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Solving Systems By Graphing Worksheet
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Solving Systems Of Equations By Graphing
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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2018.50
URN: urn:nbn:de:0030-drops-95137
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9513/
Jelínek, Vít ; Opler, Michal ; Valtr, Pavel
Generalized Coloring of Permutations
A permutation pi is a merge of a permutation sigma and a permutation tau, if we can color the elements of pi red and blue so that the red elements have the same relative order as sigma and the blue
ones as tau. We consider, for fixed hereditary permutation classes C and D, the complexity of determining whether a given permutation pi is a merge of an element of C with an element of D.
We develop general algorithmic approaches for identifying polynomially tractable cases of merge recognition. Our tools include a version of nondeterministic logspace streaming recognizability of
permutations, which we introduce, and a concept of bounded width decomposition, inspired by the work of Ahal and Rabinovich.
As a consequence of the general results, we can provide nontrivial examples of tractable permutation merges involving commonly studied permutation classes, such as the class of layered permutations,
the class of separable permutations, or the class of permutations avoiding a decreasing sequence of a given length.
On the negative side, we obtain a general hardness result which implies, for example, that it is NP-complete to recognize the permutations that can be merged from two subpermutations avoiding the
pattern 2413.
BibTeX - Entry
author = {V{\'i}t Jel{\'i}nek and Michal Opler and Pavel Valtr},
title = {{Generalized Coloring of Permutations}},
booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)},
pages = {50:1--50:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-081-1},
ISSN = {1868-8969},
year = {2018},
volume = {112},
editor = {Yossi Azar and Hannah Bast and Grzegorz Herman},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9513},
URN = {urn:nbn:de:0030-drops-95137},
doi = {10.4230/LIPIcs.ESA.2018.50},
annote = {Keywords: Permutations, merge, generalized coloring}
Keywords: Permutations, merge, generalized coloring
Collection: 26th Annual European Symposium on Algorithms (ESA 2018)
Issue Date: 2018
Date of publication: 14.08.2018
DROPS-Home | Fulltext Search | Imprint | Privacy | {"url":"http://dagstuhl.sunsite.rwth-aachen.de/opus/frontdoor.php?source_opus=9513","timestamp":"2024-11-10T14:54:33Z","content_type":"text/html","content_length":"6695","record_id":"<urn:uuid:60ffebec-1780-4c10-b17c-cc853cb0cb33>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00350.warc.gz"} |
Publications about 'bilinear systems'
Publications about 'bilinear systems'
Articles in journal or book chapters
1. A. C. B. de Oliveira, M. Siami, and E. D. Sontag. Edge selections in bilinear dynamic networks. IEEE Transactions on Automatic Control, 69(1):331-338, 2024. [PDF] [doi:10.1109/TAC.2023.3269323]
Keyword(s): bilinear systems, networks, robustness.
│ │We develop some basic principles for the design and robustness analysis of a continuous-time bilinear dynamical network, where an attacker can manipulate the strength of the │
│ │interconnections/edges between some of the agents/nodes. We formulate the edge protection optimization problem of picking a limited number of attack-free edges and minimizing the │
│ │impact of the attack over the bilinear dynamical network. In particular, the H2-norm of bilinear systems is known to capture robustness and performance properties analogous to its │
│ │linear counterpart and provides valuable insights for identifying which edges arem ost sensitive to attacks. The exact optimization problem is combinatorial in the number of edges, │
│ │and brute-force approaches show poor scalability. However, we show that the H2-norm as a cost function is supermodular and, therefore, allows for efficient greedy approximations of │
│Abstract:│the optimal solution. We illustrate and compare the effectiveness of our theoretical findings via numerical simulation │
2. E.D. Sontag, Y. Wang, and A. Megretski. Input classes for identification of bilinear systems. IEEE Transactions Autom. Control, 54:195-207, 2009. Note: Also arXiv math.OC/0610633, 20 Oct 2006,
and short version in ACC'07.[PDF] Keyword(s): realization theory, observability, identifiability, bilinear systems.
│ │This paper asks what classes of input signals are sufficient in order to completely identify the input/output behavior of generic bilinear systems. The main results are that step │
│Abstract:│inputs are not sufficient, nor are single pulses, but the family of all pulses (of a fixed amplitude but varying widths) do suffice for identification. │
3. E.D. Sontag and Y. Wang. Uniformly Universal Inputs. In Alessandro Astolfi, editor, Analysis and Design of Nonlinear Control Systems, volume 224, pages 9-24. Springer-Verlag, London, 2007. [PDF]
Keyword(s): observability, identification, real-analytic functions.
│ │A result is presented showing the existence of inputs universal for observability, uniformly with respect to the class of all continuous-time analytic systems. This represents an │
│Abstract:│ultimate generalization of a 1977 theorem, for bilinear systems, due to Alberto Isidori and Osvaldo Grasselli. │
4. E.D. Sontag. Comments on integral variants of ISS. Systems Control Lett., 34(1-2):93-100, 1998. [PDF] [doi:http://dx.doi.org/10.1016/S0167-6911(98)00003-6] Keyword(s): input to state stability,
integral input to state stability, iISS, ISS.
│ │This note discusses two integral variants of the input-to-state stability (ISS) property, which represent nonlinear generalizations of L2 stability, in much the same way that ISS │
│ │generalizes L-infinity stability. Both variants are equivalent to ISS for linear systems. For general nonlinear systems, it is shown that one of the new properties is strictly weaker │
│ │than ISS, while the other one is equivalent to it. For bilinear systems, a complete characterization is provided of the weaker property. An interesting fact about functions of type KL│
│Abstract:│is proved as well. │
5. E.D. Sontag. A Chow property for sampled bilinear systems. In C.I. Byrnes, C.F. Martin, and R. Saeks, editors, Analysis and Control of Nonlinear Systems, pages 205-211. North Holland, Amsterdam,
1988. [PDF] Keyword(s): discrete-time, bilinear systems.
│ │This paper studies accessibility (weak controllability) of bilinear systems under constant sampling rates. It is shown that the property is preserved provided that the sampling period│
│ │satisfies a condition related to the eigenvalues of the autonomous dynamics matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the │
│Abstract:│linear case, and which, for observability, results in the classical Nyquist theorem. │
6. E.D. Sontag. Bilinear realizability is equivalent to existence of a singular affine differential I/O equation. Systems Control Lett., 11(3):181-187, 1988. [PDF] [doi:http://dx.doi.org/10.1016/
0167-6911(88)90057-6] Keyword(s): identification, identifiability, observability, observation space, real-analytic functions.
│ │For continuous time analytic input/output maps, the existence of a singular differential equation relating derivatives of controls and outputs is shown to be equivalent to bilinear │
│ │realizability. A similar result holds for the problem of immersion into bilinear systems. The proof is very analogous to that of the corresponding, and previously known, result for │
│Abstract:│discrete time. │
7. E.D. Sontag. Controllability is harder to decide than accessibility. SIAM J. Control Optim., 26(5):1106-1118, 1988. [PDF] [doi:http://dx.doi.org/10.1137/0326061] Keyword(s): computational
complexity, controllability, computational complexity.
│ │The present article compares the difficulties of deciding controllability and accessibility. These are standard properties of control systems, but complete algebraic characterizations│
│Abstract:│of controllability have proved elusive. We show in particular that for subsystems of bilinear systems, accessibility can be decided in polynomial time, but controllability is NP-hard.│
8. E.D. Sontag. A remark on bilinear systems and moduli spaces of instantons. Systems Control Lett., 9(5):361-367, 1987. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(87)90064-8] Keyword(s):
bilinear systems, moduli spaces, instantons.
│ │Explicit equations are given for the moduli space of framed instantons as a quasi-affine variety, based on the representation theory of noncommutative power series, or equivalently, │
│Abstract:│the minimal realization theory of bilinear systems. │
9. E.D. Sontag. An eigenvalue condition for sample weak controllability of bilinear systems. Systems Control Lett., 7(4):313-315, 1986. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(86)90045-9]
Keyword(s): discrete-time.
│ │Weak controllability of bilinear systems is preserved under sampling provided that the sampling period satisfies a condition related to the eigenvalues of the autonomous dynamics │
│Abstract:│matrix. This condition generalizes the classical Kalman-Ho-Narendra criterion which is well known in the linear case. │
10. E.D. Sontag. Realization theory of discrete-time nonlinear systems. I. The bounded case. IEEE Trans. Circuits and Systems, 26(5):342-356, 1979. [PDF] Keyword(s): discrete-time systems, nonlinear
systems, realization theory, bilinear systems, state-affine systems.
│ │A state-space realization theory is presented for a wide class of discrete time input/output behaviors. Although In many ways restricted, this class does include as particular cases │
│ │those treated in the literature (linear, multilinear, internally bilinear, homogeneous), as well as certain nonanalytic nonlinearities. The theory is conceptually simple, and │
│ │matrix-theoretic algorithms are straightforward. Finite-realizability of these behaviors by state-affine systems is shown to be equivalent both to the existence of high-order input/ │
│Abstract:│output equations and to realizability by more general types of systems. │
1. A.C.B de Olivera, M. Siami, and E.D. Sontag. Sensor and actuator scheduling in bilinear dynamical networks. In Proc. 2022 61st IEEE Conference on Decision and Control (CDC), pages WeCT09.4, 2022.
│ │In this paper, we investigate the problem of finding a sparse sensor and actuator (S/A) schedule that minimizes the approximation error between the input-output behavior of a fully │
│ │sensed/actuated bilinear system and the system with the scheduling. The quality of this approximation is measuredby an H2-like metric, which is defined for a bilinear (time-varying) │
│ │system with S/A scheduling based on the discrete Laplace transform of its Volterra kernels. First, we discuss the difficulties of designing S/A schedules for bilinear systems, which │
│ │prevented us from finding a polynomial time algorithmfor solving the problem. We then propose a polynomial-time S/A scheduling heuristic that selects a fraction of sensors and node │
│ │actuators at each time step while maintaining a small approximation error between the input-output behavior of thefully sensed/actuated system and the one with S/A scheduling in this │
│Abstract:│H2-based sense. Numerical experiments illustrate the good approximation quality of our proposed methods. │
2. A.C.B de Olivera, M. Siami, and E.D. Sontag. Bilinear dynamical networks under malicious attack: an efficient edge protection method. In Proc. 2021 Automatic Control Conference, pages 1210-1216,
2021. [PDF] Keyword(s): Bilinear systems, adversarial attacks, robustness measures, supermodular optimization.
│ │In large-scale networks, agents and links are often vulnerable to attacks. This paper focuses on continuous-time bilinear networks, where additive disturbances model attacks or │
│ │uncertainties on agents/states (node disturbances), and multiplicative disturbances model attacks or uncertainties on couplings between agents/states (link disturbances). It │
│ │investigates network robustness notion in terms of the underlying digraph of the network, and structure of exogenous uncertainties and attacks. Specifically, it defines a robustness │
│ │measure using the $\mathcal H_2$-norm of the network and calculates it in terms of the reachability Gramian of the bilinear system. The main result is that under certain conditions, │
│ │the measure is supermodular over the set of all possible attacked links. The supermodular property facilitates the efficient solution finding of the optimization problem. Examples │
│Abstract:│illustrate how different structures can make the system more or less vulnerable to malicious attacks on links. │
3. A.C.B de Olivera, M. Siami, and E.D. Sontag. Eminence in noisy bilinear networks. In Proc. 2021 60th IEEE Conference on Decision and Control (CDC), pages 4835-4840, 2021. [PDF] Keyword(s):
Bilinear systems, H2 norm, centrality, adversarial attacks, robustness measures.
│ │When measuring importance of nodes in a network, the interconnections and dynamics are often supposed to be perfectly known. In this paper, we consider networks of agents with both │
│ │uncertain couplings and dynamics. Network uncertainty is modeled by structured additive stochastic disturbances on each agent's update dynamics and coupling weights. We then study how│
│ │these uncertainties change the network's centralities. Disturbances on the couplings between agents resul in bilinear dynamics, and classical centrality indices from linear network │
│ │theory need to be redefined. To do that, we first show that, similarly to its linear counterpart, the squared H2 norm of bilinear systems measures the trace of the steady-state error │
│ │covariance matrix subject to stochastic disturbances. This makes the H2 norm a natural candidate for a performance metric of the system. We propose a centrality index for the agents │
│ │based on the H2 norm, and show how it depends on the network topology and the noise structure. Finally, we simulate a few graphs to illustrate how uncertainties on different couplings│
│Abstract:│affect the agents' centrality rankings compared to a linearized model of the same system. │
4. E.D. Sontag, Y. Wang, and A. Megretski. Remarks on Input Classes for Identification of Bilinear Systems. In Proceedings American Control Conf., New York, July 2007, pages 4345-4350, 2007. Keyword
(s): realization theory, observability, identifiability, bilinear systems.
5. E.D. Sontag. From linear to nonlinear: some complexity comparisons. In Proc. IEEE Conf. Decision and Control, New Orleans, Dec. 1995, IEEE Publications, 1995, pages 2916-2920, 1995. [PDF] Keyword
(s): theory of computing and complexity, computational complexity, controllability, observability.
│ │This paper deals with the computational complexity, and in some cases undecidability, of several problems in nonlinear control. The objective is to compare the theoretical difficulty │
│ │of solving such problems to the corresponding problems for linear systems. In particular, the problem of null-controllability for systems with saturations (of a "neural network" type)│
│ │is mentioned, as well as problems regarding piecewise linear (hybrid) systems. A comparison of accessibility, which can be checked fairly simply by Lie-algebraic methods, and │
│Abstract:│controllability, which is at least NP-hard for bilinear systems, is carried out. Finally, some remarks are given on analog computation in this context. │
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The Stacks project
Lemma 68.8.6. Let $X$ be a quasi-compact and quasi-separated algebraic space over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. There exist an integer $n$ and open subspaces
\[ \emptyset = U_{n + 1} \subset U_ n \subset U_{n - 1} \subset \ldots \subset U_1 = X \]
with the following property: setting $T_ p = U_ p \setminus U_{p + 1}$ (with reduced induced subspace structure) there exists a quasi-compact separated scheme $V_ p$ and a surjective étale morphism
$f_ p : V_ p \to U_ p$ such that $f_ p^{-1}(T_ p) \to T_ p$ is an isomorphism.
Comments (1)
Comment #1615 by Pieter Belmans on
The notation $S_p$ is potentially confusing here: it refers to the symmetric group on $p$ elements, as is explained in tag 68.8.3. I'm not saying you should change, but if people are confused
when looking at this on the website this clears up things.
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Holding period return vs internal rate of return
Internal Rate of Return IRR is a metric for cash flow analysis, used often investments, capital acquisitions, project proposals, and business case results. By definition, IRR compares returns to
costs by finding an interest rate that yields zero NPV for the investment. However, finding practical guidance for Investors and decision makers in IRR results is a challenge.
In finance, holding period return (HPR) is the return on an asset or portfolio over the whole period during which it was held. It is one of the simplest and most important measures of investment
performance. HPR is the change in value of an investment, asset or portfolio over a particular period. Holding Period Return. In finance, holding period return (HPR) is a rate of return on an asset,
investment or portfolio over a particular investment period. HPR is the sum of income and capital gains divided by the asset value at the beginning of the period, often expressed as a percentage. It
is one of the simplest measures of investment performance. A Rate of Return (ROR) is the gain or loss of an investment over a certain period of time. In other words, the rate of return is the gain
(or loss) compared to the cost of an initial investment, typically expressed in the form of a percentage. When the ROR is positive, it is considered a gain and when the ROR is negative, IRR vs ROI
Differences. When it comes to calculating the performance of the investments made, there are a very few metrics that are used more than the Internal Rate of Return (IRR) and Return on Investment
(ROI). IRR is a metric that doesn’t have any real formula. It means that no predetermined formula can be used to find out IRR. After holding costs and your mortgage payment, your pre-tax net income
is $319 per month. So, in a 12-month period, you would receive $3,828. Internal rate of return, or yield, is forward IRR [Internal Rate of Return] But an internal investment can go up or down over
the holding period, and IRR doesn’t address what happens to capital that is taken out of the investment. That When analyzing the return of an investment, investors most often use two key metrics: The
Internal Rate of Return (IRR) and Return on Investment (ROI). The latter of which is also known as the Holding Period Return. The goal for this blog post is to not only define these metrics but to
demonstrate how they […]
The internal rate of return (IRR) is a measure of an investment's rate of return. The term internal but delays returns for one or more time periods, would have a lower IRR. NPV vs discount rate
comparison for two mutually exclusive projects . as the internal rate of return as defined above, and a holding period return.
Return on investment—sometimes called the rate of return (ROR)—is the percentage increase or decrease in an investment over a set period. It is calculated by taking the difference between current, or
expected, value and original value divided by the original value and multiplied by 100. Holding period return is the total return received from holding an asset or portfolio of assets over a period
of time, known as the holding period, generally expressed as a percentage. Holding period return is calculated on the basis of total returns from the asset or portfolio (income plus changes in
value). The Holding Period Return (HPR) is the total return on an asset or investment portfolio over the period for which the asset or portfolio has been held. The holding period return can be
realized if the asset or portfolio has been held, or expected if an investor only anticipates the purchase of the asset. The Internal Rate of Return (IRR) can be a more useful tool in evaluating the
returns on a property over the entire holding period. The IRR is calculated by taking all future cash flows, and the future sales price, and discounting back to present value. Realized return
(internal rate of return) is calculated consistently for both monthly and daily data. Suppose: = the initial market value of a portfolio = the ending market value of a portfolio = a series of interim
cash flows. then the Internal Rate of Return is the rate that equates the sum of net present value of all cash flows to zero: IRR is the annual return that makes the initial investment "turn into"
future cash flows. In the previous example – a $1,000 initial investment with projected annual cash flows of $200, $250, $300 and $400 – the internal rate of return is about 5.211 percent.
