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http://math.stackexchange.com/questions/311909/transformation-of-confidence-intervals | # Transformation of confidence intervals
I'm using Matlab to perform a linear regression. In order to prevent the prediction of negative values I used a box-cox-transformation of the dependent variable ($=y_t$) with $\lambda = 0.5$.
$y^{(\lambda)} = \frac{y_t^{\lambda} - 1}{\lambda}$
After that I perform the linear regression with $y^{(\lambda)}$ as dependent variable. To get the result in my original form I transform $y^{(\lambda)}$ back into $y_t$.
$y_t = (\lambda(\frac{1}{\lambda} + y^{(\lambda)}))^{\frac{1}{\lambda}}$
My question now is, can I transform the confidence intervals in the same way as I transform my dependent variable and how do I prove it or disprove it?
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The confidence interval for which estimator? – Learner Feb 23 '13 at 11:52
I mean the confidence interval for the dependent variable. Sorry for not clarifying that. – Portbane Feb 23 '13 at 12:05
Let's say the transformed regression equation is $E \left[ y^{\left( \lambda \right)} |x \right] = x \beta$ and let's call $\hat{y}^{\left( \lambda \right)} = x \hat{\beta}$ where $\hat{\beta}$ is the estimated parameters vector. Let's call $H$ the covariance matrix of $\hat{\beta}$. Finally let's create this new function $g \left( \hat{y}^{\left( \lambda \right)} \right)$ such that $$g \left( \hat{y}^{\left( \lambda \right)} \right) = \left( \lambda \left( \frac{1}{\lambda} + \hat{y}^{\left( \lambda \right)} \right) \right)^{\frac{1}{\lambda}}$$ Now, in order to construct the confidence interval, you need to approximate the variance of your new estimator $g \left( \hat{y}^{\left( \lambda \right)} \right)$. It is possible to do so using the delta method $$\frac{\partial g}{\partial \hat{\beta}^T} H \frac{\partial g}{\partial \hat{\beta}}$$ where $\frac{\partial g}{\partial \hat{\beta}}$ is a vector of derivatives of $g$ with each of $\hat{\beta}_1, \ldots, \hat{\beta}_k$ the different parameters you estimated and $\hat{\beta}^T$ is obviously the transpose of $\hat{\beta}$.
Thanks for your answer. But what you are saying is that I can't just simply use the confidence interval calculated by Matlab and transform each boundary with $y_t = (\lambda(\frac{1}{\lambda} + y^{(\lambda)}))^{\frac{1}{\lambda}}$? – Portbane Feb 23 '13 at 15:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.941641628742218, "perplexity": 168.73366373754206}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042987552.57/warc/CC-MAIN-20150728002307-00133-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://cob.silverchair.com/jeb/article/215/6/914/11201/Methodological-advances-in-predicting-flow-induced | The modeling of fluid–structure interactions, such as flow-induced vibrations, is a well-developed field of mechanical engineering. Many methods exist, and it seems natural to apply them to model the behavior of plants, and potentially other cantilever-like biological structures, under flow. Overcoming this disciplinary divide, and the application of such models to biological systems, will significantly advance our understanding of ecological patterns and processes and improve our predictive capabilities. Nonetheless, several methodological issues must first be addressed, which I describe here using two practical examples that have strong similarities: one from agricultural sciences and the other from nuclear engineering. Very similar issues arise in both: individual and collective behavior, small and large space and time scales, porous modeling, standard and extreme events, trade-off between the surface of exchange and individual or collective risk of damage, variability, hostile environments and, in some aspects, evolution. The conclusion is that, although similar issues do exist, which need to be exploited in some detail, there is a significant gap that requires new developments. It is obvious that living plants grow in and adapt to their environment, which certainly makes plant biomechanics fundamentally distinct from classical mechanical engineering. Moreover, the selection processes in biology and in human engineering are truly different, making the issue of safety different as well. A thorough understanding of these similarities and differences is needed to work efficiently in the application of a mechanistic approach to ecology.
### The plant canopy interacting with wind
Organisms, aerial or aquatic, natural or cultivated, often grow in the form of canopies. In such organizations, the height and density of organisms vary at a much larger scale than does the size of the organism itself. This review focuses on the case of a terrestrial canopy interacting with wind. Very distinct cases exist, ranging from the short wheat canopy to the high rain-forest canopy. Let us first consider a crop canopy, as discussed by Py and colleagues (Py et al., 2006). In terms of geometrical and mechanical characteristics, it can be stated that the typical density is approximately 100 specimens per square meter, the vertical height is 1 m and the diameter of individual stems or size of leaves is of the order of a few millimeters. The natural frequency of oscillations of plants is of the order of 1 Hz, and the flow velocity is of the order of 1 m s–1. The density of the surrounding fluid is three orders of magnitude smaller than that of the plant tissue. Wind in the presence of canopy differs significantly from wind over a flat surface (Fig. 1). The wind profile shows a double boundary-layer profile – one in the canopy, and one above the canopy – that causes the wind fluctuations to be strongly organized in the form of coherent eddies (Finnigan, 2000). Flexible plants forming the canopy are known to move in wind. This motion has many consequences (de Langre, 2008): when of large amplitude, it can induce breaking or permanent bending, but even low-amplitude motion has a strong influence on growth (Moulia and Combes, 2004), gaseous exchanges inside the canopy are modified by the motion (Farquhar and Eggleton, 2000) and, finally, seed or pollen dispersal, or photosynthesis, are affected not only by wind but also by wind-induced motion (Nathan et al., 2002). Quantifying this motion, and modeling the influence of the biomechanical characteristics of the plants is essential for the understanding of these issues. Similar phenomena are found in the case of canopies made of trees, but at different scales of time and length. It can be stated, therefore, that the problem of the mechanical interaction of a plant canopy (PC) with wind has some importance in the field of plant biomechanics and, more generally, for plant ecology.
### The steam generator bundle interacting with water
The other system considered hereafter is well known in nuclear engineering. The steam generator (SG) is a key component of most modern reactors; it allows the transfer of thermal energy initially produced by the nuclear reaction from the primary circulating loop to the secondary one. Owing to the considerable flux of energy to transfer, SGs are among the largest heat exchangers in the world. They typically convey 300 MW between the two circuits they interface. In the most widespread design (pressurized water reactors), this is made possible by a bundle of ∼5000 tubes, 20 m long. These tubes, with a diameter of ∼2 cm, are very densely packed, with a distance between them of approximately half a diameter. They allow the transfer of heat from the internal fluid, coming from the nuclear core, to the external fluid, which ultimately is boiled to form vapor that will activate the turbine, generating electricity. These long U-shaped tubes are held by supports along their span and have typical natural frequencies of 10 Hz. The external water and steam flow runs at up to 5 m s–1. In the region of interest here, the U-bend part (Fig. 1), the tubes are semi-circular and the flow is mainly radial and therefore perpendicular to the tubes (Fig. 2). As for the crop canopy, flow across such a flexible system results in a vibratory motion. Here, the possible consequence of the motion is damage by fatigue or wear at the support. Wear, and subsequent leakage of the internal flow to the external circuit, have in fact significantly affected ∼40% of the reactors in existence (Diercks et al., 1999). Approximately 10,000 tubes have been plugged around the world for this reason, and many SGs have been replaced, at a cost of ∼\$100 million each (Païdoussis, 2006). More generally, the complete fracture of a damaged tube is a safety issue. The interaction of flow with these tubes is also of interest for the understanding of particle transport and deposition in the heat exchanger (Srikantiah and Chappidi, 2000).
Fig. 1.
Air and water flows in disparate study systems. (A) The plant canopy (PC) experiencing wind, with the corresponding wind profile and structured eddies (blue) and boundary layer profile (broken red line). (B) The upper part of the bundle of U-tubes in a steam generator (SG). White arrows indicate water flow.
Fig. 1.
Air and water flows in disparate study systems. (A) The plant canopy (PC) experiencing wind, with the corresponding wind profile and structured eddies (blue) and boundary layer profile (broken red line). (B) The upper part of the bundle of U-tubes in a steam generator (SG). White arrows indicate water flow.
### Fluid–structure interactions: elementary concepts
Some general concepts of the modeling of flow-induced vibrations are necessary in order to understand the approaches used to analyze the two cases presented above. First, the frequencies involved in the flow and in the moving structure have to be compared. In plants and SG tubes, the typical frequencies of motion are above the 1 Hz value: hence any variation of flow velocity at lower frequencies only has a quasi-static effect. This defines the range of interaction with the mean flow, where it is sufficient to consider the static drag force acting on the structure. This static drag affects the static deformation of the structure, such as the mean bending of the crop canopy under wind. Under very high mean flow, static damage can occur, such as tree branch breakage in storms or SG tube bending in extreme accidental cases. This occurs in a similar way for corals (Madin and Connolly, 2006). The effect of flow fluctuations of higher frequencies is more complex; see for instance the book by Païdoussis and colleagues for a general presentation (Païdoussis et al., 2011). The flow exerts oscillating forces that can come from the fluctuating part of the flow velocity, by turbulence or large eddies, or that result from a coupling between the flow and the flexibility of the solid structure. The forces cause vibratory motion, predominantly at the natural frequency of the structure, and corresponding oscillating stress. This results in damage by fatigue or wear. The relationship between the oscillating stress and the corresponding damage differs with the material, biological or metallic. Moreover, transient dynamic effects can cause overshoots of stresses, causing fractures or local buckling; this is a typical mechanism for lodging, windbreak or windthrow of trees.
### A comparative point of view
It is often the case, both in biological and engineering sciences, that a comparative analysis of systems points out similarities and differences that allow the derivation of some general conclusions. The point of view taken here is not to compare the two systems, one from nuclear engineering and one from plant biomechanics, but to compare how they have been modeled. In fact, it happens that the author of this paper has worked on both over the years. In some sense, it is hoped that the understanding of the similarities and differences in the methods used will be useful for the development of mechanistic approaches to ecology, which is the focus of this present special issue. More specifically, the scope of the paper is the adequate use of mechanical engineering to build efficient predictive methods in ecology, with particular attention to the issue of individual-level and group-level models.
Fig. 2.
Modeling the flow using an equivalent continuum medium. (A) Instantaneous flow over and through an alfalfa canopy [reproduced, with permission, from Dupont et al. (Dupont et al., 2010)]. (B) Mean flow through the bundle of a SG (modified from Pettigrew and Taylor, 2003a).
Fig. 2.
Modeling the flow using an equivalent continuum medium. (A) Instantaneous flow over and through an alfalfa canopy [reproduced, with permission, from Dupont et al. (Dupont et al., 2010)]. (B) Mean flow through the bundle of a SG (modified from Pettigrew and Taylor, 2003a).
### Defining an equivalent continuum
In both systems, the first question that appears in the modeling of the mechanical behavior is whether to consider individuals (plants, tubes) or some equivalent a continuum. Owing to the large number of individuals, and their similarities, some kind of homogenization approach seems possible. In the SG, the classical approach is to compute the flow through a fixed porous medium (Belliard and Grandotto, 2002). The main characteristic of the porous medium is the volume porosity, which is the fraction of the volume occupied by the fluid, typically 0.5. Also needed is a measure of the quantity of fluid–solid interface per unit volume. In practice, this is done through the use of a hydraulic resistance matrix, which gives the friction exerted by the flow on the porous medium as a function of the fluid velocity. Porosity plays an important role in the mass balance equation and in the hydraulic resistance matrix used in the momentum balance equation. Similarly, in PCs, a volume porosity is defined, typically 0.99. To quantify the surface of interaction per unit volume, the leaf area index (LAI) is often used, where the surface of interaction is referred to the projected area of the plant on the ground. A typical value for a tree is 10. This also allows the definition of the flow resistance in the canopy and thereby computation of the flow (see Fig. 2 for two examples). Both approaches, in the SG and the PC, are similar in principle, although the ranges of parameters are quite different: the case of the SG is dominated by porosity effects, whereas that of the PC is dominated by LAI effects (Doare et al., 2004; Dupont et al., 2010). For both systems, this definition of an equivalent continuum allows the computation of the flow with a simple approach. Nevertheless, debates exist on the reliability of this approach to compute flow through tubes or trees close to the edges, a general caveat found in all homogenization procedures; in fact, when quantities vary on a length-scale of the order of the distance between individuals, the averaging procedure used to define the equivalent continuum fails. For instance, the fluid forces on the first tree of a forest edge or on the external tube of a steam generator might not be accurately predicted with this method, and local effects are typically ignored in this approach. Edge effects are always a specific problem in mechanical engineering. For SGs, outer tubes are a priori less prone to flow-induced vibration as the flow is less confined, but local jet effects have been observed. A corresponding issue in ecology would the differential behavior of edge plants in a crop canopy that are not mechanically constrained on one side.
Fig. 3.
Modeling the dynamic behavior of individuals. (A,B) The first modal shapes of wheat stalks (modified from Farquhar et al., 2000). (C) The modal shape of a U-tube with supports (modified from Adobes and Gaudin, 2004).
Fig. 3.
Modeling the dynamic behavior of individuals. (A,B) The first modal shapes of wheat stalks (modified from Farquhar et al., 2000). (C) The modal shape of a U-tube with supports (modified from Adobes and Gaudin, 2004).
### Variability
A second question that arises is that of variability. In fact, the next step in trying to model the response to flow is to apply the flow-induced forcing to a given individual vibrating system (Fig. 3). In the SG case, one would expect that all tubes are very similar, as they are man-made. Unfortunately, this is not the case in terms of their dynamic characteristics, which depend strongly on the support conditions: these conditions vary from tube to tube over the years owing to the build-up of deposits and local contact effects. Similarly, plants differ mechanically from one another even in well-controlled growing conditions. Vibration tests on a series of alfalfa plants showed variability in terms of frequency and damping (Doare et al., 2000). At this stage, the two questions to be answered are: first, can an average individual be defined that will allow the assessment of the average behavior and, second, can an extreme case be defined that will allow one to assess the extreme behavior? In SG tubes, no average individual can be defined as the nominal support condition is never observed, and the number of combinations of possible support conditions is huge. Only a statistical approach, using Monte-Carlo simulations for instance, can be used to derive the probability density function of the response (Payen and de Langre, 1996; Delaune et al., 2000). In a PC, a comparative analysis of alfalfa and wheat showed that, although the variability was much larger in alfalfa than in wheat, using an average plant was legitimate and efficient in both cases (Py et al., 2006). Then, the problem of what the most vulnerable plant, or tube, might be in the PC, or SG, differs somewhat between the two cases. In the SG, leakage of a single tube is itself a problem in the short term, so that understanding what combination of parameters might lead to damage is essential. In a PC, the lodging of a single plant is a different issue. It might have an insignificant instantaneous effect, for instance on forest biomass production, but a considerable long-term ecological effect – for instance, in creating a gap where a faster-growing species can grow. The Monte-Carlo approach mentioned above can probably be used to identify the rate of gap formation. More generally speaking, from an ecology point of view, it can produce useful results in terms of pointing out distinct behaviors that depend on the characteristics of populations. It should also be noted that, in both SGs and PCs, it is the average behavior that affects the efficiency, thermal or agricultural. Finally, a plant canopy can range from a uniform monoculture of wheat to a natural jungle canopy or a coral bed. Mechanical and geometrical variability is indeed very different in these cases. Using a Monte-Carlo method allows the taking into account of large ranges of variations of the parameters, but large numbers of simulations will then be needed to obtain reliable statistics.
### Time scales
An issue that also arises in both systems is that of the multiple time scales present. In the SG case, with frequencies of motion of the order of 10 Hz, the vibratory period of motion is at the scale of the inverse of the frequency, ∼0.1 s. Simultaneously operating conditions are stable over days, and the wear process occurs over months. In a PC, the plant motion is at the scale of 1 s, and wind conditions are generally stable over a day or two, and growth occurs over months or even years in trees, integrating information from the instantaneous strain over time. In both cases, a careful decoupling of the two time-scales is necessary: all flow and mechanical conditions are assumed to be constant for the modeling of flow-induced motion (short times), and cumulative models are used to take into account the slow variation of these conditions (long times). In that sense, wear or fatigue models, such as those of Delaune and colleagues (Delaune et al., 2000), and growth models such as those of Moulia and colleagues (Moulia et al., 2006), are very similar in principle.
### Length scales
The question of multiple length scales also exists in both systems. A proper modeling, as described above, requires going from the modeling of the flow pattern in the SG or PC to the modeling of the dynamic response of individual plants or tubes, taking into account local effects at even smaller scales. These slender individual components have several, and quite different, scales of length: their length-to-diameter ratio can be of the order of 100:1 or even 1000:1, and even smaller length scales are also involved, such as gaps at the supports of SG tubes, of the order of 0.1 mm (Axisa et al., 1988), or the thickness of leaves, which play a role in the stiffness of interplant contact (Doaré et al., 2004). In both systems, the ratio of large-to-small scales can exceed 10,000. This can only be taken into account, as for the time-scales, by articulating models over scales. For PCs, this requires the use of plate models for leaves, spring models for contacts and beam models for stems (Niklas, 1992). In SGs, beam models and nonlinear spring models are necessary (Axisa et al., 1988). These can also be used for nonlinear contacts between plants (Doaré et al., 2004).
### Evolution
In terms of evolution, some parallels can also be drawn, with required caution, and in a purely descriptive sense. SGs have only existed for ∼50 years, but the evolution of their design has been significant, as a result of the analysis of the problems of flow-induced wear. New designs involve simple cures, such as adding supports to the tube, called anti-vibratory bars, and using materials less prone to fretting wear. Yet, the search for higher efficiency in thermal exchange has led to a denser packing of tubes, which was not compatible with reduced vibratory risk. A trade-off issue appeared that is classical in design. For plant canopies, the reduction in the height of crops is relevant here; the selection of shorter wheat, including dwarf wheat varieties, and the use of chemical agents to reduce height, does have a positive influence on yield. It can nevertheless be counter-productive on wind-induced motion for several reasons: first, shorter plants move less and therefore the beneficial effect of plant motion on internal gaseous exchange in the canopy is lost; second, their static deformation is also reduced, so that they do not benefit from drag reduction by flexibility; and finally, as the stem constitutes a much smaller part of the plant, any motion of the tip results in higher deformation of the stem, and therefore a higher possibility of fracture or buckling. Here again, a trade-off issue appears between improved agricultural productivity and increased risk of losses.
### Transfers from mechanical engineering to plant biomechanics
The striking similarities between the two problems of flow-induced vibration and damage suggest that many models and methods developed in mechanical and nuclear engineering are somehow applicable to the biomechanics of PCs. In terms of computational methods, this has been the case in the modeling of flow, where the numerical methods for simulating high-Reynolds flow over canopies – for instance large eddy simulations – are almost identical to those used in standard engineering computation (Dupont et al., 2010). Similarly, finite-element codes developed for nuclear engineering have been used for the computation of the dynamic properties of plants (Rodriguez et al., 2008). These cases are illustrated in Fig. 4. Similarly, many results on the dynamics of flows in and above canopies have been derived by using the classical methods of hydrodynamic stability theory (Finnigan, 2000; Ghisalberti and Nepf, 2002; Py et al., 2006; Gosselin and de Langre, 2009). As an illustration, Fig. 5 shows how the interactions between a canopy and air or water flow differ by using such methods. Some other fields of applications seem to be unexplored, but promising and include: a probability approach for the analysis of the biomechanics of extreme events, optimization procedures in design in relation to evolution and models for cumulating damage of distinct origins, such as fatigue and creep. Progress in these fields requires a significant investment in terms of interdisciplinary work.
### Nonlinear mechanics
A first challenge stems from the fact that plants are soft slender structures and are generally much more deformed under fluid loading than are man-made structures. This is quantified by the dimensionless Cauchy number (de Langre, 2008). For instance, the order of deformation, defined simply as displacement divided by length, is typically 0.001 in the SG case and 0.1 in the PC case. Unfortunately, most methods and codes used in mechanical engineering are based on the simplifying assumption that deformation is small, allowing one to work with linearized equations. To model the consequent reduction of drag by elastic reconfiguration, which is an important nonlinear effect in the interaction of flow and a canopy, it is necessary to use rather advanced models of structural mechanics under large deformations (see Fig. 6) (Gosselin et al., 2010). Nevertheless, these models exist, even in standard codes used for nuclear engineering or aerospace engineering (Stanford et al., 2008). Fig. 7 shows in a more advanced approach the sequence of breakage events in a tree under increasing flow intensity, using the same code (Lopez et al., 2011).
Fig. 4.
Computations using standard models of mechanical engineering. (A) Flow-induced motion of a crop canopy, using the large eddy simulation technique (Dupont et al., 2010). (B) A plant-specific modal shape of a walnut tree based on the digitized geometry and computation using a finite-element code from nuclear engineering (Rodriguez et al., 2008). (C) Comparison between the dimensionless frequencies (f/f1, f1 being the frequency of the first mode) computed using plant-specific data (in black), frequencies measured on the actual tree (in white) and frequencies derived from the allometry parameters defining the slenderness of branches and the branching organization.
Fig. 4.
Computations using standard models of mechanical engineering. (A) Flow-induced motion of a crop canopy, using the large eddy simulation technique (Dupont et al., 2010). (B) A plant-specific modal shape of a walnut tree based on the digitized geometry and computation using a finite-element code from nuclear engineering (Rodriguez et al., 2008). (C) Comparison between the dimensionless frequencies (f/f1, f1 being the frequency of the first mode) computed using plant-specific data (in black), frequencies measured on the actual tree (in white) and frequencies derived from the allometry parameters defining the slenderness of branches and the branching organization.
Fig. 5.
Modeling the interaction between a flexible canopy and the flow using standard flow-instability theory (modified from Gosselin and de Langre, 2009). The two graphs show the magnitude of the instability (growth rate, G) as a function of the dimensionless flow velocity (reduced velocity, U). (A) In air, the strong coupling region, in gray, is limited to a small range of velocities. (B) In water, the coupling is spread over a large range of velocities.
Fig. 5.
Modeling the interaction between a flexible canopy and the flow using standard flow-instability theory (modified from Gosselin and de Langre, 2009). The two graphs show the magnitude of the instability (growth rate, G) as a function of the dimensionless flow velocity (reduced velocity, U). (A) In air, the strong coupling region, in gray, is limited to a small range of velocities. (B) In water, the coupling is spread over a large range of velocities.
### Geometries and materials
An important difficulty that arises in the development of a mechanical approach to plant motion is that of properly taking into account geometries and materials. This is a general issue in biomechanics, and more generally in the modeling of natural systems, even in geophysical sciences. Several approaches are possible. The first is the equivalent of what is referred to as the ‘patient-specific approach’ in human biomechanics (referred to as ‘plant-specific’ hereafter). In practice, samples are described in fine detail, both in their geometry and in the characteristics of materials (Py et al., 2006; Sellier et al., 2006; Rodriguez et al., 2008). This allows the building-up of a reasonable model of a given plant in a given state. When a canopy needs to be modeled, the plants are assumed to be identical. The plant-specific approach allows validations of intermediate modeling steps; for instance, the computed response to wind of a given tree can be directly compared with a measured quantity on this same tree. The second approach relies on the use of allometry laws that have been derived on a large number of samples. An average plant or canopy is then built, with idealized characteristics. In previous studies (Py et al., 2006; Dupont et al., 2010), the alfalfa canopy was modeled from the allometry laws based on the analysis of many plants (Fig. 8). In another study (Rodriguez et al., 2008), the modal characteristics of trees were shown to derive only from allometry parameters, using an invariant scaling relationship (Fig. 4C). A combined solution exists, where specific geometries and materials that satisfy allometry laws are generated randomly. But the most suitable solution is certainly probabilistic as material and geometries evolve with time and instantaneous environmental conditions, so that a deterministic biomechanical approach only gives one partial result. Considering the number of parameters involved and the nonlinear relationship between parameters and response, a probabilistic approach needs to be based on multiple simulations, for instance using a Monte-Carlo technique (see Fig. 8) (Delaune et al., 2000).
### Growth
The true limit in the transfer of mechanical-engineering methods to the modeling of plants is the difficulty of taking into account the history of growth. A plant in a given environment is shaped in its geometry and material by its history of growth. Fundamentally, plants are able to differentiate directly their phenotypes in response to environmental stimuli. In a canopy, there exist differences between plants at the canopy edge and those inside the canopy. Edge plants have been exposed to a higher level of fluid loading during their growth, so that they might have different geometrical or material characteristics. This in turn can influence the flow, as well as the mechanical response to fluid forces. Similarly, tubes at the edge of a SG tube bundle have had a distinct history of flow forces and therefore of wear, but their geometry and material were identical to the rest at the initial stage. This issue is related to the existence of several time scales, as mentioned in the preceding section. Of course, models of the effect of mechanical strain on growth are needed at this stage. This is a vast field of research that will have applications in understanding the ecology of a large variety of systems as they grow, for instance kelp (Gaylord et al., 2012), coral (Madin and Connolly, 2012) or sea grasses (Ghisalberti and Nepf, 2002).
Fig. 6.
Modeling the large deformation of plant organs under flow. (A) Schematic view of a reconfigured leaf experiencing wind (modified from Vogel, 1989). (B) Experiments on flow-induced bending of a plate and (C) a simulation of the same problem using nonlinear geometrical effects (Gosselin et al., 2010).
Fig. 6.
Modeling the large deformation of plant organs under flow. (A) Schematic view of a reconfigured leaf experiencing wind (modified from Vogel, 1989). (B) Experiments on flow-induced bending of a plate and (C) a simulation of the same problem using nonlinear geometrical effects (Gosselin et al., 2010).
Fig. 7.
The computed flow-induced pruning of a walnut, using a finite-element computation of stresses and a model of brittle behavior (Lopez et al., 2011). (A–C) Three stages of the tree: (A) initial stage, (B) after breakage of the main branches and (C) before breakage of the trunk; (D) the corresponding evolution of the moment (M) at the base at the function of the Cauchy number (Cy), which increases with the flow velocity. This sequence of breakage events results in a significant reduction of the load on the tree.
Fig. 7.
The computed flow-induced pruning of a walnut, using a finite-element computation of stresses and a model of brittle behavior (Lopez et al., 2011). (A–C) Three stages of the tree: (A) initial stage, (B) after breakage of the main branches and (C) before breakage of the trunk; (D) the corresponding evolution of the moment (M) at the base at the function of the Cauchy number (Cy), which increases with the flow velocity. This sequence of breakage events results in a significant reduction of the load on the tree.
Fig. 8.
Taking into account the variability in the properties of individual components. (A) The probability density function (p) of the wear work rate (W, the product of the impact force by the sliding velocity) following a large number of simulations of tube motions, in a Monte-Carlo approach (Delaune et al., 2000). The probability density function shows a large variety of possible responses of the system. (B) A comparison between experimental (symbols) and model (line) of the local frequency (f) of oscillation of a crop canopy under flow, as a function of the flow velocity (U) (modified from Py et al., 2006). The model is based on the characteristics of the average plant.
Fig. 8.
Taking into account the variability in the properties of individual components. (A) The probability density function (p) of the wear work rate (W, the product of the impact force by the sliding velocity) following a large number of simulations of tube motions, in a Monte-Carlo approach (Delaune et al., 2000). The probability density function shows a large variety of possible responses of the system. (B) A comparison between experimental (symbols) and model (line) of the local frequency (f) of oscillation of a crop canopy under flow, as a function of the flow velocity (U) (modified from Py et al., 2006). The model is based on the characteristics of the average plant.
The use of a biomechanical approach to understand ecological issues requires first the building-up of sound biomechanical models. In the case of the effect of wind on the canopy, this biomechanical approach is not yet well established. Considerable work remains, but many approaches used in mechanical engineering are available (Table 1). Their transfer is in some cases straightforward, for instance in the computation of flows or elementary structural mechanics effects. Advanced issues can also be addressed now by using methods developed in mechanical and nuclear engineering, as has been shown for instance in the case of nonlinear structural mechanics for flexibility effects or in the case of damage propagation. At this stage, it seems inefficient to go in the direction of a detailed plant-by-plant analysis, where the exact geometrical and material characteristics of the plant are used. Similarly, all methods based on the classical assumption of small deformations miss an essential feature of plant biomechanics, which is their large deformation under flow. Incorporating growth models directly in the biomechanics of plants is certainly a very promising direction, and in that sense plant biomechanics is going to diverge significantly from mechanical engineering. Moreover, for instance in the way that variability is analyzed, some differences have appeared that are related to the different functions involved, biological or mechanical. At a larger scale, if it is desired to address ecological questions, it seems that a true understanding of the interaction of plant systems with flow in their environment would greatly benefit from methods used in system engineering for man-made components. The true challenge is in finding a well-balanced adaptation of our engineering knowledge to these plant systems. The comparative analysis of the SG and the PC shows that this is indeed possible and fruitful. Moreover, as in many cases of applications of methods from mechanical engineering to biomechanics, the benefit might also be substantial in the field of mechanical engineering itself, in terms of the generality of the methods (Crimaldi, 2012).
Table 1.
A summary of important remaining issues where engineering methods can be applied to plant biomechanics and ecology
I am indebted to François Axisa, formerly from the French Atomic Energy Commission (CEA), and to Bruno Moulia, from the French National Institute for Agricultural Research (INRA), who led me to work on steam generators and plant canopies, respectively.
Funding
This work was partly supported by the Agence Nationale de la Recherche [grant no. ANR-09-blan-0245-03].
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https://mathschallenge.net/full/diophantine_challenge | ## Diophantine Challenge
#### Problem
Given that x, y, and b are positive integers, prove that the Diophantine equation, x2 + (bx)y = 1 in x and y, has at least four solutions for all values of b.
#### Solution
We shall begin by rearranging the equation, x21 = (xb)y.
When b=1, and taking the subtract form of LHS, we get x21 = (x+1)(x1) = (x1)y, so y = x+1. That is, we have infinitely many solutions for (x,y): (1,2), (2,3), (3,4), ... .
For b2, let us deal with a slightly more general form, x2+a = (xb)y.
Clearly xb divides x2bx, and as xb divides the RHS, it follows that it must divide x2+a. Therefore, xb divides (x2+a)(x2bx) = a+bx.
Similarly xb divides (a+bx)(bxb2) = b2+a.
So a solution exists for each value of xb that divides b21, or rather each factor of b21.
When xb = b21 or xb = b2+1, we get x = b2+b1, which are both positive integers. And as we have already established that xb divides both sides of the Diophantine equation, y = (x21)/(xb) will also be positive integers. Thus we have two positive integer solutions for x and y.
But when xb = 1, we can see that x = b+1 b2+b1 for b2, and so this solution in x will be different to the previous two. In addition, by substituting xb = 1 into the Diophantine equation, we get y = x21, which provides two more positive integer solutions for x and y.
Hence we have proved that the Diophantine equation has at least four positive integer solutions all values of b.
Prove that b=2 is the only value of b for which there are exactly four solutions.
Problem ID: 227 (04 Jun 2005) Difficulty: 4 Star
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https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/32/ | ## Results (1-50 of 174 matches)
Label Dim $A$ Field CM Traces Fricke sign $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
32.2.a.a $1$ $0.256$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$-2$$ $$0$$ $-$ $$q-2q^{5}-3q^{9}+6q^{13}+2q^{17}-q^{25}+\cdots$$
32.2.g.a $4$ $0.256$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$-4$$ $$4$$ $$q+(-\zeta_{8}-\zeta_{8}^{3})q^{2}+(\zeta_{8}+\zeta_{8}^{2})q^{3}+\cdots$$
32.2.g.b $8$ $0.256$ 8.0.18939904.2 None $$-4$$ $$-4$$ $$0$$ $$-8$$ $$q-\beta _{2}q^{2}+(\beta _{1}+\beta _{3}+\beta _{5}+\beta _{7})q^{3}+\cdots$$
32.3.c.a $2$ $0.872$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+iq^{3}+2q^{5}-2iq^{7}-7q^{9}-iq^{11}+\cdots$$
32.3.d.a $1$ $0.872$ $$\Q$$ $$\Q(\sqrt{-2})$$ $$0$$ $$2$$ $$0$$ $$0$$ $$q+2q^{3}-5q^{9}-14q^{11}+2q^{17}+34q^{19}+\cdots$$
32.3.h.a $28$ $0.872$ None $$-4$$ $$-4$$ $$-4$$ $$-4$$
32.4.a.a $1$ $1.888$ $$\Q$$ None $$0$$ $$-8$$ $$-10$$ $$-16$$ $-$ $$q-8q^{3}-10q^{5}-2^{4}q^{7}+37q^{9}+40q^{11}+\cdots$$
32.4.a.b $1$ $1.888$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$22$$ $$0$$ $+$ $$q+22q^{5}-3^{3}q^{9}-18q^{13}-94q^{17}+\cdots$$
32.4.a.c $1$ $1.888$ $$\Q$$ None $$0$$ $$8$$ $$-10$$ $$16$$ $+$ $$q+8q^{3}-10q^{5}+2^{4}q^{7}+37q^{9}-40q^{11}+\cdots$$
32.4.b.a $2$ $1.888$ $$\Q(\sqrt{-7})$$ None $$0$$ $$0$$ $$0$$ $$16$$ $$q-\beta q^{3}-2\beta q^{5}+8q^{7}-q^{9}+3\beta q^{11}+\cdots$$
32.4.g.a $44$ $1.888$ None $$-4$$ $$-4$$ $$-4$$ $$-4$$
32.5.c.a $2$ $3.308$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$-76$$ $$0$$ $$q+3iq^{3}-38q^{5}+2iq^{7}-63q^{9}+\cdots$$
32.5.c.b $2$ $3.308$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$52$$ $$0$$ $$q+iq^{3}+26q^{5}+22iq^{7}+65q^{9}+\cdots$$
32.5.d.a $1$ $3.308$ $$\Q$$ $$\Q(\sqrt{-2})$$ $$0$$ $$14$$ $$0$$ $$0$$ $$q+14q^{3}+115q^{9}+46q^{11}-574q^{17}+\cdots$$
32.5.d.b $2$ $3.308$ $$\Q(\sqrt{-15})$$ None $$0$$ $$-12$$ $$0$$ $$0$$ $$q-6q^{3}-\beta q^{5}-2\beta q^{7}-45q^{9}+26q^{11}+\cdots$$
32.5.h.a $60$ $3.308$ None $$-4$$ $$-4$$ $$-4$$ $$-4$$
32.6.a.a $1$ $5.132$ $$\Q$$ None $$0$$ $$-8$$ $$14$$ $$-208$$ $+$ $$q-8q^{3}+14q^{5}-208q^{7}-179q^{9}+\cdots$$
32.6.a.b $1$ $5.132$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$-82$$ $$0$$ $+$ $$q-82q^{5}-3^{5}q^{9}-1194q^{13}+2242q^{17}+\cdots$$
32.6.a.c $1$ $5.132$ $$\Q$$ None $$0$$ $$8$$ $$14$$ $$208$$ $-$ $$q+8q^{3}+14q^{5}+208q^{7}-179q^{9}+\cdots$$
32.6.a.d $2$ $5.132$ $$\Q(\sqrt{3})$$ None $$0$$ $$0$$ $$92$$ $$0$$ $-$ $$q+\beta q^{3}+46q^{5}-6\beta q^{7}+525q^{9}+\cdots$$
32.6.b.a $4$ $5.132$ 4.0.218489.1 None $$0$$ $$0$$ $$0$$ $$-96$$ $$q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-24-\beta _{3})q^{7}+\cdots$$
32.6.g.a $76$ $5.132$ None $$-4$$ $$-4$$ $$-4$$ $$-4$$
32.7.c.a $2$ $7.362$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$100$$ $$0$$ $$q+iq^{3}+50q^{5}+46iq^{7}+713q^{9}+\cdots$$
32.7.c.b $4$ $7.362$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$-56$$ $$0$$ $$q-\beta _{1}q^{3}+(-14-\beta _{3})q^{5}+(2\beta _{1}-5\beta _{2}+\cdots)q^{7}+\cdots$$
32.7.d.a $1$ $7.362$ $$\Q$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-46$$ $$0$$ $$0$$ $$q-46q^{3}+1387q^{9}+2338q^{11}-1726q^{17}+\cdots$$
32.7.d.b $4$ $7.362$ 4.0.3803625.2 None $$0$$ $$48$$ $$0$$ $$0$$ $$q+(12-\beta _{1})q^{3}-\beta _{2}q^{5}+(\beta _{2}-\beta _{3})q^{7}+\cdots$$
32.7.h.a $92$ $7.362$ None $$-4$$ $$-4$$ $$-4$$ $$-4$$
32.8.a.a $1$ $9.996$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$-58$$ $$0$$ $-$ $$q-58q^{5}-3^{7}q^{9}-8898q^{13}-40094q^{17}+\cdots$$
32.8.a.b $2$ $9.996$ $$\Q(\sqrt{10})$$ None $$0$$ $$-16$$ $$-180$$ $$1248$$ $+$ $$q+(-8+\beta )q^{3}+(-90+8\beta )q^{5}+(624+\cdots)q^{7}+\cdots$$
32.8.a.c $2$ $9.996$ $$\Q(\sqrt{15})$$ None $$0$$ $$0$$ $$140$$ $$0$$ $+$ $$q+\beta q^{3}+70q^{5}+18\beta q^{7}+1653q^{9}+\cdots$$
32.8.a.d $2$ $9.996$ $$\Q(\sqrt{10})$$ None $$0$$ $$16$$ $$-180$$ $$-1248$$ $-$ $$q+(8+\beta )q^{3}+(-90-8\beta )q^{5}+(-624+\cdots)q^{7}+\cdots$$
32.8.b.a $6$ $9.996$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$688$$ $$q+\beta _{2}q^{3}+(\beta _{2}+\beta _{3})q^{5}+(115+\beta _{1}+\cdots)q^{7}+\cdots$$
32.8.g.a $108$ $9.996$ None $$-4$$ $$-4$$ $$-4$$ $$-4$$
32.9.c.a $4$ $13.036$ $$\Q(i, \sqrt{39})$$ None $$0$$ $$0$$ $$-728$$ $$0$$ $$q+(\beta _{1}+\beta _{3})q^{3}+(-182+\beta _{2})q^{5}+(17\beta _{1}+\cdots)q^{7}+\cdots$$
32.9.c.b $4$ $13.036$ $$\Q(i, \sqrt{19})$$ None $$0$$ $$0$$ $$1064$$ $$0$$ $$q+\beta _{3}q^{3}+(266+3\beta _{2})q^{5}+(-7\beta _{1}+\cdots)q^{7}+\cdots$$
32.9.d.a $1$ $13.036$ $$\Q$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-34$$ $$0$$ $$0$$ $$q-34q^{3}-5405q^{9}+27166q^{11}+\cdots$$
32.9.d.b $6$ $13.036$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$36$$ $$0$$ $$0$$ $$q+(6+\beta _{1})q^{3}-\beta _{2}q^{5}+(-\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots$$
32.9.h.a $124$ $13.036$ None $$-4$$ $$-4$$ $$-4$$ $$-4$$
32.10.a.a $1$ $16.481$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$2398$$ $$0$$ $-$ $$q+2398q^{5}-3^{9}q^{9}+112806q^{13}+\cdots$$
32.10.a.b $2$ $16.481$ $$\Q(\sqrt{106})$$ None $$0$$ $$-176$$ $$1404$$ $$-2784$$ $+$ $$q+(-88+\beta )q^{3}+(702-8\beta )q^{5}+(-1392+\cdots)q^{7}+\cdots$$
32.10.a.c $2$ $16.481$ $$\Q(\sqrt{5})$$ None $$0$$ $$0$$ $$-4420$$ $$0$$ $-$ $$q-\beta q^{3}-2210q^{5}-42\beta q^{7}+26397q^{9}+\cdots$$
32.10.a.d $2$ $16.481$ $$\Q(\sqrt{7})$$ None $$0$$ $$0$$ $$-68$$ $$0$$ $+$ $$q+\beta q^{3}-34q^{5}-86\beta q^{7}-3555q^{9}+\cdots$$
32.10.a.e $2$ $16.481$ $$\Q(\sqrt{106})$$ None $$0$$ $$176$$ $$1404$$ $$2784$$ $-$ $$q+(88+\beta )q^{3}+(702+8\beta )q^{5}+(1392+\cdots)q^{7}+\cdots$$
32.10.b.a $8$ $16.481$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-4800$$ $$q-\beta _{1}q^{3}+(\beta _{1}-\beta _{2})q^{5}+(-600-\beta _{3}+\cdots)q^{7}+\cdots$$
32.10.g.a $140$ $16.481$ None $$-4$$ $$-4$$ $$-4$$ $$-4$$
32.11.c.a $4$ $20.331$ $$\Q(i, \sqrt{30})$$ None $$0$$ $$0$$ $$-1400$$ $$0$$ $$q+(-\beta _{1}+\beta _{2})q^{3}+(-350-\beta _{3})q^{5}+\cdots$$
32.11.c.b $6$ $20.331$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$0$$ $$-1716$$ $$0$$ $$q-\beta _{1}q^{3}+(-286+\beta _{2})q^{5}+(58\beta _{1}+\cdots)q^{7}+\cdots$$
32.11.d.a $1$ $20.331$ $$\Q$$ $$\Q(\sqrt{-2})$$ $$0$$ $$482$$ $$0$$ $$0$$ $$q+482q^{3}+173275q^{9}+97426q^{11}+\cdots$$
32.11.d.b $8$ $20.331$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-480$$ $$0$$ $$0$$ $$q+(-60+\beta _{1})q^{3}+\beta _{3}q^{5}+(\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots$$
32.11.h.a $156$ $20.331$ None $$-4$$ $$-4$$ $$-4$$ $$-4$$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9328762292861938, "perplexity": 661.5950790884838}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663013003.96/warc/CC-MAIN-20220528062047-20220528092047-00731.warc.gz"} |
https://en.wikibooks.org/wiki/Calculus/Euler%27s_Method | # Calculus/Euler's Method
Euler's Method is a method for estimating the value of a function based upon the values of that function's first derivative.
The general algorithm for finding a value of ${\displaystyle y(x)}$ is:
${\displaystyle y_{n+1}=y_{n}+\Delta x_{\rm {step}}\cdot f(x_{n},y_{n}),}$
where f is ${\displaystyle y'(x)}$ . In other words, the new value, ${\displaystyle y_{n+1}}$ , is the sum of the old value ${\displaystyle y_{n}}$ and the step size ${\displaystyle \Delta x_{\rm {step}}}$ times the change, ${\displaystyle f(x_{n},y_{n})}$ .
You can think of the algorithm as a person traveling with a map: Now I am standing here and based on these surroundings I go that way 1 km. Then, I check the map again and determine my direction again and go 1 km that way. I repeat this until I have finished my trip.
The Euler method is mostly used to solve differential equations of the form
${\displaystyle y'=f(x,y),y(x_{0})=y_{0}}$
## Examples
A simple example is to solve the equation:
${\displaystyle y'=x+y,y(0)=1.}$
This yields ${\displaystyle f=y'=x+y}$ and hence, the updating rule is:
${\displaystyle y_{n+1}=y_{n}+0.1(x_{n}+y_{n})}$
Step size ${\displaystyle \Delta x_{\rm {step}}=0.1}$ is used here.
The easiest way to keep track of the successive values generated by the algorithm is to draw a table with columns for ${\displaystyle n,x_{n},y_{n},y_{n+1}}$ .
The above equation can be e.g. a population model, where y is the population size and x is time. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 13, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9009354114532471, "perplexity": 250.92587584255602}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934809160.77/warc/CC-MAIN-20171124234011-20171125014011-00456.warc.gz"} |
https://www.physicsforums.com/threads/the-approach-to-project-out-certain-vector.584917/ | # The approach to project out certain vector
1. Mar 8, 2012
### onako
Given that certain nullspace is spanned by a vector v, what would be the procedure to project out the v component from certain vector u? Perhaps with the Gram-Schmidt process of orthonormalization by updating
v = v - proj(u, v)
where proj(u, v) = (<u, v>/<u, u>)u, and <u, v> denotes the inner product? If that is correct way to do it, please let me know. However, if there are alternatives, I would be happy to consider those.
2. Mar 8, 2012
### chiro
Hey onako.
Yes, you're right on the money with using Gram-Schmidt process which basically does exactly what you are trying to do by 'projecting out' all of the components that have been calculated that correspond to elements of your new orthonormal basis.
Basically it boils down to thinking about the projection really is and this boils down to standard vector geometry definitions.
We know from Gram-Schmidt that if we have a vector v with u as our chosen first component of our new basis, then we do what you have said in your formula above. It is the best way to do this and it is the standard way to attack these kinds of problems.
The only thing I wanted to add was that if you need an orthonormal basis, just be aware to normalize but this is not a strict requirement for having a basis.
3. Mar 8, 2012
### onako
Thanks. Indeed, I'll need a normalization; good that you pointed that out. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8251835703849792, "perplexity": 574.8193359119883}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815500.61/warc/CC-MAIN-20180224073111-20180224093111-00269.warc.gz"} |
https://sookocheff.com/post/fp/beta-reduction/ | # Beta Reduction
Formally, beta reduction (also written $$\beta$$-reduction) is the replacement of a bound variable in a function body with a function argument. The purpose of $$\beta$$-reduction is to compute the result of a function by function application using specific rules.
More formally, the beta reduction rule states that a function application of the form $$(\lambda x.t)s$$ reduces to the term $$t[x := s]$$. The term $$t[x := s]$$ means that all instances of $$x$$ in $$t$$ are replaced with $$s$$. The $$\rightarrow$$ syntax is used as a shorthand for beta reduction. We can specify beta-reduction explicitly using the notation $$(\lambda x.t)s \rightarrow t[x := s]$$, which means that the beta reduction of $$(\lambda x.t)s)$$ is $$t[x := s]$$. The beta reduction removes the $$\lambda$$ symbol and resolves to the function’s body with the argument $$s$$ substituted into the body.
For example, we can apply beta reduction to the simple identity function. For every $$s$$, $$(\lambda x.x)s \rightarrow x[x := s] = s$$. This reduction resolves the function application and results in the function body being returned. In this case, the function body is the same as the argument $$s$$, for any $$s$$.
There isn’t much more to beta reduction than this blog post, yet it serves as the foundation for applying functions and calculating results. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8769971132278442, "perplexity": 284.8329054927066}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627997533.62/warc/CC-MAIN-20190616022644-20190616044644-00544.warc.gz"} |
https://stacks.math.columbia.edu/tag/07TD | Lemma 36.17.3 (Normalization and smooth morphisms). Let $X \to Y$ be a smooth morphism of schemes. Assume every quasi-compact open of $Y$ has finitely many irreducible components. Then the same is true for $X$ and there is a canonical isomorphism $X^\nu = X \times _ Y Y^\nu$.
Proof. By Descent, Lemma 34.13.3 every quasi-compact open of $X$ has finitely many irreducible components. Note that $X_{red} = X \times _ Y Y_{red}$ as a scheme smooth over a reduced scheme is reduced, see Descent, Lemma 34.15.1. Hence we may assume that $X$ and $Y$ are reduced (as the normalization of a scheme is equal to the normalization of its reduction by definition). Next, note that $X' = X \times _ Y Y^\nu$ is a normal scheme by Descent, Lemma 34.15.2. The morphism $X' \to Y^\nu$ is smooth (hence flat) thus the generic points of irreducible components of $X'$ lie over generic points of irreducible components of $Y^\nu$. Since $Y^\nu \to Y$ is birational we conclude that $X' \to X$ is birational too (because $X' \to Y^\nu$ induces an isomorphism on fibres over generic points of $Y$). We conclude that there exists a factorization $X^\nu \to X' \to X$, see Morphisms, Lemma 28.52.5 which is an isomorphism as $X'$ is normal and integral over $X$. $\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9989568591117859, "perplexity": 158.9944563981813}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496664437.49/warc/CC-MAIN-20191111191704-20191111215704-00510.warc.gz"} |
https://www.mathalino.com/forum/plane-and-solid-geometry/solid-geometry-5 | # solid geometry
2 posts / 0 new
Renz Kolin Gaba...
solid geometry
A concrete dam of hieght 128 ft. was built in a gorge. One side AB of the gorge slopes at an angle of 60 degrees, the other side CD at 45 degrees. The bases of the dam are horizontal and rectangular in shape. The lower base is 1215 ft. by 152 ft., and the upper base is 32 ft. wide. How many cubic yards of concrete were required?
8eightI'sD
To answer the question above, we need some figure...hehehe...
I imagine that dam to look like this:
Delving deeper into the picture, it becomes like this:
Positioning the dam so it is easier to get the volume of this dam, it looks like this:
Notice that the it is easier to get the volume of the dam if we separate the shapes that makes up the dam, the red and the green part...
The green part's volume is pretty straightforward. We'll just use the formula $$V = BH$$, where $V$ is the volume of the figure, $B$ is the area of the base and $H$ is the height of the figure. Looking at the green part of the figure, the area of it's base is a trapezoid. The area would be $$A = \frac{1}{2}h(b_1 + b_2)$$, where $h$ is the height of plane, $b_1$ is the length of the upper base, and $b_2$ is the length of the lower base. With that in mind, the volume of the green part would be...
$$V=BH$$ $$V = \left( \frac{1}{2}h(b_1 + b_2)\right)(H)$$ $$V = \left( \frac{1}{2}(128 \space feet)(1215 \space feet + 1416.9 \space feet)\right)(32 \space feet)$$ $$V = 5390131.2 \space ft^3$$
The red part is pretty tricky, but solvable. Imagining the red part as a one part cut in halves, it becomes easier to get its volume. Getting the volume of the whole red part below:
$$V=BH$$ $$V = \left( \frac{1}{2}h(b_1 + b_2)\right)(H)$$ $$V = \left( \frac{1}{2}(128 \space feet)(1215 \space feet + 1416.9 \space feet)\right)(120 \space feet)$$ $$V = 20212992 \space ft^3$$
Now getting the volume of the red part we see, it is $\frac{V}{2}$ or $10106496\space ft^3$
The total volume of the dam would be $5390131.2 \space ft^3 + 10106496 \space ft^3$ or $15496627 \space ft^3$
It's not yet the final answer because it must be converted to cubic yards. Now converting from cubic feet to cubic yards:
$$15496627 \space ft^3\left(\frac{1 \space meters}{3.28 \space feet}\right)^3 = 439152.79 \space m^3$$ $$439152.79 \space m^3 \left( \frac{1 \space yards}{0.9144 \space meter}\right)^3 = \color{green}{574390.16 \space cubic \space yards}$$
Alternate solutions are highly encouraged...... | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8404985070228577, "perplexity": 418.071835217058}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875146714.29/warc/CC-MAIN-20200227125512-20200227155512-00150.warc.gz"} |
https://www.arxiv-vanity.com/papers/1106.6346/ | # Comment on “Small Lorentz violations in quantum gravity: do they lead to unacceptably large effects?”
Joseph Polchinski KITP, University of California, Santa Barbara, CA 93106-4030, USA
###### Abstract
A recent paper by Gambini, Rastgoo and Pullin [1] investigates the important issue of constraints from Lorentz invariance on Planck scale physics, arguing that the classic analysis of Collins, Perez, Sudarsky, Urrutia and Vucetich [2] is not generally valid. We argue that the new work is based on models that do not capture the relevant physics, and that almost all models of observable high energy Lorentz violation, and proposed Lorentz-violating theories of quantum gravity, are ruled out by low energy tests; the only known exceptions are based on supersymmetry.
The high precision with which Lorentz invariance is observed in nature places strong constraints on what can happen at much higher energies. It is a general principle of local quantum theory that physics at a high scale manifests itself at lower energies through effective local terms in the action. A local operator of dimension is induced with a coefficient of order , the 4 being the dimension of spacetime. Generically all operators allowed by symmetry are generated. Thus, operators of dimension provide a direct window onto the symmetries of the high energy theory, no matter how high the scale of symmetry breaking.
The Standard Model admits a large number of dimension 4 operators that are gauge invariant but not Lorentz invariant, for example the spatial gradient terms for each of the 19 gauge multiplets. These lead to different ‘speeds of light’ for the different multiplets, so that Lorentz breaking of order one at high energy would lead to unacceptably large breaking at low energy. This reasoning has been confirmed in a model calculation in Ref. [2] (hereafter denoted CPSUV), which also reviews related work. The effective symmetry-breaking interactions are suppressed by Standard Model coupling factors, but are still far too large. Heuristically one can think of a low energy particle mixing with a highly virtual pair with momenta near : the symmetry breaking at the high scale feeds down to low energy without suppression due to the dimensional argument above.
Ref. [1] (hereafter denoted GRP) reexamines the issue raised by CPSUV, finding in two models that the Lorentz violation is small, and concluding that this result will hold rather generally. The purpose of this comment is to note that the two models considered have special features that are not present in the theories of interest. Indeed, without these the special features the calculations of GRP support the claims of CPSUV.
The first model is a Euclidean lattice theory, where the ratios of the time and space lattice steps are taken equal in the continuum limit.111The issues discussed in this paragraph were already raised in CPSUV. A Euclidean lattice with equal steps along different axes has discrete rotational symmetries, which forbids the dimension 4 terms that would violate the Euclidean Lorentz (i.e. rotational) invariance; indeed, this is essential to the success of lattice gauge theory. However, we are interested in a Lorentzian world, and a Lorentzian lattice with equal time and space steps has no such enhanced symmetry. In terms of symmetry this is better modeled by a Euclidean lattice with unequal steps (heuristically the ratio of steps is the imaginary ), in which case the calculation of GRP confirms the large effect seen in CPSUV.
One might try to define the Lorentzian theory by continuation from the Euclidean lattice theory, and so benefit from the symmetry of the latter. However, a lattice propagator has an analytic structure not consistent with unitarity. More generally it is the requirement that a relativistic quantum theory be both Lorentz-invariant within observed limits and unitary that severely constrains the set of possible theories; if either requirement is dropped the number of possibilities multiplies enormously. We are fortunate to have these constraints as guidance in the construction of quantum gravity.
The second model is a Lorentzian continuum theory with Pauli-Villars-like propagator
1−k20+→k2+m2−1−k20+→k2+m4→k2+M2+M2. (1)
It is found that the induced Lorentz violation is of order , which can be acceptably small. However, it should be noted that the Lorentz-violating term, in the second part of the propagator, is of relative order at all scales, so the small breaking is put in by hand. For the gravitational models investigated in CPSUV, the breaking is of order one at the Planck scale. If the Lorentz-breaking term in the propagator (1) is enhanced to order one at the Planck scale, then so is the induced low-energy breaking, in agreement with the results of CPSUV.
To summarize, the question investigated by CPSUV was “O(1) Lorentz violations at the Planck scale: do they lead to unacceptably large effects?” The question addressed by the models of GRP is quite different: “Small Lorentz violations at all scales: do they lead to unacceptably large effects?”
To see why the first question is the relevant one, recall that one of the main issues is the possibility of observing nonstandard optical effects in high energy propagation [3, 4]. In order for this to be observable and consistent with standard tests of Lorentz invariance, the effect must grow as a power of , but this is precisely the possibility ruled out by CPSUV: any Lorentz breaking at high energy feeds down to the Standard Model scale essentially without suppression.
Further, we note that essentially all proposed theories of Lorentz-violating quantum gravity are of the type considered by CPSUV. For example, if one formulates the theory canonically, without a Lorentz-invariant starting point, then Lorentz breaking is generically large at the Planck scale. What CPSUV show is that one cannot then rely on dimensional analysis to guarantee consistency with observation at ordinary energies: there is a strong burden on the proponents of such a theory to demonstrate that low energy Lorentz invariance is restored to high accuracy.
We note in passing that there is one class of theories that evades the problem, namely those in which some other symmetry forbids the dimension 4 Lorentz-violating terms. Supersymmetry is the one known example [6, 7], in which both the space and time derivative kinetic terms arise from a single superfield. If supersymmetry is exact at the Planck scale and broken only at a much lower scale, then the observed Lorentz breaking may be suppressed to a sufficient degree. (Again, if the supersymmetry breaking is large at the Planck scale then the low energy Lorentz and supersymmetry breaking will be large as well). Aside from this, the argument of CPSUV applies to all known models of observable high energy Lorentz breaking, and all known models of Lorentz-breaking quantum gravity.
GRP also give another argument to suggest that their result holds more generally, around Eq. (7) and more fully in Sec. IV.B of Ref. [5], namely that if measurements are made with physical rods and clocks the observed Lorentz violation is small. However, this was not applied to the relevant observations. That is, GRP discuss the direct measurement of the Lorentz-violating propagator. Rather, CPSUV look at the violation induced on other fields due to loops of this propagator: if one applies the same analysis to these fields then the result of CPSUV is reproduced.
GRP attribute their result to a non-perturbative property of their models, but as we have seen it is in fact due to special features and not true in more general models that are equally ‘non-perturbative.’ The effective field theory argument in the first paragraph is not perturbative in nature, it is based on the Wilsonian framework that forms the basis for the non-perturbative understanding of quantum field theory. Nor is background independence an issue. The tests of Lorentz invariance take place in a specific nearly flat and classical background. Any background-independent theory must reproduce the many successful tests of effective quantum field theory in this same background, and so is subject to the same constraints.
I thank Yuri Bonder, John Collins, Rodolfo Gambini, Don Marolf, Alejandro Perez, Jorge Pullin, and Edward Witten for communications and discussions. This work was supported in part by NSF grants PHY05-51164 and PHY07-57035, and by FQXi grant RFP3-1017. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9555218815803528, "perplexity": 640.6214262658525}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243990419.12/warc/CC-MAIN-20210511214444-20210512004444-00063.warc.gz"} |
http://math.stackexchange.com/questions/174010/are-there-one-way-integrals | # Are there “one way” integrals?
If we suppose that we can start with any function we like, can we work "backwards" and differentiate the function to create an integral that is hard to solve?
To define the question better, let's say we start with a function of our choosing, $f(x)$. We can then differentiate the function with respect to $x$ do get $g(x)$: $$g(x) = f'(x)$$
This, in turn, implies, under appropriate conditions, that the integral of $g(x)$ is $f(x)$: $$\int_a^b { g(x) dx } = [f(x)]_a^b$$
I'm wondering what conditions are appropriate to allow one to easily get a $g(x)$ and $f(x)$ that assure that $f(x)$ can't be easily found from $g(x)$.
SUMMARY OF THE QUESTION
Can we get a function, $g(x)$, that is hard to integrate, yet we know the solution to? It's important that no one else should be able to find the solution, $f(x)$, given only $g(x)$. Please help!
POSSIBLE EXAMPLE
This question/integral seems like it has some potential.
DEFINITION OF HARDNESS
The solution to the definite integral can be returned with the most $n$ significant digits correct. Then it is hard to do this if the time it takes is an exponential number of this time.
In other words, if we get the first $n$ digits correct, it would take roughly $O(e^n)$ seconds to do it.
-
The ease or difficulty of integrating $g$ would likely depend on how skilled the one doing the integration is at seeing a pattern... – Ｊ. Ｍ. Jul 23 '12 at 0:13
"Hard to integrate" is not a very precise standard. What is your motivation for asking this question? – Potato Jul 23 '12 at 0:17
"No one should be able to find the solution" is not very precise either. With "one way functions", we usually talk about the complexity of finding the input given the output, and expect it to be "of high complexity" (e.g., exponential on the input). The complexity will depend on the algorithm used to attempt to integrate a given function. Of more use are trapdoor functions, in which an extra piece of information, $h$, makes is so that there is an easy algorithm for computing $f$ given $g$ and $h$, and no known easy alg. for computing $f$ given $g$ alone. I don't know any such with integral – Arturo Magidin Jul 23 '12 at 0:21
@MattGroff: Numerical methods are for computing definite integrals. Your question seems to be about computing indefinite integrals. Which is it? – Arturo Magidin Jul 23 '12 at 0:31
I know double-posting is frowned upon, but you might want to post at cstheory.SE, since some crypto & complexity theory folks reside over there. – user2468 Jul 23 '12 at 1:16
I interpret your question as follows:
Is there any differentiable function $f$ and pair of real numbers $a,b$ such that computing the integral $\int_a^b f'(x)dx$ to $n$ bits of precision given $a,b,f'$ is substantially harder than computing $f(b)-f(a)$ to the same precision?
The answer to this is no, in the sense that if $f(b)-f(a)$ can be computed in $O(h(n))$ then the integral can be computed in $O(h(n))$ as well, assuming $\lim\limits_{n\to\infty} h(n)=\infty$. One way to do this would be to simply enumerate all functions with finite-length definitions and differentiate them until one is found with derivative $f'$. One might object that it is impossible to determine whether two strings of symbols produce the same function, but there are only countably many algorithms for symbolic differentiation. Any algorithm used to differentiate $f$ must be a provably correct implementation of differentiation, and one can enumerate these by enumerating the list of all algorithms and of all proofs using the standard pairing function and checking each pair (algorithm,proof) to see if the proof proves the correctness of the algorithm. We can thus enumerate all pairs (function, provably correct differentiation algorithm). Thus we get the function $f$. This is obviously done in constant time w.r.t. $n$, and so if we can compute $f(b)-f(a)$ in $O(h(n))$ we can compute the integral in $O(h(n))+O(1)=O(h(n))$.
-
There is a trivial way to produce an infinite number of algorithms for symbolic differentiation: you tack on silly expressions like $\mathord{} + K - K$ to the answer. – Zhen Lin Jul 23 '12 at 6:44
@ZhenLin Yes, but I mean finitely many he might be using. Anyway, countably many makes no difference, since whichever algorithm is actually being used will be tried after some finite number of tries. – Alex Becker Jul 23 '12 at 8:51
I'm not convinced. One must have a computable enumeration of all algorithms that compute derivatives, and by Rice's theorem there is no such enumeration. – Zhen Lin Jul 23 '12 at 8:54
@ZhenLin True, but any algorithm that he could use would have to be provably correct, and there is a computable enumeration of all algorithms that can be proven to correctly implement differentiation. – Alex Becker Jul 23 '12 at 8:58
This seems off what you are asking, but you should know it since it changes what you want. Volterra came up with an example of a function on, as I recall, the open interval $(0,1)$ which has a derivative at every point. The derivative is bounded but discontinuous on a Cantor set of positive measure. As a result, the derivative does not have an integral: possession of a definite Riemann integral is equivalent to the set of points of discontinuity being of measure zero. The example is in Counterexamples in Analysis by Gelbaum and Olmsted. I will see if I can find something on line. HERE
EDIT: the Volterra derivative does not have an indefinite integral either. This comes late in the Beamer talk by Bressoud referenced on WickedPedia.
