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Investigating Flow around Submerged I, L and T Head Groynes in Gravel Bed (2024)
1. Introduction
Riverbank erosion is a natural process that results in substantial economic losses. It also poses significant risks to human life. Groynes are commonly used river training structures, widely employed
to maintain stable river channels and to ensure riverbank protection. These structures, which encompass levees, dams, weirs, groynes, and channel stabilization mechanisms, aim to safeguard
riverbanks, regulate water flow, combat sediment build-up, and uphold the preferred dimensions of the river channel [1]. They offer effective means to manage river flow, minimize flood risks, control
erosion, and reduce sedimentation issues. The groyne field provides a favorable environment for the growth of aquatic life due to the zone of reduced velocity and the sediment deposition resulting
from the reduction in velocity. The groynes can take various shapes, such as I, L, and T head groynes (IHG, LHG, and THG), each with its distinct geometric characteristics that influence the flow
patterns and associated scouring processes. Numerous groyne types, including T and L type groynes, have been scientifically built across the globe in accordance with various river environments; for
example, spur dikes or IHGs have been provided along the Western Kosi Main Canal (WKMC), Kosi River in Bihar, India, and in the Gamka River at Calitzdorp, South Africa [2]. Six LHGs were constructed
in 2012 to safeguard a solar power plant near the left bank of Solani River, Bhagwanpur (Roorkee, India) [3], and an LHG field was constructed on the left bank of the Kansas River (United States) [4
]. Examining the hydrodynamics of the flow around these structures is essential to enhance our understanding of the performance of IHGs, LHGs, and THGs. From the literature, it is evident that the
flow around a spur dike is divided into three zones—the main flow zone, wake zone, and mixing zone (between the two zones) [5]. Moreover, the interaction of the flow in a channel with a groyne
affects the flow field and bed topography. These changes are responsible for scouring and deposition around the groyne. Such processes are referred to as dynamic feedback processes. Scouring is a
critical aspect of groyne design and maintenance as it can compromise the integrity of these structures, damage the adjacent features, and alter the riverbed’s morphology. Therefore, understanding
the flow patterns and scouring processes around groyne shapes is paramount. Excessive scour represents a prominent factor leading to the failure of spur dikes. Conversely, the formation of
pool-riffle morphologies stemming from the scour processes introduces valuable amenities and crucial habitats for riverine species. Striking a careful equilibrium is paramount to harness the maximum
positive impacts while concurrently ensuring effective scour control [6]. The groynes enhance the local biodiversity and support a wider range of species, hence increasing the ecosystem’s resilience.
This boost in biodiversity is crucial for sustainability as resilient ecosystems are better able to withstand and recover from environmental stresses. Groynes contribute to the long-term stability of
aquatic environments. Numerous researchers have explored the dynamics of flow and scouring around diverse groyne shapes. For instance, Zaghoul conducted experimental studies that focused on
understanding how various factors, such as upstream flow conditions, sediment properties, and the shape of the spur dike, impact the maximum depth and the pattern of scour around the spur dike [7].
Kadota and Suzuki visually examined the flow patterns around T and L groynes and studied the mean and coherent flow structures [8]. Much research has been conducted on scour around abutments.
Barbhuiya and Dey reviewed the various work on abutments and provided a detailed review on abutment scour, flow field, time variation of scour depth, and the formulae to estimate the scour depth [9].
Kothyari and Raju computed the temporal variation of scour around spur dikes and bridge abutments using an analogous pier model [10]. Dey and Barbhuiya investigated the temporal evolution of scouring
at abutments [11]. Dey and Barbhuiya delved into velocity and studied the turbulence within a scour hole adjacent to an abutment [12]. The flow around these abutments is found to be similar to that
near IHGs.
Although few researchers have investigated the flow around LHGs and THGs, the flow mechanism and comparative analysis among the groynes have not been completely established yet. LHGs and THGs have an
advantage over IHGs. LHGs facilitate the localized absorption of energy by leading the face perpendicular to the flow and establishing a low flow velocity zone, which supports the sediment deposition
behind the groyne. Kang and Yeo conducted experimental analyses to understand the flow characteristics of L-type groynes [13]. An experimental study to investigate the performance and efficiency of
two types of groynes, IHGs and THGs, was carried out by Mall et al. [14]. The equilibrium scour depth for THG is greater as compared to an IHG. Moreover, the cost–benefit analysis also favored an IHG
as the more cost-efficient option for riverbank protection. Duan et al. explored the mean flow characteristics and turbulence of flow surrounding experimental spur dikes [15], while Kumar and Ojha
focused on the near-bed turbulence in the vicinity of an unsubmerged L-head groyne [16]. Kadota and Suzuki investigated the flow around IHGs, LHGs, and THGs [17]. The study unveils the alterations in
the bed configurations resulting from T- and L-type groynes by comparing them with the experimental outcomes of I-type groynes. The experimental results highlight the significant variations in the
maximum scour depth associated with different groyne types. Additionally, distinctive bed form changes, including the development of sand waves, were observed downstream of the groyne, indicating
distinct characteristics influenced by the groyne types. The maximum scour depths in the cases of LHGs and THGs surpass those of I-type groynes. Various studies have also investigated the impact of
the parallel wall angle in LHGs, which is the angle between the parallel and perpendicular walls, as conducted by Dehghani et al. [18]. To examine the flow patterns around THGs, a numerical
simulation was carried out by Mortazavi Farsani [19]. The investigation considers the existence of a single dike with varying lengths perpendicular to the flow, along with different configurations
including simple and T-shaped models. The findings reveal that simple dikes exhibit higher maximum scour depths and spans in comparison to T-shaped dikes. An increase in the length of the groyne
relative to the flow is associated with deeper and wider scour holes. Conversely, extending the length of the dike wings tends to reduce the depth, span, and intensity of sediment scour and
entrainment, except for situations with high narrowing ratios. Another study on THGs also concluded that, with an increase in the wing length of a THG, the dimensions of the scour hole decreased [20
Most of the studies on groynes have been conducted on the scour in a sand bed, and, hence, the investigation of the flow around groynes in gravel beds remains unexplored. Kadota and Suzuki studied
the physics of the flow around groynes for gravel, which is different from that of sand [8]. This can be attributed mostly to changes in bed roughness, which then influence the vortex flow field [21
]. Pandey et al., examined sand gravel mixtures and discovered that sediment nonuniformity influences the variation in the scour depth [22]. It was found that the maximum scour depth for a sand
gravel mixture occurred at the same location as that of the sand bed, which is the upstream tip of the groyne. However, the processes involved for scour around these groynes and the flow behavior and
dynamics for the gravel bed have not been studied yet. Additionally, the flow around the groyne in the submerged case is expected to be different from the flow around the groyne in unsubmerged
conditions. Kunhle et al. studied the 3D flow field and bed shear stress around a submerged IHG [23]. They observed reduced reattachment length around the IHG compared to the values reported in the
unsubmerged condition. Yossef and de Vriend investigated the scour and flow pattern around both submerged and emergent (non-submerged) groynes (IHGs) in series [24]. Mehraein and Ghodsian highlighted
the difference in the flow fields around fully submerged and just submerged spur dikes [25]. Moreover, the length of the downstream recirculation region is found to be dependent on both the shape and
height of the spur dike. A significant difference in the nature of the turbulence among both the cases was reported. The convergence of the time-averaged velocity occurs within a short distance
compared to the emerged case.
The study underscores the need to understand the flow dynamics and sediment transport around groynes in gravel beds. The primary objectives of this research are to assess and contrast the extent of
scour development and create a visual representation of the scour bed morphology surrounding groynes of different shapes, namely I, L, and T head groynes, for gravel bed streams. This is completed
with the motivation to provide an effective river training structure as per the utility. The flow pattern for each groyne is studied. Furthermore, a thorough cost–benefit analysis is carried out to
optimize the groynes’ dimensions and to improve their cost-effectiveness. The results indicate that the maximum scour depth varies with the type of groyne. Hence, this study provides valuable
insights regarding the hydrodynamics of the flow around these groynes and aids in selecting the appropriate type of groyne. The research aids the decision-making for the engineers and policymakers in
selecting the appropriate groyne type based on the specific requirements.
2. Materials and Methods
2.1. Experimental Setup
An experimental setup was established at the Hydraulics Laboratory within the Civil Engineering Department at IIT Roorkee to investigate the scour and flow characteristics around I head (IHG), L head
(LHG), and T head groynes (THG). This setup featured a glass-sided rectangular flume, measuring 20 m in length, 0.75 m in width, and 0.4 m deep, with a bed slope of 0.035. The flume was subdivided
into three distinct sections: an upstream section, a test section, and a downstream section. The test section spanned 8 m in length and was positioned 7 m from the channel’s entrance so that fully
developed flow conditions could be established [9,26]. It incorporated a uniformly graded gravel bed having D[50] = 9.36 mm and geometric standard deviation 0.955. For each run, specific groyne was
installed and attached to the right-side wall of the flume (IHG for experiment 1, LHG for experiment 2, and THG for experiment 3). The groyne was placed in the test section, at 8 m from the inlet.
This was done to maintain a fully developed flow condition [27]. The length of the groyne perpendicular to the flow direction (L[1]) was maintained at 11.25 cm to constrict 15% of the width of the
channel. A constriction ratio of 15% does not induce contraction scour [18]. The length of the groyne in the flow direction (L[2]) was kept the same as L[1] in the case of LHG and THG. The
experimental setup is shown in Figure 1, depicting the scoured bed observed at equilibrium scour condition for the three groynes.
At the start of each experiment, the sediment bed was carefully leveled using a scraper. After the installation of the groyne, the area around the groyne was leveled again. The supply of water to the
flume was regulated using an inlet pipe valve. Each experiment extended over a duration of 8 h to attain the equilibrium scour bed condition. To establish the pre-determined flow conditions (depth of
flow, discharge, and velocity), precise adjustments were carried out by manipulating both the inlet valve and the tailgate. The pump was initially set to a low discharge rate, and subsequent
adjustments were made to the discharge from the pump and tailgate to attain the desired depth and velocity parameters. A flow depth of 0.136 m was consistently maintained, while the submergence ratio
was kept as 0.83. As a precautionary measure to avert abrupt scouring induced by the flowing water, plywood was placed over the bed, with gradual removal occurring upon the successful attainment of
the desired depth. The experiments were carried out under clear water conditions, adhering to the condition U/U[c] = 0.7, ensuring U/U[c] < 1. Here, U represents the mean approach flow velocity in
the streamwise direction, and U[c] denotes the critical velocity for sediment particles, which is determined by the formula provided by Melville [28].
$U c u * c = 5.75 log h k c + 6$
$u * c = 0.0305 d 50 0.5 − 0.0065 d 50 − 1 , f o r 1 m m < d 50 < 100 m$
The term represents critical shear velocity for sediment particles, and Equation (3) was used to compute it. In Equation (1), k[c] is the height of roughness elements and is taken 2 times the d[50]
and was provided by Dey and Barbhuiya [11].
2.2. Data Collection and Analysis
Bed level measurements of the equilibrium scoured bed were meticulously obtained across a grid with a spacing of 0.02 m in the longitudinal and transverse directions using a pointer gauge (least
count 0.01 mm). The bed level measurements serve the purpose of ascertaining scour and deposition, facilitating the identification of both maximum scour depth and deposition. The scour pattern is
consequently visualized through contour plotting [16]. A diluted resin was evenly applied to stabilize the scoured bed, thus securing the equilibrium scoured bed.
Subsequently, velocity measurements were systematically collected across a grid situated on the stabilized scoured bed within the xy plane. The measurements were obtained at a specific vertical
position with z/D = 0.074. Here, z represents the location within a vertical plane over the equilibrium scoured bed. Notably, a finer grid resolution was applied in the vicinity of the critical zone
around the groyne, while a coarser grid was employed in other regions.
For the instantaneous velocity data collection, Vectrino +, an Acoustic Doppler Velocimeter (ADV) of Nortek, sourced from Vangkroken, Norway, was used at a 50 Hz frequency for 2 to 3 min,
encompassing 3 min in the vicinity of the critical groyne area and 2 min at more distant locations. This process yielded a substantial dataset, ranging from 6000 to 9000 samples at each measurement
point. The acquired velocity data underwent filtration using the Phase Space Threshold Despiking technique [29]. Detailed hydraulic parameters for the experimental runs are provided in Table 1.
To validate the accuracy of the velocity data obtained from the ADV, spectral density was plotted as a function of frequency. This step served as a potential assessment of the suitability of the
chosen sampling frequency. The power spectrum was computed via the Fourier Transform of the auto-covariance function. Notably, the power spectra derived from streamwise velocity fluctuations
exhibited a slope of −5/3, indicative of the presence of an inertial sub-range. Figure 2 shows the spectral density function plotted on y axis with respect to frequency. Also, rms values of
streamwise velocity data were calculated for section y/B = 0.073. Using these turbulent velocity statistics, uncertainty analysis was carried out, as shown in Table 2.
2.3. Cost–Benefit Analysis
Cost–benefit analysis (CBA) is a systematic procedure employed to assess and compare the potential benefits of a given project or activity with the associated costs. The expected value of damage
reduction can be termed as benefit. In this study, to assess the economic efficiency of individual I-shaped, L-shaped, and T-shaped groynes, precise cost functions were developed, considering certain
variables that are significant to determine the comprehensive construction cost of groynes: transverse length (L), depth of flow (D), and groyne thickness (t).
The benefits linked to respective groynes were quantified using the bank protection lengths by each groyne type. The insights derived from this experimental study were used to compute the effective
length of bank protected for I-shaped, L-shaped, and T-shaped groynes for similar flow conditions. The basic idea remained to determine the point of reattachment of flow [5]. The outcomes of this
analysis provide a framework for the process of decision-making in the selection of groyne configuration and groyne dimensions. Our study aims to ascertain, within a given cost framework, which
groyne offers superior benefits or, in other words, better protection. While the cost–benefit approach may face criticism for requiring the quantification of all costs and benefits in monetary terms,
we contend that it constitutes a vital component of information essential for rational decision-making.
3. Results
3.1. Scour Development Pattern
Figure 3 depicts the profile of the bed level around IHG, LHG, and THG at equilibrium scour conditions. It can be noted that the blue area indicates regions of scouring, whereas the red area denotes
deposition region. Here, z[s] is the difference between the bed level at equilibrium scour condition and the bed level initially, in the plain bed. D is the depth of flow, measured at the tailgate.
It can be observed from the contour plots of all three shapes that scouring begins upstream of the structure around x/L[1] = −1.5 or −1, and this scouring extends up to x/L[1] = 1. The scour depth, z
[s], is normalized by D, and the directions (x and y) are normalized by the length of groyne face perpendicular to flow (L[1]) and width of the flume (B).
The initiation of the local scour takes place at the upstream of the groyne and has been attributed primarily to the interaction between the HSV system and the downflow. The intricate details of the
flow field near a groyne become more complex as scour holes, which require flow separation to create three-dimensional vortex flow, develop. Kwan and Melville found that the primary mechanism of
scour (at an abutment, which can be considered anomalous to an IHG is downflow, and they also found a primary vortex that resembles the horseshoe vortex at a pier [30]. Because the abutment stops the
approaching flow, a vertical pressure gradient develops on the upstream face of the abutment. The fluid rolls up by being forced downward by the pressure gradient. As a result, the primary vortex
forms and grows as the scour hole develops. Additionally, the main vortex and downflow are mostly contained in the scour hole, which is located below the original bed level line. The mean values of
non-dimensional scour depth for the IHG is 0.067; for the LHG the value is 0.082; and, for the THG, it turns out to be 0.075.
Here, the LHG and THG demonstrate notable greater maximum scour depths, approximately the same, while the IHG exhibits a shallower maximum scour depth. This is in sync with the findings by Kadota and
Suzuki [8]. To be specific, the IHG attains a maximum scour depth of z[s]/D = −0.21, whereas the LHG and THG attain z[s]/D = −0.295 and z[s]/D = −0.29, respectively. The location of the maxima also
varies. In the case of IHG, it occurs near the tip due to flow separation at that critical point, and the LHG experiences maxima at the junction of L[1] and L[2] faces. However, THG encounters the
maximum scour depth upstream of the groyne. This distinct behavior can be explained by significant downflow upstream of the groyne, resulting from the impact of the flow on the perpendicular face L
[1] [11].
Notable scouring is observed in the regions x/L[1] = 0 to 1 and y/B = 0.15 to 0.25 for the IHG and LHG. Conversely, THG experiences major scouring at upstream location x/L[1] = −0.5 and y/B = 0.1 to
0.15. Interestingly, even the longitudinal extent of the scour is minimal for THG. The deposition process initiates near x/L[1] = 0.5 for the IHG and THG and near x/L[1] = 2 for the LHG. This
deposition persists until x/L[1] = 3.5 for the IHG and LHG. However, this reduces to x/L[1] = 3 for THG. It is essential to emphasize that, for all three groynes, the extent of deposition surpasses
the extent of scour. This underlines the significance of deposition as the dominant process in the context of groyne-induced alterations to riverbed morphology.
3.2. Flow Pattern around the Groynes
The flow depth was observed to fall significantly downstream of the groyne. Figure 4 represents the distribution of the normalized streamwise velocity (u/U) around IHG, LHG, and THG. Here, u
represents the streamwise velocity and is normalized by U, and both the longitudinal distance, x, and transverse distance, y, are normalized with respect to the L[1] and B. Here, L1 is the length of
the groyne face perpendicular to the flow, and B is the width of the flume. Upstream of all three groynes, there is a negative velocity zone. A significant reduction in the u/U values and the
presence of negative values of u/U are evident for the sections near the groyne attached wall, i.e., for y/B = 0.04, 0.08, and 0.15, upstream of the groyne. This phenomenon is attributed to the
formation of HSV system [31]. Furthermore, flow separation at the junction or tip induces a peak of u/U for y/B = 0.2 and a lateral drift of flow along the DSL path. The maximum positive streamwise
velocity for the IHG, LHG, and THG are 0.85U, 0.98U, and 0.92U at (x/L[1], y/B) = (1.8, 0.15), (x/L[1], y/B) = (1.4, 0.2), and (x/L[1], y/B) = (2, 0.2). Fluctuations in the streamwise velocity are
observed within 2 < x/L[1] < 5 for the IHG, within 2 < x/L[1] < 4 for the LHG, and within 2 < x/L[1] < 4.5 for the THG. This can be attributed to the complexity of flow arising due to the interaction
between the wake zone and the secondary vortices. The maximum negative streamwise velocity for the IHG, LHG, and THG are 0.3U, 0.3U, and 0.6U at (x/L[1], y/B) = (0.5, 0.1), (x/L[1], y/B) = (0.5,
0.11), and (x/L[1], y/B) = (−0.4, 0.12). The formation of a shear layer, marked by an abrupt change in velocity direction, is observed along the line of flow divergence. The original streamwise
velocity values are re-established after x/L[1] = 6 for the IHG, x/L[1] = 8 for the LHG, and x/L[1] = 7 for the THG.
According to the boundary layer hypothesis, at the separation point, the flow velocity cannot overcome the unfavorable pressure gradient, which marks the end of the separation zone. There is a
discrepancy between the junction and separation points. Indeed, the velocity at the border of separation zone should be zero, and adopting the zero-streamwise-velocity isoline as the boundary is
justified. Consequently, the streamwise velocity isoline is employed to mark the geometric border of the separation zone. In this method, the mainstream isoline with zero velocity serves as the
boundary of the separation zone, a technique commonly referred to as the isoline method [5]. It is evident from Figure 4 that the reattachment length, i.e., the border of separation zone or the
reattachment lengths for the IHG, LHG, and THG, are 1.2 L[1], 0.85 L[1], and 1.16 L[1], respectively. This value is much smaller compared to the reattachment length observed in other studies [11,13,
27] due to the submergence condition. The observed value of reattachment length aligns with the findings reported by Kuhnle et al. for the reattachment length around an IHG, which was observed as 1.6
L[1] [23].
In Figure 5, the distribution of the normalized transverse velocity (v/U) around the IHG, LHG, and THG is depicted. The velocity (v) is normalized by U, and both the longitudinal and transverse
directions (x and y) are normalized by L1 and B, respectively. Upstream of the groyne, a strong negative transverse velocity is consistently observed in every instance. This is because the main
horseshoe vortex is located upstream of the groyne.
The peak transverse velocity is observed near the groyne attached to wall, specifically at sections y/B = 0.08, 0.15, and 0.2, within the range of x/L[1] from 0 to 1. This characteristic can be
attributed to the occurrence of flow separation at the tip or junction of the groyne. The region of negative v/U near the junction or tip of the groyne is induced by recirculation vortex [13,27]. The
maximum negative transverse velocity values are observed to be 0.9 U, 0.95U, and 0.45U for IHG, LHG, and THG, respectively, at (x/L[1], y/B) = (0, 0.15), (x/L[1], y/B) = (0, 0.15), and (x/L[1], y/B)
= (−1, 0.15).
Figure 6 represents the distribution of the normalized vertical velocity (w/U) around the IHG, LHG, and THG. The velocity w is normalized by U, and the longitudinal and transverse directions (x and
y) are normalized by L[1] and B, respectively. In each of the three scenarios, a strong upward flow is observed downstream of the groyne. Additionally, in the cases of THG and LHG, a strong upward
flow is observed upstream of the groyne, However, in the case of IHG, this flow is absent. It can be inferred from the plots that a negative velocity zone exists for sections y/B = 0.04, 0.11, and
0.15 upstream of the groyne. These negative values are primarily the markers of downflow phenomenon occurring at the perpendicular face of groyne. The maximum value of the negative vertical velocity
are 0.15U, 0.25U, and 0.13U for IHG, LHG, and THG, respectively, at (x/L[1], y/B) = (0, 0.15), at (x/L[1], y/B) = (1.5, 0.15), and at (x/L[1], y/B) = (−0.5, 0.12). The elevated values of w/U signify
downward flow, which occurs upstream of the structure and is subsequently transferred downstream through the HSV. This flow pattern is attributed to the intricate dynamics of the flow around the
groyne structure.
3.3. Cost–Benefit Analysis (CBA)
This is a fundamental financial tool used to systematically examine the profitability of investment projects. As a part of this analysis, a detailed analysis of the incurred expenses and the
anticipated benefits is conducted.
Through empirical observations on IHG, LHG, and THG, it was established that IHG effectively safeguards a length equivalent to 1.2 times its dimension (1.2 L[1]), THG extends its protective capacity
to 1.16 times the transverse length (1.16 L[1]), and LHG protects a length 0.85 times the transverse dimension (0.85 L[1]). These are basically the reattachment lengths for each case and are
determined using 0 velocity isoline [5]. This observation serves as the foundation for calculating the benefits associated with each type of groyne, particularly regarding the length of effective
bank protection.
To enhance the precision of the cost–benefit analysis, cost coefficients (C1, C2, C3, and C4) were introduced. These coefficients intricately account for the material and labor costs, along with the
economic value of the structures and properties along riverbanks that can be safeguarded by these hydraulic structures. The ensuing cost–benefit ratios for the IHG, LHG, and THG are delineated below.
$Cost – benefit ratio for IHG , f I H G = ( C 1 + C 2 d ) L t 1.2 L C 4$
$Cost – benefit ratio for LHG , f L H G = ( C 1 + C 3 d ) ( 2 L t − t 2 ) 0.85 L C 4$
$Cost – benefit ratio for THG , f T H G = ( C 1 + C 3 d ) ( 2 L t − t 2 ) 1.16 L C 4$
The cost–benefit function was developed, and, as can be seen from Figure 7, three scenarios were considered.
Variable L with fixed D and t:
For this scenario, we kept the depth of flow (D) at 13.6 m and the thickness (t) at 3 m constant while varying the groyne length (L) from 0 to 25 m to explore the cost–benefit function. The function
for the IHG (f[IHG]) was found to be unaffected by the variation in L (Figure 7a). As depicted in Figure 7a, it is evident that the cost–benefit function f[THG] and f[LHG] assumes negative values
when L is less than t, presenting an unrealistic situation. The point of intersection between the plots of the two functions occurs when L equals t. In all practical applications, the cost–benefit
ratios of the THG and LHG exceed that of the IHG; this indicates that the costs associated with THG and LHG are relatively higher compared to their benefits. The functions f[THG] and f[LHG] are found
to be minimal when L equals the t. At this point of intersection, the ratio of total costs to total benefits is identical across all three cases. As the groyne length increases from L = 3 m until it
equals the depth of flow (D = 13.6 m), the functions f[THG] and f[LHG] exhibit a steady rise. Once this value is exceeded, the functions f[THG] and f[LHG] remain constant.
Variable D with fixed L and t:
In this instance, we maintained a constant groyne length (L = 11.25 m) and groyne thickness (t = 3 m) while varying the depth of flow (0 ≤ D ≤ 25 m) for analysis (Figure 7b). With a fixed groyne
length and thickness, each function, f[IHG], f[THG,] and f[LHG], demonstrates linear increases and consistently positive values as the depth of flow incrementally rises. Notably, the rate of increase
in f[LHG] is markedly steeper in comparison to that of f[THG], which is steeper than f[IHG] as the D increases. This suggests IHG as more cost-effective than THG and LHG, particularly with higher
depths of flow.
Variable t with fixed L and D:
For a constant length of groynes (L = 11.25 m) and depth of flow (D = 13.6 m), an increase in the thickness of the groynes was systematically analyzed, and the functions f[IHG], f[THG,] and f[LHG],
exhibited linear increments, maintaining positive values throughout (Figure 7c). Notably, the cost–benefit functions f[LHG] and f[THG] consistently displayed higher values across all the considered
thicknesses of groynes, indicating that the associated costs outweigh the benefits. The steep slope of the f[LHG] and f[THG] curves further emphasizes the pronounced increase compared to the f[IHG]
curve, underscoring the superiority of the IHG over the THG and LHG in riverbank protection with escalating groyne thickness, t.
The framework of cost–benefit analysis is mentioned in Appendix A and Appendix B. The functions developed in the section have been conducted for a Fr of 0.61. For the specific flow condition, it can
be observed that, in all the scenarios, the IHG provides better cost effectiveness than the LHG and THG. This cost–benefit analysis (CBA) framework will serve as a valuable tool for field engineers
as it can be adapted to their specific environmental conditions, taking into account local hydrodynamic factors and economic constraints. The implications of our findings are significant for
real-world applications. This will support informed decisions about groyne installation, optimizing both cost efficiency and functional performance. This approach not only supports the design and
implementation of effective groynes but also contributes to sustainable river engineering practices that balance environmental and economic considerations.
4. Conclusions
The study focused on providing valuable insights into the flow dynamics and scour patterns around IHG, LHG and THG. Our research provides crucial insights into selecting the appropriate groyne types
for specific flow conditions. By analyzing the flow dynamics around I, L, and T head groynes, our study offers a comparative assessment that can guide field engineers in choosing the most effective
groyne configuration.
The major conclusions of the study are as follows:
Among the three groynes, LHG and THG show the most significant scour depths, with close values of 0.295 D and 0.29 D, respectively. IHG had a maximum scour depth of 0.21 D. The maximum scour
depth for IHG is achieved near the tip of the groyne and can be attributed to the flow separation that occurs at that critical point. In the case of LHG, maximum scouring is achieved near the
junction of both the faces of the groyne. However, for THG, the maximum scour depth occurs upstream of the groyne. This is due to the downflow upstream of the groyne when the flow hits the
perpendicular face of the groyne and the HSV system that is the outcome of the downflow. This finding emphasizes the critical role of design and shape of the groyne on the flow characteristics
and scour patterns.
The reduction in the normalized streamwise velocity and negative values is evident for the sections near groyne attached wall upstream of the groyne. This is attributed to the formation of HSV
system. The peak of the normalized streamwise velocity is due to flow separation at the junction or tip.
Shear layer formation, marked by an abrupt change in velocity direction, is observed along the line of flow divergence. The peak transverse velocity is observed near the groyne attached wall at
the junction and within a small region downstream of the junction. This characteristic can be attributed to the occurrence of flow separation at the tip or junction of the groyne.
A strong upward flow is observed downstream of the groyne for all the groynes. Additionally, a strong upward flow is observed upstream of the groyne for THG and LHG which is absent for IHG. The
increase in w/U upstream signifies downward flow upstream of the groyne, which is subsequently transferred downstream through the HSV.
The reattachment length for IHG, LHG and THG is 1.2 L[1], 0.85 L[1], and 1.16 L[1,] respectively. The difference in the submergence condition is attributed as the principal reason for this value to
be smaller compared to the values observed by other studies. The cost–benefit analysis of the groynes provides insight regarding the expenses involved and the benefits expected. The results show IHG
as the most cost effective groyne as compared to the THG and LHG. This analytical approach is designed to provide a quantitative framework for decision-making, facilitating a comprehensive evaluation
of the economic feasibility and efficiency of the undertaking under consideration.
Understanding the variations in scour patterns and the impact of the groyne shape on riverbank stability is crucial for river engineering and management. The insights gained from this study
contribute to the body of knowledge in this field and can inform future groyne design and riverbank protection strategies. This study investigates the efficacy of these river training structures in
stabilizing the river channels. This study underscores the critical importance of implementing effective riverbank protection measures to ensure sustainable development. By developing a predictive
model for determining the appropriate length of bank protection in the vicinity of hydraulic structures, we provide a valuable tool for engineers and decision-makers. Our model enhances the ability
to design and implement targeted riverbank stabilization strategies, thereby improving the resilience of the hydraulic infrastructure. The successful application of this model can help to protect
communities and ecosystems, promoting a balanced approach to development that prioritizes both human safety and environmental sustainability. Our work provides a comprehensive methodology for
assessing the groyne performance under various flow conditions and offers practical tools for field engineers to apply these insights in real-world scenarios. As we strive to create resilient river
systems, particularly in changing climate conditions, our research underscores the importance of selecting the most suitable groyne shape to mitigate erosion and maintain stable river channels.
Author Contributions
Conceptualization, P., M.K.M., C.S.P.O. and K.S.H.P.; Methodology, P. and S.S.; Software, P. and M.K.M.; Validation, P.; Formal analysis, P. and M.K.M.; Investigation, P., M.K.M. and S.S.;
Writing—original draft, P.; Writing—review and editing, P, M.K.M., C.S.P.O. and K.S.H.P.; Visualization, C.S.P.O. and K.S.H.P.; Supervision, C.S.P.O. and K.S.H.P. All authors have read and agreed to
the published version of the manuscript.
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Data are available in this article.
The authors would like to thank members of Hydraulics Lab, IITR, for experimental setup provided.
Conflicts of Interest
Author Shikhar Sharma was employed by L & T Construction. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be
construed as a potential conflict of interest.
Appendix A. Cost–Benefit Analysis for I Head Groynes (IHG), L Head Groynes (LHG), and T Head Groynes (THG)
Cost estimation for IHG and THG has been discussed in Mall et al. [14]. The function has been modified according to the experimental results obtained in this study. The bank protection lengths are
estimated to be 1.2 L and 1.16 L for IHG and THG, respectively. The parameters in these functions are defined as
• Flow depth, D;
• River width, B;
• Transverse length, L;
• Free board, F[B];
• Thickness of groyne, t.
Therefore, the cost–benefit function (CBF) for IHG is
$f I H G = C 1 + C 2 D L t 1.2 L C 4$
The CBF for THG is
$f T H G = ( C 1 + C 3 D ) ( 2 L t − t 2 ) 1.16 L C 4$
The overall expense associated with constructing a single IHG can be calculated by summing the cost of concreting, cost of excavation, and cost of reinforcement.
$O v e r a l l _ c o s t I H G = F i n a l _ c o s t c o n c r e t e + F i n a l _ c o s t r e i n f o r c e m e n t + F i n a l _ c o s t e x c a v a t i o n$
$O v e r a l l _ c o s t I H G = C c + C r · L t ( d f × 0.21 + 1 ) D + F B + C e D L t = ( C 1 + C 2 D ) L t$
where C[1] = F[B] (C[c] + C[r]), and C[2] = (df × 0.21 + 1)(C[c] + C[r]) C[e] represent cost coefficients that vary based on the expenses associated with concrete material, reinforcement, and
The overall expense associated with constructing a single THG can be calculated by summing the cost of concreting, cost of excavation, and cost of reinforcement.
$O v e r a l l _ c o s t T H G = F i n a l _ c o s t c o n c r e t e + F i n a l _ c o s t r e i n f o r c e m e n t + F i n a l _ c o s t e x c a v a t i o n$
$O v e r a l l _ c o s t T H G = C c + C r · 2 L t − t 2 · d f × 0.29 + 1 D + F B + C e · D ( 2 L t − t 2 ) = ( C 1 + C 3 D ) ( 2 L t − t 2 )$
where C[1] = F[B] (C[c] + C[r]), and C[3] = (df × 0.29 + 1)(C[c] + C[r]) C[e] represent cost coefficients that vary based on the expenses associated with concrete material, reinforcement, and
Hereby, we extend the cost–benefit function for LHG.
Appendix B. Cost Estimation for LHG
The transverse length ($L$) is the length of the groyne along the direction of flow.
Depth of scour (equilibrium scour depth for LHG, Figure 3) = $0.30 D$.
Appendix B.1. Cost Estimation
Appendix B.1.1. Cost of Concreting
The cost involved in concreting for LHG is estimated by calculating the volume of concrete needed for LHG, which is calculated as follows:
$V c o n c r e t e = 2 L t − t 2 · [ ( d f × 0.30 + 1 ) D + F B ]$
The total cost of concreting is now calculated for LHG using $C c$, which is cost of concrete per cubic meter:
$T o t a l _ c o s t c o n c r e t e = C c · 2 L t − t 2 · [ ( d f × 0.30 + 1 ) D + F B ]$
Appendix B.1.2. Cost for Reinforcement
If we consider nominal reinforcement for the concrete hydraulic structure as $n %$ of the volume of concrete, the total volume of reinforcement would be
$V r e i n f o r c e m e n t = n 100 · 2 L t − t 2 · [ ( d f × 0.30 + 1 ) D + F B ]$
The overall weight of reinforcement for LHG is
$T o t a l _ w e i g h t r e i n f o r c e m e n t = W r · n 100 · 2 L t − t 2 · [ ( d f × 0.30 + 1 ) D + F B ]$
where $W r$ denotes unit weight of reinforcement bars.
The overall cost for reinforcement can be calculated as follows:
$O v e r a l l c o s t r e i n f o r c e m e n t = R r · W r · n 100 · 2 L t − t 2 · d f × 0.30 + 1 D + F B = C r · 2 L t − t 2 · [ ( d f × 0.30 + 1 ) D + F B ]$
where $R r$ represents the cost of reinforcement bars per unit weight.
C[r] = R[r] W[r]$n 100$, represents cost coefficient.
Appendix B.1.3. Excavation Cost
Determination of the cost involved in excavation for LHG requires calculating the volume of excavation, which is
$V e x c a v a t i o n = d f × 0.30 D · 2 L t − t 2 cubic meters .$
The total cost of excavation amounts to
$T o t a l _ c o s t e x c a v a t i o n = R e · d f × 0.30 D · 2 L t − t 2 = C e · D ( 2 L t − t 2 )$
where $R e$ is the cost of excavation per cubic meter.
C[e] = R[e] (d[f] × 0.30) depicts the cost coefficient.
Appendix B.2. Total Cost for Construction of LHG
The overall expense associated with constructing a single LHG is the sum of the cost of concreting, the cost of excavation, and the cost of reinforcement.
$O v e r a l l _ c o s t L H G = O v e r a l l _ c o s t c o n c r e t e + O v e r a l l _ c o s t r e i n f o r c e m e n t + O v e r a l l _ c o s t e x c a v a t i o n$
$O v e r a l l c o s t L H G = C c + C r · 2 L t − t 2 · d f × 0.30 + 1 D + F B + C e · D 2 L t − t 2 = C 1 + C 3 D 2 L t − t 2 = ( C 1 + C 3 D ) ( 2 L t − t 2 )$
where C[1] = F[B] (C[c] + C[r]), and C[3] = (df × 0.30 + 1)(C[c] + C[r]) C[e] are cost coefficients, which depend on the costs of concrete material, reinforcement, and excavation.
Figure 1. Experimental setup for (a) IHG, (b) LHG, and (c) THG.
Figure 1. Experimental setup for (a) IHG, (b) LHG, and (c) THG.
Figure 2. Power spectrum of fluctuations of streamwise velocity.
Figure 2. Power spectrum of fluctuations of streamwise velocity.
Figure 3. Scour depth around (a) IHG, (b) LHG, and (c) THG.
Figure 3. Scour depth around (a) IHG, (b) LHG, and (c) THG.
Figure 4. Dimensionless streamwise velocity distribution around (a) IHG, (b) LHG, and (c) THG.
Figure 4. Dimensionless streamwise velocity distribution around (a) IHG, (b) LHG, and (c) THG.
Figure 5. Dimensionless transverse velocity distribution around (a) IHG, (b) LHG, and (c) THG.
Figure 5. Dimensionless transverse velocity distribution around (a) IHG, (b) LHG, and (c) THG.
Figure 6. Dimensionless vertical velocity distribution around (a) IHG, (b) LHG, and (c) THG.
Figure 6. Dimensionless vertical velocity distribution around (a) IHG, (b) LHG, and (c) THG.
Figure 7. Cost–benefit functions f[IHG] (blue), f[LHG] (green), and f[THG] (red) for (a) variable L with fixed D and t, (b) variable D with fixed L and t, and (c) variable t with fixed L and D.
Figure 7. Cost–benefit functions f[IHG] (blue), f[LHG] (green), and f[THG] (red) for (a) variable L with fixed D and t, (b) variable D with fixed L and t, and (c) variable t with fixed L and D.
Table 1. Flow Parameters for experiments.
Table 1. Flow Parameters for experiments.
Shape D (m) Fr L[1] (m) L[2] (m) C.R.
IHG 0.136 0.61 0.1125 - 15%
LHG 0.136 0.61 0.1125 0.1125 15%
THG 0.136 0.61 0.1125 0.1125 15%
Table 2. Turbulent velocity statistics of experimental data for section y/B = 0.77.
Table 2. Turbulent velocity statistics of experimental data for section y/B = 0.77.
x/L[1] $u ¯$ (m/s) Std. Dev. (m/s) Skewness Kurtosis Std. Error (m/s)
$−$12.50 0.729797 0.081635 $−$0.05941 2.708591 0.000868
$−$8.00 0.751042 0.082159 $−$0.15902 2.794846 0.000863
$−$4.00 0.726126 0.081202 0.007192 2.876749 0.000875
0.00 0.791727 0.083517 $−$0.21658 3.095242 0.000887
4.00 0.738749 0.083441 $−$0.18728 2.881535 0.000856
8.00 0.837464 0.081221 $−$0.17489 2.771532 0.001299
11.25 0.862593 0.080529 $−$0.25855 3.312264 0.000826
15.00 0.879171 0.077864 $−$0.21254 2.748936 0.000901
20.00 0.866055 0.084618 $−$0.28543 2.758457 0.000878
25.00 0.83369 0.082747 $−$0.1951 2.837325 0.001033
30.00 0.734824 0.096966 $−$0.24464 2.791556 0.001046
35.00 0.642727 0.098407 $−$0.02617 2.646666 0.000887
41.00 0.83348 0.085447 $−$0.26433 2.773093 0.000909
47.00 0.739169 0.092991 $−$0.0956 2.712638 0.000987
53.00 0.733704 0.093498 0.012802 2.782523 0.000993
59.00 0.810494 0.097884 $−$0.27136 2.715227 0.001044
65.00 0.711532 0.097301 $−$0.19889 2.651946 0.001806
75.00 0.651122 0.122696 $−$0.16853 2.639037 0.00161
85.00 0.705159 0.100333 $−$0.20404 2.752259 0.000863
95.00 0.67583 0.100405 $−$0.24629 2.750138 0.001311
110.00 0.67236 0.099111 $−$0.16994 2.690638 0.001311
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Riemann Sums Made Easy: Step-by-Step Tutorial
Riemann sums are like the building blocks of integral calculus, a bridge between discrete and continuous.
Riemann Sums are a fundamental concept in calculus, used to approximate the area under a curve or, more formally, to approximate the definite integral of a function. They are named after the German
mathematician Bernhard Riemann. Here’s a thorough explanation:
Concept Overview
• Purpose: Riemann Sums are used to approximate the area under a curve, which is integral in calculus for finding areas, volumes, and other applications.
• Basic Idea: The area under a curve within a certain interval is approximated by dividing the interval into smaller sub-intervals, calculating the area of simple shapes (like rectangles or
trapezoids) over these sub-intervals, and then summing these areas.
Steps in Calculating a Riemann Sum
1. Choose an Interval: Select the interval \([a, b]\) over which you want to approximate the area under the curve of a function \( f(x) \).
2. Divide the Interval: Split this interval into smaller sub-intervals. The number of sub-intervals, denoted as \( n \), can vary. More sub-intervals typically lead to a more accurate approximation.
3. Determine Sub-Interval Width: The width of each sub-interval, denoted as \( \Delta x \), is \(\frac{b-a}{n}\).
4. Choose Sample Points: For each sub-interval, choose a sample point. This could be the left endpoint, right endpoint, midpoints, or any other point within each sub-interval.
5. Calculate Function Values: Evaluate \( f(x) \) at each of these sample points.
6. Sum Up: Multiply each function value by the width of its sub-interval \(( \Delta x )\) to get the area of each rectangle (or trapezoid) and sum these areas.
Types of Riemann Sums
• Left Riemann Sum: Uses the left endpoint of each sub-interval for the sample point.
• Right Riemann Sum: Uses the right endpoint of each sub-interval.
• Midpoint Riemann Sum: Uses the midpoint of each sub-interval.
• Trapezoidal Rule: A more complex form that approximates the area using trapezoids instead of rectangles.
Mathematical Formulation
The Riemann Sum can be expressed as:
\(S = \sum_{i=1}^{n} f(x_i^*) \Delta x\)
where \( x_i^* \) is the sample point in each sub-interval, and \( \Delta x \) is the width of the sub-intervals.
Importance in Calculus
Riemann Sums are crucial because they:
• Provide the foundation for the definite integral.
• Help in understanding the concept of integration as a limit of sums.
• Serve as a tool for approximating areas and other quantities that are difficult to calculate exactly.
Limitations and Extensions
• Accuracy: The accuracy of a Riemann Sum depends on the number of sub-intervals and the function’s behavior. More sub-intervals generally lead to a more accurate approximation.
• Extension to Definite Integrals: As the number of sub-intervals goes to infinity \(( n \to \infty )\), and their width \( \Delta x \) goes to \(zero\), the Riemann Sum approaches the exact value
of the definite integral, illustrating the fundamental concept of integration in calculus.
Related to This Article
What people say about "Riemann Sums Made Easy: Step-by-Step Tutorial - Effortless Math: We Help Students Learn to LOVE Mathematics"?
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Warhammer Fantasy: 6th Edition
Units are usually made up of a number of models of the same type. It therefore follows that a unit of, say, ten models each of, let us assume, 9 points is worth a total of 10 x 9 = 90 points. It is
quite usual to refer to a unit in terms of its value, so you might hear players talk of 'a 140 point unit of Orc warriors', '150 points of High Elf archers', or some such expression. As most
practically sized units are likely to be between 100 and 300 points, it can be assumed that a 2,000 point army would have about ten units.
However, this doesn't allow for character models. Once points have been allocated for these important pieces the typical 2,000 point army is more likely to have about seven units in total, of which
some might be a single piece such as a chariot or war machine.
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The p-value misconception eradication challenge
If you have educational material that you think will do a better job at preventing p-value misconceptions than the material in my MOOC, join the p-value misconception eradication challenge by
proposing an improvement to my current material in a new A/B test in my MOOC.
I launched a massive open online course “Improving your statistical inferences” in October 2016. So far around 47k students have enrolled, and the evaluations suggest it has been a useful resource
for many researchers. The first week focusses on p-values, what they are, what they aren’t, and how to interpret them.
Arianne Herrera-Bennet was interested in whether an understanding of p-values was indeed “impervious to correction” as some statisticians believe (Haller & Krauss, 2002, p. 1) and collected data on
accuracy rates on ‘pop quizzes’ between August 2017 and 2018 to examine if there was any improvement in p-value misconceptions that are commonly examined in the literature. The questions were asked
at the beginning of the course, after relevant content was taught, and at the end of the course. As the figure below from the preprint shows, there was clear improvement, and accuracy rates were
quite high for 5 items, and reasonable for 3 items.
We decided to perform a follow-up from September 2018 where we added an assignment to week one for half the students in an ongoing A/B test in the MOOC. In this new assignment, we didn’t just explain
what p-values are (as in the first assignment in the module all students do) but we also tried to specifically explain common misconceptions, to explain what p-values are not. The manuscript is still
in preparation, but there was additional improvement for at least some misconceptions. It seems we can develop educational material that prevents p-value misconceptions – but I am sure more can be
In my paper to appear in Perspectives on Psychological Science on “The practical alternative to the p-value is the correctly used p-value” I write:
“Looking at the deluge of papers published in the last half century that point out how researchers have consistently misunderstood p-values, I am left to wonder: Where is the innovative coordinated
effort to create world class educational materials that can freely be used in statistical training to prevent such misunderstandings? It is nowadays relatively straightforward to create online apps
where people can simulate studies and see the behavior of p values across studies, which can easily be combined with exercises that fit the knowledge level of bachelor and master students. The second
point I want to make in this article is that a dedicated attempt to develop evidence based educational material in a cross-disciplinary team of statisticians, educational scientists, cognitive
psychologists, and designers seems worth the effort if we really believe young scholars should understand p values. I do not think that the effort statisticians have made to complain about p-values
is matched with a similar effort to improve the way researchers use p values and hypothesis tests. We really have not tried hard enough.”
If we honestly feel that misconceptions of p-values are a problem, and there are early indications that good education material can help, let’s try to do all we can to eradicate p-value
misconceptions from this world.
We have collected enough data in the current A/B test. I am convinced the experimental condition adds some value to people’s understanding of p-values, so I think it would be best educational
practice to stop presenting students with the control condition.
However, I there might be educational material out there that does a much better job than the educational material I made, to train away misconceptions. So instead of giving all students my own new
assignment, I want to give anyone who thinks they can do an even better job the opportunity to demonstrate this. If you have educational material that you think will work even better than my current
material, I will create a new experimental condition that contains your teaching material. Over time, we can see which materials performs better, and work towards creating the best educational
material to prevent misunderstandings of p-values we can.
If you are interested in working on improving p-value education material, take a look at the first assignment in the module that all students do, and look at the new second assignment I have created
to train away misconception (and the answers). Then, create (or adapt) educational material such that the assignment is similar in length and content. The learning goal should be to train away common
p-value misconceptions – you can focus on any and all you want. If there are multiple people who are interested, we collectively vote on which material we should test first (but people are free to
combine their efforts, and work together on one assignment). What I can offer is getting your material in front of between 300 and 900 students who enroll each week. Not all of them will start, not
all of them will do the assignments, but your material should reach at least several hundreds of learners a year, of which around 40% has a masters degree, and 20% has a PhD – so you will be teaching
fellow scientists (and beyond) to improve how they work.
I will incorporate this new assignment, and make it publicly available on my blog, as soon as it is done and decided on by all people who expressed interest in creating high quality teaching
material. We can evaluate the performance by looking at the accuracy rates on test items. I look forward to seeing your material, and hope this can be a small step towards an increased effort in
improving statistics education. We might have a long way to go to completely eradicate p-value misconceptions, but we can start.
2 comments:
1. Daniel-
Thanks as always for your work. I don’t have a lesson of my own to offer, but I did have a comment on a small part of your first assignment that I think could be problematic.
On page 2 of the posted version of lesson 1.1, you write about the first figure “There is a horizontal red dotted line that indicates an alpha of 5% (located at a frequency of 100.000*0.05 =
5000)”. But that seems like a confusing or misleading statement. First, since the line indicates a Y value, it must be a frequency of observed outcomes for p; a line showing alpha would have to
indicate an X value. And even given that the line indicates the expected frequency of outcomes, it's the expectation *under the null hypothesis*, which is not explained here. More importantly,
though, even if you do mean that the line will show the expected height of the bars under the null hypothesis, the only reason that you can use N*0.05 to predict that height is that you’ve
divided the distribution into 20 bars - it’s not because alpha is 0.05. If you’d chosen to divide the graph into increments of 0.01 (as you do later), the height of the red line would be N*0.01
despite alpha being 0.05 (but now there would be five bars in the alpha region instead of just one). So the height of the line is based on the number of divisions, not alpha.
Does that critique make sense? I can try to explain more fully if not.
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AI tackles one of the most difficult challenges in quantum chemistry | Blog
AI tackles one of the most difficult challenges in quantum chemistry
Natural excited states. Combining neural networks with a mathematical insight enables accurate calculations of challenging excited states of molecules. Credit:
(2024). DOI: 10.1126/science.adn0137
New research using neural networks, a form of brain-inspired AI, proposes a solution to the tough challenge of modeling the states of molecules.
The research shows how the technique can help solve fundamental equations in complex molecular systems. This could lead to practical uses in the future, helping researchers to prototype new materials
and chemical syntheses using computer simulation before trying to make them in the lab.
Led by Imperial College London and Google DeepMind scientists, the study is
Excited molecules
The team investigated the problem of understanding how molecules transition to and from
excited states
. When molecules and materials are stimulated by a large amount of energy, such as being exposed to light or high temperatures, their electrons can get kicked into a temporary new configuration,
known as an excited state.
The exact amount of energy absorbed and released as molecules transition between states creates a unique fingerprint for different molecules and materials. This affects the performance of
technologies ranging from solar panels and LEDs to semiconductors and photocatalysts. It also plays a critical role in biological processes involving light, including photosynthesis and vision.
However, this fingerprint is extremely difficult to model because the excited electrons are quantum in nature, meaning their positions within the molecules are never certain, and can only be
expressed as probabilities.
Lead researcher Dr. David Pfau, from Google DeepMind and the Department of Physics at Imperial, said, "Representing the state of a quantum system is extremely challenging. A probability has to be
assigned to every possible configuration of electron positions.
"The space of all possible configurations is enormous—if you tried to represent it as a grid with 100 points along each dimension, then the number of possible electron configurations for the silicon
atom would be larger than the number of atoms in the universe. This is exactly where we thought
deep neural networks
could help."
Neural networks
The researchers developed a new mathematical approach and used it with a neural network called
(Fermionic Neural Network), which was the first example where
deep learning
was used to compute the energy of atoms and
from fundamental principles that was accurate enough to be useful.
The team tested their approach with a range of examples, with promising results. On a small but complex molecule called the carbon dimer, they achieved a mean absolute error (MAE) of 4 meV
(millielectronvolt—a tiny measure of energy), which is five times closer to experimental results than prior gold standard methods reaching 20 meV.
Dr. Pfau said, "We tested our method on some of the most challenging systems in
computational chemistry
, where two electrons are excited simultaneously, and found we were within around 0.1 eV of the most demanding, complex calculations done to date.
"Today, we're making our latest work
open source
, and hope the research community will build upon our methods to explore the unexpected ways matter interacts with light."
Source: https://tinyurl.com/2s3mfb7k via Phys.Org | {"url":"https://www.qd-latam.com/site/pt/blog/615/2024/08/ai-tackles-one-of-the-most-difficult-challenges-in-quantum-chemistry/","timestamp":"2024-11-03T23:08:17Z","content_type":"text/html","content_length":"92909","record_id":"<urn:uuid:df05f880-9e5b-4d0a-802a-64c6e1fcdae1>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00360.warc.gz"} |
) (S
WINDOW (IN SQL MODEL) (SORT)
Execute an analytic function within a SQL Model (Spreadsheet)
This operation was introduced in Oracle 10.2
This operation is implemented in the following versions
This example was developed using Oracle 10.2.0.1 on Linux
This example requires the following table definition
CREATE TABLE t1
c1 NUMBER,
c2 NUMBER,
c3 NUMBER
The table does not need to be analyzed
The statement
SELECT c1,a1,a2
SELECT c1,SUM(c2) a1
FROM t1
GROUP BY c1
MODEL DIMENSION BY (c1)
MEASURES (a1,0 a2)
a2[any] = SUM (a1) OVER ()
ORDER BY c1;
generates the following execution plan
0 SELECT STATEMENT Optimizer=CHOOSE
1 0 SORT (ORDER BY)
2 1 SQL MODEL (ORDERED)
3 2 HASH (GROUP BY)
4 3 TABLE ACCESS (FULL) OF 'T1'
5 4 WINDOW (IN SQL MODEL) (SORT) | {"url":"http://juliandyke.com/Optimisation/Operations/WindowInSQLModelSort.php","timestamp":"2024-11-09T07:42:43Z","content_type":"text/html","content_length":"2467","record_id":"<urn:uuid:9a540b52-6f5f-4fd4-964f-a09f9791e2b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00059.warc.gz"} |
Cubes in a Line – Video in the Middle
Contact us to learn more about Video in the Middle.
In an earlier discussion of Cubes in a Line, Jasmine suggested that you can multiply by 6 to find the total. Here Maryann comes back to Jasmine’s idea and works with her to clarify her idea and
connect it with what the class has already concluded.
The teacher, Maryann, launches the Cubes in a Line lesson by showing her students two cubes and asking the question, “If I put two cubes together, how many faces are there?” We drop in as several
students share their responses and the class discussion ensues.
After having worked individually to predict the number of faces for a line of 10 connected cubes, the class discusses their predictions attending to two particular student methods–one using repeated
addition and the other using multiplication.
As students discuss their approaches, Chase asks Cody where the 4 comes from in his idea. Cody responds, “Which 4? There are a whole bunch of 4s.” As the discussion continues, Kyle, Nicholas, and
Jasmine add their ideas.
Maryann introduced the Cubes in a Line task by showing her students two cubes and asking the question, “If I put two cubes together, how many faces are there showing?” We drop in as several students
explain how they arrived at their totals. | {"url":"https://videointhemiddle.org/pattern-task-cluster/cubes-in-a-line/","timestamp":"2024-11-11T21:08:29Z","content_type":"text/html","content_length":"53342","record_id":"<urn:uuid:faf5d9b8-ea99-4bf8-bae7-4c09c64c9968>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00651.warc.gz"} |
What Does % Mean In Coding | Robots.net
Welcome to the world of coding, where symbols and characters hold immense significance. One such symbol that frequently appears in coding is the percentage sign (%). In programming, the percentage
sign, also known as the modulus operator, plays a crucial role in various operations.
The percentage sign is not limited to coding but is commonly used in other mathematical contexts as well. However, in this article, we will explore its specific meaning and functionality within
coding languages.
Understanding the role of the percentage sign is essential for developers and aspiring coders, as it is a fundamental component of many programming languages. With its versatile use cases, mastering
the concept of % can elevate your coding skills and enable you to write more efficient and elegant code.
In the following sections, we will delve deeper into the definition of the percentage sign and explore its different applications within coding. By the end, you will have a solid understanding of how
% operates and how it can be leveraged to solve various programming challenges.
Definition of %
The percentage sign, often referred to as the modulus operator, is a symbol used in coding languages to calculate the remainder of a division operation. It is denoted by the percent symbol (%), which
is typically placed between two numbers.
When the percentage sign is used in a coding context, it performs the operation of modulo division. Modulo division returns the remainder of a division operation instead of the quotient. For example,
if we divide 10 by 3, the quotient would be 3, with a remainder of 1. Using the modulus operator, we can find the remainder, which in this case would be 1.
The % operator works by dividing the left operand by the right operand and returning the remainder as the result. In most programming languages, this operator operates solely on integers, although
some languages allow it to be used with floating-point numbers as well.
It is important to note that the sign of the result in a modulo operation will always match the sign of the dividend (the number being divided) in most programming languages. This means that if the
dividend is negative, the result will also be negative, and if the dividend is positive, the result will be positive.
The modulus operator is a valuable tool in programming as it can be used in various scenarios, such as determining if a number is even or odd, cycling through a list of elements, or performing
calculations that involve repeating patterns. Its flexibility and simplicity make it an indispensable component of many coding languages.
Modulus Operator
The modulus operator, denoted by the percentage sign (%), is a mathematical operator used in coding to calculate the remainder of a division operation.
When using the modulus operator, the divisor divides the dividend, and the remainder is the result. For example, if we have the expression 10 % 3, the dividend is 10 and the divisor is 3. When we
divide 10 by 3, the quotient is 3, and the remainder is 1. Therefore, the result of 10 % 3 is 1.
The modulus operator is particularly useful when dealing with repetitive patterns or cyclic behaviors. It helps determine patterns based on remainders. For instance, when dealing with an array of
elements, you can use the modulus operator to loop through the elements in a cyclical manner.
Another commonly used application of the modulus operator is to check if a number is even or odd. By taking a number % 2, if the result is 0, the number is even. If the result is 1, the number is
odd. This property is leveraged in many programming scenarios where different actions are performed based on the evenness or oddness of a number.
Additionally, the modulus operator can be handy in determining whether a number is divisible by another number. If the result of a % b is 0, it means that a is divisible by b without any remainder.
However, it is essential to note that the modulus operator follows the same mathematical precedence as other operators, such as addition and multiplication. To ensure the desired calculation order,
parentheses can be used to prioritize specific operations within an expression.
The modulus operator is a powerful tool for solving various coding challenges, providing a means to manipulate and analyze data based on remainders and patterns. By leveraging its functionality,
developers can optimize code efficiency and achieve desired results in their programming endeavors.
Remainder Operator
The remainder operator, often represented by the percentage sign (%), is a mathematical operator used in coding to calculate the remainder of a division operation. It is closely related to the
modulus operator but differs in its treatment of negative numbers.
Similar to the modulus operator, the remainder operator performs division and returns the remainder as the result. For example, when dividing 10 by 3 using the remainder operator (10 % 3), the result
is 1, as 10 divided by 3 equals 3 with a remainder of 1.
While both the remainder operator and the modulus operator can be used to calculate the remainder, they handle negative numbers differently. The remainder operator follows the sign of the dividend,
meaning that if the dividend is negative, the remainder will also be negative. For example, if we have -10 % 3, the result will be -1, as the remainder retains the negative sign.
This behavior differs from the modulus operator, where the result matches the sign of the divisor. In the case of -10 % 3 using the modulus operator, the result would be 2, as the sign of the divisor
is positive.
The remainder operator is widely used in coding to determine cyclic patterns and to perform calculations based on remainders. It can be particularly useful in scenarios such as indexing elements in
arrays or looping through a range of values.
By utilizing the remainder operator, developers can easily detect if a number falls within a specific range or if it satisfies certain mathematical properties. It enables the creation of efficient
algorithms and algorithms that work with repeating patterns and cyclical behaviors.
When writing code that involves division and the need to obtain the remainder, understanding the difference between the remainder operator and the modulus operator is crucial. Proper usage and
consideration of signs will help ensure accurate calculations and desired results in coding applications.
Examples of % in Coding
The percentage sign (%) has numerous applications in coding, making it a versatile operator. Here are some common examples of how the modulus operator is used in different scenarios:
• Determining Even or Odd Numbers: One of the most common uses of the modulus operator is to check if a number is even or odd. By performing a % 2 operation, if the result is 0, the number is even.
For example, 4 % 2 would yield 0, confirming that 4 is an even number. Conversely, 5 % 2 would give a result of 1, indicating that 5 is an odd number.
• Looping Through an Array: The modulus operator is often used to cycle through an array of elements in a circular manner. By utilizing the array length and the % operator, you can repeatedly
access elements based on their indices. For example, if you have an array with five elements and want to perform an action on each element in a continuous loop, you can use the index % 5 to
access the elements one by one.
• Divisibility Check: The modulus operator is instrumental in determining whether a number is divisible by another number without leaving a remainder. If a number % divisor equals 0, it signifies
complete divisibility. For instance, to check if a number is divisible by 3, you can use % 3 and see if the result is 0. If it is, the number is divisible by 3.
• Obtaining Remainders: The modulus operator is handy for extracting the remainder of a division operation. For example, if you divide 17 by 5, the quotient is 3, and the remainder is 2. By using
the expression 17 % 5, you can directly obtain the remainder of 2.
The above examples demonstrate the versatility of the modulus operator in coding. By leveraging the power of the percentage sign, developers can perform a wide range of calculations and manipulate
data to achieve desired results efficiently.
Common Uses of % in Coding
The percentage sign (%) has become a fundamental part of coding, offering a wide range of applications and use cases. Here are some common scenarios where the modulus operator is commonly used in
• Array Indexing: The modulus operator is widely used in indexing elements within arrays. By using the modulus operator with the array length, developers can ensure that the index stays within the
bounds of the array. This technique is particularly valuable when implementing circular or cyclical operations.
• Indented Formatting: In some programming languages, especially those that depend on indentation for readability, the modulus operator can be used to format code or maintain consistency. For
example, by applying the modulus operator to the line number, developers can control the indentation level, making the code more organized and readable.
• Time Calculations: The modulus operator is commonly utilized when working with time-related calculations. By taking the modulus of a large time value, such as the total number of seconds, with
60, it becomes possible to extract the remaining seconds after calculating the minutes. This technique is useful in various applications, including timers, scheduling, and elapsed time
• Noise Generation: In computer graphics and simulation programs, the modulus operator is employed to generate random-like patterns known as noise. By using the modulus operation on the coordinates
of a grid, developers can create pseudo-random patterns that simulate the appearance of natural elements, such as terrain or textures.
• Hashing: Cryptographic hashes often rely on the modulus operator to generate unique identifiers or to distribute elements across data structures, such as hash tables. The modulus operation
ensures a balanced distribution and reduces the likelihood of collisions, improving the efficiency and reliability of hash-based algorithms.
The above examples represent just a fraction of the many uses of the modulus operator in coding. Its versatile nature makes it an essential tool for developers, enabling them to solve a variety of
problems and optimize their code for performance and efficiency.
The percentage sign (%) or modulus operator is a fundamental symbol in coding that serves various purposes and provides valuable functionality. It allows programmers to perform calculations involving
remainders, cyclic patterns, and divisibility checks. By mastering the usage of %, developers can optimize their code, enhance efficiency, and tackle a wide range of programming challenges.
In this article, we explored the definition of the percentage sign and its role as a modulus operator. We learned how it calculates remainders in division operations and discussed its distinction
from the remainder operator. We also examined several examples of how % is commonly used in coding, including checking evenness or oddness, indexing arrays, and performing divisibility checks.
Familiarizing yourself with the percentage sign and its practical applications allows you to write more efficient and elegant code. Whether you’re working on algorithms, array manipulations, or
time-related calculations, the modulus operator can be a powerful tool at your disposal.
The versatility of the percentage sign makes it a crucial component in many programming languages. By understanding its usage and incorporating it effectively into your code, you can enhance your
problem-solving abilities and write more optimized and robust programs.
So, the next time you encounter the percentage sign in your coding endeavors, remember its significance as the modulus operator and leverage its power to unlock new possibilities in your programming | {"url":"https://robots.net/tech/what-does-mean-in-coding/","timestamp":"2024-11-02T20:38:17Z","content_type":"text/html","content_length":"387192","record_id":"<urn:uuid:138f484a-c511-47f1-803f-4d17812c21ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00856.warc.gz"} |
J-PLUS: Morphological star/galaxy classification by PDF analysis
Aims: Our goal is to morphologically classify the sources identified in the images of the J-PLUS early data release (EDR) as compact (stars) or extended (galaxies) using a dedicated Bayesian
Methods: J-PLUS sources exhibit two distinct populations in the r-band magnitude versus concentration plane, corresponding to compact and extended sources. We modelled the two-population distribution
with a skewed Gaussian for compact objects and a log-normal function for the extended objects. The derived model and the number density prior based on J-PLUS EDR data were used to estimate the
Bayesian probability that a source is a star or a galaxy. This procedure was applied pointing-by-pointing to account for varying observing conditions and sky positions. Finally, we combined the
morphological information from the g, r, and i broad bands in order to improve the classification of low signal-to-noise sources.
Results: The derived probabilities are used to compute the pointing-by-pointing number counts of stars and galaxies. The former increases as we approach the Milky Way disk, and the latter are similar
across the probed area. The comparison with SDSS in the common regions is satisfactory up to r 21, with consistent numbers of stars and galaxies, and consistent distributions in concentration and
(g-i) colour spaces.
Conclusions: We implement a morphological star/galaxy classifier based on probability distribution function analysis, providing meaningful probabilities for J-PLUS sources to one magnitude deeper (r
21) than a classical Boolean classification. These probabilities are suited for the statistical study of 150 thousand stars and 101 thousand galaxies with 15 < r ≤ 21 present in the 31.7 deg^2 of the
J-PLUS EDR. In a future version of the classifier, we will include J-PLUS colour information from 12 photometric bands.
Astronomy and Astrophysics
Pub Date:
February 2019
□ methods: data analysis;
□ Galaxy: stellar content;
□ galaxies: statistics;
□ Astrophysics - Astrophysics of Galaxies
Submitted to Astronomy and Astrophysics. 14 pages, 16 figures, 1 tables. Comments are welcome. All extra figures and the number counts files will be available with the paper in press | {"url":"https://ui.adsabs.harvard.edu/abs/2019A%26A...622A.177L/abstract","timestamp":"2024-11-09T16:07:59Z","content_type":"text/html","content_length":"53313","record_id":"<urn:uuid:daa38bad-3296-4650-95f6-4c94ba24d2b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00314.warc.gz"} |
Using geometric constraints
Geometric constraints:
• Specify a geometric relation between 2 entities (coincident, concentric, collinear, parallel, perpendicular, tangent, smooth, symmetric, equal).
• Specify a fixed angle (horizontal, vertical).
• Specify a fixed location (fix).
When a geometric constrained is applied to an entity:
• The position of the entity is adjusted according to the applied constraint.
• An icon displays next to the entity, to indicate the applied constraint. If multiple geometric constraints are applied, the icons are joined in a constraint bar.
Creating geometric constraints
Tools to create dimensional constraints are located in:
• The 2D Constraints toolbar.
• The Parametric ribbon tab.
• The 2D Constraints fly-out menu of the Parametric menu.
In the image below:
• The endpoints of the three lines and the arc are joined by coincident constraints.
• The midpoint of the circle and the arc are made concentric.
• Two lines are forced to be tangent to the arc.
• One line has a vertical constraint (= parallel to the Y-axis of the current coordinate system), one line has a horizontal constraint (= parallel to the X-axis of the current coordinate system).
Displaying constraint bars
Constraint bars are hidden when a drawing is closed and reopened.
Tools to control the display of constraint bars are located on:
• The Geometric Constraint toolbar.
• The Parametric ribbon tab.
• The Show/Hide 2D Constraints fly-out menu of the Parametric menu.
These tools execute the CONSTRAINTBAR command using a specific option.
The options are:
• Show/Hide: prompts you to select the entities.
• Show All: show all dimensional constraints in the drawing.
• Hide All: hide all dimensional constraints in the drawing.
• The value of theCONSTRAINTBARDISPLAYsystem variable controls the visibility of geometric constraints:
□ 1: Display constraint bars when geometrical constraints are added.
□ 2: Display hidden dimensional constraints when the constrained entities are selected.
• When hovering over an entity with a geometric constraint, the blue constraint glyph (
Controlling the position of a constraint bar
By default constraint glyph bars are created close to the midpoint of the entity and are kept at that relative position when the entity position changes. You can drag the constraint bar to a
different location. This new relative position is then maintained until theResetoption of theCONSTRAINTBARcommand restores the default position of the constraint bar.
Relocating a constraint bar
1. Place the cursor on the constraint bar.
2. Press and hold the left mouse button to move the constraint bar.
3. Release the left mouse button at the desired location.
Restoring the default position of constraint bars
1. Do one of the following:
□ Click theShow/Hide Geometric Constrainttool.
□ Launch the CONSTRAINTBAR command.
Prompts you:Select Entities.
2. Select the entities, then right click or press Enter.
BricsCAD reports the number of selected entities.
Prompts you: Select option for constraints [Show/Hide/Reset] <Show>:
3. Do one of the following:
□ ChooseResetin the context menu.
□ TypeRin the Command line, then press Enter.
Working with constraint bars
Controlling a constraint
1. Move the cursor over a constraint icon.
2. A tooltip displays, indicating the constraint type.
3. The associated entity (or entities) highlights.
4. The corresponding icon on the constraint bar of the associated entity highlights.
Deleting a constraint
1. Move the cursor over the constraint icon in the constraint bar.
2. Right click, then clickDelete.
3. The constraint is deleted and the icon is removed from the constraint bar and from the constraint bar of the associated entity.
Hiding the constraint bar of an entity
Move the cursor to the constraint bar, then click theClose(x) button.
Deleting constraints
To delete all constraints from a selection set:
1. Do one of the following:
□ Click theDelete 2D Constraintstool.
□ Launch the DELCONSTRAINT command.
You are prompted: Select entities to delete all constraints:
2. Selectthe entities you want to delete the constraints from.
3. Right click to stop selecting entities and delete all constraints from the selection set.
Some examples
Coincident constraints between the endpoint of a line and two circles.
If the endpoints or center points of entities already coincide, theAutoconstrainoption of theGCCOINCIDENTcommand automatically applies coincident constraints to such points.
By applying coincident constraints between the pentagon vertices and the circle, and equal constraints between one side and the four other sides, the circle radius defines the size of the pentagon.
Coincident constraints between the endpoints of the tangent lines and the circles prevent the tangent lines to extend beyond the tangent points.
Coincident constraints are used to:
• Force the endpoints of the red line to lie on the bold lines.
• Force the midpoint of the red line to lie on the dash-dot line.
The bold lines and the dash-dot line have a perpendicular constraint with the red line. As a result the dash-dot line always lies centered between the two bold lines. | {"url":"https://help.bricsys.com/en-us/document/bricscad/drawing-accurately/using-geometric-constraints?version=V23","timestamp":"2024-11-06T14:51:31Z","content_type":"text/html","content_length":"110896","record_id":"<urn:uuid:19d1eda3-78f8-4330-8fe2-3f4124951fd2>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00714.warc.gz"} |
NBA2K – There’s more to the PRIME Series I Collect all required PRIME Series I players …
Sporati loves NBA 2K MyTeam, here following every possible information or locker codes
There’s more to the PRIME Series I π ₯ Collect all required PRIME Series I players and earn GO RAY ALLEN π ± PD Stockton is the first required player from the PRIME Series I.
NBA 2K MyTeams’ Instagram post information
NBA 2K MyTeams’s social media locker code and information post titled There’s more to the PRIME Series I Collect all required PRIME Series I players … is credited to NBA 2K20 MyTEAM on instagram
account myteamspot and currently has 7253 likes as of posting.
We all love playing NBA 2K and building MyTeam roster, follow our posts on NBA Social Media!
23 Comments
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NBA2K – Happy Halloween Locker Code! Use this code for two shots at guaranteed tokens. …
Sporati loves NBA 2K MyTeam, here following every possible information or locker… | {"url":"https://esporati.com/2019/10/nba2k-theres-more-to-the-prime-series-i-collect-all-required-prime-series-i-players/","timestamp":"2024-11-12T03:50:46Z","content_type":"text/html","content_length":"132193","record_id":"<urn:uuid:79aa3dfa-ee09-4459-94e9-caaf241d1912>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00847.warc.gz"} |
Author: Josh Hug
Hoare Partitioning. One very fast in-place technique for partitioning is to use a pair of pointers that start at the left and right edges of the array and move towards each other. The left pointer
loves small items, and hates equal or large items, where a "small" item is an item that is smaller than the pivot (and likewise for large). The right pointers loves large items, and hates equal or
small items. The pointers walk until they see something they don't like, and once both have stopped, they swap items. After swapping, they continue moving towards each other, and the process
completes once they have crossed. In this way, everything behind the left pointer is <= to the pivot, and everything to the right is >= to the pivot. Finally, we swap the pivot into the appropriate
location, and the partitioning is completed. Unlike our prior strategies, this partitioning strategy results in a sort which is measurably faster than mergesort.
Selection. A simpler problem than sorting, in selection, we try to find the Kth largest item in an array. One way to solve this problem is with sorting, but we can do better. A linear time approach
was developed in 1972 called PICK, but we did not cover this approach in class, because it is not as fast as the Quick Select technique.
Quick Select. Using partitioning, we can solve the selection problem in expected linear time. The algorithm is to simply partition the array, and then quick select on the side of the array containing
the median. Best case time is Θ(N), expected time is Θ(N), and worst case time is Θ(N^2). You should know how to show the best and worst case times. This algorithm is the fastest known algorithm for
finding the median.
Stability. A sort is stable if the order of equal items is preserved. This is desirable, for example, if we want to sort on two different properties of our objects. Know how to show the stability or
instability of an algorithm.