5 Feb 2019 Return on investment, or ROI, simply expresses the return as a percentage of the initial investment. For example, say you invest $1,000 in a short-
3 Sep 2019 The internal rate of return (IRR) shows investors how they can expect to profit dollar you have invested in a property over the entire holding period. IRR vs. Net Present Value. What Is
the IRR for Real Estate Investments? Internal rates of return (IRR) are returns are what matter to you as an investor. an odd time period, calculating the internal rate of return becomes more
difficult. 7 Mar 2011 The table shows the internal rate of return (IRR) for different holding periods during the cycle(s). Exiting at points where the plot is above the F8 — Framing effects:
expected value of stochastic IRR vs. IRR of Internal Rate of Return” stems from the fact that the holding period rate can be viewed as a.
The internal rate of return is the discount rate that makes the future value of an Holding period return is the total return on an investment over the period it was
IRR [Internal Rate of Return] But an internal investment can go up or down over the holding period, and IRR doesn’t address what happens to capital that is taken out of the investment. That
The internal rate of return (IRR) is a measure of an investment's rate of return. The term internal but delays returns for one or more time periods, would have a lower IRR. NPV vs discount rate
comparison for two mutually exclusive projects . as the internal rate of return as defined above, and a holding period return.
It is a simple calculation that can be used to compare your rate of return against a target rate of return or to compare different investment opportunities to see which
8 May 2017 The time-weighted rate of return (“TWR”) and the internal rate of return Any return period (say one year, for example) can be broken into smaller basis, holding stale the value of those
more illiquid investments, in order to 5 Feb 2019 Return on investment, or ROI, simply expresses the return as a percentage of the initial investment. For example, say you invest $1,000 in a short-
12 Oct 2018 Use this formula to calculate returns when the holding period is less than XIRR is a function in Excel for calculating internal rate of return or 29 Jul 2016 Convert holding period
return to the effective annual rate. Description. Convert holding Computing IRR, the internal rate of return. Description. | {"url":"https://digitaloptionsgciutvn.netlify.app/winterroth37394wuny/holding-period-return-vs-internal-rate-of-return-245.html","timestamp":"2024-11-03T19:59:32Z","content_type":"text/html","content_length":"35630","record_id":"<urn:uuid:171a93e3-0adf-489c-a37b-d0f95f048890>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00258.warc.gz"} |
Ramji Venkataramanan
Apr 05, 2023
Abstract:We study the problem of regression in a generalized linear model (GLM) with multiple signals and latent variables. This model, which we call a matrix GLM, covers many widely studied problems
in statistical learning, including mixed linear regression, max-affine regression, and mixture-of-experts. In mixed linear regression, each observation comes from one of $L$ signal vectors
(regressors), but we do not know which one; in max-affine regression, each observation comes from the maximum of $L$ affine functions, each defined via a different signal vector. The goal in all
these problems is to estimate the signals, and possibly some of the latent variables, from the observations. We propose a novel approximate message passing (AMP) algorithm for estimation in a matrix
GLM and rigorously characterize its performance in the high-dimensional limit. This characterization is in terms of a state evolution recursion, which allows us to precisely compute performance
measures such as the asymptotic mean-squared error. The state evolution characterization can be used to tailor the AMP algorithm to take advantage of any structural information known about the
signals. Using state evolution, we derive an optimal choice of AMP `denoising' functions that minimizes the estimation error in each iteration. The theoretical results are validated by numerical
simulations for mixed linear regression, max-affine regression, and mixture-of-experts. For max-affine regression, we propose an algorithm that combines AMP with expectation-maximization to estimate
intercepts of the model along with the signals. The numerical results show that AMP significantly outperforms other estimators for mixed linear regression and max-affine regression in most parameter
* 44 pages. A shorter version of this paper will appear in the proceedings of AISTATS 2023 | {"url":"https://www.catalyzex.com/author/Ramji%20Venkataramanan","timestamp":"2024-11-12T22:51:52Z","content_type":"text/html","content_length":"209653","record_id":"<urn:uuid:9e221fe6-bf18-415b-87d7-ed4422969cb0>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00706.warc.gz"} |
Square Footage Calculator
Calculating Square Footage
Calculating square footage is a straightforward process essential for various applications, from home renovation projects to real estate listings and landscaping designs. Square footage represents
the area of a space and is typically measured in square feet (ft²). Whether you are calculating the size of a single room or an entire property, understanding how to calculate square footage will
help you plan and estimate material needs effectively.
The Square Footage Formula
Square footage is calculated using the formula:
\( \text{Square Footage} = \text{Length} \times \text{Width} \)
• Length is the measurement of one side of the space in feet.
• Width is the measurement of the adjacent side in feet.
Step-by-Step Guide to Calculating Square Footage
To calculate the square footage of a space, follow these steps:
• Step 1: Measure the length of the space in feet. Use a tape measure or laser measuring tool to obtain an accurate measurement.
• Step 2: Measure the width of the space in feet. Ensure that the width measurement is perpendicular to the length measurement.
• Step 3: Multiply the length by the width. The result will give you the square footage of the area.
For example, if a room is 12 feet long and 10 feet wide, the square footage is:
\( 12 \, \text{ft} \times 10 \, \text{ft} = 120 \, \text{ft}^2 \)
Calculating Square Footage for Irregular Spaces
Not all spaces are perfect rectangles or squares. For irregular spaces, divide the area into smaller, regular shapes (such as rectangles, triangles, or circles), calculate the square footage of each
shape, and then sum the results.
Example: Irregular Space Calculation
Suppose you have an L-shaped room. Break it into two rectangles, calculate the square footage for each, and then add them together.
For instance:
• Rectangle 1: Length = 15 ft, Width = 10 ft → Square Footage = 150 ft²
• Rectangle 2: Length = 8 ft, Width = 5 ft → Square Footage = 40 ft²
Total Square Footage = 150 ft² + 40 ft² = 190 ft²
Practical Uses of Square Footage Calculation
Knowing how to calculate square footage can assist in several practical situations, such as:
• Home improvement: Determine the amount of flooring, paint, or other materials needed for a renovation project.
• Real estate: Accurately list or assess the size of a property for buying or selling.
• Landscaping: Plan gardens, lawns, or other outdoor projects by calculating the area of your yard.
Example: Calculating Square Footage for a Home Addition
Let’s say you are building an addition to your home. The new room is 20 feet long and 15 feet wide. To find the square footage:
\( 20 \, \text{ft} \times 15 \, \text{ft} = 300 \, \text{ft}^2 \)
The new room has 300 square feet of space.
Frequently Asked Questions (FAQ)
1. How do I calculate square footage for a multi-level property?
To calculate square footage for a multi-level property, calculate the square footage of each floor separately and then sum the totals. Ensure that areas like staircases and unfinished basements are
included based on standard property measurement guidelines.
2. Do hallways and closets count towards square footage?
Yes, hallways and closets typically count toward the total square footage of a home, as long as they are part of the finished, livable space.
3. Can I use square footage calculations for outdoor spaces?
Square footage can be used to calculate the size of outdoor spaces like patios, gardens, and driveways. However, in real estate, outdoor areas are usually listed separately from the indoor square | {"url":"https://turn2engineering.com/calculators/square-footage-calculator","timestamp":"2024-11-13T09:53:06Z","content_type":"text/html","content_length":"224165","record_id":"<urn:uuid:49c9fdfd-8dcb-4982-8cb0-4217e7975013>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00158.warc.gz"} |
Do we assume that the rest mass of
Do we assume that the rest mass of a fundamental particle is constant in all inertial reference frames? i.e. is the rest mass of an electron if it is travelling at constant velocity c/2 (relative to
the distant stars) the same as the rest mass of the electron if it is travelling at velocity 0 relative to the distant stars?
Answered question
Do we assume that the rest mass of a fundamental particle is constant in all inertial reference frames? i.e. is the rest mass of an electron if it is travelling at constant velocity c/2 (relative to
the distant stars) the same as the rest mass of the electron if it is travelling at velocity 0 relative to the distant stars? | {"url":"https://plainmath.org/relativity/92999-do-we-assume-that-the-rest-mass-of-a-fun","timestamp":"2024-11-04T11:53:23Z","content_type":"text/html","content_length":"197018","record_id":"<urn:uuid:6b64b9fc-c098-47bb-8398-02c67994ebe4>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00463.warc.gz"} |
Multiplication Flash Cards
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Term Archives - Flirting with Models
This post is available as a PDF download here.
• Yield curve changes over time can be decomposed into Level, Slope, and Curvature changes, and these changes can be used to construct portfolios.
• Market shocks, monetary policy, and preferences of different segments of investors (e,g. pensions) may create trends within these portfolios that can be exploited with absolute and relative
• In this commentary, we investigate these two factors in long/short and long/flat implementations and find evidence of success with some structural caveats.
• Despite this, we believe the results have potential applications as either a portable beta overlay or for investors who are simply trying to figure out how to position their duration exposure.
• Translating these quantitative signals into a forecast about yield-curve behavior may allow investors to better position their fixed income portfolios.
It has been well established in fixed income literature that changes to the U.S. Treasury yield curve can be broken down into three primary components: a level shift, a slope change, and a curvature
A level change occurs when rates increase or decrease across the entire curve at once. A slope change occurs when short-term rates decrease (increase) while long-term rates increase (decrease).
Curvature defines convexity and concavity changes to the yield curve, capturing the bowing that occurs towards the belly of the curve.
Obviously these three components do not capture 100% of changes in the yield curve, but they do capture a significant portion of them. From 1962-2019 they explain 99.5% of the variance in daily yield
curve changes.
We can even decompose longer-term changes in the yield curve into these three components. For example, consider how the yield curve has changed in the three years from 6/30/2016 to 6/30/2019.
Source: Federal Reserve of St. Louis.
We can see that there was generally a positive increase across the entire curve (i.e. a positive level shift), the front end of the curve increased more rapidly (i.e. a flattening slope change) and
the curve flipped from concave to convex (i.e. an inverted bowing of the curve).
Using the historical yield curve changes, we can mathematically estimate these stylized changes using principal component analysis. We plot the loadings of the first three components below for this
three-year change.
Source: Federal Reserve of St. Louis. Calculations by Newfound Research.
We can see that –PC1– has generally positive loadings across the entire curve, and therefore captures our level shift component. –PC2– exhibits negative loadings on the front end of the curve and
positive loadings on the back, capturing our slope change. Finally, –PC3– has positive loadings from the 1-to-5-year part of the curve, capturing the curvature change of the yield curve itself.
Using a quick bit of linear algebra, we can find the combination of these three factors that closely matches the change in the curve from 6/30/2016 to 6/30/2019. Comparing our model versus the
actual change, we see a reasonably strong fit.
Source: Federal Reserve of St. Louis. Calculations by Newfound Research.
So why might this be useful information?
First of all, we can interpret our principal components as if they are portfolios. For example, our first principal component is saying, “buy a portfolio that is long interest rates across the
entire curve.” The second component, on the other hand, is better expressed as, “go short rates on the front end of the curve and go long rates on the back end.”
Therefore, insofar as we believe changes to the yield curve may exhibit absolute or relative momentum, we may be able to exploit this momentum by constructing a portfolio that profits from it.
As a more concrete example, if we believe that the yield curve will generally steepen over the next several years, we might buy 2-year U.S. Treasury futures and short 10-year U.S. Treasury futures.
The biggest wrinkle we need to deal with is the fact that 2-year U.S. Treasury futures will exhibit very different sensitivity to rate changes than 10-year U.S. Treasury futures, and therefore we
must take care to duration-adjust our positions.
Why might such changes exhibit trends or relative momentum?
• During periods where arbitrage capital is low, trends may emerge. We might expect this during periods of extreme market shock (e.g. recessions) where we might also see the simultaneous influence
of monetary policy.
• Effects from monetary policy may exhibit autocorrelation. If investors exhibit any anchoring to prior beliefs, they might discount future policy changes.
• Segmented market theory suggests that different investors tend to access different parts of the curve (e.g. pensions may prefer the far end of the curve for liability hedging purposes).
Information flow may therefore be segmented, or even impacted by structural buyers/sellers, creating autocorrelation in curve dynamics.
In related literature, Fan et al (2019) find that the net hedging or speculative position has strong cross-sectional explanatory power for agricultural and currency futures returns, but not in fixed
income markets. To quote,
“In sharp contrast, we find no evidence of a significant speculative pressure premium in the interest rate and fixed income futures markets. Thus, albeit from the lens of different research
questions, our paper reaffirms Bessembinder (1992) and Moskowitz et al. (2012) in establishing that fixed income futures markets behave differently from other futures markets as regards the
information content of the net positions of hedgers or speculators. A hedgers-to-speculators risk transfer in fixed income futures markets would be obscured if agents choose to hedge their
interest rate risk with other strategies (i.e. immunization, temporary change in modified duration).”
Interestingly, Markowitz et al. (2012) suggest that speculators may be profiting from time-series momentum at the expense of hedgers, suggesting that they earn a premium for providing liquidity.
Such does not appear to be the case for fixed income futures, however.
As far as we are aware, it has not yet been tested in the literature whether the net speculator versus hedger position has been tested for yield curve trades, and it may be possible that a risk
transfer does not exist at the individual maturity basis, but rather exists for speculators willing to bear level, slope, or curvature risk.
Stylized Component Trades
While we know the exact loadings of our principal components (i.e. which maturities make up the principal portfolios), to avoid the risk of overfitting our study we will capture level, slope, and
curvature changes with three different stylized portfolios.
To implement our portfolios, we will buy a basket of 2-, 5-, and 10-year U.S. Treasury futures contracts (“UST futures”). We will assume that the 5-year contract has 2.5x the duration of the 2-year
contract and the 10-year contract has 5x the duration of the 2-year contract.
To capture a level shift in the curve, we will go long across all the contracts. Specifically, for every dollar of 2-year UST futures exposure we purchase, we will buy $0.4 of 5-year UST futures and
$0.20 of 10-year UST futures. This creates equal duration exposure across the entire curve.
To capture slope change, we will go short 2-year UST futures and long the 10-year UST futures, holding zero position in the 5-year UST futures. As before, we will duration-adjust our positions such
that for each $1 short of the 2-year UST futures position, we are $0.20 long the 10-year UST futures.
Finally, to capture curvature change we will construct a butterfly trade where we short the 2- and 10-year UST futures and go long the 5-year UST futures. For each $1 long in the 5-year UST futures,
we will short $1.25 of 2-year UST futures and $0.25 of 10-year UST futures.
Note that the slope and curvature portfolios are implemented such that they are duration neutral (based upon our duration assumptions) so a level shift in the curve will generate no profit or loss.
An immediate problem with our approach arises when we actually construct these portfolios. Unless adjusted, the volatility exhibited across these trades will be meaningfully different. Therefore,
we target a constant 10% volatility for all three portfolios by adjusting the notional exposure of each portfolio based upon an exponentially-weighted estimate of prior 3-month realized volatility.
Source: Stevens Futures. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of
all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.
It appears, at least to the naked eye, that changes in the yield curve – and therefore the returns of these portfolios – may indeed exhibit positive autocorrelation. For example, –Slope– appears to
exhibit significant trends from 2000-2004, 2004-to 2007, and 2007-2012.
Whether those trends can be identified and exploited is another matter entirely. Thus, with our stylized portfolios in hand, we can begin testing.
Trend Signals
We begin our analysis by exploring the application of time-series momentum signals across all three of the portfolios. We evaluate lookback horizons ranging from 21-to-294 trading days (or,
approximately 1-to-14 months). Portfolios assume a 21-trading-day holding period and are implemented using 21 overlapping portfolios to control for timing luck.
Source: Stevens Futures. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of
all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.
Some observations:
• Time-series momentum appears to generate positive returns for the Level portfolio. Over the period tested, longer-term measures (e.g. 8-to-14-month horizons) offer more favorable results.
• Time-series momentum on the Level portfolio does, however, underperform naïve buy-and-hold. The returns of the strategy also do not offer a materially improved Sharpe ratio or drawdown profile.
• Time-series momentum also appears to capture trends in the Slope portfolio. Interestingly, both short- and long-term lookbacks are less favorable over the testing period than intermediate-term
(e.g. 4-to-8 month) ones.
• Finally, time-series momentum appeared to offer no edge in timing curvature trades.
Here we should pause to acknowledge that we are blindly throwing strategies at data without much forethought. If we consider, however, that we might reasonably expect duration to be a positively
compensated risk premium, as well as the fact that we would expect the futures to capture a generally positive roll premium (due to a generally upward sloping yield curve), then explicitly shorting
duration risk may not be a keen idea.
In other words, it may make more sense to implement our level trade as a long/flat rather than a long/short. When implemented in this fashion, we see that the annualized return versus buy-and-hold
is much more closely maintained while volatility and maximum drawdown are significantly reduced.
Source: Stevens Futures. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of
all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.
Taken together, it would appear that time-series momentum may be effective for trading the persistence in Level and Slope changes, though not in Curvature.
Momentum Signals
If we treat each stylized portfolio as a separate asset, we can also consider the returns of a cross-sectional momentum portfolio. For example, each month we can rank the portfolios based upon their
prior returns. The top-ranking portfolio is held long; the 2^nd ranked portfolio is held flat; and the 3^rd ranked portfolio is held short.
As before, we will evaluate lookback horizons ranging from 21-to-294 trading days (approximately 1-to-14 months) and assuming a 21-trading-day holding period, implemented with 21 overlapping
Results – as well as example allocations from the 7-month lookback portfolio – are plotted below.
Source: Stevens Futures. Calculations by Newfound Research. Past performance is not an indicator of future results. Performance is backtested and hypothetical. Performance figures are gross of
all fees, including, but not limited to, manager fees, transaction costs, and taxes. Performance assumes the reinvestment of all distributions.
Here we see very strong performance results except in the 1- and 2-month lookback periods. The allocation graph appears to suggest that results are not merely the byproduct of consistently being
long or short a particular portfolio and the total return level appears to suggest that the portfolio is able to simultaneously profit from both legs.