-
Here is an example (by Harold Davenport) of a function that is hard to integrate:
$$\int{2t^6+4t^5+7t^4-3t^3-t^2-8t-8\over (2t^2-1)^2\sqrt{t^4+4t^3+2t^2+1}}\,dt\ .$$ The primitive is given by (check it!) \eqalign{&-2\log(t^2+4t+2)-\log(\sqrt2+2t)\cr&+ \log\left(\sqrt2 t+2\sqrt2-2\sqrt{t^4+4t^3+2t^2+1}\,\right)\cr &-5\log(2t^2-1)-5\log\left(2\sqrt2 t+4\sqrt2 -t^2-4t-6\right) \cr&+ 5\log\left(\bigl(4\sqrt2+19\bigr)\sqrt{t^4+4t^3+2t^2+1}\right. \cr &\qquad\qquad\left. - 16\sqrt2 t^2 -8\sqrt2 t +6\sqrt2 -29t^2-38t+5\right)\cr} \eqalign{ &+ 2\log\left(\bigl(10\sqrt2+17\bigr)\sqrt{t^4+4t^3+2t^2+1}\right.\cr &\qquad\qquad\left.+4\sqrt2 t^2+16\sqrt2 t -2\sqrt2-11t^2-44t-39\right) \cr &+ {1\over2}\log\left(\bigl(731\sqrt2 t+71492\sqrt2-70030t-141522\bigr) \sqrt{t^4+4t^3+2t^2+1} \right.\cr &\qquad\qquad-40597\sqrt2t^3-174520\sqrt2t^2-122871\sqrt2 t+50648\sqrt2\cr&\qquad\qquad\left.+90874t^3+ 403722t^2+ 272622t-61070 \vphantom{\sqrt{t^4}}\right)\cr & + {(2t+1)\sqrt{t^4+4t^3+2t^2+1}\over 4t^2-2}\ .\cr}
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Could you reveal the source of this integral? I'm not familiar with Daveport, but I'd like to know more... – Matt Groff Jul 23 '12 at 12:54
@Matt Groff: When Harold Davenport gave a talk at ETH Zurich one day in the eighties he had this example on one of his slides. He gave me a copy afterwards, and I used it many times to terrify my calculus students. He told me that he had more terrific examples in stock. – Christian Blatter Jul 23 '12 at 18:11
Thanks for the information. I ran this through my math software, and it actually returned a pseudo-answer within a minute. The problem is, it returned the answer in a formula that needed the roots of the polynomials to be known and plugged in. So it makes me believe we could in fact create a generalized formula using a math program, and then create polynomials to plug into the equation. We would start with the roots, I guess, and then expand into polynomials. This would seem to indicate that solving these integrals is no harder than factoring polynomials. – Matt Groff Jul 23 '12 at 19:09
I posted a related question to this question and answer here: math.stackexchange.com/questions/176000/… – Matt Groff Jul 28 '12 at 3:45
Find two large primes $p$ and $q$, call their product $N$; then it's easy for you to compute $\int_{J(N)}1\,dx$, where $J(N)$ is the interval between the prime factors of $N$, but for anyone else to compute the integral, she would have to factor $N$ first.
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I seem to remember hearing something about factoring being done in polynomial time, but maybe that statement has such been retracted... – Matt Groff Jul 23 '12 at 17:09
Primality testing can be done in polynomial time (in the number of digits), but if factoring can also, that's news to me (and would be major result). – user7530 Jul 23 '12 at 19:41
Well yes, but the hard part is finding the endpoints rather than doing the integration. – Robert Israel Jul 27 '12 at 21:35 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9461800456047058, "perplexity": 361.5855683616521}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510273663.2/warc/CC-MAIN-20140728011753-00402-ip-10-146-231-18.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/18731-derivatives.html | # Math Help - derivatives
1. ## derivatives
if f(1)=2 and f'(1)=5, use the equation of the line tangent to the graph of f at x=1 to approximate f(1.2).
2. Originally Posted by asnxbbyx113
if f(1)=2 and f'(1)=5, use the equation of the line tangent to the graph of f at x=1 to approximate f(1.2).
Point-Slope Form?
f(1) = 2 gives the point
f'(1) = 5 gives the slope.
y - 2 = 5(x-1)
Substitute x = 1.2
3. Originally Posted by asnxbbyx113
if f(1)=2 and f'(1)=5, use the equation of the line tangent to the graph of f at x=1 to approximate f(1.2).
Or you can think of it as a truncated Taylor series.
$f(1.2) \approx f(1) + \frac{1}{1!}f^{\prime}(1) \cdot 0.2$
So
$f(1.2) \approx 2 + 5 \cdot 0.2$
etc.
-Dan
4. Originally Posted by topsquark
Or you can think of it as a truncated Taylor series.
$f(1.2) \approx f(1) + \frac{1}{1!}f^{\prime}(1) \cdot 0.2$
So
$f(1.2) \approx 2 + 5 \cdot 0.2$
etc.
-Dan
That is not the best explanation to give this poster. Because he will not hear the term "Taylor Series" for another year. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9428498148918152, "perplexity": 547.8050230686588}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398455246.70/warc/CC-MAIN-20151124205415-00313-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/wave-runup-average-reservoir-depth.581760/ | # Wave Runup - Average Reservoir Depth
1. Feb 27, 2012
### af_231
Hello!
In order to calculate wave run-up for a reservoir, I need to determine the average reservoir depth along fetch line (D).... Does anyone know how can I calculate this depth?
In some book I have read that D is the average reservoir level along the direction of wind blow, and it is assumed to be half of the initial reservoir, D=0.5Ho, but only for 20ft <Ho<35ft. In my case the initial reservoir level is more than 35ft.
Another books and works do mention of a Engineer Letter of the U.S. Department of Corp of Engineers, named ETL 110-2-221: Wave Runup and Wind Setup on Reservoir Embakment. But I have been unable to get this book.
I would appreciate any suggestion or advice.
Thank you very much!
Can you help with the solution or looking for help too?
Draft saved Draft deleted
Similar Discussions: Wave Runup - Average Reservoir Depth | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.89027339220047, "perplexity": 2434.262429976529}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721387.11/warc/CC-MAIN-20161020183841-00483-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://proceedings.mlr.press/v151/doddi22a.html | # Efficient and passive learning of networked dynamical systems driven by non-white exogenous inputs
Harish Doddi, Deepjyoti Deka, Saurav Talukdar, Murti Salapaka
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:9982-9997, 2022.
#### Abstract
We consider a networked linear dynamical system with p agents/nodes. We study the problem of learning the underlying graph of interactions/dependencies from observations of the nodal trajectories over a time-interval T. We present a regularized non-casual consistent estimator for this problem and analyze its sample complexity over two regimes: (a) where the interval T consists of n i.i.d. observation windows of length T/n (restart and record), and (b) where T is one continuous observation window (consecutive). Using the theory of M-estimators, we show that the estimator recovers the underlying interactions, in either regime, in a time-interval that is logarithmic in the system size p. To the best of our knowledge, this is the first work to analyze the sample complexity of learning linear dynamical systems driven by unobserved not-white wide-sense stationary (WSS) inputs. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8915041089057922, "perplexity": 705.9109990302121}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103036176.7/warc/CC-MAIN-20220625220543-20220626010543-00548.warc.gz"} |
http://mathhelpforum.com/calculus/196938-polynomial-max-min.html | # Math Help - polynomial with max $min 1. ## polynomial with max$ min
Let $p(x)$ be a real polynomials of Least Degree Which has Local Maximum at $x=1$ and Local Minimum
at $x=3$.If $p(1)=6$ and $p(3)=2$. Then $p^{'}(0)=$
2. ## Re: polynomial with max $min Originally Posted by jacks Let $p(x)$ be a real polynomials of Least Degree Which has Local Maximum at $x=1$ and Local Minimum at $x=3$.If $p(1)=6$ and $p(3)=2$. Then $p^{'}(0)=$ $p(x)=x^3-6x^2+9x+2 ~\text { so }~ p'(x)=3x^2-12x+9$ Hence : $p'(0)=9$ 3. ## Re: polynomial with max$ min
Hello, jacks!
$\text{Let }p(x)\text{ be a polynomial of least degree which has local max at }x=1$
$\text{and local minimum at }x=3.\;\text{ If }p(1)=6\text{ and }p(3)=2\text{, then }p'(0)= \_\_$
The desired polynomial is a cubic: . $p(x) \:=\:ax^3 + bx^2 + cx + d$
Its derivative is: . $p'(x) \:=\:3ax^2 + 2bx + c$
We are given:
. . $\begin{array}{ccccc}p(1) = 6\!:& a + b + c + d &=& 6 \\ \\[-4mm] p(3) = 2\!:& 27a + 9b + 3c + d &=&2 \\ \\[-4mm] p'(1) = 0\!: & 3a + 2b + c &=&0 \\ \\[-4mm] p'(3) = 0\!: & 27a + 6b + c &=&0 \end{array}$
Solve the system of equations: . $a = 1,\;b = -6,\;c = 9,\;d = 2$
The cubic is: . $p(x) \:=\:x^3 - 6x^2 + 9x + 2$
. . Its derivative is: . $p'(x) \:=\:3x^2 - 12x + 9$
Therefore: . $p'(0) \:=\:3(0^2) - 12(0) + 9 \:=\:9$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 23, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9028359651565552, "perplexity": 3294.000572940802}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398466260.18/warc/CC-MAIN-20151124205426-00054-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/184110-find-limit-print.html | # Find limit
• July 5th 2011, 06:05 AM
initM
Find limit
$\lim_{x \to 0}$ cos(pi/x)/(x - 2)
How could I find this limit? Any idea would be appreciable.
• July 5th 2011, 06:10 AM
Prove It
Re: Find limit
Quote:
Originally Posted by initM
$\lim_{x \to 0}$ cos(pi/x)/(x - 2)
How could I find this limit? Any idea would be appreciable.
The limit does not exist, because for values infinitessimally close to $\displaystyle x = 0$, the function $\displaystyle \cos{\left(\frac{\pi}{x}\right)}$ can equal any value in $\displaystyle [-1, 1]$ (you can not determine a single value that it goes to).
• July 5th 2011, 06:25 AM
Also sprach Zarathustra
Re: Find limit
Quote:
Originally Posted by initM
$\lim_{x \to 0}$ cos(pi/x)/(x - 2)
How could I find this limit? Any idea would be appreciable.
Take $x_n=\frac{1}{2\pi n}$ and $y_n=\frac{1}{(2n+1)\pi}$ and $x_n ,y_n \to 0$ when $n\to \infty$.
Could you continue?
• July 5th 2011, 06:34 AM
initM
Re: Find limit
Quote:
Originally Posted by Also sprach Zarathustra
Take $x_n=\frac{1}{2\pi n}$ and $y_n=\frac{1}{(2n+1)\pi}$ and $x_n ,y_n \to 0$ when $n\to \infty$.
Could you continue?
It is little bit difficult for me. Can I use any substitution here?
• July 5th 2011, 08:03 AM
HallsofIvy
Re: Find limit
Quote:
Originally Posted by initM
It is little bit difficult for me. Can I use any substitution here?
Actually, I think there is a slight error there- the " $\pi$" should not be in there. Try instead $x_n= \frac{1}{2n}$ and $y_n= \frac{1}{2n+1}$, 1 over the even integers and 1 over the odd integers.
Now, I am hoping that "It is a little bit difficult for me" does NOT mean "I really don't want to actually do anything my self" so I will make some suggestions: If $x_n= \frac{1}{2n}$, what is $\frac{\pi}{x_n}$? What is the cosine of that? What does that go to as n goes to infinity? If $y_n= \frac{1}{2n+1}$, what is $\frac{\pi}{y_n}$? What is the cosine of that? What does that go to as n goes to infinity? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 21, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8531628251075745, "perplexity": 1038.2981065338388}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443737929054.69/warc/CC-MAIN-20151001221849-00230-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://quant.stackexchange.com/questions/45941/equal-weight-correlation-swap-payoff-derivation | # Equal weight correlation swap payoff derivation
I am aware that the payoff for a equal weighted correlation swap is;
$$(\rho_K-\rho)*\text{Notional}$$ where $$\rho = \frac{2} {n(n-1)}\sum_{i < j} \rho_{ij}$$ I am wondering how I can derive this $$\rho$$ term.
• What do you mean by derive? It is an average of the (non-diagonal) entries in the correlation matrix. A n by n matrix has $n(n-1)/2$ elements in its upper (or lower) triangle, so to find the average we add these elements and then divide by this number. For example a 3 by 3 matrix has 3*2/2 = 3 elements above the main diagonal. The average of these elements is $(1/3)*(\rho_{12}+\rho_{13}+\rho_{23})$ – Alex C Jun 5 at 2:19
• Thanks Alex that clarifies it. The textbook I was using there was no summation and I was wondering if the coefficient was somehow capturing all of the pairwise correlations if they were assumed to be equal. Do you know are the volatilities used realised or implied for the correlation matrix? – J19 Jun 5 at 12:38
• AFAIK the payoff at maturity is based on the realized correlations during the period until the swap matures. – Alex C Jun 5 at 17:12 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 3, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.857050359249115, "perplexity": 526.2248877361689}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540525598.55/warc/CC-MAIN-20191209225803-20191210013803-00402.warc.gz"} |
https://pirsa.org/21090004 | ## Abstract
Although cold dark matter (CDM) has been established, this is only the case for measurements at large scales, which are larger than galaxy-sized structures. Even though we need to understand the important role of baryonic components, matter distribution at small scales can be the key to distinguishing different particle dark matter candidates. In fact, warm dark matter, self-interacting dark matter, and fuzzy dark matter have been proposed, yielding different matter distributions at sub-galactic scales. These small-scale distributions have been studied with numerical simulations. Whereas very reliable, numerical simulations suffer certain issues. They are limited by both numerical resolution and shot noise. They tend to take a lot of computational time. These might be a bottleneck for surveying through multi-dimensional parameter space of various dark matter models. I present semi-analytic models of small-scale structure, especially dark matter subhalos, which are based on structure formation theory and the tidal evolution of subhalos. Our models for CDM have been well tested against the results of various numerical simulations. The semi-analytic models are free from all the problems of the numerical simulations mentioned above. Therefore, they might be essential ingredients for identifying the particle nature of dark matter through gravitational (lensing, pulsar timing), astrophysical (satellite counts, stellar streams), and astroparticle probes (gamma rays).
## Details
Talk Number PIRSA:21090004
Speaker Profile Shin'ichiro Ando | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8345388770103455, "perplexity": 1410.6211434445277}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320302355.97/warc/CC-MAIN-20220120160411-20220120190411-00238.warc.gz"} |
http://scielo.sld.cu/scielo.php?script=sci_arttext&pid=S0864-084X2018000100019&lng=en&tlng=es | ## Indicators
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## Print version ISSN 0864-084XOn-line version ISSN 2075-5635
### Nucleus no.63 Ciudad de La Habana Jan.-June 2018
Nuclear Sciences
Possible effects of clustering structure in the competition between fast emission processes and compound nucleus decay
Posibles efectos de estructura de agrupamiento en la competencia entre los procesos de emisión rápida y desintegración del núcleo compuesto
2INFN Laboratori Nazionali di Legnaro, Legnaro (PD), Italy
4INFN Firenze and Dipartimento di Fisica e Astronomia, Universitá di Firenze, Firenze, Italy
5INFN Bologna and Dipartimento di Fisica e Astronomia, Universitá di Bologna, Bologna, Italy
6Grand Accelerateur National d’IonsLourds, 14076 Caen, France
7Science and Art Faculty, Physics Department, NevsehirHaciBektasVeli University, Nevsehir, Turkey
8INFN Napoli and Dipartimento di Fisica, Universitá di Napoli, Napoli, Italy
9Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow St. University, Moscow, Russia
10National Research Center "Kurchatov Institute", Moscow, Russia
11Laboratori Nazionali del Sud, Catania, Italy
12Tohoku University, Sendai, Japan
Abstract
The attention to nuclear clustering has been renewed due to the study of weakly bound nuclei at the drip lines. In particular, clustering structural properties in medium-mass systems have been studied by looking at the competition between the evaporation and pre-equilibrium particle emission in central collisions. Although for light nuclei at an excitation energy close to the particle separation value there are experimental evidence of such structure effects, this is still not the case for heavier systems since the determination of pre-formed clusters within nuclear matter is less obvious. Two systems, leading to the same 81Rb* compound nucleus, have been studied at the same beam velocity 16 AMeV: 16O + 65Cu and 19F + 62Ni. The experiment has been performed using the GARFIELD + RCo detection system installed at the Legnaro National Laboratories.Light charged particles energy distributions and multiplicities have been compared with different statistical and dynamical model calculations. From the first comparison between the two systems a difference in the fast α -decay channel has been evidenced, which can be related to the difference in the projectile structure. Recent data analysis results and comparisons with model calculations are presented in this contribution.
Key words: cluster model; nuclear decay; cluster emission model; equilibrium; comparative evaluations; Legnaro National Laboratory
Resumen
Palabras claves: modelo de haz; desintegración nuclear; modelo emisión de núcleos agrupados; equilibrio; evaluaciones comparativas; Laboratorio Nacional Legnaro
INTRODUCTION
One of the oldest model used to describe the nucleus is the α -cluster model based on the concept that clusters of nucleons might be pre-formed prior to emission from nuclei [1]. Examining the binding energies per nucleon of light nuclei in their ground state, as a function of the mass number, a systematic trend has been observed for α -conjugate nuclei (even and equal number of protons and neutrons) well described by the liquid drop model as due to a shell structure effect [2]. A nucleus is a finite quantum many-body system consisting of protons and neutrons interacting via nuclear forces. Its ground state has shell structure, in which nucleons move almost independently in a mean field. On the other hand, because of nuclear attraction, spatial correlations between nucleons can be rather strong giving rise to cluster structures in which nucleons are confined, mainly at the surface of the nucleus. Due to their strong binding energy, the α particle is the most probable cluster subsystem in nuclei and it is the main ingredient in the concept proposed by Ikeda in his diagram [3], where highly clustered states are predicted at excitation energies around the energy threshold for the decomposition into specific cluster channels. Moreover, in neutron-rich nuclei there is the possibility that additional neutrons may act as valence particles which can be exchanged between the α particles cores. These covalent neutrons stabilize the unstable multi-cluster states, giving rise to nuclear structure which can be described by molecular concepts. The extended Ikeda diagram is a new threshold diagram needed to describe the structure of these non-alpha conjugate nuclei. Recently, nuclear clustering has gained large interest due to the study of weakly bound nuclei at the drip lines, where clustering might be the preferred structural mode, especially in the case of light nuclei [1].
Many nuclear reactions involve the emission or capture of clusters of nucleons and these cluster reactions are particularly interesting to investigate the interplay between nuclear structure and reaction dynamics. In fact, such clusters can participate in nuclear reactions and this enables their properties to be studied.
The coexistence of cluster and mean-field aspects points out several phenomena in nuclear many-body systems as a function of excitation energy and isospin degree of freedom and many exotic and new features of clustering have been discovered [4].
While for light nuclei several links between cluster emission and its connection with nuclear structure and dynamics have been pointed out [1] [5], this is less obvious moving towards heavier systems. In fact, in reactions involving medium-mass nuclei the determination of pre-formed clusters in nuclear matter is more complicated.
An interesting way to investigate the structural properties of medium-mass systems is to study the competition between evaporation and pre-equilibrium particle emission in central collisions, as a function of different entrance channel parameters. In fact, the pre-formed clusters have been observed especially close to the nuclear surface, making strong the link between pre-equilibrium emission and cluster structure.
Pre-equilibrium light charged particles and/or neutrons are fast and forward focused particles emitted during the very early stages of the collision, before the formation of thermally equilibrated compound system. Two opposite mechanisms have been suggested for cluster emission in pre-equilibrium reactions: on one side, the α -particle is assumed to be pre-formed inside the nucleus and it can be treated as a single strongly coupled object. On the other side, the coalescence models assume that clusters (not only α -particles) are formed, in a dynamical way, during the course of the reaction [5].
Comparing pre-equilibrium particles with those emitted after thermal equilibration is possible to derive information on the interplay between equilibrium and non-equilibrium processes. In particular, information on structural properties, like cluster pre-formation probabilities, may be derived from the experimental comparison between different entrance channels leading to the same compound system and comparing the experimental data with model predictions.
MATERIALS AND METHODS
To investigate the possible effects of the α -cluster structure of the projectile, two different entrance channel reactions have been studied in an energy range where fast particle emission was predicted. The two fusion reactions 16O + 65Cu and 19F + 62Ni, leading to the same 81Rb* compound nucleus and with different N/Z projectile structure, have been studied at 16 AMeV incident energy. The same projectile velocity has been chosen since the pre-equilibrium emission is expected to mostly depend on this parameter [6] As a consequence, the non-equilibrium processes are predicted to be almost the same for the two systems, while some little diferences may appear in the evaporative part of the emitted particle spectra due to the slightly different initial excitation energies of the compound nucleus (E*=209 MeV and E*=240 MeV respectively for 16O and 19F induced reactions). The observation of any difference of fast α -particles in the experimental spectra between the two reaction scould bein terpreted, in a model independent way, as possible influence of the projectile α -structure effect.
The experiment has been performed at the Legnaro National Laboratories in Legnaro (Italy). The beams have been provided by the ALPI-TANDEM XTU accelerator complex and the experimental set-up used is the GARFIELD detection array implemented with the Ring Counter (RCo), at the most forward angles, fully equipped with digital electronics [7].
The GARFIELD apparatus consists of two large volume cylindrical drift chambers, each equipped with Micro Strip Gas Chambers (MSGC) as amplified ∆E stage followed by CsI (Tl) scintillators residual energy detectors. Intermediate mass fragments and light charged particles are detected in an angular range from θ = 29o to151o. The Ring Counter is a three stage annular detector, covering the θ = 5o- 17o angular range, with an Ionization Chamber (IC) as first stage, followed by reverse mounted nTD Silicon Strip detectors (Si) and CsI(Tl) scintillators. The GARFIELD plus RCo apparatus can perform complete high quality charged particle identification (both Z and A) and energy determination in a nearly 4π coverage (θ = 5o- 151o) for light charged particles and, in the most forward direction (θ = 5o- 17o), also for fragments with charge up to Z=14. Light charged particles, detected in GARFIELD and RCo, have been measured in coincidence with Evaporation Residues (ER) collected in the first two stages (IC-Si) of the RCo within the angular range θ = 8.6o- 17o, just beyond the grazing angle. The ERs have been selected setting proper gates in the reconstructed Z versus Energy distributions.
RESULTS
The selection of central events have been performed asking for the detection of the ER in the forward direction (i.e. in the RCo) in coincidence with one or more light charged particles in the all remain apparatus. The ER, characterized by a velocity close to the center of mass velocity of the reaction, have been selected looking at the correlation between the detected charge and the energy distribution in the laboratory frame.
The double differential proton and alpha energy spectra, in coincidence with ERs, have been sorted out and the spectra obtained from the two systems have been compared.
An example is shown in Fig. 1, where the comparison between proton and α -particles spectra, normalized to the maximum, at the most forward angle range of GARFIELD are reported. The two reactions show very similar proton spectra on the angular range measured, except for a small difference at the most forward angles (upper panels). This effect can be ascribed to the slightly larger excitation energy in the 19F induced reaction. On the contrary, a much larger difference is observed in the α -particles spectra. The predicted emission spectra performed with the statistical model code GEMINI++ [8] (bottom panels), which describes only statistical emission from complete fusion reactions and takes into account the difference in the compound nucleus excitation energies, confirm that the purely statistical emission spectra should be very similar for the two systems, supporting the idea that a second fast emission source for both systems is needed when comparing with experimental data.
A first estimate of the expected amount of fast emission in the two cases has been performed comparing the data with the predictions of the Moscow Pre-equilibrium Model (MPM) [9], which is a modified version of the statistical code PACE2 where a non-equilibrium stage before the complete thermalization of the compoud nucleus has been inserted. The relaxation processes occurring during the fusion reaction is accounted for by the exciton model, based on the Griffin model [10], inwhichthedescriptionoftheangulardistributionofthefastemittedparticlesisstillanintricatequestion [11]. The main parameter to be set is the initial number of excitons (n o =n particles +n holes ) in the projectile, which can be estimated from the empirical trend obtained in the work of N. Cindro et. al. [12]. The calculations donefor the 16O + 65Cu case (with an initial excit on number of no= 17 (16p + 1h))and for the 19F + 62Ni reaction (with no= 20 (19p + 1h)) show quite similar results: the shape of the α -particles spectra are reasonably described in the case of 16O + 65Cu, while in the 19F induced reaction an overproduction of fast α -particles is evident. In both cases the proton spectra are largely overstimated. A possible explanation for the extrayieldoffast α -particles in both systems, may be due to the fact that even the 19F can have an alpha structure and, in particular, that its α +15N state is characterized by an energy (4.01 MeV) evens mallerthan the α +12C (7.2MeV) of the 16O.A unique set of initial parameters of the MPM model seems not to be able to describe both proton sand α -particles decay channel sindicating that some implementations to the model has to be introduced [13].
As next step of the analysis, different theoretical approaches have been considered in order to follow the evolution of the reaction on an event-by-event basis.
First, the energy distributions of the light charged particles, in coincidence with evaporation residues, have been compared with simulations performed with the statistical model code GEMINI++ [8], which describes the decay of hot nuclei formed in fusion reactions, usinga Monte Carlo code, and generates light charged particles distribution emitted after the thermal equilibrium is reached. The Monte Carlo code generates an event file, which can be filtered through a software replica of the experimental set-up taking into account the geometry of the apparatus (energy thresholds, energy resolutions, detectors solid angles) for a realistic comparison with the experimental data.
Then, to take into account the dynamical part of the reaction, two models have been considered: the first is the Antisymmetrized Molecular Dynamics (AMD) [14] in which the dynamics is considered by the equation of motion of Gaussian wave packetsrepresenting the colliding nucleons. The clustering effects of the colliding partners can be taken into account through the nucleon-nucleon correlations term. The second model is the Stochastic Mean Field (SMF) [15], implemented in the TWINGO code [16], which considers each nucleon as composed of many test particlessubjected to a mean field. GEMINI++ has been applied, as afterburner, to the results of AMD and TWINGO to generate the secondary fragments distributions to be directly compared to the experimental data.
DISCUSSION
Protons and α -particles experimental energy spectra, in coincidence with ER, have been compared first with the predictions of GEMINI++ alone and then with AMD and TWINGO coupled with GEMINI++ for both systems and at all the detected angles. In general, the backward spectra are well reproduced by GEMINI++ alone, while a fast component is more and more evident going towards the most forward angles, this component is more pronounced for the 19F + 62Ni system.
In Fig. 2 proton spectra, for selected angles, are shown. GEMINI++ and TWINGO give similar information, apart from the temperature of the emitting thermalized source. In fact, the SMF approach does not reconstruct properly the nucleons (due to the test particle method) and in particular the clusters (due to the mean field approach) but it only defines the fragment size, number and excitation at a certain collision time. Therefore, only those particles emitted from the produced excited fragments, the decay of which is established by the after-burner, are finally implemented in the spectra. No pre-equilibrium spectra can therefore be provided. On the contrary, AMD is able to reconstruct fast nucleons and clusters as a consequence it describes better the spectra even at the forward angles.
In the case of α -particles, at forward angles, the component of fast emission is more evident in the experimental spectra. This effect is partially described by the predictions from the AMD+GEMINI code, which include a possible influence of alpha clustering structure in the projectile, as shown in Fig. 3. However, the statistics of simulated events with AMD has still to be incremented to better describe the experimental spectra, avoiding unphysical statistical fluctuations. Moreover, further calculations can be provided, varying the input parameters in the code related to the clusterization effects to look for an optimization in the description of the data.
As a further step, in Fig. 4 experimental proton and α -particles multiplicity distributions, in coincidence with ER, are shown. The distributions, for the two systems, have been normalized to the number of ER and compared with the different model predictions. The predictions of GEMINI are always slightly higher than AMD calculations and seem to reproduce correctly the α -particles multiplicityfor both systems. The AMD predictions are reproducing better the multiplicities of protons, for the two systems, while TWINGO calculations predict always less particles emission. In general, all the models are not able to reproduce the multiplicities of channels with the emission of more than 6 or 7 particles. Further analysis is ongoing looking to specific decay channels.
CONCLUSIONS
Possible α -clustering effects in medium mass nuclei have been investigated by analyzing the secondary particle emission from 16O + 65Cu and 19F + 62Ni at the same beam velocity 16 AMeV, in particular studying the competition between evaporation and pre-equilibrium particle emission. Indeed, a difference in the fast α -particles component of the emitted spectra of the two systems has been evidenced, which can be related to the difference in the projectile structure. Experimental energy spectra of protons and α -particles, in coincidence with evaporation residue, have been compared with different model predictions. The predictions of the Moscow Pre-equilibrium Model, which takes into account both the pre-equilibrium and evaporation emission, show that the model is not able to describe, with a unique set of initial parameters, protons and α -particles at the same time. The model is being upgrading, improving the introduction of clustering structure effects and a correct filter in the evaporation residues distribution. GEMINI++, which considers only complete fusion processes, and TWINGO, which considers the dynamics of the reaction but is not able to reconstruct correctly the fast emitted nucleons and clusters, both cannot describe the experimental spectra. On the contrary, AMD model, which includes cluster structure effects in the projectile, seems to have a better agreement with the experimental data. However, the input parameters need to be further adjusted.
The data analysis is still going on to study more exclusive channels, in particular particle-particle correlations in events with α -particles multiplicities greater than two. As first attempt, the minimum, medium and maximum energy distributions of events in which 3 α -particles are emitted, in coincidence with any evaporation residues, have been sorted out for the two systems. The comparison with the predictions of GEMINI++ shows that the faster particles are not completely reproduced by the code, this means that, even in events with emission of three α -particles, there is evidence of pre-equilibrium emission of α -particles. This effect is more evident for the system 19F + 62Ni also for medium velocity particles.
References
[1] von OERTZEN W, FREER M, KANADA-EN’YO Y. Nuclear clusters and nuclear molecules. Phys. Rep. 2006; 432(2): 43-113. [ Links ]
[2] FREER M. The clustered nucleus-cluster structures in stable and unstable nuclei. Rep. Prog. Phys. 2007; 70(12): 2149. [ Links ]
[3] IKEDA K, TAGIKAWA N, HORIUCHI H. The Systematic Structure-Change into the Molecule-like Structures in the Self-Conjugate 4n Nuclei. Prog. Theor. Phys. Suppl. 1968; E68: 464-475. [ Links ]
[4] KANADA-EN’YO Y, KIMURA M, ONO A. Antisymmetrized molecular dynamics and its applications to cluster phenomena. Prog. Theor. Exp. Phys. 2012; 1(1): 01A202. [ Links ]
[5] HODGSON PE, BĚTÁK E. Cluster emission, transfer and capture in nuclear reactions. Phys. Rep. 2003; 374(1): 1-89). [ Links ]
6. [6] CABRERA J, et. al. Fusion-fission and fusion-evaporation processes in 20Ne+159Tb and 20Ne+169Tm interactions between E/A=8 and 16 MeV. Phys. Rev. C. 2003; 68(3): 034613. [ Links ]
[7] BRUNO M, GRAMEGNA F, et. al. GARFIELD + RCo digital upgrade: A modern set-up for mass and charge identification of heavy-ion reaction products. Eur. Phys. J. 2013; 49: 128. [ Links ]
[8] CHARITY RJ. A systematic description of evaporation spectra for light and heavy compound nuclei. Phys. Rev. C. 2010; 82: 014610. [ Links ]
[9] FOTINA OV, et. al. Pre-equilibrium effects in the secondary particle spectra in the reactions with heavy ions. Int. Jour. Mod. Phys. E. 2010; 19(05n06): 1134. [ Links ]
[10] GRIFFIN JJ. Statistical model of intermediate structure. Phys. Rev. Lett. 1966; 17(9): 478. [ Links ]
[11] BLANN M, CHADWICK MB. Precompound Monte-Carlo model for cluster induced reactions. Phys.Rev.C. 2000; 62(3): 034604. [ Links ]
[12] CINDRO N, et. al. Early stages of nucleus-nucleus collisions: A microscopic calculation of the initial number of degrees of freedom. Phys.Rev. Lett. 1991; 66(7): 868. [ Links ]
[13] FABRIS D, et. al. Pre-equilibrium Particles Emission and Its Possible Relation to α-clustering in Nuclei. Acta Phys. Polonica B. 2015; 46(3): 447. [ Links ]
[14] ONO A, et. al. Antisymmetrized molecular dynamics with quantum branching processes for collisions of heavy nuclei. Phys. Rev. C. 1999; 59(2): 853. [ Links ]
[15] COLONNA M, et. al. Fluctuations and dynamical instabilities in heavy-ion reactions. Nucl. Phys. A. 1998; 642(3-4): 449-460. [ Links ]
[16] GUARNERA A, et. al. 3D stochastic mean-field simulations of the spinodal fragmentation of dilute nuclei. Phys. Lett. B. 1996; 373(4): 267-274. [ Links ]
Recibido: 13 de Febrero de 2018; Aprobado: 29 de Mayo de 2018 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8301015496253967, "perplexity": 2208.4819590890697}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764499470.19/warc/CC-MAIN-20230128023233-20230128053233-00527.warc.gz"} |
http://chand2like.blogspot.com/2013/05/how-to-hack-windows-7-password.html | How to Hack windows 7 Password
How to Reset Windows 7 Password From the Command Prompt
Today i want to tell you a Simple trick using which you can Reset or hack windows 7 password using only Command Prompt. Using this trick you can reset password of any windows 7 computer easily.
In the Command Prompt you can monitor all the user of the computer by typing some commands.
1: Press Win Key+R and type cmd in the Run command box and hit enter to open command prompt. And then type net user in the command prompt and hit Enter. The Command Prompt will show you all accounts on the Windows 7 PC.
2: Type net user xyz 123(xyz is the locked admin account’s name and 123 is the new password) and press Enter. Now you have successfully created a new password on the locked admin account.
3: Type shutdown -r -t 00 and press Enter.
Your Windows 7 PC should be rebooted automatically and then you can log in Windows 7 with the new password. It is a simple and easy way to change the password of any users from the Command Prompt.
But the drawback of this technique is that you can’t solve Windows 7 password recovery issues if you do not have the Administrator Privilege. So it is quite hard to change the Windows password of the users without logging in as the administrator. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8818743824958801, "perplexity": 2156.6778491182354}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500821289.49/warc/CC-MAIN-20140820021341-00038-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://randommath.net/2014/03/13/mathbio-homework-5-due-wed-march-19/?replytocom=34 | # MathBio: Homework 5, due Wed March 19
For the following assignment, you may work in groups but the homework you turn in should be your own.
Homework 5 (pdf)
UPDATE (3/17, 3:30 pm): For 1(b) you may set $m = 0.9$ and determine the stable age structure.
1. David Garcia
Hmm…I’m not getting anything near 0.9 for m in exercise 3a, I’m getting a range around -0.1 for m. I’m guessing that’s super wrong?
March 13, 2014
2. Bryan
For number 2, we tried finding the eigenvalues using Wolfram-Alpha but the solution was incoherent. We can’t use the auxiliary equation since it is a 3-D system; is there a better way to determine stability in terms of h?
March 18, 2014
3. jd
David, I also got an inequality with m near -0.1 which would make the question trivial. Perhaps we are correct and the question is somewhat faulty based on the assumed parameters.
March 18, 2014
4. jd
David, I also got a value of m near -0.1. Perhaps we are correct and the question is faulty based on the assumed parameters.
March 18, 2014
5. Vitor
I did it through wolfram, I just kept plugging in numbers between 0 and 1 until I found one that made the one of the eigenvalues hover just over 1
March 18, 2014
6. AE
For 3.3 #4 is the -dy a derivative of y or constant “d”?
March 26, 2014
• Scott Alister McKinley
The d is a constant in that problem. I agree, that’s a bad choice of letter!
March 26, 2014 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 1, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9057114720344543, "perplexity": 1762.018918572435}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107907213.64/warc/CC-MAIN-20201030033658-20201030063658-00402.warc.gz"} |
https://physics.stackexchange.com/questions/269966/newtons-second-law-confusion | # Newton's Second Law Confusion [closed]
I don't quite understand the relationship between an object's mass, acceleration and net force. For example, if a car was going to the right at a constant velocity or with no acceleration, does that mean that it would have no net force?
From my understanding, net force is directly related to acceleration, as where ever the force is applied, the acceleration follows. Can someone please explain it using a better example and answer my question?
I tried to do some research on this but can't find information pertaining to the situation I have laid out.
## closed as off-topic by sammy gerbil, ACuriousMind♦, garyp, Diracology, GertJul 27 '16 at 2:16
• This question does not appear to be about physics within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.
• I'm actually not sure what you need explained. You seem to understand it quite well. A net force causes an acceleration and an acceleration must imply a net force causing it. And mass is the "resistance" against acceleration - inertia. – Steeven Jul 26 '16 at 8:07
• I agree with @Steeven You seem to have it spot on. What is making you feel uncertain? (If you want to clarify your question, it might be better to edit the question rather than post a comment which might be missed.) – garyp Jul 26 '16 at 11:01
• I'm voting to close this question as off-topic because there is no confusion here, there is nothing to explain. – sammy gerbil Jul 26 '16 at 11:08
• @Steeven Whenever I question myself I just can't seem to recall fast enough- I keep having to think of scenarios like these but that leads to even more confusion. – Imagine Dragons Jul 27 '16 at 2:25
You are correct. The car has no net force on its environment, and the environment has no net force on the car. This is true of any object traveling with a constant velocity.
This is even true in the vertical direction. There is a force of gravity pulling down on the car, and there is a force caused by the road pushing up on the car. If the car is not accelerating up nor down, then these forces must be equal and opposite such that the net force is zero.
The one minor tweak I would make to what you are saying is in the second paragraph. You say "Wherever the force is applied, the acceleration follows," but its also valid to think of its as "wherever there is an acceleration, there must be a corresponding force." The two properties do not so much "follow" each other as they are directly linked to each other. In most cases, your intuition will lead you to correct results. However, sometimes a problem will be posed where you know the acceleration and not the forces. In solving such problems, its nice to remember that you can use the equations in either direction.
If a body is moving, this doesn't mean that a net force certainly must be exerted on it. It can move without any net force (First Law of Newton). You might say "How that body has started its motion without any net fore?" The answer is: "Equations of motion are moment equations, i.e. they are stated for moments not for a time interval ($F(t_0)=ma(t_0)$). So when we say a body is moving without any net force, we mean at that instance ($t$), net force acting on it is zero and it is moving with constant velocity at that instance.(It is obvious that net force can be zero for a time interval ($t_1 \le t \le t_2$). Then, $F(t)=ma(t)=0\;,\;t_1 \le t \le t_2$)"
If a body is moving with constant velocity, this doesn't mean that no force is exerted on it. It is possible that some forces are exerted on it but net (resultant) force is equal to zero.
You are correct in your definition of force. A car, not accelerating, has zero net force associated with it. However, if the car were to hit something--let's say it's me standing in the middle of the street--it would exert a net force on me, and by Newton's Third Law experience a net force equal in magnitude and opposite in direction.
So how can an object that is not accelerating exert a force on another object? First, consider what happens: the car hits me in this scenario, which causes it to slow down and me to speed up. If the car is slowing down, it is undergoing negative acceleration. I mentioned Newton's Third Law earlier; the forces experienced by myself and the car are equal in magnitude due to the Third Law, so adding them up using the Second Law, the system consisting of myself and the car has experienced zero net force, as we expect.
To bring math into this, let a force $F$ be equal to mass $m$ times acceleration $a$: $F=ma$. We know from basic calculus that acceleration can be defined using velocity $v$ as $a=\frac{\mathrm dv}{\mathrm dt}$. Substituting, we get $F=m\frac{\mathrm dv}{\mathrm dt}$. $m\frac{\mathrm dv}{\mathrm dt}$ represents a change in momentum, where a momentum $p$ is defined as $p=mv$, so if we want to deal with changes in mechanical force, such as when I am hit by a car, we will inevitably need to consider momentum. It follows that $F=\frac {\mathrm dp}{\mathrm dt}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8843297362327576, "perplexity": 205.13618329148088}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627998369.29/warc/CC-MAIN-20190617022938-20190617044938-00144.warc.gz"} |
https://mathoverflow.net/questions/32692/from-function-field-to-curve-non-algebraically-closed-ground-field-and-functor/32702 | # From function field to curve: non-algebraically closed ground field and functor of points
This question concerns the (re)construction of a smooth projective curve $C$ over a field $k$, using the function field $K=K(C)$ of $C$. When $k$ is algebraically closed, this is described for instance in Hartshorne, I.6.
My questions are the following:
1. At a few points in the construction, Hartshorne uses that $k$ is algebraically closed, at other points at least that $k$ is infinite. Can one get around this, and use non-algebraically closed or even finite ground fields for reconstructing a curve from its function field?
2. In constructing a smooth projective curve from a finitely generated field $K$ of transcendence degree $1$ over $k$, one takes as the underlying points of the curve the discrete valuations on $K$, defines a topology on this set by declaring closed sets to be fintie or the whole set, and then building an appropriate sheaf of rings. Then one checks that the result is a smooth projective curve. Is there a slick way to describe the functor of points for this curve, instead of the associated locally ringed space?
-
## 1 Answer
1. The construction holds for any base field $k$. But if $k$ is not perfect, you get a projective curve which is normal but not necessarily smooth. For example, if $k$ has characteristic $p>2$ and $t\in k$ is a not a $p$-th power in $k$, consider the function field $K=k(x,y)$ defined by the relation $y^2=x^P-t$. The curve you get is not smooth, and there is no projective smooth curve over $k$ whose function field is $K$. Note that smooth curves are always normal, and the converse is true if $k$ is perfect.
2. Pick any transcendental element $x\in K$. Then $K$ is finite over $k(x)$. Let $A$ be the integral closure of $k[x]$ in $K$ and let $B$ be the integral closure of $k[1/x]$ in $K$. Then the localizations $A_x$ and $B_{1/x}$ are both equal to the integral closure of $k[x, 1/x]$ in $K$. Therefore we can glue the affine curves ${\rm Spec} A$ and ${\rm Spec} B$ along ${\rm Spec} A_x$ and get a curve $C$. By constuction $C$ is normal and integral, with field of functions $K$, and there is finite morphism $C\to \mathbb P^1={\rm Spec} k[x] \cup {\rm Spec} k[1/x]$ (obtained by glueing ${\rm Spec} A\to {\rm Spec} k[x]$ and ${\rm Spec} B\to {\rm Spec} (k[1/x])$). Hence $C$ is its projective, and it is the projective normal curve associated to $K$.
3. As a bonus, one also has a nice correspondance betweeen finite morphisms of curves and extensions of function fields of one variable. If $f : C\to D$ is a finite morphism of projective normal integral curves over $k$, then it induces a finite extension $k(D)\to k(C)$. One can show that this establises a anti-equivalence from the category of integral normal projective curves over $k$ (where morphisms are finite morphisms of $k$-curves) to the category of function fields of one variable over $k$ (where morphisms are morphisms of $k$-extensions).
-
In (1), let $l$ be any prime different from $p$ (so $l = 2$ or $l = 3$ is sufficient) and use $y^l = x^p - t$. Then (1) goes through for $p = 2$. – KConrad Jul 20 '10 at 22:16
Is it also true that $C$ can be characterized by a universal property? Something like the universal example of a scheme $X$ with a proper morphism to $Spec k$ and a morphism from $Spec K$ such that the composed morphism $Spec K\to Spec k$ is what it should be? – Tom Goodwillie Jul 20 '10 at 22:17
I agree. Any morphism $f$ of $k$-schemes from ${\rm Spec } K$ to a proper $k$-scheme $X$ factorizes uniquely as the canonical morphism ${\rm Spec} K\to C$ followed by a $k$-morphism $C\to X$. – Qing Liu Jul 20 '10 at 22:33
Qing Liu's answer above is very helpful. The fact that normal implies regular in dimension 1 is independent of the ground field, right? So when the ground field is not perfect, the issue is really smoothness as opposed to regularity. Maybe the universal property mentioned above is sufficient for working with the curve. It would still be helpful to describe the functor of points, though. – A. Pascal Jul 21 '10 at 6:24
A regular scheme or variety is always normal. A locally noetherian normal scheme of dimension 1 (e.g. normal curve over any field) is always regular. To describe the points of $C$, (2) gives you a concrete method. Of course, the valuation theory as in Hartshorne also decribes the points of $C$. But I don't know how to decribe the functor of points of $C$ directly from the field $K$. If $X$ and $Y$ are birational integral varieties over $k$, you can not distinguish the dominant morphisms for $X$ and $Y$ to $C$ in terms of $K$. – Qing Liu Jul 21 '10 at 9:31 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9789600372314453, "perplexity": 134.07158742610585}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860125524.70/warc/CC-MAIN-20160428161525-00095-ip-10-239-7-51.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/magnitude-of-a-force-on-a-hatch-in-a-vessel.291389/ | # Magnitude of a Force on a hatch in a Vessel.
1. Feb 10, 2009
### Basch
1. The problem statement, all variables and given/known data
We're told the density of ever material involved here, the liquid in the vessel, called "LAB", the water on the outside, and the acryllic that the vessed is made of.
LAB: 860 kg/m^3
Water: 1000 kg/m^3
Acryllic: 1185kg/m^3
Also, the wall thickness of the vessel is 5.4 cm. The height of the neck is labeled delta h, we're also told to not take the pressure difference between the inside and the outside, due to the 5.4cm wall.
This is a three part question, but I'm only looking for help on the first part, which asks us to find the force on the hatch, which is 1.00m^2 big, when delta h = 0.
2. Relevant equations
We need to use Pressure = Force/Area for this, solving for force gives us
Force = Pressure*Area
density * gravity * height = pressure
3. The attempt at a solution
My attempt to solve the question, was to solve for the value of pressure of the inside of the system, and then use that to solve for force. Since the height of liquid in the neck is 0, we have no liquid in the neck, so we only have to consider whats inside the circular area, which has a height of 12m, density of 860kg/m^3, and gravity is 9.81 m/s^2, so which we have to add the pressure of 1 atm due to the air at the top, 1.01e-5 Pa.
Which gives us: 101239 Pa * 1.00 m^2, so the answer is 1.01e5 N?
2. Feb 10, 2009
### LowlyPion
But what about the buoyant force from the water beneath the hatch?
Isn't the pressure there going to be 12m*Δρ*A ?
3. Feb 10, 2009
### Basch
Oops, I accidentally submitted this twice, i had already solved it in another post. Sorry.
Similar Discussions: Magnitude of a Force on a hatch in a Vessel. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9193994998931885, "perplexity": 801.3175760976726}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109470.15/warc/CC-MAIN-20170821172333-20170821192333-00113.warc.gz"} |
https://cs.stackexchange.com/questions/109330/how-to-draw-an-lts-based-on-the-parallel-process-in-ccs-milners-logic | # How to draw an LTS based on the parallel process “|” in CCS Milner's logic?
I'm trying to provide a Hennessy-Milner logic formula for CCS expressions that are not (strongly) bisimilar. An example with a sketch:
For each of the following CCS expressions, decide whether they are strongly bisimilar and if no, find a distinguishing formula in Hennessy-Milner logic.
$$b.a.Nil+b.Nil$$ and $$b(a.Nil+b.Nil)$$
I first draw these two to get a better understanding as follows (excuse my awful drawing skills but I couldn't figure out how to put it in LaTeX so I used draw.io):
You can clearly see the Right Hand Side could do $$b.b$$ but the Left Hand Side can't respond to that. And my distinguishing HML formula is $$[b]tt$$. On the LHS you get: $$[b]\{b.a.Nil+b.Nil\} = \emptyset$$ and on the RHS you get: $$[b]\{b(a.Nil+b.Nil), a.Nil+b.Nil\} = b(a.Nil+b.Nil)$$. Therefore this is a valid distinguishing formula.
Hopefully, I made what the exercise is about clear. Now, the following CCS expressions that I have to distinguish are:
$$a.Nil|b.Nil$$ and $$a.b.Nil + b.a.Nil$$.
I know how to draw the RHS because the + denotes a choice but I don't have any idea of how the parallelism work in CCS and couldn't understand it after reading. My guess is the following but it doesn't make sense and is probably wrong:
Could someone help me understand how to sketch the LHS so that I can complete this exercise?
P.S My tags are probably not correct but I couldn't find any tags to Hennessy-Milner logic, so feel free to edit them.
• Your last LTS is wrong, it should be a diamond-shaped graph, where on one side you perform a then b, and on the other b then a, both sides leading to Nil. Also note that if you can't distinguish between two processes, maybe they are bisimilar, and you could try proving that instead. – chi May 14 at 12:33
• @chi Like this imgur.com/a/p78PUVj ? – Does it matter May 14 at 14:36
• @chi If my drawing is correct, then your comment is exactly what I needed. Please feel free to add your comment as an answer so that I can accept it. – Does it matter May 14 at 14:44
• And maybe it's an idea to introduce Hennessy–Milner logic tag? You can do it as you have enough reputation. – Does it matter May 14 at 14:46
The last LTS in your question is wrong. It should be a diamond-shaped graph, where on one side you perform action $$a$$ then $$b$$, and on the other you perform action $$b$$ then $$a$$, both sides leading to $$\sf Nil$$.
This is because the $$a$$ action is run in parallel w.r.t. the $$b$$ action, so we get all the possible "interleavings" between such events. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 15, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.826384961605072, "perplexity": 422.1893101966065}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575515.93/warc/CC-MAIN-20190922135356-20190922161356-00145.warc.gz"} |
http://science.sciencemag.org/content/330/6006/891.3 | Education
# Wikipedia Goes to Grad School
See allHide authors and affiliations
Science 12 Nov 2010:
Vol. 330, Issue 6006, pp. 891
DOI: 10.1126/science.330.6006.891-c
Very few graduate-level science curricula include training in communicating advanced concepts to a general audience. Moy et al. report a class project that addressed this by having chemistry students edit an entry in Wikipedia.org collaboratively. Students selected topics that were related to the course and were minimally covered on Wikipedia. Student entries contained references, an introduction aimed at the general public, and figures to enhance the explanation of the topic. Student feedback collected at the end of the project revealed increased knowledge of their topic. A specialist in writing and rhetoric concluded that the students' entries were more engaging to general readers because of the attention to real-world applications and clear explanations of vocabulary. Course professors noted that students appeared to assess the material they added to the entry more critically than when they were simply studying for the class, which is consistent with the notion of students' developing a higher level of explanatory knowledge when teaching the material is a goal.
J. Chem. Educ. 87, 1159 (2010). | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8712855577468872, "perplexity": 2105.9852041241866}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125946597.90/warc/CC-MAIN-20180424100356-20180424120356-00010.warc.gz"} |
http://www.waynetippetts.com/?tag=couture-fashion-week | Posts Tagged ‘Couture Fashion Week’
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A touch of neon sparks-off this monochrome look.
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Anya Ziourova Fashion Director of Tatler Russia, on the move in Paris for Couture Fashion Week.
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Natuka modeling yet another one of her fabulous creations, at the Trocadero.
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Paris – Elena Perminova
Friday, July 8th, 2011
Elena Perminova, taken just before Valentino couture show held at The Hotel Salomon de Rothschild. Elena manages to disguise her pregnant bump well by wearing this wonderful dress.
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Daisy Lowe Leaving Ellie Saab couture fashion show, during Paris couture fashion week.
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Monday, July 4th, 2011
Natuka Karkashadze, stylist and journalist, wearing her own designs. Shot shortly before the Christian Dior show, on the first day of Couture fashion week.
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Posted in Street Style | 1 Comment » | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9751075506210327, "perplexity": 4790.125983284964}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783403508.34/warc/CC-MAIN-20160624155003-00088-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/458750/how-does-the-generalized-effective-action-in-wetterichs-exact-rg-scheme-relate | # How does the generalized effective action in Wetterich's exact RG scheme relate to observables at different scales?
I am not familiar with Wetterich's exact RG paradigm, and cannot understand the main idea behind it. I understand that if one could have solved the model and obtained the all the n-point functions then there would not have been any need for any RG analysis.
The Wilson-Polchinski RG framework (see, for example, arxiv-0702365) extends the ordinary purturbative RG approach beyond perturbative regimes. But it stills adheres to the same general principle: lowering the UV cutoff of the model and tracking the growth of various terms.
In Wettrich's approach (See, for example, arxiv-0005122), a generalized effective action, $$\Gamma_k[\phi]$$ is used, which is the 1PI generating functional with all modes below a momentum scale $$k$$ frozen (i.e., an IR cutoff at the momentum scale $$k$$).
Why is $$\Gamma_{k}[\phi]$$ useful for extracting information about IR physics? (To be more concrete, in Eq. 2.11 on page 17 of arxiv-0005122, it seems one gets back the same source, $$J$$, and not some coarse-grained version.)
• Minor comment to the post (v3): In the future please link to abstract pages rather than pdf files. – Qmechanic Feb 4 '19 at 20:41 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 5, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9174454808235168, "perplexity": 809.7259861475818}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371880945.85/warc/CC-MAIN-20200409220932-20200410011432-00399.warc.gz"} |
https://www.physicsforums.com/threads/probably-a-really-easy-problem.70986/ | # Probably a really easy problem
1. Apr 11, 2005
### Exulus
Could anyone help me with this? I need to solve this equation:
$\frac{d^2y}{dx^2} + 4y = \sin{2x}$
Everything I seem to try for y ends up canceling itself out, as the second diffferential is always the negative of what you begin with, and the 2x part of the sin means you get a factor of 4 in the second differential, which then cancels with the 4y. Eg:
Trying $y = c \sin{2x}$
$\frac{dy}{dx} = 2c \cos{2x}$
$\frac{d^2y}{dx^2} = -4c \sin{2x}$
Substituting these into the LHS of my question gives:
$-4c \sin{2x} + 4c \sin{2x} = 0$
The only thing i can think of now is that its a complex solution..but we haven't been taught that yet!
Any help appreciated :(
(ps no idea why the itex stuff is coming out so small..sorry about that).
2. Apr 11, 2005
### estalniath
You'll have to find the homogeneous solution first before applying the method of undetermined coefficients. Let (D^2+4)y=r^2+4=0. hence r=2i,-2i
Thus y(homogenous)=Acos2x+Bsin2x
Therefore, let the trial function be y=AxSin2x+BxCos2x
You should get the answer from here. =)
3. Apr 11, 2005
### estalniath
*Ammendments:Trial solution=CxSin2x+DxCos2x
Last edited: Apr 11, 2005
4. Apr 11, 2005
### Exulus
Hi,
I tried your trial solution but it still comes out as 0 :(
$y = Cx\sin{2x} + Dx\cos{2x}$
$\frac{dy}{dx} = 2Cx\cos{2x} + C\sin{2x} - 2Dx\sin{2x} + D\cos{2x}$
$\frac{d^2y}{dx^2} = -4Cx\sin{2x} + 2C\cos{2x} - 2C\cos{2x} - 4Dx\cos{2x} - 2D\sin{2x} - 2D\sin{2x} =$
$= -4x(C\sin{2x} + D\cos{2x}) - 4D\sin{2x}$
Subbing back into question:
$-4x(C\sin{2x} + D\cos{2x}) + 4Cx\sin{2x} + 4Dx\cos{2x} = 0$
:(
5. Apr 11, 2005
### dextercioby
Try Lagrange's method (of constant variation)
Assume the inhomogenous solution
$$y_{p}=C(x)\sin 2x+D(x)\cos 2x$$
And solve for the unknown functions,by plugging it in the original ODE.
Daniel.
6. Apr 11, 2005
### Exulus
doh, it looks like i did my sums wrong i think. I've now got $y = -\frac{1}{4}\cos{2x}$ which looks like the right answer (haven't tested yet).
Cheers for the help guys! :)
Last edited: Apr 11, 2005
7. Apr 11, 2005
### Palindrom
If you multiplicate it by x I'll believe you...
8. Apr 11, 2005
### dextercioby
Anyways,there's an alternative to 'trial solutions'.Lagrange's method is suitable for these problems.
Daniel.
9. Apr 11, 2005
### estalniath
After subbing the values of d^2y/dx^2,dy/dx and y into the differential equation, the RHS of the equation should be sin2x, then you'll have to do comparing of the coefficients to determine the value of the constants =)
10. Apr 11, 2005
### Exulus
Sorry yeah i typed it wrong, whoops. I did actually have it with the x written down..honest ;)
Thanks for the other method as well, dextercioby :)
Similar Discussions: Probably a really easy problem | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9178900122642517, "perplexity": 2764.474455135013}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463609054.55/warc/CC-MAIN-20170527191102-20170527211102-00143.warc.gz"} |
http://mathhelpforum.com/trigonometry/175713-proving-trig-identities-problems-need-correction-print.html | # Proving Trig identities Problems that need correction
• Mar 24th 2011, 01:52 PM
MajorJohnson
Proving Trig identities Problems that need correction
Here are some more problems that need checking from my other thread I made.
1) $sin x * tan x + cos x$
$sin x * (sin x ) / (cos x) + cos x$
$(sin x ^ 2) / (cos x) + cos x = Answer: sin x ^ 2$
2) $(1)/ (1 + cos x) + (1 ) / (1 - cos x) =$
$(1 - cos x) / (1 + cos x) + (1 + cos x) / (1 - cos x =
Answer: (2)/(2) = 1$
)
For this problem, does it matter if the right side's answer is the same value on the left side but flipped?
3) $sin t * tan t = 1 - cos ^ 2 t / cos t$
Doing the right side:
$sin ^ 2 t/ cos t$
$(sin t) / (cos t) * sin t$
$Answer: tan t * sec t$
• Mar 24th 2011, 02:06 PM
dwsmith
Quote:
Originally Posted by MajorJohnson
Here are some more problems that need checking from my other thread I made.
1) $sin x * tan x + cos x$
$sin x * (sin x ) / (cos x) + cos x$
$(sin x ^ 2) / (cos x) + cos x = Answer: sin x ^ 2$
$\displaystyle \frac{\sin^2(x)}{\cos(x)}+\cos(x)\cdot\frac{cos(x) }{\cos(x)}=\frac{\sin^2(x)+\cos^2(x)}{\cos(x)}=\cd ots$
2 is wrong. Try obtaining the same common denominator.
3 is correct.
• Mar 24th 2011, 02:07 PM
pickslides
What are you trying to do with these?
for the first one, I get,
$\displaystyle \sin x \tan x +\cos x$
$\displaystyle \sin x \frac{\sin x }{\cos x} +\cos x$
$\displaystyle \frac{\sin^2 x }{\cos x} +\cos x$
$\displaystyle \frac{\sin^2 x }{\cos x} +\frac{\cos^2 x}{\cos x}$
$\displaystyle \frac{\sin^2 x+\cos^2 x}{\cos x}$
$\displaystyle \frac{1}{\cos x}$
$\displaystyle \sec x$
• Mar 26th 2011, 06:42 AM
MajorJohnson
My objective is to find the other side.
Quote:
Originally Posted by dwsmith
$\displaystyle \frac{\sin^2(x)}{\cos(x)}+\cos(x)\cdot\frac{cos(x) }{\cos(x)}=\frac{\sin^2(x)+\cos^2(x)}{\cos(x)}=\cd ots$
2 is wrong. Try obtaining the same common denominator.
3 is correct.
$2 / (1- cos x) (1 + cos x)$
I also want to know if my method for this problem is legal:
$sec^2 x * csc^2 x = sec ^ 2 x + csc ^ 2 x$
$(1 + tan ^ 2 x ) (1 + cot ^ 2 x)$
$1 + cot ^ 2 x + tan ^ 2 x + (tan ^ 2 x) (cos ^ 2 x)$
$(sin x / cos x) (sin x / cos x) * (cos x / sin x ) (cos x / sin x)$
Cross multipling ( sin x / cos x) * (cos x / sin x) would give me one.
$1 + cot ^ 2 x + tan ^ 2 x + 1$
Here i canceled the ones out and got ' cot ^ 2 x + tan ^ 2 x '.
$csc ^ 2 x - 1 + sec ^ 2 - 1$
Canceling the ones again would give me csc^ 2 x + sec^2x.
• Mar 26th 2011, 08:22 AM
Quacky
Edit: I misunderstood your approach, I think it works, other than the fact that you cancel the ones at the end. When you reach the stage $1+Cot^2(x)+Tan^2(x)+1$, you can just use identities to show that it is the same as $Sec^2(x)+Cosec^2(x)$. I'll leave my method here, but after rereading, I think your proof is otherwise fine.
$Sec^2(x)Cosec^2(x)=Sec^2(x)+Cosec^2(x)$
Generally, you'd start with the side which is easiest to simplify. Here, that's the right hand side. Start by putting everything over a common denominator:
$\dfrac{1}{Cos^2(x)}+\dfrac{1}{Sin^2(x)}$
$=\dfrac{Sin^2(x)}{Sin^2(x)Cos^2(x)}+\dfrac{Cos^2(x )}{Sin^2(x)Cos^2(x)}$
$=\dfrac{Sin^2(x)+Cos^2(x)}{Sin^2(x)Cos^2(x)}$
$=\dfrac{1}{Sin^2(x)Cos^2(x)}$
$=Sec^2(x)Cosec^2(x)$ which is the left hand side.
You also haven't finished 2) - you can simplify the fraction further using the identity $Sin^2(x)+Cos^2(x)=1$
• Mar 27th 2011, 09:04 AM
MajorJohnson
Thank you. In solving these problems in general, can there be more than one possible answer?
• Mar 27th 2011, 01:07 PM
Quacky
Quote:
Originally Posted by MajorJohnson
Thank you. In solving these problems in general, can there be more than one possible answer?
Depends on the problem. If you've been asked to simplify, usually not, but if it's a proof, there will often be several approaches, as we've shown here.
• Mar 27th 2011, 01:20 PM
pickslides
Quote:
Originally Posted by MajorJohnson
Thank you. In solving these problems in general, can there be more than one possible answer?
In a lot of trig identity problems you can complete the proof using a number of different methods as the common identitiies themselves are all linked together.
• Mar 27th 2011, 04:16 PM
MajorJohnson
Okay, but even if i have to find the other side of an identity, such as with some of the other ones i had posted in my other thread, there can only be one answer right?
• Mar 31st 2011, 02:59 PM
MajorJohnson
How would i go about solving this problem?
$cos (PI/2 - x) sec x$
So far I have:
cosPI/2 - cos x 1/ cos x
• Mar 31st 2011, 03:14 PM
topsquark
Quote:
Originally Posted by MajorJohnson
How would i go about solving this problem?
$cos (PI/2 - x) sec x$
So far I have:
cosPI/2 - cos x 1/ cos x
Hint: $cos(a - b) = cos(a) \cdot cos(b) + sin(a) \cdot sin(b)$
-Dan
• Apr 1st 2011, 03:52 AM
MajorJohnson
Okay that helped me a little bit, I also have another question unrelated to this topic but with dealing with Solving Right triangles. When going about solving the outer sides, how do I know whether or not I should take the given outside side and times it by whatever degree given inside the triangle, as opposed to dividing the outside side by the inside? (Example $3 * sin (45.3), 3 \div sin(45.3)$
• Apr 1st 2011, 04:20 AM
masters
Quote:
Originally Posted by MajorJohnson
Okay that helped me a little bit, I also have another question unrelated to this topic but with dealing with Solving Right triangles. When going about solving the outer sides, how do I know whether or not I should take the given outside side and times it by whatever degree given inside the triangle, as opposed to dividing the outside side by the inside? (Example $3 * sin (45.3), 3 \div sin(45.3)$
Hi MajorJohnson,
Your question is a little vague (at least to me, it is). Might I suggest you start a new thread with your new problem and be a little more specific.
This thread is getting a bit too long. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 42, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.854789137840271, "perplexity": 897.2829177786107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721027.15/warc/CC-MAIN-20161020183841-00144-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showthread.php?p=2834978 | # Random walk in spherical coordinates
by arandall
Tags: coordinates, random walk, spherical
P: 1 Hi, I'm modeling receptors moving along a cell surface that interact with proteins inside of a cell. I figured it would be easier to model the receptors in spherical coordinates, however I'm unsure of how to model a random walk. In cartesian coordinates, I basically model a step as: x = x + sqrt(6*D*timeStep)*randn y = y + sqrt(6*D*timeStep)*randn z = z + sqrt(6*D*timeStep)*randn Where D is my diffusion constant. How can I do this just using theta and phi? Modeling random walk in spherical coordinates will be really nice, because I can fix r such that the receptors can't leave the membrane of the cell, and just focus on how it moves in 2D with respect to the membrane.
