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KSEEB Solutions for Class 8 Maths Chapter 11 Congruency of Triangles Ex 11.4
Students can Download Maths Chapter 11 Congruency of Triangles Ex 11.4 Questions and Answers, Notes Pdf, KSEEB Solutions for Class 8 Maths helps you to revise the complete Karnataka State Board
Syllabus and score more marks in your examinations.
Karnataka Board Class 8 Maths Chapter 11 Congruency of Triangles Ex 11.4
Question 1.
In the given figure. If AB || DC and P is the midpoint of BD prove that P is also the midpoint of BD prove that p is also the midpoint of AC.
In ∆ PDC and ∆ PAB
PD = PB (data)
∠DPC = ∠APB [V.O.A]
∠PD∆ = ∠PBA [Alternate angles AB || CD]
∴ ∆ PDC ≅ ∆ PAB [ASA postulate]
∴ PC = PA [Corresponding sides]
∴ P is the midpoint AC
Question 2.
In the adjacent figure, CD and BE are altitudes of an isosceles triangle ABC with AC = AB prove that AE = AD.
In ∆ADC and ∆AEB
AC = AB [data]
∠ADC = ∠AEB[BE & CD altitudes]
∠DAC =∠EAB [Common angle]
∆ ADC = ∆ AEB[AS∆postulate]
AD = AE [Corresponding sides]
Question 3.
In the figure, AP and BQ are perpendiculars to the line segment AB and AP = BQ. Prove that O is the midpoint of line segment AB as well as PQ.
In ∆APO and ∆BQO
AP = BQ [data]
∠POA = ∠BOQ [ V.O.A]
∠PAO = ∠QBO = 90°
[AP & BQ are perpendiculars]
∴ ∆APO = ∆BQO [AS∆ postulate]
∴ AO = BO [Corresponding sides]
PO = OQ
∴ O is the midpoint of AB and PQ.
Question 4.
Suppose ABC is an isosceles triangle with AB = AC, BD and CE are bisectors of|fi andlC. Prove that BD = CE.
In ∆ABD and ∆ACE
∠BAD = ∠CAE [Common angle]
AB = AC [data]
∠ABD = ∠ACE [∠B =∠C and BD and CE are anglar bisectors]
∴ ∆ ABD ≅ ∆ ACE [AS∆ postulate]
BD = CE [Corresponding sides]
Question 5.
Suppose ABC is an equiangular triangle. Prove that it is equilateral. (You have seen earlier that an equilateral triangle is equiangular. Thus for triangles equiangularity is equivalent is
ABC is an equiangular triangle i.e.,
∠A =∠B =∠C
∠A = ∠B
BC = AC …(I) [Theorem2]
∠A =∠C
∴BC = AB ….(I) [Theorem2]
From (i) and (ii)
AB = BC = AC
∴ ∆ABC is equilateral. | {"url":"https://ktbssolutions.com/kseeb-solutions-class-8-maths-chapter-11-ex-11-4/","timestamp":"2024-11-11T01:02:52Z","content_type":"text/html","content_length":"87291","record_id":"<urn:uuid:ae07049f-d241-4417-bc31-acb16f5b009f>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00545.warc.gz"} |
Parallel Algorithms For The Spectral Transform Method | Request PDF
Parallel Algorithms For The Spectral Transform Method
The spectral transform method is a standard numerical technique for solving par- tial differential equations on a sphere and is widely used in atmospheric circulation models. Re- cent research has
identified several promising algorithms for implementing this method on mas- sively parallel computers; however, no detailed comparison of the different algorithms has previ- ously been attempted. In
this paper, we describe these different parallel algorithms and report on computational experiments that we have conducted to evaluate their efficiency on parallel com- puters. The experiments used a
testbed code that solves the nonlinear shallow water equations on a sphere; considerable care was taken to ensure that the experiments provide a fair compar- ison of the different algorithms and that
the results are relevant to global models. We focus on hypercube- and mesh-connected multicomputers with cut-through routing, such as the Intel iPSC/860, DELTA, and Paragon, and the nCUBE/2, but we
also indicate how the results extend to other parallel computer architectures. The results of this study are relevant not only to the spectral transform method but also to multidimensional fast
Fourier transforms (FFTs) and other parallel transforms.
No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
... For example, Figure 4 shows two variants for computing the 3D DFT. The first algorithm represents the so-called slab pencil decomposition [19][20][21], where the 3D DFT is decomposed into a 2D
DFT followed by a batch of multiple 1D DFTs. The second algorithm represents the pencil-pencil-pencil decomposition [19][20][21], where the 3D DFT is decomposed into three batches of 1D DFTs, where
each 1D DFT is applied in the three dimensions. ...
... The first algorithm represents the so-called slab pencil decomposition [19][20][21], where the 3D DFT is decomposed into a 2D DFT followed by a batch of multiple 1D DFTs. The second algorithm
represents the pencil-pencil-pencil decomposition [19][20][21], where the 3D DFT is decomposed into three batches of 1D DFTs, where each 1D DFT is applied in the three dimensions. The slab-pencil
decomposition views the input (output) column vectors x (y) as 2D matricesx (ỹ) of size (n 0 n 1 ) ×n 2 . ...
... Under this assumption, the p processors are connected via a hyper-cube topology and the communication requires loд(p) stages, where each two processors exchange n 2p data points. Equation 20
represents the lower bound of the all-to-all collective if a bucket algorithm is used. Given this implementation, the p processors are connected via ring topology, where each processor has a left and
a right neighbor. ...
Multi-dimensional discrete Fourier transforms (DFT) are typically decomposed into multiple 1D transforms. Hence, parallel implementations of any multi-dimensional DFT focus on parallelizing within or
across the 1D DFT. Existing DFT packages exploit the inherent parallelism across the 1D DFTs and offer rigid frameworks, that cannot be extended to incorporate both forms of parallelism and various
data layouts to enable some of the parallelism. However, in the era of exascale, where systems have thousand of nodes and intricate network topologies, flexibility and parallel efficiency are key
aspects all multi-dimensional DFT frameworks need to have in order to map and scale the computation appropriately. In this work, we present a flexible framework, built on the Redistribution
Operations and Tensor Expressions (ROTE) framework, that facilitates the development of a family of parallel multi-dimensional DFT algorithms by 1) unifying the two parallelization schemes within a
single framework, 2) exploiting the two different parallelization schemes to different degrees and 3) using different data layouts to distribute the data across the compute nodes. We demonstrate the
need of a versatile framework and thus a need for a family of parallel multi-dimensional DFT algorithms on the K-Computer, where we show almost linear strong scaling results for problem sizes of 1024
^3 on 32k compute nodes.
... Communication cost model. All of our analysis makes use of a commonlyused [16,31,8,3,14] communication cost model that is as useful as it is simple: each process is assumed to only be able to
simultaneously send and receive a single message at a time, and, when the message consists of n units of data (e.g., double-precision floating-point numbers), the time to transmit such a message
between any two processes is α + βn [19,2]. The α term represents the time required to send an arbitrarily small message and is commonly referred to as the message latency, whereas 1/β represents the
number of units of data which can be transmitted per unit of time once the message has been initiated. ...
... While this may seem overly restrictive, a large class of important transforms falls into this category, most notably: the Fourier transform, where Φ(x, y) = 2πx · y, backprojection [13],
hyperbolic Radon transforms [20], and Egorov operators, which then provide a means of efficiently applying Fourier Integral Operators [7]. Due to the extremely special (and equally delicate)
structure of Fourier transforms, a number of highly-efficient parallel algorithms already exist for both uniform [16,26,12] and non-uniform [27] Fourier transforms, and so we will instead concentrate
on more sophisticated kernels. We note that the high-level communication pattern and costs of the parallel 1D FFT mentioned in [16] are closely related to those of our parallel 1D butterfly
algorithm. ...
... Due to the extremely special (and equally delicate) structure of Fourier transforms, a number of highly-efficient parallel algorithms already exist for both uniform [16,26,12] and non-uniform
[27] Fourier transforms, and so we will instead concentrate on more sophisticated kernels. We note that the high-level communication pattern and costs of the parallel 1D FFT mentioned in [16] are
closely related to those of our parallel 1D butterfly algorithm. Algorithm 2.3 was instantiated in the new DistButterfly library using blackbox, user-defined phase functions, and the low-rank
approximations and translation operators introduced in [7]. ...
The butterfly algorithm is a fast algorithm which approximately evaluates a discrete analogue of the integral transform \int K(x,y) g(y) dy at large numbers of target points when the kernel, K(x,y),
is approximately low-rank when restricted to subdomains satisfying a certain simple geometric condition. In d dimensions with O(N^d) source and target points, when each appropriate submatrix of K is
approximately rank-r, the running time of the algorithm is at most O(r^2 N^d log N). A parallelization of the butterfly algorithm is introduced which, assuming a message latency of \alpha and
per-process inverse bandwidth of \beta, executes in at most O(r^2 N^d/p log N + (\beta r N^d/p + \alpha)log p) time using p processes. This parallel algorithm was then instantiated in the form of the
open-source DistButterfly library for the special case where K(x,y)=exp(i \Phi(x,y)), where \Phi(x,y) is a black-box, sufficiently smooth, real-valued phase function. Experiments on Blue Gene/Q
demonstrate impressive strong-scaling results for important classes of phase functions. Using quasi-uniform sources, hyperbolic Radon transforms and an analogue of a 3D generalized Radon transform
were respectively observed to strong-scale from 1-node/16-cores up to 1024-nodes/16,384-cores with greater than 90% and 82% efficiency, respectively. These experiments at least partially support the
theoretical argument that, given p=O(N^d) processes, the running-time of the parallel algorithm is O((r^2 + \beta r + \alpha)log N).
... The model is based on global hydrostatic primitive equations on sphere and uses the spectral transform method [11,12] in the horizontal directions with the use of the Gaussian grid and the
finite-difference method in the vertical direction with the sigma coordinates. It predicts such variables as horizontal winds, temperatures, ground surface pressure, specific humidity, and cloud
water. ...
... There have been many algorithms proposed and examined on various computers so far about the parallelization of the spectral transform method [12,13,14]. Here the outline of the Legendre transform
(LT) that makes up the core part of the spectral transform will be given by following Foster [12]. ...
... There have been many algorithms proposed and examined on various computers so far about the parallelization of the spectral transform method [12,13,14]. Here the outline of the Legendre transform
(LT) that makes up the core part of the spectral transform will be given by following Foster [12]. ...
A spectral atmospheric general circulation model called AFES (AGCM for Earth Simulator) was developed and optimized for the architecture of the Earth Simulator (ES). The ES is a massively parallel
vector supercomputer that consists of 640 processor nodes interconnected by a single stage crossbar network with its total peak performance of 40.96 Tflops was achieved for a high resolution
simulation (T1279L96) with AFES by utilizing the full 640-node configuration of the ES. The resulting computing efficiency is 64.9% of the peak performance, well surpassing that of conventional
weather/climate applications having just 25-50% efficiency even on vector parallel computers. This remarkable performance proves the effectiveness of the ES as a viable means for practical
... casting systems (Barros et al. 1995). Previous investigations of these methods have focused on the parallel aspects of either shared memory vector implementations (Gärtel et al. 1995), or on the
details of distributed memory implementations on MPPs (Foster and Worley 1994;Dent et al. 1995;Hammond et al. 1995). Perhaps because of the communications requirements of transposition-based message
passing implementations, and the poor capabilities of commodity interconnect fabrics until quite recently, little attention has been paid to studying the spherical harmonic transform method on
commodity clusters. ...
... In order to compute the Legendre polynomials coefficients on the fly using the recurrence relation (15), the paired longitudinal wavenumbers are then distributed across the processing elements
(PEs) in contiguous blocks of wavenumbers. We emphasize that, while most previous work has focused on more highly parallel 2D decompositions (Foster and Worley 1994), such fine-grain decompositions
require expensive low-latency-high-bandwidth networks, like the Cray T3E. Our strategy here is different. ...
The practical question of whether the classical spectral transform method, widely used in atmospheric modeling, can be efficiently implemented on inexpensive commodity clusters is addressed.
Typically, such clusters have limited cache and memory sizes. To demonstrate that these limitations can be overcome, the authors have built a spherical general circulation model dynamical core,
called BOB (''Built on Beowulf''), which can solve either the shallow water equations or the atmospheric primitive equations in pressure coordinates. That BOB is targeted for computing at high
resolution on modestly sized and priced commodity clusters is reflected in four areas of its design. First, the associated Legendre polynomials (ALPs) are computed ''on the fly'' using a stable and
accurate recursion relation. Second, an identity is employed that eliminates the storage of the derivatives of the ALPs. Both of these algorithmic choices reduce the memory footprint and memory
bandwidth requirements of the spectral transform. Third, a cache-blocked and unrolled Legendre transform achieves a high performance level that resists deterioration as resolution is increased.
Finally, the parallel implementation of BOB is transposition-based, employing load-balanced, one-dimensional decompositions in both latitude and wavenumber. A number of standard tests is used to
compare BOB's performance to two well-known codes—the Parallel Spectral Transform Shallow Water Model (PSTSWM) and the dynamical core of NCAR's Community Climate Model CCM3. Compared to PSTSWM, BOB
shows better timing results, particularly at the higher resolutions where cache effects become important. BOB also shows better performance in its comparison with CCM3's dynamical core. With 16
processors, at a triangular spectral truncation of T85, it is roughly five times faster when computing the solution to the standard Held-Suarez test case, which involves 18 levels in the vertical.
BOB also shows a significantly smaller memory footprint in these comparison tests.
... The horizontal resolution typically used for climate simulations in the U.S. research community is T85 with 26 vertical levels, which requires a 256 x 128 horizontal grid Worley and Drake 2005] .
The parallel algorithm used for these high resolution studies and benchmarking was given in [Foster and Worley 1997]. The FFT algorithm used was given in [Temperton 1983], a Fortran code specifically
designed for vector computation of multiple (blocked) fast Fourier transforms. ...
... A performance model of the parallel spectral transform can be developed to estimate the time for a multi-level calculation. The computational operation counts and communication cost estimates are
based on a model in [Foster and Worley 1997] for a one dimensional decomposition and modified by Rich Loft (NCAR) to reflect a simple transpose between FFT and Legendre transform phases including
vertical levels. The time for the FFT, the Legendre transform and the communication overhead are estimated using machine-dependant rate constants a,b,d, and e. ...
A collection of MATLAB classes for computing and using spherical harmonic transforms is pre-sented. Methods of these classes compute differential operators on the sphere and are used to solve simple
partial differential equations in a spherical geometry. The spectral synthesis and analysis algorithms using fast Fourier transforms and Legendre transforms with the associated Legendre functions are
presented in detail. A set of methods associated with a spectral field class provides spectral approximation to the differential operators ·, ×, , and 2 in spherical geometry. Laplace inversion and
Helmholtz equation solvers are also methods for this class. The use of the class and methods in MATLAB are demonstrated by the solution of the barotropic vorticity equa-tion on the sphere. A survey
of alternative algorithms is given and implementations for parallel high performance computers are discussed in the context of global climate and weather models.
... In both estimates and simulations, the transposition strategy appears no less efficient on realistic massively parallel computers than the best alternative static domain decomposition based
parallelization strategy. The geometric idea of transposition is illustrated in the pictures in (III), and results of comparison benchmarks are reported in Foster and Worley (1994). In that
reference, the authors also develop an elaborate communication strategy for domain decomposition that attains the same asymptotic efficiency as the transposition strategy. ...
... The transposition strategy is the parallelization strategy that in the asymptotic limit has the smallest data volume to communicate of all parallelization strategies for any implicit time
stepping scheme, for spectral and grid point models alike, as explained in subsection 5.3.1 above. Since the current research, a thorough analysis and a careful parameterized implementation has been
made of the two principal families of parallelization strategies for global atmospheric models -transposition versus static domain decomposition based and , while finding various parallel computers
on which each of the numerous versions and combinations of the strategies belonging to each family proves to be the most efficient, the authors find the transposition strategy to be a robust choice
on virtually all current computers (Foster and Worley (1994), Worley, Foster and Toonen (1994)). Model benchmarking tests initiated in (III) were expanded to a full two-dimensional version of the IFS
model in Gärtel et al. (1994-1), see also Gärtel et al. (1994-2), and eventually to the operational version of IFS (Barros et al. (1994). ...
Diss. -- Lappeenrannan teknillinen korkeakoulu.
... In this work, we use the transpose algorithm, which has performed better for large sizes in earlier work [8]. Gupta and Kumar [11], and Foster and Worley [9] review both methods. ...
We have implemented fast Fourier transforms for one, two, and three-dimensional arrays on the Cerebras CS-2, a system whose memory and processing elements reside on a single silicon wafer. The
wafer-scale engine (WSE) encompasses a two-dimensional mesh of roughly 850,000 processing elements (PEs) with fast local memory and equally fast nearest-neighbor interconnections. Our wafer-scale FFT
(wsFFT) parallelizes a $n^3$ problem with up to $n^2$ PEs. At this point a PE processes only a single vector of the 3D domain (known as a pencil) per superstep, where each of the three supersteps
performs FFT along one of the three axes of the input array. Between supersteps, wsFFT redistributes (transposes) the data to bring all elements of each one-dimensional pencil being transformed into
the memory of a single PE. Each redistribution causes an all-to-all communication along one of the mesh dimensions. Given the level of parallelism, the size of the messages transmitted between pairs
of PEs can be as small as a single word. In theory, a mesh is not ideal for all-to-all communication due to its limited bisection bandwidth. However, the mesh interconnecting PEs on the WSE lies
entirely on-wafer and achieves nearly peak bandwidth even with tiny messages. This high efficiency on fine-grain communication allow wsFFT to achieve unprecedented levels of parallelism and
performance. We analyse in detail computation and communication time, as well as the weak and strong scaling, using both FP16 and FP32 precision. With 32-bit arithmetic on the CS-2, we achieve 959
microseconds for 3D FFT of a $512^3$ complex input array using a 512x512 subgrid of the on-wafer PEs. This is the largest ever parallelization for this problem size and the first implementation that
breaks the millisecond barrier.
... The inverse Fourier transform can be easily computed using the forward FFT engine adding a 1/N scaling factor and conjugating the imaginary part; we won't debate the in-31 Chapter 3.
Three-dimensional Fast Fourier Transform → P 1 P 0 P 3 P 2 At a global level, with a 2D data domain decomposition, the X transform can proceed independently on each processing node because data on
the X dimension of the grid resides entirely in the local host memory and each one has its own assigned portion of the array. When data is non-local, that means that it is divided across processor
boundaries, the most efficient approach ( [57], [18]) is to reorganize the data array by a global transposition. This is called the transpose method, in opposition to the distributed method, where
the 1D transform is performed in parallel with data exchange occurring at 3.2. ...
In the field of High Performance Computing, communications among processes represent a typical bottleneck for massively parallel scientific applications. Object of this research is the development of
a network interface card with specific offloading capabilities that could help large scale simulations in terms of communication latency and scalability with the number of computing elements. In
particular this work deals with the development of a double precision floating point complex arithmetic unit with a parallel-pipelined architecture, in order to implement a massively parallel
computing system tailored for three dimensional Fast Fourier Transform.
... This can be done in parallel using sequential 1D FFTs. See [7] for alternative parallel methods such as the binary exchange method and a comparison between these methods. ...
We present a parallel algorithm for the fast Fourier transform (FFT) in higher dimensions. This algorithm generalizes the cyclic-to-cyclic one-dimensional parallel algorithm to a cyclic-to-cyclic
multidimensional parallel algorithm while retaining the property of needing only a single all-to-all communication step. This is under the constraint that we use at most $\sqrt{N}$ processors for an
FFT on an array with a total of $N$ elements, irrespective of the dimension $d$ or shape of the array. The only assumption we make is that $N$ is sufficiently composite. Our algorithm starts and ends
in the same distribution. We present our multidimensional implementation FFTU which utilizes the sequential FFTW program for its local FFTs, and which can handle any dimension $d$. We obtain
experimental results for $d\leq 5$ using MPI on up to 4096 cores of the supercomputer Snellius, comparing FFTU with the parallel FFTW program and with PFFT. These results show that FFTU is
competitive with the state-of-the-art and that it allows to use a larger number of processors, while keeping communication limited to a single all-to-all operation. For arrays of size $1024^3$ and
$64^5$, FFTU achieves a speedup of a factor 149 and 176, respectively, on 4096 processors.
... Handling the collective communications underlying distributed-memory FFT computation can be achieved using different approaches (refer to [77,78] for more information). The most effective
strategy already in use in many high performance FFT libraries is the so-called "the transpose transform". ...
The complexity of the physical mechanisms involved in ultra-high intensity laser-plasma interaction requires the use of particularly heavy PIC simulations. At the heart of these computational codes,
high-order pseudo-spectral Maxwell solvers have many advantages in terms of numerical accuracy. This numerical approach comes however with an expensive computational cost. Indeed, existing
parallelization methods for pseudo-spectral solvers are only scalable to few tens of thousands of cores, or induce an important memory footprint, which also hinders the scaling of the method at large
scales. In this thesis, we developed a novel, arbitrarily scalable, parallelization strategy for pseudo-spectral Maxwell's equations solvers which combines the advantages of existing parallelization
techniques. This method proved to be more scalable than previously proposed approaches, while ensuring a significant drop in the total memory use.By capitalizing on this computational work, we
conducted an extensive numerical and theoretical study in the field of high order harmonics generation on solid targets. In this context, when an ultra-intense (I>10¹⁶W.cm⁻²) ultra-short (few tens of
femtoseconds) laser pulse irradiates a solid target, a reflective overdense plasma mirror is formed at the target-vacuum interface. The subsequent laser pulse non linear reflection is accompanied
with the emission of coherent high order laser harmonics, in the form of attosecond X-UV light pulses (1 attosecond = 10⁻¹⁸s). For relativistic laser intensities (I>10¹⁹ W.cm⁻²), the plasma surface
is curved under the laser radiation pressure. And the plasma mirror acts as a focusing optics for the radiated harmonic beam. In this thesis, we investigated feasible ways for producing isolated
attosecond light pulses from relativistic plasma-mirror harmonics, with the so called attosecond lighthouse effect. This effect relies introducing a wavefront rotation on the driving laser pulse in
order to send attosecond pulses emitted during different laser optical cycles along different directions. In the case of high order harmonics generated in the relativistic regime, the plasma mirror
curvature significantly increases the attosecond pulses divergence and prevents their separation with the attosecond lighthouse scheme. For this matter, we developed two harmonic divergence reduction
techniques, based on tailoring the laser pulse phase or amplitude profiles in order to significantly inhibit the plasma mirror focusing effect and allow for a clear separation of attosecond light
pulses by reducing the harmonic beam divergence. Furthermore, we developed an analytical model to predict optimal interaction conditions favoring attosecond pulses separation. This model was fully
validated with 2D and 3D PIC simulations over a broad range of laser and plasma parameters. In the end, we show that under realistic laser and plasma conditions, it is possible to produce isolated
attosecond pulses from Doppler harmonics.
... Several approaches have been considered to efficiently parallelise spectral transforms between physical and spectral space (e.g. Foster & Worley 1997). ...
We present a new pseudo-spectral open-source code nicknamed pizza. It is dedicated to the study of rapidly-rotating Boussinesq convection under the 2-D spherical quasi-geostrophic approximation, a
physical hypothesis that is appropriate to model the turbulent convection that develops in planetary interiors. The code uses a Fourier decomposition in the azimuthal direction and supports both a
Chebyshev collocation method and a sparse Chebyshev integration formulation in the cylindrically-radial direction. It supports several temporal discretisation schemes encompassing multi-step time
steppers as well as diagonally-implicit Runge-Kutta schemes. The code has been tested and validated by comparing weakly-nonlinear convection with the eigenmodes from a linear solver. The comparison
of the two radial discretisation schemes has revealed the superiority of the Chebyshev integration method over the classical collocation approach both in terms of memory requirements and operation
counts. The good parallelisation efficiency enables the computation of large problem sizes with $\mathcal{O}(10^4\times 10^4)$ grid points using several thousands of ranks. This allows the
computation of numerical models in the turbulent regime of quasi-geostrophic convection characterised by large Reynolds $Re$ and yet small Rossby numbers $Ro$. A preliminary result obtained for a
strongly supercritical numerical model with a small Ekman number of $10^{-9}$ and a Prandtl number of unity yields $Re\simeq 10^5$ and $Ro \simeq 10^{-4}$. pizza is hence an efficient tool to study
spherical quasi-geostrophic convection in a parameter regime inaccessible to current global 3-D spherical shell models.
... An introduction and theoretical comparison can be found in [8]. In this paper, we restrict ourselves to transpose algorithms that need much less data to be exchanged [9] and have direct support
in many software libraries, e.g. FFTW [5]. ...
The 3D fast Fourier transform (FFT) is the heart of many simulation methods. Although the efficient parallelisation of the FFT has been deeply studied over last few decades, many researchers only
focused on either pure message passing (MPI) or shared memory (OpenMP) implementations. Unfortunately, pure MPI approaches cannot exploit the shared memory within the cluster node and the OpenMP
cannot scale over multiple nodes. This paper proposes a 2D hybrid decomposition of the 3D FFT where the domain is decomposed over the first axis by means of MPI while over the second axis by means of
OpenMP. The performance of the proposed method is thoroughly compared with the state of the art libraries (FFTW, PFFT, P3DFFT) on three supercomputer systems with up to 16k cores. The experimental
results show that the hybrid implementation offers 10-20% higher performance and better scaling especially for high core counts.
... The CORAL set of benchmark codes [18], intended for HPC vendors, include a set of " Skeleton Benchmarks, " but this term is used to refer to benchmarks that each focus on a specific platform
characteristic, unlike our application skeletons. Additional examples are from Kerbyson et al. [19], who used simplified version of parallel MPI applications to study Blue Gene systems; and Worley et
al. [20] [21], who studied a parallel spectral transform shallow water model by implementing the real spectral transform in what is otherwise a synthetic code that replicates a range of different
communication structures as found in different parallelizations of climate models. Similarly, Prophesy [22] is an infrastructure that helps in performance modeling of applications on parallel and
distributed systems through a relational database that allows for the recording of performance data, system features and application details. ...
Computer scientists who work on tools and systems to support eScience (a variety of parallel and distributed) applications usually use actual applications to prove that their systems will benefit
science and engineering (e.g., improve application performance). Accessing and building the applications and necessary data sets can be difficult because of policy or technical issues, and it can be
difficult to modify the characteristics of the applications to understand corner cases in the system design. In this paper, we present the Application Skeleton, a simple yet powerful tool to build
synthetic applications that represent real applications, with runtime and I/O close to those of the real applications. This allows computer scientists to focus on the system they are building; they
can work with the simpler skeleton applications and be sure that their work will also be applicable to the real applications. In addition, skeleton applications support simple reproducible system
experiments since they are represented by a compact set of parameters.
... The spectral code is parallelised using a so-called 2-D decomposition (Foster and Worley, 1997;Kanamitsu et al., 2005). In a 2-D decomposition, two of the three dimensions are divided across the
processors, and so there is a column and row of processors, with the columns divided across one dimension and the rows across another. ...
The IGCM4 (Intermediate Global Circulation Model version 4) is a global spectral primitive equation climate model whose predecessors have extensively been used in areas such as climate research,
process modelling and atmospheric dynamics. The IGCM4's niche and utility lies in its speed and flexibility allied with the complexity of a primitive equation climate model. Moist processes such as
clouds, evaporation, atmospheric radiation and soil moisture are simulated in the model, though in a simplified manner compared to state-of-the-art global circulation models (GCMs). IGCM4 is a
parallelised model, enabling both very long integrations to be conducted and the effects of higher resolutions to be explored. It has also undergone changes such as alterations to the cloud and
surface processes and the addition of gravity wave drag. These changes have resulted in a significant improvement to the IGCM's representation of the mean climate as well as its representation of
stratospheric processes such as sudden stratospheric warmings. The IGCM4's physical changes and climatology are described in this paper.
... The model is similar though not as detailed as the ones in Ayala and Wang [2013], Kerbyson and Barker [2011] and Kerbyson et al. [2013]. Since different runtime FFT algorithms on each machine are
used, and since the slab decomposition is used for a small number of process and the pencil decomposition is used for more than 512 processes, and since there are different algorithms for performing
an all to all exchange, the best of which will depend on the size of the problem being solved, the computer being used and the number and location of the processors being used on that computer (see
Foster and Worley [1997]), a more detailed model is not developed. For small p and fixed N, the runtime decreases close to linearly, and once p is large enough the runtime starts to increase again
due to communication costs. ...
The cubic Klein-Gordon equation is a simple but non-trivial partial differential equation whose numerical solution has the main building blocks required for the solution of many other partial
differential equations. In this study, the library 2DECOMP&FFT is used in a Fourier spectral scheme to solve the Klein-Gordon equation and strong scaling of the code is examined on thirteen different
machines for a problem size of 512^3. The results are useful in assessing likely performance of other parallel fast Fourier transform based programs for solving partial differential equations. The
problem is chosen to be large enough to solve on a workstation, yet also of interest to solve quickly on a supercomputer, in particular for parametric studies. Unlike other high performance computing
benchmarks, for this problem size, the time to solution will not be improved by simply building a bigger supercomputer.
... FFTW [15]. In addition, several FFT algorithms have been proposed for distributed machines (for example see [14,36,37]). For computing 3D FFTs, the key challenge is in dividing the data across
the processes. ...
We discuss the fast solution of the Poisson problem on a unit cube. We benchmark the performance of the most scalable methods for the Poisson problem: the Fast Fourier Transform (FFT), the Fast
Multipole Method (FMM), the geometric multigrid (GMG) and algebraic multigrid (AMG). The GMG and FMM are novel parallel schemes using high-order approximation for Poisson problems developed in our
group. The FFT code is from P3DFFT library and AMG code from ML Trilinos library. We examine and report results for weak scaling, strong scaling, and time to solution for uniform and highly refined
grids. We present results on the Stampede system at the Texas Advanced Computing Center and on the Titan system at the Oak Ridge National Laboratory. In our largest test case, we solved a problem
with 600 billion unknowns on 229,379 cores of Titan. Overall, all methods scale quite well to these problem sizes. We have tested all of the methods with different source distributions. Our results
show that FFT is the method of choice for smooth source functions that can be resolved with a uniform mesh. However, it loses its performance in the presence of highly localized features in the
source function. FMM and GMG considerably outperform FFT for those cases.
... The original model is based on the three- (6) dimensional global hydrostatic primitive equations. The spectral transform method [12] is applied to discretize in the horizontal direction and a
finite-difference method in the vertical direction with the use of sigma coordinates. AFES predicts such variables as horizontal winds, temperatures, ground-level pressure, specific humidity, and
cloud water at grid points generated around the entire process. ...
Two major developments in the infrastructure of the computational science and engineering research in Japan are reviewed. Both of these developments, resulting from the recent construction of a
high-speed backbone network and a huge vector parallel computer, will surely change the scene of the computational science and engineering researches. The first one is the ITBL
(Information-Technology Based Laboratory) project, where R&D are made to realize a virtual research environment over the network. Here, basic software tools for distributed environments have been
developed to solve science and engineering problems. The second one is the Earth Simulator project. In this project, a huge SMP-cluster vector parallel system was developed, which will undoubtedly
give a great impact on the numerical simulations in the areas, for example, the climate modeling. Furthermore, activities in large-scale numerical simulations, which are carried out in various
application fields and have a potential for further integration of the above systems, are presented.
... Performance models for a specific given application domain, which presents performance bounds for implicit CFD codes have also been considered [15]. The efficiency of the spectral transform
method on parallel computers has been evaluated by Foster [9]. Kerbyson et al. provide an analytical model for the application SAGE [17]. ...
Exascale systems are predicted to have approximately one billion cores, assuming Gigahertz cores. Limitations on affordable network topologies for distributed memory systems of such massive scale
bring new challenges to the current parallel programing model. Currently, there are many efforts to evaluate the hardware and software bottlenecks of exascale designs. There is therefore an urgent
need to model application performance and to understand what changes need to be made to ensure extrapolated scalability. The fast multipole method (FMM) was originally developed for accelerating
N-body problems in astrophysics and molecular dynamics, but has recently been extended to a wider range of problems, including preconditioners for sparse linear solvers. It's high arithmetic
intensity combined with its linear complexity and asynchronous communication patterns makes it a promising algorithm for exascale systems. In this paper, we discuss the challenges for FMM on current
parallel computers and future exascale architectures, with a focus on inter-node communication. We develop a performance model that considers the communication patterns of the FMM, and observe a good
match between our model and the actual communication time, when latency, bandwidth, network topology, and multi-core penalties are all taken into account. To our knowledge, this is the first formal
characterization of inter-node communication in FMM, which validates the model against actual measurements of communication time.
... As mentioned earlier, in a spectral model a full dimension in one of two horizontal directions (East-West [X] and North-South [Y] directions in Fig. 1) is needed for the FFT computation. Similar
method has been introduced in several previous studies (Oikawa, 2001;Juang et al., 2001;Foster and Worley, 1997;Barros et al, 1995;Skalin and Biorge, 1997). First, rows along X and Y directions in
the spectral space are divided depending on the number of working processors while the vertical column has a full dimension (upper left corner cube in Fig. 1). ...
... In the horizontal direction we use a one-dimensional decomposition. This is in contrast to earlier work on two-dimensional decompostions by Foster and Worley (1994). Although the 2D decomposition
allows for a finer grain decomposition this also results in considerably more network traffic, requiring expensive low-latency-high-bandwidth networks. ...
... In the rst part of this section, we discussed machine performance without reference to the algorithms being used on diierent problem size/machine type/machine size conngurations. Yet there is
considerable variability in the performance of diierent algorithms (Foster and Worley 1994; Worley and Foster 1994), and average performance would have been considerably worse if we had restricted
ourselves to a single algorithm. Factors that can aaect performance include the choice of FFT algorithm, LT algorithm, aspect ratio, the protocols used for data transfer, and memory requirements. ...
Massively parallel processing (MPP) computer systems use high-speed interconnection networks to link hundreds or thousands of RISC microprocessors. With each microprocessor having a peak performance
of 100 Mflops/sec or more, there is at least the possibility of achieving very high performance. However, the question of exactly how to achieve this performance remains unanswered. MPP systems and
vector multiprocessors require very different coding styles. Different MPP systems have widely varying architectures and performance characteristics. For most problems, a range of different parallel
algorithms is possible, again with varying performance characteristics. In this paper, we provide a detailed, fair evaluation of MPP performance for a weather and climate modeling application. Using
a specially designed spectral transform code, we study performance on three different MPP systems: Intel Paragon, IBM SP2, and Cray T3D. We take great care to control for performance differences due
to var...
... [31] The transform between physical and Fourier/ Chebyshev space is a global operation which requires communication among the processors. Different approaches have been proposed for the
parallelization of such transforms [Foster and Worley, 1997]. Here, we use a transpose-based algorithm. ...
Numerical simulations of the process of convection and magnetic field generation in planetary cores still fail to reach geophysically realistic control parameter values. Future progress in this field
depends crucially on efficient numerical algorithms which are able to take advantage of the newest generation of parallel computers. Desirable features of simulation algorithms include (1) spectral
accuracy, (2) an operation count per time step that is small and roughly proportional to the number of grid points, (3) memory requirements that scale linear with resolution, (4) an implicit
treatment of all linear terms including the Coriolis force, (5) the ability to treat all kinds of common boundary conditions, and (6) reasonable efficiency on massively parallel machines with tens of
thousands of processors. So far, algorithms for fully self-consistent dynamo simulations in spherical shells do not achieve all these criteria simultaneously, resulting in strong restrictions on the
possible resolutions. In this paper, we demonstrate that local dynamo models in which the process of convection and magnetic field generation is only simulated for a small part of a planetary core in
Cartesian geometry can achieve the above goal. We propose an algorithm that fulfills the first five of the above criteria and demonstrate that a model implementation of our method on an IBM Blue Gene
/L system scales impressively well for up to O(104) processors. This allows for numerical simulations at rather extreme parameter values.
... This transpose method is named 2-Dimensional decomposition, because one of the dimensions is fixed but the other two are distributed. It has been studied by many authors (e.g., Foster and Worley
1997;Barros et al. 1995;Skalin and Bjorge 1997), and has been widely used in many global spectral models. Since the RSM code structure is very similar to the GSM which uses the transpose method for
parallelization, the same method was adopted for RSM parallelization. ...
... Example PSTSWM Input Files a) Problem b) Algorithm c) Measurements sums of spectral harmonics and inverse real FFTs. Each of these steps is presented in mathematical detail in[34]. ...
Thesis (Ph.D.)--Illinois Institute of Technology, 2005. Includes bibliographical references (leaves 239-247).
... What distinguishes our work from prior research is our framework for providing useful, accurate performance modeling and performance understanding that is tractable for a wide variety of machines
and applications. Previous work either developed very detailed models for performance45678, concentrated on tool development910, was very specific to a given application domain111213, or focused on
integrating compilation with scalability analysis [14]. Additionally, previous work by Worley [15] evaluated specific machines via benchmarking. ...
This paper presents a performance modeling methodology that is faster than traditional cycle-accurate simulation, more sophisticated than performance estimation based on system peak-performance
metrics, and is shown to be effective on a class of High Performance Computing PETSc kernels. The method yields insight into the factors that affect performance and scalability on parallel computers.
... The 2-D decomposition data transposition strategy utilizes a 2-D model data structures, meaning a single dimension of data structure in memory for each processor. This method has been applied
successfully in several parallel atmospheric models (Foster and Worley 1997;Barros et al. 1995;Skalin and Bjorge 1997), including the Global Spectral Model. 2-D can be used up to the number of
product of two smallest dimensions in all directions and all spaces, except with any prime number of processors (Juang and Kanamitsu 2001). ...
The Regional Spectral Model (RSM) is a nested primitive equation spectral model used by U.S. operational centers and international research communities to perform weather forecasts and climate
prediction on a regional scale. In this paper, we present the development of an efficient parallel RSM with a message passing paradigm. Our model employs robust and efficient 1-D and 2-D
decomposition strategies and incorporates promising parallel algorithms to deal with complicated perturbation architecture and ensure portability using hybrid MPI and openMP. We also achieve bit
reproducibility when our parallel RSM is compared to the sequential code. Performance tests were performed on an IBM SP, Compaq, NEC SX-6 and the Earth Simulator and our results show good scalability
at over a thousand processors.
... These numerical difficulties usually lead to use of the spectral method as in [2]. But the global transposition of data on a network of processors needed in the spectral method makes them
difficult to parallelize [5]. ...
We present a nonoverlapping domain decomposition method with local Fourier basis applied to a model problem in liquid flames. The introduction of domain decomposition techniques in this paper is for
numerical and parallel efficiency purposes when one requires a large number of grid points to catch complex structures. We obtain then a high-order accurate domain decomposition method that allows us
to generalize our previous work on the use of local Fourier basis to solve combustion problems with nonperiodic boundary conditions (M. Garbey and D. Tromeur-Dervout, J. Comput. Phys.145, 316
(1998)). Local Fourier basis methodology fully uses the superposition principle to split the searched solution in a numerically computed part and an analytically computed part. Our present
methodology generalizes the Israeli et al. (1993, J. Sci. Comput. 8, 135) method, which applies domain decomposition with local Fourier basis to the Helmholtz's problem. In the present work, several
new difficulties occur. First, the problem is unsteady and nonlinear, which makes the periodic extension delicate to construct in terms of stability and accuracy. Second, we use a streamfunction
biharmonic formulation of the incompressible Navier–Stokes equation in two space dimensions: The application of domain decomposition with local Fourier basis to a fourth-order operator is more
difficult to achieve than for a second-order operator. A systematic investigation of the influence of the method's parameters on the accuracy is done. A detail parallel MIMD implementation is given.
We give an a priori estimate that allows the relaxation of the communication between processors for the interface problem treatment. Results on nonquasi-planar complex frontal polymerization
illustrate the capability of the method.
... In this method, one dimension is kept fixed while the other two dimensions are decomposed, and the spectral method can be applied in parallel in the fixed dimension. After this, the system is
transposed before applying the algorithm in another direction (13). The Vlasov solver uses periodic boundary conditions in configuration space, where a pseudo-spectral method is employed to calculate
derivatives accurately. ...
We present a parallelized algorithm for solving the time-dependent Vlasov–Maxwell system of equations in the four-dimensional phase space (two spatial and velocity dimensions). One Vlasov equation is
solved for each particle species, from which charge and current densities are calculated for the Maxwell equations. The parallelization is divided into two different layers. For the first layer, each
plasma species is given its own processor group. On the second layer, the distribution function is domain decomposed on its dedicated resources. By separating the communication and calculation steps,
we have met the design criteria of good speedup and simplicity in the implementation.
... The slave tasks contain four distinct phases: data receiving, CPU bound computing, I/O bound computing and data sending (to the master). The second one, namely B, is the PSTSWM (Parallel Spectral
Transform Shallow Water Model) [16]. The PSTSWM is a message-passing benchmark code that solves the nonlinear shallow water equations on a rotating sphere using the spectral transform method. ...
A new approach for acquiring knowledge of parallel applications regarding resource usage and for searching similarity on workload traces is presented. The main goal is to improve decision making in
distributed system software scheduling, towards a better usage of system resources. Resource usage patterns are defined through runtime measurements and a self-organizing neural network architecture,
yielding an useful model for classifying parallel applications. By means of an instance-based algorithm, it is produced another model which searches for similarity in workload traces aiming at making
predictions about some attribute of a new submitted parallel application, such as run time or memory usage. These models allow effortless knowledge updating at the occurrence of new information. The
paper describes these models as well as the results obtained applying these models to acquiring knowledge in both synthetic and real applications traces.
... Instead, a dynamics– physics coupler (dp_coupling) would be used to move data between data structures representing the dynamics state and the physics state. In previous work (Drake et al.,
1995Drake et al., , 1999 Foster et al., 1996; Foster and Worley 1997), significant effort has been expended to determine data structures and domain decompositions that work well with both the
dynamics and the physics, in order to minimize memory requirements, to avoid the cost of buffer copies, and/or to avoid the cost of interprocess communication when execution moves between the
dynamics and the physics during each time step of the algorithm. With the decision to decouple physics and dynamics data structures , a global design was no longer necessarily advantageous . ...
Community models for global climate research, such as the Community Atmospheric Model, must perform well on a variety of computing systems. Supporting diverse research interests, these
computationally demanding models must be efficient for a range of problem sizes and processor counts. In this paper we describe the data structures and associated infrastructure developed for the
physical parameterizations that allow the Community Atmospheric Model to be tuned for vector or non-vector systems, to provide load balancing while minimizing communication overhead, and to exploit
the optimal mix of distributed Message Passing Interface (MPI) processes and shared OpenMP threads.
... In the distributed memory architecture, the remapping procedure implements a global all-to-all exchange of data blocks with size (Nx, Ny/P, Nz/P), where Nx (Ny, Nz) is the number of grid points
along the ix (iy, iz) direction, and P is the total number of distributed processors. The global exchange performs essentially a pairwise block exchange where the local 3-D arrays on each processor
are viewed as 1-D array of blocks (Bokhari 1991;Foster and Worley 1997). This exchange involves all-toall communication. ...
Century-long global climate simulations at high resolutions generate large amounts of data in a parallel architecture. Currently, the community atmosphere model (CAM), the atmospheric component of
the NCAR community climate system model (CCSM), uses sequential I/O which causes a serious bottleneck for these simulations. We describe the parallel I/O development of CAM in this paper. The
parallel I/ O combines a novel remapping of 3-D arrays with the paral- lel netCDF library as the I/O interface. Because CAM history variables are stored in disk file in a different index order than
the one in CPU resident memory because of parallel decom- position, an index reshuffle is done on the fly. Our strategy is first to remap 3-D arrays from its native decomposition to z- decomposition
on a distributed architecture, and from there write data out to disk. Because z-decomposition is consist- ent with the last array dimension, the data transfer can occur at maximum block sizes and,
therefore, achieve maximum I/ O bandwidth. We also incorporate the recently developed parallel netCDF library at Argonne/Northwestern as the col- lective I/O interface, which resolves a long-standing
issue because netCDF data format is extensively used in climate system models. Benchmark tests are performed on several platforms using different resolutions. We test the perform- ance of our new
parallel I/O on five platforms (SP3, SP4, SP5, Cray X1E, BlueGene/L) up to 1024 processors. More than four realistic model resolutions are examined, e.g. EUL T85 (~1.4°), FV-B (2° × 2.5°), FV-C (1° ×
1.25°), and FV-D (0.5° × 0.625°) resolutions. For a standard single history out- put of CAM 3.1 FV-D resolution run (multiple 2-D and 3-D arrays with total size 4.1 GB), our parallel I/O speeds up by
a factor of 14 on IBM SP3, compared with the existing I/O; on IBM SP5, we achieve a factor of 9 speedup. The estimated time for a typical century-long simulation of FV D-resolution on IBM SP5 shows
that the I/O time can be reduced from more than 8 days (wall clock) to less than 1 day for daily out- put. This parallel I/O is also implemented on IBM BlueGene/ L and the results are shown, whereas
the existing sequential I/O fails due to memory usage limitation.
Numerical weather prediction (NWP) is one of the first applications of scientific computing and remains an insatiable consumer of high-performance computing today. In the face of a half-century’s
exponential and sometimes disruptive growth in HPC capability, major weather services around the world continuously develop and incorporate new meteorological research into large expensive
operational forecasting software suites. At the heart is the weather model itself: a computational fluid dynamics core on a spherical domain with physics. The mapping of a planar grid to the geometry
of the earth’s atmosphere presents one of a number of challenges for stability, accuracy, and computational efficiency that lead to development of the major numerical formulations and grid systems in
use today: grid point, globally spectral, and finite/spectral element. Significant challenges await in the exascale era: large memory working sets, overall low computational intensity, load
imbalance, and a fundamental lack of weak scalability in the face of critical real-time forecasting speed requirements. This chapter provides a history of weather and climate modeling on
high-performance computing systems, a discussion of each of the major types of model dynamics formulations, grids, and model physics, and directions going forward on emerging HPC architectures.
The IGCM4 (Intermediate Global Circulation Model version 4) is a global spectral primitive equation climate model whose predecessors have extensively been used in fields such as climate dynamics,
processes modelling, and atmospheric dynamics. The IGCM4's niche and utility lies in its parallel spectral dynamics and fast radiation scheme. Moist processes such as clouds, evaporation, and soil
moisture are simulated in the model, though in a simplified manner compared to state-of-the-art GCMs. The latest version has been parallelised, which has led to massive speed-up and enabled much
higher resolution runs than would be possible on one processor. It has also undergone changes such as alterations to the cloud and surface processes, and the addition of gravity wave drag. These
changes have resulted in a significant improvement to the IGCM's representation of the mean climate as well as its representation of stratospheric processes such as sudden stratospheric warmings. The
IGCM4's physical changes and climatology are described in this paper.
The computation on the polar regions plays an crucial role in the design of global numerical weather prediction (NWP) models, which details itself in the following two aspects: the particular
treatment of polar regions in the model's dynamic framework and the load-balancing problem caused by the parallel data partitioning strategies. The latter has become the bottleneck of massive
parallelization of NWP models. To address this problem, a novel spherical data partitioning algorithm based on the weighted-equal-area approach is proposed. The weight describes the computational
distribution across the entire sphere. The new algorithm takes the collar amount and the weight function as its parameters and performs the spherical partitioning as follows: the north and the south
polar regions are partitioned into a singular subdomain; then the remaining sphere surface is partitioned into some collars along the latitude; and finally each collar is partitioned into subdomains
along the longitude. This partitioning method can result in two polar caps plus a number of collars with increasing partition counts as we approach the equator. After a theoretical analysis of the
quality relevant to the partition performed by the algorithm, we take the PSTSWM, which is a spectral shallow water model based on the spherical harmonic transform technique, as our test-bed to
validate our method. The preliminary results indicate that the algorithm can result in good parallel load balance and a promising prospect can be expected for its application within the global
atmospheric model of GRAPES.
The development of fully parallelized regional spectral model (RSM) was described. The vertical layer bands are transposed into latitude bands using the MPI routines. The computation of the linear
dynamics such as semi-implicit and time filter was performed in the spectral space. It was found that the RSM-MPI is twice faster with respect to old RSM-MPI and could use about factor of 5 larger
An atmospheric general circulation model (AGCM) for climate studies was developed for the Earth Simulator (ES). The model is called AFES which is based on the CCSR/NIES AGCM and is a global
three-dimensional hydrostatic model using the spectral transform method. AFES is optimized for the architecture of the ES. We achieved the high sustained performance by the execution of AFES with
T1279L96 resolution on the ES. The performance of 26.58 Tflops was achieved the execution of the main time step loop using all 5120 processors (640 nodes) of the ES. This performance corresponds to
64.9% of the theoretical peak performance 40.96 Tflops. The T1279 resolution, equivalent to about 10 km grid intervals at the equator, is very close to the highest resolution in which the hydrostatic
approximation is valid. To our best knowledge, no other model simulation of the global atmosphere has ever been performed with such super high resolution. Currently, such a simulation is possible
only on the ES with AFES. In this paper we describe optimization method, computational performance and calculated result of the test runs.
Fourier and related transforms are a family of algorithms widely employed in diverse areas of computational science, notoriously difficult to scale on high-performance parallel computers with a large
number of processing elements (cores). This paper introduces a popular software package called P3DFFT which implements fast Fourier transforms (FFTs) in three dimensions in a highly efficient and
scalable way. It overcomes a well-known scalability bottleneck of three-dimensional (3D) FFT implementations by using two-dimensional domain decomposition. Designed for portable performance, P3DFFT
achieves excellent timings for a number of systems and problem sizes. On a Cray XT5 system P3DFFT attains 45% efficiency in weak scaling from 128 to 65,536 computational cores. Library features
include Fourier and Chebyshev transforms, Fortran and C interfaces, in- and out-of-place transforms, uneven data grids, and single and double precision. P3DFFT is available as open source at http://
code.google.com/p/p3dfft/. This paper discusses P3DFFT implementation and performance in a way that helps guide the user in making optimal choices for parameters of their runs.
Fast, accurate computation of geophysical fluid dynamics is often very challenging. This is due to the complexity of the PDEs themselves and their initial and boundary conditions. There are several
practical advantages to using a relatively new numerical method, the spectral-element method (SEM), over standard methods. SEM combines spectral-method high accuracy with the geometric flexibility
and computational efficiency of finite-element methods. This paper is intended to augment the few descriptions of SEM that aim at audiences besides numerical-methods specialists. Advantages of SEM
with regard to flexibility, accuracy, and efficient parallel performance are explained, including sufficient details that readers may estimate the benefit of applying SEM to their own computations.
The spectral element atmosphere model (SEAM) is an application of SEM to solving the spherical shallow-water or primitive equations. SEAM simulated decaying Jovian atmospheric shallow-water
turbulence up to resolution T1067, producing jets and vortices consistent with Rhines theory. SEAM validates the Held-Suarez primitive equations test case and exhibits excellent parallel performance.
At T171L20, SEAM scales up to 292 million floating-point operations per second (Mflops) per processor (29% of supercomputer peak) on 32 Compaq ES40 processors (93% efficiency over using 1 processor),
allocating 49 spectral elements/processor. At T533L20, SEAM scales up to 130 billion floating-point operations per second (Gflops) (8% of peak) and 9 wall clock minutes per model day on 1024 IBM
POWER3 processors (48% efficiency over 16 processors), allocating 17 spectral elements per processor. Local element-mesh refinement with 300% stretching enables conformally embedding T480 within T53
resolution, inside a region containing 73% of the forcing but 6% of the area. Thereby the authors virtually reproduced a uniform-mesh T363 shallow-water computation, at 94% lower cost.
The coupling of a semi-Lagrangian treatment of horizontal advection with a semi-implicit treatment of gravitational oscillations permits longer timesteps than those allowed by a semi-implicit
Eulerian scheme. The timestep is then limited by the stability of the explicit treatment of vertical advection. To remove this stability constraint, we propose the use of the semi-Lagrangian
advection scheme for both horizontal and vertical advection. This is done in the context of the Canadian regional finite-element weather forecast model that includes a parameterization of the most
relevant sub-grid scale processes. It is shown that the three-dimensional semi-Lagrangian scheme produces stable, accurate integrations using timesteps that far exceed the stability limit for the
Eulerian model. -Authors
The HIRLAM (high resolution limited area modelling) limited-area atmospheric model was originally developed and optimized for shared memory vector-based computers, and has been used for operational
weather forecasting on such machines for several years. This paper describes the algorithms applied to obtain a highly parallel implementation of the model, suitable for distributed memory machines.
The performance results presented indicate that the parallelization effort has been successful, and the Norwegian Meteorological Institute will run the parallel version in production on a Cray T3E.
We evaluate dynamic data remapping on cluster of SMP architectures under OpenMP, MPI, and hybrid paradigms. Traditional method of multi-dimensional array transpose needs an auxiliary array of the
same size and a copy back stage. We recently developed an inplace method using vacancy tracking cycles. The vacancy tracking algorithm outperforms the traditional 2-array method as demonstrated by
extensive comparisons. Performance of multi-threaded parallelism using OpenMP are first tested with different scheduling methods and different number of threads. Both methods are then parallelized
using several parallel paradigms. At node level, pure OpenMP outperforms pure MPI by a factor of 2.76 for vacancy tracking method. Across entire cluster of SMP nodes, by carefully choosing thread
numbers, the hybrid MPI/OpenMP implementation outperforms pure MPI by a factor of 3.79 for traditional method and 4.44 for vacancy tracking method, demonstrating the validity of the parallel paradigm
of mixing MPI with OpenMP.
Run time variability of parallel application codes continues to be a significant challenge in clusters. We are studying run time variability at the communication level from the perspective of the
application, focusing on the network. To gain insight into this problem our earlier work developed a tool to emulate parallel applications and in particular their communication. This framework,
called parallel application communication emulation (PACE) has produced interesting insights regarding network performance in NOW clusters. A parallel application run time sensitivity evaluation
(PARSE) function has been added to the PACE framework to study the run time effects of controlled network performance degradation. This paper introduces PARSE and presents experimental results from
tests conducted on several widely used parallel benchmarks and application code fragments. The results suggest that parallel applications can be classified in terms of their sensitivity to network
performance variation
Reconfigurable computing offers the promise of performing computations in hardware to increase performance and efficiency while retaining much of the flexibility of a software solution. Recently, the
capacities of reconfigurable computing devices, like field programmable gate arrays, have risen to levels that make it possible to execute 64b floating-point operations. SRC Computers has designed
the SRC-6 MAPstation to blend the benefits of commodity processors with the benefits of reconfigurable computing. In this paper, we describe our effort to accelerate the performance of several
scientific applications on the SRC-6. We describe our methodology, analysis, and results. Our early evaluation demonstrates that the SRC-6 provides a unique software stack that is applicable to many
scientific solutions and our experiments reveal the performance benefits of the system.
This book explains how to use the bulk synchronous parallel (BSP) model to design and implement parallel algorithms in the areas of scientific computing and big data. Furthermore, it presents a
hybrid BSP approach towards new hardware developments such as hierarchical architectures with both shared and distributed memory. The book provides a full treatment of core problems in scientific
computing and big data, starting from a high-level problem description, via a sequential solution algorithm to a parallel solution algorithm and an actual parallel program written in the
communication library BSPlib. Numerical experiments are presented for parallel programs on modern parallel computers ranging from desktop computers to massively parallel supercomputers. The
introductory chapter of the book gives a complete overview of BSPlib, so that the reader already at an early stage is able to write his/her own parallel programs. Furthermore, it treats BSP
benchmarking and parallel sorting by regular sampling. The next three chapters treat basic numerical linear algebra problems such as linear system solving by LU decomposition, sparse matrix-vector
multiplication (SpMV), and the fast Fourier transform (FFT). The final chapter explores parallel algorithms for big data problems such as graph matching. The book is accompanied by a software package
BSPedupack, freely available online from the author’s homepage, which contains all programs of the book and a set of test programs.
A collection of MATLAB classes for computing and using spherical harmonic transforms is presented. Methods of these classes compute differential operators on the sphere and are used to solve simple
partial differential equations in a spherical geometry. The spectral synthesis and analysis algorithms using fast Fourier transforms and Legendre transforms with the associated Legendre functions are
presented in detail. A set of methods associated with a spectral field class provides spectral approximation to the differential operators ∇�, ∇×, ∇, and ∇2 in spherical geom- etry. Laplace inversion
and Helmholtz equation solvers are also methods for this class. The use of the class and methods in MATLAB is demonstrated by the solution of the barotropic vortic- ity equation on the sphere. A
survey of alternative algorithms is given and implementations for parallel high performance computers are discussed in the context of global climate and weather models. Categories and Subject
Descriptors: F.2.1 (Analysis of Algorithms and Problem Complex- ity): Numerical Algorithms and Problems—Computation of transforms; G.4 (Mathematical Software) ; G.1.8 (Numerical Analysis) Partial
Differential Equations; J.2 (Physical Sciences and Engineering): —Earth and atmosphere science General Terms: Algorithms, Theory
The spectral transform method used in climate and weather models is known to be computationally intensive. Typically accounting for more than 90% of the execution time of a serial model, it is poised
to benefit from computational parallelism. Since dimensionally global transforms impact parallel performance, it is important to establish the realizable parallel efficiency of the spectral
transform. To this end, this paper quantitatively characterizes the parallel characteristics of the spectral transform within an atmospheric modeling context. It comprehensively characterizes and
catalogs a baseline of operations required for the spectral transform. While previous investigations of the spectral transform method have offered highly idealized analyses that are abstract and
simplified in terms of orders of computational magnitude, this research provides a detailed model of the computational complexity of the spectral transform, validated by empirical results. From this
validated quantitative analysis, an operational closed-form expression characterizes spectral transform performance in terms of general processor parameters and atmospheric data dimensions. These
generalized statements of the computational requirements for the spectral transform can serve as a basis for exploiting parallelism.
The best approach to parallelize multidimensional FFT algorithms has long been under debate. Distributed transposes are widely used, but they also vary in communication policies and hence
performance. In this work we analyze the impact of different redistribution strategies on the performance of parallel FFT, on various machine architectures. We found that some redistribution
strategies were consistently superior, while some others were unexpectedly inferior. An in-depth investigation into the reasons for this behavior is included in this work. Copyright © 2001 John Wiley
& Sons, Ltd.
SUMMARY The problem of understanding how Earth's magnetic field is generat ed is one of the foremost challenges in modern science. It is believed to be generated by a dynamo process, where the
complex motions of an electrically conducting fluid provide the inductive action to susta in the field against the effects of dissipation. Current dynamo simulations, based on the numerical
approximation to the governing equations of magnetohydrodynamics, cannot reach the very rapid rotation rates and low viscosities (i.e. low Ekman number) of Earth due to limitations in available
computing power. Using a pseudospectral method, the most widely-used method for simulating the geodynamo, computational requirements needed to run simulations in an 'Earth-like' parameter regime are
explored theoretically by ap proximating operation counts, memory requirements, and communication costs in the asymptotic limit of large problem size. Theoretical scalings are tested using numerical
calculations. For asymptotically large problems the spherical transform is shown to be the limiting step within the pseudospectral method; memory requirements and communication costs
This study measures the effects of changes in message latency and bandwidth for production-level codes on a current generation tightly coupled MPP, the Intel Paragon. Messages are sent multiple times
to study the application sensitivity to variations in bandwidth and latency. This method preserves the effects of contention on the interconnection network.Two applications are studied: PCTH, a shock
physics code developed at Sandia National Laboratories; and PSTSWM, a spectral shallow water code developed at Oak Ridge National Laboratory and Argonne National Laboratory. These codes are
significant in that PCTH is a ‘full physics’ application code in production use, while PSTSWM serves as a parallel algorithm test bed and benchmark for production codes used in atmospheric modeling.
They are also significant in that the message-passing behavior differs significantly between the two codes, each representing an important class of scientific message-passing applications. © 1998
John Wiley & Sons, Ltd.
A composite mesh finite-difference method using overlapping stereographic coordinate systems is compared to transform methods based on scalar and vector spherical harmonics. The methods are compared
in terms of total computer time, memory requirements, and execution rates for relative accuracy requirements of two and four digits in a five-day forecast. The computational requirements of the three
methods were well within an order of magnitude of one another. In most of the cases that are examined, the time step was limited by accuracy rather than stability. This problem can be overcome by the
use of a higher order time integration scheme, but at the expense of an increase in the memory requirements. -from Authors
This report presents the details of the governing equations, physical parameterizations, and numerical algorithms defining the version of the NCAR Community Climate Model designated CCM3. The
material provides an overview of the major model components, and the way in which they interact as the numerical integration proceeds. As before, it is our objective that this model provide NCAR and
the university research community with a reliable, well documented atmospheric general circulation model. This version of the CCM incorporates significant improvements to the physics package, new
capabilities such as the incorporation of a slab ocean component, and a number of enhancements to the implementation (e.g., the ability to integrate the model on parallel distributed-memory
computational platforms). We believe that collectively these improvements provide the research community with a significantly improved atmospheric modeling capability.
Conventional algorithms for computing large one-dimensional fast Fourier transforms (FFTs), even those algorithms recently developed for vector and parallel computers, are largely unsuitable for
systems with external or hierarchical memory. The principal reason for this is the fact that most FFT algorithms require at least m complete passes through the data set to compute a 2m -point FFT.
This paper describes some advanced techniques for computing an ordered FFT on a computer with external or hierarchical memory. These algorithms (1) require as few as two passes through the external
data set, (2) employ strictly unit stride, long vector transfers between main memory and external storage, (3) require only a modest amount of scratch space in main memory, and (4) are well suited
for vector and parallel computation. Performance figures are included for implementations of some of these algorithms on Cray supercomputers. Of interest is the fact that a main memory version
outperforms the current Cray library FFT routines on the CRAY-2, the CRAY X-MP, and the CRAY Y-MP systems. Using all eight processors on the CRAY Y-MP, this main memory routine runs at nearly two
In a multiprocessor with distributed storage the data structures have a significant impact on the communication complexity. In this paper we present a few algorithms for performing matrix
transposition on a Boolean n-cube. One algorithm performs the transpose in a time proportional to the lower bound both with respect to communication start-ups and to element transfer times. We
present algorithms for transposing a matrix embedded in the cube by a binary encoding, a binary-reflected Gray code encoding of rows and columns, or combinations of these two encodings. The
transposition of a matrix when several matrix elements are identified to a node by consecutive or cyclic partitioning is also considered and lower bound algorithms given. Experimental data are
provided for the Intel iPSC and the Connection Machine.
The development and use of three-dimensional computer models of the earth's climate are discussed. The processes and interactions of the atmosphere, oceans, and sea ice are examined. The basic theory
of climate simulation which includes the fundamental equations, models, and numerical techniques for simulating the atmosphere, oceans, and sea ice is described. Simulated wind, temperature,
precipitation, ocean current, and sea ice distribution data are presented and compared to observational data. The responses of the climate to various environmental changes, such as variations in
solar output or increases in atmospheric carbon dioxide, are modeled. Future developments in climate modeling are considered. Information is also provided on the derivation of the energy equation,
the finite difference barotropic forecast model, the spectral transform technique, and the finite difference shallow water waved equation model.
Implementations of climate models on scalable parallel computer systems can suffer from load imbalances because of temporal and spatial variations in the amount of computation required for physical
parameterizations such as solar radiation and convective adjustment. We have developed specialized techniques for correcting such imbalances. These techniques are incorporated in a general-purpose,
programmable load-balancing library that allows the mapping of computation to processors to be specified as a series of maps generated by a programmer-supplied load-balancing module. The
communication required to move from one map to another is performed automatically by the library, without programmer intervention. In this paper, we describe the load-balancing problem and the
techniques that we have developed to solve it. We also describe specific load-balancing algorithms that we have developed for PCCM2, a scalable parallel implementation of the Community Climate Model,
and present experimental...
This paper is a brief overview of a parallel version of the NCAR Community Climate Model, CCM2, implemented for MIMD massively parallel computers using a messagepassing programming paradigm. The
parallel implementation was developed on an Intel iPSC/860 with 128 processors and on the Intel Delta with 512 processors, and the initial target platform for the production version of the code is
the Intel Paragon with 2048 processors. Because the implementation uses a standard, portable message-passing library, the code can be easily ported to other multiprocessors supporting a
message-passing programming paradigm, or run on machines distributed across a network. The parallelization strategy used is to decompose the problem domain into geographical patches and assign each
processor to do the computation associated with a distinct subset of the patches. With this decomposition, the physics calculations involve only grid points and data local to a processor and are
performed in parallel. Using...
this paper, we review the various parallel algorithms used in PCCM2 and the work done to arrive at a validated model. 2. THE NCAR CCM Over the past decade, the NCAR Climate and Global Dynamics
Division has provided a comprehensive, three-dimensional global atmospheric model to university and NCAR scientists for use in the analysis and understanding of global climate. Because of its
widespread use, the model was designated a Community Climate Model (CCM). The most recent version of the CCM, CCM2, was released to the research community in October 1992 (Hack et al. 1992; Bath,
Rosinski, and Olson 1992). This incorporates improved physical representations of a wide range of key climate processes, including clouds, radiation, moist convection, the planetary boundary layer,
and transport. Changes to the pa- 0.1 1 10 100 1000 1 10 100 1000
Choice of an appropriate strategy for balancing load in climate models running on par-allel processors depends on the nature and size of inherent imbalances. Physics routines of the NCAR Community
Climate Model were instrumented to produce per-cell load data for each time step, revealing load imbalance resulting from surface type, polar night, weather patterns, and the earth's terminator.
Hourly, daily, and annual cycles in processor perfor-mance over the model grid were also uncovered. Data from CCM1 suggested a number of static processor allocation strategies.
The communication performance of the i860-based Intel DELTA mesh supercomputer is compared with the Intel iPSC/860 hypercube and the Ncube 6400 hypercube. Single and multiple hop communication
bandwidth and latencies are measured. Concurrent communication speeds and speed under network load are also measured. File I/O performance of the mesh-attached Concurrent File System is measured.
A one-level, global, spectral model using the primitive equations is formulated in terms of a concise form of the prognostic equations for vorticity and divergence. The model integration incorporates
a grid transform technique to evaluate nonlinear terms; the computational efficiency of the model is found to be far superior to that of an equivalent model based on the traditional interaction
coefficients. The transform model, in integrations of 116 days, satisfies principles of conservation of energy, angular momentum, and square potential vorticity to a high degree.
This book explains how many major scientific algorithms can be used on large parallel machines. Based on five years of research on hypercubes, the book concentrates on practically motivated model
problems, that serve to illustrate generic algorithmic and decomposition techniques. The authors include results for hypercube-class concurrent computers with up to 128 nodes, and the principles
behind the extrapolation to much larger systems are described.
The spectral transform method is a standard numerical technique used to solve partial differential equations on the sphere in global climate modeling. In particular, it is used in CCM1 and CCM2, the
Community Climate Models developed at the National Center for Atmospheric Research. This paper describes initial experiences in parallelizing a program that uses the spectral transform method to
solve the non-linear shallow water equations on the sphere, showing that an efficient implementation is possible on the Intel iPSC/860. The use of PICL, a portable instrumented communication library,
and Paragraph, a performance visualization tool, in tuning the implementation is also described. The Legendre transform and the Fourier transform comprise the computational kernel of the spectral
transform method. This paper is a case study of parallelizing the Legendre transform. For many problem sizes and numbers of processors, the spectral transform method can be parallelized efficiently
by parallelizing only the Legendre transform.
A suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry. The shallow water equations exhibit the
major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth. These cases are designed for use in the evaluation of numerical methods proposed
for climate modeling and to identify the potential trade-offs which must always be made in numerical modeling. Before a proposed scheme is applied to a full baroclinic atmospheric model it must
perform well on these problems in comparison with other currently accepted numerical methods. The cases are presented in order of complexity. They consist of advection across the poles, steady state
geostrophically balanced flow of both global and local scales, forced nonlinear advection of an isolated low, zonal flow impinging on an isolated mountain, Rossby-Haurwitz waves, and observed
atmospheric states. One of the cases is also identified as a computer performance/algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers.
GMD and ECMWF (European Centre for Medium-Range Weather Forecasts) joined forces some months ago in order to parallelize ECMWF's production code for medium-range weather forecasts, the Integrated
Forecasting System (IFS). Meanwhile, the first milestone of this cooperation has been reached: The 2D model of the IFS, which contains already all relevant data structures and algorithmic components
of the corresponding 3D models, has been parallelized and run successfully on quite a large variety of differential parallel machines. Performance measurements confirm the expected parallel
efficiencies of up to 80% and more. This paper discusses the parallelization strategy employed and gives a survey on the performance results obtained on the parallel systems.
We describe the design of a parallel global atmospheric circulation model, PCCM2. This parallel model is functionally equivalent to the National Center for Atmospheric Research's Community Climate
Model, CCM2, but is structured to exploit distributed memory multi-computers. PCCM2 incorporates parallel spectral transform, semi-Lagrangian transport, and load balancing algorithms. We present
detailed performance results on the IBM SP2 and Intel Paragon. These results provide insights into the scalability of the individual parallel algorithms and of the parallel model as a whole.
One issue which is central in developing a general purpose FFT subroutine on a distributed memory parallel machine is the data distribution. It is possible that different users would like to use the
FFT routine with different data distributions. Thus there is a need to design FFT schemes on distributed memory parallel machines which can support a variety of data distributions. In this paper we
present an FFT implementation on a distributed memory parallel machine which works for a number of data distributions commonly encountered in scientific applications. We have also addressed the
problem of rearranging the data after computing the FFT. We have evaluated the performance of our implementation on a distributed memory parallel machine, the Intel iPSC/860.
In a hypercube multiprocessor with distributed memory, messages have a street address and an apartment number, i.e., a hypercube node address and a local memory address. Here we describe an optimal
algorithm for performing the communication described by exchanging the bits of the node address with that of the local address. These exchanges occur typically in both matrix transposition and bit
reversal for the fast Fourier transform.
Several multiprocessor FFTs are developed in this paper for both vector multiprocessors with shared memory and the hypercube. Two FFTs for vector multiprocessors are given that compute an ordered
transform and have a stride of one except for a single ‘link’ step. Since multiple FFTs provide additional options for both vectoriation and distribution we show that a single FFT can be performed in
terms of two multiple FFTs and development of algorithms that minimize interprocessor communication. On a hypercube of dimension d the unordered FFT requires d + 1 parallel transmissions. The ordered
FFT requires from 1.5d + 2 to 2d + 1 parallel transmissions depending on the length of the sequence. It is also shown that a class of orderings called index-digit permutations which includes matrix
transposition, the perfect shuffle, and digit reversal can be performed with less than or equal to 1.5d parallel transmissions.
The performance of the Intel iPSC/860 hypercube and the Ncube 6400 hypercube are compared with earlier hypercubes from Intel and Ncube. Computation and communication performance for a number of
low-level benchmarks are presented for the Intel iPSC/1, iPSC/2, and iPSC/860 and for the Ncube 3200 and 6400. File I/O performance of the iPSC/860 and Ncube 6400 are compared.
Analytic and empirical studies are presented that allow the parallel performance, and hence the scalability, of the spectral transform method to be quantified on different parallel computer
architectures. Both the shallow-water equations and complete GCMs are considered. Results indicate that for the shallow-water equations, parallel efficiency is generally poor because of high
communication requirements. It is predicted that for complete global climate models, the parallel efficiency will be significantly better; nevertheless, projected teraflop computers will have
difficulty achieving accpetable throughput necessary for long-term regional climate studies. -from Authors
A modified version of the Fast Fourier Transform is developed and described. This version is well adapted for use in a special-purpose computer designed for the purpose. It is shown that only three
operators are needed. One operator replaces successive pairs of data points by their sums and differences. The second operator performs a fixed permutation which is an ideal shuffle of the data. The
third operator permits the multiplication of a selected subset of the data by a common complex multiplier. If, as seems reasonable, the slowest operation is the complex multiplications required,
then, for reasonably sized date sets—e.g. 512 complex numbers—parallelization by the method developed should allow an increase of speed over the serial use of the Fast Fourier Transform by about two
orders of magnitude. It is suggested that a machine to realize the speed improvement indicated is quite feasible. The analysis is based on the use of the Kronecker product of matrices. It is
suggested that this form is of general use in the development and classification of various modifications and extensions of the algorithm.
Conventional algorithms for computing large one-dimensional fast Fourier transforms (FFTs), even those algorithms recently developed for vector and parallel computers, are largely unsuitable for
systems with external or hierarchical memory. The principal reason for this is the fact that most FFT algorithms require at least m complete passes through the data set to compute a 2m-point FFT.
This paper describes some advanced techniques for computing an ordered FFT on a computer with external or hierarchical memory. These algorithms (1) require as few as two passes through the external
data set, (2) employ strictly unit stride, long vector transfers between main memory and external storage, (3) require only a modest amount of scratch space in main memory, and (4) are well suited
for vector and parallel computation. Performance figures are included for implementations of some of these algorithms on Cray supercomputers. Of interest is the fact that a main memory version
outperforms the current Cray library FFT routines on the Cray-2, the Cray X-MP, and the Cray Y-MP systems. Using all eight processors on the Cray Y-MP, this main memory routine runs at nearly two
The original Cooley-Tukey FFT was published in 1965 and presented for sequences with length N equal to a power of two. However, in the same paper they noted that their algo- rithm could be
generalized to composite N in which the length of the sequence was a pro- duct of small primes. In 1967, Bergland presented an algorithm for composite N and vari- ants of his mixed radix FFT are
currently in wide use. In 1968, Bluestein presented an FFT for arbitrary N including large primes. However, for composite N , Bluestein's FFT was not competitive with Bergland's FFT. Since it is
usually possible to select a composite N , Bluestein's FFT did not receive much attention. Nevertheless because of its minimal com- munication requirements, the Bluestein FFT may be the algorithm of
choice on multipro- cessors, particularly those with the hypercube architecture. In contrast to the mixed radix FFT, the communication pattern of the Bluestein FFT maps quite well onto the hypercube.
With P = 2d processors, an ordered Bluestein FFT requires 2d communication cycles with packet length N ∕2P which is comparable to the requirements of a power of two FFT. For fine-grain computations,
the Bluestein FFT requires 20log2N computational cycles. Although this is double that required for a mixed radix FFT, the Bluestein FFT may nevertheless be preferred because of its lower
communication costs. For most values of N it is also shown to be superior to another alternative, namely parallel matrix multiplication.
