manual'
PaddingSize: [0 0]
PaddingValue: 0
Include a 1-D average pooling layer in a layer array.
layers = [
layers =
9x1 Layer array with layers:
1 '' Sequence Input Sequence input with 12 dimensions
2 '' 1-D Convolution 96 11 convolutions with stride 1 and padding [0 0]
3 '' ReLU ReLU
4 '' 1-D Average Pooling average pooling with pool size 3, stride 1, and padding [0 0]
5 '' 1-D Convolution 96 11 convolutions with stride 1 and padding [0 0]
6 '' ReLU ReLU
7 '' 1-D Global Max Pooling 1-D global max pooling
8 '' Fully Connected 10 fully connected layer
9 '' Softmax softmax
1-D Average Pooling Layer
A 1-D average pooling layer performs downsampling by dividing the input into 1-D pooling regions, then computing the average of each region. The layer pools the input by moving the pooling regions
along a single dimension.
Layers in a layer array or layer graph pass data to subsequent layers as formatted dlarray objects. The format of a dlarray object is a string of characters in which each character describes the
corresponding dimension of the data. The formats consist of one or more of these characters:
• "S" — Spatial
• "C" — Channel
• "B" — Batch
• "T" — Time
• "U" — Unspecified
For example, you can represent vector sequence data as a 3-D array, in which the first dimension corresponds to the channel dimension, the second dimension corresponds to the batch dimension, and the
third dimension corresponds to the time dimension. This representation is in the format "CBT" (channel, batch, time).
The dimension that the layer pools over depends on the layer input:
• For time series and vector sequence input (data with three dimensions corresponding to the "C" (channel), "B" (batch), and "T" (time) dimensions), the layer pools over the "T" (time) dimension.
• For 1-D image input (data with three dimensions corresponding to the "S" (spatial), "C" (channel), and "B" (batch) dimensions), the layer pools over the "S" (spatial) dimension.
• For 1-D image sequence input (data with four dimensions corresponding to the "S" (spatial), "C" (channel), "B" (batch), and "T" (time) dimensions), the layer pools over the "S" (spatial)
Layer Input and Output Formats
Layers in a layer array or layer graph pass data to subsequent layers as formatted dlarray objects. The format of a dlarray object is a string of characters in which each character describes the
corresponding dimension of the data. The formats consist of one or more of these characters:
• "S" — Spatial
• "C" — Channel
• "B" — Batch
• "T" — Time
• "U" — Unspecified
For example, you can represent vector sequence data as a 3-D array, in which the first dimension corresponds to the channel dimension, the second dimension corresponds to the batch dimension, and the
third dimension corresponds to the time dimension. This representation is in the format "CBT" (channel, batch, time).
You can interact with these dlarray objects in automatic differentiation workflows, such as those for developing a custom layer, using a functionLayer object, or using the forward and predict
functions with dlnetwork objects.
This table shows the supported input formats of AveragePooling1DLayer objects and the corresponding output format. If the software passes the output of the layer to a custom layer that does not
inherit from the nnet.layer.Formattable class, or a FunctionLayer object with the Formattable property set to 0 (false), then the layer receives an unformatted dlarray object with dimensions ordered
according to the formats in this table. The formats listed here are only a subset. The layer may support additional formats such as formats with additional "S" (spatial) or "U" (unspecified)
Input Format Output Format
"SCB" (spatial, channel, batch) "SCB" (spatial, channel, batch)
"CBT" (channel, batch, time) "CBT" (channel, batch, time)
"SCBT" (spatial, channel, batch, time) "SCBT" (spatial, channel, batch, time)
In dlnetwork objects, AveragePooling1DLayer objects also support these input and output format combinations.
Input Format Output Format
"SC" (spatial, channel) "SC" (spatial, channel)
"CT" (channel, time) "CT" (channel, time)
"SCT" (spatial, channel, time) "SCT" (spatial, channel, time)
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
• You can generate generic C/C++ code that does not depend on third-party libraries and deploy the generated code to hardware platforms.
GPU Code Generation
Generate CUDA® code for NVIDIA® GPUs using GPU Coder™.
Usage notes and limitations:
• You can generate CUDA code that is independent of deep learning libraries and deploy the generated code to platforms that use NVIDIA^® GPU processors.
Version History
Introduced in R2021b | {"url":"https://ww2.mathworks.cn/help/deeplearning/ref/nnet.cnn.layer.averagepooling1dlayer.html","timestamp":"2024-11-07T17:11:44Z","content_type":"text/html","content_length":"127301","record_id":"<urn:uuid:3bcd218e-f411-45b3-8562-e7ad48eb101e>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00356.warc.gz"} |
Vanpoucke Danny
Author's posts
Authors: Danny E. P. Vanpoucke and Geert Brocks
Journal: Phys. Rev. B 77, 241308 (2008)
doi: 10.1103/PhysRevB.77.241308
IF(2008): 3.322
export: bibtex
pdf: <Phys.Rev.B> <arXiv> <UTwentePublications>
Pt deposited onto a Ge(001) surface gives rise to the spontaneous formation of atomic nanowires on a mixed Pt-Ge surface after high-temperature annealing. We study possible structures of the mixed
surface and the nanowires by total energy density functional theory calculations. Experimental scanning-tunneling microscopy images are compared to the calculated local densities of states. On the
basis of this comparison and the stability of the structures, we conclude that the formation of nanowires is driven by an increased concentration of Pt atoms in the Ge surface layers. Surprisingly,
the atomic nanowires consist of Ge instead of Pt atoms.
Permanent link to this article: https://dannyvanpoucke.be/paper2008_nwrapid-en/
3D f-Orbitals
3D gnuplot-gif-animations of the f-orbitals S03(θ,φ), S23(θ,φ) and
S33(θ,φ). In the images presented, the blue part represents the positive phase, and the red part the negative phase. Note that in gnuplot, the spherical coordinate θ is defined as π/2 – θ. Other than
that the definitions of φ and θ coincide with those used in Griffiths’ Introduction to Quantum Mechanics.
For those interested: animations in gnuplot are only available for gnuplot versions > 4.0 (which at the moment of making these animations, was still in beta version).
Permanent link to this article: https://dannyvanpoucke.be/f-orbitals-en/
3D d-Orbitals
3D Maple-images of the d-orbitals S02(θ,φ), S12(θ,φ) en S22(θ,φ). Note that the spherical coordinates (θ and φ) used by Maple are reversed compared to the definitions used in Griffiths’ Introduction
to Quantum Mechanics (the latter being the more standard definition in physics and mathematics courses).
> plot3d(abs(3*cos(phi)*cos(phi)-1),theta=0..Pi,phi=0..2*Pi,
> plot3d(abs(sin(phi)*cos(phi)*cos(theta)),theta=0..2*Pi,phi=0..Pi,
> plot3d(abs(sin(phi)*sin(phi)*cos(2*theta)),theta=0..2*Pi,phi=0..Pi,
Effect of exchanging θ and φ
Maple assumes the first angle given is the angle in the xy-plane; the second angle is with regard to the z-axis. This makes that you have to be very careful when giving Maple the θ and φ angles, and
make sure that their definitions are the same. If the definitions
are reversed: I.e. if we use the variable θ as the variable φ and vice versa, the resulting plots become something quite different. This goes for all available plotting programs (Maple, gnuplot…);
make sure you certain that what you think you enter is also what the program thinks you have entered. If not you could end up with surprising results. The same images as above, but now with θ and φ
Permanent link to this article: https://dannyvanpoucke.be/d-orbitals-en/ | {"url":"https://dannyvanpoucke.be/author/danny/page/20/","timestamp":"2024-11-11T16:22:14Z","content_type":"text/html","content_length":"87219","record_id":"<urn:uuid:e282389a-7e8e-4aef-991b-e5d139aa2296>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00684.warc.gz"} |
[Tutorial] Number theory — Storing information about multiples/divisors - Codeforces
Hello everyone,
here is a very simple idea that can be useful for (cp) number theory problems, especially those concerning multiples, divisors, $$$\text{GCD}$$$ and $$$\text{LCM}$$$.
Prerequisites: basic knowledge of number theory (divisibility, $$$\text{GCD}$$$ and $$$\text{LCM}$$$ properties, prime sieve).
Let's start from a simple problem.
You are given $$$n$$$ pairs of positive integers $$$(a_i, b_i)$$$. Let $$$m$$$ be the maximum $$$a_i$$$. For each $$$k$$$, let $$$f(k)$$$ be the sum of the $$$b_i$$$ such that $$$k | a_i$$$. Output
all pairs $$$(k, f(k))$$$ such that $$$f(k) > 0$$$.
An obvious preprocessing is to calculate, for each $$$k$$$, the sum of the $$$b_i$$$ such that $$$a_i = k$$$ (let's denote it as $$$g(k)$$$). Then, there are at least $$$3$$$ solutions to the
Solution 1: $$$O(m\log m)$$$
For each $$$k$$$, $$$f(k) = \sum_{i=1}^{\lfloor m/k \rfloor} g(ik)$$$. The complexity is $$$O\left(m\left(\frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{m}\right)\right) = O(m\log m)$$$.
Solution 2: $$$O(n\sqrt m)$$$
There are at most $$$n$$$ nonzero values of $$$g(k)$$$. For each one of them, find the divisors of $$$k$$$ in $$$O(\sqrt k)$$$ and, for each divisor $$$i$$$, let $$$f(i) := f(i) + g(k)$$$.
If $$$m$$$ is large, you may need to use a map to store the values of $$$f(k)$$$ but, as there are $$$O(n\sqrt[3] m)$$$ nonzero values of $$$f(k)$$$, the updates have a complexity of $$$O(n\sqrt[3] m
\log(nm)) < O(n\sqrt m)$$$.
Solution 3: $$$O(m + n\sqrt[3] m)$$$
Build a linear prime sieve in $$$[1, m]$$$. For each nonzero value of $$$g(k)$$$, find the prime factors of $$$k$$$ using the sieve, then generate the divisors using a recursive function that finds
the Cartesian product of the prime factors. Then, calculate the values of $$$f(k)$$$ like in solution 2.
Depending on the values of $$$n$$$ and $$$m$$$, one of these solutions can be more efficient than the others.
Even if the provided problem seems very specific, the ideas required to solve that task can be generalized to solve a lot of other problems.
Hint 1
Hint 2
Hint 3
Hint 1
Hint 2
Hint 3
Hint 1
Hint 2
Hint 3
Hint 4
Other problems
1493D - GCD of an Array (suggested by nor)
1436F - Sum Over Subsets (nor)
Codechef — Chefsums (nor)
We've seen that this technique is very flexible. You can choose the complexity on the basis of the constraints, and $$$f(k)$$$ can be anything that can be updated fast.
Of course, suggestions/corrections are welcome. In particular, please share in the comments other problems that can be solved with this technique.
I hope you enjoyed the blog! | {"url":"https://mirror.codeforces.com/topic/92317/?locale=en","timestamp":"2024-11-13T22:52:20Z","content_type":"text/html","content_length":"96069","record_id":"<urn:uuid:2ea5de6d-e324-4063-a1d8-60a68206f577>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00668.warc.gz"} |
Contributed Talks - QCrypt 2016
“Simple and Tight Device-Independent Security Proofs” by Rotem Arnon-Friedman, Renato Renner and Thomas Vidick
Monday, 11:25 a.m. – Slides / Video
Winner of the QCrypt Student Paper Prize
Device-independent (DI) cryptography aims at achieving security that holds irrespective of the quality, or trustworthiness, of the physical devices used in the implementation of the protocol. Such a
surprisingly high level of security is made possible due to the phenomena of quantum non-locality. The lack of any a priori characterization of the device used in a DI protocol makes proving security
a challenging task. Indeed, proofs for, e.g., DI quantum key distribution (DIQKD) were only achieved recently and result in far from optimal key rates while being quite complex.
In this work we show that a newly developed tool, the “entropy accumulation theorem” of Dupuis et al., can be effectively applied to give fully general proofs of DI security that yield essentially
tight parameters for a broad range of DI tasks. At a high level, our technique amounts to establishing a reduction to the scenario in which the untrusted device operates in an identical and
independent way in each round of the protocol. This makes the proof much simpler and allows us to achieve significantly better quantitative results for the case of general quantum adversaries.
As concrete applications we give simple and modular security proofs for DIQKD and randomness expansion protocols based on the CHSH inequality. For both tasks we establish essentially optimal key
rates and noise tolerance that are much higher than what was known before. Our results considerably decrease the gap between theory and experiments, thereby marking an important step towards
practical DI protocols and their implementations.
“Zero-Knowledge Proof Systems for QMA” by Anne Broadbent, Zhengfeng Ji, Fang Song and John Watrous
Monday, 11:45 a.m. – Video
Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made
secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge
proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA
has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations.
Our QMA proof system is sound against arbitrary quantum provers, but only requires an honest prover to perform polynomial-time quantum computations, provided that it holds a quantum witness for a
given instance of the QMA problem under consideration. The proof system relies on a new variant of the QMA-complete local Hamiltonian problem in which the local terms are described by Clifford
operations and standard basis measurements. We believe that the QMA-completeness of this problem may have other uses in quantum complexity.
“A Modulator-Free QKD Transmitter” by Zhiliang Yuan, Bernd Fröhlich, Marco Lucamarini, George Roberts, James Dynes and Andrew Shields
Monday, 2:15 p.m.
Quantum key distribution (QKD) is a powerful method for guaranteeing the confidentiality of future communication networks. It has progressed from laboratories to real-world implementations and is
gradually being integrated into existing optical networks.
However, its commercial success still requires significant innovations that will make the technology more robust and affordable. As a step toward this goal, we propose and demonstrate a novel light
source that can generate pulses modulated in phase without the aid of an external phase modulator. This allows to considerably reduce the source driving voltage and to reliably control the phase
randomization of the emitted pulses. By changing the electrical signals only, a diverse range of QKD protocols can easily be accommodated. This development makes QKD devices substantially more
compact, versatile and energy- efficient—features that are essential for widespread adoption.
“77-Day Field Trial of High Speed Quantum Key Distribution with Implementation Security” by Alexander Dixon, James Dynes, Marco Lucamarini, Bernd Fröhlich, Andrew Sharpe, Alan Plews, Simon Tam,
Zhiliang Yuan, Yoshimichi Tanizawa, Hideaki Sato, Shinichi Kawamura, Mikio Fujiwara, Masahide Sasaki and Andrew Shields
Monday, 2:35 p.m. – Slides
Quantum key distribution’s central and unique claim is information theoretic security. However, there is an increasing awareness that the security of real QKD systems rely not only on theoretical
security proofs, but also on how closely the system matches the theoretical models and resists known attacks. These hacking or side channel attacks exploit physical devices which do not necessarily
behave precisely as the theory expects. As a result, there is a need to demonstrate QKD systems providing both theoretical and implementation based security.
We report here a QKD system which has been designed to provide these features of resistance to real security issues, component monitoring and failure detection—important not only from a security
point of view, but also for reliable and robust operation. Alongside the increased security confidence level, the system operates with a high and stable secure key rate due to newly developed active
stabilization, averaging 210 kbps and producing 1.33 Tbits of secure key data over 77 days in a telecom network.
“Towards Secure QKD with Testable Assumptions on Modulation Devices” by Akihiro Mizutani, Yuichi Nagamatsu, Marcos Curty, Hoi-Kwong Lo, Koji Azuma, Rikizo Ikuta, Takashi Yamamoto, Nobuyuki Imoto and
Kiyoshi Tamaki
Monday, 2:55 p.m. – Slides / Video
In order to realize secure communication in practice, one serious problem is to establish practical security proofs to bridge the gap between theory and practice.
Currently, source devices become the only region exploitable by a potential eavesdropper (Eve). Therefore, it is urgently required to establish security proofs based on practical source devices for
realizing secure communication in practice.
In this work, we have accommodated two dominant imperfections in the source devices, i.e., phase modulation and intensity fluctuation errors. For both imperfections, we made potentially
experimentally testable assumptions, and proved the security against coherent attacks in the finite-key regime.
As a result of our security proof, even under a realistic phase modulation and intensity fluctuation errors, we show that long distance secure communication is possible with reasonable times of
signal transmission. Our result constitutes a significant step toward realizing secure quantum communication with practical devices.
“Observation of Quantum Fingerprinting Beating the Classical Limit” by Jianyu Guan, Feihu Xu, Hualei Yin, Wei-Jun Zhang, Si-Jing Chen, Xiao-Yan Yang, Li Li, Li-Xing You, Teng-Yun Chen, Zhen Wang,
Qiang Zhang and Jianwei Pan
Monday, 4:20 p.m. – Slides / Video
Quantum communication promises the remarkable advantage of an exponential reduction in the transmitted information over classical communication to accomplish distributed computational tasks. However,
to date, demonstrating this advantage in a practical setting continues to be a central challenge.
Here, we report an experimental demonstration of a quantum fingerprinting protocol that for the first time surpasses the ultimate classical limit to transmitted information. Ultra-low noise
superconducting single-photon detectors and a stable fiber-based Sagnac interferometer are used to implement a quantum fingerprinting system that is capable of transmitting less information than the
classical proven lower bound over 20 km. standard telecom fiber for input sizes of up to two Gbits. The results pave the way for experimentally exploring the advanced features of quantum
communication and open a new window of opportunity for research in communication complexity.
“24-Hour Long Relativistic Bit Commitment” by Ephanielle Verbanis, Raphaël Houlmann, Gianluca Boso, Felix Bussières, Anthony Martin and Hugo Zbinden
Monday, 4:40 p.m. – Video
We report on the first implementation of a relativistic bit commitment protocol sustained for 24 hours using high-speed optical communication and FPGA-based processing between standard computers. Our
commitment time is more than six orders of magnitude longer than what was previously achieved, and we show that it could be extended even further.
“Quantum Teleportation Over Deployed Fibres and Applications to Quantum Networks” by Venkata Ramana Raju Valivarthi, Marcel-Li Grimau Puigibert, Qiang Zhou, Gabriel H. Aguilar, Varun Verma, Francesco
Marsili, Sae Woo Nam, Daniel Oblak and Wolfgang Tittel
Monday, 5 p.m. – Slides / Video
If a photon interacts with a member of an entangled photon pair via a so-called Bell-state measurement (BSM), its state is teleported over arbitrary distances (in principle) onto the second member of
the pair. Starting in 1997, this puzzling prediction of quantum mechanics has been demonstrated many times. However, with just one very recent exception, only the photon that received the teleported
state—if any—traveled far, while the photons partaking in the BSM were always measured close to where they were created.
Here, using the Calgary Fibre Network, we report quantum teleportation from a telecommunication- wavelength photon, interacting with another telecommunication photon after both have traveled over
several kilometers in beeline, onto a photon at 795 nm. wavelength. This improves the distance over which teleportation takes place from 818 m. to 6.2 km. Our demonstration establishes an important
requirement for quantum repeater-based communications and constitutes a milestone on the path to a global quantum Internet.
“Quantum Homomorphic Encryption for Polynomial-sized Circuits” by Yfke Dulek, Christian Schaffner and Florian Speelman
Tuesday, 10 a.m. – Slides / Video
We present a new scheme for quantum homomorphic encryption that is compact and allows for efficient evaluation of arbitrary polynomial-sized quantum circuits. Building on the framework of Broadbent
and Jeffery [BJ15] and recent results in the area of instantaneous non-local quantum computation [Spe15], we show how to construct quantum gadgets that allow perfect correction of the errors that
occur during the homomorphic evaluation of T gates on encrypted quantum data. Our scheme can be based on any classical (leveled) fully homomorphic encryption (FHE) scheme and requires no
computational assumptions besides those already used by the classical scheme.
The size of our quantum gadget depends on the space complexity of the classical decryption function, which aligns well with the current efforts to minimize the complexity of the decryption function.
Our scheme (or slight variants of it) offers a number of additional advantages such as ideal compactness, the ability to supply gadgets “on demand,” circuit privacy for the evaluator against passive
adversaries, and a three-round scheme for blind delegated quantum computation, which puts only very limited demands on the quantum abilities of the client.
“Rate-distance Tradeoff and Resource Costs for All-Optical Quantum Repeaters” by Mihir Pant, Hari Krovi, Dirk Englund and Saikat Guha
Tuesday, 11:25 a.m. – Slides / Video
We present a resource-performance tradeoff of an all-optical quantum repeater that uses photon sources, linear optics, photon detectors and classical feed forward at each repeater node, but no
quantum memories.
We show that the quantum-secure key rate has the form R(t) = Dt^s bits per mode, where t is the end-to- end channel’s transmissivity, and the constants D and s are functions of various device
inefficiencies and the resource constraint, such as the number of available photon sources at each repeater node. Even with lossy devices, we show that s < 1 is possible to attain, and in turn to
outperform the maximum key rate attainable without quantum repeaters, R_direct(t) = -log_2(1-t) bits per mode for t<<1, beyond a certain total range L, where t~e^{-aL} in optical fiber.
We also propose a suite of modifications to a recently-proposed all-optical repeater protocol that ours builds upon, which lower the number of photon sources required to create photonic clusters at
the repeaters so as to outperform R_direct(t), from ~10^11 to ~10^6 photon sources per repeater node. We show that the optimum separation between repeater nodes is independent of the total range L,
and is around 1.5 km. for assumptions we make on various device losses. Our results shed light on the tradeoff between resource requirements and the end-to-end key rate achieved using any specific
repeater architecture.
“Continuous Variable Quantum Computing on Encrypted Data” by Kevin Marshall, Christian S. Jacobsen, Clemens Schafermeier, Tobias Gehring, Christian Weedbrook and Ulrik L. Andersen
Tuesday, 11:45 a.m. – Slides / Video
In today’s era of cloud and distributed computing, protecting a client’s privacy is a task of the highest priority. Performing computations in the cloud on encrypted data rather than on plain text is
a promising tool to achieve this goal.
Here, we report about a continuous variable protocol for performing computation on encrypted data on a quantum computer. We theoretically investigate the protocol and present a proof-of-principle
experiment implementing displacements and squeezing gates. We demonstrate losses of up to 10 km. both ways between the client and the server and show that security can still be achieved.
Our approach offers a number of practical benefits, which can ultimately allow for the potential widespread adoption of this quantum technology in future cloud-based computing networks.
“New Security Notions and Feasibility Results for Authentication of Quantum Data” by Sumegha Garg, Henry Yuen and Mark Zhandry
Tuesday, 2:25 p.m. – Video
We give a new class of security definitions for authentication in the quantum setting. Our definitions capture and strengthen several existing definitions, including superposition attacks on \emph
{classical} authentication, as well as full authentication of quantum data. We argue that our definitions resolve some of the shortcomings of existing definitions.
We then give several feasibility results for our strong definitions. As a consequence, we obtain several interesting results, including: the classical Carter-Wegman authentication scheme
with 3-universal hashing is secure against superposition attacks, as well as adversaries with quantum side information; quantum authentication where the entire key can be reused if verification is
successful; conceptually simple constructions of quantum authentication; and a conceptually simple QKD protocol.
Continuous-variable quantum key distribution (CV-QKD) protocols based on coherent detection have been studied extensively in both theory and experiment. While the existing security proofs of CV-QKD
are based on the assumption that the local oscillator (LO) for coherent detection is trustable, this assumption cannot be justified in most practical implementations of CV-QKD, where both the quantum
signal and the LO are generated from the same laser at the sender’s side and propagate through an insecure quantum channel.
To close the above gap between theory and experiment, we proposed an intradyne CV-QKD scheme where the LO is generated from an independent laser source at the receiver’s end (Phys. Rev. X 5, 041009,
2015). This scheme not only removes the security issues related to an untrusted LO, but also greatly simplifies QKD implementation. We demonstrate the above scheme in a coherent communication system
constructed by a spool of 25 km. single mode fiber and two independent commercial laser sources operated at free-running mode. The observed phase-noise variance is 0.04 (rad^2), which is small enough
to enable secure key distribution. This technology also opens the door for other quantum communication protocols, such as measurement-device-independent (MDI) CV-QKD
Here, using the Calgary Fibre Network, we report quantum teleportation from a telecommunication-wavelength photon, interacting with another telecommunication photon after both have traveled over
several kilometers in beeline, onto a photon at 795 nm. wavelength. This improves the distance over which teleportation takes place from 818 m. to 6.2 km. Our demonstration establishes an important
requirement for quantum repeater-based communications and constitutes a milestone on the path to a global quantum Internet.
Note: This talk is combined with the following talk.
“Theoretical Analysis and Proof-of-Principle Demonstration of Self-Referenced Continuous-Variable Quantum Key Distribution” by Constantin Brif, Daniel Soh, Patrick Coles, Norbert Lutkenhaus, Ryan
Camacho, Junji Urayama and Mohan Sarovar
Slides / Video
This work presents the theoretical analysis and proof-of-principle demonstration of a new continuous- variable quantum key distribution (CV-QKD) protocol, self-referenced CV-QKD. This protocol
eliminates the need for transmission of a high-power local oscillator between the communicating parties. Instead, each signal pulse is accompanied by a reference pulse (or a pair of twin reference
pulses), used to align Alice’s and Bob’s measurement bases.
We quantify the expected secret key rates by expressing them in terms of experimental parameters and present a proof-of-principle, fiber-based experimental demonstration of the protocol. Our analysis
of the secret key rate fully takes into account the inherent uncertainty associated with the quantum nature of the reference pulse(s) and quantifies the limit at which the theoretical key rate
approaches that of the respective conventional protocol that requires local oscillator transmission. The self-referenced protocol greatly simplifies the hardware required for CV-QKD, especially for
potential integrated photonics implementations of transmitters and receivers, with minimum sacrifice of performance. As such, it provides a pathway towards scalable integrated CV-QKD transceivers, a
vital step toward large-scale QKD networks.
“Quantum-Limited Measurements of Signals from a Satellite in Geostationary Earth Orbit” by Dominique Elser, Kevin Günthner, Imran Khan, Birgit Stiller, Ömer Bayraktar, Christian R. Müller, Karen
Saucke, Daniel Tröndle, Frank Heine, Stefan Seel, Peter Greulich, Herwig Zech, Björn Gütlich, Ines Richter, Rolf Meyer, Christoph Marquardt and Gerd Leuchs
Wednesday, 11:25 a.m. – Slides / Video
Quantum communication has been implemented in metropolitan area networks around the world. Optical satellite communication lends itself to interconnect such metropolitan networks over global
distances. For this purpose, existing Laser Communication Terminals (LCTs) can be upgraded to quantum key distribution (QKD) application. We have performed first satellite measurement campaigns to
validate this approach.
“Time-Bin Encoding Along Satellite-Ground Channels” by Giuseppe Vallone, Daniele Dequal, Marco Tomasin, Francesco Vedovato, Matteo Schiavon, Vincenza Luceri, Giuseppe Bianco and Paolo Villoresi
Wednesday, 11:45 a.m. – Slides / Video
Time-bin encoding is an extensively used technique to encode a qubit in quantum key distribution (QKD) along optical fibers. Despite its success in fibers QKD (in particular in the “plug-and-play”
systems), time- bin encoding was never implemented in long-distance free-space QKD.
Here we demonstrate that time-bin interference at the single photon level can be observed along free- space channels and in particular along satellite-ground channels. To this purpose, we used a
scheme similar to the “plug-and-play” systems: a coherent superposition between two wavepackets is generated on ground, sent on space and reflected by a rapidly moving satellite at a very large
distance with a total path length up to 5000 km. The beam returning on ground is at the single photon level and we measured the interference between the two time-bins. We will demonstrate that the
varying relative velocity of the satellite with respect to the ground introduces a modulation in the interference pattern that can be predicted by special relativistic calculations. Our results
attest the viability of time-bin encoding for quantum communications in space.
“Cross-Phase Modulation of a Probe Stored in a Waveguide for Non-Destructive Detection of Photonics Qubits” by Chetan Deshmukh, Neil Sinclair, Khabat Heshami, Daniel Oblak, Christoph Simon and
Wolfgang Tittel
Thursday, 11:25 a.m. – Slides / Video
Non-destructive detection of photonic qubits is an enabling technology for quantum information processing and quantum communication. For practical applications such as quantum repeaters and networks,
it is desirable to implement such detection in a way that allows some form of multiplexing as well as easy integration with other components such as solid-state quantum memories.
Here we propose an approach to non-destructive photonic qubit detection that promises to have all the mentioned features. Mediated by an impurity-doped crystal, a signal photon in an arbitrary
time-bin qubit state modulates the phase of an intense probe pulse that is stored during the interaction. A proof-of- principle experiment with macroscopic signal pulses has been able to demonstrate
the expected cross-phase modulation as well as the ability to preserve the coherence between temporal modes. Our findings open the path to a new key component of quantum photonics based on
rare-earth-ion doped crystals.
“Information Theoretically Secure Distributed Storage System with Quantum Key Distribution Network and Password Authenticated Secret Sharing Scheme” by Mikio Fujiwara, Atsushi Waseda, Ryo Nojima,
Shiho Moriai, Wakaha Ogata and Masahide Sasaki
Thursday, 11:45 a.m. – Slides / Video
A quantum key distribution (QKD) allows two users to share random numbers with the unconditional security based on the fundamental laws of physics. By combining a QKD with one-time pad encryption
(OTP), communication with unconditional security can be realized.
A QKD system, however, does not guarantee the security of stored data. Shamir’s (k, n)-threshold secret sharing (SS) scheme in which the data are split into n pieces (shares) for storage and at least
k pieces of them must be gathered for reconstruction, provides information theoretical security. Therefore, a combination of a QKD system and SS scheme is a combination for secure data transmission
and storage. However, assumed is authentication must be perfectly secure, which is not trivial in practice.
Here we propose a totally information theoretically secure distributed storage system based on a user-friendly single-password-authenticated secret sharing scheme and secure transmission using
quantum key distribution, and demonstrate it in the Tokyo metropolitan area (≤90 km).
“Quantum-Proof Multi-Source Randomness Extractors in the Markov Model” by Rotem Arnon-Friedman, Christopher Portmann and Volkher Scholz
Thursday, 2:15 p.m. – Slides / Video
Randomness extractors, widely used in classical and quantum cryptography as well as in device independent randomness amplification and expansion, are functions which generate almost uniform
randomness from weak sources of randomness.
In the quantum setting, one must take into account the quantum side information held by an adversary, which might be used to break the security of the extractor. In the case of seeded extractors, the
presence of quantum side information has been extensively studied. For multi-source extractors, one can easily see that high conditional min-entropy is not sufficient to guarantee security against
arbitrary side information, even in the classical case. Hence, the interesting question is under which models of side information multi-source extractors remain secure.
In this work we suggest a natural model of side information, which we call the Markov model, and prove that any multi-source extractor remains secure in the presence of quantum side information of
this type (albeit with weaker parameters). This improves on previous results in which more restricted models were considered and the security of only some types of extractors were shown.
“On Quantum Obfuscation” by Gorjan Alagic and Bill Fefferman
Thursday, 2:35 p.m. – Video
Encryption of data is fundamental to secure communication. Beyond encryption of data lies obfuscation, i.e., encryption of functionality. It has been known for some time that the most powerful
classical obfuscation, so-called “black-box obfuscation,” is impossible. In this work, we initialize the rigorous study of obfuscating programs via quantum-mechanical means. We prove quantum
analogues of several foundational results in obfuscation, including the aforementioned black-box impossibility result.
In its most powerful “quantum black-box” instantiation, a quantum obfuscator would turn a description of a quantum program f into a quantum state R_f , such that anyone in possession of R_f can
repeatedly evaluate f on inputs of their choice, but never learn anything else about the original program. We formalize this notion of obfuscation, and prove an impossibility result: such obfuscation
is only possible in a setting where the adversary never has access to more than one obfuscation (of either the same program, or of different programs). Our proof involves a novel and recently
developed technical idea: chosen-ciphertext-secure encryption for quantum states. In addition, we show that some applications of obfuscation still appear possible in spite of our impossibility
result. These include encryption for quantum states, quantum fully-homomorphic encryption, and quantum money.
We also define quantum versions of indistinguishability obfuscation and best-possible obfuscation. We then show that these notions are equivalent, and that their perfect and statistical variants are
impossible to achieve. The remaining (i.e., computational) variant would still have an application of interest: witness encryption for QMA.
“Breaking Symmetric Cryptosystems Using Quantum Period Finding” by Marc Kaplan, Gaëtan Leurent, Anthony Leverrier and María Naya-Plasencia
Thursday, 2:55 p.m. – Slides / Video
Due to Shor’s algorithm, quantum computers are a severe threat for public key cryptography. This motivated the cryptographic community to search for quantum-safe solutions. On the other hand, the
impact of quantum computing on secret key cryptography is much less understood.
In this paper, we consider attacks in the quantum chosen plaintext model, in which an adversary can query an oracle implementing a cryptographic primitive in a quantum superposition of different
states. The adversary is then very powerful, but recent results show that it is nonetheless possible to design secure cryptosystems.
We introduce new applications of a quantum procedure called Simon’s algorithm (the simplest quantum period finding algorithm) in order to attack symmetric cryptosystems in this model. Following
previous works in this direction, we show that several classical attacks based on finding collisions can be dramatically sped up using Simon’s algorithm: finding a collision requires Ω(2n/2) queries
in the classical setting, but when collisions happen with some hidden periodicity, they can be found with only O(n) queries in the quantum model.
We obtain attacks with very strong implications. First, we show that the most widely used modes of operation for authentication and authenticated encryption (e.g. CBC-MAC, PMAC, GMAC, GCM and OCB)
are completely broken in this security model. Our attacks are also applicable to many CAESAR candidates: CLOC, AEZ, COPA, OTR, POET, OMD and Minalpher.
Second, we show that slide attacks can also be sped up using Simon’s algorithm. This is the first exponential speed up of a classical symmetric cryptanalysis technique in the quantum model.
“Adaptive Versus Non-Adaptive Strategies in the Quantum Setting” by Frédéric Dupuis, Serge Fehr, Philippe Lamontagne and Louis Salvail
Friday, 11:25 a.m. – Video
We prove a general relation between adaptive and non-adaptive strategies in the quantum setting, i.e., between strategies where the adversary can or cannot adaptively base its action on some
auxiliary quantum side information. Our relation holds in a very general setting, and is applicable as long as we can control the bit-size of the side information, or, more generally, its
“information content.”
Since adaptivity is notoriously difficult to handle in the analysis of (quantum) cryptographic protocols, this gives us a very powerful tool: as long as we have enough control over the side
information, it is sufficient to restrict ourselves to non-adaptive attacks.
We demonstrate the usefulness of this methodology with two examples. The first is a quantum bit commitment scheme based on 1-bit cut-and-choose. Since bit commitment implies oblivious transfer (in
the quantum setting) and oblivious transfer is universal for two-party computation, this implies the universality of 1-bit cut-and-choose, and, thus, solves the main open problem of [10]. The second
example is a quantum bit commitment scheme proposed in 1993 by Brassard et al.
It was originally suggested as an unconditionally secure scheme, back when this was thought to be possible. We partly restore the scheme by proving it secure in (a variant of ) the bounded quantum
storage model.
In both examples, the fact that the adversary holds quantum side information obstructs a direct analysis of the scheme, and we circumvent it by analyzing a non-adaptive version, which can be done by
means of known techniques, and applying our main result.
“Computational Security of Quantum Encryption” by Gorjan Alagic, Anne Broadbent, Bill Fefferman, Tommaso Gagliardoni, Michael St. Jules and Christian Schaffner
Friday, 11:45 a.m. – Slides / Video
Quantum-mechanical devices have the potential to transform cryptography. Most research in this area has focused either on the information-theoretic advantages of quantum protocols or on the security
of classical cryptographic schemes against quantum attacks. In this work, we initiate the study of another relevant topic: the encryption of quantum data in the computational setting.
In this direction, we establish quantum versions of several fundamental classical results. First, we develop natural definitions for private-key and public-key encryption schemes for quantum data. We
then define notions of semantic security and indistinguishability and, in analogy with the classical work of Goldwasser and Micali, show that these notions are equivalent. Finally, we construct
secure quantum encryption schemes from basic primitives. In particular, we show that quantum-secure one-way functions imply IND-CCA1-secure symmetric-key quantum encryption, and that quantum-secure
trapdoor one-way permutations imply semantically-secure public-key quantum encryption.
“Integrated Silicon Photonics for Quantum Key Distribution” by Philip Sibson, Jake Kennard, Stasja Stanisic, Chris Erven and Mark Thompson
Friday, 1:40 p.m. – Slides / Video
Integrated photonics provides a compact and robust platform to implement complex photonic circuitry, and with silicon, in particular, offers extreme levels of miniaturization in a CMOS-compatible
Here we demonstrate integrated silicon photonic devices for polarization and time-bin encoded quantum key distribution protocols. These GHz clocked devices use a combination of slow but ideal
thermo-optic phase shifters and fast but non-ideal carrier-depletion phase modulators to transmit BB84 states. This work experimentally demonstrates the feasibility of QKD transmitters for high-speed
QKD based on CMOS- compatible silicon photonic integrated circuits.
Note: This talk is combined with the following talk.
“Wavelength-Division-Multiplexed QKD with Integrated Photonics” by Philip Sibson, Chris Erven and Mark Thompson
This work experimentally demonstrates Wavelength-Division-Multiplexed QKD with integrated photonics for high-rate QKD. We use two GHz rate indium phosphide transmitters and a silicon oxynitride
receiver with integrated wavelength de-multiplexing and two reconfigurable receivers for multi-protocol QKD. The increase in rates and the ability to scale up these circuits opens the way to new and
advanced integrated quantum communication technologies and larger adoption of quantum-secured communications.
“Laser Damage Creates Backdoors in Quantum Cryptography” by Shihan Sajeed, Sarah Kaiser, Poompong Chaiwongkhot, Mathieu Gagne, Jean-Philippe Bourgoin, Carter Minshull, Matthieu Legre, Thomas
Jennewein, Raman Kashyap and Vadim Makarov
Friday, 2:05 p.m. – Video
Implementations of quantum communication (QC) protocols are assumed to be secure as long as implemented devices are perfectly characterized and all side channels are identified and closed. We show
that this assumption is not always true.
We introduce a laser-damage attack that can, on-demand, create deviations in the behavior of the implemented devices from the characterized one. We test it on two different and perfectly
characterized implementations of quantum key distribution and coin-tossing protocols and successfully create deviations to render the system insecure. Our results show that in order to provide
unconditional security, quantum cryptography protocols need to be supported by additional testing and countermeasures against laser damage.
Note: This talk is combined with the following talk.
“Insecurity of Detector-Device-independent Quantum Key Distribution” by Anqi Huang, Shihan Sajeed, Shihai Sun, Feihu Xu, Vadim Makarov and Marcos Curty
It is time to close the gap between theory and practice in quantum key distribution (QKD). To bridge this gap, detector-device-independent QKD (ddiQKD) has recently been proposed. However, from our
analysis, this protocol is not as secure as expected. The main contributions of this work are two-fold. First, we show that, in contrast to mdiQKD, the security of ddiQKD cannot be based on
post-selected entanglement alone as assumed. Second, we argue that ddiQKD is actually insecure under detector side-channel attacks. | {"url":"https://qcrypt.github.io/2016.qcrypt.net/contributed-talks.1.html","timestamp":"2024-11-10T04:44:28Z","content_type":"text/html","content_length":"60286","record_id":"<urn:uuid:f35979fb-c97b-4521-8665-e38b6939883c>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00308.warc.gz"} |
Without Geometry, Life is Pointless
As a wrap-up to our Games of Probability group project, my 6th graders and I were talking about the probability of getting a "Yahtzee" if you were playing with 2 dice instead of five (if there is one
mathematical habit of mind that may be burned into the soul of each of my students this year, let it be the idea to try a simpler case). We were discussing the case where you get your Yahtzee on the
third roll. We went over the probability of not getting a Yahtzee on the first roll. I rolled 2 dice and didn't get a Yahtzee. We then went over the probability of not getting a Yahtzee on the second
roll. I rolled again and didn't get a Yahtzee. We went over the probability of getting a Yahtzee on the third roll. I rolled two dice and, what do you know, rolled a Yahtzee.
In the words of @cheesemonkeysf, "I love it whenever the Universe tosses a teacher a freebie."
My fellow 6th grade math teacher and I are trying something crazy this week. We were two weeks into our counting/probability unit and wanted a project that would:
• Help students practice the foundational skills and concepts
• Push students to explore more challenging, unfamiliar, open-ended problems
• Reiterate the importance of group work
• Reiterate the importance of writing down solutions, reasoning, and explanations in an organized fashion.
So our crazy new idea? We've put together four open ended, challenging probability games. Eight groups of 3 or 4 will spend tomorrow working on one of these games (2 groups for each game in each
class). On Tuesday, groups will rotate and work on a different game, starting where the previous group left off.
The hope is that, as an entire group, we make significant progress on each of the four problems. I'm super excited to see how this works (and I'm sure we'll all learn a lot even if it completely
bombs). Below are the games we're introducing, and the investigation questions.
Investigate the probabilities associated with this game. Think about the possible ways to make the required combinations, and then think about the points that you win for each part of the game. Are
the points that Milton Bradley assigned to each combination in the game “fair?” If you feel that they are “unfair” how you would you recommend to the makers of this game that they change the scoring
to make it “fair?”
Pig (we're calling this "On a Roll"):
When you think that you understand the game, begin to investigate different strategies that you might use or recommend to others to help players decide when to stop rolling and when to take a chance
and roll just one more time.
Deal or No Deal:
Play a version of the game at http://bit.ly/dealnodealgame. Keep track of the revealed suitcases and the banker offers and determine whether or not (with an explanation) you should take each offer.
What other questions do you have about this game?
Drunken Walk (we're calling this "Dizzy Walk"):
1 dimensional version:
Where should you put your house? Would your placement of your house change if you initially rolled a die to determine how many times to flip the coin? Explore other questions you have about this
2 dimensional version:
Let’s call the address of where Larry starts (0,0) and the address he would end up at after flipping HHTTT (2,3). How many different ways are there for Larry to get to (3,5)?
Imagine instead that Larry starts at (5,5). This time, instead of flipping a coin, you spin a spinner split into 4 equal areas marked North, East, South, & West. Now where would you put your house?
What if you can create the spinner and determine the size of the 4 areas.
Here are the nitty gritty details of how to play the games if you're interested (ie our handouts).
Spent last weekend at the California Math Council North conference (#cmcmath) in Pacific Grove, CA. 'Tis a beautiful spot, even though it was raining the whole weekend this year. The one upside to
rain, though, is that it made for an excuse to splurge on the quite pricy entrance fee to the Monterey Bay Aquarium.
At the CMC-South (the Palm Springs equivalent) conference in November, Dan Meyer and I lamented the fact that it was impossible to see every good workshop and talked about creating a site to recap
sessions (it was actually all Dan's idea but I take credit for my supportive listening). No surprise, Dan had a site up within a week: mathrecap.com.
The number of session recaps are growing, and I especially recommend Dan's recap of the Friday keynote speaker, Kyndall Brown.
So far, I've written one recap of my own.
[Amy Ellis] Laying a Foundation for Learning to Prove
By Avery Pickford | Published December 7, 2012
It’s hard not to enjoy sessions when you’re already drinking the kool-aid. That said, Amy Ellis did a fantastic job of balancing research and practice around laying a foundation for proof well
before Geometry class (on a side note, I still hope to lead a session called “Proof Doesn’t Start in Geometry” at some point in the future). She gave a convincing argument for the importance of
introducing the idea of proof in early elementary school, and more importantly discussed structures and cultures that promote proof at any age. [Read the entire recap here]
My big take away: don’t fall into the trap that proof must require algebra, two columns, and/or a less eloquent rehash of Euclid.
Henri Picciotto was also kind enough to write not just one, but two recaps of my session on student posed problems. Part one is more of a recap, while part two addresses (or at least states) some of
the challenges of implementation. | {"url":"http://www.withoutgeometry.com/2012/","timestamp":"2024-11-08T22:27:15Z","content_type":"text/html","content_length":"158202","record_id":"<urn:uuid:ad1f8e61-f40b-4b94-be4a-9b22a563cd21>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00674.warc.gz"} |
Top Amazon Data Structure Interview Questions - Algo2Ace.com
Top Amazon Data Structure Interview Questions
Welcome to our new post Top Amazon Data Structure Interview Questions. here you will get the top 10 most-asked interview questions on data structure and algorithms in Amazon. Get interview-ready for
Amazon with a tailored focus on data structure questions for candidates with around one year of experience. Elevate your preparation with a curated selection of top Amazon interview questions that
delve into fundamental data structure concepts. Whether you’re tackling linked lists, binary trees, or hash tables, these questions will hone your problem-solving skills. Be equipped to discuss array
vs. linked list distinctions, hash table operations, heap structures, and more. Strengthen your understanding of time and space complexities, dynamic programming, and recursion. This resource
provides a strategic advantage for your Amazon interview readiness, ensuring you’re poised to excel in technical discussions. Master the essentials and confidently stand out as you approach your
Amazon data structure interview.
It will help you to prepare the Top Amazon Data Structure Interview Questions.
To Read more Interview questions about JAVA, Python, Scala
Top Amazon Data Structure Interview Questions
1. What is the difference between an array and a linked list? Explain the advantages and disadvantages of each data structure.
Aspect Array Linked List
Memory Allocation Contiguous block of memory Individual nodes with scattered memory
Insertion/Deletion Costly for inserts/deletes in the middle Efficient for inserts/deletes anywhere
Access Time Constant time (O(1)) for random access Linear time (O(n)) for sequential access
Size Flexibility Fixed-size; reallocation may be required Dynamic size; can grow or shrink as needed
Memory Overhead Lower due to direct storage Higher due to storing next pointers
Cache Performance Good due to memory locality May suffer due to non-contiguous memory
Usage Best suited for frequent random access Suitable for dynamic inserts/deletes
Advantages of Arrays:
1. Constant Time Access: Arrays provide direct access to elements using an index, resulting in continuous time access (O(1)).
2. Memory Locality: Contiguous memory allocation leads to better cache performance, enhancing data retrieval speed.
3. Simplicity: Arrays are simple to use and understand, making them a natural choice for straightforward scenarios.
Advantages of Linked Lists:
1. Dynamic Size: Linked lists can grow or shrink dynamically, accommodating changing data requirements.
2. Efficient Insertion/Deletion: Insertions and deletions at any point in the list are efficient, requiring only local changes.
3. Memory Efficiency: Linked lists allow memory allocation for each node individually, minimizing memory wastage.
4. No Preallocation Required: Linked lists don’t need preallocation, unlike arrays which often require resizing.
While arrays excel at random access and memory efficiency, linked lists shine when dealing with dynamic data and frequent insertions/deletions. The choice between them depends on the specific use
case and performance requirements.
2. Explain the concept of time complexity and space complexity. How do you analyze the efficiency of an algorithm in terms of these complexities?
Time and space complexity are fundamental concepts used to analyze the efficiency of algorithms. They help us understand how an algorithm’s runtime and memory usage grow as the input size increases.
Time Complexity:
Time complexity measures the amount of time an algorithm takes to complete as a function of the input size. It’s typically expressed using Big O notation, which provides an upper bound on the growth
rate of the algorithm’s runtime. For example, an algorithm with a time complexity of O(n) indicates that its runtime grows linearly with the size of the input.
To analyze time complexity:
1. Identify the basic operations in the algorithm.
2. Count the number of times each basic operation is executed as a function of the input size.
3. Express the count in terms of Big O notation by dropping constants and lower-order terms.
Space Complexity:
Space complexity measures the amount of memory an algorithm uses as a function of the input size. It considers both the memory required for the algorithm’s instructions and the memory used by data
structures, variables, and other auxiliary components.
To analyze space complexity:
1. Identify the memory used by the algorithm’s variables, data structures, and other components.
2. Sum up the memory used by each component as a function of the input size.
3. Express the sum in terms of Big O notation.
Efficiency Analysis:
1. Worst Case vs. Average Case: Algorithms may perform differently for different inputs. Analyze the worst-case scenario, which gives an upper bound on how the algorithm performs for any
2. Dominant Terms: Focus on the most significant terms in the time and space complexity expressions. More minor terms and constants are dropped in Big O notation.
3. Comparative Analysis: Compare the complexities of different algorithms solving the same problem to determine which one is more efficient in terms of time and space.