If we return back to the graph of the stylized portfolios, we can see a significant negative correlation between the Level and Slope portfolios from 1999 to 2011. The negative correlation appears to
disappear after this point, almost precisely coinciding with a 6+ year drawdown in the cross-sectional momentum strategy.
This is due to a mixture of construction and the economic environment.
From a construction perspective, consider that the Level portfolio is long the 2-, the 5-, and the 10-year UST futures while the Slope portfolio is short 2-year and long the 10-year UST futures.
Since the positions are held in a manner that targets equivalent duration exposure, when the 2-year rate moves more than the 10-year rate, we end up in a scenario where the two trades have negative
correlation, since one strategy is short and the other is long the 2-year position. Conversely, if the 10-year rate moves more than the 2-year rate, we end up in a scenario of positive correlation,
since both strategies are long the 10-year.
Now consider the 1999-2011 environment. We had an easing cycle during the dot-com bust, a tightening cycle during the subsequent economic expansion, and another easing cycle during the 2008 crisis.
This caused significantly more directional movement in the 2-year rate than the 10-year rate. Hence, negative correlation.
After 2008, however, the front end of the curve became pinned to zero. This meant that there was significantly more movement in the 10-year than the 2-year, leading to positive correlation in the
two strategies. With positive correlation there is less differentiation among the two strategies and so we see a considerable increase in strategy turnover – and effectiveness – as momentum signals
become less differentiated.
With that in mind, had we designed our Slope portfolio to be long 2-year UST futures and short 10-year UST futures (i.e. simply inverted the sign of our allocations), we would have seen positive
correlation between Level and Slope from 1999 to 2011, resulting in a very different set of allocations and returns. In actually testing this step, we find that the 1999-2011 period is no longer
dominated by Level versus Slope trades, but rather Slope versus Curvature. Performance of the strategy is still largely positive, but the spread among specifications widens dramatically.
Taken all together, it is difficult to conclude that the success of this strategy was not, in essence, driven almost entirely by autocorrelation in easing and tightening cycles with a relatively
stable back end of the curve.^1 Given that there have only been a handful of full rate cycles in the last 20 years, we’d be reluctant to rely too heavily on the equity curve of this strategy as
evidence of a robust strategy.
In this research note, we explored the idea of generating stylized portfolios designed to isolate and profit from changes to the form of the yield curve. Specifically, using 2-, 5-, and 10-year UST
futures we design portfolios that aim to profit from level, slope, and curvature changes to the US Treasury yield curve.
With these portfolios in hand, we test whether we can time exposure to these changes using time-series momentum.
We find that while time-series momentum generates positive performance for the Level portfolio, it fails to keep up with buy & hold. Acknowledging that level exposure may offer a positive long-term
risk premium, we adjust the strategy from long/short to long/flat and are able to generate a substantially improved risk-adjusted return profile.
Time-series momentum also appears effective for the Slope portfolio, generating meaningful excess returns above the buy-and-hold portfolio.
Applying time-series momentum to the Curvature portfolio does not appear to offer any value.
We also tested whether the portfolios can be traded employing cross-sectional momentum. We find significant success in the approach but believe that the results are an artifact of (1) the
construction of the portfolios and (2) a market regime heavily influenced by monetary policy. Without further testing, it is difficult to determine if this approach has merit.
Finally, even though our study focused on portfolios constructed using U.S. Treasury futures, we believe the results have potential application for investors who are simply trying to figure out how
to position their duration exposure. For example, a signal to be short (or flat) the Level portfolio and long the Slope portfolio may imply a view of rising rates with a flattening curve.
Translating these quantitative signals into a forecast about yield-curve behavior may allow investors to better position their fixed income portfolios.
Since this study utilized U.S. Treasury futures, these results translate well to implementing a portable beta strategy. For example, if you were an investor with a desired risk profile on par with
100% equities, you could add bond exposure on top of the higher risk portfolio. This would add a (generally) diversifying return source with only a minor cash drag to the extent that margin
requirements dictate.
This post is available as a PDF download here.
• The bond risk premium is the return that investors earn by investing in longer duration bonds.
• While the most common way that investors can access this return stream is through investing in bond portfolios, bonds often significantly de-risk portfolios and scale back returns.
• Investors who desire more equity-like risk can tap into the bond risk premium by overlaying bond exposure on top of equities.
• Through the use of a leveraged ETP strategy, we construct a long-only bond risk premium factor and investigate its characteristics in terms of rebalance frequency and timing luck.
• By balancing the costs of trading with the risk of equity overexposure, investors can incorporate the bond risk premium as a complementary factor exposure to equities without sacrificing return
potential from scaling back the overall risk level unnecessarily.
The discussion surrounding factor investing generally pertains to either equity portfolios or bond portfolios in isolation. We can calculate value, momentum, carry, and quality factors for each asset
class and invest in the securities that exhibit the best characteristics of each factor or a combination of factors.
There are also ways to use these factors to shift allocations between stocks and bonds (e.g. trend and standardizing based on historical levels). However, we do not typically discuss bonds as their
own standalone factor.
The bond risk premium – or term premium – can be thought of as the premium investors earn from holding longer duration bonds as opposed to cash. In a sense, it is a measure of carry. Its theoretical
basis is generally seen to be related to macroeconomic factors such as inflation and growth expectations.^1
While timing the term premium using factors within bond duration buckets is definitely a possibility, this commentary will focus on the term premium in the context of an equity investor who wants
long-term exposure to the factor.
The Term Premium as a Factor
For the term premium, we can take the usual approach and construct a self-financing long/short portfolio of 100% intermediate (7-10 year) U.S. Treasuries that borrows the entire portfolio value at
the risk-free rate.
This factor, shown in bold in the chart below, has exhibited a much tamer return profile than common equity factors.
Source: CSI Analytics, AQR, and Bloomberg. Calculations by Newfound Research. Data from 1/31/1992 to 6/28/2019. Results are hypothetical. Results assume the reinvestment of all distributions.
Results are gross of all fees, including, but not limited to manager fees, transaction costs, and taxes. Past performance is not an indicator of future results.
Source: CSI Analytics, AQR, and Bloomberg. Calculations by Newfound Research. Data from 1/31/1992 to 6/28/2019. Results are hypothetical. Results assume the reinvestment of all distributions.
Results are gross of all fees, including, but not limited to manager fees, transaction costs, and taxes. Past performance is not an indicator of future results.
But over the entire time period, its returns have been higher than those of both the Size and Value factors. Its maximum drawdown has been less than 40% of that of the next best factor (Quality), and
it is worth acknowledging that its volatility – which is generally correlated to drawdown for highly liquid assets with non-linear payoffs – has also been substantially lower.
The term premium also has exhibited very low correlation with the other equity factors.
Source: CSI Analytics, AQR, and Bloomberg. Calculations by Newfound Research. Data from 1/31/1992 to 6/28/2019. Results are hypothetical. Results assume the reinvestment of all distributions.
Results are gross of all fees, including, but not limited to manager fees, transaction costs, and taxes. Past performance is not an indicator of future results.
A Little Free Lunch
Whether we are treating bonds as factor or not, they are generally the primary way investors seek to diversify equity portfolios.
The problem is that they are also a great way to reduce returns during most market environments through their inherently lower risk.
Anytime that an asset with lower volatility is added to a portfolio, the risk will be reduced. Unless the asset class also has a particularly high Sharpe ratio, maintaining the same level of return
is virtually impossible even if risk-adjusted returns are improved.
In a 2016 paper^2, Salient broke down this reduction in risk into two components: de-risking and the “free lunch” affect.
The reduction in risk form the free lunch effect is desirable, but the risk reduction from de-risking may or may not be desirable, depending on the investor’s target risk profile.
The following chart shows the volatility breakdown of a range of portfolios of the S&P 500 (IVV) and 7-10 Year U.S. Treasuries (IEF).
Source: CSI Analytics and Bloomberg. Calculations by Newfound Research. Data from 1/31/1992 to 6/28/2019. Results are hypothetical. Results assume the reinvestment of all distributions. Results are
gross of all fees, including, but not limited to manager fees, transaction costs, and taxes. Past performance is not an indicator of future results.
Moving from an all equity portfolio to a 50/50 equity reduces the volatility from 14.2% to 7.4%. But only 150 bps of this reduction is from the free lunch effect that stems from the lower correlation
between the two assets (-0.18). The remaining 530 bps of volatility reduction is simply due to lower risk.
In this case, annualized returns were dampened from 9.6% to 7.8%. While the Sharpe ratio climbed from 0.49 to 0.70, an investor seeking higher risk would not benefit without the use of leverage.
Despite the strong performance of the term premium factor, risk-seeking investors (e.g. those early in their careers) are generally reluctant to tap into this factor too much because of the
de-risking effect.
How do investors who want to bear risk commensurate with equities tap into the bond risk premium without de-risking their portfolio?
One solution is using leveraged ETPs.
Long-Only Term Premium
By taking a 50/50 portfolio of the 2x Levered S&P 500 ETF (SSO) and the 2x Levered 7-10 Year U.S. Treasury ETF (UST), we can construct a portfolio that has 100% equity exposure and 100% of the term
premium factor.^3
But managing this portfolio takes some care.
Left alone to drift, the allocations can get very far away from their target 50/50, spanning the range from 85/15 to 25/75. Periodic rebalancing is a must.
Source: CSI Analytics and Bloomberg. Calculations by Newfound Research. Data from 1/31/1992 to 6/28/2019. Results are hypothetical. Results assume the reinvestment of all distributions. Results are
gross of all fees, including, but not limited to manager fees, transaction costs, and taxes. Past performance is not an indicator of future results.
Of course, now the question is, “How frequently should we rebalance the portfolio?”
This boils down to a balancing act between performance and costs (e.g. ticket charges, tax impacts, operational burden, etc.).
On one hand, we would like to remain as close to the 50/50 allocation as possible to maintain the desired exposure to each asset class. However, this could require a prohibitive amount of trading.
From a performance standpoint, we see improved results with longer holding periods (take note of the y-axes in the following charts; they were scaled to highlight the differences).
Source: CSI Analytics and Bloomberg. Calculations by Newfound Research. Data from 1/31/1992 to 6/28/2019. Results are hypothetical. Results assume the reinvestment of all distributions. Results are
gross of all fees, including, but not limited to manager fees, transaction costs, and taxes. Past performance is not an indicator of future results.
The returns do not show a definitive pattern based on rebalance frequency, but the volatility decreases with increasing time between rebalances. This seems like it would point to waiting longer
between rebalances, which would be corroborated by a consideration of trading costs.
The issues with waiting longer between the rebalance are twofold:
1. Waiting longer is essentially a momentum trade. The better performing asset class garners a larger allocation as time progresses. This can be a good thing – especially in hindsight with how well
equities have done – but it allows the portfolio to become overexposed to factors that we are not necessarily intending to exploit.
2. Longer rebalances are more exposed to timing luck. For example, a yearly rebalance may have done well from a performance perspective, but the short-term performance could vary by as much as
50,000 bps between the best performing rebalance month and the worst! The chart below shows the performance of each iteration relative to the median performance of the 12 different monthly
rebalance strategies.
Source: CSI Analytics and Bloomberg. Calculations by Newfound Research. Data from 1/31/1992 to 6/28/2019. Results are hypothetical. Results assume the reinvestment of all distributions. Results are
gross of all fees, including, but not limited to, manager fees, transaction costs, and taxes. Past performance is not an indicator of future results.
As the chart also shows, tranching can help mitigate timing luck. Tranching also gives the returns of the strategies over the range of rebalance frequencies a more discernible pattern, with longer
rebalance period strategies exhibiting slightly higher returns due to their higher average equity allocations.
Under the assumption that we can tranche any strategy that we choose, we can now compare only the tranched strategies at different rebalance frequencies to address our concern with taking bets on
Pausing for a minute, we should be clear that we do not actually know what the true factor construction should be; it is a moving target. We are more concerned with robustness than simply trying to
achieve outperformance. So we will compare the strategies to the median performance of the previously monthly offset annual rebalance strategies.
The following charts shows the aggregate risk of short-term performance deviations from this benchmark.
The first one shows the aggregate deviations, both positive and negative, and the second focuses on only the downside deviation (i.e. performance that is worse than the median).^4
Both charts support a choice of rebalance frequency somewhere in the range of 3-6 months.
Source: CSI Analytics and Bloomberg. Calculations by Newfound Research. Data from 1/31/1992 to 6/28/2019. Results are hypothetical. Results assume the reinvestment of all distributions. Results are
gross of all fees, including, but not limited to manager fees, transaction costs, and taxes. Past performance is not an indicator of future results.
With the rebalance frequency set based on the construction of the factor, the last part is a consideration of costs.
Unfortunately, this is more situation-specific (e.g. what commissions does your platform charge for trades?).
From an asset manager point-of-view, where we can trade with costs proportional to the size of the trade, execute efficiently, and automate much of the operational burden, tranching is our preferred
We also prefer this approach over simply rebalancing back to the static 50/50 allocation more frequently.
In our previous commentary on constructing value portfolios to mitigate timing luck, we described how tranching monthly is a different decision than rebalancing monthly and that tranching frequency
and rebalance frequency are distinct decisions.
We see the same effect here where we plot the monthly tranched annually rebalanced strategy (blue line) and the strategy rebalanced back to 50/50 every month (orange line).
Source: CSI Analytics and Bloomberg. Calculations by Newfound Research. Data from 1/31/1992 to 6/28/2019. Results are hypothetical. Results assume the reinvestment of all distributions. Results are
gross of all fees, including, but not limited to manager fees, transaction costs, and taxes. Past performance is not an indicator of future results.
Tranching wins out.
However, since the target for the term premium factor is a 50/50 static allocation, running a simple allocation filter to keep the portfolio weights within a certain tolerance can be a way to
implement a more dynamic rebalancing model while reducing costs.
For example, rebalancing when the allocations for SSO and UST we outside a 5% band (i.e. the portfolio was beyond a 55/45 or 45/55) achieved better performance metrics than the monthly rebalanced
version with an average of only 3 rebalances per year.
The bond term premium does not have to be reserved for risk-averse investors. Investors desiring portfolios tilted heavily toward equities can also tap into this diversifying return stream as a
factor within their portfolio.
Utilizing leveraged ETPs is one way to maintaining exposure to equities while capturing a significant portion of the bond risk premium. However, it requires more oversight than investing in other
factors such as value, momentum, and quality, which are typically packaged in easy-to-access ETFs.
If a fixed frequency rebalance approach is used, tranching is an effective way to reduce timing risk, especially when markets are volatile. Aside from tranching, we find that, historically, holding
periods between 3 and 6 months yield results close in line with the median rolling short-term performance of the individual strategies. Implementing a methodology like this can reduce the risk of
poor luck in choosing the rebalance frequency or starting the strategy at an unfortunate time.
If frequent rebalances – like those seen with tranching – are infeasible, a dynamic schedule based on a drift in allocations is also a possibility.
Leveraged ETPs are often seen as risk trading instruments that are not fit for retail investors who are more focused on buy-and-hold systems. However, given the right risk management, these
investment vehicles can be a way for investors to access the bond term premium, getting a larger free lunch, and avoiding undesired de-risking along the way.
This post is available as a PDF download here.
• In this commentary, we revisit the idea of portable beta: utilizing leverage to overlay traditional risk premia on existing strategic allocations.
• While a 1.5x levered 60/40 portfolio has historically out-performed an all equity blend with similar risk levels, it can suffer through prolonged periods of under-performance.
• Positive correlations between stocks and bonds, inverted yield curves, and rising interest rate environments can make simply adding bond exposure on top of equity exposure a non-trivial pursuit.
• We rely on prior research to introduce a tactical 90/60 model, which uses trend signals to govern equity exposure and value, momentum, and carry signals to govern bond exposure.
• We find that such a model has historically exhibited returns in-line with equities with significantly lower maximum drawdown.
In November 2017, I was invited to participate in a Bloomberg roundtable discussion with Barry Ritholtz, Dave Nadig, and Ben Fulton about the future of ETFs. I was quoted as saying,
Most of the industry agrees that we are entering a period of much lower returns for stocks and fixed income. That’s a problem for younger generations. The innovation needs to be around efficient
use of capital. Instead of an ETF that holds intermediate-term Treasuries, I would like to see a U.S. Treasury ETF that uses Treasuries as collateral to buy S&P 500 futures, so you end up getting
both stock and bond exposure. By introducing a modest amount of leverage, you can take $1 and trade it as if the investor has $1.50. After 2008, people became skittish around derivatives,
shorting, and leverage. But these aren’t bad things when used appropriately.
Shortly after the publication of the discussion, we penned a research commentary titled Portable Beta which extolled the potential virtues of employing prudent leverage to better exploit
diversification opportunities. For investors seeking to enhance returns, increasing beta exposure may be a more reliable approach than the pursuit of alpha.
In August 2018, WisdomTree introduced the 90/60 U.S. Balanced Fund (ticker: NTSX), which blends core equity exposure with a U.S. Treasury futures ladder to create the equivalent of a 1.5x levered 60/
40 portfolio. On March 27, 2019, NTSX was awarded ETF.com’s Most Innovative New ETF of 2018.
The idea of portable beta was not even remotely uniquely ours. Two anonymous Twitter users – “Jake” (@EconomPic) and “Unrelated Nonsense” (@Nonrelatedsense) – had discussed the idea several times
prior to my round-table in 2017. They argued that such a product could be useful to free up space in a portfolio for alpha-generating ideas. For example, an investor could hold 66.6% of their
wealth in a 90/60 portfolio and use the other 33.3% of their portfolio for alpha ideas. While the leverage is technically applied to the 60/40, the net effect would be a 60/40 portfolio with a set
of alpha ideas overlaid on the portfolio. Portable beta becomes portable alpha.