P: 1,262 Because the receptors are so much smaller than the size of the cell, it should be fine if you treat theta and phi just like x-y; i.e. pretend its a 2D random walk in Cartesian coordinates.
Sci Advisor P: 5,937 To choose points on the surface of a sphere uniformly, the two angles should be chosen as follows (I'll call them latitude and longitude): Longitude (θ) - θ uniform between 0 and 2π. Latitude (φ) - sinφ uniform between -1 and 1.
Related Discussions Advanced Physics Homework 2 Advanced Physics Homework 3 Calculus & Beyond Homework 7 Calculus & Beyond Homework 0 Calculus & Beyond Homework 0 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8086036443710327, "perplexity": 764.6433533642862}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609538824.34/warc/CC-MAIN-20140416005218-00075-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.aanda.org/articles/aa/full_html/2017/04/aa29379-16/aa29379-16.html | Free Access
Issue A&A Volume 600, April 2017 A93 27 Interstellar and circumstellar matter https://doi.org/10.1051/0004-6361/201629379 05 April 2017
## 1. Introduction
Massive stars (M > 8 M) affect their surrounding medium due to the action of both their ionizing photons and stellar winds. They form ionized (H ii ) regions that expand, bordered by a shell of swept-up neutral material (Dyson & Williams 1997). Star formation is observed at the edges of Galactic and extragalactic H ii regions (Bernard et al. 2016). Young stars form there either spontaneously or through various mechanisms linked to the expansion of the ionized region (Deharveng et al. 2010).
Star formation observed at the edges of H ii regions has been studied in detail during the past ten years. With the GLIMPSE (Benjamin et al. 2003) and MIPSGAL (Carey et al. 2009) surveys, the Spitzer satellite has revealed that we live in a bubbling galactic disk where thousands of H ii regions have a clear impact on their environment. Anderson et al. (2011) have shown that half of all H ii regions have a bubble morphology. Studies of triggering have focused on bubble H ii regions. Deharveng et al. (2010) used Spitzer GLIMPSE and MIPSGAL data combined with ATLASGAL (Schuller et al. 2009) data on 102 bubbles. They showed that star formation observed at the edges of H ii regions is an important phenomenon in our Galaxy. Up to 25% of the ionized regions show high-mass star formation triggered on their edges. This result has been confirmed by Thompson et al. (2012) and Kendrew et al. (2012, 2016) who found an overdensity of young stellar objects (YSOs), including massive objects, around Spitzer and ATLASGAL bubbles. Simpson et al. (2012) have listed 5106 bubbles using these GLIMPSE and MIPSGAL surveys. Many studies of individual H ii regions, including numerical simulations, confirm that H ii regions impact on their surrounding, enhancing significantly the star formation there (Minier et al. 2013; Samal et al. 2014; Liu et al. 2015; Ladeyschikov et al. 2015). This impact is also observed at the waist of bipolar H ii regions as recently discovered by Deharveng et al. (2015).
However Dale et al. (2015) assessed the relevance of standard observational criteria used to decide whether the star-formation process is of spontaneous or triggered origin at the edges of H ii region. By comparing the observational criteria used to their own new numerical results they concluded that, when interpreting observations of star formation in the vicinity of feedback-driven structures in terms of triggering, one should exercise caution.
While the large and rapidly increasing bulk of knowledge tends to offer empirical evidence in support of some impact of H ii regions on the local star formation, there are still many unanswered questions on the possible influence of these regions on star formation near their edges. One way to firmly establish the causal link existing between the ionized region and the star-formation process taking place on its surrounding could be to measure a clear difference between the age of the ionizing stars, located in the central H ii region and the ones formed at its edges (Martins et al. 2010; Bik et al. 2010). However, the determination of stellar ages is challenging (Martins et al. 2010).
We are left in a situation where we observe an overdensity of young stars at the edges of these H ii regions. These young stars are highly efficient (up to 25%) at forming massive stars. (Bik et al. 2010; Ellerbroek et al. 2013; Cappa et al. 2014; Tapia et al. 2014). But we do not know how the material is assembled (uniformly distributed then collected versus pre-condensed in an inhomogeneous medium) and what are the mechanisms that control the formation of stars in these regions. For pre-existing clumps, star formation could occur spontaneously before encountering the ionization front or the ionizing radiation leaking from the H ii region. Dedicated observations can help in answering these questions. High resolution molecular spectroscopy reveals the distribution and velocity field of the material that surrounds H ii regions (Anderson et al. 2015; Liu et al. 2015). The spatial distribution, properties and evolutionary stage of YSOs are key points to address the triggering issue. We need to obtain an overview of all stages of star formation in a given region and access the distribution of the surrounding material on all spatial scales to discuss the history of star formation. The large scale distribution should help in understanding the initial distribution of the material (uniform versus clumpy, filamentary). A better knowledge of the distribution and properties (density, temperature) of the material that surrounds H ii regions could also help in better understanding how the material is assembled and how star formation occurs around ionized regions.
The Herschel satellite offers a unique opportunity to study star formation around Galactic H ii regions and helps in answering some of the pending questions. Thanks to its sensitivity and its large wavelength coverage in the far-infrared, Herschel is perfectly suited to study the earliest phases of star formation. The six measured photometric points (70, 100, 160, 250, 350, 500 μm) really help in constraining the young sources’ properties (temperature, envelope mass, luminosity). Moreover, Herschel’s wavelength range covers the peak of the YSOs’ spectral energy distribution (SED) also helping to characterize the young source’ evolutionary stage. Combined with existing infrared and molecular data, Herschel observations allow us to obtain a global view of the star-formation history (Nguyen et al. 2015).
Here we present the results obtained for young compact sources observed towards the bubble H ii region RCW 120. Using Herschel photometric PACS and SPIRE data, we re-examine this region to better determine the nature and evolutionary stage of the YSOs observed there. We aim to discuss the region’s star-formation history there using sources’ evolutionary stage. Section 2 presents the current knowledge on RCW 120. The Herschel observations are described in Sect. 3. The data analysis and sources’ extraction are presented in Sect. 4. The results are presented in Sect. 5 and discussed in Sect. 6. The main results and conclusions are given in Sect. 7.
## 2. The RCW 120 region
RCW 120 (Rodgers et al. 1960) is an egg-shaped Galactic H ii region of 3.8 pc diameter, located 0.5° above the Galactic plane. Due to its simple morphology and isolation, this region has been studied in detail during the past ten years. The main results are summarized below:
The region is ionized by an O8V star, CD− 38°11636 (Zavagno et al. 2007, hereafter ZAV07; Martins et al. 2010). An emission arc is observed at 24 μm below the star (Deharveng et al. 2009, hereafter DEH09; Martins et al. 2010, see their Fig. 3) and is interpreted as representing the upstream boundary between the wind bubble and the photoionized interstellar medium (Mackey et al. 2015).
The photometric distance of RCW 120 was computed by Russeil (2003) using UBV and Hβ photometry. The uncertainty is estimated to be 0.6 kpc and comes from the uncertainty in the spectral type estimate (around 0.3 mag).
RCW 120 and its surrounding layer have been observed in the dust continuum at 870 μm (DEH09) and 1.3 mm (ZAV07) and in CO molecular lines (Anderson et al. 2015; Torii et al. 2015). These observations show that RCW 120 is surrounded by a dense shell of gas and dust.
Torii et al. (2015) observed two molecular clouds towards RCW 120 and suggest that some collision between the clouds triggered the formation of the ionizing O star of RCW 120 in a short timescale of 0.2–0.4 Myr. An age of 0.4 Myr is also obtained by Mackey et al. (2015). Simulations from Tremblin et al. (2014a) lead to a similar age for the ionizing star of RCW 120.
Anderson et al. (2015) found no evidence for expansion of the molecular material associated with RCW 120 and therefore can make no claim about its geometry (2D or 3D). Dust emission simulations suggest that the H ii region RCW 120 is not spherical, but instead cylindrical, and that we observe the object along the axis of this cylinder (Pavlyuchenkov et al. 2013).
Using 1.3 mm continuum emission ZAV07 found eight condensations (five located at the edges of the ionized region, see their Fig. 4) and studied the young stellar content of these condensations, pointing out the possible importance of long-distance influence of the ionized region on its surrounding. This study has been completed by DEH09 who characterized the evolutionary stage by adding 24 μm data from MIPSGAL and confirmed the importance of long-distance interaction between the H ii region and its surroundings. Many YSOs, including Class I and Class II sources are observed at the edges of the ionized region. A noticeable massive Class 0 candidate is detected towards the highest density condensation (condensation 1), later confirmed with Herschel observations (Zavagno et al. 2010). A spectrophotometric study of the YSOs in the near-infrared confirm that these YSOs are associated with the RCW 120 region because they have the same velocity than that of the ionized gas (Martins et al. 2010).
DEH09 observed a series of eleven young sources aligned parallel to the ionization front towards the most massive condensation at 24 μm equally spaced by 0.1 pc and is thought to be the result of Jeans gravitational instabilities.
Tremblin et al. (2014b) studied the probability density function (PDF) of a series of Galactic H ii regions, including RCW 120 (see their Figs. 8 and 9). They found evidence for compression, and the value of the exponent derived to fit the PDF towards condensation one may indicate the role of ionization compression in the formation of this condensation and its collapse to form stars. According to numerical simulations lead by Minier et al. (2013), if the condensation had gravitationally collapsed prior to the passage of the ionization front, the condensation would be already sufficiently dense to resist the ionization front expansion. It would, for example, trigger the formation of a pillar rather than a condensation remaining in the shell.
Walch et al. (2015) performed three dimensional smoothed particle hydrodynamics (SPH) simulations of H ii regions expanding into fractal molecular clouds and then used RADMC-3D to compute the synthetic dust continuum emission at 870 μm from their simulations, applied to RCW 120. They found a hybrid form of triggering which combines elements of collect and collapse (C&C) mechanism (Elmegreen & Lada 1977) and radiation driven implosion (RDI; Kessel-Deynet & Burkert 2003).
Figure 1 presents a three-color image of the RCW 120 region as seen by Herschel. The 70 μm emission (blue part) underlines the emission of the warm dust while the 250 μm emission (red part) underlines the emission from the colder dust located in the dense material that surrounds the ionized region and that interacts with the ionizing radiation.
Fig. 1RCW 120: Herschel-PACS 70 μm (blue), 160 μm (green) and Herschel-SPIRE 250 μm (red). The field size is 21.8′ × 24.5′. North is up, east is left. Open with DEXTER
## 3. Observations and data reduction
### 3.1. Herschel observations
RCW 120 was observed with the PACS and SPIRE photometers. Details of these observations (map size, observing time, observational identification (ObsID), observational date (Obs.) operational day (OD), map center) are given in Table 1. The PACS photometer was used to make simultaneous photometric observations in two photometric bands as part of the HOBYS key program (Motte et al. 2010). Two cross-scan maps were done at angle 45° and 135° with a scanning speed of 20/s. This observing mode is described in Sect. 5.2 of the PACS Observers’ Manual1. The beam FWHM varies between 59 at 70 μm , 60 at 100 μm and 114 at 160 μm. The total observing time is 2.6 h.
RCW 120 was observed with the SPIRE photometer as part of the Evolution of Interstellar Dust key program for the Herschel Science Demonstration Phase. The SPIRE photometer was used to make simultaneous photometric observations in the three photometer bands (250, 350 and 500 μm). The map is made by scanning the telescope at a given scan speed of 30/s along lines. Cross-linked scanning is achieved by scanning at 42° (Scan A angle) and then at − 42° (Scan B angle). This ensures that the effect of 1/f noise on the map can be minimized and also leads to improved map coverage. This observing mode is described in details in the last version of the SPIRE Observers’ Manual2 in Sect. 3.1.2. One map at each scanning angle was obtained. The beam FWHM varies between 182 at 250 μm , 252 at 350 μm and 366 at 500 μm. The total observing time is 0.34 h.
Table 1
Summary of Herschel observational parameters.
The PACS maps were produced using the HIPE Level 1 data and then version 21 of the Scanamorphos software package which performs baseline and drift removal before regridding (Roussel 2012). The SPIRE images were reduced using modified pipeline scripts of Version 10 of HIPE3, the Herschel Interactive Processing Environment. Each map direction (nominal and orthogonal) was first reduced individually to Level 1 data, correcting for effects such as temperature drifts and jumps, glitches and cooler burps. The individual maps were then combined to create one map (Level 2 data). Map reconstruction was done using the SPIRE default “naive” mapmaking algorithm at the same time as a destriper module (including a median correction and in bright source mode). The default gridding of 6, 10, 14 for the SPIRE wavelengths 250, 350, 500 μm was chosen. The fits output files for each SPIRE wavelength are in units of Jy/beam. For an absolute calibration of the SPIRE maps, the zeroPointCorrection task calculates the absolute offset for a SPIRE map, based on cross-calibration with Planck HFI-545 and HFI-857 maps, color-correcting HFI to SPIRE wavebands assuming a gray-body function with fixed spectral index. The offsets determined in this way correspond well to the ones provided by J.-P. Bernard (priv. comm.).
### 3.2. Complementary data
We complement the Herschel data with data from the Two Micron All-Sky Survey (2MASS) at 1.25 μm (J), 1.65 μm (H) and 2.17 μm (Ks) with a resolution of 2 (Skrutskie et al. 2006), from the Spitzer4 GLIMPSE and MIPSGAL surveys of the Galactic Plane at 3.6 μm, 4.5 μm, 5.8 μm and 8 μm and 24 μm with a resolution of 17, 17, 19, 2 and 6 (Benjamin et al. 2003; Carey et al. 2009).
## 4. Data analysis
### 4.1. Compact sources’ extraction method
Herschel compact sources were extracted using the multi-wavelength, multi-scales getsources algorithm5 (version 1.140127; Men’shchikov et al. 2012; Men’shchikov 2013). The working method of getsources can be roughly decomposed into two steps: the detection and the measurement. While the latter is performed on all maps inserted into the algorithm, the detection can be made from a selected sample of maps depending on the aim of the study. In order to improve this step, an Herschel high-resolution density map (Hill et al. 2012; Palmeirim et al. 2013) was created (see Sect. 4.5) and added to better constrain the detection of compact sources. Moreover, some original maps were modified in order to enhance the contrast of the cooler and hence the densest regions since heated structures could be detected and misleading the final sample of sources. For this purpose and to provide valuable guidance to the detection algorithm, we use the 160 μm PACS and 250 μm SPIRE maps as they represent a good compromise between resolution and non-contamination by very small grains (VSG). The photometric offsets derived using IRAS-Planck model (Bernard et al. 2010) were added and the 160 μm map was convolved to the resolution of the 250 μm SPIRE observations (25.̋2). We assumed a modified blackbody (hereafter, MBB) model with a spectral index of 2. This value is higher than the reference for the galaxy (~1.6, Planck Collaboration Int. XLVIII 2016) but for dense regions, in the inner regions of the galactic plane for instance, β tends to increase and hence, a value of 2 should be more appropriate for compact regions (Paradis et al. 2012). Non-linear fitting of the SEDs was performed using the Levenberg-Marquardt’s algorithm (Markwardt 2009). From the SED, a color temperature can be found for each pixel by using the ratio of the two maps (1)where is the 160 μm map convolved at the 250 μm resolution.
A weight-map is then created as the ratio between the map giving the MBB flux corresponding to the color temperature and a fiducial temperature of 20 K (median temperature). Multiplying the native 160 μm PACS map by the weight-map give the 160 μm corrected map where colder regions are enhance compared to warmer regions. The 250 μm corrected map is created in the same way and both are used in replacement of the native 160 μm PACS and 250 μm SPIRE maps for the detection step.
To summarize, for extraction of the sources, we used the original 70 μm, 100 μm, 350 μm, 500 μm maps and improved the detection by including the high-resolution density map and the 160 μm and 250 μm corrected maps.
### 4.2. Pre-selection
The final getsources catalog contains many useful informations about the detected sources. In the following subsections, we will keep only a small part of them: J2000 coordinates, detection significance at each wavelength, flux at peak and integrated flux with their corresponding errors and source ellipse parameters (major axis, minor axis, position angle).
Before doing the analysis, the sources present in the catalog have to be filtered to select the well-defined ones. The selection criteria defined by the HOBYS consortium are listed below (see also Tigé et al. 2017):
Each source must have a deconvolved size (see Eq. (2)) smaller than 0.1 pc at the reference wavelength (see below) and three reliable fluxes (including the reference wavelength). A flux is considered as reliable if the detection is reliable (the detection significance is higher than 7, see Men’shchikov et al. 2012), the signal to noise of the peak and integrated flux is higher than 2 and the elongation (defined as the ratio of the major and minor axis of the ellipse footprint) lower than 2 in order to limit the sample to circular compact cores.
From these criteria, sources extracted by getsources are expected to be dense and cold. Therefore, we consider as a good assumption that thermal emission from the cores is optically thin and not contaminated by VSGs for λ100 μm. The deconvolved size at wavelength λ, , is computed as (2)where / stand for the major/minor convolved size estimate of the source at wavelength λ (given in the getsources catalog) and HPBWλ is the half-power beam width at wavelength λ. The reference wavelength was chosen to be 160 μm as a compromise between the resolution (11.̋4) and the tracer of optically thin dust emission. This trade-off allows for both the correct identification of the peak of the SED and a good scaling of the flux (Motte et al. 2010; Nguyen Luong et al. 2011). Nevertheless, in some marginal cases, the 160 μm emission may be contaminated by small grains heated in the photo-dissociation region (PDR) leading to a deconvolved size larger than the one measured at 250 μm. In such cases, the 250 μm is taken as the reference (resolution 18.̋2).
Detections complying with the above-mentionned criteria were kept for the analysis. However, the 70 μm data were systematically excluded from the SED fitting to avoid contamination from VSG’s emission even though the criteria were satisfied.
Among the 359 detections of the getsources algorithm, 80 were kept at the end of the selection (put in the pre-selected sample). Rejected sources appear to be false detections, mainly filament pieces or sources with not enough flux measurements to fit the SED. Rejected sources were visually inspected and those which look like compact sources with at least one reliable wavelength (70 μm included) were kept in a tentative sample (80 sources). The physical properties of the tentative sources were derived throughout an indirect method (see Sect. 5.2.4) since a SED fitting couldn’t be done.
### 4.3. Spectral energy distribution
Before fitting the SED for each compact source, the fluxes must be scaled since we want them to be measured within the same aperture. A full treatment of this scaling can be found in Motte et al. (2010), Nguyen Luong et al. (2011) where the relation between flux and source’s angular size is taken to be same as for protostellar cores. This aperture scaling is based on the assumptions that the source is optically thin for λ> 100μm, M(r) ∝ r and the gradient of the temperature is weak within the region (Elia et al. 2014). The scaling was done when the size at the reference wavelength and the wavelength to be scaled could be deconvolved. Following their procedure, we applied scaling factors to fluxes according to the formula (3)where represents the rescaled flux associated with scaling factor ζλ and Sλ is the original flux.
The model to be fitted is a MBB (4) using the Hildebrand relation (Hildebrand 1983) (4)with a gas-to-dust ratio R = 100, D the distance of RCW 120, 1.3 kpc, and C is introduced as a constant of the fit. The HOBYS consortium decided to use the dust opacity law κν = κ300 μm(ν/ν0)2 with κ300 μm = 10 cm2 g-1, ν0 = 1000 GHz (Beckwith et al. 1990; Motte et al. 2010). As explained before, the spectral index has been fixed to two, reducing the model space of the fitting parameters to the C-T plane. The initial errors used to weight the data have been set to the quadratic sum of the getsources and the calibration errors (3% at 100 μm, 5% at 160 μm and 7% for SPIRE bands).
Table 2
Minimum, maximum and median values for the color correction factors at Herschel wavelengths for the final sample of sources.
During the acquisition of the sources with the PACS and SPIRE instruments of Herschel, the spectrum is assumed to be flat across the bands (νSν = const.) which is not true because we expect the sources to follow a MBB model. To correct for this assumption, we apply color correction factors given in the PACS and SPIRE observer’s manual. The fitting algorithm reaches convergence when the absolute difference between two subsequent temperatures (obtained at two consecutive steps) is fainter than 0.1 K. Since the spectral index was fixed to two, these factors depends on the temperature for PACS and are constant for SPIRE. Table 2 gives the minimum, maximum and median value for the color correction factors at Herschel wavelengths. Color corrections are high for short wavelengths and low temperature. For 70 μm and 100 μm, they go up to 54% and 21% and corresponds to a source with T = 11.2 K. However, considering the median value for all wavelengths, the color correction factors are low and do not change drastically the sources’ fluxes of the final sample.
The final temperature is derived directly as one of the parameter of the MBB model and assuming optically thin emission for the dust, the envelope mass is derived as (5)The uncertainties are derived from the fitting errors and are 3% and 20% in average for the dust temperature and the envelope mass, respectively. Obviously, these error values on the physical parameters do not take into account the dependence of β with the wavelength (from 1 to 2) and the uncertainty on the opacity factor which is at least a factor of two due to the unknown properties of dust grains (Deharveng et al. 2012, see their Sect. 4.1).
Each detected YSO’s bolometric luminosity was computed by integrating the corresponding SED curve. For sources having infrared (IR) counterparts (from the 2MASS, GLIMPSE and MIPSGAL surveys) within a radius of 4, the SED was bipartitioned and partial integrations are made over it. Below 70 μm, a so-called IR luminosity is obtained using a trapezoidal integration scheme (numerical integration done by connecting the data points with straight lines) from the first IR counterpart found in the catalogs to the 70 μm flux. From 70 μm onwards a so-called “Herschel luminosity” is obtained by integrating over the Herschel SED. The bolometric luminosity is obtained by adding these two values. The SED fitting algorithm returned the error on Herschel luminosity while that affecting the IR one is obtained by computing the IR luminosity with the fluxes plus the uncertainties (higher-limit) and with the fluxes minus the uncertainties (lower-limit). On average this resulted in an overall uncertainty of 30% on bolometric luminosity. The average volume density was computed by assuming a spherically symetric core with a diameter equal to its deconvolved size at the reference wavelength(6)After the SED fitting, a second stage selection was applied to obtain the final sample of sources discussed in this paper (described below).
### 4.4. Final selection
At the end of the SED fitting procedure, the sample of sources obeying our first-stage selection scheme (pre-selected sample) was visually inspected at each Herschel wavelength, to ensure the detection of truly compact sources. The requirements for a source to pass successfully this second-stage selection scheme are two-fold: (1) The source had to be clearly seen by eye on one of the Herschel images; and (2) The source’s SED had to be well-constrained.
The first condition was checked by two different people to avoid subjective detections. The second condition allows us to eliminate dubious SED, for example SED with unconstrained peak or SED with increasing flux mostly at SPIRE wavelengths. From this second stage selection, 35 sources were kept for the study (final sample), seven sources having unconstrained SED were added to the tentative sample and 38 sources looking like small clumps or filamentary pieces were rejected.
To summarize the two-stage selection: from the 359 detections, 35 sources are included in the final sample and 87 sources are included in a tentative sample. The latter is composed of sources possessing at least one reliable wavelength (including 70 μm) clearly seen (by eye) in the Herschel images but not pre-selected due to the lack of flux measurements or pre-selected sources whose SED is unconstrained. These detections are thought to be real sources and kept in order to derive their physical properties with an indirect method since their SED cannot be used.
The second-stage selection is definitely highly non-conservative but ensures the reliability of the sources to be investigated. As discussed before, part of the remaining detections could be real sources under filaments with well-constrained SED which are eliminated in order to have a reliable sample of sources. Finally, we are left with two samples: the final one which will be discussed in the paper and the tentative one whose physical parameters will be derived with an indirect method. We assume that the selected sources are associated with RCW 120, that is, they are located at the same distance. This assumption is supported by the study by Martins et al. (2010) who showed using high resolution near-IR spectro-photometric observations that the YSOs with a IR counterpart observed towards RCW 120 are at the same velocity as that of the ionized gas. We further discuss this point in Sect. 5.2.2.
### 4.5. Dust temperature and column density maps
#### 4.5.1. Method
Following the procedure of Hill et al. (2011, 2012), a map of dust temperature at 366 can be obtained by fitting the flux at each pixel, using the MBB model (7)where Fν is the brightness, μ the mean molecular weight (2.8), mH the proton mass and N(H2), the column density. The temperature and the gas surface density Σ500 μm (the 500 μm subscript stands for the corresponding resolution) are the fitting parameters. Another direct byproduct of the fitting algorithm is a map of the H2 column density at the same low-resolution assuming the dust opacity law of Beckwith et al. (1990, see Sect. 4.3. Since most of the observed regions through the HOBYS project are not observed at 100 μm, the method considers only wavelengths higher than 160 μm for the SED fitting even if the data were available. This choice made the comparison of temperature and column density maps obtained for different regions easier.
Following the procedure described by Palmeirim et al. (2013) based on a multi-scale decomposition method, a high-resolution column-density map at 18.̋2 can be computed. The gas surface density smoothed at 250 μm resolution (Σ250 μm) can be written as a sum of gas surface density smoothed at 350 μm (Σ350 μm) and 500 μm resolution (8)The second term in parentheses of Eq. (8) represents the spatial scale structure of the region seen at 350 μm without the largest structure corresponding to the 500 μm observations. An estimate of Σ500 μm can be obtained by considering that these data are approximately equal to Σ350 μmG350−500 where G350−500 is the full width at half maximum (FWHM) of the point spread function (PSF) needed to convolve the 350 μm map to the 500 μm resolution (). The gas surface density Σ350 μm is obtained in the same way than Σ500 μm but excluding the 500 μm data from the SED fitting.
The third term of Eq. (8) represents the structure seen at 250 μm without the largest scale structure seen at 350 μm. As before, Σ350 μm can be written as Σ250 μmG250−350 and 17.̋4 is the FWHM of the PSF needed. The gas surface density Σ250 μm is obtained using the ratio of the 160 μm and 250 μm maps as explained in Sect. 4.1. Finally, Eq. (8) can be rewritten as (9)Hence, the resulting high-resolution density map can be seen as a composite map representing the multi-scale structure of RCW 120 from 250 μm to 500 μm resolution.
#### 4.5.2. Comparison with Anderson et al. (2012) maps
Fig. 2a) SED fitting for the pixels giving the highest temperature. The continuous curve represents the fit made with all Herschel fluxes and dashed curved is the fit obtained by the method of Hill et al. (2012). b) Same for the pixel giving the lowest temperature. No background is subtracted in both cases. Open with DEXTER
Fig. 3a) Ratio of temperature between the maps obtained in this paper (no 70 μm and 100 μm data included and no background subtraction) over the ones obtained by Anderson et al. (2012, see text). The yellow contours correspond to 870 μm emission at 0.1 Jy/beam. b) Same but with the column density maps. Open with DEXTER
Anderson et al. (2012) constructed temperature and column density maps for a sample of H ii regions (Sh 104, W5-E, Sh 241, RCW 71, RCW 79, RCW 82, G332.5-0.1 and RCW 120). Two differences exist between the method he used and the one we used (also described in Hill et al. 2012). In the method used by Anderson et al. (2012), the SED is fitted with all the data available (from 70 μm to 500 μm) and a flat background is subtracted in each Herschel map (see also Battersby et al. 2011). In warm regions (the ionized zone typically) where 70 μm and 100 μm fluxes are high (around 3 × 103 MJy sr-1), the inclusion of these data in the fit induces a shift of the SED towards the high-frequency region and increases the temperature. The cold regions are less affected because the 70 μm and 100 μm fluxes are lower (around 3 × 102 MJy sr-1). Spectral energy distributions representing both case for pixels giving a high and low temperature are shown in Fig. 2. In hot regions, the difference in temperature reaches 4 K (18%) while it is only 2 K (14%) for cold regions.
To make a comparison between the temperature maps obtained using each method, we resample them at 14 pix-1 to the same center and compute their ratio to see how the different methods lead to different temperature in specific regions (see Fig. 3a). The structure seen on these images clearly reproduces the egg-shaped of RCW 120 and shows that the differences occur in specific regions. We define an area in the warmest (around the ionizing star) and coldest region (defined as condensation 5 hereafter) and compute the median and the standard deviation for them and the whole map. Results are shown in Table 3 where HR and CR stand for hot and cold regions. From the first and third line, we see, as expected, that the temperature found is higher when the 70 μm, 100 μm and background subtraction are included particulary for the warmest region where the change is around 6 K. The colder region do not present significant difference (0.2 K).
To estimate the change in temperature induced by the background subtraction, we created another temperature map using the method of Anderson et al. (2012, including the 70 μm and 100 μ but without removing any background. The range of temperature for this method is listed in the second line of Table 3. We note, as expected, that hot regions are more affected by the inclusion of high-frequency maps (2 K) and by the background subtraction (5 K). The error on the fit for the temperature map has a mean value of 0.45 K hence the temperature for cold regions remains roughly the same.
The comparison between the two column density maps is less straighforward since Anderson et al. (2012) used only the 350 μm to obtain this map while our is the byproduct of the SED fitting. In Fig. 3b, the ratio of the column density map shows that warm regions are more affected than colder ones. Due to the anticorrelation between column density and temperature, we expect warm regions to be more affected with the inclusion of the 70 μm and the 100 μm fluxes. Moreover, since the flux is linear with the column density to first order, we expect the background to be roughly equal for all the regions. Table 4 presents the values of column density for the whole map, the densest region (condensation 1 defined hereafter) and an area in the north-west of RCW 120 where the density is low (empty region) using the three different methods. Trends can be seen: as we include high frequency maps and background subtraction, the median column density decreases. The 70 μm and the 100 μm fluxes lead to a difference of 4 × 1021 cm-2 for the warm region and does not change significantly the column density for cold ones. Removing the background causes a loss in column density of 1 × 1021 cm-2. The method described in Hill et al. (2012) was the choice of the HOBYS consortium for the construction of temperature and column density maps and consequently, no background subtraction is made. This rule will allow an unbiased comparison between the different regions observed in the HOBYS project.
Table 3
Range for the temperature map constructed following the method described in Hill et al. (2012, first line), with all wavelengths and no background subtraction (second line) and following Anderson et al. (2012, third line) for the whole map (first and fourth columns), the hottest region (second and fifth columns) and coldest region (third and sixth columns).
Table 4
Range for the column density map constructed following the method described in Hill et al. (2012, first line), with all wavelengths and no background subtraction (second line) and following Anderson et al. (2012, third line) for the whole map (first and fourth columns), the densest region (second and fifth columns) and empty region (third and sixth columns).
## 5. Results
### 5.1. Dust temperature and column density maps
Fig. 4Temperature map of RCW 120 at 36.̋6 resolution with 870 μm emission from LABOCA (in yellow countours) and the final sample of 35 compact sources (white dots) discussed in this paper. Condensations observed at 870 μm are identified following the labelling in DEH09. The temperature ranges from 15 K (dark) to 24 K (white). Warm regions are observed towards the ionized zone. Colder regions are located outside the ionized region and are distributed in cores, filaments and condensations. Open with DEXTER
Fig. 5a) On logarithmic scale, H2 column density map of RCW 120 at 366 resolution with 870 μm emission from LABOCA (in yellow countours), the final sample of sources (black dots) and the three prestellar clumps (red dots). Condensations observed at 870 μm are identified following the labelling in DEH09. The density values range from 7 × 1021 cm-2 to 4 × 1023 cm-2. b) High resolution H2 column density map of RCW 120 at 182 resolution (in red) and Hα emission (in blue) from the SuperCOSMOS Hα Survey. The column density values range from 7 × 1021 cm-2 to 9.4 × 1023 cm-2. Open with DEXTER
Figure 4 presents the temperature map obtained for RCW 120 with labelled condensations of DEH09 defined by yellow contours. The temperature ranges from 15 K to 24 K. Temperatures between 19 K and 24 K are observed towards the ionized region, the highest temperature being observed to the south. A colder medium with a temperature around 15–18 K surrounds the H ii region. This colder medium is highly structured, organized in clumps, filaments and condensations that correspond to the condensations defined in DEH09 where the sources are located. A remarkable feature is the sharp edge seen on the temperature map at the south-western border of the ionized region. This drop in temperature (from 21 K to 16 K) corresponds to the presence of the (sub)millimeter condensation 1.
Figure 5a presents the low resolution (36.̋6) column density map with the condensations of DEH09 and Fig. 5b the high-resolution column density map (182) together with Hα emission from the SuperCosmos Hα survey (Parker et al. 2005). The values range from 7 × 1021 cm-2 to 4 × 1023 cm-2 for the low-resolution map and goes up to 9 × 1023 cm-2 for the high-resolution one. We checked that the convolution of the high resolution column density map to 366 with the same grid agrees with the values found for the low resolution one. As expected, the ionized region with its egg-shaped corresponds to a drop in column density compared to the PDR and low column density filaments (N(H2) = 1.7 × 1022 cm-2) are observed within it. These are seen in absorption in the optical (see Fig. 1 in ZAV07) and show some compact structures that host sources (see Fig. 7 and Sect. 5.2). Around the ionized region, a highly structured material is distributed in filaments and clumps where the nine condensations already observed at 1.3 mm (ZAV07) and 870 μm (DEH09) are well seen. The leaking of the UV flux presented in DEH09 (see their Fig. 16) is also seen on Fig. 5b. It creates the extended elliptical structures observed on the southern part of the ionized region together with the structures observed on the north-eastern and north-western parts. Three pre-stellar clumps are seen on the temperature and density maps at (α,δ) = (25791, 3828), in absorption on the 70 and 100 μm images and in emission at 160 μm onward (see Fig. 5a). In general, the size and elongation of these clumps are too large to be part of the studied sample but their detection at SPIRE wavelengths (and at 870 μm, see Fig. 2 in DEH09) suggests that they are pre-stellar clumps. The contrast between the high and low density regions is equal to 60. The highest density is observed in condensation 1 located at the south-western edge of the ionized region. This condensation could have been formed due to compression from the ionization region (Tremblin et al. 2014b). Towards RCW 120, star formation is observed in column density region higher than 2.2 × 1022 cm-2.
### 5.2. Compact sources
#### 5.2.1. Compact sources’ spatial distribution
Figure 6 shows the 35 selected sources superimposed on a PACS 70 μm gradient-filtered image of RCW 120. This image was produced with a standard 3 × 3 bi-directional Sobel-Feldman convolution kernel applied to the original image. For each pixel, the derivatives along the horizontal and vertical directions are obtained and the final value for each pixel is computed as giving an approximate value of the gradient norm. This gradient-filtering cuts-off the diffuse emission and enhances the contrast of steep emission regions. In the following we define the PDR as the filamentary emission region revealed by this gradient-filtering and shown by the green dashed contour seen in Fig. 7.
Using the selection criteria described in Sect. 4 and a visual inspection as a final check, we end up with 35 sources that are discussed (i.e., sources for which the temperature, envelope mass and bolometric luminosity can be derived). 87 additional detected sources are also shown in Fig. 7 but have less than two reliable fluxes (up to three if the 70 μm is included) or have unconstrained SEDs, meaning that their properties cannot be derived using SED fitting. Their original fluxes (given by getsources without any aperture scaling or color-corrections) are given in Table A.1.
Fig. 6All 35 compact sources detected using getsources (and discussed in the text) superimposed on a 70 μm gradient image of RCW 120. The sources are color-coded depending on their location: red circles for sources observed towards the PDR, blue squares for sources outside (see text). Open with DEXTER
As seen on Fig. 6, 14 sources are located outside the PDR and 21 are inside.
Fig. 7All 87 sources detected by getsources but not part of the final sample due to the lack of reliable flux measurements, mainly at SPIRE wavelengths. Physical parameters of these sources are derived in a secondhand way explained and presented in Sect. 5.2.4. The PDR region is enclosed in the green countours (see text). Open with DEXTER
#### 5.2.2. Compact sources’ association with the region
Spectroscopic observations with SINFONI at the ESO-VLT showed the YSOs detected in the near IR towards RCW 120 have the same velocity as that of the ionized gas (8 km s-1) and are thus associated with RCW 120 (Martins et al. 2010). Even though most of the sources are thought to be part of RCW 120 because they are embedded in its filamentary region, sources located outside the PDR might not be associated with the region.
Studying the J = 0 → 1 transition of 12CO, 13CO, C18O and C17O with the ATNF Mopra 22 m radio telescope, Anderson et al. (2015) identified three emission peaks at 7 km s-1 (main temperature peak around the velocity of the ionized gas), 30 km s-1 and 60 km s-1. The J = 0 → 1 emission from the CO isotopologues (Anderson et al. 2015) integrated between 75 km s-1 and 50 km s-1, 35 km s-1 and 15 km s-1 and 15 km s-1 and +3 km s-1 are presented in Fig. 8 for 12CO, Fig. 9 for 13CO and Fig. 10 for C18O. We point out that, contrary to the other maps in this paper, these figures are given in galactic coordinates.
Condensation 6, the northern-part of condensation 5 and sources 55 and 150 (the western part, between condensation 6 and 7) are located outside the PDR but present an emission peak around 15 km s-1 and +3 km s-1 which indicates that they are part of RCW 120. Condensation 9 presents an emission peak in the same range but also between 35 km s-1 and 15 km s-1. Although we cannot rule out the fact that condensation 9 might in the foreground or background of the region, the emission peak is stronger in the main-peak velocity range and therefore, we consider this condensation to be part of RCW 120. Between 15 km s-1 and +3 km s-1, the other condensations are distributed along the strong CO emission, following the PDR that surrounds the ionized region. This strongly suggests that the 35 sources of the final sample are indeed associated with RCW 120.
Fig. 8Integrated intensity of 12CO (J = 0 → 1) between a) −75 km s-1 and −50 km s-1; b) −35 km s-1 and −15 km s-1; c) −15 km s-1 and 3 km s-1. The dots represent the 35 sources of the final sample and the contours stand for the 870 μm condensations of DEH09. The unit of the color image is in Jy km s-1 beam-1. Open with DEXTER
Fig. 9Integrated intensity of 13CO (J = 0 → 1) within the same velocity ranges. Dots and contours are the same as in Fig. 8. Open with DEXTER
Fig. 10Integrated intensity of C18O (J = 0 → 1) within the same velocity ranges. Dots and contours are the same as in Fig. 8. Open with DEXTER
#### 5.2.3. Compact source properties
We have shown that the detected sources are likely to be associated with RCW 120, that is, located at the same distance (see Sect. 5.2.2). Table 5 gives the physical properties derived for the 35 sources: the getsources identification number (identification number in DEH09 given in parenthesis if any), the envelope temperature T, envelope mass Menv, bolometric luminosity Lbol, ratio of the submillimetric luminosity (defined as the luminosity computed from 350 μm onwards) over the bolometric luminosity, ratio of the envelope mass over the bolometric luminosity, associated condensation towards which the source is observed, evolutionary class derived from the study of DEH09 and from Lλ ≥ 350 μm/Lbol, near- and mid-infrared counterparts, and volume density.
In the following, source ID refers to IDs given in Col. 1 of Tables 5 and A.1.
Among the 35 sources of the final sample, 14 match the previous list discussed in DEH09. The sources previously identified on the basis on GLIMPSE and MIPSGAL data are now identified at Herschel wavelengths, based on their spatial correspondance in a radius of 4. We discuss their evolutionary class in Sect. 6. Adopting β = 1.6 from the latest Planck results, modifies the physical parameters (envelope temperature, envelope mass and bolometric luminosity) by 10% in average throughout the MBB model and color corrections factors. A higher value of β better represents denser regions (Paradis et al. 2012).
Table 5
Properties of the 35 compact sources discussed in the text.
Figure 11 presents the distribution of dust envelope temperature for compact sources observed towards RCW 120. All but three sources (24, 28, 36) have envelope temperature lower than 25 K. As discussed in ZAV07, sources 24 and 28 are observed towards condensation 4. They are classified as Herbig Ae/Be objects and contain a central star of spectral type B4V for source 24 and B7V for source 28. Their extended nature is thought to be the result of local PDR due to radiation of the star but not massive enough to form H ii regions which is consistent with the envelope mass derived, 2 M and 1 M for source 24 and 28, respectively. Source 36 is located in a region of low density and high temperature (23 K).
Figure 12 presents the distribution of envelope mass for sources observed towards RCW 120. Twenty-seven sources have a low mass (Menv ≤ 20 M) envelope. Sources with envelope mass up to 1 M are detected here. From their Herschel study of dense cores in NGC 6334, Tigé et al. (2017) derived an envelope mass limit of 60 M (lower limit at which they detect ongoing high-mass star activity) for a core to form a high-mass star. Five sources (2, 9, 10, 39, 94) have envelope mass higher than this limit and four (2, 9, 10, 39) are located in condensation 1. Source 94 is located in condensation 6. The column density towards these condensations is higher than 1.7 × 1022 cm-2. We point out the fact that the high-mass cores represent 15% of the total number of sources in the final sample. Nevertheless, two biases could radically change the result. Firstly, it is possible that the cores are unresolved even at the best Herschel resolution (5.̋9) and represents more than one YSO then decreasing the number of possible high-mass cores. We are confident that source 2 might represent this problematic case. Secondly, our final sample represent reliable sources but is incomplete. According to our selection criteria, the non-selected sources should represent low-mass objects. Consequently, this value of 15% should represent a higher-limit to the number of high-mass cores.
Table 6 summarizes the physical properties of the final sample of sources.
#### 5.2.4. Properties of the tentative sources
Physical properties of the 87 tentative sources shown in Fig. 7 could not be obtained directly with the SED fitting due to a lack of Herschel measurements (see Sect. 4). From the final sample of 35 sources, we fitted the envelope temperature obtained from the SED versus the temperature found at the source location on the temperature map (Fig. 13a). We then used the fitted linear relation to assign a temperature to each of the tentative sources. The mass is computed using the Hildebrand formula (Hildebrand 1983) with one of the Herschel fluxes. For the bolometric luminosity, we used the relation between the flux density and the bolometric luminosity (log10(Fν) log10(Lbol)) using the final sample (Fig. 13b) (Ragan et al. 2012). The relation of Dunham et al. (2008) for embedded protostars is normalized at 1.3 kpc (10)where \begin{lxirformule}$F_{\nu}^{1.3~\rm kpc}$\end{lxirformule} is the flux density at 1.3 kpc and Fν is the flux density of Dunham et al. (2008). This relation is represented in Fig. 13 by the blue-dotted line. Results are given in Appendix B.
#### 5.2.5. Mass of the condensations using the H2 column density map
Using the H2 column density map, we derived the mass of each condensation defined by the same area as the one used by DEH09 to compute the mass from APEX 870 μm data. We used the following formula: (11)where Apixel is the area of a pixel in cm2, μ is the mean molecular weight (2.8), mH is the hydrogen atom mass and is the H2 column density value at pixels (i, j). DEH09 computed the mass using the Hildebrand formula with T = 20 K (and also with T = 30 K but this value for the condensations is too high compared to the ones derived from the Herschel temperature map). The results are given in Table 7. Column 1 gives the condensation number from DEH09. Column 2 gives the condensation’s mass derived using the H2 column density map and Eq. (11), and Col. 3 the mass derived by DEH09 using the 870 μm data, assuming a dust temperature of 20 K.
Compared to DEH09, we obtain higher masses for the condensations. At first sight, the absence of background subtraction could explain this difference but since they are massive, the background only accounts for a small amount of the total pixels value. The main difference between the two results could be explained by the extending emission filtering of the ground-based telescope at 870 μm, leading to an underestimation of the mass (Csengeri et al. 2016). We point out that the condensation mass is critical for star-formation rate and star-formation efficiency estimates (Liu et al. 2017).
Fig. 11Envelope temperature distribution for the 35 sources observed towards RCW 120. Open with DEXTER
Fig. 12Histogram of envelope mass for sources observed towards RCW 120. Open with DEXTER
Table 6
Physical properties of the final sample.
## 6. Discussion
### 6.1. Compact sources’ evolutionary stage
As the submillimetric luminosity depends on the envelope mass and the bolometric luminosity on the stellar mass, André et al. (1993) proposed to use the submillimetric to bolometric luminosity ratio as an evolutionary indicator. Bontemps et al. (2010a) used the same kind of criteria to distinguish Class 0 and Class I objects in the Aquila Rift using Herschel data but the limits used were different. Class 0 are defined with Lλ ≥ 350 μm/Lbol> 0.03 while it is higher than 0.01 for Class I. The region between 0.01 and 0.03 contains sources with uncertain classification.
Figure 14 shows the distribution of sources’ envelope temperature, color-coded according to their Lλ ≥ 350 μm/Lbol value. As expected, there is a relation between the Lλ ≥ 350 μm/Lbol value and the envelope temperature of the source. Class I objects (in red) have a higher temperature than Class 0 objects (in green) while the uncertain cases (in blue) are located in between.
Table 7
Condensations’ mass using the low resolution density map (second column) and from DEH09 at 20 K (third column).
Fig. 13a) Temperature given by the SED fitting versus temperature obtained at the source location in the temperature map for the final sample of 35 sources (red diamonds). b) ν70 μm × S70 μm versus bolometric luminosity for the final sample of 35 sources following Ragan et al. (2012) where the black continuous line represents the fit and the blue dotted one represented the relation from Dunham et al. (2008). Open with DEXTER
Figure 15 shows the sample of the 35 compact sources on the Lλ ≥ 350 μm/Lbol versus Menv diagram coded depending on their location with respect to the PDR. 80% of the sources are located in the Class 0 region with Lλ ≥ 350 μm/Lbol> 0.01. If an age gradient was at work in the region, sources towards the PDR would have been under the Class I limit and sources outside the PDR would have been above the Class 0 limit. Depending on the Class 0 limit taken, 1% for André et al. (1993) or 3% for Bontemps et al. (2010a), a weak trend in favor of very young objects out of the PDR can be seen. Towards the PDR, no trend is seen since these sources spread over the entire range of Lλ ≥ 350 μm/Lbol values.
Saraceno et al. (1996) presented the evolution of Class 0 to Class II via spherical accretion by a path in the LbolMenv diagram. The unknown mechanism of massive star formation (scaled-up analogue of low-mass star or merging of low-mass stars) and the difficulties of establishing the evolution phase for individual YSOs (d ≥ 1 kpc, non-resolved clusters) make the construction of an evolutionary scenario for high-mass objects difficult. An attempt has been made by Molinari et al. (2008) to reproduce the LbolMenv evolutionary paths for massive objects. From a sample of 42 sources characterized by their [25–12] color value, they classified them as IR-sources if the SED could be fitted with a zero age main sequence (ZAMS) model or MM-sources if a MBB model was used. This difference in the SED translates into a different location in the LbolMenv diagram well separated by a line representing IR-sources (see Molinari et al. 2008) practically equivalent to the strip of low-mass Class I objects from Saraceno et al. (1996). Assuming a scaled-up analogue of the low-mass star regime with a turbulent core, a model of time dependent accretion rate (McKee & Tan 2003) with fixed final stellar masses and core surface densities, evolutionary paths in the LbolMenv have been computed. The first sequence (indicated in Figs. 1618) represents the accretion phase where the luminosity is dominated by the accelerated accretion and the lost of envelope mass is due to accretion, outflows and possible draining by other YSOs. At the end of the first phase, the star reaches or is close to the ZAMS with a final stellar mass. During the second phase (also indicated in the figures), the envelope mass continues to decrease (the increase of stellar mass by residual accretion is neglected in this model) and the luminosity is now the sum of accretion and stellar luminosity. The final point of the paths corresponds to a lost of 90% of the envelope mass for the four low-mass tracks and to a time of 2.1 × 106 yr and 2.7 × 106 yr when the star is optically visible for the two highest-mass tracks. We warn readers that this is a simple model which cannot be used to predict accurately the evolution of YSOs but rather to obtain indication about the evolutionary class of a source.
Fig. 14Histogram of the temperature color-coded according to Lλ ≥ 350 μm/Lbol in green for Class 0, red for Class I and blue for uncertain cases. Open with DEXTER
In Fig. 16, we plot the evolutionary paths for low-mass (Saraceno et al. 1996) and high-mass stars (Molinari et al. 2008) with their corresponding stripes for Class I sources and include our sample. Sources with Menv> 10 M are all located under the Class I stripe and a qualitative analogy with Fig. 9 of Molinari et al. (2008) permits a rough classification of them: sources 1, 2, 3, 5, 8, 14 are Class I and sources 9, 10, 39, 40, 63, 94, 179 are Class 0. On the contrary, the distribution of sources with Menv< 10 M has a higher dispersion around the Class-I strip, also seen in Fig. 9 of Molinari et al. (2008). As in Fig. 15, sources located outside the PDR might tend to be younger but no evidence for a more evolved stage for sources located inside the PDR is seen, as it could have been expected if star formation progresses gradually in the surrounding medium, following the expansion of the ionization front and the leaking of the ionizing radiation.
In Fig. 17, the sources are color-coded depending on their location in the condensations. We see a clear trend for the sources’ envelope mass and evolutionary stage to be determined by their hosting condensation: sources observed towards condensation 1 have the highest envelope mass and are in low evolutionary stage while sources in condensation 5 are low mass envelope sources, and possibly in a later evolutionary stage. Sources observed towards condensation 4 (pre-existing clump) tend to be evolved and of intermediate envelope mass. Condensation 8 is observed further away from the ionized front and hosts sources in a low-evolutionary stage. Sources 50 and 155 do not belong to any condensation according to DEH09 but are spatially close, outside the PDR and in a similar evolutionary state than the sources in condensation 5. Sources in condensation 2 show a higher dispersion in this diagram compared to the other condensations. The eastern part of condensation 2 contains Class 0-Class I objects of intermediate mass and low-mass objects in the western part. The eastern-part of this condensation seems to be radiation-shielded thanks to the filament in front of sources 3, 8 and 16 while the western-part receives a significant part of the radiation through photons’ leaking. This might explain the dispersion of sources’ properties observed towards this condensation.
Fig. 15Lλ ≥ 350 μm/Lbol versus Menv. The dotted-dashed lines represents the Lλ ≥ 350 μm/Lbol limits between Class 0 and Class I from Bontemps et al. (2010a) and sources are color-coded depending on their location: red squares for sources inside the PDR, blue triangles for outside. Open with DEXTER
Fig. 16Lbol versus Menv. Evolutionary tracks are adapted from Saraceno et al. (1996) and Molinari et al. (2008). Labeled arrows indicate (1) the accreation phase and (2) envelope cleaning phase. Sources are coded as a function of their location with respect to the PDR: red squares sources are for the ones observed towards the PDR and blue triangles for those outside. Error bars for Lbol and Menv are shown by gray lines. Open with DEXTER
Fig. 17Same as Fig. 16 but sources are color-coded as a function of their hosting condensation previously identified using the 870 μm and 1.3 mm emission (ZAV07, DEH09). The condensation number refers as the one given in Fig. 5 (left). Open with DEXTER
In Fig. 18, the sources are color-coded depending on their Lλ ≥ 350 μm/Lbol value. We note that the magenta diamond sources (Lλ ≥ 350 μm/Lbol< 0.01) are above the Class I stripe of Saraceno et al. (1996), red square sources (Lλ ≥ 350 μm/Lbol> 0.03) are below the Class I stripe of Molinari et al. (2008) and blue triangle sources (0.03 >Lλ ≥ 350 μm/Lbol> 0.01) are spread around these stripes. Hence, the two methods give consistent results to derive sources’ evolutionary class.
Fig. 18Same as Fig. 16 but the sources are color-coded as a function of their Lλ ≥ 350 μm/Lbol ratio. Magenta diamonds for Class I objects and red squares for Class 0, while blue-triangle sources are uncertain. Open with DEXTER
We suggest that the main parameter that controls the star formation and the evolutionary stage of the YSOs is the column density of their hosting condensation. This means that a simple search for YSOs’ age gradient around H ii region cannot be used as a simple indicator for establishing evidence for triggered star formation.
### 6.2. Evolutionary stage derived by DEH09
Color-color diagrams using near- and mid-IR data can also be used to infer the class of a source. DEH09 used Spitzer GLIMPSE and MIPSGAL colors to discuss the evolutionary stage of YSOs observed towards RCW 120. The results are given in Cols. 8 and 9 of Table 5. Figures 19 to 25 present a zoom of the sources observed from condensation 1 to 8 on the gradient image of the Herschel PACS 70 μm emission and the 870 μm emission in countours. All the sources in these figures are detected by getsources and identified according to their getsources identification number in Tables 5 and A.1. Final-sample sources, and those identified by DEH09 are indicated in the figures. Hence, non-labelled sources are either not part of the final sample and/or not detected by DEH09. In the following we compare the evolutionary class of sources obtained from mid-IR color-color diagrams (DEH09) with the one obtained in this paper.
Condensation 1 (2530 M): this is the most massive and densest condensation observed. The classification of Source 9 (40) and 63 (37) do not agree with DEH09. Both objects do not have IR-counterpart in the J, H and Ks and following Fig. 5 of DEH09, they have a K− [ 24 ] higher than 10 mag. Hence, they are likely to be in an early evolutionary stage. Source 2 is the massive Class 0 object discussed in Zavagno et al. (2010, see also DEH09). It is located at the peak of the column density distribution (N(H2) = 4 × 1023 cm-2) and has the highest envelope mass (Menv = 174 M) and bolometric luminosity (Lbol = 1163L) of the sample. It is probably a Class 0 source, since no IR-counterpart is detected. Sources 10, 39 and 82 are not detected by DEH09 and are classified as Class 0. This condensation hosts 80% of the massive cores. Because condensation 1 is the densest and most massive in RCW 120, the core formation efficiency (CFE) is expected to be higher compared to the other condensations (Motte et al. 1998; Bontemps et al. 2010b; Liu et al. 2017).
Condensation 2 (540 M): source 3 (50) has been classified as a Class I source by DEH09 in agreement with our classification. Sources 16 and 36 are not discussed by DEH09 maybe because of the high filamentary background around these compact sources. Therefore, no IR-counterpart could be reliably detected and the sources are classified as Class 0I.
Condensation 4 (350 M): sources 6 (76), 14 (69) and 19 (67) are classified as at least Class I objects and in agreement with DEH09. Source 24 (Object A) and 28 (Object B) are surrounded by local PDR revealed as shells on the gradient 70 μm image. Because their IR counterparts are diffuse, no attempt has been made by DEH09 to classify them but are likely Class I or further. ZAV07 suggested that this condensation could be a pre-existing clump engulfed in the ionized region. A subsequent RDI process could have accelerated the collapse which might explain why the objects are in a higher evolutionary stage compared to the other condensations.
Condensation 5 (1580 M): this region is highly structured and hosts nine YSOs among the 35 discussed. Among the sources of the final sample and discussed by DEH09, source 33 is the only one whose class does not agree. In the same way as sources 9 and 63 in condensation 1, DEH09 did not measured any near IR-counterpart for this source and its K− [ 24 ] value should also be higher than ten. Therefore, this source is also in an early evolutionary stage. Sources 44 and 48 present IR-counterparts at all wavelengths except 8 μm and only at 24 μm respectively but are too weak (Menv = 1 M) to be discussed by DEH09. They probably are weak Class I sources. Sources 84 and 123 do not present IR-counterparts and are classified as Class 0.
Fig. 19Condensation 1 and 7: 870 μm emission (countours) superimposed on the gradient image of the Herschel PACS 70 μm emission. Sources are identified with their getsources identification number. Sources coded with a red circle are those discussed (among the sample of 35 sources). The green square sources are detected but not discussed due to a lack of Herschel measurements (see text). Open with DEXTER
Fig. 20Condensation 2: same as for Fig. 19. Open with DEXTER
Fig. 21Condensation 3: same as for Fig. 19. Open with DEXTER
Fig. 22Condensation 4: same as for Fig. 19. Open with DEXTER
Fig. 23Condensation 5: same as for Fig. 19. Open with DEXTER
Condensation 6 (330 M): we identify a massive YSO (source 94) of Menv = 70M with IR-counterparts but classified as Class 0. It is possible that the higher fluxes coming from source 4 contaminates source 94 at long wavelengths hence overestimating the Lλ ≥ 350 μm/Lbol value and hence, the classification.
Condensation 8 (370 M): located south of the ionized region (see Fig. 5b), this condensation was probably formed by the leaking of UV photons passing through the low density medium seen on the high resolution density map (see Fig. 5b) at (25807, 3852). Hence, sources 175 and 179 were probably formed later compared to the sources located in the PDR. This is confirmed by their low-temperature (between 11.2 K and 13.3 K), low evolutionary stage and the absence of IR counterparts.
Fig. 24Condensation 6: same as for Fig. 19. Open with DEXTER
Fig. 25Condensation 8: same as for Fig. 19. Open with DEXTER
### 6.3. Comparison with the Walch et al. (2015) model
Walch et al. (2012, 2013) show that clumpy, shell-like structures like that seen in RCW 120 are probably attributable to pre-existing density structures in the natal molecular cloud. During the expansion of the H ii region and the collection of the dense shell, the pre-existing density structures are enhanced and lead to a clumpy distribution within the shell. The masses and locations of the swept-up clumps depend on the fractal density structure of the molecular cloud, through the parameters n and ρ0, related to the fractal dimension of the cloud and the density, respectively (Walch et al. 2013, see Sect. 2).
Walch et al. (2015) compared simulations and APEX-LABOCA 870 μm observations of RCW 120. They performed three-dimensional SPH simulations of H ii regions expanding into fractal molecular clouds in order to investigate whether the formation of massive clumps in the swept-up shell necessarily requires the C&C mechanism (Elmegreen & Lada 1977). They show that a distribution of clumps similar to the one seen in RCW 120 can be explained by a non-uniform initial molecular cloud structure, implying that a shell-like configuration of massive clumps does not imply that the C&C mechanism is at work. They find a hybrid form of triggering, which combines elements of C&C mechanism and RDI.
We discuss below how the Herschel results presented here compare with their findings. The temperature map obtained from Herschel images indicates that dust temperatures lower than 30 K, the temperature used by Walch et al. (2015), are observed. This means that the mass they derived for the condensations represents a lower limit. The H2 column density maps obtained from Herschel images show that the observations better correspond with a low value of ρ0, where ρ0 is the scaling constant for the density fluctuations field caracterizing the width of the density PDF (Walch et al. 2012, 2013). However pillars are not observed on the northern, lower density part of the ionized region as obtained in their simulations (Walch et al. 2015, Fig. 2). This suggests that the numerical treatment adopted better describe higher density regions while lower density regions seem to be better represented by the higher value of ρ0.
The distribution of sources observed towards the central part of the ionized region in the simulation is also not observed (Walch et al. 2015, their Fig. 2 right). The distribution of sources in condensations is also not well reproduced by this model, as seen on Figs. 6 and 7. The number of sources they found towards the three main condensations well corresponds with our findings – nine sources towards condensation 1 (their condensation 3), three sources towards condensation 2 (their condensation 1) and six sources towards condensation 4 (their condensation 2). For the two runs, the condensation 3 formed the highest number of high-mass protostars (12.7 M and 19 M in average). This is in agreement with the observations where high-mass cores are found towards our condensation 1. We remind the reader that the mass computed in this paper is the envelope mass while Walch et al. (2015) use the protostars mass. Therefore, it is not surprising that the mass computed in condensation 1 are much higher compared to Walch et al. (2015). Nonetheless, their condensation 2 host protostars of lower masses (9 M in average) while our corresponding condensation contains low-mass cores. A new modeling of this region using Herschel results would help for discussing the star formation history and its propagation in the ambient medium. It would be interesting to discuss the parameters (and mechanisms) that lead to the formation of the high number of high-mass cores observed towards condensation 1.
### 6.4. Comparison with the model of Torii et al. (2015)
Using MOPRA observations of 12CO, 13CO and C18O in the J = 1 → 0 transition, Anderson et al. (2015) did not detect any expansion of the H ii region which means that the expansion velocity is either too low to be observed or inexistant. Considering this fact, Torii et al. (2015) explained the formation of the O star and the corresponding ring-like structure following the cloud-cloud collision (CCC) scenario from Habe & Ohta (1992) which can be described in three stages. First, a small and a large clouds are heading towards each other. Secondly, a cavity is created in the large cloud due to the collision with the small cloud. The place where the two clumps collided is compressed, leading to massive star formation. Finally, the cavity in the large cloud is filled with the ionzing radiation coming from the recently formed massive star(s). A schematic explanation can be found in Torii et al. (2015, see their Figs. 12 and 13). In the case of RCW 120, they suggest that the weak leaking of Hα emission in the northern part of the ring indicates only the beginning of the erosion by the ionizing radiation. Hence, the triggering which is assumed to take place as a consequence of the C&C mechanism cannot be seen yet. However, after the formation of the ionizing star, a triggering mechanism caused by the compression of the remaining small clump on the large clump is plausible. This could be an alternative explanation which should only affect the formation of YSOs in the southern part of the ring. This study shows that the main driver of the evolutionary stage is the density of the hosting condensation and not the (projected) distance to the ionizing star as expected earlier.
## 7. Summary and conclusions
We used Herschel PACS and SPIRE images, complemented with existing data, to study the star formation observed towards the Galactic ionized region RCW 120.
Zavagno et al. (2010) presented the first results from Herschel, however this paper is an in-depth study under the HOBYS recipe which allow us to compare the results between different regions observed in this key program. Moreover, while the first Herschel results were focused on source 2, we produced the first reliable catalog of compact sources using Herschel data in this region.
The unprecedent coverage and sensitivity in the far infrared of the Herschel data allow us to derive, for the first time, the temperature and H2 column density map for this region. The temperature ranges from 15 K to 24 K and the column density from 7 × 1021 cm-2 up to 9 × 1023 cm-2. The condensations defined by DEH09 at 870 μm corresponds to cold and dense regions where the majority of the sources are detected.
We also derive, for the first time, the envelope mass, envelope dust temperature and bolometric luminosity of compact sources detected there. The temperature ranges from 11.2 K to 34.1 K with a median of 19.1 K, from 1 M to 174 M with a median of 4 M for the envelope mass and from 5 L to 1163 L with a median of 30 L. The volume density was computed by assuming a spherical source with the size defined at the reference wavelength (160 μm or 250 μm) going from 2 × 105 cm-3 to 108 cm-3.
We use the physical parameters to discuss the star formation history in this region. We show that most of the compact sources (21 of the 35) are observed towards the PDR.
Thanks to Herschel data, we detected 21 sources, mostly in an early evolutionary stage, which were not detected and hence discussed in DEH09.
Using the Lλ ≥ 350 μm/Lbol criteria from Bontemps et al. (2010a), we classify the sources between Class 0, intermediate and Class I. We found respectively 14, 15 and 6 sources in this classification.
We find that the projected distance to the ionizing source is not the parameter which controls the evolutionary stage of the sources, contrary to what was expected before, wrongly. In fact, the main driver for this is the density of the condensation where the source is located, whatever its distance to the ionizing sources. Consequently, there is no conflict between possible triggering and projected distance because the density plays a major role in the overall picture. Despite the fact that the southern layer of the region is compressed (Tremblin et al. 2014), Herschel data do not allow us to conclude on triggering. High resolution spectroscopic data are needed to determine the structure (possible fragmentation) of the cores and the evolutionary stage of the sources in these cores.
3
HIPE is a joint development software by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS, and SPIRE consortia.