4. Trade-offs: Sometimes, optimizing time complexity may lead to higher space complexity and vice versa. Analyze the trades based on the requirements of the problem.
5. Asymptotic Analysis: Big O notation provides an asymptotic upper bound. It’s especially useful for analyzing how algorithms scale for large input sizes.
03. What is the time complexity of searching for an element in a sorted array using binary search?
The time complexity of searching for an element in a sorted array using binary search is O(log n), where “n” is the number of elements in the array.
Binary search works by repeatedly dividing the search interval in half until the desired element is found or it’s determined that the element is not present in the array. In each step of the
algorithm, the search space is effectively halved. This logarithmic behavior means that the time it takes to complete the search increases very slowly as the size of the input (the array) grows. It’s
significantly faster than linear search, which has a time complexity of O(n) for a sorted array.
04. Define a binary search tree (BST). How does it differ from a regular binary tree?
A Binary Search Tree (BST) is a specific type of binary tree data structure in which each node has at most two children, and the following properties hold:
a. Value Ordering: For each node in the BST:
• All nodes in its left subtree have values less than the node’s value.
• All nodes in its right subtree have values greater than the node’s value.
b. Unique Values:
All values stored in the BST are unique. No two nodes can have the same value.
A regular binary tree, on the other hand, does not have any specific ordering of values between nodes. In a regular binary tree, there are no restrictions on how values are organized within the tree,
and there is no requirement for unique values among nodes.
To illustrate the difference, here’s an example:
Binary Search Tree (BST):
In this BST, the value ordering property holds. For any node, all values in its left subtree are less than the node’s value, and all values in its right subtree are greater.
Regular Binary Tree:
In this regular binary tree, there is no specific ordering of values between nodes. It’s not organized in a way that satisfies the BST property.
The key difference between a BST and a regular binary tree is that a BST is designed for efficient searching, insertion, and deletion of elements with a time complexity of O(log n) in average cases
(assuming it’s reasonably balanced), while a regular binary tree does not have any inherent ordering, and its performance characteristics for these operations may not be as efficient.
05. What is the significance of a balanced binary tree, and why is it important in data structures?
A balanced binary tree, such as an AVL tree or a Red-Black tree, is a specific type of binary search tree (BST) that maintains a balance condition. In a balanced binary tree, the heights of the left
and right subtrees of any node differ by at most one. This balance condition ensures that the tree remains relatively shallow and height-balanced.
The significance of a balanced binary tree and why it’s important in data structures can be understood through the following points:
a. Efficient Searching:
A balanced binary tree guarantees that the depth of the tree is logarithmic in the number of nodes. As a result, searching for an element in a balanced tree has an average and worst-case time
complexity of O(log n), where “n” is the number of nodes. This is significantly faster than searching in an unbalanced binary tree, which can have a worst-case time complexity of O(n) in the worst
b. Efficient Insertion and Deletion:
Maintaining balance in a binary tree ensures that insertions and deletions can be performed efficiently in O(log n) time. This is crucial for data structures that require dynamic operations, such as
sets, maps, and dictionaries.
c. Prevents Worst-Case Scenarios:
Without balance, a binary tree could degenerate into a linked list-like structure, where one subtree becomes much deeper than the other. In this worst-case scenario, searching, insertion, and
deletion operations become inefficient, with a time complexity of O(n). Balanced trees prevent such worst-case scenarios.
06. Explain the concept of hashing. How is it used in data structures like hash tables?
Hashing is a technique used in computer science to map data of arbitrary size (such as keys or values) to fixed-size values, typically numerical values, known as hash codes or hash values. The
primary goal of hashing is to efficiently store, retrieve, and manage data in various data structures, with a focus on achieving constant-time average-case complexity for key operations like
insertion, deletion, and retrieval.
a. Hash Function:
• A hash function is a mathematical function that takes an input (or “key”) and returns a fixed-size hash code.
• The hash code is typically a numerical value, but it can be any fixed-size data (e.g., an integer or a bit string).
• The same input should always produce the same hash code (deterministic behavior).
b. Hashing in Data Structures:
• Hashing is commonly used in data structures, with one of the most popular applications being the hash table (also known as a hash map).
• A hash table is an array-based data structure that uses a hash function to map keys to array indices (buckets).
• Each bucket can store one or more key-value pairs.
c. Hashing Use Cases:
• Hashing is used in various data structures and algorithms, not just hash tables. For example, it’s used in hash-based sets and dictionaries, as well as in techniques like bloom filters, which
provide fast membership tests.
• Hashing is also used in security applications, such as password storage (salting and hashing) and digital signatures.
07. What is dynamic programming, and in what type of problems is it typically applied?
Dynamic programming (DP) is a powerful algorithmic technique used in computer science and mathematics to solve problems by breaking them down into smaller overlapping subproblems and storing the
solutions to these subproblems to avoid redundant computations. It’s particularly effective for optimization problems where you want to find the best solution among a set of possible solutions.
Dynamic programming is typically applied to problems falling into one of two categories:
a. Top-Down (Memoization):
In this approach, you start with the original problem and recursively break it down into smaller subproblems. You memorize (store) the solutions to these subproblems in a data structure (like a
dictionary or an array) to avoid recomputing them when needed. This approach is known as memoization.
b. Bottom-Up (Tabulation):
In this approach, you start by solving the smallest subproblems first and use their solutions to build up to the original problem. You often use an array or a table to store solutions to subproblems,
and you fill it in a systematic manner. This approach is known as tabulation.
08. What is the Big O notation, and how is it useful in analyzing algorithm efficiency?
The Big O notation is a mathematical notation used in computer science to describe the upper bound of an algorithm’s time complexity or space complexity in terms of the input size. It provides a way
to analyze and compare the efficiency of algorithms while abstracting away constant factors and lower-order terms. Big O notation is useful for understanding how an algorithm’s performance scales as
the input size grows.
Key points about Big O notation and its utility in analyzing algorithm efficiency:
a. Definition:
Big O notation, denoted as O(f(n)), represents an upper bound on the growth rate of a function in terms of the input size “n.” It describes how the algorithm’s resource usage (time or space) grows
asymptotically as the input size increases.
b. Asymptotic Analysis:
Big O notation focuses on the behavior of an algorithm as the input size approaches infinity. It doesn’t concern itself with specific constants, lower-order terms, or input sizes that are not “large
enough.” This simplification allows for a high-level understanding of efficiency trends.
c. Comparative Analysis:
Big O notation allows you to compare algorithms and make informed decisions about which one to choose for a particular problem. An algorithm with a lower-order Big O complexity is generally more
efficient for large inputs.
d. Worst-Case Analysis:
Big O notation often describes the worst-case scenario for an algorithm. It provides an upper bound on how an algorithm behaves when dealing with the most unfavorable input.
09. Describe the differences between depth-first search (DFS) and breadth-first search (BFS) traversal algorithms in graphs.
Aspect Depth-First Search (DFS) Breadth-First Search (BFS)
Traversal Order LIFO (Last In, First Out) FIFO (First In, First Out)
Data Structure (Used for Stack Queue
Nature of Traversal Depth-first (explores as far as possible along each branch before Breadth-first (explores all neighbors at the current level before moving to the next level)
Implementation Typically implemented recursively or using an explicit stack. Typically implemented using a queue data structure.
Memory Usage Can use less memory compared to BFS as it explores one branch completely Tends to use more memory, especially for wide graphs, as it stores all neighbors of a level before
before moving to the next. proceeding to the next level.
Time Complexity O(V + E) for an adjacency list representation (V: vertices, E: edges) in O(V + E) for an adjacency list representation in the worst case.
the worst case.
10. What is memoization, and how can it improve the performance of recursive algorithms?
Memoization is an optimization technique used in computer programming to improve the performance of recursive algorithms, particularly those that involve repeated calculations of the same
subproblems. It involves caching or storing the results of expensive function calls and reusing those results when the same inputs occur again, instead of recalculating them.
Here’s how memoization works and how it can improve the performance of recursive algorithms:
a. Caching Results:
When a recursive function is called with a particular set of input parameters, memoization involves checking whether the function has already computed and stored the result for those parameters. If
it has, the cached result is returned immediately instead of re-computing it.
b. Storage Mechanism:
Memoization typically uses data structures like dictionaries, arrays, or hash tables to store computed results. The input parameters serve as keys, and the corresponding function results are stored
as values.
c. Base Cases:
Recursive functions that use memoization still need to define base cases to terminate the recursion. Base cases are typically straightforward and return predefined results for simple inputs,
preventing infinite recursion.
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ZMP Seminar and Colloquium (Winter term 2021/22)
The topic of the ZMP Seminar this term will be Cluster algebras. For questions about the seminar, please contact Georgios Papathanasiou, Ingo Runkel or Volker Schomerus.
The seminar will either be on BBB or on zoom as indicated in the announcement email each week.
The ZMP Colloquium will typically be on zoom.
Seminar notes:
Some references:
• "WHAT IS ... a cluster algebra" [pdf] from the "WHAT IS..." series of the AMS [web]
• The review "Cluster algebras and derived categories" by B. Keller (arXiv:1202.4161 [math.RT])
• "Polylogarithm identities, cluster algebras and the N=4 supersymmetric theory" by C. Vergu (arXiv:1512.08113 [hep-th])
• The review "The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes" by S. Caron-Huot et al. (arXiv:2005.06735 [hep-th])
• The lecture notes "Mathematical aspects of scattering amplitudes" by C. Duhr (arXiv:1411.7538 [hep-ph])
• "Cluster mutation-periodic quivers and associated Laurent sequences" by A.P. Fordy and B. Marsh (arXiv:0904.0200 [math.CO])
• (more to come)
The plan for this term is (subject to change): | {"url":"https://www.math.uni-hamburg.de/home/runkel/ZMP/ws21/index.html","timestamp":"2024-11-03T16:28:47Z","content_type":"text/html","content_length":"7738","record_id":"<urn:uuid:ef17b3ce-b11a-49fd-8477-e46f705540fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00801.warc.gz"} |
Discovering Mathematics
Aufmann’s DISCOVERING MATHEMATICS: A QUANTITATIVE REASONING APPROACH, 2nd EDITION, with WebAssign, helps you learn mathematics in the context of the world around you. Focusing on topics relevant to
your life and on developing critical-thinking skills that you can apply beyond the course, this text provides you with exactly what you need for the world around you in an approachable, engaging and
streamlined format.
Purchase Enquiry INSTRUCTOR’S eREVIEW COPY
Module 1: VALIDITY STUDIES.
1. An Introduction to Problem Solving.
Inductive and Deductive Reasoning. Estimation and Graphs. Problem-Solving Strategies. Fair Division.
2. Sets and Logic.
Sets, Set Operations, and Applications. Categorical Logic. Propositional Logic. Investigating Fallacies.
3. Proportions and Variation.
Ratios and Rates. Measurement. Proportions. Percent. Variation.
Module 2: FINANCIAL LITERACY.
4. Managing Your Finances.
Simple Interest. Compound Interest. Future and Present Value of an Annuity. Credit Cards. Student Loans. Financial Considerations of Car Ownership. Financial Considerations of Home Ownership. Stocks,
Bonds, and Mutual Funds.
Module 3: MODELING.
5. Linear Functions.
Introduction to Functions. Properties of Linear Functions. Linear Models.
6. Nonlinear Functions.
Exponential Models. Logarithmic Functions. Quadratic Functions.
Module 4: PROBABILITY AND STATISTICS.
7. Introduction to Probability.
The Counting Principle. Permutations and Combinations. Probability and Odds. Addition and Complement Rules. Conditional Probability. Expectations.
8. Introduction to Statistics.
Measures of Central Tendency. Measures of Dispersion. Measures of Relative Position. Normal Distributions. Linear Regression and Correlation.
Module 5: ADDITIONAL TOPICS.
9. Voting and Apportionment.
Introduction to Apportionment. Introduction to Voting. Weighted Voting Systems.
10. Circuits and Networks.
Graphs and Euler Circuits. Weighted Graphs. Planarity and Euler’s Formula. Graph Coloring.
11. Geometry.
Measurement. Basic Concepts of Euclidean Geometry. Perimeter and Area of Plane Figures. Volume and Surface Area. Properties of Triangles. Right Triangle Trigonometry. Non-Euclidean Geometry.
Appendix A: A Review of Integers, Rational Numbers, and Percents.
Appendix B: Variable Expressions and Equations.
Appendix C: Review of the Rectangular Coordinates and Graphing.
• Richard N. Aufmann
Richard Aufmann is the lead author of two best-selling Developmental Math series and a best-selling College Algebra and Trigonometry series, as well as several derivative Math texts. Mr. Aufmann
taught Math, Computer Science and Physics at Palomar College in California, where he was on the faculty for 28 years. His textbooks are highly recognized and respected among college mathematics
professors. Today, Mr. Aufmann's professional interests include quantitative literacy, the developmental math curriculum and the impact of technology on curriculum development. He holds a
Bachelor of Arts in Mathematics from the University of California, Irvine, and a Master of Arts degree in Mathematics from California State University, Long Beach.
• NEW AND REFRESHED APPLICATION PROBLEMS DEMONSTRATE THE RELEVANCE OF EACH TOPIC. Updated applications include recent data from the U.S. Census Bureau and technology students constantly use.
• MORE EXPLANATION OF FAIR DIVISION GUIDES STUDENT LEARNING: More detail was added to examples to help students understand this interesting, yet challenging, topic.
• MORE EMPHASIS ON PRACTICAL APPLICATION OF NONLINEAR FUNCTIONS: CHAPTER 6, Nonlinear Functions, has been revised to put more emphasis on the practical applications of those functions to provide
relevancy to students' lives.
• EXCEL HAS BEEN INTEGRATED INTO WEBASSIGN: Students can use this tool to solve problems for finance and statistics exercises. Spreadsheet templates are provided to the student for exercises that
are data intensive.
• REIMAGINED PROBLEM TYPES PROVIDE RELEVANCE AND EMPOWER STUDENTS: Traditional skill-building problems are recreated as word problems, providing context to the content and allowing students to
practice interpreting and extracting data from sentences. This integration of skill-building and applications establishes the importance of mathematics in everyday life.
• PRELIMINARY EXERCISES PROVIDE ADDITIONAL PRACTICE: Located at the beginning of each problem set, Preliminary Exercises provide an opportunity for the student to practice an exercise that is very
similar to a worked example in the text. A complete solution for each Preliminary Exercise can be found in a solution section of the text.
• NEW END OF CHAPTER, MULTI-PART EXERCISES DEMONSTRATE "MATH AT WORK": Math in Practice exercises expand on chapter learning objectives with real-world, applicable situations students can relate to
in their daily lives, or their intended career tracks.
• NEW LEARN IT MODULES ADDRESS STUDENTS’ KNOWLEDGE GAPS: Offering scaffolded help, Learn It modules provide just-in-time instruction that meets students’ diverse learning styles. Learn Its provide
immediate, targeted instruction and practice for a topic. Explanations include clear narratives, videos and tutorials—all in one place.
Cengage provides a range of supplements that are updated in coordination with the main title selection. For more information about these supplements, contact your Learning Consultant.
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barotropic A barotropic fluid is one whose pressure and density are related by an equation of state that does not contain the temperature as a dependent variable. Mathematically, the equation
of state can be expressed as p = p(r) or r = r(p).
compressible A fluid flow is compressible if its density r changes appreciably (typically by a few percent) within the domain of interest. Typically, this will occur when the fluid velocity
exceeds Mach 0.3. Hence, low velocity flows (both gas and liquids) behave incompressibly.
density, r The mass of fluid per unit volume. For a compressible fluid flow, the density can vary from place to place.
incompressible An incompressible fluid is one whose density is constant everywhere. All fluids behave incompressibly (to within 5%) when their maximum velocities are below Mach 0.3.
inviscid Not viscous.
irrotational An irrotational fluid flow is one whose streamlines never loop back on themselves. Typically, only inviscid fluids can be irrotational. Of course, a uniform viscid fluid flow without
boundaries is also irrotational, but this is a special (and boring!) case.
laminar An organized flow field that can be described with streamlines. In order for laminar flow to be permissible, the viscous stresses must dominate over the fluid inertia stresses.
Mach Mach number is the relative velocity of a fluid compared to its sonic velocity. Mach numbers less than 1 correspond to sub-sonic velocities, and Mach numbers > 1 correspond to
super-sonic velocities.
Newtonian A Newtonian fluid is a viscous fluid whose shear stresses are a linear function of the fluid strain rate. Mathematically, this can be expressed as: t[ij] = K[ijqp]*D[pq], where t[ij]
is the shear stress component, and D[pq] are fluid strain rate components.
perfect A perfect fluid is defined as a fluid with zero viscosity (i.e. inviscid).
rotational A rotational fluid flow can contain streamlines that loop back on themselves. Hence, fluid particles following such streamlines will travel along closed paths. Bounded (and hence
nonuniform) viscous fluids exhibit rotational flow, typically within their boundary layers. Since all real fluids are viscous to some amount, all real fluids exhibit a level of
rotational flow somewhere in their domain. Regions of rotational flow correspond to the regions of viscous losses in a fluid. Inviscid fluid flows can also be rotational, but these
are special nonphysical cases. For an inviscid fluid flow to be rotational, it must be set up that way by initial conditions. The amount of rotation (called the velocity circulation)
in an inviscid fluid flow is conserved, provided that the fluid is also barotropic and subject only to conservative body forces. This conservation is known as Kelvin's Theorem of
constant circulation.
Stokesian A Stokesian (or non-Newtonian) fluid is a viscous fluid whose shear stresses are a non-linear function of the fluid strain rate.
streamline A path in a steady flow field along which a given fluid particle travels.
turbulent A flow field that cannot be described with streamlines in the absolute sense. However, time-averaged streamlines can be defined to describe the average behavior of the flow. In
turbulent flow, the inertia stresses dominate over the viscous stresses, leading to small-scale chaotic behavior in the fluid motion.
viscosity, m A fluid property that relates the magnitude of fluid shear stresses to the fluid strain rate, or more simply, to the spatial rate of change in the fluid velocity field.
Mathematically, this is expressed as: t = m*(dV/dy), where t is the shear stress in the same direction as the fluid velocity V, and y is a direction perpendicular to the fluid
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3.19: Quarter-Wavelength Transmission Line
Last updated
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Quarter-wavelength sections of transmission line play an important role in many systems at radio and optical frequencies. The remarkable properties of open- and short-circuited quarter-wave line are
presented in Section 3.16 and should be reviewed before reading further. In this section, we perform a more general analysis, considering not just open- and short-circuit terminations but any
terminating impedance, and then we address some applications.
The general expression for the input impedance of a lossless transmission line is (Section 3.15): \[Z_{in}(l) = Z_0 \frac{ 1 + \Gamma e^{-j2\beta l} }{ 1 - \Gamma e^{-j2\beta l} } \label{m0093_eZ} \]
Note that when \(l=\lambda/4\): \[2\beta l = 2 \cdot \frac{2\pi}{\lambda} \cdot \frac{\lambda}{4} = \pi \nonumber \] Subsequently: \[\begin{split} Z_{in}(\lambda/4) &= Z_0 \frac{ 1 + \Gamma e^{-j\pi}
}{ 1 - \Gamma e^{-j\pi} } \\ &= Z_0 \frac{ 1 - \Gamma }{ 1 + \Gamma } \end{split} \nonumber \] Recall that (Section 3.15): \[\Gamma = \frac{Z_L-Z_0}{Z_L+Z_0} \nonumber \] Substituting this expression
and then multiplying numerator and denominator by \(Z_L+Z_0\), one obtains \[\begin{split} Z_{in}(\lambda/4) &= Z_0 \frac{ \left( Z_L + Z_0\right) - \left(Z_L - Z_0\right) }{ \left( Z_L + Z_0\right)
+ \left(Z_L - Z_0\right) } \\ &= Z_0 \frac{ 2Z_0 }{ 2Z_L } \end{split} \nonumber \] Thus, \[\boxed{ Z_{in}(\lambda/4) = \frac{Z_0^2 }{ Z_L } } \label{m0091_eQWII} \] Note that the input impedance is
inversely proportional to the load impedance. For this reason, a transmission line of length \(\lambda/4\) is sometimes referred to as a quarter-wave inverter or simply as a impedance inverter.
Quarter-wave lines play a very important role in RF engineering. As impedance inverters, they have the useful attribute of transforming small impedances into large impedances, and vice-versa – we’ll
come back to this idea later in this section. First, let’s consider how quarter-wave lines are used for impedance matching. Look what happens when we solve Equation \ref{m0091_eQWII} for \(Z_0\): \
[Z_0 = \sqrt{Z_{in}(\lambda/4) \cdot Z_L } \label{m0091_eQWZ0} \] This equation indicates that we may match the load \(Z_L\) to a source impedance (represented by \(Z_{in}(\lambda/4)\)) simply by
making the characteristic impedance equal to the value given by the above expression and setting the length to \(\lambda/4\). The scheme is shown in Figure \(\PageIndex{1}\).
Design a transmission line segment that matches \(300~\Omega\) to \(50~\Omega\) at 10 GHz using a quarter-wave match. Assume microstrip line for which propagation occurs with wavelength 60% that of
free space.
The line is completely specified given its characteristic impedance \(Z_0\) and length \(l\). The length should be one-quarter wavelength with respect to the signal propagating in the line. The
free-space wavelength \(\lambda_0=c/f\) at 10 GHz is \(\cong 3\) cm. Therefore, the wavelength of the signal in the line is \(\lambda=0.6\lambda_0\cong 1.8\) cm, and the length of the line should be
\(l=\lambda/4 \cong 4.5\) mm.
The characteristic impedance is given by Equation ref{m0091_eQWZ0}: \[Z_0 = \sqrt{ 300~\Omega \cdot 50~\Omega } \cong 122.5~\Omega \nonumber \] This value would be used to determine the width of the
microstrip line, as discussed in Section 3.11.
It should be noted that for this scheme to yield a real-valued characteristic impedance, the product of the source and load impedances must be a real-valued number. In particular, this method is not
suitable if \(Z_L\) has a significant imaginary-valued component and matching to a real-valued source impedance is desired. One possible workaround in this case is the two-stage strategy shown in
Figure \(\PageIndex{2}\). In this scheme, the load impedance is first transformed to a real-valued impedance using a length \(l_1\) of transmission line. This is accomplished using Equation \ref
{m0093_eZ} (quite simple using a numerical search) or using the Smith chart (see “Additional Reading” at the end of this section). The characteristic impedance \(Z_{01}\) of this transmission line is
not critical and can be selected for convenience. Normally, the smallest value of \(l_1\) is desired. This value will always be less than \(\lambda/4\) since \(Z_{in}(l_1)\) is periodic in \(l_1\)
with period \(\lambda/2\); i.e., there are two changes in the sign of the imaginary component of \(Z_{in}(l_1)\) as \(l_1\) is increased from zero to \(\lambda/2\). After eliminating the imaginary
component of \(Z_L\) in this manner, the real component of the resulting impedance may then be transformed using the quarter-wave matching technique described earlier in this section.
A particular patch antenna exhibits a source impedance of \(Z_A = 35+j35~\Omega\). (See “Microstrip antenna” in “Additional Reading” at the end of this section for some optional reading on patch
antennas.) Interface this antenna to \(50~\Omega\) using the technique described above. For the section of transmission line adjacent to the patch antenna, use characteristic impedance \(Z_{01}=50~\
Omega\). Determine the lengths \(l_1\) and \(l_2\) of the two segments of transmission line, and the characteristic impedance \(Z_{02}\) of the second (quarter-wave) segment.
The length of the first section of the transmission line (adjacent to the antenna) is determined using Equation \ref{m0093_eZ}: \[Z_1(l_1) = Z_{01} \frac{ 1 + \Gamma e^{-j2\beta_1 l_1} }{ 1 - \Gamma
e^{-j2\beta_1 l_1} } \nonumber \] where \(\beta_1\) is the phase propagation constant for this section of transmission line and \[\Gamma \triangleq \frac{Z_A-Z_{01}}{Z_A+Z_{01}} \cong -0.0059+j0.4142
\nonumber \] We seek the value of smallest positive value of \(\beta_1 l_1\) for which the imaginary part of \(Z_1(l_1)\) is zero. This can determined using a Smith chart (see “Additional Reading” at
the end of this section) or simply by a few iterations of trial-and-error. Either way we find \(Z_1(\beta_1 l_1 = 0.793~\mbox{rad}) \cong 120.719-j0.111~\Omega\), which we deem to be close enough to
be acceptable. Note that \(\beta_1 = 2\pi/\lambda\), where \(\lambda\) is the wavelength of the signal in the transmission line. Therefore \[l_1 = \frac{\beta_1 l_1}{\beta_1} = \frac{\beta_1 l_1}{2\
pi} \lambda \cong 0.126\lambda \nonumber \]
The length of the second section of the transmission line, being a quarter-wavelength transformer, should be \(l_2 = 0.25\lambda\). Using Equation \ref{m0091_eQWZ0}, the characteristic impedance \(Z_
{02}\) of this section of line should be \[Z_{02} \cong \sqrt{\left(120.719~\Omega\right) \left(50~\Omega\right) } \cong 77.7~\Omega \nonumber \]
Discussion. The total length of the matching structure is \(l_1+l_2 \cong 0.376\lambda\). A patch antenna would typically have sides of length about \(\lambda/2 = 0.5\lambda\), so the matching
structure is nearly as big as the antenna itself. At frequencies where patch antennas are commonly used, and especially at frequencies in the UHF (300–3000 MHz) band, patch antennas are often
comparable to the size of the system, so it is not attractive to have the matching structure also require a similar amount of space. Thus, we would be motivated to find a smaller matching structure.
Although quarter-wave matching techniques are generally effective and commonly used, they have one important contraindication, noted above – They often result in structures that are large. That is,
any structure which employs a quarter-wave match will be at least \(\lambda/4\) long, and \(\lambda/4\) is typically large compared to the associated electronics. Other transmission line matching
techniques – and in particular, single stub matching (Section 3.23) – typically result in structures which are significantly smaller.
The impedance inversion property of quarter-wavelength lines has applications beyond impedance matching. The following example demonstrates one such application:
Transistor amplifiers for RF applications often receive DC current at the same terminal which delivers the amplified RF signal, as shown in Figure \(\PageIndex{3}\). The power supply typically has a
low output impedance. If the power supply is directly connected to the transistor, then the RF will flow predominantly in the direction of the power supply as opposed to following the desired path,
which exhibits a higher impedance. This can be addressed using an inductor in series with the power supply output. This works because the inductor exhibits low impedance at DC and high impedance at
RF. Unfortunately, discrete inductors are often not practical at high RF frequencies. This is because practical inductors also exhibit parallel capacitance, which tends to decrease impedance.
A solution is to replace the inductor with a transmission line having length \(\lambda/4\) as shown in Figure \(\PageIndex{4}\). A wavelength at DC is infinite, so the transmission line is
essentially transparent to the power supply. At radio frequencies, the line transforms the low impedance of the power supply to an impedance that is very large relative to the impedance of the
desired RF path. Furthermore, transmission lines on printed circuit boards are much cheaper than discrete inductors (and are always in stock!).
Additional Reading:
• “Quarter-wavelength impedance transformer” on Wikipedia.
• “Smith chart” on Wikipedia.
• “Microstrip antenna” on Wikipedia. | {"url":"https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_I_(Ellingson)/03%3A_Transmission_Lines/3.19%3A_Quarter-Wavelength_Transmission_Line","timestamp":"2024-11-07T02:28:51Z","content_type":"text/html","content_length":"140110","record_id":"<urn:uuid:b7ee5df6-ce6c-4476-a18e-3b24a631c784>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00173.warc.gz"} |
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If you find this tool useful, please share it with others! | {"url":"https://toolxy.com/fraction-calculator","timestamp":"2024-11-04T03:50:31Z","content_type":"text/html","content_length":"27502","record_id":"<urn:uuid:d6b72e77-9c25-4d52-9096-ce452acb1e40>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00658.warc.gz"} |
RFC 6049: Spatial Composition of Metrics
Internet Engineering Task Force (IETF) A. Morton
Request for Comments: 6049 AT&T Labs
Category: Standards Track E. Stephan
ISSN: 2070-1721 France Telecom Orange
January 2011
Spatial Composition of Metrics
This memo utilizes IP performance metrics that are applicable to both
complete paths and sub-paths, and it defines relationships to compose
a complete path metric from the sub-path metrics with some accuracy
with regard to the actual metrics. This is called "spatial
composition" in RFC 2330. The memo refers to the framework for
metric composition, and provides background and motivation for
combining metrics to derive others. The descriptions of several
composed metrics and statistics follow.
Status of This Memo
This is an Internet Standards Track document.
This document is a product of the Internet Engineering Task Force
(IETF). It represents the consensus of the IETF community. It has
received public review and has been approved for publication by the
Internet Engineering Steering Group (IESG). Further information on
Internet Standards is available in Section 2 of RFC 5741.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
Morton & Stephan Standards Track [Page 1]
RFC 6049 Spatial Composition January 2011
Copyright Notice
Copyright (c) 2011 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
(http://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
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to this document. Code Components extracted from this document must
include Simplified BSD License text as described in Section 4.e of
the Trust Legal Provisions and are provided without warranty as
described in the Simplified BSD License.
This document may contain material from IETF Documents or IETF
Contributions published or made publicly available before November
10, 2008. The person(s) controlling the copyright in some of this
material may not have granted the IETF Trust the right to allow
modifications of such material outside the IETF Standards Process.
Without obtaining an adequate license from the person(s) controlling
the copyright in such materials, this document may not be modified
outside the IETF Standards Process, and derivative works of it may
not be created outside the IETF Standards Process, except to format
it for publication as an RFC or to translate it into languages other
than English.
Morton & Stephan Standards Track [Page 2]
RFC 6049 Spatial Composition January 2011
Table of Contents
1. Introduction ....................................................4
1.1. Motivation .................................................6
1.2. Requirements Language ......................................6
2. Scope and Application ...........................................6
2.1. Scope of Work ..............................................6
2.2. Application ................................................7
2.3. Incomplete Information .....................................7
3. Common Specifications for Composed Metrics ......................8
3.1. Name: Type-P ...............................................8
3.1.1. Metric Parameters ...................................8
3.1.2. Definition and Metric Units .........................9
3.1.3. Discussion and Other Details ........................9
3.1.4. Statistic ...........................................9
3.1.5. Composition Function ................................9
3.1.6. Statement of Conjecture and Assumptions ............10
3.1.7. Justification of the Composition Function ..........10
3.1.8. Sources of Deviation from the Ground Truth .........10
3.1.9. Specific Cases where the Conjecture Might Fail .....11
3.1.10. Application of Measurement Methodology ............12
4. One-Way Delay Composed Metrics and Statistics ..................12
4.1. Name: Type-P-Finite-One-way-Delay-<Sample>-Stream .........12
4.1.1. Metric Parameters ..................................12
4.1.2. Definition and Metric Units ........................12
4.1.3. Discussion and Other Details .......................13
4.1.4. Statistic ..........................................13
4.2. Name: Type-P-Finite-Composite-One-way-Delay-Mean ..........13
4.2.1. Metric Parameters ..................................13
4.2.2. Definition and Metric Units of the Mean Statistic ..14
4.2.3. Discussion and Other Details .......................14
4.2.4. Statistic ..........................................14
4.2.5. Composition Function: Sum of Means .................14
4.2.6. Statement of Conjecture and Assumptions ............15
4.2.7. Justification of the Composition Function ..........15
4.2.8. Sources of Deviation from the Ground Truth .........15
4.2.9. Specific Cases where the Conjecture Might Fail .....15
4.2.10. Application of Measurement Methodology ............16
4.3. Name: Type-P-Finite-Composite-One-way-Delay-Minimum .......16
4.3.1. Metric Parameters ..................................16
4.3.2. Definition and Metric Units of the Minimum
Statistic ..........................................16
4.3.3. Discussion and Other Details .......................16
4.3.4. Statistic ..........................................16
4.3.5. Composition Function: Sum of Minima ................16
4.3.6. Statement of Conjecture and Assumptions ............17
4.3.7. Justification of the Composition Function ..........17
4.3.8. Sources of Deviation from the Ground Truth .........17
Morton & Stephan Standards Track [Page 3]
RFC 6049 Spatial Composition January 2011
4.3.9. Specific Cases where the Conjecture Might Fail .....17
4.3.10. Application of Measurement Methodology ............17
5. Loss Metrics and Statistics ....................................18
5.1. Type-P-Composite-One-way-Packet-Loss-Empirical-Probability 18
5.1.1. Metric Parameters ..................................18
5.1.2. Definition and Metric Units ........................18
5.1.3. Discussion and Other Details .......................18
5.1.4. Statistic:
Type-P-One-way-Packet-Loss-Empirical-Probability ...18
5.1.5. Composition Function: Composition of
Empirical Probabilities ............................18
5.1.6. Statement of Conjecture and Assumptions ............19
5.1.7. Justification of the Composition Function ..........19
5.1.8. Sources of Deviation from the Ground Truth .........19
5.1.9. Specific Cases where the Conjecture Might Fail .....19
5.1.10. Application of Measurement Methodology ............19
6. Delay Variation Metrics and Statistics .........................20
6.1. Name: Type-P-One-way-pdv-refmin-<Sample>-Stream ...........20
6.1.1. Metric Parameters ..................................20
6.1.2. Definition and Metric Units ........................20
6.1.3. Discussion and Other Details .......................21
6.1.4. Statistics: Mean, Variance, Skewness, Quantile .....21
6.1.5. Composition Functions ..............................22
6.1.6. Statement of Conjecture and Assumptions ............23
6.1.7. Justification of the Composition Function ..........23
6.1.8. Sources of Deviation from the Ground Truth .........23
6.1.9. Specific Cases where the Conjecture Might Fail .....24
6.1.10. Application of Measurement Methodology ............24
7. Security Considerations ........................................24
7.1. Denial-of-Service Attacks .................................24
7.2. User Data Confidentiality .................................24
7.3. Interference with the Metrics .............................24
8. IANA Considerations ............................................25
9. Contributors and Acknowledgements ..............................27
10. References ....................................................28
10.1. Normative References .....................................28
10.2. Informative References ...................................28
1. Introduction
The IP Performance Metrics (IPPM) framework [RFC2330] describes two
forms of metric composition: spatial and temporal. The composition
framework [RFC5835] expands and further qualifies these original
forms into three categories. This memo describes spatial
composition, one of the categories of metrics under the umbrella of
the composition framework.
Morton & Stephan Standards Track [Page 4]
RFC 6049 Spatial Composition January 2011
Spatial composition encompasses the definition of performance metrics
that are applicable to a complete path, based on metrics collected on
various sub-paths.
The main purpose of this memo is to define the deterministic
functions that yield the complete path metrics using metrics of the
sub-paths. The effectiveness of such metrics is dependent on their
usefulness in analysis and applicability with practical measurement
The relationships may involve conjecture, and [RFC2330] lists four
points that the metric definitions should include:
o the specific conjecture applied to the metric and assumptions of
the statistical model of the process being measured (if any; see
[RFC2330], Section 12),
o a justification of the practical utility of the composition in
terms of making accurate measurements of the metric on the path,
o a justification of the usefulness of the composition in terms of
making analysis of the path using A-frame concepts more effective,
o an analysis of how the conjecture could be incorrect.
Also, [RFC2330] gives an example using the conjecture that the delay
of a path is very nearly the sum of the delays of the exchanges and
clouds of the corresponding path digest. This example is
particularly relevant to those who wish to assess the performance of
an inter-domain path without direct measurement, and the performance
estimate of the complete path is related to the measured results for
various sub-paths instead.
Approximate functions between the sub-path and complete path metrics
are useful, with knowledge of the circumstances where the
relationships are/are not applicable. For example, we would not
expect that delay singletons from each sub-path would sum to produce
an accurate estimate of a delay singleton for the complete path
(unless all the delays were essentially constant -- very unlikely).
However, other delay statistics (based on a reasonable sample size)
may have a sufficiently large set of circumstances where they are
Morton & Stephan Standards Track [Page 5]
RFC 6049 Spatial Composition January 2011
1.1. Motivation
One-way metrics defined in other RFCs (such as [RFC2679] and
[RFC2680]) all assume that the measurement can be practically carried
out between the source and the destination of interest. Sometimes
there are reasons that the measurement cannot be executed from the
source to the destination. For instance, the measurement path may
cross several independent domains that have conflicting policies,
measurement tools and methods, and measurement time assignment. The
solution then may be the composition of several sub-path
measurements. This means each domain performs the one-way
measurement on a sub-path between two nodes that are involved in the
complete path, following its own policy, using its own measurement
tools and methods, and using its own measurement timing. Under the
appropriate conditions, one can combine the sub-path one-way metric
results to estimate the complete path one-way measurement metric with
some degree of accuracy.
1.2. Requirements Language
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in RFC 2119 [RFC2119].
In this memo, the characters "<=" should be read as "less than or
equal to" and ">=" as "greater than or equal to".
2. Scope and Application
2.1. Scope of Work
For the primary IP Performance Metrics RFCs for loss [RFC2680], delay
[RFC2679], and delay variation [RFC3393], this memo gives a set of
metrics that can be composed from the same or similar sub-path
metrics. This means that the composition function may utilize:
o the same metric for each sub-path;
o multiple metrics for each sub-path (possibly one that is the same
as the complete path metric);
o a single sub-path metric that is different from the complete path
o different measurement techniques like active [RFC2330], [RFC3432]
and passive [RFC5474].
Morton & Stephan Standards Track [Page 6]
RFC 6049 Spatial Composition January 2011
We note a possibility: using a complete path metric and all but one
sub-path metric to infer the performance of the missing sub-path,
especially when the "last" sub-path metric is missing. However, such
de-composition calculations, and the corresponding set of issues they
raise, are beyond the scope of this memo.
2.2. Application
The composition framework [RFC5835] requires the specification of the
applicable circumstances for each metric. In particular, each
section addresses whether the metric:
o Requires the same test packets to traverse all sub-paths or may
use similar packets sent and collected separately in each
o Requires homogeneity of measurement methodologies or can allow a
degree of flexibility (e.g., active, active spatial division
[RFC5644], or passive methods produce the "same" metric). Also,
the applicable sending streams will be specified, such as Poisson,
Periodic, or both.
o Needs information or access that will only be available within an
operator's domain, or is applicable to inter-domain composition.
o Requires synchronized measurement start and stop times in all
sub-paths or largely overlapping measurement intervals, or no
timing requirements.
o Requires the assumption of sub-path independence with regard to
the metric being defined/composed or other assumptions.
o Has known sources of inaccuracy/error and identifies the sources.
2.3. Incomplete Information
In practice, when measurements cannot be initiated on a sub-path (and
perhaps the measurement system gives up during the test interval),
then there will not be a value for the sub-path reported, and the
entire test result SHOULD be recorded as "undefined". This case
should be distinguished from the case where the measurement system
continued to send packets throughout the test interval, but all were
declared lost.
When a composed metric requires measurements from sub-paths A, B, and
C, and one or more of the sub-path results are undefined, then the
composed metric SHOULD also be recorded as undefined.
Morton & Stephan Standards Track [Page 7]
RFC 6049 Spatial Composition January 2011
3. Common Specifications for Composed Metrics
To reduce the redundant information presented in the detailed metrics
sections that follow, this section presents the specifications that
are common to two or more metrics. The section is organized using
the same subsections as the individual metrics, to simplify
Also, the index variables are represented as follows:
o m = index for packets sent.
o n = index for packets received.
o s = index for involved sub-paths.
3.1. Name: Type-P
All metrics use the "Type-P" convention as described in [RFC2330].
The rest of the name is unique to each metric.
3.1.1. Metric Parameters
o Src, the IP address of a host.
o Dst, the IP address of a host.
o T, a time (start of test interval).
o Tf, a time (end of test interval).
o lambda, a rate in reciprocal seconds (for Poisson Streams).
o incT, the nominal duration of inter-packet interval, first bit to
first bit (for Periodic Streams).
o dT, the duration of the allowed interval for Periodic Stream
sample start times.
o T0, a time that MUST be selected at random from the interval
[T, T + dT] to start generating packets and taking measurements
(for Periodic Streams).
o TstampSrc, the wire time of the packet as measured at MP(Src)
(measurement point at the source).
o TstampDst, the wire time of the packet as measured at MP(Dst),
assigned to packets that arrive within a "reasonable" time.
Morton & Stephan Standards Track [Page 8]
RFC 6049 Spatial Composition January 2011
o Tmax, a maximum waiting time for packets at the destination, set
sufficiently long to disambiguate packets with long delays from
packets that are discarded (lost); thus, the distribution of delay
is not truncated.
o M, the total number of packets sent between T0 and Tf.
o N, the total number of packets received at Dst (sent between T0
and Tf).
o S, the number of sub-paths involved in the complete Src-Dst path.
o Type-P, as defined in [RFC2330], which includes any field that may
affect a packet's treatment as it traverses the network.
In metric names, the term "<Sample>" is intended to be replaced by
the name of the method used to define a sample of values of parameter
TstampSrc. This can be done in several ways, including:
1. Poisson: a pseudo-random Poisson process of rate lambda, whose
values fall between T and Tf. The time interval between
successive values of TstampSrc will then average 1/lambda, as per
2. Periodic: a Periodic stream process with pseudo-random start time
T0 between T and dT, and nominal inter-packet interval incT, as
per [RFC3432].
3.1.2. Definition and Metric Units
This section is unique for every metric.
3.1.3. Discussion and Other Details
This section is unique for every metric.
3.1.4. Statistic
This section is unique for every metric.
3.1.5. Composition Function
This section is unique for every metric.
Morton & Stephan Standards Track [Page 9]
RFC 6049 Spatial Composition January 2011
3.1.6. Statement of Conjecture and Assumptions
This section is unique for each metric. The term "ground truth" is
frequently used in these sections and is defined in Section 4.7 of
3.1.7. Justification of the Composition Function
It is sometimes impractical to conduct active measurements between
every Src-Dst pair. Since the full mesh of N measurement points
grows as N x N, the scope of measurement may be limited by testing
There may be varying limitations on active testing in different parts
of the network. For example, it may not be possible to collect the
desired sample size in each test interval when access link speed is
limited, because of the potential for measurement traffic to degrade
the user traffic performance. The conditions on a low-speed access
link may be understood well enough to permit use of a small sample
size/rate, while a larger sample size/rate may be used on other
Also, since measurement operations have a real monetary cost, there
is value in re-using measurements where they are applicable, rather
than launching new measurements for every possible source-destination
3.1.8. Sources of Deviation from the Ground Truth
3.1.8.1. Sub-Path List Differs from Complete Path
The measurement packets, each having source and destination addresses
intended for collection at edges of the sub-path, may take a
different specific path through the network equipment and links when
compared to packets with the source and destination addresses of the
complete path. Example sources of parallel paths include Equal Cost
Multi-Path and parallel (or bundled) links. Therefore, the
performance estimated from the composition of sub-path measurements
may differ from the performance experienced by packets on the
complete path. Multiple measurements employing sufficient sub-path
address pairs might produce bounds on the extent of this error.
We also note the possibility of re-routing during a measurement
interval, as it may affect the correspondence between packets
traversing the complete path and the sub-paths that were "involved"
prior to the re-route.
Morton & Stephan Standards Track [Page 10]
RFC 6049 Spatial Composition January 2011
3.1.8.2. Sub-Path Contains Extra Network Elements
Related to the case of an alternate path described above is the case
where elements in the measured path are unique to measurement system
connectivity. For example, a measurement system may use a dedicated
link to a LAN switch, and packets on the complete path do not
traverse that link. The performance of such a dedicated link would
be measured continuously, and its contribution to the sub-path
metrics SHOULD be minimized as a source of error.
3.1.8.3. Sub-Paths Have Incomplete Coverage
Measurements of sub-path performance may not cover all the network
elements on the complete path. For example, the network exchange
points might be excluded unless a cooperative measurement is
conducted. In this example, test packets on the previous sub-path
are received just before the exchange point, and test packets on the
next sub-path are injected just after the same exchange point.
Clearly, the set of sub-path measurements SHOULD cover all critical
network elements in the complete path.
3.1.8.4. Absence of Route
At a specific point in time, no viable route exists between the
complete path source and destination. The routes selected for one or
more sub-paths therefore differ from the complete path.
Consequently, spatial composition may produce finite estimation of a
ground truth metric (see Section 4.7 of [RFC5835]) between a source
and a destination, even when the route between them is undefined.
3.1.9. Specific Cases where the Conjecture Might Fail
This section is unique for most metrics (see the metric-specific
For delay-related metrics, one-way delay always depends on packet
size and link capacity, since it is measured in [RFC2679] from first
bit to last bit. If the size of an IP packet changes on its route
(due to encapsulation), this can influence delay performance.
However, the main error source may be the additional processing
associated with encapsulation and encryption/decryption if not
experienced or accounted for in sub-path measurements.
Fragmentation is a major issue for composition accuracy, since all
metrics require all fragments to arrive before proceeding, and
fragmented complete path performance is likely to be different from
performance with non-fragmented packets and composed metrics based on
non-fragmented sub-path measurements.
Morton & Stephan Standards Track [Page 11]
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Highly manipulated routing can cause measurement error if not
expected and compensated for. For example, policy-based MPLS routing
could modify the class of service for the sub-paths and complete
3.1.10. Application of Measurement Methodology
o The methodology SHOULD use similar packets sent and collected
separately in each sub-path, where "similar" in this case means
that Type-P contains as many equal attributes as possible, while
recognizing that there will be differences. Note that Type-P
includes stream characteristics (e.g., Poisson, Periodic).
o The methodology allows a degree of flexibility regarding test
stream generation (e.g., active or passive methods can produce an
equivalent result, but the lack of control over the source,
timing, and correlation of passive measurements is much more
o Poisson and/or Periodic streams are RECOMMENDED.
o The methodology applies to both inter-domain and intra-domain
o The methodology SHOULD have synchronized measurement time
intervals in all sub-paths, but largely overlapping intervals MAY
o Assumption of sub-path independence with regard to the metric
being defined/composed is REQUIRED.
4. One-Way Delay Composed Metrics and Statistics
4.1. Name: Type-P-Finite-One-way-Delay-<Sample>-Stream
This metric is a necessary element of delay composition metrics, and
its definition does not formally exist elsewhere in IPPM literature.
4.1.1. Metric Parameters
See the common parameters section (Section 3.1.1).
4.1.2. Definition and Metric Units
Using the parameters above, we obtain the value of the Type-P-One-
way-Delay singleton as per [RFC2679].
Morton & Stephan Standards Track [Page 12]
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For each packet "[i]" that has a finite one-way delay (in other
words, excluding packets that have undefined one-way delay):
Type-P-Finite-One-way-Delay-<Sample>-Stream[i] =
FiniteDelay[i] = TstampDst - TstampSrc
This metric is measured in units of time in seconds, expressed in
sufficiently low resolution to convey meaningful quantitative
information. For example, resolution of microseconds is usually
4.1.3. Discussion and Other Details
The "Type-P-Finite-One-way-Delay" metric permits calculation of the
sample mean statistic. This resolves the problem of including lost
packets in the sample (whose delay is undefined) and the issue with
the informal assignment of infinite delay to lost packets (practical
systems can only assign some very large value).
The Finite-One-way-Delay approach handles the problem of lost packets
by reducing the event space. We consider conditional statistics, and
estimate the mean one-way delay conditioned on the event that all
packets in the sample arrive at the destination (within the specified
waiting time, Tmax). This offers a way to make some valid statements
about one-way delay, at the same time avoiding events with undefined
outcomes. This approach is derived from the treatment of lost
packets in [RFC3393], and is similar to [Y.1540].
4.1.4. Statistic
All statistics defined in [RFC2679] are applicable to the finite one-
way delay, and additional metrics are possible, such as the mean (see
4.2. Name: Type-P-Finite-Composite-One-way-Delay-Mean
This section describes a statistic based on the Type-P-Finite-One-
way-Delay-<Sample>-Stream metric.
4.2.1. Metric Parameters
See the common parameters section (Section 3.1.1).
Morton & Stephan Standards Track [Page 13]
RFC 6049 Spatial Composition January 2011
4.2.2. Definition and Metric Units of the Mean Statistic
We define
Type-P-Finite-One-way-Delay-Mean =
1 \
MeanDelay = - * > (FiniteDelay [n])
N /
n = 1
where all packets n = 1 through N have finite singleton delays.