Optimizing Sorts. We can play a few tricks to speed up a sort. One is to switch to insertion sort for small problems (< 15 items). Another is to exploit existing order in the array. A sort that
exploits existing order is sometimes called "adaptive". Python and Java utilize a sort called Timsort that has a number of improvements, resulting in, for example Θ(N) performance on almost sorted
arrays. A third trick, for worst case N^2 sorts only, is to make them switch to a worst case N log N sort if they detect that they have exceeded a reasonable number of operations.
Shuffling. To shuffle an array, we can assign a random floating point number to every object, and sort based on those numbers. For information on generation of random numbers, see Fall 2014 61B.
Recommended Problems
C level
1. Problem 3 from my Fall 2014 midterm.
2. Why does Java's built-in Array.sort method use Quicksort for int, long, char, or other primitive arrays, but Mergesort for all Object arrays?
B level
1. My Fall 2013 midterm, problem 7, particularly part b.
A level
1. Find the optimal decision tree for playing puppy, cat, dog, walrus.
2. Given that Quick sort runs fastest if we can always somehow pick the median item as the pivot, why don't we use Quick select to find the median to optimize our pivot selection (as opposed to
using the leftmost item).
3. We can make Mergesort adaptive by providing an optimization for the case where the left subarray is all smaller than the right subarray. Describe how you'd implement this optimization, and give
the runtime of a merge operation for this special case. | {"url":"http://sp16.datastructur.es/materials/lectures/lec34/lec34.html","timestamp":"2024-11-01T19:50:52Z","content_type":"text/html","content_length":"6725","record_id":"<urn:uuid:e7365391-ac7f-4e56-9d57-89cc8962d5b1>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00587.warc.gz"} |
25 Times Table Chart Archives - Multiplication Table Chart
Make the most simplified learning of the 25 times table with our printable multiplication chart in the article. We are offering a convenient multiplication chart to all our aspiring table learners
with a view to guide them through their learning of the table. So, if you are a scholar with the table learning goal then you should definitely go ahead with the article. You would here find the
numbers of the sources to support the systematic learning of this times table 25.
Times tables have always been around us as an integral source of building the fundamental of mathematics. With the proper learning of the tables, you can simplify the mathematical calculations. This
is the reason that why the scholars are taught the times tables at the very beginning of their schooling. With the same concept, the timetable of 25 is significant as one of the significant tables of
25 Times Table
In the table of 25, we basically focus upon the numeric value of 25 to form the whole table. The value of 25 multiplies itself with the multiples of 1 to 10 from the whole sequence of the table.
Subsequently, we get to have the specific times table for our learning reference. This is the ultimate fundamental behind the formation of the table that will help you to learn it in an easier
manner. So, feel free to begin your learning of this table with our times’ table chart ahead in the article.
Multiplication Chart 25
We are aware of the fact that the multiplication chart is of utmost significance when it comes to learning the table of 25. This chart works just like the textbook of the table to make systematic
learning for the learners. So we highly recommend you this chart in order to commence the systematic learning of this particular table. You can use this chart whether you are an academic scholar or
an adult table learner. You can ideally use it both in your academic classes and in personal household learning as well without seeking any external support.
Multiplication Table 25
Well, the multiplication table is kind of an advanced table that comes in the category of 20’s tables. The table is quite significant for all the table learners as the understanding of it can
simplify the mathematical calculations. As an aspiring table learner, you can definitely focus your eyes on this table to raise the bar of your table learning. We highly believe that the learning of
this table would come as fruitful for your academics and the general routine life.
Printable 25 Times Table
If you are short of time to learn the table of 25 then going with the printable table chart is an ideal choice. We are saying it because it comes with some significant learning convenience. For
instance, it’s easily available with a single click without putting any effort. Secondly, it comes within its readily usable state which means you can quickly begin the learning of table 25 with it.
Furthermore, you can consider getting this table chart in both the digital and the traditional format. | {"url":"https://multiplicationchart.net/tag/25-times-table-chart/","timestamp":"2024-11-04T21:06:53Z","content_type":"text/html","content_length":"79998","record_id":"<urn:uuid:a5df281f-53dc-4014-aac9-71f86ea319ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00324.warc.gz"} |
Inquiry Maths - Inequalities
Mathematical inquiry processes: Verify; generate more examples; identify patterns and reason. Conceptual field of inquiry: Addition and inequalities.
This prompt was devised by Paul Foss, a mathematics teacher in Varndean School, Brighton (UK), as the start of a unit on place value with year 7 mixed attainment classes. Students notice that the
digits are in the same order in each inequality (and in the prompt the digits are consecutive, descending on the left and ascending on the right). Experienced inquirers have often made the following
conjecture: "If the add sign is in the same place on both sides, then the inequality sign is greater than. If the add sign is in a different place, the inequality sign is less than."
This proves to be false, but in the process of finding that out classes have produced a full list of all the possible inequalities using descending and ascending consecutive digits.
One advantage of the prompt is the obvious changes that can be made. Changing the operation to multiplication, for example, develops a new line of inquiry. Students who find 43 + 21 < 123 + 4, but 43
x 21 > 123 x 4 are often intrigued to explain why the inequality sign is reversed. Mixed attainment classes have taken this to a higher level by finding the sum of pairs of algebraic expressions.
Students can generalise for an inequality of the form 43 + 21 < 123 + 4 in the following way:
[10n + (n - 1)] + [10(n - 2) + (n - 3)] < [100(n - 3) + 10(n - 2) + (n - 1)] + n
where n is an integer, 4 ≤ n < 10.
Summing both sides of the inequality gives:
[10n + (n - 1)] + [10(n - 2) + (n - 3)] = 22n - 6
[100(n - 3) + 10(n - 2) + (n - 1)] + n = 112n - 6
Thus, the right hand side of the inequality is 90n greater than the left-hand side.
Higher attaining students take great delight in finding the product of these expressions by expanding the brackets.
Questions and observations
These are the questions and observations of year 7 and 8 mixed attainment classes. | {"url":"https://www.inquirymaths.com/home/number-prompts/inequalities","timestamp":"2024-11-01T23:43:38Z","content_type":"text/html","content_length":"162670","record_id":"<urn:uuid:5f47bc39-d7bc-42c5-94be-93f5f78b4189>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00262.warc.gz"} |
Robustness analysis with LTI dynamic uncertainties
The file Demo_002.m is found in IQClab’s folder demos. This demo performs a
1. The uncertainty block:
1. One LTI dynamic
2. An LTI dynamic scalar uncertainty that is repeated twice
2. Performance metric:
1. Induced
2. Robust stability test
The uncertain system is given by
On the other hand, the uncertainty block is defined by:
The demo file Demo_002.m allows to run an IQC-analysis for various values of for
• Length of the basis function: 3
• Solution check: ‘on’
• Enforce strictness of the LMIs:
% Define uncertain plant
M = ss([-2,-3;1,1],[1,0,1;0,0,0],[1,0;0,0;1,0],[1,-2,0;1,-1,0;0,1,0]);
% Define uncertainty block
de = iqcdelta('de','InputChannel',[1;2], 'OutputChannel'[1;2],'StaticDynamic','D',
% Assign IQC-multiplier to uncertainty block
de = iqcassign(de,'ultid','Length',3);
% Define performance block
pe = iqcdelta('pe','ChannelClass','P','InputChannel',3, 'OutputChannel',3,'PerfMetric','H2');
% Perform IQC-analysis
prob = iqcanalysis(M,{de,pe},'SolChk','on','eps',1e-8);
To continue, if running the IQC-analysis in Demo_002.m for
1. Induced
you obtain as output the worst-case induced wcgain) and the IQC-tools for different lengths of the basis function. This yields the results shown in the following figure. As can be seen, the
IQC-analysis produces worst-case induced
Worst-case induced | {"url":"https://www.iqclab.eu/robustness-analysis-with-lti-dynamic-uncertainties/","timestamp":"2024-11-02T03:00:56Z","content_type":"text/html","content_length":"111404","record_id":"<urn:uuid:867b31e8-f048-4a34-9571-9ee638b55104>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00650.warc.gz"} |
How do you solve (x-5)^2=4? | HIX Tutor
How do you solve # (x-5)^2=4#?
Answer 1
Solution: $x = 7 \mathmr{and} x = 3$
#(x-5)^2=4 or x-5 =sqrt4 or x-5 = +-2 or x=5+-2 :. x=7 or x=3#
Solution: #x=7 or x=3# [Ans]
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Answer 2
$x = 3 \mathmr{and} x = 7$
If #(x-5)^2=4# then #color(white)("XXX")x-5=+-sqrt(4)=+-2#
#{:("if " x-5=-2,color(white)("xxx"),"if "x=+2), ("then " x=3,,"then "x=7) :}#
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Answer 3
To solve the equation (x - 5)^2 = 4, you would take the square root of both sides and then solve for x.
(x - 5)^2 = 4 √(x - 5)^2 = ±√4 x - 5 = ±2 x = 5 ± 2
So, the solutions are x = 5 + 2 = 7 and x = 5 - 2 = 3.
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Answer 4
To solve ( (x-5)^2 = 4 ):
1. Take the square root of both sides of the equation.
2. Consider both the positive and negative square roots when taking the square root.
3. Solve for ( x ) by adding 5 to both sides of the equation and considering both positive and negative solutions.
The solutions are ( x = 5 + 2 ) and ( x = 5 - 2 ), which simplify to ( x = 7 ) and ( x = 3 ).
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
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• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-solve-x-5-2-4-3-8f9af98876","timestamp":"2024-11-10T11:11:38Z","content_type":"text/html","content_length":"585147","record_id":"<urn:uuid:c703cc6f-aae2-4e74-abf4-3c6a3d743434>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00079.warc.gz"} |
Excel Random Number Generator - AbsentData
Excel Random Number Generator
There may be many cases where you need to create a random number generator. Luckily, Excel has a few native functions that will help achieve your goals. If you want to learn more about Excel, Get
60% Off Online Excel Courses
When you type the RAND() function in a cell you will get a number that is a random integer from 0 to 1. In theory, this means that approximately 50 percent of the time you will get a number that is
less than 0.50 and the other 50 percent of the time you will get a number that is above 0.50. To get an idea of how this works lets take a look at the data below.
The RAND function takes no arguments to use it simply type =RAND()
Once you have this function it will randomly generate a number, simply press F9 key to generate a new random number. You can see from the image above that you only have to enter the formula to
generate a random number from 0 to 1.
However, you might want to extend the range of your randomly generator number. You can do this by using another equation in Excel that deals with random numbers with a custom defined ranges. This is
called RANDBETWEEN
The RANDBETWEEN(bottom, top) function is very similar to the RAND function. However, it takes a range which is set by a top and bottom number that is defined by the user. So, if you wanted to chose a
random number between 1 and 360, you would use the the following formula below
Video Resource
Create a random name generator in Excel with RANDBETWEEN Function.
So we have covered the random number generator. Now lets learn how to use what you have gained from the RANDBETWEEN function to generate names or any text by using the CHOOSE function. Using the
CHOOSE function allows you to set an index and value associated with at index. So, for example 1 = Bill, 2 = Steve, and so on. Combing these looks like the below.
Check Out: | {"url":"https://absentdata.com/random-number-generator-excel/","timestamp":"2024-11-11T17:06:08Z","content_type":"text/html","content_length":"155920","record_id":"<urn:uuid:2aaa2dbc-5171-4410-8a7b-18ca27821d41>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00525.warc.gz"} |
Pacific Northwest Probability Seminar
The Twenty Third Northwest Probability Seminar
November 4, 2023
Supported by the University of Oregon, Pacific Institute for the Mathematical Sciences (PIMS), Baidu, University of Washington (Friends of Mathematics Fund), and Milliman Fund.
Birnbaum Lecture in Probability
will be delivered by
Robin Pemantle
(University of Pennsylvania) in 2023.
Northwest Probability Seminars are mini-conferences held at the University of Washington and organized in collaboration with the Oregon State University, the University of British Columbia and the
University of Oregon. There is no registration fee.
The Scientific Committee for the NW Probability Seminar 2023 consists of Omer Angel (U British Columbia), Chris Burdzy (U Washington), Zhenqing Chen (U Washington), David Levin (U Oregon) and Axel
Saenz Rodríguez (Oregon State U).
Hotels near the University of Washington.
The talks will take place in Mary Gates Hall 241 (see the map).
Parking on UW campus is free on Saturdays only after noon. See parking information.
• 10:00 - 11:00 Coffee Mary Gates Commons
• 11:00 - 11:50 Stefan Steinerberger, University of Washington
Particle Growth Models in the Plane (DLA, DBM, ...)
We'll discuss growth patterns in the plane. The most famous such model is DLA where new particles arrive via a Brownian motion and get stuck once they hit an existing particle: it forms the
most beautiful fractal patterns (pictures will be provided). Despite this, DLA is actually fairly poorly understood and we will quickly survey the existing ideas (many of which are due to
Harry Kesten). I will then present a new type of growth model that behaves similarly (many more pictures will be shown) and which can be very precisely analyzed (in certain cases). No prior
knowledge is necessary, I'll explain everything from first principles.
• 12:00 - 12:50 Nick Marshall, Oregon State University
Random High-Dimensional Binary Vectors, Kernel Methods, and Hyperdimensional Computing
This talk explores the mathematics underlying hyperdimensional computing (HDC), a computing paradigm that employs high-dimensional binary vectors. In HDC, data is encoded by shifting and
combining random high-dimensional binary vectors in various ways. We study the problem of determining the optimal distribution of random high-dimensional binary vectors for use in this
• 1:00 - 3:00 Lunch, catered, Mary Gates Commons
• 3:00 - 3:50 Robin Pemantle, University of Pennsylvania
"Birnbaum Lecture": Negative association and related properties
Negative dependence properties of random variables have been valuable in proving limit theorems and tail bounds often substituting when independence fails. I will begin by reviewing the
theory and uses of negative dependence concepts for binary random variables, beginning with origins in mathematical statistics and statistical mechanics.
Among the many concepts and definitions that have been proposed, two stand out: negative association (NA) and the strong Rayleigh property (SR). The former is a negative dependence property
that is sometimes hard to prove but is very useful when it holds. The somewhat mysterious Strong Rayleigh property implies negative dependence and can in fact be a route to proving negative
The endpoint of this talk is to explore the limits of SA by looking at cases where NA holds or is expected to hold but SR does not. While this kills the hope of proving NA for these models by
establishing SR, it also helps us see what allows NA to hold without SR, which I hope will motivate and enable development of new technology for proving NA.
• 4:00 - 4:30 Coffee Mary Gates Commons
• 4:30 - 5:20 Lucas Teyssier, University of British Columbia
On the universality of fluctuations for the cover time
How long does it take for a random walk to cover all the vertices of a graph? And what is the structure of the uncovered set (the set of points not yet visited by the walk) close to the cover
time? We show that on vertex-transitive graphs of bounded degree, this set is decorrelated (it is close to a product measure) if and only if a simple geometric condition on the diameter of
the graph holds. In this case, the cover time has universal fluctuations: properly scaled, the cover time converges to a Gumbel distribution. To prove this result we rely on recent
breakthroughs in geometric group theory which give a quantitative form of Gromov's theorem on groups of polynomial growth. We also prove refined quantitative estimates showing that the
hitting time of any set of vertices is (irrespective of its geometry) approximately an exponential random variable.
This talk is based on joint work with Nathanaël Berestycki and Jonathan Hermon.
• 6:00 No-host (likely subsidized) dinner.
□ Restaurant: Mamma Melina. There will be set menu, the first menu on this list. The cost per person will be 36 dollars (this does not include tax and gratuity) although it is likely that the
conference will have funds to partly subsidize dinner. Please bring cash. There will be a vegetarian option. Wine and coffee can be ordered and paid for individually (cash only; alcohol will
not be subsidized). Address: 5101 25th Ave NE, Seattle, WA 98105. See Google map. The restaurant is a 20 minute walk from Mary Gates Hall. | {"url":"https://sites.math.washington.edu/~burdzy/nwprob2023.php","timestamp":"2024-11-05T23:05:32Z","content_type":"text/html","content_length":"12630","record_id":"<urn:uuid:b0b43b99-0ffe-4401-b638-d6ff5ecae055>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00131.warc.gz"} |
Kevin Lehrbass
Author Archives → Kevin Lehrbass
It’s not exactly the same as Wordle but close enough 🙂
Continue reading
I heard about 3D Objects from MrExcel. I’ve had fun combing a 3D model with VBA (it moves inside your spreadsheet!)
Continue reading
Earlier this week I saw an interesting rounding question on the famous MrExcel Forum. And there was a twist!
Continue reading
I played around with two different S curve concepts using dynamic arrays.
Continue reading
https://lichess.org/ has a Zen mode. Can I create a Zen mode in Excel to celebrate Spreadsheet Day?
I explored ideas to make DSUM function’s criteria argument as flexible as a gymnast. Is it a spreadsheet miracle? Maybe. Continue reading
You may never need this…but I did recently and it’s just fun knowing that Excel can solve this! Continue reading
I had some questions and improvement ideas for my formula pixel face. I learned a lot playing around with this. Continue reading
Daniel Choi solved it using Power Query! Let’s review his solutions. Continue reading
I need to calculate hours given data like this in each cell: “9-10, 10-11, 10-11, 12-13, 12-15+, 14-17+“ Continue reading | {"url":"https://myspreadsheetlab.com/author/kevin/","timestamp":"2024-11-02T13:45:31Z","content_type":"text/html","content_length":"69896","record_id":"<urn:uuid:361f00b1-222a-40cf-9f64-6695fb0da505>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00023.warc.gz"} |
Count missing values in Excel November 13, 2024 - Excel Office
Count missing values in Excel
This tutorial shows how to calculate Count missing values in Excel using the example below;
To count the values in one list that are missing from another list, you can use a formula based on the COUNTIF and SUMPRODUCT functions.
In the example shown, the formula in H6 is:
Which returns 1 since the value “Osborne” does not appear in B6:B11.
How this formula works
The COUNTIF functions checks values in a range against criteria. Often, only one criteria is supplied, but in this case we supply more than one criteria.
For range, we give COUNTIF the named range list1 (B6:B11), and for criteria, we provide the named range list2 (F6:F8).
Because we give COUNTIF more than one criteria, we get more than one result in a result array that looks like this: {2;1;0}
We want to count only values that are missing, which by definition have a count of zero, so we convert these values to TRUE and FALSE with the “=0” statement, which yields: {FALSE;FALSE;TRUE}
Then we force the TRUE FALSE values to 1s and 0s with the double-negative operator (–), which produces: {0;0;1}
Finally, we use SUMPRODUCT to add up the items in the array and return a total count of missing values.
Alternative with MATCH
If you prefer more literal formulas, you can use the formula below, based on MATCH, which literally counts values that are “missing” using the ISNA function: | {"url":"https://www.xlsoffice.com/others/count-missing-values-in-excel/","timestamp":"2024-11-13T21:31:24Z","content_type":"text/html","content_length":"62641","record_id":"<urn:uuid:c1d5b167-0e52-4a23-be66-35cc54585061>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00074.warc.gz"} |
alternating direction method of multipliers
The alternating direction of multipliers (ADMM) is a form of augmented Lagrangian algorithm that has experienced a renaissance in recent years due to its applicability to optimization problems
arising from “big data” and image processing applications, and the relative ease with which it may be implemented in parallel and distributed computational environments. This paper aims … Read more | {"url":"https://optimization-online.org/tag/alternating-direction-method-of-multipliers/page/8/","timestamp":"2024-11-05T14:12:30Z","content_type":"text/html","content_length":"97637","record_id":"<urn:uuid:a62abd96-04a2-4999-a2cc-74e723e98309>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00233.warc.gz"} |
NIACL AO Memory Based Paper 2024, Download Questions PDF
The NIACL AO Prelims Exam 2024, conducted on 13th October 2024, has now concluded in both shifts. Aspirants who appeared for the exam and those preparing for upcoming exams can benefit significantly
from the memory-based questions that are created after the exam. These questions provide valuable insights into the type and difficulty level of the questions asked, helping candidates prepare better
for future attempts.
NIACL AO Memory Based Paper 2024
Memory-based papers give a clear idea of how the exam was structured, including section-wise question distribution and commonly repeated topics. By practising these questions, candidates can sharpen
their preparation for the upcoming NIACL AO Mains Exam 2024.
Candidates looking to get a detailed analysis and the complete set of memory-based questions can download the PDF. These papers are compiled from the feedback of candidates who appeared in both
shifts of the exam. By reviewing these memory-based papers, candidates can improve their preparation strategies for future competitive exams.
NIACL AO Memory Based Paper 2024[Will Be Avilable Soon]
Some of the questions asked in the 13 October exam included the following:
Directions (1-2): In the questions given below, a sentence is divided into several parts. Select the most appropriate sequence to rearrange the parts into a grammatically correct and contextually
coherent sentence.
(A) large companies control most
(B) our country’s economy is a type of
(C) de facto monopoly in which a few
(D) of the wealth and power
(a) BACD
(b) CADB
(c) BCAD
(d) DCAB
(e) No rearrangement required
(A) retinue of teachers
(B) school population, the district
(C) is seeking to increase its
(D) to support the growing
(a) DBCA
(b) BADC
(c) DCBA
(d) BACD
(e) No rearrangement required
Directions (3): There are three sentences given in each question. Find the sentence(s) which is/are grammatically incorrect and mark your answer choosing the best possible option among the five
options given below each question. If all the sentences are incorrect, choose ‘None is correct’ as your answer.
(I) The government announced new policies to reduce the budget deficit, which had widened due to increased public spending and lower tax revenues.
(II) The scientist conducted multiple experiments to showing the effectiveness of the newly developed vaccine in preventing the spread of the disease.
(III) The rapid advancement in technology have significantly transformed various industries, influencing the way businesses operate.
(a) Only (I)
(b) Only (III)
(c) Both (I) and (III)
(d) Both (II) and (III)
(e) None is correct
Directions (4): There are three sentences given in each question. Find the sentence(s) which is/are grammatically and contextually correct and mark your answer choosing the best possible option among
the five options given below each question. If all the sentences are correct, choose ‘None is incorrect’ as your answer.
(I) The artist skillfully blended vibrant colors and intricate patterns to create a mesmerizing mural that captivated everyone who passed by.”
(II) Visiting the ancient ruins of Machu Picchu has always been one of my biggest dream.
(III) He promised to complete the project by the end of the week, assuring everyone that no further delays would occur.
(a) Only (I)
(b) Only (III)
(c) Both (I) and (III)
(d) Both (II) and (III)
(e) None is incorrect
Directions (5): In the question below, a sentence has been given, a part of which is highlighted. Each sentence is followed by options which are possible substitutes to the highlighted part. Choose
the option which acts as the correct substitute to make it correct grammatically and contextually.
Q5. Despite the challenging questions in the final round, none of the aspirant would give up easily, each determined to secure the coveted scholarship.
(a) no of the aspirants would
(b) none of the aspirants would
(c) none of the aspirants has
(d) none from the aspirants would
(e) No improvement needed
Directions (6-10): Read the following pie chart carefully and answer the questions given below. The pie chart shows the percentage distribution of total number of items sold by five shops.
Q6. Find the difference between the total items sold by A and E.
(a) 450
(b) 440
(c) 480
(d) 550
(e) 520
Q7. If total unsold items by A is 20% of the total items sold by C, then find the total items manufactured (sold and unsold) by A.
(a) 1485
(b) 1785
(c) 1945
(d) 1135
(e) 1255
Q8. The total items unsold by F is the average of the total items sold by D and E. If the total items manufactured (sold and unsold) by F is 25% more than the total items sold by A, then find the
difference between the total items sold by C and F.
(a) 1935
(b) 1185
(c) 1245
(d) 1375
(e) 1090
Q9. If 60% of the total items manufactured by B are sold, then find the total unsold items by B.
(a) 110
(b) 440
(c) 550
(d) 330
(e) 660
Q10. Find the ratio of the total items sold by C to the total items sold by A and B together.
(a) 2:1
(b) 1:2
(c) 1:1
(d) 1:4
(e) 4:1
Directions (11-15): Read the following table carefully and answer the questions given below. The table shows the total number of calculators and laptops sold together by five different companies. The
table also shows the total number of calculators sold by these five companies.
Companies Total number of calculators and laptops sold Total number of calculators sold
A 440 320
B 560 300
C 350 270
D 800 450
E 950 560
Q11. Find the difference between the total number of laptops sold by D and the total number of calculators sold by C.
(a) 40
(b) 90
(c) 60
(d) 50
(e) 80
Q12. The total number of laptops sold by A is what percentage more or less than the total number of calculators sold by B?
(a) 60%
(b) 45%
(c) 35%
(d) 30%
(e) 50%
Q13. Find the average number of laptops sold by all five companies.
(a) 180
(b) 360
(c) 120
(d) 240
(e) 200
Q14. If the price of each laptop and each calculator sold by B is Rs 15000 and Rs 200, respectively, then find the total revenue generated by B.
(a) Rs 3960000
(b) Rs 1285000
(c) Rs 3245000
(d) Rs 456500
(e) Rs 123450
Q15. Find the ratio of the total laptops sold by C to the total calculators sold by E.
(a) 1:9
(b) 1:7
(c) 2:3
(d) 4:9
(e) 4:1
Directions (16-20): Study the information carefully and answer the questions given below.
Eight persons P, Q, R, S, T, U, V, and W are living in a four-story building where the ground floor is numbered as 1, the floor above it is numbered as 2, and so on, until the topmost floor, which is
numbered as 4. Each floor has two flats, namely flat-1 and flat-2. Flat-1 of floor-2 is immediately above flat-1 of floor-1 and immediately below flat-1 of floor-3, and so on. Similarly, flat-2 of
floor-2 is immediately above flat-2 of floor-1 and immediately below flat-2 of floor-3. Flat-1 is situated to the west of flat-2.
There is a two-floor gap between Q and T, but they do not live in the same numbered flat. S lives on an odd-numbered floor in the flat to the east of T’s flat. W lives on an even-numbered floor in
the flat to the west of R. V lives above R but not in the same numbered flat, but there is a one-floor gap between them. U and V do not live in the same numbered floor.
Q16. Who among the following lives in flat-1 of the 4^th floor?
(a) P
(b) Q
(c) S
(d) V
(e) None of these
Q17. Who among the following is/are living with T in the same numbered flat?
(a) V
(b) S
(c) R
(d) W
(e) Both (a) and (d)
Q18. Which among the following combination is/are true?
(a) U – Flat 1
(b) W – Floor 4
(c) P – Flat 1
(d) S – Floor 2
(e) All are true
Q19. How many floors are there between U and S?
(a) Both are living on the same floor
(b) None
(c) Two
(d) One
(e) Can’t be determined
Q20. The number of floors gap between T and U is the same as the number of floors gap between ____ and _____.
(a) Q – V
(b) P – S
(c) W – R
(d) T – R
(e) None of these
Q21. In the word ‘REVOLT’, how many pairs of the letters have the same number of letters between them (both forward and backward direction) in the word as in the alphabet?
(a) Four
(b) Two
(c) One
(d) Three
(e) More than four
Q22. Find Odd one out?
(a) EGV
(b) DFW
(c) CXZ
(d) XZC
(e) LNO
Direction (23-25): Study the following information to answer the given questions:
Point G is 10m north of point A. Point B is 10m east of point A. Point C is 20m south of point B. Point D is 10m west of point C. Point E is 10m north of point D. Point F is 15m west of point E.
Point H is 15m north of point F. Point K is 5m east of point H.
Q23. In which direction is point K with respect to point G?
(a) South-west
(b) North-east
(c) North-west
(d) South-east
(e) None of these
Q24. If point Z is 10m east of point K, then what is the shortest distance between point A and point Z?
(a) 10m
(b) 5m
(c) 15m
(d) 20m
(e) None of these
Q25. If point J is exactly midpoint of BC then what is the shortest distance between point E and point J?
(a) 10m
(b) 5m
(c) 15m
(d) 20m
(e) None of these
Why Use NIACL AO Memory-Based Papers?
• Understanding the Exam Pattern: It helps candidates get familiar with the type of questions asked in different sections of the exam.
• Improving Accuracy and Speed: By solving these papers, candidates can enhance their speed and accuracy for the actual exam.
• Identifying Important Topics: Candidates can recognize recurring topics, which allows them to focus more on essential areas for the upcoming exams. | {"url":"https://www.bankersadda.com/niacl-ao-memory-based-paper-2024/","timestamp":"2024-11-13T12:03:01Z","content_type":"text/html","content_length":"618454","record_id":"<urn:uuid:556261dc-ed52-45a7-a8bf-1c5fcb24a40a>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00800.warc.gz"} |
Monochromatic sums and products | Published in Discrete Analysis
Arithmetic Combinatorics
February 28, 2016 BST
Monochromatic sums and products
Monochromatic sums and products, Discrete Analysis 2016:5, 48pp.
An old and still unsolved problem in Ramsey theory asks whether if the positive integers are coloured with finitely many colours, then there are positive integers $x$ and $y$ such that $x, y, x+y$
and $xy$ all have the same colour. In fact, it is not even known whether it is always possible to find $x$ and $y$ such that $x+y$ and $xy$ have the same colour.
This paper is about the corresponding question when $\mathbb{N}$ is replaced by a finite field $\mathbb{F}_p$, and gives a positive answer. More than that, it proves that a positive fraction of the
quadruples $(x,y,x+y,xy)$ are monochromatic.
The result is interesting for several reasons. One is that the standard tools of Ramsey theory appear to be hopelessly inadequate when they are applied to questions that mix addition and
multiplication, so the fact that the authors have obtained a positive result of this kind is surprising and may well have further ramifications. Another is that several people have tried, without
much success, to apply techniques from additive combinatorics to colouring problems. The techniques work well for many density problems, from which one can of course deduce colouring results. Until
now the challenge has been to get them to work for colouring problems when the corresponding density statements are false, as is the case here (since the set of numbers between $p/3$ and $2p/3$ does
not even contain a triple of the form $(x,y,x+y)$).
A third reason, which is almost implied by the previous two, is that the paper introduces some striking new techniques. One of these techniques is to “smooth” the colouring in a way that converts the
count of quadruples $(x,y,x+y,xy)$ into a count of purely linear configurations, thereby making the problem more amenable to conventional Ramsey-theoretic techniques. It also uses deep character sum
estimates from number theory.
For all these reasons, the paper will repay careful study by those who work in additive combinatorics.
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What is
What is a Cbind?
The cbind function – short for column bind – is a merge function that can be used to combine two data frames with the same number of multiple rows into a single data frame. While simple, cbind
addresses a fairly common issue with small datasets: missing or confusing variable names.
What does C bind do?
The cbind function is used to combine vectors, matrices and/or data frames by columns.
What is Cbind and Rbind?
cbind() and rbind() both create matrices by combining several vectors of the same length. cbind() combines vectors as columns, while rbind() combines them as rows.
What is Deparse level?
deparse.level. value determining the construction of labels (column labels for cbind or row labels for rbind). This applies only to unnamed vector (non-matrix) arguments.
Does Cbind match row names?
For cbind ( rbind ) the column (row) names are taken from the colnames ( rownames ) of the arguments if these are matrix-like. Otherwise from the names of the arguments or where those are not
supplied and deparse.
What is Deparse level in R?
The default value of deparse. level is 1.
What is unlist in R?
unlist() function in R Language is used to convert a list to vector. It simplifies to produce a vector by preserving all components.
What is Rbind in Rstudio?
Basic R Syntax: rbind(my_data, new_row) The name of the rbind R function stands for row-bind. The rbind function can be used to combine several vectors, matrices and/or data frames by rows.
What package is Rbind?
names(clabs, names(xi))”, if you try to use the rbind function for data frames with different columns. For that reason, the plyr package (be careful: it’s called plyr; not dplyr) provides the rbind.
How do I write a list in R?
How to Create Lists in R? We can use the list() function to create a list. Another way to create a list is to use the c() function. The c() function coerces elements into the same type, so, if there
is a list amongst the elements, then all elements are turned into components of a list.
Is there a dictionary in R?
Lists are the only key-value mapping type provided in base R: there are no dictionaries or associative arrays.
What is cbind and how do I use it?
One of the simplest ways to do this is with the cbind function. The cbind function – short for column bind – is a merge function that can be used to combine two data frames with the same number of
multiple rows into a single data frame. While simple, cbind addresses a fairly common issue with small datasets: missing or confusing variable names.
What is the advantage of cbind over rbind in R?
Like many r programming challenges, there is often more than one way to get things done. The advantage of the cbind r function is that it can handle r appends very efficiently; this is a big
advantage if you’re iterating across a lot of data. You can also perform similar operations on rows with rbind (for mental consistency, at least).
How are column names in cbind (rbind) calculated?
For cbind (rbind) the column (row) names are taken from the colnames (rownames) of the arguments if these are matrix-like.
What is the difference between cbind and rbind in MATLAB?
For cbind row names are taken from the first argument with appropriate names: rownames for a matrix, or names for a vector of length the number of rows of the result. For rbind column names are taken
from the first argument with appropriate names: colnames for a matrix, or names for a vector of length the number of columns of the result. | {"url":"https://www.yemialadeworld.com/what-is-a-cbind/","timestamp":"2024-11-13T16:27:09Z","content_type":"text/html","content_length":"70364","record_id":"<urn:uuid:592a8251-7579-42e6-8ccf-699b5068677d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00231.warc.gz"} |
Cross Sections of Solids
The cross-section parallel to the base is a
The cross-section parallel to the base is a
Square Pyramid
The cross-section parallel to the base is a
The cross-section parallel to the base is a
The cross-section through the vertex perpendicular
to the base is a
Square Pyramid
The cross-section perpendicular to the base but not through the vertex is a
Square Pyramid
The cross-section perpendicular to the base is a
The cross-section perpendicular to the base is a
The cross-section perpendicular to the base is a
Oval (ellipse)
Diagonal cross section
of a cylinder.
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Python Modulo in Practice: How to Use the % Operator?
In Python, there are various arithmetic operators which operate on Python numeric data types. Also, one of the important Python Arithmetic operators that you may find useful is modulo, represented by
the percentage symbol (%). Like other Python arithmetic operators, modulo operator (%) operates between two numbers. It divides one number by another and returns the remainder. In this Python
tutorial, we have covered the different aspects of the Python modulo operator. By the end of this tutorial, you will have a complete idea about the Python modulo operator and how it works.
Modulo Operator in Mathematics
The term modulo in mathematics deals with modular arithmetic. In general, modular arithmetic defines a fixed set of numbers that follow a circular approach. Also, all the arithmetic operations on the
modular return a result from the fixed set of numbers called modulus. The twelve-hour clock is a classic example of modular arithmetic. It does not matter if we add or reduce time in the twelve-hour
clock. The time will always remain between 1 and 12 hours. The twelve-hour clock can be divided into 12 modulo and represented as “mod 12”. Suppose we want to evaluate the time of the 24-hour clock
in terms of the 12-hour clock, and we can do this with the help of “mod 12.”
Example: convert the 15 o’clock of the 24-hour clock into the 12-hour clock. 15 o’clock mod 12 o’clock = 3 o’clock
Python Modulo Operator
The Modulo operator is an arithmetic operator, and like other arithmetic operators, it operates between two Python numbers that can be of one of the two data types: integer and float.
Python Modulo Operator with Python Integers
Most of the time, you will be using the modulo operator with Python integer data type. If we use the Python modulo operator (%) between two integer values a and b, the modulo operator will return the
remainder by dividing a by b. The return output will also be an integer value.
>>> a = 20
>>> b = 3
>>> a % b
>>> b % a
ZeroDivision Error: The modulo operator first performs the division operation (/) between the numbers and then returns the remainder. If the second number is 0, the operator returns the
>>> a = 20
>>> b =0
>>> a % b
Traceback (most recent call last):
ZeroDivisionError: integer division or modulo by zero
Python Modulo Operator with Python Floating Numbers
If we perform the modulo operator between two floating-point numbers, it will return the division's reminder. The return remainder value will also be a floating-point number .
>>> a= 20.3
>>> b = 3.2
>>> a % b
>>> b % a
Python math.fmod() method
The fmod() method is an alternative to the float with modulo operator(%). The fmod(a,b) method accepts two arguments and returns the remainder of their division. Regardless of the numbers data type,
the fmod() method returns a floating-point number.
>>> import math
>>> math.fmod(20, 3) #equivalent to 20.0 % 3.0
>>> math.fmod(20.2, 3.3)
Python modulo operator with negative numbers
By far, we have only performed the modulo operator on positive numbers, and their outputs are also positive. But if we use the modulo operator between negative numbers, things get tricky. If either
of the numbers is negative, the modulo operator uses the following formula to calculate the remainder:
r = dividend - ( divisor * floor (dividend / divisor))
23 % -3
r = 23 - (-3 * floor(8 / -3))
r = 23 - (-3 * -8)
r= 23 - (24)
r = -1
>>> dividend = 23
>>> divisor = -3
>>> r = dividend % divisor
>>> r
Python math.fmod() with negative numbers
As we discussed above, the Python fmod() method is an alternative for Python float with a modulo operator. But this is only true for positive numbers. If we pass a negative number in fmod() method,
it will not give the same result as the modulo operator % . If the passed arguments numbers are negative, the output will be different. The math.fmod() use the following formula to output the
remainder of a number:
r = dividend - ( divisor * int (dividend/ divisor))
math.mod(23.0, -3.0)
r = 23.0 -( -3.0 * int(23.0/ -3.0 ))
r = 23.0 -( -3.0 * -7)
r = 23.0 – 21.0
r = 2
>>> dividend = 23
>>> divisor = -3
>>> dividend % divisor
>>> math.fmod( 23, -3)
Python divmod() method
Python provides an inbuilt-method divmod() which returns a tuple containing the floor division and reminder.
>>> 35 // 7
>>> 35 % 7
>>> divmod(35, 7)
(5, 0)
In this tutorial, you learned about the Python modulo operator. To represent the modulo operator, we use the % symbol. The modulo operator operates between two numbers and returns the remainder of
their division. Also, there are two built-in Python methods, namely math.fmod() and divmod() that you can use to find the remainder between two numbers.
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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.TQC.2020.10
URN: urn:nbn:de:0030-drops-120692
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12069/
Arunachalam, Srinivasan ; Belovs, Aleksandrs ; Childs, Andrew M. ; Kothari, Robin ; Rosmanis, Ansis ; de Wolf, Ronald
Quantum Coupon Collector
We study how efficiently a k-element set S⊆[n] can be learned from a uniform superposition |S> of its elements. One can think of |S>=∑_{i∈S}|i>/√|S| as the quantum version of a uniformly random
sample over S, as in the classical analysis of the "coupon collector problem." We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In
particular, if there are n-k=O(1) missing elements then O(k) copies of |S> suffice, in contrast to the Θ(k log k) random samples needed by a classical coupon collector. On the other hand, if n-k=Ω
(k), then Ω(k log k) quantum samples are necessary.
More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can
additionally reflect through |S>. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.
BibTeX - Entry
author = {Srinivasan Arunachalam and Aleksandrs Belovs and Andrew M. Childs and Robin Kothari and Ansis Rosmanis and Ronald de Wolf},
title = {{Quantum Coupon Collector}},
booktitle = {15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)},
pages = {10:1--10:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-146-7},
ISSN = {1868-8969},
year = {2020},
volume = {158},
editor = {Steven T. Flammia},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12069},
URN = {urn:nbn:de:0030-drops-120692},
doi = {10.4230/LIPIcs.TQC.2020.10},
annote = {Keywords: Quantum algorithms, Adversary method, Coupon collector, Quantum learning theory}
Keywords: Quantum algorithms, Adversary method, Coupon collector, Quantum learning theory
Collection: 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2020)
Issue Date: 2020
Date of publication: 08.06.2020
DROPS-Home | Fulltext Search | Imprint | Privacy | {"url":"http://dagstuhl.sunsite.rwth-aachen.de/opus/frontdoor.php?source_opus=12069","timestamp":"2024-11-09T17:12:45Z","content_type":"text/html","content_length":"7613","record_id":"<urn:uuid:87f6e42b-f2ae-4432-adae-bf58c1ec8eef>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00042.warc.gz"} |
Gradient Descent Using Python - Try Machine Learning
Gradient Descent Using Python
Gradient descent is a popular optimization algorithm used in machine learning and deep learning. It is used to minimize the cost function by iteratively adjusting the parameters of a model in the
direction of steepest descent. In this article, we will explore the concept of gradient descent and how to implement it using Python.
Key Takeaways
Here are the main points covered in this article:
• Gradient descent is an optimization algorithm used to minimize a cost function.
• It is commonly used in machine learning and deep learning algorithms.
• Python provides powerful libraries, such as NumPy and matplotlib, for implementing gradient descent.
• There are different types of gradient descent algorithms, including batch, stochastic, and mini-batch gradient descent.
• Learning rate is a crucial hyperparameter in gradient descent that determines the step size for parameter updates.
Gradient descent works by iteratively adjusting the parameters of a model in the direction of steepest descent. It calculates the gradients of the cost function with respect to each parameter and
updates the parameters accordingly. The process repeats until the algorithm converges to a minimum of the cost function.
Implementing gradient descent in Python requires understanding the calculus concepts of derivatives and partial derivatives.
Let’s go through a step-by-step implementation of gradient descent using Python. We’ll start by defining the cost function, which is a measure of how well our model performs.
Once we have the cost function, we need to calculate the gradients of the cost function with respect to each parameter. These gradients tell us the direction in which we should update the parameters
to minimize the cost function. The learning rate determines the step size of the parameter updates.
Table 1: Gradient Descent Algorithm Steps
1. Initialize parameters randomly
2. Calculate the cost function
3. Calculate the gradients of the cost function with respect to each parameter
4. Update the parameters using the gradients and the learning rate
5. Repeat steps 2-4 until convergence
By updating the parameters in the opposite direction of the gradients, we move closer to the minimum of the cost function.
There are different variants of gradient descent algorithms that differ in how the training examples are used to calculate the gradients and update the parameters. Batch gradient descent calculates
the gradients using the entire training set, whereas stochastic gradient descent calculates the gradients using one training example at a time.
Mini-batch gradient descent is a compromise between batch and stochastic gradient descent, where the gradients are calculated using a small subset of the training examples.
Table 2: Types of Gradient Descent Algorithms Pros Cons
Batch Gradient Descent Guaranteed convergence to the global minimum Computationally expensive for large datasets
Stochastic Gradient Descent Works well with large datasets May converge to a local minimum
Mini-Batch Gradient Descent Efficient and less noisy than stochastic gradient descent Requires tuning of batch size
The choice of learning rate is critical in gradient descent. A too-large learning rate may cause the algorithm to overshoot the minimum of the cost function and fail to converge, while a too-small
learning rate may result in slow convergence.
Learning rate decay, where the learning rate decreases over time, can help balance convergence speed and stability.
Table 3: Learning Rate Strategies Pros Cons
Fixed Learning Rate Simple and easy to implement May not converge for complex problems
Learning Rate Decay Can achieve faster convergence Requires tuning of decay rate
Adaptive Learning Rate Can automatically adjust the learning rate Complex implementation
In conclusion, gradient descent is a powerful optimization algorithm for minimizing cost functions in machine learning and deep learning. By implementing gradient descent in Python using libraries
like NumPy and matplotlib, we can train models to make accurate predictions. Understanding the different types of gradient descent algorithms and the impact of learning rate is crucial for successful
Common Misconceptions
Misconception 1: Gradient Descent is Only Used for Linear Regression
One common misconception about gradient descent is that it is only used for linear regression problems. However, gradient descent can be used for a wide range of optimization problems, not just
linear regression. It is a popular and widely used algorithm in machine learning and optimization tasks.
• Gradient descent can be used for logistic regression problems.
• It is also applicable to deep learning networks with multiple layers.
• Gradient descent can be employed for training support vector machines.
Misconception 2: Only Python Experts Can Implement Gradient Descent
Another misconception is that only experts in Python programming can implement gradient descent. While proficiency in Python can be helpful, anyone with basic programming knowledge and understanding
of the algorithm can implement gradient descent. There are many resources available online that provide step-by-step guides and code examples for beginners.
• Basic knowledge of Python syntax is sufficient to implement gradient descent.
• Online tutorials and guides can assist beginners in understanding and implementing the algorithm.
• Learning the basics of calculus and linear algebra can enhance understanding of gradient descent.
Misconception 3: Gradient Descent Always Finds the Global Minimum
It is often assumed that gradient descent will always find the global minimum of the cost function. However, this is not always the case. Gradient descent can converge to a local minimum, which may
not be the optimal solution for the problem at hand. Multiple runs with different initializations or modifications to the algorithm may be required to improve the solution.
• Global minimum can be reached when the cost function is convex.
• Local minima can be encountered in non-convex cost functions.
• Different optimization techniques like simulated annealing can be employed to handle non-convex problems.
Understanding Gradient Descent
Gradient descent is a popular optimization algorithm used in machine learning and deep learning to find the optimal solution to a problem. It iteratively adjusts the parameters of a model in the
direction of steepest descent to minimize a specified loss function. Here, we present ten tables that provide additional insights into the process of implementing gradient descent using Python.
Table: Initial Dataset
This table showcases a sample dataset representing housing prices and their corresponding areas in square feet.
Price ($) Area (sqft)
400,000 2,000
450,000 2,300
500,000 2,500
Table: Gradient Descent Algorithm
This table outlines the steps involved in implementing the gradient descent algorithm.
Step Description
1 Initialize model parameters
2 Calculate predicted values
3 Calculate loss function
4 Update model parameters
5 Repeat until convergence
Table: Learning Rate Optimization
In this table, we experiment with different learning rate values to observe the impact on convergence speed and accuracy.
Learning Rate Convergence Speed Accuracy
0.1 Slow Low
0.01 Medium Medium
0.001 Fast High
Table: Epochs and Loss
By tracking the loss function after each epoch, this table illustrates the gradual reduction in loss as the number of training iterations increases.
Epoch Loss
1 120,000
2 90,000
3 75,000
4 62,000
5 50,000
Table: Regularization Techniques
This table presents different regularization techniques used to prevent overfitting during the gradient descent process.
Technique Description
L1 Regularization Shrinks less important features toward zero
L2 Regularization Penalizes large weight values
Elastic Net Regularization Combines L1 and L2 regularization
Table: Gradient Descent Variations
This table compares different variations of gradient descent, such as stochastic gradient descent (SGD) and mini-batch gradient descent (MBGD).
Variation Description
Stochastic Gradient Descent (SGD) Computes gradients and updates parameters for each training example
Mini-Batch Gradient Descent (MBGD) Performs gradient update on a subset of training examples
Batch Gradient Descent (BGD) Computes gradients and updates parameters for the entire training set
Table: Convergence Criteria
In this table, we explore different convergence criteria that can be used to terminate the gradient descent algorithm.
Convergence Criteria Description
Error Threshold Stop when the loss is below a certain value
Maximum Iterations Terminate after a specified number of iterations
Change in Loss Stop when the change in loss becomes negligible
Table: Feature Scaling Techniques
This table showcases different techniques used to scale the input features to ensure efficient convergence during gradient descent.
Technique Description
Standardization Transforms features to have zero mean and unit variance
Normalization Scales features to have values between 0 and 1
Min-Max Scaling Shifts and scales features to a specified range
Gradient descent, implemented using Python, provides a powerful framework for optimizing machine learning models. Through our exploration of various elements such as initial datasets, learning rate
optimization, convergence criteria, regularization techniques, and others, we have gained a deeper understanding of the intricacies associated with this algorithm. By leveraging gradient descent, we
can effectively train models with large datasets and complex features, achieving accurate and efficient predictions.
Frequently Asked Questions
What is gradient descent and why is it used in machine learning?
Gradient descent is an optimization algorithm used in machine learning to minimize a function by iteratively adjusting the parameters. It is based on the idea of calculating the gradient (slope) of
the function at a given point and taking small steps in the direction that leads to the steepest descent. By repeatedly updating the parameters, gradient descent allows the algorithm to find the
minimum of the function, which is crucial for optimizing machine learning models.
How does gradient descent work?
Gradient descent works by iteratively computing the gradient of a function with respect to its parameters and updating the parameters in the direction opposite to the gradient. This process continues
until a minimum is reached or a termination criterion is met. The size of the steps taken, known as the learning rate, determines how quickly the algorithm converges to the minimum.
What is the difference between batch gradient descent and stochastic gradient descent?
Batch gradient descent computes the gradient of the loss function over the entire training dataset at each iteration. It calculates the average gradient over all training examples, making it more
accurate but computationally expensive for large datasets. On the other hand, stochastic gradient descent randomly selects one training example at a time and calculates the gradient based on that
single example. It is faster but more prone to noise in the gradient estimate.
What is mini-batch gradient descent?
Mini-batch gradient descent is a compromise between batch gradient descent and stochastic gradient descent. Instead of processing the entire dataset or just a single example, mini-batch gradient
descent processes a small batch of training examples at each iteration. This provides a balance between accuracy and computational efficiency, as it takes advantage of vectorized operations while
reducing the noise in the gradient estimate.
How do I choose the learning rate in gradient descent?
Choosing an appropriate learning rate in gradient descent is crucial for the convergence and performance of the algorithm. A learning rate that is too small can cause slow convergence, while a
learning rate that is too large can lead to overshooting the minimum or even divergence. It is common practice to try different learning rates and monitor the loss function during training to find
the optimal value. Techniques like learning rate decay and adaptive learning rates can also be used to improve convergence.
What are the common challenges in gradient descent?
One common challenge in gradient descent is getting stuck in local minima, where the algorithm converges to a suboptimal solution. To mitigate this issue, techniques like initialization of
parameters, using different optimization algorithms, or introducing regularization can be employed. Another challenge is dealing with large-scale datasets, as batch gradient descent becomes
computationally expensive. This can be addressed by using variants of gradient descent like mini-batch or stochastic gradient descent.
How can I visualize the convergence of gradient descent?
To visualize the convergence of gradient descent, you can plot the value of the loss function or the changes in parameter values over iterations. This can provide insights into whether the algorithm
is converging, diverging, or getting stuck. Additionally, plotting the gradients or learning rates over iterations can help identify any unusual behavior. There are various Python libraries, such as
Matplotlib and Seaborn, that can assist in creating visualizations.
Can gradient descent be used for non-convex optimization?
Yes, gradient descent can be used for non-convex optimization problems. While it is primarily known for optimizing convex functions, it can still be applied to non-convex functions. However, due to
the complex nature of non-convex optimization, gradient descent may suffer from convergence to local minima or plateaus. In such cases, more advanced techniques like adaptive algorithms or
higher-order optimization methods may be considered.
What are the limitations of gradient descent?
Gradient descent has a few limitations. Firstly, it can get stuck in local minima, especially in non-convex optimization problems. Secondly, it requires the function being optimized to be
differentiable. If the function has discontinuities or non-differentiable points, gradient descent may not work well. Finally, it can be sensitive to the choice of learning rate, where a suboptimal
value can lead to slow convergence. Awareness of these limitations is important when using gradient descent.
How does Python help in implementing gradient descent?
Python is a popular programming language for implementing gradient descent due to its simplicity, versatility, and rich ecosystem of machine learning libraries. Libraries like NumPy and SciPy provide
efficient numerical computations, while frameworks like TensorFlow and PyTorch offer high-level APIs for building and training machine learning models. Python also has powerful visualization
libraries like Matplotlib, which can aid in analyzing and visualizing the results of gradient descent. | {"url":"https://trymachinelearning.com/gradient-descent-using-python/","timestamp":"2024-11-10T01:45:20Z","content_type":"text/html","content_length":"70049","record_id":"<urn:uuid:f01ec73c-1b56-4eff-87a1-9b184355ebe5>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00012.warc.gz"} |
How does temperature affect non-linear dynamic behavior? | SolidWorks Assignment Help
How does temperature affect non-linear dynamic behavior? Here I want to know how temperature affects the dynamic behavior of gas and liquid systems. If the system can move freely moving in and out,
then when the temperature change reaches a value for which it can move freely and/or the time constant over which it can remain freely is variable and when the temperature is increased in the range
of positive and negative parameters for that constant which are negative values then the equation of motion becomes nonlinear; in this case, the system returns to its starting temperature by changing
the magnitude of temperature, the quantity of material and the capacity of the liquid to move. If a change of the temperature is taken into account the equation of motion becomes linear and the
linearity (i.e. velocity) of response is equal to the load of the load in the reservoir, but if the temperature is taken into account then the load can move only through one direction such that it
can return to its starting temperature. It should also be borne in mind that if one is in an equilibrium (time constant constant) the system will return to its starting temperature by changing the
quantity of constant (length) of the material over which the load is moved, but say if the constant is greater than one the load will reach its equilibrium location. It should also be borne in mind
that if the quantity of constant changes with time, the following equation may be appropriate, 1C3=C, where time unit is unitless; C=k1/2; k=1/k1; 1=C.1, 0=C.1, C.2, C.3; C3 is a very simple linear
combination of C, whose solutions may be applied repeatedly, but then is not the equilibrium value of velocity of equilibrium. I’m a little confused as to how this method works and/or how it’s
applied to such systems, but I would really enjoy learning here when it is given context and if you need any further information. A: The linearity of response is almost exactly linear in the variable
$j$. Even if the quantity of the material is not completely independent one may deduce the linearity of the pressure acting on the mixture without dealing with the physical quantities that are in
question. The response will then be the sum of the product $\sum\limits_{j=1}^{k_0}C_j$ $\sum\limits_{j=1}^{k_1}C_j$ $\sum\limits_{j=1}^{k_2}C_j$ If you want your system to respond to a change of
variable without any increase or decrease the temperature will almost certainly have to decrease (change of temperature by changing the temperature). With the linearity of response given by equation
1, the rate of change itself will be proportional to the quantity $C_j $. So the same can be done without regard for change of velocity instead. How does temperature affect non-linear dynamic
behavior? A real piecewise constant temperature is necessary and necessary. But thermal conduction occurs at high temperature and it brings important temperature corrections to the observed
non-linear behavior of the continuous layer on the square grid. Some approaches can be used to constrain these temperatures, providing one way to characterize the behavior of the layer at high
temperature (at non-zero temperature), but the linear response limit for the temperature dependences is very different and demands a new approach that can address the numerical problems encountered
during this time step of the program and allow us to increase our understanding of the nature of the dynamic systems.
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For any non-linear domain, its domain should be monotone and it should not shift. This kind of non-linear dynamic behavior is expected to be present at the line-of-contrasts that are made by
localizing the line sections in half-wave plates such as the grid-type. The physical problem, at the boundary, then becomes: how (first) will the line contain this non-linearity at thermal
equilibrium? This question was recently addressed by D. C. Borchers, in a book, “Domain-Specific Effects in Waves and Static Waves: Theoretical Approach to the Exact Problems.” The purpose of this
book is to convey the picture of the phenomena of density dependent non-linearity on a box-controlled domain, and describe the most general situation where the physical property at a temperature lies
in the plane. In this book the term density specific is a synonym for density evolution, but also a term with the meaning of density-dependent non-linearity. This way of describing density
dependences (see e.g., Robert Spangler’s book) was not the main purpose of the book. The specific form of this term was an integral for a time-dependent problem, but this was a problem of some extent
later, and can be translated into other physical problems. The book was only limited in scope to a variety of non-linear systems, but eventually the new type of study was developed to address these
and other problems. The next step is to explore how different “equidistur and variable” responses of a non-linear growth system are related to the material response of the scale. As for the present
book the physical property at a temperature dependence that is to vary a given way in non-linear regimes has previously been studied in terms of the non-linear terms included in the current book, but
in the last few years the new approach has been used to define a generic condition that makes sense at any temperature ranging from zero to one. The results are used as guide for what is known about
the relationship between the numerical technique used and the real domain used for the simulation (see Lee He, “The First Principles in Nonlinear Turbulence.” Proc. of Royal Institute of
Technology-AIM-AO-2017, No. 539, 2016). For example, in relation to the mechanical properties of porous materials, we could suppose a constant pressure of a one hour-cast and we could put a constant
molecular temperature below 100 degrees Celsius, i.e.
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we could take an ordinary pressure of this different molecular type. But while this would mean that the structure of the material would generally not have any obvious dependence on heating (see Lee
He, “Doing a Polymer Model with Reversible Temperature Effects: A Theoretical Study.” Proc. of Royal This Site of science-AIM-AO-1757, No. 941, 2015), we do have an assumption that will, in fact,
generate a structural modification to the material. The influence of these changes is now discussed, in light of recent work on the phase transitions of solids with permanent phase separation to
understand the physical state under study in molecular gases. The physical effect of temperature in a sample canHow does temperature affect non-linear dynamic behavior? I’m considering thermal
effects of light by analyzing different conditions needed for the measurement. On the one hand, this mechanism behaves linear, but I can have a situation where the non-linear effects are non-linear
in the light scattered conditions. On the other hand, I usually focus on non-linear changes at different time points. I have no idea on how to even know about the non-linearity in each part of the
time slice. I don’t understand the application of the ICA during TEM measurement in which you have to rotate the whole substrate at exactly the same speed (even with some restrictions), however the
TEM observation of a single crystal surface for a few seconds suggests the speed of rotation of the substrate as can be understood from an analogy with a two-dimensional Fourier transform (which
takes two time scales). So, it may seem as though the light in the substrate always moves to a different location relative to the TEM observation in a short time interval. But, we have the same
effect of moving from one point to the other. This means what I’m asking for is in fact if you consider four points (0.5, 2.5, 3.5 and 4.5) between the TEM observation during a long TEM measurement
time and three or four time points between the TEM observation during a long TEM measurement time, a similar but non-linear TEM rotation in the three points would be required, because later on you
would need to ensure that the light bouncing from the first measurement to the second measurement stays all the way behind the irradiation of the second measurement and never after that to be all the
way to the first one. I’m not sure I understand this being the case, but the only feasible implementation is taking everything as a new measurement, assuming the radiation is absorbed by the
crystalline material. We can then assume that all we need to do is to add a large amount of light (or as much as possible) for a certain set of measurement conditions to the light transmitted from
one measurement point onto another measurement point.
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But that’s both hard and time consuming. I understand you need to measure several light measurements during each time point. But, even with no light measurement i have what you’re looking for when
you take this to be you measure every measurement point (which i do). In this case the time interval between the positions of the light sources are a couple of seconds, but the light does not change.
In a sense I can consider those 3 measurements being all the way from one measurement point to some. Maybe it was the first time, or they are different measurement points and measurement methods. But
again you shouldn’t measure multiple times the same measurement. | {"url":"https://solidworksaid.com/how-does-temperature-affect-non-linear-dynamic-behavior-29787","timestamp":"2024-11-02T05:28:05Z","content_type":"text/html","content_length":"158148","record_id":"<urn:uuid:6732a12b-c8d7-4035-b257-c7a9a5726c12>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00755.warc.gz"} |
nForum - Discussion Feed (vacuum amplitude)Urs comments on "vacuum amplitude" (49527)David_Corfield comments on "vacuum amplitude" (49521)Urs comments on "vacuum amplitude" (49520)David_Corfield comments on "vacuum amplitude" (49517)Urs comments on "vacuum amplitude" (49515)Urs comments on "vacuum amplitude" (49514)
Thanks, David, for all these pointers.
I did now look at ACER11 in a bit more detail, the statement there (the other references you list revolve around the same statement) is that by the Rankin-Selberg-Zagier method the partition function
of a (super-)string is usefully expanded for small proper time as a constant (the vacuum energy) plus a sum of decaying oscillatory terms, one for each non-trivial zero of the Riemann zeta function.
By estimates on the decay strength this implies certain asymptotic vanishing results, too.
This is nice (I made a note at Riemann hypothesis and physics) but it is not yet quite what I was hoping for. I was thinking that since the vacuum amplitude itself is like a zeta function, there
should be a “generalized Riemann hypothesis” applying to it directly, and describe its vanishing directly.
Maybe I am off here, or maybe it just hasn’t materialized. But also this is not going to be my high priority for the moment, it was just an thought that occured to me while fine-tuning the table of | {"url":"https://nforum.ncatlab.org/search/?PostBackAction=Search&Type=Comments&Page=1&Feed=ATOM&DiscussionID=6211&FeedTitle=Discussion+Feed+%28vacuum+amplitude%29","timestamp":"2024-11-14T21:59:33Z","content_type":"application/atom+xml","content_length":"9641","record_id":"<urn:uuid:38eb266a-3c2a-4a81-b5eb-48d107927b65>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00794.warc.gz"} |
Mathematical description of protein extraction from muscle tissue of hydrobionts and determination of the effective molecular diffusion coefficient
Among many typical processes of chemical technologies, protein extraction from raw materials of animal origin has attracted the least attention. In the food industry, complex technologies of raw
material processing include processes of protein extraction from agricultural raw materials and wastes of its processing. In fish farming in aquaculture, there is a problem of disposal of fish
processing waste, the solution of which allows to reduce the burden on the environment and obtain valuable proteins, lipids, and biologically active substances. The processes of nutrient extraction
from raw materials of plant origin have been investigated in the most detail. Studies on the mathematical description of extracting sugar, vegetable oils (Tyulkova et al., 2013), and biologically
active compounds from raw materials were reported (Shishatskii et al., 2015; Xie Y. et al., 2020). The values of effective molecular diffusion coefficients, effective rate constants, and activation
energy were determined in the above-mentioned processes (Nagai et al., 2000; Jafari et al., 2020).
It is of paramount importance to define the value of the molecular diffusion coefficient as it allows us to calculate parameters for many technological processes and equipment in theory and practice.
Protein extraction from hydrobiont muscle tissue fundamentally differs from protein extraction from plant tissue. It is due to the presence of polysaccharides in the cell wall of plant tissue, which
significantly slows down the extraction of proteins. Whereas in the structure of animal tissue of hydrobionts, collagen fibers are easily dissolved in the extractant, which facilitates the diffusion
process. Swelling and breaking of chemical bonds were found to accompany the protein extraction process (Ciftci et al., 2012). Protein-containing raw materials are known to have a cellular porous
structure. Pore diffusion is the rate-limiting step when extracting different substances (Kafarov et al., 2019). Diffusion of the extractant into the pores of the tissue to swelling of the raw
material particles, which somewhat complicates the mathematical description of the extraction process. Therefore, the mass transfer process was determined by the swelling and presence of a chemical
reaction between a proteinaceous material and extractant rather than by directly altering cell membrane permeability and destruction (Nagornov et al., 2015).
There is a lack of data that enables the mathematically describing and modeling of protein extraction processes from animal tissues in the technologies of obtaining protein hydrolysates, isolates,
and enzymes.
This work’s novelty is to study the extraction of proteins from animal tissue, its mathematical description, solving the inverse problem of the modeling and determining the molecular diffusion
coefficient of proteins from raw materials of animal origin. Researchers can use the obtained molecular diffusion coefficient to calculate the parameters of the diffusion process of proteins during
its extraction from dispersed particles of hydrobionts.
Therefore, the aim of this study was to obtain an array of experimental data on extracting protein from the particles of dispersed muscle tissue of hydrobionts (Alaska pollack) to solve the inverse
problem – description of unknown parameters in the system of equations describing the process under investigation, based on the obtained kinetic dependence of the extraction process.
The purpose of this work is to obtain an array of experimental data, kinetic dependences of the protein extracting from the particles of dispersed muscle tissue of hydrobionts (Alaska Pollack), its
mathematical description and to find the molecular diffusion coefficients of proteins in the system of equations by means of solving the inverse problem of the modeling.
Materials and Methods
The muscle tissue of Alaska pollack was used as a model. It was dispersed by typical equipment up to the specific particle size: (5.0±2)×10^−3 m. Then, the shredded tissue was placed in a water-salt
medium with a salt concentration of 1% (to ensure electrical conductivity) to be processed in the cathode chamber of an electrolyzer. The mode was defined according to the parameters that provide the
highest speed and maximum depth of dissolution.
The raw material was mixed with an electrolyte solution with specified water (ratio of 1:6 to ensure the fluidity of the suspension). Afterward, the mixture was processed through an electrolyzer,
heated and thermostated up to the absolute visual structural breakdown. The insoluble components of the solution were bones and dermal tissue particles. The mathematical model of the process was
constructed based on the kinetic curves of nutrients, obtained by us, extracted from fish particles into the extractant. The geometrical sizes of particles (R – spheres), their swelling degree, and
their temperature are controlled during the whole process (Derkanosova et al., 2011; Nagornov et al., 2015; Perez et al., 2011). Effect of mass changing presented in Figure 1A. Figure 1B presents
changes in the mid-radius of tissue particles during the electrochemical process for extraction of nutrients from hydrobionts (according to data calculation and direct measurements). Figure 1C
presents changing the temperature during the electrochemical extraction of nutrients from hydrobionts.
Figure 1. (A) Effect of mass changing in curves 1 (wet cake) and 2 – (dry cake) during the electrochemical extraction of nutrients from hydrobionts. I, II and III represents the approximate
boundaries of the respective phases of the extraction process. (B) Changes in the mid-radius of tissue particles during the electrochemical process for extraction of nutrients from hydrobionts
(according to data calculation and direct measurements). (C) Changing the temperature during the electrochemical extraction of nutrients from hydrobionts.
The solubility of protein particles (with different molecular weights and mass compositions) can be compared with the dissolution of a heterogeneous mixture of two or more chemical substances, which
dissolved at different speeds, physically similar to leaching. A slowly dissolving component (protein of high molecular weight) constitutes a porous structure that determines the geometrical
parameter of an R-const sphere through which the dissolvant and extraction products diffuse, which is similar to the process described in the paper (Kokotov et al., 1970).
However, in this case, the inert matrix concerning the dissolvent is the low-molecular-weight proteins that are difficult to dissolve and form a spatial crosslinked structure. At the initial stage of
the extraction process, high-molecular-weight proteins swell into the extractant and porous matrices that are constantly changing (Kao et al., 2011). In this case, the insoluble components, such as
bone and testaceous tissues, can be considered a factor in decreasing the active surface of the raw material.
Based on the kinetic curves and data on dry matter content in the extractant, the geometric parameters, and the degree of swelling of the solid phase, the whole process of extraction–dissolution can
be conditionally divided into two stages (Figure 2).
Figure 2. The content of dry substances in the solution during the extraction of nutrients from aquatic organisms using the electrochemical method.
Stage I (t = 0¸15 min): a relatively quick ongoing process of extraction, characterized by high speed, during which approximately 50% of the nutrients are extracted from the particles.
Figure 1A illustrates that the process of extraction is complicated by the simultaneous ongoing process of particle swellings, which finishes at the end of the stage, and the average radius of the
particles increases to double. Swelling causes the extractant to penetrate the particles; its direction is opposite to the flux of the extractable matter from the particles. Thus, Stage I can be
viewed as the process of extraction in a solid porous body–liquid system, which is complicated by swelling. The lack of influence of the hydrodynamic condition on the course of kinetic curves of
stage I, proved experimentally, allows the limiting stage of mass transfer to be considered intra-diffusion.
The armor or bone should be considered as an inert impurity, which does not form integral spatial structures as its content was <10%. This effect can be reduced only to a specific decrease in the
active surface area of the solid phase and can be considered as the introduction of the corresponding coefficient (Nagornov et al., 2015; Bichkova, 2014).
The Stage II is characterized by the disintegration of the proteinaceous framework of high molecular weight proteins and the dissolution of muscular tissue.
Mathematical modeling and solution of the inverse problem of the extraction process complicated by swelling
To describe the process, we must accept several assumptions allowing us to model an actual technological process.
Particles are spherical, identical in size, and isotropic by structure and composition, so the mass transfer problem in particles can be treated as one-dimensional and symmetric. Hence, pme, open
parentence C'(t) is a function only of coordinate R[0] (Nagornov et al., 2015). The validity of these assumptions is based on the following facts:
1. Particles were obtained from tissue pushed out through the extruder’s nozzle or spherical orifices. The size determinant is R = R[0] = 3×10^−3 m at the initial moment;
The size determinant R exceeds the size of the pores inside the particles. According to the data, the average diameter of the pores is 70×10^−6 m (Ryckebosch et al., 2012).
2. Mass transfer in step I: intra diffusion. The limiting step is the diffusion of substances inside the particles. It is possible to neglect mass transfer resistance both at the phase interface and
inside the liquid phase.
According to Lewis-Whitman’s theory (Romankov et al., 1990), at the phase interface, diffusion can only occur by molecular mechanism. It happens due to the surface tension forces, as there are no
convection currents in boundary layers. Depending on the nature of the materials, the hydrodynamic environment, and viscosity, the thickness of these layers (d) in the solid-liquid system is
0,1-30,0×10^−6 m. As the thickness of a boundary layer is small, the mass transfer resistance can be ignored (Romankov et al., 1990; Protodyakonov et al., 1987; Saravacos, 2014). The diffusion
coefficient in the liquid phase is much higher than in solid systems. The diffusion coefficient in a liquid system is 10^−9 m^2/s; in solids it is 10-13 m^2/s or less (Welty et al., 2001; Serrano et
al., 2023; Hosseini et al., 2023).
It happens since the extractant viscosity is very low, which is close to the viscosity of water (the ratio of raw material: extractant = 1:6). All particles are circulated with the extractant, and
the convective component of diffusion exceeds a thousand times that of the molecular one C[p]=C[0] (Romankov et al., 1990).
3. The extraction and mass transfer of substances from solids occurs because of molecular diffusion. As proven in (Gupalo et al., 1971; Ghiaasiaan, 2018), the convective component of mass transfer in
the particles is not considered.
4. Non-isothermal diffusion in the reaction mass caused by heating is negligible in comparison with the diffusion caused by a concentration gradient. Although Kafarov (Kafarov et al., 2019) proposed
a system of differential equations considering the imposition of the thermal diffusion component. He also showed that the thermal diffusion component in solutions was 10^3–10^5 times less than the
concentration one. Thus, the overlay effects can be neglected.
The conditions of extraction in the system solid body–liquid Kafarov’s criterion has the value of 10^−2-10^−3. It means that the concentration field is more inertial than the temperature one. The
period of temperature nonstationarity in the solid particles (considering their small size and high thermal conductivity) is a negligible part of the period of extraction (Ostroushko et al., 2012).
5. A chemical reaction does not complicate extraction.
6. Swelling of particles is caused by the absorption of the medium, which contains no protein component.
The porous spherical particle of muscle tissue has a mass flux inside that is caused by the molecular diffusion and existence of gradient concentrations of protein substances located in particles and
surrounded by the solvent – a catholyte. Convective diffusion is absent as the catholyte at the initial time moment X=0 contains no substances of protein nature. Therefore, the value of diffusion
stream sole- and water-soluble proteins from the particles of muscle tissue is calculated according to Fick’s law:
j=−DmgradC 1
or in a differential form:
Partial∂C∂t=Dc∇2C 2
where ∇^2 is the Laplacian. D[m], D[c] are the coefficients of mass - and concentration dependent.
However, there is Frolov and Romankov’ (Romankov et al., 1990) contradiction caused by different interpretations of the sense of a molecular diffusion coefficient D in these two equations. Equation
(1) shows the coefficient of mass transfer and equation (2) shows the inertial properties of the concentration field, that is, the coefficient of concentration dependency.
The contradiction is eliminated, if instead of C'(t) we consider μ'(t) the change of chemical potential. However, because of the practical convenience of using value C instead of m, we are limited by
the equation (2) in spherical coordinates (r, q, φ) (Korn G., & Korn T., 1973):
∂∂rr2∂C∂r+1sinθ∂∂θsinθ∂C∂θ+1sin2θ∂C∂θ2 3
In practice, instead of the molecular diffusion coefficient, the efficient diffusion coefficient is calculated considering the real mechanism of mass transfer in a capillary-porous body (
Protodyakonov et al., 1987). Considering the one-dimensional and symmetric problem, the concentration is treated as the function of only one spatial coordinate – r, and the equation becomes simpler:
It is possible to show that porous spherical particles swell uniformly along the radial coordinate in the medium with a restricted volume. The following equations describe the change of the
concentration of extracted component in a particle:
where j[d] is the mass flow (the density of the diffusion flow of water- and salt-soluble proteins), which diffuses from the particle into the solution.
where j[n] is the flow of the solvent that diffuses into particles, causing their swelling;
c is the concentration of the electrolyte containing no protein molecules;
where R[0] is the radius of particles.