5
The getsources algorithm is publicy available and can be downloaded at http://www.herschel.fr/cea/gouldbelt/en/getsources/
## Acknowledgments
We thank the referee for his/her report which helps to improve the quality of the paper. This work is based on observations obtained with Herschel-PACS and Herschel-SPIRE photometers. PACS has been developed by a consortium of institutes led by MPE (Germany) and including UVIE (Austria); KU Leuven, CSL, IMEC (Belgium); CEA, LAM (France); MPIA (Germany); INAF-IFSI/OAA/OAP/OAT, LENS, SISSA (Italy); IAC (Spain). This development has been supported by the funding agencies BMVIT (Austria), ESA-PRODEX (Belgium), CEA/CNES (France), DLR (Germany), ASI/INAF (Italy), and CICYT/MCYT (Spain). SPIRE has been developed by a consortium of institutes led by Cardiff Univ. (UK) and including: Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC, UKSA (UK); and NASA (USA). This work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. We have made use of the NASA/IPAC Infrared Science Archive to obtain data products from the 2MASS, Spitzer -GLIMPSE, and Spitzer -MIPSGAL surveys. The Centre National d’Etudes Spatiales (CNES) is deeply acknowledged for the financial support. Part of this work was supported by the ANR (Agence Nationale pour la Recherche) project “PROBeS”, number ANR-08-BLAN-0241.
## Appendix A: Herschel original fluxes for sources detected by getsources (see text)
Table A.1
Identification, position, and fluxes for compact sources detected towards RCW 120.
## Appendix B: Properties of tentative sources
Table B.1
Tentative-source properties.
## Appendix C: Image of the sources
In this section, we present the three first sources of the final sample on the 2MASS, Spitzer GLIMPSE and MIPSGAL, Herschel and density maps (low and high resolution) together with the result of their SED fitting. The maps (1′× 1′ for IR maps and 2′× 2′ for Herschel maps) are centered on the coordinates given by getsources with the corresponding wavelength written in the upper-left part of the image.
Fig. C.12MASS, GLIMPSE, MIPSGAL, Herschel, low-resolution and high-resolution images of source 1. If a counterpart is seen in the infrared catalogs, a black circle of 4″ radius is shown to indicate the location of this counterpart otherwise, the center (position of the source given by getsources) is indicated by a cross. For Herschel images, the ellipses shown are the getsources parameters, AFWHM and BFWHM. For the representation of the SED fitting, the original fluxes are represented by a magenta diamond, the corrected fluxes (flux scaling + color correction) at the wavelength used for the fitting are represented by a black cross, and the blue triangles represent the IR counterparts, if any. The identification number of the source is given in the title of the SED. Open with DEXTER
Fig. C.2Same as Fig. C.1 for source 2. Open with DEXTER
Fig. C.3Same as Fig. C.1 for source 3. Open with DEXTER
## All Tables
Table 1
Summary of Herschel observational parameters.
Table 2
Minimum, maximum and median values for the color correction factors at Herschel wavelengths for the final sample of sources.
Table 3
Range for the temperature map constructed following the method described in Hill et al. (2012, first line), with all wavelengths and no background subtraction (second line) and following Anderson et al. (2012, third line) for the whole map (first and fourth columns), the hottest region (second and fifth columns) and coldest region (third and sixth columns).
Table 4
Range for the column density map constructed following the method described in Hill et al. (2012, first line), with all wavelengths and no background subtraction (second line) and following Anderson et al. (2012, third line) for the whole map (first and fourth columns), the densest region (second and fifth columns) and empty region (third and sixth columns).
Table 5
Properties of the 35 compact sources discussed in the text.
Table 6
Physical properties of the final sample.
Table 7
Condensations’ mass using the low resolution density map (second column) and from DEH09 at 20 K (third column).
Table A.1
Identification, position, and fluxes for compact sources detected towards RCW 120.
Table B.1
Tentative-source properties.
## All Figures
Fig. 1RCW 120: Herschel-PACS 70 μm (blue), 160 μm (green) and Herschel-SPIRE 250 μm (red). The field size is 21.8′ × 24.5′. North is up, east is left. Open with DEXTER In the text
Fig. 2a) SED fitting for the pixels giving the highest temperature. The continuous curve represents the fit made with all Herschel fluxes and dashed curved is the fit obtained by the method of Hill et al. (2012). b) Same for the pixel giving the lowest temperature. No background is subtracted in both cases. Open with DEXTER In the text
Fig. 3a) Ratio of temperature between the maps obtained in this paper (no 70 μm and 100 μm data included and no background subtraction) over the ones obtained by Anderson et al. (2012, see text). The yellow contours correspond to 870 μm emission at 0.1 Jy/beam. b) Same but with the column density maps. Open with DEXTER In the text
Fig. 4Temperature map of RCW 120 at 36.̋6 resolution with 870 μm emission from LABOCA (in yellow countours) and the final sample of 35 compact sources (white dots) discussed in this paper. Condensations observed at 870 μm are identified following the labelling in DEH09. The temperature ranges from 15 K (dark) to 24 K (white). Warm regions are observed towards the ionized zone. Colder regions are located outside the ionized region and are distributed in cores, filaments and condensations. Open with DEXTER In the text
Fig. 5a) On logarithmic scale, H2 column density map of RCW 120 at 366 resolution with 870 μm emission from LABOCA (in yellow countours), the final sample of sources (black dots) and the three prestellar clumps (red dots). Condensations observed at 870 μm are identified following the labelling in DEH09. The density values range from 7 × 1021 cm-2 to 4 × 1023 cm-2. b) High resolution H2 column density map of RCW 120 at 182 resolution (in red) and Hα emission (in blue) from the SuperCOSMOS Hα Survey. The column density values range from 7 × 1021 cm-2 to 9.4 × 1023 cm-2. Open with DEXTER In the text
Fig. 6All 35 compact sources detected using getsources (and discussed in the text) superimposed on a 70 μm gradient image of RCW 120. The sources are color-coded depending on their location: red circles for sources observed towards the PDR, blue squares for sources outside (see text). Open with DEXTER In the text
Fig. 7All 87 sources detected by getsources but not part of the final sample due to the lack of reliable flux measurements, mainly at SPIRE wavelengths. Physical parameters of these sources are derived in a secondhand way explained and presented in Sect. 5.2.4. The PDR region is enclosed in the green countours (see text). Open with DEXTER In the text
Fig. 8Integrated intensity of 12CO (J = 0 → 1) between a) −75 km s-1 and −50 km s-1; b) −35 km s-1 and −15 km s-1; c) −15 km s-1 and 3 km s-1. The dots represent the 35 sources of the final sample and the contours stand for the 870 μm condensations of DEH09. The unit of the color image is in Jy km s-1 beam-1. Open with DEXTER In the text
Fig. 9Integrated intensity of 13CO (J = 0 → 1) within the same velocity ranges. Dots and contours are the same as in Fig. 8. Open with DEXTER In the text
Fig. 10Integrated intensity of C18O (J = 0 → 1) within the same velocity ranges. Dots and contours are the same as in Fig. 8. Open with DEXTER In the text
Fig. 11Envelope temperature distribution for the 35 sources observed towards RCW 120. Open with DEXTER In the text
Fig. 12Histogram of envelope mass for sources observed towards RCW 120. Open with DEXTER In the text
Fig. 13a) Temperature given by the SED fitting versus temperature obtained at the source location in the temperature map for the final sample of 35 sources (red diamonds). b) ν70 μm × S70 μm versus bolometric luminosity for the final sample of 35 sources following Ragan et al. (2012) where the black continuous line represents the fit and the blue dotted one represented the relation from Dunham et al. (2008). Open with DEXTER In the text
Fig. 14Histogram of the temperature color-coded according to Lλ ≥ 350 μm/Lbol in green for Class 0, red for Class I and blue for uncertain cases. Open with DEXTER In the text
Fig. 15Lλ ≥ 350 μm/Lbol versus Menv. The dotted-dashed lines represents the Lλ ≥ 350 μm/Lbol limits between Class 0 and Class I from Bontemps et al. (2010a) and sources are color-coded depending on their location: red squares for sources inside the PDR, blue triangles for outside. Open with DEXTER In the text
Fig. 16Lbol versus Menv. Evolutionary tracks are adapted from Saraceno et al. (1996) and Molinari et al. (2008). Labeled arrows indicate (1) the accreation phase and (2) envelope cleaning phase. Sources are coded as a function of their location with respect to the PDR: red squares sources are for the ones observed towards the PDR and blue triangles for those outside. Error bars for Lbol and Menv are shown by gray lines. Open with DEXTER In the text
Fig. 17Same as Fig. 16 but sources are color-coded as a function of their hosting condensation previously identified using the 870 μm and 1.3 mm emission (ZAV07, DEH09). The condensation number refers as the one given in Fig. 5 (left). Open with DEXTER In the text
Fig. 18Same as Fig. 16 but the sources are color-coded as a function of their Lλ ≥ 350 μm/Lbol ratio. Magenta diamonds for Class I objects and red squares for Class 0, while blue-triangle sources are uncertain. Open with DEXTER In the text
Fig. 19Condensation 1 and 7: 870 μm emission (countours) superimposed on the gradient image of the Herschel PACS 70 μm emission. Sources are identified with their getsources identification number. Sources coded with a red circle are those discussed (among the sample of 35 sources). The green square sources are detected but not discussed due to a lack of Herschel measurements (see text). Open with DEXTER In the text
Fig. 20Condensation 2: same as for Fig. 19. Open with DEXTER In the text
Fig. 21Condensation 3: same as for Fig. 19. Open with DEXTER In the text
Fig. 22Condensation 4: same as for Fig. 19. Open with DEXTER In the text
Fig. 23Condensation 5: same as for Fig. 19. Open with DEXTER In the text
Fig. 24Condensation 6: same as for Fig. 19. Open with DEXTER In the text
Fig. 25Condensation 8: same as for Fig. 19. Open with DEXTER In the text
Fig. C.12MASS, GLIMPSE, MIPSGAL, Herschel, low-resolution and high-resolution images of source 1. If a counterpart is seen in the infrared catalogs, a black circle of 4″ radius is shown to indicate the location of this counterpart otherwise, the center (position of the source given by getsources) is indicated by a cross. For Herschel images, the ellipses shown are the getsources parameters, AFWHM and BFWHM. For the representation of the SED fitting, the original fluxes are represented by a magenta diamond, the corrected fluxes (flux scaling + color correction) at the wavelength used for the fitting are represented by a black cross, and the blue triangles represent the IR counterparts, if any. The identification number of the source is given in the title of the SED. Open with DEXTER In the text
Fig. C.2Same as Fig. C.1 for source 2. Open with DEXTER In the text
Fig. C.3Same as Fig. C.1 for source 3. Open with DEXTER In the text
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https://www.quizover.com/online/course/0-3-modelling-corruption-software-receiver-design-by-openstax?page=6 | # 0.3 Modelling corruption (Page 7/11)
Page 7 / 11
$w\left(t\right)=\frac{A}{2j}\left[{e}^{j2\pi {f}_{0}t},-,{e}^{-j2\pi {f}_{0}t}\right].$
Applying the linearity property [link] and the result of Exercise [link] gives
$\begin{array}{ccc}\hfill \mathcal{F}\left\{w\left(t\right)\right\}& =& \frac{A}{2j}\left[\mathcal{F},\left\{{e}^{j2\pi {f}_{0}t}\right\},-,\mathcal{F},\left\{{e}^{-j2\pi {f}_{0}t}\right\}\right]\hfill \\ & =& j\frac{A}{2}\left[-,\delta ,\left(f-{f}_{0}\right),+,\delta ,\left(f+{f}_{0}\right)\right].\hfill \end{array}$
Thus, the spectrum of a sine wave is a pair of $\delta$ functions with opposite signs, located symmetrically about zero frequency. The corresponding magnitude spectrum,shown in [link] , is at the heart of one importantinterpretation of the Fourier transform: it shows the frequency content of any signal by displayingwhich frequencies are present (and which frequencies are absent) from the waveform. For example, [link] (a) shows the magnitude spectrum $W\left(f\right)$ of a real-valued signal $w\left(t\right)$ . This can be interpreted as saying that $w\left(t\right)$ contains (or is made up of) “all the frequencies” up to $B$ Hz, and that it contains no sinusoids with higher frequency. Similarly,the modulated signal $s\left(t\right)$ in [link] (b) contains all positive frequencies between ${f}_{c}-B$ and ${f}_{c}+B$ , and no others.
Note that the Fourier transform in [link] is purely imaginary, as it must be because $w\left(t\right)$ is odd (see [link] ). The phase spectrum is a flat line at $-{90}^{\circ }$ because of the factor $j$ .
What is the magnitude spectrum of $\mathrm{sin}\left(2\pi {f}_{0}t+\theta \right)$ ? Hint: Use the frequency shift property [link] . Show that the spectrum of $\mathrm{cos}\left(2\pi {f}_{0}t\right)$ is $\frac{1}{2}\left(\delta \left(f-{f}_{0}\right)+\delta \left(f+{f}_{0}\right)\right)$ . Compare this analytical result to the numerical resultsfrom Exercise [link] .
Let ${w}_{i}\left(t\right)={a}_{i}\mathrm{sin}\left(2\pi {f}_{i}t\right)$ for $i=1,2,3$ . Without doing any calculations, write down the spectrum of $v\left(t\right)={w}_{1}\left(t\right)+{w}_{2}\left(t\right)+{w}_{3}\left(t\right)$ . Hint: Use linearity. Graph the magnitude spectrum of $v\left(t\right)$ in the same manner as in [link] . Verify your results with a simulation mimicking that in [link] .
Let $W\left(f\right)=\mathrm{sin}\left(2\pi f{t}_{0}\right)$ . What is the corresponding time function?
## Convolution in time: it's what linear systems do
Suppose that a system has impulse response $h\left(t\right)$ , and that the input consists of a sum of three impulses occurring at times ${t}_{0}$ , ${t}_{1}$ , and ${t}_{2}$ , with amplitudes ${a}_{0}$ , ${a}_{1}$ , and ${a}_{2}$ (for example, the signal $w\left(t\right)$ of [link] ). By linearity of the Fourier transform, property [link] , the output is a superpositionof the outputs due to each of the input pulses. The output due to the first impulse is ${a}_{0}h\left(t-{t}_{0}\right)$ , which is the impulse response scaled by the size of the input and shifted to beginwhen the first input pulse arrives. Similarly, the outputs to the second and thirdinput impulses are ${a}_{1}h\left(t-{t}_{1}\right)$ and ${a}_{2}h\left(t-{t}_{2}\right)$ , respectively, and the complete output is the sum ${a}_{0}h\left(t-{t}_{0}\right)+{a}_{1}h\left(t-{t}_{1}\right)+{a}_{2}h\left(t-{t}_{2}\right)$ .
Now suppose that the input is a continuous function $x\left(t\right)$ . At any time instant $\lambda$ , the input can be thought of as consisting of an impulse scaled by the amplitude $x\left(\lambda \right)$ , and the corresponding output will be $x\left(\lambda \right)h\left(t-\lambda \right)$ , which is the impulse response scaled by thesize of the input and shifted to begin at time $\lambda$ . The complete output is then given by integrating over all $\lambda$
$y\left(t\right)={\int }_{-\infty }^{\infty }x\left(\lambda \right)h\left(t-\lambda \right)d\lambda \equiv x\left(t\right)*h\left(t\right).$
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
how did I we'll learn this
f(x)= 2|x+5| find f(-6)
f(n)= 2n + 1
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Got questions? Join the online conversation and get instant answers! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 49, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8883959054946899, "perplexity": 771.3818485829544}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815918.89/warc/CC-MAIN-20180224172043-20180224192043-00093.warc.gz"} |
http://math.jasonbhill.com/courses/fall-2010-math-2300-005/lectures/radius-convergence | Section:
9.5
Date:
Friday, October 8, 2010 - 14:00 - 15:00
AttachmentSize
fall2010math2300_power-series.pdf59.93 KB
Math 2300 Section 005 – Calculus II – Fall 2010
Power Series and Radius of Converges – October 8, 2010
Power Series
Definition: A power series is an infinite series of the form
where
• represents the coefficient of the th term.
• is some (fixed) real number.
• varies around , and so we sometimes say that the power series is “centered” at .
Theorem: For a power series , exactly one of the following happens:
1. The series converges for a single value, when . (Zero radius of convergence)
2. The series converges for all . (Infinite radius of convergence)
3. There is a positive number such that the series converges whenever and diverges whenever . (Finite non-zero radius of convergence. Check the endpoints of your interval of convergence to determine if the endpoints converge.)
• We use the ratio test to determine the radius of convergence. Specifically, the ratio test says that the series will converge if
• Notice that this is the same as
• The key point to understand here is that we know the series will always converge when , since this makes all but the first terms in the series zero. As the series is “centered at ,” the radius of convergence is “centered at .” That is, the quantity in the calculation above denotes distance between and . If the limit is some fixed finite number (it often is), the multiplication by will determine if the inequality is satisfied…depending on how big is. Thus, the radius satisfying
will be the maximum value that may take on and still satisfy the inequality. This value of is the “radius of convergence.”
• Notice that you only need to consider the coefficients in order to determine the radius of convergence. That is you don't always need to plug in the portion of the series terms into the ratio test in order to calculate the radius of convergence.
Examples:
1. Recall that . Find the radius of convergence of
Solution: This is a power series with center zero. That is
By the ratio test (as described above), we have
This implies
where the radius is some positive number, implying that . In this situation, the radius of convergence is infinite. This implies that the series converges everywhere on the real number line.
2. Find the radius of convergence of
3. Find the radius of convergence of | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9954923391342163, "perplexity": 292.98562122017466}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267156857.16/warc/CC-MAIN-20180921053049-20180921073449-00299.warc.gz"} |
https://proceedings.mlr.press/v35/kale14b.html | # Open Problem: Efficient Online Sparse Regression
Satyen Kale
; Proceedings of The 27th Conference on Learning Theory, PMLR 35:1299-1301, 2014.
#### Abstract
In practical scenarios, it is often necessary to be able to make predictions with very limited access to the features of any example. We provide one natural formulation as an online sparse regression problem with squared loss, and ask whether it is possible to achieve sublinear regret with efficient algorithms (i.e. polynomial running time in the natural parameters of the problem). | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9069706797599792, "perplexity": 363.54384250421015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243991370.50/warc/CC-MAIN-20210515131024-20210515161024-00582.warc.gz"} |
https://physicscatalyst.com/maths/binomial_expansion.php | # Binomial Theorem
## Binomial Theorem
• An algebraic expression containing two terms is called binomial expression.
Example
$( x+a)$
$(\frac {1}{2} +x)$
$(\frac {2}{x} - \frac{1}{x^3})$
• The general form of the binomial expression is $(x+a)$ and expansion $(x+a)^n , n \in N$ is called Binomial expression
• It was developed by Sir Issac newton
• The general expression for the Binomial Theorem is
$(x+a)^n = ^{n}C_{0} x^n a^0 + ^{n}C_{1} x^{n-1} a^1 + ^{n}C_{2} x^{n-2} a^2 + .....+ ^{n}C_{r} x^{n-r} a^r ....+^{n}C_{n} x^{0} a^n$
$(x+a)^{n} = \sum_{k=0}^{n} {}^{n} C_k x^{n-k}a^{k}$
Proof:
We can prove this theorem with the help of mathematical induction
Let us assume P(n) be the statement is
$(x+a)^n = ^{n}C_{0} x^n a^0 + ^{n}C_{1} x^{n-1} a^1 + ^{n}C_{2} x^{n-2} a^2 + .....+ ^{n}C_{r} x^{n-r} a^r ....+ ^{n}C_{n} x^{0} a^n$
Step 1
Now the value of P(1)
$(x+a)^1 = ^{1}C_{0} x^1 a^0 + ^{1}C_{1} x^{1-1} a^1$
$=(x+a)$
So P(1) is true
Step 2
Now the value of P(m)
$(x+a)^m = ^{m}C_{0} x^m a^0 + ^{m}C_{1} x^{m-1} a^1 + ^{m}C_{2} x^{m-2} a^2 + ...+ ^{m}C_{m} x^{0} a^m$
Now we have to prove
$(x+a)^{m+1} = ^{m+1}C_{0} x^{m+1} a^0 + ^{m+1}C_{1} x^{m} a^1 + ^{m+1}C_{2} x^{m-1} a^2 +...+ ^{m+1}C_{m+1} x^{0} a^{m+1}$
Now
$(x+a)^{m+1} =(x+a)(x+a)^m$
$= (x+a)( ^{m+1}C_{0} x^{m+1} a^0 + ^{m+1}C_{1} x^{m} a^1 + ^{m+1}C_{2} x^{m-1} a^2 + ...+ ^{m+1}C_{m+1} x^{0} a^{m+1}$
$= ^m C_0 x^{m+1}a{0} + ( ^mC_1 + ^mC_0) x^ma^1 + (^mC_2 + ^mC_1) x^{m-1}a^2 +....$ $+(^m C_{m-1} + ^mC_{m}) x^1a^m + ^mC_m x^0a^{m+1}$
As $^mC_{r-1} + ^mC_r = ^{m+1}C_r$
So,
$= ^{m+1}C_{0} x^{m+1} a^0 + ^{m+1}C_{1} x^{m} a^1 + ^{m+1}C_{2} x^{m-1} a^2 + ...+ ^{m+1}C_{m+1} x^{0} a^{m+1}$
So by principle of Mathematical induction, P(n) is true for $n \in N$
## Important conclusion from Binomial Theorem
1 $(x+a)^n =\sum_{k=0}^{n} {}^{n} C_k x^{n-k}a^{k}$ We can easily see that $(x+a)^n$ has $(n+1)$ terms as k can have values from 0 to n 2 The sum of indices of x and a in each is equal to n $x^{n-k}a^{k}$ 3 The coefficient nCr is each term is called binomial coefficient 4 $(x-a)^n$ can be treated as $[x+(-a)]^n$ So $(x -a)^n =\sum_{k=0}^{n} {}^{n} C_k (-1)^k x^{n-k}a^k$ So the terms in the expansion are alternatively positive and negative. The last term is positive or negative depending on the values of n 5 $(1+x)^n$ can be treated as $(x+a)^1$ where x=1 and a=x So $(1 +x)^n =\sum_{k=0}^{n} {}^{n} C_k x^{k}$ This is the expansion is ascending order of power of x 6 $(1+x)^n$ can be treated as $(x+a)^n$ where x=x and a=1 So $(1 +x)^n =\sum_{k=0}^{n} {}^{n} C_k x^{n-k}$ This is the expansion is descending order of power of x 7 $(1-x)^n$ ;can be treated as $(x+a)^n$ where x=1 and a=-x So $(1 -x)^n =\sum_{k=0}^{n} {}^{n} C_k (-1)^k x^{k}$ This is the expansion is ascending order of power of x
Question -1
Expand using Binomial Theorem
$(x^2 +2)^5$
Solution
We know from binomial Theorem
$(x+a)^n = ^{n}C_{0} x^n a^0 + ^{n}C_{1} x^{n-1} a^1 + ^{n}C_{2} x^{n-2} a^2 + .....+ ^{n}C_{r} x^{n-r} a^r ....+^{n}C_{n} x^{0} a^n$
So putting values $x=x^2 ,a=2 \; and \; n=5$
We get
$(x^2 +2)^5= ^{5}C_{0} (x^2)^5 + ^{5}C_{1} (x^2)^4 2^1 + ^{5}C_{2} (x^2)^3 2^2 + ^{5}C_{3} (x^2)^2 2^3 + ^{5}C_{4} (x^2)^1 2^4 + ^{5}C_{5} (x^2)^0 2^5$ $=x^10 +20x^8 +160x^6 +640x^4 +1280x^2 +1024$
Practice Questions
• $(x +2)^6$
• $(1 -x^2)^5$
• $(1 -x)^7$
• $(z - x)^5$
## General Term in Binomial Expansion
$(x+a)^n =\sum_{k=0}^{n} {}^{n} C_k x^{n-k}a^k$
The First term would be = $^{n} C_0 x^n a^0$
The Second term would be =$^{n} C_1 x^{n-1} a^1$
The Third term would be = $^{n} C_2 x^{n-2} a^2$
The Fourth term would be = $^{n} C_3 x^{n-3} a^3$
Like wise (k+1) term would be
$T_{k+1}= ^n C_k x^{n-k}a^k$
This is called the general term also as every term can be find using this term
$T_1= T_{0+1}=^n C_0 x^n a^0$
$T_2= T_{1+1}=^n C_1 x^{n-1} a^1$
Similarly for
$(x -a)^n =\sum_{k=0}^{n} {}^{n} C_k (-1)^k x^{n-k}a^k$
$T_{k+1}= ^n C_k (-1)^k x^{n-k}a^k$
Again similarly for
$(1 +x)^n =\sum_{k=0}^{n} {}^{n} C_k x^{k}$
$T_{k+1}= ^n C_k x^{k}$
Again similarly for
$(1 -x)^n =\sum_{k=0}^{n} {}^{n} C_k (-1)^k x^{k}$
$T_{k+1}= ^n C_k (-1)^k x^{k}$
To summarize it
Binomial term (k+1) term $(x+a)^n =\sum_{k=0}^{n} {}^{n} C_k x^{n-k}a^k$ $T_{k+1}= ^n C_k x^{n-k}a^k$ $(x -a)^n =\sum_{k=0}^{n} {}^{n} C_k (-1)^k x^{n-k}a^k$ $T_{k+1}= ^n C_k (-1)^k x^{n-k}a^k$ $(1 +x)^n =\sum_{k=0}^{n} {}^{n} C_k x^{k}$ $T_{k+1}= ^n C_k x^{k}$ $(1 -x)^n =\sum_{k=0}^{n} {}^{n} C_k (-1)^k x^{k}$ $T_{k+1}= ^n C_k (-1)^k x^{k}$
## Middle Term in Binomial Expansion
• A binomial expansion contains $(n+1)$ terms
• If n is even then the middle term would $[(\frac {n}{2}+1 ]$ th term
• If n is odd,then $\frac{n+1}{2}$ and $\frac {n+3}{3}$ are the middle term
## Solved Examples
Question-1
If the coefficient of $(2k + 4)$ and $(k - 2)$ terms in the expansion of $(1+x)^{24}$ are equal then find the value of k
Solution:
The general term of $(1 + x)^n$ is
$T_{k+1}= ^n C_k x^{k}$
Hence coefficient of (2k + 4)th term will be
$T_{2k+4} = T_{2k+3+1} = ^{24} C_{2k+3}$
and coefficient or (k - 2)th term will be
$T_{k-2} = T_{k-3+1} = ^{24} C_{k-3}$
As per question both the terms are equal
24C2k+3 = 24Ck-3.
or (2k + 3) + (k-3) = 24
k = 8
Practice Question
• Prove that $^nC_0 + ^nC_1 + ^nC_2 + .....+ ^nC_n = 2^n$
• Find the Coefficent of $x^5$ in the expansion $(1+x)^3 (1-x)^6$
• Use Binomial theorem to evaluate $(96)^3$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8782758116722107, "perplexity": 794.3360092598496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038077810.20/warc/CC-MAIN-20210414095300-20210414125300-00073.warc.gz"} |
http://lampx.tugraz.at/~hadley/ss2/linearresponse/dmetal.php | ## Optical properties of a diffusive metal
It is assumed that electrons in a diffusive metal scatter so often that we can average over the scatering events. The differential equation that describes the motion of the electrons is,
### $m\frac{d\vec{v}}{dt}+\frac{e\vec{v}}{\mu} = -e\vec{E}.$
Here $m$ is the mass of an electron, $\vec{v}$ is the velocity of the electron, $-e$ is the charge of an electron, and $\vec{E}$ is the electric field. When a constant electric field is applied, the solution is,
### $\vec{v}= -\mu \vec{E}.$
Thus the (negatively charged) electrons move in the opposite direction as the electric field.
If the electric field is pulsed on, the reponse of the electrons is described by the impulse response function $g(t)$. The impulse response function satisfies the equation,
### $m\frac{dg}{dt}+\frac{eg}{\mu} = -e\delta \left(t\right).$
When the electric field is pulsed on, the electrons suddenly start moving and then their velocity decays exponentially to zero in a time $\tau = m\mu /e$.
### $g\left(t\right)=-\frac{e}{m}\exp \left(-t/\tau\right)H(t).$
Where $H(t)$ is the Heaviside step function.
The scattering time $\tau$ and the electron density $n$ are the only two parameters that are needed to describe many of the optical properties of a diffusive metal. The form below can be used to input $\tau$ and $n$ and then a script calculates and plots the impulse response function, the Fourier transform of the impulse response function, the mobility, the dc conductivity, the frequency dependent complex conductivity, the electric susceptibility, the dielectric function, the plasma frequency, the index of refraction, the extinction coefficient, the absorption coefficient, and the reflectance.
τ = [s] n = [m-3] τ and n for some metals at 273 K:
Impulse response function
The impulse response function describes the velocity of the electrons after the electric field has be pulsed on briefly.
### $g\left(t\right)=-\frac{e}{m}\exp \left(-t/\tau\right)H(t).$
The impulse response function can be decomposed into its even and odd components.
### $g\left(t\right)=E\left(t\right)+O\left(t\right).$
(m/e)g(t) t [ps]
Generalized susceptibility
The generalized susceptibility $\chi$ is the Fourier transform of the impulse response function. It can be constructed by assuming a harmonic form for the electric field and the velocity, $E\left(\omega\right)e^{i\omega t}$ and $v\left(\omega\right)e^{i\omega t}$. Substituting this into the differential equation yields,
### $\chi\left(\omega \right)=\frac{v\left(\omega\right)}{E\left(\omega\right)}=-\frac{\mu \left(1-i\omega\tau\right)}{1+\omega^2\tau^2}$
χ(ω) ω [THz]
Complex conductivity
The current density $\vec{j}$ is proportional to the average velocity $\vec{v}$, $\vec{j}=-ne\vec{v}$. The frequency dependent conductivity $\sigma(\omega)$ is the ratio of the current density to the electric field.
### $\sigma\left(\omega \right)=\frac{j\left(\omega\right)}{E\left(\omega\right)}=-ne\frac{v\left(\omega\right)}{E\left(\omega\right)}=ne\frac{\mu \left(1-i\omega\tau\right)}{1+\omega^2\tau^2}$
σ(ω)[108 Ω-1 m-1] ω [THz]
Electric susceptibility
The electric susceptibility $\chi_E$ describes the relationship between the polarization $\vec{P}$ and the electric field $\vec{E}$, $\vec{P} = \epsilon_0\chi_E\vec{E}$. The electric dipole caused by one electron is $-e\vec{r}$ where $\vec{r}$ is the displacement of the electron from its equilibrium position. The polarization is the electric dipole caused by one electron times the electron density, $\vec{P}=-ne\vec{r}$. Assuming a harmonic form for $\vec{P}$, $\vec{r}$,and $\vec{E}$; the frequency dependent electric susceptibility is,
### $\chi_E\left(\omega \right)=\frac{P\left(\omega\right)}{\epsilon_0 E\left(\omega\right)}=\frac{-ner\left(\omega\right)}{\epsilon_0 E\left(\omega\right)}=\frac{-nev\left(\omega\right)}{i\omega \epsilon_0 E\left(\omega\right)}=-\frac{ne\mu}{\omega \epsilon_0}\left(\frac{ \omega\tau+i}{1+\omega^2\tau^2} \right) .$
Comparing the electrical conductivity to the electric susceptibility we find,
### $\chi_E\left(\omega \right)=\frac{\sigma\left(\omega\right)}{i\omega \epsilon_0}.$
χE(ω) ω/ωp
Dielectric function
The relative dielectric constant describes the relationship between the electric displacement $\vec{D}$ and the electric field $\vec{E}$, $\vec{D}=\epsilon_r \epsilon_0 \vec{E}= \vec{P}+\epsilon_0 \vec{E}$.
### $\epsilon_r\left(\omega \right)=1+\chi_E=1-\frac{ne\mu}{\omega \epsilon_0}\left(\frac{ \omega\tau+i}{1+\omega^2\tau^2} \right)=1-\omega_p^2\left(\frac{ \omega\tau^2+i\tau}{\omega+\omega^3\tau^2} \right)$
$\omega_p=\sqrt{\frac{ne^2}{m\epsilon_0}}=$
$\epsilon_r\left(\omega \right)$ ω/ωp
The index of refraction n and the extinction coefficient K
The real and imaginary parts of the square root of the dielectric constant are the index of refraction and the extinction coefficient.
### $\sqrt{\epsilon_r}= n+iK$
When waves travel from vacuum into some material, the frequency remains constant. A plane wave moving to the right in vaccuum has the form $\exp\left(i\left(\omega x/c -\omega t\right)\right)$ where $c$ is the speed of light in vacuum. When this wave enters some material, $c \rightarrow c/ \left(n+iK\right)$. The speed of the electromagnetic waves is smaller than the speed of light in vacuum by a factor of $n$. The extinction coeffcient describes the exponential decay of the amplitude of the electromagnetic waves. For waves propagating in the $x$-direction, the amplitude decays like $\exp\left(-ax\right)$ where $a=\omega K/c$.
$\sqrt{\epsilon_r}$ ω/ωp
Absorption coefficient $\alpha$
The absorption coefficient describes how the intensity of the light decays. Since the intensity is proportional to the amplitude of the waves squared, the exponential decay of the intensity is $I= I_0\exp\left(-\alpha x\right)$ where,
### $\alpha =\frac{2\omega K}{c}$
α[106 m-1] ω/ωp
Reflectance
The reflectance of light striking the metal normal to the surface from vacuum ($\epsilon_r=1$) is,
### $R=\frac{\left(n-1\right)^2+K^2}{\left(n+1\right)^2+K^2}$
R ω/ωp
τ = [s] n = [m-3] τ and n for some metals at 273 K: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9845330715179443, "perplexity": 308.3687881344548}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949355.52/warc/CC-MAIN-20230330163823-20230330193823-00139.warc.gz"} |
https://eprints.lancs.ac.uk/id/eprint/123994/ | # The rigidity of infinite graphs
Kitson, D. and C. Power, S. (2018) The rigidity of infinite graphs. Discrete and Computational Geometry, 60 (3). pp. 531-557. ISSN 0179-5376
Preview
PDF (The_rigidity_of_infinite_graphs)
The_rigidity_of_infinite_graphs.pdf - Accepted Version
## Abstract
A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in the normed spaces $(\bR^d,\|\cdot \|_q)$, for $d\geq 2$ and $1 <q < \infty$. Generalisations are obtained for the Laman combinatorial characterisation of generic infinitesimal rigidity for finite graphs in $(\bR^2,\|\cdot \|_2)$. Also Tay's multi-graph characterisation of generic infinitesimal rigidity for finite body-bar frameworks in $(\bR^d,\|\cdot\|_2)$ is generalised to the non-Euclidean norms and to countably infinite graphs. For all dimensions and norms it is shown that a generically rigid countable simple graph is the direct limit $G= \varinjlim G_k$ of an inclusion tower of finite graphs $G_1 \subseteq G_2 \subseteq \dots$ for which the inclusions satisfy a relative rigidity property. For $d\geq 3$ a countable graph which is rigid for generic placements in $\bR^d$ may fail the stronger property of sequential rigidity, while for $d=2$ the properties are equivalent.
Item Type:
Journal Article
Journal or Publication Title:
Discrete and Computational Geometry
The final publication is available at Springer via http://dx.doi.org/10.1007/s00454-018-9993-0
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600/2614
Subjects:
Departments:
ID Code:
123994
Deposited By:
Deposited On:
09 Mar 2018 16:42
Refereed?:
Yes
Published?:
Published | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8954364061355591, "perplexity": 2450.528205365875}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335444.58/warc/CC-MAIN-20220930051717-20220930081717-00469.warc.gz"} |
http://mathhelpforum.com/number-theory/62311-how-do-i-show-transendental.html | # Thread: How do i show this is transendental
1. ## How do i show this is transendental
If e-pi is transendental over Q, how can i show that
e^3 -3e^2.pi + 3e.pi^2 -pi^3 is transendental over Q.
I guess you start off by assuming e-pi is algebraic, but then i dont know how to go about it. Also if i factorise e^3 -3e^2.pi + 3e.pi^2 -pi^3 i get (e-pi)^3. Need help from here, thanks.
2. Maybe I am wrong but suppose that (e-pi)^3 is not transcental
It means that (e-pi)^3 is solution of a polynomial equation with integer coefficients
$a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\;=\;0$
Then
$a_n((e-\pi)^3)^n + a_{n-1}((e-\pi)^3)^{n-1} + ... + a_1(e-\pi)^3 + a_0\;=\; 0$
$a_n(e-\pi)^{3n} + a_{n-1}(e-\pi)^{(3n-3)} + ... + a_1(e-\pi)^3 + a_0\;=\; 0$
Then e-pi is solution of
$a_nx^{3n} + a_{n-1}x^{(3n-3)} + ... + a_1x^3 + a_0\;=\; 0$
Which is not possible because e-pi is supposed transcendental
3. Can anyone check if the above is correct please. Thanks.
4. Originally Posted by thegarden
If e-pi is transendental over Q, how can i show that
e^3 -3e^2.pi + 3e.pi^2 -pi^3 is transendental over Q.
I guess you start off by assuming e-pi is algebraic, but then i dont know how to go about it. Also if i factorise e^3 -3e^2.pi + 3e.pi^2 -pi^3 i get (e-pi)^3. Need help from here, thanks.
If $e-\pi$ is transcendental so is $(e-\pi)^n,\ n \in \mathbb{N}$. Now put $n=3$ and expand.
CB
5. Originally Posted by thegarden
If e-pi is transendental over Q, how can i show that
e^3 -3e^2.pi + 3e.pi^2 -pi^3 is transendental over Q.
I guess you start off by assuming e-pi is algebraic, but then i dont know how to go about it. Also if i factorise e^3 -3e^2.pi + 3e.pi^2 -pi^3 i get (e-pi)^3. Need help from here, thanks.
Thread closed due to OP deleting questions after getting help. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9373780488967896, "perplexity": 2674.068105377032}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105326.6/warc/CC-MAIN-20170819070335-20170819090335-00158.warc.gz"} |
http://www.emis.de/classics/Erdos/cit/41710039.htm | ## Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 417.10039
Autor: Erdös, Paul; Katai, I.
Title: On the growth of some additive functions on small intervals. (In English)
Source: Acta Math. Acad. Sci. Hung. 33, 345-359 (1979).
Review: Let g: N ––> R denote a non-negative strongly additive function, and let fk(n) = max{g(n+j): j = 1,...,k}. The authors give conditions which imply that for every \epsilon > 0 and every k0 the inequality fk(n) < (1+\epsilon)fk(0) holds for k \geq k0 and all but \delta(\epsilon,k0)x integers n in [1,x], \delta(\epsilon,k0) ––> 0 for k0 ––> oo. Some questions concerning the necessity of the conditions remain open. The main part of the paper is devoted to the special case g = \omega, where \omega(n) denotes the number of distinct prime factors of n in N. Let
0k(n) = max{\omega(n+j): j = 1,...,k},ok(n) = max{\omega(n+j): j = 1,...,k}.
The authors prove, by use of Brun's sieve, that for every \epsilon > 0 the inequalities ( log2 = log log)
0k(n) \geq (1-\epsilon)\rho(\frac{log k}{log2n}) log2n, ok(n) \leq (\overline{\rho}(\frac{log k}{log2n})+\epsilon) log2n
hold for every k \geq 1 apart from a set of n's having zero density. Here \rho, \overline{\rho} are defined as the inverse functions of \Psi with \Psi(r) = r log r/e +1 for r \geq 1 resp. 0 < r \leq 1, \overline{\rho}(u) = 0 for u \geq 1. This result corresponds to similar upper resp. lower bounds obtained by I.Kátai [Publ. Math., Debrecen 18, 171-175 (1971; Zbl 261.10029)].
Reviewer: L.Lucht
Classif.: * 11N35 Sieves
11N05 Distribution of primes
11N37 Asymptotic results on arithmetic functions
Keywords: sieve methods; additive functions; growth; strongly additive function; number of distinct prime factors
Citations: Zbl.261.10029
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9889072775840759, "perplexity": 3454.3941227496643}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931006747.51/warc/CC-MAIN-20141125155646-00017-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/40013/does-the-following-dynamic-system-converge-to-a-steady-state | does the following dynamic system converge to a steady state?
This is an economics problem, but I'm pretty sure this kind of thing comes up elsewhere. I've used dynamic programming to find the optimal path of a system (law of motion), which is:
$k_{t+1}=\beta\alpha k_t^\alpha$,
where $\alpha,\beta \in R(0,1)$
now I want to find out if this expression converges to a steady state, defined as
$\lim_{t\to\infty} k_{t+1}=k_t=k^*.$
I thought about trying to tackle it as an eigenvalue problem, of the form
$\left[ \begin{matrix} \log k(t+1)\\ \log k(t+2) \end{matrix} \right] = \left[ \begin{matrix} 0 &1\\ 0 &\alpha \end{matrix}\right] \left[ \begin{matrix} \log k(t)\\ \log k(t+1)\end{matrix} \right] + \left[ \begin{matrix} 0\\ \log\beta + \log\alpha \end{matrix} \right]$
or
$U_{t+1} = AU_t +B$
My question is this: will the system converge as long as the eigenvalues of A are inside the unit circle? I'm concerned about how the contribution of B behaves over time. Am I on the right track with this thing at all?
-
Let $x_t=\log k_t$, then $x_{t+1}=\alpha x_t+b$ with $b=\log(\alpha\beta)$. Since $\alpha\ne1$, this is equivalent to $x_{t+1}-x^*=\alpha (x_t-x^*)$ with $x^*=b/(1-\alpha)$, hence $x_t=x^*+\alpha^t(x_0-x^*)$ is the explicit solution of your dynamical system. Since $|\alpha|<1$, $x_t\to x^*$ for every starting point $x_0=\log k_0$, which means that $k_t\to k^*$ with $k^*=\exp(x^*)=(\alpha\beta)^{1/(1-\alpha)}$.
Thank you very much. I think $a=\alpha$ though, not $\log\alpha$. – jefflovejapan May 20 '11 at 1:55 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9809846878051758, "perplexity": 147.31541314128611}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443736678861.8/warc/CC-MAIN-20151001215758-00165-ip-10-137-6-227.ec2.internal.warc.gz"} |
http://www.computer.org/csdl/trans/tm/2014/01/ttm2014010146-abs.html | Subscribe
Issue No.01 - Jan. (2014 vol.13)
pp: 146-158
Yu Cheng , Dept. of Electr. & Comput. Eng., Illinois Inst. of Technol., Chicago, IL, USA
Weihua Zhuang , Dept. of Electr. & Comput. Eng., Univ. of Waterloo, Waterloo, ON, Canada
ABSTRACT
The distributed nature of the CSMA/CA-based wireless protocols, for example, the IEEE 802.11 distributed coordinated function (DCF), allows malicious nodes to deliberately manipulate their backoff parameters and, thus, unfairly gain a large share of the network throughput. In this paper, we first design a real-time backoff misbehavior detector, termed as the fair share detector (FS detector), which exploits the nonparametric cumulative sum (CUSUM) test to quickly find a selfish malicious node without any a priori knowledge of the statistics of the selfish misbehavior. While most of the existing schemes for selfish misbehavior detection depend on heuristic parameter configuration and experimental performance evaluation, we develop a Markov chain-based analytical model to systematically study the performance of the FS detector in real-time backoff misbehavior detection. Based on the analytical model, we can quantitatively compute the system configuration parameters for guaranteed performance in terms of average false positive rate, average detection delay, and missed detection ratio under a detection delay constraint. We present thorough simulation results to confirm the accuracy of our theoretical analysis as well as demonstrate the performance of the developed FS detector.
INDEX TERMS
Detectors, IEEE 802.11 Standards, Markov processes, Protocols, Delay, Analytical models, Real-time systems,CUSUM test, Detectors, IEEE 802.11 Standards, Markov processes, Protocols, Delay, Analytical models, Real-time systems, Markov chain model, Selfish misbehavior, real-time detection, IEEE 802.11
CITATION
Jin Tang, Yu Cheng, Weihua Zhuang, "Real-Time Misbehavior Detection in IEEE 802.11-Based Wireless Networks: An Analytical Approach", IEEE Transactions on Mobile Computing, vol.13, no. 1, pp. 146-158, Jan. 2014, doi:10.1109/TMC.2012.227
REFERENCES | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8135416507720947, "perplexity": 3622.8251333611756}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398538144.64/warc/CC-MAIN-20151124205538-00047-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/146071/prove-or-disprove-mathbbq-is-isomorphic-to-mathbbz-times-mathb/146089 | # Prove or disprove: $(\mathbb{Q}, +)$ is isomorphic to $(\mathbb{Z} \times \mathbb{Z}, +)$?
Prove or disprove: $\mathbb{Q}$ is isomorphic to $\mathbb{Z} \times \mathbb{Z}$. I mean the groups $(\mathbb Q, +)$ and $(\mathbb Z \times \mathbb Z,+).$ Is there an isomorphism?
-
Isomorphic as what? Fields? Rings? $\mathbb{Z}$-modules? – Neal May 17 '12 at 0:37
Let $\phi: \mathbb{Q}\to \mathbb{Z}\times \mathbb{Z}$ be an isomorphism, and suppose that $\phi(a)=(1,0)$. Then $\phi(\frac{a}{2}+\frac{a}{2})=\phi(\frac{a}{2})+\phi(\frac{a}{2})=(1,0)$. However, there is no element $u$ of $\mathbb{Z}\times \mathbb{Z}$ such that $u+u=(1,0)$. – André Nicolas May 17 '12 at 1:23
AHH My answer and @André's is basically saying that there isn't an isomorphism. If such an isomorphism were to exist, then that leads to a contradiction. You don't need to know exactly what $\phi$ is for every element in $\mathbb{Q}$, you just need one thing that doesn't make sense. – Thomas May 17 '12 at 2:00
@DougSpoonwood: In general, if $X$ is an infinite set and $n$ is some finite number then $|X^n|=|X|$ (it is sufficient to prove $|X^2|=X$). It is only when you try to do things like $X^{\aleph_0}$ that you go "up" a level or too. However, the fact that $\mathbb{R}^2=\{(a, b): a, b\in\mathbb{R}\}$ and $\mathbb{C}=\{a+ib: a, b\in\mathbb{R}\}$ have the same cardinality is surely easy, as the map $(a, b)\mapsto a+ib$ is a bijection... – user1729 May 17 '12 at 9:14
Yet another way to see the two cannot be isomorphic as additive groups: if $a,b\in\mathbb{Q}$, and neither $a$ nor $b$ are equal to $0$, then $\langle a\rangle\cap\langle b\rangle\neq\{0\}$; that is, any two nontrivial subgroups intersect nontrivially. To see this, write $a=\frac{r}{s}$, $b=\frac{u}{v}$, with $r,s,u,v\in\mathbb{Z}$, $\gcd(r,s)=\gcd(u,v)=1$. Then $(su)a = (rv)b\neq 0$ lies in the intersection, so the intersection is nontrivial.
However, in $\mathbb{Z}\times\mathbb{Z}$, the elements $(1,0)$ and $(0,1)$ are both nontrivial, but $\langle (1,0)\rangle\cap\langle (0,1)\rangle = \{(0,0)\}$.
-
$\langle a\rangle\cap\langle b\rangle\neq\{0\}$ $\implies \{0\} \langle a\rangle$ and $\{0\} \langle b\rangle$ $\implies \langle a\rangle$ and $\langle b\rangle$ are not subgroups of $\mathbb{Q}$. – AHH May 17 '12 at 3:04
@AHH: No; $\langle a\rangle\cap\langle b\rangle\neq\{0\}$ means that the intersection is not just $0$. That could be either because $0$ is not in the intersection, or, in this case, because therre are things other than $0$** that are *also* in the intersection. It does not mean that $0$ is not in the intersection. **By definition, $\langle a\rangle$ means "the smallest subgroup that contains $a$", so it must be a subgroup. – Arturo Magidin May 17 '12 at 3:06
I assume that you are asking whether we have an isomorphism of additive groups.
In that case, assume that $\phi: \mathbb{Q} \to \mathbb{Z}\times \mathbb{Z}$ is such an isomorphism. So we have for example that $\phi(0) = (0,0)$. Let $a\in \mathbb{Q}$ be such that $\phi(a) = (1,0)$ and $b$ be such that $\phi(b) = (0,1)$. Then we see that $\mathbb{Q}$ is equal to $\{na + mb \lvert n,m \in \mathbb{Z}\}$. This is a contradiction...
(Hence the argument is that $\mathbb{Q}$ is not finitely generated while $\mathbb{Z}\times \mathbb{Z}$ is.)
(I will leave the details to you.)
-
Note that: $\forall a \in \mathbf{Q} \implies a = m/n : m,n \in \mathbf{Z} , n \neq zero.$ – AHH May 17 '12 at 1:43
Another argument that you can construct with the following (be sure you can prove/answer every section):
1) An abelian additive group $\,A\,$ is said to be divisible if $\,\forall a\in A\,\,n\in\mathbb{N}\,\,\exists b\in A\,\,s.t.\,\,nb=a\,$ . To be sure, $n\neq 0$
2) $\,\mathbb{Q}\,$ is a divisible group
3) Any homomorphic image of a divisible group is a divisible group
4) Is $\,\mathbb{Z}\times\mathbb{Z}\,$ divisible?
-
Let $\phi: \mathbb{Q} \to \mathbb{Z}\times \mathbb{Z}$ be a homomorphism. Fix $u/v\in\mathbb{Q}$ and let $(a_n,b_n)=\phi(u/v^n)$. Since $\phi(u/v)=v^{n-1}\phi(u/v^n)$, we get $a_1=v^{n-1}a_n$ and $b_1=v^{n-1}b_n$ for all $n\in\mathbb N$, which is clearly impossible unless $\phi(u/v)=(a_1,b_1)=(0,0)$.
So, the only homomorphism $\mathbb{Q} \to \mathbb{Z}\times \mathbb{Z}$ is the zero map, and there is no chance of an isomorphism.
-
Can we find $\phi: \mathbb{Q} \to \mathbb{Z}\times {0} \varsubsetneqq \mathbb{Z}\times \mathbb{Z}$ "homomrphism"?! – AHH May 17 '12 at 3:24
@AHH, no, by the same reason. – lhf May 17 '12 at 3:25
if $\phi(a)= \lfloor a \rfloor ,\forall a \in \mathbb{Q}$, then $\phi$ a homomrphism. – AHH May 17 '12 at 3:33
@AHH, no, it's not: $\phi(1/2)+\phi(1/2)=0$ but $\phi(1)=1$. – lhf May 17 '12 at 3:34
good counterexample :) – AHH May 17 '12 at 3:36
Another idea: suppose there is an isomorphism $\mathbb Q \to \mathbb Z \oplus \mathbb Z$, then tensor both sides $\otimes_\mathbb{Z} \mathbb Q$, and get a $\mathbb Q$-module isomorphism $\mathbb Q = \mathbb Q \otimes_\mathbb{Z} \mathbb Q \to (\mathbb Z \oplus \mathbb Z)\otimes_\mathbb{Z} \mathbb Q = \mathbb Q\oplus \mathbb Q$.
-
This also works with $\otimes_\mathbb{Z} \mathbb Z /n$ for any $n\ge 2$, giving an isomorphism $0 \to \mathbb Z/n \oplus \mathbb Z/n$, which of course is just a rephrasing of some of the arguments elsewhere. I like applying functors and seeing things are different that way, it's the algebraic topologist in me. – Justin Young May 19 '12 at 22:10
You can show that the group $(\mathbb{Q}, +)$ has no proper subgroup of finite index, but for example $\mathbb{Z} \times 2\mathbb{Z}$ has finite index in ($\mathbb{Z} \times \mathbb{Z}, +)$.
-
That's not true - every proper, non-trivial subgroup of $(\mathbb{Q}, +)$ has finite index! (Remember, every quotient is finite.) – user1729 May 17 '12 at 10:13
@user1729: If $M$ is a proper subgroup of finite index $d$, then for any $q \in \mathbb{Q}$ you have $d(q + M) = M$ so $dq \in M$. But take some $r \not\in M$ and you get $d(r/d) \not\in M$, a contradiction. – Mikko Korhonen May 17 '12 at 10:26
Sorry - my brain is slow this morning... – user1729 May 17 '12 at 10:34
Another way of seeing this (yes, there are many ways!) is to notice that two isomorphic groups have the same quotients. That is, if $G\cong H$ and $G\twoheadrightarrow K$ then $H \twoheadrightarrow K$.
Now, by this question (which was only asked the other day, which is why I am posting this answer!), we know that every proper quotient of $\mathbb{Q}$ is torsion (that is, every element has finite order). On the other hand, $\mathbb{Z}\times\mathbb{Z}$ has a torsion-free proper quotient, $\mathbb{Z}\times\mathbb{Z}\twoheadrightarrow \mathbb{Z}$. Thus, they cannot be isomorphic.
(Indeed, this actually proves that there cannot be a homomorphism from $\mathbb{Q}$ to $\mathbb{Z}\times\mathbb{Z}$, as lhf has already shown - the result we use tells us that the map cannot have non-trivial kernel, while this proves that the kernel cannot be trivial either.)
-
In ($\mathbb{Q}$, +) every element has a "square root": for any $q \in \mathbb{Q}$ there is an $x \in \mathbb{Q}$ such that $x + x = q$. But in $(\mathbb{Z} \times \mathbb{Z}, +)$ for any element $(a,b) \in \mathbb{Z} \times \mathbb{Z}$ there is an $x \in \mathbb{Z} \times \mathbb{Z}$ such that $x + x = (a,b)$ if and only if both $a$ and $b$ are even.
This motivates the following proof. If $\phi: \mathbb{Q} \rightarrow \mathbb{Z} \times \mathbb{Z}$ were an isomorphism, then $\phi(q) = (1,1)$ for some $q \in \mathbb{Q}$. Then $(1,1) = \phi(q/2 + q/2) = \phi(q/2) + \phi(q/2)$, but this is a contradiction since $1$ is not even.
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https://stacks.math.columbia.edu/tag/060U | Lemma 73.6.1. Let $X$ be an algebraic space over a scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be an fpqc covering such that each $f_ i^*\mathcal{F}$ is a finite type $\mathcal{O}_{X_ i}$-module. Then $\mathcal{F}$ is a finite type $\mathcal{O}_ X$-module.
Proof. This follows from the case of schemes, see Descent, Lemma 35.7.1, by étale localization. $\square$
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9959806799888611, "perplexity": 232.23198086272768}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500619.96/warc/CC-MAIN-20230207134453-20230207164453-00690.warc.gz"} |
https://euro-math-soc.eu/review/hypoelliptic-laplacian-and-ray-singer-metrics | # The Hypoelliptic Laplacian and Ray-Singer Metrics
In this book, analytic theory of the hypoelliptic Laplacian is developed and corresponding results on the associated Ray-Singer torsion are established. The hypoelliptic Laplacian is a second order differential operator defined on the cotangent bundle of a compact Riemannian manifold. It is supposed to interpolate the classical Laplacian and an operator related to a geodesic flow. In this way, it gives a semiclassical version of the fact that the Witten Laplacian on the corresponding loop space should interpolate between the classical Hodge Laplacian and Morse theory for the same energy functional. The authors develop Hodge theory for the studied Laplacian and the local index theory of the associated heat kernel. They adapt the theory of Ray-Singer torsion and the analytic torsion forms of Bismut-Lott and they develop an appropriate pseudodifferential calculus. They show that when the deformation parameter tends to zero, the hypoelliptic Laplacian tends to the usual Hodge Laplacian of the base space by a collapsing argument letting the fibre shrink to a point. They also obtain small time asymptotics for the supertrace of the associated heat kernel. A comparison formula between the elliptic and hypoelliptic analytic torsions is derived, studying an equivariant setting of the Ray-Singer torsion of the studied operator and the associated Ray-Singer metrics on the determinant of the cohomology ring. To obtain localisation estimates, probabilistic methods related to diffusion processes are used.
Reviewer:
skr
Book details
## Publisher:
Published:
2008
ISBN:
978-0-691-13732-2
Price:
USD 45
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https://socratic.org/questions/how-does-the-radius-affect-the-moment-of-inertia | Physics
Topics
# How does the radius affect the moment of inertia?
Mar 23, 2018
Moment of inertia is directly proportional to the square of the radius.
#### Explanation:
The moment of inertia, $I$, of a single mass, $M$, being twirled by a thread of length, $R$, is
$I = M \cdot {R}^{2}$
A body that is being rotated will closely resemble that relationship. The formulas for various geometric shapes are derived with integration. For example, for a solid sphere, moment of inertia is
$I = \left(\frac{2}{5}\right) \cdot M \cdot {R}^{2}$
I hope this helps,
Steve
##### Impact of this question
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https://www.physicsforums.com/threads/kinematics-problem-initial-acceleration.880199/ | # Kinematics problem (initial acceleration)
Tags:
1. Jul 27, 2016
### rndaryam
• Thread moved from the technical forums, so no HH Template is shown.
A ball is rolling over a soccer field with constant velocity Vb > 0 at an angle of 45° to the goal line. It starts at the corner of the field (distance d from the middle of the goal line) at the time t0 = 0. At the same time a player starts heading towards the goal on a path perpendicular to the goal line. He starts from a position at distance 2d in front of the middle of the goal line) with ap(t) = a0 (1− (t/ß))
I've solved the problems but the left thing to do is just the initial acceleration. I use integration to find these solutions:
- Distance covered by the ball sb=Vb*t
- Time when ball is in front of the middle of the goal line tA = 2√d / Vb (with distance sA = d / cos45°)
- Velocity of player vp = a0 (t − (t^1/2ß))
- Distance covered by the player sp = a0 / 6*ß (3ßt^2−t^3)
- Velocity of the player when he reach the point (the point after the ball reach distance sA) vpA = a0*d / vb (√2 − d/(vb*ß)) *I'm inserting tA into vp, am I right?
- Initial acceleration of the player in order to reach the point (the point after the ball reach distance sA) at the same time as the ball a0 = ??
Any ideas would be very appreciated. Thanks.
2. Jul 27, 2016
### haruspex
Did you mean that?
Are you saying it is $a_0(t-(\frac {t^{\frac 12}}{\beta}))$?
I do not understand how you get that from ap(t) = a0 (1− (t/ß))
3. Jul 27, 2016
### jbriggs444
When posing these problems, it is good practice to define all of your variables. It is good form to use subscripts rather than mushing letters together and expecting the reader to understand the intended meaning.
Vb becomes Vb, the velocity of the ball on its 45 degree trajectory.
t0 becomes t0, the starting time. It is unnecessary to specify that t0 = 0, but it does not hurt either. [and eventually saves a fair bit of ink].
t becomes the elapsed time such that t=t0 at the outset.
d is good as it stands. It is the width of the field and also the distance that the player stands away from the goal line at time t0
That still leaves us guessing at what ap(t), a0 and ß are supposed to mean.
Guess: ap(t) is the acceleration of the player at time t, more easily understood as ap(t).
Guess: a0 is a free parameter, the starting acceleration of the player.
Guess: This is multiplied by ${(1-\frac{t-t_0}{\beta}})$ to simulate a declining rate of acceleration (which may eventually go very negative).
It is also good form to specify what question is being asked. What is supposed to be calculated here?
So now sb is the distance covered by the ball along its 45 degree trajectory. It is a function of time and should be expressed as sb(t) if one is being picky.
So now we have tA as the time when the ball is in front of the goal. But surely that should be directly proportional to d rather than directly proportional to the square root of d? Perhaps you meant to say $t_A = \frac{d\sqrt{2}}{V_b}$
This should be vp(t). But I question the result of your integration. Surely you meant to write $a_0(t\ -\ \frac{t^2}{2\beta})$?
Note that you just broke your own case convention. You have Vb for the ball and vp for the player. Not cool.
This seems right and recovers from the previous error. Repairing the formatting and re-distributing we have:
$s_p = a_0(\frac{t^2}{2} - \frac{t^3}{6\beta})$
That should be vp(tA). Yes, you should be substituting tA into the formula for vp(t).
So you are looking for the required initial acceleration of the player in terms of $\beta$, d and Vb if the player is to arrive at the ball exactly as it crosses in front of the middle of the goal?
4. Aug 7, 2016
### rndaryam
Thank you so much for the hints, I already got all the answers. I'm so sorry because this is my first thread, will improve it for next time :)
Also thanks for haruspex, I've already got the solution as jbriqqs444 gave me the hints to solve it :)
Draft saved Draft deleted
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https://www.physicsforums.com/threads/heat-transfer-question.184620/ | # Heat transfer question
1. Sep 14, 2007
### kayjaygee_13
this question was posed by someone on a computer forum and is probably easy for some of you, so here goes :
imagine water in an open container which is perfectly insulated on the sides and bottom. the water is stationary and heat is being added from the top (assume that the surroundings above the water are at a much higher temperature than the water itself). eventually, will the complete mass of water reach a higher equilibrium temperature? if so, what mechanism of heat transfer causes this to happen?
2. Sep 14, 2007
### Cyrus
Thats wrong on many levels. First of all, the 'mass' of water wont reach one temperature, there will be a temperature gradient across the water. Second, the water will reach an equilibrium temperature. That temperature wont be higher than the source temperature, or heat would flow back into the surroundings. This is all done by conduction and convection.
3. Sep 14, 2007
### Integral
Staff Emeritus
Cyrus,
You may want to reread the problem statement. The container is insulated on the bottom and sides, no heat loss implies that the water temperature will stabilize at the same temp as the air.
The mechanism is conduction.
4. Sep 14, 2007
### kayjaygee_13
i guess my initial post was a little ambiguous. when i said "higher equilibrium temperature", i meant higher than the original water temperature before heat addition, i didn't mean higher than the ambient.
why will there be convection? from my limited knowledge, convection is a result of varying densities, however since heat is added from the top, the colder denser fluid stays at the bottom and lighter warm fluid stays at the top.
will there still be a gradient when time equals infinity or will the entire mass of water be the same temperature as the surroundings?
i'm asking because i don't know, so correct me wherever i'm wrong.
5. Sep 14, 2007
### Cyrus
Right, the top surface of the water will be at the same temperature as the air; however, the sides wont. The sides will be at room temperature, because its a perfect insulator. You will have a thermal gradient in the water, and hence convection inside the water. Conduction at the top water/air interface.
6. Sep 14, 2007
### alvaros
I think the sides will be at the same temperature of the water.
And you forgot radiation and absotions of this radiation in/of the water. ( and reflection and absortion of the radiation at the sides and bottom of the container )
And if the container is open the water will evaporate.
7. Sep 14, 2007
### Cyrus
Radiation will be negligible compared to everything else alvaros. Maybe 1% in magnitude or less. If the water at the insulator interface were the same temperature as the heated interface, there would be a temperature jump from room temperature to air temperature at that boundary, and hence an infinte slope jump. This makes no physical sense.
8. Sep 14, 2007
### f95toli
Is this a practial question of a gedanken experiment?
In a real experiment you could simply use a thermos bottle with an insulation vacuum as the container. In this case the inner sides of the container will be, to a very good approximation, at the same temperature as the water and the outer sides the temperature of the room.
So in a real experiment the "dynamics" would be almost complettely governed by the flux of energy at the surface.
9. Sep 14, 2007
### AlephZero
Trying to define what you mean by "the temperature of a perfect insulator" is more about philosophy than engineering. And it does't really matter, because if it's a perfect insulator there is no heat flow whatever its temperature is.
Re the "temperature jump" between the air and the water, it does make sense in terms of a kinetic theory model. The average KE of the the gas molecules can be different from the average KE of the water molecules, therefore the temperatures at the interface can be different - though different temperatures are not in equilibrium of course. If the temperatures at the interface were the same (on a practical size scale), Newton's law of convective cooling wouldn't make any sense.