This metric is measured in units of time in seconds, expressed in
sufficiently fine resolution to convey meaningful quantitative
information. For example, resolution of microseconds is usually
4.2.3. Discussion and Other Details
The Type-P-Finite-One-way-Delay-Mean metric requires the conditional
delay distribution described in Section 4.1.3.
4.2.4. Statistic
This metric, a mean, does not require additional statistics.
4.2.5. Composition Function: Sum of Means
The Type-P-Finite-Composite-One-way-Delay-Mean, or CompMeanDelay, for
the complete source to destination path can be calculated from the
sum of the mean delays of all of its S constituent sub-paths.
Morton & Stephan Standards Track [Page 14]
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Then the
Type-P-Finite-Composite-One-way-Delay-Mean =
CompMeanDelay = > (MeanDelay [s])
s = 1
where sub-paths s = 1 to S are involved in the complete path.
4.2.6. Statement of Conjecture and Assumptions
The mean of a sufficiently large stream of packets measured on each
sub-path during the interval [T, Tf] will be representative of the
ground truth mean of the delay distribution (and the distributions
themselves are sufficiently independent), such that the means may be
added to produce an estimate of the complete path mean delay.
It is assumed that the one-way delay distributions of the sub-paths
and the complete path are continuous. The mean of multi-modal
distributions has the unfortunate property that such a value may
never occur.
4.2.7. Justification of the Composition Function
See the common section (Section 3).
4.2.8. Sources of Deviation from the Ground Truth
See the common section (Section 3).
4.2.9. Specific Cases where the Conjecture Might Fail
If any of the sub-path distributions are multi-modal, then the
measured means may not be stable, and in this case the mean will not
be a particularly useful statistic when describing the delay
distribution of the complete path.
The mean may not be a sufficiently robust statistic to produce a
reliable estimate, or to be useful even if it can be measured.
If a link contributing non-negligible delay is erroneously included
or excluded, the composition will be in error.
Morton & Stephan Standards Track [Page 15]
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4.2.10. Application of Measurement Methodology
The requirements of the common section (Section 3) apply here as
4.3. Name: Type-P-Finite-Composite-One-way-Delay-Minimum
This section describes a statistic based on the Type-P-Finite-One-
way-Delay-<Sample>-Stream metric, and the composed metric based on
that statistic.
4.3.1. Metric Parameters
See the common parameters section (Section 3.1.1).
4.3.2. Definition and Metric Units of the Minimum Statistic
We define
Type-P-Finite-One-way-Delay-Minimum =
MinDelay = (FiniteDelay [j])
such that for some index, j, where 1 <= j <= N
FiniteDelay[j] <= FiniteDelay[n] for all n
where all packets n = 1 through N have finite singleton delays.
This metric is measured in units of time in seconds, expressed in
sufficiently fine resolution to convey meaningful quantitative
information. For example, resolution of microseconds is usually
4.3.3. Discussion and Other Details
The Type-P-Finite-One-way-Delay-Minimum metric requires the
conditional delay distribution described in Section 4.1.3.
4.3.4. Statistic
This metric, a minimum, does not require additional statistics.
4.3.5. Composition Function: Sum of Minima
The Type-P-Finite-Composite-One-way-Delay-Minimum, or CompMinDelay,
for the complete source to destination path can be calculated from
the sum of the minimum delays of all of its S constituent sub-paths.
Morton & Stephan Standards Track [Page 16]
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Then the
Type-P-Finite-Composite-One-way-Delay-Minimum =
CompMinDelay = > (MinDelay [s])
s = 1
4.3.6. Statement of Conjecture and Assumptions
The minimum of a sufficiently large stream of packets measured on
each sub-path during the interval [T, Tf] will be representative of
the ground truth minimum of the delay distribution (and the
distributions themselves are sufficiently independent), such that the
minima may be added to produce an estimate of the complete path
minimum delay.
It is assumed that the one-way delay distributions of the sub-paths
and the complete path are continuous.
4.3.7. Justification of the Composition Function
See the common section (Section 3).
4.3.8. Sources of Deviation from the Ground Truth
See the common section (Section 3).
4.3.9. Specific Cases where the Conjecture Might Fail
If the routing on any of the sub-paths is not stable, then the
measured minimum may not be stable. In this case the composite
minimum would tend to produce an estimate for the complete path that
may be too low for the current path.
4.3.10. Application of Measurement Methodology
The requirements of the common section (Section 3) apply here as
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5. Loss Metrics and Statistics
5.1. Type-P-Composite-One-way-Packet-Loss-Empirical-Probability
5.1.1. Metric Parameters
See the common parameters section (Section 3.1.1).
5.1.2. Definition and Metric Units
Using the parameters above, we obtain the value of the Type-P-One-
way-Packet-Loss singleton and stream as per [RFC2680].
We obtain a sequence of pairs with elements as follows:
o TstampSrc, as above.
o L, either zero or one, where L = 1 indicates loss and L = 0
indicates arrival at the destination within TstampSrc + Tmax.
5.1.3. Discussion and Other Details
5.1.4. Statistic: Type-P-One-way-Packet-Loss-Empirical-Probability
Given the stream parameter M, the number of packets sent, we can
define the Empirical Probability of Loss Statistic (Ep), consistent
with average loss in [RFC2680], as follows:
Type-P-One-way-Packet-Loss-Empirical-Probability =
1 \
Ep = - * > (L[m])
M /
m = 1
where all packets m = 1 through M have a value for L.
5.1.5. Composition Function: Composition of Empirical Probabilities
The Type-P-One-way-Composite-Packet-Loss-Empirical-Probability, or
CompEp, for the complete source to destination path can be calculated
by combining the Ep of all of its constituent sub-paths (Ep1, Ep2,
Ep3, ... Epn) as
Morton & Stephan Standards Track [Page 18]
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Type-P-Composite-One-way-Packet-Loss-Empirical-Probability =
CompEp = 1 - {(1 - Ep1) x (1 - Ep2) x (1 - Ep3) x ... x (1 - EpS)}
If any Eps is undefined in a particular measurement interval,
possibly because a measurement system failed to report a value, then
any CompEp that uses sub-path s for that measurement interval is
5.1.6. Statement of Conjecture and Assumptions
The empirical probability of loss calculated on a sufficiently large
stream of packets measured on each sub-path during the interval
[T, Tf] will be representative of the ground truth empirical loss
probability (and the probabilities themselves are sufficiently
independent), such that the sub-path probabilities may be combined to
produce an estimate of the complete path empirical loss probability.
5.1.7. Justification of the Composition Function
See the common section (Section 3).
5.1.8. Sources of Deviation from the Ground Truth
See the common section (Section 3).
5.1.9. Specific Cases where the Conjecture Might Fail
A concern for loss measurements combined in this way is that root
causes may be correlated to some degree.
For example, if the links of different networks follow the same
physical route, then a single catastrophic event like a fire in a
tunnel could cause an outage or congestion on remaining paths in
multiple networks. Here it is important to ensure that measurements
before the event and after the event are not combined to estimate the
composite performance.
Or, when traffic volumes rise due to the rapid spread of an email-
borne worm, loss due to queue overflow in one network may help
another network to carry its traffic without loss.
5.1.10. Application of Measurement Methodology
See the common section (Section 3).
Morton & Stephan Standards Track [Page 19]
RFC 6049 Spatial Composition January 2011
6. Delay Variation Metrics and Statistics
6.1. Name: Type-P-One-way-pdv-refmin-<Sample>-Stream
This packet delay variation (PDV) metric is a necessary element of
Composed Delay Variation metrics, and its definition does not
formally exist elsewhere in IPPM literature (with the exception of
6.1.1. Metric Parameters
In addition to the parameters of Section 3.1.1:
o TstampSrc[i], the wire time of packet[i] as measured at MP(Src)
(measurement point at the source).
o TstampDst[i], the wire time of packet[i] as measured at MP(Dst),
assigned to packets that arrive within a "reasonable" time.
o B, a packet length in bits.
o F, a selection function unambiguously defining the packets from
the stream that are selected for the packet-pair computation of
this metric. F(current packet), the first packet of the pair,
MUST have a valid Type-P-Finite-One-way-Delay less than Tmax (in
other words, excluding packets that have undefined one-way delay)
and MUST have been transmitted during the interval [T, Tf]. The
second packet in the pair, F(min_delay packet) MUST be the packet
with the minimum valid value of Type-P-Finite-One-way-Delay for
the stream, in addition to the criteria for F(current packet). If
multiple packets have equal minimum Type-P-Finite-One-way-Delay
values, then the value for the earliest arriving packet SHOULD be
o MinDelay, the Type-P-Finite-One-way-Delay value for F(min_delay
packet) given above.
o N, the number of packets received at the destination that meet the
F(current packet) criteria.
6.1.2. Definition and Metric Units
Using the definition above in Section 5.1.2, we obtain the value of
Type-P-Finite-One-way-Delay-<Sample>-Stream[n], the singleton for
each packet[i] in the stream (a.k.a. FiniteDelay[i]).
Morton & Stephan Standards Track [Page 20]
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For each packet[n] that meets the F(first packet) criteria given
above: Type-P-One-way-pdv-refmin-<Sample>-Stream[n] =
PDV[n] = FiniteDelay[n] - MinDelay
where PDV[i] is in units of time in seconds, expressed in
sufficiently fine resolution to convey meaningful quantitative
information. For example, resolution of microseconds is usually
6.1.3. Discussion and Other Details
This metric produces a sample of delay variation normalized to the
minimum delay of the sample. The resulting delay variation
distribution is independent of the sending sequence (although
specific FiniteDelay values within the distribution may be
correlated, depending on various stream parameters such as packet
spacing). This metric is equivalent to the IP Packet Delay Variation
parameter defined in [Y.1540].
6.1.4. Statistics: Mean, Variance, Skewness, Quantile
We define the mean PDV as follows (where all packets n = 1 through N
have a value for PDV[n]):
Type-P-One-way-pdv-refmin-Mean = MeanPDV =
1 \
- * > (PDV[n])
N /
n = 1
We define the variance of PDV as follows:
Type-P-One-way-pdv-refmin-Variance = VarPDV =
1 \ 2
------- > (PDV[n] - MeanPDV)
(N - 1) /
n = 1
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We define the skewness of PDV as follows:
Type-P-One-way-pdv-refmin-Skewness = SkewPDV =
--- 3
\ / \
> | PDV[n] - MeanPDV |
/ \ /
n = 1
/ \
| ( 3/2 ) |
\ (N - 1) * VarPDV /
(See Appendix X of [Y.1541] for additional background information.)
We define the quantile of the PDV sample as the value where the
specified fraction of singletons is less than the given value.
6.1.5. Composition Functions
This section gives two alternative composition functions. The
objective is to estimate a quantile of the complete path delay
variation distribution. The composed quantile will be estimated
using information from the sub-path delay variation distributions.
6.1.5.1. Approximate Convolution
The Type-P-Finite-One-way-Delay-<Sample>-Stream samples from each
sub-path are summarized as a histogram with 1-ms bins representing
the one-way delay distribution.
From [STATS], the distribution of the sum of independent random
variables can be derived using the relation:
Type-P-Composite-One-way-pdv-refmin-quantile-a =
. .
/ /
P(X + Y + Z <= a) = | | P(X <= a - y - z) * P(Y = y) * P(Z = z) dy dz
/ /
` `
z y
Morton & Stephan Standards Track [Page 22]
RFC 6049 Spatial Composition January 2011
Note that dy and dz indicate partial integration above, and that y
and z are the integration variables. Also, the probability of an
outcome is indicated by the symbol P(outcome), where X, Y, and Z are
random variables representing the delay variation distributions of
the sub-paths of the complete path (in this case, there are three
sub-paths), and "a" is the quantile of interest.
This relation can be used to compose a quantile of interest for the
complete path from the sub-path delay distributions. The histograms
with 1-ms bins are discrete approximations of the delay
6.1.5.2. Normal Power Approximation (NPA)
Type-P-One-way-Composite-pdv-refmin-NPA for the complete source to
destination path can be calculated by combining the statistics of all
the constituent sub-paths in the process described in [Y.1541],
Clause 8 and Appendix X.
6.1.6. Statement of Conjecture and Assumptions
The delay distribution of a sufficiently large stream of packets
measured on each sub-path during the interval [T, Tf] will be
sufficiently stationary, and the sub-path distributions themselves
are sufficiently independent, so that summary information describing
the sub-path distributions can be combined to estimate the delay
distribution of the complete path.
It is assumed that the one-way delay distributions of the sub-paths
and the complete path are continuous.
6.1.7. Justification of the Composition Function
See the common section (Section 3).
6.1.8. Sources of Deviation from the Ground Truth
In addition to the common deviations, a few additional sources exist
here. For one, very tight distributions with ranges on the order of
a few milliseconds are not accurately represented by a histogram with
1-ms bins. This size was chosen assuming an implicit requirement on
accuracy: errors of a few milliseconds are acceptable when assessing
a composed distribution quantile.
Also, summary statistics cannot describe the subtleties of an
empirical distribution exactly, especially when the distribution is
very different from a classical form. Any procedure that uses these
statistics alone may incur error.
Morton & Stephan Standards Track [Page 23]
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6.1.9. Specific Cases where the Conjecture Might Fail
If the delay distributions of the sub-paths are somehow correlated,
then neither of these composition functions will be reliable
estimators of the complete path distribution.
In practice, sub-path delay distributions with extreme outliers have
increased the error of the composed metric estimate.
6.1.10. Application of Measurement Methodology
See the common section (Section 3).
7. Security Considerations
7.1. Denial-of-Service Attacks
This metric requires a stream of packets sent from one host (source)
to another host (destination) through intervening networks. This
method could be abused for denial-of-service attacks directed at the
destination and/or the intervening network(s).
Administrators of source, destination, and intervening networks
should establish bilateral or multilateral agreements regarding the
timing, size, and frequency of collection of sample metrics. Use of
this method in excess of the terms agreed upon between the
participants may be cause for immediate rejection or discarding of
packets, or other escalation procedures defined between the affected
7.2. User Data Confidentiality
Active use of this method generates packets for a sample, rather than
taking samples based on user data, and does not threaten user data
confidentiality. Passive measurement MUST restrict attention to the
headers of interest. Since user payloads may be temporarily stored
for length analysis, suitable precautions MUST be taken to keep this
information safe and confidential. In most cases, a hashing function
will produce a value suitable for payload comparisons.
7.3. Interference with the Metrics
It may be possible to identify that a certain packet or stream of
packets is part of a sample. With that knowledge at the destination
and/or the intervening networks, it is possible to change the
Morton & Stephan Standards Track [Page 24]
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processing of the packets (e.g., increasing or decreasing delay),
which may distort the measured performance. It may also be possible
to generate additional packets that appear to be part of the sample
metric. These additional packets are likely to perturb the results
of the sample measurement.
To discourage the kind of interference mentioned above, packet
interference checks, such as cryptographic hash, may be used.
8. IANA Considerations
Metrics defined in the IETF are typically registered in the IANA IPPM
Metrics Registry as described in the initial version of the registry
IANA has registered the following metrics in the
ietfFiniteOneWayDelayStream OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 4.1."
::= { ianaIppmMetrics 71 }
ietfFiniteOneWayDelayMean OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 4.2."
::= { ianaIppmMetrics 72 }
ietfCompositeOneWayDelayMean OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 4.2.5."
::= { ianaIppmMetrics 73 }
ietfFiniteOneWayDelayMinimum OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 4.3.2."
::= { ianaIppmMetrics 74 }
Morton & Stephan Standards Track [Page 25]
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ietfCompositeOneWayDelayMinimum OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 4.3."
::= { ianaIppmMetrics 75 }
ietfOneWayPktLossEmpiricProb OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 5.1.4"
::= { ianaIppmMetrics 76 }
ietfCompositeOneWayPktLossEmpiricProb OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 5.1."
::= { ianaIppmMetrics 77 }
ietfOneWayPdvRefminStream OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 6.1."
::= { ianaIppmMetrics 78 }
ietfOneWayPdvRefminMean OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 6.1.4."
::= { ianaIppmMetrics 79 }
ietfOneWayPdvRefminVariance OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 6.1.4."
::= { ianaIppmMetrics 80 }
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RFC 6049 Spatial Composition January 2011
ietfOneWayPdvRefminSkewness OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 6.1.4."
::= { ianaIppmMetrics 81 }
ietfCompositeOneWayPdvRefminQtil OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 6.1.5.1."
::= { ianaIppmMetrics 82 }
ietfCompositeOneWayPdvRefminNPA OBJECT-IDENTITY
STATUS current
REFERENCE "RFC 6049, Section 6.1.5.2."
::= { ianaIppmMetrics 83 }
9. Contributors and Acknowledgements
The following people have contributed useful ideas, suggestions, or
the text of sections that have been incorporated into this memo:
- Phil Chimento <vze275m9@verizon.net>
- Reza Fardid <RFardid@cariden.com>
- Roman Krzanowski <roman.krzanowski@verizon.com>
- Maurizio Molina <maurizio.molina@dante.org.uk>
- Lei Liang <L.Liang@surrey.ac.uk>
- Dave Hoeflin <dhoeflin@att.com>
A long time ago, in a galaxy far, far away (Minneapolis), Will Leland
suggested the simple and elegant Type-P-Finite-One-way-Delay concept.
Thanks Will.
Yaakov Stein and Donald McLachlan also provided useful comments along
the way.
Morton & Stephan Standards Track [Page 27]
RFC 6049 Spatial Composition January 2011
10. References
10.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[RFC2330] Paxson, V., Almes, G., Mahdavi, J., and M. Mathis,
"Framework for IP Performance Metrics", RFC 2330,
May 1998.
[RFC2679] Almes, G., Kalidindi, S., and M. Zekauskas, "A One-way
Delay Metric for IPPM", RFC 2679, September 1999.
[RFC2680] Almes, G., Kalidindi, S., and M. Zekauskas, "A One-way
Packet Loss Metric for IPPM", RFC 2680, September 1999.
[RFC3393] Demichelis, C. and P. Chimento, "IP Packet Delay Variation
Metric for IP Performance Metrics (IPPM)", RFC 3393,
November 2002.
[RFC3432] Raisanen, V., Grotefeld, G., and A. Morton, "Network
performance measurement with periodic streams", RFC 3432,
November 2002.
[RFC4148] Stephan, E., "IP Performance Metrics (IPPM) Metrics
Registry", BCP 108, RFC 4148, August 2005.
[RFC5835] Morton, A. and S. Van den Berghe, "Framework for Metric
Composition", RFC 5835, April 2010.
10.2. Informative References
[RFC5474] Duffield, N., Chiou, D., Claise, B., Greenberg, A.,
Grossglauser, M., and J. Rexford, "A Framework for Packet
Selection and Reporting", RFC 5474, March 2009.
[RFC5481] Morton, A. and B. Claise, "Packet Delay Variation
Applicability Statement", RFC 5481, March 2009.
[RFC5644] Stephan, E., Liang, L., and A. Morton, "IP Performance
Metrics (IPPM): Spatial and Multicast", RFC 5644,
October 2009.
[STATS] Mood, A., Graybill, F., and D. Boes, "Introduction to the
Theory of Statistics, 3rd Edition", McGraw-Hill, New York,
NY, 1974.
Morton & Stephan Standards Track [Page 28]
RFC 6049 Spatial Composition January 2011
[Y.1540] ITU-T Recommendation Y.1540, "Internet protocol data
communication service - IP packet transfer and
availability performance parameters", November 2007.
[Y.1541] ITU-T Recommendation Y.1541, "Network Performance
Objectives for IP-based Services", February 2006.
Authors' Addresses
Al Morton
AT&T Labs
200 Laurel Avenue South
Middletown, NJ 07748
Phone: +1 732 420 1571
Fax: +1 732 368 1192
EMail: acmorton@att.com
URI: http://home.comcast.net/~acmacm/
Stephan Emile
France Telecom Orange
2 avenue Pierre Marzin
Lannion, F-22307
EMail: emile.stephan@orange-ftgroup.com
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Distance Problems
Subject: Math
Lesson Length: 35 - 45 mins
Topic: Measurement and Data
Grade Level: 4
Standards / Framework:
Brief Description: Students will interpret data to write and solve a real-world word problem involving distance.
Know Before You Start: Students should be familiar with the components needed to craft a word problem.
• Read and discuss the sample comic. What data does the comic provide?
• Have students brainstorm and discuss what type of problems they could create and solve based on the data.
• Have students use the data in the comic to create a three-to-four-panel comic of their own.
• The comic should present a real-world distance word problem using any of the data from the sample comic.
□ Panel 1: Present the problem.
□ Panel 2-3: Explain how to solve the problem. Which of the four basic operations will be needed?
□ Panel 4: Solve.
• Have students share their word problems with a partner.
• Tell them to cover the last panel so their partner doesn’t see the answer.
• Their partner will now solve the problem.
• Allow students to use the speech-to-text feature.
• Allow students to use the voiceover feature to read their comics aloud.
• Allow students to work in pairs or groups as needed.
• Assign specific problems to students as necessary.
• Comic to print or display | {"url":"https://ideas.pixton.com/distance-problems","timestamp":"2024-11-02T09:33:38Z","content_type":"text/html","content_length":"33005","record_id":"<urn:uuid:24f05ec5-a620-4cb3-8e3d-f179ebda6157>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00674.warc.gz"} |
Handwritten Tensor Analysis Notes pdf lecture download 2023
Tensor Analysis Notes PDF
Free Tensor Analysis notes pdf are provided here for Tensor Analysis students so that they can prepare and score high marks in their Tensor Analysis exam.
In these free Tensor Analysis notes pdf, we will study the concept of generalized mathematical constructs in terms of Algebraic Structures (mainly Vector Spaces) and Tensors to have an in-depth
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Topics in our Tensor Analysis Notes PDF
The topics we will cover in these Tensor Analysis pdf Notes will be taken from the following list:
Vector Space and Subspace: Binary Operations, Groups, Rings & Fields, Vector Space & Subspace, Examples of Vector Spaces, Euclidean Vector Spaces: Length and Distance in Rn, Matrix notation for
vectors in Rn, Four Subspaces associated with a Matrix.
Basic and Dimension: Linear Dependence and Independence of vectors, Spanning a Space, Basis and Dimensions, Rank and Nullity of a Matrix, Examples from Real Function Space and Polynomial Space,
Orthogonal Vectors and Subspaces, Orthogonal Basis, Gram Schmidt process of generating an Orthonormal Basis.
Linear Transformation: Function and Mapping, General Linear Transformations and Examples, Kernel and Range of a Matrix Transformation, Homomorphism, and Isomorphism of vector space, Singular and
Non-singular Mapping/Transformations, Algebra of Linear operator.
Invertible operators: Identity Transformation, Matrices, and Linear Operators, Matrix Representation of a Linear transformation and change of basis, Similarity.
Matrices and Matrix Operations: Addition and Multiplication of Matrices, Null Matrices, Diagonal, Scalar and Unit Matrices, Upper Triangular and Lower-Triangular Matrices, Transpose of a Matrix,
Symmetric and Skew-Symmetric Matrices, Matrices for Networks, Matrix Multiplication and System of Linear Equations, Augmented Matrix, Echelon Matrices, Gauss Elimination and Gauss-Jordan Elimination,
Inverse of a Matrix, Elementary Matrix, Conjugate of a Matrix. Hermitian and Skew-Hermitian Matrices, Determinants, Evaluating Determinants by Row Reduction, Properties of Determinants, Adjoint of a
Matrix, Singular and Non-Singular matrices, Orthogonal Matrix, Unitary Matrices, Trace of a Matrix, Inner Product.
Eigen-values and Eigenvectors: Finding Eigen-values and Eigen vectors of a Matrice. Diagonalization of Matrices. Properties of Eigen-values and Eigen Vectors of Orthogonal, Hermetian and Unitary
Matrices. Cayley- Hamiliton Theorem (Statement only). Finding inverse of a matrix using Cayley-Hamiltion Theorem. Use of Matrices in Solving Coupled Linear Ordinary Differential Equations of first
order. Functions of a Matrix.
Cartesian Tensor: Transformation of co-ordinates, Einstein’s summation convention, Relation between Direction Cosines, Tensors, Algebra of Tensors: Sum, Difference and Product of Two Tensors.
Contraction, Quotient Law of Tensors, Symmetric and Anti-symmetric Tensors, Invariant Tensors: Kronecker and Alternating Tensors, Association of Antisymmetric Tensor of Order Two and Vectors. Vector
Algebra and calculus using Cartesian Tensors: Scalar and Vector Products of 2, 3, 4 vectors. Gradient, Divergence and Curl of Tensor Fields. Vector Identities. Tensorial Character of Physical
Quantities. Moment of Inertia Tensor. Stress and Strain Tensors: Symmetric Nature. Elasticity Tensor.Generalized Hooke’s Law.
Geometrical Applications: Equation of a line, Angle between lines. Projection of a line on another line. Condition for two lines to be coplanar. Foot of the Perpendicular from a Point on a Line,
Rotation Tensor, Isotropic tensors (definition only), Moment of Inertia tensors.
General Tensors: Transformation of Co-ordinates, Contravariant & Covariant Vectors, Contravariant, Covariant and Mixed Tensors, Kronecker Delta and Permutation Tensors, Algebra of Tensors, Sum,
Difference & Product of Two Tensors, Contraction, Quotient Law of Tensors, Symmetric and Anti- symmetric Tensors, Metric Tensor.
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Physics Notes
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Author: Delhi University | {"url":"https://www.tutorialsduniya.com/notes/linear-algebra-tensor-analysis-notes/","timestamp":"2024-11-06T13:23:48Z","content_type":"text/html","content_length":"114244","record_id":"<urn:uuid:d129162b-ab64-483a-81e2-cc00fca50542>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00668.warc.gz"} |
Position-Dependent Mass Quantum systems
SciPost Submission Page
Position-Dependent Mass Quantum systems and ADM formalism
by Davood Momeni
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Davood Momeni
Submission information
Preprint Link: https://arxiv.org/abs/2008.02113v2 (pdf)
Date accepted: 2021-01-08
Date submitted: 2020-10-29 18:44
Submitted by: Momeni, Davood
Submitted to: SciPost Physics Proceedings
Proceedings issue: 4th International Conference on Holography, String Theory and Discrete Approach in Hanoi (STRHAN2020)
Ontological classification
Academic field: Physics
• Gravitation, Cosmology and Astroparticle Physics
Specialties: • High-Energy Physics - Theory
• Quantum Physics
Approach: Theoretical
The classical Einstein-Hilbert (EH) action for general relativity (GR) is shown to be formally analogous to the classical system with position-dependent mass (PDM) models. The analogy is developed
and used to build the covariant classical Hamiltonian as well as defining an alternative phase portrait for GR. The set of associated Hamilton's equations in the phase space is presented as a
first-order system dual to the Einstein field equations. Following the principles of quantum mechanics, I build a canonical theory for the classical general. A fully consistent quantum Hamiltonian
for GR is constructed based on adopting a high dimensional phase space. It is observed that the functional wave equation is timeless. As a direct application, I present an alternative wave equation
for quantum cosmology. In comparison to the standard Arnowitt-Deser-Misner(ADM) decomposition and quantum gravity proposals, I extended my analysis beyond the covariant regime when the metric is
decomposed into the 3+1 dimensional ADM decomposition. I showed that an equal dimensional phase space can be obtained if one applies ADM decomposed metric.
Author comments upon resubmission
The Editor
SciPost Physics Proceedings
Dear Editor,
We thank the editor and the referee for giving us the opportunities once to address some questions/comments and helping us to improve our presentation. We have modified the paper to accommodate the
referee's suggestions accordingly. We have also corrected possible typos/errors.
We hope that the editor does appreciate our efforts to make sure that the paper is of highest quality. We are confident that we have now answered, in detail, to all the points raised by the referee
and henceforth resubmit the paper for publication.
Our replies to the referee's comments are given below.
Best regards,
[Comment-\#1]{The manuscript “Position-Dependent Mass Quantum systems and ADM formalism”, by Davood Momeni
tries to present a Hamiltonian formulation of general relativity, inspired by position-dependent mass systems, and then it proposes a quantum version of the theory.\\
The Hamiltonian formulation of general relativity has been discussed extensively in the literature for decades. Some references are given, but many important contributions are missing (one can find
them in textbooks or Review papers). In any case, the author should tone down his claims about his findings and their significance, and remove statements such as “this is the first time in literature
when a first order Hamiltonian version of the gravitational field equations.”}
[Answer-\#1]{We thank the referee for pointing out this remark. In this revision, we have carefully corrected this point as suggested . Please see the discussion after eq. (3.10).}
[Comment-\#2]{The author’s approach is based on the “super mass tensor”. However, the author defines this quantity as a derivative on the Lagrangian, and hence expression (2.5) is a loop definition.
[Answer-\#2]{We thank the referee for pointing out this remark. In this revision, we have carefully corrected this point as suggested . Please see the discussion after eqs. (2.5), (2.6).}
[Comment-\#3]{I cannot see how the complicated form of field equations (3.10) can be helpful.}
[Answer-\#3]{The mentioned Hamilton equations is considered as a covariant generalization of the momentum evolutionary equation in the standard ADM decomposition. The only difference is here we don't
specify special time foliation of the spacetime, i.e, no need to consider $x^{0}=constant$ hypersurfaces as it is very convenient in ADM method. A part of this foliastion freedom, as we showed that
the equation reduces to the momentum time equation if one adapts a preferred time foliation.}
[Comment-\#4]{The matter sector, which is crucial in GR since it is the source of non-trivial curvature and geometry, is missing from the discussion.}
[Answer-\#4]{We thank the referee for pointing out this remark. In this revision, we have carefully corrected this point as suggested . Please see the discussion after eq. (1.1).}
[Comment-\#5]{The author faces the problem as a simple quantum mechanical problem and not as a quantum field theoretical one, and thus the discussion on renormalizability etc is missing.}
[Answer-\#5]{We should emphasis here that theory which we studied in this paper is considered as an attempt to construct quantum mechanics on a classical GR background. There is no simple field
theoretic interpretation for the Hamiltonian which we obtained in this work as well as any other brackets are simply non quantum field theoretical one. In his approach, we can't reach the
renormalizability as it has been investigated in many other alternative quantum gravity scenarios.}
[Comment-\#6]{Solutions (5.7),(5.8) do not have an obvious meaning, and in any case it is strange that the author finds non-trivial structure in the absence of matter.}
[Answer-\#6]{We added a discussion about these equations after eqs.(5.8).}
[Comment-\#7]{The English of the manuscript need editing.\\
In summary, a radical modification is needed before I will be able to reconsider the manuscript for publication.}
[Answer-\#7]{We fixed several typos and improved the presentation of the paper..}
List of changes
[Comment-\#1]{The manuscript “Position-Dependent Mass Quantum systems and ADM formalism”, by Davood Momeni
tries to present a Hamiltonian formulation of general relativity, inspired by position-dependent mass systems, and then it proposes a quantum version of the theory.\\
The Hamiltonian formulation of general relativity has been discussed extensively in the literature for decades. Some references are given, but many important contributions are missing (one can find
them in textbooks or Review papers). In any case, the author should tone down his claims about his findings and their significance, and remove statements such as “this is the first time in literature
when a first order Hamiltonian version of the gravitational field equations.”}
[Answer-\#1]{We thank the referee for pointing out this remark. In this revision, we have carefully corrected this point as suggested . Please see the discussion after eq. (3.10).}
[Comment-\#2]{The author’s approach is based on the “super mass tensor”. However, the author defines this quantity as a derivative on the Lagrangian, and hence expression (2.5) is a loop definition.
[Answer-\#2]{We thank the referee for pointing out this remark. In this revision, we have carefully corrected this point as suggested . Please see the discussion after eqs. (2.5), (2.6).}
[Comment-\#3]{I cannot see how the complicated form of field equations (3.10) can be helpful.}
[Answer-\#3]{The mentioned Hamilton equations is considered as a covariant generalization of the momentum evolutionary equation in the standard ADM decomposition. The only difference is here we don't
specify special time foliation of the spacetime, i.e, no need to consider $x^{0}=constant$ hypersurfaces as it is very convenient in ADM method. A part of this foliastion freedom, as we showed that
the equation reduces to the momentum time equation if one adapts a preferred time foliation.}
[Comment-\#4]{The matter sector, which is crucial in GR since it is the source of non-trivial curvature and geometry, is missing from the discussion.}
[Answer-\#4]{We thank the referee for pointing out this remark. In this revision, we have carefully corrected this point as suggested . Please see the discussion after eq. (1.1).}
[Comment-\#5]{The author faces the problem as a simple quantum mechanical problem and not as a quantum field theoretical one, and thus the discussion on renormalizability etc is missing.}
[Answer-\#5]{We should emphasis here that theory which we studied in this paper is considered as an attempt to construct quantum mechanics on a classical GR background. There is no simple field
theoretic interpretation for the Hamiltonian which we obtained in this work as well as any other brackets are simply non quantum field theoretical one. In his approach, we can't reach the
renormalizability as it has been investigated in many other alternative quantum gravity scenarios.}
[Comment-\#6]{Solutions (5.7),(5.8) do not have an obvious meaning, and in any case it is strange that the author finds non-trivial structure in the absence of matter.}
[Answer-\#6]{We added a discussion about these equations after eqs.(5.8).}
[Comment-\#7]{The English of the manuscript need editing.\\
In summary, a radical modification is needed before I will be able to reconsider the manuscript for publication.}
[Answer-\#7]{We fixed several typos and improved the presentation of the paper..}
Published as SciPost Phys. Proc. 4, 009 (2021)
Reports on this Submission
In the revised version the author has fulfilled all my raised points, and thus I recommend the paper for publication. Some expressions should be replaced by more formal forms (e.g. wanna→want, etc)
and moreover English editing is needed at the proof stage. | {"url":"https://www.scipost.org/submissions/2008.02113v2/","timestamp":"2024-11-01T19:34:34Z","content_type":"text/html","content_length":"39998","record_id":"<urn:uuid:3704fd96-2bee-48cb-80f5-f83080be4403>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00475.warc.gz"} |
Raise a Number to a Power in PHP
You might need to raise a number to the power of another (the exponent) in a PHP script, for example (2^3=2×2×2) (sometimes written (2^3)), or in general (x^n). This may also be used with negative
powers where (x^{−n}=\frac{1}{x^n}). Of course, you’d probably want to do this for more complicated examples!
There are two ways to do this in PHP – the exponent operator (**) and the pow() function. We’ll show you each in turn below.
Table of Contents
<span class="ez-toc-title-toggle"><a href="#" class="ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle" aria-label="Toggle Table of Content"><span class="ez-toc-js-icon-con"><span class=""><span class="eztoc-hide" style="display:none;">Toggle</span><span class="ez-toc-icon-toggle-span"><svg style="fill: #999;color:#999" xmlns="http://www.w3.org/2000/svg" class="list-377408" width="20px" height="20px" viewBox="0 0 24 24" fill="none"><path d="M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z" fill="currentColor"></path></svg><svg style="fill: #999;color:#999" class="arrow-unsorted-368013" xmlns="http://www.w3.org/2000/svg" width="10px" height="10px" viewBox="0 0 24 24" version="1.2" baseProfile="tiny"><path d="M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z"/></svg></span></span></span></a></span>
Using the Exponent Operator in PHP #
The exponent operator in PHP is represented by two asterisks (**). It’s used to raise a number to a power, which is the product of multiplying a number by itself a certain number of times. For
example, 2 to the power of 3 is 2 x 2 x 2, which equals 8.
Here’s the basic syntax of the exponent operator in PHP:
$result = $base ** $exponent;
In this example, $base is the number you want to raise to a power, and $exponent is the power you want to raise it to. $result is the result of the exponent operation.
Let’s take a look at some simple examples of using the exponent operator in PHP:
// Calculate 2 to the power of 3
$result = 2 ** 3; // Output: 8
// Calculate 4 to the power of 2
$result = 4 ** 2; // Output: 16
// Calculate 10 to the power of -2
$result = 10 ** -2; // Output: 0.01
As you can see from the examples, the exponent operator can be used with both positive and negative powers.
Using the pow() function in PHP #
In addition to using the exponent operator, you can also use the pow() function in PHP to raise a number to a power. The pow() function takes two arguments: the base number and the exponent. Here’s
an example:
// Calculate 2 to the power of 3 using the pow() function
$result = pow(2, 3); // Output: 8
The pow() function is useful when you need to calculate a power using variables or when you want to store the result of the exponent operation in a variable.
Examples of ‘Power Of’ using the pow() function #
Here are some examples, which raise a number to the power of another and display the output. In the examples, <br /> inserts an HTML line break to display the results neatly.
echo "3^5 = " . pow(3,5) . "<br />"; // 243
echo "2^9 = " . pow(2,9) . "<br />"; // 512
// Anything raised to the power of 0 is 1:
echo "99^0 = " . pow(99,0) . "<br />"; // 1
// Negative powers:
echo "3^-1 = 1/3 = " . pow(3,-1) . "<br />"; // 1/3 = 0.3333
echo "3^-2 = 1/(3^2) = " . pow(3,-2) . "<br />"; // 1/9 = 0.1111
echo "<br />";
// Loop through 2 raised to the 0,1,2,...,10:
echo "2^" . $i . " = " . pow(2,$i) . "<br />";
And here is the output of this example code:
3^5 = 243
2^9 = 512
99^0 = 1
3^-1 = 1/3 = 0.33333333333333
3^-2 = 1/(3^2) = 0.11111111111111
2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
2^10 = 1024
Below, we’ve outlined a couple of example cases where you might use these in practice.
Example: Calculating Compound Interest #
The formula for calculating compound interest involves raising the interest rate to the power of the number of compounding periods. For example, to calculate the future value of an investment with a
principal of $1,000, an interest rate of 5% compounded annually for 10 years, you would use the following formula in PHP:
$principal = 1000;
$interest_rate = 0.05;
$years = 10;
$future_value = $principal * (1 + $interest_rate) ** $years;
The code in this example performs the following steps to calculate this:
1. It sets the principal amount to $1000 and assigns it to the $principal variable.
2. It sets the interest rate to 5% and assigns it to the $interest_rate variable.
3. It sets the number of years to 10 and assigns it to the $years variable.
4. It uses the exponent operator ** to calculate the future value of the investment and assigns the result to the $future_value variable.
The future value is calculated using the formula (FV = PV * (1 + r) ^ n), where FV is the future value, PV is the present value (or principal), r is the interest rate, and n is the number of years.
The output of the code would be:
$future_value = 1628.89
This means that the future value of the investment after 10 years would be $1628.89, assuming the interest is compounded annually.
Example: Calculating Probabilities #
The exponent operator can be used to calculate the probability of an event occurring multiple times in a row. For example, to calculate the probability of flipping a coin and getting heads three
times in a row, you would use the following code in PHP:
$coin_prob = 0.5;
$num_flips = 3;
$prob_heads = pow($coin_prob, $num_flips);
This code performs the calculation as follows:
1. It the probability of flipping heads on a single coin flip to 0.5 and assigns it to the $coin_prob variable.
2. Then sets the number of coin flips to 3 and assigns it to the $num_flips variable.
3. We use the pow() function to calculate the probability of getting heads on all three coin flips and assign the result to the $prob_heads variable.
The pow() function is used to raise the probability of flipping heads to the power of the number of coin flips. The result of this calculation is the probability of getting heads on all three coin
The output of the code would be:
$prob_heads = 0.125
This means that the probability of flipping a coin and getting heads three times in a row is 0.125, or 12.5%.
<span class="ez-toc-section" id="Conclusion"></span>Conclusion<span class="ez-toc-section-end"></span>
In this tutorial you have seen how to raise a number to the power of another in PHP using two methods: the exponent operator and the <code>pow()</code> function.
You can also calculate square (and other) roots in PHP using the pow() function, which we’ve covered in a different article: <a href="/calculate-square-roots-in-php/">Calculate Square Roots in PHP</a>. | {"url":"https://www.tiposaurus.co.uk/exponent-in-php/","timestamp":"2024-11-04T15:12:44Z","content_type":"text/html","content_length":"37288","record_id":"<urn:uuid:6bda2362-630f-4209-b8c2-0ab93009a436>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00490.warc.gz"} |
Second Law Of Thermodynamics |Entropy Definition|Principles|
The Second Law of Thermodynamics provides for a precise description of a property called entropy. Entropy can be seen as a measure of how close a system is to equilibrium. It can also be seen as a
measure of (spatial and thermal) disturbance.
The second thermodynamics notes that the entropy, that is, an individual system’s disorder, will never diminish. When an isolated system reaches the maximum entropy configuration, it can no longer
change: the equilibrium has been reached. It can be shown that the second law of thermodynamics implies that transferring heat from a lower temperature region to a higher temperature region is
impossible if no work is done.
Entropy: It is a function of the state of the system since it has a unique value for each state, independent of how the system reached that state.
ΔS = ΔQ / T
Entropy is an inherent STD property primarily linked to the measurable parameters that define it.
dS = dQ / T where,
dS: entropy of STD.
dQ: exchange of thermal energy between the medium and STD.
T: the temperature at which the thermal energy exchange between the medium and STD is recorded.
The expression allows the calculation of variations but not the knowledge of absolute values.
Entropic variation in any STD and its environment considered together is positive, tending to zero in reversible processes.
ΔS Total Δ0 (irreversible process)
ΔS = 0 (reversible process)
Entropy as probability:
Entropy increases represent a rise in molecular disorder.
In thermodynamic processes, the second law of thermodynamics introduces an additional condition. It is not enough for the energy to be conserved and thus to comply with the first principle, the
system that does work in violation of the second law is called a “second-rate perpetual mobile,” Since it could continuously draw power to conduct work in a hot environment from a cold environment.
1st law: STD and environment ΔE STD + ΔE Environment = 0
1st law: STD ΔE STD = 0
2nd law: STD and environment ΔS STD + ΔS Environment Δ0
2nd law: STD ΔS STD = 0
When ΔS STD = 0 , the system is in equilibrium, and there are no transformations between the different types of energy.
When ΔS STD > 0, it is an unbalanced process and tending towards equilibrium, always with
ΔE STD = 0 .
This is one of physics’ most important laws; while they can be interpreted in several ways, they all contribute to explaining the principle of irreversibility and entropy. When viewed by other
branches of physics, particularly by statistical mechanics and information theory, this latter definition is related to the degree of disorder of matter and energy of a system. Meanwhile, there is no
physical explanation for entropy in thermodynamics, which is correlated with the sum of unusable energy in a system.
However, this merely phenomenological interpretation of entropy is entirely consistent with its statistical interpretations. So, the Second Law of Thermodynamics dictates that although matter and
energy cannot be created or destroyed, they transform and establish the sense in which this transformation occurs. However, the second law of thermodynamics capital point is that, as with all
thermodynamic theory, it refers solely to states of equilibrium.
Any definition, analogy, or concept that is extracted from it can only be applied to equilibrium states, so formally, parameters such as temperature or entropy itself will be defined only for
equilibrium states. Thus, according to the second law of thermodynamics, if you have a system that goes from the equilibrium state A to the equilibrium state B, the amount of entropy in the
equilibrium state B will be as high as possible and inevitably greater than the equilibrium state A.
Obviously, the mechanism can only operate when it is in the transition from a state of equilibrium A to B, and not in one of those states. If the system was closed, however, its energy and volume of
matter couldn’t differ, if entropy is to be maximized at each transition from one equilibrium state to another. However, when you try to fuse the Helium nuclei, you fail to release the same amount of
energy you obtained fusing the hydrogen nuclei.
Every time the star fuses the nuclei of an element, it gets another one that is more useless for energy, and therefore the star dies. In that order of ideas, The matter it leaves behind won’t produce
another star anymore. That is how the second law of thermodynamics was used to describe the end of the universe.
Introduction to the Thermodynamics of Materials 6th Edition
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Axiomatic definition
The formal definition of the second law of thermodynamics states that:
In an equilibrium state, the values ​​that the characteristic parameters of a closed thermodynamic system take are, that they maximize the value of a certain magnitude that is a function of these
parameters, called entropy.
The entropy of a system is an abstract physical quantity that statistical mechanics identifies with the degree of a physical system’s internal molecular disorder. Classical thermodynamics, on the
other hand, describes it as the relationship between the heat transferred and the temperature at which it is transmitted.
Axiomatic thermodynamics defines entropy as a certain function –a priori, in an unknown way –, This depends on the system’s so-called “characteristic parameters,” and that can be specified only
for the system’s equilibrium states.
Such characteristic parameters are defined by a postulate derived from the first principle of thermodynamics, also referred to as the State principle.
Accordingly, the equilibrium state of a system is defined by the internal energy, volume, and molar composition of the system. Any other thermodynamic parameter is known as a function of these
parameters, such as temperature or pressure. Thus entropy will be a function of these parameters as well.
The second law of thermodynamics establishes that said entropy can only be defined for states of thermodynamic equilibrium and that of all the possible states of equilibrium –which will be defined
by the characteristic parameters– only one can be given which, of all of them, maximizes the entropy.
The consequences of this statement are subtle: when considering a closed system that tends to balance, all those which are compatible with the limits or contours of the system are included in the
possible equilibrium states. For example, the starting state of equilibrium between them, if the system varies its equilibrium state from the starting state to another, it is because the entropy of
the current state is greater than that of the original state. If the system changes its equilibrium, it can only increase its entropy. The entropy of a thermodynamically insulated system can,
therefore, only be increased.
Assuming the universe started from an equilibrium state, that at every instant of time the universe does not stray too far from thermodynamic equilibrium and that the universe is an isolated system.
In the universe, the amount of entropy is continuing to increase over time.
Nevertheless, axiomatic thermodynamics does not accept time as a thermodynamic component.
Entropy and the Tao of Counting: A Brief Introduction to Statistical Mechanics and the Second Law of Thermodynamics
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Formal, entropy can be defined only for equilibrium conditions. There are no balancing states in the process that travel from one equilibrium state to another, but entropy can not be described in
these non-equilibrium states without formal contradictions within the thermodynamics themselves. Thus entropy can not be a function of time, and it is technically incorrect to talk of variations of
it in time.
As it is done, it is because it is believed that in the transition of one state of equilibrium to another, one has gone through infinite intermediate equilibrium states, a procedure that allows time
to be entered as a parameter. As long as the final equilibrium state is that of maximum possible entropy, a frontal inconsistency will not have been incurred because these intermediate equilibrium
states have not affected the only real one (the final one).
The classical formulation argues that the change in entropy S is always greater than or equal to – exclusively for reversible processes – that the heat transfer Q produced divided by the equilibrium
temperature T of the system.
General description
The axiomatic statement of the second principle immediately reveals its main characteristic. It is one of Physics’ few ontological laws, while it generally distinguishes those physical processes and
statements that are possible from those that are not; that is, the second law of thermodynamics allows the possibility of a process or state to be determined.
In reality, the second law of thermodynamics originated in a historical sense, in the context of thermal machines, in the midst of the Industrial Revolution as an empirical explanation of why they
acted in some way and not in another. Indeed, although it may seem trivial, it was always observed, for example, that to heat a boiler it was necessary to use fuel-burning at a higher temperature
than that of the boiler.
However, the boiler was never observed to heat up by taking energy from its surroundings, which in turn would cool down. It could be reasoned that, by the first principle of thermodynamics, nothing
prevents the spontaneous transfer of heat from a cold body, e.g., at 200 K, from transferring it to another hot body, e.g., at 1000 K: it suffices that the appropriate energy balance is achieved,
which would cool the cold body even more, and the hot body would heat up even more.
All this, though, runs counter to all experience. While it seems simple and even insignificant, it has had an exceptional effect on the machines used in the Industrial Revolution. If it had not been
so, for example, the machines could work without requiring fuel, because the necessary energy could be transferred spontaneously from the rest of the environment.
Thermal machines, however, seemed to follow a certain law which materialized in the second law of thermodynamics: to produce mechanical work, additional energy (fuel) had to be supplied, which in
effect was always greater than the amount of work produced. Thus the idea of the thermal machine is closely linked to the initial declaration of the second law of thermodynamics.
A thermal machine is one which,thanks to the difference in temperature between two bodies, provides effective work. Since any thermodynamic machine requires a difference in temperature, it follows
that no useful thermal equilibrium work can be extracted from an insulated system.