Even then, the idea was not new. After NTSX launched, Cliff Asness, co-founder and principal of AQR Capital Management, commented on Twitter that even though he had a “22-year head start,”
WisdomTree had beat him to launching a fund. In the tweet, he linked to an article he wrote in 1996, titled Why Not 100% Equities, wherein Cliff demonstrated that from 1926 to 1993 a 60/40 portfolio
levered to the same volatility as equities achieved an excess return of 0.8% annualized above U.S. equities. Interestingly, the appropriate amount of leverage utilized to match equities was 155%,
almost perfectly matching the 90/60 concept.
Source: Asness, Cliff. Why Not 100% Equities. Journal of Portfolio Management, Winter 1996, Volume 22 Number 2.
Following up on Cliff’s Tweet, Jeremy Schwartz from WisdomTree extended the research out-of-sample, covering the quarter century that followed Cliff’s initial publishing date. Over the subsequent 25
years, Jeremy found that a levered 60/40 outperformed U.S. equities by 2.6% annualized.
NTSX is not the first product to try to exploit the idea of diversification and leverage. These ideas have been the backbone of managed futures and risk parity strategies for decades. The entire
PIMCO’s StocksPLUS suite – which traces its history back to 1986 – is built on these foundations. The core strategy combines an actively managed portfolio of fixed income with 100% notional exposure
in S&P 500 futures to create a 2x levered 50/50 portfolio.
The concept traces its roots back to the earliest eras of modern financial theory. Finding the maximum Sharpe ratio portfolio and gearing it to the appropriate risk level has always been considered
to be the theoretically optimal solution for investors.
Nevertheless, after 2008, the words “leverage” and “derivatives” have largely been terms non gratisin the realm of investment products. But that may be to the detriment of investors.
90/60 Through the Decades
While we are proponents of the foundational concepts of the 90/60 portfolio, frequent readers of our commentary will not be surprised to learn that we believe there may be opportunities to enhance
the idea through tactical asset allocation. After all, while a 90/60 may have out-performed over the long run, the short-run opportunities available to investors can deviate significantly. The
prudent allocation at the top of the dot-com bubble may have looked quite different than that at the bottom of the 2008 crisis.
To broadly demonstrate this idea, we can examine the how the realized efficient frontier of stock/bond mixes has changed shape over time. In the table below, we calculate the Sharpe ratio for
different stock/bond mixes realized in each decade from the 1920s through present.
Source: Global Financial Data. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees, transaction costs, and taxes. Returns assume the
reinvestment of all distributions. Bonds are the GFD Indices USA 10-Year Government Bond Total Return Index and Stocks are the S&P 500 Total Return Index (with GFD Extension). Sharpe ratios are
calculated with returns excess of the GFD Indices USA Total Return T-Bill Index. You cannot invest in an index. 2010s reflect a partial decade through 4/2019.
We should note here that the original research proposed by Asness (1996) assumed a bond allocation to an Ibbotson corporate bond series while we employ a constant maturity 10-year U.S. Treasury
index. While this leads to lower total returns in our bond series, we do not believe it meaningfully changes the conclusions of our analysis.
We can see that while the 60/40 portfolio has a higher realized Sharpe ratio than the 100% equity portfolio in eight of ten decades, it has a lower Sharpe ratio in two consecutive decades from 1950 –
1960. And the 1970s were not a ringing endorsement.
In theory, a higher Sharpe ratio for a 60/40 portfolio would imply that an appropriately levered version would lead to higher realized returns than equities at the same risk level. Knowing the
appropriate leverage level, however, is non-trivial, requiring an estimate of equity volatility. Furthermore, leverage requires margin collateral and the application of borrowing rates, which can
create a drag on returns.
Even if we conveniently ignore these points and assume a constant 90/60, we can still see that such an approach can go through lengthy periods of relative under-performance compared to buy-and-hold
equity. Below we plot the annualized rolling 3-year returns of a 90/60 portfolio (assuming U.S. T-Bill rates for leverage costs) minus 100% equity returns. We can clearly see that the 1950s through
the 1980s were largely a period where applying such an approach would have been frustrating.
Source: Global Financial Data. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees, transaction costs, and taxes. Bonds are the GFD Indices
USA 10-Year Government Bond Total Return Index and Stocks are the S&P 500 Total Return Index (with GFD Extension). The 90/60 portfolio invests 150% each month in the 60/40 portfolio and -50% in the
GFD Indices USA Total Return T-Bill Index. You cannot invest in an index.
Poor performance of the 90/60 portfolio in this era is due to two effects.
First, 10-year U.S. Treasury rates rose from approximately 4% to north of 15%. While a constant maturity index would constantly roll into higher interest bonds, it would have to do so by selling old
holdings at a loss. Constantly harvesting price losses created a headwind for the index.
This is compounded in the 90/60 by the fact that the yield curve over this period spent significant time in an inverted state, meaning that the cost of leverage exceeded the yield earned on 40% of
the portfolio, leading to negative carry. This is illustrated in the chart below, with –T-Bills– realizing a higher total return over the period than –Bonds–.
Source: Global Financial Data. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees, transaction costs, and taxes. Returns assume the
reinvestment of all distributions. T-Bills are the GFD Indices USA Total Return T-Bill Index, Bonds are the GFD Indices USA 10-Year Government Bond Total Return Index, and Stocks are the S&P 500
Total Return Index (with GFD Extension). You cannot invest in an index.
This is all arguably further complicated by the fact that while a 1.5x levered 60/40 may closely approximate the risk level of a 100% equity portfolio over the long run, it may be a far cry from it
over the short-run. This may be particularly true during periods where stocks and bonds exhibit positive realized correlations as they did during the 1960s through 1980s. This can occur when
markets are more pre-occupied with inflation risk than economic risk. As inflationary fears abated and economic risk become the foremost concern in the 1990s, correlations between stocks and bonds
Thus, during the 1960s-1980s, a 90/60 portfolio exhibited realized volatility levels in excess of an all-equity portfolio, while in the 2000s it has been below.
This all invites the question: should our levered allocation necessarily be static?
Getting Tactical with a 90/60
We might consider two approaches to creating a tactical 90/60.
The first is to abandon the 90/60 model outright for a more theoretically sound approach. Specifically, we could attempt to estimate the maximum Sharpe ratio portfolio, and then apply the appropriate
leverage such that we either hit a (1) constant target volatility or (2) the volatility of equities. This would require us to not only accurately estimate the expected excess returns of stocks and
bonds, but also their volatilities and correlations. Furthermore, when the Sharpe optimal portfolio is highly conservative, notional exposure far exceeding 200% may be necessary to hit target
volatility levels.
In the second approach, equity and bond exposure would each be adjusted tactically, without regard for the other exposure. While less theoretically sound, one might interpret this approach as
saying, “we generally want exposure to the equity and bond risk premia over the long run, and we like the 60/40 framework, but there might be certain scenarios whereby we believe the expected return
does not justify the risk.” The downside to this approach is that it may sacrifice potential diversification benefits between stocks and bonds.
Given the original concept of portable beta is to increase exposure to the risk premia we’re already exposed to, we prefer the second approach. We believe it more accurately reflects the notion of
trying to provide long-term exposure to return-generating risk premia while trying to avoid the significant and prolonged drawdowns that can be realized with buy-and-hold approaches.
Equity Signals
To manage exposure to the equity risk premium, our preferred method is the application of trend following signals in an approach we call trend equity. We will approximate this class of strategies
with our Newfound Research U.S. Trend Equity Index.
To determine whether our signals are able to achieve their goal of “protect and participate” with the underlying risk premia, we will plot their regime-conditional betas. To do this, we construct a
simple linear model:
We define a bear regime as the worst 16% of monthly returns, a bull regime as the best 16% of monthly returns, and a normal regime as the remaining 68% of months. Note that the bottom and top 16^
thpercentiles are selected to reflect one standard deviation.
Below we plot the strategy conditional betas relative to U.S. equity
We can see that trend equity has a normal regime beta to U.S. equities of approximately 0.75 and a bear market beta of 0.5, in-line with expectations that such a strategy might capture 70-80% of the
upside of U.S. equities in a bull market and 40-50% of the downside in a prolonged bear market. Trend equity beta of U.S. equities in a bull regime is close to the bear market beta, which is
consistent with the idea that trend equity as a style has historically sacrificed the best returns to avoid the worst.
Bond Signals
To govern exposure to the bond risk premium, we prefer an approach based upon a combination of quantitative, factor-based signals. We’ve written about many of these signals over the last two years;
specifically in Duration Timing with Style Premia (June 2017), Timing Bonds with Value, Momentum, and Carry (January 2018), and A Carry-Trend-Hedge Approach to Duration Timing (October 2018). In
these three articles we explore various mixes of value, momentum, carry, flight-to-safety, and bond risk premium measures as potential signals for timing duration exposure.
We will not belabor this commentary unnecessarily by repeating past research. Suffice it to say that we believe there is sufficient evidence that value (deviation in real yield), momentum (prior
returns), and carry (term spread) can be utilized as effective timing signals and in this commentary are used to construct bond indices where allocations are varied between 0-100%. Curious readers
can pursue further details of how we construct these signals in the commentaries above.
As before, we can determine conditional regime betas for strategies based upon our signals.
We can see that our value, momentum, and carry signals all exhibit an asymmetric beta profile with respect to 10-year U.S. Treasury returns. Carry and momentum exhibit an increase in bull market
betas while value exhibits a decrease in bear market beta.
Combining Equity and Bond Signals into a Tactical 90/60
Given these signals, we will construct a tactical 90/60 portfolio as being comprised of 90% trend equity, 20% bond value, 20% bond momentum, and 20% bond carry. When notional exposure exceeds 100%,
leverage cost is assumed to be U.S. T-Bills. Taken together, the portfolio has a large breadth of potential configurations, ranging from 100% T-Bills to a 1.5x levered 60/40 portfolio.
But what is the appropriate benchmark for such a model?
In the past, we have argued that the appropriate benchmark for trend equity is a 50% stock / 50% cash benchmark, as it not only reflects the strategic allocation to equities empirically seen in
return decompositions, but it also allows both positive and negative trend calls to contribute to active returns.
Similarly, we would argue that the appropriate benchmark for our tactical 90/60 model is not a 90/60 itself – which reflects the upper limit of potential capital allocation – but rather a 45% stock /
30% bond / 25% cash mix. Though, for good measure we might also consider a bit of hand-waving and just use a 60/40 as a generic benchmark as well.
Below we plot the annualized returns versus maximum drawdown for different passive and active portfolio combinations from 1974 to present (reflecting the full period of time when strategy data is
available for all tactical signals). We can see that not only does the tactical 90/60 model (with both trend equity and tactical bonds) offer a return in line with U.S. equities over the period, it
does so with significantly less drawdown (approximately half). Furthermore, the tactical 90/60 exceeded trend equity and 60/40 annualized returns by 102 and 161 basis points respectively.
These improvements to the return and risk were achieved with the same amount of capital commitment as in the other allocations. That’s the beauty of portable beta.
Source: Federal Reserve of St. Louis, Kenneth French Data Library, and Newfound Research. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees,
transaction costs, and taxes. Returns assume the reinvestment of all distributions. You cannot invest in an index.
Of course, full-period metrics can deceive what an investor’s experience may actually be like. Below we plot rolling 3-year annualized returns of U.S. equities, the 60/40 mix, trend equity, and the
tactical 90/60.
Source: Federal Reserve of St. Louis, Kenneth French Data Library, and Newfound Research. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees,
transaction costs, and taxes. Returns assume the reinvestment of all distributions. You cannot invest in an index.
The tactical 90/60 model out-performed a 60/40 in 68% of rolling 3-year periods and the trend equity model in 71% of rolling 3-year periods. The tactical 90/60, however, only out-performs U.S.
equities in 35% of rolling 3-year periods, with the vast majority of relative out-performance emerging during significant equity drawdown periods.
For investors already allocated to trend equity strategies, portable beta – or portable tactical beta – may represent an alternative source of potential return enhancement. Rather than seeking
opportunities for alpha, portable beta allows for an overlay of more traditional risk premia, which may be more reliable from an empirical and academic standpoint.
The potential for increased returns is illustrated below in the rolling 3-year annualized return difference between the tactical 90/60 model and the Newfound U.S. Trend Equity Index.
Source: Federal Reserve of St. Louis, Kenneth French Data Library, and Newfound Research. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees,
transaction costs, and taxes. Returns assume the reinvestment of all distributions. You cannot invest in an index.
From Theory to Implementation
In practice, it may be easier to acquire leverage through the use of futures contracts. For example, applying portable bond beta on-top of an existing trend equity strategy may be achieved through
the use of 10-year U.S. Treasury futures.
Below we plot the growth of $1 in the Newfound U.S. Trend Equity Index and a tactical 90/60 model implemented with Treasury futures. Annualized return increases from 7.7% to 8.9% and annualized
volatility declines from 9.7% to 8.5%. Finally, maximum drawdown decreases from 18.1% to 14.3%.
We believe the increased return reflects the potential return enhancement benefits from introducing further exposure to traditional risk premia, while the reduction in risk reflects the benefit
achieved through greater portfolio diversification.
Source: Quandl and Newfound Research. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees, transaction costs, and taxes. Returns assume the
reinvestment of all distributions. You cannot invest in an index.
It should be noted, however, that a levered constant maturity 10-year U.S. Treasury index and 10-year U.S. Treasury futures are not the same. The futures contracts are specified such that eligible
securities for delivery include Treasury notes with a remaining term to maturity of between 6.5 and 10 years. This means that the investor short the futures contract has the option of which Treasury
note to deliver across a wide spectrum of securities with potentially varying characteristics.
In theory, this investor will always choose to deliver the bond that is cheapest. Thus, Treasury futures prices will reflect price changes of this so-calledcheapest-to-deliver bond, which often does
not reflect an actual on-the-run 10-year Treasury note.
Treasury futures therefore utilize a “conversion factor” invoicing system referenced to the 6% futures contract standard. Pricing also reflects a basis adjustment that reflects the coupon income a
cash bond holder would receive minus financing costs (i.e. the cost of carry) as well as the value of optionality provided to the futures seller.
Below we plot monthly returns of 10-year U.S. Treasury futures versus the excess returns of a constant maturity 10-year U.S. Treasury index. We can see that the futures had a beta of approximately
0.76 over the nearly 20-year period, which closely aligns with the conversion factor over the period.
Source: Quandl and the Federal Reserve of St. Louis. Calculations by Newfound Research.
Despite these differences, futures can represent a highly liquid and cost-effective means of implementing a portable beta strategy. It should be further noted that having a lower “beta” over the
last two decades has not necessarily implied a lower return as the basis adjustment can have a considerable impact. We demonstrate this in the graph below by plotting the returns of
continuously-rolled 10-year U.S. Treasury futures (rolled on open interest) and the excess return of a constant maturity 10-year U.S. Treasury index.
Source: Quandl and Newfound Research. Calculations by Newfound Research. Returns are hypothetical and backtested. Returns are gross of all fees, transaction costs, and taxes. Returns assume the
reinvestment of all distributions. You cannot invest in an index.
In a low return environment, portable beta may be a necessary tool for investors to generate the returns they need to hit their financial goals and reduce their risk of failing slow.
Historically, a 90/60 portfolio has outperformed equities with a similar level of risk. However, the short-term dynamics between stocks and bonds can make the volatility of a 90/60 portfolio
significantly higher than a simple buy-and-hold equity portfolio. Rising interest rates and inverted yield curves can further confound the potential benefits versus an all-equity portfolio.
Since constant leverage is not a guarantee and we do not know how the future will play out, moving beyond standard portable beta implementations to tactical solutions may augment the potential for
risk management and lead to a smoother ride over the short-term.
Getting over the fear of using leverage and derivatives may be an uphill battle for investors, but when used appropriately, these tools can make portfolios work harder. Risks that are known and
compensated with premiums can be prudent to take for those willing to venture out and bear them.
If you are interested in learning how Newfound applies the concepts of tactical portable beta to its mandates, please reach out (info@thinknewfound.com).
This post is available as a PDF here.
• From 1981 to 2017, 10-year U.S. Treasury rates declined from north of 15% to below 2%.
• Since bond prices appreciate when rates decline, many have pointed towards this secular decline as a tailwind that created an unprecedented bull market in bonds.
• Exactly how much declining rates contributed, however, is rarely quantified. An informal poll, however, tells us that people generally believe the impact was significant (explaining >50% of bond
• We find that while, in theory, investors should be indifferent to rate changes, high turnover in bond portfolios means that a structural mis-estimation of rate changes could be harvested.
• Despite the positive long-term impact of declining rates, coupon yield had a much more significant impact on long-term returns.
• The bull market in bonds was caused more by the high average rates over the past 30 years than declining rates.
On 9/30/1981, the 10-year U.S. Treasury rate peaked at an all-time high of 15.84%. Over the next 30 years, it deflated to an all-time low of 1.37% on 7/5/2016.
Source: Federal Reserve of St. Louis
It has been repeated in financial circles that this decline in rates caused a bull market in bond returns that makes past returns a particularly poor indicator of future results.
But exactly how much did those declining rates contribute?
We turned to our financial circle on Twitter[1] with a question: For a constant maturity, 10-year U.S. Treasury index, what percent of total return from 12/1981 through 12/2012 could be attributed to
declining rates?
Little consensus was found.
Clearly there is a large disparity in views about exactly how much declining rates actually contributed to bond returns over the last 30 years. What we can see is that people generally think it is a
lot: over 50% of people said over 50% of returns can be attributed to declining rates.
Well let’s dig in and find out.
Rates Down, Bonds Up
To begin, let’s remind ourselves why the bond / rate relationship exists in the first place.
Imagine you buy a 10-year U.S. Treasury bond for $100 at the prevailing 5% rate. Immediately after you buy, interest rates drop: all available 10-year U.S. Treasury bonds – still selling for $100 –
are now offering only a 4% yield.
In every other way, except the yield being offered, the bond you now hold and the bonds being offered in the market are identical. Except yours provides a higher yield.