We outline a unified approach for building a library of collective communication operations that performs well on a cross-section of problems encountered in real applications. The target architecture
is a two-dimensional mesh with worm-hole routing, but the techniques also apply to higher dimensional meshes and hypercubes. We stress a general approach, addressing the need for implementations that
perform well for various sized vectors and grid dimensions, including non-power-of-two grids. This requires the development of general techniques for building hybrid algorithms. Finally, the approach
also supports collective communication within a group of nodes, which is required by many scalable algorithms. Results from the Intel Paragon system are included
Fairness is an important issue when benchmarking parallel computers using application codes. The best parallel algorithm on one platform may not be the best on another. While it is not feasible to
re-evaluate parallel algorithms and reimplement large codes whenever new machines become available, it is possible to embed algorithmic options into codes that allow them to be “tuned” for a
particular machine without requiring code modifications. We describe a code in which such an approach was taken. PSTSWM was developed for evaluating parallel algorithms for the spectral transform
method in atmospheric circulation models. Many levels of runtime-selectable algorithmic options are supported. We discuss these options and our evaluation methodology. We also provide empirical
results from a number of parallel machines, indicating the importance of tuning for each platform before making a comparison
Two complete exchange algorithms for meshes are given. The modified quadrant exchange algorithm is based on the quadrant exchange algorithm and it is well suited for square meshes with a power of two
rows and columns. The store-and-forward complete exchange algorithm is suitable for meshes of arbitrary size. A pipelined broadcast algorithm for meshes is also presented. This new algorithm, called
the double hop broadcast, can broadcast long messages at slightly lower cost than the edge-disjoint fence algorithm because it uses routing trees of lower height. This shows that there is still room
for improvement in the design of pipelined broadcast algorithms for meshes
The scalability of the parallel fast Fourier transform (FFT) algorithm on mesh- and hypercube-connected multicomputers is analyzed. The hypercube architecture provides linearly increasing performance
for the FFT algorithm with an increasing number of processors and a moderately increasing problem size. However, there is a limit on the efficiency, which is determined by the communication bandwidth
of the hypercube channels. Efficiencies higher than this limit can be obtained only if the problem size is increased very rapidly. Technology-dependent features, such as the communication bandwidth,
determine the upper bound on the overall performance that can be obtained from a P-processor system. The upper bound can be moved up by either improving the communication-related parameters linearly
or increasing the problem size exponentially. The scalability analysis shows that the FFT algorithm cannot make efficient use of large-scale mesh architectures. The addition of such features as
cut-through routing and multicasting does not improve the overall scalability on this architecture
The authors present the scalability analysis of a parallel fast Fourier transform (FFT) algorithm on mesh and hypercube connected multicomputers using the isoefficiency metric. The isoefficiency
function of an algorithm architecture combination is defined as the rate at which the problem size should grow with the number of processors to maintain a fixed efficiency. It is shown that it is
more cost-effective to implement the FFT algorithm on a hypercube rather than a mesh despite the fact that large scale meshes are cheaper to construct than large hypercubes. Although the scope of
this work is limited to the Cooley-Tukey FFT algorithm on a few classes of architectures, the methodology can be used to study the performance of various FFT algorithms on a variety of architectures
such as SIMD hypercube and mesh architectures and shared memory architecture
Given a vector of N elements, the perfect shuffle of this vector is a permutation of the elements that are identical to a perfect shuffle of a deck of cards. Elements of the first half of the vector
are interlaced with elements of the second half in the perfect shuffle of the vector. We indicate by a series of examples that the perfect shuffle is an important interconnection pattern for a
parallel processor. The examples include the fast-Fourier transform (FFT), polynomial evaluation, sorting, and matrix transposition. For the FFT and sorting, the rate of growth of computational steps
for algorithms that use the perfect shuffle is the least known today, and is somewhat better than the best rate that is known for versions of these algorithms that use the interconnection scheme used
in the ILLIAC IV. Copyright © 1971 by The Institute of Electrical and Electronics Engineers, Inc.
This paper investigates the suitability of the spectral transform method for parallel implementation. The spectral transform method is a natural candidate for general circulation models designed to
run on large-scale parallel computers due to the large number of existing serial and moderately parallel implementations. We present analytic and empirical studies that allow us to quantify the
parallel performance, and hence the scalability, of the spectral transform method on different parallel computer architectures. We consider both the shallow-water equations and complete GCMs. Our
results indicate that for the shallow-water equations parallel efficiency is generally poor because of high communication requirements. We predict that for complete global climate models, the
parallel efficiency will be significantly better; nevertheless, projected Teraflop computers will have difficulty achieving acceptable throughput necessary for long-term regional climate studies. 1
Introduction Current ...
The spectral transform method is a standard numerical technique for solving partial differential equations on the sphere and is widely used in global climate modeling. In this paper, we outline
different approaches to parallelizing the method and describe experiments that we are conducting to evaluate the efficiency of these approaches on parallel computers. The experiments are conducted
using a testbed code that solves the nonlinear shallow water equations on a sphere, but are designed to permit evaluation in the context of a global model. They allow us to evaluate the relative
merits of the approaches as a function of problem size and number of processors. The results of this study are guiding ongoing work on PCCM2, a parallel implementation of the Community Climate Model
developed at the National Center for Atmospheric Research. 1 Introduction Parallel algorithms for computing the spectral transform method used in climate models can be divided into two general
classes. Tr...
Implementation of the NCAR CCM2 on the Connection Machine
R. D. Loft and R. K. Sato, Implementation of the NCAR CCM2 on the Connection Machine, in Parallel Supercomputing in Atmospheric Science: Proceedings of the Fifth ECMWF Workshop on Use of Parallel
Processors in Meteorology, G.-R. Ho man and T. Kauranne, eds., World Scienti c Publishing Co. Pte. Ltd., Singapore, 1993, pp. 371{ 393.
On the parallelization of global spectral Eulerian shallow-water models
S. Barros and K. T, On the parallelization of global spectral Eulerian shallow-water models, in Parallel Supercomputing in Atmospheric Science: Proceedings of the Fifth ECMWF Workshop on Use of
Parallel Processors in Meteorology, G.-R. Ho man and T. Kauranne, eds., World Scienti c Publishing Co. Pte. Ltd., Singapore, 1993, pp. 36{43.
The ECMWF model on the Cray Y-MP8, in The Dawn of Massively Parallel Processing in
D. Dent, The ECMWF model on the Cray Y-MP8, in The Dawn of Massively Parallel Processing in Meteorology, G.-R. Hooman and D. K. Maretis, eds., Springer-Verlag, Berlin, 1990.
• G C Fox
• M A Johnson
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• S W Otto
• J K Salmon
• D W Walker
G. C. Fox, M. A. Johnson, G. A. Lyzenga, S. W. Otto, J. K. Salmon, and D. W. Walker, Solving Problems on Concurrent Processors, vol. 1, Prentice-Hall, Englewood Cli s, NJ, 1988.
Scalability estimates of parallel spectral atmospheric models
T. Kauranne and S. Barros, Scalability estimates of parallel spectral atmospheric models, in Parallel Supercomputing in Atmospheric Science: Proceedings of the Fifth ECMWF Workshop on Use of Parallel
Processors in Meteorology, G.-R. Ho man and T. Kauranne, eds., World Scienti c Publishing Co. Pte. Ltd., Singapore, 1993, pp. 312{328.
On the parallelization of global spectral Eulerian shallowwater models
S. BARROS AND T. KAURANNE, On the parallelization of global spectral Eulerian shallowwater models, in Parallel Supercomputing in Atmospheric Science: Proceedings of the Fifth ECMWF Workshop on Use of
Parallel Processors in Meteorology, G.-R. Hoffman and T. Kauranne, eds., World Scientific, Singapore, 1993, pp. 36-43.
An efficient, one-level, primitive-equation spectral model, Monthly Weather Review
W. BOURKE, An efficient, one-level, primitive-equation spectral model, Monthly Weather Review, 102 (1972), pp. 687-701. | {"url":"https://www.researchgate.net/publication/2759834_Parallel_Algorithms_For_The_Spectral_Transform_Method","timestamp":"2024-11-08T09:38:40Z","content_type":"text/html","content_length":"1028460","record_id":"<urn:uuid:20195079-2dfb-4e72-8297-3a5b42f78775>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00203.warc.gz"} |
How To Begin Bringing Rich and Inclusive Math History Resources Inside K to 12 Classrooms
Say their names:
Al-Khwarizmi. Aryabhata. Bhaskara. Germain. Agnesi. Galois. Zhenyi. Mirzakhani. Shijie. Al-Haytham. Nightingale. Hypatia. Pingala. Noether. Lovelace. Easley. Brahmagupta. Uhlenbeck.
Whether you realize it or not, simply reading and saying the names of the mathematicians in the thumbnail is an important process in cultivating identity and attribution. Each of those 18 people have
a story to tell that contributes to the massive and seemingly endless braid of the history of mathematics. And yet, simultaneously — and paradoxically — they only represent a mere fraction of the
lore of mathematics.
Regardless, individually and collectively, they represent potential spaces for connection, identity, and belonging for our students and teachers. The diversity of faces, clothing, customs, and
religions not only contributes to the kaleidoscope of mathematics, it provides a powerful snapshot for students to quickly see the magnitude of time in the thematic development of mathematics.
Each of those 18 profile pictures were pulled from this beautiful “Timeline of Mathematics” found at Mathigon. The time line is populated with mathematicians and important artifacts of mathematics.
But, more importantly, it is not a static resource. It is constantly being updated with more mathematicians, more stories, and just more magic. As you explore this free resource, you will notice that
situating students immediately in some kind of rich, historical narrative — giving origin and purpose — is what anchors all the courses.
There is much mythology and mysticism in mathematics. That in itself is a tantalizing allure to dive deeply into the rich history of mathematics.
For the past few years, I have written extensively about an urgency in broadening our ongoing discussions in math equity with actual mathematics from various cultures, races, and civilizations.
My last four articles related to the topic can be found below:
This article is all about resources(I will continually update this article with them) And, regardless of what grade you teach, there is something here for any teacher to begin the long, but wonderful
journey of bringing inclusiveness and more anti-racism ideas into our classrooms.
Here is a wonderful reflection piece by author and teacher, Alice Aspinall. It is a short, but powerful read, to firmly establish the vector that all of us need to move in the direction of for the
future of math education.
We all know that when children first start school, they are enamoured with storytelling time. Just the idea of sitting on a floor, anticipating a story from your teacher, are some of our favorite
memories of school.
With much gratitude to the people at Math Minds, a wonderful flip book is available for K-1 teachers, called Disappearing Moon, based on the oldest known math artefact, the Lebombo bone. Another
story that has a historical element is Cubey Cake, a nod to how a young Gauss added up numbers.
For middle school, Buzzmath has been teaching students about math history for over 10 years, through their Missions, a gamefied part of their platform that not only immerses students in a historical
context, but also promotes resilience with challenging problems. Free access to one of their Missions can be found here.
As well,a link to their math history cards can be found here.
What is important about math history is that serves the purpose of fostering deep curiosity. Storytelling plays a part in that cycle.
Amplify is a company that is working with this immersive process in bringing the power of narrative/storytelling to math classrooms.
A link to the math profile cards pictured above can be found here.
In addition, if you would like to read more about the importance of bringing math history inside our classrooms, an ebook that I authored can be found here.
I am working closely with Amplify to ensure that compelling mathematical stories are fixtures in the platform — that they are inextricably baked right into the curriculum. A complete sample unit can
be found here. As well as a link to feedback and interest in participating in fields trials of these narrative-rich units.
What is also critical when using math history as a resource, are the multitude of unsolved — yes, unsolved — problems that can be shared with students to bring more context, relevance, and admiration
for the work that mathematicians do. Thanks to Gord Hamilton(@gamesbygord), he has curated the small fraction which are accessible to our K to 12 students. Click on the link below.
There are, of course, some wonderful books that I would recommend to start adding to your library of math history/storytelling resources.
The Crest of the Peacock
This book(third edition) is the gold standard to find the roots of mathematics and correctly relay the trajectories of where we are today — especially with scholarly attribution of mathematical
The Puzzle Universe
I was lucky to come across this book a few years ago, and it does a beautiful job of showing the history of mathematics through some well-known — and some others not as much — puzzles. There is a
rich commentary in the opening of the book, and each puzzle has an answer with a detailed solution.
The Math Book
If this book were hardcover, you would put it on your coffee table! Absolutely gorgeous with rich illustrations, the book is a chronological tour of some of the most important discoveries in
mathematics. Obviously not an all encompassing compendium, but has a nice balance in its time line of unpacking some of mathematics’ most major milestones.
Ying and the Magic Turtle
As always, where and how children get introduced to the imagination of mathematics lies is beautiful storytelling. As you can see, I was one of the reviewers of the above book, as were some highly
respectable math educators from North America. If you are an elementary teacher, this one is definitely a must have.
One of our generation’s best storytellers in mathematics is Marcus du Sautoy, a world-renowned author and professor of mathematics at Oxford University. He narrated the brilliant 4 part series, The
Story of Maths, which is a must watch.
Speaking of “must watches”, I would strongly recommend that you watch Keith Devlin, Stanford University, and his enthusiastic storytelling of Algebra. In a time where the debates regarding algebra as
a school topic are distilled down to a tasteless arguments about usefulness, Devlin tells a rich and compelling story of its birth and migration, indirectly supporting the notion that algebra —
taught in a rich, historical context — is an invaluable area of knowledge that not only progressed mathematics, but society as a whole.
As exhaustive as this article might seem with resources, it is meant only as an initial invite into our journey in this magic labyrinth of stories, that when braided together, rightfully color
mathematics with magic, mysticism, and mythology.
To be continued…! | {"url":"https://sunilsingh-42118.medium.com/how-to-begin-bringing-rich-and-inclusive-math-history-resources-inside-k-to-12-classrooms-2d0a4162e6fc","timestamp":"2024-11-08T04:43:31Z","content_type":"text/html","content_length":"172578","record_id":"<urn:uuid:7cb9b7f6-1d56-4f33-910f-7e097a83b42e>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00380.warc.gz"} |
Difference between Self induced & Mutually induced emf | Instrumentation and Control Engineering
Difference between Self induced & Mutually induced emf
If an emf is induced without moving either the conductor or the flux, such an In transformers and reactors static emf is induced emf is called statically induced emf. This is classified into two
types.1) Self induced emf ( Current changes in the coil itself ) 2) Mutually induced emf ( Action of neighbouring coil )
Self Induced EMF :
It is defined as the emf induced in the coil due to increase or decrease of the current in the same coil. If the current is constant no. emf is induced. When a current is passed to a circuit due to
self induced emf the flow of current in the circuit is opposed .
Mutually induced emf
Coil ‘B’ is connected to galvanometer. coil ‘A’ is connected to a cell. The two coils are placed close together. The coil connected to a supply is called primary coil A. The other coil ‘B’ is called
secondary coil. The coil in which emf is induced by mutual induction is called secondary coil. When current through coil ‘A’ is established by closing switch ‘S’ then its magnetic field is setup
which partly links with or threads through the coil ‘B’. As current through ‘A’ is changed the flux linked with ‘B’ is also changed. Hence mutually induced emf is produced in ‘B’ whose magnitude is
given by faraday’s law and direction by lenz’s law. The property of inducing emf in one coil due to change of current in other coil placed near the former is called mutual induction and this property
is used in Transformers and Induction coils.
Inductance :
Inductance is defined as the property of the coil due to which it opposes the change of current in the coil. This is due to lenz’s law.
Self Inductance :
Self inductance is defined as the webers turns / ampere of the coil and is denoted by the letter ‘L’ and its units as Henry( H). The expression of self Induction by definition. is
N = No. of turns of a coil.
I = Current in Amps
L = Self Inductance
φ = Flux in webers
Mutual Inductance :
When current in coil ‘A’ changes, The changing flux linking coil ‘B’. Induces emf in coil ‘B’ and is known as mutually induced emf. Mutual inductance between two coils ‘A’ and ‘B’ is the flux
linkages of one coil B due to one ampere of outset in the other coil ‘A’
Let N1 = No. of turns of coil ‘A’
N2 = No. of turns of coil ‘B’
I1 = current in coil A
I2 = current in coil B.
A = Area of cross section of coil.
2 1 φ , φ = Flux linking with coil A and B.
Hence by definition of expression of mutual inductance ( m) = N2 φ2 henry
Also read | {"url":"https://automationforum.co/difference-between-self-induced-mutually-induced-emf/","timestamp":"2024-11-06T20:23:33Z","content_type":"text/html","content_length":"214748","record_id":"<urn:uuid:695df148-f69d-4f6a-9aba-86ca314f5d0a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00142.warc.gz"} |
EE 473
Introduction to Artificial Intelligence
┃Professor: │Bob Givan ┃
┃Office: │EE 313C ┃
┃Phone: │7248-235-567 (backwards) (text message only, any time) ┃
┃Office hours: │follow this link ┃
┃Login: │givan@purdue.edu ┃
┃Web page: │http://engineering.purdue.edu/~givan/ ┃
┃Fun: │Spotify playlists ┃
┃Teaching Assistant:│Ziyu Gong ┃
┃Email: │gong123@purdue.edu ┃
┃Teaching Assistant:│Jeeyung Kim ┃
┃Email: │jkim17@purdue.edu ┃
Please follow this link for our office hour schedule: GTA/UTA office hours
---------- Students registered for ECE 473 may access the following pages using their Purdue login----------
Latex Resources
Sample LaTeX document template. To compile it into a pdf, you can use an online compiler like overleaf, or install a LaTeX compiler on your machine.
Handwritten symbol recognition: detexify
Cheat sheet for Latex symbol abbreviations: cheat sheet
The welcome video, syllabus, FAQ, and first lecture video are available below. This course will be held in a flipped-classroom style, with:
• Online lecture videos,
• Actively answered Piazza blog,
• Homework due each Friday including the first week graded mostly by completion,
• Tuesday in-person meetings for question-and-answer (recorded, optional attendance), and
□ There is no Tuesday meeting the first week
□ The Tuesday in-person meetings will be replaced by zoom sessions for the duration of the Omicron wave.
• Thursday during class timeslot — remote unproctored BrightSpace checkpoint quizzes.
□ There is no Thursday quiz the first week
Please join the class Piazza blog and monitor all postings and announcements there.
Note: Homework due in Week n on Friday is typically associated with lectures to be watched in week n-1 and a quiz in week n+1
Week 1 — January 10 - 14
There are no meetings (no quiz and no Q&A) the first week. See the welcome video and syllabus below for more detail.
Please join the class Piazza blog and monitor all postings and announcements there.
The first homework is due Friday 1/14 at noon and you must watch the welcome video and the first lecture video (about one hour) before working on that homework. Since this is a fast start, we will
waive the usual late penalty for this homework, accepting submissions without penalty until Saturday noon (NONE accepted later than that). The fast start will enable us to moderate the pace (just a
little) at more critical times of the semester.
Course Syllabus
Welcome video
lecture schedule section below)
Homework 1 Python warmup and AI teaser   (Due on Brightspace: Friday January 14 noon, but for this one time no late penalty, accepted no later than Saturday noon)
Week 2 — January 17 - January 21
lecture schedule section below)
Q&A recording
Brightspace quiz on AI History and Python warmup. You must start this quiz between 1:30pm and 2pm
Homework 2 - Dynamic programming and math preliminaries   (Due on Brightspace: Friday January 21, noon, late homework accepted WITH penalty until Saturday noon)
Week 3 — January 24 - January 28
lecture schedule section below)
Q&A recording
Brightspace quiz on dynamic programming and math preliminaries. You must start this quiz between 1:30pm and 2pm
Homework 3 - Machine learning introduction (updated Jan 22 10:02am)   (Due on Brightspace: Friday January 28, noon, late homework accepted WITH penalty until Saturday noon)
Week 4 — January 31 - February 4
lecture schedule section below)
Q&A recording
Brightspace quiz on machine learning introduction. You must start this quiz between 1:30pm and 2pm
Homework 4 - Classification and overfitting (updated Jan 31 10:21am)   (Due on Brightspace: Friday February 4, noon, late homework accepted WITH penalty until Saturday noon)
Week 5 — February 7 - February 11
lecture schedule section below)
Q&A recording
Brightspace quiz on classification and overfitting. You must start this quiz between 1:30pm and 2pm
Homework 5 - K means   (Due on Brightspace: Friday February 11, noon, late homework accepted WITH penalty until Saturday noon) Project (Extended homework) 7 released - Local search   (Due on
Brightspace: Friday February 25, noon, late homework accepted WITH penalty until Saturday noon)
• Team selection due Friday February 11, noon
Week 6 — February 14 - February 18
lecture schedule section below)
Q&A recording
Brightspace quiz on K means and Nearest neighbor. You must start this quiz between 1:30pm and 2pm
Homework 6 - Uniform cost search (minor update 2/14/22 10:50am)   (Due on Brightspace: Friday February 18, noon, late homework accepted WITH penalty until Saturday noon)
Week 7 — February 21 - February 25
lecture schedule section below)
Q&A recording
Brightspace quiz on breadth and depth-focused search methods. You must start this quiz between 1:30pm and 2pm
Project (Extended homework) 7 released - Local search   (Due on Brightspace: Friday February 25, noon, late homework accepted WITH penalty until Saturday noon)
• Team selection was already due Friday February 11, noon. If you did not submit you are working as an individual (which should be fine)
• Python template for submission (updated 2/22 1pm to add random. package indicators)
• Zip file for draft grader and sample problems (updated Sun Feb 20 11:32am)
Week 8 — February 28 - March 4
lecture schedule section below)
Brightspace quiz on informed search and adversarial search. You must start this quiz between 1:30pm and 2pm
Homework 8 - Informed and adversarial Search   (Due on Brightspace: Friday March 4, noon, late homework accepted WITH penalty until Saturday noon)
Week 9 — March 7 - March 11
lecture schedule section below)
Brightspace quiz on Homework 8 and CSP / Bayes net lectures. You must start this quiz between 1:30pm and 2pm
Homework 9 - Introduction to Markov Decision Processes   (Due on Brightspace: Friday March 11, noon, late homework accepted WITH penalty until Saturday noon) Project (Extended homework) 11
released - Sokoban   (Due on Brightspace: Friday April 1, noon, late homework accepted WITHOUT penalty until Saturday noon)
• Team selection due Thursday March 10, noon. If you do not submit you are working as an individual (which should be fine)
• Zip file of programming files (updated 7:44pm Sunday 3/20 with summary statistics printing for 'all', see Piazza post)
Week 10 — March 21 - March 25
lecture schedule section below)
Q&A recording
Brightspace quiz on Homework 9 and Markov decision processes. You must start this quiz between 1:30pm and 2pm
Homework 10 - Q learning   (Due on Brightspace: Friday March 25, noon, late homework accepted WITH penalty until Saturday noon)
Project (Extended homework) 11 released above under week 9 - Sokoban   (Due on Brightspace: Friday April 1, noon, late homework accepted WITHOUT penalty until Saturday noon)
Week 11 — March 28 - April 1
lecture schedule section below)
Q&A recording
Brightspace quiz on Homework 10 and Q learning. You must start this quiz between 1:30pm and 2pm
Extended Homework 11 - Sokoban   (Due on Brightspace: Friday April 1, noon, late homework accepted WITHOUT penalty until Saturday noon)
Week 12 — April 4 - April 8
lecture schedule section below)
Q&A recording
Brightspace quiz on Homework 11 and policy gradient / TD learning. You must start this quiz between 1:30pm and 2pm
Homework 12 Neural networks and Sokoban minimum standard   (Due on Brightspace: Friday April 8, noon, late homework accepted WITHOUT penalty until Saturday noon)
• Fifty percent of the grade for homework 12 is from passing the minimum standard for Homework 11 by Friday April 8, Noon (no late submission, all or nothing). Please note that experience shows
that passing this standard, evaluated against our hidden problems, requires you to submit your code to our evaluator many times, incorporating our feedback along the way. Submit sokoban.py and
design.txt to Homework 11b on BrightSpace for feedback via the leaderboard linked to on Piazza, early and often, in order to succeed.
Week 13 — April 11 - April 15
lecture schedule section below)
Q&A recording
Brightspace quiz on Homework 12 and neural networks introduction. You must start this quiz between 1:30pm and 2pm
Homework 13 Bayesian network inference (updated Monday 5pm, slightly simplifying problem 4) (all written problems)   (Due on GradeScope: Friday April 15, noon, late homework accepted WITH penalty
until Saturday noon)
Week 14 — April 18 - April 22
This is our last week of lectures and homeworks. The quiz for hw14 will be during the first half our of our final exam window, but will still be a self-proctored BrightSpace quiz like all the others.
lecture schedule section below)
Q&A recording
Brightspace quiz on Homework 13 and Bayesian network inference. You must start this quiz between 1:30pm and 2pm
Homework 14 Structured neural networks (updated 12:10pm Monday 4/18 to add explanation) (all written problems)   (Due on GradeScope: Friday April 22, noon, late homework accepted WITH penalty
until Saturday noon)
Weeks 15-16 — April 25 - May 6
We have only the Q&A this week, with quiz 14 during finals week in our final exam slot. Here you reap the time rewards of our fairly intense pace to date
Brightspace quiz on Homework 14 and structured neural networks. You must start this quiz between 1pm and 1:30pm
Online lecture and viewing schedule
Online Lecture 1: AI History   (1 hour, viewing recommended by Wednesday January 12)
Lecture 2: Dynamic programming — Part 1: Optimal substructure (24 minutes)   (Viewing recommended by Saturday, January 15)
Lecture 3: Dynamic programming — Part 2: Repeated subproblems (33 minutes)   (Viewing recommended by Monday, January 17)
Lecture 4: Dynamic programming — Part 3: Solution extraction (19 minutes)   (Viewing recommended by Monday, January 17)
Lecture 5: Machine learning — Part 1: Introduction (40 minutes)   (Viewing recommended by Saturday, January 22)
Lecture 6: Machine learning — Part 2: A basic loss function (26 minutes)   (Viewing recommended by Monday, January 24)
Lecture 7: Machine learning — Part 3: Linear regression (32 minutes)   (Viewing recommended by Monday, January 24)
Lecture 8: Machine learning — Part 4: Classification introduction (24 minutes)   (Viewing recommended by Saturday, January 29)
Lecture 9: Machine learning — Part 5: Decision boundaries (13 minutes)   (Viewing recommended by Saturday, January 29)
Lecture 10: Machine learning — Part 6: Logistic regression (39 minutes)   (Viewing recommended by Saturday, January 29)
Lecture 11: Machine learning — Part 7: Softmax (11 minutes)   (Viewing recommended by Monday, January 31)
Lecture 12: Machine learning — Part 8: Generalization (28 minutes)   (Viewing recommended by Monday, January 31)
Lecture 13: Machine learning — Part 9: Nearest neighbor learning (8 minutes)   (Viewing recommended by Saturday, February 5)
Lecture 14: Machine learning — Part 10: Unsupervised learning and K means (27 minutes)   (Viewing recommended by Saturday, February 5)
Lecture 15: Local search (47 minutes)   (Viewing recommended by Tuesday, February 8)
Lecture 16: State space search — Part 1: Breadth-focused methods (40 minutes)   (Viewing recommended by Saturday, February 12)
Lecture 17: State space search — Part 2: Depth-focused methods (25 minutes)   (Viewing recommended by Monday, February 14)
Lecture 18: State space search — Part 3: Informed methods (36 minutes)   (Viewing recommended by Saturday, February 19)
Lecture 19: State space search — Part 4: Adversarial methods (54 minutes)   (Viewing recommended by Monday, February 21)
Lecture 20: State space search — Constraint satisfaction (65 minutes)    (Viewing recommended by Saturday, February 26)
Lecture 21: Bayesian networks — Introduction (49 minutes)    (Viewing recommended by Monday, February 28)
Lecture 22: Markov decision processes — Introduction (22 minutes)    (Viewing recommended by Saturday, March 5)
Lecture 23: Markov decision processes — Policy evaluation (69 minutes)    (Viewing recommended by Saturday, March 5)
Lecture 24: Markov decision processes — Optimal value and optimal policies (60 minutes)    (Viewing recommended by Monday, March 7)
Lecture 25: Markov decision processes — Q learning (68 minutes)    (Viewing recommended by Monday, March 21)
Lecture 26: Markov decision processes — Policy gradient methods (17 minutes)    (Viewing recommended by Saturday, March 26)
Lecture 27: Markov decision processes — TD Learning and alpha Zero (54 minutes)    (Viewing recommended by Tuesday, March 29)
Lecture 28: Neural networks — Introduction (27 minutes)    (Viewing recommended by Monday, April 4)
Lecture 29: Bayesian networks — D-separation (41 minutes)    (Viewing recommended by Saturday, April 9)
Lecture 30: Bayesian networks — polytree inference (46 minutes)    (Viewing recommended by Monday, April 11)
Lecture 31: Neural networks — Convolution (57 minutes)    (Viewing recommended by Saturday, April 16)
Lecture 32: Neural networks — Recurrent networks (31 minutes)    (Viewing recommended by Monday, April 18)
This concludes the lectures for Spring 2022. All lectures have been released.
Maintained by Bob Givan and course staff | {"url":"https://engineering.purdue.edu/~ee473/","timestamp":"2024-11-03T15:22:44Z","content_type":"text/html","content_length":"36235","record_id":"<urn:uuid:5e06f674-6f73-4010-9d5d-9d0a8fcf6f8b>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00024.warc.gz"} |
Normal Force Calculator & Formula - Symbolab
About Normal Force Calculator
• The Normal Force Calculator is an online tool designed to help users compute the normal force acting upon an object in various situations. It encompasses the understanding of a crucial concept in
classical mechanics - the normal force, which is instrumental in the study of statics and dynamics problems. By providing essential parameters, users can quickly determine the value of the normal
force which can be pivotal in understanding other vital aspects of physics, such as friction, tension, and acceleration.
• At its core, the normal force is a force that acts perpendicular to the contact surface between two bodies. In simpler terms, it counteracts the force with which one body presses against another.
One of the most common examples is the normal force exerted by the ground on a stationary object placed on it due to gravity. In this case, the object pushes down on the ground with a force equal
to its weight, and the ground pushes back with an equal and opposite force - the normal force.
• In order to calculate the normal force, users must provide some essential input parameters to the Normal Force Calculator. These parameters may include the object's mass, gravitational
acceleration (approximately 9.81 m/s² on Earth), the angle between the contact surface and the horizontal plane (also denoted by the angle of inclination), and, in some cases, external forces
applied to the object. The calculator then processes these given inputs and delivers the normal force as output.
• The Normal Force Calculator can prove particularly helpful when dealing with inclined surfaces, as calculating the normal force in these situations can be more complicated than flat surfaces.
When an object is on a sloped plane, the gravitational force acting on the object can be decomposed into two components: one that acts parallel to the inclined surface and another that acts
perpendicular to the inclined surface. The latter component counterbalances the normal force, necessitating the consideration of the angle of inclination in the calculations.
• To compute the normal force on an incline, the Normal Force Calculator incorporates the mass of the object, the angle of inclination, and the gravitational acceleration by using the following
• Normal Force (N) = Mass (m) × Gravitational Acceleration (g) × cos(Angle)
• It is important to remember that while the calculator provides an efficient means to determine the normal force, it does not directly account for other forces that might be acting on the object
simultaneously. It may be essential to consider factors such as applied external forces, tension in connecting cables, or friction between the objects, each of which may have a direct impact on
the calculation of the normal force.
• In conclusion, the Normal Force Calculator is a valuable tool to ascertain the normal force acting on an object under various conditions quickly. It simplifies the process by allowing users to
input essential parameters, in turn delivering accurate and timely results. As it primarily focuses on the normal force, users must be wary of other forces that may come into play in certain
situations. Understanding and correctly calculating the normal force is indispensable in solving many classical mechanics problems and can pave the way for a deeper comprehension of the
fascinating field of physics. | {"url":"http://en.symbolab.com/calculator/physics/normal-force","timestamp":"2024-11-10T04:51:54Z","content_type":"text/html","content_length":"163514","record_id":"<urn:uuid:c3cec39e-c6d7-43f0-a70f-ebf63f0e5bcd>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00723.warc.gz"} |
Banach Spaces - (Variational Analysis) - Vocab, Definition, Explanations | Fiveable
Banach Spaces
from class:
Variational Analysis
A Banach space is a complete normed vector space, which means it is a vector space equipped with a norm that allows for the measurement of vector lengths and distances, and every Cauchy sequence in
the space converges to a limit within the space. This concept plays a crucial role in variational analysis as it provides a structured environment for discussing continuity, compactness, and
convergence, all of which are important in optimization and fixed point theories.
congrats on reading the definition of Banach Spaces. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Every Banach space is also a normed space, but not every normed space is complete, making completeness a defining characteristic of Banach spaces.
2. Common examples of Banach spaces include spaces of continuous functions and sequence spaces like `l^p` for `1 <= p < infinity`.
3. The completeness property allows for the application of fixed point theorems, which are essential for finding solutions to equations and optimization problems.
4. In variational analysis, Banach spaces facilitate the use of various convergence concepts, allowing practitioners to prove results involving compactness and continuity effectively.
5. Many optimization algorithms rely on the structure provided by Banach spaces to ensure convergence to optimal solutions in infinite-dimensional settings.
Review Questions
• How does the concept of completeness in Banach spaces enhance the study of convergence within variational analysis?
□ Completeness in Banach spaces ensures that every Cauchy sequence converges to an element within the same space. This characteristic allows researchers to work confidently with sequences and
series in variational analysis, knowing that limits exist within the space. This greatly enhances the robustness of theoretical results and algorithms related to optimization and fixed point
• In what ways do Banach spaces provide essential tools for optimization problems compared to other types of vector spaces?
□ Banach spaces offer a complete framework that is critical for applying various optimization techniques. Unlike general vector spaces, where limits may not exist within the space, Banach
spaces guarantee that every Cauchy sequence converges. This completeness allows for the application of fixed point theorems and variational principles that are vital in proving the existence
and uniqueness of solutions in optimization problems.
• Evaluate how current research trends in variational analysis might leverage the properties of Banach spaces to address open problems.
□ Current research trends in variational analysis often involve exploring new applications or extending existing theories related to Banach spaces. Researchers might investigate properties like
reflexivity or separability within Banach spaces to develop more robust optimization methods or fixed point results. By addressing open problems through this lens, they can potentially
discover new connections between these mathematical structures and real-world applications, particularly in fields like functional analysis and applied mathematics.
"Banach Spaces" also found in:
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website. | {"url":"https://library.fiveable.me/key-terms/variational-analysis/banach-spaces","timestamp":"2024-11-11T01:48:26Z","content_type":"text/html","content_length":"151793","record_id":"<urn:uuid:870001d2-2714-4683-a0cb-21e14135dbe5>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00497.warc.gz"} |
Linear Regression Using Stochastic Gradient Descent in Python
Anay Pant
As Artificial Intelligence is becoming more popular, there are more people trying to understand neural networks and how they work. To illustrate, neural networks are computer systems that are
designed to learn and improve, somewhat correlating to the human brain.
An example of a convolutional neural network. Each neural network takes a certain amount of inputs and returns a certain amount of outputs. In this neural network, there are 3 columns of nodes in the
middle. Each column is called a hidden layer. There is no exact answer to how many layers a neural network needs to have. To learn more about neural networks, see this video by 3Blue1Brown to learn
In machine learning, there are many different types of algorithms that are used for different tasks. The one I will show you guys today is what is known as Linear Regression. In Linear Regression,
the expected input(s) is/are a number, as well for the outputs.
One common example of Linear Regression in everyday life is finding an appropriate value for housing value prices. In this network, there would be multiple inputs (different features of the house),
and there would be only one output(the estimated price). Some inputs could be the square feet of the house, the number of bedrooms, number of bathrooms, age in days of the house, etc. The output
would then be the appropriate value of the house.
First things first, how will we use the weights (synapses) of our model to find an approximated output? We will do this by using the simple formula:
This formula is a simple yet effective one. For anyone who didn’t know, in this equation, the output(y) is equal to the input(x) times the slope(m) + a bias or y-intercept(b).
For our Linear Regression model, we will fine-tune two parameters in this equation: the slope, and the bias.
Coding a Linear Regression Model
Now, we will get to see our Linear Regression model in Python!
You will NOT need any external libraries/packages for this model. However, this is a very simplistic version, so feel free to add your own twist to the code.
Let’s start with the code:
lass linearregression():
def __init__(self, x, y, iteration):
self.x = x
self.y = y
self.epoch = iteration
self.m = 1
self.b = 0
self.learn = 0.01
This is the first part of our code. Here, we are initializing a class in python. To describe, a class is like a ‘type’ in python. From this class, one can use it to create multiple ‘objects’.
For more in-depth info on classes, visit this video by CS Dojo:
In this class, we are taking in 3 inputs from the user: the X value, the Y value, and the iterations.
If we were to perceive this from a coordinate plane, the x value would be the x values, and the y values would be the corresponding y coordinates on the plane. Each input is a list of elements, with
each corresponding pair of elements creating a coordinate (like in geometry!)
The iterations, in this case, is the number of loops we want our model to train. I have set it to 1000, but you can change that number.
We are also setting some initial values in the __init__ function here. Our most important values are the values m and b. These two values are the tunable values (they will change based on the
correction we give them.)
Let’s move on to the next part of our model!
def feed def feedforward(self):
self.output = []
for i in range(len(self.x)):
self.output.append(self.x[i] * self.m + self.b)
In this code, we are creating a feedforward function. Feedforward means to “try out” your model, even if it is not trained. Out “outputs” here are stored in the according to list. (Also notice our
equation bring put to use here!)
Feedforward sets the values and necessary steps in order to find out what our model thinks. Then, we will see how wrong we are and learn from it.
Onto the next part of our code!
def calculate_error(self):
self.totalError = 0
for i in range(len(self.x)):
self.totalError += (self.y[i] - self.output[i]) ** 2
self.error **= 1/2
self.error = float(self.totalError / float(len(self.x)))
return self.totalError / float(len(self.x))
In this code, we are calculating the error of our model!
We initialize a total error in this function. This variable will hold the error of our code. How do we do this? Well, we take the average (mean) of all the errors!
In python terms, we iterate with a “for” loop. Through each iteration, we add the error of the true value (y) minus our model’s predicted value (output). We now have the error from our model.
## In this function, you may notice that we are squaring the error value of each iteration! Why? In the function, we want all of the error values to be positive (negative * negative = positive).
Furthermore, it is not as much the “value” of the error that we seek, but the “magnitude”. When squared, the magnitude of an error increases.’
Now, we will move on to the most challenging piece of code:
Stochastic Gradient Descent!!
def gradient_descent(self):
self.b_grad = 0
self.m_grad = 0
N = float(len(self.x))
for i in range(len(self.x)):
self.b_grad += -(2/N) * (self.y[i] - ((self.m * self.x[i]) + self.b))
self.m_grad += -(2/N) * self.x[i] * (self.y[i] - ((self.m * self.x[i]) + self.b))
self.m -= self.learn * self.m_grad
self.b -= self.learn * self.b_grad
Now, I know this looks tricky (it's not when you get used to it!)
Let’s take this one step at a time.
Stochastic Gradient Descent is a function that iterates over a value and swiftly minimizes error. How? Basically, it creates a mini bath size that then gets manipulated by our model. We try out
multiple different ways to optimize them, then find the quickest one.
Here is a picture of the formula used in stochastic gradient descent:
At first, I was lost with this subject, so I referred to Wikipedia and other websites to help me :
First, we initialize two main factors: the m_grad (gradient for slope), and the b_grad (gradient for b). These will be the minimalized values computed by our function here.
To make it not as mind-boggling as it looks in the picture above, I’ll try my best to make it easier.
FOR THE BIAS (B): We have out equation y = m * x + b, aka our outputs. We then find the simple error of y minus the output and multiply it by a preset derivative found by the geniuses of the world(-2
/N, where N is the number of elements in our list).
FOR THE SLOPE (M): This one is literally the same as the formula for the bias, but we multiply the output by the inputs[x]. We iterate over EVERY input for both, so we add to each of them (this is
why we are NOT using the error variable from earlier right now, as it only has positive values.)
And that’s stochastic gradient descent! (Kind of… )
If you are still clueless (and I don’t blame you), visit this video by 3blue1brown.
def backprop(self):
self.error = self.calculate_error()
This is pretty simple code; we are just calling our earlier functions and grouping them into this backpropagation function.
Backpropagation is the generic term for finding the error of your model, and then making the necessary changes to learn from it.
Now… THE FINAL FULL (also available on my github page at the beginning)
import matplotlib.pyplot as plt
class linearregression():
def __init__(self, x, y, iteration):
# Initializing some variables
self.x = x
self.y = y
self.epoch = iteration
# These are the tuned parameters to make our model better
self.m = 1
self.b = 0
# This is how fast we want out model to grow
self.learn = 0.01
def feedforward(self):
self.output = []
for i in range(len(self.x)):
self.output.append(self.x[i] * self.m + self.b)
def calculate_error(self):
self.totalError = 0
self.trueError = 0
for i in range(len(self.x)):
self.totalError += (self.y[i] - self.output[i]) ** 2 # The reason we square is for all error values to be positive.
self.error = float(self.totalError / float(len(self.x)))
return self.totalError / float(len(self.x))
def gradient_descent(self):
self.b_grad = 0
self.m_grad = 0
N = float(len(self.x))
for i in range(len(self.x)):
self.b_grad += -(2/N) * (self.y[i] - ((self.m * self.x[i]) + self.b))
self.m_grad += -(2/N) * self.x[i] * (self.y[i] - ((self.m * self.x[i]) + self.b))
self.m -= self.learn * self.m_grad
self.b -= self.learn * self.b_grad
def backprop(self):
self.error = self.calculate_error()
def predict(self):
while True: # This can be taken off if you don't want to predict values forever
self.user = input("\nInput an x value. ")
self.user = float(self.user)
self.ret = self.user * self.m + self.b
print("Expected y value is: " + str(self.ret))
def train(self):
for i in range(self.epoch):
Long code, right? Well, not really.
Anyways, to sum it up, this linear regression model can take in many inputs and return an output with the right value! For more info look into my Github. | {"url":"https://corp.aiclub.world/post/linear-regression-in-python","timestamp":"2024-11-04T04:37:22Z","content_type":"text/html","content_length":"1050482","record_id":"<urn:uuid:8f52ccfd-f729-4617-87bf-f80fba3d302d>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00670.warc.gz"} |
Estimating Dry Weight of Composition Material
Posted on by kpmartin Posted in Composition ink rollers, Equipment Acquisition, Repair, and Maintenance, Kevin, Testing Equipment — No Comments ↓
While trying to remelt some old composition rollers recently, I found that some of the material wouldn’t melt. Neither higher temperatures nor additional water helped. All I ended up with was a lumpy
soup, from which I strained the lumps and let the liquid portion set in a measuring cup.
I was curious as to how much of the original 250g of composition material I has managed to recover, but the sample in the cup contained quite a bit of extra water.
I had two choices: let the sample air dry until it no longer lost any weight, or sample the weight through a period of consistent drying conditions and use the changing weight to estimate the final
dry weight.
The second method might require less time so I decided to try it. I cut off a about quarter of the hardened composition material and placed it on a piece of aluminum foil on the pan of my precision
scale, which has a resolution of 100μg. The scale has a glass enclosure which I hoped would provide a consistent drying rate.
(warning: Math ahead!)
Drying is a diffusion process: the water must diffuse through to the surface of the material, transfer to the air, and the moisture in the air must diffuse into the atmosphere at large. In such
processes, the rate of transfer between two regions depends on the difference in moisture content between them. The result is that the weight of a sample will approach its equilibrium value according
to a formula:
$m⁡(t) = m eq + ( m s - m eq ) ⁢ ⅇ - R ⁢ ( t - ts )$
where $m⁡(t)$ is the sample mass as a function of time, and $m s$ is the mass measured at some sampling time $ts$. The rate of drying is represented by $R$ and $m eq$ is the ultimate equilibrium
By making measurements of the weight over time this would give several sample points for $m⁡(t)$, any of which being used to provide $m s$ and $ts$. Using this set of sample points, values for $R$
and $m eq$ could be estimated, with the latter being the ultimate dry weight of the sample.
I did not have any way of fitting the equation of this form to samples, so I transformed the samples and equations into a linear problem, first by taking the time derivative of the function, and
using the differences between consecutive samples to approximate the rate of drying at the middle of the time interval:
$d m d t = -R ⁢ ( m s - m eq ) ⁢ ⅇ - R ⁢ ( t - ts )$
fitted to values $m i - m i-1 t i - t i-1$ at time $t i + t i-1 2$ for pairs of consecutive samples.
To complete the linearization, the logarithm of both the rate samples and $- d m d t$ is taken. Although the algebra doesn’t care, $m⁡(t)$ is decreasing with time and so $d m d t$ is negative and
the logarithm of $m i - m i-1 t i - t i-1$ will be a complex number which is not easy to graph. Taking the logarithm of the negated value keeps the calculations in real numbers.
$ln⁡ ( - d m d t ) = [ ln⁡ ( R ⁢ ( m s - m eq ) ) + R ⁢ ts ] - R ⁢ t$
Note that the value in square brackets is a constant. By obtaining parameters $A$ and $B$ for a linear fit to the data in the form
$ln⁡ ( - d m d t ) = A ⁢ t + B$
we see that
$R = - A$
$m eq = m s - ⅇ B - R ⁢ t s R = m s + ⅇ B + A ⁢ t s A$
Substituting these values back into the original definition of $m⁡(t)$ yields the formula in terms of the linear fit parameters $A$ and B:
$m⁡(t) = m s + ⅇ B + A ⁢ t s A - ⅇ B A ⁢ ⅇ A ⁢ t$
The sum of the first two terms is equal to $m eq$ and the last is independent of which sampling is used for $m s$ and $ts$. The choice of which sample(s) are used to determine $m eq$ is somewhat
arbitrary. In this analysis I chose to take the mean value $m eq ‾$ calculated from all the samples. A better way might be to weight the later samples more heavily as they are closer to the
ultimate equilibrium state.
I measured the sample weight more or less daily over several weeks to see how well this method worked. Here is a chart of $ln ⁡ m i - m i-1 t i - t i-1$ as a function of $t i + t i-1 2$ along with
a fitted linear function.
Note that as the sample dries, the day-to-day weight change gets smaller so factors like error in the weigh scale and drying rate variations start to dominate the measured values, causing the charted
values to become wilder as time progresses. In particular, the linear function would be a closer fit (with a steeper slope) without the last 3 samples.
The coefficients for the linear fit equation are:
$A$ = -0.102 and $B$ = 0.1
This implies that
$R$ = 0.102 and $m eq ‾$ = 20.73g
These values were plugged back into the original formula for $m⁡(t)$ and charted against the actual measured values:
Because of the even weighting used to calculate $m eq ‾$ the somewhat absurd effect can be seen that the predicted ultimate weight of 20.73g is greater than actual measured weights from
later samples (the last sample weight was 20.36g, and at the time of writing this article, the weight was 19.71g at day 81).
The slower drop of the calculated value in the first 20 days is the result of using the last three “wild” samples in the linear fit model, as discussed above. By taking a linear fit ignoring the last
3 samples, the results are $A$ = -.1525, $B$ = 0.7, and $m eq ‾$ = 21.13g (averaged including the last 3 samples). This gives a much closer fit between calculated and actual weights up to day
35, but $m eq ‾$ is even higher than the actual ultimate weight. These discrepancies are most likely due to the assumption of a single stable equilibrium weight, while in actuality we have
progressed from early fall to early winter, resulting in decreasing air humidity, and so $m eq$ has been a moving target.
Weight vs. Mass
In this analysis I appear to use “weight” and “mass” almost interchangeably. My scale measures weight, but the formulas and analysis work with mass, and I tried to keep the wording correct to reflect
this. In practice, the factor relating weight and mass is the earth’s gravity, which is not considered to vary substantially from week to week, and although the scale measures weight, it reads in
mass units assuming a standard value for gravity.
Writing this post: MathML vs. WordPress
HTML (the descriptive language for Web pages) includes extensions for displaying equations and such constructs, called MathML. Unfortunately, WordPress, which is the basis for this blog, just can’t
seem to leave unrecognized HTML alone and strips it out. As a result I can’t just write the MathML in raw form using the text-mode WP editor because it all vanishes.
Instead I’ve added a plugin, called “WordPress HTML”, which allows pristine literal HTML to be included in posts, but at the expense of having to define these literal bits as named objects, out of
line from where they’re used, giving them a (hopefully) meaningful name, and referencing that name with somewhat clunky syntax in the main text of the post. Even ignoring the fact that writing raw
MathML is somewhat painful to do, having to name things and place them out-of-line doesn’t help matters in the least!
The only plus to this is that the MathML for something like $d m d t$ only occurs once no matter how many times I mention $d m d t$ in the main article text. But even at that, it cannot be referenced
in other math definitions, so for instance $- d m d t$ includes a copy of the definition of $d m d t$.
Note that this post was updated October 2023 to account for the deprecated <mfenced> no longer being implemented in at least some web browsers, which caused important parentheses to disappear from
the visible content.
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Summary on INCR 2023
1. Introduction
In October 2023, the International Numerical Modeling Contest of Rock Mechanics and Engineering (INCR 2023) was successfully organized. For the first time, INCR implemented a rigorous
pre-registration system. Teams from various countries or regions, including China, China-Hong Kong, the United Kingdom, Singapore, Australia, and more, participated in the contest, with a total of 80
registered teams. Out of these, 47 teams submitted their final result documents, with 39 teams working on Task A (Rock Failure) and 8 teams working on Task B (Particle Flow). Various numerical
methods or software were utilized in INCR 2023, including ANSYS, ABAQUS, PFM, PD, FLAC, RFPA, DICE2D, PFC, DDA, DLSM, 4D-LSM, CDEM, FEM-SPH, among others. To evaluate the results submitted by the
participating teams, a panel of experts in rock mechanics was convened.
2. Scoring
The scoring rules are as follows:
The competition is divided into two groups, Task A and Task B. Tasks both have a maximum score of 100, and the scores of tasks are composed as follows:
Task A comprises two sections: writing (weight 20%) and results (weight 80%). The results section is further divided into calibration (weight 25%) and prediction (weight 75%). The calibration and
prediction are further divided into two parts: the failure image and the simulation data, with weights of 30% and 70% respectively. The score composition for Task B is similar to it for Task A.
However, based on the characteristics of Task B, only the simulation data is included in the calibration and prediction sections. Finally, the percentage of each item's score in the total score is as
Based on the characteristics of the scoring items, they are divided into two categories: subjective and objective scores. Among them: 1. The subjective score of Task A accounts for 44%, and the
objective score accounts for 56%; 2. The subjective score of Task B accounts for 20%, and the objective score accounts for 80%.
The scoring method for subjective scores is as follows: a review team consisting of five experienced peers will be formed to score each item of the submitted documents. The scores are mainly divided
into five levels: A (perfect), B (very good), C (good), D (acceptable), and N (no data provided). These five levels correspond to scores of 90, 80, 70, 60, and 0, respectively. The subjective score
of the document will be calculated based on the review results.
The scoring method for the objective score is as follows: 1. Calculate the relative error based on the simulated data and experimental data provided by the document; 2. Calculate the cumulative
distribution probability of the relative error for all participating teams; 3. Give 100 points for errors with cumulative distribution probability in the top 10%; For errors with a cumulative
distribution probability of 10% to 80%, linear interpolation is used to calculate the score; If data is provided, a score of at least 60 points will be given. If no data is provided, a score of 0
will be given; 4. Calculate the objective score.
3.Award list
On October 28, 2023 (Beijing Time), the award list of INCR 2023 has been announced at ICADD16 in Chengdu, China.
Thanks to the groups that provided the competition questions and awarded the Outstanding Contribution Awards.
Rank the teams of Task A and Task B separately. | {"url":"http://m.dembox.org/cn/nd.jsp?mid=351&id=184&groupId=3","timestamp":"2024-11-06T14:03:30Z","content_type":"text/html","content_length":"113030","record_id":"<urn:uuid:21c8e0b9-b2e6-4d4b-9e3b-5b4726766ecb>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00217.warc.gz"} |
How to Find Standard DeviationHow to Find Standard Deviation
Standard deviation measures how much the values in a data set vary from the mean. It helps describe the spread or dispersion of a data set and for comparing the variability of different data sets.
In this blog post, I will explain what standard deviation means, how to find or calculate it for populations and samples, and how to interpret it in different contexts.
What is Standard Deviation?
Standard deviation is a statistic that tells you how closely the values in a data set are clustered around the mean.
A low standard deviation means that most values are close to the mean, while a high standard deviation means that the values are spread over a broader range.
For example, suppose you have two data sets of test scores:
1. Data set A: 80, 82, 84, 86, 88
2. Data set B: 60, 70, 80, 90, 100
Both data sets have the same mean of 84 but different standard deviations. Data set A has a standard deviation of 2.83, while set B has a standard deviation of 14.14.
This means the values in Data Set A are more consistent and less variable than those in Data Set B.
Standard deviation can help you understand how representative the mean is of the data set and how likely it is to find values that are far from the mean.
For example, if you have a data set with a low standard deviation, you can be more confident that the mean summarizes the data well and that most values are close to the mean.
On the other hand, if you have a data set with a high standard deviation, you can expect to find more outliers and extreme values, and the mean may not be a good indicator of the typical value.
Standard Deviation Formulas for Populations and Samples
There are two formulas for calculating standard deviation, depending on whether you deal with a population or a sample.
A population is the entire group of interest, while a sample is a subset of the population used to make inferences about the population.
The formula for the standard deviation of a population is:
SD population = ∑ | x − μ | 2 N
– SD population is the standard deviation of the population
– ∑ means “sum of”
– x is a value in the population
– μ is the mean of the population
– N is the number of values in the population
The formula for the standard deviation of a sample is:
SD sample = ∑ | x − x ¯ | 2 n − 1
– SD sample is the standard deviation of the sample
– ∑ means “sum of”
– x is a value in the sample
– x ¯ is the mean of the sample
– n is the number of values in the sample
The difference between the two formulas is that the sample formula uses n − 1 instead of N in the denominator.
This is because the sample mean estimates the population mean, and using n − 1 makes the standard deviation more accurate and unbiased.
Standard Deviation Calculator
You can use a standard deviation calculator to calculate standard deviation quickly and easily. This tool takes your data as input and computes your standard deviation. You can find a standard
deviation calculator online, such as [this one] (^1^).
To use the standard deviation calculator, you must enter your data values, separated by commas, and choose whether to calculate the standard deviation for a population or a sample.
Then, click the “Calculate” button, and the calculator will display your data’s mean and standard deviation.
For example, if you enter the data set A from the previous section (80, 82, 84, 86, 88) and choose the sample option, the calculator will show you the following results:
1. Mean: 84
2. Standard deviation: 3.16
Steps for Calculating the Standard Deviation by Hand
If you want to calculate the standard deviation by hand, you can follow these steps:
1. Find the mean of your data set. To do this, add up all the values and divide by the number of values. For example, if your data set is 6, 2, 3, 1, the mean is (6 + 2 + 3 + 1) / 4 = 3.
2. For each value in your data set, find the difference between the value and the mean and square the difference. For example, for the value 6, the difference is 6 − 3 = 3, and the square is 3^2 = 9.
3. Add up all the squared differences. This is called the sum of squares. For example, if your squared differences are 9, 1, 0, 4, the sum of squares is 9 + 1 + 0 + 4 = 14.
4. Divide the sum of squares by the number of values (for a population) or by the number of values minus one (for a sample). This is called the variance. For example, if you have a sample of 4
values, the variance is 14 / (4 − 1) = 4.67.
5. Take the square root of the variance. This is the standard deviation. For example, the square root of 4.67 is 2.16.
Why is Standard Deviation a Useful Measure of Variability?
Standard deviation is a useful measure of variability because it has several advantages over other measures, such as range or interquartile range. Some of these advantages are:
• Standard deviation considers all the values in the data set, not just the extreme or middle values.
• Standard deviation is easy to interpret and compare, as it has the same unit as the original data.
• Standard deviation is widely used in statistics and other fields and is often required for specific calculations and tests.
See; How to Multiply Fractions
Limitations of Standard Deviation
The standard deviation has a lot of uses and advantages. However, standard deviation also has some limitations, such as:
• Standard deviation is sensitive to outliers, values significantly different from the rest of the data. Outliers can inflate the standard deviation and make it seem more significant than it is.
• Standard deviation is only meaningful for data following a normal distribution, a symmetrical bell-shaped curve. For skewed data or with multiple peaks, the standard deviation may not be a good
measure of variability.
Therefore, when using standard deviation, you should always check the shape of your data distribution and look for any outliers or anomalies that may affect your results. | {"url":"https://www.checkhowto.com/find-standard-deviation/","timestamp":"2024-11-01T23:12:06Z","content_type":"text/html","content_length":"148439","record_id":"<urn:uuid:f6054389-a850-4579-98c3-8715c4cab49a>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00428.warc.gz"} |
Solving 2-Step Variable Equations - ppt video online download
2 Two Step Equations Essential QuestionHow are inverse operations used to solve two step equations? Why does order matter when solving two step equations?
3 What??? I just learned 1-step!Relax. You’ll use what you already know to solve 2-step equations.
4 Solving Two-Step EquationsTo find the solution to an equation we must isolate the variable. In other words, get the variable on one side of the equation by itself. We isolate the variable by
performing operations that will eliminate (cancel) the other numbers from the expression.
5 Solving Two-Step EquationsWe have seen how to eliminate a constant (Addition Property of Equality) & how to eliminate a coefficient (Multiplication Property of Equality). What if an equation has
both a constant and a coefficient to be eliminated? This is called a Two-Step Equation.
6 So, what’s first? You need to break up the equation into its two steps.
7 + - STEP 1 The first step will always be either adding orsubtracting. (Think: Order of Operations backwards) + -
8 ÷ • STEP 2 The second step will always be eithermultiplying or dividing. ÷ •
9 8x + 5 = 61 - 5 - 5 = 56 8x Let’s try a problem.This problem has addition, so we need to subtract first. Whatever we do on one side, we have to do on the other. = 56 8x
10 Now, Step 2. 8x = 56 Whatever we do on one side, we have to do on the other. This problem has multiplication, so we need to divide now. 8 8 x 7 =
11 Let’s try another problem.+ 8 + 8 This problem has subtraction, so we need to add first. Whatever we do on one side, we have to do on the other. = 12 3a
12 Now, Step 2. 3a = 12 Whatever we do on one side, we have to do on the other. This problem has multiplication, so we need to divide now. 3 3 a 4 =
13 Let’s try another problem.z + 2 = 26 6 Whatever we do on one side, we have to do on the other. This problem has addition, so we need to subtract first. - 2 - 2 z 6 = 24
14 Now, Step 2. z = 24 6 6 • • 6 Whatever we do on one side, we have to do on the other. This problem has division, so we need to multiply now. z 144 =
15 Let’s try one more problem.5 Whatever we do on one side, we have to do on the other. This problem has subtraction, so we need to add first. +10 +10 m 5 = 25
16 Now, Step 2. • 5 m = 25 5 5 • Whatever we do on one side, we have to do on the other. This problem has division, so we need to multiply now. m 125 =
17 Trick Question Does order matter when solving two step equations? Why or Why not? Order does matter because if you do any other math operation other than addition or subtraction first, you will
end up with an incorrect solution | {"url":"https://slideplayer.com/slide/7767034/","timestamp":"2024-11-14T04:35:26Z","content_type":"text/html","content_length":"170764","record_id":"<urn:uuid:33102c1f-286f-4f4a-b46f-206249ee45b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00299.warc.gz"} |
Accrued Interest: Understanding and Calculation
Accrued interest refers to the amount of interest earned by a lender on a loan or debt that has not been received yet. It arises when a borrower fails to make payments for the interest on a loan,
resulting in accumulated interest that becomes due. Accrued interest is an accounting concept used to recognize the interest income for a period even if it's not collected yet. It's represented as a
current asset or liability on a company's balance sheet and considered as income or expenses. | {"url":"https://desklib.com/document/blog-accrued-interest/","timestamp":"2024-11-13T01:14:28Z","content_type":"text/html","content_length":"319130","record_id":"<urn:uuid:8d5832fc-04e8-455b-b8d6-8c21f95bf38e>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00036.warc.gz"} |
Number Nightmare
Question 10 of 10
10. How many pennies do I have in my piggie bank?
Question 9 of 10
9. What is the total number of answers in this quiz that are prime numbers? The answer to question 10 is prime.
Question 8 of 10
8. What number less than 60, is the product of two of the answers in this quiz? My systolic blood pressure divided by the difference between the two digits in this number is equal to the two digit
integer on the back of my softball jersey.
Question 7 of 10
7. What is the product of two of the answers in this quiz?
Question 6 of 10
6. What prime number is less than the answer to question 2?
Question 5 of 10
5. What is the answer to question 2 plus the answer to question 3 plus the answer to question 4? No answer in this quiz is the number one less than this answer.
Question 4 of 10
4. What is the answer to question 2 plus the answer to question 3?
Question 3 of 10
3. What number's digits add up to 9?
Question 2 of 10
2. What is the sum of the square roots of two of the answers in this quiz? Both numbers in the sum are integers.
Question 1 of 10
1. What is the sum of all answers in this quiz that are the square of
integer values? This answer is the third largest number in this quiz.
Quiz Answer Key and Fun Facts
1. What is the sum of all answers in this quiz that are the square of integer values? This answer is the third largest number in this quiz.
2. What is the sum of the square roots of two of the answers in this quiz? Both numbers in the sum are integers.
3. What number's digits add up to 9?
4. What is the answer to question 2 plus the answer to question 3?
5. What is the answer to question 2 plus the answer to question 3 plus the answer to question 4? No answer in this quiz is the number one less than this answer.
6. What prime number is less than the answer to question 2?
7. What is the product of two of the answers in this quiz?
8. What number less than 60, is the product of two of the answers in this quiz? My systolic blood pressure divided by the difference between the two digits in this number is equal to the two digit
integer on the back of my softball jersey.
9. What is the total number of answers in this quiz that are prime numbers? The answer to question 10 is prime.
10. How many pennies do I have in my piggie bank?
Source: Author
This quiz was reviewed by FunTrivia editor
before going online.
Any errors found in FunTrivia content are routinely corrected through our feedback system. | {"url":"https://www.funtrivia.com/trivia-quiz/BrainTeasers/Number-Nightmare-142249.html","timestamp":"2024-11-07T23:19:32Z","content_type":"text/html","content_length":"71906","record_id":"<urn:uuid:ba2bb90a-b6d5-4e27-a43e-e668b035a995>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00224.warc.gz"} |
How to find the altitude of a triangleHow to find the altitude of a triangleHow to find the altitude of a triangle
Finding area of triangle using Heron's formula
In this video you will
Learn the formula to find the area of a triangle when you don’t know the altitude
How to apply Heron's formula in order to find the area of a triangle.
Watch example math problems using Heron's formula to find the area of a triangle.
Sing a cheesy song to help you remember Heron's formula.
Learn how to find the altitude in a triangle.
Today we are going to look at Heron’s formula. This is a formula to find the area of a triangle when you don’t know the altitude but you do know the three sides. So here is our example. You have
sides of 5, 6, and 7 in a triangle but you don’t know the altitude and you don’t have a way to. I’m going to share with you the formula, and then I’m going to run through an example of it. Heron’s
formula is when you have an unknown altitude but you know all the sides. The first step is to find S, S is the variable by adding all three sides up and divide by two. Then you take that S plug it
into the formula as follows , the square root of S times S minus A ( which is the first side) times S minus B times S minus C ( which is the third side) It looks complicated but it is very easy.
Multiple all this together and take the square root. That is how you find the area of a triangle. So let’s work this out. The first step is to find S So I have to take A plus B plus C so I’m going to
going to call 5 a 6 b and 7 is c after adding those up divide by two. When I add those up I get 18 and 18 divided by 2 is 9 so 9 is my magic s I then plug this S into the formula. You will have 9
times s minus A which is( 9 minus 5) times S minus B which is (9 minus 6 )times S minus C which is( 9 minus 7 )so I’m finding the difference between S for these three sides. This works out to be 9
times 4 times 3 times 2. Now when I multiply all of this out I get 216. I next have to take the square root of 216 So grab your calculator and the square root of 216 is 14,69 or you could round this
to 14.7 square units. That is how you use Herron’s formula to find the area of a triangle. Now I’m going to do a sing song thing to remember Herron’s formula. It is the square root of s times s minus
A times S minus B times S minus C equals the area of a triangle without an altitude. So sing along and you will have that down to. Hope this was helpful.
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DISCRETE - Data Driven Optimization Models For Secure Real-Time Operation Of Renewable Dominated Power Systems - EnergyVille
DISCRETE – Data Driven Optimization Models For Secure Real-Time Operation Of Renewable Dominated Power Systems
The operational management of the Belgian transmission system is increasingly challenging due to the increased penetration of renewable energy sources and the European electricity market integration.
DISCRETE will answer fundamental research questions on the applicability of new data driven optimization models and uncertainty modelling to minimize the total cost of operation and CO[2 ]emissions.
DISCRETE will allow the development of new decision support tools, enabling secure power system operation.
The operational complexity of the power system increases steadily, imposed by the intermittency and uncertainty of renewable generation sources and the complexity associated with the trade of
electricity in different markets and across borders. Especially the uncertainty of renewable power generation has two important consequences for power system operation.
Firstly, during the determination of operational plans, usually in day-ahead, the renewable generation uncertainty needs to be considered, in order to determine the range of preventive and corrective
actions to ensure secure grid operation in real time. Classically, preventive and corrective actions are determined using optimisation models based on the Monte Carlo approach where the power
injections from renewables are represented by a number of different samples. An optional power flow (OPF) calculation for each sample is performed to determine the necessary actions under the
assumption of the failure of critical grid elements. With an increasing amount of renewable power generation in the system, the number of calculations grows in order to cover the increased
uncertainty space of renewable power generation, and the correlations between different injection sources, e.d. wind farms. Using the classical approach, we will reach the boundaries of computational
means to construct such operational plans in day-ahead. This will lead to oversimplifications in the optimisation and thus jeopardizes the secure operation of the power sytem.
Secondly, there is an inherent mismatch between the projected renwable generation used in operational plans and the actual renewable generation in real time. This requires generation redispatch to be
performed in intraday operation in order to balance the system on the one hand, and also solve congestion in the grid due to unexpected power flows, on the other. The redispatch cost related to
real-time congestion management is in the range of a few million Euros per year in Belgium (due to the presence of nuclear power). We can expect that a power system dominated by renewables as
envisaged in 2050 will result in redispatch costs which are 1-2 magnitudes higher. The German power system gives already today a good insight of how such a system would look like, as the redispatch
costs exceed one billion Euros per year (a factor 1000 compared to Belgium), although the size of the power system is only 10 times larger than the Belgian system.
DISCRETE aims to tackle challenges in optimisation models and uncertainty modelling, in order to pave the way for decision support tools for system operators. The main objectives of DISCRETE can thus
be summarized as follows:
1. Improving uncertainty quantification in operational planning models to minimize redispatch costs and CO[2] emissions stemming therefrom.
2. Providing data-driven optimisation models for decision support for congestion management and secure power system operation.
With the support of the Energy Transition Fund | {"url":"https://energyville.be/en/project/discrete-data-driven-optimization-models-for-secure-real-time-operation-of-renewable-dominated-power-systems/","timestamp":"2024-11-09T00:59:07Z","content_type":"text/html","content_length":"120072","record_id":"<urn:uuid:384a4c6e-4f31-4377-a445-090d7aa7793d>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00491.warc.gz"} |
Momentum of a photon from energy in context of photon momentum
31 Aug 2024
Title: The Momentum of a Photon: A Relativistic Perspective
Abstract: In the realm of quantum mechanics and special relativity, the concept of photon momentum has been extensively explored. This article delves into the theoretical framework that relates the
energy of a photon to its momentum, providing an in-depth analysis of the underlying principles.
The theory of special relativity, proposed by Albert Einstein, postulates that the laws of physics remain invariant under transformations between inertial frames of reference. One of the fundamental
consequences of this theory is the equivalence of mass and energy, as expressed by the famous equation E = mc^2. In the context of photons, which are massless particles, this relationship takes on a
unique form.
The Energy-Momentum Relationship:
For a photon with energy E and momentum p, the relativistic energy-momentum relation can be written as:
E^2 = (pc)^2
where c is the speed of light in vacuum. This equation demonstrates that the energy of a photon is directly proportional to its momentum.
Derivation from Special Relativity:
To derive this relationship from special relativity, we consider a photon traveling at the speed of light along the x-axis. The four-momentum of the photon can be represented as:
p = (E/c, p)
where E/c is the energy density and p is the momentum.
Using the Lorentz transformation for a boost in the x-direction, we obtain:
E’ = γ(E - vp) p’ = γ(p - vE/c^2)
where γ is the Lorentz factor. In the limit where v approaches c, the second term on the right-hand side of the momentum equation vanishes, leaving us with:
E’ = 0 p’ = p
This result implies that a photon’s energy and momentum are invariant under Lorentz transformations.
In conclusion, the relationship between the energy and momentum of a photon is a fundamental aspect of special relativity. The relativistic energy-momentum relation E^2 = (pc)^2 provides a direct
link between these two quantities, demonstrating that the energy of a photon is proportional to its momentum. This theoretical framework has far-reaching implications for our understanding of quantum
mechanics and the behavior of particles at high energies.
• Einstein, A. (1905). On the Electrodynamics of Moving Bodies. Annalen der Physik, 17(10), 891-921.
• Planck, M. (1900). Zur Theorie des Gesetzes des Energieverbrauchs bei chemischen Reaktionen. Verhandlungen der Deutschen Physikalischen Gesellschaft, 2, 202-204.
Note: The references provided are a selection of the original papers that laid the foundation for our understanding of special relativity and quantum mechanics.
Related articles for ‘photon momentum’ :
• Reading: Momentum of a photon from energy in context of photon momentum
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Find y double prime by implicit
Find y double prime by implicit differentiation8x^2+y^2=4
Find y double prime by implicit differentiation8x^2+y^2=4
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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SUM cells with formulas not working
Hello, I have a sheet with a column of cells with formulas in them. These formulas produce numbers. Then I have a second sheet, where I have a formula in a cell that references the first sheet's
column with numbers. This formula in the second sheet is asking to SUM some of the numbers in the first sheet's column and divide it by their COUNT. So, for example, I am asking to reference Sheet 1,
and sum 2+3+4+5+6 and divide that by 5. However, the totals are coming out wrong. I saw in another post that a person was saying that if one tries to SUM cells with a formula in them, the SUM will be
wrong. Is this the case? I only have numbers, no text in the cells. Thank you!
• Hi Eva,
Can you maybe share some screenshots and what formulas you're using? (Delete/replace any confidential/sensitive information before sharing) That would make it easier to help. (share
too, andree@getdone.se)
Have a fantastic day!
Andrée Starå
Workflow Consultant @ Get Done Consulting
SMARTSHEET EXPERT CONSULTANT & PARTNER
Andrée Starå | Workflow Consultant / CEO @ WORK BOLD
W: www.workbold.com | E:andree@workbold.com | P: +46 (0) - 72 - 510 99 35
Feel free to contact me for help with Smartsheet, integrations, general workflow advice, or anything else.
• "I saw in another post that a person was saying that if one tries to SUM cells with a formula in them, the SUM will be wrong. Is this the case?"
This is not necessarily true. There are a handful of different variables that come into play to include the formula producing the numbers, the data the numbers are being pulled from, column
types, etc.
• Hi Andree, Thanks so much for the quick response! I am attaching screenshots of my two sheets. I didn't have any formal training with Smartsheet so I've been self-teaching. It's not that easy
considering what my boss wants from me. :-)
Now that I see the screenshots, is it too small to read or can you enlarge it?
• SO I just figured out that it actually is working properly - I was just including the wrong days in my own calculations. I should have excluded June 17 and 18 from my addition because today is
the 25th...so 7 days worth of numbers for the report would be 25th, 24th, 23rd, 22nd, 21st, 20th, and the 19th. Wheeew. OMG. So my formulas are actually correct and I don't have a question
anymore. Now I feel dumb. I guess I needed to write it out somewhere to arrive to this conclusion. Thank you anyway for trying to help!
• Happy to help!
Easy to miss!
Glad you got it working!
SMARTSHEET EXPERT CONSULTANT & PARTNER
Andrée Starå | Workflow Consultant / CEO @ WORK BOLD
W: www.workbold.com | E:andree@workbold.com | P: +46 (0) - 72 - 510 99 35
Feel free to contact me for help with Smartsheet, integrations, general workflow advice, or anything else.
• I am having issues with sum not working with formula fields. The answer just returns 0.
• Your numbers are actually being stored as text. Can you provide the formula that is returning the numbers you are trying to sum?
• Brilliant! As soon as you said that I realized I had " " around my return values. Once I took those out, the formula worked.
• Excellent. Glad you got it working.
• Having the same problem, I assume it is because my number might show as text but not sure how to fix that.
• I'm having this same issues. I've built a summary table to count the number of "Issues" that meet certain criteria. I'm using a the COUNTIFS formula to do this.
This summary sheet feeds a dashboard that I use pie chart on to visually show the breakdown of Issue categories. Piecharts show an error in Smartsheet when they have a zero value to display, so I
am using a workaround that someone else posted to have a category that has a count of 1, when everything else Sums to zero, I'm using a simple =(IF(SUMA1:A2)=0,1,0) formula for this.
The problem is that it seems to pull a SUM of 0, even with the COUNTIFS formulas are calculating a value that is greater than zero
Any advice?
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Advent of Code Archives • Statisticelle
Day 2 of the Advent of Code provides us with a tab delimited input consisting of numbers 2-4 digits long and asks us to calculate its “checksum”. checksum is defined as the sum of the difference
between each row’s largest and smallest values. Awesome! This is a problem that is well-suited for base R.
I started by reading the file in using read.delim, specifying header = F in order to ensure that numbers within the first row of the data are not treated as variable names.
When working with short problems like this where I know I won’t be rerunning my code or reloading my data often, I will use file.choose() in my read.whatever functions for speed. file.choose() opens
Windows Explorer, allowing you to navigate to your file path.
input <- read.delim(file.choose(), header = F)
# Check the dimensions of input to ensure the data read in correctly.
After checking the dimensions of our input, everything looks good. As suspected, this is a perfect opportunity to use some vectorization via the apply function.
row_diff <- apply(input, 1, function(x) max(x) - min(x))
checksum <- sum(row_diff)
Et voilĂ , the answer is 45,972! Continue reading Advent of Code 2017 in R: Day 2
Advent of Code 2017 in R: Day 1
My boyfriend recently introduced me to Advent of Code while I was in one of my “learn ALL of the things!” phases. Every year starting December 1st, new programming challenges are posted daily leading
up to Christmas. They’re meant to be quick 5-10 minute challenges, so, wanting to test my programming skills, I figured why not try to do all of them in base R!
I went with base R because I know I can dplyr and stringr my way to victory with some of these challenges. I really want to force myself to really go back to basics and confirm that I have the
knowledge to do these things on my own without Hadley Wickham‘s (very much appreciated in any other situation) assistance.
Since I’ve started, I’ve also seen a couple of other bloggers attempt to do these challenges in R so I’m really curious how my solutions will compare to theirs.
The first day of the challenge provides you with a string of numbers and asks you to sum all of the digits that match the next digit in a circular list, i.e. the digit after the last digit is the
first digit.
My string was…
My first thought was that I would need to separate this string such that each character was the element of an object, either a vector or a list. I kept things simple and started by just copy-pasting
the string into R. I could import it as a .txt file or otherwise but I figured that was unnecessary for such a quick problem. I stored the string as a variable named input.
# Split string after each character.
input_split <- strsplit(input, "")
# As a result, input_split is a list with 1 element:
# a vector containing each character of input as an
# element. Annoying. Let's unlist() it to extract
# *just* the vector.
char_vector <- unlist(input_split)
# The problem now is that if we are going to sum
# the elements of our string, we need them to be
# numeric and not characters. Easy enough...
num_vector <- as.numeric(char_vector)
# Now lets just initialize our sum...
num_sum = 0
# And use a loop...
for(i in 1:length(num_vector)){
# If we have the last element of the input string,
# set the next number equal to the first element
# of the string, else select element i + 1.
next_num <- ifelse(i = length(num_vector),
num_vector[i + 1])
# If our current element is equal to the next element,
# update the sum.
if(num_vector[i] == next_num){
num_sum = num_sum + num_vector[i]
Our sum is 1390 which is correct, huzzah. | {"url":"https://statisticelle.com/category/advent-of-code/","timestamp":"2024-11-08T01:27:20Z","content_type":"text/html","content_length":"57129","record_id":"<urn:uuid:06b27206-08d9-4e25-9347-55e88840f0be>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00788.warc.gz"} |
Peridynamic Solutions for Timoshenko Beams
1. Introduction
Peridynamics was introduced in 2000 [1] as a non-local theory for modeling the deformation of solid materials subjected to mechanical loading. The approach is well suited for dealing with problems
containing cracks and discontinuities including the modeling of non-colinear crack growth [2] [3] . The formulation does not require spatial derivatives, naturally modeling discontinuous phenomena
where fields are integrable but not differentiable [4] (e.g. cracks and discontinuities). A vast body of literature has been developed for various problems which include composite mechanics (e.g. [5]
), thermo-mechanical problems (e.g. [6] ) and various theoretical papers (e.g. [7] [8] ). A complete bibliography of the peridynamic literature is beyond the scope of this work; however, recent
doctoral dissertations (e.g. [5] [9] ) provide a more comprehensive review of the literature.