That is that the external power supply will be required. This empirical theory, derived from the continuous study of how the universe functions, constitutes one of the first statements of the Second
law of thermodynamics: any cyclical process is impossible whose only result is the absorption of energy in the form of heat from a thermal focus (or thermal reservoir or deposit), and the conversion
of all this energy in the form of heat into energy in the form of work.
Classical statements
In several different ways, the second law of thermodynamics has been expressed. In short, it has thus been expressed by classical thermodynamics:
“It is impossible a process whose only result is the transfer of energy in the form of heat from a lower temperature body to a higher temperature body.”
Clausius statement
“It is impossible any cyclical process whose only result is the absorption of energy in the form of heat from a heat source (or reservoir or thermal deposit), and the conversion of all this energy in
the form of heat into energy in the form of work.”
Kelvin-Planck statement
“For any potentially cyclical system, a single heat transfer is impossible such that said process is reciprocal and eventually reversible.”
Statement of John De Saint
Some corollaries of the principle, sometimes used as alternative statements, would be:
“No cyclical process is such that the system in which it occurs and its environment can both return to the same state from which they started.”
“In an isolated system, no process can occur if a decrease in the total entropy of the system is associated with it.”
Corollary of the beginning, due to Clausius
Visually, the second law of thermodynamics can be expressed by imagining a boiler on a steamboat. It could not produce work if it were not because the steam is at high temperatures and pressure
compared to the surrounding environment.
Mathematically, it is expressed thus:
where S is the entropy, and the equality symbol only exists when the entropy is at its maximum value (in equilibrium).
Entropy in statistical mechanics
Thermodynamics does not give a physical explanation of what entropy is: it describes it simply as a mathematical function that takes its maximum value for each equilibrium state. The normal
definition of entropy with molecular disorder comes from a simplistic understanding of statistical mechanics; in particular, statistical mechanics so-called microcanonical formalism. It is important
to note that although related, thermodynamics and statistical mechanics are distinct branches of physics.
Microcanonical interpretation of entropy based on the second law of thermodynamics
The fundamental equation of a closed equilibrium thermodynamic system can be expressed as
Where S represents the system’s entropy – from a thermodynamic point of view- -, U the internal energy of the system, and N1, N2, etc. the number of moles of each component of the system. All these
magnitudes are macroscopic. They are represented and can be measured and estimated without taking into account the microscopic existence of the thermodynamic system (that is, of the atoms ,
molecules, etc.).
It may seem intuitively appropriate to conclude that if the system is in equilibrium, then there are also its most basic elements, its atoms, and molecules. Nevertheless, a basic consequence of
quantum mechanics states that if the system is macroscopic, then for its atoms and molecules there can be a multitude of discrete quantum states that are globally consistent with U, V, and N1, N2, …
Of the macroscopic system. In principle, however, even if there is such a potential ability for the system’s microscopic components to move from one quantum state to another since the system is
closed and in equilibrium, it could be reasoned that such transitions will not occur.
Today, no independent machine is complete. For eg, even if we can insulate the system thermally in an absolute way, we will not be able to avoid the gravitational effects that the rest of the
universe will continue to have on the matter that we have enclosed within; nor can it isolate itself perfectly from all the electromagnetic fields that surround it, no matter how weak they may be.
In short, the system may be closed to macroscopic effects, but the action of all kinds of force fields (be they gravity, electrical, …) and the system’s interaction with the walls that enclose it
will, at least from microscopically, the system is not in equilibrium. Atoms and molecules undergo continuous transformations from one quantum to another, the causes of which are,
Statistical mechanics find that a macroscopic system enables extremely rapid and spontaneous transitions between the various quantum states so that macroscopic measurements of parameters such as
temperature, energy, even volume, … are the average from the myriad of quantum or microscopic states. And since essentially random processes produce these transitions, a macroscopic system is
accepted as a principle that visits all the permissible microscopic states with equal probability. Such allowable microscopic states are called microstates.
The number of microstates allowed for each macroscopic equilibrium state is determined by the laws of physics. For instance, if a macroscopic system has 1000 joules of energy, it is unreasonable to
suppose a microstate of that system might have more than 1000 joules of energy.
If a macroscopic state of equilibrium is considered, according to the second law of thermodynamics, it will be defined by the values ​​of the thermodynamic variables U, V, N1, N2, etc. for which
the entropy S takes its maximum value among all the possible ones. Suppose we have a thermodynamic equilibrium system defined by a fundamental limitation: the system is not permitted to have a volume
greater than a given volume.
What was given at the start is the amount of matter in the system. Gas in a gas cylinder, for example, can not have a larger volume than the cylinder, nor can there be more gas inside than what has
been placed. Considering this limitation of volume and mass, the system will acquire values ​​of U such that they maximize entropy, so then you get to the macroscopic balance.
Associated with this macroscopic state of equilibrium, we have that of the microstates: within limits imposed by the system itself, the system’s molecules that present random transitions between
different micros States. For example, they can not travel beyond the boundaries of the system, nor can they vibrate with an energy greater than the macroscopic system’s total energy, etc. that is,
associated with macroscopic equilibrium.
There is a minimal, but potentially enormous, number of microscopic states that the system’s microscopic constituents can visit with equal probability. The molecules of the system will be able to
present random transitions between different microstates within limits imposed by the system itself.
They cannot, for example, move beyond the system’s barriers, nor may they vibrate with an energy greater than the macroscopic system’s total energy, etc. That is, in accordance with macroscopic
equilibrium, there is a small but probably enormous number of microstates that the system’s microscopic constituents can visit with equal probability.
The molecules of the system will be able to present random transitions between different microstates within limits imposed by the system itself. For example, they can not travel beyond the boundaries
of the system, nor can they vibrate with an energy greater than the macroscopic system’s total energy, etc. That is, in accordance with macroscopic equilibrium, there is a small but probably enormous
number of microstates that the system’s microscopic constituents can visit with equal probability.
When we now eliminate a macroscopic device constraint, such as allowing the volume now to be greater than before, two things will happen:
• From thermodynamics, that is, from the macroscopic point of view, the variables of the system will evolve towards a state of greater entropy: Volume V is now greater than before, and even though
the quantity of matter is the same, it may now take up more space. The internal energy of the system U will thus vary, so that in the new state of equilibrium, the entropy S takes the maximum
possible value.
Said value is necessarily greater than that of the previous equilibrium state. Indeed, we may conceive of the situation in which the machine stays in its previous range, while it that, with the same
internal energy and the same subject matter. The entropy should not have shifted in that situation and that case is consistent with device limitations.
We do know, however, that nature will not operate like this: the device will continue to fill the entire volume (even though it is a solid, in which case the vapour pressure of the solid will change,
or more solid will evaporate, etc.), and the equilibrium will change. The entropy function is the mathematical function in this new equilibrium that takes its maximum value, which must be greater
than in the previous equilibrium state.
• From a microscopic viewpoint, it happens that the number of microstates consistent with the system’s limits has now increased. We will, in essence, continue to have the same as before, although
these are added to new ones. For example, an atom will now be able to move not within the previous volume, but also within the entire new volume.
Thus, as the entropy increases, there is an increase in the number of possible microstates. This implies that the number of microstates that are consistent with the macroscopic limitations of the
system will define entropy. Since the microstates are the product of chance, and the probability that each of them occurs the same, it is natural to identify entropy with the microscopic disorder.
There’s only one problem: Entropy is an additive, according to thermodynamics. That is to say, the entropy of two equal systems is twice the entropy of each. However, it’s multiplicative in the
number of potential microstates. The number of microstates of the two systems is the product of each microstate number. For example, the number of two dice ‘microstates’, if the number of each one is
6 (each face of the die is a possible microstate), is 6 × 6 = 36 microstates (having a “1” in the first one, a “3” in the second, a “2” in the first, a “5” in the second, etc.). To interpret
entropy, we will need to get the number of microstates to fulfill an additive rule.
The only solution to this is to identify the entropy with the number of possible microstates’ logarithm.
Where kB is the Boltzmann constant and appears simply to determine the scale of entropy, which is usually given as energy per degree of temperature (J / K). However, according to this interpretation,
it may have no units.
Canonical interpretation
The microcanonical interpretation of entropy conceives an isolated thermodynamic system, that is, a thermodynamic system that does not exchange either matter or energy or volume with the outside: the
composition of the system, given by N1, N2, …, its internal energy U and its volume V do not change in it. The system par excellence that meets these conditions is the universe itself. However, on
many occasions, systems that exchange energy, mass, or volume with their environment are contemplated.
In such cases, the mathematical interpretations of entropy must be expanded, but globally it is the microcanonical understanding that lasts. Nevertheless, if we consider a system that exchanges
matter with its environment, for example, we can conceive of a larger system that includes the initial system and its environment such that the global system conforms to the micro-canonical
interpretation; that system would be the universe itself, at the limit. And it is precisely the entropy of the microcanon system that is subject to the second law of thermodynamics
, that is, the one that needs to increase as the global equilibrium of the system varies.
This could then be assumed that any system can be viewed by conceiving the global structure that is subject to microcanonical interpretation, regardless of the conditions of interaction with its
environment. Nonetheless, the equilibrium state should be obtainable in theory by taking into account the total number of microstates in the global system.
However, this can be very costly if not practically impossible to estimate in most circumstances: Combinatorial calculations of the number of ways in which the available energy in a system can be
distributed are often beyond mathematical knowledge. And it is to solve those deficiencies that the other interpretations of entropy arise.
The canonical interpretation, also called canonical or Helmholtz formalism concerns a thermodynamic device that can exchange energy with a thermal reservoir or thermostat. Accordingly, By having an
infinite source of energy, each energy state will be conceivable, from the lowest to the highest.
Like the microcanonical method, though, the likelihood of any of those states won’t be the same: the system won’t be the same fraction of time in each of those states. The cornerstone of canonical
formalism is to evaluate the microstate probability distribution. And this problem is solved considering that the global system formed by the thermostat and the system at issue is a closed system,
that is,
If the total energy of the global system is E tot , and that of a microstate of the local system is E j, since the local system is in a state of energy E j, the thermostat will inevitably be reduced
to one of energy E tot – E j . The probability that the global system is in a microstate such that the thermostat has energy E tot – E j and the local system E j will be:
Following Boltzmann’s definition of entropy, this equation can be written as:
The internal energy U will be the average value of the local system’s energy, so, since the entropy is additive, it can be written that:
If it is developed in series we will have to:
Thus, the probability can be expressed as:
And since it is the free energy of Helmholtz, we can express this probability as: ,
The total probability of being in any of these states is unity, so:
, where is defined
Z is the so-called canonical partition function, generally defined as:
If the partition function Z is known for a thermally balanced particle system, the entropy can be calculated by:
Where kB is the Boltzmann constant, T the temperature, and the probabilities Pj.
This is the interpretation of entropy, called Helmholtz’s canonical interpretation or entropy.
Von Neumann entropy in quantum mechanics
The definition of entropy was introduced in the 19th century to structures made up of various particles which are classically acting. Von Neumann extended the definition of entropy for quantum
particle systems at the beginning of the 20th century, defining it for a mixed state characterized by a density matrix? From quantum entropy by Neumann as the scalar magnitude:
Generalized Entropy in General Relativity
The effort to extend traditional thermodynamic analyzes to the entire universe led in the early 1970s to investigate the thermodynamic behaviour of objects such as black holes. The preliminary
outcome of this study showed something very interesting: in the case of black holes, the second law, as it had been formulated conventionally for classical and quantum systems, maybe broken.
Nonetheless, Jacob D. Bekenstein ‘s work on information theory and black holes indicated that a generalized entropy (S gen) applied to standard entropy (S Conv) would still be true if the second law
of thermodynamics were implemented.,attributable black holes entropy which depends on the total area (A) of black holes in the universe. This generalized entropy specifically has to be defined as:
Where k is the constant of Boltzmann, c is the speed of light, G is the constant of universal gravitation, and Planck is the constant of rationalization.
Bibliography www.wikipedia.org
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Facts of Universe Cycle/Oscillating Model
Leave a Comment / Mysteries & Hypothesis, Matter & Energy, Physics & Cosmos, Space & Time / By Deep Prakash / July 8, 2020 / astronomy, astrophysics, balance, cosmologist, cosmology, cosmos, cycle of
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scientist, solar system, space, surprise, symbolize, symbols, the cyclic universe theory, theoretical, understand, understand universe, universe, universe cycle, universe cycle theory, universe
cycles, universe examples, what is cyclic universe theory
Cyclic Model of Universe
Leave a Comment / Mysteries & Hypothesis, Matter & Energy, Physics & Cosmos, Space & Time, Uncommon & Remarkable / By Deep Prakash / July 10, 2020 / astronomy, astrophysics, big bang, black, black
hole, cosmology, cosmos, cycle of the universe, cycle of universe, cycle universe, cycles of the universe, cyclic model, cyclic model of the universe, cyclic model of universe, cyclic model theory,
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physics, quasar, repeating universe theory, research, science, scientist, space, the cyclic universe theory, theoretical, theory, universe, universe cycle, universe cycle theory, universe cycles,
universe examples, what is cyclic universe theory, whirling disk | {"url":"https://cosmos.theinsightanalysis.com/second-law-of-thermodynamics-entropy/","timestamp":"2024-11-08T02:43:54Z","content_type":"text/html","content_length":"211211","record_id":"<urn:uuid:a2687012-7181-4301-ba76-a7264a6de8df>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00197.warc.gz"} |
Help with Long IF,AND,OR formula
Needing some help with the below formula. Everything is working correctly but the last IF/AND/OR statement and can't figure out what I'm missing. Any help is greatly apricated.
=IF(AND([Payment Received]@row = true, OR([Water Disconnected]@row = true, AND([Water Meter #]@row = "NO Meter", [Electric Disconnected]@row = true))), "Reconnect Needed", IF(AND([Payment Received]
@row = true, OR([Water Disconnected]@row = false, AND([Water Meter #]@row = "NO Meter", [Electric Disconnected]@row = false))), "Reconnected", IF(AND([Water Disconnected]@row = true, OR([Water Meter
#]@row = "NO Meter"), OR([Electric Disconnected]@row = true), "Disconnected", ""))))
• It appears you're missing one ending parenthesis for the last AND() statement right before "Disconnected", and also have one extra closing parenthesis for the three IF() statements that needs to
be removed. Try this:
=IF(AND([Payment Received]@row = true, OR([Water Disconnected]@row = true, AND([Water Meter #]@row = "NO Meter", [Electric Disconnected]@row = true))), "Reconnect Needed", IF(AND([Payment
Received]@row = true, OR([Water Disconnected]@row = false, AND([Water Meter #]@row = "NO Meter", [Electric Disconnected]@row = false))), "Reconnected", IF(AND([Water Disconnected]@row = true, OR
([Water Meter #]@row = "NO Meter"), OR([Electric Disconnected]@row = true)), "Disconnected", "")))
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/132308/help-with-long-if-and-or-formula","timestamp":"2024-11-03T04:40:07Z","content_type":"text/html","content_length":"424145","record_id":"<urn:uuid:3580509d-b8c2-4367-a3c0-076cbca45c7c>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00014.warc.gz"} |
Quizwiz - Ace Your Homework & Exams, Now With ChatGPT AI (2024)
Three trigonometric functions for a given angle are shown below.mc023-1.jpgWhat are the coordinates of point (x, y) on the terminal ray of angle mc023-2.jpg, assuming that the values above were not
(-5, 12)
If u(x)=-2x^2+3 and v(x)=1/x, what is the range of (u*v)(x)?
The terminal side of an angle measuring pi/6 radians intersects the unit circle at what point?
(sq root3/2, 1/2
If f(x) = 3 - 2x and g(x)=1/x+5, what is the value of (f/g)(8)?
If f(x) = x2 - 2x and g(x) = 6x + 4, for which value of x does (f + g)(x) = 0?
What is the value of mc005-1.jpg?
A Cepheid star is a type of variable star, which means its brightness is not constant. The relationship between the brightness of a Cepheid star and its period, or length of its pulse, is given by M
= -2.78(log P) - 1.35, where M is the absolute magnitude, or brightness, of the star, and P is the number of days required for the star to complete one cycle. What is the absolute magnitude of a star
that has a period of 45 days? Use a calculator. Round your answer to the nearest hundredth.
If f(x)=sq rt(x)+12 and g(x)=2 sq rt2, what is the value of (f - g)(144)?
Kim solved the equation below by graphing a system of equations.mc020-1.jpgWhat is the approximate solution to the equation?
What is the value of log 43? Use the calculator. Round your answer to the nearest tenth.
If f(x) = 16x - 30 and g(x) = 14x - 6, for which value of x does (f - g)(x) = 0?
What is 3pi/4 radians converted to degrees? If necessary, round your answer to the nearest degree.
Kalon has $175 and needs to save at least $700 for a new computer. If he can save $35 per week, what is the minimum number of weeks Kalon will need to save to reach his goal?
15 weeks
What is 5pi/6 radians converted to degrees? If necessary, round your answer to the nearest degree.
Angle D has a measure between 0 and 360 and is coterminal with a -920° angle. What is the measure of angle D?
What is the amplitude of the function below? mc007-1.jpg
What is the value of mc003-1.jpg?
On a unit circle, the vertical distance from the x-axis to a point on the perimeter of the circle is twice the horizontal distance from the y-axis to the same point. What is sinmc013-1.jpg?
2 sq root 0.2
In a right triangle, cosa=0.352 and sina=0.936. What is the approximate value of tana?
Given that sin0=21/29, what is the value of cos0, for 0<! <90?
In a right triangle, angle A measures 20°. The side opposite angle A is 10 centimeters long. Approximately how long is the hypotenuse of the triangle?
29.2 centimeters
Which logarithmic equation is equivalent to 32 = 9?
If f(x) = 3x + 2 and g(x) = x2 + 1, which expression is equivalent to (f*g)(x)?
3(x 2 + 1) + 2
One leg of a right triangle measures 6 inches. The remaining leg measures mc016-1.jpg inches. What is the measure of the angle opposite the leg that is 6 inches long?
The intensity, or loudness, of a sound can be measured in decibels (dB), according to the equation mc026-1.jpg, where I is the intensity of a given sound and I0 is the threshold of hearing intensity.
What is the intensity, in decibels, [I(dB)], when mc026-2.jpg? Use a calculator. Round to the nearest whole number.
If f(x) = x2 + 1 and g(x) = x - 4, which value is equivalent to (f*g)(10)?
What is the approximate degree measure of angle A in the triangle below?mc007-1.jpg
Which of the following are in the correct order from least to greatest?
38,pi/4,80, 7pi/6, 2pi
What is 270° converted to radians?
If s(x) = x - 7 and t(x) = 4x2 - x + 3, which expression is equivalent to (t*s)(x)?
4(x - 7)2 - (x - 7) + 3
Which of the following represents the measures of all angles coterminal with a 418° angle?
58 + 360n, for any integer n
The length of one leg of an isosceles right triangle is 3 ft. What is the perimeter of the triangle?
6+3sq root 2
What is the approximate length of arc s on the circle below? Use 3.14 for mc024-1.jpg. Round your answer to the nearest tenth.mc024-2.jpg
6.3 in
Manuel has $600 in a savings account at the beginning of the summer. He wants to have at least $300 in the account at the end of the summer. He withdraws $28 each week for food. Which inequality
represents w, the number of weeks Manuel can withdraw money while not dropping below a $300 balance?
If g(x)=(x+1)/(x-) and h(x) = 4 - x, what is the value of (g*h)(-3)?
Each side of the regular hexagon below measures 8 cm. What is the area of the hexagon?
96 sq root 3 square centimeters
Keisha and David each found the same value for mc011-1.jpg, as shown below, given mc011-2.jpg.
Both procedures are correct.
Which of the following best explains why cos2pi/3 dont equal cos5pi/3?
Cosine is negative in the second quadrant and positive in the fourth quadrant.
If I=prt, which equation is equivalent to t?
An angle whose measure is -302° is in standard position. In which quadrant does the terminal side of the angle fall?
Quadrant I
An angle whose measure is 40° is in standard position. In which quadrant does the terminal side of the angle fall?
Quadrant I
An angle whose measure is 92° is in standard position. In which quadrant does the terminal side of the angle fall?
Quadrant II
An angle that shares the same sine value of an angle that measures 5pi/4 radians is located where?
Quadrant IV
Francesca drew point (-2, -10) on the terminal ray of angle mc024-1.jpg, which is in standard position. She found values for the six trigonometric functions using the steps below.
She made her first error in step 3 because the sine, cosine, and tangent ratios are incorrect, which also results in incorrect cosecant, secant, and tangent functions.
Which of the following is true for f(x) = 5cos(x) +1?
The range of the function is the set of real numbers mc018-3.jpg.
The final velocity, V, of an object under constant acceleration can be found using the formula mc009-1.jpg, where v is the initial velocity (in meters per second), a is acceleration (in meters per
second), and s is the distance (in meters). What is the formula solved for a?
Which expression is equivalent to (f + g)(4)?
f(4) + g(4)
Which function is graphed below?mc002-1.jpg
f(x) = cos(x)
Which function is graphed below?mc001-1.jpg
f(x) = sin(x)
For which pairs of functions is (f*g)(x)=x?
f(x)=2/x and g(x)=2x
Grayson charges $35 per hour plus a $35 administration fee for tax preparation. Ian charges $45 per hour plus a $15 administration fee. If h represents the number of hours of tax preparation, for
what number of hours does Grayson charge more than Ian?
Which of the following equations is equivalent to s=pir^2h
Which expression can be used to approximate the expression below, for all positive numbers a, b, and x, where a mc001-1.jpg 1 and b mc001-2.jpg 1?loga(x)
A taxi service charges a flat fee of $1.25 and $0.75 per mile. If Henri has $14.00, which of the following shows the number of miles he can afford to ride in the taxi?
What are the values of m and mc012-1.jpg in the diagram below?mc012-2.jpg
m=sq root3/20=pi/6
Which of the following is true of the location of an angle, mc025-1.jpg, whose tangent value is -sq root3/3?
mc025-3.jpg has a 30-degree reference angle and is located in Quadrant II or IV
Six times a number is greater than 20 more than that number. What are the possible values of that number?
Which graph shows the solution set of the inequality 2.9(x+8)<26.1
open circle starting at 1<------
Which expression converts pi/4 radians to degrees?
Which values for mc017-1.jpg have the same reference angles?
A circle has a central angle of 6 radians that intersects an arc of length 14 in. Which equation finds the length of the radius, r, of the circle?
Which set of transformations is needed to graph f(x) = -2sin(x) + 3 from the parent sine function?
reflection across the x-axis, vertical stretching by a factor of 2, vertical translation 3 units up
Which expression is equivalent to (st)(6)?
s(6) × t(6)
Which equation gives the length of an arc, s, intersected by a central angle of 3 radians in a circle with a radius of 4 in.?
Gavin wrote the equation mc004-1.jpg to represent p, the profit he makes from s sales in his lawn-mowing business. Which equation is solved for s?
If cos0=0.3090, which of the following represents approximate values of sin0 and tan0 , for mc008-4.jpg?
Which Pythagorean identity is correct?
Which description best explains the domain of (g*f)(x)?
the elements in the domain of f(x) for which g(f(x)) is defined
What is the domain of f(x) = cos(x)?
the set of all real numbers
What is the range of f(x) = sin(x)?
the set of all real numbers -1<=y<=1
Cary calculated the surface area of a box in the shape of a rectangular prism. She wrote the equation mc023-1.jpg to represent the width and height of the box. She solved for w and got mc023-2.jpg
Which of the following is an equivalent equation?
Paolo wrote the following equation for the perimeter of a rectangle.p=2(1+w)Which equation is equivalent to the equation Paolo wrote?
The point P(x, y) lies on the terminal side of an angle mc014-1.jpg = -60° in standard position. What are the signs of the values of x and y?
x is positive, and y is negative.
Which equation is equivalent to logx(36)=2?
Which represents the solution set of 5(x+5)<85
Which represents the solution set of the inequality 5x-9<=21
Remmi wrote the equation of the line mc021-1.jpg He solved for x and got mc021-2.jpg Which of the following is an equivalent equation for x?
Which of the following is true of the values of x and y in the diagram below?mc010-1.jpg
Which is the solution set of the inequality 15y-9<36
Which function describes the graph shown below?mc009-1.jpg
If (z*z)(x)=(1/16)x, what is z(x)?
What is the range of y = -5sin(x)?
In the triangle below, angle B measures 60° and BC is 18. What is the length of AC?
9 sq root 3
If g(x) = 2x and (f*g)(x)=3/x, what is f(x)?
If c(x)=5/x-2 and d(x) = x + 3, what is the domain of (cd)(x)?
all real values of x except x = 2
The formula mc011-1.jpg gives the partial sum of an arithmetic sequence. What is the formula solved for an?
Which graph represents the solution set of the inequality x+2>=6
closed dot start @4----->
The point (-sq root2/2,sq root2/2) is the point at which the terminal ray of angle mc021-2.jpg intersects the unit circle. What are the values for the cosine and cotangent functions for angle
cos=-sq root2/2 cot=-1
n the triangle below, which is equivalent to sinA?
Which equation is equivalent to 4s=t+2
Given that tan0=-1, what is the value of sec0, for 3pi/2<0<2pi?
sq root 2 | {"url":"https://salmonpage.com/article/quizwiz-ace-your-homework-exams-now-with-chatgpt-ai-3","timestamp":"2024-11-10T09:04:14Z","content_type":"text/html","content_length":"70554","record_id":"<urn:uuid:927f72fd-6d2b-4ee6-8332-e6fe728de6ef>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00820.warc.gz"} |
gp_Lin2d.hxx File Reference
Describes a line in 2D space. A line is positioned in the plane with an axis (a gp_Ax2d object) which gives the line its origin and unit vector. A line and an axis are similar objects, thus, we can
convert one into the other. A line provides direct access to the majority of the edit and query functions available on its positioning axis. In addition, however, a line has specific functions for
computing distances and positions. See Also GccAna and Geom2dGcc packages which provide functions for constructing lines defined by geometric constraints gce_MakeLin2d which provides functions for
more complex line constructions Geom2d_Line which provides additional functions for constructing lines and works, in particular, with the parametric equations of lines. More... | {"url":"https://dev.opencascade.org/doc/occt-7.2.0/refman/html/gp___lin2d_8hxx.html","timestamp":"2024-11-12T16:15:31Z","content_type":"application/xhtml+xml","content_length":"6291","record_id":"<urn:uuid:0b265bc8-dc06-4443-b5a8-5be4bcf7891a>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00542.warc.gz"} |
Math Awareness Month Part 2: Infinite Series
Today's page for Math Awareness Month is about a recent video that caused some huge debate. I saw the video a month or two ago, and was very intrigued. I showed it to some of my friends, and we were
arguing about the content for quite a while. It also spread rapidly around the math department at Andover, with some teachers bringing up in their classes.
Take a look at the page and try some of the exercises. You will find the outcomes very interesting and mind-boggling. The concept of infinity is difficult for any human being to grasp, making it tons
of fun to think about.
Comment below what you think of the video. Do you think it is accurate? What do you think the fallacies are? How could this be a part of string theory if it is mathematically flawed?
In math class last week, we were given the following problem:
I then did the math and determined that the limit would be -1/12. I then called my teacher over, and pointed to that answer. Recalling the video, I asked him if I could rewrite that -1/12 as
1+2+3+4+5+6+7+... as my final answer. Thankfully, he got the reference. In addition to being a funny anecdote, the fact that people got the joke shows how wide of an audience this information has
reached and captivated, which is amazing to see. | {"url":"https://coolmathstuff123.blogspot.com/2014/04/math-awareness-month-part-2-infinite.html","timestamp":"2024-11-03T23:00:12Z","content_type":"text/html","content_length":"66722","record_id":"<urn:uuid:44549597-765d-48e1-98c2-8228240eeca3>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00844.warc.gz"} |
qp2/example.irp.f at 651b99b4312c83e4c5568d7d40a949d47fefc48c
2019-01-25 11:39:31 +01:00
subroutine example_becke_numerical_grid
implicit none
include 'constants.include.F'
! subroutine that illustrates the main features available in becke_numerical_grid
integer :: i,j,k,ipoint
double precision :: integral_1, integral_2,alpha,center(3)
print*,'routine that illustrates the use of the grid'
print*,'This grid is built as the reunion of a spherical grid around each atom'
print*,'Each spherical grid contains a certain number of radial and angular points'
print*,'n_points_integration_angular = ',n_points_integration_angular
print*,'n_points_radial_grid = ',n_points_radial_grid
print*,'As an example of the use of the grid, we will compute the integral of a 3D gaussian'
! parameter of the gaussian: center of the gaussian is set to the first nucleus
! alpha = exponent of the gaussian
alpha = 1.d0
print*,'The first example uses the grid points as one-dimensional array'
print*,'This is the mostly used representation of the grid'
print*,'It is the easyest way to use it with no drawback in terms of accuracy'
integral_1 = 0.d0
! you browse all the grid points as a one-dimensional array
do i = 1, n_points_final_grid
double precision :: weight, r(3)
! you get x, y and z of the ith grid point
r(1) = final_grid_points(1,i)
r(2) = final_grid_points(2,i)
r(3) = final_grid_points(3,i)
weight = final_weight_at_r_vector(i)
double precision :: distance, f_r
! you compute the function to be integrated
distance = dsqrt( (r(1) - center(1))**2 + (r(2) - center(2))**2 + (r(3) - center(3))**2 )
f_r = dexp(-alpha * distance*distance)
! you add the contribution of the grid point to the integral
integral_1 += f_r * weight
print*,'integral_1 =',integral_1
print*,'(pi/alpha)**1.5 =',(pi / alpha)**1.5
print*,'The second example uses the grid points as a collection of spherical grids centered on each atom'
print*,'This is mostly useful if one needs to split contributions between radial/angular/atomic of an integral'
! you browse the nuclei
do i = 1, nucl_num
! you browse the radial points attached to each nucleus
do j = 1, n_points_radial_grid
! you browse the angular points attached to each radial point of each nucleus
do k = 1, n_points_integration_angular
r(1) = grid_points_per_atom(1,k,j,i)
r(2) = grid_points_per_atom(2,k,j,i)
r(3) = grid_points_per_atom(3,k,j,i)
weight = final_weight_at_r(k,j,i)
distance = dsqrt( (r(1) - center(1))**2 + (r(2) - center(2))**2 + (r(3) - center(3))**2 )
f_r = dexp(-alpha * distance*distance)
integral_2 += f_r * weight
print*,'integral_2 =',integral_2
print*,'(pi/alpha)**1.5 =',(pi / alpha)**1.5 | {"url":"https://git.irsamc.ups-tlse.fr/LCPQ/qp2/src/commit/651b99b4312c83e4c5568d7d40a949d47fefc48c/src/becke_numerical_grid/example.irp.f","timestamp":"2024-11-08T09:08:00Z","content_type":"text/html","content_length":"60107","record_id":"<urn:uuid:022513bf-feeb-43a0-a6b5-41ff2561cd70>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00449.warc.gz"} |
Eggs Per Hen-Day
There’s an old brain teaser that is posed as a word problem, something like this:
If a hen and a half lays an egg and a half in a day and a half, how many days will it take three hens to lay three eggs?
If a hen and a half lays an egg in a half in a day and a half, how many hens will it take to lay six eggs in six days?
These are essentially the same problem and there is no significant difference to “three” or “six”. I’ve seen several internet sites that go through the math to calculate the rate of egg laying to
come up with the answer, but that is not necessary.
With a little knowledge of Algebra, we realize that this is a rate problem: how many eggs are produced per hen-day? Mathematically, we would write that this way:
Eggs = Rate * Hens * Days
and then solving for “Rate” we get this:
We then see immediately that we don’t even need to know Rate.
If we multiply Eggs and Hens by the same factor N:
N * Eggs
Rate =
(N * Hens)(Days)
that factor, being in both the numerator and denominator, will cancel and Days must remain unchanged.
In the same way if we multiply Eggs and Days by the same factor N:
N * Eggs
Rate =
(Hens)(N * Days)
again that factor, being in both the numerator and denominator, will cancel and Hens must remain unchanged.
So, we can change “one and a half” to X and three (or six) to Y:
If X hens lay X eggs in X days, how many days will it take for Y hens to lay Y eggs?
If X hens lay X eggs in X days, how many hens will it take to lay Y eggs in Y days?
For all values of X and Y, the answer will always be X. | {"url":"https://pedanticdan.com/ignore-me/eggs-per-hen-day/","timestamp":"2024-11-10T21:40:05Z","content_type":"application/xhtml+xml","content_length":"27178","record_id":"<urn:uuid:678e45d7-fe5d-451c-ab32-38821f28977e>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00809.warc.gz"} |
[Solved] Test the given claim. Assume that a simpl | SolutionInn
Answered step by step
Verified Expert Solution
Test the given claim. Assume that a simple random sample is selected from a normally distributed population. Use either the P-value method or the traditional
Test the given claim. Assume that a simple random sample is selected from a normally distributed population. Use either the P-value method or the traditional method of testing hypotheses. Company A
uses a new production method to manufacture aircraft altimeters. A simple random sample of new altimeters resulted in errors listed below. Use a 0.05 level of significance to test the claim that the
new production method has errors with a standard deviation greater than 32.2 ft, which was the standard deviation for the old production method. If it appears that the standard deviation is greater,
does the new production method appear to be better or worse than the old method? Should the company take any action?
-41,78, -25, -73, -41, 11, 18,52, -6, -55, - 108, - 108
what are the null and alternative hypotheses?
Find the test statistic.
Determine the critical value(s).
The critical value(s) is/are .
Since the test statistic is the critical value(s), H0. There is evidence to support the claim that the new production method has errors with a standard deviation greater
than 32.2 ft.
The variation appears to be than in the past, so the new method appears to be , because there will be altimeters that have errors. Therefore, the company take
immediate action to reduce the variation.
There are 3 Steps involved in it
Step: 1
Hypothesis Testing for Standard Deviation Given the data 41 78 25 73 41 11 18 52 6 55 108 108 We need to test the claim that the new production method ...
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Get Started | {"url":"https://www.solutioninn.com/study-help/questions/test-the-given-claim-assume-that-a-simple-random-sample-5404352","timestamp":"2024-11-10T04:50:19Z","content_type":"text/html","content_length":"99595","record_id":"<urn:uuid:012a94cc-d7d5-4247-8eb1-c8b0f0f66842>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00489.warc.gz"} |
My overall mean teaching evaluation comes in at 8.57/9, with a median of 9/9; the latter held pointwise until 2018. I am occasionally asked to provide more motivation in tying economic themes
• details UNC ECON 510: Advanced Microeconomic Theory Offered 5 times
sessions offered
□ Fall 2020 (mean 3.17/5; median 3.00/5)
"Kyle was a great teacher who clearly is very intelligent, but the format of the class this semester was not conducive to any type of learning whatsoever."
Ed. note: Oof.
□ Spring 2019 (mean 4.20/5; median 4.00/5)
"Now I find myself in a class where I excitedly talk to my peers and family about topics from lecture over dinner about what I'm learning, completely unprompted."
□ Fall 2017 (mean 3.90/5; median 4.00/5)
"He tried his best."
□ Spring 2017 (mean 4.70/5; median 5.00/5)
□ Fall 2015 (mean 4.38/5; median 5.00/5)
"As a student who isn't particularly interested in microeconomics, I enjoyed this class quite a bit. Professor Woodward's inclusion of real world applications of game theory and auctions
(i.e.: NRMP) made the class much more useful and much more interesting. He obviously cares very much for his students and was willing to make changes to the course to benefit the class."
• details UNC ECON 701: Analytical Methods for Mathematical Economics Offered 6 times
sessions offered
□ Fall 2020 (mean 4.00/5; median 4.00/5)
"I thought Professor Woodward tried his best to maintain a 'classroom' setting via Zoom – he obtained (or had) a whiteboard in his home that he set up really close to the camera, letting
him explain theorems and write out proofs step by step, which was really helpful to me."
□ Fall 2019 (no evaluations)
□ Fall 2018 (no evaluations)
□ Fall 2017 (no evaluations)
□ Fall 2016 (no evaluations)
□ Fall 2015 (no evaluations)
• details UNC ECON 890: Contract Theory Offered 2 times
sessions offered
□ Fall 2019 (no evaluations)
□ Spring 2019 (no evaluations)
• details UCLA ECON 201C: Welfare Economics Offered 1 time; notes available
course notes
sessions offered
□ Spring 2011 (mean 8.37/9; median 9.00/9)
• details UCLA ECON 201A: Microeconomics Offered 1 time; notes available
course notes
sessions offered
□ Fall 2011 (mean 8.56/9; median 9.00/9)
"Kyle is the best TA. He was always available and responsive, came prepared, and thoroughly answered questions."
• details UCLA ECON 106D: Market Design Offered 1 time; notes available
course notes
□ Uses of revenue equivalence
Solving an all-pay auction without the revelation principle, using revenue equivalence to compute strategies, order statistics and their applications.
sessions offered
• details UCLA ECON 101: Microeconomic Theory Offered 2 times; notes available
course notes
sessions offered
□ Fall 2012 (mean 8.65/9; median 9.00/9)
"[...] Hope Kyle will teach econ here as a professor soon."
□ Winter 2011 (mean 8.23/9; median 9.00/9)
"List of top 5 people I want to be: (1) Derek Jeter; (2) Johnny Depp; (3) Andy Roddick; (4) Michael Burry; (5) Kyle Woodward"
• details UCLA ECON 41: Statistics for Economists Offered 3 times; notes available
course notes
sessions offered
□ Winter 2013 (mean 8.60/9; median 9.00/9)
"You sir are a BEAST!"
Ed. note: I happen to enjoy pepperoni pizza.
□ Winter 2012 (mean 8.51/9; median 9.00/9)
"Phenomenal job. Honestly the most comprehensive TA I have ever had. You would make one of the best professors this school offers. Good luck and thank you for the time you put into your
precision and care."
□ Fall 2010 (mean 8.27/9; median 9.00/9)
"Kyle was really clear. He was effective, straightforward, and super helpful. He also gave us cookies, which in no way influenced my evaluation. Seriously, though, he was really good. Give
him a raise, because he really works hard to make us learn."
• details UCLA ECON 97: The Economic Toolkit Offered 1 time
sessions offered
□ Summer 2013 (no evaluations)
• details UCLA ECON 11: Microeconomic Theory Offered 1 time; notes available
course notes
sessions offered
• details UCLA ECON M134A: Environmental Economics Offered 1 time
sessions offered
□ Summer 2011 (no evaluations) | {"url":"https://1.618034.com/teaching.php","timestamp":"2024-11-02T18:41:13Z","content_type":"text/html","content_length":"33456","record_id":"<urn:uuid:fabfb8e8-211d-46fd-b624-6aaea87f2549>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00634.warc.gz"} |
All Staff and Students
Dr Andrew Morris BSc (Hons) PhD
School of Mathematics
Associate Professor in Mathematical Analysis
Contact details
School of Mathematics
Watson Building
University of Birmingham
B15 2TT
Andrew is a harmonic analyst who specialises in developing functional calculus methods to investigate solutions to partial differential equations in rough geometric contexts. His current research is
partially supported by the Royal Society International Exchange grant Eliminating symmetry and permitting singularity in periodic homogenization.
• PhD in Mathematics, Australian National University, 2011
• BSc (Hons) in Mathematics, University of Queensland, 2004
Andrew obtained a Doctor of Philosophy in Mathematics from the Australian National University in 2011. He subsequently held postdoctoral positions at the University of Missouri and the University of
Oxford before joining the University of Birmingham as a Lecturer in Mathematical Analysis in 2015.
Semester 2
LC Real Analysis
Andrew is interested in supervising postgraduate research students in harmonic analysis, operator theory and partial differential equations. Some example projects include those given below.
Research Themes
Andrew’s research concerns the development of modern techniques in harmonic analysis, functional calculus and geometric measure theory for application to partial differential equations on Riemannian
manifolds and rough domains. This includes elliptic systems with rough coefficients, local T(b) techniques, first-order methods, quadratic estimates, holomorphic functional calculus, singular
integral theory, layer potentials, Hardy spaces, boundary value problems and uniform rectifiability.
• Stage 2 Director
• Director of Employability
Bailey, J., Morris, A.J., Reguera, M.C. (2021), Unboundedness of potential dependent Riesz transforms for totally irregular measures, J. Math. Anal. Appl. 494(1):to appear.
Hofmann, S., Le, P., Morris, A.J. (2019), Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations, Anal. PDE, 12(8):2095-2146.
Hofmann, S., Mitrea, D., Mitrea, M., Morris, A.J. (2017), L^p-Square function estimates on spaces of homogeneous type and on uniformly rectifiable sets, Mem. Amer. Math. Soc., 245(1159):1-108. DOI:
Hofmann, S., Mitrea, M., Morris, A.J. (2015), The method of layer potentials in L^p and endpoint spaces for elliptic operators with L^∞ coefficients, Proc. London Math. Soc., 111(3): 681-716.
Auscher, P., McIntosh, A., Morris, A.J. (2015), Calderón reproducing formulas and applications to Hardy spaces, Rev. Mat. Iberoam., 31(3):865-900.
Hofmann, S., Mitrea, D., Mitrea, M., Morris, A.J. (2014), Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets, Electron. Res. Announc. Math. Sci.,
McIntosh, A., Morris, A.J. (2013), Finite propagation speed for first order systems and Huygens' principle for hyperbolic equations, Proc. Amer. Math. Soc., 141:3515-3527.
Carbonaro, A., McIntosh, A., Morris, A.J. (2013), Local Hardy spaces of differential forms on Riemannian manifolds, J. Geom. Anal., 23(1):106-169.
Morris, A.J. (2012), The Kato square root problem on submanifolds, J. London Math. Soc. 86(3):879-910.
Morris, A. J. (2010), “Local quadratic estimates and holomorphic functional calculi”, In: The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, Proc. Centre Math. Appl. Austral. Nat. Univ.,
vol. 44, Austral. Nat. Univ., Canberra, pp. 211-231.
eprints: arxiv.org/find/math/1/au:+Morris_Andrew/0/1/0/all/0/1 | {"url":"https://www.birmingham.ac.uk/schools/mathematics/people/navigation?ReferenceId=108338&Name=dr-andrew-morris","timestamp":"2024-11-10T00:10:46Z","content_type":"text/html","content_length":"22673","record_id":"<urn:uuid:7cc305f5-8f95-487d-8199-2861a695c8c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00847.warc.gz"} |
How to make a deduction automatic but only once a week
Auto-suggest helps you quickly narrow down your search results by suggesting possible matches as you type.
Im trying to figure out how to make the insurance deduction for my drivers automatic but only once a week when I run their payroll settlement.
Hi @Dale Would you be able to share how you have your data laid out or explain how you are running the payroll settlement?
Sure - how would the best way to do that be?
You can paste screenshots directly in the reply window here.
100% honest here - I dont know how to do that
My payroll is based on loads Delivered - my drivers are paid a percentage of the gross of each load then the monthly insurance cost is deducted weekly from the drivers settlement.
example - load pays $1000, Driver gets paid 80% or $800 the the weekly insurance is deducted from the sum of the loads for the week so lets say the driver portion is $1500 then we would deduct say
$200 from that for the net to the driver to be $1300
You have field that calculates the load pay (1000) and you just need a formula to deduct a set percent. Is it always 20% for everyone? or do you have a field that sets the deduction amount per
then this formula will deduct the percentage from the sub total
I have the same basic formula as you have there, im needing a formula for how to make a deduction once a week instead out of every load
I am a little confused about how you are introducing time into the process.
Do you have a table of payroll that is linked to people and the workload? In that table are you rolling up the total loads over the week, and then deducting the insurance?
You can make the formula dependent upon a day of the week. Datetime_Format provides a way to get the ‘day of the week’.
So you could do this with something like IF(DATETIME_FORMAT(TODAY(), 'dddd') = 'Thursday',{Total Load}-{Total Load}*{Deduction amount})
This should check if today is ‘Thursday’ and then provide the ‘Total Pay Out’. Replace ‘Thursday’ with whatever day of the week you want, and replace the payout formula with whatever you already
have. Of course, the rest of the week it will show nothing. | {"url":"https://community.airtable.com/t5/formulas/how-to-make-a-deduction-automatic-but-only-once-a-week/m-p/95611/highlight/true","timestamp":"2024-11-11T01:04:54Z","content_type":"text/html","content_length":"447838","record_id":"<urn:uuid:9f1b81ec-c62f-4de0-97ed-a94e1ae35ca1>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00078.warc.gz"} |
Telegram channel "QUIZ TIME" — @quiz_time11 — TGStat
▎Grade 12 English Quiz
Question 1: Which of the following is a synonym for "diligent"?
A) Lazy
B) Hardworking
C) Careless
D) Negligent
Question 2: In the sentence "The cat sat on the mat," what part of speech is the word "sat"?
A) Noun
B) Verb
C) Adjective
D) Adverb
Question 3: What is the main theme of George Orwell's novel "1984"?
A) The importance of friendship
B) The dangers of totalitarianism
C) The beauty of nature
D) The power of love
Question 4: Identify the figurative language in the sentence: "Time is a thief."
A) Simile
B) Metaphor
C) Personification
D) Hyperbole
Question 5: Which of the following sentences is grammatically correct?
A) He go to school every day.
B) She going to the market.
C) They plays soccer on weekends.
D) I have finished my homework.
Question 6: In poetry, what is a stanza?
A) A type of rhyme scheme
B) A group of lines forming a unit
C) A literary device used to create imagery
D) A character in a poem
Question 7: What does the term "alliteration" refer to?
A) The repetition of vowel sounds
B) The repetition of consonant sounds at the beginning of words
C) The use of exaggeration for emphasis
D) A comparison using "like" or "as"
Question 8: Which of the following is an example of an oxymoron?
A) Bitter sweet
B) Bright night
C) Soft rock
D) Loud silence
Question 9: What is the purpose of a thesis statement in an essay?
A) To provide background information
B) To summarize the entire essay
C) To present the main argument or claim
D) To introduce a quote
Question 10: In literature, what is "foreshadowing"?
A) A technique used to create suspense
B) A brief reference to a historical event
C) An introduction to a character
D) A comparison between two unlike things
Take our fun and engaging quizzes!
Click here👇👇👇👇to get started:
QUIZ TIME
#QuizTime #ChallengeYourself #BrainTeaser
✨ || Physics Multiple Choice questions with answers
▎Question 1:
What is the net force acting on an object in equilibrium?
A) Equal to its mass
B) Zero
C) Equal to its weight
D) Equal to its acceleration
Answer: B) Zero
▎Question 2:
An object is moving in a circular path at a constant speed. Which of the following statements is true?
A) The object is not accelerating.
B) The object is accelerating due to the change in direction.
C) The object's velocity is constant.
D) The net force acting on the object is zero.
Answer: B) The object is accelerating due to the change in direction.
▎Question 3:
According to Newton's third law of motion, if a person pushes against a wall with a force of 50 N, the wall:
A) Pushes back with a force of 50 N.
B) Pushes back with a force of 100 N.
C) Does not push back.
D) Pushes back with a force less than 50 N.
Answer: A) Pushes back with a force of 50 N.
▎Question 4:
A car accelerates from rest at a rate of 3 m/s². How far does it travel in the first 5 seconds?
A) 15 m
B) 30 m
C) 37.5 m
D) 75 m
Answer: C) 37.5 m
▎Question 5:
What happens to the momentum of an object if no external forces act on it?
A) It increases.
B) It decreases.
C) It remains constant.
D) It can become negative.
Answer: C) It remains constant.
▎Question 6:
In the absence of a net force, a moving object will
A) slow down and eventually stop
B) stop immediately
C) turn right
D) move with constant velocity
Answer: D) move with constant velocity
▎Question 7:
The acceleration of an object is inversely proportional to
A) the net force acting on it
B) its position
C) its velocity
D) its mass
Answer: D) its mass
▎Question 8:
Which one of the following activities of the experiment is not necessary to determine the specific latent heat of vaporization of water?
A) measuring time
B) measuring mass
C) measuring volume
D) measuring temperature
Answer: C) measuring volume
▎Question 9:
If two objects collide and stick together, what type of collision is this?