10. Sep 17, 2007
### Cyrus
Can you explain that some more Aleph? It seems to me that a jump in temperature at the boundary would mean an infinite dT/dx. The only way to make delta(E) zero at the boundary is if k=0. This seems shady to me as you would have 0*infinity, which is not really zero mathematically since its indeterminate.
Even if it was a really *REALLY* good material with k=10^-23, made from unobtanium, the fact that dT/dx is infinite would make the delta(E) blow up.
I can only see the solution making any sense if the boundaries are at room temperature and the top surface is at the source temperature. That temperature difference at each face would cause convection due to the different densities.
11. Sep 18, 2007
### AlephZero
With convection, there is flow in the fluids. Think of hot solid object in cold air. The air molecules get heated by colliding with the surface and rebounding with higher energy. It doesn't follow that each individual molecule heats up the same temp of the solid in just one collision, and even if it did it would cool down when it collided with another colder air molecule. The *average* temperature of the air, very close to the surface, is less than the surface temperature of the solid.
That's what Newton's Law of Cooling says: across the interface, heat flux Q = h(T_1-T_0) where h is the heat transfer coefficient. H which depends on the surface finish, the flow configuration, and various nondimensional fluid parameters like Reynolds and Prandtl numbers.
This is different from conduction within a solid (or between two solids in contact) where the atoms only vibrate, they don't move around on a global scale. For solid conduction agree with you, the temperature gradient has to be finite and the temperature distribution has to be continuous. At least, that's true for the "classical" model of heat flow - at the atomic scale, considering the temperature (or KE) of individual particles is a different story.
For a solid that is an absolutely perfect insulator (K = 0, not 10^-23) then the diffusion/conduction equation just says dT/dt = 0 at every point. So there can (conceptually) be any temperature distribution you like in the solid, and it never changes. There's no reason why dT/dx has to be continuous because dT/dx doesn't come into the equation (or it's multiplied by 0, if you prefer). That's what I meant by saying the concept of "temperature in a perfect insulator" is not very well defined (or useful).
But of course there is no such thing as a perfect insulator.
The question of what happens in convection between a perfect insulator and a fluid is another "what is 0/0" type question I think. There can't be any finite amount of heat flux going into or out of the solid, because the heat can't go anywhere inside the solid. I think all you can really say is dT/dn is zero at the boundary of the fluid (where n is the normal vector) and the temperature in the fluid can be whatever it wants to be. There can be heat transfer along the surface, inside the fluid, so it doesn't mean the fluid temp has to be constant over the whole boundary. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8165006041526794, "perplexity": 582.9155785988835}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560280292.50/warc/CC-MAIN-20170116095120-00077-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/a-find-each-of-the-following-critical-values-i-c-2-077-77-ii-t008-88-iii-f003-133-33-b-fin-q3660204 | ## Critical values and probabilities
a) Find each of the following critical values. i. c 2 0.77 (77) ii. t0.08 (88) iii. F0.03 (133, 33) b) Find the constant k such that i. P c 2 (33) > k = 0.003 ii. P (t(6) < k) = 0.66 iii. P (F(84, 48) > k) = 0.04 c) Find each of the following probabilities. i. P 1.55 < c 2 (15) < 15.5 ii. P (F(14, 6) > 1.46) iii. P F69 96 099
• In hypothesis testing, the critical value is a number that separates a region where the null hypothisis will not be rejected from a region where it will be rejected.
Take the usual easy problem about an average of a very large number of measurements so the sample mean is approximately normally distributed. Say that the standard error is 5. We make a null and alternate hypothesis, e.g.
H0: mean=44
H1: mean not= 44
We will reject H0 (at the alpha=0.05 significance level) if the sample mean is below 44-1.96*5
or if it is above 44+5*1.96
Roughly 34 and 54 are the critical values for this test.
If you instead, convert to a standard normal, you may call
-1.96 and 1.96 the critical values. This usually is the case when asked to do a hypothesis test.
If you have a TI-83, 2nd DISTR, 3:invnorm(0.025, 44,5)=34.2
invnorm(0.975,44,5) = 53.8
invnorm(0.025,0,1) = -1.96
invnorm(0.975,0,1)= 1.96
There is no inverse on TI-83 for the t-distribution so you need tables.
For a one tail test, there is only one critical value. You really need to draw a picture of your test situation and know on which tail the reject region lies.
Get homework help
More than 200 experts are waiting to help you now... | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.900049090385437, "perplexity": 1495.7448309072092}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368705936437/warc/CC-MAIN-20130516120536-00014-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://questions.examside.com/past-years/gate/gate-ece/communications/noise-in-digital-communication | GATE ECE
Communications
Noise In Digital Communication
Previous Years Questions
## Marks 1
A sinusoidal message signal is converted to a PCM signal using a uniform quantizer. The required signal to quantization noise ratio (SQNR) at the outp...
A binary baseband digital communication system employs the signal $$p\left( t \right) = \left\{ {\matrix{ {{1 \over {\sqrt {{T_s}} }},} & {0... A sinusoidal signal of amplitude A is quantized by a uniform quantizer. Assume that the signal utilizes all the representation levels of the quantizer... The capacity of a Binary Symmetric Channel (BSC) with cross - over probability 0.5 is ______ . Consider the pulse shape s(t) as shown. The impulse response h(t) of the filter matched to this pulse is ... ## Marks 2 A digital communication system uses a repetition code for channel encoding/decoding. During transmission, each bit is repeated three times (0 is trans... An analog pulse s(t) is transmitted over an additive white Gaussian noise (AWGN) channel. The received signal is r(t) = s(t) + n(t), where n(t) is add... Consider a binary, digital communication system which uses pulses g (t) and − g (t)for transmitting bits over an AWGN channel. If the receiver uses a ... The input X to the Binary Symmetric Channel (BSC) shown in the figure is ‘1’ with probability 0.8. The cross-over probability is 1/7. If the received... A source emits bit 0 with probability$${1 \over 3}$$and bit 1 with probability$${2 \over 3}$$. The emitted bits are communicated to the receiver. T... Consider a discrete-time channel Y = X + Z, where the additive noise Z is signal- dependent. In particular, given the trasmitted symbol$$X\, \in \,\{...
Consider a communication scheme where the binary valued signal X satisfies P{X = + 1} = 0.75 and P {X = - 1} = 0.25. The received signal Y = X + Z, wh...
Coherent orthogonal binary FSK modulation is used to transmit two equiprobable symbol waveforms $${s_1}\,(t)\, = \,\alpha \,\,\cos \,\,\,2\,\pi {f_1}... Let U and V be two independent zero mean Gaussian random variables of variances$${{1 \over 4}}$$and$${{1 \over 9}}$$respectively. The probability ... A BPSK scheme operating over an AWGN channel with noise power spectral density of N02, uses equi-probable signals$$${s_1}\left( t \right) = \sqrt ... A binary symmetric channel (BSC) has a transition probability of 1/8. If the binary transmit symbol X is such that P(X =0) = 9/10, then the probabi... A four phase and an eight-phase signal constellation are shown in the figure below. For the constraint that the minimum distance between pairs of ... A four phase and an eight-phase signal constellation are shown in the figure below. Assuming high SNR and that all signals are equally probable, ... Consider a base band binary PAM receiver shown below. The additive channel noise $$n(t)$$ is white with power spectral density $${S_N}\left( f \right)... Consider a base band binary PAM receiver shown below. The additive channel noise$$n(t)$$is white with power spectral density$${S_N}\left( f \right)... The amplitude of random signal is uniformly distributed between $$-$$5V and 5V If the signal to quantization noise ratio required in uniformly quanti... The amplitude of random signal is uniformly distributed between $$-$$5V and 5V If the positive values of the signal are uniformly quantized with a st... Consider a Binary Symmetric Channel (BSC) with probability of error being 'p'. To transit a bit, say 1, we transmit a sequence of three 1s. The receiv... During transmission over a certain binary communication channel, bit errors occurs independently with probability p. The probability of at most one bi... An input to a 6-level quantizer has the probability density function f(X) as shown in the figure. Decision boundaries of the quantizer are chosen so a... An input to a 6-level quantizer has the probability density function f(X) as shown in the figure. Decision boundaries of the quantizer are chosen so a... Two 4-ray signal constellations are shown. It is given that $${\phi _1}$$ and $${\phi _2}$$ constitute an orthonormal basis for the two constellations... Two 4-ray signal constellations are shown. It is given that $${\phi _1}$$ and $${\phi _2}$$ constitute an orthonormal basis for the two constellations... In the following figure the minimum value of the constant “C”, which is to be added to y1(t) such that y1(t) and y2(t) are different, is ... Let $$g\left( t \right){\mkern 1mu} {\mkern 1mu} \,\,\,\,\,{\mkern 1mu} = {\mkern 1mu} {\mkern 1mu} p\left( t \right){}^ * p\left( t \right)$$ where... A signal as shown in figure is applied to a matched filter. Which of the following does represent the output of this matched filter? ... A symmetric three-level midtread quantizer is to be designed assuming equiprobable occurrence of all quantization levels. The quantization noise power... A symmetric three-level midtread quantizer is to be designed assuming equiprobable occurrence of all quantization levels. If the input probability de... Consider a binary digital communication system with equally likely $$0’s$$ and $$1’s$$. When binary $$0$$ is transmitted the voltage at the detector i... If Eb, the energy per bit of a binary digital signal, is 10-5 watt-sec and the one-sided power spectral density of the white noise, N0 = 10-6 W/Hz, th... A sinusoidal signal with peak-to-peak amplitude of 1.536V is quantized into 128 levels using a mid-rise uniform quantizer. The quantization-noise powe... During transmission over a communication channel, bit errors occur independently with probability 'p'. If a block of n bits is transmitted, the probab... The peak-to-peak input to an 8-bit PCM coder is 2 volts . The signal power - to -quantization noise power ratio (in dB) for an input of 0.5 $$\cos \le... The input to a matched filter is given by$$$S\left( t \right) = \left\{ {\matrix{ {10\sin \left( {2\pi \times {{10}^6}t} \right),} & {0 <...
A signal having uniformly distributed amplitude in the interval ( -V, +V ) is to be encoded using PCM with uniform quantization. The signal - to - qua...
In a digital communication system, transmissions of successive bits through a noisy channel are assumed to be independent events with error probabilit...
Companding in PCM systems leads to improved signal - to - quantization noise ratio. This improvement is for
EXAM MAP
Joint Entrance Examination | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9755268692970276, "perplexity": 4832.500206814188}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945248.28/warc/CC-MAIN-20230324051147-20230324081147-00089.warc.gz"} |
http://mathhelpforum.com/pre-calculus/110393-iverse-variation-function-wp.html | # Math Help - Iverse variation of a function WP
1. ## Iverse variation of a function WP
The weight of a body in space varies inversely as the square of its distance from the center of the earth. If something weighs 5 lbs on the surface of the earth, how much does it weigh 1000 miles from the surface of the earth? The radius of the earth is 4,000 miles.
Would it be W= k/d^2 ? Could someone work this out for me.
Thanks
2. Originally Posted by na300zx
The weight of a body in space varies inversely as the square of its distance from the center of the earth. If something weighs 5 lbs on the surface of the earth, how much does it weigh 1000 miles from the surface of the earth? The radius of the earth is 4,000 miles.
Would it be W= k/d^2 ? Could someone work this out for me.
Thanks
$5 = \frac{k}{4000^2}$
solve for $k$, then determine $W$ when $d = 5000$
3. Thanks for your reply. So the correct equation would be k= 80,000,000 W=80,000,000/5000^2 which equals 3.2 lbs? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9528908133506775, "perplexity": 346.3593906795928}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-27/segments/1435375098849.37/warc/CC-MAIN-20150627031818-00292-ip-10-179-60-89.ec2.internal.warc.gz"} |
https://www.lessonplanet.com/teachers/chemical-formulas-10th-higher-ed | # Chemical Formulas
In this chemical compounds worksheet, students write the correct chemical formula for the chemical compounds given. This worksheet has 50 fill in the blank questions. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.962137758731842, "perplexity": 2822.15970004541}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607813.12/warc/CC-MAIN-20170524112717-20170524132717-00555.warc.gz"} |
https://undergroundmathematics.org/product-rule/r5501/solution | Review question
# Can we sketch $y=x^2/(1+x^4)$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource
Ref: R5501
## Solution
Find the coordinates of the turning points on the curve with equation $y=\frac{x^2}{1+x^4}.$
Finding the derivative $y'$ using the product rule with $y=x^2\dfrac{1}{1+x^4}$: $\frac{dy}{dx}=\frac{2x}{1+x^4}-\frac{4x^3\times x^2}{(1+x^4)^2}=\frac{2x(1+x^4)-4x^5}{(1+x^4)^2}.$ So when $y'=0$, \begin{align*} 2x(1+x^4) - 4x^5 &= 0 \\ \Longrightarrow 2x-2x^5 &= 0 \\ \Longrightarrow x(1-x^4) &= 0 \\ \Longrightarrow x(1-x^2)(x^2+1) &=0\\ \quad \Longrightarrow \quad x=0, \,\, x=\pm 1. \end{align*} Substituting back into the equation we find that the coordinates of the turning points are $(0,0)$, $(\pm 1,\dfrac{1}{2})$.
Sketch the curve.
For large $x$, the $x^4$ term dominates, and so $y \rightarrow 0 \quad \text{as} \quad |x| \rightarrow \infty$.
The only root coincides with the stationary point $(0,0)$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 1.0000057220458984, "perplexity": 712.1625781300611}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662522556.18/warc/CC-MAIN-20220518215138-20220519005138-00568.warc.gz"} |
https://www.freemathhelp.com/length-line-segment/ | # Calculate the length of a line segment
Remember that a line segment is the portion of a straight line that directly connects two given points. Unlike a line, it does not extend off to infinity in both directions. To find the length, we just use the distance formula between the two points provided. For lessons like this, often the easiest way to learn is by working out an example.
### Example:
Find the distance between (-2,8) and (-7,-5). Said another way, find the length of the line segment between points (-2,8) and (-7,-5).
First, find the distance between the x-coordinates. To do this, subtract one number from the other and then take its absolute value.
We have: |-2-(-7)| = |5| = 5.
Then repeat with the y-coordinates.
We have: |8-(-5)| = |13| = 13.
NOTE: It does NOT matter which way you subtract the numbers because the absolute value of the answer would be the same anyway.
Finally, to compete the length (or distance), square BOTH values, add them, and take the square root. Here's the first part:
$$5^2 + 13^2 = 25 + 169 = 194$$
Taking the square root of 194 and rounding to TWO decimal places, we get a distance of 13.93:
$$\sqrt(194)=13.93$$
By the way, what you are actually doing is using the Pythagorean Theorem on an imaginary right triangle with the line joining the two lines being the hypotenuse.
The general formula for distance between two points is the following: $$\sqrt{x^2 + y^2}$$, where x and y are the change in x and y between the two points.
Provided by Mr. Feliz | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9104815125465393, "perplexity": 266.7468107297626}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585302.56/warc/CC-MAIN-20211020024111-20211020054111-00638.warc.gz"} |
http://math.stackexchange.com/questions/293067/2-variable-function-continuity | # 2 Variable function continuity
How do you prove that $f(x,y) = x^y$ at $[0,1]\times[1,\infty]$ I tried to prove this by definition, but got stuck when trying to show that the distance between two points approaches $\epsilon$ when they are relatively close.
-
Google Brouwer's fixed point theorem, it should be easy to proceed from there.
-
The function in question is from a subset of $\mathbb{R}^2$ to $\mathbb{R}$, so the notion of a "fixed point" doesn't exist. Moreover, the domain, $[0,1] \times [1,\infty]$ is not compact, and even if it were, the fixed point theorem requires continuity as an assumption, not as a conclusion. Your answer doesn't make any sense. – Goos May 14 '13 at 7:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8847352266311646, "perplexity": 126.8651036910466}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609526252.40/warc/CC-MAIN-20140416005206-00473-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://neurips.cc/Conferences/2016/ScheduleMultitrack?event=6934 | `
Timezone: »
Poster
Review Networks for Caption Generation
Zhilin Yang · Ye Yuan · Yuexin Wu · William Cohen · Russ Salakhutdinov
Mon Dec 05 09:00 AM -- 12:30 PM (PST) @ Area 5+6+7+8 #74 #None
We propose a novel extension of the encoder-decoder framework, called a review network. The review network is generic and can enhance any existing encoder- decoder model: in this paper, we consider RNN decoders with both CNN and RNN encoders. The review network performs a number of review steps with attention mechanism on the encoder hidden states, and outputs a thought vector after each review step; the thought vectors are used as the input of the attention mechanism in the decoder. We show that conventional encoder-decoders are a special case of our framework. Empirically, we show that our framework improves over state-of- the-art encoder-decoder systems on the tasks of image captioning and source code captioning. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8059085011482239, "perplexity": 4341.810838474004}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780056548.77/warc/CC-MAIN-20210918154248-20210918184248-00354.warc.gz"} |
http://physics.stackexchange.com/questions/14235/how-do-we-know-that-c14-decay-is-exponential-and-not-linear/14242 | # How do we know that C14 decay is exponential and not linear?
In my previous question I asked Please explain C14 half-life The OP mentioned that I was thinking of linear decay and C14 was measured in exponential decay.
As I understand it, C14 is always in a state of decay. If we know the exact rate of decay then shouldn't it be linear?
How do we know that C14 decay's exponentially compared to linear and have there been any studies to verify this?
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atomic transmutation are one of the few left out areas in physics that couldn't be answered by any theory, notable among them is QED. – Vineet Menon Sep 2 '11 at 10:03
@Vineet: that's not true, "transmutation" (a.k.a. radioactive decay, fission, and fusion) is explained quite well by the standard model. (Also, I deleted an inappropriate comment here.) – David Z Sep 2 '11 at 15:41
@David: I'm telling you by the book I read by Feynman titled QED. – Vineet Menon Sep 4 '11 at 18:40
@Vineet: yes, QED itself doesn't explain nuclear interactions. You need the weak force for that. If you'd like to discuss this further, let's take it to Physics Chat. – David Z Sep 4 '11 at 21:30
It's also worth noting that there is nothing special about atoms.
If you have any system where in every period of time an event has a certain chance of happening which only depends on internal effects of the object and no memory or communications with others - you will get the same decay curve. It's purely a matter of the statistics.
If you have a handful of coins and every minute toss them all and remove all the heads into a separate pile - the number of coins remaining in the hand will decay with a half-life of 1 minute.
What is special about carbon14 - and why it is useful for archeaology is that new carbon14 is being made all the time in the atmosphere, and while you are alive you take in this new carbon so the decays don't have any effect until you die. It's like tossing the coins, but while you are alive adding new random coins after each toss - but then when you die have somebody else start to remove the heads. If you assume you died with an equal number of heads and tails, you can work out how many tosses have happened since you died - and so how long ago the sample died.
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+1 for the handful of coins analogy. The entire reason for exponential decay is that there is less stuff left that can decay. – JollyJoker Sep 2 '11 at 5:32
Out of all of the explanations, yours makes the most sense :) Thank you. – Jonathon Byrd Sep 2 '11 at 15:37
AC's answer below is better - if the atoms/coins are independant and have no way of effecting or signaling each other to decay then exponential is the only possible function. It takes a bit of thinking about to believe it though! – Martin Beckett Sep 2 '11 at 15:47
I still desire a little bit more comprehensiveness to establish why a random event would have a pdf of occurring at time t of $p(t) = \lambda e^{-\lambda t}$. The coin analog works well as an example with discrete math, but coming up with an intuitive example of a continuous process seems a little bit harder. The examples I can think of are unsavory, like 18th century warfare when they fired in the general direction of the other army but couldn't aim at an individual. In some war examples like that your chance of dying could follow that p(t). – Alan Rominger Sep 2 '11 at 16:09
We can show this by thinking about what is happening.
Suppose we have a set of $N$ nuclei that are all radioactive. Each of these nuclei has a chance of decaying, $\lambda$. In people lifetimes, some people live longer and some live shorter than others, but there is an average lifetime; this is what $\lambda^{-1}$ represents for nuclei.
Now how many nuclei, $\Delta N$, decay over some interval of time, $\Delta t$, can be represented mathematically by (remember $\Delta N < 0$ because nuclei are decaying), $$\Delta N = - \lambda N \Delta t$$
We can rewrite this in differential form as, $$\frac{d N}{N} = -\lambda dt$$
Using standard calculus, $$\int \frac{d N}{N} =- \int \lambda dt$$
We will find that, $$\ln N =- \lambda t + C$$
Standard algebraic manipulation gives,
$$N = e^{-\lambda t}e^{C}$$
At $t=0$ we say that $N=e^C=N_0$, thus we usually write the equation: $$N = N_0 e^{-\lambda t}$$
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Re: How do we know that C14 decay's exponentially compared to linear?
Experiment.
Re: Have there been any studies to verify this?
Yes. You can read a nice writeup of some of the early studies of C14 at the nobel prize website: http://static.nobelprize.org/nobel_prizes/chemistry/laureates/1960/libby-lecture.pdf
But to expand a bit on the underlying model as to why all radioactive decay (and spontaneous emission in general) is exponential in nature:
Radioactive decay is exponential is because the nuclei have no memory. Nuclei don't grow old like people - an 80-year-old man is very different than a 2-year-old boy, and their expected remaining lifetimes are very different. But all nuclei (until the moment they decay) are perfectly identical: they have no memory of when they were born. You can't distinguish a C14 atom that was created 10 seconds ago from one lucky atom that has managed to survive for 10000 years.
Consequently, every C14 nucleus has the same probability of decaying at a given moment in time. Thus, if $N$ is the number of remaining C14 atoms in your sample, $$\frac{\partial N}{\partial t} = -\Gamma \cdot N$$ where $\Gamma$ is some constant that describes how fast the atoms tend to decay. This equation, of course, gives rise to $N(t) = N_0 \cdot e^{-\Gamma t}$, which is just exponential decay.
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Because the half-life of C-14 is long compared to a human life time it takes careful, sophisticated measurements to demonstrate the exponential decay. However, it's probably worth noting that there are many other radioactive isotopes with very short half-lives. It a standard exercise in 1-year college physics labs to measure the exponential decay of radioactive isotopes of Silver. See for example carleton.edu/departments/PHAS/P128/lab/AgDecay04.pdf – Charles E. Grant Sep 2 '11 at 1:24
@Charles E. Grant: I certainly agree: if all you wanted to do was demonstrate that radioactive decay is exponential in nature, other isotopes would be a much better choice. But because the lifetime of C-14 is so important (carbon dating, etc.), folks have gone to considerable effort to nail down the lifetime. And since the original questioner seemed fixated on C14 for some reason, I figured I'd put in a reference to data that showed the decay of C14, and tell the story in terms of that isotope. – Anonymous Coward Sep 2 '11 at 2:08
The only big experimental difficulties are insuring that you don't have gain and acceptance drifts over the experimental time scale and getting enough C-14 in one place to let you ignore the counting statistics limits (i.e. get $10^{10}$--$10^{11}$ total counts and you're good to go). Not that this is trivial or that short-lived isotopes aren't easier, of course. – dmckee Sep 3 '11 at 21:41
How do we know that C14 decay's exponentially compared to linear
Here's an argument that might help: suppose, temporarily, that radioactive decay was linear. Let's say you started out with a sample, call it sample #1, of a billion atoms in a box, 5700 years ago (that's one half-life). By the current day, half of them would have decayed, so you'd have 500 million atoms left.
Now, let's say you take a different sample (sample #2) of 500 million atoms and put it in another box. Then wait 5700 years. According to the linear decay model, sample #1 would be entirely gone, but sample #2 would still have 250 million atoms. But if you think about it, that doesn't make sense, because if you jump back to the present, there is nothing to distinguish sample #1 from sample #2. Each of them consists of 500 million radioactive ${}^{14}\mathrm{C}$ atoms in a box. So there's no reason that sample #1 should behave any differently from sample #2. It doesn't "remember" that it came from an earlier sample of a billion atoms.
and have there been any studies to verify this?
I'm not sure if it has been explicitly verified for carbon 14, but using other radioactive elements with shorter half-lives, it has been verified probably hundreds of thousands of times over the past century or so that they decay exponentially. It's a pretty simple experiment: you just measure the number of atoms that decay in a short interval of time $\Delta t$ using a Geiger counter or something similar, then wait some time $T > \Delta t$, then again measure the number of atoms that decay in the interval $\Delta t$. The second measurement will give you fewer decays than the first, which is a sign of nonlinear decay. Making more measurements of this kind will reveal that the decay is exponential.
To see this quantitatively, you can basically just reverse the math in Chris's answer. Suppose that the number of atoms remaining after time $t$ is $N(t)$. The number of atoms which decay in a short time interval $\Delta t \ll t_{\frac{1}{2}}$ is, on average, $N(t) - N(t + \Delta t)$, which means that the rate of atoms decaying (i.e decays per minute) is
$$R(t) = \frac{N(t) - N(t + \Delta t)}{\Delta t} \approx -\frac{\mathrm{d}N(t)}{\mathrm{d}t}$$
For exponential decay, this means that
$$R(t) = -\frac{\mathrm{d}}{\mathrm{d}t}N_0 e^{-\lambda t} = \lambda N_0 e^{-\lambda t}$$
and for linear decay, it would be
$$R(t) = -\frac{\mathrm{d}}{\mathrm{d}t}(N_0 - \beta t) = \beta$$
So if radioactive elements underwent linear decay, the rate of decay would be constant, which is definitely not what is observed.
It's possible (and easy) to calculate that $t_{\frac{1}{2}} = \frac{\ln 2}{\lambda}$, so for elements that have a long half-life, $\lambda$ is pretty small. This means in turn that the decay rate $R(t)$ (what you measure) is low and can be difficult to detect. Furthermore, not only is $R(t)$ small, but the change in $R(t)$ over a medium-length period of time is even smaller. This is why direct measurements of the half life of ${}^{14}\mathrm{C}$ are somewhat tricky: the decay rate drops by only a hundredth of a percent per year. But I wouldn't be too surprised if it's been done. I can try to look for a reference if you like. (In practice, there are other ways to calculate the half-life indirectly.)
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Nice explanation with the boxes. – David Sep 2 '11 at 2:13
I think your analogy with the boxes is wrong. If you assume a linear decay, then for C14 it might be that 500 million atoms decay in 5700 years. That is about 88 thousand per year. This would then be a property of C14, so if you create an other box with 500 million atoms, the two boxes would both be decayed after another 5700 years. – Ruud v A Sep 7 '11 at 15:44
@Ruud: it's not an analogy, it's a demonstration of why the linear decay model is inconsistent with the idea of a half-life. But yes, if you don't assume that radioactive decay has a constant half-life, then the explanation with the boxes doesn't require exponential decay. (I realized that after posting but I forgot to come back and clarify it) – David Z Sep 7 '11 at 17:13
Galileo would be proud! @Ruud: So 2 separate boxes with 500M atoms would decay entirely in 5700y, but if you put them together half would remain. That seems a bit strange, to say the least... – yatima2975 Sep 8 '11 at 10:55
I would say the Activity reduces exponentially, which is a mathematical way of saying the amount of actvity is proportional to the amount of material left. Decay is a process that only involves individual atoms, and when something decays is a matter decided only by that atom.
If we have an ensemble of many atoms of the same type then it has been noted that the activity reduces over time, and follows a decreasing exponential.
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So if you would allow questions for me to better understand: The atoms are not always in a state of decay? But rather you cannot determine when they will decide to start decaying. Thus when you have a hand full of these atoms you can assume that you can average the results of their decay? sorry if that doesn't make sense. – Jonathon Byrd Sep 1 '11 at 16:36
That sounds about right! Except I would say that they $are$ always in a state of decay, but different atoms have different half lives, so its only the shorter ones that we 'notice'. The activity is a function of both the number and half life. – Nic Sep 1 '11 at 16:49
It doesn't seem like this would be a very solid method over a greater stretch of time. It seems like there's a lot of assumptions that have to be made. no? – Jonathon Byrd Sep 1 '11 at 17:00
The 'Law of large numbers' somewhat guarantees that this 'averaging', or probabalistic approach is actually very accurate. Remeber we might be talking in excess of $10^20$ atoms. that is a huge number. – Nic Sep 1 '11 at 17:05
Interesting, that is a huge number. I can see with that amount of excess and the fact that the atoms are always in a state of decay that you can begin to determine their age. I guess now I'm confused back at the beginning again. If all of the atoms are in a state of constant decay, did you just say that some C14 atoms have a longer rate of decay? I wouldn't mind talking more in chat either. – Jonathon Byrd Sep 1 '11 at 17:09
The derivations of exponential decay rates above can also be rendered into a very intuitive physical idea: "The rate of decay directly depends upon the number of atoms present at the very moment"
Hence $r$ is directly proportional to amount. That is, if $a$ is the amount of carbon and $k$ is some constant,, $\frac{da}{dt}$ is proportional to $a$
Or
$r= \frac{da}{dt} = k\cdot a$
$\frac{1}{a}da = k\cdot dt$
Performing integration (indefinite, just for simplicity)
$\log(a) = kt$
Or
$a = e^{kt}$
Which is exponential decay (generally with a negative $k$)
So, the fact that decay of carbon (or for that matter any element) can be modeled using exponential equation is just common sense. Rate depends upon amount so the decay is exponential. However, this is a statistical model and applies to a huge sample of atoms. It cannot predict the behavior of an individual atom and it's decay pattern. That is still considered VERY random. And hence Schrodinger's cat experiment is born.
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https://www.computer.org/csdl/trans/tp/2011/01/ttp2011010186-abs.html | Issue No. 01 - January (2011 vol. 33)
ISSN: 0162-8828
pp: 186-193
Rama Chellappa , University of Maryland, College Park
Ashok Veeraraghavan , Mitsubishi Electric Research Labs, Cambridge
Mahesh Ramachandran , University of Maryland, College Park
ABSTRACT
In this paper, we study the benefits of the availability of a specific form of additional information—the vertical direction (gravity) and the height of the camera, both of which can be conveniently measured using inertial sensors and a monocular video sequence for 3D urban modeling. We show that in the presence of this information, the SfM equations can be rewritten in a bilinear form. This allows us to derive a fast, robust, and scalable SfM algorithm for large scale applications. The SfM algorithm developed in this paper is experimentally demonstrated to have favorable properties compared to the sparse bundle adjustment algorithm. We provide experimental evidence indicating that the proposed algorithm converges in many cases to solutions with lower error than state-of-art implementations of bundle adjustment. We also demonstrate that for the case of large reconstruction problems, the proposed algorithm takes lesser time to reach its solution compared to bundle adjustment. We also present SfM results using our algorithm on the Google StreetView research data set.
INDEX TERMS
Structure from motion, multiple view geometry, computer vision.
CITATION
Rama Chellappa, Ashok Veeraraghavan, Mahesh Ramachandran, "A Fast Bilinear Structure from Motion Algorithm Using a Video Sequence and Inertial Sensors", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 33, no. , pp. 186-193, January 2011, doi:10.1109/TPAMI.2010.163 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8666542768478394, "perplexity": 1301.321009611086}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560285337.76/warc/CC-MAIN-20170116095125-00047-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=49&t=55966&p=207549 | ## 5G.1 [ENDORSED]
ayushibanerjee06
Posts: 177
Joined: Thu Jul 11, 2019 12:16 am
Been upvoted: 1 time
### 5G.1
Can someone explain why if you start with a higher pressure of reactant, the equilibrium constant will be larger?
SnehinRajkumar1L
Posts: 101
Joined: Thu Jul 11, 2019 12:15 am
### Re: 5G.1
I don't think higher pressure means that the equilibrium constant is larger. The constant is always set for a specific reaction regardless of pressure.
Nikki Razal 1L
Posts: 116
Joined: Fri Aug 30, 2019 12:17 am
Been upvoted: 1 time
### Re: 5G.1 [ENDORSED]
for this question, the statement, "If one starts with a higher pressure of reactant, the equilibrium
constant will be larger" is false because your initial concentration increases, causing the equilibrium constant to be smaller. For example,if you had the equation N2+3H2 <-->2NH3 and an initial concentration of 0.1 for each component K=[nh3]^2/[n2][h2]^3=100, but if pressure increased (specifically if it doubles), your initial concentration would be 0.2 for each component, and K=25
Andres Merlos 2L
Posts: 46
Joined: Wed Sep 18, 2019 12:17 am
### Re: 5G.1
So for this exercise, you have to determine which statements are true or false. If I am correct, you are referring to part c of 5G.1. The statement that if you start with a higher pressure of reactant, the equilibrium will be larger, is false.
Kristina Rizo 2K
Posts: 105
Joined: Wed Sep 18, 2019 12:19 am
### Re: 5G.1
The pressure doesn't affect the equilibrium constant. However, I think the professor stated that it does alter the overall reaction. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9093431234359741, "perplexity": 2545.53308305969}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886516.43/warc/CC-MAIN-20200704170556-20200704200556-00213.warc.gz"} |
https://quant.stackexchange.com/questions/7147/derivation-of-itos-lemma | # Derivation of Ito's Lemma
My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a geometric Brownian Motion. So, if one has
$dS = a(S,t)dt + b(S,t)dZ$
the first part on the right side of the equation is the (deterministic) drift rate and the second term is the stochastic component. Now I have read that ordinary calculus cannot handle the stochastic integral because for example
$\frac{dZ}{dt} = \phi \frac{1}{\sqrt{dt}}\\ \rightarrow \infty \text{ as } dt \rightarrow 0$
But Ito came along and proposed something which culminated in his famous Ito Lemma which helped calculate the stochastic integral. What I don't understand is the step taken from realizing that ordinary calculus does not work in this context to proposing a function $G = G(S,t) \text{ of } S(t)$, Taylor-expaning it, etc.
Or more precisely: What did Ito realize and propose that helped solve the stochastic integral?
Baxter and Rennie say it better than me, so I will summarize them.
Suppose that $N_t$ is not stochastic and $f(.)$ is a smooth function then the Taylor expansion is $$df(N_t) = f'(N_t)dN_t + \frac{1}{2}f''(N_t)(dN_t)^2 + \frac{1}{3!} f'''(N_t)(dN_t)^3 + \ldots$$ and the term $(dN_T)^2$ and higher terms are zero. Ito showed that this is not the case in the stochastic case. Suppose $W_t$ follows a Brownian motion and let $$Z_{n, i} = \frac{W\left(\frac{ti}{n}\right) - W\left(\frac{ti}{n}\right)}{\sqrt{t/n}}$$ now consider $$\int_0^t (dW_t)^2 \approx t \sum_{i=1}^n \frac{Z^2_{n,i}}{n}.$$ If $n \to \infty$ then $\sum_{i=1}^n \frac{Z^2_{n,i}}{n} \to 1$ and thus $\int_0^t (dW_t)^2 = t \neq 0$. Thus the second order term does not cancel (but higher order terms do) and we have an extra term in our derivative.
• good explanation, upvoted – Matthias Wolf Jan 29 '13 at 19:17
• Why is if reading it in the book (however clear it was written) it seems to make more sense online :). – Chinny84 Aug 18 '14 at 7:28
The logic from Bob Jansen is correct. The problem is abuse of ideas and notation the integral symbol from the deterministic world gets sloppily applied to random variables. Unlike normal $dt$, which is always positive, $dW_t$ can go 'backwards'. Thus increments of terms like $W_t dW_t$ have a first element that goes up and down with the second element (which can go up and down) and the product is always positive. Mathematically it really comes down to the fact that, if you take a function of 2 variables in the normal (deterministic world) and do a Taylor expansion, all of the second order terms can be neglected when you try to integrate the expansion. But not if there is a Brownian motion...
I think the problem that gets everyone is the loose use of the $"\int"$ symbol. When I explain it I purloin the symbol $\oint$ for stochastic integrals (there is no connection to a deterministic line integral).
Quite simply, if $X_t(t,W(t))$ is a function of two variables (which it is), and $X_0=X(0)$, we can solve for $X_t$ by integrating: $X_t=X_0+\int_0^tX_t(t,W(t))$. Now, I myself am being sloppy, as we have no "$d?.$" in the integral.
What everyone does is substitute the relevant Taylor expansion terms so it starts to look like (and now I will use the other symbol):
$X_t-X_0=\oint_0^t[\frac{\partial X_t}{\partial t}dt+\frac{\partial X_t}{\partial Wt}dW_t+\frac{1}{2}\frac{\partial^2X_t}{\partial Wt^2}(dW_t)^2]$
Recall that the t on the limit of integration is the "real" t, and the ones inside the integration are all dummies (maybe using $\tau$ would help here).
Here is where it gets cute. Mathemeticians and finance people love the abstraction of things like $\frac{\partial^2X_t}{\partial Wt^2}(dW_t)^2$, whereas physicists often look at things like that and say "I don't know what is going on there but somehow it needs to turn into an integral with respect to time since the whole point of this is to model time evolution."
The first term is, indeed, done w.r.t. time, so we can use "$\int$" in place of "$\oint$" safely.
Let's jump to the third term. This term is in the stochastic case, and not the deterministic, because of the dynamics of what we call $dW_t$. We would have had $dt^2$ as well, but it got dropped. While a Taylor series is trivially correct at its 'anchor' point, we need potentially lots of terms to fit closely farther away. However, if we want to integrate a Taylor series at a neighborhood very close to its anchor point we need only $f(a)$ and the first derivative $f'(a)$ in the limit. Intuitively we need only the slope in our approximation for tiny $\Delta 's$. Imagine a $\Delta=10^{-30}$: if we add up (integrate) $10^{30}$ items on the order of this size they matter. But, even if we add up $10^{30}$ units on the order of the square of that, and thus of size on the order of $10^{-60}$ they accumulate so slowly that they are completely negligible. This is simply the usual order concept, and $o((dt)^2)<o(dt)$.
The sleight-of-hand that seems mysterious is this oddity $dW_t$. Imagine $X_t=3t+2W_t$. Now, plugging in above, we get
$X_t-X_0=\int_0^t\frac{\partial X_t}{\partial t}dt+\oint \frac{\partial X_t}{\partial Wt}dW_t$
$X_t-X_0=\int_0^t 3 dt+\oint 2 dW_t$
The first term can be seen, correctly, to be $3t$. The second by symbolic reasoning is $2W_t$. More structurally, $dW_t$ is a random variable that is the differential limit of an RV distributed as $N(0,1/k)$ as $k\rightarrow \infty$. It is, in a sense, a tiny increment of variance alone that accumulates at twice the rate of time in this instance of $X_t$.
The real oddball is $dW_t^2$. Even though it comes from something that has 'only variance' it is squared. When an RV distributed as $X\sim N(0,1)$, $X^2$ is a new RV distributed as a chi-squared distribution: $X^2\sim \chi_1^2$. Now, as we break up this one-period distribution into k sub-periods we get
$\sum _k \frac{1}{k} X^2 = \frac{1}{k}\sum _k X^2$ and $\sum_k X^2 \sim \chi_k^2$
but $E[\chi_k^2]=k \Rightarrow E[\frac{1}{k}X^2]=1$
And here is the rub. Breaking down the 'variance only' RV into smaller fragments and squaring it and summing over the pieces gives you something which doesn't shrink away any faster than $dt$ despite it seemingly (notationally) looking like $dt^2$ Thus, everyone says, $"dW_t^2=dt"$.
Deep down the problem is pretty simple, but the use of the integral sign with $DW_t$ differentials makes it mighty confusing. All that is really going on is that we have the repeated problem that a term like $W_tdW_t$ co-varies: a positive delta-tick in $W_t$, $dW_t$, means both of the items go up together in such an increment. And, if there is a negative $dW_t$ tick $W_t$ goes down so that product is also always positive. In contrast, when we write $\int t dt$ time is always going up and $dt$ is always positive as t increases; $\int -t dt$ still has $dt$ going up. $dW_t$, however, is allowed to go either way (hence a more Lebesgue-like integral varying over the range not the domain). You need to think about what $\int dW_t$ really means, which is why I like $\oint$: to minimize errant false conculsions.
My math is a little sloppy (why did I use $E[\chi^2]$? both the quadratic variation of a path and its expected value are the same for a random walk; see Shreve's book). But if you want to see it play out in front of your eyes try using $X_t=3t+2W_t^2$ instead of the above formula. Then the second derivative has a value that matters
• can you explain how one period chi square random variable be break down into K sub periods ? – Neeraj Aug 13 '15 at 6:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.940094530582428, "perplexity": 305.99225211948163}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038507477.62/warc/CC-MAIN-20210418163541-20210418193541-00238.warc.gz"} |
https://vsoch.github.io/2013/visual-how-sampling-can-approximate-a-true-relationship/ | We know from central limit theorem and the law of large numbers that when we take a bunch of random samples from some unknown distribution, and then look at the distribution of sample means, as our number of samples goes to infinity the distribution look normal, and the mean of this normal distribution approaches the expected value of the “true” mean of the distribution. My favorite explanation of this phenomenon comes from Sal Khan at the Khan Academy, and I just stumbled on a nice plot that shows how the same idea is true with regard to linear regression.
How Sampling can Approximate a True Relationship: Linear Regression
This plot is from ESL with Applications in R. The plot on the left shows our data, with the red line representing the “true” linear relationship (for the entire population), and the blue line representing an approximation made with a sample by way of minimizing the sum of squared errors. We can see that, although the two lines aren’t exact, the estimation using the sample isn’t so far off.
The plot on the right again shows our population “true” relationship (red), and the sample estimate (dark blue), however now we have added a bunch of estimates from many random samples (the light blue lines). Here is the super cool part - we can see that if we take an average of these random sample lines, we come quite close to the true relationship, the red line! I (think) this is a good example of something akin to central limit theorum applied to a learning algorithm. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9181646704673767, "perplexity": 318.7351361095413}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540501887.27/warc/CC-MAIN-20191207183439-20191207211439-00144.warc.gz"} |
http://math.stackexchange.com/questions/313832/a-combinatorial-proof-that-the-alternating-sum-of-binomial-coefficients-is-zero | # A combinatorial proof that the alternating sum of binomial coefficients is zero
I came across the following problem in a book:
Give a combinatorial proof of $${n \choose 0} + {n \choose 2} + {n \choose 4} + \, \, ... \, = {n \choose 1} + {n \choose 3} + {n \choose 5} + \, \, ...$$ using the "weirdo" method (i.e., where one of the elements is chosen as special and included-excluded -- I'm sure you get the idea).
After days of repeated effort, the proof has failed to strike me. Because every time one of the elements is excluded, the term would be ${n-1 \choose k}$ and not ${n \choose k}$, which is not the case in either of the sides of the equation.
-
HINT: Let $A$ be a set of $n$ marbles. Paint one of the marbles red; call the red marble $m$. If $S$ is a subset of $A$ that does not contain $m$, let $S'=S\cup\{m\}$, and if $m\in S\subseteq A$, let $S'=S\setminus\{m\}$. Show that the map $S\mapsto S'$ yields a bijection between the subsets of $A$ with even cardinality and those with odd cardinality.
-
Is it necessary to prefix hints with HINT? I like this answer but I always think the practice of shouting HINT before you give one is a bit strange. – Ben Millwood Feb 25 '13 at 11:42
@Ben: I prefer to give an explicit signal that I am not providing a complete answer. This is partly for the benefit of the querent, and partly because on a few occasions when I’ve inadvertently failed to do so, someone (other than the querent) has complained that I didn’t answer the question. – Brian M. Scott Feb 25 '13 at 11:45
Well, fair enough. I guess I just haven't been bitten by not signposting my hints yet. – Ben Millwood Feb 25 '13 at 16:13
This is not a direct combinatorial proof but one can make the argument combinatorial. There is an easy combinatorial proof of the following: $${n\choose {k}}={{n-1}\choose {k}}+{{n-1}\choose {k-1}}$$ Now take $k=2r$ and $k=2r+1$ and sum over all integers $r$. In the first case you will have the expression in the LHS and for the second you will get an expression for RHS and both are equal by the above identity.
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To show this you can use the the binomial theorem
which is $(x+y)^n=\sum_{k=0}^{n}\dbinom{n}{n-k}x^{n-k}y{k}$
set x=1 y=-1
and you get
$\dbinom{n}{0}-\dbinom{n}{1}+\dbinom{n}{2}.....+(-1)^{n}\dbinom{n}{0}$
$\dbinom{n}{0}+\dbinom{n}{2}=\dbinom{n}{1}+\dbinom{n}{3}$
thus in a set the number of subsets which are even equal odd subset.
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9391621947288513, "perplexity": 224.4482112999024}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257826908.63/warc/CC-MAIN-20160723071026-00176-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://aaronsreality.blogspot.com/2013/12/how-higgs-boson-must-work-in-svm.html | ## Friday, December 13, 2013
### How the Higg's Boson Must work in The SVM
Dark Energy Rulesets: Reasons, Functions and Rules.
Let's say I am wrong and there is a Higg's Boson.Of which there is a Higg's Boson and I am wrong, This posting will describe the Higg's Boson and its effects and interactions. What would the purpose of this effect (Boson)? When does this Boson interact with the Dark Energy Ruleset?
The simple version creates the basis of the Coriolis Effect.
The other version forms crystals
This problem has two solutions in this model. The simple (integer) version that will always show chaos, and the complex (irrational) that will solve these issues. Please read the paper "The Structure of Dark Energy" a basic abstract.
The Dark Energy Ruleset:
The space created by Dark Energy is not defined by any previous space; Hilbert, Einstein, or
others. Previously defined space requires a flat universe, a vacuum and single point mass.
A discrete single Dark Energy unit is called a Ruleset. A Ruleset domain is exactly the width of
one bit of information of a boson.
• A Ruleset is the medium in which vibrations transverse.
• A Ruleset is a 3 dimensional structure.
• A Ruleset is not aware of which vibration it holds or if it is void of vibrations.
• A Ruleset domain is the width of a bit of information of a boson.
A Ruleset is where the interactions of boson information occurs. In the simple solution I will describe these interactions over time using familiar equations. I will use this model to describe simple baryonic motion, boson to boson interaction and boson to baryon interaction. This model must displais changes in elemental variables ( Gluons ,W+/- Bosons, Z Bosons and Photons). This model must result in producing chemistry examples.
The Higg's Boson transmits information to the Ruleset at . On 4 July 2012, it was announced that a previously unknown particle with a mass between 125 and 127 GeV/c2 (134.2 and 136.3 amu) had been detected; physicists suspected at the time that it was the Higgs boson.[14][9][15].
In my mind this means that three bits of information is the Coriolis solution begins. Neutrinos transmitted the temperature information, frequency, and the wavelength to the next ruleset at those voltages. Information about two touching baryons are compared at these points on the ruleset.
These are the three strands of information I described in the paper about photons. The information travelling down the Higg's is the inverse of the information held in the photon. It tells the ruleset it is entering the information required. The ruleset compares the new information to current information then makes changes to the bosons involved.
This is how small information traveling over small distances can have a great impact on other baryons.
Aaron | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8915857672691345, "perplexity": 1383.0697178963576}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1405997894931.59/warc/CC-MAIN-20140722025814-00141-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/torque-with-see-saw.809506/ | # Torque with see-saw
1. Apr 19, 2015
### goonking
1. The problem statement, all variables and given/known data
2. Relevant equations
3. The attempt at a solution
I set up my problem as :
(12kg x .5m) + ( 1/3 x 4kg) = (1.05m x Mass) + (2/3 x 4kg)
then , i just solve for mass, is that correct?
actually, should I change the (1/3 x 4kg) to ( 1m x 4kg) and the (2/3 x 4kg) to ( 2m x 4kg)?
2. Apr 19, 2015
### mukundpa
Second terms on each sides is only force not torque. The best way is to take center of mass of the plank under consideration.
3. Apr 19, 2015
### goonking
which equation do you mean? the first or second one?
4. Apr 19, 2015
### sun18
You are trying to balance torques about the pivot point of the seesaw. What you have written is mass*length, which does not have units of Newton-meters.
You are close to the right idea, but the second term on each side of the equality is not correct. As mukundpa says, find the center of mass of the board and treat it as a point mass; what torque does that point mass produce, and which side of the pendulum is it on?
5. Apr 19, 2015
### goonking
the center of mass is at the middle of the board, which is 1.5 meters.
the pivot point is to the right of the center of mass.
6. Apr 19, 2015
### sun18
That's right, so now think about the problem: you have 2 masses on the left of the pivot point, and one on the right. Now try to balance the torques and solve for M.
7. Apr 19, 2015
### goonking
what do you mean 2 ? are you counter the board and the orange cat?
if you did, shouldn't there be 2 masses on the right of the pivot point too?
8. Apr 19, 2015
### sun18
Imagine that the board is completely massless, except for one concentrated mass at the center of mass of the board. Then, the only masses in the system are the orange cat on the left, the mass of the board on the left, and the blue cat on the right.
9. Apr 19, 2015
### AlephNumbers
I believe there would also be a force exerted by gravity on the mass of the portion of the board that is to the right of the pivot. Actually, I need to learn to read.
Last edited: Apr 19, 2015
10. Apr 19, 2015
### haruspex
There are two ways you can deal with the board. You can consider each side separately, exerting opposite torques, or the simpler way sun18 suggests: consider the whole board's mass to be concentrated at its mass centre. That gives you only 3 masses to worry about.
11. Apr 19, 2015
### AlephNumbers
Oh okay. I did not realize that that was what sun18 was suggesting. That is indeed simpler.
12. Apr 19, 2015
### goonking
so now I need to find how far away the center of mass of board is from the pivot point?
13. Apr 19, 2015
### haruspex
Yes.
14. Apr 19, 2015
### goonking
center of mass is 0.5m away from pivot
(orange cat mass x 1.05m) + (4kg x 0.5m) = 12kg x 0.5m
mass orange cat = 3.809 kg.
correct?
15. Apr 19, 2015
### goonking
i'm interested on how you would do this problem the 'less simple' way. can you show me the math?
16. Apr 19, 2015
### haruspex
Looks right (but you can't justify that many significant digits).
Much the same... just treat the plank as two planks, one each side. It's more complicated because you have to calculate the mass and mass centre of each piece.
17. Apr 19, 2015
### goonking
so you mean like this?
(12kg x .5m) + ( 1/3 x 4kg) = (1.05m x Mass) + (2/3 x 4kg)
18. Apr 19, 2015
### haruspex
You've missed out the distances to the mass centres of the two pieces of plank.
19. Apr 19, 2015
### goonking
ok i see
so it is :
(12kg x .5m) + ( 1/3 x 4kg x .5m) = (1.05m x Mass) + (2/3 x 4kg x 1m)
the center of mass of the plank of the right side is .5 meters away from pivot
and the center of pass of the plank of the left side is 1 meter away from pivot.
correct?
20. Apr 19, 2015
### haruspex
looks right. Do you get the same answer?
Draft saved Draft deleted
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http://mathhelpforum.com/calculus/154127-difficult-limit.html | 1. ## difficult limit
How can I solve the folowing limit using only algebric manipulation:
limit when x aproches 0 of: {[x^(1/3)]-1}/{x^(1/2)]-1}
2. Originally Posted by Mppl
How can I solve the folowing limit using only algebric manipulation:
limit when x aproches 0 of: {[x^(1/3)]-1}/{x^(1/2)]-1}
Dear Mppl,
Do you mean, $\displaystyle\lim_{x\rightarrow{0}}\frac{x^{\frac{ 1}{3}}-1}{x^{\frac{1}{2}}-1}$ ??
3. yes precisely that one. Can you help me with that?
4. Question: Why do you say this is difficult?
Do you realize that $\displaystyle \lim _{x \to 0} \sqrt x$ does not exist because $\sqrt{x}$ is not defined for $x<0$.
But $\displaystyle \lim _{x \to 0^+} \sqrt x=0$.
So you may want to double check the statement of this question.
5. well lets admit its 0+ then...whats the result then? and how to get there?
6. Originally Posted by Mppl
well lets admit its 0+ then...whats the result then? and how to get there?
$\displaystyle \lim _{x \to 0^+} \sqrt x=0$.
$\displaystyle \lim _{x \to 0^+} \sqrt[3] x=0$.
Now you finish.
7. oh my bad the limit is x aproching 1 and not 0 sorry...what would be the solution there?
8. Let $x = y^6$ so then you have
$\displaystyle \lim_{y \to 1} \dfrac{y^2-1}{y^3-1}$
which you can factor but L'Hopital's rule would be faster.
9. Let's make the problem a bit more interesting and assume that the limit is supposed to be as x goes to 1 rather than 0. You can then check using l'Hôpital's rule that the answer should be 2/3. But it has to be done using only algebraic manipulation, so l'Hôp is not allowed.
Let $u = x^{1/6}$. Then as x goes to 1, u also goes to 1. So we want to find
$\displaystyle \lim_{x\to 1}\frac{x^{1/3}-1}{x^{1/2}-1} = \lim_{u\to 1}\frac{u^2-1}{u^3-1} = \lim_{u\to 1}\frac{(u-1)(u+1)}{(u-1)(u^2+u+1)} = \lim_{u\to 1}\frac{u+1}{u^2+u+1} = \frac23.$
Edit. Beaten by Danny.
10. thanks a lot men | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9549370408058167, "perplexity": 845.151060784658}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934807044.45/warc/CC-MAIN-20171123233821-20171124013821-00708.warc.gz"} |
https://www.physicsforums.com/threads/higgs-mechanism-questions.624234/ | # Higgs Mechanism Questions?
1. Jul 29, 2012
### AbsoluteZer0
Hi,
As I understand, please correct me if i'm wrong, when a subatomic particle interacts with the Higgs field it generates mass due to the higgs mechanism. Does this have anything to do with with e = mc2? (I'm not too privy to particle physics or relativity.)
Suppose an electron and a positron pair interacts with the Higgs field. Does 1MeV of energy from the Higgs field 'congeal' into the mass of the two particles?
Thanks,
2. Jul 30, 2012
### Staff: Mentor
With E=mc^2 (better: E=γmc^2), you can calculate the energy corresponding to the mass of a particle.
The Higgs field gives an electron a mass of 9*10^(-31)kg, which is (-> special relativity) equivalent to 511 keV.
3. Jul 30, 2012
### Naty1
E = mc2 gives the equivalent energy to the rest mass of a electron. It is in the frame of the particle...that is moving with the particle. That does not change with relative velocity. Likewise the mass imparted by the Higgs field is rest mass.
Wikipedia still says:
http://en.wikipedia.org/wiki/Higgs_field
but now reflects preliminary CERN results....a Higgs particle MAY have been found.
Elsewhere in these forums, some 'experts' here think there is a single Higgs field (a single Higgs particle]. In some models, I've read different Higgs field affects different elementary particles...there are multiple Higgs fields. As you likely know, the positron is the anti particle of the electron, and being an elementary particle, is imparted a mass equal to that of the electron.
'congeal' is likely a premature designation:
Also from Wikipedia:
"The Higgs mechanism shows how some particles can gain mass by symmetry breaking without affecting parts of current physics theory that are believed approximately correct. The existence of some kind of symmetry breaking Higgs mechanism is believed proven, however there are a number of ways it could happen and physicists have not yet determined which of these takes place in nature, or whether the mechanism arises in some other way not yet identified."
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http://math.stackexchange.com/questions/76330/prove-sequence-a-n-n1-n-is-convergent | # Prove sequence $a_n=n^{1/n}$ is convergent
How to prove that the sequence $a_n=n^{1/n}$ is convergent using definition of convergence?
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Have you proven that limits respect continuous functions? Because then $$\lim_{n\rightarrow \infty} n^{\frac{1}{n}}=\lim_{n\rightarrow\infty} e^\frac{\log n}{n}=\exp\left({\lim_{n\rightarrow\infty} \frac{\log n}{n}}\right)$$ – Eric Naslund Oct 27 '11 at 12:19
I doubt that is directly using the definition of convergence. – Listing Oct 27 '11 at 12:20
No I know that proof I am trying for proof using Definition of Convergence. – Ramana Venkata Oct 27 '11 at 12:21
@EricNaslund: how do you prove convergence of $\frac1n \log n$ by the definition? – Ilya Oct 27 '11 at 12:31
@Gortaur: That depends, what is your definition of $\log n$? – Eric Naslund Oct 27 '11 at 12:32
Noticing that $n^\frac{1}{n} > 1$ for all $n$, it all comes down to showing that for any $\epsilon > 0$, there is a $n$ such that $(1+\epsilon) \geq n^\frac{1}{n}$, or by rearranging, that
$$(1+\epsilon)^n \geq n$$
Now, let's first of all choose an $m$ such that $(1+\epsilon)^{m}$ is some number bigger than 2, let's say the smallest number greater than $3$ that you can get. From here, swap $m$ for $2m$. This will make the left side a little over 3 times larger, and the right side 2 times larger. The next doubling will still double the right side, but the left side will increase roughly 9-fold. Repeating, we can easily see that the left side will at some point overtake the right side, and we have our $n$
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Where did you use the definition here? – Ramana Venkata Oct 27 '11 at 12:59
The definition is used in the first sentence. Really, the definition demands that I find an $n$ such that $|n^\frac{1}{n} - 1| \leq \epsilon$. But as noted, $n^\frac{1}{n} > 1$, so we can do without the absolute value signs. Now, adding 1 to each side gives my inequality. – Arthur Oct 27 '11 at 14:16
The definition also states that I should find an $N$ such that for all $n> N$, this holds. A small argument about monotony like in Eric Naslund's answer will take care of that. – Arthur Oct 27 '11 at 14:23
I think this is the right elementary proof of this proposition and the one that immidiately came to my mind, when I saw this question. +1 for writing it down. – Sam Oct 27 '11 at 15:05
So here is an outline of a proof:
Step 1: Notice that $n^\frac{1}{n}\geq 1$ for all $n$.
Step 2: Prove that $a_n$ is monotonically decreasing for $n\geq 3$. Equivalently we need to show that $n^{(n+1)}>(n+1)^n$.
Step 3: Show that there is a subsequence which converges to $1$. I managed to do this by considering $b_n={a_{2^{2^n}}}$. (It does not appear well in LaTeX as there are too many nested exponents. I had typed this part out, but decided to remove it)
From these three facts you can conclude that the limit is $1$.
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One possible argument to show that it is decreasing (for $n\ge 3$) is by induction. Inductive step - by contradiction: Suppose that $(n-1)^n\ge n^{n-1}$ but $(n+1)^n> n^{n+1}$. Multiplying the two inequalities we get $(n^2-1)^n > n^{2n}$, which is a contradiction. (I kind of like this argument - which is the reason I posted it here. And I posted it as a comment, since I did not want bump this thread.) – Martin Sleziak Jun 7 '12 at 9:43
This is a well known limit. An elementary proof without the use of limits with continuous functions can be found here.
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Well, the easiest proof is that the sequence is decreasing and bounded below (by 1); thus it converges by the Monotone Convergence Theorem...
The proof from definition of convergence goes like this:
A sequence $a_{n}$ converges to a limit L in $\mathbb{R}$ if and only if $\forall \epsilon > 0$, $\exists N\in\mathbb{N}$ such that $\left | L - a_{n} \right | < \epsilon$ for all $n \geq N$.
The proposition: $\lim_{n\to\infty} n^{1/n} = 1$
Proof: Let $\epsilon > 0$ be given. Then by Archimedean property of the real numbers, there exists $M \in \mathbb{N}$ such that $M < \epsilon$ then find $x\in\mathbb{R}; x>2$ such that $1+M>x^{1/x}$ and let $P = \left \lceil x \right \rceil$. Then, since $f(x)=x^{1/x}$ is decreasing (for $x>e$) (trivial and left to the reader :D) take any $x\in\mathbb{N}$ such that $x>P$ and observe that (because of our choice and $M$ and $P$) we have $n^{1/n} \leq P^{1/P} \leq M \le 1 + \epsilon$ whenever $n\geq P$ and so $\left | 1 - a_{n} \right | < \epsilon$ whenever $n\geq P$. Thus $a_{n}$ converges (to 1).
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http://mathhelpforum.com/advanced-algebra/151039-rank-t-rank-l_a-proof-print.html | # Rank(T) = Rank(L_A) proof?
• July 15th 2010, 11:54 AM
buri
Rank(T) = Rank(L_A) proof?
I'd like to have someone check whether my proof of the following is correct. First, a couple of definitions to make sure we're all on the same page:
L_A is the mapping L_A: F^n → F^m defined by L_A (x) = Ax (the matrix product of A and x) for each column vector x in F^n.
The standard representation of V with respect to β is the function φ_β: V → F^n defined by φ_β(x) = [x]_β (i.e. the vector of x relative to β).
Let V and W be finite-dimensional vector spaces and T: V → W be an isomorphism. Let V0 be a subspace of V. Then T(V0) is a subspace of W and dim(V0) = dim(T(V0)).
Let V and W be a vector spaces of dimension n and m, respectively, and let T: V → W be a linear transformation. Define A = [T] (i.e. the matrix representation of T in the ordered bases β and γ). So we have the following relationship (please ignore the dotted lines):
V------T--------->W
|.......................... |
|...........................|
φ_β......................φ_γ
|...........................|
F^n -------L_A---> F^m
So to the quesiton finally:
Let T: V → W be a linear transformation from an n-dimensional vector space V to an m-dimensional vector space W. Let β and γ be ordered bases for V and W, respectively. Prove that rank(T) = rank(L_A), where A = [T].
*********
!!!!!PROOF!!!!!
*********
I'll show that φ_γ(R(T)) = R(L_A) (R here is talking about the range..). From which then the result I'd proved already I'll have dim(R(T)) = dim(φ_γ(R(T)) = dim(R(L_A)) (remember that φ_γ(R(T) and R(L_A) are both in F^m).
So let x be in φ_γ(R(T)).
This means for some T(y) in R(T) I have x = φ_γ(T(y)) which means x = [T(y)]_γ = [T][y]_β = Ay which is in R(L_A). Since its arbitrary I've shown that φ_γ(R(T))⊂ R(L_A).
Further, let z in R(L_A). Then z = Ax for some x in F^n. Therefore, z = [T][x]_β = [T(x)]_γ which is in φ_γ(R(T)). Therefore, R(L_A) ⊂ φ_γ(R(T)) and hence, R(L_A) = φ_γ(R(T)). So by the result I'd proved earlier dim(R(T)) = dim(φ_γ(R(T))) = dim(R(L_A)).
Was this correctly done? Or am I at least on the right track? I'd really appreciate the help. I'm taking a look at linear algebra on my own this summer, so any help is REALLY appreciated! Thanks
• July 16th 2010, 08:38 PM
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http://math.stackexchange.com/questions/240888/struggle-proving-maximal-ideals-principal-in-mathbb-z-varphi | # Struggle proving maximal ideals principal in $\mathbb Z[\varphi]$
I was worried that my proof isn't right so I want to know if there are any mistakes in this and if this way can work? Thank you very much.
We want to show every maximal ideal $\mathfrak m$ of $\mathbb Z[\varphi]$ is principal. My idea is that the quotient $\mathbb Z[\varphi]/\mathfrak m$ must be a finite field, so we can use the classification of finite fields to produce a principal ideal equal to $\mathfrak m$.
Some lemmas which are used.
Lemma $\mathbb Z[\varphi]/\mathfrak m$ is isomorphic to either $\mathbb F_p$ (case 1) or $\mathbb F_{p^2}$ (case 2) (some $p$). proof: The quotient is a field since the ideal is maximal and the field must be finite (and have degree $\le 2$) because it's the quotient of a finite dimensional space ($\mathbb Z[\varphi]$ has dimension $2$) and doesn't contain $\mathbb Q$.