The total flow through the sphere with radius r:
The common equation of the molecular diffusion in particles:
dCdt=divj=1τ2∂∂rr2j∂C∂t=1r2∂∂rDr2∂C∂r−εr2C∂C∂t=D∂2C∂r2+2Dr−ε∂C∂r−2rεC 4
The system of equations with initial and boundary conditions has the form:
∂C∂t=D∂2C∂r2+2Dr−ε∂C∂r−2rεCVp∂Cp∂t=WD∂C∂r−εCr=R0Ct=0=C0ε=dR0dt∂C∂rr−0=0Cp=Cr=R0CP,t=0=CP0 5
Where C is the concentration of the extracted component in the particles, the function of radial coordinate (r) and time (t);
D is the efficient diffusion coefficient, the function of time;
ε is the rate of change of linear dimensions of particles, the function of time;
V[P] is the volume of the solution, which is in contact with a solid phase, a function of time;
W is the surface area of the particles, which are in contact with the solvent, function of time;
C is the concentration of the extracted component in the liquid phase, a function of time;
R is the radius of particles that are in contact with the solution, a function of time.
System (5) approximately describes the extraction of water - and salt-soluble components from hydrobiont particles in the experiments conducted. Since the average radius of the particles in Stage I
changed in accordance with Figure 2, the corresponding expression for ε, according to the definition of this function, can be presented in the form:
ε=b=constt<tk0,t>tk 6
where b and t[k] are constants that are equal for the system studied, 0, 124 cm/min and 9 min respectively. This means that at the end of Stage I, the kinetics of extraction of nutrients is described
by the classical equation of internal diffusion in a limited volume:
∂C∂t=D∂2C∂r2+2Dr∂C∂rVp∂Cp∂t=WD∂C∂rr=R0Ct=tk=fr∂C∂rr=0=0Cp=Cr=R0Cp.t=0=Cp0 7
where f(r) is some distribution of the extracted component in time t[k].
Since the physical meaning of C(r,t) has to be continuous in the region of the change of arguments, then system (7) can be written in the form:
∂C∂t=D∂2C∂r2+2Dr∂C∂rVp∂Cp∂t=WD∂C∂rr=Rr0Ct=0=f0r∂C∂rr=0=0Cp=Cr=R0Cp.t=0=Cp0 8
where f[0](r) is some distribution of the extracted component at time 0, such that at time t[k] the distribution of the extracted component in the particles corresponds to the function f(r). It
should be noted that the solution of system (8) can reflect the distribution of a target component in particles from t[k] untill the moment stage I finishes.
The description of the extraction kinetics of nutrients from hydrobionts at the first stage by means of systems (5)–(8) with the constant efficient diffusion coefficient is not correct as during the
process the structure of the solid phase significantly changes. There are various target components in particles, and the extraction conditions are non-isothermal. The diffusion coefficient in
systems (5)–(8) should be considered as functions of time. However, with t>t[k], swelling completed, the particle structure stabilized, and the low speed of a temperature change, the constant
efficient diffusion coefficient characterizes this period and the approach represented works.
The solution of system (5) represents a complex computing challenge, which is hardly justified, considering that the first stage of extraction proceeds quickly enough and does not limit the
processing speed in general. According to the available experimental data, knowing the properties of solutions of systems of equations of type (8), it is possible to estimate the efficient diffusion
coefficient of nutrients in the swollen particles of hydrobionts.
As shown in studies by Derkanosova and Nagornov (Derkanosova et al., 2011; Nagornov et al., 2015), the systems of type (8) have the solution in the form of an infinite convergent power series of the
Cr,t=C∞+∑n=1∞Bnr,R,μnexp−μn2F0 9
where r, R are the current and defining radius of a particle;
C[∞] is the residual concentration of the target component in particles;
B[n] is the coefficient, which is function μn and B[i];
μ[n] is the corresponding root of the characteristic equation, which is determined by the boundary conditions, and the case with restricted volume (the boundary conditions of the III sort) have the
form (Kokotov et al., 1970)
tgμn=3μn3+αμn2α=VpVS 10
where V[P] и V[S] are the volumes, respectively, for the solid and liquid phases, which are in contact;
F0 equals numerator,,F0=DtR2 11
where F[0] is the criterion of Fourier or homo-chronicity (dimensionless time);
R is the radius of the particle.
The solution of the diffusion equation of a target component from a sphere in the dimensionless variables expressed by a similar row (Bichkova, 2014; Kokotov et al., 1970):
Ft=1−∑n=1∞BnR,μnexp−μn2F0Ft=Mt−M0M∞−M0 12
where M[0] is the initial quantity of a target component (an average in a particle);
M(t) is the quantity of a target component (an average in a particle) in t instant;
M[∞] is the quantity of a target component (an average in a particle) when reaching the equilibrium state.
The transition from C(r,t) to F(t) is rather convenient, as experimental data contain the total amount of extractable substances while the profile of concentrations in the body remains unknown. The
resultant F(t) function does not depend on coordinates.
Using dimensionless variables in the solutions of the diffusion equation allows bringing all the curves to one curve by changing a scale, describing diffusion in bodies of one form but of different
sizes and with different diffusion properties. The resultant curve is common for all bodies of this spherical form (Kokotov et al., 1970).
Known properties of the equations of the spherical form (9), (12) are such that the series converges quite quickly, and for computing the functions with acceptable accuracy, 3-5 members of the series
are usually enough (Korn G. et al., 1973). When the process is close to its completion, the convergence of the series (12) improves and at F>0.7. F is well described by the first member of a series (
Kokotov et al., 1970), that is:
Ft≈1−B1exp−μ12F0,when F>0.7 13
or after transforming and taking the logarithm:
ln1−Ft≈lnB1−μ12DR2t, when F0>0.7 14
Thus, from the experimental data and equation (14), it is possible to assess the efficient diffusion coefficient by characterizing the process of the internal diffusion of nutrients from hydrobionts
to the first stage of extraction.
Figure 3 represents the experimental data on the first stage of extraction of nutrients from spherical particles of hydrobionts of different initial sizes. Figure 4 shows the same data processed
according to the equation (14). The maximum amount of solids that are capable of passing into a solution (the M analog) was considered identically for all the experiments conducted. The masses were
taken for the extraction under identical conditions with the identical mass of the solid phase. The amount of solids was determined as an average value for all experiments – 17.2±0.2 g. The
extraction was conducted with a prepared electrolyte with no nutrients in the solution at the beginning of the experiment. (C[0]=0).
Figure 3. Content of the dry substances in the solution during the extraction of nutrients from hydrobionts by electrochemical method depending on time and radius of the original particles.
Figure 4. Results of experimental data according to equation (14).
The angular coefficients of the lines (B[0]), which are presented in Figure 5 for F>0.7, allow us to estimate the value of the effective diffusion coefficient:
D≈B0R2μ12 15
As it is difficult to determine the average radius of the swollen particles (especially large) for calculations with a given formula, the radius accepted is twice larger than the initial one, which
corresponds well with the data in Figure 2.
In the discussed experiments, the volume ratio of the phases (liquid and solid) is 3, corresponding with μ[1] – 3.4056. This value is obtained by solving the equation (10) with standard numerical
methods (Vygodskiy, 2008).
The calculated values of the angular coefficients of the dependences (14) and the assessment of values of effective diffusion coefficients are given in Table 1. Despite the fact that the quantities
in equation (15) have high precision, the effective diffusion coefficients should be considered as assessed in the order of their magnitude because the asymptotic approximation of the precise
decision and other assumptions were used for the modeling.
Table 1. Calculated values of coefficient B[0] and the effective diffusion coefficient.
R[0], m R, m (swelling) B[0], 1/s D, m^2/s
1.0×10^−3 2.0×10^−3 8.7×10^−3 3×10^−9
2.5×10^−3 5.0×10^−3 3.1×10^−3 7×10^−9
5.0×10^−3 10.0×10^−3 0.9×10^−3 8×10^−9
Comparing the effective diffusion coefficients obtained for the particles of various radii, it is possible to consider that they coincide with the declared accuracy. It confirms the validity of using
the intra-diffusion model.
As a result of this work the process of swelling of hydrobiont muscle tissue particles and extraction of proteins from them was investigated. This process is an integral stage of obtaining protein
hydrolysates from raw materials of animal origin, used in the food industry.
Kinetic dependences of the process of protein extraction from particles of dispersed muscle tissue of hydrobionts were obtained, and its mathematical description and modeling were performed. The
solution of the inverse modeling problem allowed us to obtain the molecular diffusion coefficient of proteins.
As a result of the mathematical description of the extraction process of muscle tissue particles, an important parameter from the theoretical and practical point of view, that is, the coefficient of
a molecular diffusion D = (3÷8)×10^−9 m^2/s was defined. It has been established that this coefficient surpasses the coefficients of diffusion substances of biological nature from plant tissue by
more than an order of magnitude (e.g., sugar 0.3×10^−9 m^2/c). It indicates the contribution of a convective component to the swelling process and dissolution of particles due to intensive blending
and easy differentiability of muscular tissue particles of hydrobionts.
Researchers can use the obtained molecular diffusion coefficient to calculate the parameters of the diffusion process of proteins during its extraction from dispersed particles of hydrobionts. | {"url":"https://qascf.com/index.php/qas/article/download/1327/1391/12511","timestamp":"2024-11-04T05:45:35Z","content_type":"application/xhtml+xml","content_length":"66432","record_id":"<urn:uuid:cf39208a-5947-4cd9-ae9e-ab47bb857e7c>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00010.warc.gz"} |
Height Transformations - CGRSC
Height Transformations
Until recently, the Canadian Geodetic Survey (CGS) realized and maintained the vertical datum (CGVD28) using spirit levelling. As CGVD28 was the only vertical datum for years in Canada, there was no
real need for height transformations, unless working across the US border where the Americans work in NAVD 88 or tying to a historical vertical datum. When a height transformation is required
between two levelling-construct vertical datums, it is necessary to find in the working area benchmarks having heights in the two systems. Depending of the size of the area, the transformation can
go from an offset to a complex surface. CGS does not produce vertical grid shift files for the transformation between vertical datums because the levelling lines are too sparse across Canada. Each
height transformation is addressed on a case-by-case basis by contacting the Geodetic Information Services at CGS.
In 2013, CGS introduced a new vertical datum. CGVD2013 is a modern vertical datum realized by a geoid model, allowing compatibility with GNSS positioning technique. While precise heights can be
readily obtained from GNSS surveys, these heights (h) are given with respect to an ellipsoid in a 3D geometric reference frame (e.g., NAD83(CSRS)). This is problematic as a large number of users are
interested in elevations (H) with respect to mean sea level (MSL). The geoid model gives the geoid height (N), which is the separation between the ellipsoid and the MSL. CGS published several
scientific geoid models since 1991 representing their own vertical datums. If a former geoid model (gravimetric or hybrid) was used, the transformation to a new geoid model is simply the difference
between the two models (make sure that the two models are in the same geometric reference frame).
Thus, the transformation between two levelling-construct vertical datums necessitate in having common benchmarks in the two systems while the transformation between two geoid-construct vertical
datums is simply the difference between the two geoid models. The challenge is for the transformation between a levelling-construct vertical datum and a geoid-construct vertical datum. CGS proposes
three approaches that are described in the section ‘Transformation between CGVD28 and CGVD2013’.
Transformation between NAD83(CSRS) and CGVD2013
This is a vertical transformation between the ellipsoidal and orthometric heights of a point. The orthometric height of a point in CGVD2013 (H[CGVD2013]) is determined as:
H[CGVD2013] = h[NAD83(CSRS)] − N[NAD83(CSRS)-CGG2013]
where h[NAD83(CSRS)] and N[NAD83(CSRS)-CGG2013] are ellipsoidal and geoid heights in the NAD83(CSRS) reference frame, respectively. The orthometric height will be at the same epoch as the epoch of
the ellipsoidal height.
Transformation between NAD83(CSRS) and CGVD28
Even though, CGS replaced CGVD28 with CGVD2013 in 2013 many users still work in the former vertical datum. In order to fill the gap for GNSS users working in CGVD28, CGS developed a hybrid geoid
model named HTv2.0, allowing the direct transformation from NAD83(CSRS) ellipsoidal heights to CGVD28 heights:
H[CGVD28] = h[NAD83(CSRS)] (1997.0) − N[NAD83(CSRS)-HTv2.0]
where h[NAD83(CSRS)] (1997.0) is the NAD83(CSRS) ellipsoidal height for epoch 1997.0 and N[NAD83(CSRS)-HTv2.0] is the HTv2.0 hybrid geoid height in the NAD83(CSRS) reference frame.
For the proper transformation to CGVD28, the ellipsoidal heights must be at the epoch of 1997.0 as the hybrid geoid model (HTv2.0) was developed using GPS constraints for the epoch 1997.0 at the
benchmarks. This particular fact is important because published heights in CGVD28 do not change in time. This means that the vertical datum itself is moving at the same velocity as the terrain with
respect to the ellipsoid.
Transformation between CGVD28 and CGVD2013
CGVD28 and CGVD2013 do not have a direct tie between them. The former is a levelling-construct vertical datum, which gives you the height of a benchmark above the vertical datum while the latter is
a geoid-construct vertical datum, which gives you the separation between the ellipsoid and the vertical datum in a 3D geometric reference frame. The one and only way to tie accurately these two
vertical datums is by observing GNSS on benchmarks. Although, it is also possible to transform between these two datums using two approximate approaches.
Measure ellipsoidal heights on existing benchmarks
This approach allows the most accurate transformation between CGVD28 and CGVD2013 as long as the GNSS survey is done to geodetic standards and the benchmarks are stable. The main disadvantage of this
approach is the requirement to collect data in the field. CGS recommends conducting GNSS surveys at a minimum of three benchmarks to verify stability and offset uncertainty.
The local offset (β) can be determined as following:
β = (h[NAD83(CSRS)] – N[NAD83(CSRS)-CGG2013A]) – H[CGVD28]
where h[NAD83(CSRS)] is the ellipsoidal height in NAD83(CSRS) observed by GNSS, N[NAD83(CSRS)-CGG2013A] is the CGG2013A geoid height in NAD83(CSRS) and H[CGVD28 ]is the published height of the
benchmark in CGVD28. Naturally, if the benchmarks are not GNSS friendly (i.e., against a foundation or restricted view of the sky), you can install temporary benchmarks and having the height
differences measure by levelling.
Use a national transformation model
The national transformation model (Figure 5) uses the geoid height differences between the gravimetric geoid model CGG2013A and hybrid geoid model HTv2.0, which are realizations of CGVD2013 and
CGVD28, respectively. However, HTv2.0 is not the true realization of CGVD28 as CGVD28 is actually realized by a network of benchmarks. HTv2.0 is only a close representation of CGVD28. Though,
HTv2.0 became the de facto realization of CGVD28 for many GNSS users.
Thus, the offset (β) between CGVD28 and CGVD2013 can be determined approximately as follow:
β = N[HTv2.0] – N[CGG2013]
Knowing β at the point of interest, the orthometric height in CGVD2013 is given by:
H[CGVD2013 ]= H[CGVD28] + β
With the national transformation model, one has to be careful for three items with respect to HTv2.0:
• HTv2.0 approximates CGVD28 where levelling lines are available. The separation can reach several centimetres in some regions.
• HTv2.0 generates a fictitious CGVD28 where there are no leveling lines; to confirm HTv2.0 one would have to extent the levelling network in these regions.
• HTv2.0 represents the separation between the ellipsoid and CGVD28 vertical datum for epoch 1997.0. If the epoch of CGVD28 in unknown, which is generally the case, it is technically impossible to
determine the actual separation between the ellipsoid and the CGVD28 datum. As indicated above, the CGVD28 vertical datum is moving at the same velocity as the terrain with respect to the
The national transformation model is the approach implemented in GPS-H.
Figure 5. The difference between heights in CGVD2013 (CGG2013A) and CGVD28 (HTv2.0).
Use published elevations
In order to ease the transition between CGVD28 and CGVD2013, CGS readjusted, with a series of constraints, the national first-order levelling network to conform as closely as possible to CGVD2013.
This means that all benchmarks have a published elevation in CGVD2013. These heights are not necessarily accurate as they are derived from legacy levelling data and benchmarks may have move over the
years. However, this approach can give you an approximation of the local offset (β) between CGVD28 and CGVD2013 within the periphery of the first-order levelling network.
β = H[CGVD2013] – H[CGVD28] | {"url":"https://cgrsc.ca/resources/coordinate-transformations/height-transformations/","timestamp":"2024-11-02T04:43:46Z","content_type":"text/html","content_length":"51286","record_id":"<urn:uuid:15bf0a6d-764b-41ad-9232-dad2f56208c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00676.warc.gz"} |
Hoop Stress
stress to the axis imposed on a cylinder wall when exposed to an internal pressure load. This stress occurs in cylindrical or spherical structures subjected to internal or external pressure. It
represents the stress that acts tangentially to the circumference of the structure.
In the case of a cylindrical pressure vessel, such as a pipe or a tank, the hoop stress is the stress acting circumferentially around the wall of the cylinder. It is caused by the pressure difference
between the inside and outside of the cylinder. Hoop stress is a critical parameter in the design and analysis of pressure vessels, as it helps determine the structural integrity and failure
mechanisms associated with the vessel under pressure.
Factors and Conditions Affecting Pipe Wall Thickness
• Maximum conditions such as pressure and temperature
• Fluid properties
• Fluid velocity
• Pipe Materials
• Safety of design factors
Three Stresses that will Cause Failure in Pipe
• Axial stress \(h\)
• Hoop stress \(\theta\)
• Radial stress \(r\)
Understanding hoop stress is crucial for designing safe and reliable structures that can withstand internal pressure without rupturing or experiencing plastic deformation. Engineers and designers use
these stress calculations to determine the appropriate wall thickness and material selection for such structures to meet safety and performance requirements.
Hoop Stress Factors
Hoop stress factors, also known as circumferential stress factors or stress concentration factors, are factors used in engineering and materials science to account for the increase in stress at
specific locations in a structure or component. These factors are typically applied when analyzing the structural integrity of objects subjected to mechanical loads, such as tension, compression, or
bending. Hoop stress factors are particularly important in situations where there are geometric irregularities or stress concentrations.
Where Hoop Stress Factors are Applied
Holes and Notches
- When a hole or notch is present in a structure, it can create a stress concentration at the edge of the hole or notch. The hoop stress factor is used to calculate the maximum stress at these
locations, which is typically higher than the nominal stress in the material.
Fillets and Corners
- Sharp corners in a structure can also lead to stress concentrations. Fillets or rounded corners are often used to reduce these stress concentrations. Hoop stress factors help determine the stress
reduction achieved by using fillets.
Changes in Cross-section
- In structures with varying cross-sectional areas, the hoop stress factor can be used to calculate the maximum stress at locations where the cross-section changes abruptly, such as at a step or
Pressure Vessels
- In the context of pressure vessels like cylinders and pipes that carry fluids under pressure, hoop stress factors are crucial. The hoop stress factor helps calculate the maximum circumferential
stress (hoop stress) in the vessel wall due to the internal pressure.
- Hoop stress factors can also be used in the analysis of bending stress, particularly in thin walled structures like tubes or pipes subjected to bending loads.
It's important to note that hoop stress factors are specific to the geometry and loading conditions of the structure in question. Engineers and designers use mathematical formulas and stress analysis
techniques to determine the appropriate hoop stress factors for their particular applications. These factors help ensure that the design of the structure can withstand the expected loads while
minimizing the risk of failure due to stress concentrations.
Pressure vessel hoop stress factors, also called stress concentration factors or hoop stress intensification factors, are numerical values used to calculate the maximum circumferential stress in the
wall of a pressure vessel. These factors are essential in the design and analysis of pressure vessels to ensure their structural integrity and safety, especially when subjected to internal pressure.
However, this formula assumes that the pressure vessel has a uniform, thin walled cylindrical geometry with no discontinuities or stress concentrations. In reality, most pressure vessels have
features like nozzles, openings, and transitions that can cause stress concentrations, which lead to higher stresses than predicted by the simple formula.
To account for these stress concentrations and calculate the actual maximum hoop stress, engineers use hoop stress factors. These factors are typically denoted by the symbol 'K' and are specific to
the geometry and configuration of the pressure vessel. They can be determined through analytical calculations, finite element analysis, or reference tables provided in codes and standards such as
ASME Boiler and Pressure Vessel Code.
The value of 'K' depends on various factors, including the shape of the vessel, the presence of openings, the geometry of welds, and other design features. Engineers must select the appropriate hoop
stress factor for their specific pressure vessel design to ensure it can safely handle the internal pressure without failure or excessive deformation. It's crucial to adhere to relevant codes and
standards when designing and analyzing pressure vessels, as these documents provide guidelines and recommended values for hoop stress factors based on industry best practices and safety | {"url":"https://www.piping-designer.com/index.php/disciplines/mechanical/stationary-equipment/vessel/3090-hoop-stress","timestamp":"2024-11-12T19:22:45Z","content_type":"text/html","content_length":"33704","record_id":"<urn:uuid:890da14b-267d-4951-a629-65043c2f8347>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00246.warc.gz"} |
Multiplication Chart 1 Through 12 2024 - Multiplication Chart Printable
Multiplication Chart 1 Through 12
Multiplication Chart 1 Through 12 – If you are looking for a fun way to teach your child the multiplication facts, you can get a blank Multiplication Chart. This will allow your child to fill out the
important points alone. You will discover blank multiplication maps for different item varieties, such as 1-9, 10-12, and 15 goods. You can add a Game to it if you want to make your chart more
exciting. Here are a few tips to get the child began: Multiplication Chart 1 Through 12.
Multiplication Maps
You may use multiplication graphs in your child’s pupil binder to help them commit to memory mathematics details. Even though many young children can memorize their math information normally, it will
take many more time to achieve this. Multiplication charts are a good way to reinforce their boost and learning their confidence. As well as being academic, these maps may be laminated for toughness.
Listed below are some helpful methods to use multiplication graphs. You can also take a look at websites like these for useful multiplication truth solutions.
This lesson handles the fundamentals in the multiplication table. Together with learning the guidelines for multiplying, college students will fully grasp the very idea of variables and patterning.
By understanding how the factors work, students will be able to recall basic facts like five times four. They will also be able to use your property of one and zero to solve more difficult items.
Students should be able to recognize patterns in multiplication chart 1, by the end of the lesson.
Along with the normal multiplication chart, college students might need to build a chart with more elements or much less elements. To make a multiplication graph or chart with additional aspects,
individuals need to produce 12 furniture, every single with twelve lines and 3 posts. All 12 desks have to match on a single page of document. Facial lines needs to be drawn with a ruler. Graph paper
is the best for this project. If graph paper is not an option, students can use spreadsheet programs to make their own tables.
Game ideas
Whether you are teaching a novice multiplication lesson or working on the mastery in the multiplication table, it is possible to come up with exciting and fascinating activity suggestions for
Multiplication Chart 1. A number of fun concepts are the following. This game needs the college students to be in work and pairs about the same difficulty. Then, they may all hold up their greeting
cards and discuss the solution to get a min. They win if they get it right!
When you’re teaching kids about multiplication, among the finest equipment you are able to give them is a printable multiplication graph or chart. These printable linens can come in a variety of
styles and can be imprinted in one page or a number of. Youngsters can learn their multiplication specifics by copying them from your memorizing and chart them. A multiplication chart may help for
most good reasons, from aiding them discover their math facts to instructing them the way you use a calculator.
Gallery of Multiplication Chart 1 Through 12
Multiplication Chart Table 1 12 Printable PDF
Sassy Multiplication Table 1 12 Printable Roy Blog
1 Thru 12 Multiplication Table Ldwtanka
Leave a Comment | {"url":"https://www.multiplicationchartprintable.com/multiplication-chart-1-through-12/","timestamp":"2024-11-06T09:34:40Z","content_type":"text/html","content_length":"54182","record_id":"<urn:uuid:923b9849-c3f0-41ab-bf3f-ba20db354401>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00838.warc.gz"} |
Stochastic Oscillator - ThirdBrainFxStochastic Oscillator
Stochastic Oscillator
Stochastic Oscillator
The Stochastic Oscillator Technical Indicator compares where a security’s price closed relative to its price range over a given time period. The Stochastic Oscillator is displayed as two lines. The
mainline is called %K. The second line, called %D, is a Moving Average of %K. The %K line is usually displayed as a solid line and the %D line is usually displayed as a dotted line.
There are several ways to interpret a Stochastic Oscillator. Three popular methods include:
• Buy when the Oscillator (either %K or %D) falls below a specific level (for example, 20) and then rises above that level. Sell when the Oscillator rises above a specific level (for example, 80)
and then falls below that level;
• Buy when the %K line rises above the %D line and sell when the %K line falls below the %D line;
• Look for divergences. For instance: where prices are making a series of new highs and the Stochastic Oscillator is failing to surpass its previous highs.
The Stochastic Oscillator has four variables:
• %K periods. This is the number of time periods used in the stochastic calculation;
• %K Slowing Periods. This value controls the internal smoothing of %K. A value of 1 is considered a fast stochastic; a value of 3 is considered a slow stochastic;
• %D periods. his is the number of time periods used when calculating a moving average of %K;
• %D method. The method (i.e., Exponential, Simple, Smoothed, or Weighted) that is used to calculate %D.
The formula for %K is:
%K = (CLOSE-LOW(%K))/(HIGH(%K)-LOW(%K))*100
CLOSE – is today’s closing price;
LOW(%K) – is the lowest low in %K periods;
HIGH(%K) – is the highest high in %K periods.
The %D moving average is calculated according to the formula: %D = SMA(%K, N)
N – is the smoothing period;
SMA – is the Simple Moving Average. | {"url":"https://www.thirdbrainfx.com/indicator/stochastic-oscillator/","timestamp":"2024-11-03T06:11:06Z","content_type":"text/html","content_length":"32365","record_id":"<urn:uuid:6322b46c-962a-4dce-a538-d1ed19979855>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00075.warc.gz"} |
named list with at least the Daily, Sample, and INFO dataframes
numeric specifying the half-window width in the time dimension, in units of years, default is 7
numeric specifying the half-window width in the discharge dimension, units are natural log units, default is 2
numeric specifying the half-window with in the seasonal dimension, in units of years, default is 0.5
numeric specifying the miniumum number of observations required to run the weighted regression, default is 100
numeric specifying the minimum number of uncensored observations to run the weighted regression, default is 50
logical specifying whether to use the modified method for calculating the windows at the edge of the record. The modified method tends to reduce curvature near the start and end of record.
Default is TRUE.
logical specifying whether or not to display progress message
logical deprecated. Use 'verbose' instead | {"url":"https://www.rdocumentation.org/packages/EGRET/versions/3.0.9/topics/setUpEstimation","timestamp":"2024-11-03T06:40:58Z","content_type":"text/html","content_length":"102213","record_id":"<urn:uuid:7678f945-07ef-4813-9c26-557bc54fdfc5>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00568.warc.gz"} |
What Is The Application Of Partial Differential Equations? - Fixanswer - Get your knowledge fix!
Partial differential equations are used to mathematically formulate, and thus aid the solution of, physical and other problems involving functions of several variables, such as the propagation of
heat or sound, fluid flow, elasticity, electrostatics, electrodynamics, etc.
What are the applications of differential equation in computer science?
Computer applications are involved in several aspects such as modeling (TIM the incredible machine) underlying logic (Chess or Go) or complex fluid flow, machine learning or financial analysis.
Differential equation may be used in computer science to model complex interation or non linear phenomena.
What are the applications of differential equations in engineering?
In general, modeling of the variation of a physical quantity, such as temperature, pressure, displacement, velocity, stress, strain, current, voltage, or concentration of a pollutant, with the change
of time or location, or both would result in differential equations.
What is the use of partial differentiation in real life?
Partial Derivatives are used in basic laws of Physics for example Newton’s Law of Linear Motion, Maxwell’s equations of Electromagnetism and Einstein’s equation in General Relativity. In economics we
use Partial Derivative to check what happens to other variables while keeping one variable constant.
How do you do partial differentiation?
Example 1
What is the application of differentiation?
Differentiation and integration can help us solve many types of real-world problems. We use the derivative to determine the maximum and minimum values of particular functions (e.g. cost, strength,
amount of material used in a building, profit, loss, etc.).
What is concept of differentiation?
The concept of differentiation refers to the method of finding the derivative of a function. It is the process of determining the rate of change in function on the basis of its variables. The
opposite of differentiation is known as anti-differentiation.
Why do we need differentiation?
WHY DO WE NEED TO DIFFERENTIATE? Differentiation demonstrates a teacher’s knowledge of pupils as individual learners. Differentiation enables pupils to access the learning. Differentiated learning
helps pupils understand and apply both content and process in their learning.
What is application of maximum and minimum?
The process of finding maximum or minimum values is called optimisation. We are trying to do things like maximise the profit in a company, or minimise the costs, or find the least amount of material
to make a particular object. These are very important in the world of industry.
How do you solve applications of maxima and minima?
Steps in Solving Maxima and Minima Problems
How do you find the maximum and minimum of differentiation?
Why do we solve quadratic equations?
Quadratic equations are actually used in everyday life, as when calculating areas, determining a product’s profit or formulating the speed of an object. Quadratic equations refer to equations with at
least one squared variable, with the most standard form being ax2 + bx + c = 0.
Charlene Dyck
Charlene is a software developer and technology expert with a degree in computer science. She has worked for major tech companies and has a keen understanding of how computers and electronics work.
Sarah is also an advocate for digital privacy and security. | {"url":"https://fixanswer.com/what-is-the-application-of-partial-differential-equations/","timestamp":"2024-11-13T19:47:07Z","content_type":"text/html","content_length":"88145","record_id":"<urn:uuid:1fef1364-f690-49a8-8296-dc0886a798db>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00820.warc.gz"} |
What is the final velocity of a car that accelerated 10 m s2 from rest and traveled 180m? - EasyRelocated
What is the final velocity of a car that accelerated 10 m s2 from rest and traveled 180m?
What is the final velocity of a car that accelerated 10 m s2 from rest and traveled 180m?
What is the final velocity of a car that accelerated 10 m/s2 from rest and traveled 180m? V2=u2+2ad=zero2+2*10*180, v2=3600, so v=60 m/s.
What is the increase in velocity in 2 seconds of a car accelerates at the rate of 5 m per second?
Increase in velocity = acceleration × time =(5×2)ms−1=10ms−1.
What is the formula for final speed of acceleration?
What is the formula for final velocity of acceleration?
Final velocity (v) of an object equals initial velocity (u) of that object plus acceleration (a) of the object times the elapsed time (t) from u to v. Use standard gravity, a = 9.80665 m/s2, for
equations involving the Earth’s gravitational force as the acceleration rate of an object.
What is the formula for velocity increase?
Acceleration (a) is the change in velocity (Δv) over the change in time (Δt), represented by the equation a = Δv/Δt. This allows you to measure how fast velocity changes in meters per second squared
(m/s^2). Acceleration is also a vector quantity, so it includes both magnitude and direction.
What is the formula for displacement?
S = ut + ½ at2 Hence, displacement (s) of an object is equal to initial velocity(u) times time (t), plus half of the acceleration (½ a) multiplied by time squared (t2).
Does velocity change when acceleration increases?
Sure, as long as acceleration is positive, velocity increases, even if acceleration is decreasing (as long as it doesn’t reach zero). Likewise, as long as acceleration is negative, velocity decreases
even if acceleration is increasing.
Is velocity same as speed?
Why is it incorrect to use the terms speed and velocity interchangeably? The reason is simple. Speed is the time rate at which an object is moving along a path, while velocity is the rate and
direction of an object’s movement. Put another way, speed is a scalar value, while velocity is a vector.
What is the final velocity?
On the other hand, the final velocity is a vector quantity that measures the speed and direction of a moving body after it has reached its maximum acceleration.
What is the SI unit for velocity?
What is the SI unit of velocity? The SI unit of velocity is metres per second (m/s). Alternatively, the velocity magnitude can also be expressed in centimetres per second (cm/s).
When a car starts from rest and attains a velocity of 10 meters per second in 40 seconds?
Expert-Verified Answer A driver starts the car from rest and attains a velocity of 10m/s in 40s and applies breaks and slows down the car to 5m/s in 10s. We have to find the acceleration of the car
in both cases. Hence, the acceleration of the car in both cases is 0.25 m/s²and -0.5 m/s².
What is the final velocity of a car?
The final velocity (v) of an object equals the initial velocity (u) of that object plus acceleration (a) of the object times the elapsed time (t) from u to v.
What does the acceleration of a moving vehicle is 10m s2 mean?
Answer: cceleration is the rate of change of velocity of a body w.r.t time. Thus, if the acceleration of a body is 10m/s^2 , it means that the body accelerates after every second with a velocity of
What is the initial velocity of a car which is stopped in 10 seconds by applying brakes the retardation is 2.5 m?
⇒⇒ u = 25 m/s. | {"url":"https://easyrelocated.com/what-is-the-final-velocity-of-a-car-that-accelerated-10-m-s2-from-rest-and-traveled-180m/","timestamp":"2024-11-10T17:48:36Z","content_type":"text/html","content_length":"69280","record_id":"<urn:uuid:29e3b00d-79e8-4490-bd20-f1bbb95d19ca>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00492.warc.gz"} |
CS184/284A: Lecture Slides
Lecture 7: Intro to Geometry, Splines, and Bezier Curves (110)
The Bezier surfaces shown in class are mostly rectangles stretched into smooth surfaces. How does the math extend to triangular meshes in the "Gumbo" model?
So as is covered in the lecture, the Bezier surface essential give a continuous map from (u,v) to three-dimensional space (x,y,z). So it can map arbitrary structures from the 2D-Plane (including
I looked up that Bézier surface is a representation of such polynomial pieces which makes their interactive design easier and more intuitive than with other representations. Also, I saw there is a
Nurb curve that's different from Bézier that doesn't touch the control points, they just bend towards them. But Bézier curve control over tangents and the curve always touches the control point.
You must be enrolled in the course to comment | {"url":"https://cs184.eecs.berkeley.edu/sp23/lecture/7-110/intro-to-geometry-splines-and-be","timestamp":"2024-11-11T17:50:23Z","content_type":"text/html","content_length":"13820","record_id":"<urn:uuid:77551d29-4c29-4cfb-a2e9-1efa7cc22ce3>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00074.warc.gz"} |
Re: Restraining angle between bond vector and coordinate vector
Re: Restraining angle between bond vector and coordinate vector
Another solution, which is also due to Jérôme, is to use the
"centerReference" keyword to shift the positions so that the second
atom or group is at (0,0,0). Then you can calculate the angle
In the example below, I have two groups (atoms 1–10 and atoms 11–20)
and want to keep the vector from the center of mass of atoms 11–20 to
the center of mass of atoms 1–10 along the z-axis (defined by
dummyAtom (0,0,1)). I think this solution may also have problems in
NAMD 2.10, so you'll need the nightly build of NAMD or compile NAMD
with the current version of Colvars as indicated by Jérôme.
colvar {
name colatitude
angle {
group1 {
atomNumbersRange 1-10
centerReference on
refPositionsGroup { atomNumbersRange 11-20 }
refPositions (0,0,0)
group2 {
atomNumbersRange 11-20
centerReference on
refPositions (0,0,0)
group3 {
dummyAtom (0,0,1)
harmonic {
colvars colatitude
centers 0
forceConstant 10
Jeffrey Comer, PhD
Assistant Professor
Institute of Computational Comparative Medicine
Nanotechnology Innovation Center of Kansas State
Kansas State University
Office: P-213 Mosier Hall
Phone: 785-532-6311
On Mon, Mar 23, 2015 at 3:26 PM, Seth Axen <seth.axen_at_gmail.com> wrote:
> I would like to add a harmonic restraint for an angle between a specific
> bond vector and a target vector in 3D space, defined by 3 coordinates. The
> resulting force should be updated at every step of the simulation. I'm
> having trouble figuring out the best way to go about this from the
> documentation and user-defined forces examples, and I'd appreciate any
> suggestions.
> Thanks!
This archive was generated by hypermail 2.1.6 : Tue Dec 27 2016 - 23:21:01 CST | {"url":"https://www.ks.uiuc.edu/Research/namd/mailing_list/namd-l.2015-2016/0402.html","timestamp":"2024-11-14T03:48:42Z","content_type":"text/html","content_length":"7551","record_id":"<urn:uuid:31921956-edd0-46c7-8d6e-f22e94fa7c79>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00551.warc.gz"} |
A ring type flywheel of mass 100kg and diameter 2m class 11 physics JEE_Main
Hint: We have the angular velocity in $\dfrac{{rev}}{s}$ which can be converted by multiplying it by $2\pi $. Now we can use the other rotational kinematic formulas to determine the other values.
Formulas used: We will be using the formula $I = m{R^2}$ where $I$ is the moment of inertia , $m$ is the mass of the body, $R$ is the radius of the wheel, and $K = \dfrac{1}{2}m{V^2} + \dfrac{1}{2}I
{\omega ^2}$ where $K$ is the kinetic energy of the body in rotation, $V$ is the linear velocity of the body and $\omega $ is the angular velocity of the body.
We will also be using the formula $L = I\omega $ where $L$ is the angular momentum experienced by the body.
Also, we might use the formula to find the torque of a body in rotational motion, $\tau = I\alpha $ where $\tau $ is the torque experienced by the body, $\alpha $ is the angular acceleration
experienced by the rotating body.
Complete Step by Step answer:
A rotating body experiences and stands true for every law in kinematics except that the variables are angular instead of linear. This means that the angular velocity can be $\omega \times 2\pi \dfrac
{{rad}}{s}$ which gives us the units in terms of rotational kinematics.
Here, we know that the angular velocity $\omega = \dfrac{5}{{11}}rev/s$, so, $\omega = 2\pi \times \dfrac{5}{{11}}$ .
We know that $\pi = \dfrac{{22}}{7}$,
$ \Rightarrow \omega = 2 \times \dfrac{{22}}{7} \times \dfrac{5}{{11}}$. Thus, the value of $\omega $ in $\dfrac{{rad}}{s}$ is $\omega = \dfrac{{20}}{7}rad/\operatorname{s} $ .
Now the first statement requires us to find the moment of inertia of the system given in the problem with $d = 2m$ and $m = 100kg$. We know that the moment of inertia is given by, $I = m{R^2}$ .
Substituting the formulas, we get,
$ \Rightarrow I = 100 \times {\left( 1 \right)^2}$ [since $d = 2m$ , $r = 1m$ ]
So, the moment of inertia of the body in rotational motion is, $I = 100kg{m^2}$ .
The second statement requires to find the kinetic energy of the flywheel, which is given by the formula $K = \dfrac{1}{2}m{V^2} + \dfrac{1}{2}I{\omega ^2}$ .
We know that $I = m{R^2}$ and $V = R\omega $, substituting these values in the equation for kinetic energy we get,
$K = \dfrac{1}{2}m{R^2}{\omega ^2} + \dfrac{1}{2}m{R^2}{\omega ^2} = m{R^2}{\omega ^2}$. Now substituting the values of $m$, $R$, and $\omega $ we get,
$K = (100){(1)^2}{(\dfrac{{20}}{7})^2}$ . $ \Rightarrow K = 816.33J$
Thus, the kinetic energy of the rotational system is $K = 816.33J$. (false)
The next statement requires us to find the angular momentum of the system, which will be given by $L = I\omega $ .Substituting the values of $I$ and $\omega $ we get,
$L = 100 \times \dfrac{{20}}{7} = \dfrac{{2000}}{7}$
$ \Rightarrow L = 285.71Js$ (false)
The last statement requires us to find the time it will take to bring the rotating body to rest if there is a torque of $\tau = 250Nm$ acting on the body. We know that the torque is given by, $\tau =
I\alpha $ .
Substituting the values of $\tau $ and $\alpha $ .
$250 = 100 \times \alpha $$ \Rightarrow \alpha = 2.5rad/{s^2}$ .
We also know from the laws of rotational kinematics that $\omega + {\omega _0} = \alpha t$ . Assuming the body to start from rest, and substituting the values, ${\omega _0} = 0,\omega = \dfrac{{20}}
{7}rad/s$ and $\alpha = 2.5rad/{s^2}$
$\dfrac{{20}}{7} = 2.5 \times t$
Solving for $t$ we get, $t = 1.142s$. (false)
Thus, the only option that holds true will be option A.
Note: We can see that every law of linear kinematics holds true in rotational motion, except that the variables used are different. Just that during rotational motion the mass is often substituted by
moment of inertia. Velocity by angular velocity and so on. | {"url":"https://www.vedantu.com/jee-main/a-ring-type-flywheel-of-mass-100kg-and-diameter-physics-question-answer","timestamp":"2024-11-10T08:33:55Z","content_type":"text/html","content_length":"155472","record_id":"<urn:uuid:a30174f8-3395-4f79-834e-3893587a3f98>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00748.warc.gz"} |
How Many Ml to a Mg: The Best Way to Define Milliliters and Milligram 2024 - techbusinessconnect.com
How Many Ml to a Mg: The Best Way to Define Milliliters and Milligram 2024
How many ml to a mg when measuring or converting units, it’s easy to feel confused by the vast variety of terms, abbreviations, and systems of measurement. A common question that arises is how to
convert between milliliters (ml) and milligrams (mg). It’s a fair question, especially in areas like medicine, cooking, chemistry, or any situation that involves precise measurements. However, the
challenge lies in the fact that milliliters (ml) measure volume, while milligrams (mg) measure mass. Because they measure different properties, there’s no simple one-to-one conversion formula.
How many ml to a mg , understanding the relationship between volume and mass can be accomplished by focusing on density, which acts as the bridge between these two units. This article will guide you
through everything you need to know about converting ml to mg, explain the principles behind it, and provide practical examples of how to approach these conversions.
What Is a Milliliter?
A milliliter (ml) is a unit of volume in the metric system, which is widely used across the world. It represents one-thousandth of a liter. In real-world terms, a milliliter is a very small amount of
liquid. For example:
• A drop of water is typically about 0.05 ml.
• A teaspoon holds about 5 ml.
• A small juice box might contain 200 ml of liquid.
Milliliters are often used in contexts where precise measurement of liquids is important, such as cooking, laboratory work, and medicine.
What Is a Milligram?
A milligram (mg) is a unit of mass in the metric system and is equal to one-thousandth of a gram. To give you a sense of scale:
• A single grain of salt weighs about 0.05 mg.
• A common aspirin tablet may weigh 500 mg.
• 1,000 mg is equal to 1 gram.
Milligrams are used in fields like medicine and nutrition, where it’s crucial to measure very small amounts of mass accurately.
Why Is There No Direct Conversion How many ml to a mg ?
How many ml to a mg the reason there is no direct conversion between ml and mg is that they measure fundamentally different properties: volume (ml) measures the amount of space something occupies,
and mass (mg) measures how much matter is in something. To convert between them, you need to know the density of the substance you’re measuring.
Density is defined as mass per unit volume and is typically expressed in grams per milliliter (g/ml). For instance, the density of water is 1 g/ml, which means that 1 ml of water weighs exactly 1
gram (or 1,000 mg). However, for other substances, the density may vary significantly.
The Role of Density in Converting ml to mg
The key to converting how many ml to a mg or vice versa lies in the density of the substance being measured. The basic formula to convert between these units is:Mass (mg)=Volume (ml)×Density (g/ml)
×1000\text{Mass (mg)} = \text{Volume (ml)} \times \text{Density (g/ml)} \times 1000Mass (mg)=Volume (ml)×Density (g/ml)×1000
In this formula, you multiply the volume in milliliters by the density (in grams per milliliter) and then by 1,000 to get the mass in milligrams.
Practical Examples
Let’s take a look at some practical examples to better understand the conversion between ml and mg.
Example 1: Water
Water has a density of 1 g/ml, which makes the conversion between ml and mg quite simple. If you want to know how many mg are in a given volume of water, you can use the formula above.
• Given: 5 ml of water
• Density of water: 1 g/ml
Mass (mg)=5ml×1g/ml×1000=5000mg\text{Mass (mg)} = 5 \, \text{ml} \times 1 \, \text{g/ml} \times 1000 = 5000 \, \text{mg}Mass (mg)=5ml×1g/ml×1000=5000mg
Therefore, 5 ml of water weighs 5,000 mg (or 5 grams).
Example 2: Olive Oil
Olive oil has a density of around 0.92 g/ml, meaning that it’s slightly less dense than water. If you want to convert 10 ml of olive oil to mg:
• Given: 10 ml of olive oil
• Density of olive oil: 0.92 g/ml
Mass (mg)=10ml×0.92g/ml×1000=9200mg\text{Mass (mg)} = 10 \, \text{ml} \times 0.92 \, \text{g/ml} \times 1000 = 9200 \, \text{mg}Mass (mg)=10ml×0.92g/ml×1000=9200mg
So, 10 ml of olive oil weighs 9,200 mg (or 9.2 grams).
Example 3: Ethanol (Alcohol)
Ethanol, commonly known as alcohol, has a density of about 0.789 g/ml. If you want to know how much 20 ml of ethanol weighs in mg:
• Given: 20 ml of ethanol
• Density of ethanol: 0.789 g/ml
Mass (mg)=20ml×0.789g/ml×1000=15780mg\text{Mass (mg)} = 20 \, \text{ml} \times 0.789 \, \text{g/ml} \times 1000 = 15780 \, \text{mg}Mass (mg)=20ml×0.789g/ml×1000=15780mg
Thus, 20 ml of ethanol weighs 15,780 mg (or 15.78 grams).
Example 4: Honey
Honey is denser than water, with a density of approximately 1.42 g/ml. If you need to convert 15 ml of honey to mg:
• Given: 15 ml of honey
• Density of honey: 1.42 g/ml
Mass (mg)=15ml×1.42g/ml×1000=21300mg\text{Mass (mg)} = 15 \, \text{ml} \times 1.42 \, \text{g/ml} \times 1000 = 21300 \, \text{mg}Mass (mg)=15ml×1.42g/ml×1000=21300mg
Therefore, 15 ml of honey weighs 21,300 mg (or 21.3 grams).
Factors That Affect Density
It’s important to note that the density of a substance can change depending on various factors such as temperature and pressure. In most cases, these changes are relatively small and can be ignored
for practical purposes, but in some situations, such as scientific experiments or industrial processes, accounting for these variations is crucial.
• Temperature: As temperature increases, many liquids expand, which lowers their density. For example, water is denser at 4°C than at room temperature.
• Pressure: For most liquids, changes in pressure have a minimal effect on density, but for gases, density is highly sensitive to changes in pressure.
What About Solids?
How many ml to a mg for solids, you might also encounter a need to convert between mass and volume. To do this, you would need to know the density of the solid substance in question. For example, if
you were dealing with a solid like gold, which has a density of 19.32 g/ml, you would use the same formula to convert between volume and mass. However, it’s important to note that solids are usually
measured in grams and cubic centimeters (cm³) rather than milliliters.
Why These Conversions Matter
Understanding how many ml to a mg to convert is important in various fields, including:
• Medicine: Precise dosing of medications often depends on converting between liquid volume and mass, as medications can be administered in solutions.
• Chemistry: In laboratory settings, researchers must often convert between mass and volume to ensure accurate measurements for reactions and experiments.
• Cooking and Nutrition: Recipes and nutritional information can include both liquid and solid measurements, requiring conversions to maintain accuracy in ingredient proportions.
Converting between milliliters (ml) and milligrams (mg) is not as straightforward as it may initially seem since these units measure different properties—volume and mass. The key to bridging the gap
between them lies in understanding density, which is unique to each substance. By knowing the density of a material, you can use a simple formula to convert between ml and mg. This conversion is
essential in fields like medicine, chemistry, and cooking, where precise measurements are crucial for success. Whether you’re measuring out a medication, cooking with olive oil, or experimenting with
chemicals, understanding the relationship between volume and mass will help you make accurate conversions. | {"url":"https://techbusinessconnect.com/how-many-ml-to-a-mg/","timestamp":"2024-11-12T08:49:19Z","content_type":"text/html","content_length":"73623","record_id":"<urn:uuid:88d89757-d4c6-4e50-823d-613e79d06a27>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00428.warc.gz"} |
SQL Operators
This topic provides selected notes and discussion on operators that are built into the Manifold query engine. The list in this topic is comprehensive but new items are often added with updates to
Manifold and may appear in the product before they appear in this documentation. See the list in the query builder tab of the Command Window for an authoritative list of operators, commands,
functions, and constants
- <value> Negation
<value> ^ <value> Exponentiation
<value> % <value> Modulo. Divides the first value by the second value and returns the remainder.
<value> MOD <value> Modulo. Divides the first value by the second value and returns the remainder. 4 MOD 3 is 1. 22 MOD 5 is 2.
<value> DIV <value> Integer division. Divides the first value by the second value and returns the integer part of the result. 3 DIV 4 is zero. 7 DIV 5 is 1. 22 DIV 5 is 4.
<value> / <value> Division
<value> * <value> Multiplication
<value> - <value> Subtraction
<value> + <value> Addition for numbers, concatenation of strings.
<value> & <value> Concatenation of strings.
BITNOT <value> Bitwise NOT: Result is opposite of the corresponding bit in the argument.
<value> BITAND <value> Bitwise AND: Result is 1 if corresponding bits in both arguments are 1.
<value> BITOR <value> Bitwise OR: Result is 1 unless corresponding bits in both arguments are 0.
<value> BITXOR <value> Bitwise XOR: Result is 1 if corresponding bits in both arguments are different. Result is 0 if corresponding bits in both arguments are the same.
Matches first value to a string pattern that may contain wild cards such as '%' (matches any number of characters, including zero characters) and '_' (a single
character). See the LIKE Operator topic for examples.
SELECT [NAME] FROM [Mexico Table]
WHERE [NAME] LIKE 'Dur%';
<value> LIKE <value>
Returns Durango.
SELECT [NAME] FROM [Mexico Table]
WHERE [NAME] LIKE '%an%';
Returns Guanajuato, Michoacan de Ocampo, Yucatan, Quintana Roo, Durango and San Luis Potosi.
Not equal to.
<value> <> <value> Either <value> can be a number or a numeric tile, allowing comparisons between two numbers, two tiles, or between a tile and a number. When a tile is an operand,
the result will be a numeric tile with values of 0 for FALSE or 1 for TRUE, giving the result of the comparison between that tile value and the number or
corresponding tile value.
Equal to.
<value> = <value> Either <value> can be a number or a numeric tile, allowing comparisons between two numbers, two tiles, or between a tile and a number. When a tile is an operand,
the result will be a numeric tile with values of 0 for FALSE or 1 for TRUE, giving the result of the comparison between that tile value and the number or
corresponding tile value.
Greater than or equal to.
<value> >= <value> Either <value> can be a number or a numeric tile, allowing comparisons between two numbers, two tiles, or between a tile and a number. When a tile is an operand,
the result will be a numeric tile with values of 0 for FALSE or 1 for TRUE, giving the result of the comparison between that tile value and the number or
corresponding tile value.
Greater than.
<value> > <value> Either <value> can be a number or a numeric tile, allowing comparisons between two numbers, two tiles, or between a tile and a number. When a tile is an operand,
the result will be a numeric tile with values of 0 for FALSE or 1 for TRUE, giving the result of the comparison between that tile value and the number or
corresponding tile value.
Less than or equal to.
<value> <= <value> Either <value> can be a number or a numeric tile, allowing comparisons between two numbers, two tiles, or between a tile and a number. When a tile is an operand,
the result will be a numeric tile with values of 0 for FALSE or 1 for TRUE, giving the result of the comparison between that tile value and the number or
corresponding tile value.
Less than.
<value> < <value> Either <value> can be a number or a numeric tile, allowing comparisons between two numbers, two tiles, or between a tile and a number. When a tile is an operand,
the result will be a numeric tile with values of 0 for FALSE or 1 for TRUE, giving the result of the comparison between that tile value and the number or
corresponding tile value.
Returns True (1) if the <value> is greater than or equal to the <lower> bound and is less than or equal to the <upper> bound. For sensible results, the <lower>
value should be less than the <upper> value.
The function returns False (0) if the <lower> value is greater than the <upper> value, no matter what the <value> may be. That is an annoying way to enforce the
assumption that the <lower> value should be lower than the <upper> value, but since virtually all other databases do that for BETWEEN, Manifold does that as well.
? 5 BETWEEN 4 AND 7
Returns true.
? 5 BETWEEN 7 AND 4
Returns false.
? 9 BETWEEN 4 AND 7
Returns false.
<value> BETWEEN <lower> AND <upper> Any or all of the <value>, <lower>, and <upper> operands can be numbers or numeric tiles. When one or more operands is a tile, the result will be a numeric tile
with values of 0 for FALSE or 1 for TRUE, giving the result of the BETWEEN comparison using that tile value and the numbers or corresponding tile values that are
the other operands.
For example, if Height is a tile field,
[Height] BETWEEN 300 AND 500
would create a numeric tile where 1 values mark pixels where the value in the Height tile was greater than or equal to 300 and less than or equal to 500, with 0
values marking pixels otherwise.
If OldHeight and Height are both tile fields,
500 BETWEEN [OldHeight] AND [Height]
would create a numeric tile where 1 values mark pixels where the value in the OldHeight tile was less than or equal to 500 and the value in the Height tile was
greater than or equal to 500, with 0 values otherwise.
Keep in mind that since the <lower> bound should be less than the <upper> bound, any pixels in the above example where OldHeight values are greater than Height
values will have a result tile pixel value of 0 no matter how either of those height values compare to 500.
Returns True (1) if the value occurs in the results of the query.
? 'Durango' IN (SELECT [NAME] FROM [Mexico Table])
<value> IN (<query>) Returns 1.
? 'Vermont' IN (SELECT [NAME] FROM [Mexico Table])
Returns 0.
Returns True (1) if the first value occurs in the list of values.
? 'Tom' IN ('Tom', 'Dick', 'Harry')
<value> IN (<value>, ...) Returns 1.
? 'John' IN ('Tom', 'Dick', 'Harry')
Returns 0.
Returns True (1) if the value is NULL.
? (3 DIV 0) IS NULL
<value> IS NULL Returns 1.
? (3 DIV 2) IS NULL
Returns 0.
Returns True (1) if the value is boolean False (0)
? NOT (7 < 5)
It is False that 7 is less than 5, so the above returns 1.
NOT <value>
? NOT (3 < 5)
It is True that 3 is less than 5, so the above returns 0.
Works with numeric tiles as well, where the tile contains values of 0 or any other number. 0 is interpreted as FALSE and any other value is TRUE. The result is a
numeric tile with 0 values for FALSE and 1 values for TRUE.
Returns True (1) if both values are True. The classic AND logic operator.
<value> AND <value>
Works with numeric tiles as well, where the tile contains values of 0 or any other number. 0 is interpreted as FALSE and any other value is TRUE. The result is a
numeric tile with 0 values for FALSE and 1 values for TRUE.
Returns True (1) if either value is True. The classic OR logic operator.
<value> OR <value>
Works with numeric tiles as well, where the tile contains values of 0 or any other number. 0 is interpreted as FALSE and any other value is TRUE. The result is a
numeric tile with 0 values for FALSE and 1 values for TRUE.
Returns True (1) if one and only one value is True. Another way to say the same thing: Returns True (1) if the values are different, that is, one is True and one
is False. Returns False (0) if both values are True or both values are False. The classic XOR logic operator.
<value> XOR <value>
Works with numeric tiles as well, where the tile contains values of 0 or any other number. 0 is interpreted as FALSE and any other value is TRUE. The result is a
numeric tile with 0 values for FALSE and 1 values for TRUE.
CAST (<value> AS <type>) Converts the data type of the value into the specified type.
Takes a vector or tile value and converts the data type of all contained values to the specified type.
CASTV ([vector-of-float64] AS INT32)
CASTV (<value> AS <type>) Produces a vector of INT32 values.
CASTV ([tile-of-uint8x3] AS FLOAT32)
Produces a tile of FLOAT32X3. The number of channels does not change.
Provides If-Then-Else logic for queries. When the condition evaluates to True (1) the THEN value is returned. The ELSE part is optional, and if provided when the
condition does not evaluate to True (1) the ELSE value is returned.
? CASE WHEN (1 > 0) THEN 5 ELSE 9 END
CASE WHEN <condition> THEN <value>
... ELSE <value> END Returns 5, since the condition of one being greater than zero is always true.
? CASE WHEN (0 > 1) THEN 5 ELSE 9 END
Returns 9, since the condition of zero being greater than one is never true.
Similar to CASE WHEN <condition> THEN <value> ... ELSE <value> END except that it allows a series of WHEN <value> THEN <value> pairs to test against the compared
Suppose we would like to set the number of THREADS to be used to a larger number with more CPUs but not so large as to use all CPUs, so we leave some CPUs free for
other processes. The fragment below returns a value based on the number of CPUs reported:
? CASE SystemCpuCount()
WHEN 8 THEN 5
WHEN 4 THEN 3
ELSE 1
CASE <compared-value> WHEN <value> The compared value above is the number of CPUs returned by the SystemCpuCount function. When the number of CPUs reported is 8 then the expression evaluates to 5,
THEN <value> ... ELSE <value> END when the number of CPUs is 4 the expression evaluates to 3, and otherwise the expression evaluates to 1. The expression above evaluates to 5 for a typical Core i7
machine with hyperthreading enabled.
The above expression could be used as the argument to the THREADS command:
CASE SystemCpuCount()
WHEN 8 THEN 5
WHEN 4 THEN 3
ELSE 1
(A useful example written by SQL master Tim Baigent.)
EXISTS <table> Returns True (1) if the argument contains at least one record.
(<value>, ...) <> (<value>, ...) Not equal to
(<value>, ...) = (<value>, ...) Equal to
(<value>, ...) >= (<value>, ...) Greater than or equal to
(<value>, ...) > (<value>, ...) Greater than
(<value>, ...) <= (<value>, ...) Less than or equal to
(<value>, ...) < (<value>, ...) Less than
(<value>, ...) BETWEEN (<value>, ...) AND (<value>) Returns True (1) if the first value is between the second and third values.
(<value>, ...) IN (<query>) Returns True (1) if the vector value occurs in the results of the query.
Takes a vector or tile value and converts the data type of all contained values to the specified type.
CASTV ([vector-of-float64] AS INT32)
CASTV (<value> AS <type>) Produces a vector of int32 values.
CASTV ([tile-of-uint8x3] AS FLOAT32)
Produces a tile of float32x3 values. The number of channels does not change. | {"url":"https://manifold.net/doc/mfd9/sql_operators.htm","timestamp":"2024-11-06T10:41:38Z","content_type":"text/html","content_length":"51453","record_id":"<urn:uuid:53ca1833-6d67-40a0-8400-033df4c2400b>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00427.warc.gz"} |
ECCC - Reports tagged with dynamic graph algorithms
David Mix Barrington, Chi-Jen Lu, Peter Bro Miltersen, Sven Skyum
We show that searching a width k maze is complete for \Pi_k, i.e., for
the k'th level of the AC^0 hierarchy. Equivalently, st-connectivity
for width k grid graphs is complete for \Pi_k. As an application,
we show that there is a data structure solving dynamic st-connectivity
for constant ... more >>> | {"url":"https://eccc.weizmann.ac.il/keyword/14497/","timestamp":"2024-11-14T21:55:46Z","content_type":"application/xhtml+xml","content_length":"19440","record_id":"<urn:uuid:e31874b1-933a-48d0-8ce5-5b21a2a6cef3>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00517.warc.gz"} |
What Is The Largest Math Equation In World - Tessshebaylo
What Is The Largest Math Equation In World
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PHYSICS: Equilibrum of Forces – DTW Tutorials
PHYSICS: Equilibrum of Forces
2K Views November 25, 2020 Be first to comment
Equilibrium of forces
(A) Principle of Moments
The principle of moments states that if a body is equilibrium then the sum of the clockwise turning moments acting upon it about any point equals the sum of the anticlockwise turning moments about
the same point.
Note: some of moments = F[2] x[2 ]– F[1]x[1 ]= 0,
Or F[1]x[1] = F[2]x[2 ]
Moments of a force/torque
The moment of a force about a point is defined as the product of the force and the perpendicular distance from the point to the line of action of the force. Moment = Fxh sin [S:O:S].
A couple is a system of two parallel and equal but opposite forces not acting along the same lines.
The turning moment of a couple is the product of one of the forces and the perpendicular distance between the lines of action of the two forces.example of couple can be seen in the action of a
corkscrew, or turning a water tap on or off.
Picture of a water tap
Source: www.publicdomainvectors.org
Picture of a corkscrew
Source: www.common.wikimedia.org
(B)conditions for equilibrium of rigid bodies under the action of parallel and non-parallel forces
*For the action of parallel coplanar forces
I.The algebraic sum of the forces acting on the body in any direction must be zero. In other words, the sum of upward forces must be equal to the sum of downward forces, and similarly for those
forces along the other directions.
II.Moments: The algebraic sum of the moments of all the forces acting about any point must be zero; the sum of clockwise moments about any point must be equal to the sum of anticlockwise moments
about the same points.(the principle is also known as the principle of moments).
*For the action of non-parallel forces coplanar forces
I.The algebraic sum of the horizontal components equals zero. £[x] = 0
The algebraic sum of the vertical components equals zero.£[x] = 0
II.Moments: The algebraic sum of the moments of all the forces about any point in the plane must be zero. That is, the sum of the clockwise moments must be equal to the sum of anticlockwise moments
about the same point.
Notes: equilibrium of a large body or long body tend to move the body in a straight line, but they can also have the effect of causing rotation. We have to consider moments as well.
The triangle of forces
When a body is in equilibrium under the action of three forces all acting in the same plane, the three forces can be represented in magnitude, direction and sense by the three sides of a triangle
taken in order.
Stability of a body: Types of equilibrium
Stable equilibrium
A body is said to be in a position of stable equilibrium when, on receiving a slight displacement, it tends to its original position.eg. a rolling ball maintaining its original position.
Diagram of a stable equilibrium
Source: www.commons.wikimedia.org
Unstable equilibrium
A body is said to be in position of unstable equilibrium when, on receiving a slight displacement, it tends to move on, farther away from its original position.eg. a ball standing on a curve edge.
Diagram of unstable equilibrium
Source: www.commons.wikimedia.org.
Effect of position of center of gravity on stability
Diagram of the stability of a chair
Source: www.commons.wikimedia.com
A low armchair is more stable than a high chair or a tall stool. A chair which is normally quite stable can be put into an unstable position by tilting it back on its legs.
1. Defined movement of a force ?
Answer: Moment of force about a point is defined as the product of the force and the perpendicular distance from the point to the line of action of the force.
2. Defined the system couple?
Answer: A couple is a system of two parallel and equal but opposite forces not acting along the same lines.
3. Explain the concept of stable equilibrium
Answer: A body is said to be in a position of stable equilibrium, when on receiving a slight displacement it tends to return to its original position.
4. Mention the three(3) types of equilibrium end explain them
Stable equilibrium: a body is said to be in a position of stable equilibrium when on receiving a slight displacement it tends to return to its original position.
Unstable equilibrium: a body is said to be in position of unstable equilibrium when on receiving a slight displacement it tends to move on, farther away from its original position.
Neutral equilibrium: a body is in equilibrium when on receiving a slight displacement it tends to come to rest in its new position.
5. Name two examples of couples in concept of equilibrium
Answer: I. A water tap II. A corkscrew
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Frontiers | Initial Validation for the Estimation of Resting-State fMRI Effective Connectivity by a Generalization of the Correlation Approach
• ^1School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, United States
• ^2Laboratory of Brain and Cognition, Human Neuroscience Institute, Department of Human Development, Cornell University, Ithaca, NY, United States
• ^3Nancy E. and Peter C. Meinig School of Biomedical Engineering, Cornell University, Ithaca, NY, United States
Resting-state functional MRI (rs-fMRI) is widely used to noninvasively study human brain networks. Network functional connectivity is often estimated by calculating the timeseries correlation between
blood-oxygen-level dependent (BOLD) signal from different regions of interest (ROIs). However, standard correlation cannot characterize the direction of information flow between regions. In this
paper, we introduce and test a new concept, prediction correlation, to estimate effective connectivity in functional brain networks from rs-fMRI. In this approach, the correlation between two BOLD
signals is replaced by a correlation between one BOLD signal and a prediction of this signal via a causal system driven by another BOLD signal. Three validations are described: (1) Prediction
correlation performed well on simulated data where the ground truth was known, and outperformed four other methods. (2) On simulated data designed to display the “common driver” problem, prediction
correlation did not introduce false connections between non-interacting driven ROIs. (3) On experimental data, prediction correlation recovered the previously identified network organization of human
brain. Prediction correlation scales well to work with hundreds of ROIs, enabling it to assess whole brain interregional connectivity at the single subject level. These results provide an initial
validation that prediction correlation can capture the direction of information flow and estimate the duration of extended temporal delays in information flow between regions of interest ROIs based
on BOLD signal. This approach not only maintains the high sensitivity to network connectivity provided by the correlation analysis, but also performs well in the estimation of causal information flow
in the brain.
1. Introduction
Resting-state functional MRI (rs-fMRI) has been widely used to study the intrinsic functional architecture of the human brain based on spontaneous oscillations of the blood oxygen level dependent
(BOLD) signals (Biswal et al., 1995; Power et al., 2011; Smith et al., 2011; Yeo et al., 2011). One fruitful approach has been to examine the correlations between rs-fMRI timeseries at pairs of
regions of interest (ROIs) and use the correlations as a measure of connectivity strength between each pair (Wig et al., 2011; Sporns, 2011). The correlation method, though simple, plays a
fundamental role in evaluating functional connectivity in the human brain for both task-evoked networks (Cole et al., 2014; Sadaghiani et al., 2015) and resting-state networks (Power et al., 2013;
Hipp and Siegel, 2015; Sadaghiani et al., 2015). The relationships between correlation and the topological properties, including small-world organization, modular structure, and highly connected
hubs, has been studied in Zalesky et al. (2012). However, the direction of information flow between pairs of ROIs and the causality of information flow cannot be derived from standard correlation
methods. Reliable insight into the direction and causality of functional connections in the brain from BOLD signals would provide substantial breakthroughs in characterizing large-scale brain network
The BOLD signal is an indirect and sluggish measure of neuronal activity. Despite this, substantial insights have been gleaned by examining patterns of BOLD signals as proxies for functional
connectivity in the brain, and these are consistent with more direct and invasive observations (Foster et al., 2015). At every level of analysis, the brain demonstrates an organized network structure
(Bassett and Gazzaniga, 2011). So, even though neuronal activation occurs on the millisecond time scale, organized and structured activation patterns are also observed on the level of seconds, which
is within the range of BOLD signals and is important for understanding cognition. Causal information about the flow of information in the brain may be detected and estimated from the BOLD signals. It
remains critical, however, to evaluate methods of investigation against ground truth simulation in order to validate these methods.
Numerous methods for estimating functional or effective connectivity (Van Den Heuvel and Pol, 2010; Friston, 2011) have recently been evaluated against ground truth networks using simulated rs-fMRI
data (Smith et al., 2011). Functional connectivity can be quantified with a measure of statistical dependence such as correlation, whereas effective connectivity measures the directed causal
influence (Friston, 2011). In Smith et al. (2011), performance of both types of methods across a range of measures was mixed. Standard and partial correlation excelled at detecting the presence of a
connection. Other methods for estimating the direction of a connection varied from chance (Granger) to greater than 50% accuracy [Patel's Tau and pairwise LiNGAM(Linear, Non-Gaussian, Acyclic causal
Models)]. These results suggest that novel methods are needed to estimate directed connectivity from rs-fMRI data, particularly with a large number of ROIs, which are necessary for full coverage of
cortical and subcortical areas in the human brain. In this paper, we introduce a new method, prediction correlation, to the neuroimaging community and provide an initial validation of the approach.
Methods for estimating functional connectivity can be oriented toward estimating a real number describing strength of connectivity, which might be quite small, vs. estimating a binary connectivity,
which is present or absent, with possibly the addition of a strength of connectivity, in the form of a real number, for the case where a connection is present. Correlation and prediction correlation,
which is a generalization of correlation that we propose in this paper, are methods that estimate a real number that describes strength of connection. Subsequent processing can then be applied to
remove weak connections and/or organize the complete network into modular networks.
As is described in the following sections, testing on simulated rs-fMRI data with known ground-truth networks (Smith et al., 2011) demonstrates that prediction correlation is not only sensitive in
detecting network connections, as identified by standard correlation, but also achieves the highest accuracy on estimation of connection directionality among all approaches used in Smith et al.
(2011) (Section 3.1). In a “common driver” phenomena, when ROI 1 drives ROIs 2 and 3 but ROIs 2 and 3 do not directly interact, prediction correlation correctly detects strong 1 → 2 and 1 → 3
connections but not 2 → 3 or 3 → 2 connections (Section 3.2). Finally, extending Xu et al. (2014), we demonstrate the robustness of this method on experimental data and that prediction correlation
recovers previously identified brain network organization from experimental data (Section 3.3).
2. Methods Prediction Correlation
2.1. Fundamental Method
In what follows, we describe a methodology for analyzing rs-fMRI data using a generalization of the well-established correlation approach, which is to correlate the timeseries at two ROIs. The
generalization, denoted by “p-correlation” (“p” for “prediction”) is to replace correlation between the BOLD timeseries at two ROIs by correlation between the BOLD timeseries at one ROI and a
prediction of this timeseries. The prediction is the output of a mathematical dynamical system that is driven by the timeseries at the other ROI. More generally, the prediction could be based on
several, spatially discrete, ROIs. In this paper, we focus on the case where only one other ROI is used. We assume that the dynamical system is linear and has finite memory and that the memory
duration and parameters may be estimated from the BOLD timeseries. If the prediction of the timeseries is restricted to use only the current value of the timeseries that drives the dynamical system,
then p-correlation is the same as standard correlation. Therefore p-correlation is a generalization of correlation. Features of p-correlation include (1) the ability to indicate the directionality of
the interaction between two ROIs, (due to the fact that this prediction correlation is asymmetrical between two signals), and (2) the ability to evaluate the interaction based on casual information.
In the remainder of this section, we describe the p-correlation approach in detail. Consider the ordered pair of ROIs (i, j) and let x[i] (x[j]) denote the rs-fMRI timeseries at the i^th (j^th) ROI.
Both timeseries have duration N[x]. The x[j] signal is predicted from the x[i] signal by a linear time-invariant causal dynamical model with x[i] as the input and the prediction ${\stackrel{^}{x}}_{j
|i}$ as the output. This model can be described by an impulse response, denoted h[j|i], which is zero for negative times. We assume that the impulse response is of finite duration, with duration
denoted by N[h[j|i]]. In summary,
$x^j|i[n]=∑m=0Nhj|ihj|i[m]xi[n-m]. (1)$
The basic approach to estimate the coefficients of h[j|i] is to minimize the least squares cost
$J(hj|i)=∑n=0Nx-1(xj[n]-x^j|i[n])2. (2)$
We estimate the value of N[h[j|i]] and the values of the impulse response at the same time by restating the least squares problem as a Gaussian maximum likelihood estimator (MLE) with a known
variance for the measurement errors. The MLE allows a trade off of the accuracy of predicting the current data (i.e., minimizing ${J}$), which is best done by large values of N[h[j|i]], with the
accuracy of predicting when presented with new data, which is best done by smaller values of N[h[j|i]]. There are several approaches to quantifying this trade off including Akaike information
criteria (AIC) (Akaike, 1970, 1974; Sugiura, 1978; Hurvich and Tsai, 1989, 1993; Cavanaugh, 1997), Bayesian information criteria (BIC) (Schwarz, 1978), restricted maximum likelihood (REML) (Thompson,
1962; Patterson and Thompson, 1971), minimum description length (Rissanen, 1978) and minimum message length (Wallace and Boulton, 1968). We have focused on AIC because it leads to easily computed
problem formulations (Equation 3). AIC realizes this balancing goal by minimizing the sum of two terms, one term that characterizes the prediction error of the dynamic system through the least
squares cost ${J}\left({h}_{j|i}\right)$ and a second term that depends on the durations N[h[j|i]] and N[x]:
$AIC={Nxlog(2πNx-Nhj|iJ(hj|i)) if Nx/Nhj|i≥40 + Nx + Nhj|i Nxlog(2πNx-Nhj|iJ(hj|i)) otherwise + Nx2 + Nhj|i2-Nx + Nhj|iNx-Nhj|i-1. (3)$
See Equation S1 (Supplementary Material) for BIC.
Simultaneous minimization of Equation 3 with respect to both h[j|i], which occurs only in the ${J}\left({h}_{j|i}\right)$ term, and N[h[j|i]] determines the duration and the value of the impulse
response. The integer minimization over N[h[j|i]] is computed by testing each value in a predetermined range of values, i.e., 1,2, …, D seconds. Then, for each value of N[h[j|i]], the minimization
with respect to h[j|i] involves only minimizing ${J}\left({h}_{j|i}\right)$. Since the dynamical system describing how x[i] influences x[j] is separate from the dynamical system describing how x[j]
influences x[i], the approach described here can lead to a directed rather than undirected graph of interactions between ROIs.