A) Elastic collision
B) Inelastic collision
C) Perfectly elastic collision
D) Super elastic collision
Answer: B) Inelastic collision
▎Question 10:
In a frictionless environment, if a car travels at a constant speed around a circular track, which of the following statements is true?
A) The car's acceleration is zero.
B) The car's velocity is constant.
C) There is a net force acting on the car towards the center of the circle.
D) The car will eventually come to a stop.
Answer: C) There is a net force acting on the car towards the center of the circle.
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First round geography model exam.pdf
📚2017 Geography FIRST ROUND MODEL EXAMINATION
⏰TIKMIT 2017/ NOV 2024
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Let f(x) is any polynomial function, then when f(x) is divided by x - c, the remainder is always constant.
• False
• True
749 votes
🔥🔥 Chemistry multiple choice questions and answers
1. What aspect of the modern view of atomic structure was proved by Rutherford's gold foil experiment?
(A) the existence of the nucleus
(B) the charge on an electron
(C) the charge on alpha particle
(D) the existence of electrons
Answer A
2. Which of the following particle was discovered by cathode ray experiment?
(A) neutron
(B) proton
(C) electron
(D) nucleus
Answer C
3. The maximum number of electrons that can be accommodated in a sublevel for which l = 3 is:
(A) 2
(B) 10
(C) 6
(D) 14
Answer D
4. The Heisenberg Principle states that____
(A) no two electrons in the same atom can have the same set of four quantum numbers.
(B) two atoms of the same element must have the same number of protons.
(C) it is impossible to determine accurately both the position and momentum of an electron simultaneously.
(D) electrons of atoms in their ground states enter energetically equivalent sets of orbitals singly before they pair up in any orbital of the set.
Answer C
5. The neutral atoms of all of the isotopes of the same element have
(A) different numbers of protons.
(B) equal numbers of neutrons.
(C) the same number of electrons.
(D) the same mass numbers.
Answer C
6. In the Lewis structure for the OF2 molecule, the number of lone pairs of electrons around the central oxygen atom is
(A) 0
(B) 1
(C) 2
(D) 3
Answer C
7. Which is classified as nonpolar covalent?
(A) the H-I bond in HI
(B) the H-S bond in H2S
(C) the P-Cl bond in PCl3
(D) the N-Cl bond in NCl3
Answer D
8. The valence electrons of representative elements are
(A) in s orbitals only.
(B) located in the outermost occupied major energy level.
(C) located closest to the nucleus.
(D) located in d orbitals.
Answer B
9. What is the total number of electrons in the correct Lewis dot formula of the sulfite ion(SO₃²⁻)?
(A) 8
(B) 24
(C) 26
(D) 30
Answer C
10. Which one of the compounds below is most likely to be ionic?
(a) GaAs
(b) CaCl2
(c) NO2
(d) CCl4
Answer B
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1. What is the purpose of DNA replication?
A) To produce proteins
B) To create energy
C) To ensure that each new cell has a complete set of DNA
D) To repair damaged DNA
Answer: C) To ensure that each new cell has a complete set of DNA
2. What does the term "homozygous" mean?
A) Having two different alleles for a trait
B) Having two identical alleles for a trait
C) Having multiple alleles for a trait
D) Having no alleles for a trait
Answer: B) Having two identical alleles for a trait
3. Which of the following best describes a phenotype?
A) The genetic makeup of an organism
B) The physical appearance of an organism
C) The location of genes on a chromosome
D) The sequence of nucleotides in DNA
Answer: B) The physical appearance of an organism
4. What is the role of tRNA in protein synthesis?
A) To carry genetic information from DNA to ribosomes
B) To bring amino acids to the ribosome for protein assembly
C) To form the structure of ribosomes
D) To replicate DNA
Answer: B) To bring amino acids to the ribosome for protein assembly
5. Which process results in the formation of gametes?
A) Mitosis
B) Meiosis
C) Binary fission
D) Budding
Answer: B) Meiosis
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Biology MCQs about genetics:
1. What is the basic unit of heredity?
A) Chromosome
B) Gene
C) DNA
D) RNA
Answer: B) Gene
2. Which of the following structures carries genetic information?
A) Ribosome
B) Chromosome
C) Endoplasmic reticulum
D) Golgi apparatus
Answer: B) Chromosome
3. What is the term for different forms of a gene?
A) Alleles
B) Genotypes
C) Phenotypes
D) Chromatids
Answer: A) Alleles
4. In Mendelian genetics, what is the expected phenotypic ratio for a monohybrid cross?
A) 1:1
B) 3:1
C) 9:3:3:1
D) 1:2:1
Answer: B) 3:1
5. What is the function of messenger RNA (mRNA)?
A) To replicate DNA
B) To transport amino acids
C) To carry genetic information from DNA to the ribosome
D) To form the structure of ribosomes
Answer: C) To carry genetic information from DNA to the ribosome
6. Which of the following is a type of mutation?
A) Insertion
B) Deletion
C) Substitution
D) All of the above
Answer: D) All of the above
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⭕️grammar test explanation ⭕️
Why these are the correct answers:
So a bit of bit, let take a look to questions nd the answer!!!
Why today's test have these answers:
So see them and notice to explanations:👇
Number 1: B
✳️this sentences is first conditional, so regard to that we should use even if.
Number2: B
✳️why each is correct? When we use each that we isolated one of the members in one hand.
Number3: B
✳️this is simple present nd we should use stops.
Number 4: A
✳️we use simple future perfect tense from an action that u know it will be done in near future nd in the other hand u use it from an action that u expected that it will be done in a success way
So we use future perfect✅
Number 5: A
✳️we use present perfect, that we did sth in past but it still continue and one those signs are for+...... ,since+......... .
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Since the subject was unfamiliar, ___students could follow the speaker.
• many
• few
• a few
• little
682 votes
✅ #grammar_test
👨👩👧👧kings and queens👨👩
1️⃣'So you definitely won't come to the concert on Saturday?' 'No, [ _ _ _ _ _ ] you pay for my ticket.'
A. not if
B. not even if
C. not even
2️⃣'Is this a difficult game to play?' 'No, it's easy. [ _ _ _ _ _ ] has to collect as many cards as they can.'
A. Every player
B. Each player
C. All the players
3️⃣'Which bus [ _ _ _ _ _ ] near the station?' 'The number 8, I think.'
A. does stop
B. stops
C. did stop
4️⃣I think the opening night [ _ _ _ _ _ ] a great success. Lots of people say they are coming to watch the show.
A. will be
B. will have been
C. will be being
5️⃣I'm starting to play the guitar again. I [ _ _ _ _ _ ] for years.
A. haven't played
B. haven't been playing
C .wasn't playing
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## Trigonometry Challenge: Get Your Angle On!
Instructions: Choose the best answer for each question.
1. If sin θ = 3/5 and θ is in Quadrant I, what is the value of cos θ?
A. 4/5
B. -4/5
C. 3/4
D. -3/4
Answer: A
2. What is the exact value of tan (150°)?
A. √3
B. -√3
C. 1/√3
D. -1/√3
Answer: D
The correct answer is D. -1/√3. Here's how to find it:
👉150° is in the second quadrant. Its reference angle is 30° (180° - 150°). Tangent is negative in the second quadrant.vtan(30°) = 1/√3. Since tan(150°) is negative and has the same magnitude as tan
tan(150°) = -1/√3
3. If cos θ = -1/2 and θ is in Quadrant II, what is the value of sin θ?
A. √3/2
B. -√3/2
C. 1/2
D. -1/2
Answer: A
Explanation: Use the Pythagorean identity and the fact that sin θ is positive in Quadrant II.
4. Simplify the expression: (sin²x + cos²x) / tan x
A. cos x
B. sin x
C. cot x
D. tan x
Answer: C
Explanation: Use the Pythagorean identity and the definition of Cotangent.
5. Solve the equation: 2sin²x - sin x - 1 = 0 for 0 ≤ x ≤ 2π.
A. π/2, 3π/2
B. π/6, 5π/6
C. π/3, 2π/3
D. π/6, 5π/6, 3π/2
Answer: D
Explanation: Factor the quadratic equation and solve for sin x. Find the angles where sin x satisfies the solutions.
6. Simplify the expression: (1 + cot²θ) / (1 + tan²θ)
A. cot²θ
B. tan²θ
C. 1
D. sin²θ
Answer: A
Explanation: Use the Pythagorean identities for cot²θ and tan²θ.
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Cell Parts/Function Practice Test
1. Which cell feature is responsible for making proteins?
A: lysosomes
B: ribosomes
C: mitochondria
2. What is the name of the jelly-like substance that is inside the cell?
A: cytoplasm
B: ectoplasm
C: cytokinesis
3. What cell feature is responsible for powering the cell?
A: endoplasmic reticulum
B: golgi bodies
C: mitochondria
4. Where in the cell does chromatin (DNA) found?
A: ribosomes
B: nucleus
C: nucleolus
5. What are two features that plant cells have that animal cells do not?
A: lysosome and cell walls
B: cell wall and chloroplasts
C: cell membrane and nucleolus
6. What cell feature contains digestive enzymes which breaks things down?
A: lysosomes
B: ribosomes
C: vacuoles
7. Which cell feature packages and moves things around the cell?
A: endoplasmic reticulum
B: chloroplasts
C: golgi bodies
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Maths work book.pdf
✅Maths question🚩
📚This will help you to do more question in suitable way for entrance examination.
⌨Don't forget math is about doing more questions.😉
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Try these questions
Q1.Today is Monday. After 61 days, it will be:
Q2.The sum of ages of 5 children born at the intervals of 3 years each is 50 years. What is the age of the youngest child?
A.4 years
B.8 years
C.10 years
D.None of these
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1) In eukaryotic cells, MOST of cellular respiration takes place in the___
(A) Nuclei
(B) Cytoplasm
(C) Mitochondria
(D) Cell walls
Answer C
2) Sister chromatids are attached to each other at an area called
(A) Centriole
(B) Spindle
(C) Centromere
(D) Chromosome
Answer C
3) In order for the cell to divide successfully, the cell must first
(A) Duplicate its genetic information
(B) Decrease its volume
(C) Increase its number of chromosomes
(D) decrease its number of organelles
Answer A
4) The rate at which materials enter and leave the cell depends on the cell‘s
(A) Volume
(B) Weight
(C) Speciation
(D) Surface area
Answer D
5) During which phase of meiosis is chromosome number reduced?
(A) Anaphase I
(B) Metaphase I
(C) Telophase I
(D) Telophase II
Answer D
6) In a typical plant, all of the following factors are necessary for photosynthesis EXCEPT
(A) Chlorophyll
(B) Light
(C) Oxygen
(D) Water
Answer C
7) A Punnett square is used to determine the
(A) Probable outcome of a cross
(B) Result of incomplete dominance
(C) Actual outcomes of a cross
(D) Result of meiosis
Answer A
8) Which of the following does NOT describe the structure of DNA?
(A) Double helix
(B) Nucleotide polymer
(C) Contain adenine-guanine pair
(D) Bacteria contain DNA but not protein
Answer C
9) Which process always involves the movement of materials from inside the cell to outside the cell?
(A) Phagocytosis
(B) Exocytosis
(C) Endocytosis
(D) Osmosis
Answer B
10) The nucleus includes all of the following structures EXCEPT
(A) Cytoplasm
(B) A nuclear membrane
(C) DNA
(D) A nucleolus
Answer A
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2017 English First round model examinations.pdf
📚2017 ENGLISH FIRST ROUND MODEL EXAMINATION
⏰TIKMIT 2017/ NOV 2024
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Try it
Q1. Hydrocarbons are organic compounds with element __
a) Hydrogen
b) Oxygen
c) Carbon
d) Both hydrogen and carbon
Q2. Find the odd one out.
a) Aromatic
b) Alkanes
c) Alkynes
d) Alkenes
Q3. Identify the addition reaction which is not undergone by the alkenes?
a) Mercuration
b) Oxymercuration
c) Hydroboration
d) Halogenation
Q4. 4-chlorobut-1-ene is the name of which among the following alkenes?
a) CH2Cl-CH2=CH-CH2
b) CH2Cl-CH2-CH-CH2
c) CH2Cl=CH2-CH=CH2
d) CH2Cl-CH2-CH=CH2
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Try this probability question
1.Which of these numbers cannot be a probability?
a) -0.00001
b) 0.5
c) 1.001
d) 0
e) 1
f) 20%
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❓|| SAT(Scholastic aptitude test) English part multiple choice questions and answers
I. Choose the Synonym of the given words
A) dangerous
B) careful
C) slippery
D) favourable
Answer A
A) clear
B) equal
C) brand
D) ambiguous
Answer D
3) BENEVOLENT
A) cruel
B) selfish
C) generous
D) happy
Answer C
4. COSMIC
A) universal
B) estranged
C) ambiguous
D) authoritative
Answer A
5. AVARICIOUS
A) humble
B) greedy
C) irritable
D) intelligent
Answer B
II) Analogy
6. FOOT : SKATEBOARD
A) tire : automobile
B) lace : shoe
C) ounce : scale
D) pedal : bicycle
Answer D
7. HAT : HEAD
A) cold : hot
B) winter : snow
C) glove : hand
D) basic : advanced
Answer C
8. PHOBIC : FEARFUL
A) finicky : thoughtful
B) cautious : emotional
C) envious : desiring
D) ridiculous : silly
Answer D
9. TEACHER : SCHOOL
A) actor : role
B) judge : courthouse
C) mechanic : engine
D) jockey : horse
Answer B
10. PHARMACY : DRUGS
A) mall : store
B) doctor : medicine
C) bakery : bread
D) bell : church
Answer C
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1k 0 13 1 7
6.8k 3 18 6
1.1k 0 25 1 12
1.3k 0 17 4 20
5.6k 1 82 3 8
1.4k 0 14 9
5.2k 1 23 15
1.5k 0 24 9
5.9k 1 100 8
1.8k 0 39 4 23
5.5k 1 40 3
1.8k 0 18 3 10
1.7k 0 1 2
1.7k 0 44 1 9
6.2k 1 58 1 12
6.4k 1 69 9
1.7k 0 10 3 5
2k 0 51 11 | {"url":"https://et.tgstat.com/channel/@quiz_time11","timestamp":"2024-11-10T19:17:43Z","content_type":"text/html","content_length":"209884","record_id":"<urn:uuid:dc5ec928-ae12-4b14-aaa2-35c7df107b30>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00009.warc.gz"} |
6) find the area enclosed by the Curve
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6) find the area enclosed by the Curve
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Question Text 6) find the area enclosed by the Curve
Updated On Sep 12, 2023
Topic Integration
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 93
Avg. Video Duration 14 min | {"url":"https://askfilo.com/user-question-answers-mathematics/6-find-the-area-enclosed-by-the-curve-35353236363531","timestamp":"2024-11-05T04:35:25Z","content_type":"text/html","content_length":"238919","record_id":"<urn:uuid:7289cadb-5fba-4e89-b163-a499a1faaf50>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00079.warc.gz"} |
Talk:Bekenstein bound
From Scholarpedia
Reviewer B
I think this is a very nice introduction to the Bekenstein bound. I also agree with the author that all early objections to the derivation of the bound from the generalized second law were eventually
responded satisfactorily. However, there is one important point raised in (D. Marolf, D. Minic, S. F. Ross, Phys. Rev. D69, 064006 (2004), D. Marolf, arXiv:hep-th/0410168) which could also be
addressed by the article, and which may lead to substantial modifications to the content of the bound. The essential point made by these authors is that in the region exterior to the black hole one
should count also the entropy of the thermal atmosphere, which is present before and after the object falls down to the black hole. Taking into account these entropies could make the bound to hold
regardless of the object matter content. This point is related with the fact that the quantities involved in the bound, energy and entropy, are ill defined for a localized state in a relativistic
theory. This is due to particle pair production induced by the localization. In order to properly define the bound the entropy and energy of this cloud of created particles should be taken into
account (see for example my article in Class. Quant. Grav. 25, 205021 (2008), arXiv:0804.2182).
H. Casini
Reviewer A
In the article `Bekenstein bound' the problem of `many species' has been somewhat paramount. even though it is covered in the reference given (Bekenstein 2005) at the end, it seems to me more
fundamental than the other objections that he addresses in that paper. perhaps a comment on this could be included in the article (i think this problem might make te Bekenstein bound somewhat
different in principle from the holographic bound, for instance) | {"url":"http://www.scholarpedia.org/article/Talk:Bekenstein_bound","timestamp":"2024-11-10T05:23:56Z","content_type":"text/html","content_length":"20750","record_id":"<urn:uuid:0c54d5b4-e4d6-4b7c-996f-9ea07d07ab5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00848.warc.gz"} |
Questions on Integration and Vector Algebra
Integration is another aspect of calculus. It is sometimes called anti-differentiation because it is the reverse of differentiation. You can attempt up to 22 questions on differentiation by clicking
on this link.
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What is Vector Algebra?
A Vector is a quantity having both magnitude and direction. And since you can’t talk about Vector without mentioning Scalar, it would be good to note that Scalar is a quantity that has magnitude but
one direction. (kindly use the comment box below to ask a question).
Questions on Integration
1. The gradient of a curve passing through the point (3,1) is given by:
dy/dx = x^2 – 4x + 3
Find the equation of the curve and the area enclosed by the curve, the maximum and minimum ordinates and the x-axis.
2. The marginal revenue of a firm is found to be 15 – 4q. Determine the revenue function if from past sales the revenue was N1,250 when 28 units of the product were sold.
3. If the marginal cost of producing a product in a firm is:
MC = 5q^2 – 7q + 20,
Find the total variable cost of producing the first 100 units of the product in the firm.
4. Determine the customer’s surplus of a commodity at a price of N15 given that the demand function is p = 24 – q^2.
5. If the demand function for a product is p = 155 – 3q and the supply function is p = 7q – 20
• The equilibrium price and quantity.
• The consumer’s surplus.
• The producer’s surplus.
Questions on Vector Algebra
1. A security firm decides to sell four of its stocks. If the shares are 200, 150, 300 and 250 for stocks A, B, C, and D respectively and the selling price per share N200 for A, N100 for B, N300 for
C and N250 for stock D. Calculate the total receipts from the stocks.
2. Raphael Investment Limited decides to sell three of its stock. 300 shares of stock P, 400 shares of stock Q and 200 shares of stock R were sold. The selling price per share was N10, N8, and N5
respectively. Determine the total amount received from the sales of the stock.
3. If A = 3i – 2J + k and B = 2i + 3J – 4k
• A . B and angle between the vectors A and B
• A x B
• |A| and |B|
• Hence find the angle between the vectors.
4. Find the vector product of the following:
• A = 2i + J and b = I -3J + K
• (8,4,2) and (6,4,-3)
5. If r = 3i – 2J + k
• Find the magnitude r
• Find the unit vector in the direction of r
• Calculate the direction cosine of r.
6. Show that p and q are perpendiculars where:
P = 3i – 4 J + 5k and
q = 2i + J + 2k
Other Mathematics Questions
HS Tutorial on YouTube
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Current ratio vs quick ratio: Which is best? +formulas
Current ratio vs quick ratio: Which is best? +formulas
Corporate finance teams use these ratios to inform decisions about taking on more debt, paying dividends, buying back shares and managing cash flow. Compare a company’s current ratio and quick ratio
over time to identify trends. Companies usually keep most of their quick assets in the form of cash and short-term investments (marketable securities) to meet their immediate financial obligations
that are due in one year. Accounts receivable, cash and cash equivalents, and marketable securities are the most liquid items in a company. Current assets (also called short-term assets) are cash or
any other asset that will be converted to cash within one year. You can find them on the balance sheet, alongside all of your business’s other assets.
• A lower quick ratio could mean that you’re having liquidity problems, but it could just as easily mean that you’re good at collecting accounts receivable quickly.
• In the example above, the quick ratio of 1.19 shows that GHI Company has enough current assets to cover its current liabilities.
• Unlike current ratio, quick ratio calculations only use quick assets or short-term investments that can be liquidated to cash in 90 days or less.
• ProfitWell Metrics provides real-time, accurate subscription reporting and analytics in one dashboard.
• For instance, while the current ratio takes into account all of a company’s current assets and liabilities, it doesn’t account for customer and supplier credit terms, or operating cash flows.
These ratios can also provide help in analyzing the long-term solvency as well. By comparing ratios over time and against industry benchmarks, review boards can spot negative trends. For example, a
declining current ratio alongside growing inventory and accounts receivable could suggest liquidity problems ahead.
A company with a current ratio of less than one doesn’t have enough current assets to cover its current financial obligations. XYZ Inc.’s current ratio is 0.68, which may indicate liquidity problems.
In Year 1, the quick ratio can be calculated by dividing the sum of the liquid assets ($20m Cash + $15m Marketable Securities + $25m A/R) by the current liabilities ($150m Total Current Liabilities).
By dividing the current assets balance of the company by the current liabilities balance in the coinciding period, we can determine the current ratio for each year.
Current ratio vs. quick ratio: What’s the difference?
A current ratio of 1.0 means current assets perfectly cover current liabilities. Meanwhile, a ratio higher than 1.0 suggests good short-term financial health. Putting the above together, the total
current assets and total current liabilities each add up to $125m, so the current ratio is 1.0x as expected. This current ratio is classed with several other financial metrics known as liquidity
ratios. These ratios all assess the operations of a company in terms of how financially solid the company is in relation to its outstanding debt.
• Company B has more cash, which is the most liquid asset, and more accounts receivable, which could be collected more quickly than liquidating inventory.
• Start with a free account to explore 20+ always-free courses and hundreds of finance templates and cheat sheets.
• In this example, although both companies seem similar, Company B is likely in a more liquid and solvent position.
• For example, inventory build-ups before peak sales seasons can temporarily increase the ratio.
Finally, the operating cash flow ratio compares a company’s active cash flow from operating activities (CFO) to its current liabilities. This allows a company to better gauge funding capabilities by
omitting implications created by accounting entries. For example, a normal cycle for the company’s collections and payment processes may lead to a high current ratio as payments are received, but a
low current ratio as those collections ebb. Calculating the current ratio at just one point in time could indicate that the company can’t cover all of its current debts, but it doesn’t necessarily
mean that it won’t be able to when the payments are due. However, when evaluating a company’s liquidity, the current ratio alone doesn’t determine whether it’s a good investment or not. It’s
therefore important to consider other financial ratios in your analysis.
What is a good current ratio for a company?
With NetSuite, you go live in a predictable timeframe — smart, stepped implementations begin with sales and span the entire customer lifecycle, so there’s continuity from sales to services to
support. In addition, the business could have to pay high interest rates if it needs to borrow money. These typically have a maturity period of one year or less, are bought and sold on a public stock
exchange, and can usually be sold within three months on the market. Let’s say, for instance, these are the numbers from your SaaS financial statements. Licensing flexibility, unlimited growth
potential, and scalability are some of the upsides of the SaaS business model.
What is the current ratio?
At the 2022, the company reported $154.0 billion of current liabilities, almost $29 billion greater than current liabilities from the prior period. A ratio under 1.00 indicates that the company’s
debts due in a year or less are greater than its assets—cash or other short-term assets expected to be converted to cash within a year or less. A current ratio of less than 1.00 may seem alarming,
although different situations can negatively affect the current ratio in a solid company. For example, a company may have a very high current ratio, but its accounts receivable may be very aged,
perhaps because its customers pay slowly, which may be hidden in the current ratio. Analysts also must consider the quality of a company’s other assets vs. its obligations.
Current liabilities
Likewise, current liabilities are the debts your company owes that are due and payable within a year. The current ratio is called current because, unlike some other liquidity ratios, it incorporates
all current assets and current liabilities. A company can’t exist without cashflow and the ability to pay its bills as they come due. By measuring its quick ratio, a company can better understand
what resources they have in the very short-term in case they need to liquidate current assets. Though other liquidity ratios measure a company’s ability to be solvent in the short-term, the quick
ratio is among the most aggressive in deciding short-term liquidity capabilities. The quick ratio looks at only the most liquid assets that a company has available to service short-term debts and
In most companies, inventory takes time to liquidate, although a few rare companies can turn their inventory fast enough to consider it a quick asset. Prepaid expenses, though an asset, cannot be
used to pay for invoice templates 2021 current liabilities, so they’re omitted from the quick ratio. Both the current ratio and the quick ratio are considered liquidity ratios, measuring the ability
of a business to meet its current debt obligations.
The current ratio includes current assets that mature, expire, or can be converted within one year. Once both figures are ready, we can divide the quick assets by current liabilities to find the
quick ratio for the current accounting period. The current ratio is directly linked with the working capital management of a business. Both the current assets and current liabilities have the biggest
shares of items that directly relate to the working capital. The current and quick ratios are good indicators of the short-term liquidity of a business.
Write a Comment | {"url":"https://weboo.in/blog/current-ratio-vs-quick-ratio-which-is-best-2/","timestamp":"2024-11-04T04:55:33Z","content_type":"text/html","content_length":"77417","record_id":"<urn:uuid:c9923d0e-9e1f-4d49-b290-d4973457dea8>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00510.warc.gz"} |
Stock R has a beta of 2.0, Stock S has a beta of 0.95, the
Stock R has a beta of 2.0, Stock S has a beta of 0.95, the required...
Stock R has a beta of 2.0, Stock S has a beta of 0.95, the required return on an average stock is 13%, and the risk-free rate of return is 5%. By how much does the required return on the riskier
stock exceed the required return on the less risky stock? Round your answer to two decimal places.
Beta of Stock R = 2.0
Beta of Stock S = 0.95
The higher the Beta the more risker it is. Thus, Stock R is more risker as having higher Beta
As per CAPM,
where, rf = Risk free return = 5%
Rm = Market Return or average stock return = 13%
- Beta of Stock R = 2.0
Required Return of Stock R = 5% + 2.0(13%-5%)
Required Return of Stock R = 21%
- - Beta of Stock S = 0.95
Required Return of Stock S = 5% + 0.95(13%-5%)
Required Return of Stock S = 12.60%
So, the required return on the riskier stock exceed the required return on the less risky stock = Required Return of Stock R - Required Return of Stock S
= 21% - 12.60%
= 8.4%
Thus, Required Return of Risker Stock exceed less risky stock by 8.40%
If you need any clarification, you can ask in comments.
If you like my answer, then please up-vote as it will be motivating | {"url":"https://justaaa.com/finance/50996-stock-r-has-a-beta-of-20-stock-s-has-a-beta-of","timestamp":"2024-11-05T06:41:34Z","content_type":"text/html","content_length":"41430","record_id":"<urn:uuid:45bd7d28-5bd9-4d45-a55a-2ea056ca34a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00025.warc.gz"} |
vector of vectors in c++
A vector is a dynamic array that is built into a standard C++ library. This vector class can grow in size dynamically. The elements are arranged in a continuous manner so that iterators can traverse
them. To learn more about vectors, you can visit here.
In this article, we will see an introduction to vectors, 2D vectors, vector of vectors in c++, examples of vectors, and implementation in detail.
2D vectors
A vector of the vector is a 2D vector. A 2D vector is like a 2D array which can be declared and assigned values.
Now, let’s understand the 2d vectors with the help of an implementation.
Implementation 1
/* Program to demonstrate a 2D vector where
each of its elements is of a different size.*/
#include <iostream>
#include <vector>
using namespace std;
int main()
We initialize a 2D vector
named "vect" on line 16 with
different number of values
in each element.
vector<vector<int>> vec{
/* Element one with two values in it. */
{'a', 'b'},
/* Element two with three values in it. */
{'c', 'd', 'e'},
/* Element three with four values in it. */
{'f', 'g', 'h', 'i'}};
Now we will print the vector that we
just defined using the simple
nested for loops.
for (int m = 0; m < vec.size(); m++)
for (int k = 0; k < vec[m].size(); k++)
char c = (char)vec[m][k];
cout << c << " ";
cout << "\n";
return 0;
You can also try this code with Online C++ Compiler
Run Code
Time complexity
O(N*N) since N is the size of the vector.
Space complexity
O(N) as N is the size of the vector.
2D vectors are commonly depicted as a matrix with "rows" and "columns". They are 2D vector elements under the hood.
We then establish an integer variable called "row," followed by an array called "column," which will hold the size of the row. We will then proceed to initialize the memory of each row by the column
Implementation 2
// Program to insert into a vector of vectors in c++
// CPP program
#include <iostream>
#include <vector>
using namespace std;
int main()
int row;
int col;
cout << "Enter the number of row and column respectively\n";
cin >> row >> col;
Create a vector containing "n"
vectors each of size "m".
cout << "2D vector Matrix after inserting the values\n";
vector<vector<int>> vect(row, vector<int>(col));
for (int i = 0; i < row; i++)
for (int j = 0; j < col; j++)
vect[i][j] = j + i + 1;
for (int i = 0; i < row; i++)
for (int j = 0; j < col; j++)
cout << vect[i][j] << " ";
cout << endl;
return 0;
You can also try this code with Online C++ Compiler
Run Code
Time complexity
O(row*col) as we have used a nested loop to build and print a matrix.
Space complexity
The space complexity is O(N), where N is the size of the vector.
Read More - Time Complexity of Sorting Algorithms | {"url":"https://www.naukri.com/code360/library/working-with-vectors-of-vectors-in-cpp","timestamp":"2024-11-10T00:05:09Z","content_type":"text/html","content_length":"420927","record_id":"<urn:uuid:b49f82b7-70f5-4510-ab9c-cdd899f1b046>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00326.warc.gz"} |
Coin Puzzles
Here are a couple of puzzles from Henry Dudeney's article "The Best Puzzles with Coins," in the Strand Magazine, July 1909. These are maybe less interesting than the mathematical puzzles that John
and I have been doing, but for whatever reason, they caught my eye.
Kissing Coins #
If I lay a penny flat on the table, how many other pennies can I place around it, every one also lying flat on the table, so that they all touch the first one?
It needn't be pennies, of course. Any coin (or any disc) will do, as long as all the coins in question are the same size. Try to do this one by geometry, not experiment.
Making Change #
If you split the check at restaurants a lot, then you probably have practical experience with this one, or at least you did until you and all your friends got Venmo. Remember, this puzzle is from
1909, though I've reworded it slightly.
A tourist went into a shop in New York and bought goods at a cost of thirty-four cents. The only money they had was a dollar (100 cents), a three-cent piece, and a two-cent piece.Yes! The U.S.
two-cent piece and three-cent piece really existed, although by 1909 they probably weren't in circulation anymore. The seller had only a half-dollar and a quarter (25 cents). But another customer
happened to be present, and when asked to help produced two dimes (dime = 10 cents), a nickel (5 cents), a two-cent piece, and a penny (1 cent). How did the seller make change?
Solutions beneath the 3-cent nickel.
Solutions #
Kissing Coins #
Imagine three coins of radius , and lay them out so that they all just kiss each other:
Then the triangle formed by the center of the three coins is an equilateral triangle with sides . Each interior angle of the triangle is 60°.
360°/ 60° = 6, so we can lay out six such triangles around a center coin, where each coin just kisses its two neighboring coins. That's six coins (because every perimeter coin is in two triangles).
Making change #
At the start, the allocations are:
• buyer: 100¢, 3¢, 2¢, and wants to make a 34¢ purchase
• seller: 50¢, 25¢
• customer: 10¢, 10¢, 5¢, 2¢, 1¢
If the buyer gives the seller a dollar, then the change is 66¢, which obviously isn't going to work. So the buyer changes their 3¢ coin with the customer, for the 2¢ coin and the penny. This gives:
• buyer: 100¢, 2¢, 2¢, 1¢
• seller: 50¢, 25¢
• customer: 10¢, 10¢, 5¢, 3¢
Now the buyer gives the seller $1.04, and needs 70¢ back. So the seller changes their quarter with the customer for two dimes and a nickel:
• buyer: [S:100¢, 2¢, 2¢,:S] 1¢ and a 34¢ purchase
• seller: (100¢, 2¢, 2¢) 50¢, 10¢, 10¢, 5¢,
• customer: 25¢, 3¢
A half-dollar and two dimes is 70¢, so the seller can make change, and everyone is happy. | {"url":"https://ninazumel.com/blog/2024-10-08-coin-puzzles/","timestamp":"2024-11-06T05:09:46Z","content_type":"text/html","content_length":"24316","record_id":"<urn:uuid:ae5f6839-7c14-4285-bdc8-00ec09ee4319>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00646.warc.gz"} |
Multiplication Worksheets 100 Per Sheet
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Documentation/Calc Functions/MINIFS
Function name:
Statistical Analysis
Identifies the minimum value of a set of numbers in a cell range, with the cells to be considered determined using multiple criteria.
The criteria passed to MINIFS can utilize wildcards or regular expressions.
MINIFS(Min range; Range 1; Criteria 1 [ ; Range 2; Criteria 2 [;... ; [Range 127; Criteria 127]]])
Returns a real number, which is the minimum value in the relevant cells.
Min range argument
Min range specifies the cell range from which a subset will be selected for determining the minimum value. Min range is a reference to a cell range (which may not utilize the reference concatenation
operator (~)), the name of a named range, or the name of a database range.
Range arguments
Range 1 specifies the set of cells to be matched against Criteria 1 and takes one of the forms listed for Min range. Range 1 should have the same dimensions as Min range.
Range 2, ..., Range 127 have the same meaning as Range 1.
Criteria arguments
Criteria 1 is the criterion for matching against the cells in Range 1, or a cell containing that criterion. Criteria 1 can take one of the following forms:
• A number, such as 34.5. Dates and logical values (TRUE or FALSE) are treated as numbers.
• An expression, such as 2/3, SQRT($D$1), or DATE(2021; 11; 1).
• A text string, such as "golf" or "<>10".
MINIFS looks for cells in Range 1 that are equal to Criteria 1, unless Criteria 1 is a text string that starts with a comparator (>, <, >=, <=, =, or <>). In the latter case MINIFS compares the cells
in Range 1 with the remainder of the text string (interpreted as a number if possible and text otherwise). For example, the condition ">4.5" tests if the content of each cell is greater than the
number 4.5, the condition "<dog" tests if the content of each cell comes alphabetically before the text "dog", and the condition "<>2021-11-01" tests if the content of each cell is not equal to the
specified date.
Criteria 1 supports the following specific behaviors:
• The string "=" matches empty cells. For example the formula =MINIFS(A1:A10; B1:B10; "=") returns the minimum of all values in the range A1:A10 if all cells in the range B1:B10 are empty. Note
that "=0" does not match empty cells.
• The string "<>" matches non-empty cells. For example the formula =MINIFS(A1:A10; C1:C10; "<>") returns the minimum of all values in the range A1:A10 if there are no empty cells in the range
• If the value after the <> comparator is not empty, then Criteria 1 matches any cell content except that value, including empty cells.
Criteria 2, ..., Criteria 127 have the same meaning as Criteria 1.
Error conditions
• If any cell range passed as an argument contains a reference concatenation operator (~), then MINIFS reports an invalid argument error (Err:502).
• All the cell ranges passed as arguments (Min range and Range 1, …, Range 127) must occupy the same number of rows and the same number of columns. If this is not the case, then MINIFS reports an
invalid argument error (Err:502).
• If the Range n and Criteria n arguments are not correctly paired, then MINIFS reports a variable missing error (Err:511).
• If no cell is selected via the matching conditions, then MINIFS returns 0.
• If no cell selected via the matching conditions contains a number, then MINIFS returns 0.
Additional details:
Details specific to MINIFS function
• The default matching performed by MINIFS is case-insensitive. However, a case-sensitive match can be carried out when using a regular expression by including a mode modifier "(?-i)" within the
regular expression, as demonstrated by one of the examples below.
• The behavior of MINIFS is affected by several settings available on the ▸ ▸ ▸ dialog ( ▸ ▸ ▸ on macOS).
1. If the checkbox is ticked for Search criteria = and <> must apply to whole cells, then the condition "red" will match only "red"; if unticked it will match "red", "Fred", "red herring".
2. If the checkbox is ticked for Enable wildcards in formulas, the condition will match using wildcards – so for example "b?g" will match "bag", "beg", "big", "bog", and "bug".
3. If the checkbox is ticked for Enable regular expressions in formulas, the condition will match using regular expressions – so for example "r.d" will match "red", "rid", and "rod", while
"red.*" will match "red", "redraw", and "redden".
4. The setting of the Case sensitive checkbox has no impact on the operation of MINIFS.
General information about Calc's regular expressions
For convenience, the information in this subsection is repeated on all pages describing functions that manipulate regular expressions.
• A regular expression is a string of characters defining a pattern of text that is to be matched. More detailed, general background information can be found on Wikipedia’s Regular expression page.
• Regular expressions are widely used in many domains and there are multiple regular expression processors available. Calc utilises the open source Regular Expressions package from the
International Components for Unicode (ICU), see their Regular Expressions documentation for further details, including a full definition of the syntax for ICU Regular Expressions.
• In addition, the LibreOffice Help system provides a high-level list of regular expressions.
• Calc’s regular expression engine supports numbered capture groups, which allow sub-ranges within a match to be identified and used within replacement text. Parentheses are used to group
components of a regular expression together and create a numbered capture group. To insert a capture group into a replacement text, use the "$n" form, where n is the number of the capture group.
General information about Calc's wildcards
Wildcards are special characters that can be used in search strings passed as arguments to some Calc functions; they can also be used to define search criteria in the Find & Replace dialog. The use
of wildcards enables the definition of more advanced search parameters with a single search string.
Calc supports either wildcards or regular expressions as arguments depending on the current application settings. By default, wildcards are supported instead of regular expressions.
To make sure wildcards are supported, go to ▸ ▸ ▸ and check if the option Enable wildcards in formulas is selected. Note that you can use this dialog to switch to regular expressions by choosing
Enable regular expressions in formulas or choose to support neither wildcards nor regular expressions.
The following table identifies the wildcards that Calc supports.
Calc wildcards
Wildcard Description
? (question Matches any single character. For example, the search string "b?g" matches "bag" and "beg" but will not match "boog" or "mug".
Note that it will not match "bg" as well, since "?" must match exactly one character. The "?" wildcard does not correspond to a zero-character match.
Matches any sequence of characters, including an empty string. For example, the search string "*cast" will match "cast", "forecast", and “outcast”, but will not match "forecaster"
using default Calc settings.
* (asterisk)
If the option Search criteria = and <> must apply to whole cells is disabled in Tools > Options > LibreOffice Calc > Calculate, then "forecaster" will be a match using the "*cast"
search string.
Escapes the special meaning of a question mark, asterisk, or tilde character that follows immediately after the tilde character.
~ (tilde)
For example, the search string "why~?" matches "why?" but will not match "whys" nor "why~s".
Wildcard comparisons are not case sensitive, hence "A?" will match both "A1" and "a1".
These wildcards are supported in both Calc and Microsoft Excel. Therefore, if interoperability between the two applications is needed, choose to work with wildcards instead of regular expressions.
Conversely, if interoperability is not necessary, consider using regular expressions for more powerful search capabilities.
Stationery sales examples
Consider the following table showing sales and revenue information for a small stationery supplier. The string "N/A" refers to products that were not available for supply during the period covered by
the data.
Stationery sales examples
A B C
1 Product Sales Revenue
2 Pencil 20 $65
3 Pen 35 $85
4 Notebook 20 $190
5 Book 17 $180
6 Pencil case N/A N/A
In all examples based on this table, it should be noted that row 6 for pencil cases contains no numeric data and so will never contribute to the result of MINIFS, whatever criteria are specified.
Formula Description Returns
=MINIFS(B2:B6; B2:B6; ">=18") Here the function finds the minimum of the numeric Sales data that are greater than or equal to 18. 20
=MINIFS(C2:C6; B2:B6; ">=20"; C2:C6; ">70") Here the function locates entries greater than or equal to 20 in the Sales data that also have a value greater than $70 in the 85
Revenue data, and finds the minimum of the corresponding numeric entries in the Revenue data.
=MINIFS(C2:C6; B2:B6; ">"&MIN(B2:B6); B2:B6; "<"&MAX Here the function calculates the minimum of the numeric entries in the Revenue data that correspond to all values in the Sales data 65
(B2:B6)) except the minimum and maximum.
This example will only work as described here if regular expressions are enabled. Here the function locates entries in the Product
=MINIFS(C2:C6; A2:A6; "pen.*"; B2:B6; "<"&MAX(B2:B6)) data that begin with the characters "pen" and have a value in the Sales data that is not the maximum, and finds the minimum of the 65
corresponding numeric entries in the Revenue data.
=MINIFS(C2:C6; A2:A6; E2&".*"; B2:B6; "<"&MAX(B2:B6)) This example will only work as described here if regular expressions are enabled. If you need to change a criterion easily, you may
where cell E2 contains the string "pen" (entered without want to specify it in a separate cell and use a reference to this cell in the condition of the MINIFS function. Here the link to the 65
typing the double quotes). cell is substituted with its content, giving the same result as the previous example.
Sporting equipment sales examples
The examples in this subsection are based on a small database of sales data for sports equipment, with the data organized as in the following table.
Sports equipment sales data
A B C D E
1 Date Sales Value Category Region Employee
2 2021-10-02 $1,508 Golf East Hans
3 2021-10-02 $410 Tennis North Kurt
4 2021-10-02 $2,340 Sailing South Ute
5 2021-10-03 $4,872 Tennis East Brigitte
6 2021-10-06 $3,821 Tennis South Fritz
7 2021-10-06 $2,623 Tennis East Fritz
8 2021-10-07 $3,739 Golf South Fritz
9 2021-10-08 $4,195 Golf West Ute
10 2021-10-10 $2,023 Golf East Hans
Formula Description Returns
=MINIFS(B2:B10; B2:B10; ">=3000") Here the function finds the minimum of the numeric entries in the Sales Value data that are greater than or equal to $3,000. 3739
=MINIFS(B2:B10; E2:E10; "ute") Here the function locates the entries for Ute in the Employee data and finds the minimum of the corresponding numeric entries 2340
in the Sales Value data.
=MINIFS(B2:B10; CategoryData; "golf") where the named range Here the function locates the entries for Golf in the Category data and finds the minimum of the corresponding numeric 1508
CategoryData has been created to cover the cell range C2:C10. entries in the Sales Value data.
=MINIFS(B2:B10; D2:D10; F1; E2:E10; F2) where cells F1 and F2 Here the function locates the entries for South and West in the Region data that also have Ute in the Employee data, and
contain the text strings ">=south" and "ute" respectively (both finds the minimum of the corresponding numeric entries in the Sales Value data. 2340
entered without typing the double quotes).
=MINIFS(B2:B10; A2:A10; DATE(2021; 10; 2); C2:C10; "tennis") Here the function locates the entries for 2021-10-02 in the Date column that also have Tennis in the Category data, and finds 410
the minimum of the corresponding numeric entries in the Sales Value data.
=MINIFS(B2:B10; A2:A10; ">="&DATE(2021; 10; 6); E2:E10; "<>"&E8) Here the function locates the entries dated on or after 2021-10-06 in the Date column that do not have Fritz in the Employee 2023
data, and finds the minimum of the corresponding numeric entries in the Sales Value data.
=MINIFS(B2:B10; C2:C10; "tennis"; D2:D10; "east"; E2:E10; Here the function locates the entries that have Tennis in the Category data, East in the Region data and Fritz in the 2623
"fritz") Employee data. It finds the minimum of the corresponding numeric entries in the Sales Value data.
This example will only work as described here if wildcards are enabled. Here the function locates the four-character entries
=MINIFS(B2:B10; D2:D10; "????"; E2:E10; "*e") in the Region data (East and West) that also have an entry in the Employee data that ends with the letter "e" or "E" 4195
(Brigitte and Ute). It finds the minimum of the corresponding numeric entries in the Sales Value data.
This example will only work as described here if regular expressions are enabled. Here the function locates the entries in
=MINIFS(B2:B10; C2:C10; "^t.*"; D2:D10; ".*h") the Category data that start with the letter "t" or "T" (Tennis) that also have an entry in the Region data that ends with 410
the letter "h" or "H" (North and South). It finds the minimum of the corresponding numeric entries in the Sales Value data.
This example will work as described here if regular expressions are enabled. The "(?-i)" mode modifier within the regular
=MINIFS(B2:B10; E2:E10; "(?-i)ute") expression changes to a case-sensitive match and so no entries are found in the Employee data. Contrast this with the second 0
example in this table.
Additional examples
For more examples, download and view this Calc spreadsheet.
Related LibreOffice functions:
ODF standard:
Related (or similar) Excel functions: | {"url":"https://wiki.documentfoundation.org/Documentation/Calc_Functions/MINIFS","timestamp":"2024-11-06T20:18:48Z","content_type":"text/html","content_length":"54931","record_id":"<urn:uuid:b8bfaf69-6444-4fff-b4fe-724fda03114a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00000.warc.gz"} |
विद्यालय एवं अभिभावक के बीच पत्र-व्यवहार
दिनांक sin3π+i(1−cos3... | Filo
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Question Text विद्यालय एवं अभिभावक के बीच पत्र-व्यवहार दिनांक . If , find,
Updated On Nov 8, 2022
Topic Trigonometry
Subject Mathematics
Class Class 11
Answer Type Video solution: 1
Upvotes 107
Avg. Video Duration 1 min | {"url":"https://askfilo.com/user-question-answers-mathematics/vidyaaly-evn-abhibhaavk-ke-biic-ptr-vyvhaar-dinaank-if-find-33303737393333","timestamp":"2024-11-05T03:30:57Z","content_type":"text/html","content_length":"228239","record_id":"<urn:uuid:002392af-1e91-4be4-9ee6-d80015b13fde>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00681.warc.gz"} |
AR Model and Others
Single Self Correlation Analysis studies the relationship between the value and the value of previous 1 step. This is the first step of Self Correlation Analysis.
AR Model is more general formulation of self correlation.
AR Model
AR stands for "Auto Regressive."
If each step is affected from some former steps, AR model is useful. AR model is one of the Multi Regression Analysis .
Application of AR Model
If we consider the variation of AR model, there are many patterns. Examples are below. By the change and fusion of ideas of formulation, there are various patterns.
(1) x(n) = f[x(n-1)]
This is the more general formulation of SCA.
If single correlation between x(n) and x(n-1) is strong, the formulation could be
x(n) = A * (x-1).
If single correlation between x(n) and x(n-1) is not strong, we may find suitable formulation between x(n) and x(n-1) by the scatter plot of x(n) and x(n-1).
x(n) = x(n-1)
means that the value does not change.
(2) x(n) = f[x(n-1),x(n-2)]
This is the application of (1). It is a general formulation of AR model using former 2 steps.
For example,
x(n) = x(n-1) + 0.01 x(n-2)
means that the former 2 step effects by 1%.
(3) x(n) = f[u(n),v(n)]
The time of both sides of the formulation is the same.
So it seems that this formulation is not for the time series analysis. But it is used in Condition Analysis. And this formulation is used if the measurement of x(n) needs much time to get.
(3) is needed to understand (4) and (5) more easily.
(4) x(n) = f[u(n-1),v(n-1)]
u and v means causes. And x means an effect.
(5) x(n) = x(n-1) + f[y(n-1),z(n-1)]
This is same to
x(n) - x(n-1) = f[y(n-1),z(n-1)].
. It means that "difference of x is caused by other variables."
(6) x(n) = x(n - t)
This formulation means that "The value of former t hours repeats." It means periodicity.
We can use (6) for Spectrum Analysis.
Non-linear Analysis
Analysis for non-linear data is studied in chaos field.
NEXT Moving Analysis | {"url":"http://data-science.tokyo/ed-e/ede1-9-3-4-2.html","timestamp":"2024-11-02T04:57:23Z","content_type":"text/html","content_length":"3601","record_id":"<urn:uuid:712be6c4-6df7-4072-a0d5-193de3143ffb>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00633.warc.gz"} |
A Second Course in Elementary Differential Equations
• 1st Edition - May 10, 2014
• Paperback ISBN:
9 7 8 - 1 - 4 8 3 2 - 4 8 1 2 - 7
• eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 7 6 6 0 - 1
A Second Course in Elementary Differential Equations deals with norms, metric spaces, completeness, inner products, and an asymptotic behavior in a natural setting for solving… Read more
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A Second Course in Elementary Differential Equations deals with norms, metric spaces, completeness, inner products, and an asymptotic behavior in a natural setting for solving problems in
differential equations. The book reviews linear algebra, constant coefficient case, repeated eigenvalues, and the employment of the Putzer algorithm for nondiagonalizable coefficient matrix. The text
describes, in geometrical and in an intuitive approach, Liapunov stability, qualitative behavior, the phase plane concepts, polar coordinate techniques, limit cycles, the Poincaré-Bendixson theorem.