Edit, the next lemma is wrong but I don't use it anymore.
Lemma Suppose $\mathbb Z[\varphi]/\mathfrak m \simeq \mathbb Z[\varphi]/\mathfrak m'$ is an isomorphism then $\mathfrak m = \mathfrak m'$. proof: In general $R/I \simeq R/J$ implies $J$ is an automorphism of $R$ applied to $J$ but $\mathbb Z[\varphi]$ has no automorphisms.
Lemma $p$ is of the form $x^2 + xy - y^2$ iff $X^2 - X - 1$ has a solution mod $p$. proof: I don't actually prove this but I assume it could be done like this.
Note, if $p$ is of that form it can be written that way for infinitely many $(x,y)$ but $x+\varphi y$ will be an associate of $x'+\varphi y'$ or $x'+\bar\varphi y'$ if so.
The main theorem:
Lemma In case 1, $X^2 - X - 1$ has a solution $\mod p$. proof: The quotient map is a ring homomorphism.
Lemma If $X^2 - X - 1$ has a solution $\mod p$ then case 2 is impossible. proof: The basis $\{1,\varphi\}$ collapses as the image of $\varphi$ is expressible as an integer multiple of $1$, so $M$ is one dimensional.
Theorem $\mathfrak m$ is principal. proof:
In case 2, $\mathbb Z[\varphi]/(p)$ has size $p^2$ and it is the only ideal which gives a finite field of this size so $\mathfrak m = (p)$.
In case 1, Let $x$,$y$ such that $p = x^2 + xy - y^2 = (x + \varphi y)(x + \bar \varphi y)$ and both quotients like $\mathbb Z[\varphi]/(x + \varphi y)$ have size $p$ and there's no other way to quotient to get a field of the right size so either $\mathfrak m = (x + \varphi y)$ or $\mathfrak m = (x + \bar \varphi y)$ for any such associate pair $x$,$y$ (it doesn't matter which you choose).
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Do you know anything about the ideal class group? Your general approach of using the fact that the residue fields are all finite is not going to be a good strategy in general for rings like this, because such quadratic rings always have finite residue fields but many of them are not PIDs. – KCd Nov 20 '12 at 1:39
Dear sperners lemma, Your second lemma is not true. Your third lemma is the key point, and note that it uses more than just the finiteness of the residue fields (and thus gets around @KCd's objection above) --- the estimates needed to prove it will be sensitive to the particular value of the norm of $\varphi$, and related facts. Regards, – Matt E Nov 20 '12 at 1:46
@KCd, I have studied unique factorization of ideals and the class group being size 1 iff the integers have unique factorization. – sperners lemma Nov 20 '12 at 13:30
@spernerslemma: If you know about the ideal class group, and have ever done calculations of it, then I think a better way to approach your problem is to show the class number is 1 (thus showing all ideals are principal) rather than to focus only on the maximal ideals. – KCd Nov 20 '12 at 16:06
Nowhere have you said what $\phi$ means. If it means $(1+\sqrt5)/2$, then just the other day I posted an approach to proving your ring is Euclidean (a fortiori, a PID). See math.stackexchange.com/questions/240700/… – Gerry Myerson Nov 21 '12 at 5:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9516652822494507, "perplexity": 174.75630895806773}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783397562.76/warc/CC-MAIN-20160624154957-00027-ip-10-164-35-72.ec2.internal.warc.gz"} |
https://homework.cpm.org/category/CC/textbook/cc2/chapter/3/lesson/3.2.1/problem/3-38 | ### Home > CC2 > Chapter 3 > Lesson 3.2.1 > Problem3-38
3-38.
Simplify the following fraction expressions. Show all of your work. Homework Help ✎
1. $\frac{3}{4}+\frac{2}{3}$
Draw diagrams to represent each fraction.
Now change the diagram so that each whole is divided into equal portions.
1. $\frac{7}{8}-\frac{1}{4}$
Follow the steps in part (a) to solve the problem.
1. $\frac{3}{5}\cdot\frac{1}{3}$
Draw a rectangle.
1. $\frac{4}{7}\cdot\frac{2}{3}$
$\frac{8}{21}$ | {"extraction_info": {"found_math": true, "script_math_tex": 5, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9247031211853027, "perplexity": 4460.379676964523}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986653216.3/warc/CC-MAIN-20191014101303-20191014124303-00140.warc.gz"} |
http://www.ck12.org/tebook/Algebra-I-Teacher's-Edition/r1/section/9.1/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# 9.1: Addition and Subtraction of Polynomials
Difficulty Level: At Grade Created by: CK-12
## Learning Objectives
At the end of this lesson, students will be able to:
• Write a polynomial expression in standard form.
• Classify polynomial expression by degree.
• Problem-solve using addition and subtraction of polynomials.
## Vocabulary
Terms introduced in this lesson:
polynomial
term
coefficient
constant
degree
cubic term, quadratic term, linear term
nth order term
monomial, binomial
standard form
rearranging terms
like terms, collecting like terms
## Teaching Strategies and Tips
There are a large number of new terms in this lesson. Introduce new vocabulary with concrete, specific examples. It is also helpful to provide examples of what the new word does not mean.
• Polynomials consist of terms with variables of nonnegative integer powers. Polynomials can have more than one variable.
Examples:
These are polynomials:
\begin{align*}& -\sqrt{12}x^8 - x^5 + \pi\\ & x^5 + x^4 - x^3 + x^2 - x + 1\\ & -\frac{3} {7}\\ & x^2 + y^2\\ & xy\\ & 2x^2 - 4xy + 1\end{align*}
These are not polynomials:
\begin{align*}& \sqrt{x} + x + 2\\ & y^{4.2} - x^{3.5} + x^{1.1} - x + 1\\ & x + \frac{1} {x} - 3\\ & 2^x + 8\\ & x^{-2} + x^{-1} + 1\end{align*}
Have students explain their answers. Suggest that they use explanations such as:
This is not a polynomial because...
...it has a negative exponent.
...the power of \begin{align*}x\end{align*} appears in the denominator.
...it has a fractional exponent.
...it has an exponential term.
• Terms are added or subtracted “pieces” of the polynomial.
Examples:
The polynomial \begin{align*}x^5 + x^4 - x^3 + x^2 - x + 1\end{align*} has \begin{align*}6\end{align*} terms.
The polynomial \begin{align*}2x^2 - 4xy + 1\end{align*} has \begin{align*}3\end{align*} terms; it is called a trinomial.
\begin{align*}x^2 + y^2\end{align*} is a binomial because it has \begin{align*}2\end{align*} terms.
\begin{align*}xy\end{align*} and \begin{align*}-\frac{3} {7}\end{align*} are \begin{align*}1-\end{align*}term poynomials and are called monomials.
In the polynomial \begin{align*}-2x^5 + 7x^3 - x + 8\end{align*}, the \begin{align*}8\end{align*} is a term; but neither \begin{align*}-2, 5, 7,\end{align*} nor \begin{align*}3\end{align*} are terms. \begin{align*}-x\end{align*} is another term; but \begin{align*}x^3\end{align*} is not. \begin{align*}7x^3\end{align*} is a term.
• The constant term is that number appearing by itself without a variable.
Examples:
In the polynomial \begin{align*}-2x^5 + 7x^3 - x + 8\end{align*}, the \begin{align*}8\end{align*} is the only constant term.
The polynomial \begin{align*}x^2 + y^2\end{align*} has no constant terms.
• Coefficients are numbers appearing in terms in front of the variable.
Examples:
\begin{align*}2x^2 - 4xy + 1\end{align*}. The coefficient of the first term is \begin{align*}2\end{align*}. The coefficient of the second term is \begin{align*}-4\end{align*}.
\begin{align*}x^5 + x^4 - x^3 + x^2 - x + 1\end{align*}. The coefficient of each of the terms is \begin{align*}1\end{align*}.
• In standard form, a polynomial is arranged in decreasing order of powers; terms with higher exponents appear to the left of other terms.
Examples:
These polynomials are in standard form:
\begin{align*} & x^5 + x^4 - x^3 + x^2 - x + 1\\ & xy + x - y - 1\end{align*}
These polynomials are not in standard form:
\begin{align*} & 1 - x + x^2 - x^3 + x^4 + x^5\\ & xy + x^2 y^2 - 1\end{align*}
• The first term of a polynomial in standard form is called the leading term, and the coefficient of the leading term is called the leading coefficient.
Examples:
The leading term and leading coefficient of the polynomial \begin{align*}2x^2 - 4xy + 1\end{align*} are \begin{align*}2x^2\end{align*} and \begin{align*}2\end{align*}, respectively.
The leading term and leading coefficient of the polynomial \begin{align*}9x^2 + 8x^3 + x + 1\end{align*} are \begin{align*}8x^3\end{align*} and \begin{align*}8\end{align*}, respectively. Remind students to write polynomials in standard form.
• Like terms are terms with the same variable(s) to the same exponents. Like terms may have different coefficients. A polynomial is simplified if it has no terms that are alike.
Examples:
These are like terms:
\begin{align*}2x^3\end{align*} and \begin{align*}-8x^3\end{align*}
\begin{align*}-xy\end{align*} and \begin{align*}17.2xy\end{align*}
\begin{align*}2x, -4x,\end{align*} and \begin{align*}\sqrt{2}x\end{align*}
\begin{align*}-4, \pi,\end{align*} and \begin{align*}\sqrt{2}\end{align*}
These are not like terms:
\begin{align*}x^2y\end{align*} and \begin{align*}xy^2\end{align*}
\begin{align*}x^2y^2\end{align*} and \begin{align*}x^2 + y^2\end{align*}
The polynomial \begin{align*}-x^3 + 3.1x^2 - 4x^2 + x - 2\end{align*} is not simplified.
• The degree of a term is the power (or the sum of powers) of the variable(s). The constant term has a degree of \begin{align*}0\end{align*}. The degree of a polynomial is the degree of its leading term. Encourage students to name polynomials by their degrees: cubic, quadratic, linear, constant.
Examples:
The term \begin{align*}-8x^3\end{align*} has degree \begin{align*}3\end{align*}.
The term \begin{align*}7.1x^2y^2\end{align*} has degree \begin{align*}4\end{align*}.
\begin{align*}x^5 + x^4 - x^3 + x^2 - x + 1\end{align*} is a fifth-degree polynomial.
\begin{align*}9x^2 + 8x^3 + x + 1\end{align*} is a cubic polynomial. Remind students to write polynomials in standard form.
Assess student vocabulary by asking them to determine all parts (terms, leading term, coefficients, leading coefficient, constant term) of a given polynomial and have them describe it in as many ways as they can (its degree, whether it is in standard form, number of variables, etc.) See Example 1-3.
When adding or subtracting polynomials, suggest that students do so vertically. The vertical or column format helps students keep terms organized.
Example:
Subtract and simplify.
\begin{align*}4x^2 + 2x + 1 - (3x^2 + x - 4)\end{align*}
Solution: Subtract vertically. Keep like terms aligned.
\begin{align*}& \quad 4x^2 + 2x + 1\\ & -3x^2 - x + 4\\ & \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\\ & \qquad x^2 + x + 5\end{align*}
When simplifying like terms, suggest that students rearrange the terms into groups of like terms first. This is especially helpful in Review Questions 11, 12, and 16. See also Example 4.
## Error Troubleshooting
General Tip: Remind students to distribute the minus sign to every term in the second polynomial when subtracting two polynomials. See Example 6 and Review Questions 13-16.
When simplifying polynomials, such as in Example 4b and Review Questions 12 and 16, remind students that like terms must have the same variables and exponents.
In Example 6, remind students that to subtract \begin{align*}A\end{align*} from \begin{align*}B\end{align*} means \begin{align*}B - A\end{align*} and not the other way around.
Example:
Subtract \begin{align*}-2m^2 + 3n^2 + 4mn - 1\end{align*} from \begin{align*}-2n^2 - 7 + 2mn + 8m^2\end{align*}.
Hint: Setup the problem as \begin{align*}-2n^2 - 7 + 2mn + 8m^2 -(-2m^2 + 3n^2 + 4mn - 1)\end{align*}. Then distribute the negative inside the parentheses to every term. Group like terms.
General Tip: Some students will give the incorrect degree of a polynomial; remind students write polynomials in standard form and then look for the leading term.
General Tip: Students can check their answers by plugging in a simple value for the variable in the original polynomials and simplified polynomial and check if the results have the same value.
Example:
Subtract \begin{align*}-2m^2 + 3n^2 + 4mn - 1\end{align*} from \begin{align*}-2n^2 - 7 + 2mn + 8m^2\end{align*}.
Solution:
Distribute. \begin{align*}-2n^2 - 7 + 2mn + 8m^2 - (-2m^2 + 3n^2 + 4mn - 1)\end{align*}
Group like terms. \begin{align*}-2n^2 - 7 + 2mn + 8m^2 + 2m^2 - 3n^2 - 4mn + 1\end{align*}.
\begin{align*}(8m^2 + 2m^2) + (-2n^2 - 3n^2) + (2mn - 4mn) + (-7 + 1)\end{align*}
Answer: \begin{align*}10m^2 - 5n^2 - 2mn - 6\end{align*}
Check.
Let \begin{align*}m = -1\end{align*} and \begin{align*}n = 1\end{align*}.
Original: \begin{align*}-2n^2 - 7 + 2mn + 8m^2 - (-2m^2 + 3n^2 + 4mn - 1)\end{align*}
\begin{align*}& -2 \cdot 1^2 - 7 + 2 \cdot (-1) \cdot 1 + 8 \cdot (-1)^2 - (-2 \cdot (-1)^2 + 3 \cdot 1^2 + 4 \cdot (-1) \cdot 1 - 1)\\ & -3 - (-4) = 1\end{align*}
Simplified: \begin{align*}10m^2 - 5n^2 - 2mn - 6\end{align*}
\begin{align*}& 10 \cdot (-1)^2 - 5 \cdot 1^2 - 2 \cdot (-1) \cdot 1 - 6\\ & 10 - 5 + 2 - 6 = 1\end{align*}
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https://www.physicsforums.com/threads/a-question-of-rolling.799861/ | A question of rolling?
1. Feb 25, 2015
Rishavutkarsh
1. The problem statement, all variables and given/known data
A force(Fi) is acting on the top point of a disc of radius r and mass m. The disc is rolling without slipping. Angular velocity of disc after center has been displaced distance x is?
2. Relevant equations
Energy conservation; Moment of inertia of disc (MR^2)/2
3. The attempt at a solution
By energy conservation
Fx=1/2*Mr^2/2*w^2+1/2mv^2
also v=rw (Rolling)
What am I missing? Any help appreciated.
2. Feb 25, 2015
Staff: Mentor
It look like you are using work done = mechanical energy gained
That should work. Now go ahead and solve for angular velocity.
3. Feb 25, 2015
PeroK
Beaten to it by NO!
4. Feb 25, 2015
haruspex
For Fx to be the work done, what must the relationship between the force F and the distance x be? Is that their relationship in this question?
5. Feb 25, 2015
Rishavutkarsh
Well I thought so too but that didn't really work out, I got the wrong answer. The answer is root 2 times my answer.
They should be in the same direction and yes they are.
6. Feb 25, 2015
PeroK
Is F the only force acting on the disc?
7. Feb 25, 2015
Rishavutkarsh
No, friction must be acting too since it's rolling but it wont do any work during rolling.
8. Feb 25, 2015
PeroK
Even so, you could take a look at the equations of motion involving both F and the frictional force.
9. Feb 25, 2015
Rishavutkarsh
F-fr=ma
FR+frR=I*(alpha)
Using this I got the answer... Hurray!!! But I am still curious to know why did energy conservation give me wrong answer.
10. Feb 25, 2015
PeroK
First, work out over what distance F would have to act to generate the correct KE. Then, see whether you can explain why F does indeed act over that distance.
11. Feb 25, 2015
Rishavutkarsh
2x? Well yeah that gives me the answer but how does F 'indeed act over that distance'? A hint would help, I can't think of anything.
12. Feb 25, 2015
PeroK
Perhaps a couple of ways to look at it:
If you're applying a force to something that's moving, then the work done by the force in time $dt$ is $Fvdt$. The top of the disk is moving twice as fast as the centre of mass of the disk. So, the work done by a force applied at the top of the disc is $F2vdt$ (where v is the velocity of the centre of mass).
Alternatively, if you think of the force as a continuous series of impulses $Fdt$ (applied at the top of the disk), you see the same result. Each impulse acts over the same time, but twice the distance, that a similar impulse at the centre of the disk would do.
If you redo your problem with the force at the centre of the disc, you can use work-energy directly over the distance x. But, at the top of the disc, the force acts through a distance of 2x.
It's similar to why the friction force does no work, as the bottom of the disc is stationary when in contact with the ground.
13. Feb 25, 2015
haruspex
Yes, but there's a bit more to it than that, as PeroK notes, x must be the distance the force acts over:
Suppose you push on a lever with force F, and you make the lever (at the point where you push it) move distance x. The work you have done is Fx. Had you applied the force twice as far from the fulcrum point the force needed would only have been F/2, but you would have needed to advance the point of application 2x in order to achieve the same movement of the lever.
14. Feb 26, 2015
Rishavutkarsh
Thank you both of you for helping me out I was able to think it as the point above moving twice as fast as center but the analogy of lever and the ground being stationary so that friction does no work helped me understand much more clearly.
Draft saved Draft deleted
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https://www.isa-afp.org/entries/Triangle.html | # Basic Geometric Properties of Triangles
Title: Basic Geometric Properties of Triangles Author: Manuel Eberl Submission date: 2015-12-28 Abstract: This entry contains a definition of angles between vectors and between three points. Building on this, we prove basic geometric properties of triangles, such as the Isosceles Triangle Theorem, the Law of Sines and the Law of Cosines, that the sum of the angles of a triangle is π, and the congruence theorems for triangles. The definitions and proofs were developed following those by John Harrison in HOL Light. However, due to Isabelle's type class system, all definitions and theorems in the Isabelle formalisation hold for all real inner product spaces. BibTeX: ```@article{Triangle-AFP, author = {Manuel Eberl}, title = {Basic Geometric Properties of Triangles}, journal = {Archive of Formal Proofs}, month = dec, year = 2015, note = {\url{http://isa-afp.org/entries/Triangle.html}, Formal proof development}, ISSN = {2150-914x}, }``` License: BSD License Used by: Chord_Segments, Ordinary_Differential_Equations, Stewart_Apollonius | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.923066258430481, "perplexity": 443.69787778294034}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267161638.66/warc/CC-MAIN-20180925123211-20180925143611-00119.warc.gz"} |
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# Im happy I finally registered
The very next time I read a blog, Hopefully it does not fail me just as much as this one. After all, Yes, it was my choice to read, but I really thought you would probably have something interesting to say. All I hear is a bunch of complaining about something you could possibly fix if you weren't too busy seeking attention.
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/home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 546, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #11 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #14 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #15 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 561, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #11 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #14 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #15 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 562, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #11 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #14 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #15 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 661, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #11 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #14 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #15 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 671, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #11 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #14 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #15 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 671, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #11 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #14 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #15 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 680, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #11 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #14 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #15 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 681, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #11 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #14 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #15 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 1068, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #11 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #14 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #15 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 1463, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #11 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #14 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #15 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 948, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/MarkdownExtra.php(16): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #9 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #10 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #11 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #14 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #15 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #16 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #17 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #18 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #19 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 1349, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/MarkdownExtra.php(16): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #9 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #10 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #11 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #14 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #15 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #16 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #17 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #18 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #19 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 1374, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/MarkdownExtra.php(16): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #9 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #10 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #11 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #14 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #15 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #16 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #17 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #18 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #19 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 1785, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/MarkdownExtra.php(16): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #9 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #10 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #11 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #14 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #15 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #16 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #17 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #18 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #19 {main}``` Notice ```Array and string offset access syntax with curly braces is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): gdn_ErrorHandler(8192, 'Array and strin...', '/home/bdurrell/...', 1787, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #3 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #4 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/MarkdownExtra.php(16): spl_autoload_call('Michelf\\Markdow...') #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #6 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #7 [internal function]: Composer\Autoload\ClassLoader->loadClass('Michelf\\Markdow...') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.markdownvanilla.php(18): spl_autoload_call('Michelf\\Markdow...') #9 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(444): include('/home/bdurrell/...') #10 /home/bdurrell/public_html/realeducated.com/forums/vendor/composer/ClassLoader.php(322): Composer\Autoload\includeFile('/home/bdurrell/...') #11 [internal function]: Composer\Autoload\ClassLoader->loadClass('MarkdownVanilla') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): spl_autoload_call('MarkdownVanilla') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #14 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #15 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #16 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #17 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #18 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #19 {main}``` Notice ```Function create_function() is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/Markdown.php(1873): gdn_ErrorHandler(8192, 'Function create...', '/home/bdurrell/...', 1873, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/Markdown.php(156): Michelf\Markdown->_initDetab() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/MarkdownExtra.php(108): Michelf\Markdown->__construct() #3 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): Michelf\MarkdownExtra->__construct() #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #5 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1096): Gdn_Format::to('The very next t...', 'Markdown') #6 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(168): Gdn_Format::plainText('The very next t...', 'Markdown') #7 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #9 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #10 {main}``` Notice ```Function create_function() is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/Markdown.php(1873): gdn_ErrorHandler(8192, 'Function create...', '/home/bdurrell/...', 1873, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/Markdown.php(156): Michelf\Markdown->_initDetab() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/MarkdownExtra.php(108): Michelf\Markdown->__construct() #3 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): Michelf\MarkdownExtra->__construct() #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #5 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(170): Gdn_Format::to('The very next t...', 'Markdown') #6 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #7 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #8 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #9 {main}``` Notice ```Function create_function() is deprecated #0 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/Markdown.php(1873): gdn_ErrorHandler(8192, 'Function create...', '/home/bdurrell/...', 1873, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/Markdown.php(156): Michelf\Markdown->_initDetab() #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/michelf/php-markdown/Michelf/MarkdownExtra.php(108): Michelf\Markdown->__construct() #3 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(1799): Michelf\MarkdownExtra->__construct() #4 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.format.php(2198): Gdn_Format::markdown('The very next t...') #5 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/views/discussion/helper_functions.php(24): Gdn_Format::to('The very next t...', 'Markdown') #6 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/views/discussion/discussion.php(89): formatBody(Object(stdClass)) #7 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/views/discussion/index.php(31): include('/home/bdurrell/...') #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(718): include('/home/bdurrell/...') #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1316): Gdn_Controller->fetchView('', false, false) #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.pluggable.php(210): Gdn_Controller->xRender() #11 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(279): Gdn_Pluggable->__call('render', Array) #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #13 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #14 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #15 {main}``` Notice ```Trying to access array offset on value of type bool #0 /home/bdurrell/public_html/realeducated.com/forums/cache/Smarty/compile/vanilla^dfd3cb91447a990622a084888f161cbd51217ba9_0.file.default.master.tpl.php(70): gdn_ErrorHandler(8, 'Trying to acces...', '/home/bdurrell/...', 70, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_resource_base.php(123): content_5bb46009911e26_60856270(Object(Smarty_Internal_Template)) #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_compiled.php(114): Smarty_Template_Resource_Base->getRenderedTemplateCode(Object(Smarty_Internal_Template)) #3 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_template.php(216): Smarty_Template_Compiled->render(Object(Smarty_Internal_Template)) #4 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(232): Smarty_Internal_Template->render(false, 1) #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(134): Smarty_Internal_TemplateBase->_execute(Object(Smarty_Internal_Template), NULL, 'vanilla', NULL, 1) #6 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.smarty.php(161): Smarty_Internal_TemplateBase->display('/home/bdurrell/...', NULL, 'vanilla') #7 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1957): Gdn_Smarty->render('/home/bdurrell/...', Object(DiscussionController)) #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1401): Gdn_Controller->renderMaster() #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.pluggable.php(210): Gdn_Controller->xRender() #10 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(279): Gdn_Pluggable->__call('render', Array) #11 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #13 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #14 {main}``` Notice ```Trying to access array offset on value of type bool #0 /home/bdurrell/public_html/realeducated.com/forums/cache/Smarty/compile/vanilla^dfd3cb91447a990622a084888f161cbd51217ba9_0.file.default.master.tpl.php(115): gdn_ErrorHandler(8, 'Trying to acces...', '/home/bdurrell/...', 115, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_resource_base.php(123): content_5bb46009911e26_60856270(Object(Smarty_Internal_Template)) #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_compiled.php(114): Smarty_Template_Resource_Base->getRenderedTemplateCode(Object(Smarty_Internal_Template)) #3 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_template.php(216): Smarty_Template_Compiled->render(Object(Smarty_Internal_Template)) #4 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(232): Smarty_Internal_Template->render(false, 1) #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(134): Smarty_Internal_TemplateBase->_execute(Object(Smarty_Internal_Template), NULL, 'vanilla', NULL, 1) #6 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.smarty.php(161): Smarty_Internal_TemplateBase->display('/home/bdurrell/...', NULL, 'vanilla') #7 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1957): Gdn_Smarty->render('/home/bdurrell/...', Object(DiscussionController)) #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1401): Gdn_Controller->renderMaster() #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.pluggable.php(210): Gdn_Controller->xRender() #10 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(279): Gdn_Pluggable->__call('render', Array) #11 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #13 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #14 {main}``` Notice ```Trying to access array offset on value of type bool #0 /home/bdurrell/public_html/realeducated.com/forums/cache/Smarty/compile/vanilla^dfd3cb91447a990622a084888f161cbd51217ba9_0.file.default.master.tpl.php(149): gdn_ErrorHandler(8, 'Trying to acces...', '/home/bdurrell/...', 149, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_resource_base.php(123): content_5bb46009911e26_60856270(Object(Smarty_Internal_Template)) #2 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_compiled.php(114): Smarty_Template_Resource_Base->getRenderedTemplateCode(Object(Smarty_Internal_Template)) #3 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_template.php(216): Smarty_Template_Compiled->render(Object(Smarty_Internal_Template)) #4 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(232): Smarty_Internal_Template->render(false, 1) #5 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(134): Smarty_Internal_TemplateBase->_execute(Object(Smarty_Internal_Template), NULL, 'vanilla', NULL, 1) #6 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.smarty.php(161): Smarty_Internal_TemplateBase->display('/home/bdurrell/...', NULL, 'vanilla') #7 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1957): Gdn_Smarty->render('/home/bdurrell/...', Object(DiscussionController)) #8 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1401): Gdn_Controller->renderMaster() #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.pluggable.php(210): Gdn_Controller->xRender() #10 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(279): Gdn_Pluggable->__call('render', Array) #11 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #12 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #13 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #14 {main}``` Notice ```Trying to access array offset on value of type bool #0 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(241): gdn_ErrorHandler(8, 'Trying to acces...', '/home/bdurrell/...', 241, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(2812): CategoryModel->calculateUser(Array) #2 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(282): CategoryModel->{closure}(Array) #3 [internal function]: CategoryCollection->calculateDynamic(Array, 0) #4 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(490): array_walk(Array, Array) #5 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(424): CategoryCollection->getChildrenByParents(Array, 'PermsDiscussion...') #6 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(911): CategoryCollection->getTree(-1, Array) #7 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/modules/class.categoriesmodule.php(54): CategoryModel->getChildTree(NULL, Array) #8 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/modules/class.categoriesmodule.php(78): CategoriesModule->getData() #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.module.php(258): CategoriesModule->toString() #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.modulecollection.php(45): Gdn_Module->render() #11 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.modulecollection.php(69): Gdn_ModuleCollection->render() #12 /home/bdurrell/public_html/realeducated.com/forums/library/SmartyPlugins/function.asset.php(45): Gdn_ModuleCollection->toString() #13 /home/bdurrell/public_html/realeducated.com/forums/cache/Smarty/compile/vanilla^dfd3cb91447a990622a084888f161cbd51217ba9_0.file.default.master.tpl.php(169): smarty_function_asset(Array, Object(Smarty_Internal_Template)) #14 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_resource_base.php(123): content_5bb46009911e26_60856270(Object(Smarty_Internal_Template)) #15 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_compiled.php(114): Smarty_Template_Resource_Base->getRenderedTemplateCode(Object(Smarty_Internal_Template)) #16 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_template.php(216): Smarty_Template_Compiled->render(Object(Smarty_Internal_Template)) #17 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(232): Smarty_Internal_Template->render(false, 1) #18 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(134): Smarty_Internal_TemplateBase->_execute(Object(Smarty_Internal_Template), NULL, 'vanilla', NULL, 1) #19 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.smarty.php(161): Smarty_Internal_TemplateBase->display('/home/bdurrell/...', NULL, 'vanilla') #20 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1957): Gdn_Smarty->render('/home/bdurrell/...', Object(DiscussionController)) #21 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1401): Gdn_Controller->renderMaster() #22 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.pluggable.php(210): Gdn_Controller->xRender() #23 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(279): Gdn_Pluggable->__call('render', Array) #24 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #25 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #26 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #27 {main}``` Notice ```Trying to access array offset on value of type bool #0 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(241): gdn_ErrorHandler(8, 'Trying to acces...', '/home/bdurrell/...', 241, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(2812): CategoryModel->calculateUser(Array) #2 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(282): CategoryModel->{closure}(Array) #3 [internal function]: CategoryCollection->calculateDynamic(Array, 1) #4 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(490): array_walk(Array, Array) #5 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(424): CategoryCollection->getChildrenByParents(Array, 'PermsDiscussion...') #6 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(911): CategoryCollection->getTree(-1, Array) #7 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/modules/class.categoriesmodule.php(54): CategoryModel->getChildTree(NULL, Array) #8 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/modules/class.categoriesmodule.php(78): CategoriesModule->getData() #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.module.php(258): CategoriesModule->toString() #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.modulecollection.php(45): Gdn_Module->render() #11 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.modulecollection.php(69): Gdn_ModuleCollection->render() #12 /home/bdurrell/public_html/realeducated.com/forums/library/SmartyPlugins/function.asset.php(45): Gdn_ModuleCollection->toString() #13 /home/bdurrell/public_html/realeducated.com/forums/cache/Smarty/compile/vanilla^dfd3cb91447a990622a084888f161cbd51217ba9_0.file.default.master.tpl.php(169): smarty_function_asset(Array, Object(Smarty_Internal_Template)) #14 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_resource_base.php(123): content_5bb46009911e26_60856270(Object(Smarty_Internal_Template)) #15 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_compiled.php(114): Smarty_Template_Resource_Base->getRenderedTemplateCode(Object(Smarty_Internal_Template)) #16 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_template.php(216): Smarty_Template_Compiled->render(Object(Smarty_Internal_Template)) #17 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(232): Smarty_Internal_Template->render(false, 1) #18 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(134): Smarty_Internal_TemplateBase->_execute(Object(Smarty_Internal_Template), NULL, 'vanilla', NULL, 1) #19 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.smarty.php(161): Smarty_Internal_TemplateBase->display('/home/bdurrell/...', NULL, 'vanilla') #20 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1957): Gdn_Smarty->render('/home/bdurrell/...', Object(DiscussionController)) #21 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1401): Gdn_Controller->renderMaster() #22 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.pluggable.php(210): Gdn_Controller->xRender() #23 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(279): Gdn_Pluggable->__call('render', Array) #24 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #25 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #26 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #27 {main}``` Notice ```Trying to access array offset on value of type bool #0 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(241): gdn_ErrorHandler(8, 'Trying to acces...', '/home/bdurrell/...', 241, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(2812): CategoryModel->calculateUser(Array) #2 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(282): CategoryModel->{closure}(Array) #3 [internal function]: CategoryCollection->calculateDynamic(Array, 2) #4 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(490): array_walk(Array, Array) #5 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(424): CategoryCollection->getChildrenByParents(Array, 'PermsDiscussion...') #6 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(911): CategoryCollection->getTree(-1, Array) #7 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/modules/class.categoriesmodule.php(54): CategoryModel->getChildTree(NULL, Array) #8 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/modules/class.categoriesmodule.php(78): CategoriesModule->getData() #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.module.php(258): CategoriesModule->toString() #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.modulecollection.php(45): Gdn_Module->render() #11 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.modulecollection.php(69): Gdn_ModuleCollection->render() #12 /home/bdurrell/public_html/realeducated.com/forums/library/SmartyPlugins/function.asset.php(45): Gdn_ModuleCollection->toString() #13 /home/bdurrell/public_html/realeducated.com/forums/cache/Smarty/compile/vanilla^dfd3cb91447a990622a084888f161cbd51217ba9_0.file.default.master.tpl.php(169): smarty_function_asset(Array, Object(Smarty_Internal_Template)) #14 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_resource_base.php(123): content_5bb46009911e26_60856270(Object(Smarty_Internal_Template)) #15 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_compiled.php(114): Smarty_Template_Resource_Base->getRenderedTemplateCode(Object(Smarty_Internal_Template)) #16 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_template.php(216): Smarty_Template_Compiled->render(Object(Smarty_Internal_Template)) #17 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(232): Smarty_Internal_Template->render(false, 1) #18 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(134): Smarty_Internal_TemplateBase->_execute(Object(Smarty_Internal_Template), NULL, 'vanilla', NULL, 1) #19 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.smarty.php(161): Smarty_Internal_TemplateBase->display('/home/bdurrell/...', NULL, 'vanilla') #20 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1957): Gdn_Smarty->render('/home/bdurrell/...', Object(DiscussionController)) #21 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1401): Gdn_Controller->renderMaster() #22 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.pluggable.php(210): Gdn_Controller->xRender() #23 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(279): Gdn_Pluggable->__call('render', Array) #24 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #25 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #26 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #27 {main}``` Notice ```Trying to access array offset on value of type bool #0 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(241): gdn_ErrorHandler(8, 'Trying to acces...', '/home/bdurrell/...', 241, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(2812): CategoryModel->calculateUser(Array) #2 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(282): CategoryModel->{closure}(Array) #3 [internal function]: CategoryCollection->calculateDynamic(Array, 3) #4 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(490): array_walk(Array, Array) #5 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(424): CategoryCollection->getChildrenByParents(Array, 'PermsDiscussion...') #6 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(911): CategoryCollection->getTree(-1, Array) #7 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/modules/class.categoriesmodule.php(54): CategoryModel->getChildTree(NULL, Array) #8 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/modules/class.categoriesmodule.php(78): CategoriesModule->getData() #9 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.module.php(258): CategoriesModule->toString() #10 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.modulecollection.php(45): Gdn_Module->render() #11 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.modulecollection.php(69): Gdn_ModuleCollection->render() #12 /home/bdurrell/public_html/realeducated.com/forums/library/SmartyPlugins/function.asset.php(45): Gdn_ModuleCollection->toString() #13 /home/bdurrell/public_html/realeducated.com/forums/cache/Smarty/compile/vanilla^dfd3cb91447a990622a084888f161cbd51217ba9_0.file.default.master.tpl.php(169): smarty_function_asset(Array, Object(Smarty_Internal_Template)) #14 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_resource_base.php(123): content_5bb46009911e26_60856270(Object(Smarty_Internal_Template)) #15 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_template_compiled.php(114): Smarty_Template_Resource_Base->getRenderedTemplateCode(Object(Smarty_Internal_Template)) #16 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_template.php(216): Smarty_Template_Compiled->render(Object(Smarty_Internal_Template)) #17 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(232): Smarty_Internal_Template->render(false, 1) #18 /home/bdurrell/public_html/realeducated.com/forums/vendor/smarty/smarty/libs/sysplugins/smarty_internal_templatebase.php(134): Smarty_Internal_TemplateBase->_execute(Object(Smarty_Internal_Template), NULL, 'vanilla', NULL, 1) #19 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.smarty.php(161): Smarty_Internal_TemplateBase->display('/home/bdurrell/...', NULL, 'vanilla') #20 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1957): Gdn_Smarty->render('/home/bdurrell/...', Object(DiscussionController)) #21 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.controller.php(1401): Gdn_Controller->renderMaster() #22 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.pluggable.php(210): Gdn_Controller->xRender() #23 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/controllers/class.discussioncontroller.php(279): Gdn_Pluggable->__call('render', Array) #24 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(845): DiscussionController->index('159611', 'im-happy-i-fina...', '') #25 /home/bdurrell/public_html/realeducated.com/forums/library/core/class.dispatcher.php(274): Gdn_Dispatcher->dispatchController(Object(Gdn_Request), Array) #26 /home/bdurrell/public_html/realeducated.com/forums/index.php(29): Gdn_Dispatcher->dispatch() #27 {main}``` Notice ```Trying to access array offset on value of type bool #0 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(241): gdn_ErrorHandler(8, 'Trying to acces...', '/home/bdurrell/...', 241, Array) #1 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/models/class.categorymodel.php(2812): CategoryModel->calculateUser(Array) #2 /home/bdurrell/public_html/realeducated.com/forums/applications/vanilla/library/class.categorycollection.php(282): CategoryModel->{closure}(Array) #3 [internal function]: 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https://www.rdocumentation.org/packages/forecast/versions/8.7/vignettes/JSS2008.Rmd | [object Object]
library('forecast') library('expsmooth')
Introduction
Automatic forecasts of large numbers of univariate time series are often needed in business. It is common to have over one thousand product lines that need forecasting at least monthly. Even when a smaller number of forecasts are required, there may be nobody suitably trained in the use of time series models to produce them. In these circumstances, an automatic forecasting algorithm is an essential tool. Automatic forecasting algorithms must determine an appropriate time series model, estimate the parameters and compute the forecasts. They must be robust to unusual time series patterns, and applicable to large numbers of series without user intervention. The most popular automatic forecasting algorithms are based on either exponential smoothing or ARIMA models.
In this article, we discuss the implementation of two automatic univariate forecasting methods in the \pkg{forecast} package for \proglang{r} package.
The \pkg{forecast} package for the \proglang{r} package contains the 1001 time series from the M-competition [@Mcomp82] and the 3003 time series from the M3-competition [@M3comp00].
The \pkg{forecast} package implements automatic forecasting using exponential smoothing, ARIMA models, the Theta method [@AN00], cubic splines [@HKPB05], as well as other common forecasting methods. In this article, we primarily discuss the exponential smoothing approach (in Section \ref{sec:expsmooth}) and the ARIMA modelling approach (in Section \ref{sec:arima}) to automatic forecasting. In Section \ref{sec:package}, we describe the implementation of these methods in the \pkg{forecast} package, along with other features of the package.
Exponential smoothing {#sec:expsmooth}
Although exponential smoothing methods have been around since the 1950s, a modelling framework incorporating procedures for model selection was not developed until relatively recently. @OKS97, @HKSG02 and @HKOS05 have shown that all exponential smoothing methods (including non-linear methods) are optimal forecasts from innovations state space models.
Exponential smoothing methods were originally classified by Pegels' (1969)\nocite{Pegels69} taxonomy. This was later extended by @Gardner85, modified by @HKSG02, and extended again by @Taylor03a, giving a total of fifteen methods seen in the following table.
\begin{table}[!hbt] \begin{center}\vspace{0.2cm} \begin{tabular}{|ll|ccc|} \hline & &\multicolumn{3}{c|}{Seasonal Component} \ \multicolumn{2}{|c|}{Trend}& N & A & M\ \multicolumn{2}{|c|}{Component} & (None) & (Additive) & (Multiplicative)\ \cline{3-5} &&&-0.3cm] N & (None) & N,N & N,A & N,M\ &&&&\[-0.3cm] A & (Additive) & A,N & A,A & A,M\ &&&&\[-0.3cm] A\damped & (Additive damped) & A\damped,N & A\damped,A & A\damped,M\ &&&&\[-0.3cm] M & (Multiplicative) & M,N & M,A & M,M\ &&&&\[-0.3cm] M\damped & (Multiplicative damped) & M\damped,N & M\damped,A & M\damped,M\ \hline \end{tabular}\vspace{0.2cm} \end{center} \caption{The fifteen exponential smoothing methods.} \end{table} Some of these methods are better known under other names. For example, cell (N,N) describes the simple exponential smoothing (or SES) method, cell (A,N) describes Holt's linear method, and cell (A\damped,N) describes the damped trend method. The additive Holt-Winters' method is given by cell (A,A) and the multiplicative Holt-Winters' method is given by cell (A,M). The other cells correspond to less commonly used but analogous methods. Point forecasts for all methods We denote the observed time series by y_1,y_2,\dots,yn. A forecast of y{t+h} based on all of the data up to time t is denoted by \hat{y}_{t+h|t}. To illustrate the method, we give the point forecasts and updating equations for method (A,A), the Holt-Winters' additive method: \begin{subequations}\label{eq:AMmethod}\vspace{-15pt} \begin{align} \mbox{Level:}\quad &\ell_t = \alpha (yt - s{t-m}) + (1-\alpha)(\ell{t-1} + b{t-1})\hspace{1cm} \label{eq:3-44a}\ \mbox{Growth:}\quad &b_t = \beta^(\ellt - \ell{t-1}) + (1-\beta^)b_{t-1} \label{eq:3-45a}\ \mbox{Seasonal:}\quad &s_t = \gamma(yt - \ell{t-1} -b{t-1}) + (1-\gamma)s{t-m}\label{eq:3-46a}\ \mbox{Forecast:}\quad &\hat{y}_{t+h|t} = \ell_t + bth +s{t-m+h_m^+}. \label{eq:3-47a} \end{align} \end{subequations} where m is the length of seasonality (e.g., the number of months or quarters in a year), \ell_t represents the level of the series, b_t denotes the growth, st is the seasonal component, \hat{y}{t+h|t} is the forecast for h periods ahead, and h_m^+ = \big[(h-1) \mbox{ mod } m\big] + 1. To use method \eqref{eq:AMmethod}, we need values for the initial states \ell_0, b0 and s{1-m},\dots,s_0, and for the smoothing parameters \alpha, \beta^ and \gamma. All of these will be estimated from the observed data. Equation \eqref{eq:3-46a} is slightly different from the usual Holt-Winters equations such as those in @MWH3 or @BOK05. These authors replace \eqref{eq:3-46a} with s_t = \gamma^(yt - \ell{t}) + (1-\gamma^)s_{t-m}. If \ell_t is substituted using \eqref{eq:3-44a}, we obtain s_t = \gamma^(1-\alpha)(yt - \ell{t-1}-b_{t-1}) + {1-\gamma^(1-\alpha)}s_{t-m}. Thus, we obtain identical forecasts using this approach by replacing \gamma in \eqref{eq:3-46a} with \gamma^*(1-\alpha). The modification given in \eqref{eq:3-46a} was proposed by @OKS97 to make the state space formulation simpler. It is equivalent to Archibald's (1990)\nocite{Archibald90} variation of the Holt-Winters' method. \begin{sidewaystable} \begin{small} \begin{center} \begin{tabular}{|c|lll|} \hline & \multicolumn{3}{c|}{Seasonal} \ {Trend} & \multicolumn{1}{c}{N} & \multicolumn{1}{c}{A} & \multicolumn{1}{c|}{M}\ \cline{2-4} & \ell_t = \alpha yt + (1-\alpha) \ell{t-1} & \ell_t = \alpha (yt - s{t-m}) + (1-\alpha) \ell_{t-1} & \ell_t = \alpha (yt / s{t-m}) + (1-\alpha) \ell_{t-1}\ {N} & & s_t = \gamma (yt - \ell{t-1}) + (1-\gamma) s_{t-m} & s_t = \gamma (yt / \ell{t-1}) + (1-\gamma) s{t-m} \ & \hat{y}{t+h|t} = \ellt & \hat{y}{t+h|t} = \ellt + s{t-m+hm^+} & \hat{y}{t+h|t}= \ellts{t-m+h_m^+} \ \hline & \ell_t = \alpha yt + (1-\alpha) (\ell{t-1}+b_{t-1}) & \ell_t = \alpha (yt - s{t-m}) + (1-\alpha) (\ell{t-1}+b{t-1}) & \ell_t = \alpha (yt / s{t-m}) + (1-\alpha) (\ell{t-1}+b{t-1})\ {A} & b_t = \beta^ (\ellt-\ell{t-1}) + (1-\beta^) b_{t-1} & b_t = \beta^ (\ellt-\ell{t-1}) + (1-\beta^) b_{t-1} & b_t = \beta^ (\ellt-\ell{t-1}) + (1-\beta^) b_{t-1}\ & & s_t = \gamma (yt - \ell{t-1}-b{t-1}) + (1-\gamma) s{t-m} & s_t = \gamma (yt / (\ell{t-1}-b{t-1})) + (1-\gamma) s{t-m}\ & \hat{y}_{t+h|t} = \ell_t+hbt & \hat{y}{t+h|t} = \ell_t +hbt +s{t-m+hm^+} & \hat{y}{t+h|t}= (\ell_t+hbt)s{t-m+h_m^+} \ \hline & \ell_t = \alpha yt + (1-\alpha) (\ell{t-1}+\phi b_{t-1}) & \ell_t = \alpha (yt - s{t-m}) + (1-\alpha) (\ell{t-1}+\phi b{t-1}) & \ell_t = \alpha (yt / s{t-m}) + (1-\alpha) (\ell{t-1}+\phi b{t-1})\ {A\damped } & b_t = \beta^ (\ellt-\ell{t-1}) + (1-\beta^) \phi b_{t-1} & b_t = \beta^ (\ellt-\ell{t-1}) + (1-\beta^) \phi b_{t-1} & b_t = \beta^ (\ellt-\ell{t-1}) + (1-\beta^) \phi b_{t-1}\ & & s_t = \gamma (yt - \ell{t-1}-\phi b{t-1}) + (1-\gamma) s{t-m} & s_t = \gamma (yt / (\ell{t-1}-\phi b{t-1})) + (1-\gamma) s{t-m}\ & \hat{y}_{t+h|t} = \ell_t+\dampfactor bt & \hat{y}{t+h|t} = \ell_t+\dampfactor bt+s{t-m+hm^+} & \hat{y}{t+h|t}= (\ell_t+\dampfactor bt)s{t-m+h_m^+} \ \hline & \ell_t = \alpha yt + (1-\alpha) \ell{t-1}b_{t-1} & \ell_t = \alpha (yt - s{t-m}) + (1-\alpha) \ell{t-1}b{t-1} & \ell_t = \alpha (yt / s{t-m}) + (1-\alpha) \ell{t-1}b{t-1}\ {M} & b_t = \beta^ (\ellt/\ell{t-1}) + (1-\beta^) b_{t-1} & b_t = \beta^ (\ellt/\ell{t-1}) + (1-\beta^) b_{t-1} & b_t = \beta^ (\ellt/\ell{t-1}) + (1-\beta^) b_{t-1}\ & & s_t = \gamma (yt - \ell{t-1}b{t-1}) + (1-\gamma) s{t-m} & s_t = \gamma (yt / (\ell{t-1}b{t-1})) + (1-\gamma) s{t-m}\ & \hat{y}_{t+h|t} = \ell_tbt^h & \hat{y}{t+h|t} = \ell_tbt^h + s{t-m+hm^+} & \hat{y}{t+h|t}= \ell_tbt^hs{t-m+h_m^+} \ \hline & \ell_t = \alpha yt + (1-\alpha) \ell{t-1}b^\phi_{t-1} & \ell_t = \alpha (yt - s{t-m}) + (1-\alpha)\ell{t-1}b^\phi{t-1} & \ell_t = \alpha (yt / s{t-m}) + (1-\alpha)\ell{t-1}b^\phi{t-1}\ {M\damped } & b_t = \beta^ (\ellt/\ell{t-1}) + (1-\beta^) b^\phi_{t-1} & b_t = \beta^ (\ellt/\ell{t-1}) + (1-\beta^) b^\phi_{t-1} & b_t = \beta^ (\ellt/\ell{t-1}) + (1-\beta^) b^\phi_{t-1}\ & & s_t = \gamma (yt - \ell{t-1}b^\phi{t-1}) + (1-\gamma) s{t-m} & s_t = \gamma (yt / (\ell{t-1}b^\phi{t-1})) + (1-\gamma) s{t-m}\ & \hat{y}_{t+h|t} = \ell_tb_t^{\phih} & \hat{y}{t+h|t} = \ell_tb_t^{\phih} + s{t-m+hm^+} & \hat{y}{t+h|t}= \ell_tb_t^{\phih}s{t-m+h_m^+} \ \hline \end{tabular} \end{center} \end{small} \caption{Formulae for recursive calculations and point forecasts. In each case, \ell_t denotes the series level at time t, b_t denotes the slope at time t, s_t denotes the seasonal component of the series at time t, and m denotes the number of seasons in a year; \alpha, \beta^*, \gamma and \phi are constants, \phi_h = \phi+\phi^2+\dots+\phi^{h} and h_m^+ = \big[(h-1) \mbox{ mod } m\big] + 1.}\label{table:pegels} \end{sidewaystable} Table \ref{table:pegels} gives recursive formulae for computing point forecasts h periods ahead for all of the exponential smoothing methods. Some interesting special cases can be obtained by setting the smoothing parameters to extreme values. For example, if \alpha=0, the level is constant over time; if \beta^*=0, the slope is constant over time; and if \gamma=0, the seasonal pattern is constant over time. At the other extreme, naïve forecasts (i.e., \hat{y}_{t+h|t}=y_t for all h) are obtained using the (N,N) method with \alpha=1. Finally, the additive and multiplicative trend methods are special cases of their damped counterparts obtained by letting \phi=1. Innovations state space models {#sec:statespace} For each exponential smoothing method in Table \ref{table:pegels}, @expsmooth08 describe two possible innovations state space models, one corresponding to a model with additive errors and the other to a model with multiplicative errors. If the same parameter values are used, these two models give equivalent point forecasts, although different prediction intervals. Thus there are 30 potential models described in this classification. Historically, the nature of the error component has often been ignored, because the distinction between additive and multiplicative errors makes no difference to point forecasts. We are careful to distinguish exponential smoothing \emph{methods} from the underlying state space \emph{models}. An exponential smoothing method is an algorithm for producing point forecasts only. The underlying stochastic state space model gives the same point forecasts, but also provides a framework for computing prediction intervals and other properties. To distinguish the models with additive and multiplicative errors, we add an extra letter to the front of the method notation. The triplet (E,T,S) refers to the three components: error, trend and seasonality. So the model ETS(A,A,N) has additive errors, additive trend and no seasonality---in other words, this is Holt's linear method with additive errors. Similarly, ETS(M,M\damped,M) refers to a model with multiplicative errors, a damped multiplicative trend and multiplicative seasonality. The notation ETS(\cdot,\cdot,\cdot) helps in remembering the order in which the components are specified. Once a model is specified, we can study the probability distribution of future values of the series and find, for example, the conditional mean of a future observation given knowledge of the past. We denote this as \mu{t+h|t} = \E(y{t+h} \mid \bm{x}_t), where \bm{x}_t contains the unobserved components such as \ell_t, b_t and s_t. For h=1 we use \mut\equiv\mu{t+1|t} as a shorthand notation. For many models, these conditional means will be identical to the point forecasts given in Table \ref{table:pegels}, so that \mu{t+h|t}=\hat{y}{t+h|t}. However, for other models (those with multiplicative trend or multiplicative seasonality), the conditional mean and the point forecast will differ slightly for h\ge 2. We illustrate these ideas using the damped trend method of @GM85. \subsubsection{Additive error model: ETS(A,A_d,N)} Let \mu_t = \hat{y}t = \ell{t-1}+b{t-1} denote the one-step forecast of y{t} assuming that we know the values of all parameters. Also, let \varepsilon_t = y_t - \mu_t denote the one-step forecast error at time t. From the equations in Table \ref{table:pegels}, we find that\vspace{-15pt} \begin{align} \label{ss1} yt &= \ell{t-1} + \phi b_{t-1} + \varepsilon_t\ \ellt &= \ell{t-1} + \phi b_{t-1} + \alpha \varepsilon_t \label{ss2}\ bt &= \phi b{t-1} + \beta^(\ellt - \ell{t-1}- \phi b{t-1}) = \phi b{t-1} + \alpha\beta^\varepsilon_t. \label{ss3} \end{align} We simplify the last expression by setting \beta=\alpha\beta^. The three equations above constitute a state space model underlying the damped Holt's method. Note that it is an \emph{innovations} state space model [@AM79;@Aoki87] because the same error term appears in each equation. We an write it in standard state space notation by defining the state vector as \bm{x}_t = (\ell_t,b_t)' and expressing \eqref{ss1}--\eqref{ss3} as \begin{subequations}\vspace*{-15pt} \begin{align} yt &= \left[ 1 \phi \right] \bm{x}{t-1} + \varepsilon_t\label{obseq}\ \bm{x}t &= \left[\begin{array}{ll} 1 & \phi\ 0 & \phi \end{array}\right]\bm{x}{t-1} + \left[\begin{array}{l} \alpha\\ \beta \end{array}\right]\varepsilon_t.\label{stateeq} \end{align} \end{subequations} The model is fully specified once we state the distribution of the error term \varepsilon_t. Usually we assume that these are independent and identically distributed, following a normal distribution with mean 0 and variance \sigma^2, which we write as \varepsilon_t \sim\mbox{NID}(0, \sigma^2). \subsubsection{Multiplicative error model: ETS(M,A_d,N)} A model with multiplicative error can be derived similarly, by first setting \varepsilon_t = (y_t-\mu_t)/\mu_t, so that \varepsilon_t is the relative error. Then, following a similar approach to that for additive errors, we find\vspace{-15pt} \begin{align} yt &= (\ell{t-1} + \phi b_{t-1})(1 + \varepsilon_t)\ \ellt &= (\ell{t-1} + \phi b_{t-1})(1 + \alpha \varepsilon_t)\ bt &= \phi b{t-1} + \beta(\ell{t-1}+\phi b{t-1})\varepsilon_t, \end{align} or\vspace{-15pt} \begin{align} yt &= \left[ 1 \phi \right] \bm{x}{t-1}(1 + \varepsilon_t)\ \bm{x}t &= \left[\begin{array}{ll} 1 & \phi \ 0 & \phi \end{array}\right]\bm{x}{t-1} + \left[ 1 \phi \right] \bm{x}_{t-1} \left[\begin{array}{l} \alpha\ \beta \end{array}\right]\varepsilon_t. \end{align} Again we assume that \varepsilon_t \sim \mbox{NID}(0,\sigma^2). Of course, this is a nonlinear state space model, which is usually considered difficult to handle in estimating and forecasting. However, that is one of the many advantages of the innovations form of state space models --- we can still compute forecasts, the likelihood and prediction intervals for this nonlinear model with no more effort than is required for the additive error model. State space models for all exponential smoothing methods {#sec:ssmodels} There are similar state space models for all 30 exponential smoothing variations. The general model involves a state vector \bm{x}_t = (\ell_t, b_t, st, s{t-1}, \dots, s_{t-m+1})' and state space equations of the form \begin{subequations}\label{eq:ss}\vspace*{-15pt} \begin{align} yt &= w(\bm{x}{t-1}) + r(\bm{x}_{t-1})\varepsilon_t \label{eq:ss1}\ \bm{x}t &= f(\bm{x}{t-1}) + g(\bm{x}_{t-1})\varepsilon_t \label{eq:ss2} \end{align} \end{subequations} where {\varepsilon_t} is a Gaussian white noise process with mean zero and variance \sigma^2, and \mut = w(\bm{x}{t-1}). The model with additive errors has r(\bm{x}_{t-1})=1, so that yt = \mu{t} + \varepsilont. The model with multiplicative errors has r(\bm{x}{t-1})=\mu_t, so that yt = \mu{t}(1 + \varepsilon_t). Thus, \varepsilon_t = (y_t - \mu_t)/\mut is the relative error for the multiplicative model. The models are not unique. Clearly, any value of r(\bm{x}{t-1}) will lead to identical point forecasts for y_t. All of the methods in Table \ref{table:pegels} can be written in the form \eqref{eq:ss1} and \eqref{eq:ss2}. The specific form for each model is given in @expsmooth08. Some of the combinations of trend, seasonality and error can occasionally lead to numerical difficulties; specifically, any model equation that requires division by a state component could involve division by zero. This is a problem for models with additive errors and either multiplicative trend or multiplicative seasonality, as well as for the model with multiplicative errors, multiplicative trend and additive seasonality. These models should therefore be used with caution. The multiplicative error models are useful when the data are strictly positive, but are not numerically stable when the data contain zeros or negative values. So when the time series is not strictly positive, only the six fully additive models may be applied. The point forecasts given in Table \ref{table:pegels} are easily obtained from these models by iterating equations \eqref{eq:ss1} and \eqref{eq:ss2} for t=n+1, n+2,\dots,n+h, setting \varepsilon{n+j}=0 for j=1,\dots,h. In most cases (notable exceptions being models with multiplicative seasonality or multiplicative trend for h\ge2), the point forecasts can be shown to be equal to \mu{t+h|t} = \E(y_{t+h} \mid \bm{x}_t), the conditional expectation of the corresponding state space model. The models also provide a means of obtaining prediction intervals. In the case of the linear models, where the forecast distributions are normal, we can derive the conditional variance v{t+h|t} = \var(y{t+h} \mid \bm{x}_t) and obtain prediction intervals accordingly. This approach also works for many of the nonlinear models. Detailed derivations of the results for many models are given in @HKOS05. A more direct approach that works for all of the models is to simply simulate many future sample paths conditional on the last estimate of the state vector, \bm{x}_t. Then prediction intervals can be obtained from the percentiles of the simulated sample paths. Point forecasts can also be obtained in this way by taking the average of the simulated values at each future time period. An advantage of this approach is that we generate an estimate of the complete predictive distribution, which is especially useful in applications such as inventory planning, where expected costs depend on the whole distribution. Estimation {#sec:estimation} In order to use these models for forecasting, we need to know the values of \bm{x}_0 and the parameters \alpha, \beta, \gamma and \phi. It is easy to compute the likelihood of the innovations state space model \eqref{eq:ss}, and so obtain maximum likelihood estimates. @OKS97 show that\vspace{-15pt} \label{likelihood} L^(\bm\theta,\bm{x}0) = n\log\Big(\sum{t=1}^n \varepsilon^2t\Big) + 2\sum{t=1}^n \log|r(\bm{x}_{t-1})| is equal to twice the negative logarithm of the likelihood function (with constant terms eliminated), conditional on the parameters \bm\theta = (\alpha,\beta,\gamma,\phi)' and the initial states \bm{x}_0 = (\ell_0,b_0,s0,s{-1},\dots,s_{-m+1})', where n is the number of observations. This is easily computed by simply using the recursive equations in Table \ref{table:pegels}. Unlike state space models with multiple sources of error, we do not need to use the Kalman filter to compute the likelihood. The parameters \bm\theta and the initial states \bm{x}_0 can be estimated by minimizing L^*. Most implementations of exponential smoothing use an ad hoc heuristic scheme to estimate \bm{x}_0. However, with modern computers, there is no reason why we cannot estimate \bm{x}_0 along with \bm\theta, and the resulting forecasts are often substantially better when we do. We constrain the initial states \bm{x}_0 so that the seasonal indices add to zero for additive seasonality, and add to m for multiplicative seasonality. There have been several suggestions for restricting the parameter space for \alpha, \beta and \gamma. The traditional approach is to ensure that the various equations can be interpreted as weighted averages, thus requiring \alpha, \beta^=\beta/\alpha, \gamma^=\gamma/(1-\alpha) and \phi to all lie within (0,1). This suggests 0<\alpha<1,\qquad 0<\beta<\alpha,\qquad 0<\gamma < 1-\alpha,\qquad\mbox{and}\qquad 0<\phi<1. However, @HAA08 show that these restrictions are usually stricter than necessary (although in a few cases they are not restrictive enough). Model selection Forecast accuracy measures such as mean squared error (MSE) can be used for selecting a model for a given set of data, provided the errors are computed from data in a hold-out set and not from the same data as were used for model estimation. However, there are often too few out-of-sample errors to draw reliable conclusions. Consequently, a penalized method based on the in-sample fit is usually better. One such approach uses a penalized likelihood such as Akaike's Information Criterion: \mbox{AIC} = L^*(\hat{\bm\theta},\hat{\bm{x}}_0) + 2q, where q is the number of parameters in \bm\theta plus the number of free states in \bm{x}_0, and \hat{\bm\theta} and \hat{\bm{x}}_0 denote the estimates of \bm\theta and \bm{x}_0. We select the model that minimizes the AIC amongst all of the models that are appropriate for the data. The AIC also provides a method for selecting between the additive and multiplicative error models. The point forecasts from the two models are identical so that standard forecast accuracy measures such as the MSE or mean absolute percentage error (MAPE) are unable to select between the error types. The AIC is able to select between the error types because it is based on likelihood rather than one-step forecasts. Obviously, other model selection criteria (such as the BIC) could also be used in a similar manner. Automatic forecasting {#sec:algorithm} We combine the preceding ideas to obtain a robust and widely applicable automatic forecasting algorithm. The steps involved are summarized below. \begin{compactenum} \item For each series, apply all models that are appropriate, optimizing the parameters (both smoothing parameters and the initial state variable) of the model in each case. \item Select the best of the models according to the AIC. \item Produce point forecasts using the best model (with optimized parameters) for as many steps ahead as required. \item Obtain prediction intervals for the best model either using the analytical results of Hyndman, Koehler, et al. (2005), or by simulating future sample paths for {y{n+1},\dots,y{n+h}} and finding the \alpha/2 and 1-\alpha/2 percentiles of the simulated data at each forecasting horizon. If simulation is used, the sample paths may be generated using the normal distribution for errors (parametric bootstrap) or using the resampled errors (ordinary bootstrap). \end{compactenum} @HKSG02 applied this automatic forecasting strategy to the M-competition data [@Mcomp82] and the IJF-M3 competition data [@M3comp00] using a restricted set of exponential smoothing models, and demonstrated that the methodology is particularly good at short term forecasts (up to about 6 periods ahead), and especially for seasonal short-term series (beating all other methods in the competitions for these series). ARIMA models {#sec:arima} A common obstacle for many people in using Autoregressive Integrated Moving Average (ARIMA) models for forecasting is that the order selection process is usually considered subjective and difficult to apply. But it does not have to be. There have been several attempts to automate ARIMA modelling in the last 25 years. @HR82 proposed a method to identify the order of an ARMA model for a stationary series. In their method the innovations can be obtained by fitting a long autoregressive model to the data, and then the likelihood of potential models is computed via a series of standard regressions. They established the asymptotic properties of the procedure under very general conditions. @Gomez98 extended the Hannan-Rissanen identification method to include multiplicative seasonal ARIMA model identification. @TRAMOSEATS98 implemented this automatic identification procedure in the software \pkg{TRAMO} and \pkg{SEATS}. For a given series, the algorithm attempts to find the model with the minimum BIC. @Liu89 proposed a method for identification of seasonal ARIMA models using a filtering method and certain heuristic rules; this algorithm is used in the \pkg{SCA-Expert} software. Another approach is described by @MP00a whose algorithm for univariate ARIMA models also allows intervention analysis. It is implemented in the software package Time Series Expert'' (\pkg{TSE-AX}). Other algorithms are in use in commercial software, although they are not documented in the public domain literature. In particular, \pkg{Forecast Pro} [@ForecastPro00] is well-known for its excellent automatic ARIMA algorithm