The majority of the work to date has focused on solid mechanics applications where the equations of motion are developed for translational deformation in 1, 2 or 3 translational degrees of freedom
(DOF). There has been little work to date investigating peridynamic formulations for structural mechanics (beams, plates and shells) where rotational DOFs are considered. This paper provides a
complete solution for the deformation of a microelastic peridynamic Timoshenko beam. The formulation follows the traditional Timoshenko beam theory by considering extensional deformation, bending
deformation and torsional deformation independently. Peridynamic deformations, analogous to traditional beam theory, can be obtained from the superposition of these three deformation modes. The
problem is formulated for micro-elastic materials undergoing infinitesimal deformations. Work is in progress to extend the theory for finite deformations as well as inelastic material behavior.
Section 2 of this paper summarizes the theoretical formulation of an elastic rod which was introduced in the seminal paper on peridynamics by Dr. Silling in 2000 [1] . A numerical solution
considering Dirichlet boundary conditions is presented and an example verification problem is solved. The static peridynamic solution reduces to the static continuum solution independent of horizon
and grid spacing (assuming numerical convergence for all cases). Similar 1-D numerical solutions are explored in [10] . Section 3 presents a derivation for the bending of a Timoshenko beam including
transverse shear effects. As is traditional for peridynamic formulations, it is derived from an assumed micro-potential leading to the peridynamic forces. An example problem with Dirichlet boundary
conditions is solved and compared with the reference solution. Section 4 considers the more typical problem with mixed Dirichlet and Neumann boundary conditions. An example problem with reference
analytical solution is solved. While not explicitly shown, the approach for mixed boundary condition axial deformation problems is identical to the approach for bending. Section 5 provides a
discussion on the general solution for full 3-D deformation of Timoshenko beams followed by Section 6 with conclusions.
2. Peridynamic Elastic Rod—Numerical Solution
The discrete equation of motion for the infinitesimal deformations in a micro-elastic peridynamic rod with viscous damping is written as:
where [i] in the domain of the problem, A is the cross-sectional area of the rod, c is termed the micro-elastic constant, ρ is the mass density, c[d] is the damping constant and b(x[i], t) is the
body force. The integral is evaluated over a finite length of the rod termed the horizon. The distance, ε, is measured from the reference point to the integrand point as shown in Figure 1.
The rod is discretized with a uniform grid spacing of Δ, then
The Dirichlet boundary conditions are given in the form:
If we designate:
We obtain the following system of 2nd order ODEs to solve numerically:
Figure 1. Schematic for 1-D rod peridynamic formulation.
We choose to temporally integrate (6) using the Leapfrog Velocity Verlet scheme [11] with fixed time step, dt, which yields:
For the purposes of the examples in this section, we assume homogeneous initial conditions so that:
The peridynamic integral is evaluated using composite trapezoidal rule. It is most convenient to write I[i] as the sum of three integrals:
Starting with the 2nd integral term, which is included for any choice of the horizon, we obtain:
The first integral can be cast into the sum:
Similarly, the third integral can be cast into the sum:
Combining, we can compute
Examining the peridynamic integral for horizons which extend beyond nearest neighbors, it is evident the horizon extends outside the physical domain as one approaches the ends of the rod. If one
simply assumes the micro-elastic modulus drops to zero outside of the physical domain (which is a perfectly valid peridynamic assumption), the material within 1/2 the horizon length of either end are
“artificially” too hard or too soft (depending on the boundary condition) when comparing to Classical Continuum Mechanics (CCM). This is not surprising since truncating the micro-elastic modulus
introduces a mathematical discontinuity which is inconsistent with the requirement in the CCM for u(x) and
The algorithm presented above provides a formulation which yields a peridynamic solution which approximates CCM in the static limit (assuming Dirichlet boundary conditions). For problems with time
dependent Dirichlet boundary conditions the well documented dispersive characteristics of the peridynamic solution will be evident [2] [3] . These aspects will be demonstrated with a simple example
Example 1: Micro-Elastic Rod with Transient Sinusoidal Loading Consider the special case of a steel rod with the following parameters:
E = 3.0 + 7 psi
ρ = 7.324e-04 Pound-s2/in4 L = 100 inches The rod is fixed at x = 0 and the following boundary condition is applied at x = L
This provides a transient response which settles down to the steady state when damping is employed. For this example, we employed 10% critical damping at the first harmonic of the rod. The driving
frequency, f[d], is three times the first harmonic.
Figure 2 shows the transient response at the center of the rod for the full time analyzed.
As expected, the response damps to the static solution independent of horizon length (equivalent to the CCM static solution). The value, δ, refers to the horizon which spans −δ <=> δ, δ = dx
corresponds to nearest neighbor horizon where δ = 10dx corresponds to a horizon spanning 10dx (where dx is the grid spacing). The physical dimension of δ is 1 inch in this example and the grid
spacing is adjusted accordingly. Figure 3 shows an enlarged section of the early transient response.
The well-known variable dispersion characteristics of the horizon [2] are demonstrated. With a nearest neighbor horizon, the peridynamic solution is equivalent to a finite difference solution of CCM.
With δ = 10dx, the dispersion produces high frequency content during the transient response with diminishing amplitude due to the damping.
Figure 2. Center displacement for transient response of end driven rod.
Figure 3. Early time center displacement showing dispersive behavior of peridynamic solution.
The well-known variable dispersion characteristics of the horizon [2] are demonstrated. With a nearest neighbor horizon, the peridynamic solution is equivalent to a finite difference solution of CCM.
With δ = 10dx, the dispersion produces high frequency content during the transient response with diminishing amplitude due to the damping.
3. Peridynamic Formulation for Micro-Elastic Timoshenko Beam
Consider the bending deformation for a straight beam using the Peridynamic approach (Figure 4). The equilibrium equations for transverse force and bending moment are written as shown in (23) and (24)
Figure 4. Bending Behavior Of A Timoshenko Beam.
For analogy to the Timoshenko beam theory, the shear force density at a point for a linear micro-elastic material takes the form:
The rotational inertia induced in Timoshenko Beam Theory is due to two sources: rotational effects due to pure shear in the cross-section and the bending moment induced by the differential change in
normal rotation. The total moment density (note, moment density includes pure bending + the rotational inertia induced by pure shear), therefore, can be written as:
The peridynamic equilibrium equations become:
Now, for infinitesimal deformation, we can introduce the notation that:
And we define the ε extent of the horizon as:
The final form of the equations becomes:
Using the typical definitions from Timoshenko beam theory,
If we restrict ourselves to problems without body forces or moments and include viscous damping in the formulation, the equations become:
If we discretize the beam with N equally spaced points along the beam length, L, the discrete equations of motions to be solved are:
Equation (36) is integrated in time using Leapfrog-Verlet integration while equation (37) is integrated using standard Verlet integration. The computational procedure to advance from t^j to t^j^+1
At the completion of each time step,
The peridynamic integrals for the Timoshenko beam are evaluated using a composite trapezoidal rule assuming a horizon defined by
For the case of m = 1, the peridynamic equations reduce to the standard finite difference equations for the Timoshenko beam with the peridynamic constants defined as:
For those problems with Dirichlet boundary conditions, if the peridynamic horizon extends beyond the physical end points of the domain, the peridynamic integrals in (37), (38) and (39) involve points
outside the physical domain. The field variables
Example 2: Micro-Elastic Beam with Transient Sinusoidal Loading Consider the cantilevered beam with square cross-section subject to and specified displacement, d(t) and fixed rotation at the opposite
end as shown in Figure 5.
The beam has length, L = 100" and a square cross section with dimensions a = b = 4.0". The boundary conditions for the problem are:
The specific loading function for this problem is:
This drive point displacement is shown in Figure 6.
The material properties for steel (assumingFigure 7.
The motion follows the 100 Hz. driving frequency during the transient part of the loading, transitions to natural damped vibration of its first resonant frequency of ~82.4 Hz. and stabilizes at the
static response state (shown in Figure 8).
For this problem, we investigated the impact of varying the grid density with a fixed horizon length. Varying the grid density had minimal impact on the transient response and all converged to the
static CCM solution (as anticipated). The solution to static equilibrium is shown for the cases of
While the responses are very similar, the transients are subtly different. Taking a closer look at the early time the period of the response is identical but the amplitude is slightly higher with a
denser mesh. This is shown in Figure 10.
Figure 5. Timoshenko beam with dirichlet boundary conditions.
Figure 6. Driven end point transverse displacement.
Figure 7. Beam midpoint transverse displacement response.
The denser grid produces the same static solution as the coarse grid with a fixed horizon as shown in Figure 11.
4. Numerical Approach for Mixed Boundary Conditions
The application of Neumann boundary conditions in problems formulated with peridynamic theory can be problematic. Various authors have adopted approaches using boundary regions where the loading is
specified in terms of body forces which are interior to the physical domain [12] or approaches where a hypothesized exterior
Figure 8. Beam midpoint transverse displacement response.
Figure 9. Center point transverse deflection for 2 grid spacings with constant δ.
domain is used as an extension of the physical domain [10] . Problems with Neumann conditions have also been treated by developing “compensation factors” accounting for the softening of the boundary
[5] . All of these approaches require some assumption of smoothness in the boundary regions and/or iterative techniques [9] .
For the problem of the Timoshenko beam with mixed Dirichlet and Neumann boundary conditions, the approach taken here is to discretize the domain into cells of dimension Δ with a collocation point (or
node) taken at the center of the cell. If the beam length is L, there are n = L/Δ cells in the domain. The horizon must include 3 or more cells and is fully consistent with the formulation given in
Section 3. The equations of motion given in (36) and (37) are unchanged but a body force term must be included in the region of Neumann loading which approximates point load conditions. The solution
procedure follows the Leapfrog-Verlet approach of Section 3, including the forces from loaded cells in the governing equations.
Figure 10. Early time response for fixed horizon with 2 grid spacings.
Figure 11. Predicted static deflection with constant horizon and 2 mesh densities.
Consider the case of an end loaded cantilever beam shown in Figure 12.
The left end of the beam (x = 0) is fixed (w = 0; γ = 0). The loaded end of the beam is approximated by adding a body force distributed in the end cell of the beam with value b = p/(AΔ) and the
absence of a body moment. Ghost cells are added to integrate the peridynamic kernels beyond the physical domain using analytic continuation assuming smoothness of the variables. This is done using
standard Taylor series expansions in the form:
For the example problem given ahead, 2^nd order finite differences are used to approximate the derivatives at the beam ends (backward differences at x = L and forward differences at x = 0). For the
simple problem, using a first order Taylor series provided essentially identical results to problems solved with a 2nd order Taylor series. For more complex problems, higher order Taylor series
extrapolations for analytical continuation may yield
Figure 12. Cantilever Beam with Free End Shear Loaded and Moment Free.
higher accuracy.
As an example problem, consider a beam with the parameters shown in Table 1 (example problem shown schematically in Figure 12):
The problem was solved assuming horizons of 2Δ, 3Δ and 4Δ. The load was applied as a step function and the solution was continued to steady state including the same damping applied in Section 3. The
resulting static deflection curve is shown in Figure 13 and compared with the analytical solution. The peridynamic solution is almost identical to the analytic solution.
5. Discussion
The general motion of a micro-elastic beam undergoing infinitesimal deformations is found from the superposition of axial extension due to the axial load component, bending about the 2 planar axes of
the beam cross-section and torsion about the axial direction. Sections 2-4 provide the solution process for all but the torsional deformation. If we restrict the discussion to non-warping torsional
deformation, the micro-potential can be written as:
with θ as the torsional twist angle. The corresponding peridynamic moment (the torsional moment) therefore, is:
This is the same functional form as for the axial deformation of the peridynamic rod and the equations of motion are analogous to equation (1). The solution for torsion, therefore, follows the
identical procedure as used for solution for axial deformation.
Sections 2 through 5 of this paper, therefore, provide the framework for the general solution of the fully three dimensional deformation of a micro-elastic Timoshenko beam under arbitrary loading
(within the confines of beam theory).
6. Conclusions
Numerical predictions of the dynamic behavior of large, complex structures under load, are typically performed using beam, plate and shell assumptions in order to reduce the dimensionality of the
analysis due to practical computing limitations. To take advantage of the strengths of the peridynamic solution methodology, therefore, it is necessary to develop and evaluate the peridynamic
formulation for these structural elements. This paper provides the general formulation for the deformation of a Timoshenko beam using peridynamics. Numerical procedures are developed and are
applicable to problems involving both Dirichlet and Neumann boundary conditions. For all cases studied, the damped dynamic peridynamic solutions converge to the solutions derived from Classical
Continuum Mechanics (CCM). The characteristic dispersion introduced by peridynamics is noted in the dynamic solutions as has been discussed by various authors (e.g. [2] ). The paper provides both a
theoretical and numerical approach which models deformations including rotational DOF and transverse shear effects. Work is ongoing to extend the approach to address peridynamic plates and shells in
order to model the behavior of large thin-walled structures.
In the extension to plates and shells, the formulation will explore the use of State Based peridynamics as discussed in [3] [7] . This will facilitate the modeling of inelastic deformation states as
well as finite deformation
Figure 13. Comparison between the peridynamic and analytical solution for a cantilever beam with end loading.
processes. Finally the formulation will be packaged for inclusion in general purpose finite element codes. It is anticipated that solutions using peridynamic formulations will be computationally more
expensive compared to finite element solutions for problems without fracture processes. Because of the advantages of the peridynamic approach for modeling material failure, the goal is to develop a
peridynamic element which can be readily embedded in a finite element grid where fracture processes are predicted to occur. It is envisioned that a hybrid finite element-peridynamic solution for the
modeling of crack propagation problems in large structures may provide an optimal solution procedure and combine the advantages of both computational approaches.
The authors would like to acknowledge the support from the In-House Laboratory Independent Research (ILIR) program at the NSWC Carderock Division, funded by the Office of Naval Research | {"url":"https://www.scirp.org/journal/paperinformation?paperid=46262","timestamp":"2024-11-01T20:52:19Z","content_type":"application/xhtml+xml","content_length":"125713","record_id":"<urn:uuid:f2eae31d-4a9d-445d-895a-51285d7843c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00860.warc.gz"} |
Convert Year to Minute
Please provide values below to convert year [y] to minute [min], or vice versa.
Definition: A year (symbol: y, yr, or a) is a measurement of time based on Earth's orbital period around the sun. This is approximated as 365 or 366 days in the Gregorian calendar. Earth's orbit
around the sun is seen as the passing of seasons in which weather, daylight hours, vegetation, and soil fertility, among other things, change.
History/origin: Various definitions of a year have been used by different calendars throughout history. The most commonly used calendar for civil affairs today is the Gregorian calendar, a solar
calendar which consists of 365 or 366 days, depending on whether or not the year is a leap year. In astronomy, many other definitions of the year exist, such as the sidereal, draconic, lunar, and
Gaussian year, among others.
The Gregorian calendar is a calendar that was reformed from the Julian calendar, which was a calendar that itself was reformed from the ancient Roman calendar.
Current use: The definition of a year based on the Gregorian calendar is used worldwide. In some cultures, lunisolar calendars are also used. In astronomy, many different definitions of the year are
used, including the Julian year as a unit of time equal to 365.25 days based on the SI (International System of Units) base unit of seconds.
The term "year" is also used to describe periods that are loosely associated to the calendar year, such as the seasonal year, fiscal year, and academic year.
Definition: A minute (symbol: min) is a unit of time based on the second, the base unit of the International System of Units (SI). It is equal to 60 seconds. Under Coordinated Universal Time, a
minute can have a leap second, making the minute equal to 61 rather than 60 seconds.
History/origin: The term "minute" is derived from the Latin "pars minuta prima" which means the "first small part." The minute was originally defined as 1/60 of an hour (60 seconds), based on the
average period of Earth's rotation relative to the sun, known as a mean solar day.
Current use: The minute, as a multiple of the second, is used for all manner of measurements of duration, from timing races, measuring cooking or baking times, number of heart beats per minute, to
any number of other applications.
Year to Minute Conversion Table
Year [y] Minute [min]
0.01 y 5259.6 min
0.1 y 52596 min
1 y 525960 min
2 y 1051920 min
3 y 1577880 min
5 y 2629800 min
10 y 5259600 min
20 y 10519200 min
50 y 26298000 min
100 y 52596000 min
1000 y 525960000 min
How to Convert Year to Minute
1 y = 525960 min
1 min = 1.9012852688417E-6 y
Example: convert 15 y to min:
15 y = 15 × 525960 min = 7889400 min
Popular Time Unit Conversions
Convert Year to Other Time Units | {"url":"https://www.unitconverters.net/time/year-to-minute.htm","timestamp":"2024-11-12T17:15:55Z","content_type":"text/html","content_length":"11885","record_id":"<urn:uuid:1e36dd70-fafe-4002-a77f-5d9ccdb7ffb7>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00857.warc.gz"} |
Odds of beating the market over many years -- continued
An earlier
shows just how difficult it is to beat the market over many years.
For instance, with a 45 percent chance of beating the market over one year, the probability of beating the market 9 or more years out of 12 is small -- just 36 times out of 1,000, or 3.6 percent (add
the probabilities for 9, 10, 11, and 12 years in the figure below). And beating the market 11 or more years out of 12 is truly comically tiny -- just 11 times out of 10,000, or 0.11 percent.
You may object to my 45 percent assumption, but this is roughly the (optimistic) probability of a mutual fund beating the market in a given year. Here's some proof.
In the 1996 edition of Burton Malkiel's exceptional book, "A Random Walk Down Wall Street," he states:
"Over the whole 22-year period since the first edition of this book, about two-thirds of the funds proved inferior to the market as a whole. [Thus the probability of beating the market (or equaling
it) is about 33 percent. To be fair, I am not entirely sure whether this is per year. Nor am I sure of the type of return calculation.]"
However a nearby graph in his book gives plenty of detail. In the graph, Malkiel shows the probabilities of the broad market beating the typical general equity fund for each of the last 22 years.
Averaging the values over the 22 years, 1973 through 1994, I get a 56.6 percent probability of the market beating the funds each year or a
43.4 percent of the typical general equity fund beating (or equaling) the market each year
Thus, the 45 percent assumption is not silly.
In fact, as the markets have become more efficient, I suspect this 45 percent assumption gives fund managers ample benefit of the doubt. For instance, you will often read that only 25 percent, or so,
funds beat the market in the current year.
I do not know the statistics for the individual investor but I suspect 45 percent is being quite generous. In fact, typical investor performances in mutual funds have
trailed the already shoddy performance of the funds. In other words, the typical investor underperforms the typical mutual fund -- most likely because of the usual culprits, namely, greed and panic
asserting themselves at precisely the wrong times.
Finally, one last point. Some may argue it is the total compound return that matters over many years not how many times someone beats the market. That is certainly true but just remember generating
40 percent one year, far above the market (say), usually implies weak returns the next few years. And because of how investors behave, jumping in at the tail end of the 40 percent, then holding the
fund for the next few years, net, they get killed.
Ideally, what you want is consistent returns just above market with moderate risk. Do that and you make a killing.
So the next time someone tells you they beat the market, ask them how many times they have beaten the market over a long period. Only then will you get the honest truth -- in my opinion, 70 or 75
percent of the time over a long period is evidence of an extremely strong strategy and performance.
This isn't easy.
The portfolio in "Investing in Dividend Growth Stocks" has beaten the market 11 times out of the last 12, and really 12 times out of the last 13 years, or
92 percent
, a record that is matched by no more than a few, if any, (general purpose) mutual funds over this period. Available on Amazon,
The portfolio in "Dividend Growth Whisperer," published in October, 2018, handily hammered the market in the fourth quarter of 2018. The portfolio in this book is an update to the portfolio in
"Investing in Dividend Growth Stocks." It also presents a novel (and simpler) variation of the crucial concept of valuation. Available on Amazon,
The latest edition of "A Random Walk Down Wall Street," | {"url":"http://www.neocadence.com/2019/01/odds-of-beating-market-over-many-years.html","timestamp":"2024-11-12T05:49:53Z","content_type":"text/html","content_length":"66816","record_id":"<urn:uuid:65a766e5-ad6c-45b9-807c-980c2dc905e7>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00511.warc.gz"} |
Chapter 12 Reshaping and Combining Data Sets | An Introduction to R and Python For Data Analysis: A Side By Side Approach
Chapter 12 Reshaping and Combining Data Sets
12.1 Ordering and Sorting Data
Sorting a data set, in ascending order, say, is a common task. You might need to do it because
1. ordering and ranking is commonly done in nonparametric statistics,
2. you want to inspect the most “extreme” observations in a data set,
3. it’s a pre-processing step before generating visualizations.
In R, it all starts with vectors. There are two common functions you should know: sort() and order(). sort() returns the sorted data, while order() returns the order indexes.
order() is useful if you’re sorting a data frame by a particularly column. Below, we inspect the top 5 most expensive cars in an example data set (“SAS^ Viya^ Example Data Sets” 2021). Notice that we
need to clean up the MSRP (a character vector) a little first. We use the function gsub() to find patterns in the text, and replace them with the empty string.
## Make Model MSRP cleanMSRP
## 335 Porsche 911 GT2 2dr $192,465 192465
## 263 Mercedes-Benz CL600 2dr $128,420 128420
## 272 Mercedes-Benz SL600 convertible 2dr $126,670 126670
## 271 Mercedes-Benz SL55 AMG 2dr $121,770 121770
## 262 Mercedes-Benz CL500 2dr $94,820 94820
In Python, Numpy has np.argsort() and np.sort().
For Pandas’ DataFrames, most of the functions I find useful are methods attached to the DataFrame class. That means that, as long as something is inside a DataFrame, you can use dot notation.
Pandas’ DataFrames and Series have a .replace() method. We use this to remove dollar signs and commas from the MSRP column. Note that we had to access the .str attribute of the Series column before
we used it. After the string was processed, we converted it to a Series of floats with the astype() method.
Finally, sorting the overall data frame could have been done with the same approach as the code we used in R (i.e. raw subsetting by row indexes), but there is a built-in method called sort_values()
that will do it for us.
12.2 Stacking Data Sets and Placing Them Shoulder to Shoulder
Stacking data sets on top of each other is a common task. You might need to do it if
1. you need to add a new row (or many rows) to a data frame,
2. you need to recombine data sets (e.g. recombine a train/test split), or
3. you’re creating a matrix in a step-by-step way.
In R, this can be done with rbind() (short for “row bind”). Consider the following example that makes use of GIS data queried from (Albemarle County Geographic Data Services Office 2021) and cleaned
with code from (Ford 2016).
The above example was with data.frames. This example of rbind() is with matrix objects.
In Python, you can stack data frames with pd.concat(). It has a lot of options, so feel free to peruse them. You can also replace the call to pd.concat() below with test.append(train). Consider the
example below that uses the Albemarle County real estate data (Albemarle County Geographic Data Services Office 2021) (Ford 2016).
Take note of the extra square brackets when we create test. If you use real_estate.iloc[0,] instead, it will return a Series with all the elements coerced to the same type, and this won’t pd.concat()
properly with the rest of the data!
12.3 Merging or Joining Data Sets
If you have two different data sets that provide information about the same experimental units, you can put the two data sets together using a merge (aka join) operation. In R, you can use the merge
() function. In Python, you can use the .merge() method.
Merging (or joining) data sets is not the same as placing them shoulder to shoulder. Placing data sets shoulder to shoulder will not reorder the rows of your data and the operation requires that both
input data sets have the same number of rows to start off with. On the other hand, merging data takes care to match rows together in an intelligent way, and it can handle situations of missing or
duplicate matches. In both cases, the resulting data set is wider, but with merging, the output might end contain either more or fewer rows.
Here’s a clarifying example. Suppose you have to sets of supposedly anonymized data about individual accounts on some online platforms.
The first thing you need to ask yourself is “which column is the unique identifier that is shared between these two data sets?” In our case, they both have an “identification number” column. However,
these two data sets are coming from different online platforms, and these two places use different schemes to number their users.
In this case, it is better to merge on the email addresses. Users might be using different email addresses on these two platforms, but there’s a stronger guarantee that matched email addresses means
that you’re matching the right accounts. The columns are named differently in each data set, so we must specify them by name.
In Python, merge() is a method attached to each DataFrame instance.
The email addresses anotherfake@gmail.com and notreal@gmail.com exist in both data sets, so each of these email addresses will end up in the result data frame. The rows in the result data set are
wider and have more attributes for each individual.
Notice the duplicate email address, too. In this case, either the user signed up for two accounts using the same email, or one person signed up for an account with another person’s email address. In
the case of duplicates, both rows will match with the same rows in the other data frame.
Also, in this case, all email addresses that weren’t found in both data sets were thrown away. This does not necessarily need to be the intended behavior. For instance, if we wanted to make sure no
rows were thrown away, that would be possible. In this case, though, for email addresses that weren’t found in both data sets, some information will be missing. Recall that Python and R handle
missing data differently (see 3.8.2).
You can see it’s slightly more concise in Python. If you are familiar with SQL, you might have heard of inner and outer joins. This is where Pandas takes some of its argument names from.
Finally, if both data sets have multiple values in the column you’re joining on, the result can have more rows than either table. This is because every possible match shows up.
12.4 Long Versus Wide Data
12.4.1 Long Versus Wide in R
Many types of data can be stored in either a wide or long format.
The classical example is data from a longitudinal study. If an experimental unit (in the example below this is a person) is repeatedly measured over time, each row would correspond to an experimental
unit and an observation time in a data set in a long form.
A long format can also be used if you have multiple observations (at a single time point) on an experimental unit. Here is another example.
If you would like to reshape the long data sets into a wide format, you can use the reshape() function. You will need to specify which columns correspond with the experimental unit, and which column
is the “factor” variable.
reshape() will also go in the other direction: it can take wide data and convert it into long data
12.4.2 Long Versus Wide in Python
With Pandas, we can take make long data wide with pd.DataFrame.pivot(), and we can go in the other direction with pd.DataFrame.melt().
When going from long to wide, make sure to use the pd.DataFrame.reset_index() method afterwards to reshape the data and remove the index. Here is an example similar to the one above.
Here’s one more example showing the same functionality–going from long to wide format.
Here are some examples of going in the other direction: from wide to long with pd.DataFrame.melt(). The first example specifies value columns by integers.
The second example uses strings to specify value columns.
12.5 Exercises
12.5.1 R Questions
Recall the car.data data set (“Car Evaluation” 1997), which is hosted by (Dua and Graff 2017).
1. Read in the data set as carData.
2. Convert the third and fourth columns to ordered factors.
3. Order the data by the third and then the fourth column (simultaneously). Do not change the data in place. Instead store it under the name ordCarData1
4. Order the data by the fourth and then the third column (simultaneously). Do not change the data in place. Instead store it under the name ordCarData2
1. Pretend day1Data and day2Data are two separate data sets that possess the same type of measures but on different experimental units. Stack day1Data on top of day2Data and call the result
2. Pretend day1Data and day2Data are different measurements on the same experimental units. Place them shoulder to shoulder and call the result sideBySide. Put day1Data first, and day2Data second.
If you are dealing with random matrices, you might need to vectorize a matrix object. This is not the same as “vectorization” in programming. Instead, it means you write the matrix as a big column
vector by stacking the columns on top of each other. Specifically, if you have a \(n \times p\) real-valued matrix \(\mathbf{X}\), then
\[$$\text{vec}(\mathbf{X}) =\begin{bmatrix} \mathbf{X}_1 \\ \vdots \\ \mathbf{X}_p \end{bmatrix}$$\]
where \(\mathbf{X}_i\) is the \(i\)th column as an \(n \times 1\) column vector. There is another operator that we will use, the Kronecker product:
\[$$\mathbf{A} \otimes \mathbf{B} = \begin{bmatrix} a_{11} \mathbf{B} & \cdots & a_{1n} \mathbf{B} \\ \vdots & \ddots & \vdots \\ a_{m1} \mathbf{B} & \cdots & a_{mn} \mathbf{B} \\ \end{bmatrix}.$$\]
If \(\mathbf{A}\) is \(m \times n\) and \(\mathbf{B}\) is \(p \times q\), then \(\mathbf{A} \otimes \mathbf{B}\) is \(pm \times qn\).
1. Write a function called vec(myMatrix). Its input should be one matrix object. It’s output should be a vector. Hint: matrix objects are stored in column-major order.
2. Write a function called unVec(myVector, nRows) that takes in the vectorized matrix as a vector, splits that into elements with nRows elements, and then places them together shoulder-to-shoulder
as a matrix.
3. Write a function called stackUp(m, BMat) that returns \(\mathbf{1}_m \otimes \mathbf{B}\) where \(\mathbf{1}_m\) is a length \(m\) column vector of ones. You may check your work with %x%, but do
not use this in your function.
4. Write a function called shoulderToShoulder(n, BMat) that returns \(\mathbf{1}^\intercal_n \otimes \mathbf{B}\) where \(\mathbf{1}_n^\intercal\) is a length \(n\) row vector of ones. You may check
your work with %x%, but do not use this in your function.
This problem uses the Militarized Interstate Disputes (v5.0) (Palmer et al. ) data set from The Correlates of War Project. There are four .csv files we use for this problem. MIDA 5.0.csv contains the
essential attributes of each militarized interstate dispute from 1/1/1816 through 12/31/2014. MIDB 5.0.csv describes the participants in each of those disputes. MIDI 5.0.csv contains the essential
elements of each militarized interstate incident, and MIDIP 5.0.csv describes the participants in each of those incidents.
1. Read in the four data sets and give them the names mida, midb, midi, and midp. Take care to convert all instances of -9 to NA.
2. Examine all rows of midb where its dispnum column equals 2. Do not change midb permanently. Are these two rows corresponding to the same conflict? If so, assign TRUE to sameConflict. Otherwise,
assign FALSE.
3. Join the first two data sets together on the dispute number column (dispnum). Call the resulting data.frame join1. Do not address any concerns about duplicate columns.
4. Is there any difference between doing an inner join and an outer join in the previous question? If there was a difference, assign TRUE to theyAreNotTheSame. Otherwise, assign FALSE to it.
5. Join the last two data sets together by incidnum and call the result join2. Is there any difference between an inner and an outer join for this problem? Why or why not? Do not address any
concerns about duplicate columns.
6. The codebook mentions that the last two data sets don’t go as far back in time as the first two. Suppose then that we only care about the events in join2. Merge join2 and join1 in a way where all
undesired rows from join1 are discarded, and all rows from join2 are kept. Call the resulting data.frame midData. Do not address any concerns about duplicate columns.
7. Use a scatterplot to display the relationship between the maximum duration and the end year. Plot each country as a different color.
8. Create a data.frame called longData that has the following three columns from midp: incidnum (incident identification number) stabb (state abbreviation of participant) and fatalpre (precise
number of fatalities). Convert this to “wide” format. Make the new table called wideData. Use the incident number row as a unique row-identifying variable.
9. Bonus Question: identify all column pairs that contain duplicate information in midData, remove all but one of the columns, and change the column name back to its original name.
12.5.2 Python Questions
Once again, recall the "car.data" data set (“Car Evaluation” 1997).
1. Read in the data set as car_data.
2. Order the data by the third and then the fourth column. Do not change the data in place. Instead store it under the name ord_car_data1
3. Order the data by the fourth and then the third column. Do not change the data in place. Instead store it under the name ord_car_data2
Consider the following random data set.
1. Pretend d1 and d2 are two separate data sets that possess the same type of measures but on different experimental units. Stack d1 on top of d2 and call the result stacked_data_sets. Make sure the
index of the result is the numbers \(0\) through \(39\)
2. Pretend d1 and d2 are different measurements on the same experimental units. Place them shoulder to shoulder and call the result side_by_side_data_sets. Put d1 first, and d2 second.
Consider the following two data sets:
import numpy as np
import pandas as pd
dog_names1 = ['Charlie','Gus', 'Stubby', 'Toni','Pearl']
dog_names2 = ['Charlie','Gus', 'Toni','Arya','Shelby']
nicknames = ['Charles','Gus The Bus',np.nan,'Toni Bologna','Porl']
breed_names = ['Black Lab','Beagle','Golden Retriever','Husky',np.nan]
dataset1 = pd.DataFrame({'dog': dog_names1,
'nickname': nicknames})
dataset2 = pd.DataFrame({'dog':dog_names2,
1. Join/merge the two data sets together in such a way that there is a row for every dog, whether or not both tables have information for that dog. Call the result merged1.
2. Join/merge the two data sets together in such a way that there are only rows for every dog in dataset1, whether or not there is information about these dogs’ breeds. Call the result merged2.
3. Join/merge the two data sets together in such a way that there are only rows for every dog in dataset2, whether or not there is information about the dogs’ nicknames. Call the result merged3.
4. Join/merge the two data sets together in such a way that all rows possess complete information. Call the result merged4.
Let’s consider Fisher’s “Iris” data set (Fisher 1988) again.
1. Read in iris.csv and store the DataFrame with the name iris. Let it have the column names 'a','b','c', 'd' and 'e'.
2. Create a DataFrame called name_key that stores correspondences between long names and short names. It should have three rows and two columns. The long names are the unique values of column five
of iris. The short names are either 's', 'vers' or 'virg'. Use the column names 'long name' and 'short name'.
3. Merge/join the two data sets together to give iris a new column with information about short names. Do not overwrite iris. Rather, give the DataFrame a new name: iris_with_short_names. Remove any
columns with duplicate information.
4. Change the first four column names of iris_with_short_names to s_len, s_wid, p_len, and p_wid. Use Matplotlib to create a figure with 4 subplots arranged into a \(2 \times 2\) grid. On each
subplot, plot a histogram of these four columns. Make sure to use x-axis labels so viewers can tell which column is being plotted in each subplot.
5. Let’s go back to iris. Change that to long format. Store it as a DataFrame called long_iris. Make the column names row, variable and value, in that order. Last, make sure it is sorted
(simultaneously/once) by row and then variable.
“Car Evaluation.” 1997. UCI Machine Learning Repository.
Dua, Dheeru, and Casey Graff. 2017. “UCI Machine Learning Repository.” University of California, Irvine, School of Information; Computer Sciences. http://archive.ics.uci.edu/ml.
Fisher, Test, R.A. & Creator. 1988. “Iris.” UCI Machine Learning Repository.
Palmer, Glenn, Roseanne W McManus, Vito D’Orazio, Michael R Kenwick, Mikaela Karstens, Chase Bloch, Nick Dietrich, Kayla Kahn, Kellan Ritter, and Michael J Soules“The Mid5 Dataset, 2011–2014:
Procedures, Coding Rules, and Description.” Conflict Management and Peace Science 0 (0): 0738894221995743. https://doi.org/10.1177/0738894221995743. | {"url":"https://randpythonbook.netlify.app/reshaping-and-combining-data-sets","timestamp":"2024-11-07T13:23:24Z","content_type":"text/html","content_length":"116423","record_id":"<urn:uuid:8b1ec714-2c4c-4219-ba96-4a2447b6e7e1>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00792.warc.gz"} |
Moodle Gradebook Aggregation
Moodle offers a number of different ways to calculate course grades. The St. Olaf Moodle instance allows you to choose from among four different types. These different aggregation types can be mixed
and matched within a course, where the course aggregation type can be different than individual category aggregation types, which can differ from one another. Most courses will be well-served by
using Natural aggregation (the default for our Moodle) for the entire course. However, particular circumstances may call for the use of one of the other types. These aggregation types are summarized
in the table below, and detailed with information below on how each one works, what it is useful for, and what limitations it has.
Aggregation types overview
Type Based on Scale Values
Total points or Weighted avg. of % Yes (if all weighted same and same poss. points) 1, 2, 3, ... Yes
Avg. of % Yes 0, 1, 2, ... No
Weighted avg. of % Yes 0, 1, 2, ... No
Total points Yes 0, 1, 2, ... Yes
Natural weighting automatically calculates a percentage weight for each item or category based on the points that item or category contributes to the total . Each calculated weight can be optionally
Useful for:
Natural weighting is the most versatile type, and can handle the majority of what you will want to do in your gradebook.
• Weighting based on: Points or weighted percentage for each item
• Extra credit possible: Yes
• Drop lowest possible: Yes (in certain cases)
• Extra credit and dropping cannot be used together
• Reports category/course total in total points (Student user report also shows %)
By default, Moodle is set to use Natural weighting. As such, a 10-point assignment has a default worth half of a 20-point assignment. In dealing with categories, Moodle will look at the total points
in that category compared to the total overall points to determine its default percentage (e.g. a category with 600 points in a course with 1000 total points would be worth 600/1000 or 60%).
For any item or category, you may choose to check the box next to the default calculated weight and enter an override between 0 and 100 (say you had an assignment that was worth 20 points, but you
want it to be worth the same as the other 10-point items in that category). If you manually set one value, Moodle will subtract that amount from 100 and redistribute the remaining percentage amongst
the other items, according to their total possible points (e.g. assigning one of three 100-point quizzes in a category to 20 [20%] will leave 80% to distribute to the remaining two, at 40% each, as
they have the same total possible points).
This has been set as the default gradebook aggregation type as it does what most people expect (three 100-point tests in a category are each worth 33% of a Quiz category), but also allows for
adjustments within a category, and setting each category’s weight, regardless of the number of points in the category (for example, a Homework category can be set to 20% of a 1000-point course, even
though all of the assignments add up to 500 points, or 50% of the points).
Extra credit categories or grade items are possible with this aggregation type. You may also choose to drop the lowest X number of grade items in a category. However, all the items in that category
must have the same number of possible points, or this option will be excluded. Additionally, scaled grades in this type are given numerical values starting at 1 and not 0, so that a Complete/
Incomplete grade will be worth 1 or 2. If you wish to use your scaled grades in calculations with the lowest value equal to zero, you'll need to choose another type. In either of the previous cases,
you will most likely want to set the category to use Simple weighted mean of grades instead.
Mean of grades (x̄)
Mean of grades calculates every grade item as a percentage and computes the average of all items in a category for its total, or the average of all subcategories in a category or course.
Useful for:
Mean of grades is best for when you have multiple items with different point totals in a category, and you don’t want to manually set them to have equal weight.
• Weighting based on: Percentage for each item
• Extra credit possible: No
• Drop lowest possible: Yes
• Reports category/course total out of 100
Using Mean of grades means that there could be multiple exams in a test category, each with a differing point total, and each would have the same weight within the Tests category (for example, 3
tests of 90, 110 and 140 points would each be worth ⅓ of the Tests grade, with the three percentage scores added up and divided by 3). This is the same as using the Natural aggregation type and
manually setting each test to 33.33%.
This aggregation type could be useful for a category with multiple items with different point values that should be treated equally based only on the percentage score of the item. This would
eliminate the need to manually set each of the items to the same weighted value that natural weighting would require.
Mean of grades does not allow for extra credit items or categories, but does allow for the dropping of the lowest X grades in a category. In order to use extra credit, it is advisable to use Natural
grading aggregation and setting the weights manually. If you wish to have extra credit and drop the lowest grade(s) in a category, then Simple weighted mean of grades with manually set weights is
Weighted mean of grades (x̄)
Weighted mean of grades calculates every grade item and category as a percentage and initially assigns each item or category an equal weight of 1, meaning they are counted once in the average. Each
item or category can be manually assigned a lower or higher weight to be counted more or less in the average (i.e. 0.5 for half as much, 2 for twice as much).
Useful for:
Weighted mean of grades is helpful when you want to assign relative weights to items in a category, and you don’t know what the total number of assignments will be.
• Weighting based on: Percentage for each item (counted x times)
• Extra credit possible: No
• Drop lowest possible: Yes
• Reports category/course total out of 100
Using Weighted mean of grades means that each item initially carries equal weight in the category, regardless of the number of total possible points available. Without manually changing weights, this
is the same as Mean of grades. However, by adjusting the default weight of 1 for a grade item or category, you can increase or decrease the contribution to the total.
Simple weighted mean of grades (x̄) (=Natural without any weights)
Simple weighted mean of grades is the same as Natural without any weight overrides set. It simply adds up the points of each item in the category/course and divides by the total points possible. Like
Natural, it can also accommodate extra credit items, with the points earned contributing to the total points earned, but the maximum points available does not add to the total possible.
• Weighting based on: Point total for each item
• Extra credit possible: Yes
• Drop lowest possible: Yes
• Extra credit and dropping can be used together
• Calculates scales on a 0 -> X scale, not 1 -> X
• Reports category/course total out of 100
Useful for:
The best use case for Simple weighted mean of grades is for when you want to use absolute point values, but need to drop the lowest or keep the highest grade and have items with different point
totals, or if you don’t need weighting and you want to use extra credit and dropping the lowest X scores together. Additionally, this aggregation scheme assigns values to grading scales starting at
0, while others start at 1, so if you wish to do calculations with scaled grades, and want the lowest item to count for 0 points, you'll need to use Simple weighted mean of grades.
With Simple weighted mean of grades, the scores of all items are summed up and divided by the sum of all the possible points, exactly like in Natural weighting with no manual weight overrides.
Dropping capability
As noted in the summaries above, all of the aggregation types allow you to drop the lowest X score(s) in the category. There are two exceptions to this in the Natural aggregation type. The first is
that dropping can only be used with Natural when all grade items in the category have the same total possible point values. The other is that dropping can only be used in Natural aggregation when no
extra credit items exist. In both of these cases, if no weighting override is used, the alternative is to use Simple weighted mean of grades, which allows both situations above.
In the case where you have grade items with differing maximum point totals which need to be weighted equally (or simply have manual overrides) and you wish to use dropping, then the alternative is to
use Weighted mean of grades. In order to also use extra credit in this situation, you must put the differing point total items into a category using Weighted mean of grades and put that category into
another category using Natural or Weighted mean of grades with the extra credit item(s).
Dropping calculation
When determining which grade item to drop, Moodle will drop the lowest score by percentage, dropping the first item it encounters when 2 or more have equal percentage. In the case of Simple weighted
mean of grades and Weighted mean of grades, the dropped grade will affect the overall category grade to a magnitude relative to the proportion of the total points or weight it represents.
This automatic determination could theoretically be beneficial to one student (dropping a high value low grade) and detrimental to another (dropping a low value low grade). In Simple weighted mean of
grades, this occurs because grade items with a greater maximum point total represent a larger percentage of the overall grade, and in Weighted mean of grades, a grade item weighted greater than
others (i.e. with a weight of 2 or 3, versus others with a weight of one) is effectively removing two or three grade items, whereas removing an item with a weight of 1 is removing only one low score.
For this reason, it is recommended that dropping is only done where all items carry the same weight, or that some manual oversight is used when using it with unequally weighted items.
Both individual grade items as well as entire categories can be designated as extra credit, as long as they are in a category that allows for it. The easiest way to add course extra credit is to have
the course set to Natural aggregation and have a course-level (i.e. top-level) grade item marked as extra credit, and set the weight of the extra credit item to be equal to the maximum number of
points available, e.g. 5 maximum extra credit points possible, and 5.0 as the weight. As long as these two numbers are the same, the number entered into the gradebook will add that many extra credit
percentage points to the total course grade.
When using Simple weighted mean of grades, it is much more difficult to set up an item that corresponds to specific extra credit points. For example, if a category has 500 total possible points from
all grade items, an extra credit item would need to have a total maximum possible points of the following formula: # of possible extra credit points * 1% * total possible course points. For our
example of 500 with 5 extra credit percentage points available, we would have 5 * .01 * 500 = 25. This would mean that there would have to be 25 points possible, and each extra percentage point would
be entered as 25 / 5 = 5.
In cases where there is an extra credit assignment, which has similar total possible points to others in the same category, using either Natural or Simple weighted mean of grades aggregation could
yield the desired result, adding the same weight as one of the grade items in that category. | {"url":"https://stolafcarleton.teamdynamix.com/TDClient/1893/StOlaf/KB/ArticleDet?ID=142475","timestamp":"2024-11-03T22:47:19Z","content_type":"application/xhtml+xml","content_length":"76490","record_id":"<urn:uuid:370892cb-66ca-48a3-aa05-37933ea7a7d6>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00807.warc.gz"} |
Length Conversion Calculator - How to Calculate Length Conversion?
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Length Conversion Calculator
Length conversion is the procedure of converting the units of length with the help of the correct conversion factor. We deal with measurements every day and converting these units to the required
unit involves the process of multiplication or division.
What is a Length Conversion Calculator?
A 'Length Conversion Calculator' is a free online tool that helps in converting one length measuring unit into another. In this calculator, you can select a length measuring unit, and the unit in
which you want it to convert and its equivalent will be displayed within a few seconds.
Length Conversion Calculator
How to Use Length Conversion Calculator?
Follow the steps given below to use the calculator:
• Step 1: Select both the initial and the final unit from the dropdown and the value to convert in the space provided.
• Step 2: Click on the "Convert" button to find the value.
• Step 4: Click on the "Reset" button to clear the field and select new units.
How to do a Length Conversion?
A conversion factor is a value or a number that is used to change one set of units to another, either by multiplication or division. An appropriate conversion factor makes calculation quick and easy.
The table given below contains the conversion factors for length measuring units.
│1 millimeter│0.001 meter │
│1 centimeter│0.01 meter │
│1 decimeter │0.1 meter │
│1 decameter │10 meters │
│1 hectometer│100 meters │
│1 kilometer │1000 meters │
│1 inch │2.54 x 10^-2 meters │
│1 foot │0.3048 meters/ 12 inches │
│1 yard │0.9144 meters/ 3feet │
│1 mile │1.609344 kilometers/ 1760 yards/ 5280 feet/ 63,360 inches │
Want to find complex math solutions within seconds?
Use our free online calculator to solve challenging questions. With Cuemath, find solutions in simple and easy steps.
Solved Examples on Length Conversion Calculator
Example 1:
What is the equivalent of 56 yards into feet and verify it using length conversion calculator.
We know that
1 yard = 3 feet
Therefore, 56 yards = 167.94 feet
Example 2:
What is the equivalent of 24 millimeter into meters and verify it using length conversion calculator.
We know that
1 millimeter = 0.001 meter
Therefore, 24 millimeter = 0.024 meter
Example 3:
What is the equivalent of 80 foot into inches and verify it using length conversion calculator.
We know that
1 foot = 12 inches
Therefore, 80 foot = 960 inches
Similarly, you can try the length conversion calculator and try to find the following:
• Convert 20 feet to inch(es).
• Convert 82 hectometer(s) to kilometer(s).
☛ Related Articles:
☛ Math Calculators:
│Log Calculator │Fraction Calculator │
│Leap Year Calculator │Interpolation Calculator │
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/calculators/length-conversion-calculator/","timestamp":"2024-11-14T11:34:49Z","content_type":"text/html","content_length":"208578","record_id":"<urn:uuid:5d240686-c9f6-49f7-8758-6e7f8d06d5c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00011.warc.gz"} |
Best 16 Strategies to Winning Pick 3
Pick 3 lotteries are very popular in many parts of the world because of their very high probability of winning, 1 in 1000. Using the right strategy, picking 3 lottery games can become much easier.
This makes it possible to improve the odds of these lotteries, which are popular among players who like to trade better odds for smaller prizes. Since there are different patterns and bet types, it
is helpful to know some pick 3 lottery strategies to determine which numbers to choose.
These Pick 3 Strategies are available for all Pick 3 lotteries, such as Arizona Pick 3, Arkansas Cash 3, California Daily 3, Connecticut Play 3, Florida Pick 3, Indiana DAILY 3, New York NUMBERS, and
all pick 3 from 0-9. If you play Powerball, Mega Millions, Lotto Texas, Euro Millions, SuperEnalotto, Lotto Max, and Canada Lotto 6/49, etc. Please visit BEST 15 Strategies of Picking Lottery Numbers
. If you play Pick 4, please visit Best 11 Strategies to Winning Pick 4.
You should read the following do’s and don’ts before you buy your number combinations. These steps will help you choose your numbers effectively and greatly increase your chances of winning the Pick
3 games.
Do Not Use The Following Pattern Selection Number Combinations
1. Do Not Choose the Extreme Combinations
We analyzed the over 100 Pick 3 lotteries, with tens of thousands of drawings results, like 0 1 2, and 7 8 9, combinations like this rarely occur.
2. Do Not Choose Regular Combinations
Don’t use regular combinations, For example, 0 2 4, 3 6 9, 9 7 5, 6 4 2.
3. Do Not Choose the Combinations Have Been Drawn Before
Pick 3 has only 1,000 different number combinations of results and hundreds of draws per year, so it’s easy to see how the results will be repeated often. We analyzed over 100 Pick 3 lotteries. The
odds of a combination that has been drawn in the previous 50 draws being repeated are very low. So the combination of the previous 50 draws can be excluded.
4. Do Not Choose All Numbers From One Group
Split the numbers into 3 groups, 0-2, 3-5, and 6-9. We analyzed hundreds of Pick 3 lotteries draws, the odds of all 3 numbers coming from one group are only about 5%. Avoid selecting all the numbers
from one group.
Use The Following Pattern Selection Number Combinations
1. Choose Low Number and High Number Ratio
We split the numbers 0-9 into low and high numbers, with 0-4 being the low numbers and 5-9 being the high numbers. Analyzed many Pick 3 lottery draw results, most winning numbers are made up of low
field and high field. In rare cases, the winning numbers are all low field numbers or all high field numbers. The odds of all the numbers in the group being low or high is about 15%. The winning
numbers most occurrence ratio of low to high are 1:2, 2:1.
2. Use Even-Odd and Low-High Pattern
In a combination, we use the upper case E for even number, use the upper case O for odd number, Pick 3 There are 8 Even-Odd patterns: EEE EEO EOE EOO OEE OEO OOE OOO; Using the same method all
Low-High Patterns: LLL LLH LHL LHH HLL HLH HHL HHH. After our statistics and analysis, there are patterns that have a very low probability of occurring over a specific of time, so that we can filter
out these patterns.
3. Choose Odd Number and Even Number Ratio
We have analyzed many lottery draw results, most winning numbers are made up of odd numbers and even numbers. In rare cases, the winning numbers are all odd numbers or all even numbers. The winning
numbers most occurrence ratio of odd to even are 1:2, 2:1.
4. Choose Numbers From Different Groups
Split the numbers into 3 groups, 0-2, 3-5, and 6-9. We have analyzed many lottery draw results, most of them spread across different groups. However, sometimes, one of the number groups is missing.
The more groups that are split, the more groups that are missed.
5. Choose Adjacent Numbers of Previous Winning Numbers
What is the adjacent number? If number 5 has been hit in the last drawn, its adjacent numbers(3, 4, 6, 7) are likely to be drawn.
We found a pattern by looking at the drawings trend chart, a number that appeared in the previous drawn has a high probability of appearing in its adjacent numbers.
Adjacent numbers of previous winning numbers
6. Use AC Value to Exclude Some Combinations
According to our statistics, in the Pick 3 games, such as Arizona Pick 3, Arkansas Cash 3, California Daily 3, Connecticut Play 3, Florida Pick 3, Indiana DAILY 3…, 95% of winning numbers are between
AC value from 2 to 3.
Please see the chart below, we counted the 20 drawings winning numbers of California Daily 3 Midday from 8/25/2020 to 12/2/2020, the AC value of 1 occurred only once, 4%, AC value of 2 occurred 37
times, 37%. AC value of 3 occurred 59 times, 59%. So before betting on numbers, calculate the AC value of all combinations and filter out those with AC is 1.
AC values chart
7. Use Numbers Average Value to Exclude Some Combinations
We analyzed several Pick 3 lotteries and came to the conclusion that the average value of the winning numbers is basically in a range. For example, we analyzed Arizona Pick 3, Arkansas Cash 3,
California Daily 3, Connecticut Play 3, Florida Pick 3, Indiana DAILY 3, and New York NUMBERS previous 100 drawings, and the average value range was basically 2-6. So before betting on numbers,
exclude combinations where the average is too low or too high.
8. Use Numbers Sum Value to Exclude Some Combinations
We analyzed several Pick 3 lotteries and came to the conclusion that the sums of the winning numbers are basically in a range. For example, we analyzed Powerball and Mega Millions previous 100
drawings, and the sum range was basically 6-21. So before betting on numbers, exclude combinations where the sum is too low or too high.
9. Use Hot and Cold Numbers
Hot and cold numbers represent the most common (hot) and less common (cold) numbers in the past draws. Through analysis, we do not recommend choosing all popular numbers or all cold numbers.
10. Use Mirror Numbers
The mirror number of another number is the result of adding “5” to it (If the sum of the numbers is greater than or equal to 10, pick the second number). Thus, whenever this term is used in a Pick 3
strategy, it refers to a given number plus 5. We can use the mirror of the previous lottery numbers as our next betting numbers.
As an example. We want to buy 5 pick 3 combinations and the lottery numbers for the previous 5 drawings are:
Mirror number:
4 0 7 = 4+5 0+5 7+5 = 9 5 2 (“12” = “2”) = 9 5 2
2 3 9 = 2+5 3+5 9+5 = 7 8 4(“14” = “4”) = 7 8 4
3 0 1 = 3+5 0+5 1+5 = 8 5 6
6 8 2 = 6+5 8+5 2+5 = 1(“11” = “1”) 3(“13” = “3”) 7 = 1 3 7
1 5 0= 1+5 5+5 0+5 = 6 0(“10” = “0”) 5 = 6 0 5
11. Use Rundown
Search for the last winning numbers in your game state and write them down. Add each of the three numbers together as “1”. For example, “6 1 3” becomes “7 2 4”.
Do it again and keep repeating it, remembering how the science of lotteries works. Stop only when you reach the original lottery number.
For example, winning numbers from the previous draw: 613.
The above demonstration is Rundown 111, we can also use Rundown 123, Rundown 317, etc., the calculation method is similar to Rundown 111. Before using it, we can verify the effect of different
Rundown formulas and choose the one that works best in the current lottery to use.
12. You Should Attention to the Combinations That Have Not Been Drawn So Far
We know that a Pick 3 lottery has 1,000 combinations and the probability of winning the jackpot is 1 in 1,000. Each combination is drawn on average once in a thousand times. But is this actually the
case? We analyzed and counted the data of all Pick 3 lotteries in the United States since 2002. Some lotteries have been drawn more than 5,000 times, but some combinations have not been drawn so far,
or have been drawn only once. According to the probability, these combinations will be opened sooner or later, so we should pay attention to them. For more information, please visit here.
We can’t guarantee that you will win 100% using these strategies above, but we can guarantee that you will increase your odds of winning and save a lot of money.
If you have any suggestions or tips for picking lottery numbers, please send us your ideas or methods and you will have a chance to get our lottery software for FREE.
The data quoted in the article and some of the statistical charts were generated by the SamP3P4 software program. If you are interested you can download the free version, you can use most of the
features and the statistical analysis module is completely free! | {"url":"https://www.samlotto.com/best-11-strategies-to-winning-pick-3-lottery/","timestamp":"2024-11-06T19:05:27Z","content_type":"text/html","content_length":"104381","record_id":"<urn:uuid:eef87beb-ad44-416e-b8ec-896e09aecafb>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00773.warc.gz"} |
CHEM 1030 - Chemistry 1 Tutoring at Auburn
Every Topic in Your Syllabus is Covered 💯
(Note: Some professors may go out of order or skip a topic, so feel free to bounce around.)
Your Chem Tutor: Marty
Marty attended the University of Florida, where he earned a Bachelor of Science in statistics, a Bachelor of Arts in mathematics, and a Master of Science in electrical engineering.
While working as a teaching assistant for math and physics professors, he also did research on control systems. Later, he also taught high school physics and AP Computer Science.
Marty prides himself on making difficult concepts easier to understand. He’s been tutoring since he was a teenager, and he believes that anybody is capable of learning tough subjects with the right
Hit him up at [email protected].
Other Courses We Cover at Auburn
We offer tutoring for chemistry, physics, calculus, and more at Auburn, so have a look at the entire CramBetter catalog for AU, and pass your classes the easy way! 😏
Check out the answers to our most frequently asked questions, or shoot us an email at [email protected] for a quick reply! | {"url":"https://members.crambetter.com/p/chem-1030-auburn","timestamp":"2024-11-10T14:35:24Z","content_type":"text/html","content_length":"434952","record_id":"<urn:uuid:e09bff20-66ca-4f8f-bae3-f0b986ea0032>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00211.warc.gz"} |
Interpolation Problems and the Characterization of the Hilbert Function
Date of Graduation
Degree Name
Bachelor of Science in Mathematics
Degree Level
Mathematical Sciences
Mantero, Paolo
Committee Member/Reader
Harriss, Edmund
Committee Member/Second Reader
Zamboanga, Byron L.
Committee Member/Third Reader
Chapman, Kate
In mathematics, it is often useful to approximate the values of functions that are either too awkward and difficult to evaluate or not readily differentiable or integrable. To approximate its values,
we attempt to replace such functions with more well-behaving examples such as polynomials or trigonometric functions. Over the algebraically closed field C, a polynomial passing through r distinct
points with multiplicities m1, ..., mr on the affine complex line in one variable is determined by its zeros and the vanishing conditions up to its mi − 1 derivative for each point. A natural
question would then be to consider the case in higher dimensions corresponding to polynomials in several variables. For this thesis, we will classify polynomials in three variables passing through a
set of discrete points using an abstract algebraic structure known as an ideal. Then, we will analyze these ideals and specifically provide structural and numerical information. That is, we
characterize their Hilbert Functions, which in our setting, are functions describing the number of linearly independent polynomials passing through the set of up to six points with a given
multiplicity. Specifically, we will also see that in these cases, there is an expected ”maximal” Hilbert Function value, and the main goal is to determine whether these ideals have the ”maximal”
Hilbert Function or not.
Commutative Algebra; Hilbert Function; Algebraic Geometry
Xie, B. (2023). Interpolation Problems and the Characterization of the Hilbert Function. Mathematical Sciences Undergraduate Honors Theses Retrieved from https://scholarworks.uark.edu/mascuht/5 | {"url":"https://scholarworks.uark.edu/mascuht/5/","timestamp":"2024-11-09T03:42:38Z","content_type":"text/html","content_length":"38587","record_id":"<urn:uuid:fff1632e-886e-43a6-9470-b0d01600d800>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00336.warc.gz"} |
discrete topology in English - dictionary and translation
, a
discrete space
is a particularly simple example of a
topological space
or similar structure, one in which the points form a
discontinuous sequence
, meaning they are
from each other in a certain sense. The discrete topology is the
topology that can be given on a set, i.e., it defines all subsets as open sets. In particular, each
is an open set in the discrete topology. | {"url":"http://info.babylon.com/onlinebox.cgi?cid=CD566&rt=ol&tid=pop&x=20&y=4&term=discrete%20topology&tl=English&uil=Hebrew&uris=!!ARV6FUJ2JP","timestamp":"2024-11-14T14:12:19Z","content_type":"text/html","content_length":"6166","record_id":"<urn:uuid:42ea850d-4fd7-410f-a748-ca480a4d58c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00812.warc.gz"} |
K54 soon to be Blood Bath, War time!
No point in this thread even being up now, Lock please?
Too bad those happened. Asgard had a lot of potential, Comeatmebro was a great leader.
Too bad those happened. Asgard had a lot of potential, Comeatmebro was a great leader.
comeatmebro is the reason why asgard is now gone
Too bad those happened. Asgard had a lot of potential, Comeatmebro was a great leader.
He may make a good War General but unless he changes his attitude he'll never be a good leader.
He may make a good War General but unless he changes his attitude he'll never be a good leader.
What attitude are you talking about he just went offline for some reason, you can't know what have happened to him, hence you clearly can't judge his attitude?
Last edited by a moderator:
What attitude are you talking about he just went offline for some reason, you can't know what have happened to him, hence you clearly can't judge his attitude?
What does him going offline have to do with his attitude? Surely you aren't suggesting that's the only reason C² stood above Asgard, I can assure you we wouldn't relent in a fight if our Duke
What does him going offline have to do with his attitude? Surely you aren't suggesting that's the only reason C² stood above Asgard, I can assure you we wouldn't relent in a fight if our Duke
Relent? Afaik you didn't really even fight at all.. Just got lucky, that's it. And yes, that's exactly what I'm suggesting
Nice temporary raise in points when you recruitied top players from Asgard but ironicaly didn't help you staying in top 3 for long lol.
I don't get why would you use the word "attitude".. if it's not to do with him going offline..?
Relent? Afaik you didn't really even fight at all.. Just got lucky, that's it. And yes, that's exactly what I'm suggesting
Nice temporary raise in points when you recruitied top players from Asgard but ironicaly didn't help you staying in top 3 for long lol.
I don't get why would you use the word "attitude".. if it's not to do with him going offline..?
I don't know who you are or what you hope to achieve in defending ComeAtMeBro, but you obviously don't know what you're talking about. So I'm ending this argument now (this will be my last post in
this thread), but if you're in K54 we'll be seeing you soon
I liked it how you use the term "lucky", whereas we are number one in ODA and top of the continent by a long, long way. Lucky? Really? Maybe this will help:
Lucky. LOL!
Relent? Afaik 1)you didn't really even fight at all.. Just got lucky, that's it. And yes, that's exactly what I'm suggesting
Nice temporary raise in points when 2)you recruitied top players from Asgard but ironicaly didn't help you staying in top 3 for long lol.
I don't get why would you 3)use the word "attitude".. if it's not to do with him going offline..?
1)I guess that's why we are in no. 1 in ODA
2)Asgard leader quit and player jumped ship as for the ranking if you are giving too much preference to it then you are noob.
3) well read his previous post you will sense attitude in them.
What attitude are you talking about he just went offline for some reason, you can't know what have happened to him, hence you clearly can't judge his attitude?
havent you read any of his previous posts?
Oh deary me! looks like Oc gonna get theirs! We have joined, tremble oh ye warriors!:icon_evil:
1)I guess that's why we are in no. 1 in ODA
2)Asgard leader quit and player jumped ship as for the ranking if you are giving too much preference to it then you are noob.
3) well read his previous post you will sense attitude in them.
when you say we are no 1 in ODA please dont count your self since you had nothing to do with that war
when you say we are no 1 in ODA please dont count your self since you had nothing to do with that war
why would i hate someone whos playing with other players that are known for cheating on other worlds
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None of our players are known for cheating.
But to get this back on topic: We declared on [OC]. They disbanded around 12 hours later. What a war!!!
Yea, that was crazy. Turn away from the computer for 2 minutes and my tribe disbands. I knew that was gonna happen when Cuddlez gave the entire tribe duke privelages...
Yeah i heard about that lol. He gave the whole tribe duke privilages and our spy took advantage by demoting everyone and disbanding it. You may call it dirty tactics, but i didnt even instruct him to
do this he did it himself | {"url":"https://forum.tribalwars.net/index.php?threads/k54-soon-to-be-blood-bath-war-time.259404/page-13","timestamp":"2024-11-10T11:18:40Z","content_type":"text/html","content_length":"111832","record_id":"<urn:uuid:2a872a75-07d8-4af0-ae16-1f274ec95090>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00156.warc.gz"} |
How Not to Pass a Math Course
10 Easy Steps for How Not to Pass a Math Course
Related Topics:
More Math Trivia Math Worksheets
We hope you enjoy our collection of excuses for not doing Math Homework.You may want to check out our algebra math jokes, calculus math jokes, geometry math jokes etc. on our
Math Trivia
Submitted by Louette McInnes, Christchurch Boys' High School
1. Don’t bother to do homework - go out and play and count on trying to learn a whole year’s work in two weeks.
2. Talk instead of working in class - your current socializing is more important than your future.
3. Do as few problems as you can - after all, practice only counts at sport or music.
4. Don’t worry if you’re failing at mid-year. You still have half a year to learn a year's work. Anyway, sportsmen don’t need brains (unless they get injured, or dropped from the team, or need to
invest their earnings for when their short sporting career is over, and this would never happen to someone as talented as you.)
5. Leave all assignments to the last minute - so you have to spend less time worrying about them. You were going to do a poor job and get a poor grade anyway and this gives you an excuse to use.
6. Always lose the assignment sheet and claim you never got one - as an excuse for why you missed handing in the assignment, or try “the dog ate it” as a variation.
7. Never take your textbook to class - otherwise someone might expect you to do some work.
8. Spend the first 15 minutes of every period looking for your pens, pencils and book, then you only have 40 more minutes to try and waste.
9. Spend all your study time on the subjects you like and leave the rest - you plan to fail them anyway.
10. After using steps 1-9 all year, blame everyone else for your failure on the exam.
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. | {"url":"https://www.onlinemathlearning.com/math-jokes-math-course.html","timestamp":"2024-11-06T02:44:46Z","content_type":"text/html","content_length":"38315","record_id":"<urn:uuid:5353cd39-e92f-40f2-b5ac-cf0dfa8eedfb>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00751.warc.gz"} |
This research work presents the analysis of rectangular plate on Winkler’s foundation using second degree characteristic orthogonal polynomials. The plates were of three different edge conditions:
Thin Rectangular Plate pinned on all edges (SSSS), Thin Rectangular Plate clamped on all edges (CCCC) and Thin Rectangular Plate pinned on two opposite edges and clamped on the other two edges
(CSCS). The exact displacement and bending moment functions of these plates under uniformly distributed in–plane loading on their longitudinal edges on Winkler foundation were obtained by applying
work principle approach and total minimum energy method. These methods were applied on the governing differential equation of plates so as to obtain the deflection coefficient Wuv equations. These
coefficients were obtained by finding the second degree of freedom functional equation and substituting them on work done expression for plates putting into consideration the aspect ratio of the
rectangular plate. The expression gotten were later rearranged in matrix form to give the stiffness matrix of the plates which were later substituted back to the defection coefficient equation,
Wuv The bending moments were then obtained by substituting the shape functions in the moment equation expressed in non-dimensional parameters and aspect ratio. The values of midspan deflection
gotten for SSSS plate as aspect ratio varies from 1.0 to 2.0 are (0.000062590, 0.000062748, 0.000062859, 0.000062939, 0.000063000, 0.000063082, 0.000063111, 0.000063135,and 0.000063154)m. The
difference from Timoshenko and Woinowsky values reflects that second degree characteristic orthogonal polynomials produces better results when used to analyse higher degree polynomials. This also
featured on results for CCCC plate-(0.000065330, 0.000065680, 0.000065923, 0.000066098, 0.000066227, 0.000066325, 0.000066401, 0.000066460, 0.000066508, and 0.000066546) m and CSCS plate-
(0.0011881, 0.0011898, 0.0011910, 0.0011917, 0.0011923, 0.0011927, 0.0011930, 0.0011932, 0.0011934, and 0.0011935) m. In conclusion higher degrees of freedom Characteristic orthogonal
polynomials shape functions for rectangular plates are satisfactory in approximating the deformed shape of thin rectangular plates of various boundary conditions. The results from the study compares
very well with those of the literature. Though they are upper bound, they are adequate to be used in design as they are safer and straight forward.
1.1 Background of Study
There are manyhypotheses on models of elastic foundationsproven by many researchers. The simplest of them has been suggested by Winkler (Ventseland Krauthammer, 2001). It is based on the assumption
that the foundation’s reaction q(x,y) can be described by the following relationship:
� = 𝑘� 1.1 where k is a constant termed the foundation modulus, which has the dimensions of force per unit surface area of the plate per unit deflection, N/m2 (1/m) in SI units. Values of k for
various soils are given in numerous works, which is the resisting pressure of the foundation and w is the deflection of the plate.
Thin plates are initially flat structural members bounded by two parallel planes called faces, and either a plane or cylindrical surface, called an edge or boundary. The distance between the plane
faces is called the thickness (h) of the plate. The mechanical properties of plates can be isotropic or anisotropic. The classical theory of elasticity assumes the material is homogeneous and
isotropic, i.e., its mechanical (material) properties are the same in all directions and at all points. Many construction materials such as steel, aluminum, etc., fall into this category. However,
certain materials display direction-dependent properties. Consequently, they are referred to as anisotropic for examples wood, plywood, delta wood, fiber-reinforced, plastics, etc. A number of
manufactured plates made of isotropic materials also may fall into the category of anisotropic plates: examples include corrugated and stiffened plates, etc. These examples of anisotropy is referred
to as structural anisotropy plate. Practical applications of orthotropic plates in civil, marine, and aerospace engineering are numerous and include decks of contemporary steel bridges,
composite-beam grid works, and platesreinforced with closely spaced flexible ribs, and reinforced concrete plates.
The plate can also be rectangular, circular or any other polygonal shape. Rectangular plates are those plates that have four plane surfaces (edges). They have wide applications in Civil and
Mechanical engineering. They have three dimensions a, b and h. Where b and a are respectively secondary and
primary in-plane dimensions and h is the plate thicknesswhich could be uniform or varying. The ratio, a/h is used to classify a plate as thick, stiff, thin or membrane. If the ratio, a/h is less than
ten (10) then the plate is thick. If the range, 10 ≤ a/h ≤ 100 holds then the plate is thin. If the ratio, a/h is greater than hundred (100) then the plate is a membrane. In this research, emphasis
was on isotropic thin rectangular plates of constant thickness.
Thin rectangular isotropic plate has four edges and the numbering of the edges is shown in Figure 1.1. Some of the boundary conditions of the edges of a thin rectangular isotropic plate are: S –
designates simply support, C – designates clamped support and F – designates free support. A rectangular plate is unique from the other by the conditions of its four edges. These conditions can be
same or mixed for a particular plate type. Such as SSSS, CCCC, CSCS, CSSS, CCSS, CCCS, etc. Any plate is named
according to conditions at the edges in line with the order of their arrangement shown in Figure 1.1.
Figure 1.1: Rectangular plate with edge numbering
Three approaches are used in the solution of thin rectangular plate analysis. There are;
I. The equilibrium (Euler) approach, II. The numerical approach.
III. The energy (approximate) approach
The Euler approach tends to find solution of the governing differential equation by direct integration and satisfying the boundary conditions of the four edges of the plate. Numerical approach is a
good alternative to the Euler approach. Some examples of this approach include truncated double Fourier series, finite difference, finite strip, Runge-Kutta and finite element methods among others.
Energy approach is another method that can be used and is quite different from Euler and numerical approaches. The solution from it agrees approximately with the exact solution. Typical examples of
energy approaches are Ritz, Raleigh-Ritz; Garlekin, minimum potential energy etc. These methods are called variational methods. They seek to minimize the total potential energy functional in order to
get the solution matrix and accuracy of the solution is dependent on the accuracy of the approximate deflection function (shape function). Approximate shape function is substituted in the total
potential energy functional, and the resulting equation is partially differentiated. The total potential energy is said to be minimized when its partial derivative is equated to zero. This implies
that the difference between the approximate and exact solutions is zero (Iyengar, 1988).The energy approach can be of direct variational approach or by indirect variational approach. In the direct
variational approach, the energy functional is minimized to arrive at equilibrium of forces equations. That is, differentiating the energy functional with respect to displacement gives force function
and equating that function to zero. The energy functional is said to have been minimized. Indirect variational approach doesn’t convert the energy functional to force function, rather uses the
principle of conservation of energy. That is, total energy (input and output) in a continuum that is in static equilibrium is always equal to zero. Typical examples of direct energy variational
approach are Rayleigh, Ritz and Rayleigh-Ritz method. Typical examples of indirect energy variational approach are the finite difference method, the methods of boundary collocations, the boundary
element method, and the Galerkin method.
Methods of numerical approaches have the capacity of handling plates of various boundary conditions. It has been shown from past works that in most cases, the solution from numerical approach
approximate closely to those of the exact approach (Ventsel and Krauthammer, 2001). The problem with these numerical solutions is that the accuracy of the solution is dependent on the amount of work
to be done. For instance, if one is using finite element method, the more the number of elements used in the analysis, the closer the approximate solution to the exact solution. Hence, when a plate
has to be divided into several elemental plates for an accurate solution to be reached, then the extensive analysis is involved, requiring enormous time to be invested. A sound knowledge in
mathematics and skilful
experience in computer programming are inevitable in this case. At this point one will see vividly that the problem one is trying to avoid in equilibrium approach is still found in numerical
1.2. STATEMENT OF PROBLEM
The exact and explicit elastic analysis of isotropic rectangular plate have been a subject of continuous study from the conceptual time to the recent time. Just recently, attentions were moving away
from finding the solution of plates’ problem through assumption that its solutions existed in the single degree of freedom domain. Engineering members are more of multi – degrees of freedom systems
by their weight and positions. Researches on engineering members were focused presently on the behaviour and such solutions when derived would give improved expressions and enhance convergence of
their mechanical behaviour.Galerkin energy method is not an adequate tool for a continuum whose deflection function has up to two degrees of freedom. Ritz and Rayleigh-Ritz which could be adequate
for single and multi-degrees of freedoms make use of total potential energy approach that involves squares of derivatives. This approach usually culminate to large number of terms in the resulting
expression, which could lead to difficulty in the final computation and susceptible to high computational errors (Ibearugbulem, et. al; 2014). The situation is worse with trigonometric function
solutions for plates’ support other than simple supports.No researcher to the best ofmy knowledgein the course of this research have bothered to apply the recently developed work principle technique
by Ibearugbulem et. al (2014) on elastic analysis of isotropic rectangular plates. In the light of the above problems, the work undertook a closed – form analysis of elastic isotropic rectangular
plate by using characteristic orthogonal polynomial approach to solve for deflection functions of the plates for three different edge conditions. The new theories for work, termed: “work principle
and minimum work error theory” were used to evaluate the elastic behaviour of thin rectangular plates under uniformly distributed lateral loading and expressions of their critical mechanical
characteristics such as maximum deflections and moments
1.3 OBJECTIVE OF STUDY
The main objective of this study was to carry out analysis of rectangular plate on Winkler’s
Foundation using Characteristic Orthogonal Polynomials method, while the specific objectives are:
i. To explore the potentials and functionalities of work principle technique on the elastic analysis of isotropic rectangular plates’ problem of different edge conditions and subjected to uniformly
distributed transverse load using multi – degree of freedom deflection functions.
ii. To use orthogonal polynomials to obtain the shape functions for thin rectangular isotropic plate analysis
iii. To use orthogonal polynomials to obtain the defection functions for thin rectangular isotropic plate analysis.
iv. To use orthogonal polynomials to obtain the bending moment functions for thin rectangular isotropic plate analysis
v. To compare results of the study with those of literature where Euler method was used in solving rectangular plate problems; and subsequently making justifiable inferences.
1.4 SCOPE OF STUDY
This work centers on classical thin rectangular plate analysis using second degrees of freedom characteristic orthogonal polynomials on three cases of plates ‘namely: Thin Rectangular Plate pinned on
all boundaries ( SSSS), Thin Rectangular Plate clamped on all edges (CCCC) and Thin Rectangular Plate pinned on two opposite edges and clamped on the other two edges (CSCS).
1.5 JUSTIFICATION OF THE STUDY
This research is set to introduce the use of multi – degrees of freedom characteristic orthogonal polynomials displacement equations in work principle technique of thin rectangular plate problems and
to open new frontier in classical structural mechanics for plane continua analysis. It will also give clearly distinctions among the energy methods in terms of the suitability in elastic analysis of
isotropic rectangular plates’ problems.
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To connect ideas
sentences, the following adverbs can be used:
however, nevertheless, on the other hand, otherwise, though, still, all the same.
The more he talked the more silent Mary became. However he thought she had been interested.