Therefore, it should be more valuable. After all, you are getting more return for your investment. And hence we get the inverse relationship between bonds and interest rates. As rates fall,
existing bond values go up and as rates rise, existing bond values go down.
With rates falling by an average of 42 basis points a year over the last 35 years, we can imagine a pretty steady, and potentially sizable tailwind to returns.
Just How Much More Valuable?
In our example, exactly how much did our bond appreciate when rates fell? Or, to ask the question another way: how much would someone now be willing to buy our bond for?
The answer arises from the fact that markets loathe an arbitrage opportunity. Scratch that: markets love arbitrage. So much so that they are quickly wiped away as market participants jump to
exploit them.
We mentioned that in the example, the bond you held and the bonds now being offered by the market were identical in every fashion except the coupon yield they offer.
Consider what would happen if the 4% bonds and your 5% bonds were both still selling for $100. Someone could come to the market, short-sell a 4% bond and use the $100 to buy your 5% bond from you.
Each coupon period, they would collect $5 from the bond they bought from you, pay $4 to cover the coupon payment they owe from the short-sale, and pocket $1.
Effectively, they’ve created a free stream of $1 bills.
Knowing this to be the case, someone else might step in first and try to offer you $101 for your bond to sweeten the deal. Now they must finance by short-selling 1.01 shares of the 4% bonds, owing
$4.04 each period and $101 at maturity. While less profitable, they would still pocket a free $0.86 per coupon payment.[2]
The scramble to sweeten the offer continues until it reaches the magic figure of $108.11. At this price, the arbitrage disappears: the cost of financing exactly offsets the extra yield earned by the
Another way of saying this is that the yield-to-maturity of both bonds is identical. If someone pays $108.11 for the 5% coupon bond, they may receive a $5 coupon each period, but there will be a
“pull-to-par” effect as the bond matures, causing the bond to decline in value. This effect occurs because the bond has a pre-defined payout stream: at maturity, you are only going to receive your
$100 back.
Therefore, while your coupon yield may be 5%, your effective yield – which accounts for this loss in value over time – is 4%, perfectly matching what is available to other investors.
And so everyone becomes indifferent[3] to which bond they hold. The bond you hold may be worth more on paper, but if we try to sell it to lock in our profit, we have to reinvest at a lower yield and
offsets our gain.
In a strange way, then, other than mark-to-market gains and losses, we should be largely indifferent to rate changes.
The Impact of Time
One very important aspect ignored by our previous example is time. Interest rates rarely gap up or down instantaneously: rather they move over time.
We therefore need to consider the yield curve. The yield curve tells us what rate is being offered for bonds of different maturities.
Source: Federal Reserve of St. Louis.
In the yield curve plotted above, we see an upward sloping trend. Buying a 7-year U.S. Treasury earns us a 2.25% rate, while the 10-year U.S. Treasury offers 2.45%.
Which introduces an interesting dynamic: if rates do not change whatsoever, if we buy a 10-year bond today and wait three years, our bond will appreciate in value.
The answer is because it is now a 7-year bond, and compared to other 7-year bonds it is offering 0.20% more yield.
In fact, depending on the shape of the yield curve, it can continue to appreciate until the pull-to-par effect becomes too strong. Below we plot the value of a 10-year U.S. Treasury as it matures,
assuming that the yield curve stays entirely constant over time.
Source: Federal Reserve of St. Louis. Calculations by Newfound Research.
Unfortunately, like in our previous example, the amount of the bond gains in value is exactly equal to the level required to make us indifferent to holding the bond to maturity or selling it and
reinvesting at the prevailing rate. For all intents and purposes, we could simply pretend we bought a 7-year bond at 2.45% and rates fell instantly to 2.25%. By the same logic as before, we’re no
better off.
We simply cannot escape the fact that markets are not going to give us a free return.
The Impact of Choice
Again, reality is more textured than theory. We are ignoring an important component: choice.
In our prior examples, our choice was between continuing to hold our bond, or selling it and reinvesting in the equivalent bond. What if we chose to reinvest in something else?
For example:
• We buy a 2.45% 10-year U.S. Treasury for $100
• We wait three years
• We sell the now 7-year U.S. Treasury for $101.28 (assuming the yield curve did not change)
• We reinvest in 2.45% 10-year U.S. Treasuries, sold at $100
If the yield curve never changes, we can keep capturing this roll return by simply waiting, selling, and buying what we previously owned.
What’s the catch? The catch, of course, is that we’re assuming rates won’t change. If we stop for a moment, however, and consider what the yield curve is telling us, we realize this assumption may
be quite poor.
The yield curve provides several rates at which we can invest. What if we are only interested in investing over the next year? Well, we can buy a 1-year U.S. Treasury at 0.85% and just hold it to
maturity, or we could buy a 10-year U.S. Treasury for 2.45% and sell it after a year.
That is a pretty remarkable difference in 1-year return potential.
If the market is even reasonably efficient, then the expected 1-year return, no matter where we buy on the curve, should be the same. Therefore, the only way the 10-year U.S. Treasury yield should
be so much higher than the 1-year is if the market is predicting that rates are going to go up such that the extra yield is exactly offset by the price loss we take when we sell the bond.
Hence a rising yield curve tells us the market is expecting rising rates. At least, that’s what the pure expectations hypothesis (“PEH”) says. Competing theories argue that investors should earn at
least some premium for bearing term risk. Nevertheless, there should be some component of a rising yield curve that tells us rates should go up.
However, over the past 35 years, the average slope of the yield curve (measured as 10-year yields minus 2-year yields) has been over 100bp. The market was, in theory, was consistently predicting
rising rates over a period rates fell.
Source: Federal Reserve of St. Louis. Calculations by Newfound Research.
Not only could an investor potentially harvest roll-yield, but also the added bump from declining rates.
Unfortunately, doing so would require significant turnover. We would have to constantly sell our bonds to harvest the gains.
While this may have created opportunity for active bond managers, a total bond market index typically holds bonds until maturity.
Turnover in a Bond Index
Have you ever looked at the turnover in a total bond market index fund? You might be surprised.
While the S&P 500 has turnover of approximately 5% per year, the Bloomberg Barclay’s U.S. Aggregate often averages between 40-60% per year.
Where is all that turnover coming from?
• Index additions (e.g. new issuances)
• Index deletions (e.g. maturing bonds)
• Paydowns
• Coupon reinvestment
If the general structure of the fixed income market does not change considerably over time, this level of turnover implies that a total bond market index will behave very similarly to a constant
duration bond fund.
Bonds are technically held to maturity, but roll return and profit/loss from shifts in interest rates are booked along the way as positions are rebalanced.
Which means that falling rates could matter. Even better, we can test how much falling rates mattered by proxying a total bond index with a constant maturity bond index[4].
Specifically, we will look at a constant maturity 10-year U.S. Treasury index. We will assume 10-year Treasuries are bought at the beginning of each year, held for a year, and sold as 9-year
Treasuries[5]. The proceeds will then be reinvested back into the new 10-year Treasuries. We will also assume that coupons are paid annually.
We ran the test from 12/1981 to 12/2012, since those dates represented both the highest and lowest end-of-year rates.
We will then decompose returns into three components:
• Coupon yield (“Coupon”)
• Roll return (“Roll”)
• Rate changes (“Shift”)
Coupon yield is, simply, the return we get from the coupon itself. Roll return is equal to the slope between 10-year and 9-year U.S. Treasuries at the point of purchase adjusted by the duration of
the bond. Rate changes are measured as price return we achieve due to shifts in the 9-year rate from the point at which we purchased the bond and the point at which we are selling it.
This allows us to create a return stream for each component as well as identify each component’s contribution to the total return of the index.
Source: Federal Reserve of St. Louis. Calculations by Newfound Research
What we can see is that coupon return dominates roll and shift. On an annualized basis, coupon was 6.24%, while roll only contributed 0.24% and shift contributed 2.22%.
Which leaves us with a final decomposition: coupon yield accounted for 71% of return, roll accounted for 3%, and shift accounted for 26%.
We can perform a similar test for constant maturity indices constructed at different points on the curve as well.
Total Return % Contribution
Coupon Roll Shift Coupon Roll Shift
10-year 6.24% 0.24% 2.22% 71.60% 2.84% 25.55%
7-year 6.08% 0.62% 1.72% 72.16% 7.37% 20.47%
5-year 5.81% 0.65% 1.29% 75.01% 8.38% 16.61%
Conclusion: Were Declining Rates Important?
A resounding yes. An extra 2.22% per year over 30+ years is nothing to sneeze at. Especially when you consider that this was the result of a very unique period unlikely to be repeated over the next
30 years.
Just as important to consider, however, is that it was not the most important contributor to total returns. While most people in our poll answered that decline in rates would account for 50%+ of
total return, the shift factor only came in at 26%.
The honor of the highest contributor goes to coupon yield. Even though rates deflated over 30 years, the average yield was high enough to be, by far and away, the biggest contributor to returns.
The bond bull was not due to declining rates, in our opinion, but rather the unusually high rates we saw over the period.
A fact which is changing today. We can see this by plotting the annual sources of returns year-by-year.
Source: St. Louis Federal Reserve. Calculations by Newfound Research.
Note that while coupon is always a positive contributor, its role has significantly diminished in recent years compared to the influence of rate changes.
The consistency of coupon and the varying influence of shift on returns (i.e. both positive and negative) means that coupon yield actually makes an excellent predictor of future returns. Lozada
(2015)[6] finds that the optimal horizon to use yield as a predictor of return in constant duration or constant-maturity bond funds is at twice the duration.
Which paints a potentially bleak picture for fixed income investors.
Fund Asset Duration TTM Yield Predicted Return
AGG U.S. Aggregate Bonds 5.74 2.37% 2.37% per year through
IEI 3-7 Year U.S. Treasuries 4.48 1.31% 1.31% per year through
IEF 7-10 Year U.S. 7.59 1.77% 1.77% per year through
Treasuries 2032
TLT 20+ Year U.S. Treasuries 17.39 2.56% 2.56% per year through
LQD Investment Grade Bonds 8.24 3.28% 3.28% per year through
Source: iShares. Calculations by Newfound Research.
Note that we are using trailing 12-month distribution yield for the ETFs here. We do this because ETF issuers often amortize coupon yield to account for pull-to-par effects, making it an
approximation of yield-to-worst. It is not perfect, but we don’t think the results materially differ in magnitude with any other measure: it’s still ugly.
The story remains largely the same as we’ve echoed over the past year: when it comes to fixed income, your current yield will be a much better predictor of returns than trying to guess about changing
Coupon yield had 3x the influence on total return over the last 30 years than changes in rates did.
What we should be concerned about today is not rising rates: rather, we should be concerned about the returns that present low rates imply for the future.
And we should be asking ourselves: are there other ways we can look to manage risk or find return?
[1] Find us on Twitter! Newfound is @thinknewfound and Corey is @choffstein.
[2] It is $0.86 instead of $0.96 because they need to set aside $0.10 to cover the extra dollar they owe at maturity.
[3] This is a bit of a simplification as the bonds will have different risk characteristics (e.g. different durations and convexity) which could cause investors, especially those with views on future
rate changes, to prefer one bond over the other.
[4] We made the leap here from total bond index to constant duration index to constant maturity index. Each step introduces some error, but we believe for our purposes the error is de minimis and a
constant maturity index allows for greater ease of implementation.
[5] Since no 9-year U.S. Treasury is offered, we create a model for the yield curve using cubic splines and then estimate the 9-year rate.
[6] http://content.csbs.utah.edu/~lozada/Research/IniYld_6.pdf | {"url":"https://blog.thinknewfound.com/category/risk-premia-style-premia/term/","timestamp":"2024-11-11T16:47:50Z","content_type":"text/html","content_length":"758027","record_id":"<urn:uuid:2dcc4557-b071-4453-b7d8-f8bfae5c4b75>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00168.warc.gz"} |
Solve One-Step Linear Inequalities Using Addition And Subtraction Worksheets [PDF]: Algebra 1 Math
How Will This Worksheet on "Solve One-Step Linear Inequalities Using Addition and Subtraction" Benefit Your Students' Learning?
• It helps us understand how to isolate variables in an equation or an inequality laying foundation for complex algebraic expressions.
• It enhances problem-solving skills as students learn how to examine the given inequality and use appropriate operations to find the solution.
• It helps us to have a better understanding of algebraic concepts .
• Performing arithmetic operations helps in increasing the speed while executing the operations and boosts confidence.
How to Solve One-Step Linear Inequalities Using Addition and Subtraction?
• Identify the inverse operation to be used to isolate the variable.
• Apply the inverse operation to both sides of the inequality and solve for the variable.
Q. Solve for $j$.$ewline$$j + 4 < 6$ | {"url":"https://www.bytelearn.com/math-algebra-1/worksheet/solve-one-step-linear-inequalities-using-addition-and-subtraction","timestamp":"2024-11-10T04:51:04Z","content_type":"text/html","content_length":"113724","record_id":"<urn:uuid:69751b50-d64d-40b2-bbc9-d342aa11bfe2>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00740.warc.gz"} |
Rigid holography and six-dimensional (Formula presented.) theories on AdS<sub>5</sub> × S<sup>1</sup>
Field theories on anti-de Sitter (AdS) space can be studied by realizing them as low-energy limits of AdS vacua of string/M theory. In an appropriate limit, the field theories decouple from the rest
of string/M theory. Since these vacua are dual to conformal field theories, this relates some of the observables of these field theories on AdS space to a subsector of the dual conformal field
theories. We exemplify this 'rigid holography' by studying in detail the six-dimensional N = (2, 0) A(K-1) superconformal field theory (SCFT) on AdS(5)xS(1), with equal radii for AdS(5) and for S-1.
We choose specific boundary conditions preserving sixteen supercharges that arise when this theory is embedded into Type IIB string theory on AdS(5)xS(5)/Z(K). On R(4,1)xS(1), this six-dimensional
theory has a 5(K-1)-dimensional moduli space, with unbroken five-dimensional SU(K) gauge symmetry at (and only at) the origin. On AdS(5)xS(1), the theory has a 2(K-1)-dimensional 'moduli space' of
supersymmetric configurations. We argue that in this case the SU(K) gauge symmetry is unbroken everywhere in the 'moduli space' and that this five-dimensional gauge theory is coupled to a
four-dimensional theory on the boundary of AdS(5) whose coupling constants depend on the 'moduli'. This involves non-standard boundary conditions for the gauge fields on AdS(5). Near the origin of
the 'moduli space', the theory on the boundary contains a weakly coupled four-dimensional N = 2 supersymmetric SU(K) gauge theory. We show that this implies large corrections to the metric on the
'moduli space'. The embedding in string theory implies that the six-dimensional N = (2, 0) theory on AdS(5)xS(1) with sources on the boundary is a subsector of the large N limit of various
four-dimensional N = 2 quiver SCFTs that remains non-trivial in the large N limit. The same subsector appears universally in many different four-dimensional N = 2 SCFTs. We also discuss a decoupling
limit that leads to N = (2, 0) 'little string theories’ on AdS5 × S 1.
All Science Journal Classification (ASJC) codes
• Nuclear and High Energy Physics
Dive into the research topics of 'Rigid holography and six-dimensional (Formula presented.) theories on AdS[5] × S^1'. Together they form a unique fingerprint. | {"url":"https://cris.iucc.ac.il/en/publications/rigid-holography-and-six-dimensional-formula-presented-theories-o","timestamp":"2024-11-02T23:58:04Z","content_type":"text/html","content_length":"57192","record_id":"<urn:uuid:7550424e-0fbe-49c4-826b-faf61117a9a4>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00883.warc.gz"} |
CSC263H Homework Assignment #2 solved
Question 1. (10 marks)
In this question, you must use the insertion and deletion algorithms as described in the “Balanced Search
Trees: AVL trees” handout posted on the course web site.
Insert into an initially empty AVL tree each of the following keys, in the order in which they appear in the
sequence: 0, 25, 19, 5, -2, 28, 13, -5, 2, 6, 14, 7.
Show the resulting AVL tree T. (Only the final tree should be shown; any intermediate trees shown will
be disregarded, and not given partial credit.)
From AVL tree T, delete 2, and show the resulting tree.
In the two trees you must show the key and balance factor of each node.
Question 2. (20 marks)
In the following, B1 and B2 are two binary search trees such that every key in B1 is smaller than every
key in B2.
Describe an algorithm that, given pointers b1 and b2 to the roots of B1 and B2, merges B1 and B2 into a
single binary search tree T. Your algorithm should satisfy the following two properties:
1. Its worst–case running time is O(min{h1, h2}), where h1 and h2 are the heights of B1 and B2.
2. The height of the merged tree T is at most max{h1, h2} + 1.
Note that the heights h1 and h2 are not given to the algorithm (in other words, the algorithm does not
“know” the heights of B1 and B2). Note also that B1, B2 and T are not required to be balanced.
Describe your algorithm, and justify its correctness and worst-case running time, in clear and concise
Note: Partial credit may be given for an algorithm that runs in O(max{h1, h2}) time.
Question 3. (24 marks)
The task in this question is to compute the medians of all prefixes of an array. As input we are given
the array A[1 . . n] of arbitrary integers. Using a heap data structure, design an algorithm that outputs
another array M[1 . . n], so that M[i] is equal to the median of the numbers in the subarray A[1 . . i]. Recall
that when i is odd, the median of A[1 . . i] is the element of rank (i + 1)/2 in the subarray, and when i is
even, the median is the average of the elements with ranks i/2 and i/2 + 1. Your algorithm should run in
worst-case time O(n log n).
a. (20 marks) Describe your algorithm in clear and concise English, and also provide the corresponding
pseudocode. Argue that your algorithm is correct.
b. (4 marks) Justify why your algorithm runs in time O(n log n).
[The questions below will not be corrected/graded. They are given here as interesting problems
that use material that you learned in class.]