The book explores, in an analytical procedure, the existence and uniqueness theorems, metric spaces, operators, contraction mapping theorem, and initial value problems. The contraction mapping
theorem concerns operators that map a given metric space into itself, in which, where an element of the metric space M, an operator merely associates with it a unique element of M. The text also
tackles inner products, orthogonality, bifurcation, as well as linear boundary value problems, (particularly the Sturm-Liouville problem). The book is intended for mathematics or physics students
engaged in ordinary differential equations, and for biologists, engineers, economists, or chemists who need to master the prerequisites for a graduate course in mathematics.
Preface1 Systems of Linear Differential Equations 1. Introduction 2. Some Elementary Matrix Algebra 3. The Structure of Solutions of Homogeneous Linear Systems 4. Matrix Analysis and the Matrix
Exponential 5. The Constant Coefficient Case: Real and Distinct Eigenvalues 6. The Constant Coefficient Case: Complex and Distinct Eigenvalues 7. The Constant Coefficient Case: The Putzer Algorithm
8. General Linear Systems 9. Some Elementary Stability Considerations 10. Periodic Coefficients 11. Scalar Equations 12. An Application: Coupled Oscillators2 Two-Dimensional Autonomous Systems 1.
Introduction 2. The Phase Plane 3. Critical Points of Some Special Linear Systems 4. Critical Points of General TWo-Dimensional Linear Systems 5. Behavior of Nonlinear TWo-Dimensional Systems Near a
Critical Point 6. Elementary Liapunov Stability Theory 7. Limit Cycles and the Poincaré-Bendixson Theorem 8. An Example: Lotka-Volterra Competition 9. An Example: The Simple Pendulum3 Existence
Theory 1. Introduction 2. Preliminaries 3. The Contraction Mapping Theorem 4. The Initial Value Problem for One Scalar Differential Equation 5. The Initial Value Problem for Systems of Differential
Equations 6. An Existence Theorem for a Boundary Value Problem4 Boundary Value Problems 1. Introduction 2. Linear Boundary Value Problems 3. Oscillation and Comparison Theorems 4. Sturm-Liouville
Problems 5. The Existence of Eigenvalues for Sturm-Liouville Problems 6. Twο Properties of Eigenfunctions 7. An Alternate Formulation-Integral Equations 8. Eigenfunction Expansions 9. The
Inhomogeneous Sturm-Liouville Problem 10. Some Standard Applications of Sturm-Liouville Theory 11. Nonlinear Boundary Value ProblemsIndex
• Paperback ISBN: 9781483248127
• eBook ISBN: 9781483276601 | {"url":"https://shop.elsevier.com/books/a-second-course-in-elementary-differential-equations/waltman/978-0-12-733910-8","timestamp":"2024-11-08T12:29:47Z","content_type":"text/html","content_length":"178470","record_id":"<urn:uuid:0c7fd33e-da1f-455e-98bf-b87f83a902ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00613.warc.gz"} |
Finding Outstanding Math Jobs Online
Finding Outstanding Math Jobs Online
The net can be a terrific resource for selecting math assignments at every volume of cognitive require. While regular students want exposure to jobs at all levels— lower together with higher— the
emphasis needs to be placed on individuals at the bigger levels. That means we need the relevant skills to evaluate what exactly and isn’t really cognitively stressing.
To determine the excellent of on the internet activities, our research companions and I used Margaret Schwan Smith along with Mary Kay Stein’s 1998 Task Evaluation Guide (TAG), which consists of four
particular levels of intellectual demand: memory, procedures without connections, measures with contacts, and accomplishing mathematics.
Using memorization, very important thinking isn’t really necessary, certainly no connections are supposed to understanding how come the answer succeeds, and techniques are bypassed. This type of
undertaking can looks like recalling information. Procedures while not connections tend to be algorithmic; young people come up with a remedy without creating connections additional math guidelines
and certainly not required to describe their perform. Problems that adhere to simple methods, like demanding the You. S. regular algorithm to get addition, fall under this category. Memory and
treatments without internet connections are reduced cognitive desire tasks simply because they don’t require lots of thinking.
Lecturers often offer visual layouts or manipulatives like Unifix cubes as well as base 10 blocks to solve math assignments that are operations with links, which permit students in order to approach
the challenge from many angles. These types of problems apply procedures, for example the partial supplement algorithm meant for multiplication, that will help students realize why the answer works
as opposed to solely knowing how to find the answer.
A very high level concerns, doing mathematics, require non-algorithmic thinking, need self-monitoring, and permit for several strategies to be used— college students at this point tend to be
exploring mathematical concepts.
Procedures with relationships and engaging in mathematics tend to be high cognitive demand duties because scholars need to produce connections, evaluate information, along with draw findings to solve
them all, according to Jackson and Stein.
So as to present primary school students through problems each and every cognitive degree, teachers should be critical buyers of the options available. Within research, this points aided my fellow
workers and me personally evaluate the cognitive demand as well as quality of online assignments.
Age counts. The level of intellectual demand can turn depending on the involving the children issues was created just for. For example , creating a worksheet of basic one-digit add-on problems can be
coded because memorization for that fourth grader, who is likely to have them commited to memory (even also if the college student is being timed), but it is considered undertaking procedures devoid
of connections just for kindergarteners, that happen to be just studying what it means to add two sections to make a person whole.
If you are looking for higher cognitive demand from customers tasks, one that encounters any of the subsequent criteria can be viewed as a procedure having connections; to essay writer become
classified because doing arithmetic, there must be many ways to work out the task:
The matter usually involves manipulatives (e. g,. diez frames, bottom part 10 hindrances, number strains, number grids).
There are actually directions calling for students to produce explanations of how they identified the answer (through models, key phrases, or both).
The good news is high level connected with critical wondering required. Like students decide how to handle a problem that could be solved in more than the best way, make real world connections
towards the math, or maybe explain their very own mathematical pondering.
If evaluating some math activity, teachers should likewise evaluate any kind of images that accompany it. Is definitely an image involved solely to get decorative purposes, or does the unit use a
functional factor in fixing the problem? Photographs with well-designed roles can include clock people, 10 casings, and charts. If an activity has a ornamental image, it is actually significantly
more likely to end up a low cognitive demand endeavor; if it contains a functional picture, it is greatly subjected to be coded at a advanced of cognitive demand. While an activity may very well be
popular because decorative, extremely cute images, looks does not associate with high amounts of cognitive require. It’s important to target the content as opposed to the art.
There are a notably increased chance of finding math activities at a higher level of cognitive demand on websites where information are reviewed before e book as opposed to online websites like
Trainers Pay Trainers or Pinterest where any individual can submit. The following web sites publish examined resources:
Illustrative Mathematics helps teachers to look for tasks determined by content standards by area or mark for K-12 (free).
EngageNY is really a set of pre-K to quality 8 French language disciplines and mathematics curricula created by the modern York Point out Department involving Education. Additionally, it has math
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Illuminations is undoubtedly an online tool available over the National Council of Instructors of Arithmetic (NCTM) which offers materials influenced by both NCTM standards as well as Common Major
State Standards for marks pre-K that will 12. Entry requires any NCTM membership rights (cost: $49 to $139 a year). | {"url":"http://www.gpradvogados.com.br/finding-outstanding-math-jobs-online-10/","timestamp":"2024-11-13T11:49:18Z","content_type":"text/html","content_length":"34554","record_id":"<urn:uuid:51f41baa-5ea7-464c-b72f-b314ce3a3807>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00880.warc.gz"} |
1. [Linear Systems] | Linear Algebra | Educator.com
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while
watching the lecture.
Section 1: Linear Equations and Matrices
Linear Systems 39:03
Matrices 30:34
Dot Product & Matrix Multiplication 41:42
Properties of Matrix Operation 43:17
Solutions of Linear Systems, Part 1 38:14
Solutions of Linear Systems, Part II 28:54
Inverse of a Matrix 40:10
Section 2: Determinants
Determinants 21:25
Cofactor Expansions 59:31
Section 3: Vectors in Rn
Vectors in the Plane 46:54
n-Vector 52:44
Linear Transformation 48:53
Linear Transformations, Part II 34:08
Lines and Planes 37:54
Section 4: Real Vector Spaces
Vector Spaces 42:19
Subspaces 43:37
Spanning Set for a Vector Space 33:15
Linear Independence 17:20
Basis & Dimension 31:20
Homogeneous Systems 24:45
Rank of a Matrix, Part I 35:03
Rank of a Matrix, Part II 29:26
Coordinates of a Vector 27:03
Change of Basis & Transition Matrices 33:47
Orthonormal Bases in n-Space 32:53
Orthogonal Complements, Part I 21:27
Orthogonal Complements, Part II 33:49
Section 5: Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors 38:11
Similar Matrices & Diagonalization 29:55
Diagonalization of Symmetric Matrices 30:14
Section 6: Linear Transformations
Linear Mappings Revisited 24:05
Kernel and Range of a Linear Map, Part I 26:38
Kernel and Range of a Linear Map, Part II 25:54
Matrix of a Linear Map 33:21
This is the first lesson of Linear Algebra course, here at Educator.com.0004
It is a complete Linear Algebra course from beginning to end.0010
So a Linear Algebra, I am going to introduce just a couple of terms right now , just to give you an idea of what it is that you are going to expect in this course.0014
It is the study of something called Linear Mappings or Linear transformations, also known as linear functions between vector spaces.0023
And this is a profoundly important part of mathematics, because linear functions are the heart and soul of Science and Mathematics, everything that you sort of enjoy in your world today consist of
essentially a study of linear systems.0032
So, don't worry about what these terms mean vector space, linear mapping, transformation, things like that, we will get to that eventually.0047
Today's topic, our first topic is going to be linear systems and it's going to be the most ubiquitous of the topics, because we are going to use linear systems as our fundamental technique to deal
with all of the other mathematical structures that we deal with.0055
In one form or another, we are always going to be solving some set of linear equations.0070
Okay, so let's just start with something that many of you have seen already, if not, no worries.0081
If we have something like AX=B, this is a linear equation, one reason that linear is used, the term linear is because this is the equation of a straight line.0091
However as it turns out, although we use the term linear, because it comes from the straight line later on in the course, we are actually going to get the precise definition of what we mean by
And believe it or not, it actually has nothing to do with a straight line.0114
It just so happens that the equation, this AX=B, which can be represented by a straight line on a sheet of paper on a two dimensional surface.0117
It had, happens to be a straight line so we call it linear, but its, but the idea of linearity is actually a deeper algebraic property about how this function actually behaves when we start moving
from space to space.0129
Okay, so this is sort of a single variable, we have ax=b, something like for example, [inaudible].0143
Well, that's okay we will just leave it like that.0153
If I can write this, A1X1 + A2X2 + A3X3 = B, well these answer just different coefficients, 5, 4, 6, (-7).0155
These x1, x2 and x3 are the variable, so now instead of just the one variable, some equation up here.0175
We have three variables X1, X2, X3, we can have any number of them and B.0183
So a solution to something like this is a series of X's that satisfy this particular equation.0189
That's all what's going on here, linear equation, you know this linear essentially is when this exponent up here is A, that pretty much is what we are used to see when we deal with linear
But again linearity is a deeper algebraic property, which we will explore a little bit later in the class, and that's when linear algebra becomes very, very exciting.0210
Okay, so let's use a specific example, so if I had something like 6X1 - 3X2 + 4X3 = (-13).0220
...X1 = 2, X2 = 3 and X3 = (-4), well this 2, this 3, this (-4) for X1, X2 and X3 is a solution to this linear equation.0239
That's it, we are just looking, it is that, that's all we were looking for, we are looking for variable that satisfy this equality, that's all that's happening here.0256
note however that we can also have X1 = 3...0264
X2 = 1 and X3 = (-7). So if we put 3, 1, (-7) in for X1, X2 and X3 respectively, we also get this equality (-13), so as it turns out these particular variables don't necessarily have to be
Several, sometimes they can be unique, other times a whole bunch of, set of numbers can actually satisfy that equality, so we want to find as many of the solutions that satisfy that equality,
Now let's generalize this some more and talk about a system of equations, so I am going to go ahead and represent this symbolically, so see we have...0302
A11X1+ A12X2 + ... + A1N XN = b1, so this just is our first equation, we have n variable, that's what the X1 to X10, and these are just the coefficients in front of those variables X's and this is
just some number.0315
So this is just one linear equation, now we'll write another one A21X1, and I'll explain what these subscripts mean in just a moment + A22X2 + ... + A2NXn = B2.0344
Now we have our second equation and then we go down the line, so I am going to put a ... there ... means we are dealing with several equations here.0367
And then I am going to write AM1X1 + AM2X2 + ... + AMn, I know that's a little small but that's an MN right there, equals Bm, so notice we used two subscripts here, like for example we usually the
subscripts I, J.0379
And the first subscript represents the row or the equation, so in this case 1, 2,3,4,5 all the way to the nth equation, so A11 is the first equation and the second entry J represents that particular
column, that particular entry.0415
So, A11 represents the first coefficient in the first equation, if I did something like let's say I had A32, that would mean the third equation, the second entry, the second coefficient, the
coefficient for X2.0437
That's all this means, so here I have notice X all the way to n, Xn Xn all the way down, oops I forgot an Xn right here, so I have n variables....0457
...and I have as many rows M equations and this is exactly what we say when we have n equations and N variables, this many and this many.0473
We just arrange it like this, so this is a system of linear equations.0486
What this means when we are looking for a solution to a system of linear equations as supposed to just one linear equation, we are looking for...0490
We want a set of X1, X2, all the way to Xn, such that all of these equations are satisfied simultaneously...0503
... such that all equalities, I'll say equalities instead of equations, we know we are dealing with equations; we want all of these equalities to satisfied...0521
In other words we want numbers such that, that holds, that holds, that holds, that holds if one of them doesn't hold, it's not a solution.0540
Let's say you have seven equations, and let's say you found some numbers that satisfy six of them, but they don't satisfy the seventh, that system doesn't have that solution.0547
It has to satisfy all of them, that's the whole idea.0558
... okay, we are going to use a process called elimination...0571
To solve systems of linear equations, now we are going to start in with the examples to see what kind of situations we can actually come up with.0583
One solution infinitely manages solutions, no solutions, what are the things that can happen when dealing with linear system.0592
How many variables, how many equation and, what's the relationship that exists, just to get a sense of what's going on, just to get us back into the habit of working with these.0598
Now of course many of you have dealt with these in algebra.0605
You have seen the method of elimination; you have used the method of substitution.0608
Essentially elimination is turning one equation, let's say you have two equations and two unknowns, you are going to manipulate one of the equations so that you can eliminate one of the variable.0612
Because again in algebra, ultimately when you are solving an equation, you can deal with one variable at a time.0620
Lets just jump in and I think the, the technique itself will be self-explanatory...0628
...okay, so our first example is X + 2I = 8, 3X - 4Y = 4, we want to find X and Y such that both of these hold simultaneously, okay.0636
In this particular case elimination and it really doesn't matter which variable you eliminate, so a lot of times, it's a question of personal choice.0649
Some people just like one particular variable, often times you look at what look like it's easy to do, that will guide your choice.0658
In this particular case I notice that this coefficient is 1, so chances are if I multiply this by 3, by (-3), this whole equation by (-3) to transform it, and then add it to this equation, the -3X
and the 3X will disappear.0665
Let us go ahead and multiply everything by (-3) and when I do that, I tend to put a (-3) here, (-3) there to remind me.0684
-3X - 6Y= (-24) and of course this equation we just leave it alone.0698
And now we can go ahead and then, the -3X + 3X, that goes away, -6Y - 4Y gives us -10Y, -24 + 4 is -20.0717
And when we divide through by -10, we get Y = 2.0731
We are able to find our first variable Y = 2.1219 Now, I can put this Y = 2 back into any one of the original equations, you could put them in these two, it's not a problem.0735
it doesn't, multiplying by a constant doesn't change the nature of the equation, because again you are multiplying, you are retaining the equality, you are doing the same thing to both sides, so Y =
Lets go ahead and use the first equation, therefore I will go ahead and draw a little line here, we will say X + 2 times 2, which is Y = 8X + 4 = 8X oops...0757
Let us put the X on the left hand side, X = 4, so there you have it, a solution X = 4, Y = 2, if X = 4, if Y = 2, that will solve both of these simultaneously.0780
Both of these equalities will be satisfied, so in this particular case, we have one solution.0796
Now let's try X - 3Y = -7, 2X - 6Y = 7, so let's see what happens here.0826
Well, in this particular case again I notice that I have a 2 and a coefficient of 1, some we have to go ahead and eliminate the X again, so in order to eliminate the X, I need this to be a -2X, so I
am going to multiply everything by (-2) of top.0834
-2 times X is -2X, -2 times -3Y is +6Y = -2 times -7 gives 14.0849
I can pretty much guarantee you that in your just, a small digression, the biggest problem in linear algebra is not, as this is not going to be the linear algebra, it is going to be the arithmetic,
just keeping track of the negative signs or positive signs and just the arithmetic addition, subtraction, multiplication and division.0861
My recommendation of course is, you can certainly do this by hand, and it is not a problem, but at some point you are going to want to start to use the mathematical software, things like maple, math
cad, mathematica, they make life much, much easier.0881
Now, obviously you want to understand what is going on with mathematics, but now some of, as we get into the course, a lot of the computational procedures are going to be kind of tedious in the sense
that they are easy, except they are arithmetically heavy, so they are going to take time.0897
You might want to avail yourself over the mathematical software, okay.0912
Let us continue on and then this one doesn't change, so it's 2X - 6Y = 7 and then when we add these, we get +6Y and -6Y, wow these cancel too, so we end up with 0 = 14 + 7 is 21.0917
We get something like this, 0 = 21; well 0 does not equal 21, okay, so this is no solution.0936
We call this an inconsistent system, so any time you see something that is not true, that tells you that there is no solution.0946
In other words there is no way for me to pick an X and a Y that will satisfy both of these equalities simultaneously.0954
Example three, okay, now we have got three equations and three unknowns, X, Y and Z.0974
Well we deal with these two equations at a time, so let's go ahead, we see an X here, and a 2, 3.0979
I am going to go ahead and just deal with the first two equations, and I am going to multiply, I am going to go ahead and eliminate the X, so I am going to multiply by -2 here.0986
And again just be very, very systematic in what you do, write everything down, the biggest problems that I had seen with my students is that they want to do things in their head and they want to skip
Well, when you are dealing with multiple steps, let us say if you have a seven step problem, and each one of those steps requires may be three or four steps, if you skip a step in each sub portion of
the problem, you have skipped about seven steps.1007
I promise there has been a mistake, which there always will be, and when it comes to arithmetic, you are going to have a very hard time finding where you went wrong, so just write everything
You will never ever go wrong if you write everything own, and yes I am guilty of that myself.1028
Okay, so this becomes, let us write it over here, -2X - 4Y - 2 + 3Z is -6Z, -2 times 6 is -12.1035
And let us bring this equation over unchanged, that is the whole idea, 2X -3Y + 2Z = 14, let us go ahead and, so the X's eliminate, and then we end up with -4Y - 3Y is -7Y, -6Z + 2Z is -4Z, and -12 +
14 = 2, so that's our first equation.1051
And now we have reduced these two, eliminated the X, so now we have an equation in two unknowns,1078
Now, let us deal with the first and the third, so on this particular case, I am going to do this one in blue, I am going to, I want to eliminate the X again, because I eliminated the here, so I am
going to eliminate the X here.1086
I am going to multiply by a -3 this time.1100
-3 time +2 is -6Y, -3 times 3 is -9Z and -3 times 6 is -18, and I am hoping that we are going to confirm my arithmetic here.1112
And then again I leave this third one unchanged, 3X + Y - Z = -2.1127
I eliminate those -6Y + 1Y is -5y, and then I get -9 -1 is -10Z, -18 - 2, I get -20.1141
Now, I have my second equation, and this one was first equation, so now I have two equations and two variables, Y and Z, Y and Z.1155
Now, I can work with these two, so let me go ahead and bring them over and rewrite them, -7Y - 4Z = 2, and -5Y - 10Z = -20.1164
Good, so now we have a little bit of a choice to make, do we eliminate the y or do we eliminate the Z now.1184
It's again, it's a personal choice, I am going to go ahead and eliminate the Y's for no other reasons, and beside and I am just going to work from left to right, not a problem.1192
I am going to multiply, so I need the Y's to disappear and they are both negative, so I thing I am going to multiply the top equation by a -5.1201
And I am going to multiply the bottom equation, I will write that in black, no actually I will keep it in blue, the bottom equation by 7 , 7 here, 7 here.1214
This will give me a positive value here and a negative value here, this should take care of it.1225
Let me multiply the first one, what I get is 35Y right, -5 times -4 is +20Z, -5 times 2 = 1-10., 7 times -5 is -35Y.1231
So far so good, 7 times -10 is -70Z and 7 times a -20 is -140.1252
Now, when we solve this, the Y's go away and we get +20Z - 70Z for a total of -50Z = -10 - 140 - 150.1263
Okay, so now that I have Z = 3, I can go back and put it into one of these equations to find Y, so let me go ahead and use the first equation, so let me move over here next.1285
I would write -7Y - 4 times Z which was 3 = 2, I get -7Y -12 = 2, -7Y = 14, Y = -2, notice I didn't skip these steps, I wrote down everything, yes I know its basic algebra.1298
But it's always going to be the basic stuff that is going to slip you up, so Y = -2.1320
I have done my algebra correctly, my arithmetic, that's that one, now that I have a Z and I have a Y, I can go back to any one of my original equations and solve for my X.1325
Okay, I am going to go ahead and take the first one since, because that coefficient is there, so I get X + 2 times Y, which is -2 +, write it out exactly like that.1335
Don't multiply this out and make sure you actually see it like this again.1350
Write it all out, + 3 times 3 = 6, we get X - 4 + 9 = 6, get, oops, that is little straight lines here.1355
Erase these , if you guys are bothered, okay, X - 4, what is -4 + 9, that's 5 right.1371
X + 5 = 6, we get X = 1, and there you have it, you have X = 1, Y = -2, Z = 3.1381
Again one solution, notice what we did, we eliminated, we picked two equations, eliminated variable, the first and the third to eliminate the same variable, we dropped it down to now two equations
and two unknowns.1401
Now we eliminated the common variable, got down to 1 and the, we worked our way backward, very, very simple, very straight forward, nice and systematic.1413
Again nothing difficult, just a little long, that's all, okay.1421
Let's see what else we have in store here, example four, okay so we have X +2Y - 3Z = -4, 2X + Y - 3Z = 4, notice in this case we have two equations and we have 3 unknowns, so let's see what's going
to happen here.1427
Well, this is a coefficient 1, this is 2, so let's multiply this by a -2, let's go ahead and use a blue here so we will do -2 here and -2 there.1444
And let's go, now let's move over in this direction, so we have -2X - 4Y and this is going to +6Z right, equals +8 and then we will leave this one alone, because we want to eliminate the variable
Excuse me, + Y - 3Z = 4, okay let's eliminate those, now we have -4Y + Y, it should be -3Y, 6Z - 3Z is +3Z, 8, 9, 10, 11, 12, that is equal to 12, okay.1478
Now we have -3Y + 3Z = 12, we can simplify this a little bit because every number here, all the coefficients are divisible by 3, so let me go ahead and rewrite this as, let me divide by (-), actually
it doesn't really matter.1501
I am going to divide by -3 just to make this a positive, so this becomes....1520
right now let me actually do a little error out, so divide by -3, this becomes Y, this becomes a -Z, and 12 divided by 3 becomes -4, is that correct? Yes, so now we have this equation Y - Z = 4,
that's as far as we go.1529
Now let's, what we are going to do is again we need to find the solutions to this, so we need to find the X and the Y and the Z.1554
Let's go ahead and move, solve for one of the variables, so Y = Z -4, so now I have Y = Z - 4.1564
And I have this thing I can solve for X, but what do I do with this, as it turns out.1579
Whenever I have something like this, Z = any real number, so basically when you have a situation like this, you can put in any real number for Z, and whatever number you get, let's say you choose the
number 5.1584
If you put 5 in for Z, that means 5 - 4, well let's just do that as an example, so if Z = 5, well 5- 4 = 1.1600
That makes Y = 1 , and now I can go back and solve this equation, so let me just do this one quickly.1611
We get X + 2 times +2 - 15 = -4, 2 - 15 is X -13 = -4, that means X = 4 + 13 should be 9, so X = 9.1620
This is a particular solution, but it's a particular solution based on the fact that I chose Z = 5, so notice any time you have two equations three unknowns, more unknowns than equations, you are
going to end up with an infinite number of possibilities depending on how you choose Z.1644
Z can be any real number, once you choose Z you have specified why, and once you know, specified Y, you can go back and you specify X.1660
...okay, so an infinite number of solutions is also another possibility, so we have seen one solution, a system that has one solution only, we have seen system that has no solutions , that was
inconsistent and now we have seen the system that has an infinite number of solutions, okay.1683
Now let's see what we else we can do here.1698
Just want to be nice and example, happy just to get a, so make sure that every, every all the, all the steps are covered, all the bases are covered, just we know what we are dealing with.1703
Okay, this particular system is X + 2Y = 10, 2X - 2Y = -4, 3X + 5Y = 26.1711
Okay, let's start off by eliminating the X here, so I am going to multiply this by -2, -2 to give us, -2X -4Y = -20, and that of course this one stays the same, 2X -2Y = -4, when I do this I get -6Y
= -24, Y -4, okay.1720
I get Y = 4, now notice I have three equations, so this Y = 4, deals with these, this first two.1754
I need all three equations to be handled simultaneously, so now since I can't just stop here and plug back in, it's not going to work.1766
I need to make sure so now I have just done the first and the second, now I am going to do the first and the third, so this is first and second equations.1774
So now I am going to, and we do this one in red, this is X, this is 3X, so I am going to multiply by -3, so in this case I have -3X - 6Y = -3 times then actually you cross these out, -3 times that,
-30 and make sure my negative signs work here.1789
Okay, 3X's cancel, -6Y + 5Y is a -Y, and -30 + 26 is a -4, divide by -1, so we get Y = 4, okay so notice, our first and second equation we get Y = 4, our first and third equation we get Y = 4, these
equations that we come up with, we have transformed this original system.1829
Now our original system has been transformed into X + 2Y = 10, because that's what we are doing, we are just changing equations around, X + 2 = 10, and then we did this one, we got Y = 4 and we go Y
= 4, because....1858
...it worked out the same now, I can take this Y, put it in here and solve for x.1876
Let me make this a little clear, select this and we will write an X here, it is definitely a Y, so now I take X + 2 times Y, which is 4 = 10, so I get X + 8 = 10, I get X = 2.1883
And that's my solution, one solution X = 2, Y = 4, so be very, very careful with this, it's just because you end up eliminating and equation or eliminating a variable, in this particular case notice
we have three equations and two variable, you can eliminate a variable and end up with a Y = 4, which you can't stop there.1905
You can't, you have to, you have to account for the third equation, so now you do the first and the third, and if there is consistency there, you end up with this system1926
This system is equivalent to this system, that's all you are doing.1935
Every time you make the change, you are creating a new set of equations, you are just, you know, now you are dealing with this system because this and this are the same.1940
You are good, now you can go back and solve for the X, okay.1949
Let's look what we have here, again we have a system of three equations and two unknowns, so we are going to treat it same way, so let's start off by doing the first and second equations, so you
write first and second over here, so we are going to multiply this by -2, -2, so we are going to get -2X - 4Y = -20.1956
And this one we leave the 2X - 2Y = -4, when we add the X's cancel, we are left with -6Y = excuse me, and then we are left with Y = 4, again.1984
That's just the same thing that we had before, now we will take care of the first and third equation, this time to multiply again by 3.2002
Let me do this one in blue, -3, -3 and we are left with, so the first equation becomes -3X - 6Y = -30, and then this one becomes 3X + 5Y = 20.2011
Now, when I do this, the X's cancel, I am left with -Y = -30 + 20 - 10, I get Y = 10.2034
there is no way to reconcile these two to make all three equalities satisfied simultaneously, so this is no solution.2051
Again just because you found a solution here, don't stop here, don't stop here and (inaudible) into one of these equations because you just did it for the first two, and certainly don't throw it into
the third, because that won't give you anything.2062
No solution, these have to be consistent, first and second, this is first and third.2074
What we just did here is the equivalent system that we have transformed to is X + 2Y = 10, Y = 4, Y = 10.2088
All of these examples that we have done always been the same thing, we see that we either have one solution, unique solution, we have no solution or we have infinitely many solutions, those are the
only three possibilities for a linear system, one solution , no solution or infinitely many solutions.2110
Back in algebra, we are dealing with lines, again these are all just equation of lines, the ones and two variables X + Y, well the no solution case, that's when you have parallel lines, they never
... they meet at a point and the infinitely many solutions is when one line is on top of another line, infinitely many solutions.2154
But again, we are using the word linear because we have dealt with lines before we developed a mathematical theory; mathematics tends to work from specific to general.2163
And the process of going to the general, the language that they use to talk about the general is based on the stuff that we have dealt within the specifics.2173
We have dealt with line before we dealt with linear functions, once we actually came up with a precise definition for a linear function, we said let's call it, well the ones who decided to give it a
name said let's call it a linear function, a linear map, a linear transformation.2182
It actually has nothing to do with a straight line, it just so happens that the equation for a line happens to be a specific example of a linear function.2197
But linearity itself is a deeper algebraic property which we will explore and which is going to be the very heart of, well linear algebra.2204
Okay, let me just go over one more thing here, the method of elimination, so let's recap.2214
Using the method of elimination we can do three things essentially, we can interchange any two equations and interchange just means switch the order, so if I have the particular equation that has a
coefficient of 1 and one of the variable, it's usually an good idea to put that one on top.2221
But maybe you prefer it in a different location, it just means switching the order of the equations, nothing strange happening there.2234
Multiply any equation by a non-zero constant, which is really what we did most of the time here; multiply by -3, -2, 5, 7, whatever you need to do in order to make the elimination of the variables
And then third, add a multiple of one equation to another, leaving the one you multiplied by a constant in its original form, so recall when we had X + 2Y, we had X + 2Y = 8, 3X - 4Y = 4.2253
When we multiply the first equation by -3, then add it to equation 2, we ended up with the following equivalent system, so we end up converting this to -3X -6Y and end up -24 and then we brought this
one over 3X - 4Y = 4, once we actually found the answer to this, which is say, -10Y = -20.2271
we ended up with a solution, well once we get that solution, that, this is now the new equation, so that's over here, Y = 2.2305
But the original equation stays, so this, so that's what we were doing when we do this.2317
We are changing a system to an equivalent system, that's what we have to keep in mind when we are doing these eliminations.2322
Notice the first equation is unchanged, when we rewrite our entire system.2330
Okay, thank you for joining us here at educator.com, first lesson for linear algebra, we look forward to see you again, take care, bye, bye.2336
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Coining Data Science - Datascience.aero
There are incredibly vast amounts of data now available. Everywhere around us there is data being captured, for instance our location-tracked smartphones, the links we click (even to read this post)
or other ways you are accessing
through your desktop computer or laptop. Companies in almost every industry are exploiting data for competitive advantage in the so called “data era”.
In the past, firms could employ teams of statisticians, modelers, and analysts to manually explore datasets, but the volume and variety of data have far surpassed the capacity of manual analysis.
Nowadays, computers have become powerful and the data science field has had a rebirth between the boundary of the “old” statisticians and the “new” computer scientists. However, before we embark on
the present-day data science capabilities, let’s first explore the roots of data science; how has the concept been coined over the years? This post explores and documents its recent history, the
connections with other domains (statistics, computer science) and include some resourceful publications and references. Enjoy! | {"url":"https://datascience.aero/coining-data-science/","timestamp":"2024-11-02T20:10:56Z","content_type":"text/html","content_length":"65822","record_id":"<urn:uuid:11daf225-7254-4f75-b09c-969a2b2d4eaf>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00200.warc.gz"} |
Measurement of plane vibrations of a two dimensional elastic structure
Plane vibrations of a two dimensional elastic structure are analyzed in this paper. The vibrations taking place according to the eigenmode are represented by using the method of stroboscopic
geometric moiré. This requires to perform the investigation for two mutually perpendicular directions of moiré fringes. Here the superimposed moiré technique is proposed to represent both images at
the same time.
1. Introduction
In precise mechanical devices there is a great number of vibrating elastic elements. Plane vibrations of a two dimensional elastic structure are analyzed in the paper. The vibrations taking place
according to the eigenmode are represented by using the method of stroboscopic geometric moiré. This requires to perform the investigation for two mutually perpendicular directions of moiré fringes.
Thus one is to paint geometric moiré lines in the first direction and to perform the investigation. After that one is to abolish the previously painted lines and then to paint geometric moiré lines
in the perpendicular direction and perform the investigation.
In this paper the superimposed moiré technique is proposed to represent both images at the same time. Though some engineering intuition may be required in the process of interpretation of
superimposed images, but the advantage of the method is the fact that both systems of mutually perpendicular and parallel in the status of equilibrium lines are painted on the structure and remain on
it the whole time. Also vibrations in both mutually perpendicular directions are analyzed at the same time, while in the conventional moiré approach first the vibrations in the direction of the $x$
axis are analyzed and later vibrations in the direction of the $y$ axis are analyzed.
The analysis is based on the material described in [1-3] and other related papers.
2. Theoretical investigation of the proposed measurement procedure
One dimensional problem is investigated. Moiré lines in the status of equilibrium are represented as:
${I}_{1}=\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\frac{\pi }{\lambda }x,$
where $x$ is the coordinate, $\lambda$ determines the width of moiré lines, ${I}_{1}$ is the intensity of the image.
Moiré lines in the deflected state are represented as:
${I}_{2}=\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\frac{\pi }{\lambda }\left(x-u\right),$
where $u$ is the displacement, ${I}_{2}$ is the intensity of the image. In the investigation it is assumed that:
where $k$ is a constant.
Intensity of the stroboscopic image is represented as:
Investigation of moiré images of this type was performed earlier. Here gaps between moiré lines are assuned to be wider. For this purpose the following special function is introduced:
$F\left(i,\lambda ,x\right)=\left\{\begin{array}{ll}0,& \frac{1}{2}+\left(1+i\right)j<\frac{x}{\lambda }<\frac{1}{2}+i+\left(1+i\right)j,\\ \mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\frac{\pi }{\lambda }x,
& \mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e},\end{array}\right\j=0,±1,±2,\dots ,$
where $i=0,\mathrm{}1,\mathrm{}2,\dots$ is the width of the gap.
Moiré lines in the status of equilibrium are represented as:
${I}_{1}=F\left(i,\lambda ,x\right).$
Moiré lines in the deflected state are represented as:
${I}_{2}=F\left(i,\lambda ,x-u\right).$
Further it is assumed that $\lambda$ = 0.8 and $k$ = 0.1. ${I}_{1}$, ${I}_{2}$ and ${I}_{s}$ for $i=0,\mathrm{}1,\mathrm{}2,\mathrm{}3,\mathrm{}4$ are presented in Figs. 1-5.
Fig. 1I1, I2 and Is for i = 0
Results for $i$ = 0 correspond to the results presented earlier. Envelope of the stroboscopic image has 6 maximums inside the analyzed interval. From the results for $i$ = 1 it is seen that the
envelope of the stroboscopic image has 3 maximums inside the analyzed interval. From the results for $i$ = 2 it is seen that the envelope of the stroboscopic image has 2 maximums inside the analyzed
interval. From the results for $i$ = 3 it is seen that the envelope of the stroboscopic image has 1 maximum inside the analyzed interval. From the results for $i$ = 4 it is seen that the envelope of
the stroboscopic image has 1 maximum inside the analyzed interval and the distance between the maximums is bigger than for the previous value of $i$. Thus from the presented results it can be
concluded that with the increase of the width of the gap the intervals between the maximums of the envelope of intensity of the stroboscopic image increase. But it is possible to interpret the
displacements from moiré images with gaps. Those gaps enable to interpret both moiré images of parallel lines in a two dimensional problem simultaneously and this can be seen from the two dimensional
results presented further.
Fig. 2I1, I2 and Is for i = 1
Fig. 3I1, I2 and Is for i = 2
Fig. 4I1, I2 and Is for i = 3
Fig. 5I1, I2 and Is for i = 4
3. Conventional stroboscopic geometric moiré images of vibrating elastic structures
Square elastic structure with fixed lower boundary is analyzed. Stroboscopic geometric moiré images for the two conventional directions of fringes for the first eigenmode are shown in Fig. 6, for the
second eigenmode in Fig. 7, for the third eigenmode in Fig. 8, for the fourth eigenmode in Fig. 9.
4. Superimposed moiré images of vibrating elastic structures
Superimposed stroboscopic geometric moiré images for the first eigenmode are shown in Fig. 10, for the second eigenmode in Fig. 11, for the third eigenmode in Fig. 12, for the fourth eigenmode in
Fig. 13.
Fig. 6Stroboscopic geometric moiré images for the first eigenmode: a) the first direction of fringes, b) the second direction of fringes
Fig. 7Stroboscopic geometric moiré images for the second eigenmode: a) the first direction of fringes, b) the second direction of fringes
Fig. 8Stroboscopic geometric moiré images for the third eigenmode: a) the first direction of fringes, b) the second direction of fringes
Fig. 9Stroboscopic geometric moiré images for the fourth eigenmode: a) the first direction of fringes, b) the second direction of fringes
Fig. 10Superimposed stroboscopic geometric moiré image for the first eigenmode
Fig. 11Superimposed stroboscopic geometric moiré image for the second eigenmode
Fig. 12Superimposed stroboscopic geometric moiré image for the third eigenmode
Fig. 13Superimposed stroboscopic geometric moiré image for the fourth eigenmode
5. Conclusions
The superimposed moiré technique is proposed to represent both moiré images for the analysis of plane vibrations of two dimensional elastic structures at the same time. Some engineering intuition may
be required in the process of interpretation of superimposed images, but the advantage of the method is the fact that both systems of mutually perpendicular and parallel in the status of equilibrium
lines are painted on the structure and remain on it the whole time. Thus vibrations in both mutually perpendicular directions are analyzed at the same time.
The proposed technique of superimposed moiré analysis of plane vibrations of two dimensional elastic structures is applicable for the investigation of vibrations of precise mechanical devices.
• Ragulskis K., Maskeliūnas R., Zubavičius L. Analysis of structural vibrations using time averaged shadow moiré. Journal of Vibroengineering, Vol. 8, Issue 3, 2006, p. 26-29.
• Saunorienė L., Ragulskis M. Time – Averaged Moiré Fringes. Lambert Academic Publishing, 2010.
• Ragulskis M., Maskeliūnas R., Ragulskis L., Turla V. Investigation of dynamic displacements of lithographic press rubber roller by time average geometric moiré. Optics and Lasers in Engineering,
Vol. 43, 2005, p. 951-962.
About this article
elastic structure
plane vibrations
stroboscopic moiré
geometric moiré
superimposed moiré
experimental results
Copyright © 2015 JVE International Ltd.
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How do pi bonds work? | Socratic
How do pi bonds work?
1 Answer
Pi ($\pi$) bonds are made by overlapping two atomic orbitals sidelong, as shown below. In contrast, sigma ($\sigma$) bonds are made by overlapping two atomic orbitals head-on.
Either way, this overlap can either be in-phase or out-of-phase.
• The in-phase one (same colors overlapping) is lower in energy and is called the bonding $\pi$ overlap. It generates a $\pi$ molecular orbital.
• The out-of-phase one (opposite colors overlapping) is higher in energy and is called the antibonding $\pi$ overlap. It generates a ${\pi}^{\text{*}}$ molecular orbital.
A double bond has one $\sigma$ and one $\pi$ bond, while a triple bond has one $\sigma$ and two $\pi$ bonds.
The MO diagram depiction is:
• ${\pi}_{n p x}$ is the bonding molecular orbital formed by the in-phase overlap of an $n {p}_{x}$ with an $n {p}_{x}$ atomic orbital.
• ${\pi}_{n p y}$ is the bonding molecular orbital formed by the in-phase overlap of an $n {p}_{y}$ with an $n {p}_{y}$ atomic orbital.
• ${\pi}_{n p x}^{\text{*}}$ is the antibonding molecular orbital formed by the out-of-phase overlap of an $n {p}_{x}$ with an $n {p}_{x}$ atomic orbital.
• ${\pi}_{n p y}^{\text{*}}$ is the antibonding molecular orbital formed by the out-of-phase overlap of an $n {p}_{y}$ with an $n {p}_{y}$ atomic orbital.
We have three common ways that we can occupy the $\pi$ and ${\pi}^{\text{*}}$ molecular orbitals.
• When the $\pi$ molecular orbitals are filled but the ${\pi}^{\text{*}}$ ones are not, we have a $\pi$ bond.
• When both kinds of molecular orbitals are filled, those electrons are nonbonding and are lone pairs.
• When neither kind of molecular orbital is filled, there is no lone pair or bond.
The $\sigma$ overlaps are the ones that are head-on, and are not our focus (though you should know those as well).
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[Solved] With such a high base rate, you are confi | SolutionInn
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With such a high base rate, you are confident about the chance of hiring and have posted the job ad based on a prior job
With such a high base rate, you are confident about the chance of hiring and have posted the job ad based on a prior job analysis. Listed below are the final applicants and their profile of four key
KSA factors. You hired SMEs to assess the factors with behaviorally anchored rating scales; their scores are presented in parentheses.
Neal A. Donal T. Lisso J. Mitzh D. Opal W.
Associate's degree H.S. diploma Ph.D. Bachelor's degree Masters' degree
(2) (1) (5) (3) (4)
Satisfactory Satisfactory Good Satisfactory Very Strong
Communication skills
(2) (2) (3) (2) (5)
Strong Very Strong Marginal Strong Satisfactory
Managerial Skills
(4) (5) (1) (4) (2)
Very Strong Satisfactory Strong Satisfactory Good
Cognitive Ability
(5) (2) (4) (2) (3)
Total Score [Blank 1] [Blank 2] [Blank 3] [Blank 4] [Blank 5]
Instead of judging the applicants based on the descriptors (e.g., good, strong, very strong), you calculate the total score of each applicant by summing up the four factor scores (1 point). Please
fill in each participant's total score in the table above.
How to get answers for each blank? Do I just add all of them for each blank?
There are 3 Steps involved in it
Step: 1
Yes you are correct To calculate the total score for each applicant you simply add up the four KSA f...
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Sum of an integral involving Bessel functions
I am interested in the following sum:
where is a constant, and denotes the Bessel function of the first kind. This is a special case of a more general sum I'd like to consider, with :where denotes the standard Euclidean norm on , i.e. .
I've tried computing this in a few different ways using Mathematica. The first way to get rid of the Bessel functions is to use the bound for some constant depending on . However, this may be
dangerous, since by taking absolute values of the Bessel functions, we lose the ability to take advantage of any positive-negative cancellation that occurs. Mathematica doesn't seem to be able to
compute the integral for , although one can get a numerical result by replacing with numbers instead (but since I want to sum over , this is not entirely helpful). We can also use the asymptotic
expansions of , and in the case of , the Bessel function has a very simple closed form:
and in general, there are also finite sum expansions for half-integer values of Bessel functions. But I haven't managed to get any semblance of a result, using any of these methods.
Can anyone find a way of computing this sum for , or perhaps a better method for general ?
Re: Sum of an integral involving Bessel functions
Can you please show me your M code?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Sum of an integral involving Bessel functions
The input for that sum for d = 2 (restricted to positive integers) is
Sum[Abs[Integrate[((x^2 + y^2)^(-1/2))*(((b - x)^2 + (c - y)^2)^(-1/2))*
BesselJ[1, k*Sqrt[x^2 + y^2]]*
BesselJ[1, k*Sqrt[(b - x)^2 + (y - c)^2]], {x, 0, 2*Pi}, {y, 0,
2*Pi}]^2], {b, 1, Infinity}, {c, 1, Infinity}]
but that will not give an answer, I don't think. I'm currently trying to use NSum/NIntegrate to try to find some partial sums given parameters .
Last edited by zetafunc (2016-10-09 03:24:16)
Re: Sum of an integral involving Bessel functions
I am getting a syntax error out of that. Please check your brackets.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Sum of an integral involving Bessel functions
Sorry, I fixed it. There was a ] missing.
Re: Sum of an integral involving Bessel functions
That is a very tough problem and may not have a closed form. What kind of answer are you looking for?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Sum of an integral involving Bessel functions
I do not necessarily need to know the exact sum. I do know that it should converge, but I am really trying to find a bound for it in terms of .
Re: Sum of an integral involving Bessel functions
Perhaps I can get something out of this. First, I would like to test empirically your assertion that it converges.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Sum of an integral involving Bessel functions
It is possible it may diverge, but if that is the case, it means that I have done something wrong (it should be true that the above sum is actually ).
Last edited by zetafunc (2016-10-09 03:50:57)
Re: Sum of an integral involving Bessel functions
One problem is that I do not know what k is. Can you say something about k?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Sum of an integral involving Bessel functions
Sorry, the k should be a . I just entered k because it was easier than entering into M.
Last edited by zetafunc (2016-10-09 03:54:07)
Re: Sum of an integral involving Bessel functions
So, you want k to be another free variable or I am hoping we can at least bound it...
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Sum of an integral involving Bessel functions
We can use the bound mentioned in post #1 but it may cause the integral to either converge or diverge. If d = 2 then, after bounding the Bessel functions, one gets:(I left out the as neither the sum
nor integral depend on it now, if we choose to bound the Bessel functions in this way.)
Last edited by zetafunc (2016-10-09 04:08:16)
Re: Sum of an integral involving Bessel functions
Can you write that up in Mathematica speak?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Sum of an integral involving Bessel functions
Here it is:
Sum[(Integrate[((x^2 + y^2)^(-3/4))*(((b - x)^2 + (c - y)^2)^(-3/
4)), {x, 0, 2*Pi}, {y, 0, 2*Pi}]^2), {b, 1, Infinity}, {c, 1,
If it is possible to show this converges then we are done.
Last edited by zetafunc (2016-10-09 04:23:06)
Re: Sum of an integral involving Bessel functions
Bracket missing.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Sum of an integral involving Bessel functions
Sorry, fixed it again.
Re: Sum of an integral involving Bessel functions
Let me see what can be done with that now. Please hold on.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Sum of an integral involving Bessel functions
OK, thanks. I am currently waiting to see what Mathematica does with the original sum if b and c vary from 1 to 100.
Re: Sum of an integral involving Bessel functions
Are there any singularities in that integral?
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Sum of an integral involving Bessel functions
You mean the one in post #13? There is definitely a singularity at (x,y) = (0,0). Others may occur too if at any point (b,c) = (x,y). For the integral by itself though, there should only be a
singularity at (0,0).
Last edited by zetafunc (2016-10-09 04:32:01)
Re: Sum of an integral involving Bessel functions
A singularity at (0,0) is one of the endpoints of the integral can be a big problem. Is it a removable singularity?
There will of course be chances for more singularities at (b,c) = (x,y) as you point out.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Sum of an integral involving Bessel functions
Hmm, I don't think so. The limit as (x,y) tends to (0,0) blows up to infinity. That is a big problem. I may need to talk to my supervisor about this.
Re: Sum of an integral involving Bessel functions
It is not impossible but usually singularities unless they are removable cause integrals to equal infinity, in other words they do not converge and therefore do not exist.
In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.
Re: Sum of an integral involving Bessel functions
The only way out of this that I can see would be to try to use the asymptotic expansions of the Bessel functions so that we end up with something with positive powers rather than negative ones. | {"url":"https://www.mathisfunforum.com/viewtopic.php?pid=388843","timestamp":"2024-11-06T12:31:22Z","content_type":"application/xhtml+xml","content_length":"54143","record_id":"<urn:uuid:c9360936-7462-4bb7-b309-113f5bc83a42>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00618.warc.gz"} |
How to Find Range in Excel (Easy Formulas)
Normally, when I use the word range in my tutorials about Excel, it’s a reference to a cell or a collection of cells in the worksheet.
But this tutorial is not about that range.