To connect ideas
sentences the following adverbs can be used:
although, though, even though, yet, while, in spite of / despite +noun, in spite of / despite+noun or despite+verb(ing). | {"url":"http://www.designmaths.net/wims/wims.cgi?lang=en&+module=Lang%2Fenglish%2Fdoclinkwords.en","timestamp":"2024-11-13T05:17:32Z","content_type":"text/html","content_length":"17231","record_id":"<urn:uuid:1b478580-b7fa-420b-9231-a176dbda1193>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00032.warc.gz"} |
Write a Python function to find the maximum element in a queue.
Finding the maximum element in a queue can be accomplished using a straightforward approach. Below is a Python function that takes a queue (implemented as a list) as input and returns the maximum
element. If the queue is empty, the function will indicate that by returning None.
from collections import deque
def max_in_queue(queue):
Finds the maximum element in a queue.
queue (deque): A queue implemented as a deque from the collections module.
The maximum element in the queue or None if the queue is empty.
if not queue: # Check if the queue is empty
return None
max_element = queue[0] # Initialize max_element with the first element of the queue
for element in queue: # Iterate through all elements in the queue
if element > max_element: # Update max_element if a larger element is found
max_element = element
return max_element # Return the maximum element found
# Example usage
if name == "main":
# Create a queue using deque
q = deque([3, 1, 4, 1, 5, 9, 2, 6])
print("Maximum element in the queue:", max_in_queue(q))
1. Importing Needed Modules: The deque class from the collections module is used to efficiently implement the queue. 2. Function Definition: The function max_in_queue accepts a queue and checks if it
is empty. If it is, None is returned. 3. Finding the Maximum: The function initializes max_element with the first element of the queue. It then iterates through all elements, updating max_element
whenever a larger value is found. 4. Returning the Result: After examining all elements, the function returns the largest element found.
Practical Considerations:
- This implementation assumes that the input is a deque, but it can easily be modified to work with other list-like structures. - The time complexity of this function is O(n), where n is the number
of elements in the queue, as it processes each element exactly once. | {"url":"https://sourcecodeera.com/blogs/IreneSm/Write-a-Python-function-to-find-the-maximum-element-in-a-queue","timestamp":"2024-11-07T05:42:29Z","content_type":"text/html","content_length":"85056","record_id":"<urn:uuid:5739b16b-a9df-4515-afe6-cc558d5a6147>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00639.warc.gz"} |
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Alexander Scott
Alexander Scott
Professor of Mathematics, University of Oxford
Dominic Welsh Tutor in Mathematics, Merton College
Research interests: Combinatorics, probability, algorithms and related areas
Phone (Institute): 01865 615314
Phone (College): 01865 276331
Email: lastname at maths.ox.ac.uk
My CV (October 2024)
I organize the Oxford Combinatorics Seminar. With Christina Goldschmidt, I run the online Oxford Discrete Mathematics and Probability Seminar. All welcome!
I also organize the annual One-Day Meeting in Combinatorics in Oxford every May or June. This year's meeting will take place on Tuesday 21 May 2024 (please note that this is different from the usual
day; details about previous meetings can be found here). The meeting has been running since 1999, with a brief interruption for covid (in June 2021, I organized a special Round the World Relay in
Combinatorics, with 22 seminars from around the world).
204. Subdivisions and near-linear stable sets, submitted (with Tung Nguyen and Paul Seymour)
203. Trees and near-linear stable sets, submitted (with Tung Nguyen and Paul Seymour)
202. Distant digraph domination, submitted (with Tung Nguyen and Paul Seymour)
201. Graphs without a 3-connected subgraph are 4-colourable, submitted (with Édouard Bonnet, Carl Feghali, Tung Nguyen, Paul Seymour, Stéphan Thomassé and Nicolas Trotignon)
200. Lower bounds for graph reconstruction with maximal independent set queries, submitted (with Lukas Michel)
199. A note on graphs of k-colourings, submitted (with Emma Hogan, Youri Tamitegama and Jane Tan)
198. Non-homotopic drawings of multigraphs, submitted (with António Girão, Freddie Illingworth and David Wood)
197. A counterexample to the coarse Menger conjecture, submitted (with Tung Nguyen and Paul Seymour)
196. Reconstruction of shredded random matrices, submitted (with Paul Balister, Gal Kronenberg and Youri Tamitegama)
195. Induced subgraph density. VII. The five-vertex path, submitted (with Tung Nguyen and Paul Seymour)
194. Induced subgraph density. VI. Bounded VC-dimension , submitted (with Tung Nguyen and Paul Seymour)
193. Graphs with arbitrary Ramsey number and connectivity, submitted (with Isabel Ahme)
192. Superpolynomial smoothed complexity of 3-FLIP in Local Max-Cut, submitted (with Lukas Michel)
191. Game connectivity and adaptive dynamics, submitted (with Tom Johnston, Michael Savery and Bassel Tarbush)
190. Boundary rigidity of 3D CAT(0) cube complexes, European Journal of Combinatorics, to appear (with John Haslegrave, Youri Tamitegama and Jane Tan)
189. Induced subgraph density. V. All paths approach Erdős-Hajnal, submitted (with Tung Nguyen and Paul Seymour)
188. Induced subgraph density. IV. New graphs with the Erdős-Hajnal property, submitted (with Tung Nguyen and Paul Seymour)
187. Induced subgraph density. III. Cycles and subdivisions, submitted (with Tung Nguyen and Paul Seymour)
186. Some results and problems on tournament structure, submitted (with Tung Nguyen and Paul Seymour)
185. The structure and density of k-product-free sets in the free semigroup, submitted (with Freddie Illingworth and Lukas Michel)
184. Counting graphic sequences via integrated random walks, submitted (with Paul Balister, Serte Donderwinkel, Carla Groenland and Tom Johnston)
183. Perfect shuffing with fewer lazy transpositions, submitted (with Carla Groenland, Tom Johnston and Jamie Radcliffe)
182. Short reachability networks, submitted (with Carla Groenland, Tom Johnston and Jamie Radcliffe)
181. Improved bounds for 1-independent percolation on ℤ^n, submitted (with Paul Balister, Tom Johnston and Michael Savery)
180. Induced subgraphs of induced subgraphs of large chromatic number, submitted (with António Girão, Freddie Illingworth, Emil Powierski, Michael Savery, Youri Tamitegama and Jane Tan)
179. Powers of paths and cycles in tournaments submitted (with António Girão and Dániel Korándi)
178. Reconstructing the degree sequence of a sparse graph from a partial deck, submitted (with Carla Groenland, Tom Johnston, Andrey Kupavskii, Kitty Meeks and Jane Tan)
177. A logarithmic bound for the chromatic number of the associahedron, submitted (with Louigi Addario-Berry, Bruce Reed and David Wood)
176. Approximating the position of a hidden agent in a graph, submitted (with Hannah Guggiari and Alex Roberts)
to appear
175. Induced C[4]-free subgraphs with large average degree, Journal of Combinatorial Theory, Series B, to appear (with Xiying Du, António Girão, Zach Hunter and Rose McCarty)
174. Induced subgraph density. II. Sparse and dense sets in cographs, European Journal of Combinatorics, to appear (with Jacob Fox, Tung Nguyen and Paul Seymour)
173. Shotgun assembly of random graphs, Probability Theory and Related Fields, to appear (with Tom Johnston, Gal Kronenberg and Alexander Roberts)
172. Polynomial bounds for chromatic number. VIII. Excluding a path and a complete multipartite graph, Journal of Graph Theory, to appear (with Tung Nguyen and Paul Seymour)
171. Reconstructing a point set from a random subset of its pairwise distances, SIAM Journal of Discrete Mathematics, to appear (with António Girão, Freddie Illingworth, Lukas Michel and Emil
170. A multidimensional Ramsey Theorem, Discrete Analysis, to appear (with António Girão and Gal Kronenberg)
169. Pure pairs. VIII. Excluding a sparse graph, Combinatorica, to appear (with Paul Seymour and Sophie Spirkl)
168. Reconstruction from smaller cards, Israel Journal of Mathematics, to appear (with Carla Groenland, Tom Johnston and Jane Tan)
published papers
167. Product structure of graphs with an excluded minor, Transactions of the American Mathematical Society, Series B 11 (2024), 1233-1248 (with Freddie Illingworth and David Wood)
166. Induced subgraph density. I. A loglog step towards Erdős-Hajnal, International Mathematics Research Notices, Volume 2024, Issue 12, (2024), 9991-10004 (with Matija Bucić, Tung Nguyen and
Paul Seymour)
165. Asymptotic dimension of minor-closed families and Assouad-Nagata dimension of surfaces, Journal of the European Mathematical Society 26 (2024), 3739-3791 (with Marthe Bonamy, Nicolas Bousquet,
Louis Esperet, Carla Groenland, Chun-Hung Liu and François Pirot)
164. Flashes and rainbows in tournaments, Combinatorica 44 (2024), 675-690 (with António Girão, Freddie Illingworth, Lukas Michel and Michael Savery)
163. Invertibility of digraphs and tournaments, SIAM Journal on Discrete Mathematics 38 (2024), 327-347 (with Noga Alon, Emil Powierski, Michael Savery and Elizabeth Wilmer)
162. On a problem of El-Zahar and Erdős, Journal of Combinatorial Theory, Series B 165 (2024), 211-222 (with Tung Nguyen and Paul Seymour)
161. Defective colouring of hypergraphs, Random Structures and Algorithms 64 (2024), 663-675. (with António Girão, Freddie Illingworth and David Wood)
160. Polynomial bounds for chromatic number. V. Excluding a tree of radius two and a complete multipartite graph, Journal of Combinatorial Theory, Series B 164 (2024), 473-491 (with Paul Seymour)
159. Pure pairs. X. Tournaments and the strong Erdős-Hajnal property, European Journal of Combinatorics 115 (2024), 103786 (with Maria Chudnovsky, Paul Seymour and Sophie Spirkl)
158. Pure pairs. IX. Transversal trees, SIAM Journal on Discrete Mathematics 38 (2024), 645-667 (with Paul Seymour and Sophie Spirkl)
157. Bipartite graphs with no K[6] minor, Journal of Combinatorial Theory, Series B 164 (2024), 68-104 (with Maria Chudnovsky, Paul Seymour and Sophie Spirkl)
156. A note on the Gyárfás-Sumner conjecture, Graphs and Combinatorics 40 (2024), article 33 (with Tung Nguyen and Paul Seymour)
155. Clique covers of H-free graphs, European Journal of Combinatorics 118 (2024), 103909 (with Tung Nguyen, Paul Seymour and Stéphan Thomassé)
154. Induced paths in graphs without anticomplete cycles, Journal of Combinatorial Theory, Series B 164 (2024), 321-339 (with Tung Nguyen and Paul Seymour)
153. Graphs of large chromatic number, Proceedings of the International Congress of Mathematicians 2022 vol 6 (2023), 4660-4681
152. Parking on the integers, Annals of Applied Probability 33 (2023), 1076-1101 (with Michał Przykucki and Alexander Roberts)
151. Best-response dynamics, playing sequences, and convergence to equilibrium in random games, International Journal of Game Theory 52 (2023), 703-735 (with Torsten Heinrich, Yoojin Jang, Luca
Mungo, Marco Pangallo, Bassel Tarbush and Samuel Wiese)
150. Balancing connected colourings of graphs, Electronic Journal of Combinatorics 30 (2023), P1.54 (with Freddie Illingworth, Emil Powierski and Youri Tamitegama)
149. Clustered colouring of graph classes with bounded treedepth or pathwidth, Combinatorics, Probability and Computing 32 (2023), 122-133 (with Sergey Norin and David Wood)
148. Decomposing random permutations into order-isomorphic subpermutations, SIAM Journal on Discrete Mathematics 37 (2023), 1252-1261 (with Carla Groenland, Tom Johnston, Dániel Korándi, Alexander
Roberts and Jane Tan)
147. Pure pairs. VII. Homogeneous submatrices in 0/1-matrices with a forbidden submatrix, Journal of Combinatorial Theory, Series B 161 (2023), 437-464 (with Paul Seymour and Sophie Spirkl)
146. Pure pairs. V. Excluding some long subdivision, Combinatorica 43 (2023), 571-593 (with Paul Seymour and Sophie Spirkl)
145. Counting partitions of G[n,1/2] with degree congruence conditions, Random Structures and Algorithms 62 (2023), 564-584 (with Paul Balister, Emil Powierski and Jane Tan)
144. Polynomial bounds for chromatic number. I. Excluding a biclique and an induced tree, Journal of Graph Theory 102 (2023), 458-471 (with Paul Seymour and Sophie Spirkl)
143. Polynomial bounds for chromatic number. IV. A near-polynomial bound for excluding the five-vertex path, Combinatorica 43 (2023), 845-852 (with Paul Seymour and Sophie Spirkl)
142. Polynomial bounds for chromatic number. VI. Adding a four-vertex path, European Journal of Combinatorics 110 (2023), 103710 (with Maria Chudnovsky, Paul Seymour and Sophie Spirkl)
141. Polynomial bounds for chromatic number. VII. Disjoint holes, Journal of Graph Theory 104 (2023), 499-515 (with Maria Chudnovsky, Paul Seymour and Sophie Spirkl)
140. Erdős-Hajnal for graphs with no 5-hole, Proceedings of the London Mathematical Society 126 (2023), 997-1014 (with Maria Chudnovsky, Paul Seymour and Sophie Spirkl)
139. Pure pairs. IV. Trees in bipartite graphs, Journal of Combinatorial Theory, Series B 161 (2023), 120-146 (with Paul Seymour and Sophie Spirkl)
138. Strengthening Rödl's theorem, Journal of Combinatorial Theory, Series B 163 (2023), 256-271 (with Maria Chudnovsky, Paul Seymour and Sophie Spirkl)
137. Polynomial bounds for chromatic number. III. Excluding a double star, Journal of Graph Theory 102 (2022), 323-340 (with Paul Seymour and Sophie Spirkl)
136. Polynomial bounds for chromatic number. II. Excluding a star-forest, Journal of Graph Theory 101 (2022), 318-322 (with Paul Seymour and Sophie Spirkl)
135. A note on infinite antichain density, SIAM Journal on Discrete Mathematics 36 (2022), 573-577 (with Paul Balister, Emil Powierski and Jane Tan)
134. Pure pairs. VI. Excluding an ordered tree, SIAM Journal on Discrete Mathematics 36 (2022), 170-187 (with Paul Seymour and Sophie Spirkl)
133. Shotgun reconstruction in the hypercube, Random Structures and Algorithms 60 (2022), 117-150 (with Michał Przykucki and Alexander Roberts)
132. Concatenating bipartite graphs, Electronic Journal of Combinatorics 29 (2022), P2.47 (with Maria Chudnovsky, Patrick Hompe, Paul Seymour and Sophie Spirkl)
131. Active clustering for labeling training data, 35th Conference on Neural Information Processing Systems (NeurIPS 2021) (with Quentin Lutz, Élie de Panafieu, and Maya Stein)
130. A note on simplicial cliques, Discrete Mathematics 344 (2021), Article 112470 (with Maria Chudnovsky, Paul Seymour and Sophie Spirkl)
129. Optimal labelling schemes for adjacency, comparability and reachability, Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2021),1109-1117 (with Marthe Bonamy,
Louis Esperet and Carla Groenland)
128. Powers of paths in tournaments Combinatorics, Probability and Computing 30 (2021), 894-898 (with Nemanja Draganić, François Dross, Jacob Fox, António Girão, Frédéric Havet, Dániel
Korándi, William Lochet, David Munhá Correia and Benny Sudakov)
127. Exact stability for Turán's Theorem, Advances in Combinatorics, December 29, 2021 (with Dániel Korándi and Alexander Roberts)
126. Finding a shortest odd hole, ACM Transactions on Algorithms 17 (2021) Article 13, 21 pages (with Maria Chudnovsky and Paul Seymour)
125. A universal exponent for homeomorphs, Israel J. Math. 243 (2021) 141-154 (with Peter Keevash, Jason Long and Bhargav Narayanan)
124. Monochromatic components in edge-coloured graphs with large minimum degree, Electronic Journal of Combinatorics 28 (2021), P1.10 (with Hannah Guggiari)
123. Combinatorics in the exterior algebra and the Bollobás Two Families Theorem, Journal of the London Mathematical Society 104 (2021), 1812-1839 (with Elizabeth Wilmer)
122. Detecting a long odd hole, Combinatorica 41 (2021), 1-30 (with Maria Chudnovsky and Paul Seymour)
121. Maximising the number of cycles in graphs with forbidden subgraphs, Journal of Combinatorial Theory, Series B 147 (2021), 201-237 (with Natasha Morrison and Alex Roberts)
120. Lipschitz bijections between boolean functions, Combinatorics, Probability and Computing 30 (2021) 513-525 (with Tom Johnston)
119. Separation dimension and degree, Math. Proc. Camb. Phil. Soc. 170 (2021), 549-558 (with David Wood)
118. Size reconstructibility of graphs, Journal of Graph Theory 96 (2021), 326-337 (with Carla Groenland and Hannah Guggiari)
117. Pure pairs. II. Excluding all subdivisions of a graph, Combinatorica 41 (2021), 379-405 (with Maria Chudnovsky, Paul Seymour and Sophie Spirkl)
116. The component structure of dense random subgraphs of the hypercube, Random Structures and Algorithms 59 (2021), 3-24 (with Colin McDiarmid and Paul Withers)
115. Induced subgraphs of graphs with large chromatic number. V. Chandeliers and strings, Journal of Combinatorial Theory, Series B 150 (2021), 195-243 (with Maria Chudnovsky and Paul Seymour)
114. Proof of the Kalai-Meshulam Conjecture, Israel J Math 238 (2020), 639-661 (with Maria Chudnovsky, Paul Seymour and Sophie Spirkl)
113. Induced subgraphs of graphs with large chromatic number. XIII. New brooms, European Journal of Combinatorics 84 (2020), 103024 (with Paul Seymour)
112. Pure pairs. I. Trees and linear anticomplete pairs, Advances in Mathematics 375 (2 December 2020), 107396 (with Maria Chudnovsky, Paul Seymour and Sophie Spirkl)
111. Partitioning the vertices of a torus into isomorphic subgraphs, Journal of Combinatorial Theory, Series A 174 (2020), 105252 (with Marthe Bonamy and Natasha Morrison)
110. Induced subgraphs of graphs with large chromatic number. VI. Banana trees, Journal of Combinatorial Theory, Series B 145 (2020), 487-510 (with Paul Seymour)
109. A survey of χ-boundedness, Journal of Graph Theory 95 (2020), 473-504 (with Paul Seymour)
108. Pure pairs. III. Sparse graphs with no polynomial-sized anticomplete pairs, Journal of Graph Theory 95 (2020), 315-340 (with Maria Chudnovsky, Jacob Fox, Paul Seymour and Sophie Spirkl)
107. Moderate deviations of subgraph counts in the Erdős-Rényi random graphs G(n,m) and G(n,p), Transactions of the American Mathematical Society 373 (2020), 5517 5585 (with Christina Goldschmidt and
Simon Griffiths)
106. Exceptional graphs for the random walk, Annales de l Institut Henri Poincaré, 56 (2020), 2017-2027 (with Juhan Aru, Carla Groenland, Tom Johnston, Bhargav Narayanan and Alex Roberts)
105. Induced subgraphs of graphs with large chromatic number. VII. Gyárfás' complementation conjecture, Journal of Combinatorial Theory, Series B 142 (2020), 43-55 (with Paul Seymour)
104. Detecting an odd hole, Journal of the ACM 67, 1, Article 5 (January 2020), 12 pages (with Maria Chudnovsky, Paul Seymour and Sophie Spirkl)
103. Better bounds for poset dimension and boxicity, Transactions of the American Mathematical Society 373 (2020), 2157-2172 (with David Wood)
102. Induced subgraphs of graphs with large chromatic number. VIII. Long odd holes, Journal of Combinatorial Theory, Series B 140 (2020), 84-97(with Maria Chudnovsky, Paul Seymour and Sophie Spirkl)
101. Near-domination in graphs, Journal of Combinatorial Theory, Series A 165 (2019), 392-407 (with Bruce Reed and Paul Seymour)
100. Maximising H-colourings of graphs, Journal of Graph Theory 92 (2019), 172-185 (with Hannah Guggiari)
99. Towards Erdős-Hajnal for graphs with no 5-hole, Combinatorica 39 (2019), 983-991 (with Maria Chudnovsky, Jacob Fox, Paul Seymour and Sophie Spirkl)
98. Clustered colouring in minor-closed classes, Combinatorica 39 (2019), 1387-1412 (with Sergey Norin, Paul Seymour and David Wood)
97. Bad news for chordal partitions, Journal of Graph Theory 90 (2019), 5-12 (with Paul Seymour and David Wood)
96. Induced subgraphs of graphs with large chromatic number. XII. Distant stars, Journal of Graph Theory 92 (2019), 237-254 (with Maria Chudnovsky and Paul Seymour)
95. Induced subgraphs of graphs with large chromatic number. XI. Orientations, European Journal of Combinatorics 76 (2019), 53-61 (with Maria Chudnovsky and Paul Seymour)
94. Induced subgraphs of graphs with large chromatic number. X. Holes with specific residue, Combinatorica 39 (2019), 1105-1132 (with Paul Seymour)
93. Disjoint paths in unions of tournaments, Journal of Combinatorial Theory, Series B 135 (2019), 238-255 (with Maria Chudnovsky and Paul Seymour)
92. H-colouring P[t]-free graphs in subexponential time, Discrete Applied Mathematics 92 (2019), 172-185 (with Carla Groenland, Karolina Okrasa, Paweł Rzążewski, Paul Seymour and Sophie Spirkl)
91. Stability results for graphs with a critical edge, European Journal of Combinatorics 94 (2018), 27-38 (with Alexander Roberts)
90. Balancing sums of random vectors, Discrete Analysis 2018:4, 16 pp. (with Juhan Aru, Bhargav Narayanan and Ramarathnam Venkatesan)
89. Induced subgraphs of graphs with large chromatic number. IV. Consecutive holes, Journal of Combinatorial Theory, Series B 132 (2018), 180-235 (with Paul Seymour)
88. How unproportional must a graph be?, European Journal of Combinatorics 73 (2018), 138-152 (with Humberto Naves and Oleg Pikhurko)
87. Supersaturation in posets and applications involving the container method, Journal of Combinatorial Theory, Series A 154 (2018), 247-284 (with Jonathan Noel and Benny Sudakov)
86. Induced subgraphs of graphs with large chromatic number. IX. Rainbow paths, Electronic Journal of Combinatorics 24 (2017), Paper P2.53 (with Paul Seymour)
85. Induced subgraphs of graphs with large chromatic number. III. Long holes, Combinatorica 37 (2017), 1057-1072 (with Maria Chudnovsky and Paul Seymour)
84. A note on intersecting hypergraphs with large cover number, Electronic Journal of Combinatorics 24 (2017), Paper P3.26 (with Penny Haxell)
83. Maximising the number of induced cycles in a graph, Journal of Combinatorial Theory, Series B 126 (2017), 24-61 (with Natasha Morrison)
82. On a problem of Erdős and Moser, Abhandlungen des Mathematischen Seminars der Universität Hamburg 87 (2017), 213-222 (with Béla Bollobás)
81. Packing random graphs and hypergraphs, Random Structures and Algorithms 51 (2017), 3-13 (with Béla Bollobás and Svante Janson)
80. Saturation in the hypercube and bootstrap percolation, Combinatorics, Probability and Computing 26 (2017), 78-98 (with Natasha Morrison and Jonathan Noel)
79. On lower bounds for the matching number of subcubic graphs, Journal of Graph Theory 85 (2017), 336-348 (with Penny Haxell)
78. Uniform multicommodity flows in the hypercube with random edge capacities, Random Structures and Algorithms, 50 (2017), 437-463 (with Colin McDiarmid and Paul Withers)
77. Induced subgraphs of graphs with large chromatic number. I. Odd holes, Journal of Combinatorial Theory, Series B 121 (2016), 68-84 (with Paul Seymour)
76. Induced subgraphs of graphs with large chromatic number. II. Three steps towards Gyárfás' conjectures, Journal of Combinatorial Theory, Series B 118 (2016), 109 128 (with Maria Chudnovsky and
Paul Seymour)
75. Random graphs from a block-stable class, European Journal of Combinatorics 58 (2016), 96-106 (with Colin McDiarmid)
74. Disjoint dijoins, Journal of Combinatorial Theory Series B 120 (2016), 18-35 (with Maria Chudnovsky, Katherine Edwards, Ringi Kim and Paul Seymour)
73. Feedback from nature: simple randomised distributed algorithms for maximal independent set selection and greedy colouring, Distributed Computing 29 (2016), 377-393 (with Peter Jeavons and Lei Xu;
journal version of #62)
72. The parameterised complexity of list problems on graphs of bounded treewidth, Information and Computation 251 (2016), 91 103 (with Kitty Meeks)
71. Disjoint induced subgraphs of the same order and size, European Journal of Combinatorics 49 (2015), 153-166 (with Béla Bollobás, Teeradej Kittipassorn and Bhargav Narayanan)
70. Disjoint paths in tournaments, Advances in Mathematics 270 (2015), 582-597 (with Maria Chudnovsky and Paul Seymour)
69. Intersections of hypergraphs, Journal of Combinatorial Theory, Series B 110 (2015), 180 208 (with Béla Bollobás)
68. Intersections of random hypergraphs and tournaments, European Journal of Combinatorics 44A (2015), 125 139 (with Béla Bollobás)
67. Complete monotonicity for inverse powers of some combinatorially defined polynomials, Acta Mathematica 213 (2014), 323-392 (with Alan Sokal)
66. Excluding pairs of graphs, Journal of Combinatorial Theory, Series B 106 (2014), 15-29 (with Maria Chudnovsky and Paul Seymour)
65. On Saturated k-Sperner Systems, Electronic Journal of Combinatorics 21 (2014), Paper #P3.22 (with Natasha Morrison and Jonathan Noel)
64. For most graphs H, most H-free graphs have a linear homogeneous set, Random Structures and Algorithms 45 (2014), 343 361 (with Ross J. Kang, Colin McDiarmid and Bruce Reed)
63. Hypergraphs of bounded disjointness, SIAM Journal on Discrete Mathematics 28 (2014), 372-384 (with Elizabeth Wilmer)
62. Spanning trees and the complexity of flood-filling games Theory Comput. Syst. 54 (2014), 731-753; preliminary version, FUN 2012, Lecture Notes in Computer Science 7288 (2012), 282-292; (with
Kitty Meeks)
61. Feedback from nature: an optimal distributed algorithm for maximal independent set selection, PODC '13: Proceedings of the 2013 ACM symposium on Principles of distributed computing (2013),
147-156 (with Peter Jeavons and Lei Xu)
60. Substitution and χ-boundedness, Journal of Combinatorial Theory, Series B 103 (2013), 567-586 (with Maria Chudnovsky, Irena Penev and Nicolas Trotignon)
59. Tournaments and colouring, Journal of Combinatorial Theory, Series B 103 (2013), 1-20 (with Eli Berger, Krzysztof Choromanski, Maria Chudnovksy, Jacob Fox, Martin Loebl, Paul Seymour and Stephan
58. The complexity of Free-Flood-It on 2xn boards, Theoretical Computer Science 500 (2013), 25-43 (with Kitty Meeks)
57. A counterexample to a conjecture of Schwartz, Social Choice and Welfare 40 (2013), 739-743 (with Felix Brandt, Maria Chudnovsky, Ilhee Kim, Gaku Liu, Sergey Norin, Paul Seymour and Stephan
56. Excluding induced subdivisions of the bull and related graphs, Journal of Graph Theory 71 (2012), 49-68 (with Maria Chudnovsky, Irena Penev and Nicolas Trotignon)
55. Monochromatic cycles in 2-Coloured graphs, Combinatorics, Probability and Computing 21 (2012), 57-87 (with Fabricio Benevides, Tomasz Łuczak, Jozef Skokan and Matthew White)
54. The complexity of flood-filling games on graphs, Discrete Applied Mathematics 160 (2012), 959-969 (with Kitty Meeks)
53. The minimal covering set in large tournaments, Social Choice and Welfare 38 (2012), 1-9 (with Mark Fey)
52. On Ryser's Conjecture, Electronic Journal of Combinatorics 19 (2012), #P23, 10 pages (with Penny Haxell)
51. A bound for the cops and robbers problem, SIAM Journal on Discrete Mathematics 25 (2011), 1438-1442 (with Benny Sudakov)
50. Cover-decomposition and polychromatic numbers SIAM Journal of Discrete Mathematics 27 (2013), 240-256; preliminary version, Algorithms ESA 2011, Lecture Notes in Computer Science 6942 (2011),
799-810 (with Béla Bollobás, David Pritchard and Thomas Rothvoss)
49. Szemerédi's Regularity Lemma for matrices and sparse graphs, Combinatorics, Probability and Computing 20 (2011), 455-466
48. Intersections of graphs, Journal of Graph Theory 66 (2011), 261-282 (with Béla Bollobás)
47. Almost all H-free graphs have the Erdős-Hajnal property, An Irregular Mind (Szemerédi is 70), Bolyai Society Mathematical Studies, Springer, Berlin, 21 (2010) 405-414 (with Martin Loebl, Bruce
Reed, Stephan Thomassé and Andrew Thomason)
46. Max k-cut and judicious k-partitions, Discrete Math. 310 (2010), 2126-2139 (with Béla Bollobás)
45. Some variants of the exponential formula, with application to the multivariate Tutte polynomial (alias Potts model), Séminaire Lotharingien de Combinatoire 61A (2009), Article B61Ae, 33 pages
(with Alan Sokal)
44. Uniform multicommodity flow through the complete graph with random edge-capacities, Operations Research Letters 37 (2009), 299--302 (with David Aldous and Colin McDiarmid)
43. Polynomial Constraint Satisfaction Problems, Graph Bisection, and the Ising Partition Function, ACM Transactions on Algorithms (TALG) 5 (2009), Article No 45 (with Gregory Sorkin)
42. Maximum directed cuts in acyclic digraphs, Journal of Graph Theory 55 (2007), 1-13 (with Noga Alon, Béla Bollobás, András Gyárfás and Jenő Lehel)
41. Linear-programming design and analysis of fast algorithms for Max 2-CSP, Discrete Optimization 4 (2007), 260-287 (with Gregory Sorkin)
40. On separating systems, European Journal of Combinatorics 28 (2007), 1068-1071 (with B. Bollobás)
39. Computational complexity of some restricted instances of 3SAT, Discrete Applied Mathematics 155 (2007), 649-653 (with Piotr Berman and Marek Karpinski)
38. Separating systems and oriented graphs of diameter two, Journal of Combinatorial Theory Series B 97 (2007), 193-203 (with Béla Bollobás)
37. Partitions and orientations of the Rado graph, Transactions of the American Mathematical Society 359 (2007), no. 5, 2395--2405 (with Reinhard Diestel, Imre Leader and Stephan Thomassé)
36. Infinite locally random graphs, Internet Mathematics 3 (2006), 321-332 (with Pierre Charbit)
35. On dependency graphs and the lattice gas, Combinatorics, Probability and Computing 15 (2006), 253-279 (with Alan Sokal)
34. An LP-Designed Algorithm for Constraint Satisfaction, Lecture Notes in Computer Science 4168 (2006) 588-599 (with Gregory Sorkin)
33. Reconstructing under group actions, Graphs and Combinatorics 22 (2006), 399-419 (with Jamie Radcliffe)
32. Solving Sparse Random Instances of Max Cut and Max 2-CSP in Linear Expected Time, Combinatorics, Probability and Computing 15 (2006), 281-315 (with Gregory Sorkin)
31. Discrepancy in graphs and hypergraphs, More sets, graphs and numbers, Bolyai Society Mathematical Studies, Springer, Berlin, 15 (2006), 33-56 (with Béla Bollobás)
30. The repulsive lattice gas, the independent-set polynomial, and the Lovász Local Lemma, Journal of Statistical Physics 118 (2005), 1151-1261 (with Alan Sokal)
29. Judicious partitions and related problems, in Surveys in Combinatorics 2005, 95-117, London Math. Soc. Lecture Note Ser., 327, Cambridge Univ. Press, Cambridge, 2005
28. Reversals and transpositions over finite alphabets, SIAM Journal of Discrete Mathematics 19 (2005), 224-244 (with Jamie Radcliffe and Elizabeth Wilmer)
27. Max Cut for random graphs with a planted partition, Combinatorics, Probability and Computing 13 (2004), 451-474 (with Béla Bollobás)
26. Judicious partitions of bounded-degree graphs, Journal of Graph Theory 46 (2004), 131-143 (with Béla Bollobás)
25. Finite subsets of the plane are 18-reconstructible, SIAM Journal of Discrete Mathematics, 16 (2003), 262-275 (with Luke Pebody and Jamie Radcliffe)
24. Faster algorithms for MAX CUT and MAX CSP, with polynomial expected time for sparse instances, Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques, proceedings
of RANDOM 2003, Lecture Notes in Computer Science 2764 (2003), 382-395 (with Gregory Sorkin)
23. Problems and results on judicious partitions, Random Structures and Algorithms 21 (2002), 414-430 (with Béla Bollobás)
22. On cycle lengths in graphs, Graphs and Combinatorics 18 (2002), 491-496 (with Ron Gould and Penny Haxell)
21. Better bounds for Max Cut, Contemporary Combinatorics, Bolyai Society Mathematical Studies 10 (2002), 185-246 (with Béla Bollobás)
20. Alternating knot diagrams, Euler circuits and the interlace polynomial, European Journal of Combinatorics 22 (2001), 1-4 (with Paul Balister, Béla Bollobás and Oliver Riordan)
19. On induced subgraphs with all degrees odd, Graphs and Combinatorics 17 (2001), 539-553
18. Subdivisions of transitive tournaments, European Journal of Combinatorics 21 (2000), 1067-1071
17. Judicious partitions of 3-uniform hypergraphs, European Journal of Combinatorics 21 (2000), 289-300 (with Béla Bollobás)
16. Exact bounds for judicious bipartitions of graphs, Combinatorica 19 (1999) 473-486 (with Béla Bollobás)
15. Another simple proof of a theorem of Milner, Journal of Combinatorial Theory Series A 87 (1999), 379-380
14. Reconstructing subsets of reals, Electronic Journal of Combinatorics 6 (1999), Research Paper 20, 7 pages (with Jamie Radcliffe)
13. Induced cycles and chromatic number, Journal of Combinatorial Theory Series B 76 (1999), 150-154
12. Reconstructing subsets of Z[n], Journal of Combinatorial Theory Series A 83 (1998), 169-187 (with Jamie Radcliffe)
11. Judicious partitions of hypergraphs, Journal of Combinatorial Theory Series A 78 (1997), 15-31 (with Béla Bollobás)
10. Induced trees in graphs of large chromatic number, Journal of Graph Theory 24 (1997), 297-311
9. Reconstructing sequences, Discrete Mathematics 175 (1997), 231-238
8. All trees contain a large induced subgraph having all degrees 1 (mod k), Discrete Mathematics 175 (1997), 35-40 (with D.M. Berman, Jamie Radcliffe, H. Wang and L. Wargo)
7. On graph decompositions modulo k, Discrete Mathematics 175 (1997), 289-291
6. Better bounds for perpetual gossiping, Discrete Applied Mathematics 75 (1997), 189-197
5. Independent sets and repeated degrees, Discrete Mathematics 170 (1997), 41-49 (with Béla Bollobás)
4. A proof of a conjecture of Bondy concerning paths in weighted digraphs, Journal of Combinatorial Theory Series B 66 (1996), 283-292 (with Béla Bollobás)
3. Every tree has a large induced subgraph with all degrees odd, Discrete Mathematics 140 (1995), 275-279 (with Jamie Radcliffe)
2. On judicious partitions of graphs, Periodica Mathematica Hungarica 26 (1993), 127-139 (with Béla Bollobás)
1. Large induced subgraphs with all degrees odd, Combinatorics, Probability and Computing 1 (1992), 335-349
Edited book
Combinatorics and Probability: Celebrating Béla Bollobás s 60th birthday, Cambridge University Press (2007), 660 pages (edited volume, with Graham Brightwell, Imre Leader, and Andrew Thomason)
Other publications
4. The mathematics and physics of phase transitions, UCL Science 17 (2003), 12-13; this article also appeared in French translation in Quadrature, a magazine aimed at high-school and university
students of mathematics in France (with Alan Sokal)
3. The paradox of the question, Analysis 59 (1999), 331-335 (with Michael Scott)
2. Taking the Measure of Doom, Journal of Philosophy 95 (1998), 133-141 (with Michael Scott)
1. What is in the Two Envelopes Paradox?, Analysis 57 (1997), 34-41 (with Michael Scott)
Technical reports
9. Structure of random r-SAT below the pure literal threshold, CoRR abs/1008.1260 (2010) (with Gregory Sorkin)
8. Generalized Constraint Satisfaction Problems, IBM Technical Report RC23935 (2006) (with Gregory Sorkin)
7. Computational Complexity of Some Restricted Instances of 3SAT, Electronic Colloquium on Computational Complexity, Technical Report TR04-111 (with Piotr Berman and Marek Karpinski)
6. Faster exponential algorithms for Max Cut, Max 2-Sat and Max k-Cut, IBM Technical Report RC23457 (2004) (with Gregory Sorkin)
5. Solving sparse semi-random instances of Max Cut and Max 2-CSP, IBM Technical report RC23417 (2004) (with Gregory Sorkin)
4. Approximation hardness and satisfiability of bounded occurrence instances of SAT, Electronic Colloquium on Computational Complexity, Technical Report TR03-022 (2003) (with Piotr Berman and Marek
3. Approximation hardness of short symmetric instances of MAX-3SAT, Electronic Colloquium on Computational Complexity, Technical Report TR03-049 (2003) (with Piotr Berman and Marek Karpinski)
2. Arithmetic progressions of cycles, Technical Report No. 16 (1998), Matematiska Institutionen, Umea universitet (with Roland Haggkvist)
1. Cycles of nearly equal length in cubic graphs, Technical Report No. 15 (1998), Matematiska Institutionen, Umea universitet (with Roland Haggkvist)
Two joint publications
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Comparing Fractions Calculator - Easy and Free Tool - Factor Calculator
Comparing Fractions Calculator – Easy and Free Tool
Our Comparing Fractions Calculator helps you effortlessly compare fractions, decimals, and percentages to determine which is greater, lesser, or if they are equal. Enter two fractions, hit the
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Comparing Fractions Calculator
What is this calculator for?
Our Fraction Comparison Calculator is a versatile tool designed to help users compare the sizes of fractions, decimals, and percentages. Whether you’re a student learning fractions or an adult
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2. Click Compare: After entering the fractions, click the “Compare” button to generate results.
3. View Result: The result section will display whether the first fraction is greater than, lesser than, or equal to the second one. For instance, it might show “1/2 is less than 3/4”.
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Our Fraction Comparison Calculator determines the relative sizes of fractions, decimals, and percentages. It helps users understand which fraction is larger or smaller, aiding in various mathematical
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Can this calculator compare mixed numbers or improper fractions?
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Question #47267 | Socratic
Question #47267
1 Answer
The key to this problem is the density of the solution.
The density of a substance essentially tells you the mass of one unit of volume of that substance. In your case, the density of the solution is said to be $2.5$grams per milliliter, which means that
one unit of volume will be $\text{1 mL}$.
So, the density of the solution tells you that every $\text{1 mL}$ of this solution has a mass of $\text{2.5 g}$.
You can thus use density as a conversion factor to help you convert between the mass of the solution and the volume it occupies.
Now, the problem tries to put you off track a little by telling you that
1000 mL of an extremely viscous solution has a density of $2.5$grams/milliliter...
It doesn't matter how much of this solution you have, its density will always be the same.
This means that you can ignore the given $\text{1000 mL}$ altogether, that information is not important in this context.
A fluid ounce is approximately equal to
$\textcolor{p u r p \le}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\text{1 fl oz " = " 29.57 mL}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
Use this conversion factor to go from fluid ounces to milliliters
#2 color(red)(cancel(color(black)("fl oz"))) * "29.57 mL"/(1color(red)(cancel(color(black)("fl oz")))) = "59.14 mL"#
Now, if $\text{1 mL}$ of this solution has a mass of $\text{2.5 g}$, it follows that $\text{59.14 mL}$ will have a mass of
#59.14 color(red)(cancel(color(black)("mL"))) * overbrace("2.5 g"/(1color(red)(cancel(color(black)("mL")))))^(color(blue)("the density of the solution")) = "147.85 g"#
You should round this off to one sig fig, the number of sig figs you have for the volume of the solution, i.e. $2$ fluid ounces, but I'll leave it rounded to two sig figs
#"mass of 2 fl oz" = color(green)(|bar(ul(color(white)(a/a)color(black)("150 g")color(white)(a/a)|)))#
Finally, to convert this to ounces, use the conversion factor
$\textcolor{p u r p \le}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\text{1 oz " = " 28.35 g}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
You will have
#150 color(red)(cancel(color(black)("g"))) * "1 oz"/(28.35color(red)(cancel(color(black)("g")))) = color(green)(|bar(ul(color(white)(a/a)color(black)("5.3 oz")color(white)(a/a)|)))#
Impact of this question
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Operators and Operations | NUS Hackers Wiki
Operators in JavaScript are symbols that can perform operations on certain values and data types. The most basic example are the arithmetic operators: +, -, * and /. Operators each have a
corresponding operation, can be binary or unary, and can accept a fixed set of values
A binary operator is one that can operate on two values, and a unary operator can operate only on one value
A value that an operator performs an operation on is called an operand
The following section has a few tables of operators, their corresponding operations and their accepted operand(s).
Arithmetic operators
These are the classic addition, subtraction, multiplication and division operators that are nearly standard across languages:
These are all binary operators that take two numbers in as operands.
As an aside, the + operator also allows for string concatenation and array concatenation:
let str1 = "hello";
let str2 = "world";
str1 + str2; // evaluates to "helloworld"
let arr1 = [1, 2];
let arr2 = [3, 4];
arr1 + arr2; // evaluates to [1, 2, 3, 4]
Other math operators
The ** operator allows for exponents:
let x = 2;
let y = 3;
x ** y; // 8
The % operator gets the remainder after division
let x = 5;
let y = 3;
x % y; // 2
Comparison operators
These operators allow for, as you can guess, comparing two values. They are binary operators that return a boolean value. The symbols are as follows, with their usage with numbers being as expected:
a > b returns true if a is greater than b
a >= b returns true if a is greater than or equal to b
a < b returns true if a is lesser than b
a <= b returns true if a is lesser than or equal to b
a !== b returns true if a is not equal to b
The comparison operators can also be used to compare two strings; they return true or false based on a character-by-character comparison of the two strings. An example is below:
let str1 = "abcd";
let str2 = "abdc";
str1 > str2; // returns false
The way the above works is that the first character in each string are compared. If they are equal, then the next character in each string are compared. Eventually, when two characters in a string
are unequal, then they are compared according to the operator (after internally converting the characters to their numeric ASCII values) and a boolean value returned accordingly.
If one string ends before the other and all characters up till that point are the same, the longer string is deemed the "greater" value:
"abcd" > "abc"; // returns true
Equality operators
The equality operation has two possible operators, each of which function slightly differently.
The first is the "triple-equals" or the "type-strict equals" operator: ===. This operator works as you would expect:
2 === 2; // returns true
2 === 3; // returns false
2 === "abcd"; // returns false
2 === "2"; // returns false
2 === [2]; // returns false
The second is the "double-equals" or the "type-lax equals" operator: ==. This operator works similar to the type-strict equals operator, except that it implicity converts the two operands to the same
2 == 2; // returns true
2 == 3; // returns false
2 == "abcd"; // returns false
2 == "2"; // returns true, because "2" is converted to 2
2 == [2]; // returns true, because [2] is converted to 2
2 == [2, 2]; // returns false, because there is more than one value in the array now
It is good practice to stick to more strict typing in your program, to prevent the chance of errors propagating. On the other hand, sometimes it may be prefered to use the type-lax equals, such as in
cases where a value has to store the number 2 but it is irrelevant whether it is in a string, array, or as a primitive number. But such cases are rare.
Logical operators
These operators can take one or two boolean expressions and returns a boolean value. They are often combined with comparison operators.
The AND operator, && (double ampersand symbol), is a binary operator that compares two boolean values expressions and returns true only if both the expressions evaluate to true. If either one of the
expressions evaluate to false, then && returns false.
2 === 2 && 3 < 4; // true because 2 is equal to 2 AND 3 is less than 4
2 === 2 && 3 > 4; // false because 3 is not greater than 4
2 === 3 && 3 > 4; // false
The OR operator, || (double bars), is a binary operator that compares two boolean expressions and returns true if either one of them evaluates to true. If both the expressions evaluate to false, then
the operator returns false.
2 === 2 || 3 < 4; // true
2 === 2 || 3 > 4; // true because 2 is equal to 2
2 === 3 || 3 > 4; // false because 2 is not equal to 3 and 3 is not greater than 4
The NOT operator, !, is a unary operator that reverses a boolean expression's value. If the expression evaluates to true, it returns false; if the expression evalutes to false then it returns true.
It does not change the original expression's value
let value = 2 === 2 || 3 < 4; // true
!value; // false
value; // still true
Assignment operators
Value assignment operator
As seen before, this operator has the symbol = and allows you to assign a value to a variable or constant. Example:
let x = 10; // assigning 10 to x
Operation assignment operators
This class of operators are formed by combining a binary logical or mathematical operator with the value assignment operator: op=. They can then be used as an assignment operator, assigning a value
to a variable while performing the binary operation on both. Essentially: a op= b is the same as a = a op b. Some examples are below:
let x = 10;
x += 5; // same as x = x + 5; x is now 15
x -= 2; // same as x = x - 2; x is now 13
x *= 10; // same as x = x * 10; x is now 130
x **= 2; // same as x = x ** 2; x is now 16900
x; // 16900
let y = x > 1000; // 16900 is greater than 1000, so y is true
y &&= (x !== 4); // same as y = y && (x !== 4)
// x is not equal to 4, so the expression in brackets evaluates to true
y; // true && true gives true
Increment and decrement operators
These two unary operators allow to increment and decrement numeric variable values by 1. The increment operator is a double plus (++) and the decrement operator is a double minus (--). They can be
placed either behind or in front of a variable name, and are accordingly called pre- or post-increment or decrement.
let m = 4;
m++; // post-increment, m is now 5
m--; // post-decrement, m is back to 4
--m; // pre-decrement, m is now 3
++m; // pre-increment, m is back to 4 again
The difference between the pre- and post- version of these operators is to do with their return values:
let x = 10;
let y;
y = ++x; // x is 11, and y is also 11
y = x++; // x is 12, but y remains 11 because the increment happens after assignment
Bitwise operators
These operators allow you to perform bitwise operations like bitwise AND, OR and NOT on values. To see how they work in detail, visit this site.
Type conversion
Recall that JavaScript is a weakly-typed language. This means that you can end up with situations like this:
"10" + 1; // "101" because 1 gets converted to a string and JS performs string concatenation
"10" - 1; // 9 because "10" gets converted to a number and JS performs subtraction
4 / "2"; // 2 because "2" gets converted to a number
"3" * "2"; // 6 because JS converts both operands to numbers
[3] / "10"; // 0.3, same reason
This is one of the reasons it is important to maintain type consistency and stronger typing in your code. It is also wacky issues like this that contributed to the popularity of languages like
TypeScript, a strongly-typed version of JavaScript.
Next steps
Now that we have covered data types and operations concerning these datatypes, the next section will cover a few more coding constructs of JavaScript. | {"url":"https://wiki.nushackers.org/orbital/readme/operations","timestamp":"2024-11-07T15:48:33Z","content_type":"text/html","content_length":"617673","record_id":"<urn:uuid:c9e8abf9-f4e3-40f0-acdc-e82e21664cc2>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00561.warc.gz"} |
Bohemia Interactive
Exponential function with the base e. Equal to e^x.
exp x
x: Number
Return Value:
Example 1:
_e = exp 1
// Returns e (2.7182...)
Example 2:
Additional Information
See also:
Report bugs on the Feedback Tracker and/or discuss them on the Arma Discord or on the Forums.
Only post proven facts here! Add Note
Franze - c
Posted on Mar 03, 2009 - 22:04 (UTC)
Note that you cannot exponent a value greater than 88.72283554077147726999 (999 repeating), as this is beyond what the game can calculate. | {"url":"https://community.bistudio.com/wiki/exp","timestamp":"2024-11-09T15:33:50Z","content_type":"text/html","content_length":"31707","record_id":"<urn:uuid:7c4540d0-8f7f-4250-b39e-d6935451ab0e>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00816.warc.gz"} |
CN106556395A - A kind of air navigation aid of the single camera vision system based on quaternary number
- Google Patents
CN106556395A - A kind of air navigation aid of the single camera vision system based on quaternary number - Google Patents
A kind of air navigation aid of the single camera vision system based on quaternary number Download PDF
Publication number
CN106556395A CN106556395A CN201611010993.6A CN201611010993A CN106556395A CN 106556395 A CN106556395 A CN 106556395A CN 201611010993 A CN201611010993 A CN 201611010993A CN 106556395 A CN106556395
A CN 106556395A
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quaternary number
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Beijing Union University
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Beijing Union University
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Priority to CN201611010993.6A priority Critical patent/CN106556395A/en
Publication of CN106556395A publication Critical patent/CN106556395A/en
Pending legal-status Critical Current
☆ G—PHYSICS
☆ G01—MEASURING; TESTING
☆ G01C—MEASURING DISTANCES, LEVELS OR BEARINGS; SURVEYING; NAVIGATION; GYROSCOPIC INSTRUMENTS; PHOTOGRAMMETRY OR VIDEOGRAMMETRY
☆ G01C21/00—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00
☆ G01C21/26—Navigation; Navigational instruments not provided for in groups G01C1/00 - G01C19/00 specially adapted for navigation in a road network
□ Engineering & Computer Science (AREA)
□ Radar, Positioning & Navigation (AREA)
□ Remote Sensing (AREA)
□ Automation & Control Theory (AREA)
□ Physics & Mathematics (AREA)
□ General Physics & Mathematics (AREA)
□ Navigation (AREA)
The present invention discloses a kind of air navigation aid of the single camera vision system based on quaternary number, including：Step 1, constructing environment map：Step 2, preservation
map datum and tracing point coordinate information；Step 3, loading map and track point coordinates calculate corner β according to yaw angle θ, and the direction of step 4, the angle beta
obtained according to step 3 and corner sends a command to vehicle bottom control.Using technical scheme, the navigation of the single camera vision system based on quaternary number on the ROS
platforms under linux system is realized, research cost is reduced.
A kind of air navigation aid of the single camera vision system based on quaternary number
Technical field
The invention belongs to the unmanned cruiser independent navigation field of low speed, is related to a kind of merely using room in monocular cam room Outside fix and the method for navigation.
Background technology
The unmanned intelligent vehicle that China is researching and developing just, in the middle of like a raging fire carrying out, has attracted countless colleges and universities, section Grind
the extensive concern of mechanism's research worker and automobile vendor.The working environment of low speed patrol intelligent vehicle patrol is zoo, dynamic In thing garden, vegetation is
grown prosperity, and it is weaker that Jing part of detecting patrol section receives gps signal, it is impossible to is advanced according to GPS navigation vehicle.Cause This, carries out map
structuring using sensor acquisition external information and navigation is particularly important.SLAM problems are in driving intelligent in recent years Car field is constantly subjected to
the extensive concern of research worker.The sensor that it can utilize robot self-contained sets up the environment being incremented by Map, then realizes self poisoning by building map,
without the need for any external reference systems (such as:GPS) and other sensors, With potential economic worth and being widely applied prospect.
For in the research of unmanned intelligent vehicle, map structuring and positioning are vehicle DAS (Driver Assistant System)s and unmanned The key technology in intelligent vehicle field.What
the various algorithms for SLAM had developed at present is more and more ripe, mainly uses one Plant or multiple sensors information allows the robot to independently build figure
positioning.Its method for solving is divided into the side based on Kalman filtering Method, the method optimized based on particle filter method and based on figure.Method based on figure
optimization is as which is in extensive environment It is excellent to build figure performance, become at present the main method of research both at home and abroad.But at present for the
research of this respect is only limited to determine Position and figure is built, be substantially the navigation of joint other sensors information for navigation feature, such as ultrasonic
sensor, Laser Measuring Distance meter, radar, stereoscopic vision etc., for the use of multisensor increased Financial cost.
The content of the invention
For the problems referred to above that prior art is present, the present invention is proposed on a kind of ROS platforms under linux system Air navigation aid based on the single camera vision
system of quaternary number.Firstly, it is necessary to realize the map structuring based on key frame, it is critical only that The trajectory coordinates information of the image center that
preservation is built during figure；Secondly tracing point coordinate information is carried out according to certain rule Overstocked point screening；Then map and track point coordinate data
are reloaded；Last calculating according to navigational portions algorithm of the present invention is worked as Previous frame coordinate points complete navigation feature to the traveling
angle and route of the next position image center point coordinates.
For achieving the above object, the present invention is adopted the following technical scheme that：
A kind of air navigation aid of the single camera vision system based on quaternary number, comprises the following steps：
Step 1, constructing environment map, comprise the following steps：
1) carry out camera calibration；
2) photographic head is fixed to into right ahead；
3) input picture is transformed into into greyscale color space from RGB color；
4) feature extraction and matching is carried out to image；
5) map initialization；
6) closed loop detection and reorientation；
7) obtain Current camera pose, calculate quaternary number be transformed into 180 degree to 180 degree scope Eulerian angles ψ, θ,It is described ψ、θ、Respectively about the z axis, the anglec
of rotation of Y-axis, X-axis, yaw angle θ are the angle of present frame；
Step 2, preservation map datum and tracing point coordinate information；
Step 3, loading map and track point coordinates calculate corner β according to yaw angle θ,
The direction of step 4, the angle beta obtained according to step 3 and corner, sends a command to vehicle bottom control.
Preferably, ψ, θ andScope is -90 degree to 90 degree, and correspondence solution formula is：
When θ scopes for -180 degree to+180 degree when, correspondence solution part formula is：
Preferably, the process that step 3 calculates corner β is：
If (dx, dy), (gx, gy) are respectively current point and will reach point coordinates, present frame angle is θ, if current point and Impact point line with x-axis angle isScope is 0 to 90 degree
；β is corner, and bottom Body Control is from left to right 180 degree is arrived for 0；γ judges angle for middle decision-making level, and different target point has different values from
current point position relationship.Current location When being divided into following four kinds of different situations with aiming spot relative position relation：
When the direction of motion is E to F, i.e., as (dx>gx&&dy<When gy)：
If 0=<θ<=90 i.e. L1 directions when, then left-hand rotation β=| θ |+90- α, if -180=<θ<- 90 i.e. L4 directions when, then Right-hand rotation β=| θ | -90+ α, if -90=<θ<0 i.e. L2 and L3 directions
when, then judge γ=90- α-| θ |, if γ>0 i.e. L2 directions are then Left-hand rotation β=γ, otherwise i.e. then turn right β=- γ in L3 directions；
When the direction of motion is that F ' arrives E ', i.e., as (dx<gx&&dy>When gy)：
If 0=<θ<=90 i.e. L1 directions when, then right-hand rotation β=90- | θ |+α, if -180=<θ<=-90 i.e. L4 directions when, Then left-hand rotation β=180- | θ |+90- α, if 90<θ<=180 i.e. L2 and L3
directions when, then judge γ=90- α-(180- | θ |), if γ>0 i.e. L3 directions when then turn left that otherwise i.e. then turn right β=γ β=- γ in L2 directions；
When direction of motion P is to Q, i.e., as (dx<gx&&dy<When gy)：
If -90=<θ<=0 i.e. L1 directions when, then right-hand rotation β=90+ | θ |-α, if 90=<θ<=180 i.e. L4 directions when, then Left-hand rotation β=| θ | -90+ α, if 0<θ<90 i.e. L2 and L3 directions
when, then judge γ=90- α-| θ |, if γ>0 i.e. L2 directions when then Right-hand rotation β=γ is otherwise that L3 directions are then turned left β=- γ；
When the direction of motion is that Q ' arrives P ', i.e., as (dx>gx&&dy>When gy)：
If -90=<θ<=0 i.e. L1 directions when, then left-hand rotation β=90- | θ |+α, if 90=<θ<=180 i.e. L4 directions when, then Right-hand rotation β=180- | θ |+90- α, if -180=<θ<- 90 i.e. L2 and L3
directions when, then judge γ=90+ α-| θ |, if γ>0 is Then turning left during L2 directions, otherwise i.e. then turn right β=γ β=- γ in L3 directions；
The present invention is as using above technical scheme, which has advantages below：1st, the present invention utilizes monocular vision SLAM systems System calculates image center pose
coordinate during figure is built as driving trace point coordinates.2nd, the present invention proposes that a kind of utilization builds figure Image center pose after rear monocular track
point coordinates and reorientation calculates vehicle using quaternary number and the conversion of Eulerian angles Angle data is navigated, and the method is not used any other external sensor,
is completed navigation using photographic head merely, is reduced Research cost, has potential using value.
Description of the drawings
Fig. 1 is method flow diagram involved in the present invention；
Fig. 2 is the chessboard figure that Zhang Zhengyou standardizitions are used；
Fig. 3 is FAST Corner Detection schematic diagrams；
Fig. 4 is the closed loop overhaul flow chart based on bag of words；
Fig. 5 is yaw angle calculation flow chart；
Fig. 6 is the corresponding Eulerian angles broken line graph of each frame；
Fig. 7 is corner direction calculating schematic diagram；
Fig. 8 is the navigation path result figure under unencryption map；
Fig. 9 is Fig. 8 trajector deviation analysis charts；
Figure 10 is the navigation path result figure under encryption map；
Figure 11 is Figure 10 trajector deviation analysis charts；
The GPS track comparison diagram that Figure 12 is built under figure and navigation for image；
Table 1 turns Eulerian angles value result for different quaternary numbers；
Table 2 is corner decision table.
Specific embodiment
The invention will be further described with reference to the accompanying drawings and examples.
The flow chart of the method for the invention is as shown in figure 1, comprise the following steps：
Step 1, map structuring
Step 1.1, camera calibration
Camera calibration method adopts Zhang Zhengyou camera calibration methods：If Fig. 2 lineaments are 7 black and white for taking advantage of that 7 length of sides are 0.02m Alternate grid.7
plane chessboard, hand-held chessboard is taken advantage of to obtain 30 multiple checkerboard images at various orientations camera alignment 7, i.e., The intrinsic parameter and distortion
parameter of camera can be calculated.
Ultimate principle is as follows：
Wherein, Intrinsic Matrixes of the k for camera, [X Y 1]^TFor the homogeneous coordinates put on plane chessboard, [u v 1]^TPosition chess Homogeneous coordinates of the spot projection to
corresponding point on the plane of delineation, [r on disk[1] r[2] r[3]] and t be the rotation under camera coordinates system respectively Turn and translation vector, k [r[1] r[2]T] for
homography matrix H.According to matrix solving method, when picture number is more than or equal to 3, k can To obtain unique solution.
Photographic head is fixed on right ahead by step 1.2, it is ensured that during map structuring and when navigating, position and direction are protected Hold constant.
Input picture is transformed into greyscale color space, conversion formula from RGB color by step 1.3：Gray=R* 0.299+G*0.587+B*0.114；
Step 1.4, carries out feature extraction and matching to image
Feature extraction and matching adopts ORB (Oriented Brief) method, detects initially with FAST-12 algorithms Characteristic point is as shown in Figure 3：
Gray values of the I (x) for any point on circumference, gray scales of the I (p) for the center of circle, ε[d]For the threshold value of gray value differences, if greatly Then think that p is
a characteristic point in threshold value.
Then it is sub to calculate the description of a characteristic point using BRIEF algorithms：Select around characteristic point after smoothed image One Patch, by a kind of selected method
picking out 256 points pair in this Patch.Then for each point Compare its brightness value to (p, q), if I (p)>I (q) then this to generate one in two-value string value be 1, if I (p) <I (q),
then the value corresponded in two-value string is -1, is otherwise 0.All 256 points pair, between being all compared, then obtain one The binary string of individual 256 length, represents
description using binary number string.
Step 1.5, map initialization
The map initialization of monocular SLAM is carried out, the first segment distance of dollying obtains two key frames, by key frame Between characteristic matching calculate Homography models or
Fundamental models and obtain initialized cartographic model, The computational methods of Homography models are normalized DLT, and the computational methods of Fundamental matrix are normalized
8points.In visual odometry part, first to two width picture frame P[K-2]、P[K-1]Carry out feature extraction and Match somebody with somebody；Then trigonometric ratio P[K-2]、P[K-1]Feature, to
new image P[K]Feature extraction and P[K-1]Characteristic matching；Finally by employing Camera pose is estimated in matching of the PnP algorithms from 3D to 2D, is optimized by Bundle
Adjustment, can be excellent using figure Chemical industry tool g2o carries out map optimization.The model of PnP (Perspective-n-Point) problem is：
Attitudes of the R for camera, calibration matrixes of the C for camera, (u, v) is two-dimensional pixel coordinate, and (x, y, z) is three-dimensional coordinate Coordinate i.e. under world
coordinate system.
Step 1.6, closed loop detection and reorientation
Present frame is converted into image bag of words, image data base is then retrieved, view data bag of words storehouse part is using classical DBoW2 bag of words storehouse model, by the
similar scene for calculating the image that is input into and model in bag of words storehouse in monocular cam, be complete It is as shown in Figure 4 that key frame is searched in office's
reorientation.The insertion standard of key frame is when the characteristic point cloud that new frame is detected is closed with reference When key frame characteristic point cloud gap is less
than 90%, determines that present frame is key frame, preserve into map datum.It is special that we calculate ORB Seek peace each key frame map cloud point corresponding relation, RANSAC
iterative calculation is then performed to each key frame, is calculated with PnP Method is found out the i.e. camera pose quaternary number in position of the camera in world coordinate system
and is represented.Quaternary number is a scalar (w) and The combination of individual 3D vectors (x, y, z).The definition of quaternary number：
Q=[w x y z]^T
Step 1.7, obtains Current camera pose
The posture information of the Current camera that acquisition is calculated by PnP+RANSAC, that is, represent the quaternary of three dimensions rotation After number.Quaternary number is
transformed into into Eulerian angles, ψ, θ,Respectively about the z axis, the anglec of rotation of Y-axis, X-axis, yaw angle θ are current The angle of frame.Generally ψ, θ of definition
andScope is -90 degree to 90 degree, and correspondence solution formula is：
Eulerian angles turn quaternary number formula：
Driftage angle range can not uniquely represent current positional information when being -90 to 90 degree, it is therefore desirable to by yaw angle model Enclose expand to -180 degree is to 180
degree.When θ scopes for -180 degree to+180 degree when correspondence solution part formula it is as follows.It is only right For yaw angle is sought whole algorithm with reference to 1,2,3 its
flow chart of formula as shown in figure 5, table 1 turns Eulerian angles for different quaternary numbers Value result, Fig. 6 are the Eulerian angles of correspondence each frame position when
rotation takes two turns.
1 quaternary number of table turns the conversion output contrast (unit of Eulerian angles：Degree)
Present frame quaternary number True Eulerian angles θ This algorithm is exported
(0.0297,0.9994,0.0152,0.0046) -180 -180
(- 0.0275, -0.8640,0.0126,0.5025) -120 -120
(- 0.0214, -0.5709,0.0086,0.8207) -70 -70
(0.3214,0.117,0.3214,0.883) 0 0
(0.0049,0.5010, -0.0087,0.8654) 60 60
(0.0190,0.8360, -0.1951,0.5124) 120 120
(0.0348,0.9713, -0.2353, -0.0075) 180 180
Step 2, preserves map datum and tracing point coordinate information
By cloud data (three-dimensional coordinate, ORB Feature Descriptors in world coordinate system), key by the way of data flow Frame data (camera posture information) save as .bin formatted files,
and the data that point coordinates information includes x and z directions are saved in In txt file.
Step 3, loading map and track point coordinates calculate corner
Step 3.1 loads the map built up, and execution resets bit function, calculates Current camera posture information with this.
Can be derived in the navigation algorithm based on camera tracing point according to the navigation algorithm based on GPS.Propose accordingly to be based on The navigation decision making
algorithm of vision SLAM, as shown in Figure 7.Algorithm is divided into two parts：Choose pre- described point algorithm, calculate corner number of degrees β Algorithm.
Step 3.2 tracing point is encrypted, and is chosen pre- described point and is filtered pretreatment to pre- described point：Obtain by key frame approach Tracing point data division place can be
than sparse, it is therefore desirable to carries out next step operation after being encrypted again.Using average The method of filtering is to sparse point encryption：
When choosing pre- described point, if current point is B points, according to search in the threshold value 0.35 before and after method a little, search Institute of the rope apart from all
distances of B points in the range of 0.35 a little, is saved in array M [n]={ A, B, C, D ... }, then logarithm Group sorted from small to large sort (M, M+n) choose M [n] be pre- described
point.For different map track datas has different Pre- described point selected threshold, needs carry out parameter selection according to actual map datum.For the non-starting point of closed
path and terminal position Put：For the closed path of 200 tracing points：Initial piont mark 0, terminal label 199, after the point for searching is sorted most Big value chooses 0 point of
label when being 199 be that next impact point is pre- described point, it is desirable to which 0 point to 199 dot spacings from great-than search threshold Value.
Step 3.3 calculates corner：Calculate corner number of degrees β algorithms：Propose two kinds of calculating corner algorithms.(if dx, dy), (gx, gy) is respectively current point and will reach
point coordinates：
By angle, θ before Fig. 7 under the coordinate system of scope ± 180 °, if current point and impact point line with x-axis angle areScope is 0 to 90 degree；β is corner, and bottom Body Control
is from left to right 0 to 180 degree；γ determines for centre Plan layer judges angle, and different target point has different values from current point position relationship.Corner decision
table of the table 2 for correspondence Fig. 6, currently Position and aiming spot relative position relation are divided into four kinds of different situations shown in table.
2 corner decision table of table
When the direction of motion is E to F, i.e., as (dx>gx&&dy<When gy)：
If 0=<θ<=90 i.e. L1 directions when, then left-hand rotation β=| θ |+90- α, if -180=<θ<- 90 i.e. L4 directions when, then Right-hand rotation β=| θ | -90+ α, if -90=<θ<0 i.e. L2 and L3 directions
when, then judge γ=90- α-| θ |, if γ>0 i.e. L2 directions are then Left-hand rotation β=γ, otherwise i.e. then turn right β=- γ in L3 directions；
When the direction of motion is that F ' arrives E ', i.e., as (dx<gx&&dy>When gy)：
If 0=<θ<=90 i.e. L1 directions when, then right-hand rotation β=90- | θ |+α, if -180=<θ<=-90 i.e. L4 directions when, Then left-hand rotation β=180- | θ |+90- α, if 90<θ<=180 i.e. L2 and L3
directions when, then judge γ=90- α-(180- | θ |), if γ>0 i.e. L3 directions when then turn left that otherwise i.e. then turn right β=γ β=- γ in L2 directions；
When direction of motion P is to Q, i.e., as (dx<gx&&dy<When gy)：
If -90=<θ<=0 i.e. L1 directions when, then right-hand rotation β=90+ | θ |-α, if 90=<θ<=180 i.e. L4 directions when, then Left-hand rotation β=| θ | -90+ α, if 0<θ<90 i.e. L2 and L3 directions
when, then judge γ=90- α-| θ |, if γ>0 i.e. L2 directions when then Right-hand rotation β=γ is otherwise that L3 directions are then turned left β=- γ；
When the direction of motion is that Q ' arrives P ', i.e., as (dx>gx&&dy>When gy)：
If -90=<θ<=0 i.e. L1 directions when, then left-hand rotation β=90- | θ |+α, if 90=<θ<=180 i.e. L4 directions when, then Right-hand rotation β=180- | θ |+90- α, if -180=<θ<- 90 i.e. L2 and L3
directions when, then judge γ=90+ α-| θ |, if γ>0 is Then turning left during L2 directions, otherwise i.e. then turn right β=γ β=- γ in L3 directions；
Step 4, the angle beta obtained according to step 3 and the direction of corner, send a command to vehicle bottom control.
Conclusion：Fig. 8 is original camera trace information and independent navigation trace information, wherein it is not pass through to build figure track The data of encryption.Analysis finds
that leading line and ground figure line are essentially coincided.The navigation path deviation data of concrete analysis Fig. 8 As shown in figure 9, maximum deviation is 0.029 about 0.1m,
deviation is very little.Figure 10 is to build figure track and right through encryption The three circle navigation paths answered, Figure 11 is trajector deviation data, and trajector deviation
maximum is 0.017, can be considered that route is essentially coincided. Figure 12 is during monocular vision builds figure while the gps data and the monocular vision under using said method of
collection are navigated through The gps data trajectory diagram gathered in journey, is about 0.0013km at two track maximum deviations, and it is due to turning that deviation is larger herein
The turning number of degrees at place are larger, and during self-navigation, turn inside diameter can be completely superposed where the turning number of degrees are little.
In sum, the present invention utilizes merely monocular cam, it is not necessary to any other external sensor, in less error In the range of realize navigation system based on monocular vision
SLAM, reduce research and application cost, with potentially applying valency Value.
Claims (3)
1. a kind of air navigation aid of the single camera vision system based on quaternary number, it is characterised in that step includes：
Step 1, constructing environment map, comprise the following steps：
1) carry out camera calibration；
2) photographic head is fixed to into right ahead；
3) input picture is transformed into into greyscale color space from RGB color；
4) feature extraction and matching is carried out to image；
5) map initialization；
6) closed loop detection and reorientation；
7) obtain Current camera pose, calculate quaternary number be transformed into 180 degree to 180 degree scope Eulerian angles ψ, θ,The ψ, θ,Respectively about the z axis, the anglec of rotation
of Y-axis, X-axis, yaw angle θ are the angle of present frame；
Step 2, preservation map datum and tracing point coordinate information；
Step 3, loading map and track point coordinates calculate corner β according to yaw angle θ,
The direction of step 4, the angle beta obtained according to step 3 and corner, sends a command to vehicle bottom control.
2. the air navigation aid of the single camera vision system based on quaternary number as claimed in claim 1, it is characterised in that ψ, θ andModel Enclose be -90 degree to 90 degree,
correspondingly solution formula is：
When θ scopes for -180 degree to+180 degree when, correspondence solution part formula is：
3. the air navigation aid of the single camera vision system based on quaternary number as claimed in claim 2, it is characterised in that step 3 is counted Calculate corner β process be：
If (dx, dy), (gx, gy) are respectively current point and will reach point coordinates, if current point and impact point line and x-axis angle ForScope is 0 to 90 degree；β is corner, and bottom
Body Control is from left to right 0 to 180 degree；γ is centre Decision-making level judges angle, and different target point has different values from current point position
relationship.Current location and aiming spot are with respect to position When the relation of putting is divided into following four kinds of different situations：
When the direction of motion is E to F, i.e., as (dx>gx&&dy<When gy)：
If 0=<θ<=90 i.e. L1 directions when, then left-hand rotation β=| θ |+90- α, if -180=<θ<- 90 i.e. L4 directions when, then turn right β =| θ | -90+ α, if -90=<θ<0 i.e. L2 and L3 directions when,
then judge γ=90- α-| θ |, if γ>Then turn left β in 0 i.e. L2 directions =γ, otherwise i.e. then turn right β=- γ in L3 directions；
When the direction of motion is that F ' arrives E ', i.e., as (dx<gx&&dy>When gy)：
If 0=<θ<=90 i.e. L1 directions when, then right-hand rotation β=90- | θ |+α, if -180=<θ<=-90 i.e. L4 directions when, then it is left Turn β=180- | θ |+90- α, if 90<θ<=180 i.e. L2 and L3
directions when, then judge γ=90- α-(180- | θ |), if γ>0 I.e. L3 directions when then turn left that otherwise i.e. then turn right β=γ β=- γ in L2 directions；
When direction of motion P is to Q, i.e., as (dx<gx&&dy<When gy)：
If -90=<θ<=0 i.e. L1 directions when, then right-hand rotation β=90+ | θ |-α, if 90=<θ<=180 i.e. L4 directions when, then turn left β=| θ | -90+ α, if 0<θ<90 i.e. L2 and L3 directions when, then
judge γ=90- α-| θ |, if γ>0 i.e. L2 directions when then turn right β =γ is otherwise that L3 directions are then turned left β=- γ；
When the direction of motion is that Q ' arrives P ', i.e., as (dx>gx&&dy>When gy)：
If -90=<θ<=0 i.e. L1 directions when, then left-hand rotation β=90- | θ |+α, if 90=<θ<=180 i.e. L4 directions when, then turn right β=180- | θ |+90- α, if -180=<θ<- 90 i.e. L2 and L3 directions
when, then judge γ=90+ α-| θ |, if γ>0 i.e. L2 side To when then turn left that otherwise i.e. then turn right β=γ β=- γ in L3 directions.
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CN107369181A (en) * 2017-06-13 2017-11-21 华南理工大学 Cloud data collection and processing method based on bi-processor architecture
CN107680133A (en) * 2017-09-15 2018-02-09 重庆邮电大学 A kind of mobile robot visual SLAM methods based on improvement closed loop detection algorithm
CN108051002A (en) * 2017-12-04 2018-05-18 上海文什数据科技有限公司 Transport vehicle space-location method and system based on inertia measurement auxiliary vision
CN109900272A (en) * 2019-02-25 2019-06-18 浙江大学 Vision positioning and build drawing method, device and electronic equipment
CN110174892A (en) * 2019-04-08 2019-08-27 北京百度网讯科技有限公司 Processing method, device, equipment and the computer readable storage medium of vehicle direction
CN111353941A (en) * 2018-12-21 2020-06-30 广州幻境科技有限公司 Space coordinate conversion method
CN111798574A (en) * 2020-06-11 2020-10-20 广州恒沙数字科技有限公司 Corner positioning method for three-dimensional field
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CN101067557A (en) * 2007-07-03 2007-11-07 北京控制工程研究所 Environment sensing one-eye visual navigating method adapted to self-aid moving vehicle
CN101441769A (en) * 2008-12-11 2009-05-27 上海交通大学 Real time vision positioning method of monocular camera
CN101598556B (en) * 2009-07-15 2011-05-04 北京航空航天大学 Unmanned aerial vehicle vision/inertia integrated navigation method in unknown environment
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CN101067557A (en) * 2007-07-03 2007-11-07 北京控制工程研究所 Environment sensing one-eye visual navigating method adapted to self-aid moving vehicle
CN101441769A (en) * 2008-12-11 2009-05-27 上海交通大学 Real time vision positioning method of monocular camera
CN101598556B (en) * 2009-07-15 2011-05-04 北京航空航天大学 Unmanned aerial vehicle vision/inertia integrated navigation method in unknown environment
Cited By (9)
* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN107369181A (en) * 2017-06-13 2017-11-21 华南理工大学 Cloud data collection and processing method based on bi-processor architecture
CN107369181B (en) * 2017-06-13 2020-12-22 华南理工大学 Point cloud data acquisition and processing method based on dual-processor structure
CN107680133A (en) * 2017-09-15 2018-02-09 重庆邮电大学 A kind of mobile robot visual SLAM methods based on improvement closed loop detection algorithm
CN108051002A (en) * 2017-12-04 2018-05-18 上海文什数据科技有限公司 Transport vehicle space-location method and system based on inertia measurement auxiliary vision
CN111353941A (en) * 2018-12-21 2020-06-30 广州幻境科技有限公司 Space coordinate conversion method
CN109900272A (en) * 2019-02-25 2019-06-18 浙江大学 Vision positioning and build drawing method, device and electronic equipment
CN109900272B (en) * 2019-02-25 2021-07-13 浙江大学 Visual positioning and mapping method and device and electronic equipment
CN110174892A (en) * 2019-04-08 2019-08-27 北京百度网讯科技有限公司 Processing method, device, equipment and the computer readable storage medium of vehicle direction
CN111798574A (en) * 2020-06-11 2020-10-20 广州恒沙数字科技有限公司 Corner positioning method for three-dimensional field
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The Stacks project
Lemma 70.5.9. Notation and assumptions as in Situation 70.5.5. If $X$ is separated, then $X_ i$ is separated for some $i \in I$.