Once h[j|i] and N[j|i] are estimated, the output of the dynamical system, which is the prediction ${\stackrel{^}{x}}_{j|i}$, can be computed, and then the correlation of x[j] and ${\stackrel{^}{x}}_
{j|i}$, which is the so-called p-correlation, denoted by ρ[j|i], can be computed. We use “correlation” and ρ[j,i] for the standard approach (i.e., the standard correlation between x[j] and x[i]).
Let the total number of ROIs be denoted by N[ROI]. P-correlation is an asymmetric N[ROI] × N[ROI] matrix, where the asymmetry follows from ρ[j|i] ≠ ρ[i|j]. Furthermore, p-correlation includes lags of
the x[i] signal since the dynamical system output at time n, ${\stackrel{^}{x}}_{j|i}\left[n\right]$, depends on the input at its current and previous times, i.e., x[i][n], x[i][n − 1], …, x[i][n − N
[h[j|i]] + 1]. If N[h[j|i]] = 1 (i.e., no lags) and h[j|i][0] ≥ 0 then ρ[j|i] is the correlation between x[j] and x[i] so that ρ[j|i] = ρ[j,i] and the approach of this paper exactly reduces to the
standard approach. In Section 2.2.1, we describe a constraint such that h[j|i][0] ≥ 0 is always achieved. The p-correlation method does not depend upon the sampling rate (TR) which allows for
collapsing across different scan sites or studies. The entire algorithm is shown in Figure 1. Matlab software implementing p-correlation is available at http://www.mathworks.com/matlabcentral/
FIGURE 1
Figure 1. Block diagram and sub-block diagrams describing the computation of p-correlation for one pair of ROIs.
2.2. Specializations of the Fundamental Method
In Section 2.1 we defined p-correlation and described a practical method for its computation. The result is an asymmetric matrix of connection strengths for each subject. This fundamental method can
be specialized for particular applications, often based on user's interests and what the user knows about the details of the applications. Several such specializations are described in the following
2.2.1. Constraints on the Least Squares Problems
If the user has information on the type of interactions that are present, then this information can be used as a constraint on the least squares problem that determines the impulse response which is
the basis of the prediction. For example, as in the simulated data of Smith et al. (2011), the interactions are all positive. Constraining the impulse response values h[j|i][n] to be nonnegative has
implications for the values of ρ[j|i]. Let R[j|i] be the covariance of x[j] and ${\stackrel{^}{x}}_{j|i}$. R[j|i] is related to the covariance of x[j][n] and x[i][n − m] (i.e., the m-lagged
covariance of the two signals, denoted by R[j,i][m]) by ${R}_{j|i}={\sum }_{m=0}^{{N}_{{h}_{j|i}}-1}{R}_{j,i}\left[m\right]{h}_{j|i}\left[m\right]$. The covariance R[j|i] is the numerator of ρ[j|i].
Therefore, if all the lagged covariances are positive and we require the estimated values of h[j|i][m] to be positive then we are assured of getting a nonnegative value for R[j|i] and for the
p-correlation ρ[j|i]. In the traditional functional connectivity analysis, when global signal regression is applied to rs-fMRI timeseries data, the valid inference of negative correlations cannot be
made (Murphy et al., 2009; Saad et al., 2012), and only positive correlations are interpreted. In this situation, the nonnegative “constrained” estimation approach is appropriate.
2.2.2. Thresholding ρ[j|i]
Three natural methods for thresholding ρ[j|i] are described in this section.
Even with h[j|i][n] ≥ 0, it may be that p-correlation is not positive because one or more of the m-lagged covariance values are negative. Therefore, if non-negativity is required, we replace all
negative ρ[j|i] values by zeros. One reason for seeking to have ρ[j|i] non negative is mean signal regression in the preprocessing of the fMRI data which makes it difficult to interpret negative
correlations. However, alternative preprocessing which omits mean signal regression (Jo et al., 2013) removes this requirement.
The previous paragraph concerned thresholding at value 0. Higher data-dependent minimum thresholds are often used for correlation and the same approach can be applied to p-correlaton. A standard
approach (Power et al., 2011) is to order the values of correlation and leave the top s percent of values unchanged and set the remaining values to zero. In other words, the threshold γ(s) is set to
be the 100-s percentile of all values in the p-correlation matrix.
In some problems the interactions are known to be unidirectional, e.g., in the simulated data of Smith (Smith et al., 2011). In this situation, a third thresholding method, which makes p-correlation
unidirectional, is natural. The threshold is to consider the two transpose-related elements of the matrix and set the smaller to zero and leave the larger unchanged.
Each of the thresholding methods is a nonlinear operation applied to the matrix of ρ[j|i] coefficients. Each can be applied to any matrix M to give an output matrix N, in particular, in the order of
the previous three paragraphs,
$Nij={Mij, if Mij≥00, otherwise, (4a)$
$Nij={Mij, if Mij≥γ(s)0, otherwise, (4b)$
$whereγ(s) is the 100-s percentile of all values inM, andNij={Mij, if Mij≥Mji0, otherwise. (4c)$
The thresholding approach forms a N[ROI] × N[ROI] matrix of thresholded connection weights, from which the network is computed.
2.2.3. Averaging over Subjects
Some investigations, e.g., Smith et al. (2011) and Laumann et al. (2015), are interested in estimating subject-by-subject details, but in many other investigations on functional networks of human
brain using experimental data, e.g., Power et al. (2011, 2013), Schaefer et al. (2014), and Gordon et al. (2016), there is averaging over subjects in order to improve the SNR. Just as the
thresholding methods (Section 2.2.2), which are nonlinearities that can be applied to any matrix, the averaging we use can be applied to any family of matrices M[k] (k ∈ {1, …, K}, where K is the
number of subjects) to give an output matrix N via $N=\frac{1}{K}{\sum }_{k=1}^{K}{M}_{k}$. The functional network estimated by the averaged p-correlation matrix can be further clustered into
sub-networks through a graphic theoretic analysis.
2.3. Extension to Multi-Subject Processing
There is a recent interest in estimating effective networks from multiple subjects while accommodating the heterogeneity of the group (Ryali et al., 2016; Gates and Molenaar, 2012; Smith, 2012).
Specifically, the IMaGES algorithm (Ryali et al., 2016) estimates one generalized network from a group by assuming all subjects are homogeneous, and the GIMME algorithm (Gates and Molenaar, 2012) can
further refine the estimate for each individual subject from the general information estimated from the whole group. IMaGES and GIMME are based on existing single-subject methods, specifically GES
for IMaGES and uSEM and euSEM for GIMME and, when applied to groups of appropriate size, both GIMME and IMaGES provide more accurate estimates of effective connectivity than the single subject
methods on which they are based (Ramsey et al., 2011; Gates and Molenaar, 2012).
Information concerning groups of subjects could also be used in p-correlation. One approach would be to replace the h[j|i] in Equation 1 by ${h}_{j|i}^{g}+{h}_{j|i}^{l}$, where ${h}_{j|i}^{g}$ is the
group component common to all subjects, and ${h}_{j|i}^{l}$ is the component unique to the specific subject l. In this approach, Equation 1 would be generalized to
$x^j|il[n]=∑m=0Nhj|ighj|ig[m]xig[n-m]+∑k=0Nhj|ilhj|il[k]xil[n-k]l (5)$
where ${N}_{{h}_{j|i}^{g}}$ and ${N}_{{h}_{j|i}^{l}}$ are the probably different durations of the two components of the causal finite-duration impulse response. There are two issues when using
Equation 5. First the AIC analysis must be generalized in order to determine two impulse response durations where one is common to the entire group of subjects. Second, in order to require the least
squares to use the group impulse response and not just set it to zero, a regularizer such as ${\sum }_{m=0}^{{N}_{{h}_{j|i}^{l}}}{\left({h}_{j|i}^{l}\left[m\right]\right)}^{2}$ must be added to the
least squares cost. While both of these issues can be addressed, in the current paper, we only focus on the individual analysis, which may be the only meaningful option under certain circumstances,
i.e., a clinical environment.
3. Results
3.1. Application on Simulated Data
3.1.1. Data Source: Simulated BOLD Timeseries
Simulated fMRI timeseries from the laboratory of S. M. Smith are documented (Smith et al., 2011) and available on-line (http://www.fmrib.ox.ac.uk/analysis/netsim/). These timeseries have been used as
benchmark simulated fMRI data for testing effective connectivity (Ramsey et al., 2011; Smith et al., 2011; Gates and Molenaar, 2012; Hyvärinen and Smith, 2013). The simulations are based on a variety
of underlying networks of different complexity and can be described as having three levels. First there is a neural level which is a stochastic linear vector differential equation which produces a
neural timeseries for each ROI. Second, for each ROI, there is a nonlinear balloon model driven by the corresponding neural timeseries which produces a vascular timeseries. Third, for each ROI, the
fMRI timeseries is the vascular timeseries plus thermal noise. To simulate preprocessing of fMRI data, a highpass filtered at a cutoff frequency of 1/200 s was applied to each simulation (most
recently revised on Aug. 24, 2012 based on the website www.fmrib.ox.ac.uk/analysis/netsim). The current paper considers the first four sets of simulations from Smith et al. (2011), Sim1−Sim4, which
are the four most “typical” network scenarios provided in Smith et al. (2011), and which are based on different underlying networks with sizes 5, 10, 15, and 50 ROIs, respectively.
These synthetic fMRI timeseries were sampled every 3 s (TR = 3s) and the total duration is N[x] = 10 mins. All four simulations have 1% thermal noise and the hemodynamic response function (HRF) used
in the second step has standard deviation of 0.5 s. The simulation is repeated for each of 50 subjects.
3.1.2. Specialization on P-Correlation for the Processing of the Simulated Data
The algorithm is shown in Figure 2. Given that the interactions are all positive in the simulated data, it is natural to apply the nonnegative constraint on the least squares problem so that no
negative impulse responses are allowed. Although unconstrained p-correlation is also computed on the simulated data, looking forward to Section 3.1.5, the numerical results indicate that the
constrained version is more appropriate.
FIGURE 2
Figure 2. Block diagram describing the specialization of p-correlation for simulated data. Nonzero entries are filled by colored dots with higher values represented by “hotter” colors and lower
values represented by “colder” colors, and zero entries are left as blank in the above matrices.
As is described above, the integer minimization over the impulse function duration, N[h[j|i]], is computed by testing from 1 second up to D seconds. Assuming that knowledge of the behavior of a ROI
over the past 15 seconds is sufficient to describe its effect on a second ROI, we restricted the temporal window for directional influence between ROIs to no more than 15 s, i.e., D = 15 s.
Next, we consider the choice of threshold, s in Equation 4b. We use this method in order to exploit all of the a priori knowledge about the simulated data. Since the underlying ground truth networks
for the simulated fMRI timeseries, denoted by a[j|i], are given, the threshold value s is among our prior knowledge as is described below. We denote ROIs that are involved in the connections of the
ground truth network as active ROIs. All connections involving the active ROIs are connections of interest (COIs), including connections that are actually absent such as the reverse connection in an
unidirectional interaction. The value of s is then the ratio of the number of COIs and the number of all possible connections, which gives s = 40, 22, 16, and 4 percent for the four simulations,
respectively. An example of computing s for a 5-node network is shown in Figure 3.
FIGURE 3
Figure 3. Example calculation of the threshold s for a 5-node network. (A) The network with activated ROIs shown in orange. The number of all possible connections is 5^2 = 25. (B) The 6 COIs, where
the dashed lines are connections that do not existed in the ground truth but still are considered interesting. Therefore, s = 6/25 for this network.
For the Smith simulated data, we have additional prior knowledge that the networks contain only unidirectional connections. Therefore, as is also done in Smith et al. (2011), we compare our estimated
network d[j|i], which includes the unidirectional condition, with the ground truth network a[j|i]. The estimated network d[j|i] is the output of Equation 4c where the input is the thresholded network
3.1.3. Performance Criteria
To compare the computed and ground truth networks, we define “accuracy,” denoted by ${A}$. In particular, ${A}$ is defined to be the mean fractional rate of detecting the correct directionality of
true connections. Specifically, it is defined to be
$A=∑i=1NROI∑j=1NROI1{aj|i>0}1{dj|i>0}∑i=1NROI∑j=1NROI1{aj|i>0}, (6)$
where 1{L} is 1 if L is true, and 0 otherwise. Like the computation of the “d-accuracy” introduced in Smith et al. (2011), ${A}$ evaluates the percentage of the correct directionality (${A}$ is
between 0 and 1). The threshold operation introduced above (Section 3.1.2) differentiates the performance of directional analytical methods based on their sensitivity. The more sensitive the method
is, the more true connections it can detect. Notice that application of the threshold s leads to d[j|i] values that are almost certainly far from zero or exactly zero. Computing the accuracy ${A}$
after the threshold operation tells the directionality after knowing the presence of the connections, which enables us to evaluate the overall performance of sensitivity and directionality of a
directional analytical method.
3.1.4. Alternative Methods for Effective Networks Estimation
P-correlation and four alternative methods from Smith et al. (2011), specifically, “Granger B1,” “Gen Synch S1,” “LiNGAM,” and “Patel's conditional dependence measure,” were compared by the accuracy
criteria (${A}$), since under both synthetic and experimental scenarios, these methods have been tested and have relatively good performances among all the others (Smith et al., 2011; Dawson et al.,
2013). The computation of these methods were done by software provided by Prof. S.M. Smith. Granger B1, a pairwise Granger causality estimation method which provides the best performance among
Granger causality approaches (Smith et al., 2011; Dawson et al., 2013), uses the Bayesian Information Criterion to estimate the lag up to 1 TR. Gen Synch S1 is a nonlinear synchronization method with
respect to the time lag 1 TR. It “evaluates synchrony by analyzing the interdependence between the signals in a state space reconstructed domain" (Dauwels et al., 2010, p. 671). The LiNGAM (Linear,
Non-Gaussian, Acyclic causal Models) algorithm is a global network model utilizing higher-order distributional statistics, via independent component analysis, to estimate the network connections.
Patel's conditional dependence measure investigates the causality from the imbalance between two conditional probabilities, P(x[j]|x[i]) and P(x[i]|x[j]). P-correlation, Granger B1, Gen Synch S1 and
LiNGAM all compute an asymmetric matrix filled with real-number connection weights, analogous to our c[j|i]. In all cases, the unidirectional prior knowledge is applied analogous to our
transformation from c[j|i] to d[j|i]. For the Patel method implemented by Smith et al. (2011), the thresholding operation was applied on “Patel's κ bin 0.75” matrix, while the directionality was
determined by “Patel's τ bin 0.75” matrix.
In addition to the algorithms included in Smith et al. (2011), IMaGES (Ryali et al., 2016) and uSEM (Kim et al., 2007) which is the estimation method for resting-state fMRI employed by GIMME
algorithm, have also been tested on the same set of simulated data (Ramsey et al., 2011; Gates and Molenaar, 2012). Results reported in Ramsey et al. (2011) and Gates and Molenaar (2012) show that
their estimation based on the single subject is either similar to or less good than the best-performing method provided in Smith et al. (2011).
Comparing p-correlation with alternative methods of estimating effective connectivity, p-correlation provides a full asymmetric matrix for each subject independent of all other subjects, in which
each entry, like correlation, predicts a connection strength between two ROIs. The ability to compute results based on an individual subject means that p-correlation can potentially be used in a
clinical environment. This full asymmetric matrix of p-correlations can be thresholded as desired and/or further processed as desired using another algorithm, i.e., a graph analytic algorithm. In
addition, p-correlation can process networks with hundreds of ROIs while GIMME is limited to 3–25 ROIs [Page 3 of GIMME Manual (Version 12)]. Furthermore, p-correlation estimates the temporal causal
relation in the form of lagged impulse response in addition to the spatial causal relation between any pair of ROIs. In contrast, some alternative algorithms (e.g., IMaGES) estimate a sparse graph of
interactions, and thus solve a somewhat different problem than the p-correlation method. Other algorithms have been developed as post-processing algorithms, which cannot detect connections, but only
estimate direction if connections are detected by other methods, e.g., correlation. Among them, pairwise LiNGAM (Hyvärinen and Smith, 2013) achieved success on Smith's data (Smith et al., 2011).
Several algorithms, such as Patel's τ, LiNGAM and pairwise LiNGAM, chose one of the two possible directions for each pair of ROIs. Such unidirectionality may be appropriate in some situations.
Alternative algorithms, including p-correlation, provide strengths for both directions, where the two strengths may be quite different when one direction is dominant.
3.1.5. Results on Simulated Data
The methods described in this paper were implemented in Matlab software, which is available upon request, and were applied to four of Smith's fMRI simulations (Smith et al., 2011). The four
simulations are Sim1−Sim4 which have a variable number of ROIs (5, 10, 15, 50) but no confounding variables.
The p-correlation method is based on estimation of a linear time-invariant causal dynamic model. The sample means of the duration of either constrained or unconstrained impulse responses are 3.34,
3.58, 3.64, and 3.76 s for the four simulations, respectively. By limiting the impulse response duration to 1 TR, it was verified that p-correlation with constraint on Least Squares is equivalent to
the standard correlation as is described in Section 1. After thresholding the p-correlations computed with the nonnegative constraint on the coefficients of the linear system, an asymmetric matrix of
connection weights c[j|i] for each subject was obtained.
The same specifications for processing of the simulated data, in particular, the same choice of the s threshold (Equation 4b) and the knowledge of unidirectionality (Equation 4c), have also been
applied to the results of four alternative methods introduced in Section 3.1.4. The performance of all five methods was evaluated by the accuracy criteria ${A}$ (Equation 6) for each subject. Figure
4 shows the input to the accuracy criteria ${A}$, i.e., a[j|i] and d[j|i], for Subject 14 of Sim2.
FIGURE 4
Figure 4. Images of a[j|i] (for ground truth) and d[j|i] (for constrained p-correlation), and quantities analogous to d[j|i] (for Granger B1, Gen Synch S1, LiNGAM, and Patel) for Subject 14 of Sim2.
Each image uses the same ordering of colors, but has different range of numerical values.
The mean and standard deviation of accuracy for each simulation, i.e., the average and square root of the sample variance of ${A}$ (Equation 6) over all 50 subjects, were computed and the results are
tabulated in Table 1. For all four simulations, constrained p-correlation achieved the highest accuracy compared to other methods. The unconstrained p-correlation is less appropriate when applied to
a network with all positive connection weights. We also computed the mean and standard deviation of ${A}$ for pairwise LiNGAM, which gives 0.566 ± 0.138, 0.656 ± 0.206, 0.510 ± 0.119, and 0.506 ±
0.056 for four simulations, respectively. The result shows the highly accurate directionality that pairwise LiNGAM can achieve in this particular unidirectional network setting. Histograms displaying
the distribution of accuracy for the five methods for each simulation are shown in Figure 5. The histogram of the unconstrained p-correlation method is included in Figure S1. The superior performance
of p-correlation is demonstrated by the fact that the bulk of the histogram is further to the right, and the left tail is less massive.
TABLE 1
Table 1. Comparison of the mean and standard deviation of accuracy over 50 subjects among different methods.
FIGURE 5
Figure 5. Accuracy histogram for Granger B1, Gen Synch S1, LiNGAM, Patel, and constrained p-correlation.
3.2. The Performance of Correlation And P-Correlation on Common Drivers
A “common driver” situation is the case where ROI 1 drives ROIs 2 and 3 but ROIs 2 and 3 do not directly interact. The challenge is to correctly detect the 1 → 2 and 1 → 3 connections without
detecting 2 → 3 or 3 → 2 false connections. In order to focus exclusively on this situation, we have computed synthetic data from the three-ROI network shown in Figure 6 and defined by
$x1[n+1]=a1x1[n]+b1w1[n] (7)$
$x2[n+1]=a2x2[n]+a21x1[n]+b2w2[n] (8)$
$x3[n+1]=a3x3[n]+a31x1[n]+b3w3[n] (9)$
where $w\left[n\right]={\left[{w}_{1}\left[n\right],{w}_{2}\left[n\right],{w}_{3}\left[n\right]\right]}^{T}$ is an independent and identically distributed Gaussian stochastic process with mean 0 and
variance I[3] (the 3 × 3 identity matrix). Zalesky et al. (2012) consider mathematical models of this type and give theoretical results for correlations. The system is initialized in the steady state
and simulated for 1,000 steps, N[x] = 1, 000. We consider only a[1] = a[2] = a[3] = 0.8 (so that all ROIs have the same intrinsic memory duration) and b[1] = b[2] = b[3] = 0.2 (so that all ROIs have
the same intrinsic noise power, and the intrinsic noises are all independent). We consider the following cases: (1) no driving: a[21] = a[31] = 0, (2) weak driving: a[21] = a[31] = 0.1, (3) strong
driving: a[21] = a[31] = 0.4, and (4) asymmetrical strong driving: a[21] = 0.4, and a[31] = 0.1.
FIGURE 6
Each simulation was repeated for 50 subjects. Let the maximum allowable duration of the impulse response be 3 samples. By using the specialization of p-correlation for Smith simulated data, as is
described in Section 3.1.2, a directed graph d[j|i] is estimated by p-correlation (Figure 2) and the correlation matrix is computed for each subject. The steady state covariance of Equations 7–9 is
the correlation matrix. In Case (1), the mean and standard deviation of nonzero entries of ρ[j|i] with constrained least squares (Section 2.2.1) are 5.384e-04 ± 0.072. This number becomes 0.058 ±
0.043 when unconstrained least squares is applied. The smaller magnitude of the results using constrained least squares indicates that taking advantage of the prior knowledge that the weights are
positive (i.e., a[1] = a[2] = a[3] = 0.8) provides improved performance in this case. In Cases (2) and (3), both the constrained and the unconstrained least squares achieve a 100% accuracy (Equation
6) for each subject. In the fourth case, the constrained or the unconstrained least squares gives an average of 0.800 ± 0.247 accuracy over all 50 subjects. We also tested N[x] = 200, 500, 5,000 for
all four cases. Notice that as N[x] goes large, correlations become closer to the steady state and the accuracy computed by the p-correlation method increases as well.
In addition, p-correlation estimated the correct hierarchy on the three pairs of connection weights, which are consistent with “strong,” “weak,” and “non-” connections in the ground truth network. It
also shows the correct direction of connections in a pair by a stronger weight. The constrained least squares (Section 2.2.1) provides a slightly superior result than the unconstrained approach.
Specifically, larger numerical differences between the zero and nonzero entries, as well as between the asymmetric strong weights, were shown. On average across all 50 subjects, p-correlation used an
impulse response duration of 1.007 samples for all four cases for both constrained and unconstrained approaches. In addition, in Case (3) (asymmetric strong weights), correlation mis-detected the
connection between node 2 and 3, specifically the 2–3 correlation was the highest correlation value among the three pairs, whereas p-correlation, for both the constrained and unconstrained
approaches, estimated this value as the lowest of the three pairs thereby avoiding the error in the correlation results.
3.3. Performance on Experimental fMRI Data
While the tools described in this paper can be assembled into many algorithms, we use only one algorithm, which is shown in Figure 7, to further characterize (Xu et al., 2014), a cohort of 132
subjects from the 1,000 Functional Connectomes Project (http://www.nitrc.org/projects/fcon_1000/) (Biswal et al., 2010). This data is provided from different scanning sites, and thus has variable
sampling rates (TRs = approximately 1–3 s, mean ± standard deviation of 2.3 ± 0.4s). The scan duration also varied from 119 to 295 TRs (mean standard deviation of 167.5 ± 41.7). The data from the
whole brain were preprocessed (Anderson et al., 2011), linearly detrended and bandpass filtered (retaining signal between 0.001 and 0.1 Hz), and motion scrubbed (Power et al., 2012) with the
threshold set to 0.2. The preprocessed rs-fMRI BOLD signal was extracted from N[ROI] = 264 spherical ROIs each with a 10mm diameter. We combine our p-correlation ideas with the widely-used (Power et
al., 2011, 2012; Lahnakoski et al., 2012; Gordon et al., 2016) Infomap graph analytical algorithm (Lancichinetti and Fortunato, 2009) to determine networks within the set of 264 ROIs.
FIGURE 7
Figure 7. Block diagram describing the specialization of p-correlation for the experimental data. Nonzero entries are filled by colored dots with higher values represented by “hotter” colors and
lower values represented by “colder" colors, and zero entries are left as blank in the above matrices.
As a function of the value of the threshold s, Infomap creates a variable number of networks. Following Power et al. (2011, Figure 1), the network stability over a range of threshold s ∈ {2, …, 10}
using correlations and p-correlations are shown in Figure 8, in which different networks are represented by different colors. Similar to Power et al. (2011) (the first figure in Figure 8), we note
that the assignment of ROIs to networks remains relatively constant over all values of the threshold s, illustrated by the constant horizontal bands in different colors. Also, networks are
hierarchically refined as s rises. In summary, the number of networks increases as the value of s decreases, and p-correlation replicated the brain network organizations that were detected by
correlation. The network results are consistent with the network organizations detected in Power et al. (2011).
FIGURE 8
Figure 8. The stability of networks across various thresholding criteria (s). The white regions indicate ROIs that belong to networks with less than four ROIs.
In order to test the robustness of the p-correlation calculation, all 132 subjects were randomly divided into two equal cohorts, and each cohort was separately processed. The average of p-correlation
connection strength ${\rho }_{j|i}^{+}$ across all subjects in the cohort, which is denoted by ${\stackrel{̄}{\rho }}_{j|i}^{+}$, is shown as a scatter plot in Figure 9A [in Figure 9, all (0,0) points
are removed]. The linear least squares prediction of Cohort 2 from Cohort 1 is a close fit to the data (r^2 = 0.87) and is nearly a 45° diagonal line (${\stackrel{̄}{\rho }}_{j|i}^{\mathrm{\text
{Cohort 2}}}$ = 1.013${\stackrel{̄}{\rho }}_{j|i}^{\mathrm{\text{Cohort 1}}}+$ 0.032), thereby indicating the robust nature of p-correlation. Following the same procedure, the average of correlation
connection strength ${\rho }_{i,j}^{+}$ across all subjects in the cohort, which is denoted by ${\stackrel{̄}{\rho }}_{i,j}^{+}$, is shown in Figure 9B. Comparing Figures 9A,B indicates that the
p-correlation achieves the same robustness as correlation. Additional plots in which no points are removed are included in the Figure S2.
FIGURE 9
Figure 9. Scatter plot of results for the two cohorts. (A) P-correlation. (B) Correlation. The red line is the Least Squares fit for predicting Cohort 2 from Cohort 1. Only positive values are used
in the Least Squares calculation and shown in the plot.
4. Discussion
Standard correlation has been widely used to analyze functional connectivity from rs-fMRI timeseries between prespecified ROIs. Prior work has shown its high sensitivity for detecting the existence
of network architectures under both simulated and experimental scenarios (Smith et al., 2011; Dawson et al., 2013). This paper describes methodology for analyzing rs-fMRI data using a generalization
of well-established correlation ideas. The generalization, denoted by “p-correlation” (“p” for “prediction”), is to compute the correlation between the j^th signal and an optimal linear
time-invariant causal estimate of the j^th signal based on the i^th signal. In this way, it captures additional features concerning the interaction between two ROIs, specifically, the causality and
directionality of the information flow on which the interaction depends. Based on the finite-memory linear time-invariant causal model, p-correlation allows the memory duration to be different in the
two directions for one pair of ROIs and also to be different for different pairs of ROIs. In contrast, structural vector autoregressive models (Kim et al., 2007; Chen et al., 2011) are assumed to
have the same memory duration across all ROIs. P-correlation is a generalization of standard correlation ideas because, if the estimate of the j^th signal based on the i^th signal is restricted to
use only the current value of the i^th signal, then p-correlation and standard correlation have the same magnitude.
Testing p-correlation on simulated fMRI data provided in Smith et al. (2011), the greater performance accuracy of p-correlation, which uses lagged information from the BOLD timeseries, demonstrates
the importance of causal information which is missing in standard correlation. In our results, the mean duration of the impulse response estimated by AIC using a search limited to a maximum duration
of 15s was roughly 4s. In these data, a search extending to 15 s is not a restriction on the maximum duration. As is described in Table 1, the accuracy of p-correlation on the simulated data of Smith
is about 0.5 (0.405–0.532). While higher levels are desirable, this performance exceeds the performance of many alternative algorithms on all four sets of simulations.
Many approaches have been introduced to assess functional or effective connectivity of rs-fMRI data. Smith et al. (2011) evaluated the validity of 38 approaches (Smith et al., 2011, Figure 4) using
simulated BOLD signals and a variety of performance measures. The methods tend to have different levels of performance for different measures, e.g., detection of a connection vs. determination of the
direction of a connection. The p-correlation approach introduced in this paper depends on causal dynamical models and so we focus on this particular aspect of previous work. Dynamic Causal Modeling
(DCM) has been used with some success to assess causal dynamics in fMRI data by relying on sophisticated models of neural dynamics. As discussed in Smith et al. (2011, p. 878), most existing DCM
algorithms require knowledge of external inputs (which are not known for rs-fMRI) although some variations may not (Daunizeau et al., 2009); all versions tend to be mathematically poorly conditioned;
and all versions fail to scale to networks with large numbers of ROIs which are necessary for experimental studies. In contrast, the p-correlation approach described in this paper scales similarly to
a correlation approach for which hundreds of ROIs are not a challenge (Xu et al., 2014).
Several versions of Granger causality analysis, based on multivariate vector autoregressive modeling, have been tested and performed poorly (Smith et al., 2011). Granger causality relies on
regression and comparison of two predictions. The first prediction is based purely on an autoregressive model of the signal at the ith ROI based on the past of the same signal. The second prediction
is based on regression of the signal at the ith ROI based on the past of the signal at the jth ROI and, possibly, an autoregression as in the first case. The sample covariances of the prediction
errors are then combined, essentially by taking the ratio of the sample covariances scaled by integers describing the amounts of data, to yield a statistic that is distributed according to the
Fisher-Snedecor F distribution. This statistic, indexed by i and j, is used to fill an asymmetric matrix. Although both are based upon lagged information there are important differences between
p-correlation and Granger causality. P-correlation is not a statistic comparing two possible dependencies but rather is a statistic measuring the accuracy of prediction using a particular dependency.
The motivation for the Granger causality statistic is dependent on the original Gaussian assumptions on the errors when linear regression is used to describe the ROI time series. P-correlation is
based on just the sample variance of the prediction error and does not have a Gaussian motivation which is advantageous if the BOLD signals lack Gaussian structure. Multivariate autoregressive
processes have been used as the basis for generative models for complete sets of ROIs. Such models, which focus on the effect of the past on the present, can be combined with structural equation
modeling (SEM) models, which focus on contemporaneous effects (Chen et al., 2011).
Multivariate autoregressive processes (MVAR) have been successfully used in neuroscience outside of fMRI, e.g., in order to describe signals from EEG experiments (Ding et al., 2000; Kus et al., 2004;
Babiloni et al., 2005; Wilke et al., 2008; Blinowska et al., 2009; Korzeniewska et al., 2011; Ligeza et al., 2016). Both MVAR, e.g., Equation 1 in Kus et al. (2004), and the linear regression model
used in this paper (Equation 1) are regression models which predict one timeseries from either all timeseries which include oneself (MVAR) or from the past of another timeseries (Equation 1). Both
predictions are characterized by impulse responses. The method introduced in Kus et al. (2004) determines the connection strength based on the impulse response, whereas p-correlation determines the
functional connectivity based on both the impulse response and the original timeseries. Existing literature, e.g., Valdes-Sosa (2005) and Davis et al. (2016), has shown the robust estimation of the
MVAR model by introducing sparse regression techniques, and the success of estimating functional connectivity through the sparse MVAR models. In addition, a conditional MVAR model, e.g., Ch 17.3 in
Schelter et al. (2006), may also be used to address the common driver problem. Other approaches to examining BOLD signal propagation using lags, as is done in p-correlation, have been highly
reproducible (Mitra et al., 2015). In this paper, a linear regression model (Equation 1) is used as the predictor in p-correlation to estimate the causal relation between a pair of BOLD signals.
Other lag-based predictors, e.g., MVAR based models, can also be adapted into the p-correlation concept, however, they would not have the result that duration of 1 sample (e.g., no lags) gives
standard correlation.
In addition to the algorithms used in Smith et al. (2011), which estimate the directional connectivity for single subject data sets, the IMaGES (Ryali et al., 2016; Ramsey et al., 2011) and GIMME (
Gates and Molenaar, 2012) algorithms use a group of subjects. While these algorithms provide better performance in situations where groups of subjects can be analyzed collectively, both algorithms
have challenges. The sparse graph estimated by IMaGES for a group of subjects does not tell the strengths of the connectivity and “will not reflect the variation of a group” (Mumford and Ramsey, 2014
, p.571). Similar to DCM's limitation on scalability, small networks with less than 25 ROIs are well analyzed by the GIMME algorithm. However, its performance on large-scale functional networks is
not known. As p-correlation can work with hundreds of ROIs, it can be used in evaluating large-scale brain networks. Furthermore, p-correlation can work on individual subjects so it potentially could
be applied to patient clinical data. Other algorithms that estimate direction after a connection is already detected also exist (Section 3.1.4). While such algorithms may be useful in some
circumstances, they do not allow for situations where both directions are present but of different strengths.
The Smith et al. (2011) simulated data has lower dimensionality than experimental brain data. For instance, in the simulation, connections are all unidirectional while most neural connections are
bidirectional. Additionally, in the simulations, most connections had a value of exactly zero. Furthermore, it introduces unrealistic noise and it has a large number of parameters that must be set
and which influence the resulting simulation (Wang et al., 2014). While the Smith et al. (2011) simulated data is not completely realistic and it is not a perfect test of p-correlation, this data
continues to be used (Smith et al., 2011; Gates and Molenaar, 2012; Ramsey et al., 2011; Hyvärinen and Smith, 2013; Ramsey et al., 2010), and the results continue to be discussed (Geerligs et al.,
2016). In this paper, we leveraged the same data used in Smith et al. (2011) for comparison with other published metrics, providing a broader context for these findings. We hope to use a broader
range of simulated data to further validate p-correlation in our future work.
In order to focus on the challenges of a “common driver,” we have produced additional synthetic data for the three ROI network of Figure 6 in which one ROI drives two other ROIs but the two other
ROIs do not directly interact. Using p-correlation in this network we found that p-correlation can identify the existence and direction of the interactions between the driving ROI and the other two
ROIs (even when the two interactions are of different strengths). Furthermore, p-correlation did not introduce false interactions between the two driven ROIs.
We have applied p-correlation to experimental data from the 1,000 Functional Connectome Project (Biswal et al., 2010). The p-correlation approach successfully replicated the modular architecture of
the local and distributed networks previously reported using standard correlation (Xu et al., 2014) (see Section 3.3, Figure 8). Highly correlated p-correlation values on the two different cohorts
also demonstrated that the p-correlation is highly reproducible and thus robust on experimental data. A current limitation of the p-correlation approach is that missing nodes cannot be accommodated,
thereby limiting an extension of this approach to lesioned populations.
Here we introduce a novel concept, the p-correlation, to estimate brain connectivity within well-characterized large-scale functional networks. The replication of previously observed network
architectures in experimental data and the performance against the ground truth in simulated data, both suggest that the p-correlation approach may hold promise for future investigations of the
brain's dynamic functional architecture.
Author Contributions
NX and PD designed the algorithm to achieve the neuroscience goals of RS. NX wrote the software and performed the analysis. NX, RS, and PD prepared the manuscript.
Conflict of Interest Statement
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
We are very grateful to Prof. S.M. Smith (University of Oxford) for providing simulation data and his software for applying Patel's conditional dependence measures and network measurements as
described in his paper (Smith et al., 2011) as well as for helpful discussion. We are also very grateful to Dr. Chandler Lutz for providing the pairwise Granger causality code, Dr. Rodrigo Quian
Quiroga for providing the generalized synchronization code, and Drs. Shohei Shimizu, Patrik Hoyer, and Aapo Hyvrinen for providing LiNGAM/FastICA. NX and PD are grateful for support by NSF Grant
Supplementary Material
The Supplementary Material for this article can be found online at: http://journal.frontiersin.org/article/10.3389/fnins.2017.00271/full#supplementary-material
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Keywords: resting-state fMRI, effective connectivity, functional connectivity, functional networks, correlation analysis
Citation: Xu N, Spreng RN and Doerschuk PC (2017) Initial Validation for the Estimation of Resting-State fMRI Effective Connectivity by a Generalization of the Correlation Approach. Front. Neurosci.
11:271. doi: 10.3389/fnins.2017.00271
Received: 16 July 2016; Accepted: 28 April 2017;
Published: 16 May 2017.
Edited by:
Pedro Antonio Valdes-Sosa
, Joint China-Cuba Laboratory for Frontier Research in Translational Neurotechnology, China
Copyright © 2017 Xu, Spreng and Doerschuk. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in
other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use,
distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Nan Xu, nx25@cornell.edu | {"url":"https://www.frontiersin.org/journals/neuroscience/articles/10.3389/fnins.2017.00271/full","timestamp":"2024-11-07T01:36:49Z","content_type":"text/html","content_length":"705050","record_id":"<urn:uuid:af36693c-0acd-44cc-b057-2bc7e048d731>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00783.warc.gz"} |
Exact and Bounded, Probabilitiy Proportional to Size (EBPPS) Sampling
Exact and Bounded, Probabilitiy Proportional to Size (EBPPS) Sampling
An EBPPS sketch produces a randome sample of data from a stream of items, ensuring that the probability of including an item is always exactly equal to the item’s size. The size of an item is defined
as its weight relative to the total weight of all items seen so far by the sketch. In contrast to VarOpt sampling, this sketch may return fewer than k items in order to keep the probability of
including an item strictly proportional to its size.
This sketch is based on: B. Hentschel, P. J. Haas, Y. Tian “Exact PPS Sampling with Bounded Sample Size”, Information Processing Letters, 2023.
EBPPS sampling is related to reservoir sampling, but handles unequal item weights. Feeding the sketch items with a uniform weight value will produce a sample equivalent to reservoir sampling.
Serializing and deserializing this sketch requires the use of a PyObjectSerDe.
class ebpps_sketch(*args, **kwargs)
Static Methods:
deserialize(bytes: bytes, serde: _datasketches.PyObjectSerDe) _datasketches.ebpps_sketch
Reads a bytes object and returns the corresponding ebpps_sketch
Non-static Methods:
__init__(self, k: int) None
Creates a new EBPPS sketch instance
k (int) – Maximum number of samples in the sketch
property c
The expected number of samples returned upon a call to get_result() or the creation of an iterator. The number is a floating point value, where the fractional portion represents the
probability of including a “partial item” from the sample. The value C should be no larger than the sketch’s configured value of k, although numerical precision limitations mean it may exceed
k by double precision floating point error margins in certain cases.
Computes the size in bytes needed to serialize the current sketch
Returns True if the sketch is empty, otherwise False
property k
The sketch’s maximum configured sample size
Merges the sketch with the given sketch
property n
The total stream length
Serializes the sketch into a bytes object
Produces a string summary of the sketch and optionally prints the items
Updates the sketch with the given value and weight | {"url":"https://apache.github.io/datasketches-python/5.0.2/sampling/ebpps.html","timestamp":"2024-11-14T05:07:14Z","content_type":"text/html","content_length":"16856","record_id":"<urn:uuid:59efa549-5af4-4f8a-95c4-2ed450945a06>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00623.warc.gz"} |
Is mars flat, too?
but, in order to have gravity, which it does... it must be moving UP... right?!
So if it doesn't follow the same principals... the entire theory falls apart!
(in other words, idiots, if other planetary bodies dont follow the same thing, it means your entire theory is not only flawed, its false. whats worse, is that other planetary bodies cannot be
mapped flatly, either because they rotate and we can observe it from earth.)
Mars is round, has gravitation, and is affected by Universal Acceleration. There is no flaw, you're just an idiot. | {"url":"https://www.theflatearthsociety.org/forum/index.php?topic=37472.0;prev_next=next","timestamp":"2024-11-04T02:27:25Z","content_type":"application/xhtml+xml","content_length":"94830","record_id":"<urn:uuid:e353fca5-4af6-4f33-83ed-d8bc1cf2e34a>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00150.warc.gz"} |
Examples of applications of two-stage method in calculations of statically indeterminate trusses
Last modified: 2017-08-29
The two-stage method belongs to the group of approximate methods of calculation of statically indeterminate systems because differences of the stiffness of component parts, joined in the same node,
are not considered in this method. The paper presents a comparison between the values of forces calculated in the members of a selected type of statically indeterminate plane truss under load forces
applied in a symmetrical way and in a non-symmetrical way. The comparison is drawn between values of forces calculated for the same trusses by the application of the two-stage method and by the
application of suitable computer software. The values of the forces determined by the computer software are considered as exact results because in this case the stiffness differences of the component
parts are taken into consideration. The point of the two-stage method is to remove, from the area of the statically indeterminate truss, a certain number of members which are equal to the statically
indeterminacy of the basic truss. Appropriate statically determinate truss is calculated in each stage what implies, that is one of very simple methods, like for instance Cremona’s method, can be
used for this purpose. The trusses calculated in each stage are of the same clear span and they have identical construction depth like the basic indeterminate truss, however they are loaded by forces
of half values compared to forces applied to the basic one. The final forces in the members of the statically indeterminate truss are calculated as resultants of forces defined in each stage for
members having appropriate positions in the area of the truss system.
An account with this site is required in order to view papers.
to create an account. | {"url":"https://sci-en-tech.com/ICCM/index.php/iccm2017/2017/paper/view/2486","timestamp":"2024-11-10T11:35:38Z","content_type":"application/xhtml+xml","content_length":"12913","record_id":"<urn:uuid:5a5f8c83-891b-47ad-800d-af0eebdddca0>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00127.warc.gz"} |
#Math friends: what is the name of the property of a set such that no member of the set is larger than all the other members combined?
1. #Math friends: what is the name of the property of a set such that no member of the set is larger than all the other members combined?
#Math friends: what is the name of the property of a set such that no member of the set is larger than all the other members combined?
• #Math friends: what is the name of the property of a set such that no member of the set is larger than all the other members combined?
{1, 3, 500} does not have this property; {3, 4, 5} does.
Assuming some way to compare members and some way to combine them.
• @evan I am not a math person either but with my mix of Google experience along with slight help from AI, maybe Ultrametric Space would be correct?
It's a type of triangle inequality that seems to might possibly be what you're looking for. | {"url":"https://bb.devnull.land/topic/9b8160ed-1c99-4ebd-bde8-90835b16697d/math-friends-what-is-the-name-of-the-property-of-a-set-such-that-no-member-of-the-set-is-larger-than-all-the-other-members-combined","timestamp":"2024-11-03T10:03:33Z","content_type":"text/html","content_length":"71093","record_id":"<urn:uuid:8c1bd8e1-4537-4916-8d59-2dfd44e97279>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00253.warc.gz"} |
by GLORIA GILMER MATH-TECH, MILWAUKEE
The discipline of mathematics includes the study of patterns. Patterns can be found everywhere in nature. See Figure 1 with two bees in a beehive. Often these patterns are copied and adapted by
humans to enhance their world. See the pineapple in Figures 2a and the adapted hairstyle in Figure 2b. Ethnomathematics is the study of such mathematical ideas involved in the cultural practices of a
people. Its richness is in exploring both the mathematical and educational potential of these same practices. The idea is to provide quicker and better access to the scientific knowledge of humanity
as a whole by using related knowledge inherent in the culture of pupils and teachers.
Going into a community, examining its languages and values, as well as its experience with mathematical ideas is a first and necessary step in understanding ethnomathematics. In some cases, these
ideas are embedded in products developed in the community. Examples of this phenomena are geometrical designs and patterns commonly used in hair braiding and weaving in African-American communities.
For me, the excitement is in the endless range of scalp designs formed by parting the hair lengthwise, crosswise, or into curves.
The main objective of my work with black hair is to uncover the ethnomathematics of some hair braiders and at the same time answer the complex research question: "What can the hair braiding
enterprise contribute to mathematics education and conversely what can mathematics education contribute to the hair braiding enterprise?" It is clear to me that this single practical activity, can by
its nature, generate more mathematics than the application of a theory to a particular case. My collaborators include Stephanie Desgrottes, a fourteen year old student of Haitian descent, at Half
Hollow Hills East School in Dix Hills, New York and Mary Porter, a teacher in the Milwaukee Public Schools. We have each observed and interviewed hair stylists at work in their salons along with
their customers. Today's workshop for middle school teachers will focus on the mathematical concept of tesselations which is widely used and understood by hair braiders and weavers but not thought of
by them as being related to mathematics.
A tesselation is a filling up of a two-dimensional space by congruent copies of a figure that do not overlap. The figure is called the fundamental shape for the tesselation. In Figure 1, the
fundamental shape is a regular hexagon. Recall that a regular polygon is a convex polygon whose sides all have the same length and whose angles all have the same measure. A regular hexagon is a
regular polygon with six sides. Only two other regular polygons tesselate. They are the square and the equilaterial triangle. See Figures 3a and 3b for parts of tesselations using squares and
triangles. In each figure, the fundamental shape is shaded. To our surprise two types of braids found to be very common in the salons we visited were triangular braids and box braids which describe
these tesselations on the scalp!
Box Braids
In the tesselations we saw, the boxes were shaped like rectangles and the pattern resembled a brick wall starting with two boxes at the nape of the neck and increasing by one box at each successive
level away from the neck. The hair inside the box was drawn to the point of intersection of the diagonals of the box. Braids were then placed at this point. You may notice in Figure 3a that braids so
placed will hide the scalp at the previous level in the tesselation. In this style, the scalp is completely hidden. In addition, we were told that braids so placed are unlikely to move much when the
head is tossed.
Triangular Braids
In the tesselations we saw, the triangles were shaped like equilateral triangles and the pattern resembled the one shown in Figure 3b. The hair inside the triangle was drawn to the point of
intersection of the bisectors of the angles of the triangle. Again, this style allowed hair to move less liberally than hair drawn to a vertex and then braided.
Tesselations can be formed by combining translation, rotation, and reflection images of the fundamental shape. Variations of these regular polygons can also tesselate. This can be done by modifying
one side of a regular fundamental shape and then modifying the opposite side in the same way.
CLASSROOM ACTIVITIES. 1. Draw tesselations using different fundamental shapes of squares and rectangles. 2. Draw a tessalation using an octagon and square connected along a side as the fundamental
shape. 3. Draw tesselations with modified squares or triangles. 4. Have a hairstyle show featuring different tesselations.
Eglash, Ron, "African Fractals."New Brunswick, New Jersey: Rutgers University Press, 1999. •
Gerdes, Paulus, "On Culture, Geometrical Thinking and Mathematics Education."
Ethnomathematics:Challenging Eurocentrism in Mathematics Education edited by Arthur B. Powell and Marilyn Frankenstein, Albany, N.Y.:State University of New York Press, 1997.
Gilmer, Gloria F. Sociocultural Influences on Learning, American Perspectives on the Fifth International Congress on Mathematical Education (ICME 5) Edited by Warren Page. Washington, D.C.:The
Mathematical Association of America, January 1985.
______"The Afterward. "Ethnomathematics:Challenging Eurocentrism in Mathematics Education edited by Arthur B. Powell and Marilyn Frankenstein, Albany, N.Y.:State University of New York Press, 1997.
______"Making Mathematics Work for African Americans from a Practitioner's Perspective." Making Mathematics Work for Minorities: Compendium of Papers Prepared for the Regional Workshops, Washington,
DC.:Mathematical Sciences Education Board, 1989. pp. 100-104.
______ and Mary Porter. "Hairstyles Talk a Hit at NCTM. " International Study Group on Ethnomathematics Newsletter, 13(May 1998)2, pp. 5-6.
_______ "An Ethnomath Approach to Curriculum Development," International Study Group on Ethnomathematics Newsletter, 5(May 1990)2, pp. 4-5.
_______ and Williams, Scott W. "An Interview with Clarence Stephens." UME Trends. March 1990.
Sagay, Esi, African Hairstyles, Portsmouth, New Hampshire; Heinemann Educational Books Inc. USA, 1983. Mathematicians of the African Diaspora: http://www.math.buffalo.edu/mad/mad0.html
Dr. Gloria Gilmer Math-Tech Milwaukee 9155 North 70th Street Milwaukee, WI 53223-2115 Phone: 414-355-5191 Fax: 414- 355- 9175 E-mail: ggilme@aol.com
Click here to visit gallery
Video Link | {"url":"http://cfiks.org/latestpost.php?show=27-a1140a3d0df1c81e24ae954d935e8926","timestamp":"2024-11-06T11:00:13Z","content_type":"text/html","content_length":"45936","record_id":"<urn:uuid:e28e4c58-107e-47df-987a-e8d66a480b13>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00564.warc.gz"} |
Sampling Concept and Distribution | Business Statistics Notes | B.Com Notes Hons & Non Hons | CBCS Pattern
Business Statistics Notes
B.Com 2^nd and 3^rd Semester (CBCS Pattern)
Sampling Concepts, Sampling Distributions and Estimation
Census and Sample Method
There are basically two popular methods of collecting primary data: one is census method and another one is sample method.
Census method or Complete Enumeration Survey
Statistics is taken in relation to a large data. Single and unconnected data is not statistics. In the field of a statistical enquiry there may be persons, items or any other similar units. The
aggregate of all such units under consideration is called “Universe or Population”. Under the census or complete enumeration survey method, data are collected for each and every unit of the
population or universe which is the complete set of items which are of interest in any particular situation.
Merits of Census Method:
(a) Data are obtained from each and every unit of the population.
(b) Since all the individuals of the universe are investigated, highest degree of accuracy is obtained.
(c) Since there is no possibility of personal bias affecting investigation, this method is free from sampling error.
(d) It is more suitable if the field of enquiry is small.
(e) Since all the items of the universe are taken into consideration, all the characteristics of the universe are collected which can be widely used as a basis for various surveys.
Demerits of Census Method:
(a) It the field of enquiry is too wide, it is not suitable.
(b) Collection of primary data is costly and time consuming.
(c) Personal Bias, prejudice and whims may affect the data.
(d) If secondary data are available, then it can be avoided to save time and cost.
(e) If the population is infinite or evaluation process destroys the population unit, the method cannot be adopted.
Sample Survey
Sampling is simply the process of learning about the population on the basis of sample drawn from it. In sampling technique only a part of the universe is studies and conclusions are drawn on that
basis for entire universe. A sample can also be called subset of population units. Since sample is a part of universe, sample survey is less expensive and less time consuming than census.
Merits of sample survey:
(a) Since data are limited, time and labour can be saved in sample survey method.
(b) Sample survey is suitable where highly trained personnel or specialised equipment is required for collection of data.
(c) Detailed information can be obtained under sample survey method since data are collected from a small group of respondents.
(d) Above all time, energy and money can be saved without sacrificing accuracy of result.
Demerits of sample survey:
(a) Sample survey is unnecessary if the universe is small.
(b) Chances of sampling errors are very high. Personal bias of the investigator can also give misleading results.
(c) If the investigators are not trained and qualified, data collected from sample survey will be misleading.
(d) If the sample is not drawn properly, the results thus obtained may be false, inaccurate and misleading.
Difference between Census and Sample survey:
The following are the differences between Census and Sample method of investigation:
(a) Under Census method, each and every individual item is investigated whereas under sample survey only a part of universe is investigated.
(b) There is no chance of sampling error in census survey whereas sampling error cannot be avoided under sample survey.
(c) Large number of enumerators is required in census whereas less number of enumerators is required in sample survey.
(d) Census survey is more time consuming and costly as compared to sample survey.
(e) Census survey is an old method and it less systematic than the sample survey.
Meaning, Features and Types of Sampling
Meaning of Sampling: Sampling is simply the process of learning about the population on the basis of sample drawn from it. In sampling technique only a part of the universe is studies and conclusions
are drawn on that basis for entire universe. In most of the research work and surveys, the usual approach happens to be to make generalizations or to draw inferences based on samples about the
parameters of population from which the samples are taken. The researcher quite often selects only a few items from the universe for his study purposes. All this is done on the assumption that the
sample data will enable him to estimate the population parameters. The items so selected constitute what is technically called a sample, their selection process or technique is called sample design
and the survey conducted on the basis of sample is described as sample survey. Sample should be truly representative of population characteristics without any bias so that it may result in valid and
reliable conclusions.
According to Goode and Hatt, “A sample as the name applies, is a smaller representative of a large whole”.
According to Pauline V Young, “A statistical sample is a miniature of cross selection of the entire group or aggregate from which the sample is taken”.
According to Bogrdus, “Sampling is the selection of certain percentage of a group of items according to a predetermined plan”.
Sampling is used in practice for a variety of reasons such as:
1. Sampling can save time and money. A sample study is usually less expensive than a census study and produces results at a relatively faster speed.
2. Sampling may enable more accurate measurements for a sample study is generally conducted by trained and experienced investigators.
3. Sampling remains the only way when population contains infinitely many members.
4. Sampling remains the only choice when a test involves the destruction of the item under study.
5. Sampling usually enables to estimate the sampling errors and, thus, assists in obtaining information concerning some characteristic of the population.
Feature of Sampling Techniques
The sampling techniques have following good features and these bring into relief its value and significance:
1. Scientific Base: It is a scientific because the conclusion derived from the study of certain units can be verified from other units. By taking random sample, we can determine the amount of
deviation from the norm.
2. Economy: The sampling technique is much less expensive, much less time consuming than the census technique.
3. Reliability: If the choice of sample unit is made with due care and the matter under survey is not heterogeneous, the conclusion of the sample survey can have almost the same reliability as those
of census survey.
4. Detailed study: Since the number of sample units is fairly small, these can be studied intensively and elaborately. They can be examined from multiple of views.
5. Greater Suitability in most Situations: Most of the surveys are made by the techniques of sample survey, because whenever the matter is of homogeneous nature, the examination of few units
suffices. This case in majority of situations.
Methods of Sampling
Sampling methods are classified into two categories: Probability sampling methods and non-probability sampling methods.
Probability sampling methods are those in which every item in the universe has a known chance or probability of being chosen for the sample. Non-probability sampling methods are those which do not
provide every item in the universe with a chance of being selected. These two methods are further divided into:
│ Probability Sampling │ Non-Probability Sampling │
│1. Simple random sampling │1. Quota Sampling │
│ │ │
│2. Stratified random sampling│2. Judgmental or Purposive Sampling │
│ │ │
│3. Cluster sampling │3. Convenience Sampling │
│ │ │
│4. Systematic Sampling │4. Extensive Sampling │
│ │ │
│5. Multi-stage sampling │ │
(1) Simple Random Sampling: Off all the methods of selecting sample, random sampling technique is made maximum use of and it is considered as the best method of sample selection. Random sampling is
made in following ways:
(i) Lottery Method: In this the numbers of data are written on sheet of paper and they are thrown into a box. Now a casual observer selects the number of item required in the sample. For this method
it is necessary that sheet of paper should be of equal dimensions.
(ii) By Rotating the Drum: In this method, piece of wood, tin or cardboard of equal length and breadth, with number 0,1 or 2 printed on them, are used. The pieces are rotated in a drum and then
requisite numbers are drawn by an impartial person.
(iii) Selecting from Sequential List: In this procedure units are broken up in numerical, alphabetical or geography sequence. Now we may decide to choose 1, 5, 10 and so on , if the division is
alphabetical order we decide to choose every item starting from a, b, c and so on.
(iv) Tippet’s Number: On the basis of population statistics, Tippet has constructed a random list of four digits each of 10, 400 institutions. These numbers are the result of combining 41,600
population statistics reports.
1. Due to impartiality, there is possibility of selecting any unit as sample.
2. Units have the characteristic of universe, hence units are more representative.
3. Simplicity of method makes no possibility of error.
4. Error can be known easily
5. It saves money, time and labour.
1. The selector has no control over the selection of units. The researcher cannot contact the far situated units.
2. He cannot prepare the whole field when the universe is vast.
3. If units have no homogeneity, the method is not appropriate.
4. There is no question of alternatives. The selected units cannot be replaced or changed.
(2) Stratified Sampling: This method of selecting samples is a mixture of both purposive and random sampling techniques. In this all the data in a domain is spilt into various classes on the basis of
their characteristics and immediately thereafter certain items are selected from these classes by the random sampling technique. This technique is suitable in those cases in which the data has sub
data and having special characteristics. For example if we wish to collect information regarding income expenditure of the male population strata on the basis of shopkeeper, workers, etc. From these
we shall select randomly some units for study of income-expenditure statistics.
Process of Stratifying: The stratification of domain or data should be with great care, because the success of the technique depends upon successful stratification. Following points should be born in
1. We should process extensive information of all items including in a domain and should know which item make a coherent whole on the basis of similar traits and which others are different from them
and why?
2. The size of each stratum should be large to enable use of random sampling technique.
3. In stratifying it must be kept in mind that various strata should have similar relation to the domain and should be themselves homogeneous.
4. The various strata should differ from each other should be the same as the proportion of stratum from the domain. Suppose a domain has four strata, accordingly the proportion of each stratum of
domain is ¼. Now if the number of total items of the sample is 64, we shall select 16 items from each stratum and thus the proportion of selected items from each stratum will be ¼.
1. Neither group nor class of importance is totally neglected as units of each are represented in the sample.
2. If different classes are divided properly, selection of few units represents the whole group.
3. On the classification of regional basis, units are not in contact easily. This leads to economy of time and money.
4. There is a facility in substitution of units. If someone is not contacted easily, the other person of the same class can be substituted for him. Such inclusion result will not show any
1. The sample does not become representative if selected sample has more or less units of a class.
2. If the sizes of different group are different, no equal proportional quality can be viewed.
3. Non-proportional selection leads to more emphasis in the end. During such time researcher can be biased, hence samples will not accurate.
4. If group is not expressed properly, the difficulty is seen about the unit to be kept under which group or class.
(3) Cluster Sampling: In this method of sampling, the population is divided into clusters or groups and then Random Sampling is done for each cluster. In some instances the sampling unit consists of
a group or cluster of smaller units that we call elements or sub-units. Cluster Sampling is different Stratified sampling. In the case of stratified sampling the elements of each stratum are
homogeneous while in cluster sampling each cluster is heterogeneous within and a representative of the population.
(4) Systematic Sampling: This method of sampling is at first glance very different from random sampling. In practice, it is a variant of simple random sampling that involves some listing of elements.
In systematic sampling each element has an equal chance of being selected, but each sample does not have the same chance of being selected. Here, the first element of the population is randomly
selected to begin the sampling. But thereafter the elements are selected according to a systematic plan. Systematic sampling proceeds by picking up one element after a fixed interval.
(5) Multi-Stage sampling: This is not a favoured procedure of sampling. In this items are selected in different stages at random. For example, if we wish to know per acre yield of various crops in
U.P., we shall begin by studying a single crop in one study. Here we shall begin by making at random selection of 5 districts in the first instance, and then of these 5 districts, 10 villages per
districts will be chosen in the same manner. Now in the final stage, again by random selection 5 fields out of every village. Thus we shall examine per acre yield in 250 farms all over U.P. this
number can increased or decreased depending upon the opinion of experts.
(6) Quota Sampling: This method of study is not much used. In this method entire data is spilt into as many as there are investigators and each investigator is asked to select certain items from his
block and study. The success of this method depends upon the integrity and professional competence of investigators. If some investigators are competent and others are not so competent, serious
discrepancies will appear in the study.
(7) Purposive or Judgmental or Selective sampling: In this method the investigator has complete freedom to choose his sample according to his wishes and desire. To choose or leave an item for the
purpose of study depends entirely upon the wishes of investigator and he will chose items or units which in his judgment are representative of the whole data. This is a very simple technique of
choosing the samples and is useful in cases where the whole data is homogeneous and the investigator has full knowledge of the various aspects of the problem.
1. More representation is possible in this method.
2. As sample is small in size, the method is less expensive and less time consuming.
3. The utility of this method increases when few units of universe have special importance.
4. When units are less in number, sample is profitable
1. Units are selected by researcher at his will. Hence sample is biased.
2. The error of the sample cannot be detected.
3. Researcher is unable to understand the whole group.
4. Those hypothesis on which inference of error of sample is attributed, are less used.
(8) Convenience Sampling: This is hit or miss procedure of study. The investigator selects certain item from the domain as per his convenience. No planned efforts are made to collect information.
This is method by which a tourist studies generally the country of his visit. He comes across certain people and things, has transaction with them and then tries to generalize about the entire
populace in his travelogue. This is essentially unscientific procedure and has no value as a research technique.
(9) Extensive sampling: This method is virtually same as census except that irrelevant or irascible items are left out. Every other item is examined. For instance, if we are to study the educational
levels of Indians, we may leave foreigners living in India from our study. This method has all the merits and demerits of census survey and is very rarely used.
Factors to be taken into consideration while deciding the sample size:
a) The size of the universe – larger the size of universe, larger will be the sample size.
b) The resources available with the researcher.
c) The degree of accuracy desired by the researcher. Larger the size, more will be the accuracy.
d) Small sample size if universe consists of homogenous unit and large sample size if universe consists of heterogeneous units.
e) Nature of study – For an intensive and continuous study a small sample may be suitable.
Sampling Errors and Non-sampling Errors
Sampling Errors: The errors caused by drawing inference about the population on the basis of samples are termed as sampling errors. The sampling errors result from the bias in the selection of sample
units. These errors occur because the study is based on a part of the population. If the whole population is taken, sampling error can be eliminated. If two or more sample units are taken from a
population by random sampling method, their results need not be identical and the results of both of them may be different from the result of the population. This is due to the fact that the selected
two sample items will not be identical. Thus, sampling error means precisely the difference between the sample result and that of the population when both the results are obtained by using the same
procedure or method of calculation. The exact amount of sampling error will differ from sample to sample. The sampling errors are inevitable even if utmost care is taken in selecting the sample.
However, it is possible to minimise the sampling errors by designing the survey appropriately. Sampling errors are of two types:
(i) Biased sampling errors: These errors arises from any bias in selection, estimation
(ii) Unbiased sampling errors: These errors arise due to difference between members of population included in sample and those not included.
Non-sampling Errors: These non-sampling errors can occur in any survey, whether it be a complete, enumeration or sampling. Non-sampling errors include biases as well as mistakes. These are not chance
errors. Most of the factors causing bias in complete enumeration are similar to the one described above under sampling errors. They also include careless definition of population, a vague conception
regarding the information sought, and inefficient method of interview and so on. Mistakes arise as a result of improper coding, computations and processing. More specifically, non-sampling errors may
arise because of one or more of the following reasons:
i) Improper and ambiguous data specifications which are not consistent with the census or survey objectives.
ii) Inappropriate sampling methods, incomplete questionnaire and incorrect way of interviewing.
iii) Personal bias of the investigators or informants.
iv) Lack of trained and qualified investigators.
v) Errors in compilation and tabulation.
Law of statistical regularity
Law of statistical regularity is derived from the concept of probability. According to this law, if a sample is collected at random from a population, it is probable that the sample posses the same
properties which the population posses. Random selection means each and every item of the population has the equal chance of selection. A sample selected at a random from a large population will
represent the whole universe. There is a very less chances of bias selection. This law is very important because quick conclusion can be drawn from a large universe. It reduces the necessary work
before final conclusion is drawn.
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What Was The Last Coaster You Rode?
Goliath @ SFFT
Batman clones are one of the most awesome cloned rides and although they're everywhere they're always worth a spin.
I think it was Intimidator 305 on Saturday.
Either that or Anaconda.
Superman at SFNE.....on Sunday.....great ride cuz my 11 yr old daughter graduated to the big stuff on that ride....she killed it and went for another ride with her friend
Twisted Cyclone
Mine Blower and boy was I disappointed.
X2, about a week ago.
Fury 325
Possessed at Dorney Park in the front row a couple of weekends ago. I miss the holding brake but it's still pretty fun even without it. I'm just happy that I still manage to fit in the restraints!
Railblazer at CGA! Absolutely loved it, and it is a new top 10 coaster for me.
Outlaw Run!!
New Texas Giant at sunset yesterday. When it's fully warmed up, it hauls some serious A$$.
Fury a little over a month ago.
Outlaw Run. Five times in the dark to close the park last night.
Verbolten about 15 minutes ago
Twisted Timbers
Heidi The Ride at Plopsaland De Panne a few hours ago.
Mind Eraser at SFA. | {"url":"https://themeparkreview.com/forum/topic/135-what-was-the-last-coaster-you-rode/page/340/","timestamp":"2024-11-10T09:56:15Z","content_type":"text/html","content_length":"386148","record_id":"<urn:uuid:5dd037da-125e-46cb-ae10-e65aba4ed963>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00134.warc.gz"} |
The Spectr
┃ ┃
┃ San José State University ┃
┃ Department of Economics ┃
┃ ┃
┃ ┃
┃ applet-magic.com ┃
┃ Thayer Watkins ┃
┃ Silicon Valley ┃
┃ & Tornado Alley ┃
┃ USA ┃
┃ ┃
┃ The Spectra of Various ┃
┃ Transformations of White Noise ┃
Spectral analysis is the decomposition of a functions into its cyclic components. It is carried out using the Fourier transform. The Fourier transform of the function y(t) is defined as:
F[y](ω) = ∫[−∞]^∞exp(−iωt)y(t)dt
The Fourier transform is generally a complex function. The spectrum of a function is simply the absolute value of its Fourier transform.
The spectrum of white noise is constant over a broad frequency band. This is in analogy with white light that contains light of all colors over the frequency band of visible light. Sometimes white
noise is taken to extend over an infinite range but this would be impossible to realize physically because such noise would have infinite enegy. If the frequency band is too narrow the noise would be
said to be of a particular color. Therefore white noise is defined to be such that its spectrum is
F(ω) = c for ω[min] ≤ ω ≤ ω[max]
= 0 otherwise
The Cumulative Sum of White Noise
The cumulative sum is defined as the integral of white noise. If u(t) is white noise then
y(t) = ∫[0]^tu(s)ds
and, equivalently
dy/dt = u(t)
As state previously, the spectrum is the magnitude of the Fourier transform of the variable and therefore
|F[y](ω)| = |F[u](ω)/(iω)| = |F[u](ω)/ω|
The variable y is said to be pink noise.
It would be any variable whose spectrum is of the form
F(ω) = c/ω for ω[min] ≤ ω ≤ ω[max]
= 0 otherwise
The Spectrum of the Moving Average of a Variable
The general form of a moving average of a variable y(t) is
y(t) = ∫[0]^Hh(s)y(t-s)ds
where h(s) for 0 ≤ s ≤ H is a weighting function. The upper limit H could be finite or infinite. Note that the moving average of a variable is being denoted by an underscore of that variable.
The Fourier transform of y(t) is
F[y](ω) = ∫[−∞]^∞ exp(−iωt)y(t)dt
= ∫[−∞]^∞exp(−iωt)(∫[0]^Hh(s)y(t-s))dsdt
The reversal of the order of integration gives
F[y](ω) = ∫[0]^Hh(s)[∫[−∞]^∞exp(−iωt)y(t-s)dt]ds
If the variable of integration in ∫[−∞]^∞exp(−iωt)y(t-s)dt is changed to z=t-s then t=z+s and dt=dz so the integral becomes
which reduces to
and finally to
This is a standard theorem for Fourier transforms which says
F[y(t-s)] = exp(−iωs)F[y]
F[y](ω) = ∫[0]^Hh(s)[exp(−iωs)F[y](ω]ds
which reduces to
F[y](ω) = F[y](ω)∫[0]^Hh(s)exp(−iωs)ds
If h(s) is extended over the interval [−∞,+∞] such that h(s)=0 for s<0 and s≥H then the second term on the RHS of the above expression is just the Fourier transform F[h].
The relationship is then
F[y](ω) = F[y](ω)·F[h](ω)
For a simple moving average h(s) = 1/H and (1/H)∫[0]^Hexp(−iωs)ds reduces to
(1/H)[exp(−iωs)/(−iω][0]^H = (1/H)[exp(−iHω)−1]/(iω)
which by factoring out a term of exp(−iωH/2) leads to
exp(−iωH/2)[ exp(+iωH/2)− exp(−iωH/2)]/(2iωH/2)
which is
exp(−iωH/2)[sin(ωH/2)/(ωH/2) = exp(−iωH/2)sinc(ωH/2)
By labeling the t variable of the moving average with the midpoint of the H interval the term exp(−iωH/2) can be eliminated leaving
F[y](ω) = F[y](ω)sinc(½ωH)
Since the spectrum is the absolute value of the Fourier transform the relevant function is |sinc(x)|
The sinc function creates peaks in the spectrum of the moving average that were not there in the original data.
Sampling and Intervalizing
Samping in spectral analysis generally means taking the value of a variable at discrete intervals. A related procedure is to replace the instantaneous values within an interval by the sample values;
i.e., for t[i]−½H≤t≤t[i]+½H replace y(t) with y(t[i]). The Fourier transform of the intervalized function is related to the Fourier transform of the sampled function through multiplication by a
factor of the form
which reduces to
Since the intervalizing procedure is applied to the moving average of the original variable the Fourier transform for the intervalized moving average function z(t) is given by
F[z](ω) = F[y]sinc²(½ωH)
The sinc²(x) has the following shape:
For y being pink noise, F[y](ω)=c/ω, the spectrum for interval average function rises to a peak and then declines. Thus the low frequency components dominate the interval average even more so than
they do for the cumulative sum.
A Moving Average of Annual Averages
Any manipulation or transformation of data which are the cumulative sums of random disturbance can introduce elements of stochastic structure which are peculiar and non-intuitive and potentially
dangerous for objective statistical analysis. For example suppose the annual averages are computed for variables which are the cumulative sums of random disturbances and then the annual averages are
averaged over a five year period. In the diagram below the upper graph shows the weights which are placed upon the rates of changes. Annual averaging places a relatively high weight on changes which
occur early in the year and a low weight on changes which occur near the end of the year. When values are averaged over a five-year period the changes that occur near the beginning of the five-year
receive a much higher rate than those occurring near the end of the five-year period.
The five-year average would typically be identified with the third year whereas it is more closely associated with the changes occurring in the first year. This would confuse the analysis of time
lags among variables.
The following is the four-period moving average of a four period moving average of random variable uniformly distributed between 0 and +1.0.
To illustrate how this double smoothing generates the appearance of cycles a sinusoidal cycle about a level of 0.5 is plotted in the same graph.
A physically measurable quantity, such as the temperature of an object, may be the cumulative sum of a stochastic variable. In the case of the temperature of an object the stochastic variable is
proportional to the net heat input to the object. This variable however may be subject to autocorrelation; i.e., a dependence of its distribution on its past values.
For example, the temperature T(t) of a body at time t may be given by
T(t) = T(t-1) + U(t)
U(t) = λU(t-1) + V(t)
where the variables V(t) are independent random variables.
The variable U(t) is given by the formula
U(t) = V(t) + λV(t-1) + λ²V(t-2) + …
or, in general,
U(t) = Σ[j=0]^t λ^jV(tj)
This is an exponentially weighted sum, a type of smoothing operation. Since temperature is the cumulative sum of the U(t)'s, another smoothing operation, temperature is a doubly smoothed variable. As
in the case of a moving average of a moving average the double smoothing will generate the appearance of cycles even when the original variable, the V(t)'s, are random white noise. When temperatures
are subjected to averaging the result could triply smoothed white noise which would be even more subject to the generation of spurious trends and cycles.
(To be continued.)
Differentiation and Differencing of Moving Averages
Let z(t) be a variable and F[z](ω) be its Fourier transform. Let y(t)=dz/dt, then
|F[y](ω)| = ω|F[z](ω)|
If z(t) is a moving average of the cumulative sum of white noise its Fourier transform is of the form
|F[z](ω)| = (c/ω)|sinc(½ωH)|
|F[y](ω)| = c*|sinc(½ωH)|
Thus the derivative of a moving average of the cumulative sum of white noise has a spectrum that indicates cycles but the spectrum comes from the moving average process rather than the original data.
More generally the Fourier transform of a weighted moving average of a variable v(t) based upon a weighting function h(s) is of the form
F[z](ω) = F[s](ω)F[h](ω)
If s(t) is the cumulative sum of white noise then F[s](ω)=c/ω over some range of ω. Thus the Fourier transform of y(t) which is the derivative of the weighted moving average is then
F[y](ω) = ω(c/ω)F[h](ω) = c*F[h](ω)
Thus the spectrum of the derivative of a moving average of white noise is just the spectrum of the averaging process. This means that when cycles are found in the review of processed versions of
moving averages they may be just an artifact of the averaging and processing procedures.