Question 4. (0 marks) This question is about the cost of successively inserting k elements into a binomial
heap of size n.
a. Prove that a binomial heap with n elements has exactly n − α(n) edges, where α(n) is the number
of 1’s in the binary representation of n.
b. Consider the worst-case total cost of successively inserting k new elements into a binomial heap H
of size |H| = n. In this question, we measure the worst-case cost of inserting a new element into H as the
maximum number of pairwise comparisons between the keys of the binomial heap that is required to do
this insertion. It is clear that for k = 1 (i.e., inserting one element) the worst-case cost is O(log n). Show
that when k > log n, the average cost of an insertion, i.e., the worst-case total cost of the k successive
insertions divided by k, is bounded above by constant.
Hint: Note that the cost of each one of the k consecutive insertions varies — some can be expensive, other
are cheaper. Relate the cost of each insertion, i.e., the number of key comparisons that it requires, with
the number of extra edges that it forms in H. Then use part (a). | {"url":"https://codeshive.com/questions-and-answers/csc263h-homework-assignment-2-solved/","timestamp":"2024-11-05T03:21:47Z","content_type":"text/html","content_length":"102118","record_id":"<urn:uuid:61fd033c-d906-4bbe-87e9-c662fda9913e>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00017.warc.gz"} |
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Ancient Mathematics
Mathematics of the ancient world, prior to 500 AD. I know that's a little late, but the works of 3rd C. Diophantus are really a remnant of the ancient world.
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Mathematics History
Mathematics historiography. Any period is acceptable. Just keep in mind that we're finitists, so we're not going to be that interested in Cantor, Dedekind, Russel, Peano, Tarski, and that
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Documents For An Access Point
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# Author Title Accn# Year Item Type Claims
1 Tura i Brugu??s, Jordi Characterizing Entanglement and Quantum Correlations Constrained by Symmetry I10461 2017 eBook
2 Hertel, Peter Quantum Theory and Statistical Thermodynamics I10295 2017 eBook
3 Aidelsburger, Monika Artificial Gauge Fields with Ultracold Atoms in Optical Lattices I10237 2016 eBook
4 Pinheiro, Fernanda Multi-species Systems in Optical Lattices I10125 2016 eBook
5 Barenghi, Carlo F A Primer on Quantum Fluids I10045 2016 eBook
6 Satz, Helmut Extreme States of Matter in Strong Interaction Physics I10023 2018 eBook
7 Lewis-Swan, Robert J Ultracold Atoms for Foundational Tests of Quantum Mechanics I09967 2016 eBook
8 Iftikhar, Zubair Charge Quantization and Kondo Quantum Criticality in Few-Channel Mesoscopic Circuits I09929 2018 eBook
9 Sugiura, Sho Formulation of Statistical Mechanics Based on Thermal Pure Quantum States I09901 2017 eBook
10 Nolting, Wolfgang Theoretical Physics 7 I09888 2017 eBook
(page:1 / 8) [#74] Next Page Last Page
Title Characterizing Entanglement and Quantum Correlations Constrained by Symmetry
Author(s) Tura i Brugu??s, Jordi
Publication Cham, Springer International Publishing, 2017.
Description XXV, 237 p. 17 illus., 8 illus. in color : online resource
Abstract This thesis focuses on the study and characterization of entanglement and nonlocal correlations constrained under symmetries. It includes original results as well as detailed methods and
Note explanations for a number of different threads of research: positive partial transpose (PPT) entanglement in the symmetric states; a novel, experimentally friendly method to detect
nonlocal correlations in many-body systems; the non-equivalence between entanglement and nonlocality; and elemental monogamies of correlations. Entanglement and nonlocal correlations
constitute two fundamental resources for quantum information processing, as they allow novel tasks that are otherwise impossible in a classical scenario. However, their elusive
characterization is still a central problem in quantum information theory. The main reason why such a fundamental issue remains a formidable challenge lies in the exponential growth in
complexity of the Hilbert space as well as the space of multipartite correlations. Physical systems of interest, on the other hand, display symmetries that can be exploited to reduce
this complexity, opening the possibility that some of these questions become tractable for such systems
ISBN,Price 9783319495712
Keyword(s) 1. Condensed materials 2. EBOOK 3. EBOOK - SPRINGER 4. MATHEMATICAL PHYSICS 5. Phase transformations (Statistical physics) 6. QUANTUM COMPUTERS 7. Quantum Gases and Condensates 8.
Quantum Information Technology, Spintronics 9. QUANTUM PHYSICS 10. SPINTRONICS
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Title Quantum Theory and Statistical Thermodynamics : Principles and Worked Examples
Author(s) Hertel, Peter
Publication Cham, Springer International Publishing, 2017.
Description XIV, 368 p. 27 illus., 17 illus. in color : online resource
Abstract This textbook presents a concise yet detailed introduction to quantum physics. Concise, because it condenses the essentials to a few principles. Detailed, because these few principles
Note ??? necessarily rather abstract ??? are illustrated by several telling examples. A fairly complete overview of the conventional quantum mechanics curriculum is the primary focus, but the
huge field of statistical thermodynamics is covered as well. The text explains why a few key discoveries shattered the prevailing broadly accepted classical view of physics. First,
matter appears to consist of particles which, when propagating, resemble waves. Consequently, some observable properties cannot be measured simultaneously with arbitrary precision.
Second, events with single particles are not determined, but are more or less probable. The essence of this is that the observable properties of a physical system are to be represented
by non-commuting mathematical objects instead of real numbers. Chapters on exceptionally simple, but highly instructive examples illustrate this abstract formulation of quantum physics.
The simplest atoms, ions, and molecules are explained, describing their interaction with electromagnetic radiation as well as the scattering of particles. A short introduction to many
particle physics with an outlook on quantum fields follows. There is a chapter on maximally mixed states of very large systems, that is statistical thermodynamics. The following chapter
on the linear response to perturbations provides a link to the material equations of continuum physics. Mathematical details which would hinder the flow of the main text have been
deferred to an appendix. The book addresses university students of physics and related fields. It will attract graduate students and professionals in particular who wish to systematize
or refresh their knowledge of quantum physics when studying specialized texts on solid state and materials physics, advanced optics, and other modern fields
ISBN,Price 9783319585956
Keyword(s) 1. Condensed materials 2. EBOOK 3. EBOOK - SPRINGER 4. Phase transformations (Statistical physics) 5. Quantum Gases and Condensates 6. QUANTUM PHYSICS 7. STATISTICAL PHYSICS 8.
Statistical Physics and Dynamical Systems
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I10295 On Shelf
Title Artificial Gauge Fields with Ultracold Atoms in Optical Lattices
Author(s) Aidelsburger, Monika
Publication Cham, Springer International Publishing, 2016.
Description XIII, 172 p. 76 illus., 2 illus. in color : online resource
Abstract This work reports on the generation of artificial magnetic fields with ultracold atoms in optical lattices using laser-assisted tunneling, as well as on the first Chern-number
Note measurement in a non-electronic system. It starts with an introduction to the Hofstadter model, which describes the dynamics of charged particles on a square lattice subjected to strong
magnetic fields. This model exhibits energy bands with non-zero topological invariants called Chern numbers, a property that is at the origin of the quantum Hall effect. The main part of
the work discusses the realization of analog systems with ultracold neutral atoms using laser-assisted-tunneling techniques both from a theoretical and experimental point of view.
Staggered, homogeneous and spin-dependent flux distributions are generated and characterized using two-dimensional optical super-lattice potentials. Additionally their topological
properties are studied via the observation of bulk topological currents. The experimental techniques presented here offer a unique setting for studying topologically non-trivial systems
with ultracold atoms
ISBN,Price 9783319258294
Keyword(s) 1. Condensed materials 2. EBOOK 3. EBOOK - SPRINGER 4. LOW TEMPERATURE PHYSICS 5. LOW TEMPERATURES 6. Phase transformations (Statistical physics) 7. QUANTUM COMPUTERS 8. Quantum Gases
and Condensates 9. Quantum Information Technology, Spintronics 10. SPINTRONICS
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Title Multi-species Systems in Optical Lattices : From Orbital Physics in Excited Bands to Effects of Disorder
Author(s) Pinheiro, Fernanda
Publication Cham, Springer International Publishing, 2016.
Description XVI, 126 p. 44 illus., 16 illus. in color : online resource
Abstract This highly interdisciplinary thesis covers a wide range of topics relating to the interface of cold atoms, quantum simulation, quantum magnetism and disorder. With a self-contained
Note presentation, it provides a broad overview of the rapidly evolving area of cold atoms and is of interest to both undergraduates and researchers working in the field. Starting with a
general introduction to the physics of cold atoms and optical lattices, it extends the theory to that of systems with different multispecies atoms. It advances the theory of many-body
quantum systems in excited bands (of optical lattices) through an extensive study of the properties of both the mean-field and strongly correlated regimes. Particular emphasis is given
to the context of quantum simulation, where as shown here, the orbital degree of freedom in excited bands allows the study of exotic models of magnetism not easily achievable with the
previous alternative systems. In addition, it proposes a new model Hamiltonian that serves as a quantum simulator of various disordered systems in different symmetry classes that can
easily be reproduced experimentally. This is of great interest, especially for the study of disorder in 2D quantum systems.
ISBN,Price 9783319434643
Keyword(s) 1. Condensed materials 2. EBOOK 3. EBOOK - SPRINGER 4. LOW TEMPERATURE PHYSICS 5. LOW TEMPERATURES 6. Phase transformations (Statistical physics) 7. QUANTUM COMPUTERS 8. Quantum Gases
and Condensates 9. Quantum Information Technology, Spintronics 10. QUANTUM PHYSICS 11. SPINTRONICS
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Title A Primer on Quantum Fluids
Author(s) Barenghi, Carlo F;Parker, Nick G
Publication Cham, Springer International Publishing, 2016.
Description XIII, 119 p. 56 illus., 34 illus. in color : online resource
Abstract The aim of this primer is to cover the essential theoretical information, quickly and concisely, in order to enable senior undergraduate and beginning graduate students to tackle
Note projects in topical research areas of quantum fluids, for example, solitons, vortices and collective modes. The selection of the material, both regarding the content and level of
presentation, draws on the authors analysis of the success of relevant research projects with newcomers to the field, as well as of the students feedback from many taught and self-study
courses on the subject matter. Starting with a brief historical overview, this text covers particle statistics, weakly interacting condensates and their dynamics and finally superfluid
helium and quantum turbulence. At the end of each chapter (apart from the first) there are some exercises. Detailed solutions can be made available to instructors upon request to the
ISBN,Price 9783319424767
Keyword(s) 1. Condensed materials 2. EBOOK 3. EBOOK - SPRINGER 4. Fluid- and Aerodynamics 5. FLUIDS 6. LOW TEMPERATURE PHYSICS 7. LOW TEMPERATURES 8. Phase transformations (Statistical physics) 9.
Quantum Gases and Condensates
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Title Extreme States of Matter in Strong Interaction Physics : An Introduction
Author(s) Satz, Helmut
Publication Cham, Springer International Publishing, 2018.
Description XV, 288 p : online resource
Abstract This book is a course-tested primer on the thermodynamics of strongly interacting matter ??? a profound and challenging area of both theoretical and experimental modern physics.
Note Analytical and numerical studies of statistical quantum chromodynamics provide the main theoretical tool, while in experiments, high-energy nuclear collisions are the key for extensive
laboratory investigations. As such, the field straddles statistical, particle and nuclear physics, both conceptually and in the methods of investigation used. The book addresses, above
all, the many young scientists starting their scientific research in this field, providing them with a general, self-contained introduction that highlights the basic concepts and ideas
and explains why we do what we do. Much of the book focuses on equilibrium thermodynamics: first it presents simplified phenomenological pictures, leading to critical behavior in
hadronic matter and to a quark-hadron phase transition. This is followed by elements of finite temperature lattice QCD and an exposition of the important results obtained through the
computer simulation of the lattice formulation. It goes on to clarify the relationship between the resulting critical behavior due to symmetry breaking/restoration in QCD, before turning
to the QCD phase diagram. The presentation of bulk equilibrium thermodyamics is completed by studying the properties of the quark-gluon plasma as a new state of strongly interacting
matter. The final chapters of the book are devoted to more specific topics that arise when nuclear collisions are considered as a tool for the experimental study of QCD thermodynamics.
This second edition includes a new chapter on the hydrodynamic evolution of the medium produced in nuclear collisions. Since the study of flow for strongly interacting fluids has gained
ever-increasing importance over the years, it is dealt with it in some detail, including comments on gauge/gravity duality. Moreover, other aspects of experimental studies are brought up
to date, such as the search for critical behavior in multihadron production, the calibration of quarkonium production in nuclear collisions, and the relation between strangeness
suppression and deconfinement
ISBN,Price 9783319718941
Keyword(s) 1. ASTROPHYSICS 2. Astrophysics and Astroparticles 3. COMPLEX SYSTEMS 4. Condensed materials 5. DYNAMICAL SYSTEMS 6. EBOOK 7. EBOOK - SPRINGER 8. Elementary particles (Physics) 9.
Elementary Particles, Quantum Field Theory 10. Heavy ions 11. NUCLEAR PHYSICS 12. Nuclear Physics, Heavy Ions, Hadrons 13. Phase transformations (Statistical physics) 14. QUANTUM FIELD
THEORY 15. Quantum Gases and Condensates 16. STATISTICAL PHYSICS 17. Statistical Physics and Dynamical Systems
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Title Ultracold Atoms for Foundational Tests of Quantum Mechanics
Author(s) Lewis-Swan, Robert J
Publication Cham, Springer International Publishing, 2016.
Description XVI, 156 p. 35 illus., 14 illus. in color : online resource
Abstract This thesis presents a theoretical investigation into the creation and exploitation of quantum correlations and entanglement among ultracold atoms. Specifically, it focuses on these
Note non-classical effects in two contexts: (i) tests of local realism with massive particles, e.g., violations of a Bell inequality and the EPR paradox, and (ii) realization of quantum
technology by exploitation of entanglement, for example quantum-enhanced metrology. In particular, the work presented in this thesis emphasizes the possibility of demonstrating and
characterizing entanglement in realistic experiments, beyond the simple ???toy-models??? often discussed in the literature. The importance and relevance of this thesis are reflected in a
spate of recent publications regarding experimental demonstrations of the atomic Hong-Ou-Mandel effect, observation of EPR entanglement with massive particles and a demonstration of an
atomic SU(1,1) interferometer. With a separate chapter on each of these systems, this thesis is at the forefront of current research in ultracold atomic physics.
ISBN,Price 9783319410487
Keyword(s) 1. Condensed materials 2. EBOOK 3. EBOOK - SPRINGER 4. Phase transformations (Statistical physics) 5. QUANTUM COMPUTERS 6. Quantum Gases and Condensates 7. Quantum Information
Technology, Spintronics 8. QUANTUM PHYSICS 9. SPINTRONICS
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Title Charge Quantization and Kondo Quantum Criticality in Few-Channel Mesoscopic Circuits
Author(s) Iftikhar, Zubair
Publication Cham, Springer International Publishing, 2018.
Description XIII, 137 p. 73 illus., 47 illus. in color : online resource
Abstract This thesis explores several fundamental topics in mesoscopic circuitries that incorporate few electronic conduction channels. The reported results establish a new state of the art in a
Note field that has been waiting for this kind of experiments for decades. The first experiments address the quantized character of charge in circuits. The thesis discusses the charge
quantization criterion, observes the predicted charge quantization scaling, and demonstrates a crossover toward a universal behavior as temperature is increased. In turn, the second set
of experiments explores the unconventional quantum critical physics that arises in the multichannel Kondo model. At the symmetric quantum critical point, the predicted universal Kondo
fixed points and scaling exponents are observed, and the full numerical renormalization group scaling curves validated. In addition, the thesis explores the crossover from quantum
criticality: direct visualization of the development of a quantum phase transition, the parameter space for quantum criticality, as well as universality and scaling behaviors
ISBN,Price 9783319946856
Keyword(s) 1. Condensed materials 2. EBOOK 3. EBOOK - SPRINGER 4. ELECTRONIC CIRCUITS 5. Electronic Circuits and Devices 6. LOW TEMPERATURE PHYSICS 7. LOW TEMPERATURES 8. Phase transformations
(Statistical physics) 9. Quantum Gases and Condensates 10. QUANTUM PHYSICS
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Title Formulation of Statistical Mechanics Based on Thermal Pure Quantum States
Author(s) Sugiura, Sho
Publication Singapore, Springer Singapore, 2017.
Description XII, 73 p. 4 illus. in color : online resource
Abstract This thesis introduces the concept of "thermal pure quantum (TPQ) states", which are pure quantum states in equilibrium. The author establishes a new formulation of statistical mechanics
Note based on the TPQ states. This formulation allows us to obtain not only mechanical variables but also thermodynamic variables such as entropy and free energy from a single TPQ state.
Furthermore, the formulation provides a new physical description in which all fluctuations including thermally driven ones are uniquely identified to be quantum fluctuations. The use of
TPQ formulation has practical advantages in its application to numerical computations and allows for significant reduction in computation cost in numerics. For demonstration purposes, a
numerical computation based on TPQ formulation is applied to a frustrated two-dimensional quantum spin model, and the result is also included in this book.
ISBN,Price 9789811015069
Keyword(s) 1. COMPLEX SYSTEMS 2. Condensed materials 3. DYNAMICAL SYSTEMS 4. EBOOK 5. EBOOK - SPRINGER 6. Phase transformations (Statistical physics) 7. Quantum Gases and Condensates 8. QUANTUM
PHYSICS 9. STATISTICAL PHYSICS 10. Statistical Physics and Dynamical Systems 11. Strongly Correlated Systems, Superconductivity 12. SUPERCONDUCTIVITY 13. SUPERCONDUCTORS
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Title Theoretical Physics 7 : Quantum Mechanics - Methods and Applications
Author(s) Nolting, Wolfgang
Publication Cham, Springer International Publishing, 2017.