Proof. Choose an affine scheme $U_0$ and a surjective étale morphism $U_0 \to X_0$. For $i \geq 0$ set $U_ i = U_0 \times _{X_0} X_ i$ and set $U = U_0 \times _{X_0} X$. Note that $U_ i$ and $U$ are
affine schemes which come equipped with surjective étale morphisms $U_ i \to X_ i$ and $U \to X$. Set $R_ i = U_ i \times _{X_ i} U_ i$ and $R = U \times _ X U$ with projections $s_ i, t_ i : R_ i \
to U_ i$ and $s, t : R \to U$. Note that $R_ i$ and $R$ are quasi-compact separated schemes (as the algebraic spaces $X_ i$ and $X$ are quasi-separated). The maps $s_ i : R_ i \to U_ i$ and $s : R \
to U$ are of finite type. By definition $X_ i$ is separated if and only if $(t_ i, s_ i) : R_ i \to U_ i \times U_ i$ is a closed immersion, and since $X$ is separated by assumption, the morphism $
(t, s) : R \to U \times U$ is a closed immersion. Since $R \to U$ is of finite type, there exists an $i$ such that the morphism $R \to U_ i \times U$ is a closed immersion (Limits, Lemma 32.4.16).
Fix such an $i \in I$. Apply Limits, Lemma 32.8.5 to the system of morphisms $R_{i'} \to U_ i \times U_{i'}$ for $i' \geq i$ (this is permissible as indeed $R_{i'} = R_ i \times _{U_ i \times U_ i}
U_ i \times U_{i'}$) to see that $R_{i'} \to U_ i \times U_{i'}$ is a closed immersion for $i'$ sufficiently large. This implies immediately that $R_{i'} \to U_{i'} \times U_{i'}$ is a closed
immersion finishing the proof of the lemma. $\square$
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Block feedback and points system
Hi all,
OS (Windows 11)
PsychoPy version (v2021.2.2)
I am very new to PsychoPy and have been mostly working with the builder view inserting code components where necessary. I have managed to do everything needed with the help of posts on this forum and
others. There is one thing that I just can’t get my head around as to how to even begin implementing. Below is a snapshot of my flow:
I would like the routine after before the ISI (Block_Feedback) to show accruement of points across the course of the block (in text elements), that vary according to the response type, judgement type
and player type. I found the following post: Giving feedback adding points collected in every round however my response-feedback is not binary and I am struggling to even adapt the code for my needs.
Here is a snapshot of my excel file to demonstrate what I am referencing:
In experimental terms I want to implement the following:
1. Player 1 (Participant) - text component 1, total of the following points over the course of the 64 trials per block:
• For where $match == NOMATCH
**For correct answer, truth == 1, this trial is worth 0 points.
**For incorrect answer, truth == 0: if response == $low_lie this trial is worth +1 point WHERE $judgement == Believed, and -1 Point WHERE $judgement == Not Believed; if response == $high_lie this
is worth +2 points WHERE $judgement == Believed, and -2 points WHERE judgement == Not Believed.
*For where $match == COLOURMATCH
**If correct answer, truth == 1, this trial is worth +1 point.
**If incorrect answer, truth == 0: if response == $high_lie, this is worth +2 points WHERE $judgement == Believed, and -2 points WHERE $judgement == Not Believed; where response is not the truth or
high value lie, this is worth 0 points (perceived as a behavioural error).
• For where $match == SUITMATCH
**If correct answer, truth == 1, this trial is worth +2 points.
**if incorrect, truth == 0 , this is perceived as a behavioural error worth 0 points.
2. Player 2 (Computer) - text component 2, total of the following points over 64 trials per block:
*For where player 1 response is correct, truth == 1, this is worth +1 point WHERE $judgement == Believed, and -1 point WHERE $judgement == Not Believed.
*For where participants report the incorrect answer, truth == 0, this is worth +1 point WHERE $judgement == Not Believed, and -1 point WHERE $judgement == Believed.
Hopefully this communicates effectively what I am trying to do, I am just so stumped how to write a loop with so many conditional statements. I am also worried that the lack of input for some of the
cells in the excel files (low_lie/high_lie) may screw up the code- but not sure how to express that for these cells if they make any other response than those held within the truth, low_lie, high_lie
variables then this is a behavioural error.
Thank you in advance for any aid you can give me!
Hello Tigan,
an ifelse-construction would do the job. If you don’t know how to set up a ifelse-construction, google “if elif python”. Start with a simpler example and expand later.
Best wishes Jens
Hi Jens,
Thank you for this suggestion! I have written the following elif construction which currently isn’t working but given I know barely any python I think it’s at least halfway there:
# needs to be declared in the project at start: @ begin experiment
# The below variable values will need to be replaced with the specific project variables: already have $truth (correct answer), $match (match type) and $judgement (beleieved or not believed) encoded within experiment.
def CalcFunction(match,truth): # Main function to score and increment the totals for correct responses to all types of match
global runningTotalParticipant
global low_lie_trial
if match == "SUITMATCH": # Only correct adds to total, any other response taken as behavioural error
if truth:runningTotalParticipant+=2
elif match == "NOMATCH": # Only incorrect adds to total
if not truth:CalcNoMatch(low_lie_trial,judgement)
elif matchType == "COLOURMATCH": # Both options can add to total
if truth:runningTotalParticipant+=1
elif not truth:CalcColourMatch(low_lie_trial,judgement)
def CalcNoMatch(low_lie_trial,judgement): # Function to calculate low or high value lies on a NOMATCH trial, need to debug to define low_lie_trial is only where response = COLOURMATCH (i.e.worth +/- 1)- how?
global runningTotalParticipant
if low_lie_trial and judgement == "Believed":runningTotalParticipant+=1 #low lie and believed
elif low_lie_trial and judgement == "Not Believed":runningTotalParticipant-=1 #low lie and not believed
elif not low_lie_trial and judgement == "Believed":runningTotalParticipant+=2 #high lie and believed
elif not low_lie_trial and judgement == "Not Believed":runningTotalParticipant-=2 #high lie and not believed
def CalcColourMatch(low_lie,judgement): # Function to calculate the high lie on a COLOURMATCH trial
global runningTotalParticipant
if not low_lie_trial and judgement == "Believed":runningTotalParticipant+=2 #high lie and believed
elif not low_lie_trial and judgement == "Not Believed":runningTotalParticipant-=2 #high lie and not believed
def CalcComputerScore (judgement, truth):# Function to calcluate the computer score
global runningTotalComputer
if truth and judgement == "Believed": runningTotalComputer+=1
elif truth and judgement == "Not Believed": runningTotalComputer-=1
elif not truth and judgement == "Believed": runningTotalComputer+=1
elif not truth and judgement == "Not Believed": runningTotalComputer-=1
# Call the main function
# debug code
print (str(runningTotalParticipant) + " " + str(runningTotalComputer))
The experiment crashes when I try to implement it. The reason for this I assume is potentially two things:
• I haven’t defined the boolean variable low_lie_trial fully. I was advised to use it and understand the premise of the variable is that it is always either true or false. At the beginning I state
that it is true (does this mean always true throughout the experiment?) but I need it to be false on certain occasions and I do not know where exactly in the code I should state it to be false
although I know I need it to be for high lies only.
• I am calling variables that are previously/not defined in the routine for code. PsychoPy obviously works by insert code at certain places within the experimental routine (before or end of
routine). The $truth variable value will be known at the end of the trials_exp routine, then the $judgement variable value will be known at the end of P2_judgement routine (see image above). It
is almost like I need to have the main bulk of code at the end of each loop run to have all the information to calculate the points, but PsychoPy doesn’t seem to have an option for this. I am
unsure whether I have the same code at the end of both routines it will synchronise appropriately if you follow what I mean.
I would greatly appreciate any help anyone can give me!
Kind regards,
Hello Tigan,
well I never used to define a function in PsychoPy. I use the builder to build the experiment and use code-elements to tweak it. So I assume that I can’t be offer you much help.
I always initialize/define variables before I use them. Given that they are variables you can change them throughout the experiment. So, your boolean variable will not always be true. But is the
variable “truth” ever initialized?
I noticed one thing in your CalcFunction: the first two if/elif compare match, the third matchType. Does this work?
Best wishes Jens
Hi Tigan,
Please could you share the error message that you get?
Sorry I am having a little trouble understanding exactly what you need here, is it that you have one running point counter (that is increased differentially depending on different roles) or several
point counters increased independantly?
I edited the elif construction and then set the Boolean variable to be based on response types and it works!
Thank you
#Calculate Participant score
if match == "SUITMATCH": # Only correct adds to total, any other response taken as behavioural error
if pr_match_resp == 1:TrialParticipant+=2
elif match == "NOMATCH": # Only incorrect adds to total
if ((key == 'left') and (expInfo == "A")) or ((key == 'right') and (expInfo == "B")): low_lie_trial=False
else: low_lie_trial = True
if pr_match_resp == 0:
if low_lie_trial and judgement == "Believed":TrialParticipant+=1 #low lie and believed
elif low_lie_trial and judgement == "Not Believed":TrialParticipant-=1 #low lie and not believed
elif not low_lie_trial and judgement == "Believed":TrialParticipant+=2 #high lie and believed
elif not low_lie_trial and judgement == "Not Believed":TrialParticipant-=2 #high lie and not believed
elif match == "COLOURMATCH": # Both options can add to total
if (key == 'left' and expInfo == "A") or (key == 'right' and expInfo == "B"): low_lie_trial=False
else: low_lie_trial = True
if pr_match_resp == 1:TrialParticipant+=1
elif pr_match_resp == 0:
if not low_lie_trial and judgement == "Believed":TrialParticipant+=2 #high lie and believed
elif not low_lie_trial and judgement == "Not Believed":TrialParticipant-=2 #high lie and not believed
#Calculate Computer score
if pr_match_resp == 1 and judgement == "Believed": TrialComputer+=1
elif pr_match_resp == 1 and judgement == "Not Believed": TrialComputer-=1
elif pr_match_resp == 0 and judgement == "Believed": TrialComputer-=1
elif pr_match_resp == 0 and judgement == "Not Believed": TrialComputer+=1 | {"url":"https://discourse.psychopy.org/t/block-feedback-and-points-system/26146","timestamp":"2024-11-15T04:39:55Z","content_type":"text/html","content_length":"43896","record_id":"<urn:uuid:c8e3fe67-cb3e-4ffb-8cdd-063ef31f2c3c>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00865.warc.gz"} |
Version 0.19
Version 0.19#
Version 0.19.2#
July, 2018
This release is exclusively in order to support Python 3.7.
Version 0.19.1#
October 23, 2017
This is a bug-fix release with some minor documentation improvements and enhancements to features released in 0.19.0.
Note there may be minor differences in TSNE output in this release (due to #9623), in the case where multiple samples have equal distance to some sample.
API changes#
• Reverted the addition of metrics.ndcg_score and metrics.dcg_score which had been merged into version 0.19.0 by error. The implementations were broken and undocumented.
• return_train_score which was added to model_selection.GridSearchCV, model_selection.RandomizedSearchCV and model_selection.cross_validate in version 0.19.0 will be changing its default value from
True to False in version 0.21. We found that calculating training score could have a great effect on cross validation runtime in some cases. Users should explicitly set return_train_score to
False if prediction or scoring functions are slow, resulting in a deleterious effect on CV runtime, or to True if they wish to use the calculated scores. #9677 by Kumar Ashutosh and Joel Nothman.
• correlation_models and regression_models from the legacy gaussian processes implementation have been belatedly deprecated. #9717 by Kumar Ashutosh.
Bug fixes#
Regressions in 0.19.0 fixed in 0.19.1:
Code and Documentation Contributors#
With thanks to:
Joel Nothman, Loic Esteve, Andreas Mueller, Kumar Ashutosh, Vrishank Bhardwaj, Hanmin Qin, Rasul Kerimov, James Bourbeau, Nagarjuna Kumar, Nathaniel Saul, Olivier Grisel, Roman Yurchak, Reiichiro
Nakano, Sachin Kelkar, Sam Steingold, Yaroslav Halchenko, diegodlh, felix, goncalo-rodrigues, jkleint, oliblum90, pasbi, Anthony Gitter, Ben Lawson, Charlie Brummitt, Didi Bar-Zev, Gael Varoquaux,
Joan Massich, Joris Van den Bossche, nielsenmarkus11
Version 0.19#
August 12, 2017
Changed models#
The following estimators and functions, when fit with the same data and parameters, may produce different models from the previous version. This often occurs due to changes in the modelling logic
(bug fixes or enhancements), or in random sampling procedures.
Details are listed in the changelog below.
(While we are trying to better inform users by providing this information, we cannot assure that this list is complete.)
New features#
Classifiers and regressors
Other estimators
Model selection and evaluation
Trees and ensembles
Linear, kernelized and related models
Other predictors
Decomposition, manifold learning and clustering
Preprocessing and feature selection
Model evaluation and meta-estimators
Bug fixes#
Trees and ensembles
Linear, kernelized and related models
Other predictors
Decomposition, manifold learning and clustering
Preprocessing and feature selection
Model evaluation and meta-estimators
API changes summary#
Trees and ensembles
Linear, kernelized and related models
Other predictors
Decomposition, manifold learning and clustering
Preprocessing and feature selection
Model evaluation and meta-estimators
Code and Documentation Contributors#
Thanks to everyone who has contributed to the maintenance and improvement of the project since version 0.18, including:
Joel Nothman, Loic Esteve, Andreas Mueller, Guillaume Lemaitre, Olivier Grisel, Hanmin Qin, Raghav RV, Alexandre Gramfort, themrmax, Aman Dalmia, Gael Varoquaux, Naoya Kanai, Tom Dupré la Tour,
Rishikesh, Nelson Liu, Taehoon Lee, Nelle Varoquaux, Aashil, Mikhail Korobov, Sebastin Santy, Joan Massich, Roman Yurchak, RAKOTOARISON Herilalaina, Thierry Guillemot, Alexandre Abadie, Carol
Willing, Balakumaran Manoharan, Josh Karnofsky, Vlad Niculae, Utkarsh Upadhyay, Dmitry Petrov, Minghui Liu, Srivatsan, Vincent Pham, Albert Thomas, Jake VanderPlas, Attractadore, JC Liu,
alexandercbooth, chkoar, Óscar Nájera, Aarshay Jain, Kyle Gilliam, Ramana Subramanyam, CJ Carey, Clement Joudet, David Robles, He Chen, Joris Van den Bossche, Karan Desai, Katie Luangkote, Leland
McInnes, Maniteja Nandana, Michele Lacchia, Sergei Lebedev, Shubham Bhardwaj, akshay0724, omtcyfz, rickiepark, waterponey, Vathsala Achar, jbDelafosse, Ralf Gommers, Ekaterina Krivich, Vivek Kumar,
Ishank Gulati, Dave Elliott, ldirer, Reiichiro Nakano, Levi John Wolf, Mathieu Blondel, Sid Kapur, Dougal J. Sutherland, midinas, mikebenfield, Sourav Singh, Aseem Bansal, Ibraim Ganiev, Stephen
Hoover, AishwaryaRK, Steven C. Howell, Gary Foreman, Neeraj Gangwar, Tahar, Jon Crall, dokato, Kathy Chen, ferria, Thomas Moreau, Charlie Brummitt, Nicolas Goix, Adam Kleczewski, Sam Shleifer, Nikita
Singh, Basil Beirouti, Giorgio Patrini, Manoj Kumar, Rafael Possas, James Bourbeau, James A. Bednar, Janine Harper, Jaye, Jean Helie, Jeremy Steward, Artsiom, John Wei, Jonathan LIgo, Jonathan Rahn,
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Laughlin, Iván Vallés, Aurélien Bellet, JPFrancoia, Jacob Schreiber, Asish Mahapatra | {"url":"https://scikit-learn.qubitpi.org/whats_new/v0.19.html","timestamp":"2024-11-13T06:09:34Z","content_type":"text/html","content_length":"181433","record_id":"<urn:uuid:838694ce-54f6-4a23-8f63-beefe322ade0>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00613.warc.gz"} |
What is: Bayes' Rule
What is Bayes’ Rule?
Bayes’ Rule, also known as Bayes’ Theorem, is a fundamental principle in probability theory and statistics that describes how to update the probability of a hypothesis based on new evidence. It
provides a mathematical framework for reasoning about uncertainty and is widely used in various fields, including data science, machine learning, and artificial intelligence. The rule is named after
Thomas Bayes, an 18th-century statistician and theologian, who developed the concept of conditional probability.
The Mathematical Formula of Bayes’ Rule
The mathematical expression of Bayes’ Rule is given by the formula: P(H|E) = [P(E|H) * P(H)] / P(E), where P(H|E) is the posterior probability, P(E|H) is the likelihood, P(H) is the prior
probability, and P(E) is the marginal probability. This formula allows statisticians and data analysts to calculate the probability of a hypothesis H given the evidence E, by incorporating prior
knowledge and new data.
Understanding the Components of Bayes’ Rule
Each component of Bayes’ Rule plays a crucial role in the updating process of probabilities. The prior probability, P(H), represents the initial belief about the hypothesis before observing any
evidence. The likelihood, P(E|H), quantifies how probable the evidence is, assuming the hypothesis is true. The marginal probability, P(E), serves as a normalizing constant that ensures the total
probability sums to one. Together, these components enable a systematic approach to updating beliefs in light of new information.
Applications of Bayes’ Rule in Data Science
Bayes’ Rule is extensively applied in data science for various tasks, including classification, regression, and decision-making under uncertainty. In machine learning, Bayesian methods are used to
create probabilistic models that can incorporate prior knowledge and adapt as new data becomes available. For instance, Bayesian classifiers, such as Naive Bayes, leverage Bayes’ Rule to predict
class labels based on feature probabilities, making them effective for text classification and spam detection.
Bayesian Inference and Its Importance
Bayesian inference is a statistical method that utilizes Bayes’ Rule to update the probability distribution of a parameter or hypothesis as more data becomes available. This approach contrasts with
frequentist statistics, which relies on fixed parameters and does not incorporate prior beliefs. Bayesian inference is particularly valuable in scenarios where data is scarce or uncertain, allowing
analysts to make informed decisions based on both prior knowledge and observed evidence.
Challenges and Limitations of Bayes’ Rule
Despite its powerful applications, Bayes’ Rule is not without challenges. One significant limitation is the need for accurate prior probabilities, which can be subjective and may lead to biased
results if not chosen carefully. Additionally, calculating the marginal probability, P(E), can be computationally intensive, especially in high-dimensional spaces. These challenges necessitate a
thorough understanding of the underlying assumptions and careful consideration of the data being analyzed.
Bayes’ Rule in Real-World Scenarios
In real-world applications, Bayes’ Rule has been successfully employed in various domains, such as medical diagnosis, finance, and marketing. For example, in healthcare, Bayes’ Rule can help
determine the probability of a disease given specific symptoms, allowing for more accurate diagnoses. In finance, it can be used to assess the risk of investment portfolios based on historical
performance and market trends. These practical applications highlight the versatility and importance of Bayes’ Rule in decision-making processes.
Bayesian Networks and Their Significance
Bayesian networks are graphical models that represent a set of variables and their conditional dependencies using directed acyclic graphs. They leverage Bayes’ Rule to perform probabilistic inference
and reasoning in complex systems. By modeling relationships between variables, Bayesian networks enable analysts to capture uncertainty and make predictions based on incomplete data. This capability
is particularly valuable in fields such as artificial intelligence, bioinformatics, and social sciences.
Conclusion: The Relevance of Bayes’ Rule Today
Bayes’ Rule remains a cornerstone of modern statistics and data analysis, providing a robust framework for reasoning under uncertainty. Its applications span numerous fields, and its principles
continue to influence advancements in machine learning and artificial intelligence. As data becomes increasingly abundant and complex, the relevance of Bayes’ Rule in guiding decision-making
processes and enhancing predictive models will only grow. | {"url":"https://statisticseasily.com/glossario/what-is-bayes-rule-understanding-the-concept/","timestamp":"2024-11-13T22:35:44Z","content_type":"text/html","content_length":"138787","record_id":"<urn:uuid:ca7067f3-ac85-42fc-b9cb-0876cb6d4ef4>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00094.warc.gz"} |
The ‘Waverider’ R package uses the continuous wavelet transform to analyze cyclostratigraphic data. The continuous wavelet transform enables the observation of transient/non-stationary cyclicity
in time-series. The goal of cyclostratigraphic studies is to define frequency/period in the depth/time domain. By conducting the continuous wavelet transform on cyclostratigraphic data series one can
observe and extract cyclic signals/signatures from signals. These results can then be visualized and interpreted enabling one to identify/interpret cyclicity in the geological record, which can be
used to construct astrochronological age-models and identify and interpret cyclicity in past en present climate systems.
You can install the development version of WaverideR from GitHub with: | {"url":"https://cran.hafro.is/web/packages/WaverideR/readme/README.html","timestamp":"2024-11-12T13:48:42Z","content_type":"application/xhtml+xml","content_length":"5799","record_id":"<urn:uuid:0e8573ae-1901-4ca3-b986-8011d789b48e>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00452.warc.gz"} |
Microgravity: living on the International Space Station
5 Quantum energy levels
You probably won’t be surprised to know that there is an equation for energy levels! Equation 2 is used to calculate the energy of a certain level (E[n]) when you know the energy level (n). This n
can take any integer value, that is 1, 2, etc.
You should note the minus sign. It means that the electron is bound to an atom. You need to give it this amount of energy to release it. You can practise calculating energy levels in Activity 7.
Activity 7 Calculating energy values
Timing: Allow approximately 15 minutes
You will now look at probably the most famous experiment in physics: the double slit experiment. | {"url":"https://www.open.edu/openlearn/mod/oucontent/view.php?id=77557§ion=6","timestamp":"2024-11-12T10:24:30Z","content_type":"text/html","content_length":"149171","record_id":"<urn:uuid:f3baee43-5da3-42e8-92b1-8b17a6bb7602>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00141.warc.gz"} |
Chemical equation balancer solve using algebra solver
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• Using Domain and Range on TI-84 Plus Calculator | {"url":"https://www.softmath.com/math-com-calculator/quadratic-equations/chemical-equation-balancer.html","timestamp":"2024-11-14T13:29:29Z","content_type":"text/html","content_length":"167103","record_id":"<urn:uuid:1a681fe1-43ec-4c90-b9b8-2a5a3518b864>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00061.warc.gz"} |
Physics - Standards
I am very keen to have consistent standards of notation and terminology across this site. Its hard enough to learn these things without being confused by inconsistent standards. It may not be perfect
yet so, if you find any inconsistencies across the site please let me know.
The units used are based on the international system of units (SI units) which has 3 primary units as discussed on this page. I have also tried to apply standards to the mathematical notation and
terminology (see mathematical standards).
Here are some of the notations for physical quantities: | {"url":"http://www.euclideanspace.com/physics/other/standards/index.htm","timestamp":"2024-11-10T11:26:32Z","content_type":"text/html","content_length":"23178","record_id":"<urn:uuid:6d0530a9-fb82-4ae9-b42d-a9e710cc7b54>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00533.warc.gz"} |
Velocity-Time Graphs in context of velocity to acceleration
06 Oct 2024
Title: Exploring the Relationship between Velocity and Acceleration through Velocity-Time Graphs
This article delves into the fundamental connection between velocity and acceleration, focusing on the role of velocity-time graphs in illustrating this relationship. By examining the shape and
characteristics of these graphs, we can gain a deeper understanding of how velocity and acceleration are intertwined. This analysis will provide a comprehensive overview of the mathematical
underpinnings and physical implications of velocity-time graphs, shedding light on the intricate dance between velocity and acceleration.
Velocity-time graphs are a powerful tool for visualizing the relationship between an object’s velocity and time. By plotting velocity against time, we can gain insights into the object’s
acceleration, deceleration, or constant motion. In this article, we will explore the mathematical framework underlying velocity-time graphs and examine their implications for understanding the
connection between velocity and acceleration.
Mathematical Framework:
The relationship between velocity (v) and acceleration (a) is governed by the following equation:
v(t) = v0 + ∫[0,t] a(t’) dt’
where v0 is the initial velocity, t is time, and a(t’) is the acceleration at time t’.
Velocity-Time Graphs:
A velocity-time graph represents the relationship between an object’s velocity and time. The shape of this graph can be used to infer information about the object’s acceleration.
• Constant Velocity: A horizontal line on the graph indicates constant velocity, implying zero acceleration.
• Acceleration: An upward-sloping line represents increasing velocity, indicating positive acceleration (a > 0). Conversely, a downward-sloping line indicates decreasing velocity and negative
acceleration (a < 0).
• Deceleration: A downward-sloping line with a finite slope represents deceleration, where the object’s velocity is decreasing but not coming to rest.
Physical Implications:
The shape of a velocity-time graph has significant physical implications for understanding the relationship between velocity and acceleration. For instance:
• Acceleration: An upward-sloping section of the graph indicates an increase in velocity, which can be attributed to an external force or a change in the object’s mass.
• Deceleration: A downward-sloping section implies a decrease in velocity, potentially due to frictional forces or air resistance.
Velocity-time graphs provide a powerful tool for visualizing the relationship between velocity and acceleration. By examining the shape and characteristics of these graphs, we can gain insights into
the physical implications of changes in velocity and acceleration. This analysis has demonstrated the mathematical framework underlying velocity-time graphs and highlighted their significance in
understanding the intricate dance between velocity and acceleration.
• [1] Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of Physics. John Wiley & Sons.
• [2] Serway, R. A., & Jewett, J. W. (2014). Physics for Scientists and Engineers. Cengage Learning.
Note: The article does not provide numerical examples, as per your request.
Related articles for ‘velocity to acceleration ‘ :
• Reading: **Velocity-Time Graphs in context of velocity to acceleration **
Calculators for ‘velocity to acceleration ‘ | {"url":"https://blog.truegeometry.com/tutorials/education/998b05e7866796c6b208f8f1ffad3082/JSON_TO_ARTCL_Velocity_Time_Graphs_in_context_of_velocity_to_acceleration_.html","timestamp":"2024-11-11T10:42:47Z","content_type":"text/html","content_length":"17654","record_id":"<urn:uuid:cde0d6d1-7cb9-4908-8a2c-6c494e0b3ad9>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00829.warc.gz"} |
Fast Compact Sparse Bit Sets
Imagine you need a relatively compact data structure for quickly checking membership of mostly-consecutive non-negative integers. (If this sounds really specific, it is because it is precisely what I
needed for a particular project.)
The Ruby standard library contains a Set class which may be a good starting point. Set is actually implemented as a Hash with the Set elements as keys and true as the values. Thus the overhead for
storing a value in the Set is essentially only the value itself since all keys point to the same true object. Assuming a 64-bit machine, the overhead will be 64 bits per value. This seems reasonable,
but given the specific limitations of the values we wish to store, perhaps we can do better?
Bit Sets
A bit set is a compact data structure of binary values where membership is indicated by setting a bit to 1. The position of the bit indicates the element value. For example, the second bit from the
right might be used to indicate whether or not the value 1 is in the set.
One method to determine membership is to AND the bit set with a mask with only the desired bit set to 1. If the result is 0, the value is not in the set. If it is any other result (actually the mask
itself, but the zero check is sufficinet), the value is a member of the set.
In Ruby, this looks like
For example, to check if the value 4 is in the set, we use the mask 00010000 (the 5th bit from the right is set to 1) which is the decimal value 8:
Since the result is zero, we know the value 4 is not in the set.
If we check for the value 6, the result is not zero, indicating the value is a member of the set:
Now, instead of 64 bits per value, it only requires a single bit! Now we just need to put a lot of bits together, either by using a long string or a bunch of integers in an array.
Sparse Bit Sets
The problem with a long binary string or an array of integers is that membership is entirely position-based. To store the value 1000, the data structure requires 1001 bits, all but one of which is
set to 0. This is quite inefficient, especially for very large values.
One solution is to create a sparse bit set by combining a hash table with bit sets as values. The hash table keys provide fast look up of the correct bit set, then the bit set is checked for the
desired element. The keys indicate the lowest value stored in the bit set (e.g., the decimal key 4 pointing to the binary bit set 00000001 would mean the value 4 is in the set).
Below is an example of a hash table using integer keys and 8 bit integers for the bit sets:
The average overhead is ⌊(m * n) / w⌋ + m bits, where m is the number of values (assumed to be consecutive), w is the number of bits per bit set, and n is the number of bits per key. In 64-bit Ruby,
if we use integers for the bit sets, n = 64 and w = 62*. This works out to an average of 2 bits per value in the set. Of course, a single value incurs the overhead of both the key and the bit set:
128 bits! But if there are many consecutive values, the cost per value begins to shrink. For example, the numbers 0 to 61 can be stored in a single bit set, so 62 values can be stored in the 128 bits
and we are back to about 2 bits per value.
Note that while it is best to use consecutive values which fit neatly into the bit sets (in this case, runs of 62 integers), the sequences can start and end at arbitrary points with only a little
“wasted” overhead. To store just the number 1000, we now only need 128 bits, not 1001.
On top of the space savings, the membership checks remain fast. Still assuming 64-bit Ruby, to determine if a value is in the table look up index i = value / 61. Then check the bit set with bitset &
(1 << (value % 61) != 0 as previously. (The divisor is 61 because there are 62 bits, but the values are 0 to 61).
Space Efficiency
I have implemented a Ruby version of the data structure described above which I call the Dumb Numb Set (DNS).
To measure the space used by the bit sets, we compare the Marshal data size for the bit sets versus regular Hashes (using true for all values, just like a Ruby Set).
These are the results for perfectly ordered data on a 64-bit version of Ruby 1.9.3 (size is number of bytes):
Items Hash DNS %reduction
1 | 7 | 41 |-486%
100 | 307 | 61 | 80%
1k | 4632 | 253 | 95%
10k | 49632 | 2211 | 96%
100k | 534098 | 24254 | 95%
1M | 5934098 | 245565 | 96%
10M | 59934098 | 2557080 | 96%
100M | 683156884 | 26163639 | 96%
1B | ? | 262229211 | ?
At 1 billion items, my machine ran out of memory.
For a single item, as expected, overhead in the DNS is quite high. But for as little as 100 items in the set, the DNS is considerably more compact.
This is, however, the best case scenario for the DNS. Less perfectly dense values cause it to be less efficient. For very sparse values, a Hash/Set is probably a better choice.
Even Better Space Efficiency
It may not surprise you to find out I was very interested in minimizing the serialized version of the sparse bit set for sending it over a network. In investigating easy but compact ways of doing so,
I realized the Marshal data for Hashes and integers is not very compact, especially for large integers.
Fortunately, there is an existing solution for this scenario called MessagePack. For storing 1 million values, serialized size is reduced from 245,565 to 196,378 bytes (20%). The DNS will use
MessagePack automatically if it is installed.
Somewhat surprisingly, the DNS is quite fast even when compared to MRI Ruby’s Hash implementation.
With MRI Ruby 1.9.3p448 (x86_64) and 1 million values:
user system total real
Hash add random 0.540000 0.020000 0.560000 ( 0.549499)
DumbNumbSet add random 0.850000 0.020000 0.870000 ( 0.864700)
Hash add in order 0.540000 0.020000 0.560000 ( 0.556441)
DumbNumbSet add in order 0.490000 0.000000 0.490000 ( 0.483713)
Hash add shuffled 0.570000 0.020000 0.590000 ( 0.589316)
DumbNumbSet add shuffled 0.540000 0.010000 0.550000 ( 0.538420)
Hash look up 0.930000 0.010000 0.940000 ( 0.940849)
DNS look up 0.820000 0.000000 0.820000 ( 0.818728)
Hash remove 0.980000 0.030000 1.010000 ( 0.999362)
DNS remove 0.950000 0.000000 0.950000 ( 0.953170)
The only operation slower than a regular Hash is inserting many random values. All other operations are roughly equal.
For my specific scenario, a simple custom data structure was just as fast as a built-in data structure, but required significantly less space for the expected use case.
There are other solutions for this type of problem, but it should be noted I only really care about fast insertion, fast membership checks, and compact representation. Additionally, values may be
very large, although I attempt to keep them within the Fixnum range for Ruby (i.e. less than 2^62 - 1). This rules out some implementations which require arrays the size of the maximum value!
I also did not want to deal with compression schemes, of which there are quite a few, since my sets were going to be dynamic. I imagine there are very efficient implementations for fixed data sets.
Footnote: Integer Size in Ruby
Integers in 32-bit MRI Ruby only have 30 bits available, and in 64-bit MRI Ruby they only have 62 bits available:
$ irb
1.9.3p448 :001 > ("1" * 62).to_i(2).class
=> Fixnum
1.9.3p448 :002 > ("1" * 63).to_i(2).class
=> Bignum | {"url":"https://blog.presidentbeef.com/blog/2013/09/02/fast-compact-sparse-bitsets/","timestamp":"2024-11-11T16:29:46Z","content_type":"text/html","content_length":"15910","record_id":"<urn:uuid:2a5be666-ceb0-4fbc-8953-c87663f53d58>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00752.warc.gz"} |
CRISP-DM methodology in technical view
On this paper discuss about CRISP-DM (Cross Industry Standard Process for data mining) methodology and its steps including selecting technique to successful the data mining process. Before going to
CRISP-DM it is better to understand what data mining is? So, here first I introduce the data mining and then discuss about CRISP-DM and its steps for any beginner (data scientist) need to know.
1 Data Mining
Data mining is an exploratory analysis where has no idea about interesting outcome (Kantardzic, 2003). So data mining is a process to explore by analysis a large set of data to discover meaningful
information which help the business to take a proper decision. For better business decision data mining is a way to select feature, correlation, and interesting patterns from large dataset (Fu, 1997;
SPSS White Paper, 1999).
Data mining is a step by step process to discover knowledge from data. Pre-processing data is vital part for a data mining. In pre-process remove noisy data, combining multiple sources of data,
retrieve relevant feature and transforming data for analysis. After pre-process mining algorithm applied to extract data pattern so data mining is a step by step process and applied algorithm to find
meaning full data pattern. Actually data mining is not only conventional analysis it is more than that (Read, 1999).
Data mining and statistics closely related. Main goal of data mining and statistic is find the structure of data because data mining is a part of statistics (Hand, 1999). However, data mining use
tools, techniques, database, machine learning which not part of statistics but data mining use statistics algorithm to find a pattern or discover hidden decision.
Data mining objective could be prediction or description. On prediction data mining considering several features of dataset to predict unidentified future, on the other hand description involve
identifying pattern of data to interpreted (Kantardzic, 2003).
From figure 1.1 shows data mining is the only one part of getting unknown information from data but it is the central process of whole process. Before data mining there are several processes need to
be done like collecting data from several sources than integrated data and keep in data storage. Stored unprocessed data evaluated and selected with pre-processed activity to give a standard format
than data mining algorithm to analysis for hidden pattern.
2 CRISP-DM Methodologies
Cross Industry Standard Process for data mining (CRISP-DM) is most popular and widely uses data mining methodology. CRISP-DM breaks down the data mining project life cycle into six phases and each
phase consists of many second-level generic tasks. Generic task cover all possible data mining application. CRISP-DM extends KDD (Knowledge Discovery and Data Mining) into six steps which are
sequence of data mining application (Martínez-Plumed 2019).
Data science and data mining project extract meaningful information from data. Data science is an art where a lot of time need to spend for understanding the business value and data before applying
any algorithm then evaluate and deployed a project. CRISP-DM help any data science and data mining project from start to end by giving step by step process.
Present world every day billions of data are generating. So organisations are struggling with overwhelmed data to process and find a business goal. Comprehensive data mining methodology, CRISP-DM
help business to achieve desirable goal by analysing data.
CRISP-DM (Cross Industry Standard Process for Data Mining) is well documented, freely available, data mining methodology. CRISP-DM is developed by more than 200 data mining users and many mining tool
and service providers funded by European Union. CRISP-DM encourages organization for best practice and provides a structure of data mining to get better, faster result.
CRISP-DM is a step by step methodology. Figure-2.1 show the phases of CRISP-DM and process of data mining. Here one side arrow indicates the dependency between phases and double side arrow represents
repeatable process. Six phases of CRISP-DM are Business understanding, Data understanding, Modelling, Evaluation and Deployment.
2.1 Business Understanding
Business Understanding or domain understanding is the first step of CRISP-DM methodology. On this stage identify the area of business which is going to transform into meaningful information by
analysing, processing and implementing several algorithms. Business understanding identifies the available resource (human and hardware), problems and set a goal. Identification of business objective
should be agreed with project sponsors and other unit of business which will be affected. This step also focuses about details business success criteria, requirements, constraints, risk, project plan
and timeline.
2.2 Data Understanding
Data understanding is the second and closely related with the business understanding phase. This phase mainly focus on data collection and proceeds to get familiar with the data and also detect
interesting subset from data. Data understanding has four subsets these are:-
2.2.1 Initial data collection
On this subset considering the data collection sources which is mainly divided into two categories like outsource data or internal source data. If data is from outsource then it may costly, time
consuming and may be low quality but if data is collected form internal source it is an easy and less costly, but it may be contain irrelevant data. If internal source data does not fulfil the
interest of analysis than it is necessary to move outsource data. Data collection also give an assumption that the data is quantitative (continuous, count) or qualitative (categorical). It also
gives information about balance or imbalanced dataset. On data collection should avoid random error, systematic error, exclusion errors, and errors of choosing.
2.2.2 Data Description
Data description performs initial analysis about data. On this stage it is going to determine about the source of data like RDBMS, SQL, NoSQL, Big data etc. then analysis and describe the data about
size (large data set give more accurate result but time consuming), number of records, tables, database, variables, and data types (numeric, categorical or Boolean). On this phase examine the
accessibility and availability of attributes.
2.2.3 Exploratory data analysis (EDA)
On exploratory data analysis describe the inferential statistics, descriptive statistics and graphical representation of data. Inferential statistics summarize the entire population from the sample
data to perform sampling and hypothesis testing. On Parametric hypothesis testing (Null or alternate – ANOVA, t-test, chi square test) perform for known distribution (based on population) like mean,
variance, standard deviation, proportion and Non-parametric hypothesis testing perform when distribution is unknown or sample size is small. On sample dataset, random sampling implement when dataset
is balance but for imbalance dataset should be follow random resampling (under and over sampling), k fold cross validation, SMOTE (synthetic minority oversampling technique), cluster base sampling,
ensemble techniques (bagging and boosting – Add boost, Gradient Tree Boosting, XG Boost) to form a balance dataset.
On descriptive statistics analysis describe about the mean, median, mode for measures of central tendency on first moment business decision. On second moment business decision describe the measure of
dispersion about the variance, standard deviation and range of data. On third and fourth moment business decision describe accordingly skewness (Positive skewness – heavier tail to the right,
negative skewness – heavier tail to the left, Zero skewness – symmetric distribution) and Kurtosis (Leptokurtosis – heavy tail, platykurtosis – light tail, mesokurtic – normal distribution).
Graphical representation is divided into univariate, bivariate and multivariate analysis. Under univariate whisker plot, histogram identify the outliers and shape of distribution of data and Q-Q plot
(Quantile – Quantile) plot describe the normality of data that means data is normally distribution or not. On whisker plot if data present above of Q3 + 1.5 (IQR) and below of Q1 – 1.5 (IQR) is
outlier. For Bivariate correlations identify with scatter plot which describe positive, negative or no correlation and also identify the data linearity or non-linearity. Scatter plot also describe
the clusters and outliers of data. For multivariate has no graphical analysis but used to use regression analysis, ANOVA, Hypothesis analysis.
2.2.4 Data Quality analysis
This phase identified and describes the potential errors like outliers, missing data, level of granularity, validation, reliability, bad metadata and inconsistency. On this phase AAA (attribute
agreement analysis) analysed discrete data for data error. Continuous data analysed with Gage repeatability and reproducibility (Gage R & R) which follow SOP (standard operating procedures). Here
Gage R & R define the aggregation of variation in the measurement data because of the measurement system.
2.3 Data Preparation
Data Preparation is the time consuming stage for every data science project. Overall on every data science project 60% to 70% time spend on data preparation stage. Data preparation stapes are
described below.
2.3.1 Data integration
Data integration involved to integrate or merged multiple dataset. Integration integrates data from different dataset where same attribute or same columns presents but when there is different
attribute then merging the both dataset.
2.3.2 Data Wrangling
On this subset data are going to clean, curate and prepare for next level. Here analysis the outlier and treatment done with 3 R technique (Rectify, Remove, Retain) and for special cases if there are
lots of outliner then need to treat outlier separately (upper outliner in an one dataset and lower outliner in another dataset) and alpha (significant value) trim technique use to separate the
outliner from the original dataset. If dataset has a missing data then need to use imputation technique like mean, median, mode, regression, KNN etc.
If dataset is not normal or has a collinearity problem or autocorrelation then need to implement transformation techniques like log, exponential, sort, Reciprocal, Box-cox etc. On this subset use the
data normalization (data –means/standard deviation) or standardization (min- max scaler) technique to make unitless and scale free data. This step also help if data required converting into
categorical then need to use discretization or binning or grouping technique. For factor variable (where has limited set of values), dummy variable creation technique need to apply like one hot
encoding. On this subset also help heterogeneous data to transform into homogenous with clustering technique. Data inconsistencies also handle the inconsistence of data to make data in a single
2.3.3 Feature engineering and selection/reduction
Feature engineering may called as attribute generation or feature extraction. Feature extraction creating new feature by reducing original feature to make simplex model. Feature engineering also do
the normalized feature by producing calculative new feature. So feature engineering is a data pre-process technique where improve data quality by cleaning, integration, reduction, transformation and
Feature selections reduce the multicollinearity or high correlated data and make model simple. Main two type of feature selection technique are supervised and unsupervised. Principal Components
Analysis (PCA) is an unsupervised feature reduction/ feature selection technique and LDA is a Linear Discriminant analysis supervised technique mainly use for classification problem. LDA analyse by
comparing mean of the variables. Supervised technique is three types filter, wrapper and ensemble method. Filter method is easy to implement but wrapper is costly method and ensemble use inside a
2.4 Model
2.4.1 Model Selection Technique
Model selection techniques are influence by accuracy and performance. Because recommendation need better performance but banking fraud detection needs better accuracy technique. Model is mainly
subdivided into two category supervised learning where predict an output variable according to given an input variable and unsupervised learning where has not output variable.
On supervised learning if an output variable is categorical than it is classification problem like two classes or multiclass classification problem. If an output variable is continuous (numerical)
then the problem is called prediction problem. If need to recommending according to relevant information is called recommendation problem or if need to retrieve data according to relevance data is
called retrieval problem.
On unsupervised learning where target or output variable is not present. On this technique all variable is treated as an input variable. Unsupervised learning also called clustering problem where
clustering the dataset for future decision.
Reinforcement learning agent solves the problem by getting reward for success and penalty for any failure. And semi-supervised learning is a process to solve the problem by combining supervised and
unsupervised learning method. On semi-supervised, a problem solved by apply unsupervised clustering technique then for each cluster apply different type of supervised machine learning algorithm like
linear algorithm, neural network, K nearest neighbour etc.
On data mining model selection technique, where output variable is known, then need to implement supervised learning. Regression is the first choice where interpretation of parameter is important.
If response variable is continuous then linear regression or if response variable is discrete with 2 categories value then logistic regression or if response variable is discrete with more than 2
categorical values then multinomial or ordinal regression or if response variable is count then poission where mean is equal to variance or negative binomial regression where variance is grater then
mean or if response variable contain excessive zero values then need to choose Zero inflated poission (ZIP) or Zero inflated negative binomial (ZINB).
On supervised technique except regression technique all other technique can be used for both continuous or categorical response variable like KNN (K-Nearest Neighbour), Naïve Bays, Black box
techniques (Neural network, Support vector machine), Ensemble Techniques (Stacking, Bagging like random forest, Boosting like Decision tree, Gradient boosting, XGB, Adaboost).
When response variable is unknown then need to implement unsupervised learning. Unsupervised learning for row reduction is K-Means, Hierarchical etc., for columns reduction or dimension reduction PCA
(principal component analysis), LDA (Linear Discriminant analysis), SVD (singular value decomposition) etc. On market basket analysis or association rules where measure are support and confidence
then lift ration to determine which rules is important. There are recommendation systems, text analysis and NLP (Natural language processing) also unsupervised learning technique.
For time series need to select forecasting technique. Where forecasting may model based or data based. For Trend under model based need to use linear, exponential, quadratic techniques. And for
seasonality need to use additive, multiplicative techniques. On data base approaches used auto regressive, moving average, last sample, exponential smoothing (e.g. SES – simple exponential smoothing,
double exponential smoothing, and winters method).
2.4.2 Model building
After selection model according to model criterion model is need to be build. On model building provided data is subdivided with training, validation and testing. But sometime data is subdivided
just training and testing where information may leak from testing data to training data and cause an overfitting problem. So training dataset should be divided into training and validation whereas
training model is tested with validation data and if need any tuning to do according to feedback from validation dataset. If accuracy is acceptable and error is reasonable then combine the training
and validation data and build the model and test it on unknown testing dataset. If the training error and testing error is minimal or reasonable then the model is right fit or if the training error
is low and testing error is high then model is over fitted (Variance) or if training error is high and testing error is also high then model is under fitted (bias). When model is over fitted then
need to implement regularization technique (e.g. linear – lasso, ridge regression, Decision tree – pre-pruning, post-pruning, Knn – K value, Naïve Bays – Laplace, Neural network – dropout, drop
connect, batch normalization, SVM – kernel trick)
When data is balance then split the data training, validation and testing and here training is larger dataset then validation and testing. If data set is imbalance then need to use random resampling
(over and under) by artificially increases training dataset. On random resampling by randomly partitioning data and for each partition implement the model and taking the average of accuracy. Under K
fold cross validation creating K times cross dataset and creating model for every dataset and validate, after validation taking the average of accuracy of all model. There is more technique for
imbalance dataset like SMOTH (synthetic minority oversampling technique), cluster based sampling, ensemble techniques e.g. Bagging, Boosting (Ada Boost, XGBoost).
2.4.3 Model evaluation and Tuning
On this stage model evaluate according to errors and accuracy and tune the error and accuracy for acceptable manner. For continuous outcome variable there are several way to measure the error like
mean error, mean absolute deviation, Mean squared error, Root mean squared error, Mean percentage error and Mean absolute percentage error but more acceptable way is Mean absolute percentage error.
For this continuous data if error is known then it is easy to find out the accuracy because accuracy and error combining value is one. The error function also called cost function or loss function.
For discrete output variable model, for evaluation and tuning need to use confusion matrix or cross table. From confusion matrix, by measuring accuracy, error, precision, sensitivity, specificity, F1
help to take decision about model fitness. ROC curve (Receiver operating characteristic curve), AUC curve (Area under the ROC curve) also evaluate the discrete output variable. AUC and ROC curve plot
of sensitivity (true positive rate) vs 1-specificity (false positive rate). Here sensitivity is a positive recall and recall is basically out of all positive samples, how sample classifier able to
identify. Specificity is negative recall here recall is out of all negative samples, how many sample classifier able to identify. On AUC where more the area under the ROC is represent better
accuracy. On ROC were step bend it’s indicate the cut off value.
2.4.4 Model Assessment
There is several ways to assess the model. First it is need to verify model performance and success according to desire achievement. It needs to identify the implemented model result according to
accuracy where accuracy is repeatable and reproducible. It is also need to identify that the model is scalable, maintainable, robust and easy to deploy. On assessment identify that the model
evaluation about satisfactory results (identify the precision, recall, sensitivity are balance) and meet business requirements.
2.5 Evaluation
On evaluation steps, all models which are built with same dataset, given a rank to find out the best model by assessing model quality of result and simplicity of algorithm and also cost of
deployment. Evaluation part contains the data sufficiency report according to model result and also contain suggestion, feedback and recommendation from solutions team and SMEs (Subject matter
experts) and record all these under OPA (organizational process assets).
2.6 Deployment
Deployment process needs to monitor under PEST (political economical social technological) changes within the organization and outside of the organization. PEST is similar to SWOT (strength weakness
opportunity and thread) where SW represents the changes of internal and OT represents external changes.
On this deployment steps model should be seamless (like same environment, same result etc.) from development to production. Deployment plan contain the details of human resources, hardware, software
requirements. Deployment plan also contain maintenance and monitoring plan by checking the model result and validity and if required then implement retire, replace and update plan.
3 Summaries
CRISP-DM implementation is costly and time consuming. But CRISP-DM methodology is an umbrella for data mining process. CRISP-DM has six phases, Business understanding, Data understanding, Modelling,
Evaluation and Deployment. Every phase has several individual criteria, standard and process. CRISP-DM is Guideline for data mining process so if CRISP-DM is going to implement in any project it is
necessary to follow each and every single guideline and maintain standard and criteria to get required result.
4 References
1. Fu, Y., (1997), “Data Mining: Tasks, Techniques and Applications”, Potentials, IEEE, 16: 4, 18–20.
2. Hand, D. J., (1999), “Statistics and Data Mining: Intersecting Disciplines”, ACM SIGKDD Explorations Newsletter, 1: 1, 16 – 19.
3. Kantardzic, M., (2003), “Data Mining: Concepts, Models, Methods, and Algorithms” John Wiley and Sons, Inc., Hoboken, New Jersey
4. Martínez-Plumed, F., Contreras-Ochando, L., Ferri, C., Orallo, J.H., Kull, M., Lachiche, N., Quintana, M.J.R. and Flach, P.A., 2019. CRISP-DM Twenty Years Later: From Data Mining Processes to
Data Science Trajectories. IEEE Transactions on Knowledge and Data Engineering.
5. Read, B.J., (1999), “Data Mining and Science? Knowledge discovery in science as opposed to business”, 12th ERCIM Workshop on Database Research.
About Author
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data-science-blog-logo-en-300x292.png Ahamed Al Farabi2021-01-06 09:21:562020-11-22 19:58:38CRISP-DM methodology in technical view
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Bernoulli numbers as examples
Ada Lovelace had selected the calculation of Bernoulli numbers as the most advanced example in her notes to the description of the analytical engine.
One outstanding feature of this example is the necessity to have indexed access to memory cells, because the calculation of the next number is a weighted sum of all previous ones (in the form used by
It is clear now that indexing was not in the plans that Babbage had created until then, and it is even unsure if it was included in later versions.
So one might consider using the method given by A.A.L. as an example for other early machines.
Hoever, it was never mentioned before that the example has a significant disadvantage: The numbers soon get very large and grow super-exponentially:
Thus, the calculation – as decimal fractions – demands for floating point arithmetic, and many early computers were restricted to integer and (binary) fractional numbers (except Zuse's machines);
floating-point calculations sometimes provided as libaries.
While Babbage planned for a large (30) number of decimal digits, also his machine would soon have to stop because of number overflow.
So, unfortunately, the Bernoulli numbers are not an appropriate example for early computers. | {"url":"https://rclab.de/analyticalengine/bernoullinumbers","timestamp":"2024-11-02T08:02:11Z","content_type":"application/xhtml+xml","content_length":"9700","record_id":"<urn:uuid:bb1916b5-615c-4d0b-abd2-470f00a12515>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00619.warc.gz"} |
Raku Land - POSIX::PWDENT
This module creates two new dynamic variables for easy access to information in the /etc/passwd and /etc/group files in POSIX systems.
To query data in /etc/passwd use the $*PWDENT variable:
need POSIX::PWDENT; # No "use" required as no exports available.
# To get the UID of user named 'auser' use $*PWDENT variable as an Associative
my $uid = $*PWDENT<auser>.uid;
# Or
my $uid = +$*PWDENT<auser>;
# To get the name of the user with UID zero use the variable as a Positional
say $*PWDENT[0].name;
# Or
say ~$*PWDENT[0];
To query data in /etc/group use the $*GRPENT variable
need POSIX::GRPENT; # No "use" required as no exports available.
# To get the GID of a gruop named 'agroup' use $*GRPENT variable as an Associative
my $gid = $*PWDENT<auser>.gid;
# Or
my $gid = +$*PWDENT<agroup>;
# To get the name of the group with GID zero use the variable as a Positional
say $*GRPENT[0].name;
# Or
say ~$*GRPENT[0];
The $*PWDENT dynamic variable
When you request the POSIX::PWDENT module to be loaded the $*PWDENT dynamic variable is installed and is available anywhere in your code.
To get an entry by name use the variable as an Associative and to get an entry by UID use the variable as a Positional.
Accessing $*PWDENT by any of the roles, if the entry exists, returns a PwdEnt object with the following methods to access the corresponding fields of the entry:
• name Str
• passwd Str
• uid Int
• gid Int
• change Int Valid only in BSD and Darwin, 0 otherwise
• class Str Valid only in BSD and Darwin, '' otherwise
• gecos Str Valid only in Linux, BSD and Darwin, '' otherwise
• dir Str
• shell Str
• expire Int Valid only in BSD and Darwin, 0 otherwise
• fields Int Valid only in BSD and Darwin, 0 otherwise
Similar to how Raku´s allomorphs works, evaluating a PwdEnt object in Str context returns the value of .name and in Numeric context the value of .uid.
So to get the user name with UID 123, you can:
my $uname = ~$*PWDENT[123]; # Or $*PWDENT[123].Str
Or to get the UID of a user named auser all you need to do is:
my $uid = +$*PWDENT<auser>; # Or $*PWDENT<auser>.Numeric
Even you can coerce the PwdEnt object to IO to get the value of dir as an IO::Path.
The $*PWDENT variable can also be used as an Iterable, so to get a list of the user names available in the system you can:
my @users is List = (~$_ for $*PWDENT);
The $*GRPENT dynamic variable
When you request the POSIS::GRPENT module to be loaded, the $*GRPENT dynamic variable is installed
To get an entry by name use the variable as an Associative and to get an entry by GID use the variable as a Positional.
Accessing $*GRPENT by any of the roles, if the entry exists, returns a GrpEnt object with the following methods to access the corresponding fields of the entry:
• name Str
• passwd Str
• gid Int
• members List of Str
Similar to how Raku´s allomorphs works, evaluating a GrpEnt object in Str context returns the value of .name and in Numeric context the value of .gid.
So, to get the group name with GID 123, you can:
my $uname = ~$*GRPENT[123]; # Or $*GRPENT[123].Str
Or to get the GID of a group named agroup all you need to do is:
my $gid = +$*GRPENT<auser>; # Or $*GRPENT<auser>.Numeric
The $*GRPENT variable can also be used as an Iterable, so to get a list of all group names available in the system you can:
my @group is List = (~$_ for $*GRPENT);
Non existing entries
Requesting an inexistent entry from any of $*PWDENT or $*GRPENT returns Nil. So, you can use with
with $*PWDENT<apache> {
# User 'apache' exists, can access its methods
say .gecos;
say "Apache dir is {.dir}";
with $*GRPENT<wheel> {
# User 'wheel' exists, can access its methods
say .members; # The list of users with access to 'sudo' in (some?) Linux
As with other positional or associative you can use the :exists adverb to check existence.
say $*PWDENT<root>:exists; # True (root user almost always exists)
say $*PWDENT[0]:exists; # True
if $*GRPENT<kvm>:exists {
# The group 'kvm' exists
Convert PwdEnt or GRPENT objects to other types
If you need to access the returned data as other kind of structure, you can coerce it to Map, Hash or List objects (whose keys will be the names of the methods above):
my $foouser = $*PWDENT<foo>.Map;
say $foouser<name>; # Use name as a key, not as a method
my $bargroup = $*GRPENT<bar>.List;
Salvador Ortiz sortiz@cpan.org | {"url":"https://raku.land/zef:sortiz/POSIX::PWDENT","timestamp":"2024-11-07T06:41:02Z","content_type":"text/html","content_length":"22070","record_id":"<urn:uuid:79a56230-88ca-40ff-8d06-ad2c77239f63>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00267.warc.gz"} |
655 research outputs found
We perform a numerical analysis of recently proposed scaling functions for the single electron box. Specifically, we study the ``magnetic'' susceptibility as a function of tunneling conductance and
gate charge, and the effective charging energy at zero gate charge as a function of tunneling conductance in the strong tunneling limit. Our Monte Carlo results confirm the accuracy of the
theoretical predictions.Comment: Published versio
We calculate the linear response conductance of electrons in a Luttinger liquid with arbitrary interaction g_2, and subject to a potential barrier of arbitrary strength, as a function of temperature.
We first map the Hamiltonian in the basis of scattering states into an effective low energy Hamiltonian in current algebra form. Analyzing the perturbation theory in the fermionic representation the
diagrams contributing to the renormalization group (RG) \beta-function are identified. A universal part of the \beta-function is given by a ladder series and summed to all orders in g_2. First
non-universal corrections beyond the ladder series are discussed. The RG-equation for the temperature dependent conductance is solved analytically. Our result agrees with known limiting
cases.Comment: 6 pages, 5 figure
We establish a one-to-one correspondance between the ''composite particles'' with $N$ particles and the Young tableaux with at most $N$ rows. We apply this correspondance to the models of
Calogero-Sutherland and Ruijsenaars-Schneider and we obtain a momentum space representation of the ''composite particles'' in terms of creation operators attached to the Young tableaux. Using the
technique of bosonisation, we obtain a position space representation of the ''composite particles'' in terms of products of vertex operators. In the special case where the ''composite particles'' are
bosons and if we add one extra quasiparticle or quasihole, we construct the ground state wave functions corresponding to the Jain series $u =p/(2np\pm 1)$ of the fractional quantum Hall
effect.Comment: latex calcomp2.tex, 5 files, 30 pages [SPhT-T99/080], submitted to J. Math. Phy
We proposed an analytical expression for the amplitude defining the long distance asymptotic of the correlation function .Comment: 5 pages, harvmac.tex, one epsf figur
A phenomenological optical potential is generalized to include the Coulomb and nuclear interactions caused by the dynamical deformation of its surface. In the high-energy approach analytical
expressions for elastic and inelastic scattering amplitudes are obtained where all the orders in the deformation parameters are included. The multistep effect of the 2$^+$ rotational state excitation
on elastic scattering is analyzed. Calculations of inelastic cross sections for the $^{17}$O ions scattered on different nuclei at about hundred Mev/nucleon are compared with experimental data, and
important role of the Coulomb excitation is established.Comment: 9 pages; 3 figures. Submitted to the Physics of Atomic Nucle
We study the problem of a quantized elastic string in the presence of an impenetrable wall. This is a two-dimensional field theory of an N-component real scalar field $\phi$ which becomes interacting
through the restriction that the magnitude of $\phi$ is less than $\phi_{\rm max}$, for a spherical wall of radius $\phi_{\rm max}$. The N=1 case is a string vibrating in a plane between two straight
walls. We review a simple nonperturbative argument that there is a gap in the spectrum, with asymptotically-free behavior in the coupling (which is the reciprocal of $\phi_{\rm max}$) for N greater
than or equal to one. This scaling behavior of the mass gap has been disputed in some of the recent literature. We find, however, that perturbation theory and the 1/N expansion each confirms that
these theories are asymptotically free. The large N limit coincides with that of the O(N) nonlinear sigma model. A theta parameter exists for the N=2 model, which describes a string confined to the
interior of a cylinder of radius $\phi_{\rm max}$.Comment: Text slightly improved, bibilography corrected, more typos corrected, still Latex 7 page
We complement a recent exact study by L. Samaj on the properties of a guest charge $Q$ immersed in a two-dimensional electrolyte with charges $+1/-1$. In particular, we are interested in the behavior
of the density profiles and electric potential created by the charge and the electrolyte, and in the determination of the renormalized charge which is obtained from the long-distance asymptotics of
the electric potential. In Samaj's previous work, exact results for arbitrary coulombic coupling $\beta$ were obtained for a system where all the charges are points, provided $\beta Q<2$ and $\beta <
2$. Here, we first focus on the mean field situation which we believe describes correctly the limit $\beta\to 0$ but $\beta Q$ large. In this limit we can study the case when the guest charge is a
hard disk and its charge is above the collapse value $\beta Q>2$. We compare our results for the renormalized charge with the exact predictions and we test on a solid ground some conjectures of the
previous study. Our study shows that the exact formulas obtained by Samaj for the renormalized charge are not valid for $\beta Q>2$, contrary to a hypothesis put forward by Samaj. We also determine
the short-distance asymptotics of the density profiles of the coions and counterions near the guest charge, for arbitrary coulombic coupling. We show that the coion density profile exhibit a change
of behavior if the guest charge becomes large enough ($\beta Q\geq 2-\beta$). This is interpreted as a first step of the counterion condensation (for large coulombic coupling), the second step taking
place at the usual Manning--Oosawa threshold $\beta Q=2$ | {"url":"https://core.ac.uk/search/?q=author%3A(Lukyanov%20S%20L)","timestamp":"2024-11-10T06:52:24Z","content_type":"text/html","content_length":"151433","record_id":"<urn:uuid:f65ed4d8-6ca4-4d38-b407-aca11f9c6910>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00713.warc.gz"} |
Circumcentre and Incentre of a Triangle | Radius of Circumcircle
Circumcentre and Incentre of a Triangle
We will discuss circumcentre and incentre of a triangle.
In general, the incentre and the circumcentre of a triangle are two distinct points.
Here in the triangle XYZ, the incentre is at P and the circumcentre is at O.
A special case: an equilateral triangle, the bisector of the opposite side, so it is also a median.
In the ∆XYZ, XP, YQ and ZR are the bisectors of ∠YXZ, ∠XYZ and ∠YZX respectively; they are also the perpendicular bisectors of YZ, ZX and XY respectively; they are also the medians of the triangle.
So, their point of intersection, G, is the incentre, circumcentre as well as the centroid of the triangle. So, in an equilateral triangle, these three points are coincident.
If XY = YZ = ZX = 2a then in ∆XYP, YP = a and XP = \(\sqrt{3}\)a.
Now, XG = \(\frac{}{}\) = \(\frac{2}{3}\)XP = \(\frac{2\sqrt{3}a}{3}\), and GP = \(\frac{1}{3}\)XP = \(\frac{\sqrt{3}a}{3}\).
Therefore, radius of the circumcircle is XG = \(\frac{2\sqrt{3}a}{3}\) = \(\frac{2a}{\sqrt{3}}\) = \(\frac{Any side of the equilateral triangle}{\sqrt{3}}\).
The radius of the incircle = GP = \(\frac{a}{\sqrt{3}}\) = \(\frac{2a}{2\sqrt{3}}\) = \(\frac{Any side of the equilateral triangle}{2\sqrt{3}}\).
Therefore, radius of the circumcircle of an equilateral triangle = 2 × (Radius of the incircle).
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Plasma stability in a dipole magnetic field
Simakov, Andrei N., 1974-
Other Contributors
Massachusetts Institute of Technology. Dept. of Physics.
Peter J. Catto and Miklos Porkolab.
Terms of use
M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided
URL for inquiries about permission. http://dspace.mit.edu/handle/1721.1/7582
The MHD and kinetic stability of an axially symmetric plasma, confined by a poloidal magnetic field with closed lines, is considered. In such a system the stabilizing effects of plasma compression
and magnetic field compression counteract the unfavorable field line curvature and can stabilize pressure gradient driven magnetohydrodynamic modes provided the pressure gradient is not too steep.
Isotropic pressure, ideal MHD stability is studied first and a general interchange stability condition and an integro-differential eigenmode equation for ballooning modes are derived, using the MHD
energy principle. The existence of plasma equilibria which are both interchange and ballooning stable for arbitrarily large beta = plasma pressure / magnetic pressure, is demonstrated. The MHD
analysis is then generalized to the anisotropic plasma pressure case. Using the Kruskal-Oberman form of the energy principle, and a Schwarz inequality, to bound the complicated kinetic compression
term from below by a simpler fluid expression, a general anisotropic pressure interchange stability condition, and a ballooning equation, are derived. These reduce to the usual ideal MHD forms in the
isotropic limit. It is typically found that the beta limit for ballooning modes is at or just below that for either the mirror mode or the firehose.
(cont.) Finally, kinetic theory is used to describe drift frequency modes and finite Larmor radius corrections to MHD modes. An intermediate collisionality ordering in which the collision frequency
is smaller than the transit or bounce frequency, but larger than the mode, magnetic drift, and diamagnetic frequencies, is used for solving the full electromagnetic problem. An integro-differential
eigenmode equation with the finite Larmor radius corrections is derived for ballooning modes. It reduces to the ideal MHD ballooning equation when the mode frequency exceeds the drift frequencies. In
addition to the MHD mode, this ballooning equation permits an entropy mode solution whose frequency is of the order of the ion magnetic drift frequency. The entropy mode is an electrostatic flute
mode, even in equilibrium of arbitrary beta. Stability boundaries for both modes, and the influence of collisional effects on these boundaries has also been investigated.
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2001.
Includes bibliographical references (p. 137-141).
Date issued
Massachusetts Institute of Technology. Department of Physics
Massachusetts Institute of Technology | {"url":"https://dspace.mit.edu/handle/1721.1/60756","timestamp":"2024-11-15T03:54:55Z","content_type":"text/html","content_length":"23397","record_id":"<urn:uuid:b0b190a2-2314-4288-af90-ab3e47a6684d>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00295.warc.gz"} |
Let ${\displaystyle B_{t}}$ be a Brownian motion on a standard filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},{\mathcal {F}}_{t},P)}$ and let ${\displaystyle {\mathcal {G}}_{t}}$
be the augmented filtration generated by ${\displaystyle B}$ . If X is a square integrable random variable measurable with respect to ${\displaystyle {\mathcal {G}}_{\infty }}$ , then there exists a
predictable process C which is adapted with respect to ${\displaystyle {\mathcal {G}}_{t},}$ such that
${\displaystyle X=E(X)+\int _{0}^{\infty }C_{s}\,dB_{s}.}$
${\displaystyle E(X|{\mathcal {G}}_{t})=E(X)+\int _{0}^{t}C_{s}\,dB_{s}.}$
Application in finance
The martingale representation theorem can be used to establish the existence of a hedging strategy. Suppose that ${\displaystyle \left(M_{t}\right)_{0\leq t<\infty }}$ is a Q-martingale process,
whose volatility ${\displaystyle \sigma _{t}}$ is always non-zero. Then, if ${\displaystyle \left(N_{t}\right)_{0\leq t<\infty }}$ is any other Q-martingale, there exists an ${\displaystyle {\mathcal
{F}}}$ -previsible process ${\displaystyle \varphi }$ , unique up to sets of measure 0, such that ${\displaystyle \int _{0}^{T}\varphi _{t}^{2}\sigma _{t}^{2}\,dt<\infty }$ with probability one, and
N can be written as:
${\displaystyle N_{t}=N_{0}+\int _{0}^{t}\varphi _{s}\,dM_{s}.}$
The replicating strategy is defined to be:
• hold ${\displaystyle \varphi _{t}}$ units of the stock at the time t, and
• hold ${\displaystyle \psi _{t}B_{t}=C_{t}-\varphi _{t}Z_{t}}$ units of the bond.
where ${\displaystyle Z_{t}}$ is the stock price discounted by the bond price to time ${\displaystyle t}$ and ${\displaystyle C_{t}}$ is the expected payoff of the option at time ${\displaystyle t}$
At the expiration day T, the value of the portfolio is:
${\displaystyle V_{T}=\varphi _{T}S_{T}+\psi _{T}B_{T}=C_{T}=X}$
and it is easy to check that the strategy is self-financing: the change in the value of the portfolio only depends on the change of the asset prices ${\displaystyle \left(dV_{t}=\varphi _{t}\,dS_{t}+
\psi _{t}\,dB_{t}\right)}$ .
See also
• Montin, Benoît. (2002) "Stochastic Processes Applied in Finance"
• Elliott, Robert (1976) "Stochastic Integrals for Martingales of a Jump Process with Partially Accessible Jump Times", Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 36, 213–226 | {"url":"https://www.knowpia.com/knowpedia/Martingale_representation_theorem","timestamp":"2024-11-10T09:29:02Z","content_type":"text/html","content_length":"104096","record_id":"<urn:uuid:2f31f6e1-e0e7-43e2-b539-f4ebbb4d3f6c>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00326.warc.gz"} |
Reinforcement learning - Wikiwand
Reinforcement learning (RL) is an interdisciplinary area of machine learning and optimal control concerned with how an intelligent agent should take actions in a dynamic environment in order to
maximize a reward signal. Reinforcement learning is one of the three basic machine learning paradigms, alongside supervised learning and unsupervised learning.
Q-learning at its simplest stores data in tables. This approach becomes infeasible as the number of states/actions increases (e.g., if the state space or action space were continuous), as the
probability of the agent visiting a particular state and performing a particular action diminishes.
Reinforcement learning differs from supervised learning in not needing labelled input-output pairs to be presented, and in not needing sub-optimal actions to be explicitly corrected. Instead, the
focus is on finding a balance between exploration (of uncharted territory) and exploitation (of current knowledge) with the goal of maximizing the cumulative reward (the feedback of which might be
incomplete or delayed).^[1] The search for this balance is known as the exploration-exploitation dilemma.
The environment is typically stated in the form of a Markov decision process (MDP), as many reinforcement learning algorithms use dynamic programming techniques.^[2] The main difference between
classical dynamic programming methods and reinforcement learning algorithms is that the latter do not assume knowledge of an exact mathematical model of the Markov decision process, and they target
large MDPs where exact methods become infeasible.^[3]
The typical framing of a Reinforcement Learning (RL) scenario: an agent takes actions in an environment, which is interpreted into a reward and a state representation, which are fed back to the
Due to its generality, reinforcement learning is studied in many disciplines, such as game theory, control theory, operations research, information theory, simulation-based optimization, multi-agent
systems, swarm intelligence, and statistics. In the operations research and control literature, RL is called approximate dynamic programming, or neuro-dynamic programming. The problems of interest in
RL have also been studied in the theory of optimal control, which is concerned mostly with the existence and characterization of optimal solutions, and algorithms for their exact computation, and
less with learning or approximation (particularly in the absence of a mathematical model of the environment).
Basic reinforcement learning is modeled as a Markov decision process:
• A set of environment and agent states (the state space), ${\displaystyle {\mathcal {S}}}$;
• A set of actions (the action space), ${\displaystyle {\mathcal {A}}}$, of the agent;
• ${\displaystyle P_{a}(s,s')=\Pr(S_{t+1}=s'\mid S_{t}=s,A_{t}=a)}$, the transition probability (at time ${\displaystyle t}$) from state ${\displaystyle s}$ to state ${\displaystyle s'}$ under
action ${\displaystyle a}$.
• ${\displaystyle R_{a}(s,s')}$, the immediate reward after transition from ${\displaystyle s}$ to ${\displaystyle s'}$ under action ${\displaystyle a}$.
The purpose of reinforcement learning is for the agent to learn an optimal (or near-optimal) policy that maximizes the reward function or other user-provided reinforcement signal that accumulates
from immediate rewards. This is similar to processes that appear to occur in animal psychology. For example, biological brains are hardwired to interpret signals such as pain and hunger as negative
reinforcements, and interpret pleasure and food intake as positive reinforcements. In some circumstances, animals learn to adopt behaviors that optimize these rewards. This suggests that animals are
capable of reinforcement learning.^[4]^[5]
A basic reinforcement learning agent interacts with its environment in discrete time steps. At each time step t, the agent receives the current state ${\displaystyle S_{t}}$ and reward ${\
displaystyle R_{t}}$. It then chooses an action ${\displaystyle A_{t}}$ from the set of available actions, which is subsequently sent to the environment. The environment moves to a new state ${\
displaystyle S_{t+1}}$ and the reward ${\displaystyle R_{t+1}}$ associated with the transition ${\displaystyle (S_{t},A_{t},S_{t+1})}$ is determined. The goal of a reinforcement learning agent is to
learn a policy:
${\displaystyle \pi$ :{\mathcal {S}}\times {\mathcal {A}}\rightarrow [0,1]} , ${\displaystyle \pi (s,a)=\Pr(A_{t}=a\mid S_{t}=s)}$
that maximizes the expected cumulative reward.
Formulating the problem as a Markov decision process assumes the agent directly observes the current environmental state; in this case, the problem is said to have full observability. If the agent
only has access to a subset of states, or if the observed states are corrupted by noise, the agent is said to have partial observability, and formally the problem must be formulated as a partially
observable Markov decision process. In both cases, the set of actions available to the agent can be restricted. For example, the state of an account balance could be restricted to be positive; if the
current value of the state is 3 and the state transition attempts to reduce the value by 4, the transition will not be allowed.
When the agent's performance is compared to that of an agent that acts optimally, the difference in performance yields the notion of regret. In order to act near optimally, the agent must reason
about long-term consequences of its actions (i.e., maximize future rewards), although the immediate reward associated with this might be negative.
Thus, reinforcement learning is particularly well-suited to problems that include a long-term versus short-term reward trade-off. It has been applied successfully to various problems, including
energy storage,^[6] robot control,^[7] photovoltaic generators,^[8] backgammon, checkers,^[9] Go (AlphaGo), and autonomous driving systems.^[10]
Two elements make reinforcement learning powerful: the use of samples to optimize performance, and the use of function approximation to deal with large environments. Thanks to these two key
components, RL can be used in large environments in the following situations:
• A model of the environment is known, but an analytic solution is not available;
• Only a simulation model of the environment is given (the subject of simulation-based optimization);^[11]
• The only way to collect information about the environment is to interact with it.
The first two of these problems could be considered planning problems (since some form of model is available), while the last one could be considered to be a genuine learning problem. However,
reinforcement learning converts both planning problems to machine learning problems.
The exploration vs. exploitation trade-off has been most thoroughly studied through the multi-armed bandit problem and for finite state space Markov decision processes in Burnetas and Katehakis
Reinforcement learning requires clever exploration mechanisms; randomly selecting actions, without reference to an estimated probability distribution, shows poor performance. The case of (small)
finite Markov decision processes is relatively well understood. However, due to the lack of algorithms that scale well with the number of states (or scale to problems with infinite state spaces),
simple exploration methods are the most practical.
One such method is ${\displaystyle \varepsilon }$-greedy, where ${\displaystyle 0<\varepsilon <1}$ is a parameter controlling the amount of exploration vs. exploitation. With probability ${\
displaystyle 1-\varepsilon }$, exploitation is chosen, and the agent chooses the action that it believes has the best long-term effect (ties between actions are broken uniformly at random).
Alternatively, with probability ${\displaystyle \varepsilon }$, exploration is chosen, and the action is chosen uniformly at random. ${\displaystyle \varepsilon }$ is usually a fixed parameter but
can be adjusted either according to a schedule (making the agent explore progressively less), or adaptively based on heuristics.^[13]
Algorithms for control learning
Even if the issue of exploration is disregarded and even if the state was observable (assumed hereafter), the problem remains to use past experience to find out which actions lead to higher
cumulative rewards.
Criterion of optimality
The agent's action selection is modeled as a map called policy:
${\displaystyle \pi$ :{\mathcal {A}}\times {\mathcal {S}}\rightarrow [0,1]}
${\displaystyle \pi (a,s)=\Pr(A_{t}=a\mid S_{t}=s)}$
The policy map gives the probability of taking action ${\displaystyle a}$ when in state ${\displaystyle s}$.^[14]^:61 There are also deterministic policies.