which was used in the M3-forecasting competition [@M3comp00]. Another proprietary algorithm is implemented in \pkg{Autobox} [@Reilly00]. @OL96 provide an early review of some of the commercial software that implement automatic ARIMA forecasting. Choosing the model order using unit root tests and the AIC A non-seasonal ARIMA(p,d,q) process is given by \phi(B)(1-B^d)y_{t} = c + \theta(B)\varepsilon_t where {\varepsilon_t} is a white noise process with mean zero and variance \sigma^2, B is the backshift operator, and \phi(z) and \theta(z) are polynomials of order p and q respectively. To ensure causality and invertibility, it is assumed that \phi(z) and \theta(z) have no roots for |z|<1 [@BDbook91]. If c\ne0, there is an implied polynomial of order d in the forecast function. The seasonal ARIMA(p,d,q)(P,D,Q)m process is given by \Phi(B^m)\phi(B)(1-B^{m})^D(1-B)^dy{t} = c + \Theta(B^m)\theta(B)\varepsilon_t where \Phi(z) and \Theta(z) are polynomials of orders P and Q respectively, each containing no roots inside the unit circle. If c\ne0, there is an implied polynomial of order d+D in the forecast function. The main task in automatic ARIMA forecasting is selecting an appropriate model order, that is the values p, q, P, Q, D, d. If d and D are known, we can select the orders p, q, P and Q via an information criterion such as the AIC: \mbox{AIC} = -2\log(L) + 2(p+q+P+Q+k) where k=1 if c\ne0 and 0 otherwise, and L is the maximized likelihood of the model fitted to the \emph{differenced} data (1-B^m)^D(1-B)^dy_t. The likelihood of the full model for y_t is not actually defined and so the value of the AIC for different levels of differencing are not comparable. One solution to this difficulty is the diffuse prior'' approach which is outlined in @DKbook01 and implemented in the \code{arima()} function [@Ripley:2002] in \R. In this approach, the initial values of the time series (before the observed values) are assumed to have mean zero and a large variance. However, choosing d and D by minimizing the AIC using this approach tends to lead to over-differencing. For forecasting purposes, we believe it is better to make as few differences as possible because over-differencing harms forecasts [@SY94] and widens prediction intervals. [Although, see @Hendry97 for a contrary view.] Consequently, we need some other approach to choose d and D. We prefer unit-root tests. However, most unit-root tests are based on a null hypothesis that a unit root exists which biases results towards more differences rather than fewer differences. For example, variations on the Dickey-Fuller test [@DF81] all assume there is a unit root at lag 1, and the HEGY test of @HEGY90 is based on a null hypothesis that there is a seasonal unit root. Instead, we prefer unit-root tests based on a null hypothesis of no unit-root. For non-seasonal data, we consider ARIMA(p,d,q) models where d is selected based on successive KPSS unit-root tests [@KPSS92]. That is, we test the data for a unit root; if the test result is significant, we test the differenced data for a unit root; and so on. We stop this procedure when we obtain our first insignificant result. For seasonal data, we consider ARIMA(p,d,q)(P,D,Q)_m models where m is the seasonal frequency and D=0 or D=1 depending on an extended Canova-Hansen test [@CH95]. Canova and Hansen only provide critical values for 2<m<13. In our implementation of their test, we allow any value of m>1. Let C_m be the critical value for seasonal period m. We plotted C_m against m for values of m up to 365 and noted that they fit the line C_m = 0.269 m^{0.928} almost exactly. So for m>12, we use this simple expression to obtain the critical value. We note in passing that the null hypothesis for the Canova-Hansen test is not an ARIMA model as it includes seasonal dummy terms. It is a test for whether the seasonal pattern changes sufficiently over time to warrant a seasonal unit root, or whether a stable seasonal pattern modelled using fixed dummy variables is more appropriate. Nevertheless, we have found that the test is still useful for choosing D in a strictly ARIMA framework (i.e., without seasonal dummy variables). If a stable seasonal pattern is selected (i.e., the null hypothesis is not rejected), the seasonality is effectively handled by stationary seasonal AR and MA terms. After D is selected, we choose d by applying successive KPSS unit-root tests to the seasonally differenced data (if D=1) or the original data (if D=0). Once d (and possibly D) are selected, we proceed to select the values of p, q, P and Q by minimizing the AIC. We allow c\ne0 for models where d+D < 2. A step-wise procedure for traversing the model space Suppose we have seasonal data and we consider ARIMA(p,d,q)(P,D,Q)_m models where p and q can take values from 0 to 3, and P and Q can take values from 0 to 1. When c=0 there is a total of 288 possible models, and when c\ne 0 there is a total of 192 possible models, giving 480 models altogether. If the values of p, d, q, P, D and Q are allowed to range more widely, the number of possible models increases rapidly. Consequently, it is often not feasible to simply fit every potential model and choose the one with the lowest AIC. Instead, we need a way of traversing the space of models efficiently in order to arrive at the model with the lowest AIC value. We propose a step-wise algorithm as follows. \begin{description} \item[Step 1:] We try four possible models to start with. \begin{itemize} \item ARIMA(2,d,2) if m=1 and ARIMA(2,d,2)(1,D,1) if m>1. \item ARIMA(0,d,0) if m=1 and ARIMA(0,d,0)(0,D,0) if m>1. \item ARIMA(1,d,0) if m=1 and ARIMA(1,d,0)(1,D,0) if m>1. \item ARIMA(0,d,1) if m=1 and ARIMA(0,d,1)(0,D,1) if m>1. \end{itemize} If d+D \le 1, these models are fitted with c\ne0. Otherwise, we set c=0. Of these four models, we select the one with the smallest AIC value. This is called the current'' model and is denoted by ARIMA(p,d,q) if m=1 or ARIMA(p,d,q)(P,D,Q)_m if m>1. \item[Step 2:] We consider up to seventeen variations on the current model: \begin{itemize} \item where one of p, q, P and Q is allowed to vary by \pm1 from the current model; \item where p and q both vary by \pm1 from the current model; \item where P and Q both vary by \pm1 from the current model; \item where the constant c is included if the current model has c=0 or excluded if the current model has c\ne0. \end{itemize} Whenever a model with lower AIC is found, it becomes the new current'' model and the procedure is repeated. This process finishes when we cannot find a model close to the current model with lower AIC. \end{description} There are several constraints on the fitted models to avoid problems with convergence or near unit-roots. The constraints are outlined below. \begin{compactitem}\itemsep=8pt \item The values of p and q are not allowed to exceed specified upper bounds (with default values of 5 in each case). \item The values of P and Q are not allowed to exceed specified upper bounds (with default values of 2 in each case). \item We reject any model which is close'' to non-invertible or non-causal. Specifically, we compute the roots of \phi(B)\Phi(B) and \theta(B)\Theta(B). If either have a root that is smaller than 1.001 in absolute value, the model is rejected. \item If there are any errors arising in the non-linear optimization routine used for estimation, the model is rejected. The rationale here is that any model that is difficult to fit is probably not a good model for the data. \end{compactitem} The algorithm is guaranteed to return a valid model because the model space is finite and at least one of the starting models will be accepted (the model with no AR or MA parameters). The selected model is used to produce forecasts. Comparisons with exponential smoothing There is a widespread myth that ARIMA models are more general than exponential smoothing. This is not true. The two classes of models overlap. The linear exponential smoothing models are all special cases of ARIMA models---the equivalences are discussed in @HAA08. However, the non-linear exponential smoothing models have no equivalent ARIMA counterpart. On the other hand, there are many ARIMA models which have no exponential smoothing counterpart. Thus, the two model classes overlap and are complimentary; each has its strengths and weaknesses. The exponential smoothing state space models are all non-stationary. Models with seasonality or non-damped trend (or both) have two unit roots; all other models---that is, non-seasonal models with either no trend or damped trend---have one unit root. It is possible to define a stationary model with similar characteristics to exponential smoothing, but this is not normally done. The philosophy of exponential smoothing is that the world is non-stationary. So if a stationary model is required, ARIMA models are better. One advantage of the exponential smoothing models is that they can be non-linear. So time series that exhibit non-linear characteristics including heteroscedasticity may be better modelled using exponential smoothing state space models. For seasonal data, there are many more ARIMA models than the 30 possible models in the exponential smoothing class of Section \ref{sec:expsmooth}. It may be thought that the larger model class is advantageous. However, the results in @HKSG02 show that the exponential smoothing models performed better than the ARIMA models for the seasonal M3 competition data. (For the annual M3 data, the ARIMA models performed better.) In a discussion of these results, @Hyndman01 speculates that the larger model space of ARIMA models actually harms forecasting performance because it introduces additional uncertainty. The smaller exponential smoothing class is sufficiently rich to capture the dynamics of almost all real business and economic time series. The forecast package {#sec:package} The algorithms and modelling frameworks for automatic univariate time series forecasting are implemented in the \pkg{forecast} package in \R. We illustrate the methods using the following four real time series shown in Figure \ref{fexamples}. \begin{compactitem} \item Figure \ref{fexamples}(a) shows 125 monthly US government bond yields (percent per annum) from January 1994 to May 2004. \item Figure \ref{fexamples}(b) displays 55 observations of annual US net electricity generation (billion kwh) for 1949 through 2003. \item Figure \ref{fexamples}(c) presents 113 quarterly observations of passenger motor vehicle production in the U.K. (thousands of cars) for the first quarter of 1977 through the first quarter of 2005. \item Figure \ref{fexamples}(d) shows 240 monthly observations of the number of short term overseas visitors to Australia from May 1985 to April 2005. \end{compactitem} \begin{figure}[!htb] \centering par(mfrow = c(2,2)) mod1 <- ets(bonds) mod2 <- ets(usnetelec) mod3 <- ets(ukcars) mod4 <- ets(visitors) plot(forecast(mod1), main="(a) US 10-year bonds yield", xlab="Year", ylab="Percentage per annum") plot(forecast(mod2), main="(b) US net electricity generation", xlab="Year", ylab="Billion kwh") plot(forecast(mod3), main="(c) UK passenger motor vehicle production", xlab="Year", ylab="Thousands of cars") plot(forecast(mod4), main="(d) Overseas visitors to Australia", xlab="Year", ylab="Thousands of people") \caption{Four time series showing point forecasts and 80\% \& 95\% prediction intervals obtained using exponential smoothing state space models.\label{fexamples}} \end{figure} etsnames <- c(mod1method, mod2method, mod3method, mod4method) etsnames <- gsub("Ad","A\\\\damped",etsnames) Implementation of the automatic exponential smoothing algorithm The innovations state space modelling framework described in Section \ref{sec:expsmooth} is implemented via the \code{ets()} function in the \pkg{forecast} package. (The default settings of \code{ets()} do not allow models with multiplicative trend, but they can be included using \code{allow.multiplicative.trend=TRUE}.) The models chosen via the algorithm for the four data sets were: \begin{compactitem} \item r etsnames[1] for monthly US 10-year bonds yield\ (\alpha=r format(coef(mod1)['alpha'], digits=4, nsmall=4), \beta=r format(coef(mod1)['beta'], digits=4, nsmall=4), \phi=r format(coef(mod1)['phi'], digits=4, nsmall=4), \ell_0 = r format(coef(mod1)['l'], digits=4, nsmall=4), b_0=r format(coef(mod1)['b'], digits=4, nsmall=4)); \item r etsnames[2] for annual US net electricity generation\ (\alpha=r format(coef(mod2)['alpha'], digits=4, nsmall=4), \beta=r format(coef(mod2)['beta'], digits=4, nsmall=4), \ell_0 = r format(coef(mod2)['l'], digits=4, nsmall=4), b_0=r format(coef(mod2)['b'], digits=4, nsmall=4)); \item r etsnames[3] for quarterly UK motor vehicle production\ (\alpha=r format(coef(mod3)['alpha'], digits=4, nsmall=4), \gamma=r format(coef(mod3)['gamma'], digits=4, nsmall=4), \ell0 = r format(coef(mod3)['l'], digits=4, nsmall=4), s{-3}=r format(-sum(coef(mod3)[c('s0','s1','s2')]), digits=4, nsmall=4), s{-2}=r format(coef(mod3)['s2'], digits=4, nsmall=4), s{-1}=r format(coef(mod3)['s1'], digits=4, nsmall=4), s_0=r format(coef(mod3)['s0'], digits=4, nsmall=4)); \item r etsnames[4] for monthly Australian overseas visitors\ (\alpha=r format(coef(mod4)['alpha'], digits=4, nsmall=4), \beta=r format(coef(mod4)['beta'], digits=2, nsmall=4), \gamma=r format(coef(mod4)['gamma'], digits=4, nsmall=4), \ell_0 = r format(coef(mod4)['l'], digits=4, nsmall=4), b0 = r format(coef(mod4)['b'], digits=4, nsmall=4), s{-11}=r format(12-sum(tail(coef(mod4),11)), digits=4, nsmall=4), s{-10}=r format(coef(mod4)['s10'], digits=4, nsmall=4), s{-9}=r format(coef(mod4)['s9'], digits=4, nsmall=4), s{-8}=r format(coef(mod4)['s8'], digits=4, nsmall=4), s{-7}=r format(coef(mod4)['s7'], digits=4, nsmall=4), s{-6}=r format(coef(mod4)['s6'], digits=4, nsmall=4), s{-5}=r format(coef(mod4)['s5'], digits=4, nsmall=4), s{-4}=r format(coef(mod4)['s4'], digits=4, nsmall=4), s{-3}=r format(coef(mod4)['s3'], digits=4, nsmall=4), s{-2}=r format(coef(mod4)['s2'], digits=4, nsmall=4), s{-1}=r format(coef(mod4)['s1'], digits=4, nsmall=4), s_0=r format(coef(mod4)['s0'], digits=4, nsmall=4)). \end{compactitem} Although there is a lot of computation involved, it can be handled remarkably quickly on modern computers. Each of the forecasts shown in Figure \ref{fexamples} took no more than a few seconds on a standard PC. The US electricity generation series took the longest as there are no analytical prediction intervals available for the ETS(M,M\damped,N) model. Consequently, the prediction intervals for this series were computed using simulation of 5000 future sample paths. To apply the algorithm to the US net electricity generation time series \code{usnetelec}, we use the following command. etsfit <- ets(usnetelec) The object \code{etsfit} is of class \code{ets}'' and contains all of the necessary information about the fitted model including model parameters, the value of the state vector \bm{x}_t for all t, residuals and so on. Printing the \code{etsfit} object shows the main items of interest. etsfit Some goodness-of-fit measures [defined in @HK06] are obtained using \code{accuracy()}. accuracy(etsfit) There are also \code{coef()}, \code{plot()}, \code{summary()}, \code{r} and \code{simulate()} methods for objects of class \code{ets}''. The \code{plot()} function shows time plots of the original time series along with the extracted components (level, growth and seasonal). The \code{forecast()} function computes the required forecasts which are then plotted as in Figure \ref{fexamples}(b). fcast <- forecast(etsfit) plot(fcast) Printing the \code{fcast} object gives a table showing the prediction intervals. fcast The \code{ets()} function also provides the useful feature of applying a fitted model to a new data set. For example, we could withhold 10 observations from the \code{usnetelec} data set when fitting, then compute the one-step forecast errors for the out-of-sample data. fit <- ets(usnetelec[1:45]) test <- ets(usnetelec[46:55], model = fit) accuracy(test) We can also look at the measures of forecast accuracy where the forecasts are based on only the fitting data. accuracy(forecast(fit,10), usnetelec[46:55]) The HoltWinters() function There is another implementation of exponential smoothing in \R\ via the \code{HoltWinters()} function [@Meyer:2002] in the \pkg{stats} package. It implements only the (N,N), (A,N), (A,A) and (A,M) methods. The initial states \bm{x}_0 are fixed using a heuristic algorithm. Because of the way the initial states are estimated, a full three years of seasonal data are required to implement the seasonal forecasts using \code{HoltWinters()}. (See @shortseasonal for the minimal sample size required.) The smoothing parameters are optimized by minimizing the average squared prediction errors, which is equivalent to minimizing \eqref{likelihood} in the case of additive errors. There is a \code{predict()} method for the resulting object which can produce point forecasts and prediction intervals. Although it is nowhere documented, it appears that the prediction intervals produced by \code{predict()} for an object of class \code{HoltWinters} are based on an equivalent ARIMA model in the case of the (N,N), (A,N) and (A,A) methods, assuming additive errors. These prediction intervals are equivalent to the prediction intervals that arise from the (A,N,N), (A,A,N) and (A,A,A) state space models. For the (A,M) method, the prediction interval provided by \code{predict()} appears to be based on @CY91 which is an approximation to the true prediction interval arising from the (A,A,M) model. Prediction intervals with multiplicative errors are not possible using the \code{HoltWinters()} function. Implementation of the automatic ARIMA algorithm \begin{figure}[!b] \centering mod1 <- auto.arima(bonds, seasonal=FALSE, approximation=FALSE) mod2 <- auto.arima(usnetelec) mod3 <- auto.arima(ukcars) mod4 <- auto.arima(visitors) par(mfrow = c(2,2)) plot(forecast(mod1), main="(a) US 10-year bonds yield", xlab="Year", ylab="Percentage per annum") plot(forecast(mod2), main="(b) US net electricity generation", xlab="Year", ylab="Billion kwh") plot(forecast(mod3), main="(c) UK passenger motor vehicle production", xlab="Year", ylab="Thousands of cars") plot(forecast(mod4), main="(d) Overseas visitors to Australia", xlab="Year", ylab="Thousands of people") \caption{Four time series showing point forecasts and 80\% \& 95\% prediction intervals obtained using ARIMA models.\label{arimaexamples}} \end{figure} The algorithm of Section \ref{sec:arima} is applied to the same four time series. Unlike the exponential smoothing algorithm, the ARIMA class of models assumes homoscedasticity, which is not always appropriate. Consequently, transformations are sometimes necessary. For these four time series, we model the raw data for series (a)--(c), but the logged data for series (d). The prediction intervals are back-transformed with the point forecasts to preserve the probability coverage. To apply this algorithm to the US net electricity generation time series \code{usnetelec}, we use the following commands. arimafit <- auto.arima(usnetelec) fcast <- forecast(arimafit) plot(fcast) # Convert character strings to latex arimanames <- c(as.character(mod1), as.character(mod2), as.character(mod3), as.character(mod4)) arimanames <- gsub("\\[([0-9]*)\", "$_{\\1}$", arimanames)
The function \code{auto.arima()} implements the algorithm of Section \ref{sec:arima} and returns an object of class \code{Arima}. The resulting forecasts are shown in Figure \ref{arimaexamples}. The fitted models are as follows: \begin{compactitem} \item r arimanames[1] for monthly US 10-year bonds yield\ ($\theta_1= r format(coef(mod1)['ma1'], digits=4, nsmall=4)$);
\item r arimanames[2] for annual US net electricity generation\ ($\phi_1= r format(coef(mod2)['ar1'], digits=4, nsmall=4)$; $\phi_2= r format(coef(mod2)['ar2'], digits=4, nsmall=4)$; $\theta_1= r format(coef(mod2)['ma1'], digits=4, nsmall=4)$; $\theta_2= r format(coef(mod2)['ma2'], digits=4, nsmall=4)$; $c= r format(coef(mod2)['drift'], digits=4, nsmall=4)$);
\item r arimanames[3] for quarterly UK motor vehicle production\ ($\phi_1= r format(coef(mod3)['ar1'], digits=4, nsmall=4)$; $\phi_2= r format(coef(mod3)['ar2'], digits=4, nsmall=4)$; $\Phi_1= r format(coef(mod3)['sar1'], digits=4, nsmall=4)$; $\Phi_2= r format(coef(mod3)['sar2'], digits=4, nsmall=4)$);
\item r arimanames[4] for monthly Australian overseas visitors\ ($\phi_1= r format(coef(mod4)['ar1'], digits=4, nsmall=4)$; $\theta_1= r format(coef(mod4)['ma1'], digits=4, nsmall=4)$; $\Theta_1= r format(coef(mod4)['sma1'], digits=4, nsmall=4)$; $\Theta_2= r format(coef(mod4)['sma2'], digits=4, nsmall=4)$; $c= r format(coef(mod4)['drift'], digits=4, nsmall=4)$). \end{compactitem} Note that the \R\ parameterization has $\theta(B) = (1 + \theta_1B + \dots + \theta_qB)$ and $\phi(B) = (1 - \phi_1B + \dots - \phi_qB)$, and similarly for the seasonal terms.
A summary of the forecasts is available, part of which is shown below.
Forecast method: ARIMA(2,1,2) with drift
Series: usnetelec
Coefficients:
ar1 ar2 ma1 ma2 drift
-1.3032 -0.4332 1.5284 0.8340 66.1585
s.e. 0.2122 0.2084 0.1417 0.1185 7.5595
sigma^2 estimated as 2262: log likelihood=-283.34
AIC=578.67 AICc=580.46 BIC=590.61
Error measures:
ME RMSE MAE MPE MAPE MASE ACF1
Training set 0.046402 44.894 32.333 -0.61771 2.1012 0.45813 0.022492
Forecasts:
Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
2004 3968.957 3908.002 4029.912 3875.734 4062.180
2005 3970.350 3873.950 4066.751 3822.919 4117.782
2006 4097.171 3971.114 4223.228 3904.383 4289.959
2007 4112.332 3969.691 4254.973 3894.182 4330.482
2008 4218.671 4053.751 4383.591 3966.448 4470.894
2009 4254.559 4076.108 4433.010 3981.641 4527.476
2010 4342.760 4147.088 4538.431 4043.505 4642.014
2011 4393.306 4185.211 4601.401 4075.052 4711.560
2012 4470.261 4248.068 4692.455 4130.446 4810.077
2013 4529.113 4295.305 4762.920 4171.535 4886.690
The training set error measures for the two models are very similar. Note that the information criteria are not comparable.
The \pkg{forecast} package also contains the function \code{Arima()} which is largely a wrapper to the \code{arima()} function in the \pkg{stats} package. The \code{Arima()} function in the \pkg{forecast} package makes it easier to include a drift term when $d+D=1$. (Setting \code{include.mean=TRUE} in the \code{arima()} function from the \pkg{stats} package will only work when $d+D=0$.) It also provides the facility for fitting an existing ARIMA model to a new data set (as was demonstrated for the \code{ets()} function earlier).
One-step forecasts for ARIMA models are now available via a \code{fitted()} function. We also provide a new function \code{arima.errors()} which returns the original time series after adjusting for regression variables. If there are no regression variables in the ARIMA model, then the errors will be identical to the original series. If there are regression variables in the ARIMA model, then the errors will be equal to the original series minus the effect of the regression variables, but leaving in the serial correlation that is modelled with the AR and MA terms. In contrast, \code{r} provides true residuals, removing the AR and MA terms as well.
The generic functions \code{summary()}, \code{print()}, \code{fitted()} and \code{forecast()} apply to models obtained from either the \code{Arima()} or \code{arima()} functions.
The forecast() function
The \code{forecast()} function is generic and has S3 methods for a wide range of time series models. It computes point forecasts and prediction intervals from the time series model. Methods exist for models fitted using \code{ets()}, \code{auto.arima()}, \code{Arima()}, \code{arima()}, \code{ar()}, \code{HoltWinters()} and \texttt{StructTS()}.
There is also a method for a \code{ts} object. If a time series object is passed as the first argument to \code{forecast()}, the function will produce forecasts based on the exponential smoothing algorithm of Section \ref{sec:expsmooth}.
In most cases, there is an existing \code{predict()} function which is intended to do much the same thing. Unfortunately, the resulting objects from the \code{predict()} function contain different information in each case and so it is not possible to build generic functions (such as \code{plot()} and \code{summary()}) for the results. So, instead, \code{forecast()} acts as a wrapper to \code{predict()}, and packages the information obtained in a common format (the \code{forecast} class). We also define a default \code{predict()} method which is used when no existing \code{predict()} function exists, and calls the relevant \code{forecast()} function. Thus, \code{predict()} methods parallel \code{forecast()} methods, but the latter provide consistent output that is more useable.
\subsection[The forecast class]{The \code{forecast} class}
The output from the \code{forecast()} function is an object of class \code{forecast}'' and includes at least the following information: \begin{compactitem} \item the original series;
\item point forecasts;
\item prediction intervals of specified coverage;
\item the forecasting method used and information about the fitted model;
\item residuals from the fitted model;
\item one-step forecasts from the fitted model for the period of the observed data. \end{compactitem}
There are \code{print()}, \code{plot()} and \code{summary()} methods for the \code{forecast}'' class. Figures \ref{fexamples} and \ref{arimaexamples} were produced using the \code{plot()} method.
The prediction intervals are, by default, computed for 80\% and 95\% coverage, although other values are possible if requested. Fan charts [@Wallis99] are possible using the combination \verb|plot(forecast(model.object, fan = TRUE))|.
Other functions {#sec:other}
We now briefly describe some of the other features of the \pkg{forecast} package. Each of the following functions produces an object of class \code{forecast}''.
\code{croston()} : implements the method of @Croston72 for intermittent demand forecasting. In this method, the time series is decomposed into two separate sequences: the non-zero values and the time intervals between non-zero values. These are then independently forecast using simple exponential smoothing and the forecasts of the original series are obtained as ratios of the two sets of forecasts. No prediction intervals are provided because there is no underlying stochastic model [@SH05].
\code{theta()} : provides forecasts from the Theta method [@AN00]. @HB03 showed that these were equivalent to a special case of simple exponential smoothing with drift.
\code{splinef()} : gives cubic-spline forecasts, based on fitting a cubic spline to the historical data and extrapolating it linearly. The details of this method, and the associated prediction intervals, are discussed in @HKPB05.
\code{meanf()} : returns forecasts based on the historical mean.
\code{r} : gives naïve'' forecasts equal to the most recent observation assuming a random walk model. This function also allows forecasting using a random walk with drift.
In addition, there are some new plotting functions for time series.
\code{tsdisplay()} : provides a time plot along with an ACF and PACF.
\code{seasonplot()} : produces a seasonal plot as described in @MWH3.
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http://mathhelpforum.com/calculus/222410-need-help-partial-fraction.html | # Math Help - need help with partial fraction
1. ## need help with partial fraction
So we are having a holiday right now and I decide to go over the math which I'm bad at "partial fraction"
The question is my exercise book is $(x^3+3x+7)sin^2(y)\frac{dy}{dx}=x^2+1 for, y(-1)=0$
so I work around with the algebra a bit and got the equation to become
$\int sin^2(y)dy=\int\frac{x^2+1}{x^3+3x+7}dx$
the problem is that I wasn't sure how to integrate $\frac{x^2+1}{x^3+3x+7}$ this since I cannot factorise the denominator into somethign like (x^2+A)(x+B)
so how does one set up $\frac{x^2+1}{x^3+3x+7}$ for partial fractions
Best Regards
Junks
2. ## Re: need help with partial fraction
see the derivative of denominator is 3x^2 + 3 = 3(x^2+1) Thus you can write
(x^2+1) / ( x^3 + 3x + 7 ) = 1/3 [ 3(x^2+1) + 7] /( x^3 + 3x + 7 )
But still I observe that you will have a problem integrating 7 / ( x^3 + 3x + 7 )
3. ## Re: need help with partial fraction
Originally Posted by junkwisch
So we are having a holiday right now and I decide to go over the math which I'm bad at "partial fraction"
The question is my exercise book is $(x^3+3x+7)sin^2(y)\frac{dy}{dx}=x^2+1 for, y(-1)=0$
so I work around with the algebra a bit and got the equation to become
$\int sin^2(y)dy=\int\frac{x^2+1}{x^3+3x+7}dx$
the problem is that I wasn't sure how to integrate $\frac{x^2+1}{x^3+3x+7}$ this since I cannot factorise the denominator into somethign like (x^2+A)(x+B)
so how does one set up $\frac{x^2+1}{x^3+3x+7}$ for partial fractions
Best Regards
Junks
One probably wouldn't set it up for partial fractions. One would probably use $u$ substitution.
4. ## Re: need help with partial fraction
Originally Posted by SlipEternal
One probably wouldn't set it up for partial fractions. One would probably use $u$ substitution.
Thank you slipeternal
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http://math.stackexchange.com/questions/281213/confusion-regarding-the-concept-of-a-function/281232 | # confusion regarding the concept of a function
I was reading principles of mathematical analysis by Walter Rudin chapter 2 when a confusion about the definition of a function cropped up. (Read definition in comment below) I had thought that functions were maps from a set called the domain to another called the codomain. From what i knew before, these maps should not be one-many. Otherwise it would simply be a Relation not a function. But the book in its definition does not talk about the fact that functions should not be one-many. Infact it doesnt differentiate between a general mapping and a function and puts no restriction on what a function can be. So my question is where exactly is the mistake?
UPDATE: The discussion till now points that maybe the "an" word in the definition points towards uniqueness of image $f(x)$ of an element $x$ in the domain. I have also commented below regarding "well-defined functions" and "not well-defined functions". I think the notion of "well-defined"ness has got to do with differences in the output for the same input in various "forms" rather than just an element in the domain mapping to various elements in the range. Hence it does not in anyway interfere with the notion that a "function" cannot be one-many. Also to make things clear, Rudin uses the word "or" in conjunction with the word "mapping" and "function". Hence he is essentially saying that the mapping is a function and a function is a mapping. Which also implies that relations in general are not mapping. Please correct me if I've been wrong.
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As I don't have the book handy, I don't know the exact words Rudin uses. Knowing that might make it easier to answer the question. – Gerry Myerson Jan 18 '13 at 4:50
Definition 2.1 Consider two sets $A$ and $B$, whose elements may be any objects whatsoever, and suppose that with each element $x$ of $A$ there is associated, in some manner, an element of $B$, which we denote by $f(x)$. Then $f$ is said to be a function from $A$ to $B$ (or a mapping of $A$ into $B$). The set $A$ is called the domain of $f$ (we also say $f$ is defined on $A$), and the elements $f(x)$ are called the values of $f$. The set of all values of $f$ is called the range of $f$. – Erik G. Jan 18 '13 at 5:06
@swanar You appear to be correct - it does not seem that Rudin gives the exact definition. That being said, it is not a good idea to get hung up on this sort of thing. Being precise is only good until it begins to interfere with actual learning. Do catch the little imprecisions Rudin makes as you read, but if the intended meaning is obvious like here (when he says function, he means function the way you mean function) then it is better to acknowledge it and move on rather than come to a complete stop. Mathematical literature is rarely as precise as your "intro to mathematics" textbook. – neuguy Jan 18 '13 at 5:32
"an element of $B$" --- why do you say Rudin does not rule out one-many? I think it's pretty clear that he does. – Gerry Myerson Jan 18 '13 at 6:25
i agree with gerry. Maybe i misinterpreted the language. When i asked someone about this, he said that a "well defined function" must not be one many but function which is not "well defined" maybe one to many. is he wrong by saying that? – swanar Jan 18 '13 at 7:22
Copying from the definition given in Rudis above: suppose that for each element $x$ of $A$ there is associated, in some manner, an element $b$ of $B$ ...
the linguistic interpretation rests on the word 'an' indicating that with each element in the domain $A$ there is associated just one (precisely one, exactly one, more than zero and less than two) element in $B$. Thus, it is to be understood that Rudin excludes multivalued functions as well as functions allowed to not attain a value at a point.
The more rigorous definition of function in terms of a relation specifying a certain condition might appeal to some since it is more rigorous but it also serves to obscure the process-like nature that one usually associates with the word 'function'. This is especially true in analysis where the subject matter is the study of the analytic properties of functions and not the set-theoretic issues of functions.
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Rudin gives the following definition:
Definition 2.1 Consider two sets $A$ and $B$, whose elements may be any objects whatsoever, and suppose that with each element $x$ of $A$ there is associated, in some manner, an element of $B$, which we denote by $f(x)$. Then $f$ is said to be a function from $A$ to $B$ (or a mapping of $A$ into $B$). The set $A$ is called the domain of $f$ (we also say $f$ is defined on $A$), and the elements $f(x)$ are called the values of $f$. The set of all values of $f$ is called the range of $f$.
In contrast:
Definition: Let $A$ and $B$ be two arbitrary sets. We say their cartesian product, denoted, $A\times{B}$, is the set $\{(a,b)\mid a\in{A}\ \mbox{and }b\in{B}\}$. That is the cartesian product is the set of all ordered pairs of elements in $A$ and $B$. A (binary) relation, $R$, is any subset of $A\times{B}$.
Thus the definitions are quite distinct. So, because I have a relation between two sets, does not imply I have a function. Whereas, a function is a special kind of relation.
See here, a function is described as functional, elements in the domain are related to distinct elements in the codomain and left-total: http://en.wikipedia.org/wiki/Relation_(mathematics)
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I agree with you. Rudin should really have said, "there is associated, in some manner, a unique element $b \in B$". Without inserting this key word, the definition allows $f$ to be multivalued.
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I disagree, Rudin's definition is unambiguous (and correct). – Did Jan 18 '13 at 7:35
@did Well what part of Rudin's definition ensures that $f(x)$ is unique? If I say, "For every cause, there is an effect", do you interpret that as meaning for a given cause there is one and only one effect? – Vectk Jan 18 '13 at 7:39
The part which says with each element x of A there is associated, in some manner, an element of B. If some elements in A had no associated element of B, one would say one or zero. If some had more than one, one would write one or more. – Did Jan 18 '13 at 7:49
@did Are you telling me that $\forall x \exists! y$ and $\forall x \exists y$ are the same? – Vectk Jan 18 '13 at 7:55
In normal English usage the relative clause in ‘an element of $B$, which we denote by $f(x)$’ implies that the element is unique. If this is not the case in your idiolect, you’re not speaking the same language as most of us. – Brian M. Scott Jan 18 '13 at 15:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8990144729614258, "perplexity": 204.91807305279872}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802777002.150/warc/CC-MAIN-20141217075257-00027-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://mathhelpforum.com/discrete-math/141537-help-equivalence-relations.html | # Thread: Help with Equivalence Relations
1. ## Help with Equivalence Relations
Here is my problem,
Let A be a set of all numbers divisible by 3. Show that A is equipotent to B. Important: When you find the bijection from A to Z (all integers) you need to prove it is a bijection.
2. Originally Posted by Nach1855
Let A be a set of all numbers divisible by 3. Show that A is equipotent to B. Important: When you find the bijection from A to Z (all integers) you need to prove it is a bijection.
From the given $A=\{3j:j\in \mathbb{Z}\}~~$. Define $\Phi:A\to \mathbb{Z}$ by $3j\mapsto j$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8062645196914673, "perplexity": 400.91557582433455}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501172050.87/warc/CC-MAIN-20170219104612-00189-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/102918-how-write-area-integral.html | # Math Help - How to write area as integral?
1. ## How to write area as integral?
The question is:
Find the area of the region between y=sin(x) and y=x
for 0 ≤ x ≤ pi/2.
It's going to be a definite integral with bounds from 0 to pi/2, right? But what is the equation and how do I get it?
Thank you!
2. Originally Posted by maiamorbific
The question is:
Find the area of the region between y=sin(x) and y=x
for 0 ≤ x ≤ pi/2.
It's going to be a definite integral with bounds from 0 to pi/2, right? But what is the equation and how do I get it?
Thank you!
$x$ is greater than $\sin(x)$ for all $x>0$, so you want to find the area under $y=x$ and subtract the area under $y=\sin(x)$. The value you want is:
$\int_0^{\pi/2}x\,dx-\int_0^{\pi/2}\sin x\,dx$
This can be rewritten as a single integral: $\int_0^{\pi/2}(x-\sin x)\,dx$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.843298614025116, "perplexity": 294.96145188623}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398461132.22/warc/CC-MAIN-20151124205421-00047-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/86533/how-many-times-more-than-0/86537 | # How many times more than $0$?
If I have $10$ apples, but you have $5$ apples, then I have $2$ times more apples than you. But what if I have $10$ apples, but you don't have any apples?
If you look at the graph $f(x)=\frac{10}{x}$, it shows that when $x$ approaches $0$, $f(x)$ approaches infinity. So it means I have infinity times more apples than you?
-
Cobold, I presume you meant $y = \frac{10}{x}$ instead of $y = \frac{10}{0}$, and I edited appropriately. Hope it's ok. – Srivatsan Nov 28 '11 at 21:43
Re "If I have 10 apples, but you have 5 apples, then I have 2 times more apples than you.": English-usage sticklers, including yours truly, will disagree: you have two times as many, or one time more. – msh210 Nov 28 '11 at 21:47
@msh210: "two times as many", or "twice as many"? – Arturo Magidin Nov 28 '11 at 21:55
That is to say, nothing wrong with "two times", even if there's a shorter alternative (and even if I personally happen to be one of those people who also like using "thrice" instead of "three times"). – Ilmari Karonen Nov 28 '11 at 21:59
@msh210, so, if I have 10 apples, and you don't have any, then I have infinity-minus-one times more apples than you. Good, I'm glad we got that settled. – Gerry Myerson Nov 28 '11 at 22:10
It would be more accurate simply to say that the question has no answer -- your ten apples cannot be described as a multiple of my zero apples.
You can choose to call that "infinity times", but for most purposes that is just a way to fool yourself into thinking that you have answered the problem when in fact you haven't. More specifically, if you know you have "two times as many as 5" apples, you can use this to compute the exact number of apples -- but knowing that you have "infinity times as many as 0 apples" tells you nothing useful.
It's better just to leave the result of the division-by-zero undefined. You don't have any times zero apples.
-
You have to be careful because infinity ($\infty$) is not a number, so saying "I have infinity apples" doesn't really make sense.
It would be better to say "I have impossibly-many apples" perhaps. And that should match your intuition from the example you gave: it is impossible to multiply zero by any number that would be large enough to make it equal to 10.
Zero is unique in this respect: any other number, no matter how big or small, can be multiplied by some other number to get 10. But not zero.
-
OP never used the phrase "infinitely-many apples", but rather was trying to make a case for the phrase "infinity times as many apples" as a person with zero apples. – Austin Mohr Nov 28 '11 at 21:53
I wasn't claiming OP used this phrase. I was giving an example of a sentence that made no sense, then one that perhaps did. I then attempted to segue back to OP's example. Sorry if it was confusing. – Fixee Nov 28 '11 at 22:29
You have indeed given an example of a sentence that makes no sense, though maybe not the one you intended! – The Chaz 2.0 Nov 28 '11 at 22:49
@TheChaz Which sentence did I write that makes no sense to you? My intention was to highlight "infinitely many apples" as nonsensical, but you are confused by something else? Or you believe my intention is otherwise? – Fixee Nov 28 '11 at 23:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.822282612323761, "perplexity": 623.986293537393}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931007501.20/warc/CC-MAIN-20141125155647-00036-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://iclr.cc/virtual/2021/poster/3219 | ## Learning advanced mathematical computations from examples
### François Charton · Amaury Hayat · Guillaume Lample
Keywords: [ deep learning ] [ differential equations ] [ computation ] [ transformers ]
[ Abstract ] [ Paper ]
[
Abstract:
Using transformers over large generated datasets, we train models to learn mathematical properties of differential systems, such as local stability, behavior at infinity and controllability. We achieve near perfect prediction of qualitative characteristics, and good approximations of numerical features of the system. This demonstrates that neural networks can learn to perform complex computations, grounded in advanced theory, from examples, without built-in mathematical knowledge. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9290663003921509, "perplexity": 4089.942704958265}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046153816.3/warc/CC-MAIN-20210729043158-20210729073158-00524.warc.gz"} |
https://wikieducator.org/Openphysics/Waves | # Openphysics/Waves
Jump to: navigation, search
Doppler Effect
The Doppler effect is observed whenever the source of waves is moving with respect to an observer. The Doppler effect can be described as the effect produced by a moving source of waves in which there is an apparent upward shift in frequency for observers towards whom the source is approaching and an apparent downward shift in frequency for observers from whom the source is receding. It is important to note that the effect does not result because of an actual change in the frequency of the source.
The Doppler effect can be observed for any type of wave - water wave, sound wave, light wave, etc. We are most familiar with the Doppler effect because of our experiences with sound waves. Perhaps you recall an instance in which a police car or emergency vehicle was traveling towards you on the highway. As the car approached with its siren blasting, the pitch of the siren sound (a measure of the siren's frequency) was high; and then suddenly after the car passed by, the pitch of the siren sound was low. That was the Doppler effect - an apparent shift in frequency for a sound wave produced by a moving source. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8126620650291443, "perplexity": 545.5877067139327}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107907213.64/warc/CC-MAIN-20201030033658-20201030063658-00706.warc.gz"} |
https://core.ac.uk/display/2181773 | Location of Repository
## Applications of AdS/QCD and Light-Front Holography to Baryon Physics
### Abstract
The correspondence between theories in anti-de Sitter space and field theories in physical space-time leads to an analytic, semiclassical model for strongly-coupled QCD which has scale invariance at short distances and color confinement at large distances. These equations, for both mesons and baryons, give a very good representation of the observed hadronic spectrum, including a zero mass pion. Light-front holography allows hadronic amplitudes in the AdS fifth dimension to be mapped to frame-independent light-front wavefunctions of hadrons in physical space-time, thus providing a relativistic description of hadrons at the amplitude level. The meson and baryon wavefunctions derived from light-front holography and AdS/QCD also have remarkable phenomenological features, including predictions for the electromagnetic form factors and decay constants. The approach can be systematically improved using light-front Hamiltonian methods. Some novel features of QCD for baryon physics are also discussed.Comment: Presented by SJB at the International Conference on the Structure of Baryons, BARYONS'10, December 7-11, 2010, Osaka, Japa
Topics: High Energy Physics - Phenomenology
Year: 2011
DOI identifier: 10.1063/1.3647346
OAI identifier: oai:arXiv.org:1103.1186 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9363564848899841, "perplexity": 4033.2322116083724}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376828056.99/warc/CC-MAIN-20181217020710-20181217042710-00418.warc.gz"} |
http://clay6.com/qa/129792/www.clay6.com/qa/129792/let-the-tangents-drawn-to-the-circle-x-2-y-2-16-from-the-point-p-0-h-meet-t | # Let the tangents drawn to the circle, $x^2+y^2 = 16$ from the point $P(0,h)$ meet the $x$-axis at point A and B. If the area of $\Delta APB$ is minimum, then $h$ is equal to :
( A ) $4\sqrt{2}$
( B ) $3\sqrt{3}$
( C ) $3\sqrt{2}$
( D ) $4\sqrt{3}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9608555436134338, "perplexity": 287.91184379946935}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703549416.62/warc/CC-MAIN-20210124141945-20210124171945-00125.warc.gz"} |
https://www.ic.sunysb.edu/Class/phy141md/doku.php?id=phy131studiof15:lectures:error | # Guide to Estimating Uncertainty
## Uncertainty in measurements
In physics, as in every other experimental science, one cannot make any measurement without having some degree of uncertainty. A proper experiment must report for each measured quantity both a “best” value and an uncertainty. Thus it is necessary to learn the techniques for estimating them. Although there are powerful formal tools for this, simple methods will suffice in this course. To a large extent, we emphasize a “common sense” approach based on asking ourselves just how much any measured quantity in our experiments could be “off”.
One could say that we occasionally use the concept of “best” value and its “uncertainty” in everyday speech, perhaps without even knowing it. Suppose a friend with a car at Stony Brook needs to pick up someone at JFK airport and doesn't know how far away it is or how long it will take to get there. You might have made this drive yourself (the “experiment”) and “measured” the distance and time, so you might respond, “Oh, it's 50 miles give or take a few, and it will take you one and a half hours give or take a half-hour or so, unless the traffic is awful, and then who knows?” What you'll learn to do in this course is to make such statements in a more precise form about real experimental data that you will collect and analyze.
Semantics: It is better (and easier) to do physics when everyone taking part has the same meaning for each word being used. Words often confused, even by practicing scientists, are “uncertainty” and “error”. We hope that these remarks will help to avoid sloppiness when discussing and reporting experimental uncertainties and the inevitable excuse, “Oh, you know what I mean (or meant).” that attends such sloppiness.
We rarely carry out an experiment by measuring only one quantity. Typically we measure two or more quantities and then “fold” them together in some equation(s), which may come from theory or even be assumed or guessed, to determine some other quantity(ies) that we believe to depend on them. Typically we compare measured result(s) with something – previous measurement(s) or theory(ies) or our assumption(s) or guess(es) – to find out if they do or do not agree. Since we never know exactly results being compared, we never obtain “exact agreement”. If two results being compared differ by less/more than the combined uncertainties (colloquially, the “sum” of their respective uncertainties), we say that they agree/disagree, but the dividing line is fuzzy. Without uncertainties, you can't say anything about agreement or disagreement, which is why uncertainties are so important in experimental science. We say that there is a “discrepancy” between two results when they “disagree” in the above sense.
Though we may assume that some quantity has an exact “true” result, we cannot know it; we can only estimate it. Now think this way about the agreement/disagreement comparison. If both compared values were known exactly, agreement would mean that the difference between them is zero. Since you don't know them exactly, the actual compared difference is never exactly zero. We may summarize this by the simple statement, worth remembering, “You cannot measure zero.” What you can say is that if there is a difference between them, it's less than such-and-such amount. If that amount is less than the combined uncertainty, then we say, “We do not find a discrepancy. The difference between them is consistent with zero.” The difference can never be exactly zero in a real experiment.
A frequent misconception is that the “experimental error” is the difference between our measurement and the accepted “official” value. (Who accepts it? Why? Not just because someone tells you without any evidence why it should be accepted.) What we mean by experimental uncertainty/error is the estimate of the range of values within which the true value of the quantity we're trying to measure is likely to lie. This range is determined from what we know about our lab instruments and methods. It is conventional to choose the uncertainty/error range as that which would comprise 68% of the results if we were to repeat the measurement a very large number of times.
In fact, we seldom make enough repeated measurements to calculate the uncertainty/error precisely, so we are usually given an estimate for this range. Note, however, that the range is established to include most of the likely outcomes, but not all of them. You might think of the process as a wager: pick the range so that if you bet on the outcome being within this range, you will be right about 2/3 of the time. If you underestimate the uncertainty, you will eventually lose money after repeated bets. (Now that's an error you probably don't want to make!) If you overestimate the range, few will be willing to take your bet!
## Error
Since nearly everyone refers to “Error Analysis” and not “Uncertainty Analysis” in measurement science, we bow to custom and will use “error” even if we really mean “uncertainty”.
If we denote a quantity that is determined in an experiment as $X$, we can call the error $\Delta X$. If, for example, $X$ represents the length of a book measured with a meter stick we might say the length $l=25.1\pm0.1$ cm where the “best” (also called “central”) value for the length is 25.1 cm and the error, $\Delta l$, is estimated to be 0.1 cm. To repeat, both the best value and its error must be quoted when reporting your experimental results. Note that in this example the best value is given with just three significant figures. Do not write significant figures beyond the first digit of the error on the quantity. Giving more precision than this to a value is misleading and irrelevant. If you're told you're using (way) too many digits, please do not try to use the excuse, “That's what the computer gave.” You're in charge of presenting your results, not the computer!
### Absolute Error
An error such as that quoted above for the book length is called the absolute error; it has the same units as the quantity itself (cm in the example). Note that if the quantity $X$ is multiplied by a constant factor $a$, the absolute error of $(aX)$ is
$\Delta (aX)=a\Delta X$
(E.1)
### Relative Error
We will also encounter relative error, defined as the ratio of the error to the best value of the quantity, so that the
relative error of $X= \Large \frac{\Delta X}{X}$
(E.2)
Thus the relative error of the book length is $\Delta l/l = (0.1/25.1) = 0.004$. (If a decimal number is in the range $-1 < x < 1$, always write it with the “leading zero”, e.g., 0.004 in the previous sentence.) The relative error is dimensionless, and should be quoted with as many significant figures as are known for the absolute error. Note that if the quantity $X$ is multiplied by a constant factor $a$ the relative error of $(aX)$ is the same as the relative error of $X$,
$\Large \frac{\Delta (aX)}{aX}=\frac{\Delta X}{X}$
(E.3)
since the constant factor $a$ cancels in the relative error of $(aX)$. Note that quantities with errors assumed to be negligible are treated as constants.
You are probably used to the percentage error from everyday life. The percentage error is the relative error multiplied by 100. In the example above, it is $0.004 = 0.4\%$.
Changing from a relative to absolute error:
Often in your experiments you have to change from a relative to an absolute error by multiplying the relative error by the best value,
$\Delta X=\Large \frac{\Delta X}{X}\normalsize \times X$
(E.4)
### Random Error
Random error occurs because of small, uncorrelated variations in the measurement process. For example, measuring the period of a pendulum with a stopwatch will give different results in repeated trials for one or more reasons. One reason could be that the watch is defective, and its ticks don't come at regular intervals. Let's assume that you have a “good” stopwatch, and this isn't a problem. (How do “you know for certain” that it isn't a problem? Think about this!) A more likely reason would be small differences in your reaction time for hitting the stopwatch button when you start the measurement as the pendulum reaches the end point of its swing and stop the measurement at another end point of the swing. If this error in reaction time is random, the average period over the individual measurements would get closer to the correct value as the number of trials $N$ is increased. The correct reported result would begin with the average for this best value,
$\Large \overline{t}=\frac {\sum t_{i}}{N}$,
(E.5)
and it would end with your estimate of the error (or uncertainty) in this best value. This usually taken as the standard deviation of the measurements. (In practice, because of time limitations we seldom make a very large number of measurements of a quantity in this lab course.) An estimate of the random error for a single measurement $t_{i}$ is
$\Large \Delta t=\sqrt{\frac {\sum (t_{i}-\overline{t})^2}{N-1}}$,
(E.5a)
$\hspace 5em$ and an estimate for the error of the average $\overline{t}$ is
$\Large \Delta \overline{t}=\sqrt{\frac {\sum (t_{i}-\overline{t})^2}{N(N-1)}}$
(E.5b)
where the sum denoted by the $\Sigma$ symbol is over the $N$ measurements $t_{i}$ . Note in equation (E.5b) the “bar” over the letter $t$ ($\bar t$ is pronounced “tee bar”) indicates that the error refers to the error in the average time $\bar t$. (Each individual measurement $t_i$ has its own error $\Delta t_i$.)
In the case that we only have one measurement but somehow know (from, say, a previous set of measurements) what the error of the average is, we can use this error of the average $\overline{t}$, $\Delta \overline{t}$, multiplied by $\sqrt{N}$ as the error of this single measurement (which you see when you divide equation (E.5a) by equation (E.5b).) [this paragraph updated 9/19/12 because of update of Eq. (5a)]
If you don’t have a value $\Delta \overline{t}$ for the error of $\overline{t}$, you must do something! Better than nothing is a “guesstimate” for the likely variation based on your experience with the equipment being used for the measurements. For example, for measurements of the book length with a meter stick marked off in millimeters, you might guess that the random error would be about the size of the smallest division on the meter stick (0.1 cm).
### Systematic Error
Some sources of uncertainty are not random. For example, if the meter stick that you used to measure the book was warped or stretched, you would never get an accurate value with that instrument. More subtly, the length of your meter stick might vary with temperature and thus be good at the temperature for which it was calibrated, but not others. When using electronic instruments such voltmeters and ammeters, you obviously rely on the proper calibration of these devices. But if the student before you dropped the meter and neglected to tell anyone, there could well be a systematic error for someone unlucky enough to be the one using it the next time. Estimating possible errors due to such systematic effects really depends on your understanding of your apparatus and the skill you have developed for thinking about possible problems. For example if you suspect a meter stick may be miscalibrated, you could compare your instrument with a 'standard' meter, but, of course, you have to think of this possibility yourself and take the trouble to do the comparison. In this course, you should at least consider such systematic effects, but for the most part you will simply make the assumption that the systematic errors are small. However, if you get a value for some quantity that seems rather far off what you expect, you should think about such possible sources more carefully. If an instrument is so broken it doesn't work at all, you would not use it. The difficult situation is when an instrument appears to be ok but, in fact, is not. You could end up trusting a device that you do not know is faulty. This happens all the time.
When it does and you report incorrect results to other scientists, you can't “blame” the meter (or buggy computer program or whatever). If it's your name associated with the results being presented, it's your responsibility to make sure the results are as free from errors as you can make them. It you later discover an error in work that you reported and that you and others missed, it's your responsibility to to make that error known publicly. This why (at least some of) the original authors of scientific papers may submit an “Erratum” to a previous publication of theirs, to alert others to errors they have discovered, after the fact, and need to correct publicly. This is much better than having other scientists publicly question the validity of published results done by others that they have reason to believe are wrong. Occasionally, if authors realize that their work in a published paper was “completely” wrong, they may ask the journal editors to publish a “retraction” of their paper. When scientific fraud is discovered, journal editors can even decide on their own to publish a retraction of fraudulent paper(s) previously published by the journal they edit. This does happen, and in this way “science corrects itself.”
### Visual Comparison of Types of Error
A figure like the one below is often used to make a visual comparison of types of error, and it allows us to introduce additional terminology that is often used (incorrectly!) when discussing measurements. You want to be sure you understand the terminology and use it correctly.
Think of the round object as an archery target. The archer shoots some number of arrows at it, and each dot shows where one landed. Now think of the “bull's eye” – the larger black dot in the center – as the “true” value of some quantity that's being measured, and think of each arrow-dot as a measurement of that quantity. The problem is that the one doing the measurements does not know the “true” value of the quantity; s/he's trying to determine it experimentally, and this means there must be uncertainty associated with the experimentally determined value. Note that each archery target – we'll call them 1,2,3,4 from left to right – shows a different distribution of arrow-hit/measurements.
In number 1 the measurements cluster pretty tightly: we say that the statistical (random) error is small, and the terminology we introduce for that is, “These measurements are precise.” However, the center of their distribution is far from the bull's eye: we say that there is a large systematic error, and the terminology we introduce for that is, “These measurements are not accurate.” In a few words, “These measurements are precise but inaccurate.”
In number 2 the measurements do not cluster tightly, but one can see that the center of their distribution is not far from the bull's eye. These measurements have a large statistical error but a small systematic error. In a few words, “These measurements are imprecise but accurate.”
In number 3 the measurements do not cluster tightly, and one can see that the center of their distribution is not close to the bull's eye. These measurements have a large statistical error and a large systematic error. In a few words, “These measurements are imprecise and inaccurate.”
In number 4 the measurements cluster tightly, and one can see that the center of their distribution is very close to the bull's eye. These measurements have a small statistical error and a small systematic error. In a few words, “These measurements are precise and accurate.”
Here is a crucial point: You can always know your measurements achieve a high level of precision if they cluster tightly, and you can quantify “how precise” they are. But this tells you nothing about how accurate they are. To aim properly, an archer needs to know where the bull's eye is, but suppose, in our analogy, a white sheet is put up to block view of the target. Not knowing where the bull's eye is, the archer's shots could still cluster tightly but there's no way of the archer knowing without additional information where they are with respect to the bull's eye. The accuracy is unknown.
To achieve high experimental accuracy requires that all measuring instruments and all measurement procedures need to be thoroughly understood and calibrated, quantitatively, against relevant “standards”, e.g,, the length standard, the time standard, the voltage standard, etc. The average laboratory, and certainly our undergraduate teaching laboratories, lack such standards. They are expensive to acquire and maintain. Periodically they should be compared with “the” (primary) standards maintained, say, by NIST, the National Institute of Standards and Technology, or by similar organizations in other countries. Section 8 of the U.S. Constitution specifies that is the duty of the Federal Government “To coin Money, regulate the Value thereof, and of foreign Coin, and fix the Standard of Weights and Measures…”. It's not optional; it's the law.
Make sure you now know the difference between “precision” and “accuracy”.
### Propagation of Errors
Often in the lab, you need to combine two or more measured quantities, each of which has an error, to get a derived quantity. For example, if you wanted to know the perimeter of a rectangular field and measured the length $l$ and width $w$ with a tape measure, you would then have to calculate the perimeter, $p =2(l+w)$, and would need to get the error of $p$ from the errors you estimated for $l$ and $w$, $\Delta L$ and $\Delta w$. Similarly, if you wanted to calculate the area of the field, $A = lw$, you would need to know how to do this using $\Delta L$ and $\Delta w$. There are simple rules for calculating errors of such combined, or derived, quantities. Suppose that you have made primary measurements of quantities $A$ and $B$, and want to get the best value and error for some derived quantity $S$.
Case 1: For addition or subtraction of measured quantities the absolute error of the sum or difference is the ‘addition in quadrature’ of the absolute errors of the measured quantities; if $S=A\pm B$
$\Delta S=\sqrt{(\Delta A)^2+(\Delta B)^2}$.
(E.6)
This rule, rather than the simple linear addition of the individual absolute errors, incorporates the fact that random errors (equally likely to be positive or negative) partly cancel each other in the error $\Delta S$
Case 2: For multiplication or division of measured quantities the relative error of the product or quotient is the ‘addition in quadrature’ of the relative errors of the measured quantities; if $S=A\times B$ or $\Large \frac{A}{B}$
$\Large \frac{\Delta S}{S}=\sqrt{(\frac{\Delta A}{A})^2+(\frac{\Delta B}{B})^2}$.
(E.7)
Due to the quadratic addition in (E.6) and (E.7) one can often neglect the smaller of two errors. For example, if the error of $A$ is 2 (in arbitrary units) and the error of B is $1$, then
the error of $S=A+B$ is $\Delta S=\sqrt{(\Delta A)^2+(\Delta B)^2}=\sqrt{2^2+1^2}=\sqrt{5}=2.23$.
Thus, if you don’t want to be more precise in your error estimate than ~12% (which in most cases is sufficient, since errors are an estimate and not a precise calculation) you can simply neglect the error in B, although it is is 1/2 of the error of A.
Case 3: When you're interested in a measured quantity $A$ that must be raised to the n-th power in a formula ($n$ doesn't have to be an integer, and it can be positive or negative), viz., you're interested in $A^n$, the relative error of the quantity $A^n$ is the relative error of $A$ multiplied by the magnitude of the exponent $n$ :
$\Large \frac{\Delta S}{S}=|n|\times \frac{\Delta A}{A}$.
(E.8)
As an example for the application of (E.8) to an actual physics problem, let's take the formula relating the period $T$ and length $L$ of a pendulum:
$T=2 \pi \Large \sqrt{\frac{L}{g}}$
(E.9a)
where $g=9.81$ m/s$^{2}$ is the constant acceleration of gravity. We rewrite (E.9a) as
$T=\left({\Large \frac{2 \pi}{g^{1/2}}} \right) L^{1/2}$
(E.9b)
to put all the constants between the parentheses. We now identify $S$ in (E.8) with $T$ and identify $A^n$ with $L^{1/2}$. Therefore, we identify $A$ with $L$ and see that ${\Large n=+\frac{1}{2}}$ for our example. Since $|n|$ appears in (E.8) [the vertical bars around $n$ mean “absolute value”], only the magnitude of $n$ is important, so we don't have to worry about the sign of $n$: we get the same result whether the exponent $n$ is positive or negative, as long as it's ${\large \frac{1}{2}}$ in our example. If we're interested in evaluating $\frac{\Delta T}{T}$, we see from (E.3) that the constant $\alpha$, which in our case equals ${\large \left(\frac{2 \pi}{g^{1/2}}\right) }$, “drops out”.
Therefore, we find that ${\Large \frac{\Delta T}{T} = \frac{1}{2}\left(\frac{\Delta L}{L}\right)}$. This example should help you apply (E.8) to cases having values of the exponent $n$ different from the particular value used in this example. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8866382837295532, "perplexity": 479.6364462065028}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370503664.38/warc/CC-MAIN-20200331181930-20200331211930-00111.warc.gz"} |
http://mathhelpforum.com/algebra/280467-how-simplify-1-1-1-exp-x.html | # Thread: How to simplify 1-(1/1+exp(x))
1. ## How to simplify 1-(1/1+exp(x))
How to simplify the term
$$1 - \frac{1}{1+\exp(1/x)}$$
2. ## Re: How to simplify 1-(1/1+exp(x))
The only "simplifying" I see is to write "1" as a fraction with denominator 1+ exp(1/x) and do the subtraction.
3. ## Re: How to simplify 1-(1/1+exp(x))
You mean
$$\frac{1+\exp(1/x) - 1}{1+\exp(1/x)}=\frac{\exp(1/x)}{1+\exp(1/x)}$$
doesn't it simplify to the sigmoid function?
4. ## Re: How to simplify 1-(1/1+exp(x))
Originally Posted by brianx
You mean
$$\frac{1+\exp(1/x) - 1}{1+\exp(1/x)}=\frac{\exp(1/x)}{1+\exp(1/x)}$$
doesn't it simplify to the sigmoid function?
No. The sigmoid function deals with $\displaystyle e^{-x} = \frac{1}{e^x}$. This one deals with $\displaystyle e^{1/x}$ which is not the same thing.
-Dan | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9855515360832214, "perplexity": 3935.318882212133}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376827252.87/warc/CC-MAIN-20181216025802-20181216051802-00102.warc.gz"} |
http://mathoverflow.net/questions/39664/random-walk-inside-a-random-walk-inside?sort=oldest | # Random walk inside a random walk inside…
Let $G=(V,E)$ be a graph and consider a random walk on it. Let $G'=(V',E')$ be a subgraph consisting of the vertices and edges that are visited by the random walk.
Question 0: Is there a standard name for $G'$?
Intuitively $G'$ is a thin subgraph, so for instance, even when $G$ is transient, $G'$ can be recurrent.
Question 1: Is there a counterexample? So, Is there a transient graph $G$ so that $G'$ is transient with positive probability?
I'm also curious to know what happens when one iterates this procedure, $G,G',G'',\dots$. Does it eventually look like a path graph?
Question 2: What can one say about $G^{(n)}$ as $n\to \infty$?
-
Question 0: $G'$ is known as the trace of the random walk.
Question 1: $G'$ is always recurrent with probability one. This is a result of Benjamini, Gurel-Gurevich, and Lyons from 2007.
Question 2: Since $G'$ is recurrent, with probability one we have $G^{(n)}=G'$ for all $n \geq 1$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9110009074211121, "perplexity": 189.26241024106426}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802768724.86/warc/CC-MAIN-20141217075248-00144-ip-10-231-17-201.ec2.internal.warc.gz"} |
https://www.ias.ac.in/listing/bibliography/pram/K._Senapati | • K Senapati
Articles written in Pramana – Journal of Physics
• Appearance of an inhomogeneous superconducting state in La0.67Sr0.33MnO3–YBa2Cu3O7–La0.67Sr0.33MnO3 trilayers
An experimental study of proximity effect in La0.67Sr0.33MnO3–YBa2Cu3O7–La0.67Sr0.33MnO3 trilayers is reported. Transport measurements on these samples show clear oscillations in critical current ($I_{c}$) as the thickness of La0.67Sr0.33MnO3 layers ($d_{F}$) is scanned from $\sim 50$ Å to $\sim 1100$ Å. In the light of existing theories of ferromagnet–superconductor (FM–SC) heterostructures, this observation suggests a long range proximity effect in the manganite, modulated by its weak exchange energy ($\sim 2$ meV). The observed modulation of the magnetic coupling between the ferromagnetic LSMO layers as a function of $d_{F}$, also suggests an oscillatory behavior of the SC order parameter near the FM–SC interface.
• # Pramana – Journal of Physics
Volume 96, 2022
All articles
Continuous Article Publishing mode
• # Editorial Note on Continuous Article Publication
Posted on July 25, 2019 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8669008016586304, "perplexity": 4191.032606181526}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103626162.35/warc/CC-MAIN-20220629084939-20220629114939-00720.warc.gz"} |
https://www.physicsforums.com/threads/v-of-gas-from-decomposition.92020/ | # V of gas from decomposition
1. Oct 3, 2005
### Pengwuino
So heres the problem I am faced with... dum dum dummmm
So first I figured that there will be 30.6g of hydrogen peroxide. Then I figured that there are 0.4498 moles of O2. Then using the ideal gas law...
V=nRT/P
I got ((0.4498)(0.08206)(27+273.15))/(746/760) = 11.3 Liters of O2
But supposedly im wrong. Where did I go so horribly horribly wrong?