Description XIV, 565 p. 32 illus., 14 illus. in color : online resource
Abstract This textbook offers a clear and comprehensive introduction to methods and applications in quantum mechanics, one of the core components of undergraduate physics courses. It follows on
Note naturally from the previous volumes in this series, thus developing the understanding of quantized states further on. The first part of the book introduces the quantum??theory of angular
momentum and approximation methods.??More complex themes are covered in the second part of the book, which describes??multiple particle systems and scattering theory. Ideally suited to
undergraduate students with some grounding in the basics of quantum mechanics, the book is enhanced throughout with learning features such as boxed inserts and chapter summaries, with
key mathematical derivations highlighted to aid understanding. The text is supported by numerous worked examples and end of chapter problem sets.?? About the Theoretical Physics series
Translated from the renowned and highly successful German editions, the eight volumes of this series cover the complete core curriculum of theoretical physics at undergraduate level.
Each volume is self-contained and provides all the material necessary for the individual course topic. Numerous problems with detailed solutions support a deeper understanding. Nolting
is famous for his refined didactical style and has been referred to as the "German Feynman" in reviews
ISBN,Price 9783319633244
Keyword(s) 1. Condensed materials 2. EBOOK 3. EBOOK - SPRINGER 4. Phase transformations (Statistical physics) 5. Quantum Gases and Condensates 6. QUANTUM PHYSICS
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Universal Sorting: Finding a DAG using Priced Comparisons
We resolve two open problems in sorting with priced information, introduced by [Charikar, Fagin, Guruswami, Kleinberg, Raghavan, Sahai (CFGKRS), STOC 2000]. In this setting, different comparisons
have different (potentially infinite) costs. The goal is to sort with small competitive ratio (algorithmic cost divided by cheapest proof).
1) When all costs are in $\{0,1,n,\infty\}$, we give an algorithm that has $\widetilde{O}(n^{3/4})$ competitive ratio. Our algorithm generalizes the algorithms for generalized sorting (all costs are
either $1$ or $\infty$), a version initiated by [Huang, Kannan, Khanna, FOCS 2011] and addressed recently by [Kuszmaul, Narayanan, FOCS 2021].
2) We answer the problem of bichromatic sorting posed by [CFGKRS]: The input is split into $A$ and $B$, and $A-A$ and $B-B$ comparisons are more expensive than an $A-B$ comparisons. We give a
randomized algorithm with a O(polylog n) competitive ratio.
These results are obtained by introducing the universal sorting problem, which generalizes the existing framework in two important ways. We remove the promise of a directed Hamiltonian path in the
DAG of all comparisons. Instead, we require that an algorithm outputs the transitive reduction of the DAG. For bichromatic sorting, when $A-A$ and $B-B$ comparisons cost $\infty$, this generalizes
the well-known nuts and bolts problem. We initiate an instance-based study of the universal sorting problem. Our definition of instance-optimality is inherently more algorithmic than that of the
competitive ratio in that we compare the cost of a candidate algorithm to the cost of the optimal instance-aware algorithm. This unifies existing lower bounds, and opens up the possibility of an $O
(1)$-instance optimal algorithm for the bichromatic version.
• Data Structures and Algorithms
• Comparison based algorithms
• Sorting
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POV-Ray: Newsgroups: povray.advanced-users: Making Patterns with functions: Re: Making Patterns with functions
"Bald Eagle" <cre### [at] netscape> Hammering out that many equations one after the other ... has allowed me to learn
some new things and gain valuable i
nsights ...
OK, just to list the ones that I've written down so far:
(Questions, comments, and constructive criticism welcome as always)
Important tips to generate quality patterns from equations
Be very careful with the use of parentheses - especially when + or - is in an
equation. Group terms.
You may want to declare different parts of a complex equation as separate
functions, then combine them in the final function
Use fmod instead of mod to get a regular repetition of a pattern across the
HOWEVER, sometimes using mod in certain functions gives a better (more
expected) result - so try both
Use Mike Williams' "shell" trick for isosurfaces to give an infinitely thin line
a visible thickness
Change the line thickness to give definition to patterns that may not show up
with smaller values.
Some equations don't cross the zero/integer/mod threshold often enough to show
finer details. Multiply the
WHOLE equation (use parentheses) by a factor to increase the number of
Zoom in or out to see pattern attributes that may not be visible or clearly
If you're graphing y as a function of x, then you must write it in the form of:
function {y - formula} since when y=formula, the value is 0
If you're graphing a 2-argument function of both x and y, then you must write it
in the form of: function {(x - x_formula) + (y - y_formula)}
If you're graphing a polar equation, then write it in the form of: function {r -
POV-Ray's internal method of wrapping function values winds up hiding negative
vs positive parts,
and won't be revealed until made as an isosurface, or the function is shifted
by + 0.5 and visualized with a color map
If you're using functions that return values in the 0-255 range, you're going to
have to divide the result by 255
If the function repeats over a range of 0-255, then you're going to have to use
mod (Function (x, y, z), 256)/255
You're going to need function versions of HSV to RGB to perform color
For full-color patterns, you're going to separate functions for r, g, and b,
plus an average texture using all 3 functions
There are common functions and operations that the function parser doesn't
recognize, and you'll have to write your own versions of
identifiers previously declared in a scene (even for loop iterators!) cannot
then be used in a function
If you're experiencing a blank pattern (all black/white/gray) then you have
likely forgotten to explicitly include x, y, and z into your function
argument list. If your function takes some value N as an argument, but also
uses x, y, or z, then you must #declare F = function (x, y, z, N) {}
If your pattern only appears in the first quadrant (upper right, both x and y
are positive), then you will likely need to use abs(X) and abs(y) in your
Post a reply to this message | {"url":"http://news.povray.org/povray.advanced-users/message/%3Cweb.65aec611d81b84791f9dae3025979125%40news.povray.org%3E/#%3Cweb.65aec611d81b84791f9dae3025979125%40news.povray.org%3E","timestamp":"2024-11-02T02:00:42Z","content_type":"text/html","content_length":"10953","record_id":"<urn:uuid:d2704675-1f6c-4393-9186-fdb00c08a226>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00697.warc.gz"} |
The Importance of Teaching Fractions - The Teaching Couple
The Importance of Teaching Fractions
Written by Dan
Last updated
Fractions play a crucial role in the development of mathematical understanding and skills. As a fundamental building block, fractions offer an essential stepping stone to understanding more advanced
topics in mathematics such as decimals, percentages, and algebra.
Mastering fractions in the early years is imperative for students as it builds their arithmetic foundations and helps alleviate math anxiety that may develop when faced with complex concepts later on
in their education.
Related: For more, check out our article on The Importance Of Teaching Decimals here.
Due to the significance of this topic, teachers must employ effective teaching methods and adapt their curriculum standards accordingly.
The proper instruction of fractions will help students develop a conceptual understanding, enabling them to grasp the real-world applications and practical relevance of fractions in various
day-to-day situations.
In addition, teachers should also focus on the proper tools and resources to further strengthen their students’ knowledge and assessment of fractions.
Overcoming the challenges related to fraction instruction ensures students excel in their mathematical journey.
Key Takeaways
• Fractions are fundamental to understanding advanced math concepts and alleviating math anxiety
• Effective teaching methods and curriculum adaptations are crucial for students’ conceptual understanding
• Proper tools, resources, and assessments help to overcome challenges and improve students’ fraction knowledge
The Role of Fractions in Mathematics
Understanding the Basics
Fractions play a crucial role in understanding the basics of mathematics. They are a foundation for later mathematical learning, including advanced math and science classes like algebra, geometry,
statistics, chemistry, and physics.
Fractions are relational numbers that represent a part of a whole. Understanding fractions allows students to comprehend better other number concepts, such as decimals and percentages.
A fraction is often represented as a ratio of two integers, with the top number (numerator) indicating the parts being considered and the bottom number (denominator) representing the total number of
equal parts the whole is divided into.
One of the key concepts in fractions is the number line. It helps students visualize fractions by placing them between whole numbers on a horizontal line.
Fractions in Number Theory
Number theory is an area of mathematics that delves into the properties of whole and fractional numbers. In this branch, fractions play a significant role in understanding and solving problems
related to the divisibility of numbers, factors, multiples, and more.
Fractions can be used in various ways to represent different relationships between numbers. For example, they can describe part-to-whole relationships, such as the shaded portion of a circle, or
part-to-part relationships, like comparing the number of apples to the total number of fruits.
Additionally, fractions are used to explore and express ratios, proportions, and rates, essential for solving real-life problems.
In summary, the study of fractions is an essential building block in mathematics education. Their role in math and number theory basics allows students to grasp more advanced mathematical concepts
and solve a wide range of problems.
Furthermore, understanding fractions is a foundation for success in algebra and other higher-level math courses.
Teaching Methods for Fractions
Introducing Fractions Concepts
Introducing fraction concepts confidently and clearly is essential to provide students with a strong foundation in mathematics. Teachers can begin by helping students recognize that fractions are
numbers expanding the number system beyond whole numbers.
Using number lines as a visual aid can significantly improve students’ understanding of fractions and their relationships to whole numbers.
To familiarize students with fraction terminology, teachers can introduce numerator and denominator concepts and guide students in identifying and comparing fractions.
Another valuable teaching method involves using real-life examples, like dividing objects (e.g., pizzas or chocolate bars) into equal parts to illustrate the concept of fractions.
When teaching the equivalent fractions concept further, teachers can employ visual aids such as pie charts and grids to highlight the relationships between fractions with the same value.
This can help students become more comfortable with equivalent fractions and simplify fractions effectively.
Advanced Fraction Operations
Once students grasp basic fraction concepts well, they can explore more advanced fraction operations. This includes addition, subtraction, multiplication, and division of fractions, which can be
challenging as they involve multiple steps and often appear counterintuitive.
Integrating fraction bars or fraction circles into the instruction can aid students in understanding the need to find a common denominator before adding or subtracting fractions.
Teachers can also demonstrate the connection between multiplication and division of fractions through various approaches, such as the area model and the invert-and-multiply rule.
Another crucial topic in advanced fraction operations is the conversion of fractions to decimals and percentages.
Teachers should emphasize the significance of these conversions, as they extend students’ understanding of the relationships between different number forms and lay the foundation for more advanced
mathematical studies.
Teaching fractions is essential for students to develop confidence in their mathematical abilities and ensure they have a solid foundation to build upon for more advanced topics.
By using a variety of visual aids, real-life examples, and engaging teaching methods, educators can help students understand and excel in working with fractions.
Curriculum Standards and Fractions
Common Core and Beyond
The importance of teaching fractions cannot be overstated, as they play a crucial role in students’ understanding of mathematics and its applications.
Common Core State Standards are one such example of nationwide educational guidelines used by most states in the United States that have been developed to emphasize the importance of learning about
Under Common Core, fractions are introduced to students early, with a strong focus on the concept of a fraction as a number.
The Common Core Standards require students to view fractions as divided wholes and as numbers on a number line and reason about a fraction’s size. This foundational understanding enables students to
perform arithmetic operations with fractions, like: successfully
• Addition
• Subtraction
• Multiplication
• Division
These operations with fractions are further illustrated through real-life examples and problem-solving situations, ensuring that the knowledge gained is practical and relevant.
Evaluating Mathematics Curricula
When evaluating a mathematics curriculum, it is essential to consider how well it addresses fraction concepts, as fractions form the basis of several other important mathematical topics such as
decimals and percentages.
In ensuring that students are able to approach these foundational concepts with confidence, the curriculum should focus on the following key aspects:
1. Understanding fractions as numbers
2. Reasoning about the size of fractions
3. Representing fractions on number lines
4. Performing arithmetic operations with fractions
By addressing these crucial aspects of fractions, a curriculum can ensure that students build a solid foundational understanding of mathematics, preparing them for more advanced concepts in algebra
and other mathematical subjects.
Teaching fractions is essential, and curriculum standards must emphasize the effective instruction of these foundational concepts for students’ success in their mathematical journey.
Conceptual Understanding of Fractions
Linking Fractions to Quantities
A solid conceptual understanding of fractions is vital for students to work with these mathematical representations effectively.
One crucial aspect of developing this understanding is to link fractions with the notion of quantities in the real world. Fractions consist of a numerator and a denominator, where the numerator
represents the parts of interest, and the denominator represents the whole.
By connecting fractions to physical or tangible representations, students can better comprehend the relationship between the part and the whole and the individual roles of the numerator and the
For example, using manipulatives such as fraction tiles or pizza slices can allow students to visualize fractions and how they are connected to real-world quantities.
This hands-on approach to understanding fractions can make learning more engaging and long-lasting for students, as mentioned in a Yale University curriculum initiative.
Visual and Abstract Representations
In addition to linking fractions to quantities, visual and abstract representations play a significant role in the conceptual understanding of fractions.
These representations allow students to make connections between concrete and abstract concepts, which is crucial for mastering fractions and minimizing the potential for math anxiety later on,
according to ThoughtCo.
Visual representations can take the form of:
• Diagrams
• Number lines
• Fraction bars
• Circle models
On the other hand, abstract representations include:
• Mathematical notation (e.g., 3/4)
• Word problems involving fractions
• Relationships between fractions, decimals, and percentages
Incorporating visual and abstract representations in teaching fractions helps students develop a well-rounded understanding.
When students can move seamlessly between the concrete and the abstract, they are more likely to succeed in higher-level mathematics, such as algebra and calculus, as suggested by a study in
Evidence-Based Teaching.
Practical Applications of Fractions
Fractions in Real Life
Fractions play an essential role in everyday life, and mastering this concept is crucial for students to develop a solid mathematical foundation.
They are used in various real-life scenarios, such as cooking, sewing, and construction, where measurement of ingredients, fabric, and materials is required.
For instance, in cooking, it is expected to encounter measurements like half a cup, one-third of a teaspoon, or two-thirds of a stick of butter.
Also, understanding fractions enhances a person’s ability to comprehend the magnitude of quantities, whether they are comparing a piece of pie to the whole or calculating the remaining balance on a
Another common application of fractions is the area model – a method used to visually represent and solve problems involving multiplication and division of fractions.
This technique employs grids to represent various parts of the whole, which can help individuals better understand how fractions relate to one another.
Problem Solving with Fractions
A significant aspect of teaching fractions is helping students develop problem-solving skills using fractions. One popular approach is using word problems.
Word problems allow students to apply their knowledge of fractions in practical situations, requiring them to identify relevant information, perform calculations, and justify their reasoning.
Examples of word problems involving fractions:
1. Sarah has 9 feet of ribbon. She wants to cut it into 3/4 of a foot long pieces. How many pieces can she make?
2. Peter can type 120 words per minute. He takes a 10-minute break after typing for 45 minutes. How many words can he type in 2 1/2 hours?
By introducing students to real-life applications and problem-solving involving fractions, educators can equip them with essential skills and confidence to tackle more complex mathematical ideas in
the future, such as decimals and percentages.
This practical approach to teaching fractions can significantly impact their success in learning and understanding mathematics.
Remember some practical applications mentioned earlier and stay aware of their importance in building a solid foundation in mathematics.
Tools and Resources for Fraction Education
Manipulatives and Models
One effective way to teach fractions is by using manipulatives and visual models. Manipulatives are hands-on tools that help make abstract concepts, like fractions, more concrete for students.
Examples of manipulatives include fraction tiles, fraction strips, and pattern blocks.
Visual models, such as number lines, can also be essential in helping students understand fraction equivalence, fraction density, and negative fractions. For example, students can use number lines to
compare and order fractions and to add or subtract them.
Teachers can introduce algebraic concepts in upper elementary grades by connecting fractions and ratios. According to ThoughtCo, understanding fractions is the foundation for advanced math and
science classes such as algebra, geometry, statistics, chemistry, and physics.
Digital Tools and Applications
Many digital tools and applications are available to assist with teaching fractions creatively and effectively.
These tools often include interactive games, simulations, and adaptive learning platforms that cater to individual student needs. Examples of digital tools that help students practice and master
fractions are:
• Khan Academy: A comprehensive platform covering various topics, including fractions, with video lessons, quizzes, and interactive challenges.
• Prodigy Game: An engaging, adaptive math game that covers various topics, including fractions, and adjusts to each student’s learning level.
• Fraction Wall App: A digital version of a fraction wall that helps students visualize and compare different fractions.
These digital tools provide flexible and engaging learning experiences and can open up new ways for teachers and students to explore the world of fractions. By utilizing various tools, resources, and
strategies, educators can make teaching fractions more effective and enjoyable for their students.
Assessing Fraction Knowledge
Formative and Summative Approaches
Assessing students’ understanding of fractions is crucial for future mathematics success. Educators typically rely on two primary approaches to evaluate fraction knowledge: formative and summative
Formative assessments are used during the learning process, focusing on ongoing feedback that helps teachers and students identify areas for improvement. Examples of formative assessments include:
• Observations
• Quizzes
• Peer reviews
• Concept maps
Summative assessments, on the other hand, are conducted after learning has taken place. They provide a more comprehensive evaluation of the students’ understanding of fractions. Some examples of
summative assessments are:
• Standardized tests
• End-of-unit exams
• Portfolios
Both formative and summative approaches are vital in monitoring students’ progress in learning fractions and adapting teaching strategies to their needs.
National and International Research
Mathematics achievement is a topic of interest for both national and international research. Studies on fraction knowledge reveal its importance as a foundation for higher-level math skills, such as
algebra and geometry. Such research includes Fraction Learners: Assessing Understanding and studies from the National Council of Teachers of Mathematics (NCTM).
International organizations, like the Programme for International Student Assessment (PISA), have also examined mathematics achievement among students across countries.
These large-scale studies often highlight the need for more effective teaching methods and standards that promote a deeper understanding of fractions and other essential mathematical concepts.
Through a combination of formative and summative assessments and national and international research, educators can gain valuable insights on best practices for teaching fractions and improving
overall mathematics achievement.
By focusing on effective assessment strategies, teachers can ensure their students develop a strong foundation in fractions, setting them up for success in more advanced math courses.