State-value function
The state-value function ${\displaystyle V_{\pi }(s)}$ is defined as, expected discounted return starting with state ${\displaystyle s}$, i.e. ${\displaystyle S_{0}=s}$, and successively following
policy ${\displaystyle \pi }$. Hence, roughly speaking, the value function estimates "how good" it is to be in a given state.^[14]^:60
${\displaystyle V_{\pi }(s)=\operatorname {\mathbb {E} } [G\mid S_{0}=s]=\operatorname {\mathbb {E} } \left[\sum _{t=0}^{\infty }\gamma ^{t}R_{t+1}\mid S_{0}=s\right],}$
where the random variable ${\displaystyle G}$ denotes the discounted return, and is defined as the sum of future discounted rewards:
${\displaystyle G=\sum _{t=0}^{\infty }\gamma ^{t}R_{t+1}=R_{1}+\gamma R_{2}+\gamma ^{2}R_{3}+\dots ,}$
where ${\displaystyle R_{t+1}}$ is the reward for transitioning from state ${\displaystyle S_{t}}$ to ${\displaystyle S_{t+1}}$, ${\displaystyle 0\leq \gamma <1}$ is the discount rate. ${\
displaystyle \gamma }$ is less than 1, so rewards in the distant future are weighted less than rewards in the immediate future.
The algorithm must find a policy with maximum expected discounted return. From the theory of Markov decision processes it is known that, without loss of generality, the search can be restricted to
the set of so-called stationary policies. A policy is stationary if the action-distribution returned by it depends only on the last state visited (from the observation agent's history). The search
can be further restricted to deterministic stationary policies. A deterministic stationary policy deterministically selects actions based on the current state. Since any such policy can be identified
with a mapping from the set of states to the set of actions, these policies can be identified with such mappings with no loss of generality.
Brute force
The brute force approach entails two steps:
• For each possible policy, sample returns while following it
• Choose the policy with the largest expected discounted return
One problem with this is that the number of policies can be large, or even infinite. Another is that the variance of the returns may be large, which requires many samples to accurately estimate the
discounted return of each policy.
These problems can be ameliorated if we assume some structure and allow samples generated from one policy to influence the estimates made for others. The two main approaches for achieving this are
value function estimation and direct policy search.
Value function
Value function approaches attempt to find a policy that maximizes the discounted return by maintaining a set of estimates of expected discounted returns ${\displaystyle \operatorname {\mathbb {E} }
[G]}$ for some policy (usually either the "current" [on-policy] or the optimal [off-policy] one).
These methods rely on the theory of Markov decision processes, where optimality is defined in a sense stronger than the one above: A policy is optimal if it achieves the best-expected discounted
return from any initial state (i.e., initial distributions play no role in this definition). Again, an optimal policy can always be found among stationary policies.
To define optimality in a formal manner, define the state-value of a policy ${\displaystyle \pi }$ by
${\displaystyle V^{\pi }(s)=\operatorname {\mathbb {E} } [G\mid s,\pi ],}$
where ${\displaystyle G}$ stands for the discounted return associated with following ${\displaystyle \pi }$ from the initial state ${\displaystyle s}$. Defining ${\displaystyle V^{*}(s)}$ as the
maximum possible state-value of ${\displaystyle V^{\pi }(s)}$, where ${\displaystyle \pi }$ is allowed to change,
${\displaystyle V^{*}(s)=\max _{\pi }V^{\pi }(s).}$
A policy that achieves these optimal state-values in each state is called optimal. Clearly, a policy that is optimal in this sense is also optimal in the sense that it maximizes the expected
discounted return, since ${\displaystyle V^{*}(s)=\max _{\pi }\mathbb {E} [G\mid s,\pi ]}$, where ${\displaystyle s}$ is a state randomly sampled from the distribution ${\displaystyle \mu }$ of
initial states (so ${\displaystyle \mu (s)=\Pr(S_{0}=s)}$).
Although state-values suffice to define optimality, it is useful to define action-values. Given a state ${\displaystyle s}$, an action ${\displaystyle a}$ and a policy ${\displaystyle \pi }$, the
action-value of the pair ${\displaystyle (s,a)}$ under ${\displaystyle \pi }$ is defined by
${\displaystyle Q^{\pi }(s,a)=\operatorname {\mathbb {E} } [G\mid s,a,\pi ],\,}$
where ${\displaystyle G}$ now stands for the random discounted return associated with first taking action ${\displaystyle a}$ in state ${\displaystyle s}$ and following ${\displaystyle \pi }$,
The theory of Markov decision processes states that if ${\displaystyle \pi ^{*}}$ is an optimal policy, we act optimally (take the optimal action) by choosing the action from ${\displaystyle Q^{\pi ^
{*}}(s,\cdot )}$ with the highest action-value at each state, ${\displaystyle s}$. The action-value function of such an optimal policy (${\displaystyle Q^{\pi ^{*}}}$) is called the optimal
action-value function and is commonly denoted by ${\displaystyle Q^{*}}$. In summary, the knowledge of the optimal action-value function alone suffices to know how to act optimally.
Assuming full knowledge of the Markov decision process, the two basic approaches to compute the optimal action-value function are value iteration and policy iteration. Both algorithms compute a
sequence of functions ${\displaystyle Q_{k}}$ (${\displaystyle k=0,1,2,\ldots }$) that converge to ${\displaystyle Q^{*}}$. Computing these functions involves computing expectations over the whole
state-space, which is impractical for all but the smallest (finite) Markov decision processes. In reinforcement learning methods, expectations are approximated by averaging over samples and using
function approximation techniques to cope with the need to represent value functions over large state-action spaces.
Monte Carlo methods
Monte Carlo methods^[15] are used to solve reinforcement learning problems by averaging sample returns. Unlike methods that require full knowledge of the environment’s dynamics, Monte Carlo methods
rely solely on actual or simulated experience—sequences of states, actions, and rewards obtained from interaction with an environment. This makes them applicable in situations where the complete
dynamics are unknown. Learning from actual experience does not require prior knowledge of the environment and can still lead to optimal behavior. When using simulated experience, only a model capable
of generating sample transitions is required, rather than a full specification of transition probabilities, which is necessary for dynamic programming methods.
Monte Carlo methods apply to episodic tasks, where experience is divided into episodes that eventually terminate. Policy and value function updates occur only after the completion of an episode,
making these methods incremental on an episode-by-episode basis, though not on a step-by-step (online) basis. The term “Monte Carlo” generally refers to any method involving random sampling; however,
in this context, it specifically refers to methods that compute averages from complete returns, rather than partial returns.
These methods function similarly to the bandit algorithms, in which returns are averaged for each state-action pair. The key difference is that actions taken in one state affect the returns of
subsequent states within the same episode, making the problem non-stationary. To address this non-stationarity, Monte Carlo methods use the framework of general policy iteration (GPI). While dynamic
programming computes value functions using full knowledge of the Markov decision process (MDP), Monte Carlo methods learn these functions through sample returns. The value functions and policies
interact similarly to dynamic programming to achieve optimality, first addressing the prediction problem and then extending to policy improvement and control, all based on sampled experience.^[14]
Temporal difference methods
The first problem is corrected by allowing the procedure to change the policy (at some or all states) before the values settle. This too may be problematic as it might prevent convergence. Most
current algorithms do this, giving rise to the class of generalized policy iteration algorithms. Many actor-critic methods belong to this category.
The second issue can be corrected by allowing trajectories to contribute to any state-action pair in them. This may also help to some extent with the third problem, although a better solution when
returns have high variance is Sutton's temporal difference (TD) methods that are based on the recursive Bellman equation.^[16]^[17] The computation in TD methods can be incremental (when after each
transition the memory is changed and the transition is thrown away), or batch (when the transitions are batched and the estimates are computed once based on the batch). Batch methods, such as the
least-squares temporal difference method,^[18] may use the information in the samples better, while incremental methods are the only choice when batch methods are infeasible due to their high
computational or memory complexity. Some methods try to combine the two approaches. Methods based on temporal differences also overcome the fourth issue.
Another problem specific to TD comes from their reliance on the recursive Bellman equation. Most TD methods have a so-called ${\displaystyle \lambda }$ parameter ${\displaystyle (0\leq \lambda \leq
1)}$ that can continuously interpolate between Monte Carlo methods that do not rely on the Bellman equations and the basic TD methods that rely entirely on the Bellman equations. This can be
effective in palliating this issue.
Function approximation methods
In order to address the fifth issue, function approximation methods are used. Linear function approximation starts with a mapping ${\displaystyle \phi }$ that assigns a finite-dimensional vector to
each state-action pair. Then, the action values of a state-action pair ${\displaystyle (s,a)}$ are obtained by linearly combining the components of ${\displaystyle \phi (s,a)}$ with some weights ${\
displaystyle \theta }$:
${\displaystyle Q(s,a)=\sum _{i=1}^{d}\theta _{i}\phi _{i}(s,a).}$
The algorithms then adjust the weights, instead of adjusting the values associated with the individual state-action pairs. Methods based on ideas from nonparametric statistics (which can be seen to
construct their own features) have been explored.
Value iteration can also be used as a starting point, giving rise to the Q-learning algorithm and its many variants.^[19] Including Deep Q-learning methods when a neural network is used to represent
Q, with various applications in stochastic search problems.^[20]
The problem with using action-values is that they may need highly precise estimates of the competing action values that can be hard to obtain when the returns are noisy, though this problem is
mitigated to some extent by temporal difference methods. Using the so-called compatible function approximation method compromises generality and efficiency.
Direct policy search
An alternative method is to search directly in (some subset of) the policy space, in which case the problem becomes a case of stochastic optimization. The two approaches available are gradient-based
and gradient-free methods.
Gradient-based methods (policy gradient methods) start with a mapping from a finite-dimensional (parameter) space to the space of policies: given the parameter vector ${\displaystyle \theta }$, let $
{\displaystyle \pi _{\theta }}$ denote the policy associated to ${\displaystyle \theta }$. Defining the performance function by ${\displaystyle \rho (\theta )=\rho ^{\pi _{\theta }}}$ under mild
conditions this function will be differentiable as a function of the parameter vector ${\displaystyle \theta }$. If the gradient of ${\displaystyle \rho }$ was known, one could use gradient ascent.
Since an analytic expression for the gradient is not available, only a noisy estimate is available. Such an estimate can be constructed in many ways, giving rise to algorithms such as Williams'
REINFORCE method^[21] (which is known as the likelihood ratio method in the simulation-based optimization literature).^[22]
A large class of methods avoids relying on gradient information. These include simulated annealing, cross-entropy search or methods of evolutionary computation. Many gradient-free methods can achieve
(in theory and in the limit) a global optimum.
Policy search methods may converge slowly given noisy data. For example, this happens in episodic problems when the trajectories are long and the variance of the returns is large. Value-function
based methods that rely on temporal differences might help in this case. In recent years, actor–critic methods have been proposed and performed well on various problems.^[23]
Policy search methods have been used in the robotics context.^[24] Many policy search methods may get stuck in local optima (as they are based on local search).
Model-based algorithms
Finally, all of the above methods can be combined with algorithms that first learn a model of the Markov Decision Process, the probability of each next state given an action taken from an existing
state. For instance, the Dyna algorithm^[25] learns a model from experience, and uses that to provide more modelled transitions for a value function, in addition to the real transitions. Such methods
can sometimes be extended to use of non-parametric models, such as when the transitions are simply stored and 'replayed'^[26] to the learning algorithm.
Model-based methods can be more computationally intensive than model-free approaches, and their utility can be limited by the extent to which the Markov Decision Process can be learnt.^[27]
There are other ways to use models than to update a value function.^[28] For instance, in model predictive control the model is used to update the behavior directly.
Both the asymptotic and finite-sample behaviors of most algorithms are well understood. Algorithms with provably good online performance (addressing the exploration issue) are known.
Efficient exploration of Markov decision processes is given in Burnetas and Katehakis (1997).^[12] Finite-time performance bounds have also appeared for many algorithms, but these bounds are expected
to be rather loose and thus more work is needed to better understand the relative advantages and limitations.
For incremental algorithms, asymptotic convergence issues have been settled. Temporal-difference-based algorithms converge under a wider set of conditions than was previously possible (for example,
when used with arbitrary, smooth function approximation).
This section
needs additional citations for verification
(October 2022)
Research topics include:
• actor-critic architecture^[29]
• actor-critic-scenery architecture ^[3]
• adaptive methods that work with fewer (or no) parameters under a large number of conditions
• bug detection in software projects^[30]
• continuous learning
• combinations with logic-based frameworks^[31]
• exploration in large Markov decision processes
• interaction between implicit and explicit learning in skill acquisition
• intrinsic motivation which differentiates information-seeking, curiosity-type behaviours from task-dependent goal-directed behaviours large-scale empirical evaluations
• large (or continuous) action spaces
• modular and hierarchical reinforcement learning^[33]
• multiagent/distributed reinforcement learning is a topic of interest. Applications are expanding.^[34]
• occupant-centric control
• optimization of computing resources^[35]^[36]^[37]
• partial information (e.g., using predictive state representation)
• reward function based on maximising novel information^[38]^[39]^[40]
• sample-based planning (e.g., based on Monte Carlo tree search).
• securities trading^[41]
• TD learning modeling dopamine-based learning in the brain. Dopaminergic projections from the substantia nigra to the basal ganglia function are the prediction error.
• value-function and policy search methods
Comparison of key algorithms
Algorithm Description Policy Action space State space Operator
Monte Carlo Every visit to Monte Carlo Either Discrete Discrete Sample-means of state-values or action-values
TD learning State–action–reward–state Off-policy Discrete Discrete State-value
Q-learning State–action–reward–state Off-policy Discrete Discrete Action-value
SARSA State–action–reward–state–action On-policy Discrete Discrete Action-value
DQN Deep Q Network Off-policy Discrete Continuous Action-value
DDPG Deep Deterministic Policy Gradient Off-policy Continuous Continuous Action-value
A3C Asynchronous Advantage Actor-Critic Algorithm On-policy Discrete Continuous Advantage (=action-value - state-value)
TRPO Trust Region Policy Optimization On-policy Continuous or Discrete Continuous Advantage
PPO Proximal Policy Optimization On-policy Continuous or Discrete Continuous Advantage
TD3 Twin Delayed Deep Deterministic Policy Gradient Off-policy Continuous Continuous Action-value
SAC Soft Actor-Critic Off-policy Continuous Continuous Advantage
DSAC^[43]^[44]^[45] Distributional Soft Actor Critic Off-policy Continuous Continuous Action-value distribution
Associative reinforcement learning
Associative reinforcement learning tasks combine facets of stochastic learning automata tasks and supervised learning pattern classification tasks. In associative reinforcement learning tasks, the
learning system interacts in a closed loop with its environment.^[46]
Adversarial deep reinforcement learning
Adversarial deep reinforcement learning is an active area of research in reinforcement learning focusing on vulnerabilities of learned policies. In this research area some studies initially showed
that reinforcement learning policies are susceptible to imperceptible adversarial manipulations.^[49]^[50]^[51] While some methods have been proposed to overcome these susceptibilities, in the most
recent studies it has been shown that these proposed solutions are far from providing an accurate representation of current vulnerabilities of deep reinforcement learning policies.^[52]
Fuzzy reinforcement learning
By introducing fuzzy inference in reinforcement learning,^[53] approximating the state-action value function with fuzzy rules in continuous space becomes possible. The IF - THEN form of fuzzy rules
make this approach suitable for expressing the results in a form close to natural language. Extending FRL with Fuzzy Rule Interpolation ^[54] allows the use of reduced size sparse fuzzy rule-bases to
emphasize cardinal rules (most important state-action values).
Inverse reinforcement learning
In inverse reinforcement learning (IRL), no reward function is given. Instead, the reward function is inferred given an observed behavior from an expert. The idea is to mimic observed behavior, which
is often optimal or close to optimal.^[55] One popular IRL paradigm is named maximum entropy inverse reinforcement learning (MaxEnt IRL). ^[56] MaxEnt IRL estimates the parameters of a linear model
of the reward function by maximizing the entropy of the probability distribution of observed trajectories subject to constraints related to matching expected feature counts. Recently it has been
shown that MaxEnt IRL is a particular case of a more general framework named random utility inverse reinforcement learning (RU-IRL). ^[57] RU-IRL is based on random utility theory and Markov decision
processes. While prior IRL approaches assume that the apparent random behavior of an observed agent is due to it following a random policy, RU-IRL assumes that the observed agent follows a
deterministic policy but randomness in observed behavior is due to the fact that an observer only has partial access to the features the observed agent uses in decision making. The utility function
is modeled as a random variable to account for the ignorance of the observer regarding the features the observed agent actually considers in its utility function.
Safe reinforcement learning
Safe reinforcement learning (SRL) can be defined as the process of learning policies that maximize the expectation of the return in problems in which it is important to ensure reasonable system
performance and/or respect safety constraints during the learning and/or deployment processes.^[58] An alternative approach is risk-averse reinforcement learning, where instead of the expected
return, a risk-measure of the return is optimized, such as the Conditional Value at Risk (CVaR).^[59] In addition to mitigating risk, the CVaR objective increases robustness to model uncertainties.^
[60]^[61] However, CVaR optimization in risk-averse RL requires special care, to prevent gradient bias^[62] and blindness to success.^[63]
Self-reinforcement learning
Self-reinforcement learning (or self learning), is a learning paradigm which does not use the concept of immediate reward Ra(s,s') after transition from s to s' with action a. It does not use an
external reinforcement, it only uses the agent internal self-reinforcement. The internal self-reinforcement is provided by mechanism of feelings and emotions. In the learning process emotions are
backpropagated by a mechanism of secondary reinforcement. The learning equation does not include the immediate reward, it only includes the state evaluation.
The self-reinforcement algorithm updates a memory matrix W =||w(a,s)|| such that in each iteration executes the following machine learning routine: 1. in situation s perform action a 2. receive a
consequence situation s' 3. compute state evaluation v(s') of how good is to be in the consequence situation s' 4. update crossbar memory w'(a,s) = w(a,s) + v(s')
Initial conditions of the memory are received as input from the genetic environment. It is a system with only one input (situation), and only one output (action, or behavior).
Self reinforcement (self learning) was introduced in 1982 along with a neural network capable of self-reinforcement learning, named Crossbar Adaptive Array (CAA).^[64]^[65] The CAA computes, in a
crossbar fashion, both decisions about actions and emotions (feelings) about consequence states. The system is driven by the interaction between cognition and emotion. ^[66]
Statistical comparison of reinforcement learning algorithms
Efficient comparison of RL algorithms is essential for research, deployment and monitoring of RL systems. To compare different algorithms on a given environment, an agent can be trained for each
algorithm. Since the performance is sensitive to implementation details, all algorithms should be implemented as closely as possible to each other.^[67] After the training is finished, the agents can
be run on a sample of test episodes, and their scores (returns) can be compared. Since episodes are typically assumed to be i.i.d, standard statistical tools can be used for hypothesis testing, such
as T-test and permutation test.^[68] This requires to accumulate all the rewards within an episode into a single number - the episodic return. However, this causes a loss of information, as different
time-steps are averaged together, possibly with different levels of noise. Whenever the noise level varies across the episode, the statistical power can be improved significantly, by weighting the
rewards according to their estimated noise.^[69]
See also
Further reading
External links
Wikiwand in your browser!
Seamless Wikipedia browsing. On steroids.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent. | {"url":"https://www.wikiwand.com/en/articles/Reinforcement_learning?ref=rolisz.ro","timestamp":"2024-11-02T18:56:41Z","content_type":"text/html","content_length":"774768","record_id":"<urn:uuid:6cfdfe6e-dff1-47cd-8ce5-ed72960de803>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00318.warc.gz"} |
countif column values based on a monthly value
I am creating a table of count of values in a column based on what month is reported in another. the table I want is attached. The formula I am using is as below. Can anyone advise what need to
change in the planned formula
=SUMIF{V2.0 Eirabot - Gantt Range 1}},"12", {V2.0 Eirabot - Gantt Range 1}
I also want to have an actual row beneath. I have found issues where the two different calcs are providing the same values. The fact that smartsheets demotes a random range name rather than a
pre-defined column name makes it difficult to reconsiliate. Any advice?
• If all you need is a count then try something like this...
=COUNTIFS({V2.0 Eirabot - Gantt Range 1}, MONTH(@cell) = 12)
In regards to your formula posted:
=SUMIF{V2.0 Eirabot - Gantt Range 1}},"12", {V2.0 Eirabot - Gantt Range 1}
You are missing an opening parenthesis, you have an additional curly bracket after the first cross sheet reference, you are not specifying that the MONTH needs to be 12 so the formula is looking
for the EXACT text of "12", the second cross sheet reference being the same as the first means that you are trying to sum Range 1 which only needs to be specified if different from the criteria
range (won't break anything by repeating it as long as proper data is in there, just means a few extra unnecessary steps), and there is no closing parenthesis.
To sum based on dates, you would need something like this...
=SUMIFS({Cross Sheet Reference Range to be Summed}, {Cross Sheet Reference Range containing Dates}, MONTH(@cell) = 12)
• thanks @paul
I tried the following for sumif
=SUMIFS({V2.0 Eirabot - Effort}, {V2.0 Eirabot - date}, MONTH(@cell) = 12)
does this look correct?
• Yes. That would be correct if you are trying to add values together based on the month in the date column being that of December.
• Hi Paul,
thanks for the continued support. I got the table of FTE's per month working with the following formula
=[SUMIF([V2.0 Eirabot - month1],12,{V2.0 Eirabot - Effort})]/160
• Excellent. Happy to help.
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/42211/countif-column-values-based-on-a-monthly-value","timestamp":"2024-11-06T07:40:10Z","content_type":"text/html","content_length":"407673","record_id":"<urn:uuid:9fa25f6c-3be8-4a60-89db-9ea224e95dfc>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00855.warc.gz"} |
ecimeters to Femtometers
Decimeters to Femtometers Converter
Enter Decimeters
β Switch toFemtometers to Decimeters Converter
How to use this Decimeters to Femtometers Converter π €
Follow these steps to convert given length from the units of Decimeters to the units of Femtometers.
1. Enter the input Decimeters value in the text field.
2. The calculator converts the given Decimeters into Femtometers in realtime β using the conversion formula, and displays under the Femtometers label. You do not need to click any button. If the
input changes, Femtometers value is re-calculated, just like that.
3. You may copy the resulting Femtometers value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Decimeters to Femtometers?
The formula to convert given length from Decimeters to Femtometers is:
Length[(Femtometers)] = Length[(Decimeters)] × 1e+14
Substitute the given value of length in decimeters, i.e., Length[(Decimeters)] in the above formula and simplify the right-hand side value. The resulting value is the length in femtometers, i.e.,
Calculation will be done after you enter a valid input.
Consider that a luxury watch has a case diameter of 5 decimeters.
Convert this diameter from decimeters to Femtometers.
The length in decimeters is:
Length[(Decimeters)] = 5
The formula to convert length from decimeters to femtometers is:
Length[(Femtometers)] = Length[(Decimeters)] × 1e+14
Substitute given weight Length[(Decimeters)] = 5 in the above formula.
Length[(Femtometers)] = 5 × 1e+14
Length[(Femtometers)] = 500000000000000
Final Answer:
Therefore, 5 dm is equal to 500000000000000 fm.
The length is 500000000000000 fm, in femtometers.
Consider that a premium sound system's speaker has a woofer diameter of 3 decimeters.
Convert this diameter from decimeters to Femtometers.
The length in decimeters is:
Length[(Decimeters)] = 3
The formula to convert length from decimeters to femtometers is:
Length[(Femtometers)] = Length[(Decimeters)] × 1e+14
Substitute given weight Length[(Decimeters)] = 3 in the above formula.
Length[(Femtometers)] = 3 × 1e+14
Length[(Femtometers)] = 300000000000000
Final Answer:
Therefore, 3 dm is equal to 300000000000000 fm.
The length is 300000000000000 fm, in femtometers.
Decimeters to Femtometers Conversion Table
The following table gives some of the most used conversions from Decimeters to Femtometers.
Decimeters (dm) Femtometers (fm)
0 dm 0 fm
1 dm 100000000000000 fm
2 dm 200000000000000 fm
3 dm 300000000000000 fm
4 dm 400000000000000 fm
5 dm 500000000000000 fm
6 dm 600000000000000 fm
7 dm 700000000000000 fm
8 dm 800000000000000 fm
9 dm 900000000000000 fm
10 dm 1000000000000000 fm
20 dm 2000000000000000 fm
50 dm 5000000000000000 fm
100 dm 10000000000000000 fm
1000 dm 100000000000000000 fm
10000 dm 1000000000000000000 fm
100000 dm 10000000000000000000 fm
A decimeter (dm) is a unit of length in the International System of Units (SI). One decimeter is equivalent to 0.1 meters or approximately 3.937 inches.
The decimeter is defined as one-tenth of a meter, making it a convenient measurement for intermediate lengths.
Decimeters are used worldwide to measure length and distance in various fields, including science, engineering, and everyday life. They provide a useful scale for measurements that are larger than
centimeters but smaller than meters, and are commonly used in educational settings and certain industries.
A femtometer (fm) is a unit of length in the International System of Units (SI). One femtometer is equivalent to 0.000000000001 meters or 1 Γ 10^(-15) meters.
The femtometer is defined as one quadrillionth of a meter, making it a very small unit of measurement used for measuring atomic and subatomic distances.
Femtometers are commonly used in nuclear physics and particle physics to describe the sizes of atomic nuclei and the ranges of fundamental forces at the subatomic level.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Decimeters to Femtometers in Length?
The formula to convert Decimeters to Femtometers in Length is:
Decimeters * 1e+14
2. Is this tool free or paid?
This Length conversion tool, which converts Decimeters to Femtometers, is completely free to use.
3. How do I convert Length from Decimeters to Femtometers?
To convert Length from Decimeters to Femtometers, you can use the following formula:
Decimeters * 1e+14
For example, if you have a value in Decimeters, you substitute that value in place of Decimeters in the above formula, and solve the mathematical expression to get the equivalent value in | {"url":"https://convertonline.org/unit/?convert=decimeters-femtometers","timestamp":"2024-11-09T10:17:56Z","content_type":"text/html","content_length":"90779","record_id":"<urn:uuid:24b11202-af75-40cf-9f44-b153b55894bf>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00244.warc.gz"} |
Axel Ljungström: A Cubical Formalisation of Cohomology Theory and $\pi_4(S^3)$ = Z/2Z | KTH
Axel Ljungström: A Cubical Formalisation of Cohomology Theory and $\pi_4(S^3)$ = Z/2Z
Time: Tue 2023-05-16 13.00 - 14.00
Location: Classroom 29, Albano campus, house 4, floor 2
Doctoral student: Axel Ljungström
Opponent: Eric Finster
Supervisor: Anders Mörtberg
Examiner: Alexander Berglund
The primary goal of this thesis is to provide a computer formalisation on the main result of Guillaume Brunerie's 2016 PhD thesis, namely that $\pi_4(S^3)$ = Z/2Z in Homotopy Type Theory (HoTT). The
formalisation is carried out in Cubical Agda, an extension of the Agda proof assistant based on cubical type theory.
The thesis consists of two papers. In the first paper, we provide a formalisation of integral cohomology theory in Cubical Agda. This includes many fundamental constructions crucially used in
Brunerie's proof. In particular, we provide a novel construction of the cup product in HoTT which both avoids smash products and is easier to work with. This allows us to provide the first complete
construction and formalisation of the fact that the cup product satisfies the graded-commutative ring laws. We also compute the cohomology groups of some simple spaces. While the proofs of the main
results in this paper are often easy to translate into standard HoTT, all constructions are carefully chosen to exhibit optimal computational behaviour in Cubical Agda.
The second paper is a summary of three formalisations of \(\pi_4(S^3) = \mathbb{Z}/2\mathbb{Z}\). The primary formalisation follows Brunerie's thesis closely but diverges in some details. In
particular, we only work with a special case of the so called James construction, providing direct proofs of the relevant theorems. We also completely avoid the smash product by using the cohomology
theory of the first paper. In addition, we present two new formalisations of the main result. In these, we circumvent the second half of Brunerie's thesis by providing direct computations of the so
called Brunerie number, i.e. the number β such that \(\pi_4(S^3) = \mathbb{Z}/β\mathbb{Z}\), as defined in the first half of Brunerie's thesis. The first of these formalisations uses a new direct
proof of the fact that β = -2. In the second one, we attempt to show that \(β = \pm 2\) by simply normalising it in Cubical Agda. While this normalisation still fails to terminate (in reasonable
time), we can, by following the proof strategy in the second formalisation, rewrite β until we arrive at a definition which actually normalises. | {"url":"https://www.kth.se/math/kalender/axel-ljungstrom-a-cubical-formalisation-of-cohomology-theory-and-pi-4-s-3-z-2z-1.1255335?date=2023-05-16&orgdate=2023-03-19&length=1&orglength=0","timestamp":"2024-11-04T14:23:30Z","content_type":"text/html","content_length":"57831","record_id":"<urn:uuid:bd79e931-0819-481b-9871-8bb712d83026>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00331.warc.gz"} |
Determine the equation from graph
Determine the equation that represents the relationship between and in the graph below.
1. Determine which type of equation, linear or exponential, will fit the graph.
2. Identify at least three points from the graph.
3. Find the slope of the line, using any two of the points.
4. Find the -intercept of the line.
5. Use the slope and -intercept to find an equation of the line. | {"url":"https://stage.geogebra.org/m/Z62rtJx3","timestamp":"2024-11-02T10:50:47Z","content_type":"text/html","content_length":"91440","record_id":"<urn:uuid:59ee5a7f-a849-4ace-93a2-fef5f6a6e68e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00230.warc.gz"} |
How big is a square meter?
Calculator Use
To use this converter, just choose a unit to convert from, a unit to convert to, and then type the value you want to convert. The result will be shown immediately.
This converter accepts decimal, integer and fractional values as input, so you can input values like: 1, 4, 0.5, 1.9, 1/2, 3 1/2, etc.
Note that to enter a mixed number like 1 1/2, you should leave a space between the integer and the fraction
The numerical result exactness will be according to the number of significant figures that you choose.
When the result shows one or more fractions, you should consider its colors according to the table below:
Exact fraction or 0% 1% 2% 5%10%15%
These colors represent the maximum approximation error for each fraction. If the error does not fit your need, you should use the decimal value and possibly increase the number of significant
Please, if you find any issues in this calculator, or if you have any suggestions, please contact us. | {"url":"https://ezunitconverter.com/area/square-meter/","timestamp":"2024-11-05T05:40:43Z","content_type":"text/html","content_length":"42737","record_id":"<urn:uuid:73d98e76-e99f-4652-85e3-91c77bb1b481>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00557.warc.gz"} |
Kepler’s Laws
The meaning of the laws of Kepler is well described in their statements, which you can find in any textbook:
1. First Law, or law of the elliptical orbits:
The orbit of a planet is an ellipse with the Sun at one of the two foci.
2. Second Law, or law of the areas:
A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
3. Third Laws, or laws of the periods:
The square of a planet’s orbital period is proportional to the cube of the length of the semi-major axis of its orbit.
It that all? Well, yes… just as Leonardo’s Mona Lisa might be described as a small painting depicting a girl with folded arms and a half smile.
In fact, behind both products of the human genius, there is much more than that. As for Kepler’s laws, over and above their astronomical value, their importance extends to several fields, such as:
• classical Mathematics (Geometry and Trigonometry)
• Physics (they inspired Newton’s Law of universal gravitation, and constitute clear examples of the preservation of physical quantities, such as energy and the angular moment)
• differential or infinitesimal calculus (also invented by Newton, and Leibniz, just to solve those laws of motion),
• philosophy, as well as the history of human progress.
It goes without saying that, without Kepler’s Laws and the scientific progress they inspired, humankind would not have been able to successfully send missions to explore our Solar System, and more
broadly, would have a much narrower and incorrect view of the Universe.
The activity we are introducing here does not pretend to be a “course”, but rather a “path” which, with the idea of learning the meaning of these three laws better, will touch many mathematical and
physical concepts. At the same time, there will be the chance to face programming challenges for increasingly complex problems. To this aim, we will use the Scratch language, which certainly is not
the quickest and most efficient one for numerical calculus, but offers big advantages, such as the intuitive approach to coding and the easy management of interactive interfaces.
To the various stages of this path will be associated programs, not only to be run passively, but whose code should be read and especially understood (that’s why we shall try to make them more
understandable, with some explanatory comment) and, in our intentions, these programs should also be used as a starting point for experiments of coding.
Here are the stages, one after the other, with the codes in order to carry them out:
• Draw an ellipse, centered with respect to the intersection of its axes. Identify some significant elements of an ellipse (major axis, minor axis, eccentricity, focal distance).
• Draw an ellipse, centered with respect to one of its foci. Identify other elements of the orbit (right half-side), describe the relationships among them, with rispect to certain dynamical
quantities, in the case of an orbit.
• Calculate the orbital motion of a planet around a star, while calculating dynamical equations, first by using a simplified algorithm, then a more sophisticated one (Runge-Kutta method). By
calculating several dynamic and geometric quantities, we can carry out a series of tests, such as for example on the total energy constancy and angular momentum, and on the effetcive dependence
of orbit parameters on physical constants, in accordance with the formulas introduced in the previus step.
Future Steps (work in progress)
• Dynamical calculus of the orbital motions of two bodies with comparable masses. Introducing the concept of mass centre, and system’s reduced mass.
• Comparison between the law of gravitational attraction, which scales with radial distance 1/R2, and laws scaling as 1/Rn, with n other than 2. Analysis of the precession of the perihelion, as
well as of the orbital stability (orbits reaching the central star).
• Reducing the problem of the orbit to a one-dimensional problem. Introducing the concept of Effective Potential, and extending it to cases with n other than 2.
• Open orbits: parabolic and hyperbolic.
• Impact of meteorites upon the Earth. Dependence of the Earth impact section according to relative velocity.
• Motion under the gravitational effect of two major bodies: planets’ orbits around a binary stellar system; the orbits of a satellite subject to the attraction of a star and a planet (Lagrangian
points; Greek and Trojan asteroids).
• Motions of 2 planets around a star, and mutual disturbance of their orbits.
• Shepherd moons and Saturn’s rings.
• “Non”-gravitational motions. Double tail of comets.
• Rocket travel in a star-planet system.
• Gravitational slingshot method.
Leave a Reply Cancel reply | {"url":"https://play.inaf.it/en/keplers-laws/","timestamp":"2024-11-04T11:09:06Z","content_type":"text/html","content_length":"78698","record_id":"<urn:uuid:b54db480-670d-4d5a-b40c-13f313be4783>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00705.warc.gz"} |
Introduction of Qcad Window, Drawing Tools, Snap Tools, Drawing Area, Status
Line, List Docking Area, Loading and Naming Files, Saving Files, Don't
Overwrite, Fileload Auto, Zoom, Grid Scale Adjusts to File, Pen Toolbar, Zoom
Auto Tool, Help Menu, Grid Dots Control, Coordinate Display, Mouse
Status.Coordinate System- Types of Coordinates, Center of Origin, Drawing Area
Rulers, X-Y Coordinates, Polar Coordinates, Polar Angle Measurement, Relative
Reference Point | {"url":"http://www.irdtuttarakhand.org.in/ubter/PaperDetail.aspx?code=742&bcode=28&scode=991006","timestamp":"2024-11-02T14:00:19Z","content_type":"application/xhtml+xml","content_length":"14233","record_id":"<urn:uuid:f6831eef-15ec-43a4-8045-630e21bd7089>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00868.warc.gz"} |
Middle School Math Moments (now Cognitive Cardio Math)
Click to go to post I'm so excited - I was given the opportunity to guest post on Rachel Lynette's blog, Minds in Bloom . The post went up on her blog today, if you'd like to check it out:) Many
thanks to Rachel!
Today, as my students were working on a color by number in math class (which I thought was a fun, different way to practice math), one of them asked "How does coloring help with math?" The question
was asked with a "there's no reason I should have to do this" attitude. I explained that it helped with motor skills and helped one to use the brain in a different way, and that exercising the brain
in different ways could help in all things that require thinking (not just math). I don't think he really appreciated my answer:) Integer Operations Color by Number - freebie So, I decided to do a
little research, to see what I could find. Most of what I found (not a super-long time of searching, because I didn't have that much time!) was mostly related to the benefits of coloring for young
children (and did relate to math skills) and for adults. Here are a few things that I found, as coloring relates to adults: According to the Huffington Post (10/13/14
Yesterday was my birthday, and it was a great day! I got up early and went out for a walk/run, worked in the garden and had a great dinner with my husband and kids. Among my wonderful gifts were
these fantastic "coloring books for adults" from my daughter. They are AMAZING! They were created by Johanna Basford (maybe you've already seen them) and are just incredibly detailed. The picture
above is the one I started to color....it's a little hard to see the detail, but I think it's just incredible. Now all I want to do is color.....:)
Chapter 3 - Examples of Differentiated Planning for Achievable Challenge This is a continuation of Chapter 3, from a couple of weeks ago (I had my notes written, but it has taken me a while to type
them!!). In the previous Chapter 3 post , I reviewed a couple of the examples of differentiated planning and activities that the author offered. In each example, students are learning the same basic
concepts, but at different levels of challenge, which should lead to maximum success and should minimize their frustration. This example is called Exploring Number Lines, and the author states that
it is a helpful activity for both "explorers" and "map readers." As a preliminary activity, students explore number lines without any specific assignment; the author suggests using large number lines
that can be rolled out on the floor. Students meet in groups and create KWL charts. In working with the number line, students will predict where they will end up with certai
I have not been doing a great job keeping up with my blog lately! Too many things going on! Click here to go to the free bundle! But, I did need to jump on here and type up a quick post about the
Winner's Bundle that went along with March Madness tournament! If you read my "March Mayhem" post last month, you read that my collaborators and I ( Tools for Teaching Teens group) put some of our
paid products into bundles that went along with the teams in the tournament. Since Duke won, our Duke bundle is now free in our stores! The bundle includes: History Gal 's Printable Greek Gods
Technology Integration Depot 's Ancient Rome Bingo Colorado Classroom 's Parent and Student Communication Middle School Math Moments ' Decimal Division Color by Number Math Giraffe 's Triangle
Classification Leah Cleary 's Fall of Rome That's 6 paid products for FREE! The download will only be available Wednesday and Thursday. On Friday (April 10), | {"url":"https://www.middleschoolmathmoments.com/2015/04/","timestamp":"2024-11-12T10:41:54Z","content_type":"text/html","content_length":"165445","record_id":"<urn:uuid:3e215d71-b9b9-4dbe-83c9-abac464bf3ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00067.warc.gz"} |
Cobra Commander & Blowing Up the Moon!
This wonderful headline showed up in my Facebook feed: U.S. 'planned to blow up Moon' with nuke during Cold War era to show Soviets might. This was a bit eyebrow-raising. Dr. Strangelove was a
slightly unfair caricature, but the Cold War really did have a better-than-average share of nuttiness. Still - blowing up the moon? Surely not.
And it was in fact surely not. The article itself says there was a never-pursued idea to detonate a nuke on the lunar surface. To an earthbound observer, it would have meant a bright, brief pinprick
of light on the face of our celestial neighbor and not much else. It wouldn't have even left a naked-eye visible mark. (At the distance of the moon, the smallest features resolvable by the naked eye
are more than 60 miles across).
To their credit, a lot of other news outlets jumped on this. Forbes and PC Magazine are a few of the ones on the top of Google News with more accurate takes on the issue. Forbes even gave a quite
accurate estimate for the energy required. I myself did something similar on this blog a couple years back with a post on the physics of the Death Star*, with a derivation of the gravitational
binding energy. Rather than go through that again, let's do a more handwaving but possibly more intuitive derivation of the energy required to blow up the moon.
Ok, imagine you're standing on the moon. You want to turn it into dust, spread diffusely throughout the solar system. In this thought experiment, you might imagine grabbing a handful of the lunar
soil, take it into space, and letting it go at some very large distance from the moon. The energy required to do this is going to be related to the gravitational force the moon exerts on your handful
of dirt. which according to Newton is proportional to the gravitational constant G and the product of their masses. After the first handful is gone, you come back to the moon and repeat the process.
This time it's infinitesimally easier, because the gravitational pull of the moon is infinitesimally smaller, because it's less massive by the handful of dirt that's already gone. If you repeat this
process a bazillion times until the moon is gone, you'll have spent an amount of energy that's proportional to the mass of the moon and the mass of all the handfuls (also the mass of the moon). So
there's probably going to be a GM^2 term in the equation.
Well, that's nice but it won't yet give us a number. We need to be able to put a number on the energy required per handful. We might imagine that if the moon were compressed into a much smaller and
denser ball of the same mass, it would be harder to remove a handful. The gravity would be stronger. Again according to Newton, the force is inversely proportional to the square distance between the
objects, in this case, between the handful and the moon's center of mass. So maybe there's a 1/R^2 term too, where R is the radius of the moon. This is almost right. The thing we're interested isn't
quite the force, but the energy required to move an object against that force. The energy required to move an object against a uniform force is just the force multiplied by the distance you move it
through. Here the distance is enormous but the force isn't uniform either - it falls off with distance. So maybe as a first approximation we can say that we have to move the object "about" a distance
R to get it away from the moon's gravity. Multiplying R by 1/R^2 gives 1/R.
Great! Now we know the approximate energy required to blow up the moon. It's
[latex]E = \frac{GM^2}{R}[/latex].
Well, at least to the accuracy of our hand-waving dimensional arguments. In reality there's a factor of 3/5 and a more careful calculation shows the gravitational binding energy is
[latex]E = \frac{3GM^2}{5R}[/latex].
In any case, the gravitational binding energy for the moon is about 10^29 joules. The biggest thermonuclear weapon the US ever made was on the order of a 10^14 joule device, so you'd have needed at
least a quadrillion of them to blow up the moon. Even in the crazier days of the Cold War, our lunar neighbor didn't have a whole lot to worry about.
*Most of the images and equations on my old posts were borked in the transition to WordPress and NatGeo. It's not their fault - our old system was a disaster area. We're working on getting it fixed.
More like this
Standing on the edge of Niagra Falls you can watch the water pour over. Falling down the gravity of the earth, it exchanges its potential energy for kinetic energy by picking up speed. Some of that
energy is extracted by turbines and lights the homes and businesses of Yankees and Canucks alike.…
I recently learned about a great blog by S.C. Kavassalis of the University of Toronto called The Language of Bad Physics. She discusses, among other things, the way language is used in physics. She's
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Cobra Commander: [Irritated] What _is_ it, Destro?
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Cobra Commander: [Best Inspirational Leadership Tone] Do something, you fools!!! | {"url":"https://scienceblogs.com/builtonfacts/2012/12/03/cobra-commander-blowing-up-the-moon","timestamp":"2024-11-05T15:27:58Z","content_type":"text/html","content_length":"41735","record_id":"<urn:uuid:46ff5507-9b1a-4de5-a289-ee92bcff7b88>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00853.warc.gz"} |
Dell Inc. Bonds: A Risk and Return Analysis - SAS Risk Data and Analytics
In this note we analyze the current levels and past history of default probabilities for Dell Inc. (DELL) and we compare those default probabilities to credit spreads on 292 bond trades in 8
different bond issues on July 22, 2013. Trading volume in Dell Inc. bonds on that day totaled $37.3 million. Assuming the recovery rate in the event of default would be the same on all bond issues,
a sophisticated investor who has moved beyond legacy ratings seeks to maximize revenue per basis point of default risk from each incremental investment, subject to risk limits on macro-factor
exposure on a fully default-adjusted basis. We analyze the maturities where the credit spread/default probability ratio is highest for Dell Inc. We also consider whether or not a reasonable investor
would judge the firm to be “investment grade” under the June 2012 rules mandated by the Dodd-Frank Act of 2010.
Definition of Investment Grade
On June 13, 2012, the Office of the Comptroller of the Currency published the final rules defining whether a security is “investment grade,” in accordance with Section 939A of the Dodd-Frank Act of
2010. The new rules delete reference to legacy credit ratings and replace them with default probabilities. The web page explaining the Office of the Comptroller of the Currency’s new rules defining
investment grade and related guidance can be found here.
Term Structure of Default Probabilities
Maximizing the ratio of credit spread to matched maturity default probabilities requires that default probabilities be available at a wide range of maturities. The graph below shows the current
default probabilities for Dell Inc. ranging from one month to 10 years on an annualized basis. The default probabilities range from 0.36% at one month to 0.17% at 1 year and 0.78% at ten years.
We explain the source and methodology for the default probabilities below.
Summary of Recent Bond Trading Activity
The National Association of Securities Dealers launched the TRACE (Trade Reporting and Compliance Engine) in July 2002 in order to increase price transparency in the U.S. corporate debt market. The
system captures information on secondary market transactions in publicly traded securities (investment grade, high yield and convertible corporate debt) representing all over-the-counter market
activity in these bonds. TRACE data for Dell Inc. included 301 trades in the bonds of the firm on July 22, 2013. After eliminating data errors, 292 trades were analyzed on 8 different bond issues.
The graph below shows 5 different yield curves that are relevant to a risk and return analysis of Dell Inc. bonds. The lowest curve, in dark blue, is the yield to maturity on the benchmark U.S.
Treasury bonds most similar in maturity to the traded bonds of Dell Inc. The second lowest curve, in the lighter blue, shows the yields that would prevail if investors shared the default probability
views outlined above, assumed that recovery in the event of default would be zero, and demanded no liquidity premium above and beyond the default-adjusted risk-free yield. The third line from the
bottom (in orange) graphs the lowest yield reported by TRACE on that day on Dell Inc. bonds. The fourth line from the bottom (in green) displays the average yield reported by TRACE on the same day.
The highest yield is obviously the maximum yield in each Dell Inc. issue recorded by TRACE.
The data makes it very clear that there is a very large liquidity premium built into the yields of Dell Inc. above and beyond the “default-adjusted risk free curve” (the risk-free yield curve plus
the matched maturity default probabilities for the firm).
The high, low and average credit spread at each maturity are graphed below. While there is a fair amount of volatility in spread prevailing on the longer maturities, credit spreads are gradually
increasing with the maturity of the bonds.
Because we have default probabilities in addition to credit spreads, we can analyze the number of basis points of credit spread per basis point of default risk at each maturity. This ratio of spread
to default probability is shown in the following table for Dell Inc. At all maturities, the reward from holding the bonds of Dell Inc., relative to the matched maturity default probability, is at
least 4 basis points of credit spread reward for every basis point of default risk incurred. The ratio of spread to default probability generally declines as the maturity of the bonds lengthens.
The next graph plots the ratio of credit spread to default probability at each maturity.
For Dell Inc., the spread/default probability ratios are highest for maturities under three years. The reward for bearing a basis point of default risk declines from levels of 10-16 times to ratios
generally in the 4 to 8 times default risk range.
The Depository Trust & Clearing Corporation reports weekly on new credit default swap trading volume by reference name. For the week ended July 12, 2013 (the most recent week for which data is
available), the credit default swap trading volume on Dell Inc. showed 68 contracts trading with a notional principal of $390.6 million. The next graph shows the weekly number of credit default swaps
traded on Dell Inc. in the 155 weeks ended June 28, 2013.
The table below summarizes the volatile nature of credit default swap trading in Dell Inc. during this three year period.
On a cumulative basis, the default probabilities for Dell Inc. range from 0.17% at 1 year to 7.52% at 10 years, as shown in the following graph.
Over the last 10 years, the 1 year and 5 year default probabilities for Dell Inc. have varied as shown in the following graph. The one year default probability peaked at just under 1.40% and the 5
year default probability peaked just short of 0.60% in the first half of 2009 at the peak of the credit crisis.
Over the same decade, the legacy credit ratings (those reported by credit rating agencies like McGraw-Hill (MHP) unit Standard & Poor’s and Moody’s (MCO)) for Dell Inc. have changed once since the
company was first rated in 2012.
The macro-economic factors driving the historical movements in the default probabilities of Dell Inc. over the period from 1990 to the present include the following factors of those listed by the
Federal Reserve in its 2013 Comprehensive Capital Analysis and Review:
• BBB-rated corporate bond yields
• 30 year fixed rate mortgage yields
• The Dow Jones Industrials stock price index
• Commercial real estate price index
• 5 international macro factors
These macro factors explain 74.9% of the variation in the default probability of Dell Inc. since 1990.
Dell Inc. can be compared with its peers in the same industry sector, as defined by Morgan Stanley and reported by Compustat. For the USA technology, hardware and equipment sector, Dell Inc. has the
following percentile ranking for its default probabilities among its peers at these maturities:
│1 month │83rd percentile │
│1 year │65th percentile │
│3 years │47th percentile │
│5 years │35th percentile │
│10 years │31st percentile │
Dell Inc. is in the bottom half of creditworthiness among its peers for maturities of one year or less. Dell Inc. ranks in the 2nd quartile by riskiness for maturities from 3 to 10 years. A
comparison of the legacy credit rating for Dell Inc. with predicted ratings indicates that the company is rated equally by both statistical means and by legacy credit rating agencies.
Dell Inc. has experienced fairly minimal variation in its default probabilities over the last decade, particularly in comparison with financial institutions. Current default probabilities are fairly
high, however, and they reflect both an increased risk of a more aggressive capital structure and a substantial upheaval in Dell’s core markets. At current default probability levels, we believe that
a majority sophisticated analysts would rate Dell Inc. as investment grade by the Comptroller of the Currency definition, but it would be ranked near the bottom of the investment grade range.
Background on Default Probabilities Used
The Kamakura Risk Information Services version 5.0 Jarrow-Chava reduced form default probability model makes default predictions using a sophisticated combination of financial ratios, stock price
history, and macro-economic factors. The version 5.0 model was estimated over the period from 1990 to 2008, and includes the insights of the worst part of the recent credit crisis. Kamakura default
probabilities are based on 1.76 million observations and more than 2000 defaults. The term structure of default is constructed by using a related series of econometric relationships estimated on this
data base. An overview of the full suite of related default probability models is available here.
General Background on Reduced Form Models
For a general introduction to reduced form credit models, Hilscher, Jarrow and van Deventer (2008) is a good place to begin. Hilscher and Wilson (2013) have shown that reduced form default
probabilities are more accurate than legacy credit ratings by a substantial amount. Van Deventer (2012) explains the benefits and the process for replacing legacy credit ratings with reduced form
default probabilities in the credit risk management process. The theoretical basis for reduced form credit models was established by Jarrow and Turnbull (1995) and extended by Jarrow (2001). Shumway
(2001) was one of the first researchers to employ logistic regression to estimate reduced form default probabilities. Chava and Jarrow (2004) applied logistic regression to a monthly database of
public firms. Campbell, Hilscher and Szilagyi (2008) demonstrated that the reduced form approach to default modeling was substantially more accurate than the Merton model of risky debt. Bharath and
Shumway (2008), working completely independently, reached the same conclusions. A follow-on paper by Campbell, Hilscher and Szilagyi (2011) confirmed their earlier conclusions in a paper that was
awarded the Markowitz Prize for best paper in the Journal of Investment Management by a judging panel that included Prof. Robert Merton. | {"url":"https://www.kamakuraco.com/dell-inc-bonds-a-risk-and-return-analysis/","timestamp":"2024-11-02T11:50:01Z","content_type":"text/html","content_length":"153685","record_id":"<urn:uuid:d5886f29-12f8-45c8-864e-bf9120957816>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00520.warc.gz"} |
Gaussian elimination to row echelon form, not reduced row echelon form?
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Gaussian elimination to row echelon form, not reduced row echelon form?
Hi all.
How can I get Mathematica to perform Gaussian elimination on a matrix to get it to row echelon form, but not reduced row echelon form? I don't want all the leading variables to be 1. Mathematica
seems to only have the "RowReduced" function, which results in a reduced row echelon form.
Thanks in advance.
7 Replies
You can use LUDecomposition. I'll show a simple example.
mat = RandomInteger[{-10, 10}, {3, 5}]
{lu, p, c} = LUDecomposition[mat]
u = Normal[lu*SparseArray[{i_, j_} /; j >= i -> 1, Dimensions[mat]]]
(* Out[417]= {{-8, 0, 2, 6, -7}, {1, -1, 10, 4, -2}, {9, -1, 0, -7, 4}}
Out[418]= {{{1, -1, 10, 4, -2}, {-8, -8, 82,
38, -23}, {9, -1, -8, -5, -1}}, {2, 1, 3}, 0}
Out[419]= {{1, -1, 10, 4, -2}, {0, -8, 82, 38, -23}, {0,
0, -8, -5, -1}} *)
The result will be dependent on the permutation that was used in forming the LU decomposition.
--- edit 2024-06-14 ---
We now have UpperTriangularDecomposition in the Wolfram Function Repository. It returns an upper-triangularized form and the corresponding conversion matrix. Example:
mat = RandomInteger[{-10, 10}, {3, 5}]
{uu, conv} = ResourceFunction["UpperTriangularDecomposition"][mat]
(* Out[125]= {{-10, 3, 8, 3, 9}, {8, 9, -3, 1, -5}, {-1, -8, 2, 2, -7}}
Out[126]= {{{-10, 3, 8, 3, 9}, {0, -114, -34, -34, -22}, {0, 0, -419, -476, 718}}, {{1, 0, 0}, {-8, -10, 0}, {-55, -83, -114}}} *)
conv . mat == uu
(* Out[127]= True *)
--- end edit ---
Thanks a lot for your reply, Daniel.
It would be great if Mathematica had a Gaussian elimination function, as well as a Gauss-Jordan elimination function. Maple has both of these options.
Gaussian elimination stops once the matrix is in row echelon form(back substitution can be used to solve for variables, if necessary), Gauss-Jordan continues until the matrix is in reduced row
echelon form.
But why might such a function be useful? It's not a unique form and moreover I'm not sure what one might do with the result. The RREF, by contrast, can be (and, internally, is) used for finding
solutions to linear equations, null space generators, inverses, and maybe more.
I guess what I am asking is in what way does the non-reduced echelon form offer any advantage over an LU decomposition?
Sometimes it is not necessary to have a matrix in RREF, only in REF. So if I am doing a matrix reduction by hand and stop at REF, then I would like Mathematica to be able to also stop at REF, so that
I can check my calculations.
In what instances would I stop at REF when doing calculations by hand? 1. Sometimes it is faster for me to use back substitution(to get the solution) than to get a matrix to RREF. 2. Another example
is for finding the basis for a column space or row space . 3. To deduce whether a system of equations has 0, 1 or infinitely many solutions
I want to use Mathematica to check calculations that I've done by hand(I don't want Mathematica give me a reduced answer). But if I stop at REF, and Mathematica stops at RREF, then I can't use it to
check whether my row reductions are correct(without me doing further unnecessary calculations). Much of my coursework involves matrices in REF. Maple has the option to get a matrix to REF and/or
RREF, and I found this feature very useful.
As a new user of Mathematica, the solution in your first reply, while it did generate a REF, it is not a trivial solution for a beginner . I don't even know what "LU decomposition" is. If your
solution can be packaged as a clickable function in Mathematica, then that would be much easier and quicker to implement.
Anyway, that's all the time I want to devote to this topic. I think that I can probably explain myself a bit better, but I have neither the time or the inclination. Thanks for your replies. :)
I realize this is an old thread, but I agree with D P. I am learning linear algebra and would like to just see the matrix in Upper triangular or REF form -- not in the complete row-reduced (RREF)
form. Many textbooks also make this distinction (such as Strang's textbook).
One can obtain an upper triangular form from the LU decomposition.
I would like for a row echelon function to be implemented.
Be respectful. Review our Community Guidelines to understand your role and responsibilities. Community Terms of Use | {"url":"https://community.wolfram.com/groups/-/m/t/475750?sortMsg=Replies","timestamp":"2024-11-06T11:57:31Z","content_type":"text/html","content_length":"126361","record_id":"<urn:uuid:01464150-c8c0-418c-8c23-7dc674f7d0ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00648.warc.gz"} |
[ASK] Equation of a Circle in the First Quadrant
• MHB
• Thread starter Monoxdifly
• Start date
In summary, the center of circle L lies on the line y = 2x in the first quadrant and is located at (3, 6). The equation of circle L is x^2 + y^2 - 6x - 12y + 36 = 0. This is determined by the fact
that the y-axis is tangent to the circle at (0, 6) and the center of the circle is at (x, 2x).
The center of circle L is located in the first quadrant and lays on the line y = 2x. If the circle L touches the Y-axis at (0,6), the equation of circle L is ...
a. \(\displaystyle x^2+y^2-3x-6y=0\)
b. \(\displaystyle x^2+y^2-12x-6y=0\)
c. \(\displaystyle x^2+y^2+6x+12y-108=0\)
d. \(\displaystyle x^2+y^2+12x+6y-72=0\)
e. \(\displaystyle x^2+y^2-6x-12y+36=0\)
Since the center (a, b) lays in the line y = 2x then b = 2a.
\(\displaystyle (x-a)^2+(y-b)^2=r^2\)
\(\displaystyle (0-a)^2+(6-b)^2=r^2\)
\(\displaystyle (-a)^2+(6-2a)^2=r^2\)
\(\displaystyle a^2+36-24a+4a^2=r^2\)
\(\displaystyle 5a^2-24a+36=r^2\)
What should I do after this?
circle center at $(x,2x)$
circle tangent to the y-axis at $(0,6) \implies x=3$
$(x-3)^2+(y-6)^2 = 3^2$
$x^2-6x+9+y^2-12y+36 =9$
skeeter said:
circle center at $(x,2x)$
circle tangent to the y-axis at $(0,6) \implies x=3$
How did you get x = 3 from (0, 6)?
Monoxdifly said:
How did you get x = 3 from (0, 6)?
The fact that the y-axis is
to the circle at (0, 6) means that the line y= 6 is a radius so the y coordinate of the center of the circle is 6. And since the center is at (x, 2x), y= 2x= 6 so x= 3.
(And the center of the circle "
on the line y= 2x", not "lays".)
HallsofIvy said:
The fact that the y-axis is tangent to the circle at (0, 6) means that the line y= 6 is a radius so the y coordinate of the center of the circle is 6. And since the center is at (x, 2x), y= 2x= 6
so x= 3.
(And the center of the circle "lies on the line y= 2x", not "lays".)
Well, I admit I kinda suck at English. I usually use "lies" as "deceives" and "lays" as "is located". Thanks for the help, anyway. Your explanation is easy to understand.
FAQ: [ASK] Equation of a Circle in the First Quadrant
What is the equation of a circle in the first quadrant?
The equation of a circle in the first quadrant is (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center of the circle and r is the radius.
How do you find the center of a circle in the first quadrant?
To find the center of a circle in the first quadrant, you can use the formula (h,k) = (a,b), where a and b are the coordinates of the center point.
How do you find the radius of a circle in the first quadrant?
The radius of a circle in the first quadrant can be found by taking the square root of the right side of the equation (x - h)^2 + (y - k)^2 = r^2.
Can the equation of a circle in the first quadrant have negative values for h and k?
No, the center of a circle in the first quadrant must have positive values for h and k, as it is located in the first quadrant where all values are positive.
How do you graph a circle in the first quadrant?
To graph a circle in the first quadrant, plot the center point (h,k) and then use the radius r to plot points around the center point. Connect these points to create a circle. | {"url":"https://www.physicsforums.com/threads/ask-equation-of-a-circle-in-the-first-quadrant.1042190/","timestamp":"2024-11-10T15:08:52Z","content_type":"text/html","content_length":"95986","record_id":"<urn:uuid:e600e0b1-e075-4ec7-aff1-e1ae0d2399bb>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00265.warc.gz"} |
Optimal modularity and memory capacity of neural reservoirs
The neural network is a powerful computing framework that has been exploited by biological evolution and by humans for solving diverse problems. Although the computational capabilities of neural
networks are determined by their structure, the current understanding of the relationships between a neural network’s architecture and function is still primitive. Here we reveal that a neural
network’s modular architecture plays a vital role in determining the neural dynamics and memory performance of the network of threshold neurons. In particular, we demonstrate that there exists an
optimal modularity for memory performance, where a balance between local cohesion and global connectivity is established, allowing optimally modular networks to remember longer. Our results suggest
that insights from dynamical analysis of neural networks and information-spreading processes can be leveraged to better design neural networks and may shed light on the brain’s modular organization.
Author Summary
Understanding the inner workings of the human brain is one of the greatest scientific challenges. It will not only advance the science of the human mind, but also help us build more intelligent
machines. In doing so, it is crucial to understand how the structural organization of the brain affects functional capabilities. Here we reveal a strong connection between the modularity of a neural
network and its performance in memory tasks. Namely, we demonstrate that there is optimal modularity for memory performance. Our results suggest a design principle for artificial recurrent neural
networks as well as a hypothesis that may explain not only the existence but also the strength of modularity in the brain.
Neural networks are the computing engines behind many living organisms. They are also prominent general-purpose frameworks for machine learning and artificial intelligence applications (LeCun,
Bengio, & Hinton, 2015). The behavior of a neural network is determined by the dynamics of individual neurons, the topology and strength of individual connections, and large-scale architecture. In
both biological and artificial neural networks, neurons integrate input signals and produce a graded or threshold-like response. While individual connections are dynamically trained and adapted to
the specific environment, the architecture primes the network for performing specific types of tasks. The architecture of neural networks varies from organism to organism and between brain regions
and is vital for functionality. The orientation columns of the visual cortex that support low-level visual processing (Hubel & Wiesel, 1972) or the looped structure of hippocampus that consolidates
memory (Otmakhova, Duzel, Deutch, & Lisman, 2013) are two examples. In machine learning, feed-forward convolutional architectures have achieved superhuman visual recognition capabilities (Ioffe &
Szegedy, 2015; LeCun et al., 2015), while recurrent architectures exhibit impressive natural language processing and control capabilities (Schmidhuber, 2015).
Yet, identifying systematic design principles for neural architecture is still an outstanding question (Legenstein & Maass, 2005; Sussillo & Barak, 2013). Here, we investigate the role of modular
architectures on memory capacity of neural networks, where we define modules (communities) as groups of nodes that have stronger internal versus external connectivity (Girvan & Newman, 2002).
We focus on modularity primarily because of the prevalence of modular architectures in the brain. Modularity can be observed across all scales in the brain and is considered a key organizing
principle for functional division of brain regions (Bullmore & Sporns, 2009) and brain dynamics (Kaiser & Hilgetag, 2010; Moretti & Muñoz, 2013; Müller-Linow, Hilgetag, & Hütt, 2008; Villegas,
Moretti, & Muñoz, 2015; Wang, Hilgetag, & Zhou, 2011), and is also considered as a plausible mechanism for working memory through ensemble-based coding schemes (Boerlin & Denève, 2011), bistability
(Constantinidis & Klingberg, 2016; Cossart, Aronov, & Yuste, 2003; Klinshov, Teramae, Nekorkin, & Fukai, 2014), gating (Gisiger & Boukadoum, 2011), and through metastable states that retain
information (Johnson, Marro, & Torres, 2013).
Here we study the role of modularity based on the theories of information diffusion, which can inform how structural properties affect spreading processes on a network (Mišić et al., 2015). Spreading
processes can include diseases, social fads, memes, random walks, or the spiking events transmitted by biological neurons (Boccaletti, Latora, Moreno, Chavez, & Hwang, 2006; Newman, 2003;
Pastor-Satorras, Castellano, Van Mieghem, & Vespignani, 2015), and they are studied in the context of large-scale network properties like small-worldness, scale-freeness, core periphery structure,
and community structure (modularity; Boccaletti et al., 2006; Newman, 2003; Strogatz, 2001).
Communities’ main role in information spreading is restricting information flow (Chung, Baek, Kim, Ha, & Jeong, 2014; Onnela et al., 2007). However, recent work showed that communities may play a
more nuanced role in complex contagions, which require reinforcement from multiple local adoptions. It turns out that under certain conditions community structure can facilitate spread of complex
contagions, mainly by enhancing initial local spreading. As a result, there is an optimal modularity at which both local and global spreading can occur (Nematzadeh, Ferrara, Flammini, & Ahn, 2014).
In the context of neural dynamics, this result suggests that communities could offer a way to balance and arbitrate local and global communication and computation. We hypothesize that an ideal
computing capacity emerges near the intersection between local cohesion and global connectivity, analogous to the optimal modularity for information diffusion.
We test whether this can be true in reservoir computers. Reservoir computers are biologically plausible models for brain computation (Enel, Procyk, Quilodran, & Dominey, 2016; Soriano, Brunner,
Escalona-Moran, Mirasso, & Fischer, 2015; Yamazaki & Tanaka, 2007) as well as a successful machine learning paradigm (Lukoševičius & Jaeger, 2009). They have emerged as an alternative to the
traditional recurrent neural network (RNN) paradigm (Jaeger & Hass, 2004; Maass, Natschlager, & Markram, 2002).
Instead of training all the connection parameters as in RNNs, reservoir computers train only a small number of readout parameters. Reservoir computers use the implicit computational capacities of a
neural reservoir—a network of model neurons. Compared with other frameworks that require training numerous parameters, this paradigm allows for larger networks and better parameter scaling. Reservoir
computers have been successful in a range of tasks including time series prediction, natural language processing, and pattern generation, and have also been used as biologically plausible models for
neural computation (Deng, Mao, & Chen, 2016; Enel et al., 2016; Holzmann & Hauser, 2010; Jaeger, 2012; Jalalvand, De Neve, Van de Walle, & Martens, 2016; Rössert, Dean, & Porrill, 2015; Soriano et
al., 2015; Souahlia, Belatreche, Benyettou, & Curran, 2016; Triefenbach, Jalalvand, Schrauwen, & Martens, 2010; Yamazaki & Tanaka, 2007).
Reservoir computers operate by taking an input signal(s) into a high-dimensional reservoir state space where signals are mixed. We use echo state networks (ESN)—a popular implementation of reservoir
computing—where the reservoir is a collection of randomly connected neurons and the inputs are continuous or binary signals that are injected into a random subset of those neurons through randomly
weighted connections. The reservoir’s output is read via a layer of read-out neurons that receive connections from all neurons in the reservoir. They have no input back into the reservoir and they
act as the system’s output on tasks.
The reservoir weights and input weights are generally drawn from a given probability distribution and remain unchanged, while the readout weights that connect the reservoir and readouts are trained
(see Figure 1A). Readout neurons can be considered as “tuning knobs” into the desired set of nonlinear computations that are being performed within the reservoir. Therefore, the ability of a
reservoir computer to learn a particular behavior depends on the richness of the dynamical repertoire of the reservoir (Lukoševičius & Jaeger, 2009; Pascanu & Jaeger, 2011).
Many attempts have been made to calibrate reservoirs for particular tasks. In echo state networks this usually entails the adjustment of the spectral radius (largest eigenvalue of the reservoir
weight matrix), the input and reservoir weight scales, and reservoir size (Farkas, Bosak, & Gergel, 2016; Jaeger, 2002; Pascanu & Jaeger, 2011; Rodan & Tio, 2011). In memory tasks, performance peaks
sharply around a critical point for the spectral radius, whereby the neural network resides within a dynamical regime with long transients and “echos” of previous inputs reverberating through the
states of the neurons preserving past information (Pascanu & Jaeger, 2011; Verstraeten, Schrauwen, D’Haene, & Stroobandt, 2007). Weight distribution has also been found to play an important role in
performance (Ju, Xu, Chong, & VanDongen, 2013), and the effects of reservoir topology have been studied using small-world (Deng & Zhang, 2007), scale-free (Deng & Zhang, 2007), columnar (Ju et al.,
2013; Li, Zhong, Xue, & Zhang, 2015; Maass et al., 2002; Verstraeten et al., 2007), Kronecker graphs (Leskovec, Chakrabarti, Kleinberg, Faloutsos, & Ghahramani, 2010; Rad, Jalili, & Hasler, 2008),
and ensembles with lateral inhibition (Xue, Yang, & Haykin, 2007), each showing improvements in performance over simple random graphs.
Echo state networks provide a compelling substrate for investigating the relationship between community structure, information diffusion, and memory. They can be biologically realistic and are simple
to train; the separation between the reservoir and the trained readouts means that the training process does not interfere in the structure of the reservoir itself (see the Supporting Information,
Table S1; Rodriguez, Izquierdo, & Ahn, 2019).
Here, we take a principled approach based on the theory of network structure and information diffusion to test a hypothesis that the best memory performance emerges when a neural reservoir is at the
optimal modularity for information diffusion, where local and global communication can be easily balanced (see the Supporting Information, Figure S1; Rodriguez et al., 2019). We implement neural
reservoirs with different levels of community structure (see Figure 1A) by fixing the total number of links and communities while adjusting a mixing parameter μ that controls the fraction of links
between communities. Control of this parameter lets us explore how community structure plays a role in performance on two memory tasks (see Figure 1B). Three simulations are performed. The first
tests for the presence of the optimal modularity phenomena in the ESNs. The second uses the same ESNs to perform a memory capacity task to determine the relationship between the optimal modularity
phenomena and task performance. Lastly, we investigate the relationship between community structure and the capacity of the ESN to recall unique patterns in a memorization task.
For the tasks we use a threshold-like activation function (see Figure 1C), which is a more biologically plausible alternative to the tanh or linear neurons often used in artificial neural networks.
The key distinction between the threshold-like activation function and tanh activation functions is that threshold-like functions only excite postsynaptic neurons if enough presynaptic neurons
activate in unison. On the other hand, postsynaptic tanh neurons will always activate in proportion to presynaptic neurons, no matter how weak those activations are.
Optimal Modularity in Reservoir Dynamics
We first test whether the optimal modularity phenomenon found in the linear threshold model can be generalized to neural reservoirs by running two simulations. Nodes governed by the linear threshold
model remain active once turned on, and are not good units for computing. Instead we use a step-like activation function (see Figure 1C). First, we assume a simple two-community configuration as in
the original study (Nematzadeh et al., 2014; see Figure 2A), where the fraction of bridges μ controls the strength of community structure in the network. When μ = 0, the communities are maximally
strong and disconnected, and when μ ≈ 0.5 the community structure vanishes. The average degree and the total number of edges remain constant as μ is varied. An input signal is injected into a random
fraction of the neurons (r[sig]) in a seed community and the activity response of each community is measured. The results confirm the generalizability of the optimal modularity phenomenon for neural
At low μ, strong local cohesion activates the seed community, while the neighboring community remains inactive as there are too few bridges (see Figure 2B). At high μ there are enough bridges to
transmit information globally but not enough internal connections to foster local spreading, resulting in a weak response. An optimal region emerges where local cohesion and global connectivity are
balanced, maximizing the response of the whole network, as was demonstrated in Nematzadeh et al. (2014) for linear threshold models. The fraction of neurons that receive input (r[sig]) modulates the
behavior of the communities. The phase diagram in Figure 2C shows how the system can switch from being inactive at low r[sig], to a single active community, to full network activation as the fraction
of activated neurons increases. The sharpness of this transition means the community behaves like a threshold-like function as well. Though we control r[sig] as a static parameter in this model, it
can represent the fraction of active neural pathways between communities, which may vary over time. Communities could switch between these inactive and active states in response to stimuli based on
their activation threshold, allowing them to behave as information gates.
Our second study uses a more general setting, a reservoir with many communities similar to ones that might be used in an ESN or observed in the brain (see Figure 2D). The previous study examined
input into only a single community; here we extend that to many communities. In Figure 2E we record the response of a 50-community network that receives a signal that is randomly distributed across
the whole network. The result shows that even when there is no designated seed community, similar optimal modularity behavior arises. At low μ the input signal cannot be reinforced because of the
lack of bridges, and is unable to excite even the highly cohesive communities. At high μ the many global bridges help to consolidate the signal, but there is not enough local cohesion to continue to
facilitate a strong response. In the optimal region there is a balance between the amplifying effect of the communities and the global communication of the bridges that enables the network to take a
subthreshold, globally distributed signal and spread it throughout the network. In linear and tanh reservoirs, no such relationship is found (see the Supporting Information, Figure S2 and Figure S3;
Rodriguez et al., 2019); instead communities behave in a more intuitive fashion, restricting information flow.
Optimal Modularity in a Memory Capacity Task
We test whether optimal modularity provides a benefit to the ESN’s memory performance by a common memory benchmark task developed by Jaeger (2002; see Figure 3A). The task involves feeding a stream
of random inputs into the reservoir and training readout neurons to replay the stream at various time lags. The coefficient of determination between the binomially distributed input signal and a
delayed output signal for each delay parameter is used to quantify the performance of the ESN. The memory capacity (MC) of the network is the sum of these performances over all time lags as shown by
the shaded region in Figure 3B.
Reservoirs with strong community structure (low μ) exhibit the poorest performance; the reservoirs are ensembles of effectively disconnected reservoirs, with little to no intercommunity
communication. Performance improves substantially with μ as the fraction of global bridges grows, facilitating intercommunity communication. A turnover point is reached beyond which replacing
connections with bridges compromises local cohesion. After a certain point, larger μ leads to performance loss. The region of elevated performance corresponds to the same region of optimal modularity
on a reservoir with the same properties and inputs as those used in the task (see the Supporting Information, Figure S4; Rodriguez et al., 2019).
We also examine the impact of input signal strength. In Figure 3C we show that this optimal region of performance holds over a wide range of r[sig], and that there is a narrow band near r[sig] ≈ 0.3
where the highest performance is achieved around μ ≈ 0.2. As expected, we also see a region of optimal r[sig] for reservoirs, because either under- or overstimulation is disadvantageous. Yet, the
added benefit of community structure is due to more than just the amplification of the signal. If communities were only amplifying the input signal, then increasing r[sig] in random graphs should
give the same performance as that found in the optimal region, but this is not the case. Figure 3C shows that random graphs are unable to meet the performance gains provided near optimal μ regardless
of r[sig]. Additionally, this optimal region remains even if we control for changes in the spectral radius of the reservoir’s adjacency matrix, which is known to play an important role in ESN memory
capacity for linear and tanh systems (Farkas et al., 2016; Jaeger, 2002; Verstraeten et al., 2007; see the Supporting Information, Figures S5–S7; Rodriguez et al., 2019). In such systems modularity
reduces memory capacity, as communities create an information bottleneck (see the Supporting Information, Figures S8–S9; Rodriguez et al., 2019). However, weight scale still plays a larger role in
determining the level of performance for ESNs in our simulations (see the Supporting Information, Figure S5; Rodriguez et al., 2019). There is also a performance difference between the increasingly
nonlinear activation functions, with linear performing best, and tanh and sigmoid performing worse, illustrating a previously established trade-off between memory and nonlinearity (Dambre,
Verstraeten, Schrauwen, & Massar, 2012; Verstraeten, Dambre, Dutoit, & Schrauwen, 2010; Verstraeten et al., 2007). Lastly, ESN performance has been attributed to reservoir sparsity in the past
(Jaeger & Hass, 2004; Lukoševičius, 2012), however as node degree, average node strength, and total number of edges remain constant as μ changes such effects are controlled for.
Optimal Modularity in a Recall Task
We employ another common memory task that estimates a different feature of memory: the number of unique patterns that can be learned. This requires a rich attractor space that can express and
maintain many unique sequences. From here out we consider an attractor to be a basin of state (and input) configurations that lead to the same fixed point in the reservoir state space. In this task,
a sequence of randomly generated 0s and 1s are fed to the network as shown in Figure 4A. For the simulation, we use sets of 4 × 5 dimensional binary sequences as input. The readouts should then learn
to recall the original sequence after an arbitrarily long delay ΔT and the presentation of a recall cue of 1 (for one time step) through a separate input channel.
By varying μ we can show how recall performance changes with community structure. Figure 4B, top, shows the average performance measured by the fraction of perfectly recalled sequences, for a set of
200 sequences. Well-performing reservoirs are able to store the sequences in attractors for arbitrarily long times. Similar to the memory capacity task, we see the poorest performance for random
networks and networks with low μ. There is a sharp spike in performance near μ ≈ 0.1. The average performance over the number of sequences (when ΔT = 80) show that optimal performance at μ starts to
drop off after ≈ 230 sequences (Figure 4B, bottom).