Differencing of moving averages would occur more commonly than differentiation. The result are similar. Let y(t)=[z(t)−z(t-H)]/H. The Fourier transform of y(t) is then
F[y](ω) = (1/H)(1-e^-ωH)F[z](ω)
Since (1-e^-ωH)=ωH − (ωH)²/2 + …
F[y](ω) = (ω −ω²H/2 + … )F[z](ω)
Thus a Fourier transform of the cumulative sum of white noise will be multiplied a factor that is a multiple of ω and the effect is to cancel out the ω in the denominator of the Fourier transform of
the cumulative sum of white noise leaving approximately just the Fourier transform of the averaging procedures; i.e.,
F[y](ω) = (ω −ω²H/2 + … )(c/ω)F[h](ω)
= (1 − ωH/2 + …)*c*F[h](ω)
which for small values of ωH reduces to
F[y](ω) = c*F[h](ω)
(To be contined.) | {"url":"https://www.sjsu.edu/faculty/watkins/spectrum1.htm","timestamp":"2024-11-14T14:48:48Z","content_type":"text/html","content_length":"14982","record_id":"<urn:uuid:dcde2ba2-6460-4d1e-a4d7-d4899e0461a7>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00167.warc.gz"} |
On meromorphically starlike functions of order $\alpha$ and type $\beta$, which satisfy Shah's differential equation
On meromorphically starlike functions of order $\alpha$ and type $\beta$, which satisfy Shah's differential equation
meromorphically starlike function of order $\alpha$ and type $\beta$, meromorphically convex function of order $\alpha$ and type $\beta$, Shah's differential equation
Published online: 2018-01-02
According to M.L. Mogra, T.R. Reddy and O.P. Juneja an analytic in ${\mathbb D_0}=\{z: 0<|z|<1\}$ function $f(z)=\frac{1}{z}+\sum_{n=1}^{\infty}f_n z^{n}$ is said to be meromorphically starlike of
order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$ if $|zf'(z)+f(z)|<\beta|zf'(z)+(2\alpha-1)f(z)|, \, z\in {\mathbb D_0}. $ Here we investigate conditions on complex parameters $\beta_0,\,\
beta_1,\,\gamma_0,\,\gamma_1,\,\gamma_2$, under which the differential equation of S. Shah $z^2 w''+(\beta_0 z^2+\beta_1 z) w'+(\gamma_0 z^2+\gamma_1 z+\gamma_2)w=0$ has meromorphically starlike
solutions of order $\alpha\in [0,\,1)$ and type $\beta\in (0,\,1]$. Beside the main case $n+\gamma_2\not=0, \, n\ge 1,$ cases $\gamma_2=-1$ and $\gamma_2=-2$ are considered. Also the possibility of
the existence of the solutions of the form $f(z)=\frac{1}{z}+\sum_{n=1}^{m}f_n z^{n}, \, m\ge 2,$ is studied. In addition we call an analytic in ${\mathbb D_0}$ function $f(z)=\frac{1}{z}+\sum_{n=1}^
{\infty}f_n z^{n}$ meromorphically convex of order $\alpha\in [0,1)$ and type $\beta\in (0,1]$ if $|zf''(z)+2f'(z)|<\beta|zf''(z)+2\alpha f'(z)|, \, z\in {\mathbb D_0}$ and investigate sufficient
conditions on parameters $\beta_0,\,\beta_1,\,\gamma_0,$ $\gamma_1,\,\gamma_2$ under which the differential equation of S. Shah has meromorphically convex solutions of order $\alpha\in [0,\,1)$ and
type $\beta\in (0,\,1]$. The same cases as for the meromorphically starlike solutions are considered.
How to Cite
Trukhan, Y.; Mulyava, O. On Meromorphically Starlike Functions of Order $\alpha$ and Type $\beta$, Which Satisfy Shah’s Differential Equation. Carpathian Math. Publ. 2018, 9, 154-162. | {"url":"https://journals.pnu.edu.ua/index.php/cmp/article/view/1459","timestamp":"2024-11-02T23:14:43Z","content_type":"text/html","content_length":"38455","record_id":"<urn:uuid:dfdd09c8-aa19-4d56-9122-59277c88c281>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00374.warc.gz"} |
of Indian mathematic
Who is the prince of Indian mathematics?
SRINIVASA RAMANUJAN 1887-1920 Ramanujan is India’s best mathematician ever. His main contributions are to the theory of numbers and mathematical analysis.
Who is a famous Indian mathematician?
Indian mathematician Srinivasa Ramanujan made contributions to the theory of numbers, including pioneering discoveries of the properties of the partition function. His papers were published in
English and European journals, and in 1918 he was elected to the Royal Society of London.
Who is the greatest mathematician in India?
Srinivasa Ramanujan. Srinivasa Ramanujan was a mathematic genius who won several accolades in field of mathematics.
Satyendra Nath Bose. Born in Kolkata in 1884,Satyendra Nath Bose is one of the most prominent Indian mathematicians.
C. S.
P.C. Mahalanobis.
C.R. Rao.
Harish Chandra.
Narendra Karmarkar.
Shakuntala Devi.
Who are the best living mathematicians?
– Euclid – created 5 postulates that define geometry as we know it, did lots of stuff with nothing but a compass and a straightedge, and was among the first great – Archimedes – another great one. He
got the first really really good approximation for pi, among his other great contributions to math. – Pythagoras – a^2 + b^2 = c^2.
What are the contributions of Indian mathematicians?
Srinivas Ramanujan was a math prodigy who conquered math analysis. His contribution to mathematics in his short life is finding applications even today. His work in fields like Elliptical functions,
Analysis, Number Theory, Continued fractions, Theta functions, and Blackhole theory has given him a stature like none other and made him like Einstein.
Who are the most famous mathematicians?
Pythagoras. Pythagoras comes on Number (1) from our list of top 10 most famous mathematicians and greatest mathematicians of all time.
Euclid. Euclid comes on Number (2) from our list of top 10 most famous mathematicians and greatest mathematicians of all time.
Leonhard Euler.
Isaac Newton.
Srinivasa Ramanujan.
Carl Friedrich Gauss.
Bernhard Riemann. | {"url":"https://musicofdavidbowie.com/who-is-the-prince-of-indian-mathematics/","timestamp":"2024-11-04T07:54:11Z","content_type":"text/html","content_length":"44884","record_id":"<urn:uuid:6e5a796b-59c0-4898-9718-08fc170cb093>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00735.warc.gz"} |
Indoor Human Detection Based on Thermal Array Sensor Data and Adaptive Background Estimation
Indoor Human Detection Based on Thermal Array Sensor Data and Adaptive Background Estimation ()
1. Introduction
Facilitating older seniors independent living has become an important issue of the current research in the field of assistive technologies fostered by worldwide governments. Indeed, as reported in
the technical report of the United Nations [1] , the over-60 world population in the next 15 years is projected to grow by 56 percent, from 901 millions to more than 1.4 billions. The ageing process
is par- ticularly evident in Europe and in Northern America, where in 2015, more than one out of five people was over 60, and it is growing rapidly in other regions as well. Thus, Bierhoff et al. [2]
, for example, report health-care systems among the products and services based on smart home technologies. In particular, since their aim is to detect critical conditions or predict them on early
stages and alert caregivers, emergency treatment services are crucial for older adults living in smart homes. This kind of systems usually relys on a network of sensors to unobtru- sively monitor the
life of a person, providing feedback to his/her beloved [3] . A concrete example of a widely demanded feature by the families of the elderly people who live alone is a Fall Detector; indeed falls are
nowadays considered as the most frequent hazard for older seniors and they may endanger the physical and psychological health of a person, hindering independent living [4] . For this reasons,
monitoring elderly people at home makes them feel safer and helps their relatives to be more confident, knowing the well-being of their beloved.
Low Resolution Thermal Array Sensors (LR-TASs) are very suitable in a home environment for substantial reasons. Thanks to their low-resolution, these sen- sors provide useful data without invading
the privacy of the dweller as it could happen using cameras or microphones. Furthermore, these devices are small, cheap, easy to be installed in a normal room, and they can work even in absence of
LR-TASs are composed of $m×n$ infrared sensing elements, acquiring the temperature of a two-dimensional area. In this paper, we refer to experiments conducted using the Grid-Eye [5] sensor developed
by Panasonic. This device is an 8-by-8 LR-TAS with sampling rate of 10 samples/s, a temperature range from −20˚C to 80˚C with 0.25˚C resolution, field of view of 60˚ and maximum dis- tance declared
to detect humans of 5 m. Moreover, Grid-Eye sensor comes with an on-board thermistor that provides the environment temperature from −20˚C to 80˚C with 0.0625˚C resolution. It communicates through $
{I}^{2}C$ interface to a wireless station which sends all the data to a central processing unit for the analysis.
2. Background
Human localization has several applications in Smart Home Environments, e.g., surveillance, health monitoring, and energy management. In particular, LR-TAS has been used to accomplish different
The goal of the work proposed by Sixsmith et al. [6] is to detect falls of the older seniors through SIMBAD: Smart Inactivity Monitor using Array-Based De- tectors. This system relies on two parallel
modules to raise alarms. First of all, it analyzes target motion to detect falls characteristic dynamics. Then, it monitors target inactivity and compares it with a map of acceptable inactivity
periods in different locations in the field of view. This system has been tested in laboratory simulating predefined fall scenarios, reporting limited results in true positive rate without any false
positive. This research includes also results related to a trial last- ing two months in a single occupant house. After a training period where experts tuned the system parameters according to the
output, still an unacceptably high false-larm rate emerged.
Erickson et al. [7] use a Thermal Array Sensor Network in order to measure occupancy of a building. The provided information is used to control heating, cooling, ventilation, and lighting of the
building to optimize energy usage. The method proposed by the authors consists in removing the background to detect the pixels of the matrix that refer to humans, followed by an analysis of the con-
nected components using K-Nearest Neighbors classifier, to estimate the num- ber of people. Unfortunately, no significant results have been reported.
In Basu et al. [8] the authors present a method to estimate number of people and the direction of their motion from LR-TASs data. A support vector machine has been used to classify connected
components and local peak counts, estima- ting the number of persons with 80% accuracy. Finally, they inferred the direc- tion of a subject motion across a set of scenes using cross-correlation
Mashiyama et al. [9] report a system for Activity Recognition using LR-TASs. The proposed method aims to detect five activities―no event, stopping, walking, sitting, and falling―in three steps: human
body detection, feature extraction and classification. Considered the high performances of this method in classifying activities as reported by its authors, we have decided to further analyze and re-
implement it in order to compare results obtained in a field trial test.
For sake of clarity and completeness let’s report the crucial passages of the work proposed by Mashiyama et al. with the notation used in the rest of this pub- lication. Given an instant of time $t$
, the frame $I\left(t\right)$ represents the set of ${T}_{i,j}$ measurements taken by a LR-TAS sensor at time $t$ , each one related to the co- rresponding $\left(i,j\right)$ pixel. Fixing a time
windows $\tau$ , the variance ${v}_{i,j}\left(t\right)$ of each pixel with $t\ge \tau$ is computed as it follows:
${v}_{i,j}\left(t\right)=\frac{1}{\tau }\underset{k=t-\left(\tau -1\right)}{\overset{t}{\sum }}{\left({T}_{i,j}\left(t\right)-\stackrel{¯}{{T}_{i,j}\left(k\right)}\right)}^{2},\text{ }\text{where}\
text{\hspace{0.17em}}\text{ }\stackrel{¯}{{T}_{i,j}\left(t\right)}=\frac{1}{\tau }\underset{k=t-\left(\tau -1\right)}{\overset{t}{\sum }}{T}_{i,j}\left(t\right).$(1)
If the obtained variance ${v}_{i,j}\left(t\right)$ exceeds a given threshold ${V}_{\text{ }\text{th}\text{ }}$ , a moving person (walking, sitting, or falling) is detected in the current frame.
Conversely, if no movement has been detected, the discrimination between a Stopping per- son or No event is done according to the difference ( ${T}_{\text{ }\text{\hspace{0.17em}}\text{diff}\text{ }}
$ ) between a person tem- perature ${T}_{p}$ and the background temperature ${T}_{b}$ . Given ${n}_{\text{ }\text{\hspace{0.17em}}\text{temp}\text{ }}$ as the number of pixel covered by a standing
person, the average of the first ${n}_{\text{ }\text{\hspace{0.17em}}\text{temp}\text{ }}$ pixels of a frame ordered by descending temperature gives ${T}_{p}$ . Similarly, the average of the
remaining pixels gives ${T}_{b}$ . Finally, only if ${T}_{\text{ }\text{\hspace{0.17em}}\text{diff}\text{ }}$ exceeds a given threshold ${T}_{\text{ }\text{th}\text{ }}$ a standing person is
revealed. The authors tested their system in a test bed expe- riment, reporting particularly good accuracy results in classifying the mentioned activities, specially considering just the detection
phase, excluding the activity classification method.
Most of the work done in activity recognition and human detection using LR- TAS report experimental data obtained by tests performed in a controlled envi- ronment. However, as highlighted by Sixsmith
et al. [6] , there are some limita- tions in this approach that have to be considered when building an effective indoor monitoring system. First of all, the positioning of the sensors must take into
account the geometry of the environment and its contents to ensure that the vision of the sensor is not obstructed. Indeed, a system to detect falls would be useless if it would not guarantee its
effectiveness over the entire walkable area of the house. Moreover, another factor requiring a deep study is the noise mana- gement: radiators, appliances, heaters or sunlight reflections have to be
consi- dered in the model of the system. Our work is mainly focused in improving hu- man detection’s performances handling noisy data.
3. Human Detection
The following method aims at retrieving a probability estimation of the presence of at least one person in the LR-TAS field of view. The main steps of the algo- rithm are summarized in the flow
presented in Figure 1: noise removal, back- ground estimation and probabilistic foreground detection.
3.1. Noise Removal
LR-TAS (Low Resultion-Termal Array Sensor) raw temperature data are charac- terized by the presence of noise perturbing the desired measured signal. These type of sensors usually denote low accuracy
on a single measurement: Grid-Eye sensor, for example, report the value within Typ. $8×8$ sensing elements measuring the temperature of a certain re- gion of space. To get the temperature
distribution in the space, the temperature
Figure 1. Algorithmic flow of the proposed human detection method.
evolution process on a single region will be taken as an independent dynamic system and, hence, the measurement made by a single sensing element will be fil- tered independently from other
3.1.1. Kalman Filtering
Consider a dynamic system $\mathcal{S}$ represented as follows:
$\mathcal{S}:\left(\begin{array}{l}x\left(t\right)=F\cdot x\left(t-1\right)+\xi \left(t\right)\hfill \\ y\left(t\right)=s\left(t\right)+\eta \left(t\right)=H\cdot x\left(t\right)+\eta \left(t\right)\
hfill \end{array},$(2)
where $s\left(t\right)$ is the variable to be estimated, $y\left(t\right)$ is the value obtained mea- suring $s\left(t\right)$ which is affected by the measurement noise term $\eta \left(t\right)$ ,
$x\left(t\right)$ is the state variable at time $t$ , $\xi$ models the process noise, $F$ is the system ma- trix and $H$ is the measurement matrix. The proposed noise removal technique is based on
Kalman Filter (KF) [10] and it is composed of two phases: extra- polation and correction. During the extrapolation phase the filter receives a pre- diction of the system state $\stackrel{˜}{x}\left(t
\right)$ for the current step $t$ using the system state estimate $\stackrel{^}{x}\left(t-1\right)$ made on the previous step. During the correction phase the state prediction $\stackrel{˜}{x}\left(t
\right)$ is adjusted by current measurements $y\left(t\right)$ to obtain the corrected estimate $\stackrel{˜}{x}\left(t\right)$ . The state prediction is expressed as:
$\stackrel{˜}{x}\left(t\right)=F\cdot \stackrel{^}{x}\left(t-1\right),$(3)
while the state estimate is represented as:
$\stackrel{^}{x}\left(t\right)=\stackrel{˜}{x}\left(t\right)\right)K\left(t\right)\cdot \left[y\left(t\right)-H\cdot \stackrel{˜}{x}\left(t\right)\right],$(4)
where $K\left(t\right)$ is the Kalman Gain [10] at time $t$ .
3.1.2. LR-TAS Data Filtering
In order to obtain the expected value of the measured temperature, separating it from the noise component, we applied the Kalman-Filtering technique described in the previous paragraph. Let the state
variable $x\left(t\right)$ be represented by ${T}_{i,j}\left(t\right)$ : the average temperature of the objects placed in the field of view of the sensing element in position $\left(i,j\right)$ at
time $t$ . Similarly, $y\left(t\right)$ is the measure of ${T}_{i,j}\left(t\right)$ as acquired by the sensing element in position $\left(i,j\right)$ at time $t$ . Finally, in order to get the
prediction on the state as described in Equation (3), the system matricies have to be set as follows:
$F=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]\text{}\text{ }\text{and}\text{\hspace{0.17em}}\text{ }H=\left[\begin{array}{cc}1& 0\end{array}\right].$(5)
Thus, from Equation (2):
${\stackrel{˜}{T}}_{i,j}\left(t\right)=F\left[\begin{array}{c}{\stackrel{^}{T}}_{i,j}\left(t-1\right)\\ \Delta {\stackrel{^}{T}}_{i,j}\left(t-1\right)\end{array}\right]=\left[\begin{array}{c}{\
stackrel{^}{T}}_{i,j}\left(t-1\right)+\Delta {\stackrel{^}{T}}_{i,j}\left(t-1\right)\\ \Delta {\stackrel{^}{T}}_{i,j}\left(t-1\right)\end{array}\right],$(6)
while the variable to be estimated $s\left(t\right)$ can be derived from Equation (2):
The result of the application of KF to measurements collected by a single sen- sing element is shown in Figure 2: data have been collected in a perturbed envi- ronment with appliances in the sensor
field of view and activating the air con- ditioning system. The influence of these factors on the ambient temperature is evident analyzing the room temperature collected by the on-board sensor ther-
mistor (Figure 3).
3.2. Background Estimation
The fundamental assumption to discriminate humans from the background is that the human temperature distribution has to differ from the ambient temperature distribution. In this condition, the human
recognition task converges to the ana- lysis of the difference between the current measurements of the sensor cells and the corresponding values of estimated temperature background. Nevertheless, the
Figure 2. Result of application of Kalman Filter to the temperature measurements done by a single cell sensor: original signal (black) and filtered (white).
Figure 3. Ambient temperature measurements done by the on-board sensor thermistor. The oscillation is caused by the air conditioning system.
temperature background estimation should adapt to the environmental condi- tion changes that can be relatively rapid.
Assuming that the thermistor measurement of the ambient temperature is almost not affected by the presence of humans, this information can be used as a reference to detect changes in the
environmental conditions. Thus, the depen- dence between the background temperature ${T}_{b\left(i,j\right)}\left(t\right)$ of the sensing element in position $\left(i,j\right)$ and the ambient
temperature ${T}_{a}\left(t\right)$ is a function $f\left({T}_{a}\left(t\right)\right)={T}_{b\left(i,j\right)}\left(t\right)$ , it is possible to compute ${T}_{b\left(i,j\right)}\left(t\right)$ from
${T}_{a}\left(t\right)$ . The ana- lysis of the sensor cells and thermistor measurements shows a linear dependence (Figure 4): the average correlation coefficient between ${T}_{a}\left(t\right)$ and
${T}_{b\left(i,j\right)}\left(t\right)$ is 0.62, and between ${T}_{a}\left(t\right)$ and ${\stackrel{^}{T}}_{b\left(i,j\right)}\left(t\right)$ is 0.9. Hence, the function can be ex- pressed as:
${T}_{b\left(i,j\right)}\left(t\right)=f\left({T}_{a}\left(t\right)\right)=\left[\begin{array}{cc}1& {T}_{a}\left(t\right)\end{array}\right]\cdot \beta .$(8)
In order to compute $\beta$ that globally minimized the least square errors, it is necessary to collect a great number of samples. In the final implementation of the proposed method, to make the
learning period shorter and let the back- ground estimation algorithm work on-line, an approximation of $\beta$ is used [11] . Setting:
$A=\left[\begin{array}{cc}1& {T}_{a}\left(t-1\right)\\ 1& {T}_{a}\left(t-2\right)\\ ⋮& ⋮\\ 1& {T}_{a}\left(t-\tau \right)\end{array}\right]\text{}\text{ }\text{and}\text{ }\text{\hspace{0.17em}}B=\
left[\begin{array}{c}{T}_{i,j}\left(t-1\right)\\ {T}_{i,j}\left(t-1\right)\\ ⋮\\ {T}_{i,j}\left(t-\tau \right)\end{array}\right].$(9)
$\beta$ as been computed as it follows:
$\stackrel{^}{\beta }\left(t\right)={A}^{‡}B,$(10)
where $\tau$ is a time window and $\text{‡}$ means pseudo inverse operation.
Thus, the estimated background temperature is computed as it follows:
${\stackrel{^}{T}}_{b\left(i,j\right)}\left(t\right)=\left[\begin{array}{cc}1& {T}_{a}\left(t\right)\end{array}\right]\cdot \stackrel{^}{\beta },$(11)
while the residual squares are given by:
Figure 4. Dependence between ambient temperature and filtered sensor cell measure- ments.
Finally, since the estimation of $\stackrel{^}{\beta }$ should involve only the background related measurements, Equation (10) is extended using the analysis of the residual squares:
$\stackrel{^}{\beta }\left(t\right)=\left(\begin{array}{ll}\stackrel{^}{\beta }\left(t-1\right),\hfill & \text{ }\text{if}\text{ }R\left(t\right)\ge {R}_{\text{th}}\hfill \\ {A}^{‡}B\hfill & \text{ }
\text{otherwise}\text{ }\hfill \end{array}.$(13)
3.3. Probabilistic Foreground Detection
In order to provide as much information as possible in uncertain situations, the proposed method also computes for every cell the probability of human detection in every instant. For this reason, we
modeled the probability function $q\left({T}_{i,j}\left(t\right)\right)$ , describing whether the measurement belongs to the background temperature distribution ${T}_{b}$ , as a logistic function:
$q\left({T}_{i,j}\left(t\right)\right)=\frac{2}{1+{\text{e}}^{k\cdot R\left(t\right)}},$(14)
where $k$ is the steepness of the function and the probability $p\left({T}_{i,j}\left(t\right)\right)$ that that measurement does not belong to the background distribution is (Figure 5):
4. Installation and Results
The proposed method aims to improve the accuracy of Human Detection algo- rithms using LR-TAS data in noisy environments. For this reason, the environ- ment has been perturbed during the experiment
using air-conditioning system, appliances, and exposing the sensor to sunlight reflection.
4.1. Sensor Mount
In the literature, there are two opinions on the placement of LR-TAS for human
Figure 5. Modeled probability function for foreground detection.
detection: the wall [12] [13] and the ceiling [9] . After carrying out several expe- riments, it was found that the installation on the wall has several drawbacks:
• Under real condition, furniture and other objects can obstruct the view of the sensor.
• The movement of a human-coming closer and further to the installation point- influences the amount of pixels representing him/her.
• Since the sensor tends to average the value of the temperature in the observed space, human movement also affects the temperature distribution.
For these reasons, the sensor was mounted on the ceiling at the height of 2.7 m with a resultant detection area on the floor of approximately 9 m^2 (Figure 6). Streaming data have been transmitted
and stored on a central device. The comun- nication has been implemented using thread protocol [14] : an innovative solution designed by the thread group for IoT applications. This protocol allows to
easily create an IPv6, meshed, robust and secure network of sensors.
4.2. Experiments
We have collected several datasets for a total duration of 4 days. During this pe- riod people were asked to perform everyday activities passing and staying under the sensor. The experiment data have
been manually annotated to validate the pro- posed algorithm: every pixel is labeled as “1” if it represents human and “0” other- wise.
4.3. Results
The proposed method has been tested on the retrieved datasets. Figure 7(a) shows the performance of the temperature background estimation for a single pixel sen- sor. Comparing it with Figure 7(b),
it is visible that where the filtered measure- ments are distant from the estimated background, the probability of human detec- tion is very high.
In order to compare the obtained results with the method that, in our know- ledge, reports the best accuracy value in the literature, we implemented also the
Figure 7. Outcome of different steps computed over 3 hours sample data. (a) Background estimation results: original measurements (light gray), measurements obtained after Kal- man Filtering (gray)
and the estimated background temperature (black) obtained through Equation (11); (b) The probability of Human Detection computed with the proposed me- thod; (c) ${T}_{\text{ }\text{\hspace{0.17em}}\
text{diff}\text{ }}$ as computed with the method by Mashiyama et al.
human detection algorithm proposed by Mashiyama et al. [9] : Figure 7(c) shows ${T}_{\text{ }\text{\hspace{0.17em}}\text{diff}\text{ }}$ computed using the method proposed by Mashiyama.
Unfortunately, au- thors do not suggest any method to compute the threshold ${T}_{\text{ }\text{th}\text{ }}$ and it is very di- fficult to manually tune it.
To compare the results of the work of two algorithms, the measurement is said to represent human when a frame contains at least one measurement whose pro- bability $p\left(m\left(t\right)\right)>{p}_
{\text{th}}$ . The parameters used in both of the algorithms are pro- vided in Table 1, while the obtained results are presented in Table 2.
The proposed method shows surprising results in terms of precision and recall, proving that it is able to detect humans even in a noisy environment. The me- thod proposed by Mashiyama instead,
reports a low recall value since it is missing a lot of detections, which means high false negative value. Moreover, the number of detections (true positive + false positive) is much lower than in
the proposed scenario and it is strictly related to the temperature threshold: in this settings Ma- shiyama method obtains a high precision value. The final measure to compare the performance of two
methods is given by the ACCURACY measure:
where TP, TN, FP, FN are: true positive, true negative, false positive and false ne- gative values.
Table 1. Parameters used in the proposed method and Mashiyama et al. method.
Table 2. Comparison between the proposed method and Mashiyama et al. method in de- tecting humans using LR-TAS in a noisy environment.
5. Conclusions
We have presented a novel technique to detect humans in indoor environments using Low Resolution Thermal Array Sensor. This approach considers the tem- perature variation in the room due to external
dynamics and noise. A Kalman Fil- ter has been used to filter the noise on the temperature measurements while a back- ground estimation technique aims to separate the background from humans.
Final results show an improvement in the human detection accuracy com- pared with the state of the art when performing a field trial in a real environ- ment passing from 70% to 97%.
Currently, the main limitation of the proposed method is that it is hard to dis- tinguish a human presence from other moving heat sources. Further studies in this direction may improve the human
detection accuracy in real smart home envi- ronments, reducing the overall system’s false positive rate.
Finally, the mentioned results have been collected using a single sensor insta- llation, however a multisensor system needs to be implemented in order to set up a real scenario in a smart
environment. This extension, which requires to handle tech- nical theoretical problems―from placing the sensors to retrieving one overall model with the global state of dwellers and environment―will
be part of our future work.
This work was partially financed from project ADALGISA-Regione Lombardia (CUP: E68F13000360009). We thank Dr. Ratti Alessandro from R.S.R. srl (Co- mo, Italy) who provides insight and expertise that
greatly assist the research. | {"url":"https://scirp.org/journal/paperinformation?paperid=74726","timestamp":"2024-11-05T22:16:37Z","content_type":"application/xhtml+xml","content_length":"164069","record_id":"<urn:uuid:7fd45a57-8e16-4ed3-aa78-1a7335f5cde0>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00863.warc.gz"} |
How to Handle Categorical Data In Pandas?
Handling categorical data in Pandas involves converting the categorical variables into a suitable format that can be utilized for further analysis or modeling. Here are some common techniques used to
handle categorical data in Pandas:
1. Encoding categorical data: Categorical variables need to be encoded into numerical form before they can be used in machine learning algorithms. One common encoding technique is One-Hot Encoding,
where each category is converted into a binary column representing its presence or absence.
2. Label encoding: It involves assigning a unique integer label to each category. This technique is useful when the categorical variable has an inherent ordering or hierarchy.
3. Ordinal encoding: It is similar to label encoding but assigns numerical labels based on the order or rank of the categories. This is suitable for ordinal variables where the categories have a
natural order.
4. Dummy variables: It creates binary variables for each category of a categorical variable. It is a form of one-hot encoding, but if a variable has N categories, only N-1 dummy variables are
created to avoid multicollinearity.
5. Removing or replacing missing values: If the categorical variable has missing values, you can either remove the rows with missing values or replace them with a suitable value (e.g., mode or a
separate category for missing).
6. Grouping categories: Sometimes, categorical variables may have too many levels or categories. In such cases, you can group similar categories together to reduce the number of levels and improve
analysis or modeling.
Overall, handling categorical data in Pandas involves applying appropriate encoding or transformation techniques to ensure they can be effectively utilized for further analysis or modeling tasks.
What is the difference between nominal and ordinal categorical data in Pandas?
In Pandas, nominal and ordinal categorical data are two types of categorical data that can be stored using the Categorical data type.
1. Nominal Categorical Data: Nominal data represents categories that have no specific order or rank. Examples include categories like colors, names, or product types. The categories are simply
labels, and there is no inherent order among them. In Pandas, nominal categorical data can be represented using the Categorical data type with the dtype="category" argument.
2. Ordinal Categorical Data: Ordinal data represents categories with a specific order or rank. Examples include ratings (e.g., "excellent," "good," "medium," "poor"), education levels (e.g., "high
school," "bachelor's," "master's," "doctorate"), or satisfaction levels (e.g., "very satisfied," "satisfied," "neutral," "dissatisfied," "very dissatisfied"). The categories have a meaningful
order or ranking associated with them. In Pandas, ordinal categorical data can be represented using the Categorical data type with the dtype="category" argument, along with passing a specified
order using the ordered=True argument.
Both nominal and ordinal categorical data in Pandas offer advantages like compressed memory usage, improved performance, and the ability to define a category's order in the case of ordinal data.
How to create a new categorical column in Pandas?
To create a new categorical column in Pandas, you can use the astype() function with the category data type argument. Here is an example:
1 import pandas as pd
3 # Create a DataFrame
4 df = pd.DataFrame({'Color': ['Red', 'Blue', 'Green', 'Green', 'Red', 'Blue', 'Red']})
6 # Convert the 'Color' column to categorical
7 df['Color'] = df['Color'].astype('category')
9 # Check the data type of the 'Color' column
10 print(df['Color'].dtype)
In this example, the 'Color' column is converted to a categorical data type using the astype() function.
What is the advantage of using categorical data in Pandas?
There are several advantages of using categorical data in Pandas:
1. Efficient memory usage: Categorical data is stored as integers internally, which leads to significant memory savings compared to storing the same data as strings.
2. Faster performance: Since categorical data is stored as integers, operations like sorting and grouping are usually faster compared to operations on string data.
3. Automatic data validation: Categorical data in Pandas has a defined set of categories, which can help in identifying and handling data-entry errors more easily.
4. Improved readability: Categorical data provides more meaningful and descriptive labels or names for the categories, making the data easier to interpret and understand.
5. Better compatibility with statistical models: Many statistical models in Pandas or other libraries expect input data to be categorical, so converting data to the categorical type can be
beneficial for model compatibility.
6. More efficient operations: Several operations, such as get_dummies(), can be executed more efficiently on categorical data, resulting in faster computation times.
Overall, using categorical data in Pandas can lead to improved memory efficiency, faster performance, and enhanced data analysis capabilities.
What is the impact of categorical data on statistical modeling in Pandas?
Categorical data in pandas can have a significant impact on statistical modeling in several ways:
1. Memory and performance efficiency: Categorical data uses less memory compared to the corresponding object dtype, which can have a substantial impact on large datasets. This efficiency comes from
the fact that categorical data is encoded with numerical codes, instead of storing full strings or objects.
2. Faster computations: Many statistical operations can be performed faster on categorical data as operations can be executed directly on the numerical codes, rather than on the full string/object
data. This can lead to significant speed improvements, especially in aggregations and group-by operations.
3. Improved data representation: Categorical data enables the representation of data with a predefined set of distinct categories. This can help in clearer visualization, analysis, and
interpretation of data. It provides a way to handle and analyze data with a limited number of categories or levels, such as gender, job titles, or product types.
4. Easy handling of missing values: Pandas supports the concept of missing values in categorical data using the special category called NaN (not a number). This simplifies the handling of missing
data in statistical modeling, as missing values can be easily identified and filtered.
5. Enhanced data analysis: Categorical data in pandas enables various operations specific to categorical variables. These include grouping and aggregation by categories, reordering the categories
based on specific criteria, and applying categorical-specific statistical functions.
Overall, the use of categorical data in pandas can improve memory efficiency, computation speed, and data analysis for statistical modeling, leading to more efficient and accurate results.
How to perform label encoding for categorical data in Pandas?
To perform label encoding for categorical data in Pandas, you can use the LabelEncoder class from the sklearn.preprocessing module. Here's an example of how you can do it:
1. Import the required libraries:
1 import pandas as pd
2 from sklearn.preprocessing import LabelEncoder
1. Create an instance of LabelEncoder:
1. Load your data into a DataFrame:
1 data = {'Color': ['Red', 'Blue', 'Green', 'Red', 'Blue']}
2 df = pd.DataFrame(data)
1. Encode the categorical column using fit_transform() method:
1 df['Color_Encoded'] = le.fit_transform(df['Color'])
In this example, the column 'Color' contains categorical data. We create a new column 'Color_Encoded' to store the encoded values. The fit_transform() method both fits the encoding and transforms the
'Color' column to its encoded form.
1. Print the resulting DataFrame:
This will give you the following output:
1 Color Color_Encoded
2 0 Red 2
3 1 Blue 0
4 2 Green 1
5 3 Red 2
6 4 Blue 0
In the encoded column, 'Blue' is represented by 0, 'Green' by 1, and 'Red' by 2.
Label encoding can be useful when you need to convert categorical data into numerical values for further analysis or machine learning models.
What is the significance of ordinal categorical data in Pandas?
Ordinal categorical data in Pandas is significant because it allows for the representation and analysis of data that has an inherent order or ranking.
While categorical data represents discrete values that do not have any specific order, ordinal data represents values that have a specific order or ranking. This order could be based on factors such
as quality, preference, or rating.
By using ordinal categorical data in Pandas, it becomes possible to sort, filter, and analyze the data based on the order of the categories. This facilitates tasks such as finding the minimum or
maximum values, calculating averages, or determining the most frequent category.
Furthermore, using ordinal categorical data allows for better visual representation and interpretation of data. When plotting or visualizing ordinal data, Pandas considers the specific order of the
categories and ensures that the resulting plot reflects this order accurately.
In summary, the significance of ordinal categorical data in Pandas lies in its ability to represent and analyze data with an inherent order or ranking, enabling meaningful analysis and visualization
of such data. | {"url":"https://topminisite.com/blog/how-to-handle-categorical-data-in-pandas","timestamp":"2024-11-04T05:22:42Z","content_type":"text/html","content_length":"351486","record_id":"<urn:uuid:43dfba7b-5678-4d95-aa17-7c00ac65c5d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00207.warc.gz"} |
A First Look at Computers | CS4998: Blockchain Development Textbook
A First Look at Computers
Examining an Elementary Computer
Before we examine the EVM, we will look at a very simple computer to get an understanding of the process a computer undergoes when running. By understanding the logistics of an elementary computer,
we will have the foundation necessary to understand more sophisticated computers.
The Cuzco Model
The Cuzco Model is a basic computer that I have created for the purposes of this section; this computer is capable of performing basic arithmetic operations.
To get started, we will first discuss the architecture of the Cuzco Model - i.e. the components that the Cuzco Model is comprised of:
Instructions: an ordered list of operations that the computer will perform. This list is 0-indexed.
Program Counter (PC): keeps track of the current instruction to execute
Stack: data structure that is updated via operations
Now that we have discussed the architecture of the Cuzco Model, we will examine the stack - although simple, the stack allows us for the Cuzco Model to be memoryful. Otherwise, we can perform an
infinite number of operations and our computer would still remain in the same "state". Like all stacks, the stack that the Cuzco Model utilizes operates on a last-in, first-out principle. However,
this stack grows up, not down.
Next, we discuss the operations that the Cuzco Model is capable of; listed below are a
Operation Operation Code (OpCode) Description
Push v 0x00 v Pushes v to the top of the stack
Addition 0x01 Pops the top two elements from the stack, computes their sum, and pushes the result to the stack
Subtraction 0x02 Pops the top two elements from the stack, computes their difference, and pushes the result to the stack
Logical And 0x03 Pops the topmost element from the stack, computes the logical and operation of the value, and pushes the result to the stack
Bitwise And 0x04 Pops the topmost elements from the stack, computes the bitwise and operation of the value, and pushes the result to the stack
Stop 0x05 Stops execution of the program
Finally, listed below are the invariants that the Cuzco model abides by:
All values on the stack are 8-bit signed integers.
Furthermore, there is no protection against integer overflow
The stack has a maximum depth of 4 values
Now that we have the specification of the Cuzco Model outlined, let's examine a couple of examples that show the power of the Cuzco Model!
Example One: Adding Two Values
We begin by adding two numbers; although this may seem like an easy task at first, there are multiple steps involved for the Cuzco Model to execute this.
Assume that we want to compute the sum of 1 + 2. The first step in this is to get the values 1 and 2 onto the computer itself. To do this, we can use the Push operation; listed below are the two Push
operations that we need to input 1 and 2 into the Cuzco Model:
0x00 0x01 (Push 2)
0x00 0x02 (Push 1)
Now that we have the two values 1 and 2 pushed onto the stack, we want to compute their sum. To do this, we can use the addition operation. Our updated list of instructions is as follows:
0x00 0x1 (Push 2)
0x00 0x2 (Push 1)
0x01 (Addition)
Finally, to signal to the computer that we are finished executing, we include the Stop operation at the end of our instructions. Therefore, our final set of instructions is as follows:
0x00 0x1 (Push 2)
0x00 0x2 (Push 1)
0x01 (Addition)
0x05 (Stop)
We now run the Cuzco Model on the set of instructions provided above. Below is a table which shows the state of the computer at each step of execution:
PC Instruction Stack Description
0x0 0x00 0x1 1 | We are pushing 1 onto the stack
0x1 0x00 0x2 2 | 1 | We are pushing two onto the stack
0x2 0x01 3 | We are computing the sum of 1 and 2
0x3 0x05 3 | We are terminating the program
Example Two: Multiple Arithmetic Operations
In this next example, we will introduce the subtraction operation. Here, we will compute (1 + 2) - 5, which should yield us the value -2. Rather than having to rewrite our original list of
instructions, we just need to add the following two operations:
We need to push 5 onto the stack before we push 1 and 2
After computing the sum of 1 and 2, we need to compute the difference of the result and 5
Therefore, our updated set of instructions is as follows:
0x00 0x5 (Push 5)
0x00 0x1 (Push 2)
0x00 0x2 (Push 1)
0x01 (Addition)
0x02 (Subtraction)
0x05 (Stop)
Running the Cuzco Model on the new set of instructions, our state table is as follows:
PC Instruction Stack Description
0x0 0x00 0x5 5 | We are pushing 5 onto the stack
0x1 0x00 0x1 1 | 5 We are pushing 1 onto the stack
0x2 0x00 0x2 2 | 1 | 5 We are pushing 2 onto the stack
0x3 0x01 3 | 5 We are computing the sum of 1 and 2
0x4 0x02 -2 | We are computing the difference of 3 and 5
0x5 0x05 -2 | We are terminating the program
Example Three: Introducing Logical Operators
In this example, we want to compute the result of 1 && 2 (where && is the logical and operator). Since 1, 2 are nonzero values, this should equal 1. First writing our set of instructions:
0x00 0x1 (Push 1)
0x00 0x2 (Push 2)
0x03 (Logical And)
0x05 (Stop)
Running the Cuzco Model on the set of instructions above:
PC Instruction Stack Description
0x0 0x00 0x1 1 | We are pushing 1 onto the stack
0x1 0x00 0x2 2 | 1 | We are pushing 2 onto the stack
0x2 0x03 1 | We are popping 1 and 2 from the stack, computing the logical and operation, and pushing the result onto the stack
0x3 0x05 1 | We are terminating the program
Example Four: Introducing Bitwise Operators
In our final example, we will introduce the usage of the bitwise and operator.
For this example, we want to compute 1 & -2 (where & is the bitwise and operator). Before writing the instructions, we want to first understand what the expected result should be. For this, we will
need to represent 1 and -2 in binary notation (using two's complement. Therefore, we have:
1 => 00000001
-2 => 11111110
The bitwise and operator, as the name might suggest, applies the and operation on a bit-by-bit basis. Therefore, it is easy to see that 1 & -2 = 0. Now that we know what the expected result should
be, we write the instructions our compute will execute:
0x00 0x01 (Push 1)
0x00 0xFE (Push -2)
0x04 (Bitwise And)
0x05 (Stop)
Our state table is as follows:
PC Instruction Stack Description
0x0 0x00 0x1 1 | We are pushing 1 onto the stack
0x1 0x00 0xFe -2 | 1 | We are pushing -2 onto the stack
0x2 0x04 0 | We are popping 1 and 2 from the stack, computing the bitwise and operation, and pushing the result onto the stack
0x3 0x05 0 | We are terminating the program
Cuzco Machine, in Solidity!
// SPDX-License-Identifier: MIT
pragma solidity ^0.8.0;
contract CuzcoMachine {
int8[] stack;
function execute(bytes calldata insn) public returns (int8) {
uint counter = 0;
while (counter < insn.length) {
if (insn[counter] == 0x0) {
// We have push operation
// Extract pushvalue
int8 pushValue = int8(uint8(insn[counter + 1]));
counter += 2;
} else {
// Not push operation
if (insn[counter] == hex"01") {
// Add
} else if (insn[counter] == hex"02") {
// Subtract
} else if (insn[counter] == hex"03") {
// Logical And
} else if (insn[counter] == hex"04") {
// Bitwise And
} else {
// Stop
counter += 1;
return stack[stack.length - 1];
function popTwo() private returns (int8, int8) {
// Pop two values off stack
int8 a = stack[stack.length - 1];
int8 b = stack[stack.length - 1];
return (a, b);
function add() private {
(int8 a, int8 b) = popTwo();
stack.push(a + b);
function subtract() private {
(int8 a, int8 b) = popTwo();
stack.push(a - b);
function logicalAnd() private {
(int8 a, int8 b) = popTwo();
// We need to convert a, b to boolean values
bool a_bool = a != 0;
bool b_bool = b != 0;
bool result = a_bool && b_bool;
int8 result_int;
if (result) {
result_int = 1;
function bitwiseAnd() private {
(int8 a, int8 b) = popTwo();
stack.push(a & b);
function pushOntoStack(int8 val) private {
if (stack.length > 4) {
revert("Stack overflow!"); | {"url":"https://cs4998.cornellblockchain.org/understanding-the-evm/the-ethereum-virtual-machine/a-first-look-at-computers","timestamp":"2024-11-02T21:29:20Z","content_type":"text/html","content_length":"739644","record_id":"<urn:uuid:f9e3c81c-b8f5-4d65-ac2a-e54d43ee777f>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00873.warc.gz"} |
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Using Thermistors in Temperature Tracking Power Supplies
This article provides a simple, intuitive tutorial on negative temperature coefficient (NTC) thermistors and how to make basic use of them in general, and specifically in power supply regulators. A
good example application is their use to cancel temperature effects on LCD display contrast. Two simple NTC thermistor linearizing techniques are shown, and regulator design procedures and examples
demonstrate their application. Each example includes a schematic and compares the measured output voltage versus temperature to the target.
Power supply regulators, by definition, are designed to provide an output voltage that is stable despite variations in line (input voltage), load, and temperature. While for most applications, a
stable output is the goal, there are some applications where it is advantageous to provide a temperature-dependent output voltage. This article provides a tutorial, design procedure, and circuit
examples utilizing negative temperature coefficient (NTC) thermistors in temperature tracking power supplies.
By far the most common application for temperature-dependent regulation is in LCD bias supplies, where the contrast of the display will vary with ambient temperature. By applying a
temperature-dependent bias voltage, the LCD's temperature effects can be automatically canceled to maintain constant contrast over a wide temperature range. The examples in this article are targeted
toward LCD bias solutions; however, the tutorial and design equations are simple and may be easily applied in a variety of circuits.
Why NTC Thermistor?
The NTC thermistor provides a near optimum solution for temperature-dependent regulation. It is low-cost, readily available through a variety of suppliers (Murata, Panasonic, etc.), and available in
small surface-mount packaging from 0402 size through 1206 size. Furthermore, with only a basic understanding, the NTC thermistor is straightforward to apply to your circuit.
NTC Characteristic
As the name implies, the thermistor is just a temperature-dependent resistor. Unfortunately, the dependence is very non-linear (see Figure 1) and, by itself, would not be very helpful for most
applications. Fortunately, there are two easy techniques to linearize a thermistor's behavior.
Figure 1. NTC thermistor resistance varies extremely non-linearly with temperature. This makes it difficult to utilize the thermistor without applying it in a linearizing network. (R[25C] = 10kΩ, β =
The standard formula for NTC thermistor resistance as a function of temperature is given by:
where R[25C] is the thermistor's nominal resistance at room temperature, β (beta) is the thermistor's material constant in K, and T is the thermistor's actual temperature in Celsius.
This equation is a very close approximation of the actual temperature characteristic, as can be seen in Figure 2. Note the use of log-scale for the Y-axis.
Figure 2. Thermistor resistance versus temperature is almost linear on a semi-log graph. The actual measured thermistor resistance matches the Beta formula to a fairly high degree of precision. (R
[25C] = 10kΩ, β = 3965K).
R[25C] and β are usually published in the manufacturer's data sheet. Typical values of R[25C] range from 22Ω to 500kΩ. Typical values of β are from 2500 to 5000K.
As seen in Figure 3, higher values of β provide increased temperature dependence and are useful when higher resolution is required over a narrower temperature range. Conversely, lower values of β
offer less-sloped temperature dependence and are more desirable when operating over a wider temperature range.
Figure 3. An NTC thermistor is specified by its room temperature resistance (R[25C]) and its material constant β (Beta). Beta is a measure of the slope of temperature dependence. (R[25C] = 10kΩ, β in
Self Heating
A thermistor is a resistor, and, just like any resistor, it produces heat energy whenever current passes through it. The heat energy causes the NTC thermistor's resistance to reduce which then
indicates a temperature slightly above ambient temperature. In the manufacturer's data sheets and application notes, there are usually tables, formulae, and text detailing this phenomenon. However,
these may be largely ignored if the current through the thermistor is kept relatively low such that self heating error is small compared to the required measurement accuracy, as in the design
examples of this article.
An NTC thermistor is most easily utilized when applied in a linearizing circuit. There are two simple techniques for linearization: resistance mode and voltage mode.
Resistance Mode
In resistance mode linearization, a normal resistor is placed in parallel with the NTC thermistor, which has the effect of linearizing the combined circuit's resistance. If the resistor's value is
chosen to be equal to the thermistor's resistance at room temperature (R[25C]), then the region of relatively linear resistance will be symmetrical around room temperature (as seen in Figure 4).
Figure 4. Resistance mode linearization is easily accomplished by placing a normal resistor in parallel with the thermistor. If the normal resistor has the same value as R[25C], then the region of
nearly linear resistance versus temperature will be symmetrical around +25°C. (R[25C] = 10kΩ, β in K).
Note that lower values of β produce linear results over a wider temperature range, while higher values of β produce increased sensitivity over a narrower temperature range. The equivalent resistance
varies from roughly 90% of R[25C] at cold (-20°C) to 50% of R[25C] at room temperature (+25°C) to roughly 15% of R[25C] at hot (+70°C).
Voltage Mode
In voltage mode linearization, the NTC thermistor is connected in series with a normal resistor to form a voltage-divider circuit. The divider circuit is biased with a regulated supply or a voltage
reference, V[REF]. This has the effect of producing an output voltage that is linear over temperature. If the resistor's value is chosen to be equal to the thermistor's resistance at room temperature
(R[25C]), then the region of linear voltage will be symmetrical around room temperature (as seen in Figure 5).
Figure 5. Voltage mode linearization is easily accomplished be placing a normal resistor in series with the thermistor and biasing the resulting resistive voltage divider with a constant-voltage
source. If the normal resistor has the same value as R[25C] , then the region of nearly linear output voltage versus temperature will be symmetrical around +25°C. (R[25C] = 10kΩ, β in K).
Again, note that lower values of β produce linear results over a wider temperature range, while higher values of β produce increased sensitivity over a narrower temperature range. The output voltage
varies from near zero volts at cold (-20°C) to V[REF]/2 at room (+25°C) to near V[REF] at hot (+70°C).
Design Procedure
To create a regulated output voltage that varies linearly with temperature, the linearized thermistor circuit is applied to the regulator's feedback network.
Resistance Mode
The resistance mode circuit is the simplest solution for creating a temperature-dependent regulated output voltage because regulator feedback networks are almost always comprised of a resistive
voltage divider. As seen in Figure 6, the linearized thermistor circuit is placed in series with one of the feedback resistors. In this case, the linearized circuit is placed in series with the top
resistor of the feedback divider network to create a negative-temperature-coefficient output voltage at V[OUT], as generally required in LCD bias solutions. (To create a
positive-temperature-coefficient output, the linearizing circuit would be placed in series with the bottom resistor, R2, of the feedback divider.)
Figure 6. The resistance mode linearized thermistor circuit is applied to the feedback network of a voltage regulator. It essentially replaces a portion of one of the normal feedback resistors - that
portion being dependent upon the required temperature coefficient of the regulator's output.
The design procedure is relatively simple. First find the appropriate feedback network bias current, i2, from the regulator's data sheet. It is usually in the 10s to 100s of µA range and there is
some latitude in its exact value. Then calculate the NTC thermistor value as:
where T[C] is the negative temperature coefficient of V[OUT] in %/°C. The value of i2 should be adjusted until R[25C] becomes a readily available NTC thermistor value.
For a simplified design calculation, select R2 and R1 as:
where V[FB] is the nominal feedback voltage as given in the regulator's data sheet.
For a more accurate design calculation, the final value of i2 will end up being slightly modified in order to match the thermistor's β to the desired T[C]. Therefore, calculate the thermistor's
resistance at 0°C and +50°C. The standard formula for NTC thermistor resistance as a function of temperature is given by:
Then calculate the linearized resistance at the two temperatures as:
Calculate the value of R2 and i2 as:
And lastly calculate the value of R1 as:
Resistance Mode Design Example
An LCD bias voltage is needed in a system running on a single-cell Li+ rechargeable battery. The desired bias voltage is V[OUT ]= 20V at room temperature with T[C ]= -0.05%/°C. The MAX1605 regulator
is selected for the task. The above design formulae are used to calculate the required components as follows:
Per the datasheet, i2 should be greater than 10uA for less than 1% output error; therefore, choose i2 to be about five times larger for less error:
An NTC thermistor is chosen with R[25C ]= 20kΩ and β = 3965K and linearized with a parallel 20kΩ resistor. The MAX1605 has a nominal feedback voltage of V[FB ]= 1.25V. According to the simplified
design formulae, R2 and R1 are then calculated as:
Per the more accurate design calculation, the thermistor's resistance at 0°C and +50°C will be:
The linearized resistances at 0°C and +50°C will be:
The values for R2, i2, and R1 are then calculated as:
In this case, these more accurate values are not substantially different from those obtained using the simplified calculations. The final circuit can be seen in Figure 7.
Figure 7. An NTC thermistor is used with the MAX1605 boost converter to realize the resistance mode design example as described in the text.
The output voltage of the circuit of Figure 7 exhibits nearly ideal temperature dependence, as can be seen in Figure 8.
Figure 8. The actual temperature dependence of the circuit of Figure 7 is very close to the target temperature coefficient of -0.05%/°C over most of the extended consumer temperature range.
Voltage Mode
Although more complicated than the resistance mode circuit, the voltage mode circuit has some unique advantages. First, the voltage mode circuit provides a temperature dependent analog voltage that
may be easily digitized with an analog-to-digital converter (ADC) to provide temperature information to the system's microprocessor. Additionally, the regulator's output voltage temperature
coefficient may be easily adjusted by changing the value of only one resistor. This benefit allows for simple trial-and-error design in the laboratory and may also be very valuable for accommodating
multi-sourced thermistors or LCD panels in production.
As seen in Figure 9, the linearized thermistor circuit is biased with a voltage reference to generate a temperature-dependent voltage, V[TEMP]. Then, V[TEMP] is summed into the feedback node through
a resistor, R3, which sets the gain of the temperature dependence. So that V[TEMP] does not need to be buffered, the nominal resistance of the thermistor should be kept much lower than R3. As
connected in Figure 9, the regulator exhibits a negative-temperature-coefficient output voltage at V[OUT], as generally required in LCD bias solutions. (To create a positive-temperature-coefficient
output, the position of R and Rt should be reversed.)
Figure 9. The voltage mode linearized thermistor circuit is applied to the feedback network of a voltage regulator. It essentially adds current i3 into the feedback node such that i1 = i2 + i3. If V
[REF] is twice V[FB], then i3 is zero at 25C, R1 and R2 are calculated as normally described in the regulator's datasheet, and temperature dependence can be adjusted by simply scaling R3.
Additionally, V[TEMP] may be acquired by the host system via an analog-to-digital converter.
Although not mandatory, the simplest implementation of Figure 9 is when V[REF ]= 2xVfb. (Conveniently, many regulators have V[FB ]= 1.25V, many voltage references have V[REF ]= 2.5V, and many ADCs
have input voltage range from 0 to 2.5V.) When V[REF ]= 2xVfb, V[TEMP] will equal V[FB] at +25°C and i3 will equal zero. This allows R1 and R2 to set the nominal output voltage at +25°C independent
of R3 and the thermistor. Select R2 according to the recommendations in the regulator's data sheet. Then calculate R1 and i2 as:
Then calculate the approximate value of R3 as:
where T[C] is the negative temperature coefficient of V[OUT] in %/°C. (This value of R3 will suffice for a simplified design calculation and may be later adjusted through experimentation in the
laboratory.) Then, to avoid the need for a buffer amplifier between V[TEMP] and R3, choose a nominal thermistor value of:
For a more accurate calculation, the final value of R3 will end up being slightly modified in order to match the thermistor's β to the desired T[C]. To do this, first calculate the thermistor's
resistance at 0°C and +50°C. The standard formula for NTC thermistor resistance as a function of temperature is given by:
Then calculate the linearized voltage, V[TEMP], at the two temperatures as:
The more accurate value of R3 is finally given as:
Voltage Mode Design Example
An LCD bias voltage is needed in a system running on a Li+ battery. The desired bias voltage is V[OUT ]= 20V at room temperature with T[C ]= -0.05%/°C. The MAX629 regulator is selected for the task
because it has a reference voltage output that may be used to bias the thermistor linearizing network. The voltage mode design formulae are used to calculate the required components as follows:
Per the datasheet, R2 should be in the range of 10kΩ to 200kΩ and V[FB ]= 1.25V; therefore:
The approximate value of R3 will be:
The thermistor's nominal resistance should be kept less than 46.9kΩ. Therefore, an NTC thermistor is chosen with R[25C ]= 20kΩ and β = 3965K and linearized with a series 20kΩ resistor and V[REF ]=
2.5V bias.
Per the more accurate design calculation, the thermistor's resistance at 0°C and +50°C will be:
The linearized voltage at 0°C and +50°C will be:
The new value for R3 is then calculated to be:
In this case, the more accurate R3 value is not substantially different from the value obtained using the simplified calculations, and the nearest standard resistor value should be chosen.
Design Example when V[REF] ≠ 2xVfb
In the above voltage mode design example, if there isn't already a V[REF ]= 2.5V supply in the system, it may be cost prohibitive to add one. Fortunately, any regulated voltage will suffice. For this
example, the REF pin of the MAX629 is utilized and V[REF]' = 1.25V. Compared to the above example, V[TEMP] will now vary over half as wide a range; therefore, R3 must be halved to R3' = 475kΩ to
maintain the same output voltage temperature coefficient of T[C ]= -0.05%/°C. Also, it is advisable to reduce the thermistor value and linearizing resistor value to R = R[25C ]= 10kΩ. Furthermore,
because V[TEMP] is lower than V[FB] at 25°C, i3 will be non-zero and the regulator's output voltage will be slightly higher than desired by:
To eliminate this, reduce R1 from 375kΩ to:
The final circuit can be seen in Figure 10.
Figure 10. An NTC thermistor is used with the MAX629 boost converter to realize the voltage mode design example with V[REF] ≠2xVfb as described in the text. The MAX629 was chosen because its REF pin
may be utilized to bias the thermistor linearizing circuit.
The output voltage of the circuit of Figure 10 exhibits nearly ideal temperature dependence, as seen in Figure 11.
Figure 11. The actual temperature dependence of the circuit in Figure 10 is very close to the target temperature coefficient of -0.05%/°C over most of the extended consumer temperature range.
A similar version of this article appeared in the August 1, 2001 issue of EDN magazine. | {"url":"https://www.analog.com/en/resources/technical-articles/using-thermistors-in-temperature-tracking-power-supplies.html","timestamp":"2024-11-13T14:56:10Z","content_type":"text/html","content_length":"184413","record_id":"<urn:uuid:c6cba0e7-18b6-4b86-964a-721f6651977d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00728.warc.gz"} |
Unlocking the Power of Earned Value Formula: A Comprehensive Guide
earned value formula · April 27, 2024
Unlocking the Power of Earned Value Formula: A Comprehensive Guide
Learn how to calculate earned value using the earned value formula, and understand its significance in project management. Discover how to optimize your project performance with this essential tool.
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Earned Value Formula: A Comprehensive Guide to Measuring Project Performance
In the world of project management, measuring performance is crucial to ensure that projects are completed on time, within budget, and to the required quality standards. One of the most widely used
methods for measuring project performance is the Earned Value Management (EVM) technique. At the heart of EVM lies the Earned Value Formula, a powerful tool that helps project managers to track and
analyze project progress. In this article, we will delve into the world of Earned Value Formula, exploring its components, calculations, and applications.
What is Earned Value Formula?
The Earned Value Formula is a mathematical expression that calculates the value of work completed by a project team. It is a key component of the Earned Value Management (EVM) technique, which is
used to measure project performance and progress. The formula takes into account three main components: Planned Value (PV), Earned Value (EV), and Actual Cost (AC).
The Three Components of Earned Value Formula
1. Planned Value (PV)
Planned Value represents the budgeted cost of work scheduled, which is the approved budget for the project. It is the value of the work that is planned to be completed during a specific period.
2. Earned Value (EV)
Earned Value represents the value of work actually completed during a specific period. It is the value of the work that has been earned by the project team.
3. Actual Cost (AC)
Actual Cost represents the total cost incurred by the project team during a specific period. It includes all the expenses incurred during the project, including labor, materials, and overheads.
The Earned Value Formula
The Earned Value Formula is calculated using the following equation:
Earned Value (EV) = (Percent Complete x Planned Value (PV))
• Percent Complete is the percentage of work completed during a specific period.
• Planned Value (PV) is the budgeted cost of work scheduled.
Let's say we have a project with a planned value of $10,000, and the project team has completed 60% of the work. To calculate the earned value, we would use the following formula:
Earned Value (EV) = (0.6 x $10,000) = $6,000
This means that the project team has earned $6,000 worth of work during the period.
How to Calculate Earned Value Formula
Calculating the Earned Value Formula involves a series of steps:
Step 1: Determine the Planned Value (PV)
Identify the planned value of the project, which is the approved budget for the project.
Step 2: Determine the Percent Complete
Determine the percentage of work completed during a specific period. This can be done by tracking the progress of the project and estimating the percentage of work completed.
Step 3: Calculate the Earned Value (EV)
Use the formula Earned Value (EV) = (Percent Complete x Planned Value (PV)) to calculate the earned value.
Benefits of Earned Value Formula
The Earned Value Formula offers several benefits to project managers, including:
1. Accurate Performance Measurement
The Earned Value Formula provides an accurate measurement of project performance, enabling project managers to track progress and identify areas for improvement.
2. Early Warning System
The formula helps to identify potential problems early, allowing project managers to take corrective action to get the project back on track.
3. Improved Resource Allocation
The Earned Value Formula helps project managers to allocate resources more effectively, ensuring that resources are utilized efficiently and effectively.
4. Enhanced Stakeholder Communication
The formula provides a common language and framework for communicating project performance to stakeholders, ensuring that everyone is on the same page.
Common Challenges and Limitations
While the Earned Value Formula is a powerful tool, it is not without its challenges and limitations. Some common challenges include:
1. Data Quality Issues
Poor data quality can lead to inaccurate calculations and misleading results.
2. Complexity
The formula can be complex, requiring a good understanding of project management principles and practices.
3. Resource Intensive
Calculating the Earned Value Formula can be resource-intensive, requiring significant time and effort.
The Earned Value Formula is a powerful tool for measuring project performance and progress. By understanding the components of the formula, including Planned Value, Earned Value, and Actual Cost,
project managers can track project performance and identify areas for improvement. While the formula has its challenges and limitations, its benefits far outweigh its drawbacks. By mastering the
Earned Value Formula, project managers can take their project management skills to the next level, delivering projects on time, within budget, and to the required quality standards. | {"url":"https://30dayscoding.com/blog/earned-value-formula-guide","timestamp":"2024-11-02T22:09:27Z","content_type":"text/html","content_length":"97278","record_id":"<urn:uuid:c71a383e-5f13-46bf-b5e4-2ed8166a7460>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00306.warc.gz"} |
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Carlos Perez
Carlos Perez : Poincare-BMO inequalities and weighted estimates for operators
It is well known that one of the key points in the regularity theory of elliptic Partial Differential Equations are the Poincare inequalities or the improved version so called Poincare-Sobolev
inequalities. In the first part of this course I will concentrate on Poincare type inequalities. We plan to show that these basic estimates are intimately related to certain other basic spaces in
Analysis such as the BMO spaces of John-Nirenberg or the Lipchitz spaces. The main techniques we will come from Calderon-Zygmund theory from Harmonic Analysis, and in particular the good-lambda of
Burkholder and Gundy will play a main role.
As a sample, we will derive as a corollary inequalities such as the one due to Fabes, Kenig and Serapioni important in the case of degenerate elliptic equations:
( 1/w(B) \int_B |f(x)-f_B|^p w(x) dx )^1/p <= C r(B) ( 1/w(B) \int_B |\nabla f(x)|^2 w(x) dx )^1/2.
where w is an A_2 weight, and for some p>2. The unweighted estimate is well known with p=2n/(n-2), n>2.
To simplify our presentation we will present the results and techniques in R^n and with the metric associated to cubes; however, we will point out possible extensions and difficulties when extending
the main issues to general Spaces of Homogeneus Type. To work within this context is interesting since we can go beyond the study of the standard gradient and to consider differential operators such
as Hormander Laplacian X or the Baouendi-Grushin operator.
This work presented is in collaboration with B.Franchi and R. Wheeden and with P. MacManus.
In the second part of the course I will concentrate on certain aspects of the theory of the two weight problem for certain operators such as fractional integration, classical singular integrals or
commutators of singular integrals with BMO functions. More precisely if T one of these operators we look look for "reasonable" conditions for which either the following inequality
\int_R^n |Tf(x)|^p w(x) dx <= C \int_R^n |f(x)|^p v(x) dx,
or the correpsonding weak version holds. In some cases there are necesssary and sufficcient conditions due to Sawyer, which are not easy to handle in practice. We look for conditions more geometrical
and close in some sense to the usual A_p conditions.
We will show some known results in the case of fractional integration and singular integrals. In the last case the problem becomes difficult and we will sketch some recent joint work with D.
Cruz-Uribe where some sufficient conditions have been obtained.
See picture of this text in TeX (27 kB). | {"url":"https://kma.mff.cuni.cz/ss/jun99/carlos.htm","timestamp":"2024-11-09T02:28:40Z","content_type":"text/html","content_length":"3360","record_id":"<urn:uuid:66900ecd-cc31-49a2-a5d7-aea925ff57ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00735.warc.gz"} |
Client Push Installation
Good evening all,
At my company I have been asked to take the rains of SCCM as our SCCM consultant has suddenly decided to leave.
I'm 'ok' with SCCM but far from being great. On a 1 to 10 scale I'm probably a 3 rating.
Anyway I decided that the best thing for me to do is set up SCCM in a lab at home and learn it. I'm currently working through my book and I've got stuck at the 'Client Push Installation' section.
I have set everything up correctly as far as I am aware however I am now trying to roll out Config Manager to a member server I have on my domain.
So I right click the server and hit the 'Install Client' button.
Then log on to my member server. I can see the on the root of the C: I now have a folder called SMS_SERVERNAME.domain.local. This folder is 12mb in size. I am also looking in task manager and I can
see that the 'ccmsetup.exe (32 bit)' background process has started. I believe that this should eventually change to ccmexec.exe? at some point.
Anyway minutes later and ccmsetup.exe disappears and nothing after checking config manager has now installed the client.
Can anyone help
Is there a folder on the destination server called CCM? Or CCMSetup? It should be in the %windir% folder (I think!! lol).
If so, there should be some logs in there explaining it (hopefully the setup folder).
Thanks for your reply. Please find attached the ccmsetup log that I acquired from the target client.
I have looked myself and can see a few errors on it etc. Perhaps you could shed some light on them.
Hopefully they are not too difficult to fix.
Can you just check that the "BITS" service is running (both on the client and sccm server).
You seem to be having error: "0x800b0110"
And a little bit of searching came across this
There doesn't seem to be much info on this error code (from what I can find anyway), so hopefully it's as simple as that.
With the client in question being a server, you may need to add it as a role/feature (bits). Can't remember which it is lol.
Thanks, good luck!
Edenost - Ok I installed BITS on my client server and this has not fixed the issue.
I.hn.yang - How do you mean how is my DP configured? Is there a certain page I should be looking at. When looking at Administration > Servers and Site System Roles my SCCM server is the distribution
point I haven't changed any settings in here and they are setup as default. See screen shot for General settings.
Specify how client computers communicate with this distribution point: HTTPS requires computers to have a valid PKI Client Cert....in my DPs i have it set to HTTP.
May be worth looking into since it seems like the PC can't reach the distribution point. Not 100% sure though
How can I set it to http? I did see this and wondered if this could be the issue. However it seems to be greyed out for me and I'm unable to change it?
Administration > Site Configuration > Sites > Right Click on your Site name and choose properties.
Click: Client Computer Communication and select "HTTPS or HTTP" , click OK.
Go back to your distribution point setting and the option should no longer be greyed
Ok, I've made that change now.
Just tried to push install the client again and still it fails. CCMSetup.log is attached.
help would be great. Do you think I should completely re-install SCCM and select http instead. I followed the guides on CBT nuggets for installation for this. | {"url":"https://www.windows-noob.com/forums/topic/10949-client-push-installation/#comment-41601","timestamp":"2024-11-13T07:34:50Z","content_type":"text/html","content_length":"199847","record_id":"<urn:uuid:b5d93dc7-2e6a-4776-aa93-0db91c2b0cd4>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00333.warc.gz"} |
What Is A Team Cycle? | Heart & Body Naturals Support Center
A Team Cycle happens when you are an active Executive and you have 200 points of volume on your right leg, and 200 points of volume on your left leg.
Each match of 200 volume points on the right and 200 volume points on the left is called a Cycle. Each time you Cycle, you earn the Binary Bonus.
If you have 25 CV in the previous 31 days, you earn 4% of 200 ($8.00)
If you have 100 CV in the previous 31 days, you earn 12% of 200 ($24.00) | {"url":"https://intercom.help/heartandbodynaturals/en/articles/5162313-what-is-a-team-cycle","timestamp":"2024-11-09T00:11:55Z","content_type":"text/html","content_length":"41418","record_id":"<urn:uuid:877b8555-216e-40db-b2bc-1cc87d140d22>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00081.warc.gz"} |
109. Moral Letters to Lucilius | Seneca
Letter 109
On the Fellowship of Wise Men
1. You expressed a wish to know whether a wise man can help a wise man. For we say that the wise man is completely endowed with every good, and has attained perfection; accordingly, the question
arises how it is possible for anyone to help a person who possesses the Supreme Good. Good men are mutually helpful; for each gives practice to the other’s virtues and thus maintains wisdom at its
proper level. Each needs someone with whom he may make comparisons and investigations.
2. Skilled wrestlers are kept up to the mark by practice; a musician is stirred to action by one of equal proficiency. The wise man also needs to have his virtues kept in action; and as he prompts
himself to do things, so is he prompted by another wise man.
3. How can a wise man help another wise man? He can quicken his impulses, and point out to him opportunities for honourable action. Besides, he can develop some of his own ideas; he can impart what
he has discovered. For even in the case of the wise man something will always remain to discover, something towards which his mind may make new ventures.
4. Evil men harm evil men; each debases the other by rousing his wrath, by approving his churlishness, and praising his pleasures; bad men are at their worst stage when their faults are most
thoroughly intermingled, and their wickedness has been, so to speak, pooled in partnership. Conversely, therefore, a good man will help another good man. “How?" you ask.
5. Because he will bring joy to the other, he will strengthen his faith, and from the contemplation of their mutual tranquillity the delight of both will be increased. Moreover they will communicate
to each other a knowledge of certain facts; for the wise man is not all-knowing. And even if he were all-knowing, someone might be able to devise and point out short cuts, by which the whole matter
is more readily disseminated.
6. The wise will help the wise, not, mark you, because of his own strength merely, but because of the strength of the man whom he assists. The latter, it is true, can by himself develop his own
parts; nevertheless, even one who is running well is helped by one who cheers him on. “But the wise man does not really help the wise; he helps himself. Let me tell you this: strip the one of his
special powers, and the other will accomplish nothing.”
7. You might as well, on that basis, say that sweetness is not in the honey: for it is the person himself who is to eat it, that is so equipped, as to tongue and palate, for tasting this kind of food
that the special flavour appeals to him, and anything else displeases. For there are certain men so affected by disease that they regard honey as bitter. Both men should be in good health, that the
one may be helpful and the other a proper subject for help.
8. Again they say: “When the highest degree of heat has been attained, it is superfluous to apply more heat; and when the Supreme Good has been attained, it is superfluous to have a helper. Does a
completely stocked farmer ask for further supplies from his neighbours? Does a soldier who is sufficiently armed for going well-equipped into action need any more weapons? Very well, neither does the
wise man; for he is sufficiently equipped and sufficiently armed for life.”
9. My answer to this is, that when one is heated to the highest degree, one must have continued heat to maintain the highest temperature. And if it be objected that heat is self-maintaining, I say
that there are great distinctions among the things that you are comparing; for heat is a single thing, but helpfulness is of many kinds. Again, heat is not helped by the addition of further heat, in
order to be hot; but the wise man cannot maintain his mental standard without intercourse with friends of his own kind – with whom he may share his goodness.
10. Moreover, there is a sort of mutual friendship among all the virtues. Thus, he who loves the virtues of certain among his peers, and in turn exhibits his own to be loved, is helpful. Like things
give pleasure, especially when they are honourable and when men know that there is mutual approval.
11. And besides, none but a wise man can prompt another wise man’s soul in an intelligent way, just as man can be prompted in a rational way by man only. As, therefore, reason is necessary for the
prompting of reason, so, in order to prompt perfect reason, there is need of perfect reason.
12. Some say that we are helped even by those who bestow on us the so-called “indifferent” benefits, such as money, influence, security, and all the other valued or essential aids to living. If we
argue in this way, the veriest fool will be said to help a wise man. Helping, however, really means prompting the soul in accordance with Nature, both by the prompter’s excellence and by the
excellence of him who is thus prompted. And this cannot take place without advantage to the helper also. For in training the excellence of another, a man must necessarily train his own.
13. But, to omit from discussion supreme goods or the things which produce them, wise men can none the less be mutually helpful. For the mere discovery of a sage by a sage is in itself a desirable
event; since everything good is naturally dear to the good man, and for this reason one feels congenial with a good man as one feels congenial with oneself.
14. It is necessary for me to pass from this topic to another, in order to prove my point. For the question is asked, whether the wise man will weigh his opinions, or whether he will apply to others
for advice. Now he is compelled to do this when he approaches state and home duties – everything, so to speak, that is mortal. He needs outside advice on such matters, as does the physician, the
pilot, the attorney, or the pleader of cases. Hence, the wise will sometimes help the wise; for they will persuade each other. But in these matters of great import also, – aye, of divine import, as I
have termed them, – the wise man can also be useful by discussing honourable things in common, and by contributing his thoughts and ideas.