A ‘Range’ is also a mathematical term that refers to the range in a data set (i.e., the range between the minimum and the maximum value in a given dataset)
In this tutorial, I will show you really simple ways to calculate the range in Excel.
What is a Range?
In a given data set, the range of that data set would be the spread of values in that data set.
To give you a simple example, if you have a data set of student scores where the minimum score is 15 and the maximum score is 98, then the spread of this data set (also called the range of this data
set) would be 73
Range = 98 – 15
‘Range’ is nothing but the difference between the maximum and the minimum value of that data set.
How to Calculate Range in Excel?
If you have a list of sorted values, you just have to subtract the first value from the last value (assuming that the sorting is in the ascending order).
But in most cases, you would have a random data set where it’s not already sorted.
Finding the range in such a data set is quite straightforward as well.
Excel has the functions to find out the maximum and the minimum value from a range (the MAX and the MIN function).
Suppose you have a data set as shown below, and you want to calculate the range for the data in column B.
Below is the formula to calculate the range for this data set:
The above formula finds the maximum and the minimum value and gives us the difference.
Quite straightforward… isn’t it?
Also read: Calculate Interquartile Range (IQR) in Excel
Calculate Conditional Range in Excel
In most practical cases, finding the range would not be as simple as just subtracting the minimum value from the maximum value
In real-life scenarios, you might also need to account for some conditions or outliers.
For example, you may have a data set where all the values are below 100, but there is one value that is above 500.
If you calculate arrange for this data set, it would lead to you making misleading interpretations of the data.
Thankfully, Excel has many conditional formulas that can help you sort out some of the anomalies.
Below I have a data set where I need to find the range for the sales values in column B.
If you look closely at this data, you would notice that there are two stores where the values are quite low (Store 1 and Store 3).
This could be because these are new stores or there were some external factors that impacted the sales for these specific stores.
While calculating the range for this data set, it might make sense to exclude these newer stores and only consider stores where there are substantial sales.
In this example, let’s say I want to ignore all those stores where the sales value is less than 20,000.
Below is the formula that would find the range with the condition:
In the above formula, instead of using the MIN function, I have used the MINIFS function (it’s a new function in Excel 2019 and Microsoft 365).
This function finds the minimum value if the criteria mentioned in it are met. In the above formula, I specified the criteria to be any value that is more than 20,000.
So, the MINIFS function goes through the entire data set, but only considers those values that are more than 20,000 while calculating the minimum value.
This makes sure that values lower than 20,000 are ignored and the minimum value is always more than 20,000 (hence ignoring the outliers).
Note that the MINIFS is a new function in Excel is available only in Excel 2019 and Microsoft 365 subscription. If you’re using prior versions, you would not have this function (and can use the
formula covered later in this tutorial)
If you don’t have the MINIF function in your Excel, use the below formula that uses a combination of IF function and MIN function to do the same:
Just like I have used the conditional MINIFS function, you can also use the MAXIFS function if you want to avoid data points that are outliers in the other direction (i.e., a couple of large data
points that can skew the data)
So, this is how you can quickly find the range in Excel using a couple of simple formulas.
I hope you found this tutorial useful.
Other Excel tutorials you may like:
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Safe Haskell Safe-Inferred
Language Haskell98
data Counter a Source
A Counter a maps bijectively between a subset of values of type a and some possibly empty or infinite prefix of [0..].
cCount is Just n when the counter is finite and manages n values, or Nothing when infinite.
cToPos converts a managed value to its natural number (starting from 0).
cFromPos converts a natural number to its managed value.
cToPos c . cFromPos c must be the identity function. This invariant is maintained using the combinators below.
listCounter :: Counter a -> Counter [a] Source
Counter for all lists of all values in given counter.
The count is 1 (the only value being the empty list) if the given counter is empty, infinite otherwise. | {"url":"https://hackage.haskell.org/package/count-0.0.1/candidate/docs/Data-Count-Counter.html","timestamp":"2024-11-13T01:49:08Z","content_type":"application/xhtml+xml","content_length":"13267","record_id":"<urn:uuid:91529bd4-19f3-432a-a328-8dc869475234>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00748.warc.gz"} |
Chapter 9 Statistical Models | Modern Data Visualization with R
Chapter 9 Statistical Models
A statistical model describes the relationship between one or more explanatory variables and one or more response variables. Graphs can help to visualize these relationships. In this section we’ll
focus on models that have a single response variable that is either quantitative (a number) or binary (yes/no).
This chapter describes the use of graphs to enhance the output from statistical models. It is assumed that the reader has a passing familiarity with these models. The book R for Data Science (
Wickham and Grolemund 2017) can provide the necessary background and freely available on-line.
9.1 Correlation plots
Correlation plots help you to visualize the pairwise relationships between a set of quantitative variables by displaying their correlations using color or shading.
Consider the Saratoga Houses dataset, which contains the sale price and property characteristics of Saratoga County, NY homes in 2006 (Appendix A.14). In order to explore the relationships among the
quantitative variables, we can calculate the Pearson Product-Moment correlation coefficients.
In the code below, the select_if function in the dplyr package is used to select the numeric variables in the data frame. The cor function in base R calculates the correlations. The use=
“complete.obs” option deletes any cases with missing data. The round function rounds the printed results to 2 decimal places.
data(SaratogaHouses, package="mosaicData")
# select numeric variables
df <- dplyr::select_if(SaratogaHouses, is.numeric)
# calulate the correlations
r <- cor(df, use="complete.obs")
The ggcorrplot function in the ggcorrplot package can be used to visualize these correlations. By default, it creates a ggplot2 graph where darker red indicates stronger positive correlations, darker
blue indicates stronger negative correlations and white indicates no correlation.
From the graph, an increase in number of bathrooms and living area are associated with increased price, while older homes tend to be less expensive. Older homes also tend to have fewer bathrooms.
The ggcorrplot function has a number of options for customizing the output. For example
• hc.order = TRUE reorders the variables, placing variables with similar correlation patterns together.
• type = "lower" plots the lower portion of the correlation matrix.
• lab = TRUE overlays the correlation coefficients (as text) on the plot.
These, and other options, can make the graph easier to read and interpret. See ?ggcorrplot for details.
9.2 Linear Regression
Linear regression allows us to explore the relationship between a quantitative response variable and an explanatory variable while other variables are held constant.
Consider the prediction of home prices in the Saratoga Houses dataset from lot size (square feet), age (years), land value (1000s dollars), living area (square feet), number of bedrooms and bathrooms
and whether the home is on the waterfront or not.
data(SaratogaHouses, package="mosaicData")
houses_lm <- lm(price ~ lotSize + age + landValue +
livingArea + bedrooms + bathrooms +
data = SaratogaHouses)
Table 9.1: Linear Regression results
term estimate std.error statistic p.value
(Intercept) 139878.80 16472.93 8.49 0.00
lotSize 7500.79 2075.14 3.61 0.00
age -136.04 54.16 -2.51 0.01
landValue 0.91 0.05 19.84 0.00
livingArea 75.18 4.16 18.08 0.00
bedrooms -5766.76 2388.43 -2.41 0.02
bathrooms 24547.11 3332.27 7.37 0.00
waterfrontNo -120726.62 15600.83 -7.74 0.00
From the results, we can estimate that an increase of one square foot of living area is associated with a home price increase of $75, holding the other variables constant. Additionally, waterfront
home cost approximately $120,726 more than non-waterfront home, again controlling for the other variables in the model.
The visreg (http://pbreheny.github.io/visreg) package provides tools for visualizing these conditional relationships.
The visreg function takes (1) the model and (2) the variable of interest and plots the conditional relationship, controlling for the other variables. The option gg = TRUE is used to produce a ggplot2
# conditional plot of price vs. living area
visreg(houses_lm, "livingArea", gg = TRUE)
The graph suggests that, after controlling for lot size, age, living area, number of bedrooms and bathrooms, and waterfront location, sales price increases with living area in a linear fashion.
How does visreg work? The fitted model is used to predict values of the response variable, across the range of the chosen explanatory variable. The other variables are set to their median value
(for numeric variables) or most frequent category (for categorical variables). The user can override these defaults and chose specific values for any variable in the model.
Continuing the example, the price difference between waterfront and non-waterfront homes is plotted, controlling for the other seven variables. Since a ggplot2 graph is produced, other ggplot2
functions can be added to customize the graph.
# conditional plot of price vs. waterfront location
visreg(houses_lm, "waterfront", gg = TRUE) +
scale_y_continuous(label = scales::dollar) +
labs(title = "Relationship between price and location",
subtitle = paste0("controlling for lot size, age, ",
"land value, bedrooms and bathrooms"),
caption = "source: Saratoga Housing Data (2006)",
y = "Home Price",
x = "Waterfront")
There are far fewer homes on the water, and they tend to be more expensive (even controlling for size, age, and land value).
The vizreg package provides a wide range of plotting capabilities. See Visualization of regression models using visreg (Breheny and Burchett 2017) for details.
9.3 Logistic regression
Logistic regression can be used to explore the relationship between a binary response variable and an explanatory variable while other variables are held constant. Binary response variables have two
levels (yes/no, lived/died, pass/fail, malignant/benign). As with linear regression, we can use the visreg package to visualize these relationships.
The CPS85 dataset in the mosaicData package contains a random sample of from the 1985 Current Population Survey, with data on the demographics and work experience of 534 individuals.
Let’s use this data to predict the log-odds of being married, given one’s sex, age, race and job sector. We’ll allow the relationship between age and marital status to vary between men and women by
including an interaction term (sex*age).
# fit logistic model for predicting
# marital status: married/single
data(CPS85, package = "mosaicData")
cps85_glm <- glm(married ~ sex + age + sex*age + race + sector,
Using the fitted model, let’s visualize the relationship between age and the probability of being married, holding the other variables constant. Again, the visreg function takes the model and the
variable of interest and plots the conditional relationship, controlling for the other variables. The option gg = TRUE is used to produce a ggplot2 graph. The scale = "response" option creates a plot
based on a probability (rather than log-odds) scale.
# plot results
visreg(cps85_glm, "age",
gg = TRUE,
scale="response") +
labs(y = "Prob(Married)",
x = "Age",
title = "Relationship of age and marital status",
subtitle = "controlling for sex, race, and job sector",
caption = "source: Current Population Survey 1985")
## Conditions used in construction of plot
## sex: M
## race: W
## sector: prof
For professional, white males, the probability of being married is roughly 0.5 at age 25 and decreases to 0.1 at age 55.
We can create multiple conditional plots by adding a by option. For example, the following code will plot the probability of being married by age, separately for men and women, controlling for race
and job sector.
# plot results
visreg(cps85_glm, "age",
by = "sex",
gg = TRUE,
scale="response") +
labs(y = "Prob(Married)",
x = "Age",
title = "Relationship of age and marital status",
subtitle = "controlling for race and job sector",
caption = "source: Current Population Survey 1985")
In this data, the probability of marriage for men and women differ significantly over the ages measured.
9.4 Survival plots
In many research settings, the response variable is the time to an event. This is frequently true in healthcare research, where we are interested in time to recovery, time to death, or time to
If the event has not occurred for an observation (either because the study ended or the patient dropped out) the observation is said to be censored.
The NCCTG Lung Cancer dataset in the survival package provides data on the survival times of patients with advanced lung cancer following treatment. The study followed patients for up 34 months.
The outcome for each patient is measured by two variables
• time - survival time in days
• status - 1 = censored, 2 = dead
Thus a patient with time = 305 & status = 2 lived 305 days following treatment. Another patient with time = 400 & status = 1, lived at least 400 days but was then lost to the study. A patient with
time = 1022 & status = 1, survived to the end of the study (34 months).
A survival plot (also called a Kaplan-Meier Curve) can be used to illustrates the probability that an individual survives up to and including time t.
# plot survival curve
sfit <- survfit(Surv(time, status) ~ 1, data=lung)
title="Kaplan-Meier curve for lung cancer survival")
Roughly 50% of patients are still alive 300 days post treatment. Run summary(sfit) for more details.
It is frequently of great interest whether groups of patients have the same survival probabilities. In the next graph, the survival curve for men and women are compared.
# plot survival curve for men and women
sfit <- survfit(Surv(time, status) ~ sex, data=lung)
legend.labs=c("Male", "Female"),
palette=c("cornflowerblue", "indianred3"),
title="Kaplan-Meier Curve for lung cancer survival",
xlab = "Time (days)")
The ggsurvplot function has many options (see ?ggsurvplot). In particular, conf.int provides confidence intervals, while pval provides a log-rank test comparing the survival curves.
The p-value (0.0013) provides strong evidence that men and women have different survival probabilities following treatment. In this case, women are more likely to survive across the time period
9.5 Mosaic plots
Mosaic charts can display the relationship between categorical variables using rectangles whose areas represent the proportion of cases for any given combination of levels. The color of the tiles can
also indicate the degree relationship among the variables.
Although mosaic charts can be created with ggplot2 using the ggmosaic package, I recommend using the vcd package instead. Although it won’t create ggplot2 graphs, the package provides a more
comprehensive approach to visualizing categorical data.
People are fascinated with the Titanic (or is it with Leo?). In the Titanic disaster, what role did sex and class play in survival? We can visualize the relationship between these three categorical
variables using the code below.
The dataset (titanic.csv) describes the sex, passenger class, and survival status for each of the 2,201 passengers and crew. The xtabs function creates a cross-tabulation of the data, and the ftable
function prints the results in a nice compact format.
# input data
titanic <- read_csv("titanic.csv")
# create a table
tbl <- xtabs(~Survived + Class + Sex, titanic)
## Sex Female Male
## Survived Class
## No 1st 4 118
## 2nd 13 154
## 3rd 106 422
## Crew 3 670
## Yes 1st 141 62
## 2nd 93 25
## 3rd 90 88
## Crew 20 192
The mosaic function in the vcd package plots the results.
The size of the tile is proportional to the percentage of cases in that combination of levels. Clearly more passengers perished, than survived. Those that perished were primarily 3rd class male
passengers and male crew (the largest group).
If we assume that these three variables are independent, we can examine the residuals from the model and shade the tiles to match. The shade = TRUE adds fill colors. Dark blue represents more cases
than expected given independence. Dark red represents less cases than expected if independence holds.
The labeling_args, set_labels, and main options improve the plot labeling.
shade = TRUE,
labeling_args =
list(set_varnames = c(Sex = "Gender",
Survived = "Survived",
Class = "Passenger Class")),
set_labels =
list(Survived = c("No", "Yes"),
Class = c("1st", "2nd", "3rd", "Crew"),
Sex = c("F", "M")),
main = "Titanic data")
We can see that if class, gender, and survival are independent, we are seeing many more male crew perishing, and 1^st, 2^nd and 3^rd class females surviving than would be expected. Conversely, far
fewer 1^st class passengers (both male and female) died than would be expected by chance. Thus the assumption of independence is rejected. (Spoiler alert: Leo doesn’t make it.)
For complicated tables, labels can easily overlap. See ?labeling_border for plotting options.
Breheny, Patrick, and Woodrow Burchett. 2017.
“Visualization of Regression Models Using Visreg.”
Journal Article.
The R Journal
9 (2): 56–71.
Wickham, Hadley, and Garrett Grolemund. 2017. R for Data Science : Import, Tidy, Transform, Visualize, and Model Data. Book. Beijing: O’Reilly. | {"url":"https://rkabacoff.github.io/datavis/Models.html","timestamp":"2024-11-05T12:14:59Z","content_type":"text/html","content_length":"79308","record_id":"<urn:uuid:6b35a567-fd6c-4263-ba0a-64b0a84fe190>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00057.warc.gz"} |
Aaltodoc Repository :: Browsing by Author "Gionis, Aristides, Associate Prof., Aalto University, Department of Computer Science, Finland"
Browsing by Author "Gionis, Aristides, Associate Prof., Aalto University, Department of Computer Science, Finland"
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• Advances in Analysing Temporal Data
(Aalto University, 2017) Kostakis, Orestis; Tietotekniikan laitos; Department of Computer Science; Data Mining Group; Perustieteiden korkeakoulu; School of Science; Gionis, Aristides, Associate
Prof., Aalto University, Department of Computer Science, Finland
Modern technical capabilities and systems have resulted in an abundance of data. A significant portion of all that data is of temporal nature. Hence, it becomes imperative to design effective and
efficient algorithms, and solutions that enable searching and analysing large databases of temporal data. This thesis contains several contributions related to the broad scientific field of
temporal-data analysis. First, we present a distance function for pairs of event-interval sequences, together with proofs of important properties, such as that the function is a metric, and a
lower-bounding function. An embedding-based indexing method is proposed for searching through large databases of event-interval sequences, under this distance function. Second, we study the
problem of subsequence search for event-interval sequences. This includes hardness results, an exact worst-case exponential-time algorithm, two upper bounds and a scheme for approximation
algorithms. In addition, an equivalence is established between graphs and event-interval sequences. This equivalence allows to derive hardness results for several problems of event-interval
sequences. Most importantly, it raises the question which techniques, results, and methods from each of the fields of graph mining and temporal data mining can be applied to the other that would
advance the current state of the art. Third, for the problem of subsequence search, we propose an indexing method based on decomposing event-interval sequences into 2-interval patterns. The
proposed indexing method is benchmarked against other approaches. In addition, we examine different variations of the problem and propose exact algorithms for solving them. Fourth, we describe a
complete system that enables the clustering of a stream of graphs. The graphs are clustered into groups based on their distances, via approximating the graph edit distance. The proposed
clustering algorithm achieves a good clustering with few graph comparisons. The effectiveness and usefulness of the systems is demonstrated by clustering function call-graphs of binary executable
files for the purpose of malware detection. Finally, we solve the problem of summarising temporal networks. We assume that networks operate in certain modes and that the total sequence of
interactions can be modelled as a series of transitions between these modes. We prove hardness results and provide heuristic procedures for finding approximate solutions. We demonstrate the
quality of our methods via benchmarking and performing case-studies on datasets taken from sports and social networks.
• Sampling from scarcely defined distributions: Methods and applications in data mining
(Aalto University, 2016) Kallio, Aleksi; Mannila, Heikki, Prof., Aalto University, Department of Computer Science, Finland; Puolamäki, Kai, Docent, Aalto University, Department of Computer
Science, Finland; Tietotekniikan laitos; Department of Computer Science; Perustieteiden korkeakoulu; School of Science; Gionis, Aristides, Associate Prof., Aalto University, Department of
Computer Science, Finland
The importance of data is widely acknowledged in the modern society. Increasing volumes of information and growing interest in data driven decision making are creating new demands for analytical
methods. In data mining applications, users are often required to operate with limited background knowledge. Specifically, one needs to analyze data and derived statistics without exact
information on underlying statistical distributions. This work introduces the term scarcely defined distributions to describe such statistical distributions. In traditional statistical testing
one often makes assumptions about the source of data, such as those related to normal distribution. If data are produced by a controlled experiment and originate from a well-known source, these
assumptions can be justified. In data mining strong presuppositions about the data source typically cannot be made, as the data source is not under the control of the analyst, is not well known
or is too complex to understand. The present research discusses methods and applications of data mining, in which scarcely defined distributions emerge. Several strategies are put forth that
allow to analyze the dataset even when distributions are not well known, both in frequentist and information-theoretic statistical frameworks. A recurring theme is how to employ controls at the
analysis phase, if the data were not produced in a controlled experiment. In most cases presented, control is achieved by adopting randomization and other empirical sampling methods that rely on
large data sizes and computational power. Data mining applications reviewed in this work are from several fields. Biomedical measurement data are explored in multiple cases, involving both
microarray and high-throughput sequencing data types. In ecological and paleontological domains the analysis of presence-absence data of taxa is discussed. A common factor for all of the
application areas is the complexity of the underlying processes and the biased error sources of the measurement process. Finally, the study discusses the future trend of growing data volumes and
the relevance of the proposed methods and solutions in that context. It is noted that the growing complexity and the needs for quickly adaptable methods favor the general approach taken in the
thesis, while increasing data volumes and computational power makes it practically feasible. | {"url":"https://aaltodoc.aalto.fi/browse/author?value=Gionis,%20Aristides,%20Associate%20Prof.,%20Aalto%20University,%20Department%20of%20Computer%20Science,%20Finland","timestamp":"2024-11-04T15:22:24Z","content_type":"text/html","content_length":"426549","record_id":"<urn:uuid:2cd498de-3309-4028-8cb1-70efa2be03ea>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00172.warc.gz"} |
School Math
Submitted by Atanu Chaudhuri on Sun, 26/06/2016 - 23:05
Instead of routine procedures, solve a problem in many ways and apply less facts and more procedures approach
Solve a problem in many ways and apply less facts and more procedures approach, both working towards improving the general problem solving skills of the student.
Often we find at high school level, math problems solved following a routine series of steps and in only one predictable way. The students learn to believe that generally problems are to be solved in
only one mechanical routine way. Especially this we find in case of problems of the type,
Prove that, "Some expression " = "Some other expression".
In school math terminology, the "Some expression" is called LHS (short form of Left Hand Side) and the "Some other expression" as RHS (short form of Right Hand Side). This type of problems occur
abundantly in Elementary Trigonometry of proving Identities.
These solutions use a conventional approach of going towards the solution from LHS to RHS (or initial state to goal state) usually through many steps using the expansion of the LHS expression and
then simplification or consolidation of the numerous expanded terms towards the form of expression on the right hand side, that is, the RHS.
The most important element absent in this type of of conventional solutions is the element of Analysis, and the reasons why a particular path of solution is followed.
This approach has two important disadvantages,
1. Not only does this approach take considerable amount of time and effort, but because of large number of steps, chances of error is much higher in this approach.
2. This mechanical approach relies heavily on manipulation of terms using low level mathematical constructs without using the problem solving abilities of the student. In fact, if students follow
only this approach of solving problems, they may tend to become used to mechanical and procedural thinking suppressing their inherent creative and innovative out-of-the-box thinking abilities.
While solving the problem example we have chosen this time, our objectives will be two and both important. We will,
• show and encourage you to solve a problem in many ways. This is practice of Many Ways Technique, one of the most powerful problem solving skill improving techniques, and,
• show an example of problem solver's reasoning in devising an unusual method of solution that uses only the simplest of the relevant formulas. This is application of the Less facts more procedures
approach that is crucial to expand your general problem solving skillsets or abilities.
Problem example
Prove the identity:
$\displaystyle\frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1} = \frac{1 + \sin \theta }{\cos \theta}$
First try to solve this problem yourself and then only go ahead.
Try to solve the problem in as many different ways as you can and judge and compare the advantages and disadvantages of the different methods of solution.
There is no better method than learning by yourself.
Conventional solution 1
We will see the first possible way this problem may be solved conventionally.
$\displaystyle\frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1}$
$=\displaystyle\frac{\sec \theta + \tan \theta - 1}{\tan \theta - \sec \theta + 1}$
$=\displaystyle\frac{(\sec \theta + \tan \theta) - (\sec^2\theta - \tan^2\theta)}{\tan \theta - \sec \theta + 1}$, as $\sec^2\theta - \tan^2\theta = 1$
$=\displaystyle\frac{(\sec \theta + \tan \theta)(1 - sec \theta + \tan \theta)}{\tan \theta - \sec \theta + 1}$
$=\displaystyle\frac{(\sec \theta + \tan \theta)(\tan \theta - \sec \theta + 1)}{\tan \theta - \sec \theta + 1}$
$=\sec \theta + \tan \theta$
$=\displaystyle\frac{1}{\cos \theta} + \displaystyle\frac{\sin \theta}{\cos \theta}$
$=\displaystyle\frac{1 + \sin \theta}{\cos \theta}$.
Conventional solution 2 using rich trigonometric concept
The second possible way this problem may be solved conventionally,
$\displaystyle\frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1}$
$=\displaystyle\frac{\sec \theta + \tan \theta - 1}{\tan \theta - \sec \theta + 1}$
$=\displaystyle\frac{\displaystyle\frac{1}{\sec \theta - \tan \theta} - 1}{\tan \theta - \sec \theta + 1}$, as $\sec^2\theta - \tan^2\theta = 1$, or, $\sec \theta + \tan \theta = \displaystyle\frac
{1}{\sec \theta - \tan \theta}$, this we call a rich Trigonometric concept derived from basic concepts
$=\displaystyle\frac{1 - \sec \theta + \tan \theta}{(\sec \theta - \tan \theta)(\tan \theta - \sec \theta + 1)}$
$=\displaystyle\frac{\tan \theta - \sec \theta + 1}{(sec\theta - tan\theta)(\tan \theta - \sec \theta + 1)}$
$=\displaystyle\frac{1}{\sec \theta - \tan \theta}$
$=\sec \theta + \tan \theta$, applying the rich concept formula a second time
$=\displaystyle\frac{1}{\cos \theta} + \displaystyle\frac{\sin \theta}{\cos \theta}$
$=\displaystyle\frac{1 + \sin \theta}{\cos \theta}$.
A new way to the solution in a few steps
As a part of efficient problem solving process, the very first step that you must take is to analyze the problem.
In any problem solving, math or otherwise, this must be the first step. You must start with analyzing the problem statement.
Without the first step of Problem analysis, no efficient problem solving is possible.
A corollary,
In competitive exams, and also in competitive work environment, the first step of problem analysis is crucial for success.
The better and quicker you are able to analyze a problem, the faster you would reach the desired solution.
Problem analysis
The first step in analyzing this type of problem is to see how close are the goal state and the initial state. In this case as there are no similarities between the RHS and the LHS at all, we
Inherently the given expression has the simplification resources in itself which if used will lead us automatically to the simple goal state quickly.
Secondly we find the goal state RHS is in terms of $sin$ and $cos$ functions while the given expression is in one level higher $tan$ and $sec$ functions. So we take the clear decision to transform
the given expression in terms of $sin$ and $cos$ in one single step. This should make further simplification easier. In trigonometry we can encapsualte this concept as a general technique, the Goal
form matching technique,
Transform the level of the input or given expression to the level of the goal or target expression first.
Based on the analysis at the first stage and after applying goal form matching technique, now we take the bold step of multiplying and dividing the given expression by the target expression. The
reasoning for this unusual action is:
As the target expression has no similarity with the given expression, when we multiply and divide the given expression by the target expression, especially as both the given and the target
expressions are fractional terms, the multiplied part will remain untouched while the divided part has to resolve itself to unity, thus leaving the multiplied part as the RHS or the solution
Aside: Psychology and process of problem solving by End State Analysis and deductive reasoning: The desired goal to reach undoubtedly rank highest in importance in your mind among all other
information about the problem as your natural tendency is to reach the goal state in quickest possible time.
This pre-eminence of importance of the desired end state or goal state focuses your attention naturally on this end state when you know it. This is the case of proving identities..
What would you look for in the end state?
If it is a journey from one city to another, you study the distance to the destination from your starting point. You try to judge what kind of transportation along which path would take you to the
destination in shortest possible time, isn't it? We assume here the importance of optimal journey, which is the case of any important problem solving.
The same happens in this case. You judge the end state (or RHS expression) with respect to the initial given state (or LHS expression). If somehow you find significant similarities between the two,
it would be easy for you to span the gap between the two states quickly.
In majority of cases though there would be significant dissimilarity between the initial starting point and the desired end point. The similarity if at all there, would be hidden from casual
This is where the ability of key information discovery plays its prime role in solving the problem.
More often than not, ability to recognize useful common pattern, even if hidden, results in key information discovery.
If you don't know the desired goal state, from initial problem analysis you have to form possible desired goal states.
This is application of one of the most powerful problem solving resources that we are aware of - the End State Analysis. If you want to know more you can refer to it here.
Though usually there will be inherent similarities between the goal state and the initial state, in some cases there will be total dissimilarity. In such cases, the dissimilarity itself will be a key
information for deciding the efficient course of action in solving the problem.
Problem solver's solution:
$\displaystyle\frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1}$
$=\displaystyle\frac{\displaystyle\frac{\sin \theta}{\cos \theta} + \displaystyle\frac{1}{\cos \theta} - 1}{\displaystyle\frac{\sin \theta}{\cos \theta} - \displaystyle\frac{1}{\cos \theta} + 1}$
$=\displaystyle\frac{\sin \theta - \cos \theta + 1}{\sin \theta + \cos \theta - 1}$
$=\displaystyle\frac{\sin \theta - \cos \theta + 1}{\sin \theta + \cos \theta - 1}\times{\frac{\cos \theta}{1+ \sin \theta}}$
$\hspace{35mm}\times{\displaystyle\frac{1+ \sin \theta}{\cos \theta}}$
$=\displaystyle\frac{\sin \theta - \cos \theta + 1}{(\sin \theta - 1) + \cos \theta}\times{\frac{\cos \theta}{(\sin \theta + 1)}}$
$\hspace{35mm}\times{\displaystyle\frac{1+ \sin \theta}{\cos \theta}}$
$=\displaystyle\frac{\sin \theta cos\theta -\cos^2\theta + \cos \theta}{(sin^2\theta - 1) + \sin\theta \cos\theta +\cos \theta }$
$\hspace{35mm}\times{\displaystyle\frac{1+ \sin \theta}{\cos \theta}}$
$=\displaystyle\frac{\sin \theta \cos\theta - \cos^2\theta + \cos \theta}{\sin \theta \cos \theta - \cos^2\theta + \cos \theta }$
$\hspace{35mm}\times{\displaystyle\frac{1+ \sin \theta}{\cos \theta}}$
$=\displaystyle\frac{1+ \sin \theta}{\cos \theta}$.
This is a method that directly reaches the solution breaking down the barrier in a way that is not usual, but yes it reaches the destination with assurance and in a few steps notwithstanding the
seemingly involved deduction in between.
Less facts more procedures approach
You would notice that in this solution we have used the simplest level of concepts or formulas in Trigonometry along with problem analysis and deductive reasoning. If in general while you solve a
problem in any situation, you rely more on the simplest concepts along with your problem solving techniques and strategies that are more general with wide applicability, finally you would find that
you need to remember least amount of basic concepts and a few powerful problem solving techniques or procedures in solving a large variety of problems.
Practice and ability to use this approach enables you to quickly arrive at the most elegant solution with least memory load and least effort.
End note: There is at least one more solution to this problem. If you can, find it out. More importantly, our recommendation as always, would be to evaluate the solutions yourself and choose the one
that suits you best. Be a learner and judge yourself
And Always think: is there any other shorter better way to the solution?
Guided help on Trigonometry in Suresolv
To get the best results out of the extensive range of articles of tutorials, questions and solutions on Trigonometry in Suresolv, follow the guide,
Reading and Practice Guide on Trigonometry in Suresolv for SSC CHSL, SSC CGL, SSC CGL Tier II Other Competitive exams.
The guide list of articles is up-to-date. | {"url":"https://suresolv.com/efficient-math-problem-solving/how-solve-school-math-problem-few-steps-and-many-ways-trigonometry-4","timestamp":"2024-11-04T06:00:52Z","content_type":"text/html","content_length":"42929","record_id":"<urn:uuid:0e893b23-362a-4ef4-b911-ef4193ef4a9a>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00240.warc.gz"} |
Introduction to Pyepidemics - epidemiological modeling in Python | Eki.Lab
During the first wave of COVID19 in 2020, Ekimetrics joined the CoData movement, a coalition of data and artificial intelligence specialists whose goal was to pool their skills to provide answers and
solutions on the evolution of the pandemics. We had the chance to work with many epidemiological experts, and as we went along we built a toolbox to facilitate our modeling of the current pandemics.
We then put this toolbox in open source under the name pyepidemics to contribute to the community on this scientific discipline difficult to apprehend for Data Scientists but with obvious bridges
facilitating innovation. Today, with the resurgence of the epidemic in Europe, it seemed important to present this library more widely to democratize these analyses on a larger scale.
• The library is available on Github at this link
• The documentation is available at this link
This article will serve as a synthetic presentation of what can be done with the library, please refer to the documentation for more details. Do not hesitate to post issues on Github and to
contribute with new proposals, the development is still in experimental version.
Introduction to Pyepidemics
Pyepidemics allows to simply create compartmental epidemiological models (also used in system dynamics) and to solve the differential equations that model the phenomenon. The different features
implemented today are:
• Creation of classical compartmental models (SIR, SEIR, SEIDR, etc...)
• Creation of COVID19 related model (with ICU and different levels of symptoms)
• Creation of custom compartmental model
• Implementation of policies (lockdown, tracing, testing, etc...)
• Calibration of epidemiological parameters on real-world data using Bayesian optimization
You can simply install pyepidemics using the command
Introduction to epidemiological modeling in Python
The images, the reasoning and the construction of the first bricks of the library are largely inspired by the exceptional work of Henri Froese with his series of articles on epidemiology, in
particular the first article Infectious Disease Modelling: Beyond the Basic SIR Model.
Compartmental models
An epidemic is modeled with the different possible states for the population, for example: unaffected, immunized, vaccinated, symptomatic patient, asymptomatic patient, hospitalized, in intensive
care ... Each state will be modeled with a compartment, and the population will transition between the different compartments according to parameters of probability and transition duration. In
concrete terms, this means solving a system of differential equations as a function of time.
For example the simplest compartmental model is the SIR model (for the 3 states Susceptible - Infected - Removed). It is possible to write the following compartments with their transition:
The transition parameters being given by :
Building a SIR model with pyepidemics
This section is detailed in this tutorial which is also available directly on Colab.
Using the bank of models
As the SIR model is a standard model, you can find an already coded version in the model bank. We will learn in the next section how this abstraction is built to allow you to add details in the
With pyepidemics:
from pyepidemics.models import SIR
N = 1000 # Thousand persons
beta = 3/4 # One person contaminates 3/4 person per day
gamma = 1/4 # One person stay infected for 4 days
sir = SIR(N,beta,gamma)
It is then possible to solve the system of differential equations simply with the method .solve() (more parameters are available, see the tutorial quoted above :
states = sir.solve()
states.show(plotly = False)
What is commonly called an epidemic "wave" is observed.
Reimplementing the SIR model
Let's now go into the internal workings of this abstraction to reimplement its operation in a few lines of code. Concretely we build a graph between the different states by detailing the transitions.
Pyepidemics then translates this graph into a system of differential equations in order to solve it.
from pyepidemics.models import CompartmentalModel
class SIR(CompartmentalModel):
def __init__(self,N,beta,gamma):
# Define compartments name and number
compartments = ["S","I","R"]
# Parameters
self.N = N # Total population
self.beta = beta # How many person each person infects per day
self.gamma = gamma # Rate of infection, duration = 1/gamma
# Add transition
self.add_transition("S","I",lambda y,t: self.beta * y["S"] * y["I"] / self.N)
self.add_transition("I","R",lambda y,t: self.gamma * y["I"])
So it is possible to build any kind of compartmental model with this operation. You can see more complex examples in this tutorial.
Create a realistic epidemiological model for COVID19
In the first section we saw how to build a SIR model, however we are far from being able to use this model to simulate an epidemic like COVID19 and for governments to be able to take actions using
those analyses. For this we would need to:
• Have a more complete description taking into account the specifics of the disease states
• Take into account the mitigation strategies provided by governments
• Calibrate the model to describe the evolution of the epidemic as closely as possible
Build the right compartmental model
Indeed, the COVID19 is more specific than simply the 3 SIR states described above. During the year 2020, we used the following compartmental model inspired by the work of the Pasteur Institute and
the INSERM (taking into account the different levels of symptoms, the incubation phase, and the passages in hospital or in intensive care)
You can look in the library at the implementation of the COVID19 model to see the implementation (very close to what has been described above for the SIR model, one of the strengths of pyepidemics).
Describe the mitigation solutions implemented (containment, testing)
Some parameters, such as the famous "R0", have varied enormously during the epidemic depending on behaviors (distancing, masks), policies (testing, lockdowns, vaccination), or variants of the
epidemic. Thus, to create a realistic model, we need to take these variations into account. We will take the example of the first lockdowns to understand how to implement these variations with
Let's create an evolution of the parameter R(t) according to the different lockdowns (during the first wave for example)
from pyepidemics.policies.utils import make_dynamic_fn
policies = [
fn = make_dynamic_fn(policies,sigmoid = False)
# Create our time-dependent parameter
beta = lambda y,t : fn(t)
# Visualize policies
x = np.linspace(0,100)
y = np.vectorize(fn)(x)
plt.figure(figsize = (15,4))
And that's it! We can use this time-dependent parameter like we used a constant before. To see more complete examples and results, you can redirect to this tutorial.
Find the most appropriate parameters for calibration
Finally, in the previous example, as in those before, we chose arbitrarily the values of the different probabilities and transition times. But these are precisely what epidemiologists want to
estimate. How does this work? Simply by testing different parameters until the solution of the compartmental model matches as much as possible the real evolution of the epidemic. Of course, we are
not going to test all the combinations but rather :
• Start with an a priori on the different parameter values from the most recent epidemiological studies
• Use a Bayesian optimization algorithm to go through the parameter space efficiently without having to calculate everything.
These two methods are easy to implement with pyepidemics if we have a pyepimemics model with a .reset() method that recreates a compartmental model with the new set of parameters. First we create a
parameter space to explore:
space = {
Then we will simply use the method .fit() by giving as input a dataframe with the real values of the epidemic for the compartments for which we would have succeeded in obtaining data (in general the
deaths and the hospital stays which are the most reliable).
n = 200,
early_stopping = 100,
This step is not easy, takes time and often also requires iterations to change the definition of the compartmental model. But once solved it is what allows epidemiologists to follow the dynamics of
the pandemic.
More details on calibration are available in this tutorial.
With this article we could introduce you to our epidemiological modeling library pyepidemics - simple modeling, flexible compartmental models, calibration, dynamic parameters, etc... The details of
the different features are available in the library documentation.
As previously mentioned, this library is still under experimental development, we are still using it to model the 5th wave of the epidemic, in particular by adding compartments for vaccination - we
will write an article on the modeling in the coming months. Do not hesitate to contribute to democratize this discipline to the Data Science community.
These few references helped us greatly during the development of the library | {"url":"https://ekimetrics.github.io/blog/introduction-pyepidemics/","timestamp":"2024-11-08T01:23:52Z","content_type":"text/html","content_length":"76558","record_id":"<urn:uuid:aefdcdc1-3755-4d15-b780-1cd7310bc2cc>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00819.warc.gz"} |
Span (cloth) to Handbreadth Converter
Enter Span (cloth)
β Switch toHandbreadth to Span (cloth) Converter
How to use this Span (cloth) to Handbreadth Converter π €
Follow these steps to convert given length from the units of Span (cloth) to the units of Handbreadth.
1. Enter the input Span (cloth) value in the text field.
2. The calculator converts the given Span (cloth) into Handbreadth in realtime β using the conversion formula, and displays under the Handbreadth label. You do not need to click any button. If the
input changes, Handbreadth value is re-calculated, just like that.
3. You may copy the resulting Handbreadth value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Span (cloth) to Handbreadth?
The formula to convert given length from Span (cloth) to Handbreadth is:
Length[(Handbreadth)] = Length[(Span (cloth))] / 0.3333333333384133
Substitute the given value of length in span (cloth), i.e., Length[(Span (cloth))] in the above formula and simplify the right-hand side value. The resulting value is the length in handbreadth, i.e.,
Calculation will be done after you enter a valid input.
Consider that a piece of cloth is measured to be 3 spans in width.
Convert this width from spans to Handbreadth.
The length in span (cloth) is:
Length[(Span (cloth))] = 3
The formula to convert length from span (cloth) to handbreadth is:
Length[(Handbreadth)] = Length[(Span (cloth))] / 0.3333333333384133
Substitute given weight Length[(Span (cloth))] = 3 in the above formula.
Length[(Handbreadth)] = 3 / 0.3333333333384133
Length[(Handbreadth)] = 9
Final Answer:
Therefore, 3 span is equal to 9 handbreadth.
The length is 9 handbreadth, in handbreadth.
Consider that a scarf is 2 spans long.
Convert this length from spans to Handbreadth.
The length in span (cloth) is:
Length[(Span (cloth))] = 2
The formula to convert length from span (cloth) to handbreadth is:
Length[(Handbreadth)] = Length[(Span (cloth))] / 0.3333333333384133
Substitute given weight Length[(Span (cloth))] = 2 in the above formula.
Length[(Handbreadth)] = 2 / 0.3333333333384133
Length[(Handbreadth)] = 6
Final Answer:
Therefore, 2 span is equal to 6 handbreadth.
The length is 6 handbreadth, in handbreadth.
Span (cloth) to Handbreadth Conversion Table
The following table gives some of the most used conversions from Span (cloth) to Handbreadth.
Span (cloth) (span) Handbreadth (handbreadth)
0 span 0 handbreadth
1 span 3 handbreadth
2 span 6 handbreadth
3 span 9 handbreadth
4 span 12 handbreadth
5 span 15 handbreadth
6 span 18 handbreadth
7 span 21 handbreadth
8 span 24 handbreadth
9 span 27 handbreadth
10 span 30 handbreadth
20 span 60 handbreadth
50 span 150 handbreadth
100 span 300 handbreadth
1000 span 3000 handbreadth
10000 span 30000 handbreadth
100000 span 300000 handbreadth
Span (cloth)
A span (cloth) is a unit of length used historically in textiles and cloth measurement. One span (cloth) is approximately equivalent to 24 inches or 0.6096 meters.
The span (cloth) is based on the width of a person's outstretched hand from thumb to little finger, providing a practical measure for fabric lengths and textile work.
Spans (cloth) were used in the textile industry for measuring and cutting fabric. While less common today, the unit remains of historical interest and reflects traditional practices in cloth
measurement and tailoring.
A handbreadth is a historical unit of length used to measure small distances, typically based on the width of a hand. One handbreadth is approximately equivalent to 4 inches or about 0.1016 meters.
The handbreadth is defined as the width of a person's hand, measured from the edge of the thumb to the edge of the little finger when the hand is spread out. This unit was used for practical
measurements in various contexts, including textiles and construction.
Handbreadths were used in historical measurement systems for assessing lengths and dimensions where precise tools were not available. Although less common today, the unit provides historical context
for traditional measurement practices and everyday use in different cultures.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Span (cloth) to Handbreadth in Length?
The formula to convert Span (cloth) to Handbreadth in Length is:
Span (cloth) / 0.3333333333384133
2. Is this tool free or paid?
This Length conversion tool, which converts Span (cloth) to Handbreadth, is completely free to use.
3. How do I convert Length from Span (cloth) to Handbreadth?
To convert Length from Span (cloth) to Handbreadth, you can use the following formula:
Span (cloth) / 0.3333333333384133
For example, if you have a value in Span (cloth), you substitute that value in place of Span (cloth) in the above formula, and solve the mathematical expression to get the equivalent value in | {"url":"https://convertonline.org/unit/?convert=span_cloth-handbreadths","timestamp":"2024-11-10T12:16:44Z","content_type":"text/html","content_length":"90874","record_id":"<urn:uuid:9d93ac98-1090-416f-9ca9-c5a577b78b3f>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00729.warc.gz"} |
Deduce facts about shapes (5G2a) - Reasoning: Geometry Maths Worksheets for KS2 Maths SATs Booster by URBrainy.com
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Calculate the lengths of shapes from given information.
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Online scheduling with a buffer on related machines
Online scheduling with a buffer is a semi-online problem which is strongly related to the basic online scheduling problem. Jobs arrive one by one and are to be assigned to parallel machines. A buffer
of a fixed capacity K is available for storing at most K input jobs. An arriving job must be either assigned to a machine immediately upon arrival, or it can be stored in the buffer for unlimited
time. A stored job which is removed from the buffer (possibly, in order to allocate a space in the buffer for a new job) must be assigned immediately as well. We study the case of two uniformly
related machines of speed ratio s ≥ 1, with the goal of makespan minimization. Two natural questions can be asked. The first question is whether this model is different from standard online
scheduling, that is, is any size of buffer K > 0 already helpful to the algorithm, compared to the case K = 0. The second question is whether there exists a constant K, so that a larger buffer is no
longer beneficial to an algorithm, that is, increasing the size of the buffer above this threshold would not change the best competitive ratio further. Previous work (Kellerer et al., Oper. Res.
Lett. 21, 235-242, 1997; Zhang, Inf. Process. Lett. 61, 145-148, 1997; Englert et al., Proc. 48th Symp. Foundations of Computer Science (FOCS), 2008) shows that in the case s = 1, already K = 1
allows to design a 4/3-competitive algorithm, which is best possible for any K ≥ 1, whereas the best possible ratio for K = 0 is 3/2. Similar results have been show for multiple identical machines
(Englert et al., Proc. 48th Symp. Foundations of Computer Science (FOCS), 2008). We answer both questions affirmatively, and show that a buffer of size K = 2 is sufficient to achieve the a
competitive ratio which matches the lower bound for K → ∞ for any s > 1. In fact, we show that a buffer of size K = 1 can evidently be exploited by the algorithm for any s > 1, but for a range of
values of s, it is still weaker than a buffer of size 2. On the other hand, in the case s ≥ 2, a buffer of size K = 1 already allows to achieve optimal bounds.
• Scheduling
• Semi-online algorithms
• Uniformly related machines
ASJC Scopus subject areas
• Computer Science Applications
• Discrete Mathematics and Combinatorics
• Control and Optimization
• Computational Theory and Mathematics
• Applied Mathematics
Dive into the research topics of 'Online scheduling with a buffer on related machines'. Together they form a unique fingerprint. | {"url":"https://cris.haifa.ac.il/en/publications/online-scheduling-with-a-buffer-on-related-machines","timestamp":"2024-11-02T05:08:52Z","content_type":"text/html","content_length":"57310","record_id":"<urn:uuid:6ba4d85f-85e8-460d-9538-287295603cb3>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00397.warc.gz"} |
Niuke practice 89
Source: Niuke network
Time limit: 1 second for C / C + + and 2 seconds for other languages
Space limitation: C / C + + 26244k, other languages 524288K
64bit IO Format: %lld
A Title Description
Niuniu likes rice very much. His meal consists of s grains of rice. By chance, he found a straight line. There are n grids on this straight line. The first grid has 1 rice grain, the second grid has
2 rice grains, the third grid has 4 rice grains... And so on. The nth grid has 2n-1 rice grains. However, due to the problems of K grids, all the rice grains placed on these grids will be lost. Now I
want to select some of the remaining n-k grids and take all the rice grains on the selected grid. Is there a plan to select exactly s rice grains?
Enter description
Output description
It can form a rice grain pile with a quantity of s and output YES, otherwise output NO
#include <bits/stdc++.h>
using namespace std;
const int maxn = 2e+5;
int main(){
int n, k, s;
cin >> n >> k >> s;
int i,flag=0;
int str[maxn];
memset(str, 0, sizeof(str));
for (i = 1; i <= n;i++)
str[i] = int(pow(2.0, float(i - 1)));
int rem,t = 0,sum = 0;
rem = n - k;
for (i = 1; i <= n; i++)
sum += str[i];
flag = 1;
//The mistake is that we can't make up a sum of several random values in 1-n
cout << "YES" << endl;
cout << "NO" << endl;
return 0;
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
int main(){
ll n, k, s;
cin >> n >> k >> s;
ll a;
cin >> a;
return puts("NO"), 0;
return 0;
Bit logic operation
&Operation -------------- - both are 1 - < 1 >
0&1 =0;
0&0 =0;
1&0 =0;
1&1 =1;
& =00100
&Operations are usually used for binary bit fetching operations. For ex amp le, the result of a number & 1 is to fetch the last bit of binary.
This can be used to judge the parity of an integer. The last bit of binary is 0, indicating that the number is even, and the last bit is 1, indicating that the number is odd
Similar to < < a > > b, a > > b means that the binary is shifted b bits to the right (excluding the last B bit), which is equivalent to a divided by 2 to the power of B (rounded). We often use > > 1
instead of / 2 (div 2), such as binary search, heap insertion, etc. finding a way to use > > instead of division can greatly improve the efficiency of the program. The binary algorithm of the maximum
common divisor uses the divide by 2 operation to replace the slow and surprising% (mod) operation, and the efficiency can be improved by 60%.
int a =100;
a/4 ==a>>2;
Source: Niuke network
Time limit: 1 second for C / C + + and 2 seconds for other languages
Space limitation: C / C + + 26244k, other languages 524288K
64bit IO Format: %lld
B Title Description
Niu Niu felt very thirsty after he was full, so he found his favorite cococola!