Oops, and i just punched in 22.6 Liters and it says I'm right.... so where did I divide by 2 where I shouldn't have?
Last edited: Oct 3, 2005
2. Oct 3, 2005
### Tom Mattson
Staff Emeritus
It's impossible to locate your error since you have not shown your work.
3. Oct 3, 2005
### Bystander
You didn't read/remember the stoichiometry of the decomposition.
22.6? Gotta be a TA solving the problem sets --- and getting wrong answers, as is usual. You've been told where you didn't divide by two when you should have, and it's probably the same place the TA multiplied mistakenly.
4. Oct 3, 2005
### Pengwuino
Everything was already given, all I had to do was punch in the numbers. All i needed was moles and the temperature conversion and both were done correctly. The R constant is correct as well.
5. Oct 3, 2005
### Pengwuino
So 11.3 was correct?
I've been noticing a few of these problems are absolutely wrong lately in the homework. One problem had a very simple PV=nRT problem, EVERYTHING except 1 variable was missing and I did it and couldn't figure out the right answer. I show it to 4 other people... one about to graduate with his bs in physics, one his masters in physics, and 2 other people and no one could figure out what was wrong. The answer made sense (small volume, low pressure, normal temperature meant even smaller moles value) and it wasn't acording to the homework program.
6. Oct 3, 2005
### Bystander
Mole of peroxide yields half mole of O2. It's called a sitting duck.
Collateral duties of TAs include solving problem sets, checking answer keys, and keeping keys up to date when instructors change the numbers from year to year to defeat the frat-rat files --- these duties are usually performed in lackluster fashion, if at all.
Your job as a student is to bring necessary corrections to TAs' and instructors' attentions in as diplomatic a fashion as possible --- if they don't take it gracefully, you've got end of term evaluation forms.
7. Oct 3, 2005
### Pengwuino
Well was I right with the first calculation (11.3)?? Im getting a little confused by everyone here :P
8. Oct 3, 2005
### Bystander
Half mole, 11 liters, yes.
Similar Discussions: V of gas from decomposition | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8253456354141235, "perplexity": 3349.425715178026}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917125881.93/warc/CC-MAIN-20170423031205-00118-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/31929/the-consequence-of-overlap-sharing-for-the-length-distribution-of-rods-randomly?sort=newest | # The consequence of overlap sharing for the length-distribution of rods randomly placed on a line
Please imagine that one populates a finite line of unit length, or circle with unit length contour (to avoid edge-effects), with $N$ one-dimensional 'rods' such that their LHS-ends, at positions $(p_1, ..., p_k, ..., p_N) \in P$, are placed in accordance with a uniform random distribution over [0, 1]. Here, the rod lengths, $(l_1, ..., l_k, ..., l_N) \in L$, are exponentially distributed according to some rate parameter $\lambda$ - i.e. the random variable $l_k$ has distribution $l_k$ ~ Exp($\lambda$), giving a probability density function for rod length of $\lambda e^{\lambda l}$. One would similarly expect an exponential distribution for the distances between adjacent points in the set $P$.
We have the following two rules for handling overlaps between rods:
(1) - If the 'contour' of one rod (say, 'Rod A') completely covers another (say, 'Rod B'), i.e. where (Rod A-LHS) < (Rod B-LHS) and (Rod A-RHS) > (Rod B-RHS), we remove 'Rod B' from from the line and no longer consider it.
(2) - If there is only a partial overlap in the contours of two rods, 'Rod A' and 'Rod B', the length of this overlap is split evenly and each half is added to the contours of 'Rod A' and 'Rod B', respectively.
Starting from our initial exponential distribution of rod lengths, $(l_1, ..., l_k, ..., l_N)$, after this overlap-splitting process what is the new probability distribution for the length of some rod, $l_k$?
A few observations:
As $\lambda \rightarrow \infty$, the number of rods left on the line (after overlaps are handled) should increase, and the mean rod length should decrease.
As $N \rightarrow \infty$, the number of overlap-processed rods left on the line should increase, and the mean rod length should decrease. Intuitively I would expect that the number of rods remaining on the line after overlap processing will increase ever more slowly with $N$ after some threshold/'saturation' value is reached (presumably where the line is completely covered with rods).
As $\lambda \rightarrow -\infty$, there should be fewer rods left remaining on the line after overlap processing, and the mean rod length should increase. At some sufficiently large value of $\lambda$, we should be left with only a single rod on the line which has the left-most/smallest LHS-side. If we also have that $N \rightarrow \infty$, the mean length of the rod should approach the unit length of the line.
As $N \rightarrow 0$, there should be fewer rods, and an increasing mean rod length.
Inspired by Joseph O'Rourke's answer, and some simulation results of mine, if one fixes $\lambda$ and lets $N \rightarrow \infty$, one appears to converge to a rod length distribution centered around a mean value somewhere between $\frac{L}{2}$ and $L$, where $L$ is the original mean length of the rods before overlap processing. However, this distribution appears to be Gaussian, not uniform.
Do we actually converge to a Gaussian distribution? How does the distribution and its variance change with increasing $N$?
-
Am I correct in these two consequences of your rules?: (a) After each step, the rods have disjoint interiors; (b) eventually, the unit interval/circle is entirely covered end-to-end by rods (of various lengths). – Joseph O'Rourke Jul 15 '10 at 1:03
Dear Joseph, Yes about (a), however, (b) is not necessarily true for small 'N' and/or short rod lengths. There can be gaps. – Rob Grey Jul 15 '10 at 1:21
@Rob: I have some trouble understanding the model itself. Assume for instance that some rods A, B and C are such that LHS(A) < LHS(B) < LHS(C) < RHS(A) < RHS(B) < RHS(C). Then what happens? If one applies your rule to A and B first, getting A' and B', and then (if necessary), to B' and C, getting B'' and C', one gets a configuration A', B'' and C' which could be different from the configuration one gets if one applyes the rule to B and C first and then (if necessary) to A and the modified B. And this is only one configuration amongst many where this kind of ambiguity arises. Or am I mistaken? – Did Dec 13 '10 at 8:42
This is not an answer, only a simplification and conjecture concerning that simplification. First, only consider $N$ large enough so that the interval/circle is fully covered (the "eventually" in my comment). Second, rather than your exponential distribution, fix all rods to the same (small) length $L$, perhaps $L < \frac{1}{2}$ suffices. Retain your assumption that the left endpoint of each rod is chosen uniformly in $[0,1]$. Then I conjecture that the limiting distribution is uniform with mean rod length $L/2$. I have only heuristic arguments for this (shorter rods get absorbed by newly added ones, existing longer rods get chopped from the ends). Perhaps you could alter your simulation to this simplified circumstance to see if this is empirically true?
If this conjecture holds, then perhaps it holds even for an exponential distribution, with $L$ now the mean length of that distribution.
Addendum: I verified this myself, and indeed it seems to hold empirically. Here are results of a simulation adding 10 million rods of length $L=\frac{1}{10}$ to $[0,1]$. Only lengths of rods within $[L,1-L]$ are averaged in the graph (to exclude edge effects).
-
Dear Joseph, Thanks for your answer! However, the most interesting part of this problem for me (which I accidently stumbled upon while simulating another system) is the dramatic effect this had on smoothing out the exponential distribution of the rods with the right $\lambda$. – Rob Grey Jul 15 '10 at 14:26
Joseph, very cool, thanks for running the simulation! From my own simulations, it appears that if we fix $\lambda$ and let $N \rightarrow \infty$, we converge to a Gaussian-looking distribution centered around a mean of $\frac{L}{2}$, where $L$ is the mean length of the rods before overlap processing. – Rob Grey Jul 16 '10 at 23:19
Actually, to be more accurate, the mean seems to be somewhere between L/2 and L, not strictly at L/2... – Rob Grey Jul 17 '10 at 0:19
@Rob: My simulation converged to $L/2$ after $10^7$ iterations. I used $[0,1]$ but had to remove end-effects. I did not look at the distribution, however, just the mean. Cannot post data now... – Joseph O'Rourke Jul 17 '10 at 0:26
Dear Joseph, It looks like my simulation might converge to L/2 as well... but I'm still looking at higher values of 'N' and wanted to be conservative with my statement. My guess is that you're not going to have a uniform distribution! – Rob Grey Jul 17 '10 at 0:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9135291576385498, "perplexity": 608.2356564771585}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609537864.21/warc/CC-MAIN-20140416005217-00326-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://infoscience.epfl.ch/record/163996 | Infoscience
Journal article
Non-isothermal tensile tests during solidification of Al–Mg–Si–Cu alloys: Mechanical properties in relation to the phenomenon of hot tearing
An original set-up has been used to study the mechanical properties of aluminium alloys in tension during solidification with a high cooling rate (70 K/s). The mechanical behaviour of 6056 aluminium alloy with and without grain refiner has been investigated as well as that of mixtures between AA6056 and AA4047. The results show that the alloys exhibit a viscoplastic behaviour in the mushy state. A transition is observed between fracture in the mushy state and fracture in the solid state as a function of the displacement rate. This displacement rate at the transition depends on the cooling rate and on the composition of the alloy. The displacement before fracture is observed to be independent of displacement rate but to depend on the composition and on the solidification rate. Based on the observations a criterion for fracture in the mushy state is proposed. A simple rheological law describing the mechanical behaviour of the alloys is coupled to a finite element calculation giving the thermal field during the tensile test. This simulation is able to reproduce the mechanical response of the solidifying alloy during a non-isothermal test. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9382932782173157, "perplexity": 1004.8419213344689}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917119225.38/warc/CC-MAIN-20170423031159-00191-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://www.all-dictionary.com/sentences-with-the-word-magnetic%20flux | # Sentence Examples with the word magnetic flux
The magnetic flux per square centimetre at any point (B, B, or 0) is briefly called the induction, or, especially by electrical engineers, the flux-density.
In a uniform magnetic field of unit intensity formed in empty space the induction or magnetic flux across an area of I square centimetre normal to the direction of the field is arbitrarily taken as the unit of induction.
Based upon Faraday's fundamental law of induction, that the rate of change of the total magnetic flux linked with a conductor is a measure of the electromotive force created in it (see Electrokinetics).
View more
The whole of Faraday's investigations on this subject can be summed up in the single statement that if a conducting circuit is placed in a magnetic field, and if either by variation of the field or by movement or variation of the form of the circuit the total magnetic flux linked with the circuit is varied, an electromotive force is set up in that circuit which at any instant is measured by the rate at which the total flux linked with the circuit is changing.
When induction or magnetic flux takes place in a ferromagnetic metal, the metal becomes magnetized, but the magnetization at any point is proportional not to B, but to B - H. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9499365091323853, "perplexity": 317.0262748167749}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218205046.28/warc/CC-MAIN-20170322213005-00571-ip-10-233-31-227.ec2.internal.warc.gz"} |
http://www.aanda.org/articles/aa/full_html/2009/19/aa11590-08/aa11590-08.html | Free access
Issue A&A Volume 499, Number 1, May III 2009 129 - 135 Galactic structure, stellar clusters, and populations http://dx.doi.org/10.1051/0004-6361/200811590 08 April 2009
## Quantifying the contamination by old main-sequence stars in young moving groups: the case of the Local Association
J. López-Santiago1,2 - G. Micela2 - D. Montes1
1 - Departamento de Astrofísica y Ciencias de la Atmósfera, Universidad Complutense de Madrid, 28040 Madrid, Spain
2 - INAF - Osservatorio Astronomico di Palermo Giuseppe S. Vaiana, Piazza Parlamento 1, 90134 Palermo, Italy
Received 23 December 2008 / Accepted 23 February 2009
Abstract
Context. The associations and moving groups of young stars are excellent laboratories for investigating stellar formation in the solar neighborhood. Previous results have confirmed that a non-negligible fraction of old main-sequence stars is present in the lists of possible members of young stellar kinematic groups. A detailed study of the properties of these samples is needed to separate the young stars from old main-sequence stars with similar space motion, and identify the origin of these structures.
Aims. Our intention is to characterize members of the young moving groups, determine their age distribution, and quantify the contamination by old main-sequence stars, in particular, for the Local Association.
Methods. We used stars possible members of the young (10-650 Myr) moving groups from the literature. To determine the age of the stars, we used several suitable age indicators for young main sequence stars, i.e., X-ray fluxes from the Rosat All-sky Survey database, photometric data from the Tycho-2, Hipparcos, and 2MASS database. We also used spectroscopic data, in particular the equivalent width of the lithium line Li I 6707.8 Å and H, to constrain the range of ages of the stars.
Results. By combining photometric and spectroscopic data, we were able to separate the young stars (10-650 Myr) from the old (>1 Gyr) field ones. We found, in particular, that the Local Association is contaminated Conclusions. Among the candidate members of the classical moving groups, there is a non-negligible fraction of old field stars that should be taken into account when studying the stellar birthrate in the solar neighborhood. Our results are consistent with a scenario in which the moving groups contain both groups of young stars formed in a recent star-formation episode and old field stars with similar space motion. Only by combining X-ray and optical spectroscopic data is it possible to distinguish between these two age populations.
Key words: Galaxy: stellar content - Galaxy: solar neighborhood - stars: kinematics - stars: activity - stars: coronae
## 1 Introduction
It is well known that star formation takes place inside giant molecular clouds and, indeed, stellar class I and class II objects (classical T Tauri stars, cTTS) are found mainly in these regions. Young stars are strong X-ray emitters. In particular, cTTS and weak-line T Tauri stars (wTTS) have emission levels above those observed for main-sequence (MS) stars. Thus, X-ray surveys should preferentially detect cTTS and wTTS inside the forming regions. In a study of the spatial distribution of X-ray coronal emitters detected with ROSAT, Guillout et al. (1998) found different over density regions of X-ray active stars coinciding with the position of the nearby molecular complexes: Chamaleontis I and II ( pc), Taurus-Auriga ( pc), Scorpius-Centaurus-Lupus ( pc), and Ophiuchus ( pc). In contrast, other regions exhibited the average Galactic plane characteristics. It is logical to expect post-T Tauri stars to be found in the proximity of these stellar complexes, but not many post-T Tauris are observed near the molecular clouds (Herbig 1978). Mamajek & Feigelson (2001) pointed out that the main process by which stars are dispersed, i.e., the evaporation of stellar clusters and/or stellar complexes, is a slow process, but high dispersal velocities are required to transport stars some tens of parsecs from their parental clouds on short timescales. A dispersion velocity of 1-2 km s-1 is sufficient to separate a star from its formation locus by as much as 10 pc in 5-10 Myr, and, thus, many young stars may travel to considerable distances from their parental clouds. Since many of the nearby star-forming regions are situated in the southern hemisphere, many young pre-main-sequence stars (PMS) stars with low declinations are found in the solar neighborhood (Torres et al. 2006).
A series of associations of late-type stars with ages ranging from 8 to 50 Myr and having similar space motion have been discovered in our neighborhood: TW Hya, Pic, AB Dor, and Cha, Octans, and Argus associations, and the Great Austral complex (GAYA), which includes the Tucana-Horologium, Columba, and Carina associations (Torres et al. 2008; Zuckerman & Song 2004). A detailed study of the space motion of these young associations has shown that many of them were close to a molecular cloud in the past, in particular the Scorpius-Centaurus-Lupus complex (Zuckerman & Song 2004) and the dense clouds in Ophiuchus and Corona Australis (Fernández et al. 2008; Makarov 2007). Thus, a significant portion of the young stellar population in the solar vicinity is a remnant of the star formation that took place in the past in the region of the Galaxy occupied by Sco-Cen-Lup.
Slightly older groups of stars (50-650 Myr) with similar space motion have also been detected in the solar neighborhood. They are the classical stellar kinematic groups, or moving groups (see Montes et al. 2001a; López-Santiago et al. 2006, and references therein). In contrast to the young stellar associations, the stars belonging to a moving group are situated all over the sky. The dispersion velocity and the differential Galactic rotation, acting together over millions of years, have caused the wide separation of their members. A large spread in the velocity space of these stars is also observed (Skuljan et al. 1999). The idea of moving groups consisting of coeval stars is rather controversial. The over density of stars in some regions of the UV-plane could also be the result of dynamical perturbations caused by spiral waves (e.g. Famaey et al. 2005). However, several works showed that different age subgroups are situated in the same region of the Galactic velocity plane as the classical moving groups (see Asiain et al. 1999). Famaey et al. (2007) studied a large sample of stars sharing the space motion of the Hyades cluster, and determined that part of these stars were surely associated to it in the past, while the remainder are older stars trapped at resonance. Together with the result afore mentioned this suggests that those regions of the UV-plane consist of both field-like stars and young coeval ones (Francis & Anderson 2009; Famaey et al. 2008; Klement et al. 2008; Zhao et al. 2009; Famaey et al. 2007; Antoja et al. 2008). This problem is not exclusive to moving groups, but also to stars in young T associations. Thus, Bertout & Genova (2006) found that among the stars with similar proper motions in the Taurus region, there are field stars and pre-main-sequence ones. These authors also noticed that it is impossible to distinguish kinematically between pre-main sequence and field stars in their sample and that it is crucial to remove possible interlopers before searching for a moving group of young stars. In particular, distinguishing between members - i.e., coeval stars - of the moving groups and other field-like stars with similar space motion is necessary to investigate properly their contribution to the stellar population of the solar neighborhood.
Studies of the stellar content in flux-limited shallow X-ray surveys (Favata et al. 1993; López-Santiago et al. 2007; Micela et al. 2007) have detected an excess of yellow stars in observations that cannot be reproduced by standard galactic models using any form of continuous star-formation rate. A possible explanation in terms of binary systems with a yellow primary star and a secondary M dwarf, which responsible for the X-ray emission, was proposed by Micela et al. (2007). However, this does not explain the apparent excess of stars with high X-ray fluxes detected in the diagram (López-Santiago et al. 2007). This group of high X-ray emitters also have low scale height, which is typical of young stars. The optical follow-up of the coronal sources in the Einstein Medium Sensitivity Survey (Sciortino et al. 1995) demonstrated that many of them are indeed young lithium-rich stars. A similar result was obtained for the North Ecliptic Pole (NEP) survey (Affer et al. 2008). It agrees with a scenario in which the solar neighborhood consists of a standard population of stars formed with a constant star-formation rate, and an additional young stellar population. In this case, we should be able to detect this young population from their X-ray emission, since young stars are high X-ray emitters. A different question is how to explain their presence in the solar vicinity and their origin.
Table 1: Young moving groups.
In this work, we investigate the contamination by old main-sequence stars in samples of possible members of the young moving groups. Our main goal is to quantify the contribution of old stars in the list of candidates. We use optical photometric and spectroscopic data, as well as X-ray data from the ROSAT satellite. We attempt to determine the nature of these stars using the information given by different age indicators such as the lithium line at 6707.8 Å, the X-ray emission level, and the chromospheric activity. Isochrone fitting is used as well to place constraints on the age spread in the young moving groups. In particular, we explore the Local Association age spread caused by the higher number of candidates in our sample.
## 2 Data compilation
Figure 1: Left: vs. V-I of the sample of stars possible members of the young moving groups in the Hipparcos catalogue. Filled circles are the stars with reliable photometry. Crosses are stars with uncertain mag. Pre-main sequence isochrones of Siess et al. (2000) are plotted as dashed lines ( from top to bottom: 10, 20, 50, and 120 Myr), while the continuous line represents the ZAMS. The dot situated over the 10 Myr isochrone is AT Mic, which is a visual binary. Hipparcos gives the visual magnitude of the system. Assuming that both stars in the system have spectral type M 4.5, the correction of would be 0.75, situating AT Mic just on the 10 Myr isochrone, which is coherent with its membership in the Pictoris moving group. Right: near-IR color-color diagram of the stars in our sample. Filled circles are stars with good quality 2MASS photometry (quality flag = AAA''). The star-like symbols represent FK Ser and HD 142764. The filled grey diamonds are stars with precise photometry situated out of the main sequence track. Stars with error-bars larger than 0.1 mag are plotted as crosses. Dashed and dotted lines are the main-sequence track and the giant branch defined by Bessell & Brett (1988), transformed here to the 2MASS system. The dotted-dashed lines represent the reddening vector for A0 and M0 dwarfs. Open with DEXTER
In our study, we considered late-type stars proposed to be members of the young stellar kinematic groups of Montes et al. (2001a). This sample includes only late-type (spectral types F-M) field stars in the solar neighborhood - typically at distances pc - selected by the authors from several compilations of the literature (see Montes et al. 2001a, for a complete bibliographical list and the selection criteria). Although the sample is biased towards active stars because it was compiled mainly from studies of magnetic (chromospherical and coronal) activity and surveys of young late-type stars, many stars were taken from works in which no differentiation was made between old and young stars, such as the catalog of the 100 nearby stellar systems given by the Research Consortium on Nearby Stars (RECONS), or the search for kinematic groups in the solar neighborhood by Orlov et al. (1995). After the compilation, Montes et al. (2001a) used only kinematical criteria to assign each star to a moving group. A list of the young moving groups is given in Table 1, together with the stellar clusters historically associated with each group. We also give the age - or range of ages - of each moving group given in the literature, i.e., the Local Association (Asiain et al. 1999), Hyades supercluster (Skuljan et al. 1999), Ursa Major moving group (Soderblom & Mayor 1993; King et al. 2003; Asiain et al. 1999), IC 2391 supercluster (Eggen 1991), and Castor moving group (Barrado y Navascués 1998). In general, the ages were determined by isochrone fitting of the members of the moving groups, together with some spectroscopic criteria, such as the lithium abundance and chromospheric activity. To the initial sample of 535 stars, we added the 21 members of the moving groups from the spectroscopic survey of late-type stars in the solar neighborhood of López-Santiago (2005) and López-Santiago et al. (2009) that were not included in Montes et al. (2001a). Therefore, our sample contains a total of 556 late-type stars. López-Santiago et al. (2009) determined the lithium abundance, rotational and radial velocity, and level of magnetic activity of a sample of 144 late-type stars members of the moving groups, using data from echelle spectrographs. For these 144 stars, we applied different age indicators to our data from high resolution ( ) optical observations.
Optical photometric information ( and V-I colors) was taken from the Tycho-2 and Hipparcos catalogues (H$ø$g et al. 2000; ESA 1997). The Tycho-2 colors ( ) were used to determine V magnitudes following the Sect. 1.3 of the Hipparcos Introduction and Guide to the Data (ESA 1997). We note that the values of the V-I colors in the Hipparcos catalogue are given in the Cousin photometric system. Of the 556 stars in the initial sample, only 12 do not have Tycho-2 entries, while 62 are not included in the Hipparcos catalogue. The information on the distance provided by Hipparcos was combined with the visual magnitude (V) of each star to determine its absolute visual magnitude . For the stars not included in the Hipparcos catalogue, we used the distances given in Montes et al. (2001a), who either took them from the literature or determined spectroscopic parallaxes. We searched for IR counterparts by cross-correlating our sample with the 2MASS. We initially used a search radius of r = 10 arcsec, although only 9 IR sources (2% of the sample) were then found at r > 5 arcsec. It is remarkable that no IR source was found for 6 stars with the chosen radius, all of which are faint dwarfs (V > 11 mag). One hundred and six IR counterparts show large errors in the 2MASS colors ( mag). This was taken into account when studying their position in the near-IR color-color diagram (see Sect. 3).
We also searched for X-ray counterparts of the stars in both the ROSAT All-Sky Survey Bright Source Catalogue (RASS-BSC) and the Faint Source Catalogue (RASS-FSC). A search radius of 30 arcsec was adopted, bearing in mind the ROSAT X-ray object coordinate determination accuracy. A total of 341 stars (61% of the moving-groups sample) are matched with X-ray sources, out of which about 6% are expected to be spurious (see Guillout et al. 1998, for a detailed discussion). Since we are concerned with statistical studies, a contamination of 6% ought not to affect any of the conclusions drawn in this paper. To determine the X-ray fluxes, we used the count rate-to-energy flux conversion factor (CF) relation found by Schmitt et al. (1995):
where HR is the hardness-ratio of the star in the ROSAT energy band 0.1-2.4 keV, A = 5.30, and B = 8.31. We note that this CF relation is valid for main-sequence stars. Hünsch et al. (1996) found that for late-type giants and super-giants, B = 8.7. X-ray fluxes were determined by multiplying the CF by the count-rate of the sources in the same band. Fluxes were later transformed into luminosity using the distances of the stars. Since the CF and the count rate (CR) are defined for the ROSAT energy band 0.1-2.4 keV, the X-ray luminosity is also defined in this band.
## 3 Evidences for the presence of an old population
In Fig. 1 (left panel), we plot versus V-I for the possible members of the moving groups. Pre-main sequence (PMS) isochrones of 10, 30, 50, and 120 Myr of Siess et al. (2000) are overplotted as dashed lines, while the continuous line represents the ZAMS. Our stars are situated mainly on the main-sequence (MS) locus, but a large quantity are located above the ZAMS. Evolved (giant) stars are distinctive because of their high luminosity. The results are compatible with the sample containing a mixture of MS, PMS and some evolved stars. The PMS population shows a range of ages of approximately 10-120 Myr, as deduced by comparing the data with the isochrones.
Only one T Tauri star (FK Ser) is known to be present in the sample, but it is not plotted in the versus V-I diagram because we did not find any value of V-I in the literature. In the near-infrared color-color diagram (Fig. 1, right panel), there are only two stars with precise 2MASS photometry, i.e., the already mentioned FK Ser and HD 142764, a K5 dwarf with AV = 1.8 mag (Eiroa et al. 2001) that are situated above the MS. The remaining stars are on or close to the MS. Some stars with photometry of lower accuracy (plotted as crosses in the figure) are outside the MS track, but their position could be an artifact of the imprecise photometry. It is interesting to note that two of the four stars near the reddening vector (filled diamonds) host hot jupiters'', while the other two are known to be PMS stars.
In Fig. 2, we plot versus V-I for the sources in our sample. For comparison, we also plot the stars in the Hipparcos catalogue at pc with a ROSAT counterpart (small dots). In a similar way to Zickgraf et al. (2005), we observe a clear trend of decreasing with decreasing stellar temperature in our sample, but we also observe that the sources are divided into two strips (or branches): the lower envelope of the first one coincides with the locus of the relation (dashed line); in contrast, the sources in the upper strip are situated in the region delimited by the relation (grey area), corresponding with the saturation value for active stars. A similar behavior was noticed by Daemgen et al. (2007) using other colors in a more restricted sample of field stars (see their Fig. 2). It has been observed that the gap in the versus V-I diagram is a consequence of the decrease in X-ray emission with age (see again Daemgen et al. 2007, for a complete discussion). The upper envelope of the distribution is populated by stars exhibiting lithium absorption lines, typical of young stars, while the lower branch is typically populated by older stars without lithium absorption line. The combination of both X-ray emission and lithium abundances allows us to distinguish between old and young stars in the main sequence.
## 4 Old star population contamination in the Local Association
Figure 2: X-ray luminosity vs. V-I color diagram for the stars in our sample (circles). Filled circles are the stars with high-resolution spectroscopic observations. Small dots are the stars from the Hipparcos catalogue with pc. The grey area represents the region of , with the lower edge corresponding with the ZAMS of the pre-main sequence models of Siess et al. (2000), and the upper edge with the 1 Myr isochrone. The dashed line represents the relation. Open with DEXTER
Figure 3: Lithium ( line) equivalent width of the stars of the Local Association. The upper envelopes of the stars of some stellar clusters are plotted: dashed line for IC 2391 (10-35 Myr; Montes et al. 2001b), continuous line for the Pleiades (80-120 Myr; Soderblom et al. 1993), doted-dashed line for the Coma Berenices cluster (400 Myr; López-Santiago 2005), and dotted line for the Hyades (650 Myr; Soderblom et al. 1993). The lower envelope of the Pleiades is also plotted as a continuous line. The filled circles are stars with . Open with DEXTER
Due to the large number of possible members in our sample, the Local Association is the most suitable moving group for quantifying the fraction of old stars among the candidate members of the moving groups. It is also the moving group that contains candidates with a higher spread in ages and differences in the stellar properties. We used the information compiled by us from the literature (see Sect. 2) to determine the physical properties and, in particular, the age of each candidate.
Figure 4: Left: (H) vs. V-I of the members of the Local Association for which we have a measure of the equivalent width of H. The filled circles are stars with . Right: vs. (H) of the members of the Local Association. The symbols are the same as in the left panel. Open with DEXTER
In Fig. 3, we plot the equivalent width of the lithium line 6707.8 Å versus the V-I color of the stars considered to be members of the Local Association. For comparison, we also plot the upper envelopes of some clusters with accurately determined ages. From the figure, we can identify two populations: the first one has equivalent widths similar to or even higher than those of the stars of the Pleiades, and the second one has equivalent widths that are below the Hyades upper envelope. Most of the stars in the first group exhibit also strong X-ray emission, the filled circles representing stars with . They also have high levels of chromospheric activity. Figure 4 presents the flux in the H line in the candidates of the Local Association and its relation with the X-ray luminosity. Moreover, the stars with high X-ray emission levels exhibit a saturation in the H flux at each spectral type (Fig. 4, left panel). All these results are compatible with the stars with higher equivalent widths of the lithium line Li I in our sample being, at least, as young as the Pleiades (indicated by the upper and lower envelopes in Fig. 3). Although it is impossible to estimate their age, its range must be 10-120 Myr. The stars with equivalent widths of Li I below the lower envelope of the Pleiades show levels of chromospheric and X-ray emission typical of field stars. This group of stars is dominated by non-members of the moving group.
Assuming that all stars with low X-ray and chromospheric emission levels (and low equivalent widths of lithium in FGK stars) are non-members of the Local Association, we find that the contamination of the initial sample of candidates is 50%. This number, nevertheless, must be taken as an upper limit since our sample is incomplete. In particular, the number of matches between our sample and the RASS decreases with increasing distance. At pc, approximately 70% of our stars are cross-identified, while at pc, only a half of them have a RASS counterpart. This suggests that some X-ray emitters at large distances are undetected because of the flux limit of the RASS, producing a bias in our sample. We cannot reject these stars a priori as members of the moving groups. Using only the stars at pc, for which we are more complete, the contamination of our sample by non-members is 30%, which we propose to be a more reliable value. We note that our result is valid only for the candidates identified by Montes et al. (2001a) and is closely related to the adopted selection criteria. Famaey et al. (2008) observed a higher percentage of field stars in a sample of K and M giant stars in the Pleiades Moving Group. Bertout & Genova (2006) also presented a detailed study of the Taurus-Auriga T association, where the authors identified 53% of field stars among 217 possible pre-main sequence stars in this region.
## 5 Contamination of the other moving groups
We cannot apply the above analysis to the other moving groups since their number of candidates is too small to draw robust conclusions. Furthermore, the other moving groups are, on average, older than the Local Association and the age indicators that we have used are effective only for ages less than few hundreds million years. The lithium abundance, for instance, decays very rapidly for late-K and M dwarfs and may be barely detectable at the age of the Pleiades for late-type stars, while X-ray and chromospheric activity indicators may even be used at slightly older ages. In agreement with their age, only the members of the Castor Moving Group show higher values of equivalent width of Li I, but the number of stars for which we have data is not high enough to reach robust conclusions.
Studying the X-ray emission can still help us to separate stars of different age. In Fig. 5, we plot versus V-I for all stars of our sample with RASS counterparts. Each moving group is represented by a different symbol. The two continuous lines indicate the median X-ray emission of Pleiades and Hyades members determined by us from ROSAT data. The high X-ray emitters of the Local Association exhibit X-ray emission levels above those of the Pleiades members. A similar behavior is observed in the Castor Moving Group. Both the Local Association and the Castor candidates seem to show a similar fraction of contaminating old field stars. The dK and dM of the Ursa Major Moving Group (300-500 Myr) have , between the Castor Moving Group and the Hyades Supercluster, which have low ratios. However, for dF and dG stars, it is more difficult to identify this effect, because of the poor sensitivity X-ray luminosity to age. Finally, the late-type stars in the IC 2391 Supercluster show high X-ray emission. This would agree with the IC 2391 being as young as the Pleiades. However, the number of candidates in this moving group is quite small and we cannot draw significant conclusions.
## 6 Summary and conclusions
We have attempted to quantify the contamination by old field stars among candidate members of the young stellar kinematic groups, or moving groups. Our sample was selected from possible members of the young stellar kinematic groups studied by Montes et al. (2001a). We compiled photometric, spectroscopic, and X-ray data from the literature. We cross-correlated the sample with the Tycho-2, Hipparcos, 2MASS, and RASS databases. Spectroscopic data were taken from López-Santiago (2005) and López-Santiago et al. (2009).
The general properties of the stars for which we analyzed data indicate that there are probably two different age populations. We used the equivalent width of the lithium line 6708.8 Å and the X-ray luminosities to constrain the ages of the candidates of the Local Association, which is the moving group containing the youngest stars. The analysis indicates that the contribution of old field stars to the sample of candidates of this moving group is 50% if we used the whole sample. A more accurate study using only those stars at pc, in which our sample of X-ray sources is nearly complete, showed that the contribution of non-members is, in fact, approximately 30%. This number must be considered to represent the contamination by old main sequence stars of the Montes et al. (2001a) sample of candidate members of the young moving group. A higher percentage of interlopers among possible members of the Pleiades Moving Group was determined in Famaey et al. (2008). The age of the other moving groups does not permit members to be separated univocally from non-members, since the properties of the candidates are similar to those of field stars. However, we found that the X-ray data can be used to distinguish, at least, between stars of age <200 Myr and older stars.
Figure 5: vs. V-I of the members of the Local Association (squares), Castor Moving Groups (filled circles), Ursa Major Moving Groups (triangles), Hyades Supercluster (x symbols), and IC 2391 Supercluster (asterisks). The upper continuous line is the median of the Pleiades members. The continuous line at the bottom is the median of the stars of the Hyades. Open with DEXTER
Our result is consistent with that found for the Hyades Supercluster by Famaey et al. (2007). In that study, the authors concluded that the moving group consist of stars coeval with the Hyades cluster, which presumably formed in the same star formation episode, and old stars with similar space motion. The samples of candidate members of the classical young moving groups (<650 Myr) contain a non-negligible amount of old (>1 Gyr) field stars. With regard to this matter, several studies suggest that the three arm-like shape observed in the UV-plane is caused by a non-axisymmetric potential (Famaey et al. 2005) and that the same non-axisymmetric motion affecting old-field stars should also perturb the molecular clouds where star formation occurs (Xu et al. 2006). This possibly explains for the location of the young stellar associations such as TW Hya or inside the Local Association in the UV-plane. In addition, this scenario could explain the problem of the apparent lack of post-T Tauri stars close to star-forming regions (Soderblom et al. 1998; Herbig 1978), which is inferred from the increasing number of them identified in loose associations and moving groups (Torres et al. 2008; Montes et al. 2009; Bubar et al. 2007).
We have extended our conclusions for the Local Association to the other young moving groups. The contamination by old stars in the moving groups should be considered in studies of the history of stellar birthrate in the solar neighborhood. Only by using data of different wavelengths, in particular X-ray and optical spectroscopic data, is it possible to distinguish between the two age populations. More detailed studies in this area will enable us to determine univocally the contribution of young stars to the stellar population in the solar vicinity, to compare with the predictions of Galactic models.
Acknowledgements
J. López-Santiago acknowledges financial support by the Marie Curie Fellowship contract No. MTKD-CT-2004-002769 and financial contribution by MERG-CT-2007-046535. The Madrid group acknowledges financial contribution by the Universidad Complutense de Madrid and the Programa Nacional de Astronomía y Astrofísica of the Spanish Ministerio de Educación y Ciencia (MEC), under grants AYA2005-02750 and AYA2008-000695 and to the PRICIT project S-0505/ESP-0237 (ASTROCAM) of the Comunidad de Madrid. G. Micela acknowledges financial contribution from contract ASI-INAF I/088/06/0. We thanks the referee for useful comments that allowed to improve our manuscript. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research made use of the SIMBAD database, operated at the CDS, Strasbourg, France.
## Footnotes
... Association
On-line Tables with the data are only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/499/129
... (RECONS
RECONS: http://www.recons.org/
... 2MASS
The Two Micron All Sky Survey (2MASS) is available at http://www.ipac.caltech.edu/2mass/database.
## All Tables
Table 1: Young moving groups.
## All Figures
Figure 1: Left: vs. V-I of the sample of stars possible members of the young moving groups in the Hipparcos catalogue. Filled circles are the stars with reliable photometry. Crosses are stars with uncertain mag. Pre-main sequence isochrones of Siess et al. (2000) are plotted as dashed lines ( from top to bottom: 10, 20, 50, and 120 Myr), while the continuous line represents the ZAMS. The dot situated over the 10 Myr isochrone is AT Mic, which is a visual binary. Hipparcos gives the visual magnitude of the system. Assuming that both stars in the system have spectral type M 4.5, the correction of would be 0.75, situating AT Mic just on the 10 Myr isochrone, which is coherent with its membership in the Pictoris moving group. Right: near-IR color-color diagram of the stars in our sample. Filled circles are stars with good quality 2MASS photometry (quality flag = AAA''). The star-like symbols represent FK Ser and HD 142764. The filled grey diamonds are stars with precise photometry situated out of the main sequence track. Stars with error-bars larger than 0.1 mag are plotted as crosses. Dashed and dotted lines are the main-sequence track and the giant branch defined by Bessell & Brett (1988), transformed here to the 2MASS system. The dotted-dashed lines represent the reddening vector for A0 and M0 dwarfs. Open with DEXTER In the text
Figure 2: X-ray luminosity vs. V-I color diagram for the stars in our sample (circles). Filled circles are the stars with high-resolution spectroscopic observations. Small dots are the stars from the Hipparcos catalogue with pc. The grey area represents the region of , with the lower edge corresponding with the ZAMS of the pre-main sequence models of Siess et al. (2000), and the upper edge with the 1 Myr isochrone. The dashed line represents the relation. Open with DEXTER In the text
Figure 3: Lithium ( line) equivalent width of the stars of the Local Association. The upper envelopes of the stars of some stellar clusters are plotted: dashed line for IC 2391 (10-35 Myr; Montes et al. 2001b), continuous line for the Pleiades (80-120 Myr; Soderblom et al. 1993), doted-dashed line for the Coma Berenices cluster (400 Myr; López-Santiago 2005), and dotted line for the Hyades (650 Myr; Soderblom et al. 1993). The lower envelope of the Pleiades is also plotted as a continuous line. The filled circles are stars with . Open with DEXTER In the text
Figure 4: Left: (H) vs. V-I of the members of the Local Association for which we have a measure of the equivalent width of H. The filled circles are stars with . Right: vs. (H) of the members of the Local Association. The symbols are the same as in the left panel. Open with DEXTER In the text
Figure 5: vs. V-I of the members of the Local Association (squares), Castor Moving Groups (filled circles), Ursa Major Moving Groups (triangles), Hyades Supercluster (x symbols), and IC 2391 Supercluster (asterisks). The upper continuous line is the median of the Pleiades members. The continuous line at the bottom is the median of the stars of the Hyades. Open with DEXTER In the text | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8789094090461731, "perplexity": 1950.9549920468949}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719843.44/warc/CC-MAIN-20161020183839-00314-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://proofwiki.org/wiki/Oscillation_at_Point_(Infimum)_equals_Oscillation_at_Point_(Limit) | # Oscillation at Point (Infimum) equals Oscillation at Point (Limit)
## Theorem
Let $f: D \to \R$ be a real function where $D \subseteq \R$.
Let $x$ be a point in $D$.
Let $N_x$ be the set of open subset neighborhoods of $x$.
Let $\map {\omega_f} x$ be the oscillation of $f$ at $x$ as defined by:
$\map {\omega_f} x = \inf \set {\map {\omega_f} I: I \in N_x}$
where $\map {\omega_f} I$ is the oscillation of $f$ on a real set $I$:
$\map {\omega_f} I = \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$
Let $\map {\omega^L_f} x$ be the oscillation of $f$ at $x$ as defined by:
$\map {\omega^L_f} x = \displaystyle \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} }$
Then:
$\map {\omega_f} x \in \R$ if and only if $\map {\omega^L_f} x \in \R$
and, if $\map {\omega_f} x$ and $\map {\omega^L_f} x$ exist as real numbers:
$\map {\omega_f} x = \map {\omega^L_f} x$
## Proof
### Lemma
Let $f: D \to \R$ be a real function where $D \subseteq \R$.
Let $x$ be a point in $D$.
Let $N_x$ be the set of open subset neighborhoods of $x$.
Let $\map {\omega_f} x$ be the oscillation of $f$ at $x$ as defined by:
$\map {\omega_f} x = \displaystyle \inf \set {\map {\omega_f} I: I \in N_x}$
where $\map {\omega_f} I$ is the oscillation of $f$ on a real set $I$:
$\map {\omega_f} I = \displaystyle \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$
Let $\map {\omega^L_f} x$ be the oscillation of $f$ at $x$ as defined by:
$\map {\omega^L_f} x = \displaystyle \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} }$
Let $\map {\omega^L_f} x \in \R$.
Let $\map {\omega_f} x \in \R$.
Then $\map {\omega^L_f} x = \map {\omega_f} x$.
$\Box$
### Necessary Condition
Let $\map {\omega_f} x \in \R$.
We need to prove:
$\map {\omega^L_f} x \in \R$
$\map {\omega^L_f} x = \map {\omega_f} x$
where:
$\map {\omega^L_f} x = \displaystyle \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} }$
$\map {\omega_f} {\openint {x - h} {x + h} } = \sup \set {\size {\map f y - \map f z}: y, z \in \openint {x - h} {x + h} \cap D}$
$\map {\omega_f} x = \inf \set {\map {\omega_f} I: I \in N_x}$
$\map {\omega_f} I = \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$
Let $\epsilon \in \R_{>0}$.
Then an $I \in N_x$ exists such that:
$\map {\omega_f} I - \map {\omega_f} x < \epsilon$ by Infimum of Set of Oscillations on Set is Arbitrarily Close
Let $I$ be such an element of $N_x$.
We observe in particular that $\map {\omega_f} I \in \R$.
A neighborhood in $N_x$ contains an open subset that contains the point $x$.
So, $I$ contains such an open subset as $I \in N_x$.
Therefore, a $\delta \in \R_{>0}$ exists such that $\openint {x - \delta} {x + \delta}$ is a subset of $I$.
Let $h$ be a real number that satisfies: $0 < h < \delta$.
We observe that $\openint {x - h} {x + h} \subset I$.
Also, $\openint {x - h} {x + h} \in N_x$.
We have:
$I \in N_x$
$\openint {x - h} {x + h} \in N_x$
$\openint {x - h} {x + h} \subset I$
$\map {\omega_f} I \in \R$
from which follows by Oscillation on Subset:
$\map {\omega_f} {\openint {x - h} {x + h} } \in \R$
$\map {\omega_f} {\openint {x - h} {x + h} } \le \map {\omega_f} I$
We have that:
$\map {\omega_f} {\openint {x - h} {x + h} } \in \set {\map {\omega_f} {I'}: I' \in N_x}$
as $\openint {x - h} {x + h} \in N_x$.
Also, $\map {\omega_f} x$ is a lower bound for $\set {\map {\omega_f} {I'}: I' \in N_x}$ by the definition of $\map {\omega_f} x$.
Therefore:
$\map {\omega_f} {\openint {x - h} {x + h} } \ge \map {\omega_f} x$
We find:
$\displaystyle \map {\omega_f} x \le \map {\omega_f} {\openint {x - h} {x + h} }$ $\le$ $\displaystyle \map {\omega_f} I$ $\displaystyle \leadsto \ \$ $\displaystyle 0 \le \map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x$ $\le$ $\displaystyle \map {\omega_f} I - \map {\omega_f} x$ $\displaystyle \leadsto \ \$ $\displaystyle 0 \le \map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x$ $\le$ $\displaystyle \map {\omega_f} I - \map {\omega_f} x < \epsilon$ as $\map {\omega_f} I - \map {\omega_f} x < \epsilon$ is true $\displaystyle \leadsto \ \$ $\displaystyle 0 \le \map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x$ $<$ $\displaystyle \epsilon$ $\displaystyle \leadsto \ \$ $\displaystyle \size {\map {\omega_f} {\openint {x - h} {x + h} } - \map {\omega_f} x}$ $<$ $\displaystyle \epsilon$
which means that $\displaystyle \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} }$ exists and equals $\map {\omega_f} x$ by the definition of limit.
In other words, $\map {\omega^L_f} x \in \R$ and $\map {\omega^L_f} x = \map {\omega_f} x$.
$\Box$
### Sufficient Condition
Let $\map {\omega^L_f} x \in \R$.
We need to prove:
$\map {\omega_f} x \in \R$
$\map {\omega_f} x = \map {\omega^L_f} x$
where:
$\map {\omega_f} x = \inf \set {\map {\omega_f} I: I \in N_x}$
$\map {\omega_f} I = \sup \set {\size {\map f y - \map f z}: y, z \in I \cap D}$
$\map {\omega^L_f} x = \displaystyle \lim_{h \mathop \to 0^+} \map {\omega_f} {\openint {x - h} {x + h} }$
We have:
$\displaystyle \lim_{h \mathop \to 0^+} \map{\omega_f} {\openint {x - h} {x + h} } \in \R$ as $\map {\omega^L_f} x \in \R$
Therefore, $\map {\omega_f} {\openint {x - h} {x + h} } \in \R$ for a small enough $h$ in $\R_{>0}$ by the definition of limit.
Let $h$ be such a real number.
We observe that $\openint {x - h} {x + h}$ is a neighborhood in $N_x$.
We have:
$\openint {x - h} {x + h} \in N_x$
$\map {\omega_f} {\openint {x - h} {x + h} } \in \R$
Accordingly:
$\map {\omega_f} x \in \R$ by Infimum of Set of Oscillations on Set
$\map {\omega_f} x = \map {\omega^L_f} x$ follows by Lemma.
This finishes the proof of the theorem.
$\blacksquare$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9872041344642639, "perplexity": 147.6234953667248}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371803248.90/warc/CC-MAIN-20200407152449-20200407182949-00151.warc.gz"} |
http://mathhelpforum.com/calculus/181191-application-greens-theorem-print.html | # Application of Green's Theorem
• May 20th 2011, 09:01 PM
jonmondalson
Application of Green's Theorem
Hi, I'm trying to answer the following question:
http://imageshack.us/m/849/9551/greenstheorem.png
Basically, I know that you have to use Green's Theorem:
$\displaystyle\oint_C P(x)dx +Q(x)dy = \displaystyle\iint_A (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y})\ dx\,dy$
Which makes our integral:
$\displaystyle\iint_A x \ dx\,dy$
My problem is, I don't know how to calculate the limits for the area A. Do we use a parameterisation, or are the limits simply the points of intersection of the functions.
Any help is immensely appreciated.
Thanks :)
• May 20th 2011, 09:35 PM
FernandoRevilla
The intersection points of $y=x$ and $y=x^2-2x$ are $(0,0)$ and $(3,3)$ . You'll obtain:
$\displaystyle\oint_C 3xy\;dx +2x^2\;dy=\iint_Ax\;dxdy= \int_0^3\;dx\int_{x^2-2x}^xx\;dy=\ldots$
• May 21st 2011, 03:17 AM
HallsofIvy
IF you were to integrate around the boundary, you would need to parameterize the curve. You don't use parametric equations for an area in 2 dimensions. That's why it is easier to use Green's theorem. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9033012390136719, "perplexity": 431.3965911034276}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541525.50/warc/CC-MAIN-20161202170901-00255-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://proceedings.mlr.press/v119/arunachalam20a.html | Quantum Boosting
Srinivasan Arunachalam, Reevu Maity
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:377-387, 2020.
Abstract
Boosting is a technique that boosts a weak and inaccurate machine learning algorithm into a strong accurate learning algorithm. The AdaBoost algorithm by Freund and Schapire (for which they were awarded the G{ö}del prize in 2003) is one of the widely used boosting algorithms, with many applications in theory and practice. Suppose we have a gamma-weak learner for a Boolean concept class C that takes time R(C), then the time complexity of AdaBoost scales as VC(C)poly(R(C), 1/gamma), where VC(C) is the VC-dimension of C. In this paper, we show how quantum techniques can improve the time complexity of classical AdaBoost. To this end, suppose we have a gamma-weak quantum learning algorithm for a Boolean concept class C that takes time Q(C), we introduce a quantum boosting algorithm whose complexity scales as sqrt{VC(C)}poly(Q(C),1/gamma); thereby achieving quadratic quantum improvement over classical AdaBoost in terms of VC(C). | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9019514322280884, "perplexity": 1153.4899899156974}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943695.23/warc/CC-MAIN-20230321095704-20230321125704-00483.warc.gz"} |
http://koreascience.or.kr/search.page?keywords=%EC%84%A0%EB%9F%89%EB%8B%B9%EB%9F%89%ED%99%98%EC%82%B0%EA%B3%84%EC%88%98 | • Title, Summary, Keyword: 선량당량환산계수
### A New Approach for the Calculation of Neutron Dose Equivalent Conversion Coefficients for PMMA Slab Phantom (PMMA 평판형 팬텀에서의 중성자 선량당량 환산계수의 새로운 계산법)
• Kim, Jong-Kyung;Kim, Jong-Oh
• Journal of Radiation Protection and Research
• /
• v.21 no.4
• /
• pp.297-311
• /
• 1996
• ANSI decided PMMA slab phantom as a calibration phantom and introduced a conversion coefficient calculation method for it. For photon, the conversion coefficient can be obtained by using backscatter factor and conversion coefficient of the ICRU tissue cube and backscatter factor of the PMMA slab. For neutron, however, the ANSI has not introduced any conversion coefficient calculation method for the PMMA slab. In this work, the ANSI method for the photon conversion coefficient calculation was applied to the neutron conversion coefficient calculation of the PMMA slab. Quality weighted tissue kerma of neutron was applied to calculate the backscatter factors on the ICRU cube and the PMMA slab. The dose conversion coefficient of the ICRU cube was also calculated by using MCNP code. Then, the dose conversion coefficient of the PMMA slab was calculated from two backscatter factors and the dose conversion coefficient of the ICRU cube. The discrepancies of the dose conversion coefficients of the PMMA slab and the ICRU cube were less than 10% except 1eV(20%), 1keV(17%), and 4 MeV(16%).
### Assessment of Effective Doses in the Radiation Field of Contaminated Ground Surface by Monte Carlo Simulation (몬테칼로 시뮬레이션에 의한 지표면 오염 방사선장에서의 유효선량 평가)
• Chang, Jai-Kwon;Lee, Jai-Ki;Chang, Si-Young
• Journal of Radiation Protection and Research
• /
• v.24 no.4
• /
• pp.205-213
• /
• 1999
• Effective dose conversion coefficients from unit activity radionuclides contaminated on the ground surface were calculated by using MCNP4A rode and male/female anthropomorphic phantoms. The simulation calculations were made for 19 energy points in the range of 40 keV to 10 MeV. The effective doses E resulting from unit source intensity for different energy were compared to the effective dose equivalent $H_E$ of previous studies. Our E values are lower by 30% at low energy than the $H_E$ values given in the Federal Guidance Report of USEPA. The effective dose response functions derived by polynomial fitting of the energy-effective dose relationship are as follows: $f({\varepsilon})[fSv\;m^2]=\;0.0634\;+\;0.727{\varepsilon}-0.0520{\varepsilon}^2+0.00247{\varepsilon}^3,\;where\;{\varepsilon}$ is the gamma energy in MeV. Using the response function and the radionuclide decay data given in ICRP 38, the effective dose conversion coefficients for unit activity contamination on the ground surface were calculated with addition of the skin dose contribution of beta particles determined by use of the DOSEFACTOR code. The conversion coefficients for 90 important radionuclides were evaluated and tabulated. Comparison with the existing data showed that a significant underestimates could be resulted when the old conversion coefficients were used, especially for the nuclides emitting low energy photons or high energy beta particles.
### Characterization of Radiation Field in the Steam Generator Water Chambers and Effective Doses to the Workers (증기발생기 수실의 방사선장 특성 및 작업자 유효선량의 평가)
• Lee, Choon-Sik;Lee, Jai-Ki
• Journal of Radiation Protection and Research
• /
• v.24 no.4
• /
• pp.215-223
• /
• 1999
• Characteristics of radiation field in the steam generator(S/G) water chamber of a PWR were investigated and the anticipated effective dose rates to the worker in the S/G chamber were evaluated by Monte Carlo simulation. The results of crud analysis in the S/G of the Kori nuclear power plant unit 1 were adopted for the source term. The MCNP4A code was used with the MIRD type anthropomorphic sex-specific mathematical phantoms for the calculation of effective doses. The radiation field intensity is dominated by downward rays, from the U-tube region, having approximate cosine distribution with respect to the polar angle. The effective dose rates to adults of nominal body size and of small body size(The phantom for a 15 year-old person was applied for this purpose) appeared to be 36.22 and 37.06 $mSvh^{-1}$) respectively, which implies that the body size effect is negligible. Meanwhile, the equivalent dose rates at three representative positions corresponding to head, chest and lower abdomen of the phantom, calculated using the estimated exposure rates, the energy spectrum and the conversion coefficients given in ICRU47, were 118, 71 and 57 $mSvh^{-1}$, respectively. This implies that the deep dose equivalent or the effective dose obtained from the personal dosimeter reading would be the over-estimate the effective dose by about two times. This justifies, with possible under- or over- response of the dosimeters to radiation of slant incidence, necessity of very careful planning and interpretation for the dosimetry of workers exposed to a non-regular radiation field of high intensity. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9611647725105286, "perplexity": 4570.03982701612}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600402131412.93/warc/CC-MAIN-20201001112433-20201001142433-00090.warc.gz"} |
http://www.math-only-math.com/worksheet-on-fundamental-operation.html | # Worksheet on Fundamental Operation
In worksheet on fundamental operation, the questions are based on performing operations. This exercise sheet on precedence of operations of addition, subtraction, multiplication and division has different types of questions that can be practiced by the students to get more ideas to learn fundamental operation.
Find the value of:
1. 36 ÷ 6 + 3
2. 24 + 15 ÷ 3
3. 120 - 20 ÷ 4
4. 32 - (3 x 5) + 4
5. 3 - (5 - 6 ÷ 3)
6. 21 – 12 ÷ 3 x 2
7. 16 + 8 ÷ 4 – 2 x 3
8. 28 – 5 x 6 + 2
9. (- 20) x (-1) + (-28) ÷ 7
10. (-2) + (-8) ÷ (-4)
11. (-15) + 4 ÷ (5 - 3)
12. (-40) x (-1) + (-28) ÷ 7
13. (-3) + (-8) ÷ (-4) - 2 x (-2)
14. (-3) x (-4) ÷ (-2) ÷ (-1)
15. (-7) x (-5) x (-2) + 3
Students can check the answers of the worksheet on fundamental operation given below to make sure that the answers are correct.
1. 9
2. 29
3. 115
4. 21
5. 0
6. 13
7. 12
8. 0
9. 16
10. 0
11. -13
12. 36
13. 3
14. -7
15. -67
If students have any queries regarding the questions given in the worksheet on fundamental operation, please fill-up the comment box so that we can help you.
However, suggestions for further improvement, from all quarters would be greatly appreciated.
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Numbers - Integers
Integers
Multiplication of Integers
Properties of Multiplication of Integers
Examples on Multiplication of Integers
Division of Integers
Absolute Value of an Integer
Comparison of Integers
Properties of Division of Integers
Examples on Division of Integers
Fundamental Operation
Examples on Fundamental Operations
Uses of Brackets
Removal of Brackets
Examples on Simplification
Numbers - Worksheets
Worksheet on Multiplication of Integers
Worksheet on Division of Integers
Worksheet on Fundamental Operation
Worksheet on Simplification | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9057413935661316, "perplexity": 3618.305821989023}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257651780.99/warc/CC-MAIN-20180325025050-20180325045050-00345.warc.gz"} |
https://dsp.stackexchange.com/questions/23962/log-distance-path-loss-model/23965 | # Log Distance Path Loss Model
While simulating a MISO system, I am trying to consider the effect of varying the distance between the transmitter and receiver on the BER.
My search for the effect of varying distance has pointed me to Log Distance Path Loss.
My query is where should I include this path loss during the Matlab implementation. Should the received signal be convoluted with the path loss?
I am completely clueless as to how this can be included. Any resources or pointers to theory would be appreciated.
There are direct and indirect effects of distance to the Bit Error Rate (BER) between two transceivers.
The direct effect that distance has on the BER is through the attenuation of the signal. As the transmitter get's further away from the receiver, the power of the received signal is diminished which makes the Signal to Noise Ratio lower which makes signal reception more difficult and the likelihood for a misinterpreted symbol higher.
This is the simplest case with the transceivers operating in free space where theoretically there are no other objects or sources of "noise".
The indirect effect of distance on the BER in a realistic environment (i.e. one containing surfaces that the signal can bounce off, such as buildings) is the introduction of multipath propagation. This means that multiple copies of the same signal arrive at the receiver at slightly different times having followed slightly different paths. Depending on the frequency of the signals, these slight delays may lead to destructive interference. In other words, two versions of the same signal cancel out each other. This condition leads, again, to a decrease in the SNR and signal distortions which leads to a higher probability for errors in symbol reception. A similar (indirect) effect is introduced due to the Doppler shift of the frequencies if the velocity of movement of the transmitter is comparable to the wavelength of the used frequencies.
There are multiple ways by which you can include these effects to a simulation.
When simulating the full transceiver stack, i.e. Source, encoder, modulator, channel, demodulator, decoder, Target then Free Space Loss can be applied at the output of the modulator (as part of the "channel" building block) as an attenuation of the produced signal given a distance L.
Multipath propagation can also be applied at the "channel" stage, either by convolving the modulator's output with a suitable impulse response or by generating an "envelope" signal from a Rayleigh distribution.
• For a specific application in my project, I am working with a multiple transmitters and a single receiver (MISO) system. I am trying to analyze the effect of the distance between the various transmitters and the receiver on the BER, which is the reason I have asked about the Log Distance Path loss model. For the purpose of simulation, I intend to use an AWGN channel and include the path loss due to varying distance. For now, I do not intend to use any other channel. Even then, should I apply the log distance path loss at the channel? – smyslov Jun 8 '15 at 10:29
• The short answer is "yes", the long answer depends a lot on the details of the experiment. – A_A Jun 9 '15 at 9:31
• Ideally, including the log distance path loss would mean that I convolve the channel output with the path loss factor in dB. Is this right? – smyslov Jun 9 '15 at 9:33
• No. FSL is a simple multiplication of your signal vector with a single value. Convolution with the impulse response of the channel is something quite different (please see en.wikipedia.org/wiki/Convolution). I hope we don't start arguing about convolving with h=[0.001] (!) :) – A_A Jun 9 '15 at 9:49
• My bad. I meant the multiplication of path loss value in dB with the channel output. – smyslov Jun 9 '15 at 9:53
The log-distance path loss model is a simplified model that tries to model fluctuations in the received signal power. It is generally used in dense urban environments (i.e. a city center or suburbs) or inside a building (where transmitter and receiver are not in the same room). It also assumes large distances (much more than tens of feet).
In this model, the path loss is divided in three separate processes. First, you have the free-space loss, due to the distance between antennas, and which changes with the square of the distance. Then, you have a path loss due to the environment, which changes with a different exponent (not square). The exponent depends on the kind of materials the signal has to travel through. Generally, this exponent is calculated empirically for each propagation environment.
These firt two losses are deterministic. The third loss process is random. In an urban environment, you'll have shadowing, which is caused by the receiver antenna moving behind buildings or trees. The logarithm of this loss is Gaussian distributed. You will also have signal attenuation due to fading caused by multipath; assuming flat, slow fading, this will follow a Rayleigh distribution. You can find the equation in wikipedia (https://en.wikipedia.org/wiki/Log-distance_path_loss_model). This model is also very well explained in "Wireless Communications" by Andrea Goldsmith, in chapter 3 if I remember correctly. The book provides many examples.
Regarding how to implement this model in Matlab: the model allows you to calculate an overall path loss $PL_{dB}$. If your transmit power is $T_{dB}$, then the received power is $R_{dB}=T_{dB}-PL_{dB}$. In general, $PL_{dB}$ is random; you would generate appropriate random numbers in Matlab and change the received signal's power to account for the loss.
Having said all this, if you're running your experiments in a lab, I don't think this model is appropriate. What I would do is to try to set up a fixed environment (no people or objects moving around when you're doing the experiment). If your signal is narrowband, the loss will be a single number that you can determine empirically. This number is random but does not change with time, as would be the case in a dynamic environment. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8774169683456421, "perplexity": 444.0380699156422}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250589861.0/warc/CC-MAIN-20200117152059-20200117180059-00555.warc.gz"} |
http://math.stackexchange.com/questions/209056/how-do-i-show-that-the-limit-is-to-the-600th-places-and-is-it-too-hasty-to-take | # How do I show that the limit is to the 600th places and is it too hasty to take the limit?
I am a little unsure about (b) and (d)
For (b), is it appropriate to just take the limit directly? (do I have to show that the sequence is bounded and monotone?) from $x_{n+1} = \frac{1}{2}(x_n+5/x_n)$? That is $\ell = \frac{1}{2}(\ell+5/\ell)$? Or is the question implying I should take the one in (a) instead (even though it is equivalent). I got it to $x_{n+1}^2-5 = \dfrac{1}{4x_n}(x_n^2-5)^2$
For (d), I was able to compute the tenth term on Mathematica as a fraction and and evaluated it using the numerical command.
So I thought that to show I would get the 600th decimal place, I would do $|x_{10}-\sqrt{5}| = 0.\underbrace{000...}_{600 \; \text{zeros}}1$. On Mathematica, I got $|x_{10}-\sqrt{5}| = 0 \times 10^{600}$. Does this show I got it to the 600th place? Because I tried 601, 602, and 605 and they seem all to be $0 \times 10^{x}$. I am terrible with counting, so should I actually take the 601th place of $x_{10}$ and subtract $\sqrt{5}$'s 600th place?
EDIT
$x_{n+1}^2-5 = \dfrac{1}{4x_n}(x_n^2-5)^2 < \dfrac{(x_n^2 - 5)^2}{20}$ for all n $\geq 2$
-
"Hence" means using result (a) to do (b)? For (d), I think you got it. – Patrick Li Oct 8 '12 at 4:11
You are supposed to use (a), together with the easily calculated value of $x_1^2-5$, to work out the value of $x_{10}^2-5$, thereby answering (d). – Gerry Myerson Oct 8 '12 at 4:26
Why would the expression in (a) be any easier for a computer? – Hawk Oct 8 '12 at 4:28
"Why would the expression in (a) be any easier for a computer?" I don't understand this. Who said anything about a computer? By the way, I think you got the wrong answer for (a). – Gerry Myerson Oct 8 '12 at 6:23
I'll underestimate $x_n$, wait for edit – Hawk Oct 8 '12 at 18:51
Your teacher will not want to compare two 600+ digit numbers you computed with a device of finite accuracy digit by digit to have a proof of (d). Rather use (a) and $x_0^2-5$ to show that $x_{10}^2-5$ is how small? Then use $x_{10}^2-5=(x_{10}-\sqrt 5)(x_{10}+\sqrt5)$.
Where are you getting $x_0$ from? – Hawk Oct 8 '12 at 18:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8552784323692322, "perplexity": 344.7587401654342}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246643088.10/warc/CC-MAIN-20150417045723-00103-ip-10-235-10-82.ec2.internal.warc.gz"} |
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