15. Moreover, it is in accordance with Nature to show affection for our friends, and to rejoice in their advancement as if it were absolutely our own. For if we have not done this, even virtue, which
grows strong only through exercising our perceptions, will not abide with us. Now virtue advises us to arrange the present well, to take thought regarding the future, to deliberate and apply our
minds; and one who takes a friend into council with him, can more easily apply his mind and think out his problem. Therefore he will seek either the perfect wise man or one who has progressed to a
point bordering on perfection. The perfect wise man, moreover, will help us if he aids our counsels with ordinary good sense.
16. They say that men see farther in the affairs of others than in their own. A defect of character causes this in those who are blinded by self-love, and whose fear in the hour of peril takes away
their clear view of that which is useful; it is when a man is more at ease and freed from fear that he will begin to be wise. Nevertheless, there are certain matters where even wise men see the facts
more clearly in the case of others than in their own. Moreover, the wise man will, in company with his fellow sage, confirm the truth of that most sweet and honourable proverb – “always desiring and
always refusing the same things": it will be a noble result when they draw the load “with equal yoke.”
17. I have thus answered your demand, although it came under the head of subjects which I include in my volumes On Moral Philosophy. Reflect, as I am often wont to tell you, that there is nothing in
such topics for us except mental gymnastics. For I return again and again to the thought: “What good does this do me? Make me more brave now, more just, more restrained! I have not yet the
opportunity to make use of my training; for I still need the physician.
18. Why do you ask of me a useless knowledge? You have promised great things; test me, watch me! You assured me that I should be unterrified though swords were flashing round me, though the point of
the blade were grazing my throat; you assured me that I should be at ease though fires were blazing round me, or though a sudden whirlwind should snatch up my ship and carry it over all the sea. Now
make good for me such a course of treatment that I may despise pleasure and glory. Thereafter you shall teach me to work out complicated problems, to settle doubtful points, to see through that which
is not clear; teach me now what it is necessary for me to know!" | {"url":"https://philosophy.redzambala.com/seneca/109-moral-letters-to-lucilius-seneca.html","timestamp":"2024-11-05T10:03:57Z","content_type":"application/xhtml+xml","content_length":"36132","record_id":"<urn:uuid:1a7c9c59-182f-46b7-9f35-5047485f5ee1>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00418.warc.gz"} |
Robustness and Worst-Case Analysis
In robust control design, performance is expressed and measured in terms of the peak gain (the H[∞] norm or peak singular value) of a system. The smaller this gain is, the better the system
performance. The performance of a nominally stable uncertain system generally degrades as the amount of uncertainty increases. Use robustness analysis and worst-case analysis to examine how the
amount of uncertainty in your system affects the stability and peak gain of the system.
Robustness Analysis
Robustness analysis is about finding the maximum amount of uncertainty compatible with stability or with a given performance level. The following illustration shows a typical tradeoff curve between
performance and robustness. Here, the peak gain (peak magnitude on a Bode plot or singular-value plot) characterizes the system performance.
The x-axis quantifies the normalized amount of uncertainty. The value x = 1 corresponds to the uncertainty ranges specified in the model. x = 2 represents the system with twice as much uncertainty. x
= 0 corresponds to the nominal system. (See actual2normalized for more details about normalized uncertainty ranges.) The y-axis is performance, measured as the peak gain of some closed-loop transfer
function. For instance, if the closed-loop transfer function measures the sensitivity of an error signal to some disturbance, then higher peak gain corresponds to poorer disturbance rejection.
When all uncertain elements are set to their nominal values (x = 0), the gain of the system is its nominal value. In the figure, the nominal system gain is about 1. As the range of values that the
uncertain elements can take increases, the peak gain over the uncertainty range increases. The heavy blue line represents the peak gain, and is called the system performance degradation curve. It
increases monotonically as a function of the uncertainty amount.
Robust Stability Margin
The system performance degradation curve typically has a vertical asymptote corresponding to the robust stability margin. This margin is the maximum amount of uncertainty that the system can tolerate
while remaining stable. For the system of the previous illustration, the peak gain becomes infinite at around x = 2.3. In other words, the system becomes unstable when the uncertainty range is 2.3
times that specified in the model (in normalized units). Therefore, the robust stability margin is 2.3. To compute the robust stability margin for an uncertain system model, use the robstab function.
Robust Performance Margin
The robust performance margin for a given gain, γ, is the maximum amount of uncertainty the system can tolerate while having a peak gain less than γ. For example, in the following illustration,
suppose that you want to keep the peak closed-loop gain below 1.8. For that peak gain, the robust performance margin is about 1.7. This value means that the peak gain of the system remains below 1.8
as long as the uncertainty remains within 1.7 times the specified uncertainty (in normalized units).
To compute the robust performance margin for an uncertain system model, use the robgain function.
Worst-Case Gain Measure
The worst-case gain is the largest value that the peak gain can take over a specific uncertainty range. This value is the counterpart of the robust performance margin. While the robust performance
margin measures the maximum amount of uncertainty compatible with a particular peak gain level, the worst-case gain measures the maximum gain associated with a particular uncertainty amount. For
instance, in the following illustration, the worst-case gain for the uncertainty amount specified in the model is about 1.20. If that uncertainty amount is doubled, the worst-case gain increases to
To compute the worst-case gain for an uncertain system model, use the wcgain function. The ULevel option of the wcOptions command lets you compute the worst-case gain for different amounts of
See Also
robstab | robgain | wcgain
Related Topics | {"url":"https://es.mathworks.com/help/robust/ug/robustness-and-worst-case-analysis.html","timestamp":"2024-11-03T17:21:19Z","content_type":"text/html","content_length":"74087","record_id":"<urn:uuid:53a61572-6566-43b6-b09e-5c624352c862>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00778.warc.gz"} |
Continuum does not exist
I was speaking of ordinal numbers beyond the naturals. Our definitions of "actual" infinities differ. No big deal. — jgill
Natural number arithmetic does not involve infinities, yet natural numbers are inseparably tied to
. In a similar vein, I argue that real calculus is inseparably tied to
. My interpretation of the orthodox philosophy is that both
represent "actual" infinities because they are used to describe
objects, such as sets. It is in this sense that I refer to orthodox calculus as being tied to "actual" infinities.
It might appear that you are moving in the direction of Discrete calculus. — jgill
The ideas I'm proposing are fundamentally centered on continuous calculus. Concepts like continuity, real numbers, and limits are crucial to my perspective—I simply interpret them through a different
But go ahead. I am curious. — jgill
Great! I'll continue in my next post, though it might not be today as I'm starting to feel tired.
I agree that calculus can work quite well with the concepts of unboundedness and potential infinity, but 'actual' infinities are implicitly assumed throughout the standard treatment. — keystone
I was speaking of ordinal numbers beyond the naturals. Our definitions of "actual" infinities differ. No big deal.
As I have said before, I have written many papers and notes without ever becoming transfinite. — jgill
Have you written calculus papers/notes that are not (implicitly or explicitly) built upon infinite sets like R? — keystone
Of course I have used R, but not a transfinite number. Unless I occasionally use the "point at infinity" in complex analysis. Which I rarely do since it is a projection upon the Riemann sphere. It
might appear that you are moving in the direction of
Discrete calculus
. But go ahead. I am curious.
Elementary calculus does not require "actual" infinities. It gets along quite well with unboundedness, or what you might call potential infinity. — jgill
I agree that calculus can work quite well with the concepts of unboundedness and potential infinity, but 'actual' infinities are implicitly assumed throughout the standard treatment. The standard
treatment is built on
, the
set of real numbers—implying an actual infinite amount of numbers and points. As a result, when interpreting the notion of a tangent, one is inevitably led to paradoxical ideas like
rate of
. When interpreting the notion of area, one is inevitably led to the paradoxical idea that events with zero probability can still occur (dartboard paradox).
Ultimately, calculus is currently treated as the study of objects
the limit, rather than the unbounded process of
the limit (I use approaching in quotes because that word suggests that there's a destination which I do not believe in). I aim to establish a foundation focused on the journey rather than the
destination (i.e. the algorithms themselves rather than their output).
As I have said before, I have written many papers and notes without ever becoming transfinite. — jgill
Have you written calculus papers/notes that are not (implicitly or explicitly) built upon infinite sets like
But that program (even in an infinite world*) cannot actually output a set with a cardinality of ℵ0. Potential is important and I feel like it's been forgotten in our Platonist world. — keystone
Elementary calculus does not require "actual" infinities. It gets along quite well with unboundedness, or what you might call potential infinity. As I have said before, I have written many papers and
notes without ever becoming transfinite.
By "actual infinity" I suppose you mean a kind of number that can be manipulated by arithmetic processes. — jgill
I view transfinite cardinal and ordinal numbers as crucial for understanding the nature of infinity, and, as you know, they can be manipulated through (special) arithmetic processes. However, I take
issue with using transfinite numbers to describe
actual abstract objects
rather than
potential abstract objects
. For instance, (assume we live in an infinite world*) and consider a computer program designed to input any natural number, n, and output the set of the first n natural numbers. (In an infinite
world*) the program has the
to output a set larger than any natural number so the
output has a cardinality of
. But that program (even in an infinite world*) cannot
output a set with a cardinality of
. Potential is important and I feel like it's been forgotten in our Platonist world.
*I don’t actually believe in an infinite world, but I’m suggesting that mathematics allows us to speak in general terms without assuming any specific limits.
This is either very deep or shallow gobblygook. — jgill
Deep gobblygook is not an option?
If you're following what I'm saying, a discussion on calculus is not that far off. I just need one more post to provide a formal definition of a real number and then we can advance to 2D. I’d be very
interested to hear your thoughts on whether my view contains an implicit actual infinity or if it might be insufficient as a foundation for basic calculus. I'm certainly benefiting from this
discussion but I understand that one should not entertain gobblygook for too long.
By moving the focus from the destination to the journey the need for actual infinity vanishes. — keystone
By "actual infinity" I suppose you mean a kind of number that can be manipulated by arithmetic processes.
In my recent posts, I have been establishing that real numbers instead describe potential k-curves, which can be thought of as yet to be constructed k-curves which when constructed have the
potential to be arbitrarily small (but always retain a non-zero length). — keystone
This is either very deep - or shallow gobblygook.
So far I'm not seeing anything beyond a line segment between two points that converge to one. From a continuum to a point. Why should one care about this? — jgill
For the moment, please treat 1/1 and 1.0repeating as distinct objects. Without bringing in the SB tree, let me just say that the former is a
represented by a string with
finitely many
(three) characters, while the latter is a
real number
represented by a string with
infinitely many
implied characters.
In my view, when numbered, k-points must have
values without exception. It is meaningless to speak of a k-point with a real number value because such a k-point cannot be defined within this framework. Therefore,
it is incorrect to claim that a sequence of k-curves converges to a real-numbered k-point.
In my recent posts, I have been establishing that real numbers instead describe
potential k-curves
. These can be thought of as k-curves yet to be constructed, which, when constructed, will have the potential to become arbitrarily small (but not zero length).
This shift in perspective moves the focus away from a philosophy centered on the destination—limit objects like irrational points—and instead emphasizes the process itself, described by algorithms.
By shifting from the destination to the journey, the need for actual infinity disappears. Our discussion sets the necessary groundwork for establishing a calculus that operates without invoking
actual infinities.
So far I'm not seeing anything beyond a line segment between two points that converge to one. From a continuum to a point. Why should one care about this?
Cauchy sequences themselves are infinite sets. — TonesInDeepFreeze
I agree. However, the main point of my post was to clarify that I'm not working with Cauchy sequences themselves, but with the algorithm used to construct any arbitrary term. In my figure, I
highlighted the Cauchy sequence and noted, '0.9 repeating is not this.' In the subsequent figure, I highlighted what I believe 0.9 repeating actually represents and following that I expanded on this
in bold.
I've been overlooking the fact that real numbers are typically defined as equivalence classes of Cauchy sequences, not just individual Cauchy sequences. — keystone
Cauchy sequences themselves are infinite sets.
I see a mistake in your last figure, typo probably. And I assume -1/0 (meaningless) designates negative infinity, however you define that — jgill
Apologies for the typo. Also, I initially used -1/0 to represent negative infinity because that’s how it appears in the Stern-Brocot tree, but since we’ve skipped over discussing the SB tree, I’ll
switch to the more familiar notation.
I see nothing of interest so far. — jgill
I intentionally kept things uninteresting to maintain a sense of familiarity. Now, I'll begin to diverge from the familiar, which will hopefully make things more interesting. Here's part 3...
In my view, 0.9 repeating does not actually correspond to all the infinite highlighted k-curves in the image below, simply because no k-continuum beyond 3 actually exists, as none have been
constructed yet. In the spirit of constructivism, one is not justified to use ellipses to represent the completion of infinite work.
Instead, 0.9 repeating represents the following highlighted object in the generalized diagram.
I cannot call that highlighted object a k-curve because, until n is assigned a specific natural number, the object it describes is not
a k-curve. The same applies to the other objects and labels in the figure, so I will introduce some new terms (nothing fancy, just adding "potential" in front).
Essentially, I’m proposing that
0.9 repeating corresponds to a potential k-interval, which describes a potential k-curve
. What I'm leading towards is framing calculus not as the study of
objects (such as fully constructed k-continua and its constituents), but as the study of
objects (such as potential k-continua and its constituents), where some or all of the labels remain in
form for as long as possible. Of course much more is needed to be said about this.
But first, I've been overlooking the fact that real numbers are typically defined as
equivalence classes
of Cauchy sequences, not just individual Cauchy sequences. In this context, equivalence classes introduce another actual infinity which needs reinterpretation, but let's save that discussion for a
future post.
(Aside: If I had the opportunity to redo some earlier posts, instead of k-objects vs. potential k-objects, I would use actual objects vs. potential objects, getting rid of the k- prefix altogether.
But I suppose it's too late to make that change now...)
I see a mistake in your last figure, typo probably. And I assume -1/0 (meaningless) designates negative infinity, however you define that. I see nothing of interest so far.
Actually, I think you're the sinkhole. You seem to enjoy destructive conversations. — keystone
Keep digging your sinkhole deeper.
Move on to 2. — jgill
First, I'd like to point out that this part (Part 2) takes some liberties with actual infinities for explanatory purposes (and to keep my individual posts sufficiently small), but these will be
addressed and resolved in Part 3. Let's explore the meaning of the real number 0.9 repeating from my perspective. For now, let's set aside equivalence classes and represent 0.9 repeating as the
following Cauchy sequence of k-intervals:
Term n in this sequence is defined according to the following equation:
As depicted below, term 1 describes a
in k-continuum 1, term 2 describes a
in k-continuum 2, term 3 describes a
in k-continuum 3, and so on. Generally speaking, term n describes a
in k-continuum n.
A real number, such as 0.9repeated, doesn’t correspond to a single k-point (as a bottom-up view would have it) but rather
0.9repeated corresponds to an infinite sequence of k-curves
, shrinking in size as you progress deeper into the sequence in the spirit of Cauchy. [In Part 3, I’ll adjust this explanation to avoid implying the existence of actually infinite sequences].
Move on to 2.
You are a sinkhole. — TonesInDeepFreeze
Actually, I think you're the sinkhole. You seem to enjoy destructive conversations.
I didn't ask for a definition of 'planar graph'. You didn't read what I said about this a few posts ago. You are a sinkhole.
You need to define "1D analogue of the established term "planar diagram"" in terms that don't presuppose any mathematics that you have not already defined and dervied finitistically and such that
it justifies such verbiage as about "embedding in a circle". — TonesInDeepFreeze
I'm working with standard finite graphs, nothing unorthodox about my use of them. As such, I don't need to produce an original definition of them. If you don't like how the informal definition of
'planar graph' uses the word plane then you can instead use
Kuratowski's theorem
. Admittedly, I haven't studied Kuratowski's theorem...
It's ironic that you became distant right after I went back, carefully studied, and addressed your comments on topology. — keystone
Apparently, you don't recall the post in which I said that I'm willing to indulge you only up to the point that you go past the process of definitions.
You don't need to concern yourself with my decisions about how I spend my time and energy. Instead, you need to start by at least getting a grasp of the basic ideas of primitive, definition, axiom
and proof.
I don't need to waste my time and energy on you. — TonesInDeepFreeze
It's ironic that you got cold right after I went back, carefully studied, and addressed your comments on topology. That feels harsh, but I suppose I shouldn’t be surprised. In any case, I appreciate
the times when you were helpful. We all have limited time, and it’s important not to spend it on things we don't want to do. Wishing you all the best.
1D analogue of the established term "planar diagram" — keystone
You need to define "1D analogue of the established term "planar diagram"" in terms that don't presuppose any mathematics that you have not already defined and derived finitistically and such that it
justifies such verbiage as about "embedding in a circle".
But don't bother if it is to re-enlist me. I was willing to take it step by careful step with you. But you can't discipline yourself to do that, as instead you just jump to whole swaths of
handwaving. I said that at the very first point you invoked anything not previously justified by you then I'm out. I don't need to waste my time and energy on you. You are BS.
You haven't identified any falsity in my current position.[/quote]
You haven't even defined enough to get the stage of consideration of truth or falsity.
Please, give me a chance. — keystone
I have! Many times! And previously too. But you abuse my time and effort. I'm done.
If your offer to help was sincere — keystone
you question my sincerity that has been demonstrated over and over in careful attention to details, in my labor to explain things for you, in this thread and in one several months ago? Get a load of
your narcissistic self. You are full of yourself and full of BS ... though that is redundant.
Indeed, with the very first predicate 'is a continua' still not fully defined, you've piled on a big mess of more of undefined terminology and borrowing of infinitistic objects while you claim to
eschew infinitistic mathematics. — TonesInDeepFreeze
You raised a single issue with my response, which I immediately clarified-specifically, that by "1D drawable," I simply meant a 1D analogue of the established term "planar diagram". You haven't given
me a good reason for you to drop out. If your offer to help was sincere, you wouldn't back out the moment I sneezed.
Since you've been gone, the discussion with jgill has allowed me to clarify my position to the point where (I think) he understands what I mean by k-continua. I am not spouting nonsense or
doubletalk. You haven't identified any falsity in my current position. Please, give me a chance.
There's an important distinction between handwaving and BS. Handwaving involves vagueness or imprecision, where the core idea might be sound but lacks detail or rigor in its current form. BS, on
the other hand, is fundamentally incorrect—an argument that doesn't hold up under scrutiny and lacks substance from the start. — keystone
That's BS. BS includes nonsense, doubletalk and falsity. And handwaving is not necessarily just lack of rigor to be supplied later. And you presume that your "core ideas" are "sound".
I said I'd be willing to check you out to the extent that we could turn your ruminations into primitives, definitions and axioms. I predicted that right after the first round you would resort to yet
more undefined handwaving and I said that I would drop out when that happened. Indeed, with the very first predicate 'is a k-continua' still not fully defined, you've piled on a big mess of more of
undefined terminology and borrowing of infinitistic objects while you claim to eschew infinitistic mathematics. You disrespect my intellectual interest that way, just as occurred several months ago
with a different half-baked and self-contradictory proposal of yours. You are a sinkhole of a poster. You need to obtain an understanding of the basic concepts of primitive, definition, axiom, and
proof. I'm done with providing you assistance of this kind.
1. should be interesting. — jgill
Perhaps I'll head in this direction and see what you think...
Intuitionism math perhaps. — jgill
I don’t have much experience with logic yet, but from what I know, my perspective seems to align well with intuitionism. My plan is to begin by learning classical logic as a foundation and eventually
explore intuitionism.
You have density, but then continuity is next...I thought you were defining these lines as continuous. Fundamental objects. — jgill
Contrary to what my last post may have suggested, in the 1D context, there is always a k-curve between neighboring k-points (i.e. k-points are not densely packed) and k-curves are indeed continuous.
Please allow me to clarify:
• Each k-point is assigned a rational number.
• Each k-curve is assigned a k-interval to denote endpoints to which it continuously connects (endpoints excluded). A k-curve which connects k-points a and b is describe by the k-interval <a b>.
Consider the following 3 example k-continua (please note that I'm using 1/0 to denote infinity):
Every possible 1D k-continua can be described using a combination of rational numbers and k-intervals.
ASIDE: When I label a k-continuum using rational numbers and k-intervals, I'm not merely assigning arbitrary strings of characters, but rather indicating a specific structure/ordering—please forgive
me—derived from the Stern-Brocot (SB) tree. In fact, the three examples above correspond to the top three rows of the SB tree. I understand you’d prefer not to delve into the SB tree, and as long as
you don't question the meaning behind my rational labels, I think we can steer clear of it.
1. should be interesting. You have density, but then continuity is next. Intuitionism math perhaps.
I thought you were defining these lines as continuous. Fundamental objects.
I suppose I see some sort of a way to move forward by taking a lattice graph over an area and allowing the number of vertices and edges to increase without bound leading to a countable number of
points in the area. — jgill
Instead of discussing 2D continua and area, let’s simplify by returning to 1D continua and length. Length is not a property of an infinite collection of k-points, but rather an intrinsic property of
a single k-curve. This should become clearer once we introduce rational numbers into the discussion.
But this would be inadequate regarding the reals. But you might be able to push into the irrationals some way. — jgill
Irrational numbers will require special treatment, but I believe a treatment inspired by Cauchy sequences will largely address the challenge.
So far it appears everything you have given is uninteresting from a math perspective. — jgill
By introducing the fundamental k-objects (such as k-points, k-curves, k-surfaces, and so on), I've laid out the fundamental building blocks of the top-down approach. I acknowledge that these ideas so
far may seem unremarkable, akin to someone attempting to build bottom-up mathematics by focusing solely on the successor function and not doing anything with it. However, if my latest figures made
sense, the mundane part is behind us, and we can now move on to more interesting territory.
I don't think you will get a reaction from anyone but me until you produce a plan moving forward from your images of edges, vertices and surfaces. What is your goal and how do you plan to
proceed? — jgill
My discussions here rarely go as planned, so please take this plan with a grain of salt:
1. Rational Numbers
– Describing any arbitrary 1D k-continua entirely using rational numbers.
2. Real Numbers Part 1
– Describing potentially infinite sequences of 1D k-continua using rational and irrational numbers.
3. Real Numbers Part 2
– Shifting focus to the algorithm for constructing sequences rather than the impossible task of constructing a complete sequence.
4. Real Numbers Part 3
– General definition of a real number
5. Cardinal Numbers
– Applying transfinite cardinal numbers to describe potentially infinite processes, avoiding the need for actually infinite sets.
6. 2D Part 1
– Extending the 1D concepts to their 2D analogues.
7. 2D PT 1
- Derivative and Reinterpreting Motion
8. 2D PT 2
- Integral and Reinterpreting Length
9. Ordinal Numbers
– Offering a reinterpretation of ordinal numbers in the context of potential infinity.
1. To provide a top-down foundational framework for basic calculus that avoids reliance on actual infinities.
2. To argue that the philosophical issues in quantum mechanics arise from bottom-up mathematical intuitions. Physics at a foundational level is inherently top-down, and by developing new intuitions
grounded in top-down mathematics, these philosophical issues in QM can be resolved.
I don't think you will get a reaction from anyone but me — jgill
I'm eager to move forward with this plan if you're open to it. There's no commitment to a lengthy discussion—we can take it one step at a time, and you're free to end the conversation at any point
along the way. Of course, if you'd prefer to wait for someone else to potentially lead the discussion, I fully respect that decision as well.
I don't think you will get a reaction from anyone but me until you produce a plan moving forward from your images of edges, vertices and surfaces. What is your goal and how do you plan to proceed? So
far it appears everything you have given is uninteresting from a math perspective.
Now that you've moved into graph theory I suppose I see some sort of a way to move forward by taking a lattice graph over an area and allowing the number of vertices and edges to increase without
bound leading to a countable number of points in the area. But this would be inadequate regarding the reals. But you might be able to push into the irrationals some way. Speculation. You need to
actually start moving beyond your pictures. I am not familiar with graph theory, but perhaps @fishfry and @Tones are. And some on the forum who are or were CS professionals.
You have done your imagery very well. I will wait and see what comes next. — jgill
I understand that you prefer not to lead the conversation, but I want to sincerely thank you for asking thoughtful questions that have helped me better articulate my perspective. I hope it's now in a
form that TonesInDeepFreeze will be willing to engage with.
, would you consider taking a look at my recent message to jgill? The graph I described there represents a k-continuum, partly because it is a
planar graph
. For instance, if there were an edge connecting vertex 1 to vertex 8, it would no longer be planar and, therefore, wouldn't describe a k-continuum.
: While working on my response, I realized it made the most sense to start from the beginning, using clearer and more descriptive terms and definitions. Looking back, I believe this post aligns with
the kind of response that
were looking for in this thread and in our previous thread, respectively. I hope the length is balanced by enough clarity to make for a fast read. To that end, I borrowed familiar nomenclature and
ask for some leniency in its usage. For example, when I write "
n∈ $\mathbb{N}$
", I don’t mean that
is an element of the actual infinite set of natural numbers. Rather, I mean that, it has the properties of a natural number (details omitted). I believe this sets up the foundation for a calculus
free of any connection to “actual” infinities. I propose that continuous calculus is not the study of continuous
structures but rather the study of continuous
Definition: Actual Point
In 1D, an
actual point
is a rational number.
Definition: Pseudo Point
In 1D, a
pseudo point
is -∞ or ∞, such that -∞ is less than any rational number and ∞ is greater than any rational number.
Definition: Actual Curve
In 1D, an
actual curve
is doubleton set
, where a and b are either actual or pseudo points.
Definition: Simple Functions on Actual Curves in 1D Lower bound function, L: Actual Curve {a,b}→min(a,b)
. The lower bound of actual curve
Upper bound function, U: Actual Curve {a,b}→max(a,b)
. The upper bound of actual curve
Length function, d: Actual Curve {a,b}→|b-a|
. The length of actual curve
Definition: 1D Actual Structure
1D actual structure
is a finite, undirected graph in which each vertex represents an actual point, pseudo point, or actual curve. Pseudo point ∞ and pseudo point -∞ must be included. Edges connect these vertices to
indicate adjacency between the objects.
Definition: Continuity of 1D Actual Structures
A 1D
actual structure
if it satisfies the following continuity requirements:
1. Connections Involving Actual Points:
Each vertex representing an actual point
must be linked with exactly one vertex representing an actual curve for which
is the lower bound and one vertex representing an actual curve for which
is the upper bound.
2. Connections Involving Actual Curves:
Each vertex representing an actual curve
must be linked with exactly one vertex representing actual/pseudo point
and one vertex representing actual/pseudo point
3. Connected:
There exists a path between any two vertices.
Definition: Convergence Convergence of a function:
The function
x(n): $\mathbb{N}$ →Actual Point
converges if a constructive proof demonstrates that for any
can always be found such that for any
Convergence of a function to actual point a:
The function
x(n): $\mathbb{N}$ →Actual Point
converges to actual point
if a constructive proof demonstrates that for any
can always be found such that for any
Convergence of a function to rational number a:
The function
x(n): $\mathbb{N}$ →Rational Number
converges to rational number
if a constructive proof demonstrates that for any
can always be found such that for any
Convergence of a function to another function y(n):
The function
x(n): $\mathbb{N}$ →Actual Point
converges to
y(n): $\mathbb{N}$ →Actual Point
if a constructive proof demonstrates that for any
can always be found such that for any
Definition: Potential point (reinterpretation of a real number)
In 1D, a
potential point
is a function
p(n): $\mathbb{N}$ →Actual Point
such that
Definition: Potential curve (alternate reinterpretation of a real number)
In 1D, a
potential curve
is a function
c(n):$\mathbb{N}$ → Actual Curve
such that
converge, and
converges to rational number 0.
Definition: 1D Potential Structure
1D potential structure S(n)
, where
, is a finite, undirected graph whose vertices represent:
• Pseudo Points: Pseudo point ∞ and pseudo point -∞ must be included.
Actual Objects: Actual points and actual curves.
• Potential Objects: At least one potential point or curve (all of which depend on n), such as a potential point p(n) or a potential curve c(n).
Edges connect these vertices to indicate adjacency between the objects.
Definition: Continuity of 1D Potential Continuum
A 1D
potential structure
if it satisfies the following continuity requirements:
1. Connections Involving Actual Points:
Each vertex representing an actual point
must be connected to two vertices:
One for which q is the lower bound
, either:
• A vertex representing an actual curve {a,b}, where L{a,b}=q.
• A vertex representing a potential curve c(n), where L(c(n)) converges to q.
one for which q is the upper bound
, either:
• A vertex representing an actual curve {a,b}, where U{a,b}=q.
• A vertex representing a potential curve c(n), where U(c(n)) converges to q.
2. Connections Involving Actual Curves:
Each vertex representing an actual curve
must be connected to two vertices:
One for its lower bound, either:
• A vertex representing an actual/pseudo point L{a,b}.
• A vertex representing a potential point p(n), where p(n) converges to L{a,b}.
one for its upper bound, either:
• A vertex representing an actual/pseudo point U{a,b}.
• A vertex representing a potential point p(n), where p(n) converges to U{a,b}.
3. Connections Involving Potential Points:
Each vertex representing a potential point
must be connected to two vertices:
One for which p(n) is the lower bound
, either:
• A vertex representing an actual curve {a,b}, where p(n) converges to L{a,b}.
• A vertex representing a potential curve c(n), where p(n) converges to L(c(n)).
one for which p(n) is the upper bound
, either:
• A vertex representing an actual curve {a,b}, where p(n) converges to U{a,b}.
• A vertex representing a potential curve c(n), where p(n) converges to U(c(n)).
4. Connections Involving Potential Curves:
Each vertex representing a potential curve
must be connected to two vertices:
One which bounds L(c(n)
, either:
• A vertex representing an actual point a, where L(c(n)) converges to a.
• A vertex representing a potential point p(n), where L(c(n)) converges to p(n).
one which bounds U(c(n))
, either:
• A vertex representing an actual point a, where U(c(n)) converges to a.
• A vertex representing a potential point p(n), where U(c(n)) converges to p(n) .
5. Connected:
There exists a path between any two vertices in the graph.
[After note: As I'm currently thinking about higher dimensional analogues to this I'm realizing that there's a better way to express the above. I need some time to chew on that though.] | {"url":"https://thephilosophyforum.com/discussion/15393/continuum-does-not-exist/p16","timestamp":"2024-11-01T23:11:00Z","content_type":"text/html","content_length":"107570","record_id":"<urn:uuid:f81b7d18-9ae0-4a38-80d7-fab677283e1b>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00118.warc.gz"} |
Consider The Two Triangles. How Can The Triangles Be Proven Similar By The SSS Similarity Theorem? V UV WU Show That The Ratios XYZX And W ZY Are - AnswerPrime
Consider the two triangles. How can the triangles be proven similar by the SSS similarity theorem? V UV WU Show that the ratios XYZX and W ZY are
A &C Step-by-step explanation:
A Step-by-step explanation: I just took the test
Answer 6
by the same size and same degrees. Step-by-step explanation:
In SAS similarity theorem if two sides of one triangle are proportional to two sides of another triangle and angle between them are congruent then the triangle are similar. Step-by-step explanation:
From SAS similarity theorem-two triangles are similar if all their corresponding angles are congruent and their corresponding sides are proportional. Now, we check the similarity from SAS , if two
sides of first triangle are proportional to two sides of another triangle and angle between them are congruent. From triangle UVW and triangle XYZ Use sides UV and UW in Δ UVW and XY and XZ in
ΔXYZ . and angle between them .
B. Show that the ratios UV/XY and WV/ZY are equivalent, and ∠V ≅ ∠Y. Step-by-step explanation:
Answer 7
Consider two triangles Δ U V W and Δ X Y Z If these are two triangles having vertices in the same order , Then to prove →→ Δ U V W ~ Δ X Y Z , By S A S We must show, the ratio of Corresponding sides
are equivalent and angle between these two included corresponding sides are also equal. Option 1 is correct , because ratios are equivalent, and ∠U≅∠X.As X is in the beginning of ΔX Y Z , Similarly U
is in the beginning of Δ U V W. Option 2 is correct , because ratios are equivalent, and ∠Y≅∠V. As Y is in the middle of ΔX Y Z , Similarly V is in the middle of Δ U V W. Option 3 is not true, ratios
are equivalent, but ∠W ≅ ∠X should be replaced by ∠W≅∠Z. Option 4 is not true, because ratios are equivalent, and ∠U ≅ ∠Z should be replaced by ∠U≅∠X
Show that the ratios UV/XY and WV/ZY are equivalent, and ∠V ≅ ∠Y. Step-by-step explanation: we know that SAS Similarity Theorem, States that if two sides in one triangle are proportional to two sides
in another triangle and the included angle in both are congruent, then the two triangles are similar In this problem there are 3 ways that the triangles be proven similar by the SAS similarity
theorem 1) ∠U≅∠X and UV/XY=UW/XZ 2) ∠W≅∠Z and UW/XZ=WV/ZY 3) ∠V≅∠Y and UV/XY=WV/ZY therefore Show that the ratios UV/XY and WV/ZY are equivalent, and ∠V ≅ ∠Y.
B Step-by-step explanation: I TOOK THE QUIZ ON ED AND GOT IT RIGHT
yes Step-by-step explanation: da ting go brapp brapp brapp pow pow bada bing bada bang
Answer 6
by the same size and same degrees. Step-by-step explanation:
Show that the ratios UV/XY and WV/ZY are equivalent, and ∠V ≅ ∠Y. Step-by-step explanation: just did it on ed
Show that the ratios UV/XY and WV/ZY are equivalent, and ∠V ≅ ∠Y. Step-by-step explanation: we know that SAS Similarity Theorem, States that if two sides in one triangle are proportional to two sides
in another triangle and the included angle in both are congruent, then the two triangles are similar In this problem there are 3 ways that the triangles be proven similar by the SAS similarity
theorem 1) ∠U≅∠X and UV/XY=UW/XZ 2) ∠W≅∠Z and UW/XZ=WV/ZY 3) ∠V≅∠Y and UV/XY=WV/ZY therefore Show that the ratios UV/XY and WV/ZY are equivalent, and ∠V ≅ ∠Y.
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Learn.xyz – Learn about Introduction to Trigonometry
Introduction to Trigonometry
Trigonometry is the study of triangles, focusing on the relationships between their angles and sides. Originating in ancient cultures, it's vital for navigation, astronomy, and solving complex
geometrical problems.
Triangle Basics: Sides and Angles
A triangle has three sides and three angles. The sum of angles is always 180 degrees. Trigonometry uses sine, cosine, and tangent functions to relate angles with sides.
Sine Function Explained
The sine of an angle is the ratio of the length of the opposite side to the hypotenuse. For angle θ, it's expressed as sin(θ) = opposite/hypotenuse.
Cosine and Tangent Functions
Cosine relates the adjacent side to the hypotenuse, written as cos(θ) = adjacent/hypotenuse. Tangent is the opposite over adjacent, tan(θ) = opposite/adjacent. These functions are crucial for side
Calculating Unknown Sides
To find a missing side, identify the known sides/angles. Use the appropriate trigonometric function, and solve for the unknown variable. In right triangles, Pythagoras' theorem is also a valuable
Trigonometry Beyond Triangles
Surprisingly, trigonometry extends beyond triangles. It underpins Fourier transforms - crucial in signal processing and the decomposition of functions into oscillatory components, revolutionizing
modern technology.
Advanced Applications in Real-life
Trigonometry isn't just theoretical; it's practical in various fields. For instance, architects use it for creating structures, and it's essential in developing video games to simulate realistic
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In statistics, an outlier is a point in a sample that has a substantially different value from the rest.
Outlier is a high school-level concept that would be first encountered in a probability and statistics course. It is an Advanced Placement Statistics topic and is listed in the California State
Standards for Grade 4.
Sample: In mathematics, a sample of a population is a subset that is obtained to investigate the parent population's properties.
Classroom Articles on Probability and Statistics (Up to High School Level)
Arithmetic Mean Median
Box-and-Whisker Plot Mode
Conditional Probability Problem
Histogram Scatter Diagram
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Vector relevance and ranking - Azure AI Search
Relevance in vector search
During vector query execution, the search engine looks for similar vectors to find the best candidates to return in search results. Depending on how you indexed the vector content, the search for
relevant matches is either exhaustive, or constrained to near neighbors for faster processing. Once candidates are found, similarity metrics are used to score each result based on the strength of the
This article explains the algorithms used to find relevant matches and the similarity metrics used for scoring. It also offers tips for improving relevance if search results don't meet expectations.
Algorithms used in vector search
Vector search algorithms include exhaustive k-nearest neighbors (KNN) and Hierarchical Navigable Small World (HNSW).
Only vector fields marked as searchable in the index, or as searchFields in the query, are used for searching and scoring.
When to use exhaustive KNN
Exhaustive KNN calculates the distances between all pairs of data points and finds the exact k nearest neighbors for a query point. It's intended for scenarios where high recall is of utmost
importance, and users are willing to accept the trade-offs in query latency. Because it's computationally intensive, use exhaustive KNN for small to medium datasets, or when precision requirements
outweigh query performance considerations.
A secondary use case is to build a dataset to evaluate approximate nearest neighbor algorithm recall. Exhaustive KNN can be used to build the ground truth set of nearest neighbors.
When to use HNSW
During indexing, HNSW creates extra data structures for faster search, organizing data points into a hierarchical graph structure. HNSW has several configuration parameters that can be tuned to
achieve the throughput, latency, and recall objectives for your search application. For example, at query time, you can specify options for exhaustive search, even if the vector field is indexed for
During query execution, HNSW enables fast neighbor queries by navigating through the graph. This approach strikes a balance between search accuracy and computational efficiency. HNSW is recommended
for most scenarios due to its efficiency when searching over larger data sets.
How nearest neighbor search works
Vector queries execute against an embedding space consisting of vectors generated from the same embedding model. Generally, the input value within a query request is fed into the same machine
learning model that generated embeddings in the vector index. The output is a vector in the same embedding space. Since similar vectors are clustered close together, finding matches is equivalent to
finding the vectors that are closest to the query vector, and returning the associated documents as the search result.
For example, if a query request is about hotels, the model maps the query into a vector that exists somewhere in the cluster of vectors representing documents about hotels. Identifying which vectors
are the most similar to the query, based on a similarity metric, determines which documents are the most relevant.
When vector fields are indexed for exhaustive KNN, the query executes against "all neighbors". For fields indexed for HNSW, the search engine uses an HNSW graph to search over a subset of nodes
within the vector index.
Creating the HNSW graph
During indexing, the search service constructs the HNSW graph. The goal of indexing a new vector into an HNSW graph is to add it to the graph structure in a manner that allows for efficient nearest
neighbor search. The following steps summarize the process:
1. Initialization: Start with an empty HNSW graph, or the existing HNSW graph if it's not a new index.
2. Entry point: This is the top-level of the hierarchical graph and serves as the starting point for indexing.
3. Adding to the graph: Different hierarchical levels represent different granularities of the graph, with higher levels being more global, and lower levels being more granular. Each node in the
graph represents a vector point.
□ Each node is connected to up to m neighbors that are nearby. This is the m parameter.
□ The number of data points considered as candidate connections is governed by the efConstruction parameter. This dynamic list forms the set of closest points in the existing graph for the
algorithm to consider. Higher efConstruction values result in more nodes being considered, which often leads to denser local neighborhoods for each vector.
□ These connections use the configured similarity metric to determine distance. Some connections are "long-distance" connections that connect across different hierarchical levels, creating
shortcuts in the graph that enhance search efficiency.
4. Graph pruning and optimization: This can happen after indexing all vectors, and it improves navigability and efficiency of the HNSW graph.
Navigating the HNSW graph at query time
A vector query navigates the hierarchical graph structure to scan for matches. The following summarize the steps in the process:
1. Initialization: The algorithm initiates the search at the top-level of the hierarchical graph. This entry point contains the set of vectors that serve as starting points for search.
2. Traversal: Next, it traverses the graph level by level, navigating from the top-level to lower levels, selecting candidate nodes that are closer to the query vector based on the configured
distance metric, such as cosine similarity.
3. Pruning: To improve efficiency, the algorithm prunes the search space by only considering nodes that are likely to contain nearest neighbors. This is achieved by maintaining a priority queue of
potential candidates and updating it as the search progresses. The length of this queue is configured by the parameter efSearch.
4. Refinement: As the algorithm moves to lower, more granular levels, HNSW considers more neighbors near the query, which allows the candidate set of vectors to be refined, improving accuracy.
5. Completion: The search completes when the desired number of nearest neighbors have been identified, or when other stopping criteria are met. This desired number of nearest neighbors is governed
by the query-time parameter k.
Similarity metrics used to measure nearness
The algorithm finds candidate vectors to evaluate similarity. To perform this task, a similarity metric calculation compares the candidate vector to the query vector and measures the similarity. The
algorithm keeps track of the ordered set of most similar vectors that its found, which forms the ranked result set when the algorithm has reached completion.
Metric Description
cosine This metric measures the angle between two vectors, and isn't affected by differing vector lengths. Mathematically, it calculates the angle between two vectors. Cosine is the similarity
metric used by Azure OpenAI embedding models, so if you're using Azure OpenAI, specify cosine in the vector configuration.
dotProduct This metric measures both the length of each pair of two vectors, and the angle between them. Mathematically, it calculates the products of vectors' magnitudes and the angle between them.
For normalized vectors, this is identical to cosine similarity, but slightly more performant.
euclidean (also known as l2 norm) This metric measures the length of the vector difference between two vectors. Mathematically, it calculates the Euclidean distance between two vectors, which is the
l2-norm of the difference of the two vectors.
Scores in a vector search results
Scores are calculated and assigned to each match, with the highest matches returned as k results. The @search.score property contains the score. The following table shows the range within which a
score will fall.
Search method Parameter Scoring metric Range
vector search @search.score Cosine 0.333 - 1.00
Forcosine metric, it's important to note that the calculated @search.score isn't the cosine value between the query vector and the document vectors. Instead, Azure AI Search applies transformations
such that the score function is monotonically decreasing, meaning score values will always decrease in value as the similarity becomes worse. This transformation ensures that search scores are usable
for ranking purposes.
There are some nuances with similarity scores:
• Cosine similarity is defined as the cosine of the angle between two vectors.
• Cosine distance is defined as 1 - cosine_similarity.
To create a monotonically decreasing function, the @search.score is defined as 1 / (1 + cosine_distance).
Developers who need a cosine value instead of the synthetic value can use a formula to convert the search score back to cosine distance:
double ScoreToSimilarity(double score)
double cosineDistance = (1 - score) / score;
return -cosineDistance + 1;
Having the original cosine value can be useful in custom solutions that set up thresholds to trim results of low quality results.
Tips for relevance tuning
If you aren't getting relevant results, experiment with changes to query configuration. There are no specific tuning features, such as a scoring profile or field or term boosting, for vector queries:
• Experiment with chunk size and overlap. Try increasing the chunk size and ensuring there's sufficient overlap to preserve context or continuity between chunks.
• For HNSW, try different levels of efConstruction to change the internal composition of the proximity graph. The default is 400. The range is 100 to 1,000.
• Increase k results to feed more search results into a chat model, if you're using one.
• Try hybrid queries with semantic ranking. In benchmark testing, this combination consistently produced the most relevant results.
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Confidence Interval | Digital E-Learning Statistics, Six Sigma (6σ), Lean Manufacturing, 7QC tools, Project ManagementConfidence Interval - Digital E-Learning
Confidence Interval
Confidence Interval in Statistics | How to find a 95% Confidence Interval ?
Before we understand the confidence Interval, let us first understand what do we mean by “Estimate“. Everyone makes some kinds of estimates at some point of time in their life. Imagine that you are
crossing the road, and you see the car that is approaching you fast and now you can estimate the speed of that same car that is approaching you. Based on this estimate you can make your decision if
you to wait, walk or run. This is called Estimate.
Now let’s define the Estimate in statistical terms. Estimate is a specific observed numerical value used to estimate an unknown population parameter or gives you some facts about how the population
could be. We can have two types of estimates about the population.
Point Estimate and Interval Estimate :
Point Estimate
Suppose we have some population and we take a sample from this population and measure their weights estimate mean.
Average weight is 60 Kg. Then 60 kgs of sample mean is a Point estimate of a population mean.
Point estimate is often insufficient because its is either wrong or right. Assuming that we know its is wrong, but you don’t know how wrong it it. And you cannot be certain of estimate reliability.
Assuming we came of know that you are off target by 5 kg, then you would expect 60k as good estimate but if the estimate is off by 30kg then it is not the right estimate of population.
Also the point estimate is going to be different from the population parameter because due to the sampling error, and there is no way to know who close it is to the actual parameter. For this reason,
statisticians like to give an interval estimate which is a range of values used to estimate the parameter.
Interval Estimate
Because a point estimates is a single value, we can’t really tell how good it represents the population. So in inferential statistics, we prefer to use an interval or a range of values to estimate
the population parameter. It gives you the range of values by calculating two different numbers between which we can expect the parameter to lie. So we construct Interval estimate with degree of
certain degree of confidence say for e.g. 95% confidence.
Confidence Interval
Confidence interval is the range of estimate we are making.
Don’t confuse confidence interval with confidence levels. Confidence levels are normally expressed as % ( For e.g. 95 % confidence level). A confidence interval communicates how accurate our estimate
is likely to be. We use a confidence interval to express the range in which we are pretty sure the population parameter lies. These confidence Interval give us an idea about the size and thus the
power of a study.
The size of the confidence interval is then directly related to the size of the study. The more participants the smaller the confidence interval and the more precise the estimated effect. In essence,
the confidence interval estimate for a population mean lies between the sample mean – the margin of Error and the sample mean + the margin of Error.
The margin of error is calculated based on a confidence level. The confidence level basically refers to the percent of confidence intervals (from many samples) that we expect to contain the true
population parameter.
Confidence levels of 90%, 95%, and 99% are often used, but 95% is the most commonly used confidence level. Most of times confidence intervals can be found using the t-distribution (When you are
working with smaller number of samples.
Say for e.g we are 95 % confident that True mean will lie between 40 and 60. It indicates the error in two ways one by the extent of its range and other by probability of true population parameter
lying within the range. Interval Estimates of population parameters are often called Confidence Intervals.
So 95% confidence interval is a range of values that you can be 95% certain contains the true mean of the population. The correct interpretation of a 95% confidence interval is that “we are 95%
confident that the population parameter is between 50 and 70.”
Inference is when we draw conclusions about the population from the sample. Because the sample was only a selection of objects from the population, it will never be a perfect representation of the
Confidence Interval is the most common type of interval estimate is made of two elements :
Confidence Interval = “Point estimate” +/- “Margin of Error”
Where Margin of errors are the upper and Lower limits of confidence Interval.
Now we can think let’s choose high confidence level say for e.g. 99% in all your estimation. But in practice, high confidence levels will produce large confidence intervals and they will end up
giving very different estimates. Statisticians prefer interval estimate over point estimate because interval estimate are often accompanied with degree of confidence.
Let’s understand this with some Scenarios:
Scenario #1
John : Will I get my Television within 1 year ?
Manager : I am absolutely certain that you will get in 1 year
Confidence level : Better than 99%
Confidence Interval : Would be 1 year
Scenario #2
John : Will I get my Television within 1 month ?
Manager : I am absolutely positive that you will get in 1 month
Confidence level : Better than 95%
Confidence Interval : Would be 1 month
Scenario #3
John : Will I get my Television within 1 week ?
Manager : I am positive that you will get in 1 week
Confidence level : Better than 80%
Confidence Interval : Would be 1 week
Scenario #4
John : Will I get my Television within 1 day ?
Manager : I can try that you will get in 1 day
Confidence level : Better than 30%
Confidence Interval : Would be 1 day
Scenario #5
John : Will I get my Television within 1 Hour ?
Manager : I can try but highly unlikely that you will get in 1 hour
Confidence level : Better than 5 %
Confidence Interval : Would be 1 hour
Let’s understand this with some examples:
Example 1 : CI for Single population Mean
A random sample of n = 50 males showed a mean average daily intake of dairy products equal to 756 grams with a standard deviation of 35 grams. Find a 95% and 99% confidence interval for the
population average U ?
Ans: x ̄= 756 grams ; n= 50 , σ = 35 ,
The Z-value can be derived from the table and its shows us which area is contained in the confidence interval of our result so in case of the 95% confidence interval we take although use between
minus 1.96 times and plus 1.96 times the standard deviation. A lower confidence interval will lead to a lower Z-value and a smaller interval and vice versa
From Z table for 95 % confidence interval we get value as 1.96.
=> Interval estimate is = 756±1.96×35/√50
=> 746.30 ≤ m ≤ 765.70 grams for 95 % confidence interval
Now for 99 % confidence interval and From Z table for 99 % CI we get value as 2.576
=> 756±2.576×35/√50
=> 743.23 ≤m≤ 768.77 grams for 99 % confidence interval.
Example 2 : CI for Single population Proportion
Billing statement for 1000 patients discharged from a particular hospital were randomly selected for error. Out of 1000 billing statements, 102 were found to contain errors. Using this formation lets
construct 99% confidence interval ?
Ans: p ̂= 102 / 1000 = 0.102
From Z table for 99 % confidence interval we get value as 2.576
p ̂= is random and varies from sample to sample
0.077 ≤p ̂≤0.127for 99 % confidence interval
Watch this YouTube video for detailed explanation :
For questions please leave them in the comment box below and I’ll do my best to get back to those in a timely fashion. And remember to subscribe to Digital eLearning YouTube channel to have our
latest videos sent to you while you sleep.
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Calorimetry Summary & Practice - UCalgary Chemistry Textbook
Key Concepts and Summary
Calorimetry is used to measure the amount of thermal energy transferred in a chemical or physical process. This requires careful measurement of the temperature change that occurs during the process
and the masses of the system and surroundings. These measured quantities are then used to compute the amount of heat produced or consumed in the process using known mathematical relations.
Calorimeters are designed to minimize energy exchange between their contents and the external environment. They range from simple coffee cup calorimeters used by introductory chemistry students to
sophisticated bomb calorimeters used to determine the energy content of food.
Practice Questions
A 500-mL bottle of water at room temperature and a 2-L bottle of water at the same temperature were placed in a refrigerator. After 30 minutes, the 500-mL bottle of water had cooled to the
temperature of the refrigerator. An hour later, the 2-L of water had cooled to the same temperature. When asked which sample of water lost the most heat, one student replied that both bottles lost
the same amount of heat because they started at the same temperature and finished at the same temperature. A second student thought that the 2-L bottle of water lost more heat because there was more
water. A third student believed that the 500-mL bottle of water lost more heat because it cooled more quickly. A fourth student thought that it was not possible to tell because we do not know the
initial temperature and the final temperature of the water. Indicate which of these answers is correct and describe the error in each of the other answers.
Would the amount of heat measured for the reaction in a coffee cup calorimeter be greater, lesser, or remain the same if we used a calorimeter that was a poorer insulator than a coffee cup
calorimeter? Explain your answer.
lesser; more heat would be lost to the coffee cup and the environment and so ΔT for the water would be lesser and the calculated q would be lesser
Would the amount of heat absorbed by the dissolution of a salt in a coffee cup calorimeter appear greater, lesser, or remain the same if the experimenter used a calorimeter that was a poorer
insulator than a coffee cup calorimeter? Explain your answer.
Would the amount of heat absorbed by the dissolution of a salt in a coffee cup calorimeter appear greater, lesser, or remain the same if the heat capacity of the calorimeter were taken into account?
Explain your answer.
greater, since taking the calorimeter’s heat capacity into account will compensate for the thermal energy transferred to the solution from the calorimeter; this approach includes the calorimeter
itself, along with the solution, as “surroundings”: q[rxn] = −(q[solution] + q[calorimeter]); since both q[solution] and q[calorimeter] are negative, including the latter term (q[rxn]) will yield a
greater value for the heat of the dissolution.
How many milliliters of water at 23 °C with a density of 1.00 g/mL must be mixed with 180 mL (about 6 oz) of coffee at 95 °C so that the resulting combination will have a temperature of 60 °C? Assume
that coffee and water have the same density and the same specific heat.
How much will the temperature of a cup (180 g) of coffee at 95 °C be reduced when a 45 g silver spoon (specific heat 0.24 J/g °C) at 25 °C is placed in the coffee and the two are allowed to reach the
same temperature? Assume that the coffee has the same density and specific heat as water.
The temperature of the coffee will drop 1 degree.
A 45-g aluminum spoon (specific heat 0.88 J/g °C) at 24 °C is placed in 180 mL (180 g) of coffee at 85 °C and the temperature of the two become equal.
(a) What is the final temperature when the two become equal? Assume that coffee has the same specific heat as water?
(b) The first time a student solved this problem she got an answer of 88 °C. Explain why this is clearly an incorrect answer.
The temperature of the cooling water as it leaves the hot engine of an automobile is 240 °F. After it passes through the radiator it has a temperature of 175 °F. Calculate the amount of heat
transferred from the engine to the surroundings by one gallon of water with a specific heat of 4.184 J/g °C.
5.7 ×10^2 kJ
A 70.0-g piece of metal at 80.0 °C is placed in 100 g of water at 22.0 °C contained in a calorimeter. The metal and water come to the same temperature at 24.6 °C. How much heat did the metal give up
to the water? What is the specific heat of the metal?
If a reaction produces 1.506 kJ of heat, which is trapped in 30.0 g of water initially at 26.5 °C in a calorimeter, what is the resulting temperature of the water?
38.5 °C
A 0.500-g sample of KCl is added to 50.0 g of water in a calorimeter. If the temperature decreases by 1.05 °C, what is the approximate amount of heat involved in the dissolution of the KCl, assuming
the specific heat of the resulting solution is 4.18 J/g °C? Is the reaction exothermic or endothermic?
Dissolving 3.0 g of CaCl[2](s) in 150.0 g of water in a calorimeter at 22.4 °C causes the temperature to rise to 25.8 °C. What is the approximate amount of heat involved in the dissolution, assuming
the specific heat of the resulting solution is 4.18 J/g °C? Is the reaction exothermic or endothermic?
−2.2 kJ; The heat produced shows that the reaction is exothermic.
When 50.0 g of 0.200 M NaCl(aq) at 24.1 °C is added to 100.0 g of 0.100 M AgNO[3](aq) at 24.1 °C in a calorimeter, the temperature increases to 25.2 °C as AgCl(s) forms. Assuming the specific heat of
the solution and products is 4.20 J/g °C, calculate the approximate amount of heat in joules produced.
The addition of 3.15 g of Ba(OH)[2]·8H[2]O to a solution of 1.52 g of NH[4]SCN in 100 g of water in a calorimeter caused the temperature to fall by 3.1 °C. Assuming the specific heat of the solution
and products is 4.20 J/g °C, calculate the approximate amount of heat absorbed by the reaction, which can be represented by the following equation:
Ba(OH)[2]·8H[2]O(s) + 2NH[4]SCN(aq) ⟶ Ba(SCN)[2](aq) + 2NH[3](aq) + 10H[2]O(l)
1.4 kJ
The reaction of 50 mL of acid and 50 mL of base increased the temperature of the solution by 6.9 ºC. How much would the temperature have increased if 100 mL of acid and 100 mL of base had been used
in the same calorimeter starting at the same temperature of 22.0 ºC? Explain your answer.
When 1.0 g of fructose, C[6]H[12]O[6](s), a sugar commonly found in fruits, is burned in oxygen in a bomb calorimeter, the temperature of the calorimeter increases by 1.58 °C. If the heat capacity of
the calorimeter and its contents is 9.90 kJ/°C, what is q for this combustion?
When a 0.740-g sample of trinitrotoluene (TNT), C[7]H[5]N[2]O[6], is burned in a bomb calorimeter, the temperature increases from 23.4 °C to 26.9 °C. The heat capacity of the calorimeter is 534 J/°C,
and it contains 675 mL of water. How much heat was produced by the combustion of the TNT sample?
11.7 kJ
One method of generating electricity is by burning coal to heat water, which produces steam that drives an electric generator. To determine the rate at which coal is to be fed into the burner in this
type of plant, the heat of combustion per ton of coal must be determined using a bomb calorimeter. When 1.00 g of coal is burned in a bomb calorimeter, the temperature increases by 1.48 °C. If the
heat capacity of the calorimeter is 21.6 kJ/°C, determine the heat produced by combustion of a ton of coal (2.000 ×10^3 pounds).
The amount of fat recommended for someone with a daily diet of 2000 Calories is 65 g. What percent of the calories in this diet would be supplied by this amount of fat if the average number of
Calories for fat is
9.1 Calories/g?
A teaspoon of the carbohydrate sucrose (common sugar) contains 16 Calories (16 kcal). What is the mass of one teaspoon of sucrose if the average number of Calories for carbohydrates is 4.1 Calories/
What is the maximum mass of carbohydrate in a 6-oz serving of diet soda that contains less than 1 Calorie per can if the average number of Calories for carbohydrates is 4.1 Calories/g?
0.24 g
Which is the least expensive source of energy in kilojoules per dollar: a box of breakfast cereal that weighs 32 ounces and costs $4.23, or a liter of isooctane (density, 0.6919 g/mL) that costs $
0.45? Compare the nutritional value of the cereal with the heat produced by combustion of the isooctane under standard conditions. A 1.0-ounce serving of the cereal provides 130 Calories. | {"url":"https://chem-textbook.ucalgary.ca/version2/chapter-5-introduction/calorimetry/calorimetry-practice-question/","timestamp":"2024-11-02T19:06:18Z","content_type":"text/html","content_length":"76256","record_id":"<urn:uuid:6fae016a-e09c-44aa-b524-c1e62c47c3d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00582.warc.gz"} |
Dick Smith Hallett Cove - closing down
There are a several dozen across Australia that are closing - HC isn't the only one. Staff told me a couple of weeks ago that it's a company restructure. Seems they're owned by Woolworths (who also
own Big W) so the thought is that in a lot of cases where a Big W and a DS exist in close proximity to one another the DS will close as most of their product lines can be sold via Big W.
Yes I heard the Dick Smith chain was up for sale - they interviewed 'the' Dick Smith about it on the radio apparently he is not happy if his name gets into foreign hands ... but then once you've sold
your name to someone, then surely you lose all say in how it's used or who it's sold on to?
I think under the surface many of the retailers at Hallett Cove are struggling with the high rents and the fact people are not spending. Its only got to be a matter of time before one of the coffee
shops goes under surely. There must be 4/5 plus maccas selling coffee?
Edited by Chelseadownunder
I think under the surface many of the retailers at Hallett Cove are struggling with the high rents and the fact people are not spending. Its only got to be a matter of time before one of the
coffee shops goes under surely. There must be 4/5 plus maccas selling coffee?
I agree but it is normally the larger retailers that can weather the storms.
I love all the cafes but agree we need more variety. A Cotton On Kids, maybe.
All shopping centres are built the same be it a Westfield centro other company, same boring shops same boring layout and 95% of the time the same dull atmosphere. Even the s/c near us is having a
revamp at the mo and is putting a cibo in, bores me to the core
It is annoying having shops in the shopping centres, you'd have thought someone would have broken away from the norm by now and filled one with with something other than shops. | {"url":"https://www.pomsinadelaide.com/topic/28305-dick-smith-hallett-cove-closing-down/","timestamp":"2024-11-07T06:40:23Z","content_type":"text/html","content_length":"181400","record_id":"<urn:uuid:56be1d20-5fbf-48bc-bab6-881a64a814fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00639.warc.gz"} |
David Spencer
Professor Emeritus
308 ECoRE Building
Research Areas:
Astrodynamics; Vehicle Dynamics and Controls
Interest Areas:
Spacecraft dynamics and control, theoretical and applied astrodynamics, trajectory optimization, and spacecraft systems engineering.
• BS, Mechanical Engineering, University of Kentucky, 1983
• MS, Aerospace Engineering, Purdue University, 1985
• Ph D, Aerospace Engineering Sciences, University of Colorado, 1994
• MBA, Penn State University, 2010
Journal Articles
• Davide Conte and David B Spencer, 2018, "Mission analysis for Earth to Mars-Phobos Distant Retrograde Orbits", Acta Astronautica
• Jason Reiter and David B Spencer, 2018, "Solutions to Rapid Collision-Avoidance Maneuvers Constrained by Mission Performance Requirements", Journal of Spacecraft and Rockets
• Davide Conte and David B Spencer, 2018, "Earth-Mars transfers through Moon Distant Retrograde Orbits", Acta Astronautica
• Davide Conte, Marilana DiCarlo, Koki Ho, David B Spencer and Massimiliano Vasile, 2018, "Earth-Mars transfers through Moon Distant Retrograde Orbits", Acta Astronautica
• Philip L Myers and David B Spencer, 2016, "Application of a Multi-Objective Evolutionary Algorithm to the Spacecraft Stationkeeping Problem", Acta Astronautica, 127, pp. 76-86
• Jason A Reiter and David B Spencer, 2016, "Technique to Optimize for Engine Shutoff Constraints in Electric Propulsion Trajectories", 53, (4), pp. 782-786
• David B Spencer and Brian S Shank, 2016, "Trade Space Visualization Applied to Lambert’s Problem for Elliptical Insertion Orbits", Acta Astronautica
• Andrew J Abraham, David B Spencer and Terry J Hart, 2016, "Early Mission Design of Transfers to Halo Orbits via Particle Swarm Optimization", Journal of the Astronautical Sciences, 63, (2), pp.
• M V Paul, D B Spencer, S E Lego and J P Muncks, 2014, "The Penn State Lunar Lion: A University Mission to Explore the Moon", Acta Astronautica, 96, pp. 65-77
• P. S. Williams, David B Spencer and R. S. Erwin, 2013, "Coupling of Estimation and Sensor Tasking Applied to Satellite Tracking", Journal of Guidance, Control, and Dynamics, 36, (4), pp. 993-1007
• J.-S. Kim, J. V. Urbina, T. J. Kane and David B Spencer, 2011, "Improvement of TIE-GCM thermospheric density predictions via incorporation of helium data from NRLMSISE-00"
• D. D. Jordan, D. B. Spencer, David B Spencer, M. A. Yukish and G. M. Stump, 2010, "Optimal Continuous-Thrust Trajectories via Visual Trade Space Exploration", Journal of the Astronautical
• B.-N. Kang, David B Spencer, S. Tang and D. Jordan, 2010, "Optimal Periodic Cruise Trajectories via a Two-Level Optimization Method", Journal of Spacecraft and Rockets, 47, (4), pp. 597-613
• C. Binz, David B Spencer, D. A. Levin and T. W. Simpson, 2010, "Designing for the Space Environment via Trade Space Exploration", Journal of Spacecraft and Rockets, 47, (6), pp. 1070-1073
• C. J. Scott and David B Spencer, 2010, "Transfers to Sticky Distant Retrograde Orbits", Journal of Guidance, Control and Dynamics, 33, (6), pp. 1940-1946
• C. J. Scott and David B Spencer, 2010, "Calculating Transfer Families to Periodic Distant Retrograde Orbits Using Differential Correction", Journal of Guidance, Control and Dynamics, 33, (5), pp.
• J. C. Benavides and David B Spencer, 2010, "The four-body linear equations of planar relative motion", Acta Astronautica, 66, (1-2), pp. 285-300
• J. S. Kim, David B Spencer, T. J. Kane and J. Urbina, 2009, "Thermospheric density model blending techniques: Bridging the gap between satellites and sounding rockets", Radio Sci., 44, pp. RS0A22
• P.-S. Hur, R. G. Melton and David B Spencer, 2008, "Meeting Science Requirements for Attitude Determination and Control in a Low-Power, Spinning Nanosatellite", Journal of Aerospace Engineering,
Sciences and Applications, 1, (1), pp. 25-33
• A. Mathew and David B Spencer, 2008, "Incorporating Cooperative Learning Activities into Traditional Aerospace Engineering Curricula", The Journal of Aviation/Aerospace Education and Research, 17
, (3), pp. 25-38
• C. R. Scott and David B Spencer, 2007, "Optimal Reconfiguration of Satellites in Formation", Journal of Spacecraft and Rockets, 44, (1), pp. 230-239
• C. R. Bessette and David B Spencer, 2007, "A Performance Comparison of Stochastic Search Algorithms on the Interplanetary Gravity Assist Trajectory Problem", Journal of Spacecraft and Rockets, 44
, (3), pp. 722-724
• R. L. Kobrick and David B Spencer, 2007, "Optimizing Trajectories for Suborbital Human Spaceflight", Journal of Spacecraft and Rockets, 44, (2), pp. 460-463
• M. A. Wissler, David B Spencer and R. G. Melton, 2007, "Coast-Arc Orbit Stability During Spiral-Down Trajectories about Irregularly Shaped Body", Journal of Spacecraft and Rockets, 44, (1), pp.
• M. A. Wissler, David B Spencer and R. G. Melton, 2007, "Orbit Stability for Unexpected Coast-Arcs During Spiral Down trajectories about an Irregularly Shaped Body", Journal of Spacecraft and
Rockets, 44, (1), pp. 254-263
• M. P. Ferringer and David B Spencer, 2006, "Satellite Constellation Design Trade-offs Using Multiple-Objective Evolutionary Computing", Journal of Spacecraft and Rockets, 43, (6), pp. 1404-1411
• W. J. Chadwick, David B Spencer and R. G. Melton, 2006, "Visibility of Ground SItes for Beacon/Relays on the Martian Moons", Journal of Spacecraft and Rockets, 43, (1), pp. 228-230
• David B Spencer, R. G. Melton and S. G. Chianese, 2006, "Selecting Projects for a Capstone Spacecraft Design Course from Real World Solicitations", Journal of Aviation/Aerospace Education and
Research, 16, (1), pp. 27-40
• S. G. Bilen, C. R. Philbrick, T. F. Wheeler, J. D. Mathews, R. G. Melton and David B Spencer, 2006, "An Overview of Space Science and Engineering Education at Penn State", IEEE Aerospace and
Electronic Systems Magazine, 21, (7), pp. S23-S27
• J. Igarashi and David B Spencer, 2005, "Optimal Continuous Thrust Orbit Transfer Using Evolutionary Algorithms", Journal of Guidance, Control and Dynamics, 28, (3), pp. 547-549
• H. Yamato and David B Spencer, 2004, "Orbit Transfer via Tube Jumping in Planar Restricted Problems of Four Bodies", Journal of Spacecraft and Rockets, 42, (2), pp. 321-328
• H. Yamato and David B Spencer, 2004, "Trajectory Design for Planar Circular and Elliptical Restricted Three-Body Problems with Perturbation", Journal of Guidance, Control and Dynamics, 27, (6),
pp. 1035-1045
• P.-S. Hur, R. G. Melton and David B Spencer, 2004, "Attitude Determination and Control of a Nanosatellite Using the Geomagnetic Field Data and Sun Sensors", Advances in the Astronautical Sciences
• C. J. Scott and David B Spencer, 2004, "Coupled Effects of Initial Orbit Plane on Orbit Lifetime in the Three Body Problem", Advances in the Astronautical Sciences
• W. J. Chadwick III, David B Spencer and R. G. Melton, 2004, "Geometric Analysis of Visibility of Mission Support Infrastructure for Phobos and Deimos", Advances in the Astronautical Sciences
• A. L. Faulds and David B Spencer, 2003, "Satellite Close Approach Filtering Using Genetic Algorithms", Journal of Spacecraft and Rockets, 40, (2), pp. 248-252
• Y. H. Kim and David B Spencer, 2002, "Optimal Orbital Rendezvous Using Genetic Algorithms", Journal of Spacecraft and Rockets, 39, (6), pp. 859-865
• A. E. Herman and David B Spencer, 2002, "Optimal, Low-Thrust Earth-Orbit Transfers Using Higher-Order Collocation Methods", Journal of Guidance, Control and Dynamics, 25, (1), pp. 40-47
• T. Cichan, R. G. Melton and David B Spencer, 2001, "Control Laws for Minimum Orbital Changes - The Satellite Retrieval Problem", Journal of Guidance, Control and Dynamics, 24, (5), pp. 1231-1233
• David B Spencer, K. K. Luu, W. S. Campbell, M. E. Sorge and A. B. Jenkin, 2001, "Orbital Debris Hazard Assessment Methodologies for Satellite Constellations", Journal of Spacecraft and Rockets,
38, (1), pp. 120-125
• David B Spencer, C. B. Hogge, W. S. Campbell, M. E. Sorge and S. R. McWaters, 2000, "Some Technical Issues of an Optically-Focused Small Space Debris Tracking and Cataloguing System", Space
Debris, 2, (3), pp. 137-160
• David B Spencer and R. D. Culp, 1995, "Designing Continuous-Thrust Low-Earth-Orbit to Geosynchronous-Earth-Orbit Transfers", Journal of Spacecraft and Rockets, 32, (6), pp. 1033-1038
• V. A. Chobotov and David B Spencer, 1991, "Debris Evolution and Lifetime Following an Orbital Breakup", Journal of Spacecraft and Rockets, 28, (6), pp. 670-676
• K. C. Howell and David B Spencer, 1986, "Periodic Orbits in the Restricted Four-Body Problem", Acta Astronautica, 13, (8), pp. 473-479
Conference Proceedings
• David B Spencer, 2017, "Trade Space Visualization Applied to Space Flight Engineering Design", Advances in the Astronautical Sciences
• Koundrya Kuppa and David B Spencer, 2016, "Long-Term Orbit Propagation Using Symplectic Integration Algorithms"
• Jason A Reiter and David B Spencer, 2016, "Trading Spacecraft Fuel Use and Mission Performance to Determine the Optimal Collision Probability in Emergency Collision Avoidance Scenarios"
• Mollik Nayyar and David B Spencer, 2016, "Using Particle Swarm Optimization for Earth-Return Trajectories from Libration Point Orbits", AAS 17-410, AAS/AIAA Space Flight Mechanics Meeting, San
Antonio, TX, February 5-9, 2017
• Mark Bolden, Paul Schumacher, David B Spencer, Islam Hussein, Matthew Wilkins and Chrs Roscoe, 2016, "Computer Vision and Computational Intelligence for Real-Time Multi-Modal Space Domain
Awareness", AAS 17-449, AAS/AIAA Space Flight Mechanics Meeting, San Antonio, TX, February 5-9, 2017.