We investigate the discrepancy in performance between modular and nonmodular networks by examining the reservoir attractor space. We measure the number of unique available attractors that the
reservoirs would be exposed to by initializing the reservoirs at initial conditions associated with the sequences we use. We find a skewed response from the network as shown in Figure 4C where the
number of available attractors is maximized when μ > 0. Many of these additional attractors between 0.0 < μ < 0.2 are limit cycles that result from the interaction between the communities in the
The attractor space provides insights about the optimal region. At higher μ the whole reservoir behaves as a single system, leaving very few attractors for the network to utilize for information
storage. The reservoir has to rely on short-lived transients for storage. With extremely modular structure (μ ≈ 0), reservoirs have the most available attractors, but they are not readily
discriminated by the linear readouts. Surprisingly, these attractors are more readily teased apart as communities become more interconnected. However, there is a clear trade-off, as too much
interconnection folds all the initial conditions into a few large attractor basins.
Biological neural networks are often modeled using neurons with threshold-like behavior, such as integrate-and-fire neurons, the Grossberg-Cohen model, or Hopfield networks. Reservoirs of
threshold-like neurons, like those presented here, provide a simple model for investigating the computational capabilities of biological neural networks. By adopting and systematically varying
topological characteristics akin to those found in brain networks, such as modularity, and subjecting those networks to tasks, we can gain insight into the functional advantages provided by these
We have demonstrated that ESNs exhibit optimal modularity in the context of both signal spreading and memory capacity, and they are closely linked to the optimal modularity for information spreading.
Through dynamical analysis we found that balancing local and global cohesion enabled modular reservoirs to spread information across the network and consolidate distributed signals, although
alternative mechanisms may also be in play, such as cycle properties (Garcia, Lesne, Hilgetag, & Hütt, 2014). We then showed that such optimal regions coincide with the optimal community strength
that exhibit the best memory performance. Both the memory capacity and recall task benefited by adopting modular structures over random networks, despite performing in different dynamical regimes
(equilibrium versus nonequilibrium).
A key component of our hypothesis is the adoption of a threshold-like (or step-like) activation function for our ESNs, which is a more biologically plausible alternative to the tanh or linear neurons
often used in artificial neural networks. The optimal modularity phenomenon emerges only for neural networks of threshold-like neurons and does not exist for neural networks of linear or tanh neurons
(i.e., simple contagions) used in traditional ESNs, and so many developed intuitions about ESN dynamics and performance may not readily map to ESNs driven by complex contagions like the ones here.
Indeed, the relationship between network topology and performance is known to vary with the activation function, with threshold-like or spiking neurons (common in liquid state machines; Maass et al.,
2002) being more heavily dependent on topology (Bertschinger & Natschläger, 2004; Haeusler & Maass, 2007; Schrauwen, Buesing, & Legenstein, 2009). Because the effects of modularity vary depending
upon the activation function, a suitable information diffusion analysis should be chosen to explore the impact of network topology for a given type of spreading process. Moreover, because the
benefits of modularity are specific to threshold-like neurons, distinct network design principles are needed for biological neural networks and the artificial neural networks used in machine
learning. Additionally, as we have seen that the choice of architecture can have a profound impact on the dynamical properties that can emerge from the neural network, there may be value in applying
these insights to the architectural design of recurrent neural networks in machine learning, where all weights in the network undergo training but where architecture is usually fixed.
While weight scale remains the most important feature of the system in determining performance, our results suggest significant computational benefits of community structure, and contributes to
understanding the role it plays in biological neural networks (Bullmore & Sporns, 2009; Buxhoeveden & Casanova, 2002; Constantinidis & Klingberg, 2016; Hagmann et al., 2008; Hilgetag, Burns, O’Neill,
Scannell, & Young, 2000; Meunier, Lambiotte, & Bullmore, 2010; Shimono & Beggs, 2015; Sporns, Chialvo, Kaiser, & Hilgetag, 2004), which are also driven by complex contagions and possess modular
topologies. The dynamical principles of information spreading mark trade-offs in the permeability of information on the network that can promote or hinder performance. While this analysis provides us
some insight, it remains an open question as to whether our results can be generalized to the context of more realistic biological neural networks where spike-timing-dependent plasticity and
neuromodulation play a key role in determining the network’s dynamical and topological characteristics.
In addition to the optimal region and the ability of communities to foster information spreading and improved performance among threshold-like neurons, modularity may play other important roles. For
instance, it offers a way to compartmentalize advances and make them robust to noise (e.g., the watchmaker’s parable; Simon, 1997). Modularity also appears to confer advantages to neural networks in
changing environments (Kashtan & Alon, 2005), under wiring cost constraints (Clune, Mouret, & Lipson, 2013), when learning new skills (Ellefsen, Mouret, & Clune, 2015), and under random failures
(Kaiser & Hilgetag, 2004). These suggest additional avenues for exploring the computational benefits of modular reservoirs and neural networks. And it is still an open question how community
structure affects performance on other tasks like signal processing, prediction, or system modeling.
Neural reservoirs have generally been considered “black-boxes,” yet through combining dynamical, informational, and computational studies it maybe possible to build a taxonomy of the functional
implications of topological features for both artificial and biological neural networks. Dynamical and performative analysis of neural networks can afford valuable insights into their computational
capabilities as we have seen here.
Our ESN architecture with community structure is shown in
Figure 1A
. The inputs are denoted as
), which is a
-dimensional vector. Each dimension of input is connected to a random subset of neurons in the reservoir.
) is the
-dimensional state vector of the reservoir, where
is the number of reservoir neurons.
) represents the states of the
readout neurons. The
inputs are connected by an
to the
neurons. The network structure of the reservoir is represented by an
weight matrix
, and the output weights are represented by an
. The reservoirs follow the standard ESN dynamics without feedback or time constants:
is the reservoir activation function,
is the readout activation function, and [
] denotes the concatenation of two vectors. Often
is chosen to be a sigmoid-like function such as tanh, while
is often taken to be linear (Lukoševičius & Jaeger,
). However in our case we use a general sigmoid function:
with parameters
= 1,
= 1,
= 1,
= 10, and
= 0 giving a nonlinear threshold-like activation function, making it step-like in shape and a
complex contagion
like other neuron models (e.g., integrate-and-fire, Hopfield, or Wilson-Cowan models). For the readout neurons,
is chosen to be a step function:
Linear regression is used to solve for
is an
matrix of target outputs over a time course
, and
is the pseudoinverse of the history of the reservoir state vector (where
∈ ℝ
; Lukoševičius & Jaeger,
). To generate the reservoirs we use the LFR benchmark model (Lancichinetti, Fortunato, & Radicchi,
), which can generate random graphs with a variety of community structures. The LFR benchmark model uses a configuration model to generate random graphs. The configuration model works by imposing a
degree sequence to the nodes and randomly wiring the edge “stubs” (Newman,
). The LFR model extends this by including community assignment and rewiring steps to constrain the fraction of bridges in the network. Because of its relationship with the configuration model, LFR
graphs exhibit low average shortest path length and low average clustering coefficient in contrast to the Wattz-Strogatz models that have low average shortest path length and high clustering. For
small graphs like the ones we use for building reservoirs, the average shortest path length increases monotonically with decreasing
. This is due to the sparseness of directed links between communities. As
approaches 0 the communities become disconnected. In our case we vary the fraction of bridges (
) in the network while holding the degree distribution and total number of edges the same, controlling for the density of connections in the network. Weights for the network are drawn separately from
a uniform distribution and described in following sections. Code for all the simulations and tasks is available online (Rodriguez,
Reservoir Dynamics
We used reservoirs with N = 500 nodes, with every node having a degree of 6. Reservoir states were initialized with a zero vector, x(0) = {0, …, 0}. The first experiment uses a two-community cluster
of 250 nodes each, matching the scenario from Nematzadeh et al. (2014). Input was injected into r[sig] fraction of neurons into the seed community. The input signal lasted for the duration of the
task until the system reached equilibrium at time t[e]. The final activation values of the neurons were summed within each community and used to calculate the fractional activation of the network for
each community shown in Figure 2B, where the mean over 48 reservoir realizations is shown. All activations were summed and divided by the size of the network to give the total fractional activation 1
/N$∑i=1N$x[i](t[e]) as shown in Figure 2C.
In the following experiment, a reservoir of the same size but with 50 communities with 10 nodes each was used. This time, however, the input signal was not limited to a single community but applied
randomly to nodes across the network. Again the signal was active for the full duration of the task until the system reached equilibrium when the final activation values of the neurons were summed
within each community. Figure 2E shows the activation for each community averaged over 48 reservoir realizations, and the total fractional activity in the network is then shown in Figure 2F.
Different measures for information spreading produce similar results. Also, optimal spreading can be observed in the transitory dynamics of the system, such as in networks that receive short input
bursts and return to an inactive equilibrium state. Optimality for step-like activations has been shown to emerge regardless of community or network size using message-passing approximations
(Nematzadeh, Rodriguez, Flammini, & Ahn, 2018). For many-community cases with distributed input, optimality existence in infinite networks depends upon community variation (e.g., size, edge density,
number of inputs).
Memory Capacity task
The memory capacity task involves the input of a random sequence of numbers that the readout neurons are then trained on at various lags (see
Figure 3
). There is just one input dimension and values of 0 and 1 are input into a fraction of the reservoir’s neurons
. For each time lag there is a set of readout neurons that are trained independently to remember the input at the given time lag. The readout neurons that maximize the coefficient of determination
(or the square of the correlation coefficient) between the input signal and lagged output are used as the
th delayed short-term memory capacity of the network MC
. The MC of the ESN becomes the sum over all delays:
We operationalize this sum as the memory capacity of the network. Unlike Jaeger’s task, we input a binomial distribution of 1s and 0s rather than continuous values (see
Figure 3A
). We try to keep the network small enough and sparse enough to reduce computational load, while still being large enough to solve the task. A reservoir of
= 500 nodes and 50 communities of size 10 were used. Every node has a degree of 6. The degree was chosen to be sparse enough to help reduce computing time, while high enough to support a wide range
of modularities, which are partly constrained by degree. Reservoir parameters were not fitted to the task, rather a grid search was executed to find parameter sets that performed well, as the focus
of the experiment is not to break records on memory performance, but rather to see how it changes with modularity. Among the parameters adjusted were the upper and lower bounds of the weight
distribution and the weight scale (
), which adjusts the strengths of all the reservoir weights by a scalar value. Performance over the full range of
values was evaluated at each point on the grid. Well-performing reservoirs were found with weights between −0.2 and 1 and with a weight scale parameter of
= 1.13. The same was done for the input weight matrix, where
also varies from −0.2 to 1 with an input gain of
= 1.0. Many viable parameters existed throughout the space that exhibit optimality. This is partly due to parameter coupling, where changing multiple parameters results in the same dynamics.
Each reservoir’s readouts were trained over a 1,500-step sequence following the first 500 steps that are removed to allow initial transients to die out. Once trained, a new validation sequence of the
same length is used to evaluate the performance of the ESN. Results averaged over 64 reservoir samples are shown in Figures 3B and 3C. We also show the contour over r[sig], which is an important
parameter in determining the performance of the reservoir. Performance peaks between r[sig] = 0.3 and r[sig] = 0.4 at a μ ≈ 0.25.
Recall Task
The recall task is a simplified version of the memory task developed by Jaeger (Jaeger, 2012). A pattern of 0s and 1s is input into the network, which must recall that pattern after a distractor
period. The ESN is trained on the whole set of unique sequences and the performance of the ESN is determined from its final output during the recall period, which occurs after the distractor period.
We do this to estimate the total number of sequences that an ESN can remember. So unlike the memory capacity task that estimates memory duration given an arbitrary input sequence, the recall task
quantifies the number of distinct signals an ESN can differentiate. This involves training an ESN on a set of sequences and then having it recall the sequences perfectly after a time delay ΔT. The
input is a random 4 × 5 binary set of 0s and 1s. At a single time step just one of the four input dimensions are active. This is in order to maintain the same level of external excitation per time
step, as we are not testing the network’s dynamic range. The reservoir is initialized to a zero vector and provided with a random sequence. Following the delay period, a binary cue with value 1.0 is
presented via a fifth input dimension. After this cue, the reservoir’s readout neurons must reproduce the input sequence. The readout weights are trained on this sequence set. Figures 4B shows the
average performance over 48 reservoir samples. Many networks around the optimal μ value can retain the information for arbitrarily long times, as the task involves storing the information in a unique
attractor. Figures 4B shows the average performance when ΔT = 80 as we vary the number of sequences. In Figures 4C we determine the average number of available attractors given inputs drawn from the
full set of 4 × 5 binary sequences where only one dimension of the input is active at a given time. For each of the 4 × 5 binary sequences, the system was run until it reached the cue time, where a
decision would be made by the readout layer. At this point converged trajectories would result in a failure to differentiate patterns. Two converged trajectories are determined to fall into the same
attractor if the Euclidean distance between the system’s states are smaller than a value ϵ = 0.1. The number of attractor states is the number of these unique groupings and was robust to changes in ϵ
. Parameters for the reservoir are chosen via a grid search, as before, to find reasonable performance from which to start our analysis. Here reservoirs of size N = 1,000 with node degree 7 and
community size 10 are used. A larger reservoir was necessary in order to attain high performance on the task. Similarly, the weight distribution parameters are included in the search and reasonable
performing reservoirs were found with weights drawn between −0.1 and 1.0 with W[s] = 1.0, r[sig] = 0.3, an input gain of W[I] = 2.0, and uniform input weights of 1.0.
Nathaniel Rodriguez: Conceptualization; Formal analysis; Methodology; Software; Validation; Visualization; Writing - Original Draft; Writing - Review & Editing. Eduardo Izquierdo: Conceptualization;
Methodology; Supervision; Writing - Original Draft; Writing - Review & Editing. Yong-Yeol Ahn: Conceptualization; Methodology; Supervision; Writing - Original Draft; Writing - Review & Editing.
We would like to thank John Beggs, Alessandro Flamini, Azadeh Nematzadeh, Pau Vilimelis Aceituno, Naoki Masuda, and Mikail Rubinov for helpful discussions and valuable feedback. This research was
supported in part by Lilly Endowment, Inc., through its support for the Indiana University Pervasive Technology Institute, and in part by the Indiana METACyt Initiative. The Indiana METACyt
Initiative at IU was also supported in part by Lilly Endowment, Inc. The Indiana University HPC infrastructure (Big Red II) helped make this research possible.
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Author notes
Competing Interests: The authors have declared that no competing interests exist.
Handling Editor: Alex Fornito
© 2019 Massachusetts Institute of Technology Published under a Creative Commons Attribution 4.0 International (CC BY 4.0) license
Massachusetts Institute of Technology Published under a Creative Commons Attribution 4.0 International (CC BY 4.0) license
This is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited. For a full description of the license, please visit | {"url":"https://direct.mit.edu/netn/article/3/2/551/2217/Optimal-modularity-and-memory-capacity-of-neural","timestamp":"2024-11-06T10:26:33Z","content_type":"text/html","content_length":"339916","record_id":"<urn:uuid:40df5513-7b89-4307-bab1-1701617eb487>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00625.warc.gz"} |
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Classical Algebraic Geometry & Modern Computer Algebra: Innovative Software Design and its Applications
Aim and Scope
Recent years have shown new applications of symbolic methods in computer algebra at scale. For instance, previously deemed impossible classification algorithms from toric and tropical geometry result
in terabyte sized data sets. This calls for the development of new algorithms, allowing –for example– for parallelization, but also new high level user interfaces. The aim is to bring together
experts in computational algebraic geometry and those who want to use newly provided methods and algorithms in their research endeavors. We encourage speakers to highlight the computational aspects
of their work and to augment their presentation with code examples. Our focus includes valuable application of these methods in physics, particularly in string theory and F-theory.
Accepted Talks
All talks in this session will take place on Thursday, July 25. The schedule is as follows:
Chebyshev varieties
Chiara Meroni
Chebyshev varieties are algebraic varieties parametrized by Chebyshev polynomials or their multivariate generalizations. They play the role of toric varieties in sparse polynomial root finding, when
monomials are replaced by Chebyshev polynomials. We will introduce these objects and discuss their main properties, including dimension, degree, singular locus and defining equations, as well as some
computational experiments.
Exceptional sequences of line bundles on toric varieties
Luca Remke
We want to investigate whether every maximal exceptional sequence of line bundles is full for a smooth projective toric variety, given the existence of a full exceptional sequence of line bundles.
This property has already been proved for toric varieties of Picard ranks 1 and 2. We therefore turn our attention to toric varieties of Picard rank 3, focusing on the del Pezzo surface of degree 7.
Localization in Gromov–Witten theory of toric varieties in a computer algebra system
Giosuè Muratore (Universidade de Lisboa)
Atiyah–Bott localization formula is a powerful tool for calculating the degree of equivariant classes of the moduli space of rational stable maps $\overline{M}_{0,n}(X,\beta)$, where $X$ denotes a
smooth toric variety, $n$ is a non negative integer, and $\beta$ is an effective $1$-cycle. Implementation of the formula entails intricate computational challenges, involving graph theory,
colorations, partitions, and other discrete objects. Furthermore, the computed solution is a large summation of rational numbers, underscoring the imperative nature of computational efficiency. This
formula has been applied in very specific cases for computing Gromov–Witten invariants, addressing enumerative problems, and determining the small quantum ring of $X$, among other applications.
A comprehensive implementation as a Julia package has been recently presented by the author.We show the features of the package with a particular emphasis to the noteworthy contribution of the
package \verb!Oscar.jl!. Finally, we delve into the fundamental prerequisites for extending the implementation to encompass Algebraic GKM Manifolds.
Andrew Turner
I will discuss an in-development computational tool for the analysis of singular elliptic fibrations called FTheoryTools. This tool seeks to automate many of the intricate calculations involved in
the analysis of F-theory models, and additionally to catalogue and make available many of the constructions that appear throughout the F-theory literature. FTheoryTools is a component of the
open-source computer algebra system OSCAR.
Computational analysis of logarithmic singularities
Simon Felten
In a smooth and proper family of algebraic varieties, the Hodge–de Rham spectral sequence degenerates at the first page, and the Hodge sheaves are locally free. Similarly, in logarithmic geometry, in
a proper, log smooth, and saturated family of log schemes, the Hodge–de Rham spectral sequence degenerates at the first page, and the Hodge sheaves are locally free. This is a partial generalization
of the classical theorem to degenerations. Filip, Ruddat, and the speaker have achieved a further generalization which applies to degenerations whose logarithmic singularities are toroidal. This
generalization has important applications in the construction of new varieties. In an ongoing long-term project, we apply the method to the construction of Fano manifolds in arbitrary dimension.
Unfortunately, in its current form, the method is not strong enough to give the desired result. To overcome the current challenges, we have to analyze more general classes of log singularities than
toroidal log singularities. Any log singularity which is useful in this context has the following three properties:
1. its local deformation theory is under sufficient control to construct globally consistent local deformations,
2. a proper family with such log singularities has constant log Betti numbers, and
3. the Hodge–de Rham spectral sequence of a proper family with such log singularities degenerates at the first page.
Candidates for such log singularities arise from explicit constructions of singular log schemes. An important approach to establish or refute (1) for an explicit log singularity is via studying the
cohomological properties of an explicit resolution of log singularities. To this end, we need to compute the derived direct images of coherent sheaves on the resolution such as the pieces of the log
de Rham complex. An important approach to (2) and (3) is to test for the constancy of the log Hodge numbers in explicitly given families with the relevant singularities. Here, we need to compute the
derived direct images of the pieces of the log de Rham complex of the family. In this talk, we discuss examples of these two analyses.
Algebraic sparse factor analysis
İrem Portakal
Factor analysis is a statistical technique that explains correlations among observed random variables with the help of a smaller number of unobserved factors. In traditional full factor analysis,
each observed variable is influenced by every factor. However, many applications exhibit interesting sparsity patterns i.e. each observed variable only depends on a subset of the factors. We study
such sparse factor analysis models from an algebro-geometric perspective. Under a mild condition on the sparsity pattern, we compute the dimension of the set of covariance matrices that corresponds
to a given model. Moreover, we study algebraic relations among the covariances in sparse two-factor models. In particular, we identify cases in which a Gröbner basis for these relations can be
derived via a 2-delightful term order and joins of toric edge ideals. The talk will be accompanied with our Macaulay2 code respecting FAIR principles. This is a joint-work with Mathias Drton, Alex
Grosdos and Nils Sturma.
The AI Mathematician
Yang-Hui He
We summarize how AI can approach mathematics in three ways: theorem-proving, conjecture formulation, and language processing. Inspired by initial experiments in geometry and string theory, we present
a number of recent experiments on how various standard machine-learning algorithms can help with pattern detection across disciplines ranging from algebraic geometry to representation theory, to
combinatorics, and to number theory. At the heart of the programme is the question how does AI help with mathematical discovery.
Moderated Discussion: “If I had the software to”
Within the OSCAR project we have another opportunity to reach out for new goals and challenge the boundaries of what’s currently accessible for computations in algebraic geometry. While the “engines”
of the system such as Singular and Polymake have been around for quite some time and are well tuned and performant, it is our believe that many scientific projects are postponed or laid aside because
of the difficulties to make such specialized software packages talk to each other, or because of the difficulties to run computations on a big scale, or… In this discussion we would like to encourage
all participants in the room to lay out some ideas about such loose ends in scientific projects and set the stage for envisioning joint directions for future developments of OSCAR and potential
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Numerical comparisons of exponential expressions: The saliency of the base component
Exponential expressions represent series that grow at a fast pace such as carbon pollution and the spread of disease. Despite their importance, people tend to struggle with these expressions. In two
experiments, participants chose the larger of two exponential expressions as quickly and accurately as possible. We manipulated the distance between the base/power components and their compatibility.
In base-power compatible pairs, both the base and power of one expression were larger than the other (e.g., 2^3 vs. 3^4), while in base-power incompatible pairs, the base of one expression was larger
than the base in the other expression but the relation between the power components of the two expressions was reversed (e.g., 3^2 vs. 2^4). Moreover, while in the first experiment the larger power
always led to the larger result, in the second experiment we introduced base-result congruent pairs as well. Namely, the larger base led to the larger result. Our results showed a base-power
compatibility effect, which was also larger for larger power distances (Experiments 1–2). Furthermore, participants processed the base-result congruent pairs faster and more accurately than the
power-result congruent pairs (Experiment 2). These findings suggest that while both the base and power components are processed when comparing exponential expressions, the base is more salient. This
exemplifies an incorrect processing of the syntax of exponential expressions, where the power typically has a larger mathematical contribution to the result of the expression.
• Base-power compatibility
• Exponential expressions
• Multi-digit numbers
• Numerical comparisons
Dive into the research topics of 'Numerical comparisons of exponential expressions: The saliency of the base component'. Together they form a unique fingerprint. | {"url":"https://cris.ariel.ac.il/en/publications/numerical-comparisons-of-exponential-expressions-the-saliency-of-","timestamp":"2024-11-13T02:07:10Z","content_type":"text/html","content_length":"56682","record_id":"<urn:uuid:640c2ef7-3273-483f-9022-3d6ba526ae2a>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00173.warc.gz"} |
Multibody Grouping from Motion Images
We want to deduce, from a sequence of noisy two-dimensional images of a scene of several rigid bodies moving independently in three dimensions, the number of bodies and the grouping of given feature
points in the images to the bodies. Prior processing is assumed to have identified features or points common to all frames and the images are assumed to be created by orthographic projection (i.e.,
perspective effects are minimal). We describe a computationally inexpensive algorithm that can determine which points or features belong to which rigid body using the fact that, with exact
observations in orthographic projection, points on a single body lie in a three or less dimensional linear manifold of frame space. If there are enough observations and independent motions, these
manifolds can be viewed as a set linearly independent, four or less dimensional subspaces. We show that the row echelon canonical form provides direct information on the grouping of points to these
subspaces. Treatment of the noise is the most difficult part of the problem. This paper uses a statistical approach to estimate the grouping of points to subspaces in the presence of noise by
computing which partition has the maximum likelihood. The input data is assumed to be contaminated with independent Gaussian noise. The algorithm can base its estimates on a user-supplied standard
deviation of the noise, or it can estimate the noise from the data. The algorithm can also be used to estimate the probability of a user-specified partition so that the hypothesis can be combined
with others using Bayesian statistics.
All Science Journal Classification (ASJC) codes
• Software
• Computer Vision and Pattern Recognition
• Artificial Intelligence
• Clustering
• Rigid body motion
• Vision
Dive into the research topics of 'Multibody Grouping from Motion Images'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/multibody-grouping-from-motion-images","timestamp":"2024-11-06T19:11:07Z","content_type":"text/html","content_length":"50121","record_id":"<urn:uuid:245e09a8-8f82-4afe-b53a-84115764f895>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00401.warc.gz"} |
A Joint Characterization of Belief Revision Rules
Dietrich, Franz and List, Christian and Bradley, Richard (2012): A Joint Characterization of Belief Revision Rules.
Preview Download (361kB) | Preview
This paper characterizes different belief revision rules in a unified framework: Bayesian revision upon learning some event, Jeffrey revision upon learning new probabilities of some events, Adams
revision upon learning some new conditional probabilities, and `dual-Jeffrey' revision upon learning an entire new conditional probability function. Though seemingly different, these revision rules
follow from the same two principles: responsiveness, which requires that revised beliefs be consistent with the learning experience, and conservativeness, which requires that those beliefs of the
agent on which the learning experience is `silent' (in a technical sense) do not change. So, the four revision rules apply the same revision policy, yet to different kinds of learning experience.
Item Type: MPRA Paper
Original Title: A Joint Characterization of Belief Revision Rules
Language: English
Keywords: Subjective probability, Bayes's rule, Jeffrey's rule, axiomatic foundations, unawareness
C - Mathematical and Quantitative Methods > C0 - General > C00 - General
Subjects: D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D83 - Search ; Learning ; Information and Knowledge ; Communication ; Belief ; Unawareness
D - Microeconomics > D8 - Information, Knowledge, and Uncertainty > D80 - General
D - Microeconomics > D0 - General > D00 - General
Item ID: 41240
Depositing Franz Dietrich
Date Deposited: 12 Sep 2012 12:50
Last Modified: 02 Oct 2019 17:20
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URI: https://mpra.ub.uni-muenchen.de/id/eprint/41240
Available Versions of this Item
• A Joint Characterization of Belief Revision Rules. (deposited 12 Sep 2012 12:50) [Currently Displayed] | {"url":"https://mpra.ub.uni-muenchen.de/41240/","timestamp":"2024-11-12T06:56:20Z","content_type":"application/xhtml+xml","content_length":"33821","record_id":"<urn:uuid:300a596b-ad7b-49e2-a323-5bf9b7aa7fdc>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00335.warc.gz"} |
TY - JOUR T1 - A Numerical Study of Blowup in the Harmonic Map Heat Flow Using the MMPDE Moving Mesh Method AU - Ronald D. Haynes, Weizhang Huang & Paul A. Zegeling JO - Numerical Mathematics:
Theory, Methods and Applications VL - 2 SP - 364 EP - 383 PY - 2013 DA - 2013/06 SN - 6 DO - http://doi.org/10.4208/nmtma.2013.1130nm UR - https://global-sci.org/intro/article_detail/nmtma/5909.html
KW - Heat flow, harmonic map, blowup, moving mesh method, finite difference. AB -
The numerical solution of the harmonic heat map flow problems with blowup in finite or infinite time is considered using an adaptive moving mesh method. A properly chosen monitor function is derived
so that the moving mesh method can be used to simulate blowup and produce accurate blowup profiles which agree with formal asymptotic analysis. Moreover, the moving mesh method has finite time blowup
when the underlying continuous problem does. In situations where the continuous problem has infinite time blowup, the moving mesh method exhibits finite time blowup with a blowup time tending to
infinity as the number of mesh points increases. The inadequacy of a uniform mesh solution is clearly demonstrated. | {"url":"https://global-sci.org/intro/article_detail/getRis?article_id=5909","timestamp":"2024-11-12T08:45:49Z","content_type":"text/html","content_length":"1499","record_id":"<urn:uuid:30f2c5bc-e8dc-4e38-9ab5-5efe82876335>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00545.warc.gz"} |
Thermal Performance Analysis of Cooling Tower Fill Materials
05 Oct 2024
Popularity: ⭐⭐⭐
Cooling Tower Fills Material Calculation
This calculator provides the calculation of heat transfer rate for cooling tower fills material.
Calculation Example: The heat transfer rate for cooling tower fills material is given by the formula Q = ρ * V * A * C * (T1 - T2) * η / 100, where ρ is the density of the air, V is the velocity of
the air through the tower, A is the surface area of the tower, C is the specific heat of air, T1 is the inlet temperature of the air, T2 is the outlet temperature of the air, and η is the efficiency
of the tower.
Related Questions
Q: What is the importance of heat transfer rate in cooling tower design?
A: The heat transfer rate is a crucial factor in cooling tower design as it determines the effectiveness of the tower in removing heat from the air.
Q: How does the efficiency of the tower affect the heat transfer rate?
A: The efficiency of the tower directly influences the heat transfer rate. A higher efficiency indicates that the tower is more effective in transferring heat from the air, resulting in a higher heat
transfer rate.
Symbol Name Unit
A Surface Area m^2
V Velocity m/s
ρ Density kg/m^3
η Efficiency %
C Specific Heat kJ/kg
T1 Inlet Temperature °C
T2 Outlet Temperature °C
Calculation Expression
Cooling Tower Heat Transfer Rate: The heat transfer rate is given by Q = ρ * V * A * C * (T1 - T2) * η / 100
ρ * V * A * C * (T1 - T2) * η / 100
Calculated values
Considering these as variable values: A=1000.0, ρ=1.2, C=1.005, V=1.0, η=80.0, T1=30.0, T2=20.0, the calculated value(s) are given in table below
Derived Variable Value
Cooling Tower Heat Transfer Rate 9600.0*C
Sensitivity Analysis Graphs
Cooling Tower Heat Transfer Rate: The heat transfer rate is given by Q = ρ * V * A * C * (T1 - T2) * η / 100
Impact of null on Cooling Tower Heat Transfer Rate
Impact of null on Cooling Tower Heat Transfer Rate
Impact of null on Cooling Tower Heat Transfer Rate
Impact of null on Cooling Tower Heat Transfer Rate
Impact of null on Cooling Tower Heat Transfer Rate
Impact of null on Cooling Tower Heat Transfer Rate
Impact of null on Cooling Tower Heat Transfer Rate
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Calculator Apps | {"url":"https://blog.truegeometry.com/calculators/cooling_tower_fills_material_calculation_for_Calculations.html","timestamp":"2024-11-13T21:14:22Z","content_type":"text/html","content_length":"29031","record_id":"<urn:uuid:95482f85-8718-48ac-a898-3cbb374e0c31>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00860.warc.gz"} |
Estimated Diameter - TigerGraph Graph Data Science Library
Estimated Diameter
The diameter of a graph is the worst-case length of a shortest path between any pair of vertices in a graph. It is the farthest distance to travel, to get from one vertex to another, if you always
take the shortest path. Finding the diameter requires calculating (the lengths of) all shortest paths, which can be quite slow.
This algorithm uses a simple heuristic to estimate the diameter. Rather than calculating the distance from each vertex to every other vertex, it selects \$K\$ vertices randomly, where \$K\$ is a
user-provided parameter. It calculates the distances from each of these \$K\$ vertices to all other vertices. So, instead of calculating \$V*(V-1)\$ distances, this algorithm only calculates \$K*
(V-1)\$ distances. The higher the value of \$K\$, the greater the likelihood of finding the true diameter.
This algorithm query employs a subquery called max_BFS_depth. Both queries must be installed to run the algorithm.
The current version of this algorithm only computes unweighted distances.
tg_estimate_diameter ( SET<STRING> v_type_set, SET<STRING> e_type_set, INT seed_set_length,
BOOL print_results = TRUE, STRING file_path = "", BOOL display = FALSE)
Parameter Description Default
SET<STRING> v_type Names of vertex types to use (empty set of strings)
SET<STRING> e_type Names of edge types to use (empty set of strings)
INT seed_set_length The number \$k\$ of random seed vertices to use 10
BOOL print_results If True, output JSON to standard output False
STRING file_path If not empty, write output to this file (empty string)
We can estimate the diameter of the graph included in the Shortest Path Algorithms TGCloud Starter Kit.
This graph contains data for 7,927 Airport vertices and 19,257 flight_route edges.
With seed_set_length set to 10, the estimated diameter returned is 9. With a larger seed_set_length of 100, the new, more accurate, estimated diameter returned is 12, because the greater number of
randomly selected vertices happened to capture a larger worst-case scenario of graph diameter. | {"url":"https://docs.tigergraph.com/graph-ml/3.10/pathfinding-algorithms/estimated-diameter","timestamp":"2024-11-02T05:56:57Z","content_type":"text/html","content_length":"37754","record_id":"<urn:uuid:ffd9e364-33e4-49c0-9c49-ba4982cbbcc0>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00734.warc.gz"} |
Resonance Analyses for a Noisy Coupled Brusselator Model Chin. Phys. Lett. 2017, 34 (7): 070201 . DOI: 10.1088/0256-307X/34/7/070201
We discuss the dynamical behavior of a chemical network arising from the coupling of two Brusselators established by the relationship between products and substrates. Our interest is to investigate
the coherence resonance (CR) phenomena caused by noise for a coupled Brusselator model in the vicinity of the Hopf bifurcation, which can be determined by the signal-to-noise ratio (SNR). The CR in
two coupled Brusselators will be considered in the presence of the Gaussian colored noise and two uncorrelated Gaussian white noises. Simulation results show that, for the case of single noise, the
SNR characterizing the degree of temporal regularity of coupled model reaches a maximum value at some optimal noise levels, and the noise intensity can enhance the CR phenomena of both subsystems
with a similar trend but in different resonance degrees. Meanwhile, effects of noise intensities on CR of the second subsystem are opposite for the systems under two uncorrelated Gaussian white
noises. Moreover, we find that CR might be a general phenomenon in coupled systems.
Bright-Dark Mixed $N$-Soliton Solution of the Two-Dimensional Maccari System Chin. Phys. Lett. 2017, 34 (7): 070202 . DOI: 10.1088/0256-307X/34/7/070202
The general bright-dark mixed $N$-soliton solution of the two-dimensional Maccari system is obtained with the KP hierarchy reduction method. The dynamics of single and two solitons are discussed in
detail. Asymptotic analysis shows that two solitons undergo elastic collision accompanied by a position shift. Furthermore, our analysis on mixed soliton bound states shows that arbitrary
higher-order soliton bound states can take place.
Fermionic Covariant Prolongation Structure for a Super Nonlinear Evolution Equation in 2+1 Dimensions Chin. Phys. Lett. 2017, 34 (7): 070203 . DOI: 10.1088/0256-307X/34/7/070203
The integrability of a (2+1)-dimensional super nonlinear evolution equation is analyzed in the framework of the fermionic covariant prolongation structure theory. We construct the prolongation
structure of the multidimensional super integrable equation and investigate its Lax representation. Furthermore, the Bäcklund transformation is presented and we derive a solution to the super
integrable equation.
General Single-Mode Gaussian Operation with Two-Mode Entangled State Chin. Phys. Lett. 2017, 34 (7): 070301 . DOI: 10.1088/0256-307X/34/7/070301
Realizing the logic operations with small-scale states is pursued to improve the utilization of quantum resources and to simplify the experimental setup. We propose a scheme to realize a general
single-mode Gaussian operation with a two-mode entangled state by utilizing only one nondegenerate optical parametric amplifier and by adjusting four angle parameters. The fidelity of the output
mode can be optimized by changing one of the angle parameters. This scheme would be utilized as a basic efficient element in the future large-scale quantum computation.
Implementing Classical Hadamard Transform Algorithm by Continuous Variable Cluster State Chin. Phys. Lett. 2017, 34 (7): 070302 . DOI: 10.1088/0256-307X/34/7/070302
Measurement-based one-way quantum computation, which uses cluster states as resources, provides an efficient model to perform computation. However, few of the continuous variable (CV) quantum
algorithms and classical algorithms based on one-way quantum computation were proposed. In this work, we propose a method to implement the classical Hadamard transform algorithm utilizing the CV
cluster state. Compared with classical computation, only half operations are required when it is operated in the one-way CV quantum computer. As an example, we present a concrete scheme of four-mode
classical Hadamard transform algorithm with a four-partite CV cluster state. This method connects the quantum computer and the classical algorithms, which shows the feasibility of running classical
algorithms in a quantum computer efficiently.
Phase Dissipation of an Open Two-Mode Bose–Einstein Condensate Chin. Phys. Lett. 2017, 34 (7): 070303 . DOI: 10.1088/0256-307X/34/7/070303
We study the dynamics of a two-mode Bose–Hubbard model with phase dissipation, based on the master equation. An analytical solution is presented with nonzero asymmetry and phase noise. The effects
of asymmetry and phase noise play a contrasting role in the dynamics. The asymmetry makes the oscillation fast, while phase noise enlarges the period. The conditions for the cases of fast decay and
oscillation are presented. As a possible application, the dynamical evolution of the population for cold atomic gases with synthetic gauge interaction, which can be understood as two-mode dynamics
in momentum space, is predicted.
Floquet Bound States in a Driven Two-Particle Bose–Hubbard Model with an Impurity Chin. Phys. Lett. 2017, 34 (7): 070304 . DOI: 10.1088/0256-307X/34/7/070304
We investigate how the driving field affects the bound states in the one-dimensional two-particle Bose–Hubbard model with an impurity. In the high-frequency regime, compared with the static lattice
[Phys. Rev. Lett. 109(2012)116405], a new type of Floquet bound state can be obtained even for a weak particle–particle interaction by tuning the driving amplitude. Moreover, the localization
degree of the Floquet bound molecular state can be adjusted by tuning the driving frequency, and even the Floquet bound molecular state can be changed into the Floquet extended state when the
driving frequency is below a critical value. Our results provide an efficient way to manipulate bound states in the many-body systems.
A Four-Phase Improvement of Grover's Algorithm Chin. Phys. Lett. 2017, 34 (7): 070305 . DOI: 10.1088/0256-307X/34/7/070305
When applying Grover's algorithm to an unordered database, the probability of obtaining correct results usually decreases as the quantity of target increases. A four-phase improvement of Grover's
algorithm is proposed to fix the deficiency, and the unitary and the phase-matching condition are also proposed. With this improved scheme, when the proportion of target is over 1/3, the probability
of obtaining correct results is greater than 97.82% with only one iteration using two phases. When the computational complexity is $O(\sqrt{M/N})$, the algorithm can succeed with a probability no
less than 99.63%.
Accretion onto the Magnetically Charged Regular Black Hole Chin. Phys. Lett. 2017, 34 (7): 070401 . DOI: 10.1088/0256-307X/34/7/070401
We investigate the accretion process for static spherically symmetric geometry, i.e., magnetically charged regular black hole with isotropic fluid. We obtain generalized expressions for the velocity
($u(r)$), speed of sound ($c^2_{\rm s}$), energy density ($\rho(r)$) and accretion rate ($\dot{M}$) at the critical point near the regular black hole during the accretion process. We also plot these
physical parameters against fixed values of charge, mass and different values of equation of state parameter to study the process of accretion. We find that radial velocity and energy density of the
fluid remain positive and negative as well as rate of change of mass is increased and decreased for dust, stiff, quintessence fluid and phantom-like fluid, respectively.
A Unified Approach to the Thermodynamics and Quantum Scaling Functions of One-Dimensional Strongly Attractive $SU(w)$ Fermi Gases Chin. Phys. Lett. 2017, 34 (7): 070501 . DOI: 10.1088/0256-307X/34/7
We present a unified derivation of the pressure equation of states, thermodynamics and scaling functions for the one-dimensional (1D) strongly attractive Fermi gases with $SU(w)$ symmetry. These
physical quantities provide a rigorous understanding on a universality class of quantum criticality characterized by the critical exponents $z=2$ and correlation length exponent $\nu=1/2$. Such a
universality class of quantum criticality can occur when the Fermi sea of one branch of charge bound states starts to fill or becomes gapped at zero temperature. The quantum critical cone can be
determined by the double peaks in specific heat, which serve to mark two crossover temperatures fanning out from the critical point. Our method opens to further study on quantum phases and phase
transitions in strongly interacting fermions with large $SU(w)$ and non-$SU(w)$ symmetries in one dimension.
A High-Sensitivity Terahertz Detector Based on a Low-Barrier Schottky Diode Chin. Phys. Lett. 2017, 34 (7): 070701 . DOI: 10.1088/0256-307X/34/7/070701
A low-barrier Schottky barrier diode based on the InGaAs/InP material system is designed and fabricated with a new non-destructive dry over-etching process. By using this diode, a high-sensitivity
waveguide detector is proposed. The measured maximum responsivity is over 2000mV/mW at 630GHz. The measured noise effective power (NEP) is less than 35pW/Hz$^{0.5}$ at 570–630GHz. The minimum
NEP is 14pW/Hz$^{0.5}$ at 630GHz. The proposed high-sensitivity waveguide detector has the characteristics of simple structure, compact size, low cost and high performance, and can be used in a
variety of applications such as imaging, molecular spectroscopy and atmospheric remote sensing.
A Photonuclear Reaction Model Based on IQMD in Intermediate-Energy Region Chin. Phys. Lett. 2017, 34 (7): 072401 . DOI: 10.1088/0256-307X/34/7/072401
A photonuclear reaction transport model based on an isospin-dependent quantum molecular dynamics model (IQMD) is presented in the intermediate energy region, which is named as GiQMD in this study.
Methodology to simulate the course of the photonuclear reaction within the IQMD frame is described to study the photo-absorption cross section and $\pi$ meson production, and the simulation results
are compared with some available experimental data as well as the Giessen Boltzmann–Uehling–Uhlenbeck model.
Constructions and Preliminary HV Conditioning of a Photocathode Direct-Current Electron Gun at IHEP Chin. Phys. Lett. 2017, 34 (7): 072901 . DOI: 10.1088/0256-307X/34/7/072901
As one of the most important key technologies for future advanced light source based on the energy recovery linac, a photocathode dc electron gun is supported by Institute of High Energy Physics
(IHEP) to address the technical challenges of producing very low emittance beams at high average current. Construction of the dc gun is completed and a preliminary high voltage conditioning is
carried out up to 440kV. The design, construction and preliminary HV conditioning results for the dc gun are described.
Theoretical Analysis of Rydberg and Autoionizing State Spectra of Antimony Chin. Phys. Lett. 2017, 34 (7): 073101 . DOI: 10.1088/0256-307X/34/7/073101
We calculate the Rydberg and autoionization Rydberg spectra of antimony (Sb) from first principles by relativistic multichannel theory within the framework of multichannel quantum defect theory. Our
calculation can be used to classify and assign the atomic states described in recently reported three Rydberg series and four autoionizing states. The perturbation effects on line intensity,
variation and line profile are discussed. Assignments of the perturber states and autoionizing states are presented.
State Preparation in a Cold Atom Clock by Optical Pumping Chin. Phys. Lett. 2017, 34 (7): 073201 . DOI: 10.1088/0256-307X/34/7/073201
We implement optical pumping to prepare cold atoms in our prototype of the $^{87}$Rb space cold atom clock, which operates in the one-way mode. Several modifications are made on our previous
physical and optical system. The effective atomic signal in the top detection zone is increased to 2.5 times with 87% pumping efficiency. The temperature of the cold atom cloud is increased by 1.4$
\mu$K. We study the dependences of the effective signal gain and pumping efficiency on the pumping laser intensity and detuning. The effects of $\sigma$ transition are discussed. This technique may
be used in the future space cold atom clocks.
Ion Photon Emission Microscope for Single Event Effect Testing in CIAE Chin. Phys. Lett. 2017, 34 (7): 073401 . DOI: 10.1088/0256-307X/34/7/073401
Ion photon emission microscopy (IPEM) is a new ion-induced emission microscopy. It employs a broad ion beam with high energy and low fluence rate impinging on a sample. The position of a single ion
is detected by an optical system with objective lens, prism, microscope tube and charge coupled device (CCD). A thin ZnS film doped with Ag ions is used as a luminescent material. Generation
efficiency and transmission efficiency of photons in the ZnS(Ag) film created by irradiated Cl ions are calculated. A single Cl ion optical microscopic image is observed by high quantum efficiency
CCD. The resolution of a single Cl ion given in this IPEM system is 6μm. Several factors influencing the resolution are discussed. A silicon diode is used to collect the electrical signals caused
by the incident ions. Effective and accidental coincidence of optical images and electronic signals are illustrated. A two-dimensional map of single event effect is drawn out according to the data
of effective coincidence.
High Optical Magnification Three-Dimensional Integral Imaging of Biological Micro-organism Chin. Phys. Lett. 2017, 34 (7): 074201 . DOI: 10.1088/0256-307X/34/7/074201
A high optical magnification three-dimensional imaging system is proposed using an optic microscope whose ocular (eyepiece) is retained and the structure of the transmission mode is not destroyed.
The elemental image array is captured through the micro lens array. Due to the front diffuse transmission element, each micro lens sees a slightly different spatial perspective of the scene, and a
different independent image is formed in each micro lens channel. Each micro lens channel is imaged by a Fourier lens and captured by a CCD. The design translating the stage in $x$ or $y$ provides
no parallax. Compared with the conventional integral imaging of micro-objects, the optical magnification of micro-objects in the proposed system can enhanced remarkably. The principle of the
enhancement of the image depth is explained in detail and the experimental results are presented.
Intracavity Spontaneous Parametric Down-Conversion in Bragg Reflection Waveguide Edge Emitting Diode Chin. Phys. Lett. 2017, 34 (7): 074202 . DOI: 10.1088/0256-307X/34/7/074202
A four-wavelength Bragg reflection waveguide edge emitting diode based on intracavity spontaneous parametric down-conversion and four-wave mixing (FWM) processes is made. The structure and its
tuning characteristic are designed by the aid of FDTD mode solution. The laser structure is grown by molecular beam epitaxy and processed to laser diode through the semiconductor manufacturing
technology. Fourier transform infrared spectroscopy is applied to record wavelength information. Pump around 1.071μm, signal around 1.77μm, idler around 2.71μm and FWM signal around 1.35μm are
observed at an injection current of 560mA. The influences of temperature, carrier density and pump wavelength on tuning characteristic are shown numerically and experimentally.
Frequency Stabilization of a Microsecond Pulse Sodium Guide Star Laser with a Tilt- and Temperature-Tuned Etalon Chin. Phys. Lett. 2017, 34 (7): 074203 . DOI: 10.1088/0256-307X/34/7/074203
A frequency stabilization approach is introduced for the microsecond pulse sodium beacon laser using an intra-cavity tilt- and temperature-tuned etalon based on a computer-controlled feedback system
connected with a fast high-precision wavelength meter. The frequency stability of the sodium beacon lasers is compared with and without feedback loop controlling. The output wavelength of the laser
is locked to the sodium D$_{2a}$ absorption line (589.159nm) over 12h with the feedback loop controlling technology. As a result, the sodium laser guide star is successfully observed by the
telescope of National Astronomical Observatories at Xinglong. This approach can also be used for other pulses and continuous-wave lasers for the frequency stabilization.
High-Efficiency Generation of 0.12mJ, 8.6Fs Pulses at 400nm Based on Spectral Broadening in Solid Thin Plates Chin. Phys. Lett. 2017, 34 (7): 074204 . DOI: 10.1088/0256-307X/34/7/074204
We demonstrate efficient generation of continuous spectrum centered at 400nm from solid thin plates. By frequency doubling of 0.8mJ, 30fs Ti:sapphire laser pulses with a BBO crystal, 0.2mJ,
33fs laser pulses at 400nm are generated. Focusing the 400-nm pulses into 7 thin fused silica plates, we obtain 0.15mJ continuous spectrum covering 350–450nm. After compressing by 3 pairs of
chirped mirrors, 0.12mJ, 8.6fs pulses are achieved. To the best of our knowledge, this is the first time that sub-10-fs pulses centered at 400nm are generated by solid thin plates, which shows
that spectral broadening in solid-state materials works not only at 800nm but also at different wavelengths.
High Coupling Efficiency of the Fiber-Coupled Module Based on Photonic-Band-Crystal Laser Diodes Chin. Phys. Lett. 2017, 34 (7): 074205 . DOI: 10.1088/0256-307X/34/7/074205
The coupling efficiency of the beam combination and the fiber-coupled module is limited due to the large vertical divergent angle of conventional semiconductor laser diodes. We present a high
coupling efficiency module using photonic-band-crystal (PBC) laser diodes with narrow vertical divergent angles. Three PBC single-emitter laser diodes are combined into a fiber with core diameter of
105μm and numerical aperture of 0.22. A high coupling efficiency of 94.4% is achieved and the brightness is calculated to be 1.7MW/(cm$^{2}\cdot$sr) with the injection current of 8A.
Effect of Phase Modulation on Electromagnetically Induced Grating in a Five-Level M-Type Atomic System Chin. Phys. Lett. 2017, 34 (7): 074206 . DOI: 10.1088/0256-307X/34/7/074206
We theoretically investigate the phenomena of electromagnetically induced grating in an M-type five-level atomic system. It is found that a weak field can be effectively diffracted into high-order
directions using a standing wave coupling field, and different depths of the phase modulation can disperse the diffraction light into different orders. When the phase modulation depth is
approximated to the orders of $\pi$, $2\pi$ and $3\pi$, the first-, second- and third-order diffraction intensity reach the maximum, respectively. Thus we can take advantage of the phase modulation
to control the probe light dispersing into the required high orders.
Generation of 47fs Pulses from an Er:Fiber Amplifier Chin. Phys. Lett. 2017, 34 (7): 074207 . DOI: 10.1088/0256-307X/34/7/074207
We demonstrate a self-starting erbium fiber oscillator-amplifier system based on the nonlinear polarization rotation mode-locked mechanism. The direct output pulse from the amplifier is 47fs with
an average power of 1.22W and a repetition rate of 50MHz, corresponding to a pulse energy of 24nJ. The full width at half-maximum of the spectrum of the output pulses is approximately 93nm at a
central wavelength of 1572nm so that the transform-limited pulse duration is as short as 39fs. Due to the imperfect dispersion compensation, we compress the pulses to 47fs in this experiment.
A 526mJ Subnanosecond Pulsed Hybrid-Pumped Nd:YAG Laser Chin. Phys. Lett. 2017, 34 (7): 074208 . DOI: 10.1088/0256-307X/34/7/074208
A hybrid-pumped Nd:YAG pulse laser with a double-pass two-rod configuration is presented. The focal length of offset lens is particularly studied to compensate for the thermal lens effect and
depolarization. For input pulse energy of 141$\mu$J with pulse duration of 754ps, the pulse laser system delivers 526mJ pulse energy and 728ps pulse width output at 10Hz with pulse profile
shape preservation. The energy stability of the laser pulse is less than 3%, and the beam quality factor $M^2$ is less than 2.26.
Tight Focusing Properties of Azimuthally Polarized Pair of Vortex Beams through a Dielectric Interface Chin. Phys. Lett. 2017, 34 (7): 074209 . DOI: 10.1088/0256-307X/34/7/074209
Tight focusing properties of an azimuthally polarized Gaussian beam with a pair of vortices through a dielectric interface is theoretically investigated by vector diffraction theory. For the
incident beam with a pair of vortices of opposite topological charges, the vortices move toward each other, annihilate and revive in the vicinity of focal plane, which results in the generation of
many novel focal patterns. The usable focal structures generated through the tight focusing of the double-vortex beams may find applications in micro-particle trapping, manipulation, and material
processing, etc.
Enhanced Luminescence of InGaN-Based 395nm Flip-Chip Near-Ultraviolet Light-Emitting Diodes with Al as N-Electrode Chin. Phys. Lett. 2017, 34 (7): 074210 . DOI: 10.1088/0256-307X/34/7/074210
High-reflectivity Al-based n-electrode is used to enhance the luminescence properties of InGaN-based 395nm flip-chip near-ultraviolet (UV) light-emitting diodes. The Al-only metal layer could form
the Ohmic contact on the plasma etched n-GaN by means of chemical pre-treatment, with the lowest specific contact resistance of $2.211\times10^{-5}$$\Omega\cdot$cm$^{2}$. The Al n-electrodes
enhance light output power of the 395nm flip-chip near-UV light-emitting diodes by more than 33% compared with the Ti/Al n-electrodes. Meanwhile, the electrical characteristics of these chips with
two types of n-electrodes do not show any significant discrepancy. The near-field light distribution measurement of packaged chips confirms that the enhanced luminescence is ascribed to the high
reflectivity of the Al electrodes in the UV region. After the accelerated aging test for over 1000h, the luminous degradation of the packaged chips with Al n-electrodes is less than 3%, which
proves the reliability of these chips with the Al-based electrodes. Our approach shows a simplified design and fabrication of high-reflectivity n-electrode for flip-chip near-UV light emitting
A 420nm Blue Diode Laser for the Potential Rubidium Optical Frequency Standard Chin. Phys. Lett. 2017, 34 (7): 074211 . DOI: 10.1088/0256-307X/34/7/074211
We report a 420nm external cavity diode laser with an interference filter (IF) of 0.5nm narrow-bandwidth and 79% high transmission, which is first used for Rb optical frequency standard. The IF
and the cat-eye reflector are used for selecting wavelength and light feedback, respectively. The measured laser linewidth is 24kHz when the diode laser is free running. Using this narrow-linewidth
IF blue diode laser, we realize a compact Rb optical frequency standard without a complicated PDH system. The preliminary stability of the Rb optical frequency standard is $2\times10^{-13}$ at 1s
and decreases to $1.9\times10^{-14}$ at 1000s. The narrow-linewidth characteristic makes the IF blue diode laser a well suited candidate for the compact Rb optical frequency standard.
Source Range Estimation Based on Pulse Waveform Matching in a Slope Environment Chin. Phys. Lett. 2017, 34 (7): 074301 . DOI: 10.1088/0256-307X/34/7/074301
An approach of source range estimation in an ocean environment with sloping bottom is presented. The approach is based on pulse waveform correlation matching between the received and simulated
signals. An acoustic propagation experiment is carried out in a slope environment. The pulse signal is received by the vertical line array, and the depth structure can be obtained. For the
experimental data, the depth structures of pulse waveforms are different, which depends on the source range. For a source with unknown range, the depth structure of pulse waveform can be first
obtained from the experimental data. Next, the depth structures of pulse waveforms in different ranges are numerically calculated. After the process of correlating the experimental and simulated
signals, the range corresponding to the maximum value of the correlation coefficient is the estimated source range. For the explosive sources in the experiment with two depths, the mean relative
errors of range estimation are both less than 7%.
Coupled Perturbed Modes over Sloping Penetrable Bottom Chin. Phys. Lett. 2017, 34 (7): 074302 . DOI: 10.1088/0256-307X/34/7/074302
In environments with water depth variations, one-way modal solutions involve mode coupling. Higham and Tindle developed an accurate and fast approach using perturbation theory to locally determine
the change in mode functions at steps. The method of Higham and Tindle is limited to low frequency ($\le$250Hz). We extend the coupled perturbation method, thus it can be applied to higher
frequencies. The approach is described and some examples are given.
Simulation of Double-Front Detonation of Suspended Mixed Cyclotrimethylenetrinitramine and Aluminum Dust in Air Chin. Phys. Lett. 2017, 34 (7): 074701 . DOI: 10.1088/0256-307X/34/7/074701
The two-phase detonation of suspended mixed cyclotrimethylenetrinitramine (i.e., RDX) and aluminum dust in air is simulated with a two-phase flow model. The parameters of the mixed RDX-Al dust
detonation wave are obtained. The double-front detonation and steady state of detonation wave of the mixed dust are analyzed. For the dust mixed RDX with density of 0.565kg/m$^{3}$ and radius of
10μm as well as aluminum with density of 0.145kg/m$^{3}$ and radius of 4μm, the detonation wave will reach a steady state at 23m. The effects of the size of aluminum on the detonation are
analyzed. For constant radius of RDX particles with radius of 10μm, as the radius of aluminum particles is larger than 2.0μm, the double-front detonation can be observed due to the different
ignition distances and reaction rates of RDX and aluminum particles. As the radius of aluminum particles is larger, the velocity, pressure and temperature of detonation wave will be slower. The
pressure at the Chapman–Jouguet (CJ) point also becomes lower. Comparing the detonation with single RDX dust, the pressure and temperature in the flow field of detonation of mixed dust are higher.
Natural Frequency of Oscillating Gaseous Bubbles in Ventilated Cavitation Chin. Phys. Lett. 2017, 34 (7): 074702 . DOI: 10.1088/0256-307X/34/7/074702
An improved formula is proposed for the prediction of natural frequency of oscillating gaseous bubbles in the ventilated cavitation by considering the liquid compressibility and the thermal effects.
The differences between the previous formula and ours are quantitatively discussed in terms of both dimensional parameters (e.g., frequency and bubble radius) and non-dimensional parameters (e.g.,
the Péclet number). Our analysis reveals that our formula is superior to the existing formula in the low-frequency excitation regions.
Oscillatory and Chaotic Buoyant-Thermocapillary Convection in the Large-Scale Liquid Bridge Chin. Phys. Lett. 2017, 34 (7): 074703 . DOI: 10.1088/0256-307X/34/7/074703
To cooperate with Chinese TG-2 space experiment project, the transition process from steady to regular oscillatory flow, and finally to chaos is experimentally studied in buoyant-thermocapillary
convection. The onset of oscillation and further transitional convective behavior are detected by measuring the temperature in large-scale liquid bridge of 2cSt silicone oil. To identify the various
dynamical regimes, the Fourier transform and fractal theory are used to reveal the frequency and amplitude characteristics of the flow motion. The experimental results indicate the co-existence of
quasi-periodic and the Feigenbaum bifurcation in chaos.
Jeans Gravitational Instability with $\kappa $-Deformed Kaniadakis Distribution Chin. Phys. Lett. 2017, 34 (7): 075101 . DOI: 10.1088/0256-307X/34/7/075101
The Jeans instabilities in an unmagnetized, collisionless, isotropic self-gravitating matter system are investigated in the context of $\kappa$-deformed Kaniadakis distribution based on kinetic
theory. The result shows that both the growth rates and critical wave numbers of Jeans instability are lower in the $\kappa$-deformed Kaniadakis distributed self-gravitating matter systems than the
Maxwellian case. The standard Jeans instability for a Maxwellian case is recovered under the limitation $\kappa=0$.
Linear Growth of Rayleigh–Taylor Instability of Two Finite-Thickness Fluid Layers Chin. Phys. Lett. 2017, 34 (7): 075201 . DOI: 10.1088/0256-307X/34/7/075201
The linear growth of Rayleigh–Taylor instability (RTI) of two superimposed finite-thickness fluids in a gravitational field is investigated analytically. Coupling evolution equations for
perturbation on the upper, middle and lower interfaces of the two stratified fluids are derived. The growth rate of the RTI and the evolution of the amplitudes of perturbation on the three
interfaces are obtained by solving the coupling equations. It is found that the finite-thickness fluids reduce the growth rate of perturbation on the middle interface. However, the finite-thickness
effect plays an important role in perturbation growth even for the thin layers which will cause more severe RTI growth. Finally, the dependence of the interface position under different initial
conditions are discussed in some detail.
Forming of Space Charge Wave with Broad Frequency Spectrum in Helical Relativistic Two-Stream Electron Beams Chin. Phys. Lett. 2017, 34 (7): 075202 . DOI: 10.1088/0256-307X/34/7/075202
We elaborate a quadratic nonlinear theory of plural interactions of growing space charge wave (SCW) harmonics during the development of the two-stream instability in helical relativistic electron
beams. It is found that in helical two-stream electron beams the growth rate of the two-stream instability increases with the beam entrance angle. An SCW with the broad frequency spectrum, in which
higher harmonics have higher amplitudes, forms when the frequency of the first SCW harmonic is much less than the critical frequency of the two-stream instability. For helical electron beams the
spectrum expands with the increase of the beam entrance angle. Moreover, we obtain that utilizing helical electron beams in multiharmonic two-stream superheterodyne free-electron lasers leads to the
improvement of their amplification characteristics, the frequency spectrum broadening in multiharmonic signal generation mode, and the reduction of the overall system dimensions.
Effect of Particle Number Density on Wave Dispersion in a Two-Dimensional Yukawa System Chin. Phys. Lett. 2017, 34 (7): 075203 . DOI: 10.1088/0256-307X/34/7/075203
Effect of the particle number density on the dispersion properties of longitudinal and transverse lattice waves in a two-dimensional Yukawa charged-dust system is investigated using molecular
dynamics simulation. The dispersion relations for the waves are obtained. It is found that the frequencies of both the longitudinal and transverse dust waves increase with the density and when the
density is sufficiently high a cutoff region appears at the short wavelength. With the increase of the particle number density, the common frequency tends to increase, and the sound speed of the
longitudinal wave also increases, but that of the transverse wave remains low.
Recrystallization Phase in He-Implanted 6H-SiC Chin. Phys. Lett. 2017, 34 (7): 076101 . DOI: 10.1088/0256-307X/34/7/076101
The evolution of the recrystallization phase in amorphous 6H-SiC formed by He implantation followed by thermal annealing is investigated. Microstructures of recrystallized layers in 15keV He$^{+}$
ion implanted 6H-SiC (0001) wafers are characterized by means of cross-sectional transmission electron microscopy (XTEM) and high-resolution TEM. Epitaxial recrystallization of buried amorphous
layers is observed at an annealing temperature of 900$^{\circ}\!$C. The recrystallization region contains a 3C-SiC structure and a 6H-SiC structure with different crystalline orientations. A high
density of lattice defects is observed at the interface of different phases and in the periphery of He bubbles. With increasing annealing to 1000$^{\circ}\!$C, 3C-SiC and columnar epitaxial growth
6H-SiC become unstable, instead of [0001] orientated 6H-SiC. In addition, the density of lattice defects increases slightly with increasing annealing. The possible mechanisms for explanation are
also discussed.
Anisotropic Migration of Defects under Strain Effect in BCC Iron Chin. Phys. Lett. 2017, 34 (7): 076102 . DOI: 10.1088/0256-307X/34/7/076102
The basic properties of defects (self-interstitial and vacancy) in BCC iron under uniaxial tensile strain are investigated with atomic simulation methods. The formation and migration energies of
them show different dependences on the directions of uniaxial tensile strain in two different computation boxes. In box-1, the uniaxial tensile strain along the $\langle 100\rangle$ direction
influences the formation and migration energies of the $\langle 110 \rangle$ dumbbell but slightly affects the migration energy of a single vacancy. In box-2, the uniaxial tensile strain along the $
\langle 111\rangle$ direction influences the formation and migration energies of both vacancy and interstitials. Especially, a $\langle 110 \rangle$ dumbbell has a lower migration energy when its
migration direction is the same or close to the strain direction, while along these directions, a vacancy has a higher migration energy. All these results indicate that the uniaxial tensile strain
can result in the anisotropic formation and migration energies of simple defects in materials.
Magnetic and Electronic Properties of Double Perovskite Ba$_{2}$SmNbO$_{6 }$ without Octahedral Tilting by First Principle Calculations Chin. Phys. Lett. 2017, 34 (7): 076103 . DOI: 10.1088/
The structural, magnetic and electronic properties of the double perovskite Ba$_{2}$SmNbO$_{6}$ (for the simple cubic structure where no octahedral tilting exists anymore) are studied using the
density functional theory within the generalized gradient approximation as well as taking into account the on-site Coulomb repulsive interaction. The total energy, the spin magnetic moment, the band
structure and the density of states are calculated. The optimization of the lattice constants is 8.5173Å, which is in good agreement with the experimental value 8.5180Å. The calculations reveal
that Ba$_{2}$SmNbO$_{6}$ has a stable ferromagnetic ground state and the spin magnetic moment per molecule is 5.00$\mu$B/f.u. which comes mostly from the Sm$^{3+}$ ion only. By analysis of the band
structure, the compound exhibits the direct band gap material and half-metallic ferromagnetic nature with 100% spin-up polarization, which implies potential applications of this new lanthanide
compound in magneto-electronic and spintronic devices.
An Increase in TDDB Lifetime of Partially Depleted SOI Devices Induced by Proton Irradiation Chin. Phys. Lett. 2017, 34 (7): 076104 . DOI: 10.1088/0256-307X/34/7/076104
The effects of proton irradiation on the subsequent time-dependent dielectric breakdown (TDDB) of partially depleted SOI devices are experimentally investigated. It is demonstrated that heavy-ion
irradiation will induce the decrease of TDDB lifetime for many device types, but we are amazed to find a measurable increase in the TDDB lifetime and a slight decrease in the radiation-induced
leakage current after proton irradiation at the nominal operating irradiation bias. We interpret these results and mechanisms in terms of the effects of radiation-induced traps on the stressing
current during the reliability testing, which may be significant to expand the understanding of the radiation effects of the devices used in the proton radiation environment.
Growth and Characterization of InSb Thin Films on GaAs (001) without Any Buffer Layers by MBE Chin. Phys. Lett. 2017, 34 (7): 076105 . DOI: 10.1088/0256-307X/34/7/076105
We report the growth of InSb layers directly on GaAs (001) substrates without any buffer layers by molecular beam epitaxy (MBE). Influences of growth temperature and V/III flux ratios on the crystal
quality, the surface morphology and the electrical properties of InSb thin films are investigated. The InSb samples with room-temperature mobility of 44600cm$^{2}$/Vs are grown under optimized
growth conditions. The effect of defects in InSb epitaxial on the electrical properties is researched, and we infer that the formation of In vacancy (V$_{\rm In})$ and Sb anti-site (Sb$_{\rm In})$
defects is the main reason for concentrations changing with growth temperature and Sb$_{2}$/In flux ratios. The mobility of the InSb sample as a function of temperature ranging from 90K to 360K is
demonstrated and the dislocation scattering mechanism and phonon scattering mechanism are discussed.
Temperature-Dependent Photoluminescence Analysis of 1.0MeV Electron Irradiation-Induced Nonradiative Recombination Centers in n$^{+}$–p GaAs Middle Cell of GaInP/GaAs/Ge Triple-Junction Solar Cells
Chin. Phys. Lett. 2017, 34 (7): 076106 . DOI: 10.1088/0256-307X/34/7/076106
The effects of irradiation of 1.0MeV electrons on the n$^{+}$–p GaAs middle cell of GaInP/GaAs/Ge triple-junction solar cells are investigated by temperature-dependent photoluminescence (PL)
measurements in the 10–300K temperature range. The appearance of thermal quenching of the PL intensity with increasing temperature confirms the presence of a nonradiative recombination center in
the cell after the electron irradiation, and the thermal activation energy of the center is determined using the Arrhenius plot of the PL intensity. Furthermore, by comparing the thermal activation
and the ionization energies of the defects, the nonradiative recombination center in the n$^{+}$–p GaAs middle cell acting as a primary defect is identified as the E5 electron trap located at $E_{\
rm c}-0.96$eV.
LaB$_{6}$ Work Function and Structural Stability under High Pressure Chin. Phys. Lett. 2017, 34 (7): 076201 . DOI: 10.1088/0256-307X/34/7/076201
The work functions of the (110) and (100) surfaces of LaB$_{6}$ are determined from ambient pressure to 39.1GPa. The work function of the (110) surface slowly decreases but that of the (100)
surface remains at a relatively constant value. To determine the reason for this difference, the electron density distribution (EDD) is determined from high-pressure single-crystal x-ray diffraction
data by the maximum entropy method. The EDD results show that the chemical bond properties in LaB$_{6}$ play a key role. The structural stability of LaB$_{6}$ under high pressure is also
investigated by single-crystal x-ray diffraction. In this study, no structural or electronic phase transition is observed from ambient pressure to 39.1GPa.
An Analysis of Structural-Acoustic Coupling Band Gaps in a Fluid-Filled Periodic Pipe Chin. Phys. Lett. 2017, 34 (7): 076202 . DOI: 10.1088/0256-307X/34/7/076202
A periodic pipe system composed of steel pipes and rubber hoses with the same inner radius is designed based on the theory of phononic crystals. Using the transfer matrix method, the band structure
of the periodic pipe is calculated considering the structural-acoustic coupling. The results show that longitudinal vibration band gaps and acoustic band gaps can coexist in the fluid-filled
periodic pipe. The formation of the band gap mechanism is further analyzed. The band gaps are validated by the sound transmission loss and vibration-frequency response functions calculated using the
finite element method. The effect of the damp on the band gap is analyzed by calculating the complex band structure. The periodic pipe system can be used not only in the field of vibration reduction
but also for noise elimination.
Coupled Two-Dimensional Atomic Oscillation in an Anharmonic Trap Chin. Phys. Lett. 2017, 34 (7): 076701 . DOI: 10.1088/0256-307X/34/7/076701
In atomic dynamics, oscillation along different axes can be studied separately in the harmonic trap. When the trap is not harmonic, motion in different directions may couple together. In this work,
we observe a two-dimensional oscillation by exciting atoms in one direction, where the atoms are transferred to an anharmonic region. Theoretical calculations are coincident to the experimental
results. These oscillations in two dimensions not only can be used to measure trap parameters but also have potential applications in atomic interferometry and precise measurements.
Topological Nodal Line Semimetal in Non-Centrosymmetric PbTaS$_2$ Chin. Phys. Lett. 2017, 34 (7): 077101 . DOI: 10.1088/0256-307X/34/7/077101
Topological semimetals are a new type of matter with one-dimensional Fermi lines or zero-dimensional Weyl or Dirac points in momentum space. Here using first-principles calculations, we find that
the non-centrosymmetric PbTaS$_2$ is a topological nodal line semimetal. In the absence of spin-orbit coupling (SOC), one band inversion happens around a high symmetrical $H$ point, which leads to
forming a nodal line. The nodal line is robust and protected against gap opening by mirror reflection symmetry even with the inclusion of strong SOC. In addition, it also hosts exotic drumhead
surface states either inside or outside the projected nodal ring depending on surface termination. The robust bulk nodal lines and drumhead-like surface states with SOC in PbTaS$_2$ make it a
potential candidate material for exploring the freakish properties of the topological nodal line fermions in condensed matter systems.
Analytic Continuation with Padé Decomposition Chin. Phys. Lett. 2017, 34 (7): 077102 . DOI: 10.1088/0256-307X/34/7/077102
The ill-posed analytic continuation problem for Green's functions or self-energies can be carried out using the Padé rational polynomial approximation. However, to extract accurate results from this
approximation, high precision input data of the Matsubara Green function are needed. The calculation of the Matsubara Green function generally involves a Matsubara frequency summation, which cannot
be evaluated analytically. Numerical summation is requisite but it converges slowly with the increase of the Matsubara frequency. Here we show that this slow convergence problem can be significantly
improved by utilizing the Padé decomposition approach to replace the Matsubara frequency summation by a Padé frequency summation, and high precision input data can be obtained to successfully
perform the Padé analytic continuation.
Boundary Hamiltonian Theory for Gapped Topological Orders Chin. Phys. Lett. 2017, 34 (7): 077103 . DOI: 10.1088/0256-307X/34/7/077103
We report our systematic construction of the lattice Hamiltonian model of topological orders on open surfaces, with explicit boundary terms. We do this mainly for the Levin-Wen string-net model. The
full Hamiltonian in our approach yields a topologically protected, gapped energy spectrum, with the corresponding wave functions robust under topology-preserving transformations of the lattice of
the system. We explicitly present the wavefunctions of the ground states and boundary elementary excitations. The creation and hopping operators of boundary quasi-particles are constructed. It is
found that given a bulk topological order, the gapped boundary conditions are classified by Frobenius algebras in its input data. Emergent topological properties of the ground states and boundary
excitations are characterized by (bi-) modules over Frobenius algebras.
Modelling Magnetoresistance Effect in Limited Anisotropic Semiconductors Chin. Phys. Lett. 2017, 34 (7): 077201 . DOI: 10.1088/0256-307X/34/7/077201
A macroscopic model of the magnetoresistance effect in limited anisotropic semiconductors is built. This model allows us to solve the problem of measurement of physical magnetoresistance components
of crystals and films. Based on a unified mathematical model the method is worked out enabling us to measure tensor components of the specific electrical resistance and the relative
magnetoresistance of anisotropic semiconductors simultaneously.
Spin Noise Spectroscopy in N-GaAs: Spin Relaxation of Localized Electrons Chin. Phys. Lett. 2017, 34 (7): 077202 . DOI: 10.1088/0256-307X/34/7/077202
Spin noise spectroscopy (SNS) of electrons in n-doped bulk GaAs is studied as functions of temperature and the probe-laser energy. Experimental results show that the SNS signal comes from localized
electrons in the donor band. The spin relaxation time of electrons, which is retrieved from the SNS measurement, depends on the probe light energy and temperature, and it can be ascribed to the
variation of electron localization degree.
BiPh-$m$-BiDPO as a Hole-Blocking Layer for Organic Light-Emitting Diodes: Revealing Molecular Structure-Properties Relationship Chin. Phys. Lett. 2017, 34 (7): 077203 . DOI: 10.1088/0256-307X/34/7/
We report a simple hole-blocking material (biphenyl-3,3'-diyl)bis(diphenylphosphine oxide) (BiPh-$m$-BiDPO) based on our recent advance. The bis(phosphine oxide) compound shows HOMO/LUMO levels of $
\sim$$-6.71/-2.51$eV. Its phosphorescent spectrum in a solid film features two major emission bands peaking at 2.69 and 2.4eV, corresponding to 0–0 and 0–1 vibronic transitions, respectively. The
measurement of the electron-only devices reveals that BiPh-$m$-BiDPO possesses electron mobility of $2.28\times10^{-9}$–$3.22\times10^{-8}$cm$^{2}$V$^{-1}$s$^{-1}$ at $E=2$–$5\times10^{5}$V/cm.
The characterization of the sky blue fluorescent and red phosphorescent pin organic light-emitting diodes (OLEDs) utilizing BiPh-$m$-BiDPO as the hole blocker shows that its shallow LUMO level as
well as the low electron mobility affects significantly the power efficiency and hence operational stability, relative to the luminous efficiency, especially at high luminance. In combination with
our recent results, the present study provides an indepth insight on the molecular structure-property correlation in the organic phosphinyl-containing hole-blocking materials.
The Efficiency Droop of InGaN-Based Green LEDs with Different Superlattice Growth Temperatures on Si Substrates via Temperature-Dependent Electroluminescence Chin. Phys. Lett. 2017, 34 (7): 077301 .
DOI: 10.1088/0256-307X/34/7/077301
InGaN-based green light-emitting diodes (LEDs) with different growth temperatures of superlattice grown on Si (111) substrates are investigated by temperature-dependent electroluminescence between
100K and 350K. It is observed that with the decrease of the growth temperature of the superlattice from 895$^{\circ}\!$C to 855$^{\circ}\!$C, the forward voltage decreases, especially at low
temperature. We presume that this is due to the existence of the larger average size of V-shaped pits, which is determined by secondary ion mass spectrometer measurements. Meanwhile, the sample with
higher growth temperature of superlattice shows a severer efficiency droop at cryogenic temperatures (about 100K–150K). Electron overflow into p-GaN is considered to be the cause of such
phenomena, which is relevant to the poorer hole injection into multiple quantum wells and the more reduced effective active volume in the active region.