Niu Niu has a strong obsessive-compulsive disorder. Although they are all cocacola, if the order of some letters is reversed or the position is interchanged, he doesn't want to drink it.
In order to get cocacola as soon as possible, he asked you this question. I hope you can tell him how many times the character position needs to be exchanged at least (only one letter can be
exchanged at a time) to get cocacola
Enter description
A string with a length of 8 and guaranteed to have a solution
Output description
A number indicating the minimum number of exchanges to obtain cocacola
#include <bits/stdc++.h>
using namespace std;
int main(){
char ch[8] = {'c', 'o', 'c', 'a', 'c', 'o', 'l', 'a'};
char str[8];
cin >> str;
int i,t=0;
for (i = 0; i < 8;i++){
cout << t / 2 << endl;
return 0;
#include <bits/stdc++.h>
using namespace std;
string s, t = "cocacola";
int cnt;
int main(){
int c = 0, p = 0;
cin >> s;
for (int i = 0; i < 8;i++)
p = i;
swap(s[6], s[p]);
for (int i = 0; i <8;i++){
cout << cnt + (c + 1) / 2 << endl;
return 0; | {"url":"https://www.fatalerrors.org/a/niuke-practice-89.html","timestamp":"2024-11-10T21:06:58Z","content_type":"text/html","content_length":"14076","record_id":"<urn:uuid:aeb35da8-f66b-408b-91d7-f3a9077a5b4c>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00506.warc.gz"} |
Fake Kill Wand
1 KB, 13 years ago, submitted by
A simple tool that turns all connected bricks near the one you hit into physics bricks for 9 seconds, letting you make neat contraptions, Rube Goldberg-like sequences or just watch things fall to the
- Used by the command /killwand
- Bricks painted with "Pearl" FX are unaffected, for supports or solid areas
- Can only fake-kill once per second
- Limit 40 bricks at a time for non-admins, 240 for admins
- Only adds one datablock (A yellow Destructo Wand)
Won "Best Add-On" of Truce's 1024 Byte Scripting Contest (it's been changed a little since then but still comes in at 935 characters) | {"url":"https://blockland.online/rtb/addons/3088/fake-kill-wand","timestamp":"2024-11-08T21:40:27Z","content_type":"text/html","content_length":"60772","record_id":"<urn:uuid:294dbcc4-1347-475a-a9f4-10b76865b0e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00622.warc.gz"} |
Chapter 12
Financial Planning Exercise 7
Calculating key stock performance metrics
The Morton Company recently reported...
Chapter 12 Financial Planning Exercise 7 Calculating key stock performance metrics The Morton Company recently reported...
Chapter 12
Financial Planning Exercise 7
Calculating key stock performance metrics
The Morton Company recently reported net profits after taxes of $14.7 million. It has 5 million shares of common stock outstanding and pays preferred dividends of $1 million a year. The company's
stock currently trades at $67 per share.
1. Compute the stock's earnings per share (EPS). Round the answer to two decimal places.
$ per share
2. What's the stock's P/E ratio? Round the answer to two decimal places.
$ times
3. Determine what the stock's dividend yield would be if it paid $3.22 per share to common stockholders. Round the answer to two decimal places. | {"url":"https://justaaa.com/finance/235126-chapter-12-financial-planning-exercise-7","timestamp":"2024-11-01T20:53:22Z","content_type":"text/html","content_length":"42407","record_id":"<urn:uuid:df35fff0-c7b1-4c18-b721-eb7914eb48df>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00298.warc.gz"} |
The probably most popular Zernike based graph is amount of higher order RMS (HO RMS) over time. It shows whether or not higher order aberrations have been induced.
The graphs plots the mean value of HO RMS ±1 Standard deviation in microns (y-axis) for different follow up points (x-axis). The number of eyes per follow up is given in the lower part of the graph
Datagraph-med calculates the HO RMS value of all Zernike Terms in 3rd to 6th order.
Typically HO RMS values for a 6 mm pupil should be in the range of 0.3 to 0.4 µm. | {"url":"http://help-version-4.datagraph.eu/ho_rms.htm","timestamp":"2024-11-12T08:28:51Z","content_type":"text/html","content_length":"3091","record_id":"<urn:uuid:135aeb71-567c-46f7-9bb4-7ea418575306>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00414.warc.gz"} |
is a visual representation of an area symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes.
The above text is a snippet from Wikipedia: Map
and as such is available under the Creative Commons Attribution/Share-Alike License.
1. A visual representation of an area, whether real or imaginary.
2. A function.
The discrete topology is always continuous, therefore functions with discrete domains are always maps.
3. A continuous function.
4. A diagram of components of an item.
5. The butterfly .
6. Someone's face.
7. A predefined and confined imaginary area where a game session takes place.
"I don't want to play this map again!"
The above text is a snippet from Wiktionary: map
and as such is available under the Creative Commons Attribution/Share-Alike License.
Need help with a clue?
Try your search in the crossword dictionary! | {"url":"https://crosswordnexus.com/word/MAP","timestamp":"2024-11-03T23:34:22Z","content_type":"application/xhtml+xml","content_length":"11160","record_id":"<urn:uuid:61d9eda8-b4bb-4fe5-8d68-55f1005ee449>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00694.warc.gz"} |
Author(s) of this documentation:
by Jacques-Olivier Lachaud
Part of the Graph package.
This module shows how to use the Boost Graph library in DGtal.
The Boost Graph library.
The Boost Graph Library (http://www.boost.org/doc/libs/1_52_0/libs/graph/doc/index.html) is a very rich library for handling graph concepts, structures and algorithms. It uses a lot generic
programming to define a typology of graphs through a hierarchy of concept, and then it provides many generic algorithms on these graphs. Standard implementation of graphs are also provided (adjacency
list, incidence matrix). Furthermore this library uses the Boost Property Map Library (http://www.boost.org/doc/libs/1_52_0/libs/property_map/doc/property_map.html) to associate data with vertices or
edges in a very efficient and generic way.
For these reasons, it would be interesting that DGtal graphs match boost graph concepts. However, this cannot be done in full genericity with the present DGtal graph concepts. For instance, only
finite graphs are handled by boost graphs. Furthermore, it is tricky to have light boost graphs (i.e. graphs constructed on-the-fly), because boost graphs require multipass iterators on vertices and
For now, the only models that are wrapped to satisfy boost graph concepts are:
A natural question is why DGtal graph concepts do not match exactly boost graph concepts. This is for mainly three reasons:
□ DGtal graph concepts are very light compared to boost graph, and new models are thus easier to define.
□ boost graphs only handle finite graphs and we would like to see the adjacency graph of digital spaces as a graph.
□ boost graphs do not handle implicitly defined graphs, like graphs discover on-the-fly. If you really need boost graph, a copy / conversion is still possible.
Wrapping DigitalSurface as a boost graph
The file DigitalSurfaceBoostGraphInterface.h defines the boost graph traits for any kind of digital surface (see DigitalSurface). With these definitions, a DigitalSurface is a model of
boost::VertexListGraphConcept, boost::AdjacencyGraphConcept, boost::IncidenceGraphConcept, boost::EdgeListGraphConcept. You may use a DigitalSurface as any boost graph instance in boost graph
algorithms (see http://www.boost.org/doc/libs/1_52_0/libs/graph/doc/table_of_contents.html).
Making DigitalSurface a boost graph model
To use a DigitalSurface as a boost graph, you must include the header file DigitalSurfaceBoostGraphInterface.h before including boost graph headers (!).
#include "DGtal/topology/DigitalSurface.h"
#include "DGtal/graph/DigitalSurfaceBoostGraphInterface.h"
#include <boost/graph/graph_concepts.hpp>
#include <boost/graph/breadth_first_search.hpp>
A model of boost graph must satisfy a given number of traits (to define types) as well as functions acting on these types. This is done through specialization of boost::graph_traits. This is done for
concrete realizations of template class DigitalSurface. The following snippet shows the boost graph way to get the types associated to a graph.
typedef DigitalSurface< .... > Graph; // your preferred model of digital surface
typedef boost::graph_traits<Graph>::vertex_descriptor vertex_descriptor; // ie DigitalSurface::Vertex
typedef boost::graph_traits<Graph>::edge_descriptor edge_descriptor; // ie DigitalSurface::Arc
typedef boost::graph_traits<Graph>::vertices_size_type vertices_size_type; // ie DigitalSurface::Size
typedef boost::graph_traits<Graph>::vertex_iterator vertex_iterator; // the iterator for visiting all vertices
typedef boost::graph_traits<Graph>::out_edge_iterator out_edge_iterator; // the iterator for visiting out edges of a vertex
typedef boost::graph_traits<Graph>::edge_iterator edge_iterator; // the iterator for visiting all edges
You may check that a DigitalSurface satisfies several graph concepts
Go to http://www.boost.org/doc/libs/1_52_0/libs/graph/doc/AdjacencyGraph.html.
Go to http://www.boost.org/doc/libs/1_52_0/libs/graph/doc/EdgeListGraph.html.
Go to http://www.boost.org/doc/libs/1_52_0/libs/graph/doc/IncidenceGraph.html.
Go to http://www.boost.org/doc/libs/1_52_0/libs/graph/doc/VertexListGraph.html.
The boost graph way of visiting vertices
For any boost graph, there is a function boost::vertices that returns a pair of multipass input iterator on vertices representing the range of vertices of the graph. The following snippet shows how
it works.
Graph g(...); // your instance of digital surface
vp.first != vp.second; ++vp.first )
vertex_descriptor v1 = *vp.first;
() << v1 << std::endl;
// displays each vertex
std::pair< typename graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::vertex_iterator, typename graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::vertex_iterator >
vertices(const DGtal::DigitalSurface< TDigitalSurfaceContainer > &digSurf)
The boost graph way of visiting edges
For models of EdgeListGraphConcept, there is a function boost::edges that returns a pair of multipass input iterator on edges representing the range of (oriented) edges of the graph. The following
snippet shows how it works.
unsigned int nbEdges = 0;
( std::pair<edge_iterator, edge_iterator> ve =
( g );
ve.first != ve.second; ++ve.first, ++nbEdges )
edge_descriptor e = *ve.first;
<< v1 << " -> " << v2 << std::endl;
std::pair< typename graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::edge_iterator, typename graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::edge_iterator > edges
(const DGtal::DigitalSurface< TDigitalSurfaceContainer > &digSurf)
graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::vertex_descriptor source(typename graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::edge_descriptor edge, const
DGtal::DigitalSurface< TDigitalSurfaceContainer > &digSurf)
graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::vertex_descriptor target(typename graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::edge_descriptor edge, const
DGtal::DigitalSurface< TDigitalSurfaceContainer > &digSurf)
The boost graph way of getting adjacent vertices
For models of IncidenceGraphConcept, there is a function boost::out_edges that returns a pair of multipass input iterator on the edges that starts at the given vertex and ends on adjacent vertices.
The following snippet shows how it works.
vp.first != vp.second; ++vp.first )
vertex_descriptor v1 = *vp.first;
() <<
"Neighbors of "
<< v1 <<
" are"
( std::pair<out_edge_iterator, out_edge_iterator> ve =
( v1, g );
ve.first != ve.second; ++ve.first )
std::pair< typename graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::out_edge_iterator, typename graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::out_edge_iterator
> out_edges(typename graph_traits< DGtal::DigitalSurface< TDigitalSurfaceContainer > >::vertex_descriptor u, const DGtal::DigitalSurface< TDigitalSurfaceContainer > &digSurf)
Property maps for more elaborate algorithms
If you wish to use algorithms of the Boost Graph Library, most of them requires mapping from vertex or edge to some value (for instance a color for marking already visited vertices or a scalar for
storing a distance or a weight). This is done very generically in Boost Graph through property maps. The system is rather complex but allows you to use indifferently in your algorithms an external
map (for instance a std::map< vertex_descriptor, int >) or an embedded value in the vertex_descriptor type.
Standard boost graphs models offer simple mechanism to get a given property map for a graph. In DGtal, graph models do not integrate – for now – property maps. Therefore, only external property maps
can be used. The snippet below shows how to create two property maps for the digital surface g, using standard property map wrappers given in the Boost Property Map Library.
#include <boost/property_map/property_map.hpp>
// get the property map for coloring vertices (used for not visiting twice the same vertex).
typedef std::map< vertex_descriptor, boost::default_color_type > StdColorMap; // the container type
StdColorMap colorMap; // the container instance (will store computations).
boost::associative_property_map< StdColorMap > propColorMap( colorMap ); // a facade aroundcolorMap
// get the property map for labelling vertices (the mapping Vertex -> Size that stores the component label for each vertex)
typedef std::map< vertex_descriptor, vertices_size_type > StdComponentMap;
StdComponentMap componentMap;
boost::associative_property_map< StdComponentMap > propComponentMap( componentMap );
We may afterwards use this property maps in boost graph algorithms. This snippet extracts the connected components of the graph g, and labels each vertex with its component (result is stored in
componentMap, hence is also accessible with propComponentMap).
// g must be a model of VertexListGraph
vertices_size_type nbComp =
boost::connected_components // boost graph connected components algorithm.
( g, // the graph
propComponentMap, // the mapping vertex -> label
boost::color_map( propColorMap ) // this map is used internally when computing connected components.
() <<
"- nbComponents = "
<< nbComp << std::endl;
Note that propColorMap is given a named parameter with a call to boost::color_map. This is the method used in the Boost Graph Library to give handle parameters, especially when the algorithm requires
a lot of parameters, some of them being optionnal. This is explained in section (http://www.boost.org/doc/libs/1_52_0/libs/graph/doc/bgl_named_params.html).
You may of course use the VertexMap rebind mechanism when creating your property map, as follows.
// Works if VertexMap is a correct model of boost::UniqueAssociativeContainer and boost::PairAssociativeContainer.
typedef typename Graph::VertexMap< vertices_size_type > MyComponentMap;
MyComponentMap componentMap;
boost::associative_property_map< MyComponentMap > propComponentMap( componentMap );
A breadth-first visit with the Boost Graph Library
We need to store distances to the start vertex, therefore we create a dedicated property map (here propDistanceMap). The algorithm also requires a queue (here Q) and a first vertex (start).
// get the property map for storing distances
typedef std::map< vertex_descriptor, unsigned int > StdDistanceMap;
StdDistanceMap distanceMap;
boost::associative_property_map< StdDistanceMap > propDistanceMap( distanceMap );
boost::queue< vertex_descriptor > Q; // std::queue does not have top().
vertex_descriptor start = *( g.begin() );
boost::breadth_first_visit // boost graph breadth first visiting algorithm.
( g, // the graph
start, // the starting vertex
Q, // the buffer for breadth first queueing
boost::make_bfs_visitor( boost::record_distances( propDistanceMap, boost::on_tree_edge() ) ), // only record distances
propColorMap // necessary for visiting vertices
The following snippet computes a vertex that is as far away as possible from start.
unsigned int maxD = 0;
vertex_descriptor furthest = start;
vp.first != vp.second; ++vp.first )
unsigned int d = boost::get( propDistanceMap, *vp.first );
if ( d > maxD )
maxD = d;
furthest = *vp.first;
() <<
"- d[ "
<< furthest <<
" ] = "
<< maxD << std::endl;
More complex algorithms
You may have a look at graph/testDigitalSurfaceBoostGraphInterface.cpp to see a few more examples of using Boost Graph algorithms (max-flow and min-cut).
Wrapping Object as a boost graph
The file ObjectBoostGraphInterface.h defines the boost graph traits for any kind of Object (see Object). With those definitios, an Object is a model of VertexListGraphConcept, AdjacencyGraphConcept,
IncidenceGraphConcept, EdgeListGraphConcept. You may use an Object as a graph in any Boost Graph Library algorithm that satisfies the mentioned concepts.
You may have a look at graph/testObjectBoostGraphInterface.cpp for examples on how to use Object as a graph. Also see DigitalSurface section, as the interfaces are similar. | {"url":"https://dgtal-team.github.io/doc-nightly/moduleBoostGraphWrapping.html","timestamp":"2024-11-11T12:35:47Z","content_type":"application/xhtml+xml","content_length":"38346","record_id":"<urn:uuid:020d1cb6-eaf0-4b24-9a7c-b3b6c0dc0407>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00605.warc.gz"} |
Early-Tocharian D folklore
https://en.wikipedia.org/wiki/Proto-Germanic_folklore からそのまま引き写した奴。実際のトカラ語話者も仏教受容前はなんかよくわからん印欧系多神教信者だろうしちょうどいいでしょ
Tocharian D Meaning
firkwəña 'mountain' Cognate with or borrowed into Slav. as *per(g)ynja ('wooded hills'). See Perkwunos for further discussion.
halya 'the concealed' Attested as an afterlife location throughout Germanic languages and personified as a female entity. See Hel (being) and Hel (location).
halyayca 'hell-knowledge' A poetic name for an underworld location.
tsmina 'heaven' See Perkwunos#Heavenly vault of stone for further discussion.
tsminonka '(heaven-)meadow' A term denoting an afterlife heavenly meadow. PGmc *wangaz occurs as a gloss for 'paradise', implying an early Germanic concept of an afterlife field in the heaven.
See Fólkvangr and Neorxnawang.
micana-karþa 'middle-enclosure' See Midgard for further discussion.
yralc 'man-age' The inhabited world, the realm of humankind.
Tocharian D Meaning
alha 'temple'
osre-miena 'Osre-month' associated with a festival held around April. See Ēostre and Ēosturmōnaþ for further discussion.
platana 'to sacrifice' Source of plosra ('sacrifice') and plota ('sacrifice, worship').
plata-husa 'house of worship, house of sacrifice' Place of worship, temple.
yältsa 'evil'
fryata 'Friday' See *Frijjō above.
kaltra 'magic song, spell, charm' See galdr for further discussion.
kwäca 'priest'
haylaka 'holy' Source of haylakayana ('to make holy, consecrate').
haylaka-miena 'holy-month' equivalent to modern 'September' or 'December'. See Hāliġmōnaþ for further discussion.
harpist-miena 'autumn-month, harvest-month' roughly equivalent to modern 'August–November'.
haräka 'holy stone', perhaps 'sacrificial mound'[162] See hörgr for further information
halya-runa 'witch, sorceress'
hwänsla 'sacrifice'
hwäts-runa 'secret of the mind, magical rune'
yähwla 'Yule' Name of a festival organized at the end of each year. Cf. also yähwla-taka ('Yule-day'). See Yule for further discussion.
yähwla-miena 'Yule-month'
yera-miena 'year-month' equivalent to modern 'January'.
kwäña 'omen'
lyetsäya 'healer, physician' Source of ĺetsäna ('cure, remedy') and ĺetsnayana ('to heal').
läpia 'herbal medicine, magic potion' Medicinal herb associated with magic.
miltäña 'lightning', 'hammer' Thor's hammer. See Mjǫllnir for further discussion.
mienanta 'Monday' See *Mēnōn above.
ñmita 'sacred grove' See sacred trees and groves in Germanic paganism and mythology
runa 'secret, mystery; secret counsel; rune' Source of runa ('counsellor'), ruña ('mystery'), rona ('trial, inquiry, experiment'). See runes for further discussion.
runa-stapa 'runic letter'
sayäta 'spell, charm, magic' See also the šitana ('to work charms') .
saywala 'soul'
skalta 'poet' See skald for further discussion.
sämla 'banquet, symposium' See symbel for further discussion.
sännanta 'Sunday' See Sowēlo ~ Sōel above.
tafna 'sacrificial meat'
tofra 'sorcery, magic'
cäpra 'sacrifice, animal offering'
ciwasta 'Tuesday' See *Tīwaz above.
þenaryästa 'Thursday' See *Þun(a)raz above.
yiha 'holy, divine' Source of yitsena～yitsana ('to consecrate'), yitsäsla ('consecration'), and yitsäþa ('holiness, sanctity').
yiha 'sanctuary' See Vé (shrine) for further discussion.
yiha 'priest' See Vili and Vé for further discussion.
yäkkana 'to practice sorcery'
yitaka 'wizard, prophet' Source of the yitakayana ('to prophesy').[202]
watañästa 'Wednesday' See *Wōdanaz above. | {"url":"https://rentry.co/uc4gr","timestamp":"2024-11-06T23:41:24Z","content_type":"text/html","content_length":"16568","record_id":"<urn:uuid:1fb8f72d-9721-4a73-8b15-fe149a3e4050>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00192.warc.gz"} |
Right Angles
A right angle is an internal angle which is equal to 90°
This is a right angle
See that special symbol like a box in the corner? That says it is a right angle. The 90° is rarely written in. If we see the box in the corner, we are being told it is a right angle.
All the angles below are right angles:
A right angle can be in any orientation or rotation as long as the internal angle is 90°
Never argue with a 90° angle, it is always right!
Types of Angles
Read more about Angles
See the angle types yourself:
8984, 3288, 3289, 3291, 8981, 3290, 808, 1771, 8988, 8982 | {"url":"http://wegotthenumbers.org/rightangle.html","timestamp":"2024-11-03T16:22:23Z","content_type":"text/html","content_length":"4359","record_id":"<urn:uuid:8e4cf59f-2261-4fd1-bba2-2be3dd8e22b0>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00763.warc.gz"} |
LIFE ITSELF: Organisms vs Mechanisms, Part 2
Crosby_M (CrosbyM@po1.cpi.bls.gov)
Sun, 23 Mar 1997 18:43:40 -0500
In Ch.3 of _Life Itself_ Robert Rosen descends to the "epistemological
basement", examining the basic dualisms of self (or observer) and other
(what he calls "ambience"), which is then partitioned into system &
environment, and asserts that, even this, is a "fateful step [where]
fundamental trouble begins to creep in."=20
3B, "The First Basic Dualism": "Science is built on dualities. Indeed,
every mode of discrimination creates one. But the most fundamental
dualism, which all others presuppose, is of course the one a
discriminator makes between self and everything else.... At any rate, =
know our self with ultimate certainty, even though this knowledge is
subjective ... What else is there? Whatever it is, I shall call it the
/ambience/.... this is the external world, the world of objective
reality, the world of phenomena.... Science, in fact, requires both; it
requires an external, objective world of phenomena, and an internal,
subjective world of the self, which perceives, organizes, acts and
3C, "The Second Basic Dualism": "Our second basic dualism concerns the
way we partition our ambiences, the way we manage our perceptions of =
external world.... It rests on a consensus /imputed/ to the ambience =
It is the dualism between systems and their environments.... The
partition of ambience into system and environment ... is a basic though
fateful step for science.... Systems and environments are thenceforth
perceived in entirely different ways ... system gets described by =
... environment is characterized rather by its effects on system.
Indeed, it is precisely at this point that, as we shall see, =
trouble begins to creep in; already here."
3D, "Language": "An essential part of the inner world of any self is
one's language.... Language itself creates, or embodies, new dualism =
The first basic dualism inherent in language is that (1) it is a thing
in itself and (2) permits, even requires, referents external to itself.
These embody respectively what we call the syntactic aspects of =
and its semantic aspects... Syntax involves its own inherent dualism =
between proposition and production rules.... The syntactical production
rules of a language are its internal vehicles for what I shall call
inferential entailment.... We shall understand by a formalism any such
'sublanguage' of a natural language, defined by syntactic qualities
alone. That is, a formalism is a finite list of production rules,
together with a generating family of propositions on which they can =
without any specification or consideration of extralinguistic
referents.... As we shall see, the extraction of a formalism from a
natural language has many of the properties of extracting a system from
the ambience.... The idea of formalization, that the semantic aspects =
language can always be effectively replaced by purely syntactic ones,
will turn out to be another place where really serious trouble creeps
Rosen concludes this section "by pointing out two aspects of natural
language that will play key roles in what follows but that never end up
as part of formalisms. These are (1) the use of the interrogative ...
and (2) the use of the imperative.... imperatives constitute recipes,
protocols, blueprints, and the like, which govern /fabrication/. But, =
will become apparent, the entailment process /embodied/ by algorithms =
recipes is very different than that governing their /application/. The
difference, indeed, is precisely the difference between fabrication and
3E, "On Entailment in Formal Systems". Rosen proposes: "suppose that we
step outside our formalism and contemplate one of its theorems P....
>From that perspective, we can interrogate ... we can ask: why is P true
in the system?" Rosen proposes three operations we can perform on such =
system: "We can change an axiom, without touching the inferential rules
... We can change an inferential rule, without changing either the
axioms or the list of which that constitutes our algorithm. Finally, we
may change the algorithm, without affecting either the axioms or the
rules themselves.... the kinds of changes we have contemplated all come
from outside the formalism ... from the standpoint of the formalism,
anything that happens outside is accordingly unentailed.... This is our
first glimpse of a peculiar thing ... namely, that though formal =
allow us to talk about entailment in a coherent way, from their
standpoint everything important that affects them is itself =
Rosen says that this discussion should remind us of Aristotle: "we have
paralleled three of his four categories of causation; specifically, if
we call theorem P an /effect/, we may identify his idea of material
cause of P with the axioms of a formalism, his idea of efficient cause
of P with its production rules, and his idea of formal cause of P with
the specification of a particular sequence or algorithm of production
rules.... The reader may not be surprised to note that we do not see a
formal analog of Aristotle's fourth causal category, which he held to =
the most significant; namely, the category of final cause.... In any
formalism, there is a kind of natural flow from axioms to theorems, =
much like the familiar unidirectional flow of time.... The three
'traditional' causal categories (formal, material, and efficient
causation) always respect this flow of 'formal time' ... Final =
gives the /appearance/, at any rate, of violating this flow."
3F. "On the Comparison of Formalisms." Rosen argues that "mathematics,
in the broadest sense, is the study of formalisms and that formalisms,
in their turn, are parts of natural language." Rosen asks: "When does
one formalism subsume another, so that the second can be in some sense
generated from the first, or embedded in it? And above all, is the
machinery for dealing with such questions, i.e., with the comparison of
formalisms, itself a formalism?"
Rosen goes on to discuss coordinate systems, transformations and
modeling relations. Rosen notes: "In order to compare [two formalisms],
we need to ... express what each formalism says to itself in the
language of the other." We need a pair of dictionaries: an encoding
dictionary and a decoding dictionary. If these can establish a modeling
relation between the inferential structures of the two formalisms such
that one is a model of the other and one is a realization of the other.
Rosen's primary point here is that the encodings and decodings are
unentailed within the formalisms themselves: "The comparison of two
inferential structures ... thus inherently involves something outside
the formalisms, in effect, a /creative act/, resulting in a new kind of
formal object, namely the modeling relation itself. It involves /art/".
3G. "Entailment in the Ambience: Causality". "The fundamental question
for us, at this point, is the following: is there, in this external
world, any kind of /entailment/, analogous to the inferential =
we have seen between propositions in a language or formalism? =
if there is not, we can all go home; science is not only impossible but
also inconceivable."
3H. "The Modeling Relation and Natural Law". Using models, Rosen says,
"We can compare inferential entailment in a formal system with causal
entailments, relating a bundle of phenomena that we extract from our
ambience and identify as a natural system.... the causal entailments
manifested by a natural system provide the orderliness required of the
ambience. Inferential entailment in a formal system is a way of
providing the orderliness required of the self. The act of bringing the
two into correspondence ... is the articulation of the former within =
latter; it is in effect science itself.... It is not generally
appreciated, especially by experimentalists ... that any measurement,
however comprehensive, is an act of abstraction ... From this
standpoint, it is ironic indeed that a mere observer regards oneself as
being in direct contact with reality and that it is 'theoretical
science' alone that deals with abstractions."
3I. "Metaphor". Rosen continues: "This modeling relation between two
natural systems N1, N2 is of the most profound importance; I shall call
it analogy.... This is another way of seeing, what I alluded to =
that reduction to a common set of material constituents is not the only
way, nor even a very good way, of comparing natural systems.... As we
have seen, the modeling relation is intimately tied up with the notion
of prediction.... insofar as the entailment structure itself is =
in a congruent model, we can actually, in a sense, pull the future of
our natural system into the present.... A large part of the cost =
by Natural Law, in return for the benefit of prediction, lies in =
the right encodings. But to what extent do we really need these
encodings? Perhaps we can presume a little on Natural Law and get away
without them.... This is the essence of /metaphor/: decoding without
encoding ... Perhaps the most important for our purposes in the machine
metaphor of Descartes ... It asserts that things about machines can be
decoded into predictions about organisms ... Another one of enormous
current importance ... is what may be called the open system
metaphor.... [However] to proceed metaphorically in the above sense [we
must remember that] by giving up encoding, we also give up
/verifiability/ in any precise sense.... Hence the general =
if not active hostility, manifested by empiricists to theory couched in
metaphorical terms."
(End 03/24/97 detailed extract from _Life Itself_ ch.3, originally read
between Jan 16 and 20, 1997.)=20
(to be continued) | {"url":"http://extropians.weidai.com/extropians.1Q97/3892.html","timestamp":"2024-11-09T07:27:30Z","content_type":"application/xml","content_length":"12612","record_id":"<urn:uuid:0470ed79-a6f3-4098-bb27-406b59bfd9c1>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00192.warc.gz"} |
Can a postulate be used to prove a theorem?
Can a postulate be used to prove a theorem?
A postulate is a statement that is assumed to be true without a proof. It is considered to be a statement that is “obviously true”. Postulates may be used to prove theorems true. A theorem is a
statement that can be proven to be true based upon postulates and previously proven theorems.
What is the difference between a theorem and an axiomatic postulate?
Axioms or postulates are universal truths. They cannot be proved. Theorem are statements which can be proved.
Is postulate the same as theory?
As nouns the difference between postulate and theory is that postulate is something assumed without proof as being self-evident or generally accepted, especially when used as a basis for an argument
while theory is (obsolete) mental conception; reflection, consideration.
Why are postulates or theorems important in geometry?
Postulates and theorems are the building blocks for proof and deduction in any mathematical system, such as geometry, algebra, or trigonometry. By using postulates to prove theorems, which can then
prove further theorems, mathematicians have built entire systems of mathematics.
In mathematics, a corollary is a theorem connected by a short proof to an existing theorem. In many cases, a corollary corresponds to a special case of a larger theorem, which makes the theorem
easier to use and apply, even though its importance is generally considered to be secondary to that of the theorem.
What is the difference between corollary and theorem?
a theorem is a more important statement than a proposition which says something definitive on the subject, and often takes more effort to prove than a proposition or lemma. A corollary is a quick
consequence of a proposition or theorem that was proven recently.
Why are postulates not proven in geometry?
A postulate (also sometimes called an axiom) is a statement that is agreed by everyone to be correct. Postulates themselves cannot be proven, but since they are usually self-evident, their acceptance
is not a problem. Here is a good example of a postulate (given by Euclid in his studies about geometry).
How do postulates work?
A postulate is an assumption, that is, a proposition or statement that is assumed to be true without any proof. Postulates are the fundamental propositions used to prove other statements known as
theorems. Once a theorem has been proven it is may be used in the proof of other theorems.
Are postulates accepted without proof?
Postulates are accepted as true without proof. A logical argument in which each statement you make is supported by a statement that is accepted as true. In a conditional statement, the statement that
immediately follows the word if.
What are other differences of postulate and theorem?
Postulates and theorems are two common terms that are often used in mathematics. A postulate is a statement that is assumed to be true, without proof. A theorem is a statement that can be proven
true. This is the key difference between postulate and theorem. Theorems are often based on postulates.
What is a real world example of postulate?
An example of postulate is the fact that the world is not flat to support the argument of strong scientific development over the centuries. Postulate is defined as to claim, demand or assert
something as truth. An example of postulate is to require equality. An example of postulate is to defend the existence of God.
Does a postulate need to be proved?
Postulate is a true statement, which does not require to be proved. More About Postulate Postulate is used to derive the other logical statements to solve a problem. Postulates are also called as
What is the different between postulate and theory?
Postulate is a see also of theory. As nouns the difference between postulate and theory is that postulate is something assumed without proof as being self-evident or generally accepted, especially
when used as a basis for an argument while theory is (obsolete) mental conception; reflection, consideration. As a verb postulate is to assume as a truthful or accurate premise or axiom, especially | {"url":"https://teacherscollegesj.org/can-a-postulate-be-used-to-prove-a-theorem/","timestamp":"2024-11-11T04:37:21Z","content_type":"text/html","content_length":"143727","record_id":"<urn:uuid:c9fe1ac1-71e4-4a37-b79f-6dea47a2baf3>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00429.warc.gz"} |
Grading of Student Submissions
In general, it is recommended that the teacher assesses a fair proportion of the work submitted by the students. The assessments are shown to the students and will provide important feedback on their
The assessments from the teacher are used in two ways in the workshop module. Firstly, they are used in the calculations to determine the "grading grades", the grades given the student assessments.
Secondly they are used in the calculation of the submission grades. These assessments can be given extra weight (the "Weight of Teacher Assessments" option), this weighting effects both the grading
grade and the submission grade calculations. If it felt that the student assessments are too high (or too low), increasing this weighting factor should be considered as that will help stabilise the
grades to a degree. | {"url":"https://aesines.edu.gov.pt/moodle/help.php?module=workshop&file=gradingsubmissions.html","timestamp":"2024-11-05T05:35:32Z","content_type":"application/xhtml+xml","content_length":"6232","record_id":"<urn:uuid:f8e4c336-27d1-4fc0-8106-54fdaa4972d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00431.warc.gz"} |
What is: Permutation Test
What is a Permutation Test?
A permutation test is a non-parametric statistical method used to determine the significance of an observed effect by comparing it to a distribution of effects generated by rearranging the data. This
approach is particularly useful when the assumptions of traditional parametric tests, such as t-tests or ANOVA, are not met. By utilizing the actual data to create a reference distribution,
permutation tests provide a robust alternative that is less sensitive to outliers and non-normality, making them a valuable tool in the fields of statistics, data analysis, and data science.
How Does a Permutation Test Work?
The fundamental principle behind a permutation test involves calculating a test statistic for the observed data and then comparing it to a distribution of test statistics generated by permuting the
data. The steps typically include defining a null hypothesis, calculating the observed test statistic, and then repeatedly shuffling the data to create a distribution of test statistics under the
null hypothesis. By comparing the observed statistic to this distribution, researchers can assess the likelihood of observing such an effect by random chance, ultimately leading to a p-value that
indicates statistical significance.
Applications of Permutation Tests
Permutation tests are widely applicable across various fields, including psychology, biology, and economics, where researchers often deal with small sample sizes or non-standard data distributions.
For instance, in clinical trials, permutation tests can be employed to evaluate the effectiveness of a new treatment compared to a control group without relying on the normality assumption.
Additionally, permutation tests are useful in machine learning for feature selection, where they can help determine the importance of specific variables by assessing the impact of randomizing those
variables on model performance.
Advantages of Using Permutation Tests
One of the primary advantages of permutation tests is their flexibility. Unlike traditional parametric tests, which require specific assumptions about the data distribution, permutation tests can be
applied to a wide range of data types and structures. This makes them particularly advantageous in real-world scenarios where data may not conform to theoretical distributions. Furthermore,
permutation tests provide exact p-values, which can enhance the interpretability of results, especially in small sample studies where asymptotic approximations may not be reliable.
Limitations of Permutation Tests
Despite their advantages, permutation tests do have some limitations. The computational intensity required to perform permutation tests can be a drawback, especially with large datasets or complex
models. Generating a sufficient number of permutations to achieve reliable results may lead to long processing times. Additionally, while permutation tests are robust against violations of
assumptions, they may still be sensitive to the design of the study, such as the presence of confounding variables or the choice of test statistic, which can influence the validity of the results.
Choosing the Right Test Statistic
Selecting an appropriate test statistic is crucial when conducting a permutation test. Common choices include the difference in means, the median, or other measures of central tendency, depending on
the nature of the data and the research question. The choice of test statistic should align with the hypothesis being tested and the characteristics of the data. For example, if the data contains
outliers, using the median as a test statistic may provide a more robust measure compared to the mean, which can be heavily influenced by extreme values.
Permutation Tests in R and Python
Both R and Python offer packages and libraries that facilitate the implementation of permutation tests. In R, the `coin` package provides functions for conducting permutation tests, while the `perm`
package offers a more general framework for permutation-based analyses. In Python, the `scipy` library includes tools for performing permutation tests, and the `pingouin` library provides a
user-friendly interface for statistical analysis, including permutation tests. Utilizing these tools can streamline the process of conducting permutation tests and enhance reproducibility in
Interpreting Results from Permutation Tests
Interpreting the results of a permutation test involves examining the p-value derived from the comparison of the observed test statistic to the permutation distribution. A low p-value (typically
below a threshold of 0.05) indicates that the observed effect is unlikely to have occurred by random chance, leading to the rejection of the null hypothesis. However, researchers should also consider
the effect size and confidence intervals to provide context for the results. Effect size measures can help quantify the magnitude of the observed effect, while confidence intervals can offer insights
into the precision of the estimate.
Permutation tests represent a powerful and flexible statistical tool that can be applied across various disciplines. By leveraging the actual data to create a reference distribution, these tests
provide a robust alternative to traditional parametric methods, particularly in situations where assumptions may be violated. Understanding the mechanics, applications, and limitations of permutation
tests is essential for researchers aiming to draw valid conclusions from their data analyses. | {"url":"https://statisticseasily.com/glossario/what-is-permutation-test/","timestamp":"2024-11-05T06:48:51Z","content_type":"text/html","content_length":"139165","record_id":"<urn:uuid:2ff55ddc-9fac-4df9-8373-f9b17fb7836c>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00168.warc.gz"} |
Buscador de Personas - SIGEVA - Universidad Nacional de Córdoba
Congresos y reuniones científicas
A new look at an Old Formula: Reilly´s Law
Otro; Southern Economic Developer (semiannual publication); 2002
Institución organizadora:
Georgia Southern University
The study looks at how a simple adjustment to the traditional formula of Reilly´s Law of Retail Gravitation can enhance the usefulness of this rule of thumb for determining the market boundary
between cities in an urban hierarchy. The paper proposes to adjust the relative population variable by weighting each city´s population by its respective per capita income. The second section of the
paper describes Reilly´s Law and discusses the theoretical relationship it attempts to proxy. The reasoning behind the proposed modification to the formula is explained in the second section. To
illustrate the impact of the proposed change to the formula, the paper analyzes the growth of the Atlanta market area relative to southern cities of comparable size in 1940. Section three contains
the calculations for the market area of Atlanta relative to six other cities in the southeast using both the standard formulation of Reilly´s Law and the modified one. Section four discuses the
findings. The example shows that the breakpoint between cities is definitely influenced by the inclusion of a variable measuring the relative purchasing power of the population. | {"url":"https://sigeva-scp.unc.edu.ar/detalle.php?keywords=&id=10750&congresos=yes&detalles=yes&congr_id=120401","timestamp":"2024-11-13T05:03:58Z","content_type":"application/xhtml+xml","content_length":"6710","record_id":"<urn:uuid:203e6907-0f41-4fba-84de-81a04eeed56c>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00197.warc.gz"} |
Electromagnetic EM Spectrum by Raphael Lucas
Students watch a short time lapse video of a sunrise and complete a T-chart (I Notice/I Wonder), and the question, “How does light from the Sun and other stars travel through space to reach the
Students watch a second video entitled, “What are Electromagnetic (EM) wave properties?” They explain properties of EM waves and learn about the mathematical equation to describe the relationship
between wave properties and energy.
Students then construct a spectroscope, which they use to analyze light. They also use the EM wave equation (v = f X lambda). They will observe UV beads that glow in the dark and then make bracelets
with those beads. | {"url":"https://outreach.gi.alaska.edu/nasa-heliophysics/heliophysics/electromagnetic-em-spectrum-raphael-lucas","timestamp":"2024-11-08T11:02:06Z","content_type":"text/html","content_length":"24697","record_id":"<urn:uuid:40c93c85-856c-4631-8f89-e72e443b9381>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00257.warc.gz"} |
Geometric Topology Books Archives - Youngscientist Awards
Assist Prof Dr. University of Pisa, Italy
👩🔬 A dynamic mathematician specializing in (higher) Teichmüller theory, hyperbolic and anti-de Sitter geometry, and Higgs bundles. Holds a Ph.D. from the University of Luxembourg and currently
serves as a tenure-track Assistant Professor at the University of Pisa. Previously, held positions at Rice University as Lovett Instructor and Adjunct Assistant Professor. Recognized with prestigious
grants and awards, including NSF-DMS and Italian National Habilitation. Actively contributes to the field through workshops, publications, and teaching experiences spanning diverse mathematical
subjects. Passionate about exploring geometric and topological structures in mathematics and their applications.
Education 📚
University of Luxembourg
• Ph.D. in Mathematics, June 2018.
□ Thesis: Anti-de Sitter geometry: convex domains, foliations and volume.
□ Advisor: Jean-Marc Schlenker.
University of Pisa
• M.S. in Mathematics, April 2015.
• B.S. in Mathematics, May 2013.
Employment 💼
University of Pisa
• Tenure-track Assistant Professor, March 2022 – present.
Rice University
• Lovett Instructor, July 2018 – December 2022.
• Adjunct Assistant Professor, January 2022 – June 2025.
Research Interests 🧠
• (Higher) Teichmüller theory
• Hyperbolic and anti-de Sitter geometry
• Bounded cohomology and simplicial volume
• Higgs bundles
Grants and Awards 🏅
Member of PRIN: Geometry and Topology of manifolds (PI B. Martelli), 187,500€NSF-DMS 2005501 Award: Geometric limits in higher Teichmüller Theory, PI, $136,500Italian National Habilitation to
Associate Professor (June 2022 – June 2033)Mario Baldassarri Prize (UMI) for a paper published under the age of 30Franco Tricerri Prize (UMI) for the best Ph.D. thesis in Differential GeometrySimons
travel grant (AMS): $4,000Graduate Internship grant (GEAR)
Workshops Attended 🛠️
Geometric Group Theory and Low-Dimensional Topology, Trieste, 2016.Geometry, Topology and Dynamics of Moduli Spaces, Singapore, 2016.Days in representation theory and harmonic analysis, Luxembourg
2016.Third retreat of the GEAR network, Stanford, 2017.Geometry and physics of Higgs bundles II, Chicago, 2017.Current trends on spectral data for Higgs bundles III, Chicago 2017.Texas Geometry and
Topology Conference, Houston, 2018.Holomorphic differentials in Mathematics and Physics, MSRI, 2019.Geometric, Algebraic, and Physical structures of MQD, Stony Brook, 2024.
Teaching Experience 📝
University of Pisa
Spring 2025: Calculus 1.Fall 2024: Linear algebra, Calculus 1.Spring 2024: Calculus 1.Fall 2023: Hyperbolic geometry, Calculus 1.Spring 2023: Calculus 1.Fall 2022: Linear Algebra, Calculus 1.Rice
Spring 2022: Multivariable Calculus (Math 212).Fall 2021: Multivariable Calculus (Math 212).Summer 2021: Summer Undergraduate Research (Math 479).Spring 2021: Multivariable Calculus (Math 212), Lie
Theory (Math 371).Fall 2020: Calculus 1 (Math 101).Spring 2020: Calculus 1 (Math 101), Multivariable Calculus (Math 212).Spring 2019: Geometry (Math 366), Elements of Analysis (Math 302).Fall 2018:
ODE and Linear Algebra (Math 211).University of Luxembourg
Fall 2017: Teaching assistant for Calculus 1a, 1b.Fall 2016: Teaching assistant for Calculus 1a, 1b.Spring 2016: Teaching assistant for Calculus 2a, 2b.Fall 2015: Teaching assistant for Calculus 1a,
1b, 1c.
Publications Top Notes 📝
Title: Polynomial Quadratic Differentials on the Complex Plane and Light-like Polygons in the Einstein Universe
Author: A. Tamburelli
Year: 2019
Title: On the Volume of Anti-de Sitter Maximal Globally Hyperbolic Three-Manifolds
Authors: F. Bonsante, A. Seppi, A. Tamburelli
Year: 2017
Title: Limits of Blaschke Metrics
Authors: C. Ouyang, A. Tamburelli
Year: 2019
Title: Planar Minimal Surfaces with Polynomial Growth in the Sp (4, R)-Symmetric Space
Authors: A. Tamburelli, M. Wolf
Year: 2020
Title: Prescribing Metrics on the Boundary of Anti-de Sitter 3-Manifolds
Author: A. Tamburelli
Year: 2018
Title: Constant Mean Curvature Foliation of Domains of Dependence in 𝐴𝑑𝑆₃
Author: A. Tamburelli
Year: 2019
Title: Regular Globally Hyperbolic Maximal Anti‐de Sitter Structures
Author: A. Tamburelli
Year: 2020
Title: Length Spectrum Compactification of the SO0 (2, 3)-Hitchin Component
Authors: C. Ouyang, A. Tamburelli
Year: 2023
Title: Constant Mean Curvature Foliation of Globally Hyperbolic (2+1)-Spacetimes with Particles
Authors: Q. Chen, A. Tamburelli
Year: 2017 | {"url":"https://youngscientistawards.com/tag/geometric-topology-books/","timestamp":"2024-11-04T21:58:51Z","content_type":"text/html","content_length":"298121","record_id":"<urn:uuid:de3aa3e6-8efb-4a2d-8dc0-345c7d559efa>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00388.warc.gz"} |
JEE MAIN & ADVANCE 12th PCM Physics Demo Video Lectures -Alternating Current-1-RobomateplusJEE MAIN & ADVANCE 12th PCM Physics -Alternating Current Module-1 Demo Videos
JEE MAIN & ADVANCE 12th PCM Physics -Alternating Current Module-1 Demo Videos
Hello, students, welcome back to the chapter of alternating current. In the first module we have already discussed, what do we call as a direct current. A current that is constant or variable but
does not change its direction in the interval of time is called as a direct current. Whereas an alternating current is one which can be positive for some amount of time and can be negative for some
amount of time, is called as an alternating current. Further we have discussed, how can we construct something which is called as an AC generator. In this module we will discuss, how an AC generator
Let’s first revise how we have constructed an AC generator. It consists of two magnetic poles, one north the other south, in between which is placed a rotatable coil which is connected to slip rings
and which are connected to an external circuit through two carbon brushes.
Let’s discuss how this combination works to provide us an alternating current. If we see in the figure as the coil rotates the magnetic flux through the coil, that is the number of field lines
crossing the area in a unit time or at any instant changes and thus producing an EMF which can be read according to the relation Epsilon is equal to minus d pi by dt, where pi is the magnetic flux
through the coil. Now, if we recall, the magnetic flux through a coil can be changed in three ways. First, by changing the magnetic field, second by changing the area of the coil that means either
reducing the area or extending the area. But in this particular case the AC generator works on the principle that the magnetic flux can also be changed by changing the angle between the magnetic
field and the area vector. Let’s discuss this phenomena in detail. Let’s say that t equal to zero the area vector is anti parallel to the magnetic field. Therefore the flux in this particular case is
negative. Now, if the time taken to rotate one complete circle that is 360 degrees is T, then what would be the time to rotate by an angle of 90 degree? Yes, it’s T by 4. So, at T by 4, the coil
rotates by an angle of 90 degree, therefore we can see that the magnetic field and the area vector are perpendicular to each other. Therefore the magnetic flux through the coil is zero. Now a further
rotation of 90 degrees the time elapsed would have been T by 2. But as the coil rotates in the following manner then the magnetic flux and the area vector become parallel. And therefore the flux
through the coil becomes positive. Now, if you note from T equal to zero, at T equal to zero the flux was negative. At T equal to T by 4 it becomes zero and at T by 2 it becomes positive. Therefore
the flux from T equal to zero to T equal to T by 2 increases. Let’s discuss what happens after the time instant T by 2. On further rotation of 90 degree angle, the coil is placed like as shown in the
figure. Again we can see that the area vector is perpendicular to the magnetic field and therefore the flux is zero again. On further rotation of 90 degree angle the time elapsed is T and the area
vector is again anti parallel to the magnetic field, thus if we note from T equal to T by 2, the flux was positive. At T equal to 3T by 4 it is zero and T equal to T it becomes negative. So, we can
say from T equal to T by 2 to T equal to T it is decreasing. And thus the coil rotates and the flux increases for half of the revolution and decreases for the other half. This change in magnetic flux
thus produces an EMF, changing the direction of EMF itself which produces an alternating current. Now, this alternating current is produced through the area vector changing its angle with the
magnetic field.
In the next module we will discuss the working and construction of what is called as a DC generator, thank you.
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