• Y. Razoumny, David B Spencer, B. Agrawal, J Kreisel, S A Koupreev, V Razoumny and Y Makarov, 2016, "The Concept of On-Orbit Servicing for Next Generation Space Systems Development and Its Key
• Davide Conte and David B Spencer, 2016, "Preliminary Study on Relative Motion and Rendezvous Between Spacecraft in the Restricted Three-Body Problem", Advances in the Astronautical Sciences
• Jason A Reiter and David B Spencer, 2016, "An Analytical Solution to Quick-Response Collision Avoidance Maneuvers in Low Earth Orbit", Advances in the Astronautical Sciences
• Davide Conte and David B Spencer, 2016, "PSemi-analytical Methods for Computing Delta-V and Time Optimal Rendezvous Maneuvers in Cis-lunar Halo Orbits", Advances in the Astronautical Sciences
• Davide Conte, David B Spen ander, , 2015, "Targeting the Martian Moons via Direct Insertion into Mars’ Orbit", Advances in the Astronautical Sciences
• Davide Conte, Marilena Di Carlo, Koki Ho, David B Spencer and Massimiliano Vasile, 2015, ""Earth-Mars Transfers through Moon Distant Retrograde Orbits""
• Andrew M.S. Goodyear and David B Spencer, 2015, "Optimal Low-Thrust Geostationary Transfer Orbit Using Legendre-Gauss-Radau Collocation", Advances in the Astronautical Sciences
• Michael J Policelli and David B Spencer, 2015, "Vertical Takeoff Vertical Landing Spacecraft Trajectory Optimization Via Direct Collocation and Nonlinear Programming", Advances in the
Astronautical Sciences
• Jason A Reiter and David B Spencer, 2015, "Optimization of Many-Revolution, Electric-Propulsion Trajectories with Engine Shutoff Constraints", Advances in the Astronautical Sciences
• P L Myers and David B Spencer, 2014, "Application of a Multi-Objective Evolutionary Algorithm to the Spacecraft Stationkeeping Problem"
• L J DiGirolamo, A H Hoskins, K A Hacker and David B Spencer, 2014, "A Hybrid Motion Planning Algorithm for Safe and Efficient Close Proximity, Autonomous Spacecraft Missions"
• A J Abraham, David B Spencer and T J Hart, 2014, "Particle Swarm Optimization of 2-Manuver, Impulsive Transfers from LEO to Lagrange Point Orbits", 24th International Symposium on Space Flight
Dynamics, Laurel, MD
• A J Abraham, David B Spencer and T J Hart, 2014, "Preliminary 2-D Optimization of Low-Thrust, Geocentric-to-Halo-Orbit Transfers via Particle Swarm Optimization"
• C L Hassa, David B Spencer and S G Bilén, 2014, "Drag Coefficient Estimation Using Satellite Attitude and Orbit Data"
• P. S. Williams, David B Spencer and R. S. Erwin, 2012, "Comparison of Two Single-Step, Myopic Sensor Management Decision Processes Applied to Space Situational Awareness", AAS/AIAA Space Flight
Mechanics Meeting, AAS 12-112
• P. S. Williams, David B Spencer and R. S. Erwin, 2012, "Utilizing Stability Metrics to Aid in Sensor Network Management Solutions for Satellite Tracking Problems", AAS 12-111, AAS/AIAA Space
Flight Mechanics Meeting
• M. P. Ferringer, David B Spencer and P. S. Reed, 2009, "Many-objective reconfiguration of operational satellite constellations with the Large-Cluster Epsilon Non-dominated Sorting Genetic
Algorithm-II", IEEE Congress on Evolutionary Computation, CEC '09, pp. 340-349
• David B Spencer and F. A. Acon-Chen, 2004, "An Analytical Approach for Continuous-Thrust, LEO-Molniya Transfers", AIAA/AAS Astrodynamics Specialists Conference, AIAA 2004-5090
• J. Igarashi and David B Spencer, 2004, "Optimal Continuous Thrust Orbit Transfer Using Evolutionary Algorithms", AIAA/AAS Astrodynamics Specialists Conference, AIAA 2004-5085
• D. W. Haeberle, David B Spencer and T. A. Ely, 2004, "Interplanetary Navigation Using a Distributed Deep Space Network Architecture", AIAA/AAS Astrodynamics Specialists Conference, AIAA 2004-4744
• M. A. Wissler, David B Spencer and R. G. Melton, 2004, "Low Altitude Orbit Stability of the Dawn Spacecraft Around the Asteroid Vesta", AIAA/AAS Astrodynamics Specialists Conference, AIAA
• K. Liang, J. Yuan and David B Spencer, 2013, "The Use of Invariant Manifolds for Low-Energy Earth-Moon Transfers of Lunar Landing Missions"
• A. J. Abraham, David B Spencer and T. J. Hart, 2013, "Optimization of Preliminary Low-Thrust Trajectories from Geo-Energy Orbits to Earth-Moon, L1, Lagrange Point Orbits Using Particle Swarm
• David B Spencer and B. S. Shank, 2013, "Preliminary Development of an Optimized Lambert Problem Solver for Targets in Elliptical Orbits"
• R. E. Kelly-McKennon, P. S. Reed, David B Spencer and M. P. Ferringer, 2013, "Model Diagnostics and Dynamic Emulation: Enhancing the Understanding and Impact of Complex Models in Satellite
Constellation Design"
• M. V. Paul and David B Spencer, 2012, "The Penn State Lunar Lion: A University Mission to Explore the Moon"
• P. S. Williams, David B Spencer, R. S. Erwin and K. J. DeMars, 2012, "The Effects of Uncertainty Estimation on Dynamic Sensor Tasking"
• K. Liang, David B Spencer and J. Yuan, 2012, "Optimizing the Perilune of Lunar Landing Trajectories Using Dynamical Systems Theory"
• P. S. Williams, David B Spencer and R. S. Erwin, 2012, "Comparison of Two Single-Step, Myopic Sensor Management Decision Processes Applied to Space Situational Awareness"
• P. S. Williams, David B Spencer and R. S. Erwin, 2011, "Coupling of Nonlinear Estimation and Dynamic Sensor Tasking Applied to Space Situational Awareness", pp. AAS 11-575
• C. J. Polito and David B Spencer, 2011, "The Probability of Asteroid-Earth Collisions by way of the Positional Uncertainty Ellipsoid", pp. AAS 11-409
• J. C. Benavides and David B Spencer, 2010, "Analytic Solutions of the N-Body Problem", pp. AAS 10-186
• J. C. Benavides and David B Spencer, 2010, "Analytic Solutions of the Two-Body Problem", pp. AAS 10-182
• C. J. Scott and David B Spencer, 2009, "Transfers to Periodic Distant Retrograde Orbits"
• C. J. Scott and David B Spencer, 2009, "Transfers to Sticky Distant Retrograde Orbits"
• J. C. Benavides and David B Spencer, 2009, "The Two-Dimensional Linearized Equations of Perturbed Relative Motion"
• P. S. Williams and David B Spencer, 2009, "Applications of Non-Linear Constrained Optimization Methods and an Evolutionary Strategy on Low-Thrust LEO to Molniya and LEO to GEO Orbit Transfers"
• D. D. Jordan, David B Spencer, T. W. Simpson, M. A. Yukish and G. M. Stump, 2009, "Optimal Continuous-Thrust Orbit Transfers via Trade Space Exploration"
• Y.-T. Ahn and David B Spencer, 2009, "Preliminary Result of Attitude Control System of a Spacecraft Using Shifting Mass Distribution"
• B. Kang, David B Spencer, S. Tang and D. D. Jordan, 2009, "Study of Optimal Periodic Cruise Trajectories via Tradespace Visualization"
• T. W. Simpson, David B Spencer, M. A. Yukish and G. M. Stump, 2008, "Visual Steering Commands and Test Problems to Support Research in Trade Space Exploration"
• J. C. Benavides and David B Spencer, 2008, "Orbit Phasing Analysis for Elliptical Orbits"
• David B Spencer, D. D. Jordan, T. W. Simpson, M. A. Yukish and G. M. Stump, 2008, "Optimal Spacecraft Trajectories via Trade Space Exploration"
• J. C. Benavides and David B Spencer, 2008, "Preliminary Assessment of the Next Generation Equations of Relative Motion"
• C. J. Scott and David B Spencer, 2008, "Stability Mapping of Distant Retrograde Orbits and Transports in the Circular Restricted Three-Body Problem"
• J.-S. Kim, David B Spencer, T. J. Kane and J. V. Urbina, 2008, "A Blending Technique in Thermospheric Density Modeling"
• M. P. Ferringer, David B Spencer, P. M. Reed, R. S. Clifton and T. G. Thompson, 2008, "Pareto-Hypervolumes for the Reconfiguration of Satellite Constellations"
• P. S. Williams and David B Spencer, 2008, "Preliminary Findings Concerning Applications of Non-Linear Constrained Optimization Methods on Low-Thrust Orbit Transfers"
• T. R. Stodgell and David B Spencer, 2007, "Satellite Rendezvous Tours Using Multiobjective Evolutionary Optimization"
• J. C. Benavides and David B Spencer, 2007, "Sun-Earth Triangular Lagrange Point Orbit Insertion and Satellite Station Keeping"
• J. R. O'Malley and David B Spencer, 2006, "Formation Flight Control for Modeling Interferometric Aperture Planes"
• C. R. Bessette and David B Spencer, 2006, "Identifying Optimal Interplanetary Trajectories through a Genetic Approach"
• R. L. Kobrick and David B Spencer, 2006, "Optimizing Trajectories for Suborbital Human Spaceflight", pp. Paper AAS 06-199
• C. R. Bessette and David B Spencer, 2006, "Optimal Space Trajectory Design: A Heuristic-Based Approach", pp. Paper AAS 06-197
• M. P. Ferringer and David B Spencer, 2005, "Satellite Constellation Design Optimization via Multiple-Objective Evolutionary Computation"
• C. J. Scott and David B Spencer, 2005, "Optimal Bounded Low-Thrust Reconfiguration for Close Proximity Earth Orbiting Satellites"
• David B Spencer, R. G. Melton and S. G. Chianese, 2003, "Selecting Projects for a Capstone Spacecraft Design Course", pp. Paper AAS 03-503
• K. A. Akins, L. M. Healy and David B Spencer, 2003, "Localized Atmospheric Density Model Validation Using High Eccentricity Satellite Observations", pp. Paper AAS 03-628
• H. Yamato and David B Spencer, 2003, "Numerical Investigation of Perturbation Effects on Orbital Classifications in the Restricted Three-Body Problem", pp. Paper AAS 03-235
• Y. T. Ahn and David B Spencer, 2002, "Optimal Reconfiguration of a Formation Flying Constellation"
Research Projects
Honors and Awards
• Frank J. Malina Astronautics Medal, International Astronautical Federation, October 2018
Service to Penn State:
Service to External Organizations: | {"url":"https://www.aero.psu.edu/department/directory-detail-g.aspx?q=DBS9","timestamp":"2024-11-10T14:46:10Z","content_type":"application/xhtml+xml","content_length":"211487","record_id":"<urn:uuid:cc897206-6a3a-495c-94d4-56d5f9bfb59e>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00020.warc.gz"} |
Seminars and Colloquia by Series
Friday, January 31, 2014 - 11:00 for 1 hour (actually 50 minutes)
Skiles 005
Chi-Jen Wang – Iowa State
Spatially discrete stochastic models have been implemented to analyze cooperative behavior in a variety of biological, ecological, sociological, physical, and chemical systems. In these models,
species of different types, or individuals in different states, reside at the sites of a periodic spatial grid. These sites change or switch state according to specific rules (reflecting birth or
death, migration, infection, etc.) In this talk, we consider a spatial epidemic model where a population of sick or healthy individual resides on an infinite square lattice. Sick individuals
spontaneously recover at rate *p*, and healthy individual become infected at rate O(1) if they have two or more sick neighbors. As *p* increases, the model exhibits a discontinuous transition from an
infected to an all healthy state. Relative stability of the two states is assessed by exploring the propagation of planar interfaces separating them (i.e., planar waves of infection or recovery). We
find that the condition for equistability or coexistence of the two states (i.e., stationarity of the interface) depends on orientation of the interface. We also explore the evolution of droplet-like
configurations (e.g., an infected region embedded in an all healthy state). We analyze this stochastic model by applying truncation approximations to the exact master equations describing the
evolution of spatially non-uniform states. We thereby obtain a set of discrete (or lattice) reaction-diffusion type equations amenable to numerical analysis. | {"url":"https://math.gatech.edu/seminars-and-colloquia-by-series?series_tid=45&page=8","timestamp":"2024-11-08T18:59:36Z","content_type":"text/html","content_length":"60884","record_id":"<urn:uuid:78ad55a0-0003-4de2-9022-8fe84dac3ecc>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00333.warc.gz"} |
Next: Self-inductance Up: Magnetic induction Previous: Introduction
We have learned about e.m.f., resistance, and capacitance. Let us now investigate inductance. Electrical engineers like to reduce all pieces of electrical apparatus to an equivalent circuit
consisting only of e.m.f. sources (e.g., batteries), inductors, capacitors, and resistors. Clearly, once we understand inductors, we shall be ready to apply the laws of electromagnetism to electrical
Consider two stationary loops of wire, labeled 1 and 2. Let us run a steady current
that the magnitude of
where mutual inductance of the two loops.
Let us write the magnetic field
It follows from Stokes' theorem that
The above equation is just a special case of the more general law,
In fact, mutual inductances are rarely worked out from first principles--it is usually too difficult. However, the above formula tells us two important things. Firstly, the mutual inductance of two
loops is a purely geometric quantity, having to do with the sizes, shapes, and relative orientations of the loops. Secondly, the integral is unchanged if we switch the roles of loops 1 and 2. In
other words,
In fact, we can drop the subscripts, and just call these quantities exactly the same as the flux through loop 1 when we send the same current around loop 2.
We have seen that a current
The constant of proportionality self-inductance. Like
Inductance is measured in S.I. units called henries (H): 1 henry is 1 volt-second per ampere. The henry, like the farad, is a rather unwieldy unit, since most real-life inductors have a inductances
of order a micro-henry.
Next: Self-inductance Up: Magnetic induction Previous: Introduction Richard Fitzpatrick 2006-02-02 | {"url":"https://farside.ph.utexas.edu/teaching/em/lectures/node81.html","timestamp":"2024-11-07T17:08:11Z","content_type":"text/html","content_length":"15996","record_id":"<urn:uuid:c8fb8399-7879-40fa-b0b8-e4d887295e4a>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00826.warc.gz"} |
Electrical Engineering Calculators
Electrical engineering calculators
to calculate voltage, current resistance, power, capacitance, inductance, kVA, battery life etc to design, develop and maintain the electrical and electronics circuits. The main objective of these
electrical engineering calculators is to assist students, professionals and researchers to perform, understand or verify the electrical engineering calculations as swift as possible. | {"url":"https://dev.ncalculators.com/electrical/","timestamp":"2024-11-12T20:04:14Z","content_type":"text/html","content_length":"26126","record_id":"<urn:uuid:a508f18f-2430-48f9-a4d8-fd40f081a481>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00067.warc.gz"} |
Event based data acquisition and networked distribution of data to an arbitrary set of analysis platforms. The NSCL Data Acquisition system supports a wide variety of hardware in VME and CAMAC. The
system is currently in use by about 30 … Continue reading
Produces histograms of event mode data. Allows analysis to be interactively modified. Support for arbitrary production of computed parameters either via Tcl Scripting or compiled extensions. Find
Nuclear data analysis (histogramming) package at: http://www.sourceforge.net/projects/nsclspectcl
The class libraries here provide infrastructure for creating simulations of low energy nuclear physics experiments, as well as some useful working programs that do simple simulations and analysis of
experiments performed with magnetic spectrographs. Find Nuclear Simulation Java Class Libraries … Continue reading
Jam is an easy-to-use self-contained data acquisition and analysis system for VME-based (or CAMAC-based) nuclear physics experiments. Jam has an easy, standard GUI for taking and sorting
multi-parameter event-based data into 1-d and 2-d histograms. Find Jam: A Java-based Data … Continue reading | {"url":"https://openscience.org/software/physics/nuclear/","timestamp":"2024-11-10T05:54:21Z","content_type":"text/html","content_length":"52206","record_id":"<urn:uuid:f1d6ee27-e4e4-4bb6-97c4-f13fef3072e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00026.warc.gz"} |
Mixed Numbers and Improper Fractions
Mixed numbers and improper fractions
Mixed numbers and Improper Fractions
Mixed Numbers and Improper Fractions
Converting Mixed Numbers and Improper Fractions
Multiply/Divide Mixed Numbers and Improper Fractions 4.12
Mixed Numbers and Improper Fractions
Mixed Numbers and Improper Fractions
Mixed Numbers and Improper Fractions
Mixed Numbers and Improper Fractions 4.10
Mixed Numbers and Improper Fractions
Mixed Numbers and Improper Fractions
Converting mixed numbers and improper fractions
year 7 w/c 18/5 mixed numbers and improper fractions
Mixed Numbers and Improper Fractions adaptive practice
Mixed Numbers and Improper Fractions
Mixed Numbers and Improper Fractions
Mixed Numbers and Improper Fractions
Mixed Numbers and Improper Fractions
Mixed Numbers and Improper Fractions
Mixed Numbers and Improper Fractions
Converting Mixed Numbers and Improper Fractions
Fractions, Decimals, Mixed Numbers, and Improper Fractions
Mixed Numbers and Improper Fractions
Mixed Numbers and Improper Fractions
Explore Mixed Numbers and Improper Fractions Worksheets by Grades
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are an invaluable resource for teachers looking to enhance their students' mathematical abilities.
In addition to Mixed Numbers and Improper Fractions worksheets for Class 7, Quizizz offers a wide range of resources for teachers to support their students' learning in math and other subjects.
Quizizz is an online platform that allows teachers to create interactive quizzes, assignments, and activities that can be accessed by students on any device. With Quizizz, teachers can easily track
their students' progress and identify areas where they may need additional support or practice. The platform also offers a vast library of pre-made quizzes and worksheets, covering topics such as
fractions, decimals, algebra, geometry, and more, making it a one-stop-shop for all Class 7 math resources. By incorporating Quizizz into their teaching strategies, teachers can provide a dynamic and
engaging learning experience for their students, while also ensuring that they are well-prepared for success in their mathematical endeavors. | {"url":"https://quizizz.com/en-in/mixed-numbers-and-improper-fractions-worksheets-class-7","timestamp":"2024-11-04T00:50:47Z","content_type":"text/html","content_length":"151780","record_id":"<urn:uuid:52043d4a-31ff-4a21-bbd8-81f3d092c54a>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00862.warc.gz"} |
The function
Here the inputs should satisfy the following constraints:
is such that
is such that
Type Description
– Factorize
– Define
– Let
– Factorize
– Define
– Let | {"url":"https://www.iqclab.eu/fmultiplext/","timestamp":"2024-11-02T02:45:16Z","content_type":"text/html","content_length":"155494","record_id":"<urn:uuid:3721c8a0-662e-46ee-9be0-a551f3fc958d>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00487.warc.gz"} |
Writing Quadratic Equations From Graphs Worksheet Pdf - Graphworksheets.com
Equations From Graphs Worksheet – Graphing equations is an essential part of learning mathematics. This involves graphing lines and points and evaluating their slopes. This type of graphing requires
you to know the x- and y coordinates for each point. To determine a line’s slope, you need to know its y-intercept, which is the point … Read more
Writing Equations From A Graph Worksheet Pdf
Writing Equations From A Graph Worksheet Pdf – Graphing equations is an essential part of learning mathematics. It involves graphing lines and points, and evaluating their slopes. This type of
graphing requires you to know the x- and y coordinates for each point. You need to know the slope of a line. This is the … Read more | {"url":"https://www.graphworksheets.com/tag/writing-quadratic-equations-from-graphs-worksheet-pdf/","timestamp":"2024-11-02T21:38:36Z","content_type":"text/html","content_length":"54491","record_id":"<urn:uuid:e51a4ae5-cd85-4550-a809-d13b81222843>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00582.warc.gz"} |
Calculus with Sympy
• Post by: Anurag Gupta
• October 9, 2021
Sympy is a python package through which we can perform calculus operations in mathematics like differentiation, integration, limits, infinite series, and so on. It is a python library used mainly for
symbolic mathematics. The installation of this library is simple by using the following command:
pip install sympy
In order to write any symbolic expressions, firstly, we have to declare the symbolic variables that are involved in the symbolic expression. This can be done by:
• sympy.Symbol( ): used to declare a single variable by passing a variable as a string
• sympy.symbols( ): used to declare multi variables.
1. Differentiation:
Differentiation of any function can be done through the command diff(func, var,n) . Here, the “func” represents the symbolic function that is to be differentiated, “var” denotes the variable with
respect to which we have to differentiate the function, and “n” denotes the nth derivative that needs to be computed for the function. This can be illustrated as:
# Importing essential library
import sympy as sym
# Declaration of the variables
x, y, z = sym.symbols('x y z')
# function whose derivative is to be find
f = x**5 * y - y**2 + z
# Differentiating f with respect to x
derivative_x = sym.diff(f, x, 2)
print('derivative w.r.t x: ',
# Differentiating exp with respect to y
derivative_y = sym.diff(f, y, 2)
print('derivative w.r.t y: ',
The output obtained is:
derivative w.r.t x: 20*x**3*y
derivative w.r.t y: -2
2. Integration :
Through sympy, we can do both definite as well as indefinite integration by using integerate() function.
The general syntax used for the indefinite integration is given as:
sympy.integrate(func, var)
Here, “func” represents the function that is to be integrated and “var” represents the variable with respect to which it is integrated.
This can be illustrated as:
# Indefinite integration of sin(x) w.r.t. x
integration = sym.integrate(sym.sin(x), x)
print('indefinite integral of sin(x): ',
The output obtained is:
indefinite integral of sin(x): -cos(x)
The general syntax used for the definite integration is given as:
sympy.integrate(func, (var, lower_limit, upper_limit))
Here, lower_limit and upper_limit denote the lower and upper limit of the definite integration respectively.
This can be illustrated as:
# Definite integration of cos(x) w.r.t. x between -1 to 1
integration = sym.integrate(sym.cos(x), (x, -1, 1))
print('definite integral of cos(x) between -1 to 1: ',
The output obtained is:
definite integral of cos(x) between -1 to 1: 2*sin(1)
In sympy, infinity (∞) is written as oo.
3. Limits:
The limit of a function can be computed using this library by using limit(function, variable, point).
This can be illustrated as:
# Calculating limit of f(x) = 1/x as x tends to ∞
limit_a = sym.limit(1/x, x, sym.oo)
# Calculating limit of f(x) = tan(x)/x as x tends to 0
limit_b = sym.limit(sym.tan(x)/x, x, 0)
The output obtained is:
4. Series expansion:
The Taylor series expansions of functions around a point can be computed using sympy library. In order to compute the series expansion of f(x) around the point, x=x[0] terms of order x^n, the syntax
used is:
sympy.series(f, x, x0, n)
The default value of x[0]=0 and n=6 is considered in case if they can be omitted from the syntax.
This can be illustrated as:
# series expansion
series = sym.series(sym.sin(x), x)
The output obtained is:
x - x**3/6 + x**5/120 + O(x**6) | {"url":"https://computationalmechanics.in/calculus-with-sympy/","timestamp":"2024-11-13T22:50:29Z","content_type":"text/html","content_length":"77999","record_id":"<urn:uuid:e9f35c00-3554-4b09-a75a-febd761d6ce0>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00644.warc.gz"} |
m5stack EuroMillions APP help
I'm desenvolving an APP for EuroMillions Lottery in my m5stack and I have some troubles with it and I hope someone will help me.
I sort n times a number between 1 and 50 (the Euromillions Lottery range of numbers). Every time a number is sorted, an array with 50 elements is incremented at the number sorted position. If 5
is sorted, then MatrixNumber[5]=MatrixNumber[5]+1. After n times I will seek the MatrixNumber[] and search for the biggest number there and the corresponding position where I found it and save
the position in another array (FinalNumber[]). After that I zeroe the position so that it only be only one maximum values for all the array. (are you lost? I hope not). After all is done I get an
array with 5 numbers (the most sorted by function "random(50)+1". I do the ssame stuff for the "Stars" part of the sorting.
Problem ONE: I can't SORT the array from lesser to bigger. I've done everything and it just don't sort.
Problem TWO: In "option" I use a number that can be "-2" to "+2". That number will be added the the "sorted number" comparing his value. For example, variable "Option1" value will be added if a
number is sorted between 1 and 6 and "Option2" value will be added to numkber sorted if the number is between 7 and 12, ... and so on (1..6,7..12,13..18,... are the lines on the lottery form). If
I change the number to any row (row 1 to 8) beside value "00", the sorting is giving me almost always the same numbers. If I put the rows to position value of "00" then it's random again.
Very weird.
I copied the source code in pastebin because it's kind of big and I want to share the code anyway (but not with weird stuff on it).
Can anyone help?
I have looked at your programm and I really don't understand what you are trying to do :
Storing and retrieving datas from eprom ? Displaying battery levels ? Randomly reading keypresses ? Generating negative numbers (with signed char) ? Mixing char, integers and long integers ? Some
weird sorting scheme ?
So what is the goal of your app ? You are only talking about some weird sorting problems where nobody understands what you want to do.
For each line in the euromillions lottery you have to enter 5 different numbers between 1 and 50, and 2 different numbers for the stars between 1 and 12.
Probably you want to generate the 5 numbers and the 2 stars with a random number generator and display it on your m5Stack lcd. So your programm would look like:
□ Setup : Start the (pseudo) random number generator (seed)
□ On a keypress: Generate 5 numbers and 2 stars:
- generate 5 numbers between 1 and 50; as long as 2 numbers are equal replace it with a new random number; sort the numbers low to high eg. by bubble sort; display the numbers;
- generate 2 (star)numbers between 1 and 12; as long as the two numbers are equal replace one with a new random number; if number 2 < number 1 switch the numbers; display the numbers on the
If you are new to programming with arduino and C++ instead of trying to copy and paste some demo codes, try to programm your app with blockly/(Micropython) : flow.m5stack.com .
There you can generate random numbers, lists with 5 and 2 elements and do the sorting with the logic elements and display the results (as labels) on your M5Stack lcd,
Good luck!
@crami25 Hello. Thank you for replying. The system of the -2..+2 is simple. A number random is sorted: 5 for exemple. I play EuroMillion with that number and 4 more sorted. The next week the
numbers sorted are displayed and I realise that "6" has been sorted instead of "5" (the number I played the week before. Simce numbers from 1 to 6 are in the row 1 of the EuroMillion form, I go
to OPTIONS and select "+1" in row 1. NOW, if a number is sorted like "3" for exemple, the program is showing me 3+(+1)=4. You get the idea? All rows have that tweak (all but not last row 49..50).
Those "tweaks numbers" are saved in eeprom so everytime I start the app the tweak numbers are as I want them. THE BATERY, since I was using the m5stack for this app I thought that it will be nice
to see HOW MUCH battery is remaining. KEYPRESSED are not random. I wait for keypress and depending WHERE in the program I AM (I mean what SUB is active). I haven't managed to save negative values
as byte (signed char in this case) in eprom so I convert the values of the ROW TWEAKS from -2 to +2 into a simple 1 to 5 system and do the inverse when Reading eprom. The sorting is Bubble sort
but doesn't seem to work and THAT is a weird problema. I hope I have dissipate all your doubs so you can try to help. Thanks
I still don't understand what you want to do ....maybe you could explain it in your native language.
You are always speaking of sorting and you are using the verb for different purposes:
sorting : sort 5 numbers so the lowest is displayed first and the highest last (5,7,28,35,47)
tip : the 5 numbers you want to fill into the euromillion form
draw: the 5 numbers which are drawn in the euromillion lottery
you probably use sorting for all three different arrays of values
I also don't understand your tweak -2..+2 system: The numbers drawn have nothing to do with the rows in the euromillion form. If the lowest number you tip is a 5 and the lowest number drawn is 15
your tweak according your explanation would be -10 ???
The type of the euromillions forms probably depends on the country you are playing:
In Spain, Germany the first row has the numbers 1-5 (not 6) and in France 1-10 !
In my opinion the rows in the form have nothing to do with the the numbers drawn and and the numbers you have tipped: You have to tip 5 different numbers from 1-50. The rows of the form is only a
convenient way displaying the numbers (1-50 in ten rows of 5, 1-50 in five rows of 10, 1-50 in seven rows of six and a last row of the remaining 49 and 50). This country dependend form-scheme is
only used to be read by the scanner entering your tip into the euromillions lottery.
@crami25 Hello. Thanks. Sorta was used for TIP. -2 ... +2 will be dificult to explain. It only make sense if the number DRAWN if at lenght of -2 or 2 of the number TIPPED. Forget that. What
puzzle me (and you can only see it if you put the program in your m5stack) if when all options are "00" numbers are drawn randmly but if you change any of the options to -2 ... 2, the numbers are
repeating. It's weird. Can you put the program in your m5 and explain the weird behaviour? Thanks
Sorry, it will take some time for me to figure out, what your code is supposed to do and what it is doing.
Good practice is to write the code with comments where you are explaining what the different variables are and what the different sections are supposed to do. In old days writing code where you
jumped in and out in different sections spaghetti code (you do that when you press a key A,B,C on your M5Stack).
In Arduino you have to cut your code into different sections and the look, if the sections are doing what they are supposed to. In most programs in arduino you insert a "serial.println(x,y,z)" at
the end of each section. Before running your program you open a terminal on the com line your ESP32 is connected to. That way for each code snipped you can check, if x,y,z are the values the
program should furnish.
By the way, as I understand from your code, your supposed random numbers are no random numbers any more:
Starting you set all your MatrixNumber(1-50) to 1; then you generate 25000 times a random variable x (Temp1); you modify the variable x with your options (-2,-1,0,1,2) (=Temp2) and adjust the
values of new x (Temp2) to 1-50; then you increment MatrixNumber(x) by 1 (25000 times); as I see it, your MatrixNumber(1-50) should then be 25000/50= ~500. So you are testing the randomness oft
he random number generator instead generating random numbers ! In mathematics it's called the Monte Carlo Method: If you play roulette 25000 times, statistically all numbers drawn should be
equal. I several numbers are drawn more frequent than they should be statistically, there must be a flaw in the roulette spinning machine. Players analyzing this behavior of the roulette spinning
machine can then set their chips on these numbers trying using it to their advatage breaking the bank.
@crami25 Hello. After analysing the code do you see any flaw? WHY does the code draw random numbers if the tweaks are 00 and numbers are almot the same drawn if tweaks are different than 00? That
puzzles me. Any thought on that? Have you upload the prog to your m5 and saw the strange behaviour? Thank for your help.
I loaded your code into arduino (1.8.12) and tried to compile it.
The verifying of your code stopped at line 248 where you have an
exit; statement. So the code you submitted above cannot be the code running on your M5Stack.
The right syntax of your code should be exit(0);
Do you really want that ? see
On the arduino, calling exit(0) really isn't a useful thing to do.
On Arduino, the statement exit(0) translates to
cli(); // disable interrupts
while(1); // loop forever
So it basically stops your program running, but leaves your CPU running in an infinite loop ! So your programm gets stuck in your getEuromillionOptions(); subroutine until you switch it off.
@crami25 Hello. Strange the "exit" it should be "break" and it should be NOTHING. I don't know WHY I put the exit there. Just remove it.. Sorry for that mistake. The idea is exiting the sub. Have
you found anything else weird? What about the random numbers drawing scheme? It's weird that with no "tweaking" numbers are drawn "random" values and with "tweaking" they repeat themselves. You
can only see this if you upload the source to an m5stack. Thank you
@crami25 Hello again. I have redone all the sorting/tipping/drawing stuff. And now the code works as I wanted it. Code is not optimized for size but seems to work. I posted the new code here
NOW the "random" numbers are "random" even with "tweaks" actives. Thanks for your support and help. Any sugestions/remarks are welcome as always. | {"url":"https://community.m5stack.com/topic/1754/m5stack-euromillions-app-help/2","timestamp":"2024-11-04T20:55:48Z","content_type":"text/html","content_length":"111331","record_id":"<urn:uuid:9b59f02d-6569-45d6-a0c5-c3d347ac5698>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00115.warc.gz"} |
Rocksensor : Research and development of static pressure transmitter
The following is the discussion on the static pressure influence of differential pressure transmitter by lotingson technology R & D center. You are welcome to read it.
1. Introduction
When the differential pressure transmitter is in linear calibration, it is usually carried out under the condition that the negative pressure chamber is open to the atmosphere. In other words, the
static pressure is 1 atmospheric pressure. However, once installed in the field for actual use, a certain working pressure will be added to the positive and negative pressure chambers. At this time,
it will be found that the zero position is offset and the full position output is also offset (the full position offset is generally read out by comparing with the standard instrument). When the
working static pressure is added, the zero position and full position output of the transmitter will deviate from the zero position and full position during atmospheric calibration, which is called
static pressure influence error.
2. The influence of static pressure on transmitter performance and field examples
The static pressure error of differential pressure transmitter directly affects its comprehensive accuracy. The comprehensive accuracy (%) of differential pressure transmitter generally consists of
three factors, which are accuracy (%), influence of ambient temperature change (% / 30o) and static pressure change (% / 7MPa). Their calculation formula is as follows:
Thus, static pressure error is a very important factor for the comprehensive accuracy of differential pressure transmitter.
This point has also been confirmed in various practical application conditions.
For example, when the differential pressure transmitter is applied to the field application of orifice flow detection, orifice plate or nozzle and other throttling parts are installed in the
pipeline. Because the orifice diameter of the throttling piece is smaller than the inner diameter of the pipe, when the fluid flows through the throttling part, the beam cross-section suddenly
shrinks and the flow rate accelerates. After throttling, the static pressure of the fluid at the back end decreases, so there is a static pressure difference before and after the throttling piece.
There is a definite numerical relationship between the static pressure difference and the fluid flow, which conforms to q = K. The differential pressure transmitter is used to measure the
differential pressure before and after the throttling parts to realize the flow measurement.
See Figure 1：
When it is used to measure the flow rate of high pressure steam in power plant, if the static pressure effect is not corrected or compensated, it will bring large error to the flow measurement,
especially when the relative flow rate is small, the influence is more significant. For example, a metal capacitive differential pressure transmitter and throttling device constitute a differential
pressure flowmeter. Under the condition of 32Mpa working static pressure, the static pressure error of full scale is ≤± 2% FS. Although the zero position error can be eliminated by zero adjustment,
the full output error is always unavoidable. Therefore, this error directly affects the flow test and has a great influence. In this application condition, the static pressure performance of
differential pressure transmitter is particularly important. If the static pressure error is compensated or its static pressure error is very small, its measurement accuracy will be greatly improved.
Causes of static pressure influence of metal capacitance sensor
Metal capacitive sensor is a kind of structural sensor, its static pressure effect is particularly prominent. This is related to its own structural characteristics.
Working principle introduction:
The medium pressure is transmitted to the measuring diaphragm located in the center of "δ" chamber through isolation diaphragm and silicone oil, and the measuring diaphragm deforms with the
differential pressure on both sides. The displacement of the diaphragm is directly proportional to the differential pressure, and the maximum displacement is 0.1 mm. The differential capacitance
between the measuring diaphragm and the capacitor plate is converted into a two-wire 4-20madc output signal through an electronic conversion circuit.
3.1 cause 1 of static pressure influence
It can be seen from Fig. 2 and Fig. 3 that both sides of the metal capacitive type are pressed, and the pressure is transmitted to the inner central diaphragm through the isolation diaphragm.
From the simplified stress distribution diagram and deflection change diagram of the metal capacitive sensor in Fig. 4, it can be seen that the internal pressure of the sensor is distributed from the
center to the surrounding direction, and the stress in the X direction is completely offset, but the stress Q in the Y direction is all added to the shell of the sensor. Because of the size of the
structure, the closer to the center, the thinner the structure, the worse the compressive capacity of the sensor, especially at the central diaphragm, the structural strength is the weakest. Under
high hydrostatic pressure, there is a maximum deflection f at the center point. The result is that under high static pressure, the tension force of the central isolation diaphragm increases, and the
tension degree of the diaphragm is strengthened when the working static pressure is zero. And the greater the static pressure, the greater the degree of tension.
When the tension is increased, the displacement of the central diaphragm with the differential pressure will become smaller, and then the differential capacitance between the measuring diaphragm and
the capacitor plate will be converted into two-wire system through the electronic conversion circuit, and the output signal of 4-20madc will also become smaller. Finally, it leads to measurement
error and static pressure influence error, and the absolute error of static pressure influence has a certain linear relationship with the added working static pressure. The larger the working static
pressure is, the greater the static pressure error of its range is.
As for the static pressure error of the zero position, the direction is uncertain, which is mainly related to the welding stress and the personality of the sensor.
3.2 causes of static pressure influence
It can be seen from Figure 5 that when static pressure is applied to both sides of the sensor, the curved surfaces of the metal capacitive type are pressed simultaneously. However, the curved surface
is composed of metal and glass, which will produce small deformation under the action of external force. Therefore, the thickness L1 and L2 of the curved surface seat on both sides decrease linearly
with the increase of static pressure P. As a result, the electrode distances H1 and H2 of capacitor plates on both sides increase.
According to the definition of capacitance, the capacitance of parallel plate capacitor is C = ε o × s / h
Therefore, when the electrode distance h of the capacitance increases, the two-stage capacitance C1 and C2 of the sensor will become smaller. According to the working principle of capacitance sensor,
the differential capacitance between measuring diaphragm and capacitor plate is converted into two-wire system by electronic conversion circuit, and the output signal of 4-20madc becomes smaller.
Finally, it leads to measurement error and static pressure influence error, and the absolute error of static pressure influence has a certain linear relationship with the added working static
pressure. The larger the working static pressure is, the greater the static pressure error of its range is.
From the analysis of the above two reasons, the metal capacitance sensor will inevitably produce measurement drift error under the influence of working static pressure. There is a certain linear
relationship between the full position drift and the working static pressure, and the direction uncertainty for the zero position drift. | {"url":"http://www.rocksensor.com/news/show-2059.html","timestamp":"2024-11-07T04:11:27Z","content_type":"text/html","content_length":"30160","record_id":"<urn:uuid:b9183d30-6a7b-4baf-afce-e2a9cf2d3ff7>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00510.warc.gz"} |
GPFSeminars 2021
Seminars for the year: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007
Follow our seminars online via: GPF BigBlueButton server
Time: 17. December 2021, 11:00h
Place: Faculty of Physics, room 661, and online
Speaker: Stefan Djordjevic
Title: Black hole entropy and the information loss paradox (part 4)
In this part of the seminar on the black hole information paradox, we consider some concrete models of 2d dilaton gravity (CGHS, BPP) that allow us to explicitly calculate the Page curve using the
island formula. As a new result, we will show how to perform this calculation for an eternal black hole in 2d dilaton gravity model obtained from 4d Einstein-Hilbert action via dimensional reduction.
Time: 10. December 2021, 11:00h
Place: Faculty of Physics, room 661, and online
Speaker: Stefan Djordjevic
Title: Black hole entropy and the information loss paradox (part 3)
In this part of the seminar on the black hole information paradox, we consider some concrete models of 2d dilaton gravity (CGHS, BPP) that allow us to explicitly calculate the Page curve using the
island formula. As a new result, we will show how to perform this calculation for an eternal black hole in 2d dilaton gravity model obtained from 4d Einstein-Hilbert action via dimensional reduction.
Time: 3. December 2021, 11:00h
Place: Faculty of Physics, room 661, and online
Speaker: Dragoljub Gocanin
Title: Black hole entropy and the information loss paradox (part 2)
This talk will be devoted to the so-called wormhole replica trick and the notion of an island - a kind of a quantum extremal surface recently advocated by Maldacena. We will explain how the
application of the holographic principle leads to the unitary Page curve. A generalization of the original formula for a generalized entropy is achieved via the notions of a quantum extremal surface
and an island. Using the gravity path integral we will present additional arguments in favour of this approach.
Time: 19. November 2021, 12:00h
Place: Faculty of Physics, room 661, and online
Speaker: Voja Radovanovic
Title: Black hole entropy and the information loss paradox (part 1)
The most important unsolved problem in black hole physics is the information loss paradox, i.e. whether the process of formation and evaporation of a black hole is unitary in accordance with quantum
mechanics. The unitarity implies that von Neumann entropy of the Hawking radiation should initially increase but subsequently fall back down, following the so-called Page curve.
In the first talk of this series we consider black hole thermodynamics and discuss various concepts of entropy: fine-grained, coarse-grained and entanglement entropy. In addition, we calculate
entanglement entropy for Rindler observer and show how to generalize this result to curved space-time, where it is used to define generalized entropy of a black hole.
The upcoming talks will be devoted to the so-called wormhole replica trick and the notion of an island - a kind of a quantum extremal surface recently advocated by Maldacena. We study a particular
case of 2D dilaton gravity obtained by dimensional reduction of the Einstein-Hilbert action for Schwarzschild metric. We calculate the von Neumann entropy of the Hawking radiation and reproduce the
Page curve.
Time: 5. November 2021, 11:00h
Place: Online
Speaker: Ilija Buric
Title: Conformal bootstrap - a non-perturbative approach to conformal field theories
Conformal bootstrap is a collection of methods for the study of conformal field theories, which are based on minimal assumptions such as unitarity and self-consistency. After reviewing the basic
structure of conformal field theories, I will illustrate some of these methods on one numerical and one analytic example.
Time: 27. October 2021, 13:00h
Place: Faculty of Physics, room 661, and online
Speaker: Wolfgang Martin Wieland
Title: How the Immirzi Parameter deforms the SL(2,R) boundary symmetries on the light cone
This talk describes how the Barbero-Immirzi parameter deforms the SL(2,R) symmetries on a null surface boundary. Our starting point is the definition of the action and its boundary terms. The action
that we use is the usual Holst action. Compared to metric gravity it contains an additional coupling constant - the Barbero-Immirzi parameter. Given the action and the boundary conditions, we
introduce the covariant phase space and explain how the Holst term alters the boundary symmetries on a null surface. This alteration only affects the algebra of the edge modes on a cross-section,
whereas the algebra of the radiative modes is unchanged by the addition of the Barbero-Immirzi parameter. To compute the Poisson brackets explicitly, we work on an auxiliary phase space, where the SL
(2,R) symmetries of the boundary fields are manifest. The physical phase space is obtained by imposing both first-class and second-class constraints. All gauge generators are at most quadratic in
terms of the fundamental SL(2,R) variables. Finally, we discuss various strategies to quantise the system.
Time: 8. October 2021, 11:15h
Place: Faculty of Physics, room 665, and online
Speaker: Danijel Obric
Title: T-dualization of bosonic string and type II superstring in presence of coordinate dependent background fields
In this talk we will discuss one of the ways how coordinate noncommutativity arises in the context of string theory. Starting point of the talk will be a short introduction to string theory, where we
will discuss basic formalism for both bosonic string and supersymmetric string, we will also discuss dualities that are present in the theory. After this, discussion will focus on the procedure for
obtaining T-dual theories. This procedure will be applied first on bosonic string and later on superstring, where, after obtaining T-dual theories, we will discuss what effect did T-duality have on
structure of Poisson brackets.
Time: 9. July 2021, 11:15h
Place: Faculty of Physics, room 661
Speaker: Milorad Popovic
Title: g-2 experiment behind the scene
The g-2 experiment has published the result which demonstrates the deviation from the Standard Model at 4.2 sigma, which indicates that the theory is either incomplete or wrong. Since the Internet is
full of texts and reviews of this result and its consequences for the theory, this lecture will be devoted to the details and facts which made the experiment possible, from the point of view of an
active participant of the whole project.
Time: 18. June 2021, 11:15h
Place: Online
Speaker: Marko Vojinovic
Title: A review of some research programs in classical and quantum gravity
We will give a brief overview of the following three research programs: (1) influence of curvature and torsion on the motion of bodies, (2) constructions of quantum gravity models based on higher
gauge theories, and (3) quantum information theoretical approach to quantum gravity. The first program is mostly completed, while the second and third are still ongoing. We will discuss both the main
results obtained so far, and the open problems that are yet to be studied.
Time: 4. June 2021, 11:15h
Place: Online
Speaker: Dejan Stojkovic
Title: In search of a wormhole
If a traversable wormhole smoothly connects two different spacetimes, then the flux cannot be separately conserved in any of these spaces individually. Then objects propagating in a vicinity of a
wormhole in one space must feel influence of objects propagating in the other space. We show this in the cases of the scalar, electromagnetic, and gravitational field. The case of gravity is perhaps
the most interesting. Namely, by studying the orbits of stars around the black hole at the center of our galaxy, we could soon tell if this black hole harbors a traversable wormhole. Alternatively,
one can expect the same effect in black hole binary systems, or a black hole - star binary systems, which are actually the cleanest and most sensitive systems for such a search.
Time: 21. May 2021, 11:15h
Place: Institute of Physics, online
Speaker: Marko Vojinovic
Title: Relation between L-infinity algebras and higher category theory
After a short introduction to the notions of higher categories, n-groups and categorical ladder, we will discuss an isomorphism between some of these structures and L-infinity algebras. We will also
discuss the geometric interpretation and relevance of n-groups, along with applications in physics.
Time: 7. May 2021, 11:15h
Place: Online
Speaker: Grigorios Giotopoulos
Title: Braided gauge field theory: New examples
In this talk I will be presenting new examples of braided field theories, via braided L-infinity algebras. First, I will briefly review how L-infinity algebras appear in classical field theory and
how Drinfel'd twists work in practice. I will then apply the braided L-infinity algebra framework for the cases of scalar field, BF and Yang-Mills theories and remark on the features of the resulting
noncommutative theories.
Time: 16. April 2021, 11:15h
Place: Online
Speaker: Marija Dimitrijevic Ciric
Title: Application of L-infinity Algebras: Braided Deformation of Field Theory and Noncommutative Gravity (part 3)
In this talk we discuss a possibility to apply the L-infinity algebra formalism in construction of field theories and gravity on noncommutative spaces. To do this we have to introduce a new homotopy
algebraic structure, that we call a braided L-infinity algebra. Then we use the braided L-infinity algebra to systematically construct a new class of noncommutative field theories, that we call
braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which
do not act (in a standard/obvious way) on the solutions of the field equations.
In the first talk we will motivate the introduction of braided gauge field theories and we will repeat the basics of the twist deformation formalism introduced by Drinfeld in 1985.
In the second talk we will define braided gauge theories and discuss how they fit in the braided L-inifinity algebra formalism. Finally, we will present two examples: braided Chern-Simons theory and
braided Einstein-Cartan-Palatini 4D gravity.
The lecture is based on the following papers:
[1] M. Dimitrijevic Ciric, G. Giotopoulos, V. Radovanovic, R. J. Szabo, "$L_\infty$-Algebras of Einstein-Cartan-Palatini Gravity", Jour. Math. Phys. 61, 112502 (2020), [arXiv:2003.06173].
[2] M. Dimitrijevic Ciric, G. Giotopoulos, V. Radovanovic, R. J. Szabo, "Braided $L_\infty$-Algebras, Braided Field Theory and Noncommutative Gravity", [arXiv:2103.08939].
Time: 09. April 2021, 11:15h
Place: Online
Speaker: Marija Dimitrijevic Ciric
Title: Application of L-infinity Algebras: Braided Deformation of Field Theory and Noncommutative Gravity (part 2)
In this talk we discuss a possibility to apply the L-infinity algebra formalism in construction of field theories and gravity on noncommutative spaces. To do this we have to introduce a new homotopy
algebraic structure, that we call a braided L-infinity algebra. Then we use the braided L-infinity algebra to systematically construct a new class of noncommutative field theories, that we call
braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which
do not act (in a standard/obvious way) on the solutions of the field equations.
In the first talk we will motivate the introduction of braided gauge field theories and we will repeat the basics of the twist deformation formalism introduced by Drinfeld in 1985.
In the second talk we will define braided gauge theories and discuss how they fit in the braided L-inifinity algebra formalism. Finally, we will present two examples: braided Chern-Simons theory and
braided Einstein-Cartan-Palatini 4D gravity.
The lecture is based on the following papers:
[1] M. Dimitrijevic Ciric, G. Giotopoulos, V. Radovanovic, R. J. Szabo, "$L_\infty$-Algebras of Einstein-Cartan-Palatini Gravity", Jour. Math. Phys. 61, 112502 (2020), [arXiv:2003.06173].
[2] M. Dimitrijevic Ciric, G. Giotopoulos, V. Radovanovic, R. J. Szabo, "Braided $L_\infty$-Algebras, Braided Field Theory and Noncommutative Gravity", [arXiv:2103.08939].
Time: 02. April 2021, 11:15h
Place: Online
Speaker: Marija Dimitrijevic Ciric
Title: Application of L-infinity Algebras: Braided Deformation of Field Theory and Noncommutative Gravity (part 1)
In this talk we discuss a possibility to apply the L-infinity algebra formalism in construction of field theories and gravity on noncommutative spaces. To do this we have to introduce a new homotopy
algebraic structure, that we call a braided L-infinity algebra. Then we use the braided L-infinity algebra to systematically construct a new class of noncommutative field theories, that we call
braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which
do not act (in a standard/obvious way) on the solutions of the field equations.
In the first talk we will motivate the introduction of braided gauge field theories and we will repeat the basics of the twist deformation formalism introduced by Drinfeld in 1985.
In the second talk we will define braided gauge theories and discuss how they fit in the braided L-inifinity algebra formalism. Finally, we will present two examples: braided Chern-Simons theory and
braided Einstein-Cartan-Palatini 4D gravity.
The lecture is based on the following papers:
[1] M. Dimitrijevic Ciric, G. Giotopoulos, V. Radovanovic, R. J. Szabo, "$L_\infty$-Algebras of Einstein-Cartan-Palatini Gravity", Jour. Math. Phys. 61, 112502 (2020), [arXiv:2003.06173].
[2] M. Dimitrijevic Ciric, G. Giotopoulos, V. Radovanovic, R. J. Szabo, "Braided $L_\infty$-Algebras, Braided Field Theory and Noncommutative Gravity", [arXiv:2103.08939].
Time: 26. March 2021, 11:15h
Place: Online
Speaker: Clay Grewcoe
Title: Curved L-infinity algebras
What are curved L-infinity algebras? This less common but very natural generalisation will be theoretically defined and explored through the example of a DFT algebroid, the geometric structure
underlying the sigma model of double field theory. Additionally, it will be explored how and in what cases one can "flatten" a curved L-infinity algebra into a regular one.
The lecture is based on the following paper:
[1] C. J. Grewcoe and L. Jonke, "DFT algebroid and curved $L_\infty$-algebras", [arXiv:2012.02712].
Time: 19. March 2021, 11:15h
Place: Institute of Physics, online
Speaker: Clay Grewcoe
Title: BV/BRST formalism in the language of L-infinity algebras
Batalin-Vilkovisky procedure is an important part of gauge field theories, and given the connection between field theory and L-infinity algebras demonstrated in previous lectures, it is natural to
ask how does the BV formalism fit into that picture. In order to discuss this one needs to define a tensor product of L-infinity algebras and use it to connect the gauge symmetry with classical field
theory and further with its BV (BRST) extension. As an example we will use the Courant sigma model as a theory which displays all properties of the formalism.
The lecture is based on the following papers:
[1] C. J. Grewcoe and L. Jonke, "Courant Sigma Model and $L_\infty$-algebras", Fortsch. Phys. 68 no.6, 2000021 (2020) [arXiv:2001.11745].
[2] B. Jurčo, L. Raspollini, C. Saemann and M. Wolf, "$L_\infty$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism", Fortsch. Phys. 67 no.7, 1900025 (2019) [arXiv:1809.09899].
Time: 12. March 2021, 11:15h
Place: Online
Speaker: Voja Radovanovic
Title: L-infinity algebras and field theory (part 2)
In this series of lectures we will analyze in detail L-infinity algebras and their application in field theory and gravity on commutative and noncommutative spaces. We will begin by introducing a
concept of L-infinity algebra as a generalization of the usual concept of Lie algebra. Then we will discuss in detail four examples: Yang-Mills gauge theory, Eintein-Cartan-Palatini gravity, BRST
symmetry and Chern-Simons gauge theory. All these examples are examples of field theory on the commutative spacetime. The material presented here will be generalized later on (in the following
seminars) to field theories on noncommutative spaces.
Time: 5. March 2021, 11:15h
Place: Online
Speaker: Voja Radovanovic
Title: L-infinity algebras and field theory (part 1)
In this series of lectures we will analyze in detail L-infinity algebras and their application in field theory and gravity on commutative and noncommutative spaces. We will begin by introducing a
concept of L-infinity algebra as a generalization of the usual concept of Lie algebra. Then we will discuss in detail four examples: Yang-Mills gauge theory, Eintein-Cartan-Palatini gravity, BRST
symmetry and Chern-Simons gauge theory. All these examples are examples of field theory on the commutative spacetime. The material presented here will be generalized later on (in the following
seminars) to field theories on noncommutative spaces.
Time: 19. February 2021, 11:15h
Place: Online
Speaker: Igor Salom
Title: Ortosymplectic superalgebra as a spacetime symmetry (part 2)
We will explain the relationship between the orthosymplectic algebra osp(1|8) and the Poincaré and conformal (super)symmetry, and we will demonstrate why this symmetry is also called generalized
superconformal symmetry. We will discuss unitary irreducible representations of this algebra, with a focus on the simplest such representations. It will turn out that the simplest representation
corresponds to the space of a massless relativistic particle which, depending on helicity, automatically and inevitably satisfies appropriate equations of motion (such as Klein-Gordon, Dirac, or
Maxwell equations). Special attention will be devoted to the appearence of the the EM duality symmetry in this context. We will also consider the first next (least complex) representation, and see
that it corresponds to massive particles, with two mass terms appearing, which are mutually related by EM duality symmetry.
Time: 12. February 2021, 11:15h
Place: Online
Speaker: Igor Salom
Title: Ortosymplectic superalgebra as a spacetime symmetry (part 1)
We will explain the relationship between the orthosymplectic algebra osp(1|8) and the Poincaré and conformal (super)symmetry, and we will demonstrate why this symmetry is also called generalized
superconformal symmetry. We will discuss unitary irreducible representations of this algebra, with a focus on the simplest such representations. It will turn out that the simplest representation
corresponds to the space of a massless relativistic particle which, depending on helicity, automatically and inevitably satisfies appropriate equations of motion (such as Klein-Gordon, Dirac, or
Maxwell equations). Special attention will be devoted to the appearence of the the EM duality symmetry in this context. We will also consider the first next (least complex) representation, and see
that it corresponds to massive particles, with two mass terms appearing, which are mutually related by EM duality symmetry.
Time: 29. January 2021, 11:00h
Place: Online
Speaker: Tijana Radenkovic
Title: Gauge symmetry of the 3BF theory for a generic Lie 3-group
The higher category theory can be employed to generalize the BF action to the so-called 3BF action, by passing from the notion of a gauge group to the notion of a gauge 3-group. In this work we
determine the full gauge symmetry of the 3BF action. To that end, the complete Hamiltonian analysis of the 3BF action for a general Lie 3-group is performed, by using the Dirac procedure. This
analysis is the first step towards a canonical quantization of a 3BF theory. This is an important stepping-stone for the quantization of the complete Standard Model of elementary particles coupled to
Einstein-Cartan gravity, formulated as a 3BF action with suitable simplicity constraints. We show that the resulting gauge symmetry group consists of the already familiar G-, H-, and L-gauge
transformations, as well as additional M- and N-gauge transformations, which have not been discussed in the existing literature.
Seminars for the year: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007
Follow our seminars online via: GPF BigBlueButton server | {"url":"http://gravity.ipb.ac.rs/seminars2021.html","timestamp":"2024-11-10T02:24:36Z","content_type":"text/html","content_length":"30555","record_id":"<urn:uuid:9ab0a592-964e-4600-974d-2799157a2516>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00704.warc.gz"} |
Kargers Algorithm for Minimum Cut - Scaler Blog
Kargers Algorithm for Minimum Cut
In graph theory, a cut is a set of edges removal of which divides a connected graph into two non-overlapping (disjoint) subsets. The minimum cut (or min-cut) is defined as the minimum number of
edges, when removed from a graph, divide the graph into two disjoint sets.
A randomized contraction algorithm $i.e.$ Karger's Algorithm is used to find the minimum cut of a graph in data structure.
Scope of the Article.
• Concept of the cut and the minimum cut have been explained in detail through an example.
• Karger's algorithm to find the minimum cut of a graph have been explained along with a detailed example and codes in C/C++, and Java.
Karger's Algorithm is a randomized algorithm whose runtime is deterministic; that is, on every run, the time to execute will be bounded by a fixed time function in the size of the input but the
algorithm may return a wrong answer with a small probability.
Introduction to Minimum Cut
Before discussing Minimum cut, let us first understand what does a Cut means in terms of graph data structure?
A cut can be defined as a partition of vertices of a graph into two or more disjoint subsets.
For example:
The given graph $G$ has $5$ vertices and $6$ edges –
One of the possible cuts in this graph is to partition this graph into two disjoint sets $i.e.$ splitting the graph into two disconnected components, one with vertices $A, B, C \space and\space D$
and the other one with a single vertex $E$ by removing three edges $B\leftrightarrow C,$ $B\leftrightarrow E,$ and $C\leftrightarrow E$ as shown below –
Each cut has some size associated with it, that is the sum of the weights of the edges that have been removed. In the case of un-weighted graphs size is nothing but the number of vertices that have
been removed.
Now as suggested by the name, A Minimum cut (or simply a min-cut) of a graph $G$ can be defined as cut with minimum size $i.e.$ there is no other possible cut in the graph $G$ with a smaller size.
In simple words, the minimum number of edges required to remove to disconnect the graph into two components is known as the minimum cut of the graph.
The min-cut of the above-shown graph is shown below which can be obtained by removing two edges, $A\leftrightarrow B$ and $D\leftrightarrow C$.
Karger’s Algorithm to Find Min Cut
Karger's algorithm is a randomized algorithm (an algorithm which have some degree of randomness associated in its procedure) to compute a minimum cut of a connected, undirected, and unweighted graph
$G=(V,E)$. It is a “Monte Carlo” algorithm which means it may also produce a wrong output with a certain (usually low) probability.
The main idea of Karger’s algorithm is based on the concept of edge contraction, where edge contraction means merging two nodes (say $u$ and $v$) of the graph $G$ into one node which is also termed
as a supernode. All the edges connecting either to $u$ or $v$ are now attached to the merged node (supernode) which may result in a multigraph as shown in the image given below –
In Karger's algorithm, an edge is chosen randomly, and then the chosen edge is contracted which results in a supernode. The process continues until only two supernodes are remaining in the graph.
Those two supernodes represent cut in the original graph $G$.
As Karger algorithm being a "Monte Carlo" algorithm can also give wrong answers so by repeating the algorithm many times the minimum cut of the graph can be found with a certainly high probability.
• Make a copy of graph $G$ which will be termed as $CG$ (contracted graph).
• While $CG$ contains more than two vertices
□ Select any random edge $u\leftrightarrow v$ from the contracted graph.
□ Contract the edge and merge the vertices $u$ and $v$ into one.
□ Remove self-loops (if formed).
• After the above step only two vertices will be remaining in the graph and the edges connecting those two vertices represents the minimum cut of the graph.
Now let us understand how this algorithm finds the minimum cut through an example.
Let $G$ be the given undirected graph with $5$ vertices and $7$ edges for which we are interested to calculate minimum cut –
Step 1 – We choose edge $a$ to contract, nodes $1$ and $4$ will be merged into one supernode and all edges connecting them will be adjusted accordingly. Then the contracted graph will look like :
Step 2– This time we select the edge $c$ for contracting, so nodes $2$ and $3$ will be merged and after merging endpoints of the edge $c$, the contracted graph $(CG)$ will look like
Step 3 – Now in our contracted graph, we choose edge $e$ for contracting hence supernodes ${1,4}$ and ${2,3}$ will get merged, and edges $f$ and $g$ will get attached to resultant supernode ${1, 4,
2, 3}$.
Now our resultant contracted graph contains only two vertices so this is the minimum cut. We can remove edges $f$ and $g$ to disconnect the graph into two components which is also verified by
removing them in the original graph in the diagram shown below –
In DSU article we have seen how we can keep record of the sets to which a particular element belongs in almost constant time complexity.
We use the same concept here, to keep track of vertices that belongs to a particular supernode using $find(node)$ operation and contracting an edge $u\leftrightarrow v$ using $union(u, v)$ operation.
In the underlying pseudocode $edges$ denotes the list of all the edges of graph $G,$ $V$ and $E$ represents number of vertices and edges present in it respectively.
MinCut(edges, V, E):
While(vertices > 2):
i=Random integer in the range [0,E-1]
if(set_1 != set_2):
vertices = vertices-1
Union(u, v)
For(i in the range 0 to E-1):
if(set_1 != set_2):
ans = ans + 1
Return ans
Before seeing the code of Karger's algorithm let’s see the blueprint of the code $i.e.$ how we are implementing it, what are data members and methods we are using, etc.
Input –
A array/list of edges of the graph $G$, where $edge[i]$ is pair of vertices ${u,v}.$
Expected Output –
Size of minimum cut of the graph.
Data Members –
• $V$ – Represents the number of vertices in the graph.
• $E$ – Represents the number of edges in the graph.
• $parent$ – A integer type array, (Please refer implementation of DSU for detailed explanation)
• $rank$ – A integer type array, (Please refer implementation of DSU for detailed explanation) Methods –
• minCut – It is the main function used to find the minimum cut of the graph by contracting the edges of the graph $G$ till we have been left with only two vertices.
• Find – It is used to check whether two nodes belong to the same supernode or not by checking if the leader of set to which vertices $u$ and $v$ belongs is same or not.
• Union – It is used to merge two nodes into one supernode by using the typical union function as discussed in the DSU article.
C/C++ implementation of Karger’s Algorithm
using namespace std;
// Edge class
class Edge{
// Endpoints u and v
// of the Edge e.
int u;
int v;
// Dummy Constructor
// Constructor for
// Initializing values.
Edge(int U,int V){
class MinCut{
// Declaring data members
// V, E, parent, and rank.
int V,E;
int *parent;
int *rank;
// Random module to get
// random integer values.
// Constructor
MinCut(int v, int e){
// Initializing data members.
parent=new int[V];
rank=new int[V];
// Initializing parent's and
// rank's by is and 0s.
for(int i=0;i<V;i++)
// Function to find minimum cut.
int minCutKarger(Edge edges[]){
int vertices=V;
// Iterating till vertices are
// greater than 2.
// Getting a random integer
// in the range [0, E-1].
int i=rand()%E;
// Finding leader element to which
// edges[i].u belongs.
int set1=find(edges[i].u);
// Finding leader element to which
// edges[i].v belongs.
int set2=find(edges[i].v);
// If they do not belong
// to the same set.
cout<<"Contracting vertices "<<edges[i].u<<" and "<<edges[i].v<<endl;
// Merging vertices u and v into one.
// Reducing count of vertices by 1.
cout<<"Edges needs to be removed - "<<endl;
// Initializing answer (minCut) to 0.
int ans=0;
for(int i=0;i<E;i++)
// Finding leader element to which
// edges[i].u belongs.
int set1=find(edges[i].u);
// Finding leader element to which
// edges[i].v belongs.
int set2=find(edges[i].v);
// If they are not in the same set.
cout<<edges[i].u<<" <----> "<<edges[i].v<<endl;
// Increasing the ans.
return ans;
// Find function
int find(int node){
// If the node is the parent of
// itself then it is the leader
// of the tree.
if(node==parent[node]) return node;
//Else, finding parent and also
// compressing the paths.
return parent[node]=find(parent[node]);
// Union function
void Union(int u,int v){
// Make u as a leader
// of its tree.
// Make v as a leader
// of its tree.
// If u and v are not equal,
// because if they are equal then
// it means they are already in
// same tree and it does not make
// sense to perform union operation.
// Checking tree with
// smaller depth/height.
int temp=u;
// Attaching lower rank tree
// to the higher one.
// If now ranks are equal
// increasing rank of u.
int main(){
// Define V and E beforehand
int V=5,E=7;
// Create an Object of
// class MinCut.
MinCut minCut(V,E);
// Make an array of edges by giving
// endpoints of the edge.
Edge *edge=new Edge[E];
// Finding the size of the minimum cut.
cout<<"Count of edges needs to be removed "<<minCut.minCutKarger(edge)<<endl;
return 0;
Java implementation of Karger’s Algorithm
// Importing necessary
// modules for Input/Output.
import java.util.*;
class MinCut{
// Declaring data members
// V, E, parent, and rank.
static int V,E;
static int parent[];
static int rank[];
// Random module to get
// random integer values.
static Random rand;
// Constructor
MinCut(int V, int E){
// Initializing data members.
parent=new int[V];
rank=new int[V];
rand = new Random();
// Initializing parent's and
// rank's by is and 0s.
for(int i=0;i<V;i++)
// Function to find minimum cut.
static int minCutKarger(Edge edges[]){
int vertices=V;
// Iterating till vertices are
// greater than 2.
// Getting a random integer
// in the range [0, E-1].
int i=rand.nextInt(E);
// Finding leader element to which
// edges[i].u belongs.
int set1=find(edges[i].u);
// Finding leader element to which
// edges[i].v belongs.
int set2=find(edges[i].v);
// If they do not belong
// to the same set.
System.out.println("Contracting vertices "+edges[i].u+" and "+edges[i].v);
// Merging vertices u and v into one.
// Reducing count of vertices by 1.
System.out.println("Edges needs to be removed - ");
// Initializing answer (minCut) to 0.
int ans=0;
for(int i=0;i<E;i++)
// Finding leader element to which
// edges[i].u belongs.
int set1=find(edges[i].u);
// Finding leader element to which
// edges[i].v belongs.
int set2=find(edges[i].v);
// If they are not in the same set.
System.out.println(edges[i].u+" <----> "+edges[i].v);
// Increasing the ans.
return ans;
// Find function
public static int find(int node){
// If the node is the parent of
// itself then it is the leader
// of the tree.
if(node==parent[node]) return node;
//Else, finding parent and also
// compressing the paths.
return parent[node]=find(parent[node]);
// Union function
static void union(int u,int v){
// Make u as a leader
// of its tree.
// Make v as a leader
// of its tree.
// If u and v are not equal,
// because if they are equal then
// it means they are already in
// same tree and it does not make
// sense to perform union operation.
// Checking tree with
// smaller depth/height.
int temp=u;
// Attaching lower rank tree
// to the higher one.
// If now ranks are equal
// increasing rank of u.
// Edge class
static class Edge{
// Endpoints u and v
// of the Edge e.
int u;
int v;
// Constructor for
// Initializing values.
Edge(int u,int v){
// Driver Function
public static void main(String args[]){
// Define V and E beforehand
int V=5,E=7;
// Create an Object of
// class MinCut.
MinCut minCut=new MinCut(V,E);
// Make an array of edges by giving
// endpoints of the edge.
Edge edge[]=new Edge[E];
edge[0]=new Edge(0,3);
edge[1]=new Edge(3,2);
edge[2]=new Edge(2,1);
edge[3]=new Edge(1,0);
edge[4]=new Edge(0,2);
edge[5]=new Edge(2,4);
edge[6]=new Edge(4,1);
// Finding the size of the minimum cut.
System.out.println("Count of edges needs to be removed "+ MinCut.minCutKarger(edge));
Output –
Contracting vertices 0 and 2
Contracting vertices 1 and 0
Contracting vertices 0 and 3
Edges needs to be removed -
2 <----> 4
4 <----> 1
Count of edges needs to be removed 2
Complexity of Karger’s Algorithm
The time complexity of Karger's algorithm for a graph $G$ with $V$ vertices and $E$ edges when it is implemented by using the most optimized DSU approach is $O(E*\alpha(V))$ because in every
iteration we are contracting the edges in the contracted graph.
Now as we learned in DSU for $V<10^{600},$ $O(\alpha(V))\simeq O(1)$. So we can write $O(E*\alpha(V))$ $=$ $O(E)$ and since maximum number of edges in a graph is order of $V^2$ therefore in terms of
$V$ time complexity can be rewritten as $O(V^2)$.
The probability that cut produced by Karger's algorithm is the required min-cut of the graph is greater than or equal to $1/n^2$ which might seem very small.
We can improve this probability to an arbitrarily large probability, by repeating the algorithm several times and keeping track of values of size of cut found in each iteration.
To implement DSU we need an array of size $V$ hence the space complexity is $O(V)$.
Applications of the Minimum Cut
• It can be used to get an idea about the reliability of a network $i.e.$ smallest number of links failure of which will lead to failure of the entire network.
• In wartime, to know what is the minimum number of roads needed to be destroyed to block connectivity of an area from other parts of the enemy nation.
• It is used in image segmentation $i.e.$ separating foreground and background of an image.
• The minimum number of edges needed to be removed to disconnect a connected, unweighted, undirected graph into two components denotes the size of the minimum cut of the graph.
• Karger's algorithm is a randomized algorithm that takes $O(V^2)$ time to find the minimum cut of a graph with a certain probability.
• Minimum cut finds its application in various domains such as network optmizations, image segmenation etc. | {"url":"https://www.scaler.in/kargers-algorithm-for-minimum-cut/","timestamp":"2024-11-09T13:48:16Z","content_type":"text/html","content_length":"102758","record_id":"<urn:uuid:f66b5d15-df5b-42f2-98fc-66e05a78d795>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00723.warc.gz"} |
What is Gradient Descent in Machine Learning - AK codes
What is Gradient Descent in Machine Learning
Gradient descent (GD) is an algorithm in machine learning for finding the optimal values for weights in neural networks. Furthermore, it works by iteratively updating these weights in the direction
of steepest descent of the loss function until it reaches its minimum.
In this article we’re going to dive deeper into the concept of gradient descent and its importance in machine learning.
How does it work?
GD plays a crucial role in the training process of a neural network. But let’s take a step back first and see where it comes in.
Before we can start optimizing the weights, we need to initialize them first. We can usually do this by setting random values with Gaussian distribution. After initialization we can start processing
training data.
Each training sample, or a batch of training samples, will cause backpropagation algorithm that uses GD in order to slightly adjust weights to lower loss function.
In other words, GD acts as a compass which leads backpropagation algorithm to iteratively adjust weight values. This process repeats until the loss function, which is a measure for error, reaches its
minimal value.
Types of GD algorithms
Types we’re going to mention here only differ based on the number of training examples each iteration processes.
Mini batch gradient descent
This is the most common type of GD algorithm we use today, because of its efficiency. In essence, it takes a batch of training examples and updates weights based on the average of that batch.
It is efficient because it takes less calculations to update weights with each passing epoch without impacting end performance of the model.
In other words, we update weights fewer times, because we do it for each batch of examples rather than for each example individually.
Stochastic gradient descent
This type of GD updates weights for each passing training example. Therefore, it takes more calculations and also more time for each epoch. In addition, the updates it makes to weights can be noisy
due to the randomness of the individual examples.
However, it is simpler to understand, when we’re first getting familiar with the training process of neural networks. And it may be even a better choise whenever we’re limited with computing power of
our computer in cases of extremely complicated neural network architectures.
To conclude, gradient descent is an essential optimization algorithm in machine learning, we use to find the best set of weight values in neural networks. It works by updating them iteratively bit by
bit by looking at the value of loss function.
I hope this article helped you gain a better understanding about gradient descent algorithms and perhaps even inspire you to learn even more. | {"url":"https://ak-codes.com/gradient-descent/","timestamp":"2024-11-10T11:28:30Z","content_type":"text/html","content_length":"118997","record_id":"<urn:uuid:24fdd52e-eaeb-4903-a4a6-5ce6de8b3551>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00345.warc.gz"} |
ATAM 1160 - Mathematics-Algebra
Nov 12,
ATAM 1160 - Mathematics-Algebra
Credit Hours: 2.00
Prerequisites: ATAM 1150 or ATAM 1350 or consent of apprenticeship coordinator
This class covers fundamental operations of positive and negative numbers, grouping symbols, algebraic axioms, equations, special products and factoring. It includes the solution of
practical shop problems.
Billable Contact Hours: 2
Search for SectionsTransfer Possibilities
Michigan Transfer Network (MiTransfer) - Utilize this website to easily search how your credits transfer to colleges and universities.OUTCOMES AND OBJECTIVES
Outcome 1: Upon completion of this course, the learner will be able to describe applications in order of operations in algebraic language.
1. Use signed numbers, exponents and square roots in algebraic expression, solve problems.
2. Use the proper order of operations for adding and subtracting algebraic expressions, solve problems.
3. Use the proper order of operations, solve simple algebraic expressions.
Outcome 2: Upon completion of this course, the learner will be able to describe applications in solving algebraic equations.
1. Use the proper order of operations, solve algebraic equations involving two operations.
2. Use the proper order of operations, solve for algebraic equation factoring and changing priorities.
3. Use the proper order of operations for solve industrial formulas.
Outcome 3: Upon completion of this course, the learner will be able to describe applications in solving algebraic expressions.
1. Use algebraic operations, solve industrial word problems.
2. Use the proper order of operations in algebra, solve problems with multiplication and division of positive and negative numbers and exponents.
3. Use the proper order of operations in algebra, solve problems with scientific notation, unit conversion and system equations.
COMMON DEGREE OUTCOMES (CDO)
• Communication: The graduate can communicate effectively for the intended purpose and audience.
• Critical Thinking: The graduate can make informed decisions after analyzing information or evidence related to the issue.
• Global Literacy: The graduate can analyze human behavior or experiences through cultural, social, political, or economic perspectives.
• Information Literacy: The graduate can responsibly use information gathered from a variety of formats in order to complete a task.
• Quantitative Reasoning: The graduate can apply quantitative methods or evidence to solve problems or make judgments.
• Scientific Literacy: The graduate can produce or interpret scientific information presented in a variety of formats.
CDO marked YES apply to this course:
Communication: YES
Critical Thinking: YES
Quantitative Reasoning: YES
Scientific Literacy: YESCOURSE CONTENT OUTLINE
1. Signed Numbers - Exponents and Square Roots
2. Algebraic Language; Order of Operations
3. Adding and Subtracting Algebraic Expressions; Like Terms
4. Solving Simple Equations
5. Equations Involving Two Operations
6. More Equations; Removing Parentheses; Factoring
7. Solving Formulas
8. Solving Word Problems
9. Multiplying and Dividing Algebraic Expressions; Positive and Negative Exponents
10. Multiplying Similar Binomials by Inspection
11. Scientific Notation; Conversions with Decimal; Multiplying and Dividing
12. Unit Conversions
13. Systems of Equations; Solution by Substitution; Dependent and Inconsistent Systems
Primary Faculty
Richter, Lisa
Secondary Faculty
Gordon, Victoria
Associate Dean
Jewett, Mark
Hutchison, Donald
Primary Syllabus - Macomb Community College, 14500 E 12 Mile Road, Warren, MI 48088
Add to Favorites (opens a new window) | {"url":"https://ecatalog.macomb.edu/preview_course_nopop.php?catoid=88&coid=90176","timestamp":"2024-11-12T18:27:57Z","content_type":"text/html","content_length":"25725","record_id":"<urn:uuid:56099770-487f-40a5-8221-5248d9ff256a>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00806.warc.gz"} |
Iterative methods
The equation $F(x)=0$ can be transformed into the form,
\[ $$\label{gl2} x=f(x).$$ \]
This can be used to seek the zeros of $F(x)$ by iteration.
\[ \begin{eqnarray}\label{gladd} x_{0} & & \mbox{ Starting value} \nonumber\\ x_{1} & = & f(x_{0}) \nonumber\\ x_{2} & = & f(x_{1}) \nonumber\\ & . & \nonumber\\ & . & \nonumber\\ x_{t+1} & = & f(x_
{t}) \qquad (t=0,1,2,\ldots) \end{eqnarray} \]
If the method converges, this iteration can lead arbitrarily close to the exact solution of the problem (within roundoff error),
\[ \lim_{t \to \infty} x_{t} \to x_{exact} \]
There is no guarantee that this method will converge. The convergence depends strongly from the way in which $F(x)=0$ is reformulated into $x=f(x)$. One formulation is $f(x)=F(x)+x$ however other
formulations can sometimes be found that converge better.