Revisiting the Electron-Doped SmFeAsO: Enhanced Superconductivity up to 58.6K by Th and F Codoping Chin. Phys. Lett. 2017, 34 (7): 077401 . DOI: 10.1088/0256-307X/34/7/077401
In the iron-based high-$T_{\rm c}$ bulk superconductors, $T_{\rm c}$ above 50K was only observed in the electron-doped 1111-type compounds. Here we revisit the electron-doped SmFeAsO polycrystals
to make a further investigation for the highest $T_{\rm c}$ in these materials. To introduce more electron carriers and less crystal lattice distortions, we study the Th and F codoping effects into
the Sm-O layers with heavy electron doping. Dozens of Sm$_{1-x}$Th$_{x}$FeAsO$_{1-y}$F$_{y}$ samples are synthesized through the solid state reaction method, and these samples are carefully
characterized by the structural, resistive, and magnetic measurements. We find that the codoping of Th and F clearly enhances the superconducting $T_{\rm c}$ more than the Th or F single-doped
samples, with the highest record $T_{\rm c}$ up to 58.6K when $x=0.2$ and $y=0.225$. Further element doping causes more impurities and lattice distortions in the samples with a weakened
In Situ Electronic Structure Study of Epitaxial Niobium Thin Films by Angle-Resolved Photoemission Spectroscopy Chin. Phys. Lett. 2017, 34 (7): 077402 . DOI: 10.1088/0256-307X/34/7/077402
High-quality single crystalline niobium films are grown on a-plane sapphire in molecular beam epitaxy. The film is single crystalline with a (110) orientation, and both the rocking curve and the
reflection high-energy electron diffraction pattern demonstrate its high-quality with an atomically smooth surface. By in situ study of its electronic structure, a rather weak electron-electron
correlation effect is demonstrated experimentally in this $4d$ transition metal. Moreover, a kink structure is observed in the electronic structure, which may result from electron-phonon interaction
and it might contribute to the superconductivity. Our results help to understand the properties of niobium deeply.
Proximity-Induced Superconductivity in New Superstructures on 2H-NbSe$_2$ Surface Chin. Phys. Lett. 2017, 34 (7): 077403 . DOI: 10.1088/0256-307X/34/7/077403
Using scanning tunneling microscopy we observe a stripe phase smoothly interfacing with a triangular $2\times2$ superstructure on the surface of 2H-NbSe$_2$ single crystal. Proximity-induced
superconductivity is demonstrated in these new ordered structures by measurements of low-temperature tunneling spectra. The modulation of superconductivity by the reconstruction provides an
opportunity to understand the interplay between superconductivity and charge orders.
Superconducting (Li,Fe)OHFeSe Film of High Quality and High Critical Parameters Chin. Phys. Lett. 2017, 34 (7): 077404 . DOI: 10.1088/0256-307X/34/7/077404
A superconducting film of (Li$_{1-x}$Fe$_{x})$OHFeSe is reported for the first time. The thin film exhibits a small in-plane crystal mosaic of 0.22$^{\circ}$, in terms of the full width at half
maximum of the x-ray rocking curve, and an excellent out-of-plane orientation by x-ray $\varphi $-scan. Its bulk superconducting transition temperature $T_{\rm c}$ of 42.4K is characterized by both
zero electrical resistance and diamagnetization measurements. The upper critical field $H_{\rm c2}$ is estimated to be 79.5T and 443T for the magnetic field perpendicular and parallel to the $ab$
plane, respectively. Moreover, a large critical current density $J_{\rm c}$ of a value over 0.5MA/cm$^{2}$ is achieved at $\sim $20K. Such a (Li$_{1-x}$Fe$_{x})$OHFeSe film is therefore not only
important to the fundamental research for understanding the high-$T_{\rm c}$ mechanism, but also promising in the field of high-$T_{\rm c}$ superconductivity application, especially in
high-performance electronic devices and large scientific facilities such as superconducting accelerator.
Weyl and Nodal Ring Magnons in Three-Dimensional Honeycomb Lattices Chin. Phys. Lett. 2017, 34 (7): 077501 . DOI: 10.1088/0256-307X/34/7/077501
We study the topological properties of magnon excitations in a wide class of three-dimensional (3D) honeycomb lattices with ferromagnetic ground states. It is found that they host nodal ring magnon
excitations. These rings locate on the same plane in the momentum space. The nodal ring degeneracy can be lifted by the Dzyaloshinskii–Moriya interactions to form two Weyl points with opposite
charges. We explicitly discuss these physics in the simplest 3D honeycomb lattice and the hyperhoneycomb lattice, and show drumhead and arc surface states in the nodal ring and Weyl phases,
respectively, due to the bulk-boundary correspondence.
Gapped Spin-1/2 Spinon Excitations in a New Kagome Quantum Spin Liquid Compound Cu$_3$Zn(OH)$_6$FBr Chin. Phys. Lett. 2017, 34 (7): 077502 . DOI: 10.1088/0256-307X/34/7/077502
We report a new kagome quantum spin liquid candidate Cu$_3$Zn(OH)$_6$FBr, which does not experience any phase transition down to 50mK, more than three orders lower than the antiferromagnetic
Curie-Weiss temperature ($\sim$200K). A clear gap opening at low temperature is observed in the uniform spin susceptibility obtained from $^{19}$F nuclear magnetic resonance measurements. We
observe the characteristic magnetic field dependence of the gap as expected for fractionalized spin-1/2 spinon excitations. Our experimental results provide firm evidence for spin fractionalization
in a topologically ordered spin system, resembling charge fractionalization in the fractional quantum Hall state.
Abnormal Polarity Effects of Streamer Discharge in Propylene Carbonate under Microsecond Pulses Chin. Phys. Lett. 2017, 34 (7): 077701 . DOI: 10.1088/0256-307X/34/7/077701
Propylene carbonate (PC) has a great potential to be used as an energy storage medium in the compact pulsed power sources due to its high dielectric constant and large resistivity. We investigate
both the positive and negative breakdown characteristics of PC. The streamer patterns are obtained by ultra-high-speed cameras. The experimental results show that the positive breakdown voltage of
PC is about 135% higher than the negative one, which is abnormal compared with the common liquid. The shape of the positive streamer is filamentary and branchy, while the negative streamer is
tree-like and less branched. According to these experimental results, a charge layer structure model at the interface between the metal electrode and liquid is presented. It is suggested that the
abnormal polarity effect basically arises from the electric field strength difference in the interface between both electrodes and PC. What is more, the recombination radiation and photoionization
also play an important role in the whole discharge process.
Fluorescence Intermittency in Monolayer WSe$_{2}$ Chin. Phys. Lett. 2017, 34 (7): 077801 . DOI: 10.1088/0256-307X/34/7/077801
Fluorescence intermittent dynamics of single quantum emitters in monolayer WSe$_{2}$ are investigated via measuring spectrally resolved time traces and time-dependent fluorescence intensity
trajectories. Analysis of fluorescence trajectories and spectral shifting reveal a correlation between the fluorescence intermittency and spectral diffusion. A model of an inverse power law can be
used to understand the observed blinking dynamics.
X-Ray Radiation Sensing Properties of ZnS Thin Film: A Study on the Effect of Annealing Chin. Phys. Lett. 2017, 34 (7): 077802 . DOI: 10.1088/0256-307X/34/7/077802
Chemically synthesized ZnS thin film is found to be a good x-ray radiation sensor. We report the effect of annealing on the x-ray radiation detection sensitivity of a ZnS thin film synthesized by a
chemical bath deposition technique. The chemically synthesized ZnS films are annealed at 333, 363 and 393K for 1h. Structural analyses show that the lattice defects in the films decrease with
annealing. Further, the band gap is also found to decrease from 3.38 to 3.21eV after annealing at 393K. Current-voltage characteristics of the films are studied under dark and x-ray irradiation
conditions. Due to the decrease of lattice defects and band gap, the conductivity under dark conditions is found to increase from $2.06\times10^{-6}$ to $1.69\times10^{-5}$S/cm, while that under
x-ray irradiation increases from $4.13\times10^{-5}$ to $5.28\times10^{-5}$S/cm. On the other hand, the x-ray radiation detection sensitivity of the films is found to decrease with annealing. This
decrease of detection sensitivity is attributed to the decrease of the band gap as well as some structural and surface morphological changes occurring after annealing.
Dependence of Nonlinear Optical Response of Anatase TiO$_{2}$ on Shape and Excitation Intensity Chin. Phys. Lett. 2017, 34 (7): 077803 . DOI: 10.1088/0256-307X/34/7/077803
Nonlinear optical (NLO) properties of anatase TiO$_{2}$ with nanostructures of nanoparticle (NP), nanowire (NW) and annealed nanowire (NWA) are studied by open-aperture and closed-aperture $Z$-scan
techniques with a femtosecond pulsed laser at wavelengths of 532nm and 780nm simultaneously. At 532nm, when increasing excitation intensity, NLO absorption of TiO$_{2}$ NPs transforms from
saturable absorption to reverse-saturable absorption. However, NWs and NWAs exhibit the opposite change. At 780nm, all samples show reverse-saturable absorption, but have different sensitivities to
excitation intensity. Due to the larger surface-to-volume ratio of NPs and less defects of NWAs by annealing, nonlinear optical absorption coefficients follow the order NPs$\ge$NWs$\ge$NWAs. The
results also show that these shape and annealing effects are dominant at low excitation intensity, but do not exhibit at the high excitation intensity. The NLO refractive index of NPs shows a
positive linear relationship with the excitation intensity, whereas NW and NWAs exhibit a negative linear relationship. The results could provide some foundational guidance to applications of
anatase TiO$_{2}$ in optoelectronic devices or other aspects.
Observation of Temperature Induced Plasma Frequency Shift in an Extremely Large Magnetoresistance Compound LaSb Chin. Phys. Lett. 2017, 34 (7): 077804 . DOI: 10.1088/0256-307X/34/7/077804
We report an optical spectroscopy study on LaSb, a compound recently identified to exhibit extremely large magnetoresistance. Our optical measurement indicates that the material has a low carrier
density. More interestingly, the study reveals that the plasma frequency increases with decreasing temperature. This phenomenon suggests either an increase of the conducting carrier density or/and a
decrease of the effective mass of carriers with decreasing temperature. We attribute it primarily to the latter effect. Two possible scenarios on its physical origin are examined and discussed. The
study offers new insight into the electronic structure of this compound.
Experimental $I$–$V$ and $C$–$V$ Analysis of Schottky-Barrier Metal-Oxide-Semiconductor Field Effect Transistors with Epitaxial NiSi$_{2}$ Contacts and Dopant Segregation Chin. Phys. Lett. 2017, 34
(7): 078501 . DOI: 10.1088/0256-307X/34/7/078501
We present an experimental analysis of Schottky-barrier metal-oxide-semiconductor field effect transistors (SB-MOSFETs) fabricated on ultrathin body silicon-on-insulator substrates with a steep
junction by the dopant implantation into the silicide process. The subthreshold swing of such SB-MOSFETs reaches 69mV/dec. Emphasis is placed on the capacitance-voltage analysis of p-type
SB-MOSFETs. According to the measurements of gate-to-source capacitance $C_{\rm gs}$ with respect to $V_{\rm gs}$ at various $V_{\rm ds}$, we find that a maximum occurs at the accumulation regime
due to the most imbalanced charge distribution along the channel. At each $C_{\rm gs}$ peak, the difference between $V_{\rm gs}$ and $V_{\rm ds}$ is equal to the Schottky barrier height (SBH) for
NiSi$_{2}$ on highly doped silicon, which indicates that the critical condition of channel pinching off is related with SBH for source/drain on channel. The SBH for NiSi$_{2}$ on highly doped
silicon can affect the pinch-off voltage and the saturation current of SB-MOSFETs.
Hetero-Epitaxy and Self-Adaptive Stressor Based on Freestanding Fin for the 10nm Node and Beyond Chin. Phys. Lett. 2017, 34 (7): 078502 . DOI: 10.1088/0256-307X/34/7/078502
A promising technology named epitaxy on nano-scale freestanding fin (ENFF) is firstly proposed for hetero-epitaxy. This technology can effectively release total strain energy and then can reduce the
probability of generating mismatch dislocations. Based on the calculation, dislocation defects can be eliminated completely when the thickness of the Si freestanding fin is less than 10nm for the
epitaxial Ge layer. In addition, this proposed ENFF process can provide sufficient uniaxial stress for the epitaxy layer, which can be the major stressor for the SiGe or Ge channel fin field-effect
transistor or nanowire at the 10nm node and beyond. According to the results of technology computer-aided design simulation, nanowires integrated with ENFF show excellent electrical performance for
uniaxial stress and band offset. The ENFF process is compatible with the state of the art mainstream technology, which has a good potential for future applications.
Optical Field Confinement Enhanced Single ZnO Microrod UV Photodetector Chin. Phys. Lett. 2017, 34 (7): 078503 . DOI: 10.1088/0256-307X/34/7/078503
ZnO microrods are synthesized using the vapor phase transport method, and the magnetron sputtering is used to decorate the Al nanoparticles (NPs) on a single ZnO microrod. The micro-PL and $I$–$V$
responses are measured before and after the decoration of Al NPs. The FDTD stimulation is also carried out to demonstrate the optical field distribution around the decoration of Al NPs on the
surface of a ZnO microrod. Due to an implementation of Al NPs, the ZnO microrod exhibits an improved photoresponse behavior. In addition, Al NPs induced localized surface plasmons (LSPs) as well as
improved optical field confinement can be ascribed to an enhancement of ultraviolet (UV) response. This research provides a method for improving the responsivity of photodetectors.
Energy Conditions and Constraints on the Generalized Non-Local Gravity Model Chin. Phys. Lett. 2017, 34 (7): 079801 . DOI: 10.1088/0256-307X/34/7/079801
We study and derive the energy conditions in generalized non-local gravity, which is the modified theory of general relativity obtained by adding a term $m^{2n-2}R\Box^{-n}R$ to the Einstein–Hilbert
action. Moreover, to obtain some insight on the meaning of the energy conditions, we illustrate the evolutions of four energy conditions with the model parameter $\varepsilon$ for different $n$. By
analysis we give the constraints on the model parameters $\varepsilon$.
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{a = 1, \ldots , n} j_{U_ a!}\underline{S_ a} } \right)$$
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ROOPSD | CRAN/E
R Object Oriented Programming for Statistical Distribution
CRAN Package
Statistical distribution in OOP (Object Oriented Programming) way. This package proposes a R6 class interface to classic statistical distribution, and new distributions can be easily added with the
class AbstractDist. A useful point is the generic fit() method for each class, which uses a maximum likelihood estimation to find the parameters of a dataset, see, e.g. Hastie, T. and al (2009) .
Furthermore, the rv_histogram class gives a non-parametric fit, with the same accessors that for the classic distribution. Finally, three random generators useful to build synthetic data are given: a
multivariate normal generator, an orthogonal matrix generator, and a symmetric positive definite matrix generator, see Mezzadri, F. (2007) .
• Version0.3.9
• R version≥ 3.3
• Needs compilation?No
• Last release09/11/2023
Last 30 days
Last 365 days
The following line graph shows the downloads per day. You can hover over the graph to see the exact number of downloads per day.
Data provided by cranlogs
• Depends1 package
• Imports4 packages
• Reverse Imports1 package | {"url":"https://cran-e.com/package/ROOPSD","timestamp":"2024-11-03T23:32:34Z","content_type":"text/html","content_length":"64431","record_id":"<urn:uuid:1d1608d3-4b33-4338-89d0-7cdc4bd18d43>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00628.warc.gz"} |
Use of Ti 84 for binomial distribution - Emily Learning
Use of Ti 84 for binomial distribution
Let’s talk about the use of the Ti 84 for binomial distribution in this post. The graphic calculator that we refer to is the Ti 84 which most junior colleges in Singapore use for the H2 A Level Math
Two common functions of Ti 84 for binomial distribution
In the binomial distribution topic in H2 A Level Math Statistics, the two most important functions that we will use from the Ti 84 graphic calculator are:
How to find binompdf and binomcdf in Ti 84
To find binompdf and binomcdf, do these:
Step 1: Press [2nd]
Step 2: Press [Vars]. This will bring you to Distr.
Step 3: To go to binompdf, it is [A] for most Ti-84. To go to binomcdf, it is [B] for most Ti 84.
When do you use binompdf from Ti 84 for binomial distribution questions?
Use binompdf when you are looking at probability of X being a certain value. For example, P(X=3).
Inputs for binompdf
There are 3 inputs required to use binompdf:
• trials (n)
• p : the probability of success of each trial
• x
Sample question on use of binompdf in Ti 84
For instance, if X ~ B(10, 0.3), and we are interested in P(X=3), then we will input these into binompdf:
trials = 10; p = 0.3; x= 3
P(X=3) = 0.267 (3 s.f.)
When do you use binomcdf from Ti 84 for binomial distribution questions?
Use binomcdf when you are looking at probability of X being a certain range. Do note that by default, binomcdf is used to find P(X≤ x). Hence, in order to use binomcdf, make sure you have probability
written in terms of P(X≤ x).
For example if you have P(X<3), you’ll have to change it to P(X≤2) in order to use binomcdf.
Similarly, if you have P(X≥ 5), you’ll have to change it to 1- P(X≤4) in order to use binomcdf.
Inputs for binomcdf
There are 3 inputs required to use binomcdf are the same as binompdf:
• trials (n)
• p : the probability of success of each trial
• x
Key difference is that for binomcdf, we are finding P(X≤x) but for binompdf, we are finding P(X=x).
Sample question on use of binomcdf in Ti 84
Sample question 1
For instance, if X ~ B(10, 0.3), and we are interested in P(X≤3), then we will input these into binomcdf:
trials = 10; p = 0.3; x= 3
P(X≤3) = 0.650 (3 s.f.)
Sample question 2
For instance, if X ~ B(10, 0.3), and we are interested in P(X>5), here are the steps on how to find this probability using Ti-84 binomcdf function.
Since what is required is P(X>5), we have to re-write it in terms of P(X≤x)
P(X>5) = 1- P(X≤5)
Now, we can find P(X≤5) using binomcdf function of Ti-84, with the following inputs:
trials = 10; p = 0.3; x= 5
P(X≤5) = 0.95265
P(X>5) = 1- P(X≤5) = 1 – 0.95265 = 0.473 (3 s.f.)
Want a complete course on H2 A Level Math Statistics?
Check out our course here, where we cover all the topics tested in statistics.
The course includes concepts, types of questions, and how to apply them, so that you’ll score for your exam. | {"url":"https://emilylearning.com/use-of-ti-84-for-binomial-distribution/","timestamp":"2024-11-06T08:25:30Z","content_type":"text/html","content_length":"74543","record_id":"<urn:uuid:7c2c683e-9634-4a5f-aa1e-3911d01dd47b>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00810.warc.gz"} |
How to calculate annual growth rate percentage
Get or set growth rates of a tis time series in annual percent terms. number of lags to use in calculating the growth rate as outlined in the details below. simple.
To calculate growth rate, start by subtracting the past value from the current value. Then, divide that number by the past value. Finally, multiply your answer by 100 to express it as a percentage…
We measure growth in terms of percentage, and it is calculated by AAGR or Annual Average Growth Rate and CAGR that is Compound Average Growth Rate. How to calculate Growth Percentage. Calculating
growth percentage may sound intimidating if you are not aware of the process. Do not worry this mathematical procedure is simple and easy to use. Calculating the annual percentage growth rate is the
easiest and the quickest way to compare the yearly growth of any trend. These trends can be in the form of any comparable thing, which can include the yearly growth of company, rise in the salary of
an employee, population growth of an urban or rural area and increase in the height and weight of a person. The annual rate is equivalent to the growth rate over a year if GDP kept growing at the
same quarterly rate for three more quarters (or the same average rate). Calculating the real GDP growth rate
Calculating the annual percentage growth rate is the easiest and the quickest way to compare the yearly growth of any trend. These trends can be in the form of any comparable thing, which can include
the yearly growth of company, rise in the salary of an employee, population growth of an urban or rural area and increase in the height and weight of a person.
'compound annual growth rate' (CAGR) formula, which assesses the pace from SD objectives. : Calculation of trend not possible (for example, time series too short) Method for calculating average
scores at the goal level. The calculation of 27 Dec 2019 This article will help you learn how to calculate your growth year over year. First can help you analyze different aspects of yearly growth
and see how your Multiply it by 100 to convert this growth rate into a percentage rate. Enter the investment amount, the compound annual growth percentage rate, and the number of periods (months or
years) and the calculator will calculate the Definition: Annual growth rate of real Gross Domestic Product (GDP) per capita is calculated as the percentage change in the real GDP per capita between
two CAGR stands for Compound Annual Growth Rate, which is the annual average rate of return for an investment over a period of time. The formula for calculating
Learn how to to calculate the Compound Annual Growth Rate (CAGR) in Excel with these 4 easy examples. Includes Apply Percentage Formatting to CAGR.
To find the Percentage Growth Rate, the formula is. Percentage Growth Rate = ( Ending Value / Beginning Value ) – 1. Average Annual Growth Rate is the rise in your investment over some time as it
estimates the average growth rate over a constant period. Remember, the growth rate will fluctuate for every year.
Learn how to to calculate the Compound Annual Growth Rate (CAGR) in Excel with these 4 easy examples. Includes Apply Percentage Formatting to CAGR.
Calculating the annual percentage growth rate is the easiest and the quickest way to compare the yearly growth of any trend. These trends can be in the form of any comparable thing, which can include
the yearly growth of company, rise in the salary of an employee, population growth of an urban or rural area and increase in the height and weight of a person. The annual rate is equivalent to the
growth rate over a year if GDP kept growing at the same quarterly rate for three more quarters (or the same average rate). Calculating the real GDP growth rate 1. Calculating Percent (Straight-Line)
Growth Rates. The percent change from one period to another is calculated from the formula: Where: PR = Percent Rate V Present = Present or Future Value V Past = Past or Present Value. The annual
percentage growth rate is simply the percent growth divided by N, the number of years. Example Few workers receive raises in consistent percentages each and every year. It may be helpful to calculate
an annual rate of growth of a salary to determine the average annual increase from one Interpret your result as a percentage. The growth rate formula provides you with a final result as a decimal
number. To convert this to a percentage form that makes sense to economists, multiply by 100%. You can then report the annual growth rate as a percentage figure.
Calculating DCF Growth Rates. Since I show a lot of valuations
The Compound Annual Growth Rate formula requires only the ending value of is not influenced by percentage changes within the investment horizon that may
Over 10 years, however, the average annual rate of growth is much smaller than 20%, let alone 25%. Here's how to calculate the annual rate of growth, using the example above. Step 1. Find the
percentage change in your salary. The example starts with a $40,000 salary. It is now $60,000. | {"url":"https://binaryoptionssmjs.netlify.app/feela66067wo/how-to-calculate-annual-growth-rate-percentage-wyxu.html","timestamp":"2024-11-10T02:57:04Z","content_type":"text/html","content_length":"30065","record_id":"<urn:uuid:c5214114-a96d-40ca-92f8-58804f5e36e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00268.warc.gz"} |
SUMSQ() Formula in Google Sheets
Returns the sum of the squares of a series of numbers and/or cells.
Common questions about the SUMSQ formula:
- What does the SUMSQ formula do?
- What are the syntax requirements for the SUMSQ formula?
- How can I use SUMSQ to sum the squares of a range of cells?
- How can I use SUMSQ to sum multiple cells within a range?
How can the SUMSQ formula be used appropriately:
- The SUMSQ formula can be used to add up the squares of a range of cells.
- To use the formula correctly, the syntax should follow the pattern SUMSQ(cell_range).
- This formula is useful when calculating statistics such as population variance and standard deviation.
How can the SUMSQ formula be commonly mistyped:
- The formula can be mistyped as SUMSQUE, SUMQUE, SUMSQU, or SUMQ.
- If the syntax of the formula is incorrect, you may receive an #NAME? error.
What are some common ways the SUMSQ formula is used inappropriately:
- Adding up the squares of a range of cells when the SUM formula is more appropriate.
- Using SUMSQ when you should actually be using SUM.
- Not correctly referencing the cell range when calculating the sums.
What are some common pitfalls when using the SUMSQ formula:
- Accidentally passing a single cell reference rather than a range of cells.
- Forgetting to add parenthesis to the argument of the formula (SUMSQ(cell_range)).
- Not considering mixed data types within the range of cells you’re passing to the formula.
What are common mistakes when using the SUMSQ Formula:
- Not including a comma in between two cell references passed to the formula.
- Passing a range of cells that isn't an exact square to the formula, as this won't return the expected result.
- Forgetting to use the $ symbol to lock a column or row when referencing a range in a formula.
What are common misconceptions people might have with the SUMSQ Formula:
- Thinking the SUMSQ formula works the same as the SUM formula, when in fact they are two different formulas.
- Believing that the SUMSQ formula will return the sum of the cells in the range when it actually returns the sum of the squares of the cells in the range.
- Thinking that the SUMSQ formula only works with numerical values, when it can in fact work with other data types. | {"url":"https://www.bettersheets.co/formulas/sumsq","timestamp":"2024-11-14T08:01:40Z","content_type":"text/html","content_length":"31753","record_id":"<urn:uuid:aa4fdc08-449b-431b-82eb-73da5101bbd6>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00259.warc.gz"} |
Start Exploring Keyword Ideas
Use Serpstat to find the best keywords for your website
PPC , August 26, 2020 | 8626 106 1 | 17 min read – Read later
How To Test Bids In PPC Advertising Without Applying Them
Andrey Belousov
Growth Hacker at Serpstat
A/B tests are an integral part of any marketing strategy because you can increase its effectiveness in several times. In contextual advertising such testing also takes place, for example, to create a
high-quality, attractive ad. But are they really that effective? Let's deal with them, and also find out another secret of successful testing.
They require a huge number of clicks for reliable results.
Almost always, bids are tested only on the part of ad campaigns. The success of one variant over the other on the part of the account doesn't mean its advantage on the entire account.
Unlike testing changes on a website, it is not enough to simply compare the conversion rate of two options. You need to align the budget or CPA or CRR and compare the number of conversions or
revenue. However, the alignment of budget and CPA occurs with some margin of error.
In general, even with an infinite amount of data, we will be able to determine which option is better only if the options have a difference in the efficiency of more than 3-5%. Let's call this
theoretical sensitivity.
The following factors influence theoretical sensitivity:
We don't test the entire account, but a specific part of it. Moreover, the keywords we test are not randomly selected.
We align the budget / CPA / CRR for the two options with some margin of error.
Checkerboard order is not the same as randomly distributing clicks between options.
In practice, since we don't have an infinite amount of data, even for large sites, the sensitivity threshold of the A/B test is about 10-20%. If the difference in effectiveness between the two
options is less than this threshold, then doing A/B is not much better than flipping a coin. Let's call this practical sensitivity.
As a result, A/B testing:
• is not accurate;
• takes much time;
• requires a lot of work and specific knowledge;
• cannot be fully automated.
And after all, there is always a possibility that new rates will work worse.
Equal Bid Improvement Factor
Therefore, there may be a metric that is more sensitive (even theoretical) than the A/B test. Moreover, this metric can be simply calculated using historical data without taking into account bids. If
we automate this, then it can be done in a couple of clicks.
However, since we don't know the dependence of the number of clicks and CPC on the rate, then this metric will produce quantitative results with a rather large margin of error, but possibly quite
accurate qualitative results.
In other words, this metric will answer the question "Which option is better?" Rather well, but rather poorly on the question "How much better?".
Intermediate calculations may seem voluminous to you, but the final formula will be short.
For the sake of simplicity, let's assume that each keyword has:
CPC and number of clicks are proportional to the bid;
conversion probability is independent of the bid.
Let's set the bids in proportion to the estimate:
f - forecast the probability of conversion
L is a constant to fit our constraints (budget, target CPA)
Let p be the probability of conversion, then the expected number of conversions is:
The expenditure for a keyword is:
Let our goal be the maximum conversions with a fixed CPA = TargetCPA.
The expected CPA is:
To find L, let's equate the target CPA with the expected one:
From here you can calculate L:
We introduce the weighted mean operator:
From which it follows that:
You can also easily see that the constant can be taken out of the weighted mean:
For simplicity, we will assume that c and f are not correlated.
Let's calculate the expected number of conversions:
Let's calculate what will happen at equal rates. Let our forecast for all keys be the same and equal to some constant m. Without loss of generality, we can say that m is equal to the weighted average
Now let's divide the number of conversions for arbitrary bids by the number of conversions for equal bids. Let's get how many times bids bring more conversions than equal bids for the same CPA. Let's
call this metric the Equal Bid Improvement Factor (IBIF).
With equal rates, IBIF should be equal to 1. For example, all rates are equal to the constant S:
W [p] is just a multiplier that doesn't depend on the estimate.
W [fp] - grows with an increasing correlation between the estimate and the probability of conversion. It also grows with bid variance (spread).
W [ff] - grows with increasing variance of bids.
IBIF can be expressed in terms of rate variance and the correlation between bids and estimates.
C is Pearson's weighted correlation coefficient between bids and predictions. The measure of the adequacy of the bids doesn't exceed 1. Equal to 1 for an ideal estimate, equal to 0 for random rates.
V [] - weighted relative variance, the measure of the spread. Greater than or equal to zero. For example, if V [f] = 0, it means that all bids are the same. V [p] <1 is almost certainly true. Usually
V [p] is about 0.5. V [f] is usually not very different from V [p].
It can be seen that, in general, the larger V [f], the stronger the IBIF depends on the correlation. It is also easy to see that at high correlations (more than 0.95), the IBIF is practically
independent of V [f].
The more different the conversions for keywords, the higher the IBIF can be obtained with an ideal estimate.
From the fact that it is almost certain that V [p] <1, we get that the IBIF is almost always less than 2. The opposite indicates, most likely, that a significant number of keywords have almost zero
conversion (much less than the average for the site).
This can happen, for example, when you buy most of the clicks on products you don't have in stock, on 404 pages, or irrelevant pages; or some of the pages don't have a conversion tracking code.
In other words, IBIF > 2 may indicate that rates solve some specific problem.
Our specialists will contact you and discuss options for further work. These may include a personal demonstration, a trial period, comprehensive training articles, webinar recordings, and custom
advice from a Serpstat specialist. It is our goal to make you feel comfortable while using Serpstat.
Finding IBIF from historical data
You can calculate IBIF from historical data using the formula:
r - keyword conversion rates
n - number of clicks on keywords
Note that f should not directly depend on r. That is, f and r must be calculated at different periods.
In other words, you need to run the test like this:
For example, on the 1st day we withdraw bids or calculate the forecast. Remember f.
After a couple of weeks, for example on the 16th, we take data from the 2nd to the 15th. And we get n and r.
We calculate the IBIF, the variant with the higher IBIF is considered the winner.
Since we don't apply bids, we just look at the statistics, we will call this test passive.
If we multiply f by some constant factor S that is the same for all keywords, then the IBIF will not change:
Therefore, instead of f, you can take not only the forecast but also a set of rates that are proportional to the original estimates. That makes it possible to compare not only forecasts but also sets
of bids with each other or a set of bids with a forecast.
You can give the following assessment of the maximum IBIF (with an ideal forecast):
r - conversion rate for the first period (regular rates are considered in this period)
n - number of clicks in the first period
R - conversion rate for the second period (periods should not overlap)
If you don't have a second period, then you can divide the first period into 2 parts (equal) then:
r - conversion rate in the first part
R - conversion rate in the second part
n - number of clicks for the entire period
This formula doesn't depend on uniform seasonality when the conversion of all keywords decreases or increases by the same number of times.
The accuracy of calculating the limiting IBIF for small data is several times less than that of a conventional IBIF.
We have examined the problem "maximum number of conversions with a fixed CPA". However, IBIF can be applied to almost all tasks.
If we face the task of "maximum turnover with a fixed CRR", then the IBIF formula will not change. You just need to substitute the ratio of the turnover to the number of clicks instead of r.
However, the rule V [p] <1 and IBIF <2, in this case, may not be observed.
If we have a fixed budget, then after calculations we get:
Which will not lead to qualitative changes. If one forecast has a higher IBIF, then it has a higher root from the IBIF.
You can similarly calculate the IBIF not only for bids but also for bid modifiers by time of day, day of the week, and so on.
Seasonality accounting can also be evaluated.
To compare 2 sets of bids or forecasts, you need to calculate this parameter:
r - keyword conversion rates or revenue-to-click ratio
n - number of clicks on keywords
f - bids or forecast for conversion or revenue per click. When calculating them, data for the period for which r was taken should not be used.
W [] average weight operator
Equal bids give IBIF = 1. The maximum IBIF can be calculated as follows:
Where R is the conversion rate of keywords or the ratio of revenue to clicks for another period that does not overlap with the first.
To calculate this indicator in Excel, first you need to get rid of the weighted mean.
n (clicks) are in column A
r (conversion) are in column B
f (forecast/bids) are in column C
Then the formula will look like this:
However, the formula can be simplified if column D contains conversions and revenue:
Let R be in column G. Then the formula is:
SUM(A1:A)*SUMPRODUCT(A1:A,B1:B,G1:G)/(SUMPRODUCT(A1:A,B1:B) *SUMPRODUCT(A1:A,G1:G))
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MATH 230 - Calculus 3 - mVirtualOffice.com
MATH 230 – Calculus 3
Course Title: Calculus 3 (High School Level)
Course Description: Calculus 3 is an advanced calculus course that builds upon Calculus 1 and 2. This course introduces students to multivariable calculus, including topics such as vectors,
three-dimensional space, multiple integrals, and vector calculus.
Course Objectives: By the end of this course, students should be able to:
1. Understand and work with vectors in three-dimensional space.
2. Calculate partial derivatives and gradients.
3. Compute multiple integrals and evaluate line integrals.
4. Apply calculus concepts to problems involving three-dimensional space.
5. Analyze and solve problems using vector calculus.
• [Insert Calculus 3 Textbook Title and Author(s)]
• Notebook or binder for class notes and assignments.
• Graphing calculator (if required by the school or teacher).
• Pencils, erasers, and a ruler.
• Homework/Classwork: XX%
• Quizzes: XX%
• Tests: XX%
• Projects: XX%
• Final Exam: XX%
Course Outline:
Unit 1: Vectors in Three-Dimensional Space
• Introduction to vectors and vector operations.
• Dot product and cross product.
• Lines and planes in three-dimensional space.
Unit 2: Multivariable Functions and Partial Derivatives
• Multivariable functions and their graphs.
• Partial derivatives and their applications.
• Gradients and directional derivatives.
Unit 3: Multiple Integrals
• Double integrals and their applications.
• Triple integrals and volume calculations.
• Change of variables in multiple integrals.
Unit 4: Line Integrals and Vector Fields
• Line integrals and work.
• Green’s theorem and its applications.
• Conservative vector fields.
Unit 5: Vector Calculus
• Curl and divergence of vector fields.
• Stokes’ theorem and the divergence theorem.
• Applications of vector calculus.
Unit 6: Review and Final Exam Preparation
Note: This is a general example of a high school Calculus 3 syllabus. It’s important to adapt it to meet the specific needs and standards of your school or district. The percentages for grading, the
textbook, and materials may vary. Additionally, consult with your school’s curriculum guidelines and any state or district standards that may apply. Calculus 3 is typically a college-level course, so
the level of rigor and depth may vary based on the high school’s curriculum.
Free Textbooks
Finding completely free high school-level Calculus 3 textbooks can be challenging, as Calculus 3 is typically taught at the college level. However, there are open educational resources (OER) and free
online resources that can help you study Calculus 3 concepts. Here are some options:
While these resources may not be specifically labeled as “high school Calculus 3,” they cover the relevant topics and can be used for self-study at a high school level. Keep in mind that high
school-level calculus courses may vary in depth and content from one school or district to another, so it’s essential to align your studies with your specific curriculum.
Finding a free Massive Open Online Course (MOOC) specifically focused on high school-level Calculus 3 can be challenging, as Calculus 3 is typically a college-level course. However, you may find
college-level calculus courses on MOOC platforms that cover topics relevant to Calculus 3. Here are some MOOC platforms where you can explore calculus courses:
While these platforms offer calculus courses that cover topics relevant to Calculus 3, keep in mind that the level of depth and rigor may vary, and they may be more suitable for advanced high school
students. Be sure to explore the course descriptions and offerings on each platform to find the one that aligns best with your learning objectives and needs. | {"url":"https://mvirtualoffice.com/math/math-230-calculus-3/","timestamp":"2024-11-09T22:35:35Z","content_type":"application/xhtml+xml","content_length":"37724","record_id":"<urn:uuid:2dfeeef0-1d2e-4f47-932c-dd4b50dba809>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00125.warc.gz"} |
Downscaling Solar Power Output to 4-Seconds for Use in Integration Studies (Presentation): NREL (National Renewable Energy Laboratory)
High penetration renewable integration studies require solar power data with high spatial and temporal accuracy to quantify the impact of high frequency solar power ramps on the operation of the
system. Our previous work concentrated on downscaling solar power from one hour to one minute by simulation. This method used clearness classifications to categorize temporal and spatial variability,
anditerative methods to simulate intra-hour clearness variability. We determined that solar power ramp correlations between sites decrease with distance and the duration of the ramp, starting at
around 0.6 for 30-minute ramps between sites that are less than 20 km apart. The sub-hour irradiance algorithm we developed has a noise floor that causes the correlations to approach0.005. Below
oneminute, the majority of the correlations of solar power ramps between sites less than 20 km apart are zero, and thus a new method to simulate intra-minute variability is needed. These intra-minute
solar power ramps can be simulated using several methods, three of which we evaluate: a cubic spline fit to the one-minute solar power data; projection of the power spectral density toward the
higherfrequency domain; and average high frequency power spectral density from measured data. Each of these methods either under- or over-estimates the variability of intra-minute solar power ramps.
We show that an optimized weighted linear sum of methods, dependent on the classification of temporal variability of the segment of one-minute solar power data, yields time series and ramp
distributionssimilar to measured high-resolution solar irradiance data.
Original language American English
Number of pages 24
State Published - 2013
Publication series
Name Presented at the 2013 International Workshop on Integration of Solar Power into Power Systems, 20 - 22 October 2013, London, United Kingdom
• downscaling
• high frequency solar data
• solar integration studies
Dive into the research topics of 'Downscaling Solar Power Output to 4-Seconds for Use in Integration Studies (Presentation): NREL (National Renewable Energy Laboratory)'. Together they form a unique | {"url":"https://research-hub.nrel.gov/en/publications/downscaling-solar-power-output-to-4-seconds-for-use-in-integratio-2","timestamp":"2024-11-04T21:10:38Z","content_type":"text/html","content_length":"48758","record_id":"<urn:uuid:d0be589a-ab42-4322-aaee-3369591cd200>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00577.warc.gz"} |
P-51 Mustang & F-82 Twin Mustang Proposals and Variants
All hail the God of Frustration!!!
Staff member
Senior Member
15 April 2006
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Hi folks,
A new thread for and unknown, secret or rare North American P-51 Mustang variants. I'll start off by including what may have been the ultimate Mustang:
This was supposedly a proposal that North American had come up with in case the war had continued. As you can see, it had forward swept wings as well as an advanced (specifically how I don't know), a
Westinghouse jet engine in the rear fuselage and a tricycle undercarriage.
ACCESS: Above Top Secret
Senior Member
13 August 2007
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i found link to Italian blog
with picture of a Gunpod for F-82
Il pod sperimentale progettato per aumentare la potenza di fuoco del caccia a lungo raggio North American F-82 Twin Mustang. Esso conteneva 8 mitragliatrici Browning M2 calibro 50, che andavano ad …
after page the pod supply 8 Browning M2 (each with 440 shots, caliber 50)
6 gun in wing 2 under the pod (make in total 14 guns in F-82)
23 May 2009
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14 !! .50 caliber BMG...now THAT's an attention step! Wonder how much the aircraft slowed as these were being fired? If memory serves, we lost about 5 kt per second when firing the GPU-5 pods (30mm,
same round as A-10's GAU-8) off our F-4Es.
30 January 2008
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Weasel Pilot said:
Wonder how much the aircraft slowed as these were being fired? If memory serves, we lost about 5 kt per second when firing the GPU-5 pods (30mm, same round as A-10's GAU-8) off our F-4Es.
No contest! A (probably over-) simplified, back-of-the-envelope calculation shows that:
Force = the time derivative of the impulse.
Impulse = mass X velocity
Force (recoil): mass X velocity X rate-of-fire
This ignores the recoil produced by the propellant gasses leaving the barrel together with the projectile (but I said it was simplified
So, for the GAU-13 in the GPU-5 pod:
Projectile mass: 15,1 oz (for the API) = 430 g = 0,430 kg
Muzzle velocity: 3.600 ft/sec = 1.030 m/s
rate of fire: 2.400 rds/min = 40 rds/sec
Recoil: 0,430 X 1.030 X 40 = 17.716 N
For 14 Browning M2 .50 cal's:
Projectile mass: 622,5 grains (for the M8 API) = 40,3 g = 0,0403 kg
Muzzle velocity: 3.050 ft/sec = 930 m/s
Rate of fire (per gun): 750 rds/min = 12,5 rds/sec
Rate of fire (combined): 14 X 12,5 = 175 rds/sec
Recoil: 0,0403 X 930 X 175 = 6.558 N
or roughly 1/3 that produced by the GAU-13.
The source for the above data is Wikipedia and Janes Infantry Weapons, 2009 edition.
The above calculation, as stated, ignores the recoil force produced by the propellant gasses (both the extra mass being ejected, as well as the propulsive "rocket" effect of the gas leaving the
muzzle), and it tells you little about the "felt" recoil, since the mass of the gun itself is ignored. However, since 14 Browning M2's mass about 530 kg (38 kg pr. gun) and the GAU-13 comes in at 151
kg, "felt" recoil of the GAU-13 (1/3 the mass and 3 times the generated recoil force) should be significantly more than that of the 14 M2's.
There's probably a couple of 30mm-size holes in my calculations, so corrections are welcome (provided they are suitably polite :
Regards & all,
Thomas L. Nielsen
27 April 2007
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ı remember reading the A-10 was getting about 4500 kg of drag through gun . It fired faster at 4200 rpm (?) but are the guns same , do they have the same velocity out of the barrel ?
30 January 2008
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r16 said:
....do they have the same velocity out of the barrel ?
As I read it, the difference between the GAU-13 (gunpod) and the GAU-8 (in the A10) was the number of barrels. 4 in the GAU-13 vs. 7 in the GAU-8. Muzzle velocity should therefore be the same for the
two weapons.
PS, then a drag force of 4.500 kg corresponds to about 45.000 N. Adjusting the calculation above for increased rate of fire (4.200 rds/min instead of 2.400 rds/min) gives a recoil force of 31.000 N.
The difference up to the 45.000 N would be due to the over-simplified nature of the calculation, such as ignoring the contribution from the propellant gases*.
Regards & all,
Thomas L. Nielsen
*) Actually, I seem to remember reading somewhere (don't as me for a source) that roughly 1/3 of the recoil force from a firearm comes from the ejected propellant gases. Adjusting for this brings us
right up to the 45.000 N...... B)
16 January 2008
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Here some additional info ...
30 June 2009
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Probably a dumb question: Where would the .50's go?
This wing doesn't seem to have the required space to store the then standard complement of 6*.50 Browning guns.
19 May 2006
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Interesting that the prop was to rotate 'the wrong direction'!
ACCESS: Above Top Secret
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13 August 2007
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your sure it a jet engine ?
look more a motorjet (aka thermojet)
there was also a P-51 Ramjet version
fantasic art with P-51 ramjet
more on Jet Mustangs
F-82 Twin Mustang
I have found a Schmued patent of 1944/46 representing the XP-82 Twin-Mustang (at http://www.adventurelounge.com/aircraft/design/087.html and http://www.adventurelounge.com/aircraft/full/design/
087.html ) but I am not sure:
- if the canopy is the same as the P-51D/P-82, this would be a tiny Twin-Mustang (left below)
- if the engine/nose is the same as the P-51D/P-82, this would be a Twin-Mustang with giant canopy (right below)
Dimensions or details would help but I do not find the 3 pages patent both at Google Patents and at Espacenet… May someone help me?
The explanation may be the big canopy of the XP-51F/G/J that would have been considered also for the P-82, before prefering the P-51D/H:
1 September 2007
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In the case of the P-82 Twin Mustang, it's important to note that the Twin shared far more in common with the P-51H than the P-51D. The lightweight Mustangs (F/G/J/H and P-82) used a new airfoil
shape, new wing planform, and essentially represented an all-new airplane (aside from the powerplant.)
Thanks for this detail, while the canopy size issue remains the same D/H/82-size or F/G/J?
2 September 2008
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Michel Van said:
fantasic art with P-51 ramjet
Thanks, glad you like it!
Attached a crop of this image but edited to look like an old magazine photo.
Here's an older piece with the same aircraft but with MTO markings. Needless to say it's all fictional.
The aircraft in the background is a Junkers EF 100 transport, dressed up as "Immelmann V"...
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27 May 2007
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Great work! That postcard is especially well done, as most artificially aged stuff is not convincing at all.
6 January 2013
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GTX said:
Hi folks,
A new thread for and unknown, secret or rare North American P-51 Mustang variants. I'll start off by including what may have been the ultimate Mustang:
This was supposedly a proposal that North American had come up with in case the war had continued. As you can see, it had forward swept wings as well as an advanced (specifically how I don't
know), a Westinghouse jet engine in the rear fuselage and a tricycle undercarriage.
Can you send me a high res image of this drawing. Thanks!
Steve Pace
4 December 2009
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What did the prototype of this AC look like? Did they attempt splicing two Mustangs together before they went to a new design?
19 May 2006
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No. It looked basically like the F-82 was built.
verner said:
What did the prototype of this AC look like? Did they attempt splicing two Mustangs together before they went to a new design?
4 December 2009
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6 November 2010
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Arjen said:
Stargazer2006 said:
richard B said:
In Green's "Fighters" vol 4 p 141 ,there is a photo of the mock-up of a Mustang fitted with a RR Merlin aft of the cockpit : " y " is not a what-if , but a serious project.
I have a copy of William Green's "Fighters" vol 4, will scan image. Real project.
<edit> Found an image
<edit>Added William Green's image.
3 May 2006
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It's not a Merlin, but a Griffon engine. This was a Rolls Royce testbed.
6 November 2010
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It's not a Merlin, but a Griffon engine.
Looking at the size of the engine that might well be true. Do you have more information?
I found this
The Merlin conversion was very promising, but the director of the Hucknall establishment, Ray Dorey, had an even more ambitious idea. He wanted to mate the Mustang to a Griffon engine, mounted
behind the pilot as was the Allison in the Bell P-39 Airacobra. Rolls-Royce engineers believed this aircraft would be capable of a top speed of 800 KPH (500 MPH), but it never progressed beyond
the mock-up stage.
This was a Rolls Royce testbed.
William Green's image caption: "The mock-up of Rolls-Royce's proposal to mount a Merlin engine in the Mustang aft of the cockpit."
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17 February 2006
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Try the searchfunction for Rolls Royce FTB Mustang
and /or FTB Mustang...
Good luck.
6 November 2010
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I knew about the F.T.B. Mustang project, but I thought the one depicted in the model was different. It uses a P-51B-type fuselage and canopy instead of the later P-51D-type used in your graphics,
6 November 2010
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lark said:
Try the searchfunction for Rolls Royce FTB Mustang
and /or FTB Mustang...
Good luck.
Knowing what to look for helps
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5 April 2006
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ACCESS: Top Secret
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31 May 2009
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North American Aviation FSW Mixed Propulsion P-51 Mustang (proposed) factory model.
Flight-Sim Game Developer
25 September 2013
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Fascinating. What books describe the forward-swept P-51 prototype or other prototypes?
I have found one paragraph in
Mustang Designer: Edgar Schmued and the P-51
, Ray Wagner.
This book does have 1 or 2 pages of details and pictures about a
-powered P-51.
North American F-86 Sabre Owners' Workshop Manual
mentions advanced P-51 prototypes
before those evolved into the FJ/F-86.
I've been collection information about the exotic variants of the P-51 Mustang here:
(comments welcome)
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3 June 2006
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Re: Mustang Variants / North American D-118
Orionblamblam said:
North American D-118
A North American Aviation concept for a highly modified P-82, dating from 1949. The piston engines would be removed and replaced with Allison XT-38 turboprops. The engines would be located
mid-fuselage, necessitating that the cockpit would have to be moved well forward of their normal position. The end result would be a plane that weighed the same, gave the pilots better views and
went substantially faster.
This one may show up in a future issue of USBP, as the intended role was ground attack.
Nice find, Orionblamblam! B)
2 September 2008
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Staff member
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Senior Member
11 March 2006
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Not a 3-view, rather a very basic general arrangement drawing:
Jemiba said:
Not a 3-view, rather a very basic general arrangement drawing:
Thanks, Jens. I almost did one myself but thought I was lacking some solid reference for the XT38 in that configuration.
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11 March 2006
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There's no solid reference, of course, just that artist impression !
As the engine was to be positioned in the mid fuselage, there would have been hardly
much impact on the shape of fuselage, besides the repositioned cockpit and the deletion of
the belly scoop, I think. At least that's my interpretation of the drawing and of what seems
to be an intake directly in the nose, but I admit, that it would be appropriate to the "Theoritical
and Speculative" section, too.
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5 April 2006
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Jemiba said:
There's no solid reference, of course, just that artist impression !
Not true. There are two three-views, one detailed inboard profile, one overview of removable panels... | {"url":"https://www.secretprojects.co.uk/threads/p-51-mustang-f-82-twin-mustang-proposals-and-variants.2397/","timestamp":"2024-11-09T13:31:43Z","content_type":"text/html","content_length":"330547","record_id":"<urn:uuid:ad498fbd-0c52-4e97-8453-df4750a55244>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00741.warc.gz"} |
Crystal Prison Zone
When we sit down to play a game involving dice, we understand that the results are influenced by a combination of strategy and luck. However, it's not always clear which is more important. While we'd
like to think our results are chiefly the result of good strategy, and that the role of luck was fair and minimal, it's often difficult to judge. How can we make games which incorporate luck while
rewarding strategy?
In order to add some excitement and variety, many game developers like to add dice rolls to their games to introduce an element of randomness. Dice rolls, the argument goes, add an element of chance
that keeps the game from becoming strictly deterministic, forcing players to adapt to good and bad fortune. While some dice rolls are objectively better than others, potentially causing one player to
gain the upper hand over another through luck alone, developers claim that things will "average out" in the long run, with a given player eventually experiencing just as much good luck as bad luck.
Most outcomes are near the average, with equal amounts of "good luck" (area above green line) and "bad luck" (area below red line).
Luck should average out
With the effect of luck averaging out, the player with the better strategy (e.g., the player who obtained better modifiers on their rolls) should still be able to reliably perform better. However,
players and developers alike do not often realize
just how many rolls are necessary
before the effect of strategy can be reliably detected as something above and beyond the effect of luck.
Forums are full of players describing which build has the better average, which is often plain to see with some math. For many players, this is all they need to concern themselves with: they have
done the math and determined which build is most effective. The question for the designer, however, is whether the players can expect to see a difference within a single game or session. As it turns
out, many of these modifiers are so small compared to the massive variance of a pass-fail check that it takes surprisingly long for luck to "average out".
An example: Goofus and Gallant
For the following example, I'll use
Dungeons & Dragons
, since that's one most gamers are likely familiar with.
uses a 20-sided die (1d20) to check for success or failure, and by adjusting the necessary roll, probability of success ranges from 0% to 100% by intervals of 5%. (In future posts I hope to examine
other systems of checks, like those used in 2d6 or 3d6-based RPGs or wargames.)
Consider two similar level-1 characters, Goofus and Gallant. Gallant, being a smart player, has chosen the Weapon Expertise feat, giving him +1 to-hit. Goofus copied all of Gallant's choices but
instead chose the Coordinated Explosion feat because he's some kind of dingus. The result is we have two identical characters, one with a to-hit modifier that is +1 better than the other. So, we
expect that, in an average session, Gallant should hit 5% more often than Goofus. But how many rolls do we need before we reliably see Gallant outperforming Goofus?
For now, let's assume a base accuracy of 50%. So, Goofus hits if he rolls an 11 or better on a 20-sided die (50% accuracy), and Gallant hits on a roll of 10 or better(55% accuracy). We'll return to
this assumption later and see how it influences our results.
I used the statistical software package
to simulate the expected outcomes for sessions involving 1 to 500 rolls. For each number of rolls, I simulated 10,000 different D&D sessions. Using R for this stuff is easy and fun! Doing this lets
us examine the proportion of sessions in which Gallant outperforms Goofus and vice-versa. So, how many trials are needed for Gallant to outperform Goofus?
Goofus hits on 11, Gallant hits on 10 thanks to his +1 bonus.
One intuitive guess would be that you need 20 rolls, since that 5% bonus is 1 in 20. It turns out, however, that even at 20 trials, Gallant only has a 56% probability of outperforming Goofus.
In order to see Gallant reliably (75%) outperform Goofus requires more than a hundred rolls. Even then, Goofus will still surpass him about 20% of the time. It's difficult to see the modifier make a
reliable difference compared to the wild swings of fortune caused by a 50% success rate.
Reducing luck through a more reliable base rate
It turns out these probabilities depend a lot on the base probability of success. When the base probability is close to 50%, combat is "swingy" -- the number of successes may be centered at 50% times
the number of trials, but it's also very probable that the number of successes may be rather more or rather less than the expected value. We call this range around the expected value
. When the base probability is closer to 0% or 100%, the variance shrinks, and the number of successes tends to hang closer to the expected value.
This time, let's assume a base accuracy of 85%. Now, Goofus hits on 4 or better (85%), and Gallant hits on 3 or better (90%). How many trials are now necessary to see Gallant reliably outperform
This time, things are more stable. For very small numbers of rolls, they're more likely to tie than before. More importantly, the probability of Gallant outperforming Goofus increases more rapidly
than before, because successes are less variable at this probability.
Comparing these two graphs against each other, we see the advantages of a higher base rate. For sessions involving fewer than 10 rolls, it is rather less likely that Goofus will outperform Gallant --
they'll tie, if anything. For sessions involving more than 10 rolls, the difference between Goofus and Gallant also becomes more reliable when the base rate is high. Keep in mind that we haven't
increased the size of the difference between Goofus and Gallant, which is still just a +1 bonus. Instead, by making a more reliable base rate, we've reduced the influence of luck somewhat. In either
case, however, keep in mind that it takes at least 10 rolls before we see Gallant outperform Goofus in just
of sessions. If you're running a competitive strategy game, you'd probably want to see a more pronounced difference than that!
In conclusion
To sum it all up, the issue is that players and developers expect luck to "average out", but they may not realize how many rolls are needed for this to happen. It's one thing to do the math and
determine which build has the better expected value; it's another to actually observe that benefit in the typical session. It's my opinion that developers should seek to make these bonuses as
reliable and noticeable as possible, but your mileage may vary. This may be more important for certain games & groups than others, after all.
My advice is to center your probabilities of success closer to 100% than to 50%. When the base probability is high, combat is less variable, and it doesn't take as long for luck to average out. Thus,
bonuses are more reliably noticed in the course of play, making players observe and enjoy their strategic decisions more.
Less variable checks also have the advantage of allowing players to make more involved plans, since individual actions are less likely to fail. However, when an action does fail, it is more
surprising and dramatic than it would otherwise have been when failure is common. Finally, reduced variability allows the party to feel agentic and decisive, rather than being buffeted about by the
whims of outrageous fortune.
Another option is to reduce the variance by dividing the result into more fine-grained categories than "success" and "failure" such as "partial success". Some tabletop systems already do this, and
even D&D will try to reduce the magnitude of difference between success and failure by letting a powerful ability do half-damage on a miss, again making combat less variable. Upcoming Obsidian
Software RPG
Pillars of Eternity
plans to replace most "misses" with "grazing attacks" that do half-damage instead of no damage, again reducing the role of chance -- a design decision we'll examine in greater detail in next week's
Future directions
Next time, we'll go one step further and see how hard it can be for that +1 to-hit bonus to actually translate into an increase in damage output. To do this, I made my work PC simulate forty million
attack rolls. It was fun as heck. I hope to see you then! | {"url":"https://crystalprisonzone.blogspot.com/2013/","timestamp":"2024-11-06T02:55:03Z","content_type":"text/html","content_length":"57558","record_id":"<urn:uuid:14b69050-9f40-4c88-b234-75f98819a240>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00128.warc.gz"} |
Gen: How to combine multiple generative function traces in a higher-order generative function?
I'm going through the "Introduction to Modeling in Gen" Notebook at https://github.com/probcomp/gen-quickstart
Section 5 (Calling other generative functions) asks to "Construct a data set for which it is ambiguous whether the line or sine wave model is best"
I'm having a hard problem understanding how I work with the traces (and returns) of the component functions to create a meaningful higher-order trace that I can use.
To me the most straightforward "ambiguous" model is line(xs).+sine(xs). So I Gen.simulateed line and sine to get the traces and adding them together, like this:
@gen function combo(xs::Vector{Float64})
my_sin = simulate(sine_model_2,(xs,))
my_lin = simulate(line_model_2,(xs,))
if @trace(bernoulli(0.5), :is_line)
@trace(normal(get_choices(my_lin)[:slope], 0.01), :slope)
@trace(normal(get_choices(my_lin)[:intercept], 0.01), :intercept)
@trace(normal(get_choices(my_lin)[:noise], 0.01), :noise)
@trace(normal(get_choices(my_sin)[:phase], 0.01), :phase)
@trace(normal(get_choices(my_sin)[:period], 0.01), :period)
@trace(normal(get_choices(my_sin)[:amplitude], 0.01), :amplitude)
@trace(normal(get_choices(my_sin)[:noise], 0.01), :noise)
combo = [get_choices(my_sin)[(:y, i)] + get_choices(my_lin)[(:y, i)] for i=1:length(xs)]
for (i, c) in enumerate(combo)
@trace(normal(c, 0.1), (:y, i))
This is clearly wrong and I know I'm missing something fundamental in the whole idea of traces and prob programming in Gen.
I'd expect to be able to introspect sine/line_model's trace from within combo, and do element-wise addition on the traces to get a new trace. And not have to randomly pick a number close to
:intercept, :phase, etc. so I can include it in my trace later on.
By the way, when I do:
traces = [Gen.simulate(combo,(xs,)) for _=1:12];
grid(render_combined, traces)
Please help thanks!
Hi there — thanks for your interest in Gen! :)
Addresses of the combined model's trace
The combined model from the tutorial looks like this:
@gen function combined_model(xs::Vector{Float64})
if @trace(bernoulli(0.5), :is_line)
Its traces will have the following addresses:
• :is_line, storing a Boolean indicating whether the generated dataset was linear or not.
• Any addresses from line_model_2 or sine_model_2, depending on which was called.
Note that traces of both line_model_2 and sine_model_2 contain the addresses (:y, i) for each integer i between 1 and length(xs). Because of this, so will combined_model's traces: these are the
addresses representing the final sampled y values, regardless of which of the two processes generated them.
Constructing a new dataset
The question to "construct a data set for which it is ambiguous whether the line or sine wave model is best" does not require writing a new generative function (with @gen), but rather, constructing a
list of xs and a list of ys (in plain Julia) that you think might make a difficult-to-disambiguate dataset. You can then pass your xs and ys into the do_inference function defined earlier in the
notebook, to see what the system concludes about your dataset. Note that the do_inference function constructs a constraint choicemap that constrains each (:y, i) to the value ys[i] from the dataset
you passed in. This works because (:y, i) is always the name of the ith datapoint, no matter the value of :is_line.
Updating / manipulating traces
You write:
I'd expect to be able to introspect sine/line_model's trace from within combo, and do element-wise addition on the traces to get a new trace. And not have to randomly pick a number close to
:intercept, :phase, etc. so I can include it in my trace later on.
You can certainly call simulate twice to get two traces, outside a generative function like combo. But traces cannot be manipulated in arbitrary ways (e.g. "elementwise addition"): as data
structures, traces maintain certain invariants, like always knowing the exact probability of their current values under the model that generated them, and always holding values that actually could
have been generated from the model.
The dictionary-like data structure you're looking for is a choicemap. Choicemaps are mutable and can be built up to include arbitrary values at arbitrary addresses. For example, you can write:
observations = Gen.choicemap()
for (i, y) in enumerate(ys)
observations[(:y, i)] = y
Choicemaps can be used as constraints to generate new traces (using Gen.generate), as arguments to Gen's low-level Gen.update method (with allows you to update a trace while recomputing any relevant
probabilities, and erroring if your updates are invalid), and in several other places.
Hope that helps :) | {"url":"https://coderapp.vercel.app/answer/56928729","timestamp":"2024-11-02T07:46:49Z","content_type":"text/html","content_length":"105395","record_id":"<urn:uuid:bcf396af-7847-410d-aa9c-0f7de9ddff42>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00390.warc.gz"} |
In this chapter professor Conway will cover types of variables. It is very important to understand what type of variable you are dealing with when conducting a particular type of statistical
analysis. You will cover variables such as nominal, ordinal, interval and ratio, and you will experiment with these via interactive exercises in R. | {"url":"https://campus.datacamp.com/courses/intro-to-statistics-with-r-introduction/chapter-five-measures-of-variability?ex=7","timestamp":"2024-11-02T08:45:58Z","content_type":"text/html","content_length":"200261","record_id":"<urn:uuid:07132a1e-e34d-4064-8db0-e7f2e0ffdb71>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00752.warc.gz"} |
Creation of flood damage functions for cultivated plots
Welcome to floodam.agri
floodam.agri is an R library for producing flood damage functions for cultivated plot (agriculture).
This library has been developed within the French working group “GT AMC” which aims at developing methodologies for the economic appraisal of flood management policies (Agenais et al. 2013 ; Brémond
et al. 2022). In this context, the development of floodam.agri has received the support of the French Ministry in charge of the Environment. Further development of the library was supported by the
Labex AGRO 2011-LABX-002 as part of the CAFRUA project n°2101-027.
What floodam.agri can do
floodam.agri is based on a model of agricultural plots.
floodam.agri includes a collection of elementary damage components that describe how elementary components may be impacted in terms of monetary damage when flooded in specific conditions.
From this information, floodam.agri can calculate the following outputs:
• different type of damage functions:
□ absolute damage functions to specific crops at plot level
□ absolute damage functions to type of crops at plot level
What floodam.agri cannot do
floodam.agri cannot (and is not intended to):
• calculate damage from the intersection of hydraulic modeling, land use modeling and damage functions. This is what floodam.spatial does.
• calculate damage at the farm level by taking into account interactions between plots, buildings and equipment. This is what AVA does.
floodam.agri cannot (but should at some point be able to) :
• calculate the uncertainty of damage functions
• perform a sensitivity analysis of damage functions
• calculate damage functions at farm level
How to get floodam.agri?
You can download and install it from this archive: www.floodam.org/library/floodam.agri_1.0.3.0.tar.gz.
For instance, from a terminal, the command you can use is:
Agenais, Anne-Laure, Frédéric Grelot, Pauline Brémond, and Katrin Erdlenbruch. 2013. “Dommages Des Inondations Au Secteur Agricole. Guide Méthodologique Et Fonctions Nationales.” Groupe de travail
national ACB Inondation. IRSTEA. https://hal.inrae.fr/hal-02600061.
Brémond, Pauline, Anne-Laurence Agenais, Frédéric Grelot, and Claire Richert. 2022. “Process-Based Flood Damage Modelling Relying on Expert Knowledge: A Methodological Contribution Applied to the
Agricultural Sector.” Natural Hazards and Earth System Sciences 22 (10): 3385–3412. https://doi.org/10.5194/nhess-22-3385-2022. | {"url":"https://www.floodam.org:443/floodam.agri/","timestamp":"2024-11-03T13:48:07Z","content_type":"text/html","content_length":"11901","record_id":"<urn:uuid:6663d193-8f6d-47c6-90c2-abb5cf0ac779>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00557.warc.gz"} |
Vahid Shahverdi: Algebraic geometry with "random" flavor | SMC
Vahid Shahverdi: Algebraic geometry with "random" flavor
Time: Wed 2022-12-07 14.15 - 15.00
Location: KTH, E52
Participating: Vahid Shahverdi (KTH)
In this talk, I will introduce some basic tools and notions in algebraic geometry, measure theory, and representation theory. Then, I will talk about two concepts of degree and volume of an algebraic
set to answer the following questions:
1. How many zeros of a random polynomial are real?
2. What is the topology of random hypersurfaces?
This is a pre-colloquium targeted towards master and PhD students as a preparation for Antonio Lerario's talk at 15:15. | {"url":"https://www.math-stockholm.se/en/kalender/smc-pre-colloquium/vahid-shahverdi-algebraic-geometry-with-random-flavor-1.1209566","timestamp":"2024-11-08T15:38:29Z","content_type":"text/html","content_length":"24171","record_id":"<urn:uuid:61244dac-b32c-4d4f-91e6-e649b21fc795>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00431.warc.gz"} |
hoFlip Side is a useful analytic tool built directly into Datameer X workbooks. This feature gives you instant measurements from every sheet in your workbooks. It's a way to get a quick visualized
data overview with a click rather than having to perform the calculations manually.
Accessing Flip Side
1. Open a workbook.
2. Navigate to the sheet.
3. Open the Flip Side analytics by going to View and selecting Flip Sheet, or click the Flip Sheet button.
The Flip Side analytics from the sheet are displayed.
Find if the Flip Side is Running from Full or Preview Data
When viewing a sheet from a datasource, the Flip Side calculates the full data from the data source. Column metrics are provided by the import job.
Calculation sheet
When viewing a Flip Side sheet that is a calculation, join, or union the Flip Side calculates preview data until the workbook has been processed. Column statistics are based on preview data only.
Run the workbook to see full data.
Worksheet linked from a separate workbook
When viewing a Flip Side sheet that is a sheet linked from a different workbook, the Flip Side calculates preview data until the external workbook has been processed. Column statistics are based on
preview data only. Run the workbook to see full data.
Data link
When viewing a Flip Side sheet from a data link source, the Flip Side calculates preview data only. Full data is not available from a data link source. Column statistics are based on preview data
only. Full data statistics are not available for unkept data link sheets.
Reading Flip Side charts
The Flip Side charts show pre-calculated analytics of a worksheet based on either the preview data or full records. These metrics give users insight into a quick summary of the worksheet. The tilda
symbol () indicates that the number shown is an approximation.
Categories of worksheet metrics
The Distribution row displays a histogram of each column separating data into relevant groups. Running the mouse cursor over the histogram displays the group size and count.
The Count row displays a rounded count of records for easy visibility, the exact number of records, and a bar showing the number of empty records - the fraction of empty records is shown as a white
colored section of the bar.
The Unique row displays an estimated, rounded count of unique records and an estimated number of unique records of a column.
The Min row displays the lowest number of a row. If the data field is a date, the earliest date is displayed.
The Max row displays the highest number of a row. If the data field is a date, the most recent date is displayed.
The Mean row displays the average number of a row by adding the records and dividing by the number of records (including empty records). If the data field is a date, the date directly between the
earliest and most recent date is displayed.
Flip Side Calculations (Advanced)
The following information gives greater detail on how Flip Side calculations are created.
Numerical histograms
Histograms on a Flip Side sheets that represent numbers with very few unique values are displayed categorically as opposed to a range. This change better represents the data so analysts are given a
clear view of the data's shape. These are approximate, not exact, values. (To compute the exact count, use the GROUPBYBIN and GROUPCOUNT functions in a workbook.)
How the x-axis is calculated for numerical data
• Up to 32 most frequent bins.
• Breaks are found by splitting from minimum to maximum into 32 equal width bin bars.
How the y axis is calculated for numerical data
• Shows the count of how many records fall in the range of that bin.
• Shown is an estimate of the true count, (not sampling) due to single pass over data.
Non-numerical (categorical data) histograms
These are approximate values. Datameer X populates the histograms using a single pass over the data. A single pass top k algorithm is used to determine the categories for the bins.
How the x-axis is calculated for categorical data
• Uses all possible different values to make the bins.
• If greater than 32 values exist, the 32 most frequent are used to make the bins.
How the y-axis is calculated for categorical data
• Shows the count of the number of records of the bins.
Empties and nulls
In the histogram, null values are ignored. Empty strings can be used as a bin.
Records with null or empty values contribute to the count. Hovering over the bar shows how many were null (but not empty even though it says "empty").
Unique (cardinality)
Null values don't contribute to unique. Empty strings do contribute.
Minimum, maximum, average
Null is ignored for minimum, maximum, and average.
How estimates are calculated
Data shown on the Flip Side is based off the full data if the workbook has been run. Otherwise, it is based on the sample data.
Flip Side permissions in a workbook
If you have view access to a workbook, you can see any Flip Side in the workbook.
Special Flip Side Data for Smart Analytic Sheets
Administrators have the ability to disable full metric calculations for reasons such as controlling performance characteristics.
Click here to expand...
Disable individual job metric calculations
When configuring an import job or workbook, open the advanced settings and enter the property das.compute.column-metrics=false in the Custom Properties field.
Disable all job metric calculations
Edit Hadoop Properties under the Admin tab. Enter the property das.compute.column-metrics=false in the Custom Properties field.
The Flip Side is available, but only from preview data. Column statistics are based on preview data only. The statistics computation have been disabled for the import job (Contact system | {"url":"https://datameer.atlassian.net/wiki/spaces/DAS110/pages/9645138045/Flipping+a+Sheet","timestamp":"2024-11-13T11:46:46Z","content_type":"text/html","content_length":"983267","record_id":"<urn:uuid:00b5ec63-3eef-4c93-85cf-ee9faafa5543>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00755.warc.gz"} |
Estimating Percentiles from Lognormal Quantile Plots
Example 4.32 Estimating Percentiles from Lognormal Quantile Plots
This example, which is a continuation of Example 4.31, shows how to use a Q-Q plot to estimate percentiles such as the 95th percentile of the lognormal distribution. A probability plot can also be
used for this purpose, as illustrated in Example 4.26.
The point pattern in Output 4.31.4 has a slope of approximately 0.39 and an intercept of 5. The following statements reproduce this plot, adding a lognormal reference line with this slope and
symbol v=plus;
title 'Lognormal Q-Q Plot for Diameters';
proc univariate data=Measures noprint;
qqplot Diameter / lognormal(sigma=0.5 theta=5 slope=0.39)
vref = 5.8 5.9 6.0
The result is shown in Output 4.32.1:
The PCTLAXIS option labels the major percentiles, and the GRID option draws percentile axis reference lines. The 95th percentile is 5.9, because the intersection of the distribution reference line
and the 95th reference line occurs at this value on the vertical axis.
Alternatively, you can compute this percentile from the estimated lognormal parameters. The
A sample program for this example, uniex18.sas, is available in the SAS Sample Library for Base SAS software. | {"url":"http://support.sas.com/documentation/cdl/en/procstat/63104/HTML/default/procstat_univariate_sect087.htm","timestamp":"2024-11-07T04:20:49Z","content_type":"application/xhtml+xml","content_length":"12656","record_id":"<urn:uuid:8b1849af-1b15-4f9c-947e-e97f352ce67b>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00241.warc.gz"} |
IISc physicists find a new way to represent ‘pi’
While investigating how string theory can be used to explain certain physical phenomena, scientists at the Indian Institute of Science (IISc) have stumbled upon on a new series representation for the
irrational number pi. It provides an easier way to extract pi from calculations involved in deciphering processes like the quantum scattering of high-energy particles.
Credit: Manu Y
While investigating how string theory can be used to explain certain physical phenomena, scientists at the Indian Institute of Science (IISc) have stumbled upon on a new series representation for the
irrational number pi. It provides an easier way to extract pi from calculations involved in deciphering processes like the quantum scattering of high-energy particles.
The new formula under a certain limit closely reaches the representation of pi suggested by Indian mathematician Sangamagrama Madhava in the 15th century, which was the first ever series for pi
recorded in history. The study was carried out by Arnab Saha, a post-doc and Aninda Sinha, Professor at Centre for High Energy Physics (CHEP), and published in Physical Review Letters.
“Our efforts, initially, were never to find a way to look at pi. All we were doing was studying high-energy physics in quantum theory and trying to develop a model with fewer and more accurate
parameters to understand how particles interact. We were excited when we got a new way to look at pi,” Sinha says.
Sinha’s group is interested in string theory – the theoretical framework that presumes that all quantum processes in nature simply use different modes of vibrations plucked on a string. Their work
focuses on how high energy particles interact with each other – such as protons smashing together in the Large Hadron Collider – and in what ways we can look at them using as few and as simple
factors as possible. This way of representing complex interactions belongs to the category of “optimisation problems.” Modelling such processes is not easy because there are several parameters that
need to be taken into account for each moving particle – its mass, its vibrations, the degrees of freedom available for its movement, and so on.
Saha, who has been working on the optimisation problem, was looking for ways to efficiently represent these particle interactions. To develop an efficient model, he and Sinha decided to club two
mathematical tools: the Euler-Beta Function and the Feynman Diagram. Euler-Beta functions are mathematical functions used to solve problems in diverse areas of physics and engineering, including
machine learning. The Feynman Diagram is a mathematical representation that explains the energy exchange that happens while two particles interact and scatter.
What the team found was not only an efficient model that could explain particle interaction, but also a series representation of pi.
In mathematics, a series is used to represent a parameter such as pi in its component form. If pi is the “dish” then the series is the “recipe”. Pi can be represented as a combination of many numbers
of parameters (or ingredients). Finding the correct number and combination of these parameters to reach close to the exact value of pi rapidly has been a challenge. The series that Sinha and Saha
have stumbled upon combines specific parameters in such a way that scientists can rapidly arrive at the value of pi, which can then be incorporated in calculations, like those involved in deciphering
scattering of high-energy particles.
“Physicists (and mathematicians) have missed this so far since they did not have the right tools, which were only found through work we have been doing with collaborators over the last three years or
so,” Sinha explains. “In the early 1970s, scientists briefly examined this line of research but quickly abandoned it since it was too complicated.”
Although the findings are theoretical at this stage, it is not impossible that they may lead to practical applications in the future. Sinha points to how Paul Dirac worked on the mathematics of the
motion and existence of electrons in 1928, but never thought that his findings would later provide clues to the discovery of the positron, and then to the design of Positron Emission Tomography (PET)
used to scan the body for diseases and abnormalities. “Doing this kind of work, although it may not see an immediate application in daily life, gives the pure pleasure of doing theory for the sake of
doing it,” Sinha adds.
Physical Review Letters
Article Title
Field Theory Expansions of String Theory Amplitudes
Article Publication Date
Discover more from Science
Subscribe to get the latest posts sent to your email. | {"url":"https://scienmag.com/iisc-physicists-find-a-new-way-to-represent-pi/","timestamp":"2024-11-07T13:47:11Z","content_type":"text/html","content_length":"137930","record_id":"<urn:uuid:9ab7980e-6ff6-4277-965d-c1b2884f3981>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00884.warc.gz"} |
Exogeneous Versus Endogeneous Variables: A General Approach to Mixed Systems
BibTeX reference
The problem of exogenity in economics is a highly relative one. For example in microeconomics (production theory or consumer theory) quantities are exogeneous with respect to prices (and income in
the case of consumer theory) and conversely. In other words, as long as there is an invertible relation between quantities and prices (and income), whether we consider prices (and income) or
quantities as the signal to which the agents react is rather a matter of taste or opportunity. The question of exogenity really arises when we want to know if, say, the prices of the first k goods
together with the quantities of the last n-k ones may be considered exogeneous with respect to the other variables (e.g. the quantities of the first k goods and the prices of the n-k last ones,
together with income). The mathematical setting of the problem can be given the following form: Given a functional relation φ between x = (x^1, ..., x^n) ∈ ℝ^n and y = (y^1, ..., y^n): y = φ
(x), what are the properties of φ which allow for the existence of a map X such that
i) (y^1, y^2, ..., y^k, x^k+1, ..., x^n) = X(x^1, ..., x^k, y^k+1, ..., y^n)
ii) X and φ have same graph in ℝ^2n.
In consumer theory, the problem has been dealt with recently to give rise to what is known as "mixed demand" (C. Bronsard and L. Salvas Bronsard (1979), J.P. Chavas (1983).
Published May 1984 , 14 pages | {"url":"https://www.gerad.ca/en/papers/G-84-08","timestamp":"2024-11-08T01:48:56Z","content_type":"text/html","content_length":"15650","record_id":"<urn:uuid:e3df58fe-1a64-4293-b277-51a957692524>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00894.warc.gz"} |