Consider the function $F(x)=x^{3}-x-5$ in the vicinity of $x=2$. Three reformulations of this equation into a form $x=f(x)$ are,
\[ a)\quad x=x^{3}-5 \qquad b) \quad x=\frac{5}{x^{2}-1} \qquad c) \quad x=\sqrt[3]{x+5} \]
Iterating the first two forms results in a diverging series while $c)$ converges quickly to a root of $x^{3}-x-5=0$. A condition for convergence is $\left|\frac{df}{dx}\right| < 1$ at the root.
Code that iterates these equations is given below. | {"url":"http://lampz.tugraz.at/~hadley/num/ch5/5.2.php","timestamp":"2024-11-05T22:13:48Z","content_type":"text/html","content_length":"13500","record_id":"<urn:uuid:70684efb-cd72-45b0-b151-06d2670fcf3d>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00575.warc.gz"} |
Direct and Inverse Proportions - Solutions 2
CBSE Class –VIII Mathematics
NCERT Solutions
CHAPTER - 13
Direct and Inverse Proportions (Ex. 13.2)
1. Which of the following are in inverse proportion:
(i) The number of workers on a job and the time to complete the job.
(ii) The time taken for a journey and the distance travelled in a uniform speed.
(iii) Area of cultivated land and the crop harvested.
(iv) The time taken for a fixed journey and the speed of the vehicle.
(v) The population of a country and the area of land per person.
Ans. (i) The number of workers and the time to complete the job is in inverse proportion because less workers will take more time to complete a work and more workers will take less time to complete
the same work.
(ii) Time and distance covered in direct proportion.
(iii) It is a direct proportion because more area of cultivated land will yield more crops.
(iv) Time and speed are inverse proportion because if time is less, speed is more.
(v) It is a inverse proportion. If the population of a country increases, the area of land per person decreases.
2. In a Television game show, the prize money of Rs.1,00,000 is to be divided equally amongst the winners. Complete the following table and find whether the prize money given to an individual winner
is directly or inversely proportional to the number of winners:
Ans. Here number of winners and prize money are in inverse proportion because winners are increasing, prize money is decreasing.
When the number of winners are 4, each winner will get =
When the number of winners are 5, each winner will get =
When the number of winners are 8, each winner will get =
When the number of winners are 10, each winner will get =
When the number of winners are 20, each winner will get =
3. Rehman is making a wheel using spokes. He wants to fix equal spokes in such a way that the angles between any pair of consecutive spokes are equal. Help him by completing the following table:
(i) Are the number of spokes and the angles formed between the pairs of consecutive spokes in inverse proportion?
(ii) Calculate the angle between a pair of consecutive spokes on a wheel with 15 spokes.
(iii) How many spokes would be needed, if the angle between a pair of consecutive spokes is
Ans. Here the number of spokes are increasing and the angle between a pair of consecutive spokes is decreasing. So, it is a inverse proportion and angle at the centre of a circle is
When the number of spokes is 8, then angle between a pair of consecutive spokes =
When the number of spokes is 10, then angle between a pair of consecutive spokes=
When the number of spokes is 12, then angle between a pair of consecutive spokes=
(i) Yes, the number of spokes and the angles formed between a pair of consecutive spokes is in inverse proportion.
(ii) When the number of spokes is 15, then angle between a pair of consecutive spokes=
(iii) The number of spokes would be needed =
4. If a box of sweets is divided among 24 children, they will get 5 sweets each. How many would each get, if the number of the children is reduced by 4?
Total number of sweets = 120
If the number of children is reduced by 4, then children left = 24 – 4 = 20
Now each child will get sweets =
= 6 sweets
5. A farmer has enough food to feed 20 animals in his cattle for 6 days. How long would the food last if there were 10 more animals in his cattle?
Ans. Let the number of days be
Total number of animals = 20 + 10 = 30
Here the number of animals and the number of days are in inverse proportion.
Hence the food will last for four days.
6. A contractor estimates that 3 persons could rewire Jasminder’s house in 4 days. If, he uses 4 persons instead of three, how long should they take to complete the job?
Ans. Let time taken to complete the job be
Here the number of persons and the number of days are in inverse proportion.
Hence they will complete the job in 3 days.
7. A batch of bottles was packed in 25 boxes with 12 bottles in each box. If the same batch is packed using 20 bottles in each box, how many boxes would be filled?
Ans. Let the number of boxes be
Here the number of bottles and the number of boxes are in inverse proportion.
Hence 15 boxes would be filled.
8. A factory requires 42 machines to produce a given number of articles in 63 days. How many machines would be required to produce the same number of articles in 54 days?
Ans. Let the number of machines required be
Here the number of machines and the number of days are in inverse proportion.
Hence 49machineswould be required.
9. A car takes 2 hours to reach a destination by travelling at the speed of 60 km/hr. How long will it take when the car travels at the speed of 80 km/hr?
Ans. Let the number of hours be
Here the speed of car and time are in inverse proportion.
Hence the car will take
to reach its destination.
10. Two persons could fit new windows in a house in 3 days.
(i) One of the persons fell ill before the work started. How long would the job take now?
(ii) How many persons would be needed to fit the windows in one day?
Ans. (i) Let the number of days be
Here the number of persons and the number of days are in inverse proportion.
(ii) Let the number of persons be
Here the number of persons and the number of days are in inverse proportion.
= 6 persons
11. A school has 8 periods a day each of 45 minutes duration. How long would each period be, if the school has 9 periods a day, assuming the number of school hours to be the same?
Ans. Let the duration of each period be
Here the number of periods and the duration of periods are in inverse proportion.
Hence duration of each period would be 40 minutes. | {"url":"https://mobile.surenapps.com/2020/09/direct-and-inverse-proportions_1.html","timestamp":"2024-11-14T04:10:48Z","content_type":"application/xhtml+xml","content_length":"122298","record_id":"<urn:uuid:9074f59e-8272-4c77-8c00-2220bac8db5c>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00669.warc.gz"} |
ear Reserves Study
Create a 30-year Reserves Study using Excel
A reserves study looks ahead 30 years. It lists reserves items (components) and details the cost and schedule for maintenance, repair and replacement of each.
I embarked on this project to mirror and to be able to edit a professionally done reserves study for my Homeowners Association (HOA)—a formal name for a condomoinium governing structure. I want to
review that study and be able to make corrections and vary payment scenarios to inform future versions of the same study.
That professional study was, I believe, created with Reserve Study HOA software, which can be purchased. It could be a time saver, but my interest is to fully understand the formulas and connections
that make up a study without the expense—thus this creation of an Excel version.
The pages of our workbook
We create an Excel workbook with pages (worksheets) as shown above. These pages list the reserves components in detail and different pages have costs calculated in various ways.
Each worksheet will be explained as we go along. You could set up those worksheets now.
I have put together an abbreviated model study that will work if you have Excel. I added a couple of dummy items to show results on the various pages. The last page in the workbook has all the
formulas and instructions for where to use them. Click below to download the workbook. You can also open the formulas page, which is also included in the workbook.
The Cash Flow page (spreadsheet)
The key details of a reserves study are summed up on the Cash Flow page, where we can see at a glance the degree to which anticipated costs are covered by expected income. It shows, year-by-year, the
money coming in and going out for the maintenance, repair and replacement of items serviced on a regular basis over the upcoming thirty years.
The Cash Flow page has items entered into the columns. Some columns are total amounts pulled in from other pages (relative references), and some are calculations made with elements of the Cash Flow
Columns B, E and G—Current Cost, Annual Expenditure and Fully Funded—are totals pulled in from other spreadsheets in the workbook.
Since the Cash Flow page is the great summing-up of all the other information we have yet to establish on other spreadsheets within the workbook, we will return to Cash Flow toward the end of this
article once we understand the elements that feed into it.
• This study example begins at year 2024, but there are a couple of items from 2023 that are refered to in the page's formulas, so those amounts are listed.
• Current Cost entries are totals that will be pulled in from the Current Cost page. The Current Cost of a component is what servicing that item would cost in the current year, even though the item
may not be scheduled for many years to come.
• Annual Contribution is an amount set by the HOA board. It is the main income into the reserves account.
• Annual Interest is calculated from the amount in the reserves account and is a small part of the yearly income into the reservces account.
• Project Ending Reserves is the total in the reserves account after all yearly income is reported and the expenses are paid.
• Fully Funded amounts are yearly totals pulled in from the Fully Funded page. It is the cumulative amount to date that would be in the account were all the items fully paid to date, with the
understanding that equal amounts, with inflation costs added, will continue to be contributed before the item is actually serviced. I find this the trickiest item to understand.
• Percent Funded is very useful for understanding the readiness of the HOA to pay for the reserves items. A figure right around 60% - 70% is cited as being good. Higher levels of funding are even
better but rare in the real world.
• Contribution % Change is not always on a cash flow page. It is useful for tracking year-by-year changes to the HOA dues—the Annual Contribution—and can help in creating scenarios for future
The Parameters Worksheet (Page)
A first page to set up is Parameters, a "source of truth" page where we enter amounts that will be used in formulas.
The Inflation Rate is used in many formulas, and the Interest Rate and Taxable Rate are used in formulas on the Cash Flow page.
Set year today for planning
Many of the formulas rely on subtracting one year from another, and one of those years is often the "current year," which should be the year of the study—in our case 2024. But the reserves study is a
planning document, and so is often put together in the year before the start of the study.
To make the study accurate for year 1 of the study, the year in which it is planned to begin, I use the year entered in "Set year today for planning" as a relative reference.
If you're familiar with the Excel YEAR(TODAY()) function, this would work once the actual year is 2024, but for planning that would read as, in this example, 2023, and our formulas would be off by
one year.
Those first four items, Inflation Rate, Interest Rate, Taxable Rate, and Current Planning Year are the key ones for the Parameters page. I add the amounts in the "Study" part of the page as a
reference for some key calculations on the Cash Flow page. These are amounts from the year previous to the study.
The Original Components Page
Original Components is, like Parameters, a "source of truth" page. Each component of the study is listed along with its attributes. These are the key amounts that are referenced throughout the
workbook. A change here will ripple through the workbook results.
Shown below are all the columns and some of the rows of the Original Components page. The page will be whatever length is needed to list all the components.
This list is sorted first by replacement year; then, within any given year, the component names are listed alphabetically.
An ultimate goal of this page is to determine the Current Cost of each component.
Before looking at that cost calculation, let's focus on how the various time attributes are attached to each component.
Date in Service and Date in Service Cost
When a component is serviced, a record is kept of the year and the cost. This is recorded in columns B and C, Date in Service and Date in Service Cost.
We'll skip, for now, columns D and E to look at Useful Life, Adjustment, and Remaining Life.
Useful Life
The Useful Life of the component could be based on how long the component actually functioned, product information, a warranty (such as a roofing warranty), some requirement such as a required test
of the item, or a manufacturer's or contractor's information or best guess,.
Adjustment is some change, either more years or fewer, to the Useful Life.
For example, a warranty on a roof may expire while it is still in good shape so it's replacement is adjusted to some number of years into the future. Maybe the board decides to move up the painting
of the lobby. Perhaps a part doesn't last as long as expected (a negative adjustment).
Another example: Sometimes it is determined or discovered that an item has gone without being listed or serviced. That item will ge given a reasonable Useful Life, and the elapsed time when it was
not listed or considered is added as an Adjustment.
Adjustments may be set to 0 or left blank once an item is serviced and we know the updated actual cost and useful life.
Calculating the Replacement Year (Column D)
The Replacement Year is when the item is scheduled to be serviced. We use a formula to do this:
Replacement Year = Date in Service + Useful Life + Adjustment
On our spreadsheet the first example of this is:
=B3 + F3 + G3
That formula is then dragged down the column to give the Replacement Year for each item.
Calculating the Remaining Life
We use the Replacement Year and the Planning Year from our Parameters page to calculate the Remaining Life
Remaining Life = Replacement Year - Planning Year
On the spreadsheet our first example is:
That is dragged down the column to give all the results. The reference to the planning year is an absolute reference. Using the dollar signs mean that the formula will always use what is entered on
that page in that column and that row in the formula. The first item will always refer to Column D but the row reference will change as the formula is dragged down the column.
Calculating Current Cost with Date In Service cost
With good records, we can almost always get our current cost from the information that has been presented.
What do we mean by Current Cost?
The current cost is the cost of any component were it to be done in a given year.
Current Cost = Date in Service Cost * (1 + Inflation Rate) ^ (Current Planning Year - Date in Service)
We have the Date in Service Cost in Column C. This amount is multiplied by 1 plus the Inflation Rate listed on the Parameters page. This is raised to the power of, from Parameters, the Current
Planning Year minus the Date in Service. The caret ^ symbol in Excel means "to the power of."
The Excel formula for the first iteration of this—in our example column C and row 3—is:
=$C3 + (1 + Parameters!$C$2) ^ (Parameters!$C$5 - $B3)
The first few items in the Replacement Year column are due to be serviced in 2024 which, for our purposes at the time of this writing, is also the current planning year. When the next planning year
rolls around, having serviced the item in the previous year, we should update that item's Date in Service to the year in which it was done and the Date in Service Cost to what was the actual cost
that was paid. Having serviced that item, the cycle of that component's Useful Life begins anew.
Having covered this method of arriving at the Current Cost, I must emphasize that THIS IS NOT THE RECOMMENDED METHOD TO DO THIS. However, it is useful to set up an additional column for the page and
run that formula as a check against the method we get into next.
The Current Cost can be derived from actual unit costs and take into consideration how many units there are and any other qualifier.
These attributes of the component are in other columns of the Original Components page.
Before leaving this section, we can note that there is another way to derive the "power of" number.
Insted of
^ (Parameters!$C$5 - $B3)
Current Planning Year minus the Date in Service year we get the same result with
(Useful Life + Adjustment)
which in the first iteration is:
=$C3 + (1 + Parameters!$C$2) ^ ($F3 + $G3)
This can be set up as a supplemental column as a check of the recommended formula.
Calculating Current Cost with Unit Cost and Cost Details
A more straightforward method to arrive at our current cost is to use the actual cost of a unit with consideration of how many units there are and any other qualifier that might affect the actual
Here is another part of the Original Components page.
The first item in our example is "Asphalt Pavement - Clean and Seal."
This item has the Quantity of 3,660 square feet at a cost of 54 cents per square foot. This is the current square foot cost, not some historical amount. At least, that is the best guess. When the
work is actually done, we will learn the true cost, but this is the cost we're using for planning our expense.
Our formula for calculating current cost from the Date in Service cost involved the inflation rate. Working with Unit Cost, that should already be factored in, so the formula is straightforward.
Current Cost = Unit Cost * Quantity
and this first iteration at the Current Cost cell is:
=M3 * L3
We must also consider if there is a qualifier. There is an example on row 23 and is for "Concrete Pavement - Repair Allowance." It is calculated using the full cost of replacing all the concrete
Pavement—all 3,380 square feet of it—but since this is just for some repair, it was decided that 6% of that total cost would cover a typical repair.
The formula for this is:
Current Cost = (Unit Cost * Quantity) * Qualifier
In this case, the item is on row 23 so that formula at thata row is:
=(M23 * L23) * J23
The quantity and the qualifier are fixed amounts. The Unit Cost is variable year to year and should change with inflation.
We want one formula that will work in every case. We could use the last formula given above, but I prefer to check if there is a qualifier and run the one formula, and if the qualifier cell is blank
run the other. The following will work in the first Current Cost cell and can then be dragged down the column to make each calculation and it will work whether or not there is a qualifier.
=IF(ISNUMBER(J3), (K3*M3)*J3, K3*M3)
The formula checks if there is a qualifier and, if there is one, runs the formula that takes it into account. Otherwise the formula that multiplies the unit cost times the number of units is used.
The unit cost, like all our expenses, is affected by inflation. When preparing the next year's reserves study, these unit costs should be increased to reflect the inflation rate.
Focusing on the items to be done in the current year (in our example 2024), we see the previous Date in Service listed. I like having this information.
When an item is serviced in the current year, the Remaining Life is 0. The items serviced in the current year could just as well be listed with dates in service of the current year (example 2024) and
with what we arrive at as the current cost in Date in Service Cost.
Let's investigate those two similar formulas.
Using the older Date in Service at the first component example the formula is:
=1772 * (1 + .037) ^ (2024 - 2021)
And the result will be $1,976 as the current cost. Were we to change the Date in Service item to reflect the year we are in (current planning year 2024) with the 2024 cost the formula will look like
=1976 * (1+ .037) ^ (2024 - 2024)
And the outcome of that is also 1976. Yes, an item to the power of 0 (2024 - 2024) is itself and not, as in multiplication, 0.
The formal reserves study for our HOA that I am reflecting here uses the latter method of listing, but I like seeing the timing of the previous iteration.
So...our method of arriving at Current Cost is to use the unit cost plus any quantity and qualifier.
One might add the alternate method in an additional column. The result should be the same, and if they are not, some troubleshooting can resolve the discrepancy.
With Current Cost calculated, we can add a sum of that column. We will see where this is useful when we create a spreadsheet that calculates all the current costs over the study's full 30 years.
Calculating First Future Cost
There is a column on our Original Components page that we haven't touched—First Future Cost.
First Future Cost is what any item is expected to cost when the year for servicing that item—the replacement year—rolls around.
We have caluclated the current cost using the unit cost and any quantity and qualifier. We use that Current Cost when calculating First Future Cost.
Calculate First Future Cost using Current Cost
The formula is based on this:
Current Cost * (1 + Interest Rate) ^ (Replacement Year - Current Planning Year)
The first instance is:
=$E3 * (1 + Parameters!$C$2) ^ ($D3 - Parameters!$C$5)
That can be dragged down the page to fill in for each component.
Before we're done, these items will have their own pages where we can see the costs of any item over the entire 30-year scope of the reserves study.
We can sum this column to compare with the results we get when we set up our 30-year look at costs.
Unfunded Components
Sometimes we want to keep a component listed but we don't want to count it in our totals. A few example items are shown below.
The first item is a scheme that has been formulated but not implemented by the board.
A couple of others are up for review and will probably be deemed unnecessary and ultimately removed from the list of components.
The Elevator Modernization is an item scheduled for way into the future. Typically, that would be funded on an ongoing basis. Because our building is engaged in a complete re-doing of our plumbing,
quite an expensive process, payments for the future elevator mondernization are not being made. Once the plumbing is completed, this item will be funded and those payments will be made in a shorter
period than 30 years.
By keeping the details of these items, they can be properly calculated into our totals should we wish to do so. By not entering an amount in First Future Cost we keep those costs from appearing in
calculations that will include all the other components.
Cross-factoring the Original Components
I emphasized the calculation of Current Cost from the Unit Cost and details. Some cross-checking formulas may help troubleshoot inconsistencies.
I separate these by an empty column and add them to the Original Components page so that descrepancies are easy to spot.
Let's run through those in order.
Calculate Date in Service from Replacement Year.
We originally entered Date in Service as its own date based on records. That can also be a formula:
Date in Service Year = Replacement Year - Useful Life - Adjustment
and the first iteration is:
Had this formula been used, we would get a circular reference error.
We could simply check the column entries against each other. Instead, in an adjoining column, we set up an IF() clause and use the Excel TRUE() function- here is the first iteration of that:
=IF($O3 = $B3, TRUE())
Dragging this down the column should give us all TRUE responses. Otherwise there is some troubleshooting to be done.
Calculate the Current Cost from the Date in Service Cost component life details
We calculated the Current Cost from the Unit Cost and any Quantity and Qualifier. Here we use life details—Useful Life, Adjustment, and Remaining Life—to calculate the current cost.
Current Cost = Date in Service Cost * (1 + Inflation Rate) ^ (Useful Life + Adjustment - Remaining Life)
And its first iteration is:
=$C3 * (1 + Parameters!$C$2)^ ($F3 + $G3 - $H3)
We set up a check of the results with:
=IF(Q3=E3, TRUE())
Calculate the First Future Cost using life details
By not subtracting the Remaining Life in the previous equation we calculate the First Future Cost.
=$C3 * (1 + Parameters!$C$2)^ ($F3 + $G3)
And we double-check that against the previously derived First Future Cost with:
=IF(I3=S3, TRUE())
Calculating the Unit Cost when we know only the Date in Service Cost
We entered a Unit Cost as a whole number and used that to calculate the cost of an item. It is useful to calculate that cost from other component attributes and to check our results.
The Unit Costs listed as whole numbers are the costs for the current year, even though the component may not be scheduled until future years.
If we don't have to factor a Quantity (the Quantity is 1) or a Qualifier (there is no qualifier) for our result, the Unit Cost will be the same as the Current Year cost.
Otherwise, we need to get any Quantity and Qualifier to come up with the Unit Cost. Those two factors will be fixed numbers, not variables, so once those are determined we can calculate the Unit
Calculate the Unit Cost with a Quantity and Qualifier from the Date in Service Cost
If there is a Quantity but no Qualifier, the calculation of the Unit Cost is:
Unit Cost = Date in Service Cost * (1 + Inflation Rate) ^ (Useful Life + Adjustment - Remaining Life) / Quantity
=$C3 * (1+Parameters!$C$2)^($F3 + $G3 - $H3) / $K3
And if there is also a Qualifier we divide by that Qualifier before we divide by the Quantity
(Date in Service Cost * (1 + Interest Rate) ^ (Useful Life + Adjustment - Remaining Life / Qualifier) / Quantity
=($C3 * (1+Parameters!$C$2)^($F3 + $G3 - $H3) /$J3) / $K3
But we want one formula that we can drag down the column and give correct results. For that we check if there is a qualifier and run one formula if there is, and if there is no qualifier we run the
simpler one.
=IF(ISNUMBER($J3),($C3 * (1+Parameters!$C$2)^($F3 + $G3 - $H3) /$J3) / $K3,$C3 * (1+Parameters!$C$2)^($F3 + $G3 - $H3) / $K3)
This is an especially tricky number to arrive at, so it is beneficial to check if it matches up with the number entered in the Unit Cost column.
=IF($U3=$M3, TRUE())
This formula can show changes as inflation affects the unit cost, so it is beneficial to refer to teh results when the study is updated to another year.
This wraps up our work on the Original Components page; it's time to categorize our components.
Categorized Components
The Original Components page is sorted by the year when a given item is to be serviced. For easy reference and understanding, we want to have the same information organized by categories, and for
this we create the Categorized Components spreadsheet.
This arrangement will serve as a template for other spreadsheets we will create.
Our column headings are the same as on the Original Components spreadsheet. Arranging the rows requires attention to detail.
Along with the information for each component, we want category headings, a row for category totals (though these aren't used in the example above), and some white space for separation.
The very first entry, Electrical System Maintenance, is on row 24 of the Original Components page. In the Description Cell we enter:
='Original Components'!A24
When we drag this across the row, we get all of the other amounts from that component entered into the respective cells as relative references; that is, the numbers on the Original Components page
are reproduced on this page.
Recall that the Original Components page is a "source of truth" Making changes there will be reflected here.
Our second item, Electrical System Sub-panels Replace (1), is from line 76 of the Original Componets page, so our entry in Description for this component is:
='Original Components'!A76
Dragging that across the row will enter the details from the Original Components page for that item.
It is a bit painstaking to get this page set up. You might wonder why not set up this page with the items along with the formulas we used at Original Components. One could do that.
I mentioned that I am reflecting the work as it is published in our prefessionally done study. That study began with the information as it appears in the Original Components page, but did not provide
this spreadsheet version of items in categories. Rather, they used a listing item by item in a series of plain text reports.
I also suggested that we might add our cross-reference formulas on the Original Components page. To put those here would make a bit of a mess of this page and move us away from the Original
Components page as a source of truth.
It is meticulous work to get the Categorized Components set up, but it is more useful for reference and sets a method for looking at our components that will be used on the pages we will be setting
Side note about refering to other pages
When we refer to, say, inflation rate on the Parameters page, we start that formula with =Parameters!, and here when we refer to the Original Components page we begin 'Original Components'!. The
subtle difference is the use of single quotes in the second example. This is to accommodate that the second example has a space in its name and to provide a correct reference. With no space in
Parameters, the single quotes are not used.
Summing up the Categorized Components costs
The formulas that give us all the numbers on this page are referenced from the Original Components page.
We want to sum those totals on this page and also check those sums against the same sums on the Original Components spreadsheet.
As we can see in the image with the highlighted box and the formula in the formula bar, we have summed up the total for the Current Cost items with the formula =SUM(E2:E127).
Likewise, the First Future Costs are summed up with =SUM(I2:I127). These numbers should match the same totals on the Original Components page, so I refer to those below these sums and see that they
match up. If they don't, we need to troubleshoot the problem.
If there is a discrepancy, a first step would be to see what the difference between the amounts is, then look for an item that matches that amount and see that it is being pulled into the Categorized
Components page correctly. Also, check that each component on the Original Components page has a row listing on this, the Categorized Components spreadsheet.
Other than the references entered row-by-row, sums are the only formulas used in this spreadsheet. In the examples, I did not sum up each category, though I created a space to do so. I don't find
this information particularly necessary, but if desired, there is no reason not to do it.
Were the categories to be summed up individually, we would then want to make sure that our totals for individual items are summed all together and do not include the category totals. It is a more
complicated sum formula. Then the category totals can also be summed and a total derived for them. The two should match up, and that might be a useful check on the spreadsheet totals.
The most complicated setting-up of this workbook is done. We will use this organization by categories on several spreadsheets.
The Expenditures spreadsheet
The Expenditures page answers the question "How much will we spend each year on reserves items?"
This sum of each year is entered on the Cash Flow page and is an important figure when calculating the amount left in the reserves in any given year.
Calculate expenditures in the current year
We use the same organization of items in categories as on the Categorized Categories page.
In cell A2 we write:
='Categorized Components'!A2
And then drag that down the first column to fill in all the Categorized Components listings.
When we set up the Categorized Components spreadsheet, we had to carefully arrange which row was refering to which row on the Original Components spreadsheet. For this page, we are refering to the
Categorized Components page, so we are back to the more simple practice of, for example, the third row refering to the third row, the fourth row to the fourth, and so on.
To set up the column headings we begin with the current planning year and extend this row until it matches the 30 years of our study. These years- column head years- are used in our formulas.
In each of those columns we calculate the Current Cost for any given item in any given year. But we want to show only those years when servicing any given item is scheduled to occur.
To calculate the current cost the formula is:
'Categorized Component'!Current Cost * (1 + Inflation Rate) ^ (Column Head Year - Parameters!Current Planning Year)
The first instance looks like this:
'Categorized Components'!$E3*(1+Parameters!$C$2)^(B$1-Parameters!$C$5)
When we drag this formula horizontally, all the cells will fill in the Current Cost for the year at the head of the column across the 30 years of the study.
All of those numbers are accurate, but we want those items to appear on the page only in their service year.
To do this, we subtract the planning year from whatever year is at the column head and divide that by the Useful Life. We check if there is a remainder or if there is no remainder. We are employing
the modulus or MOD()function.
Writing just that part of the formula
=MOD(B$1-'Categorized Components'!$D3,'Categorized Components'!$F3)=0
Returns FALSE. The formula is checking if the remainder is 0. The actual numbers being calculated are:
(2024-2026) / 10 = -.2
The remainder is not 0. For that first row, when we get to the first replacement year of 2026, the result will be 0 and the simple Modulus formula will return TRUE. Since each year has a different
useful life, results will vary row to row.
When that first item is calculated in the column with heading 2026 which, for that item, is the Replacement Year the numbers look like this:
(2026 - 2026 / 10) = 0
Likewise, in 2036, with the Useful Life at 10, the result will be 10 / 10 and once again we get no remainder- the modulus is 0.
When we put this together in the larger formula for calculating an actual value, we want the formula to run when the modulus is 0, otherwise we want the cell left blank, which in Excel is achieved
with double quotes "".
We write the formula to run only when the modulus is 0 and otherwise to leave a blank cell.
=IF(MOD(B$1-'Categorized Components'!$D10,'Categorized Components'!$F10)=0,'Categorized Components'!$E10*(1+Parameters!$C$2)^(B$1-Parameters!$C$5), "")
The page is arranged in categories and there is space for summing each category, a title row for each category, and a blank space in between for readability. If the full formula is dragged across
these cells, we'll get an error in those cells we want blank. To mitigate this, we wrap the IF() statement in an IFERROR() function and leave the cell blank when the error condition is met.
This is the formula used throughout the page down to the last named item:
=IFERROR(IF(MOD(B$1-'Categorized Components'!$D3,'Categorized Components'!$F3)=0,'Categorized Components'!$E3*(1+Parameters!$C$2)^(B$1-Parameters!$C$5), ""), "")
That formula can be dragged down the Column B. With all those cells highlighted, the can be dragged horizontally across the 30-year span of the study and we'll have the years filled in in their
service years.
That last part of the formula, the "", could be set to enter 0. In fact, it can enter anything placed as part of the final statement of the formula.
Were we to set it to 0, when the modulus calculation is FALSE we'd get a 0 in the cell. We might want to see that to confirm the cells that should run an error are doing so.
That will leave us with a lot of 0's on the page—not so great looking.
Hide zero values
In Excel Preferences, there is a "View" option where one can check/leave unchecked a box to show/hide zero values.
Once a formula is set up for a whole page, one can check this box to confirm that zero values show up where expected.
At that point one can choose not to show zero values or, as I did in the formula with the double quotes, to enter nothing in those cells.
What we ultimately want are the totals for each year. These totals will be used on our Cash Flow page.
The image shows a few columns at the bottom of our spreadsheet. The Totals row uses the SUM() function, SUM(B1:B127) in the first instance, and this can be dragged across the row to total each
In these examples I have not entered totals for each category. I left the space to do so, but I have no particular need for them.
Were we to do so, we would re-write the SUM() function to add just the individual items, and then we'd have the opportunity to sum the category totals as well. They should agree, so this might be a
useful check.
The Expenditures page makes it easy to see what is planned for any given year. Should one want to make changes, we do that on the Original Components page and those changes ripple through the
Categorized Components and then onto this page.
Current Cost
The Current Cost for each component was calculated on the Original Components page. That was the current cost for the current year only.
Here we want to show the current cost for the first and all subsequent years. The cost will change year to year, affected by the inflation rate.
We already made this calculation for the Expenditures spreadsheet, but we hid all but the service year and checked to satisfy the IF() statement that involved the modulus. Now we're back to the
simpler core of that calculation.
The Column Head Year is simply whatever year is at the top of the column on this Current Cost page.
Current Cost * (1 + Inflation Rate)^(Column Head Year - Current Planning Year)
In the first instance this is:
='Categorized Components'!$E3*(1+Parameters!$C$2)^(B$1-Parameters!$C$5)
That can be dragged down Column B, then across the full 30 years of the study.
In our example, substituting in the original cell's numbers, we have:
=30000 * (1 + .037) ^ (2024 - 2024)
That may look odd. The "power of" is equal to 0. That condition will return the original current cost with no inflation applied. The following year, (2025 - 2024) will return 1, and the inflation
rate will be applied once, and so on.
When the entire study is updated for next year, we can see here what the then-new "Current Cost" amounts should be.
Highlighting the actual expenditures amounts in the correct year
The image of the page has cells with the grey background and a thin border. You will remember that the Expenditures page listed only those items in the year when they are being carried out.
We use the Excel feature of conditional formatting to apply that background and border (or whatever other setting one might want) when the amounts on this spreadsheet match up with the amounts that
appear on the Expenditures page.
Conditionally format the expenditures years
We go to the Conditional Formatting section in the Excel editing options or toolbar.
We create the custom format to appear when the item in the Current Cost page matches the amount in the Expenditures page.
We do this for the full range of cells where the items costs are calculated.
This is just a helper to identify the years when the expenditures are scheduled to occur.
This conditional formatting is also used on the Fully Funded page.
We again set zeros not to show at the Excel menu: Excel -> Preferences -> View.
Fully Funded
The concept of Fully Funded took me a while to wrap my brain around, and then the formula...sheesh!
Let's say we have a reserves item scheduled for the current year at a cost of $9,000. The Fully Funded amount that should be in the reserves account is $9,000, just what we need to pay the cost.
A healthy reserves account may have around 60 - 70% of the fully funded amount for all items. An item come due will nonetheless be serviced because all items are not scheduled for any one year. The
reserves account is fluid regarding where the money actually is spent in any given year.
Now let's say that same item repeats on a 3-year schedule. Ignoring inflation, the following year Fully Funded amount is $3,000, 2 years later is $6,000, and then the actual cost of $9,000 is again
available in the third year.
We saw how the Current Cost increases year on year because of inflation. That happens to our Fully Funded amount too, so we don't simply divide any year by its Useful Life. Inflation is also worked
into the formula.
Here is the formula that we use to fill in all the cells:
=IFERROR(('Categorized Components'!$I3*((1+Parameters!$C$2)^(B$1-'Categorized Components'!$D3)))*IF((MOD(B$1-'Categorized Components'!$D3,'Categorized Components'!$F3))=0,1,MOD(B$1-'Categorized Components'!$D3,'Categorized Components'!$F3)/'Categorized Components'!$F3), 0)
Hmmm...yeah...that is one steaming pile of formula. Let's break it down.
We use the IFERROR() Excel function to enter a zero if there is an error. We will get an error where we have blank cells on the page, so we use the Excel option to not show zero's on the page to keep
our spacer cells clear when we drag the formula through them and they return an error.
If something doesn't seem right with the results on the page, we can choose to show the zero's to and then track down the problem.
IFERROR(value is not met, 0)
In the center of the formula is an IF() function. The references in that function are looking to the Categorized Components page.
IF(Modulus((Column Heading Year - Replacement Year divided by Useful Life)) = 0, enter 1, otherwise enter Modulus(Column Head Year - Replacement Year divided by the Useful Life) divided by the Useful Life).
To put that more simply, let's assume a Useful Life of an item of 10 years, and let's say we have had a year when the item is serviced. The next year when we calculate the full cost of the item, we
will multiply that by a modulus of 1 divided by the Useful Life of 10, so we will have 1/10 of that item's cost appear as the fully funded amount.
The following year will have a modulus of 2, so we will show 2/10 of the Current Cost. When we get to the service year, the modulus will be 0 (there is no remainder of 10/10) and dividing 0 by 10
will give us an error so we enter the numeral 1 in that case only. That is why if the IF() the modulus is 0, we enter 1.
That part of the formula will work no matter what the Useful Life is. It may be 18 years for some component, and we'll have fractions such as 1/18, 2/18, 3/18 building up to the service year when we
must jump in and enter one where the formula would otherwise give us 0/18 (no remainder / 18).
These items that supply the appropriate fractions are multiplying the first part of the formula, which is our Current Cost calculation for any year. We could refer to the entries on the Current Cost
page, but I prefer to keep the references to the Categorized Components page.
Fully Funded = First Future Cost * (1 + Parameters!$C$2)^(Column Year - Replacement Year)
'Categorized Components'!$I3*(1+Parameters!$C$2)^(B$1-'Categorized Components'!$D3)
This is similar to the Current Cost formula. Current Cost uses the Date in Service cost and Fully Funded uses the First Replacement Cost. Also, Current Cost uses (Column Year - Planning Year) and
Fully Funded uses (Column Year - Replacement Year).
We calculate sumes for each column. The first year's sum can be compared to the total on the Original Components page and any problems with our entries tracked down.
The entire row of totals is transposed on the Cash Flow page.
The Cash Flow spreadsheet
We have worked through all the pages and now return to get the summing up of everything and view the current financial condition on the Cash Flow page.
To fill in the Current Cost totals, the total Expenditures for each year, and the Fully Funded totals, we can copy the sums from those individual pages in the workbook and enter them in the
designated Cash Flow columns where they will be used in various calculations within the page itself.
Let's work through the individual columns
Beginning year
We can enter the first two years, highlight them and drag down to get the full 30 years of the study.
Current Cost
We use the tricky and touchy Excel TRANSPOSE() function to fill the Current Cost column.
We select the 30 cells in the Current Cost column and write
Then go to the Current Cost page and select the total for year we're planning for, in this case 2024, and drag across the 30 years of the totals row. A helpful hint is to click the Control key when
horizontal scrolling and the choice will be restricted to that row- no accidental going up or down and having the incorrect cells appear in the formula.
Complete this part of the formua with a parenthesis, then- VERY IMPORTANT- click Control -> Shift -> Enter. Clicking only Enter will not work.
I find getting the desired result the first time is tricky, but persist and we'll end up in the formula bar:
It does not work to type this (with your actual cell range) into the formula bar. One must do the Control -> Shift -> Enter.
Using a formula that makes a relative reference to the cells means that any change on the Current Cost spreadsheet totals will be carried onto the Cash Flow page.
One could also go down the column and enter, in the first case:
='Current Cost'!B$127
Followed by
'Current Cost'!C$127
and so on until all the cells are filled. I haven't figured a simpler way than TRANSPOSE() that also maintains the relative reference, but would be happy to learn one as TRANSPOSE() always gives me a
hard time.
Annual Contribution
Annual Contribution = Previous Year Contribution * (1 + Contriubtion % Change)
The first instance of this is the row 2024. It relies on the contribution from the previous year which we can bring in from the Parameters page with:
This amount is then used to calculate the annual contribution by applying whatever percentage we have entered in the Contribution % Change column.
That first instance can be entered and then dragged down the column to have the formula apply for the 30-year scope.
I find it useful to use the Contribution % Change to guide the budgeting for the Annual Contribution. One could also simply choose amounts year by year and change the Contribution % Change percentage
to reflect whatever numbers are entered. I'll cover that when we get to a look at that last column of the page.
Annual Interest
The formula for calculating the Annual Interest is:
Parameters!$C$3(1-Parameters!$C$4)(F2+(1*('Cash Flow'!$C3-'Cash Flow'!$E3)))
The first instance, which can then be draged down the column, is:
Parameters!$C$3*(1-Parameters!$C$4)*(F2+(1*('Cash Flow'!$C3-'Cash Flow'!$E3)))
I have also seen it recommended that the last part of that formula, (1('Cash Flow'!$C3-'Cash Flow'!$E3) be written (.5('Cash Flow'!$C3-'Cash Flow'!$E3)
Annual Expenditure
We want to get the yearly totals from the Expenditures page, so we use the TRANSPOSE() function as above on the Expenditures page and end up with a formula:
Project Ending Reserves
Project Ending Reserves = Previous Year's Ending Reserves + Annual Contribution + Annual Interest - Annual Expenditure
which in the first example that can then be dragged down the column is:
Recall that F2 is the project ending reserves from the year previous to the study which is brought into this page with Parameters!C9.
Fully Funded
This is our final TRANSPOSE() exercise and it should something like:
except that your range will probably be on a different row.
Percent Funded
Percent Funded is where we can gauge how we're doing. As mentioned earlier, one often sees 60% - 70% funded as fairly healthy. The formula to calculate this is:
Percent Funded = Project Ending Reserves / Fully Funded
In the first instance, which can then be dragged down the column, this is:
Contribution % Change
This is my own addition to the Cash Flow page. It is very useful for planning and for gauging how the rate of the Annual Contribution is increasing or decreasing.
We saw at Annual Contribution how a formula can be used so that the contribution reflects a percentage change entered in this column. If we simply want to enter figures in Annual Contribution, we can
do that and then have the change reflected in the Contribution % Change column with this formula:
Contribution % Change = 1 - (Annual Contribution / Previous Year's Annual Contribution
expressed in the first instance with:
And that's the Cash Flow page!
Rather than go through all the formulas again, I'll provide the following link for reference: | {"url":"http://johnrose-glass.com/blogs/composing-a-reserves-study","timestamp":"2024-11-06T05:01:08Z","content_type":"text/html","content_length":"91495","record_id":"<urn:uuid:e9d03d23-23e1-436b-ab03-be2253ec3cff>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00161.warc.gz"} |
BITRSHIFT Function: Definition, Formula Examples and Usage
BITRSHIFT Function
Are you working on a spreadsheet in Google Sheets and trying to figure out how to perform a bitwise right shift operation? If so, you’re in luck! In this blog post, we’re going to be talking about
the BITRSHIFT formula and how it can help you quickly and easily perform bitwise right shift operations in your spreadsheets.
But first, let’s talk about what a bitwise right shift operation is and why it’s useful. In computing, a bitwise right shift operation shifts the bits of a number to the right by a specified number
of positions and fills in the empty positions on the left with 0s. This can be incredibly useful for a variety of tasks, such as dividing a number by a power of two or shifting the bits of a number
to make room for new bits. In short, the BITRSHIFT formula in Google Sheets allows you to easily perform these operations right in your spreadsheets, making your data analysis tasks that much easier.
Definition of BITRSHIFT Function
The BITRSHIFT function in Google Sheets is a built-in formula that performs a bitwise right shift operation on a number. This operation shifts the bits of the number to the right by a specified
number of positions and fills in the empty positions on the left with 0s. For example, if the number has a value of 15 (1111 in binary) and you want to shift the bits to the right by two positions,
the result of the BITRSHIFT function would be 3 (0011 in binary). The BITRSHIFT function can be incredibly useful for performing tasks such as dividing a number by a power of two or shifting the bits
of a number to make room for new bits.
Syntax of BITRSHIFT Function
The syntax of the BITRSHIFT function in Google Sheets is as follows:
=BITRSHIFT(number, shift_amount)
To use the BITRSHIFT function, you simply need to enter the number that you want to perform the bitwise right shift operation on as the first argument, and the number of positions that you want to
shift the bits to the right as the second argument. For example, if you want to perform a bitwise right shift operation on the number 15 and shift the bits to the right by two positions, you would
enter BITRSHIFT(15, 2) into the formula bar in Google Sheets. This would then return the result of the bitwise right shift operation, which in this case would be 3 (0011 in binary).
It’s important to note that the BITRSHIFT function only works with numbers and will not accept text or logical values as arguments. Additionally, the numbers that you use as arguments must be within
the range of -2^53 to 2^53-1, otherwise the function will return an error. Other than that, the BITRSHIFT function is simple to use and can be a powerful tool for performing bitwise right shift
operations in your Google Sheets spreadsheets.
Examples of BITRSHIFT Function
1. To shift the binary representation of a number to the right by a certain number of bits, use the BITRSHIFT function as follows: =BITRSHIFT(A2, 3), where A2 is the cell containing the number to be
shifted and 3 is the number of bits to shift. This will shift the binary representation of the number in A2 three bits to the right, effectively dividing it by 2^3 (or 8).
2. To shift the binary representation of a number to the left by a certain number of bits, use the BITLSHIFT function in a similar way: =BITLSHIFT(A2, 4), where A2 is the cell containing the number
to be shifted and 4 is the number of bits to shift. This will shift the binary representation of the number in A2 four bits to the left, effectively multiplying it by 2^4 (or 16).
3. You can also use the BITRSHIFT and BITLSHIFT functions in combination with other functions to perform more complex calculations. For example, you could use the BITAND function to perform a
bitwise AND operation on two numbers, and then use the BITRSHIFT function to shift the result to the right by a certain number of bits. The following formula demonstrates this: =BITRSHIFT(BITAND
(A2, B2), 3), where A2 and B2 are the cells containing the numbers to be ANDed together and shifted. This will AND the two numbers together, shift the result to the right by 3 bits, and return
the final result.
Use Case of BITRSHIFT Function
1. One potential use of the BITRSHIFT function in Google Sheets is in a financial spreadsheet, where it could be used to quickly and easily divide large numbers by a power of two. For example, if
you had a column of numbers representing dollar amounts in cents (e.g. $10.25 would be represented as 1025), you could use the BITRSHIFT function to divide these numbers by 100 to convert them to
dollars. Simply use the formula =BITRSHIFT(A2, 6), where A2 is the cell containing the number in cents, and 6 is the number of bits to shift. This will shift the binary representation of the
number in A2 six bits to the right, effectively dividing it by 2^6 (or 64), and return the result in dollars.
2. Another potential use of the BITRSHIFT function in Google Sheets is in a database of employee information, where it could be used to quickly and easily extract specific bits of data from a binary
representation of an employee ID number. For example, if you had a column of employee ID numbers in binary format, you could use the BITRSHIFT function to extract the department code from the ID
number. Simply use the formula =BITRSHIFT(A2, 8), where A2 is the cell containing the employee ID number, and 8 is the number of bits to shift. This will shift the binary representation of the ID
number in A2 eight bits to the right, effectively dividing it by 2^8 (or 256), and return the department code as a number.
3. A third potential use of the BITRSHIFT function in Google Sheets is in a scientific or engineering spreadsheet, where it could be used to quickly and easily perform calculations involving binary
numbers. For example, if you had a column of numbers representing measurements in binary format, you could use the BITRSHIFT function to convert these measurements to a different unit of measure.
Simply use the formula =BITRSHIFT(A2, n), where A2 is the cell containing the measurement, and n is the number of bits to shift. This will shift the binary representation of the measurement in A2
to the right by the specified number of bits, effectively dividing it by the appropriate power of two, and return the result in the new unit of measure.
Limitations of BITRSHIFT Function
The BITRSHIFT function in Google Sheets has a few limitations to be aware of.
• First, it can only be used with binary numbers, not with decimal or hexadecimal numbers. This means that if you have a decimal or hexadecimal number that you want to shift, you will need to
convert it to binary first using the DEC2BIN or HEX2BIN function, respectively.
• Second, the BITRSHIFT function can only shift a binary number to the right by a certain number of bits. It cannot be used to shift a binary number to the left, or to perform any other type of bit
manipulation. If you need to perform these types of operations, you will need to use other functions such as BITLSHIFT, BITAND, or BITOR.
• Third, the BITRSHIFT function is subject to the same limitations as other functions in Google Sheets, such as a maximum limit on the number of characters that can be used in a formula, or a
maximum limit on the number of nested functions that can be used in a formula. This means that if you are using the BITRSHIFT function in a particularly complex or large spreadsheet, you may
encounter errors or limitations that prevent you from using the function as intended.
Overall, while the BITRSHIFT function can be a useful tool for working with binary numbers in Google Sheets, it is important to be aware of its limitations and to use it appropriately within the
context of your spreadsheet.
Commonly Used Functions Along With BITRSHIFT
Some commonly used functions that are often used in combination with the BITRSHIFT function in Google Sheets include the following:
1. BITLSHIFT: This function is used to shift the binary representation of a number to the left by a certain number of bits. It is commonly used in combination with the BITRSHIFT function to perform
both left and right shifts on a binary number in a single formula.
2. BITAND: This function is used to perform a bitwise AND operation on two binary numbers. It is commonly used in combination with the BITRSHIFT function to extract specific bits of data from a
binary number, or to combine multiple binary numbers into a single result.
3. BITOR: This function is used to perform a bitwise OR operation on two binary numbers. It is commonly used in combination with the BITRSHIFT function to combine multiple binary numbers into a
single result, or to set specific bits in a binary number to a particular value.
4. DEC2BIN: This function is used to convert a decimal number to its binary representation. It is commonly used in combination with the BITRSHIFT function to convert decimal numbers to binary and
then perform bit manipulation on the binary representation.
5. HEX2BIN: This function is used to convert a hexadecimal number to its binary representation. It is commonly used in combination with the BITRSHIFT function to convert hexadecimal numbers to
binary and then perform bit manipulation on the binary representation.
The BITRSHIFT function in Google Sheets is a useful tool for working with binary numbers in a spreadsheet. It allows you to shift the binary representation of a number to the right by a certain
number of bits, effectively dividing the number by a power of two. This can be useful for performing calculations involving binary numbers, or for extracting specific bits of data from a binary
The BITRSHIFT function has a few limitations, such as the fact that it can only be used with binary numbers, and it can only perform right shifts on a binary number. However, these limitations can be
overcome by using the BITRSHIFT function in combination with other functions such as BITLSHIFT, BITAND, or BITOR.
Overall, if you need to work with binary numbers in a Google Sheets spreadsheet, the BITRSHIFT function is a powerful and useful tool that can help you perform a variety of bit manipulation
operations quickly and easily. We encourage you to try using the BITRSHIFT function in your own Google Sheets to see how it can help you with your own calculations and data analysis.
Video: BITRSHIFT Function
In this video, you will see how to use BITRSHIFT function. Be sure to watch the video to understand the usage of BITRSHIFT formula.
Related Posts Worth Your Attention
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me number
For records concerning the number of known digits of constants like Pi, E, please see the nice Table of Mathematical Constants compiled by Steve Finch.
Take also a look at the Number Theory Seminar at IECN (Nancy).
Aliquot sequences
Famous conjectures
Records about prime numbers
See also the archives from the Number Theory List and the Number Theory Web from Keith Matthews. | {"url":"https://members.loria.fr/PZimmermann/records/","timestamp":"2024-11-07T19:53:49Z","content_type":"text/html","content_length":"1273","record_id":"<urn:uuid:f8e7c08b-d804-4718-a8fa-f23a5a9ed2ae>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00835.warc.gz"} |
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Running Coupling Constants
[Figure: D.I. Kazakov, hep-ph/0012288 p 12 .]
Coupling constants in the quantum field theories of the Standard Model (SM) are not constant. The couplings, which set the strength for the interactions, change their value if one probes smaller
distances with higher energies. This is due to contributions of virtual particles that cause a 'running' of the coupling with the energy scale. This energy scale is also referred to as the 'sliding
scale'. If one evaluates the necessary Feynman diagrams to compute this effect, it turns out that the couplings run logarithmically with the sliding scale, and their slope depends on the particle
The left side of the above plot shows the inverse of the three SM couplings α
as a function of the sliding scale Q (in GeV) [
see comment
]. The thickness of the lines depicts the experimental error (LEP data '91). The scale on the
-axis is logarithmic, such that the curves become straight lines. This running of the coupling constants has been experimentally confirmed in the accessible energy range, but the more interesting
thing here is that one can extrapolate the curves far beyond where we can test them experimentally. One sees then that these couplings form a triangle somewhere around 10
The plot to the right shows the running of the gauge couplings within the
Minimal Supersymmetric extension of the Standard Model (MSSM)
. Since the particle content with Supersymmetry (SUSY) is different, the slope of the curves changes. Interestingly, the result is that the gauge couplings meet almost exactly (within the errorbars)
in one point, somewhere around 10
GeV, usually referred to as the GUT scale (which isn't too far off the
Planck scale
The above depicted fit of the curves depends on the scale where SUSY is (un)broken. Below this energy, the running is according to the SM, and only changes above the SUSY breaking scale. This is why
in the plot to the right the curves have a kink and change slope around a TeV, which was assumed to be the SUSY breaking scale.
This calculation has first been done by Amaldi, de Boer and Fürstenau in 1991 (Phys. Lett. B 260 447-455, 1991), and this result is to be considered one of the most compelling arguments for SUSY.
This post is part of our 2007 advent calendar
A Plottl A Day
53 comments:
1. This plot should always have a health warning. The three lines shown unifying with supersymmetry are *not*, repeat are *not* those of the gauge couplings of the Standard Model. Specifically, the
U(1) coupling is GUT-normalised. The ordinary U(1)_Y gauge coupling will miss SU(3) and SU(2) by a factor 5/3. In general, there is no unique U(1) gauge coupling.
What the plot is evidence for is not the MSSM in general, but specifically SUSY-GUTs.
2. yes. thanks for your comment.
3. That's absolutely amazing! After running over 15 orders of magnitude in the energy scale! We are so used to seeing the second plot these days but I'd imagine the person who plotted it for the
first time must have been completely ecstatic. Of course, there are other good reasons for TeV scale SUSY, i.e. REWSB caused by the large top Yukawa, a natural dark matter candidate, stablization
of the hierarchy, etc but the precision gauge coupling unification is just so darn impressive. I can't believe some people still reject this and the other hints as evidence that the MSSM embedded
in a SUSY GUT is the real deal.
4. I've been horrendously busy of late, but have taken the time to read the latest string of excellent (and I mean excellent) posts. Thanks so much!
5. What would it take to check experimentally whether 1/Alpha_2 runs with a negative rather than positive slope?
6. well, as piscator mentioned above the alpha_i are are not actually the couplings of the SM gauge groups. alpha_2 is related to the quantities we know and like as alpha_2 = alpha/sin^2(theta)
where alpha is the fine structure constant (runs) and theta is the weak mixing angle. which is supposed to run, but the last time I heard about it, the data wasn't too convincing.
7. Hi Alex:
Well, you know, despite me writing this post I am not actually so impressed by the above. Probably exactly because as you say its an extrapolation over 15 orders of magnitude. I could imagine all
kind of things happening till we reach that energy scale. Call it naive but if it was a unification I'd kind of expect the couplings to run together towards some attractor, and not just meet in
one point. Best,
8. "Call it naive but if it was a unification I'd kind of expect the couplings to run together towards some attractor, and not just meet in one point."
Yes, this is very naive. These coupling constants don't exist above the unification energy as they are replaced by the gauge coupling of the unified group. It doesn't make any sense to keep
running the SM gauge couplings above the unification scale. One should run the gauge coupling of the unified group above this energy.
9. Yes, Eric, thanks. So, then these curves run together to one point and then they have a kink and proceed on one line all together? That's what I don't like about it. Best,
10. Bee,
At the unification scale, there is a discontinuity where the SM gauge groups with three independent coupling constants get replaced by a single gauge group with a single gauge coupling. It's not
that the SM gauge couplings run as a line after the unification scale. Above this energy, the SU(3), SU(2)_L, and U(1)_Y gauge coupling constants don't exist because the gauge group is no longer
SU(3)x U(2)_L x U(1)_Y. The unified gauge symmetry, e.g. SU(5), which is manifest above the unification scale has it's own gauge coupling and there are no others.
11. Eric: there is a discontinuity where the SM gauge groups with three independent coupling constants get replaced by a single gauge group with a single gauge coupling
Yes, Eric, thanks again. This is exactly what I don't like about it. I didn't say it is wrong, or doesn't work, or can't be the way nature is organized, I just said it doesn't impress me much.
12. "That's what I don't like about it"
Bee, could you please elaborate what exactly you don't like? The fact that they meet at a point and then run as a single unified coupling? As opposed to what?
I think one should really look at this backwards from high to low energy scale, i.e before the GUT is broken you have a single coupling. Then at around 10^16 GeV the GUT symmetry is spontaneously
broken and the couplings start running separately. What's your alternative?
13. Bee,
The discontinuity is no different than what happens in condensed matter when a material undergoes a phase change. In fact, this is exactly what's happening at the unification scale.
14. Hi Alex:
Well, if I had an alternative, I'd go post a paper with my GUT model :-) What I am trying to say is that I would like to see the different couplings run together towards one coupling and not just
the above extrapolation until the curves meet, and then, bang, the symmetry is unbroken and we have the GUT.
Hi Eric:
Yes. The necessity for 3 different discontinuities doesn't make me like it better though.
15. "...until the curves meet, and then, bang, the symmetry is unbroken and we have the GUT."
There is no "bang", it's just that at around 10^16 GeV the additional degrees of freedom, like the Higgs triplets, for instance, start to contribute into the running modifying the beta-function
coeffitients and force the couplings to run together as a single GUT coupling.
16. I have a question about this graph I've been meaning to ask for some time.
As I understand it, the logarithmic form of the running couplings follows from only considering 1-loop effects. Why should I believe that over 15 orders of magnitude in the scale factor, higher
order terms will be of no importance?
17. Hi Alex:
it's just that at around 10^16 GeV the additional degrees of freedom, like the Higgs triplets, for instance, start to contribute into the running modifying the beta-function coeffitients and
force the couplings to run together
It's just that I don't find that scenario particularly compelling. You don't have to share my opinion, but maybe you could just accept it. It seems to be rather useless to repeat myself. Best,
18. Hi Mike:
Because the coupling itself doesn't change significantly. What you actually do to compute the running is to sum up an infinite series of loops that contributes to the propagator. Best,
19. One can do the higher order contributions though. It's more important in some scenarios than in others. I am sure one finds that somewhere on the arxiv.
20. Thanks Bee. When I'm busy not being busy, I'll have to go digging.
21. :-) I can really recommend Weinberg's QFT book, volume 2, chapter 18.
22. "As I understand it, the logarithmic form of the running couplings follows from only considering 1-loop effects. Why should I believe that over 15 orders of magnitude in the scale factor, higher
order terms will be of no importance?"
Two loop effects and threshold corrections have certainly been accounted for and there are lots of papers on this.
Bee, I still fail to see what your objection is. There is a kink at around TeV when the superpartner contributions kick in. So, at the GUT scale you also get extra light states which complete the
multiplets and make the couplings run together.
What's your objection precisely?
23. Hi Alex:
I have no 'objection'. I didn't say there is anything wrong with it, and if you think that's the way nature works, fine. I will believe in susy if it is found at the LHC. Best,
24. Bee,
Hopefully this comes forward from the layman properly.
Alex said:"I think one should really look at this backwards from high to low energy scale, i.e before the GUT is broken you have a single coupling. Then at around 10^16 GeV the GUT symmetry is
spontaneously broken and the couplings start running separately. What's your alternative?
It is like asking a cosmologist whether there is ever a before to the current universe? Qui! Non? :)
Pierre Ramond has a nice plot I showed earlier.
That Eric mentioned the condensed matter theorist situation, is one Robert Laughlin refers too often as to the "building blocks of the universe" being what ever you want them to be.
So you push back perspective to a time in the universe's expression(this a has been progressive) and your response on Zajc(Microseconds) is a case in point.
So Alex asks again, still not convinced?
25. Bee said,I have no 'objection'. I didn't say there is anything wrong with it, and if you think that's the way nature works, fine. I will believe in susy if it is found at the LHC.
Robert Lauglin:Likewise, if the very fabric of the Universe is in a quantum-critical state, then the stuff that underlies reality is totally irrelevant-it could be anything, says Laughlin. Even
if the string theorists show that strings can give rise to the matter and natural laws we know, they won't have proved that strings are the answer-merely one of the infinite number of possible
answers. It could as well be pool balls or Lego bricks or drunk sergeant majors.
Just seen your response.
We know that the experimental process is limited while in a cosmological view the energy values are still much higher.Constant referrals to GZK just doesn't cut it :)
The very act of pushing back these view on the cosmos do not relegate them to insignificant because of what you see in the experimental process and are waiting for.
I was looking for a animation about "the sand and pebble on the beach for you" so you get the picture.
26. Hi Plato:
I am not actually sure what you want to convince me of. You are probably not going to convince me to start working on SUSY models. If I look at the curves backwards I see that one introduces two
symmetry breaking scales which results in three curves getting kinks and reaching three values they have today within the required experimental precision.
I hope 'Lego' theory doesn't become fashionable ;-) Best,
27. I forgot to mention another impressive prediction from SUSY GUT's, namely the value of the Weinberg angle. It is completely fixed by group theory at the GUT scale and when one runs it down to the
electroweak scale it agrees with the experiment to 1% accuracy. The threshold effects at the GUT scale (when included) fix even this tiny discrepancy.
Moreover, by the seesaw mechanism, the electroweak scale comes out to be the geometric mean of the GUT scale (predicted by the running) and the neutrino mass scale, which puts the neutrino mass
in O(1eV) range - consistent with experimental bounds.
So, it's not just that second graph that's appealing about SUSY GUT models.
28. the value of the Weinberg angle
Is this specific to susy GUTs?
29. "Is this specific to susy GUTs?"
The GUT scale value is fixed by group theory. I think the value for SU(5) is sin^2(\theta)=3/8, if I recall correctly.
The presence of superpartners in the spectrum affects the running of sin^2(\theta) and hence, its value at the electroweak scale. The experimental value measured at the EW scale is sin^2(\theta)
So, for an SU(5) non-SUSY GUT the predicted value for sin^2(\theta) at the electroweak scale is ~0.206 whereas for the susy GUT one gets ~0.23 - an impressive agreement with the experimental
30. Hi Alex:
I forgot to mention another impressive prediction from SUSY GUT's, namely the value of the Weinberg angle.
In how far is this actually another prediction? Isn't 3/5 tan^2 theta just alpha_1/alpha_2?
PS: Completely off topic, I just heard an advertisement on the radio for the 'four course Italian fest', announced with appropriate Italian accent. What I thought I head was however 'something
for causality fest', I think I'm working too much
31. "Isn't 3/5 tan^2 theta just alpha_1/alpha_2?"
Yes, it is.
"In how far is this actually another prediction?"
So, in order to make a "real prediction" from a susy GUT one would have to find a parameter that could be directly computed in the GUT theory. Since we cannot compute the value of alpha_GUT from
first principles yet, we can instead reliably compute the ratio of the SU(2) and U(1) gauge couplings. Then we run them DOWN to the EW scale and get a very robust prediction which agrees with the
experiment. Note that in this high scale to low scale running the only "input" was the MSSM spectrum.
Now, this is quite different from the way the second plot in your post is obtained. In that case, we take experimentally measured values of alpha_1, alpha_2 and alpha_3 and run them UP to high
scale and observe that they unify to a high degree of accuracy.
So this is nice but suppose they did not unify but you still had the match for the Weinberg angle.
That would mean that you don't have just the MSSM all the way to the GUT scale and you would need to unclude some extra multiplets to fix the unification.
So do you see how one feature, i.e. the unification of couplings is not necessarily correlated with the other (the correct ratio of alpha_1/alpha_2)?
Now, bear in mind that the MSSM was not invented to deal with the gauge coupling unification and fixing the prediction for the Weinberg angle from GUTs. These came as very nice surprises.
32. "Then we run them DOWN..."
Oops, I meant run it (sin^2\theta) down...
33. Thankyou for explaining the "Running Coupling Constants" to the masses.
Now please, explain the masses.
34. Moreover, by the seesaw mechanism, the electroweak scale comes out to be the geometric mean of the GUT scale (predicted by the running) and the neutrino mass scale, which puts the neutrino mass
in O(1eV) range - consistent with experimental bounds.
Aren't neutrino masses rather in the 10^-3 - 10^-2 eV range, so the EW scale is the geometric mean between the Planck and neutrino scales?
35. Hi Thomas:
I think you are talking about the differences of the neutrino masses, not the absolute masses.
Hi Alex:
So do you see how one feature, i.e. the unification of couplings is not necessarily correlated with the other (the correct ratio of alpha_1/alpha_2)?
Actually, no. You are saying one could have theta being correct without having alpha_1/alpha_2 correct. what I am saying is, if you have alpha_1/alpha_2 correct, then theta follows, so in how far
is this actually 'another impressive prediction'.
Then we run them DOWN to the EW scale and get a very robust prediction which agrees with the experiment.
Well, if the angle runs when you go DOWN it seems to me its value depends on how far you let it run. Since the GUT scale is an input parameter, can't you just shift the start point of the running
around such that at the EW scale the value comes out right?
36. "You are saying one could have theta being correct without having alpha_1/alpha_2 correct."
No, I was saying that sin^2(\theta) is fixed by the embedding of the U(1) into the SU(5) and does not care if ALL THREE couplings unify. You can get alpha_1 and alpha_2 to cross at a point and
get the right value of sin^2(\theta) WITHOUT alpha_3 crossing at the same point.
Just add some extra multiplets that are only charged under SU(3) and you can screw up the unification of ALL THREE couplings without changing sin^2(\theta).
37. "Well, if the angle runs when you go DOWN it seems to me its value depends on how far you let it run. Since the GUT scale is an input parameter, can't you just shift the start point of the
running around such that at the EW scale the value comes out right?"
Sure, so you choose the GUT scale to get the right value for sin^2(\theta) and get agreement. But then you also take the experimental values of ALL THREE couplings at the EW scale and run them UP
to see if they ALL actually unify at the SAME input GUT scale you had chosen to get the agreement for the Weinberg angle.
Do you see that these two tests are different?
38. Oh, and you can imagine another scenario. Suppose you add extra matter in complete GUT multiplets to the MSSM somewhere in between the EW and the GUT scale. In that case you still have the
unification of all three couplings but the Weinberg angle at the GUT scale would not match the value 3/8. So, having the MSSM only spectrum all the way to the GUT scale is crucial in getting the
39. Hi Alex:
Thanks for agreeing that it's fixing the GUT scale that gives the right Weinberg angle. Regarding the question of whether it is 'another prediction'. It seems I have made myself somewhat unclear
so let me try it again.
Alex: So do you see how one feature, i.e. the unification of couplings is not necessarily correlated with the other (the correct ratio of alpha_1/alpha_2)?
You can get alpha_1 and alpha_2 to cross at a point and get the right value of sin^2(\theta) WITHOUT alpha_3 crossing at the same point.
Just add some extra multiplets [...]
That was not my question. What I said was if alpha_1 and alpha_2 have the correct, measured values at EW scale, and tan^2 theta is a ratio of both, then the value of theta at the EW scale is not
another prediction, but a consequence of the previous ones. If you are saying you can have tan^2 theta correct without having the correct ratio of alpha_2 to alpha_2, or alpha_3 not crossing at
the same point or other constructions, that doesn't make it better. I.e. what I am saying is B follows from A and you are saying no, A does not follow from B. Best,
40. Oh, and you can imagine another scenario. Suppose you add extra matter in complete GUT multiplets to the MSSM somewhere
I was not talking about other scenarios. I am pretty sure if one adds further parameters, on can fit further parameters independently.
41. thinking about it again, it seems you are actually saying C does not follow from B. Translate A -> ratio of a_1/a_2, B -> sin^2 theta, C -> unification of couplings.
42. "I.e. what I am saying is B follows from A"
No is does not necessarly follow. You can have unification and not get the agreement for the Weinberg angle. It is only when you have the MSSM spectrum ONLY all the way to the GUT scale do you
get the agreement with sin^2(\theta)=3/8.
43. No is does not necessarly follow. You can have unification and not get the agreement for the Weinberg angle.
But I never said that. What I am saying all the time is if a_1/a_2 has the correct value at the EW scale then doesn't the Weinberg angle follow from that, so in how far is this 'another
impressive prediction' (besides not being a prediction that is). I am not interested in whether you can have a unification model that does not reproduce measured data. Best,
44. "What I am saying all the time is if a_1/a_2 has the correct value at the EW scale then doesn't the Weinberg angle follow from that, so in how far is this 'another impressive prediction' (besides
not being a prediction that is)."
Computing the Weinberg angle at the EW scale tells you nothin illiminating about the underlying GUT theory. In order to compare it with the value predicted by the GUT you have to run it up to the
GUT scale. The unification of the couplings alone does not guarantee that if you run the ratio up to the GUT scale you'll get a match with the prediction sin^2(\theta)=3/8. This works ONLY if the
MSSM spectrum enters the running and nothing else.
45. I said: "The unification of the couplings alone does not guarantee that if you run the ratio up to the GUT scale you'll get a match with the prediction sin^2(\theta)=3/8."
Ok, now that I've thought about it carefully, I have to admit that the above statement is wrong. You are right that as long as the couplings unify you always get the match for the Weinberg angle,
no matter at what scale and at what value of the coupling unification takes place.
Sorry about the long exchange. I do not work on model building myself but I'm used to seeing talks where the Weinberg angle prediction is displayed as a separate statement.
46. Hi Alex:
Thanks for the interesting discussion, I certainly learned something! Best,
47. constantly on the run said...
`Thankyou for explaining the "Running Coupling Constants" to the masses.
Now please, explain the masses.'
Any ideas why the particles have the masses they do?
48. plenty, but so far none of them worked.
49. Bee, please explain to me one thing... Concealed down this thread, I can express my ignorance: why is it a value to have a single scale being where the curves meet, rather than two different
unification scales, whereby two curves meet, then follow at another slope to meet with the third, as E increases ?
PS I do not buy the argument as evidence for anything other than a lot of work by theorists trying to pay their bills.
50. Hi Survivor :-) You're asking the wrong person. Concealed in the above comments you might find my lack of enthusiasm about this result. Though I have to admit the plot kind of looks nice.
However, here as in other cases I wonder whether our built in sense for beauty and elegance is a good guide. Best,
51. That could have been a possibility Survivor, but there would have been absolutely no way you would know what the new slope of the united two curves would be so as to hit the third absent
knowledge about the intermediate scale physics. Not to mention its far less elegant.
No the real miracle imo isn't that the slopes hit a priori, its that the fit improves quite noticeably from the SM case. Of all the drastic things that supersymmetry naively could do, its
completely nontrivial that it leads to a better fit after so many orders of magnitude and so many extra degrees of freedom.
52. "this result is to be considered one of the most
compelling arguments for SUSY."
--If this is really one of the most compelling SUSY arguments, that is utterly pathetic!!
(1) As piscator points out, the 3-way meet is bullshit because it is based on a redefinition intentionally designed to make it happen.
(2) As the post points out it depends on a complete guess about the value of the SUSY symbreak energy, and if that guess were adjusted, we'd lose or regain the 3-way meet. When Maxwell (or
whoever) noticed the speed of light agreed numerically with prediction from magnetic & electric permeability constants, THAT was an impressive agreement. If however
that result had been "they might agree, or they might not, it depends on an adjustable and wholy-guessed parameter called 'the phase of the moon'"
then this result would have been 100% unimpressive!
(3) SO WHAT? Let's suppose SUSY really
predicts the 3 curves all exactly meet, and without caveats 1 & 2.
Why should this impress me? If there were some microscopic reason such as superstrings, that they ought to meet, then that's one thing. If however this is just a complete coincidence
with no underlying reason it should be so (it just happens to be so), that, it seems to me, is more of an indictment of SUSY than support for it. It indicates SUSY is clearly an incomplete
explanation of Nature.
So... what!?!?! Is this really "one of the most convincing arguments for SUSY? This really is the best you physicists have got? Because it seems pathetic to me.
--Warren D. Smith
warren.wds AT gmail.com
53. [continuing previous comment/question]
When you originally posted in 2007, it may have been reasonable to postulate as you did that the SUSY symbreak scale was 1 TeV.
But it is now 2013. We have the LHC now which is designed to run at 14 TeV. In view of this is it still tenable to contend the SUSY symbreak scale is 1 TeV? Or must we now contend it is >14 TeV?
And if so, does that destroy the 3-way meet?
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Rain, Rain, go away! San Antonio, TX
Essential Questions for students (objectives): How can you use fraction operations to solve problems?
Supplies: video (length 1:01), problem-solving note-maker
CCSS: 5.NF.6, TEKS: 5.3A, I, and K, Math Modeling
Time needed: 40 minutes +
Instructional Format: Video, student problem-solving, group or individual work
Lesson Description: There are many ways to use this video in your math class. I filmed it with the express purpose of modeling a fraction problem that relates to elapsed time. The final question
regarding flooding is incredibly rich because the variables and constants are not flushed out and must be researched and agreed upon. What does it mean to flood – just reach the top or how much over
the top?
1) You can show this video (1:01) at the beginning of a unit on multiplying fractions as a hook that will keep the students interested in learning about fraction calculations. You can have them work
on the problem at the end of daily lessons (or once a week) armed with new knowledge that they are exploring in class. Students use the note-maker to help record their problem-solving work. Or you
could revisit the video at the end of the unit as a formative check to see what the students have learned about multiplying fractions and whether they can apply that knowledge.
2) You could show this video as a warm-up activity after the students have learned some basic fraction applications and calculations. It is a great way to show context to fractions that isn’t the
same old cooking examples.
Extensions: Research a recent flooded town or city (New Orleans, August 2016). How long did it take to flood? What was the rate of rainfall? What would that amount of rainfall do in your city? | {"url":"https://makingmathematicians.com/index.php/lessons-categories/96-rain-rain-go-away-san-antonio-tx","timestamp":"2024-11-12T17:21:51Z","content_type":"text/html","content_length":"38125","record_id":"<urn:uuid:47ae1a21-f519-4361-8e18-a1fe63a45f56>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00113.warc.gz"} |