Challenges in Understanding Fractions
Common Misconceptions
One of the primary challenges in understanding fractions is many students’ misconceptions.
These misconceptions may stem from inadequate explanations or a lack of clarity in teaching methods. Examples of common misconceptions revolve around denominators, equivalence, and comparison.
• Denominators: Students sometimes believe that the larger the denominator, the larger the incorrect fraction. In reality, it depends on the relationship between the numerator and the denominator.
• Equivalence: Another misconception is that fractions with different numerators and denominators cannot be equal, which is also false. For instance, 1/2 and 2/4 are equivalent as they represent
the same quantity.
• Comparison: Students often struggle to compare fractions with different denominators. They might compare only the numerators without considering the denominators, leading to incorrect
• Semantics of Fractions: The language used to describe fractions can also contribute to confusion. For example, students may hear “four tenths” and mistakenly understand it as “four sets of ten”
instead of the fraction 4/10 `.
Addressing Fraction Difficulties
Teachers can employ various methods to address these challenges and support students’ understanding of fractions. Some strategies include:
1. Visual Representations: Using number lines, fraction bars, or pie charts can help students visualize fractions and understand the relationships between numerators and denominators `.
2. Concrete Examples: Relating fractions to real-life situations and using physical objects can make the concept more tangible, reinforcing understanding.
3. Clear Language: Ensuring precise and accurate language is used when describing fractions is crucial to avoid further misconceptions.
4. Systematic Practice: Regularly practising fraction concepts can help students develop a solid foundation in this area.
By addressing these common misconceptions and employing strategies to target difficulties, teachers can support students in overcoming challenges in understanding fractions, which is critical for
their future success in mathematics.
Advancements in Fraction Pedagogy
New Teaching Techniques
Advancements in teaching fractions have led to the development of innovative instructional strategies. These techniques focus on building a strong foundation for students’ understanding of fractions
as the basis of more complex mathematical concepts.
These instructional practices include using multiple representations to help students visualize and conceptualize the abstract concept of fractions. Teachers integrate concrete models, such as number
lines, area models, and sets, to make fractions more tangible for students.
Another teaching strategy involves scaffolding unit fractions, allowing students to grasp and apply fractional constructs in building non-unit fractions.
This method emphasizes the progression from simple to intricate ideas and enables students to comprehend the concept of fractions at a deeper level thoroughly.
Moreover, researchers have found that applying the idea of units from whole numbers to fractions significantly improves students’ understanding and success in higher-level math courses.
Integrating Technology
Incorporating technology into fraction instruction has also proven to be a successful avenue for enhancing student learning. Modern tools and software provide students interactive platforms to
practice and refine their fractional skills.
These tools facilitate individualized learning experiences by addressing students’ varying levels of confidence and understanding with fractions.
Examples of technology integration include:
1. Dynamic visualizations: The use of digital manipulatives that allow students to create, manipulate, and observe the representations of fractions in real-time, fostering a deeper understanding of
the concept.
2. Adaptive learning systems: Platforms that provide personalized practice in fraction skills and concepts based on the student’s needs and progress.
3. Collaborative learning tools: Applications that enable students to engage in learning activities with their peers, allowing them to exchange ideas, assist each other, and discuss fractional
The enhancements in fraction pedagogy, including new teaching techniques and technology integration, have contributed to a more effective and engaging curriculum for students and instructors alike.
As research in this field continues, it is expected that even more advancements will emerge, further refining the instructional practices for teaching fractions and developing a stronger foundation
for students’ mathematical knowledge and abilities.
About The Author
I'm Dan Higgins, one of the faces behind The Teaching Couple. With 15 years in the education sector and a decade as a teacher, I've witnessed the highs and lows of school life. Over the years, my
passion for supporting fellow teachers and making school more bearable has grown. The Teaching Couple is my platform to share strategies, tips, and insights from my journey. Together, we can shape a
better school experience for all. | {"url":"https://theteachingcouple.com/the-importance-of-teaching-fractions/","timestamp":"2024-11-03T19:03:36Z","content_type":"text/html","content_length":"150939","record_id":"<urn:uuid:19e806e0-cea2-434e-ae31-55df5e680b3a>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00851.warc.gz"} |
Beast Mode Help: Summary number to show % of transaction type
Beast Mode Help: Summary number to show % of transaction type
I have a dataset where there is a column name 'Ship Via Group' that contains three distinct values. One of which, "Counter / WC" I need to show in the summary number for the card - "What percentage
of orders are Counter / WC?"
Sample data is attached, note that order number may appear multiple times because orders may have 1 or more items.
Here is the beast mode I tried, but am getting no value returned. I'm a noob... I admit it...
(CASE when `Ship Via Group`='%Counter%' then COUNT(DISTINCT `Order Number`) else 0
END) / COUNT(DISTINCT `Order Number`)
Any help would be greatly appreciated. Thank you in advance!
Best Answer
• Hi, lets fix your beastmode first to see if that gets you where you want to be. Do this instead: COUNT (DISTINCT CASE WHEN `Ship Via Group` LIKE '%Counter%' THEN `Order Number` ELSE 0 END) /
COUNT (DISTINCT `Order Number
Domo Arigato!
**Say 'Thanks' by clicking the thumbs up in the post that helped you.
**Please mark the post that solves your problem as 'Accepted Solution'
• Hi, lets fix your beastmode first to see if that gets you where you want to be. Do this instead: COUNT (DISTINCT CASE WHEN `Ship Via Group` LIKE '%Counter%' THEN `Order Number` ELSE 0 END) /
COUNT (DISTINCT `Order Number
Domo Arigato!
**Say 'Thanks' by clicking the thumbs up in the post that helped you.
**Please mark the post that solves your problem as 'Accepted Solution'
I have an issue that I can't seem to remedy... I have a pie chart that is displaying three different order modes.
There are a total of 173 unique order numbers:
☆ 96 are "Our Truck"
☆ 58 are "Counter / WC"
☆ 19 are "UPS / MFT"
I am focused on the "Counter / WC" piece which by dividing the values stated above is 33.5% (58 / 173).
☆ PIE VALUE is Beast Mode: COUNT(DISTINCT `Order Number`)
☆ PIE NAME is "Ship Via Group" that contains the text values noted above.
Everything checks out in the pie chart (% and value for each item), all is well.
The Beast Mode for the summary number (which started this discussion) is overstating the count by 1 which causes the summary number and the chart value to not match. The summary number is
displaying 34.1% (59 / 173).
Is this a bug? I have looked at my data, and don't see any issues - there are 58 orders.
Is there a way to subtract 1 in the Beast Mode?
• Based on the spreadsheet with the data you attached , Can you check the order number 14178051 , it looks like that one is both Counter / WC and Our Truck.
Domo Arigato!
**Say 'Thanks' by clicking the thumbs up in the post that helped you.
**Please mark the post that solves your problem as 'Accepted Solution'
• @swagner, I recommend testing a slight change to the beastmode to remove the ELSE 0 statement in the numerator because the COUNT DISTINCT will actually count the 0 as a distinct value and thus
inflate your top number by 1.
COUNT (DISTINCT CASE WHEN `Ship Via Group` LIKE '%Counter%' THEN `Order Number` END) / COUNT (DISTINCT `Order Number)
Jacob Folsom
**Say “Thanks” by clicking the “heart” in the post that helped you.
**Please mark the post that solves your problem by clicking on "Accept as Solution"
• Awesome! This works! ? Thank you!
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Ordinal Numbers Chart 1-100 - OrdinalNumbers.com
Ordinal Numbers Chart 1-10 – You can enumerate unlimited sets by using ordinal numbers. They can also be used as a method to generalize ordinal figures. 1st The ordinal number is one of the
fundamental concepts in math. It is a numerical value that indicates the location of an object in a list. The ordinal … Read more
Ordinal Numbers Chart 1 100
Ordinal Numbers Chart 1 100 – A vast array of sets can be counted using ordinal numbers to serve as an instrument. These numbers can be used as a way to generalize ordinal figures. 1st One of the
basic concepts of mathematics is the ordinal numbers. It is a numerical number that shows where an … Read more | {"url":"https://www.ordinalnumbers.com/tag/ordinal-numbers-chart-1-100/","timestamp":"2024-11-13T22:00:32Z","content_type":"text/html","content_length":"52031","record_id":"<urn:uuid:b8ec798c-7b4c-4ad4-aec2-375a71cbc736>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00633.warc.gz"} |
Current Skill Estimate
Table of Contents
Current Skill Estimate
What is it?
It can take some time until the trainer has revealed all the relevant skills and so we can help you by estimating, in advance, a player's current skill levels.
In the example shown, on this page, the Playmaking skill is already known and then by playing the player as a forward and producing a star rating, we can then estimate the current skill in Wing,
Passing and Scoring. In this example, we can't estimate goalkeeper or defence skills because the star ratings are too old.
Basically - the more that actual skills that are known and the more star ratings, from different positions, are also known, then the higher the accuracy of the estimate. How this is actually
calculated is shown here.
Why is there only one player who has current skill estimations?
The process to calculate the current skill is very complex and so, we do this for only one player. It is not possible to choose which player is chosen for this and so a player is chosen randomly and
then the calculation for that player's unknown current skills is made.
If we tried to estimate the current skill, for every player, for every hy user we would use up all the processor time of our server. Therefore, if you would like to find out the current skill
estimates of more than this one player, then you have to use some credits. By doing this, it gives us the opportunity - if more people want to know more skills - to increase our server capacity
How does this work?
The calculation is very complex and hard to explain because it takes into account many different aspects of both your player and others. However, we feel it is a very good estimate and therefore
explain what will help your chances to predict what skills you need to train etc.
When we calculate the skills for a player we look at which positions he has played in the last 14 days. We then compare his star ratings with other players, who have played in the same position in
the last 14 days and where we already know the skills. We look at as many similar players as possible to calculate the current skill level, of your player. Only players that are nearly identical are
used for the calculation and so the estimate should be within half a level. If we can't predict a skill to this accuracy then we will not show it.
These points briefly explains what is needed for an actual estimate to be made:
• The player must have played in a position relevant for the skill to calculate, within the last 14 days
• The more relevant skills and star ratings that are known, then the higher the accuracy of the estimate
• Only users who invest credits can get estimations for more than one player
Current skill estimate - what is the difference between estimate and the prognosis?
Maybe you are asking yourself why you need this actual skill estimate, when there is already a skill prognosis section. This is because this new actual-skill predictor is far more accurate. The new
calculator takes far more positions and skills into account, whereas the prognosis calculator uses only a single position and does not take the skill development of the players into account.
For the Manual click | {"url":"https://wiki.hattrick-youthclub.org/en/hy/skill_estimation","timestamp":"2024-11-09T03:56:49Z","content_type":"application/xhtml+xml","content_length":"20858","record_id":"<urn:uuid:94ebf477-8d43-4107-a343-88689d7a4c4b>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00460.warc.gz"} |
A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60 0 with the wall, then find the height of the wall.
Given: length of ladder = 15 m
Angle between the ladder and the top of the wall = 60°
We have to find the height of the wall, h.
Thus, the height of the wall is 7.5 m. | {"url":"https://philoid.com/question/29020-a-ladder-15-m-long-just-reaches-the-top-of-a-vertical-wall-if-the-ladder-makes-an-angle-of-60-0-with-the-wall-then-find-the-heig","timestamp":"2024-11-13T22:40:07Z","content_type":"text/html","content_length":"34301","record_id":"<urn:uuid:13951ba7-1e15-4907-af99-6e9091cfa59d>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00802.warc.gz"} |
Second Order Elliptic Equations and Elliptic Systemssearch
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Second Order Elliptic Equations and Elliptic Systems
Ya-Zhe Chen : Peking University, Peking, People’s Republic of China
Lan-Cheng Wu : Peking University, Peking, People’s Republic of China
Softcover ISBN: 978-0-8218-1924-1
Product Code: MMONO/174.S
List Price: $165.00
MAA Member Price: $148.50
AMS Member Price: $132.00
eBook ISBN: 978-1-4704-4589-8
Product Code: MMONO/174.E
List Price: $155.00
MAA Member Price: $139.50
AMS Member Price: $124.00
Softcover ISBN: 978-0-8218-1924-1
eBook: ISBN: 978-1-4704-4589-8
Product Code: MMONO/174.S.B
List Price: $320.00 $242.50
MAA Member Price: $288.00 $218.25
AMS Member Price: $256.00 $194.00
Click above image for expanded view
Second Order Elliptic Equations and Elliptic Systems
Ya-Zhe Chen : Peking University, Peking, People’s Republic of China
Lan-Cheng Wu : Peking University, Peking, People’s Republic of China
Softcover ISBN: 978-0-8218-1924-1
Product Code: MMONO/174.S
List Price: $165.00
MAA Member Price: $148.50
AMS Member Price: $132.00
eBook ISBN: 978-1-4704-4589-8
Product Code: MMONO/174.E
List Price: $155.00
MAA Member Price: $139.50
AMS Member Price: $124.00
Softcover ISBN: 978-0-8218-1924-1
eBook ISBN: 978-1-4704-4589-8
Product Code: MMONO/174.S.B
List Price: $320.00 $242.50
MAA Member Price: $288.00 $218.25
AMS Member Price: $256.00 $194.00
• Translations of Mathematical Monographs
Volume: 174; 1998; 246 pp
MSC: Primary 35
□ Second Order Elliptic Equations
□ $L^2$ theory
□ Schauder theory
□ $L^p$ theory
□ De Giorgi-Nash-Moser estimates
□ Quasilinear equations of divergence form
□ Krylov-Safonov estimates
□ Fully nonlinear elliptic equations
□ Second Order Elliptic Systems
□ $L^2$ theory for linear elliptic systems of divergence form
□ Schauder theory for linear elliptic systems of divergence form
□ $L^p$ theory for linear elliptic systems of divergence form
□ Existence of weak solutions of nonlinear elliptic systems
□ Regularity for weak solutions of nonlinear elliptic systems
□ Sobolev spaces
□ Sard’s theorem
□ Proof of the John-Nirenberg theorem
□ Proof of the Stampacchia interpolation theorem
□ Proof of the reverse Hölder inequality
• Permission – for use of book, eBook, or Journal content
• Book Details
• Table of Contents
• Requests
Volume: 174; 1998; 246 pp
MSC: Primary 35
• Second Order Elliptic Equations
• $L^2$ theory
• Schauder theory
• $L^p$ theory
• De Giorgi-Nash-Moser estimates
• Quasilinear equations of divergence form
• Krylov-Safonov estimates
• Fully nonlinear elliptic equations
• Second Order Elliptic Systems
• $L^2$ theory for linear elliptic systems of divergence form
• Schauder theory for linear elliptic systems of divergence form
• $L^p$ theory for linear elliptic systems of divergence form
• Existence of weak solutions of nonlinear elliptic systems
• Regularity for weak solutions of nonlinear elliptic systems
• Sobolev spaces
• Sard’s theorem
• Proof of the John-Nirenberg theorem
• Proof of the Stampacchia interpolation theorem
• Proof of the reverse Hölder inequality
Permission – for use of book, eBook, or Journal content
Please select which format for which you are requesting permissions. | {"url":"https://bookstore.ams.org/MMONO/174","timestamp":"2024-11-04T07:36:26Z","content_type":"text/html","content_length":"90501","record_id":"<urn:uuid:f7345ad7-74b9-4e8b-8731-7ee22e9bddac>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00127.warc.gz"} |
Formula for 3 square root
formula for 3 square root Related topics: numerically solve nonlinear equations maple
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Author Message
Incicdor Posted: Tuesday 18th of Mar 14:35
Hi, can anyone please help me with my algebra homework? I am weak in math and would be grateful if you could help me understand how to solve formula for 3 square root problems. I also
would like to find out if there is a good website which can help me prepare well for my upcoming math exam. Thank you!
From: THE
Back to top
IlbendF Posted: Wednesday 19th of Mar 14:40
Although I understand what your problem is, but if you could explain in greater detail the areas in which you are facing struggling, then I might be in a better position to help you out .
Anyhow I have some advice for you, try Algebrator. It can solve a wide range of questions, and it can do so within minutes. And that’s not it , it also gives a detailed step-by-step
description of how it arrived at a particular answer. That way you don’t just solve your problem but also get to understand how to go about solving it. I found this program to be
particularly useful for solving questions on formula for 3 square root. But that’s just my experience, I’m sure it’ll be good for other topics as well.
Back to top
3Di Posted: Thursday 20th of Mar 07:59
Hi guys I agree, Algebrator is the best . I used it in Intermediate algebra, Remedial Algebra and Pre Algebra. It helped me learn the hardest math problems. I'm grateful to it.
45°26' N,
09°10' E
Back to top
EY Posted: Thursday 20th of Mar 11:23
I can’t believe it. You really mean it? Is it as effortless as that? Excellent . I am greatly relieved to hear this. Do tell me where can I locate this program?
Back to top
Bet Posted: Thursday 20th of Mar 13:18
You can find all the information about the software here https://softmath.com/comparison-algebra-homework.html.
From: kµlt
øƒ Ø™
Back to top | {"url":"https://softmath.com/algebra-software/subtracting-exponents/formula-for-3-square-root.html","timestamp":"2024-11-14T08:37:44Z","content_type":"text/html","content_length":"40627","record_id":"<urn:uuid:dc3331d5-eb1f-425b-bb2d-9c956a95f6cf>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00727.warc.gz"} |
EMI Moratorium Makes Your No-Cost EMI a Costly EMI - EMI Calculator
Did you recently purchase a product on the No-cost EMI? Are you planning to avail moratorium on such a loan? If the answers to both the questions is YES, how will the moratorium affect the No-cost
EMI structure? Will this still be a No-cost EMI? Let’s find out.
How Do the No-Cost EMIs Work?
• You get an upfront discount on the listed price of the product.
• The bank provides you the loan for the discounted price at a certain rate of interest (It is not a zero-interest rate loan). The interest rate and the loan tenure impact the discount. Longer the
tenure, higher the discount. Higher the interest rate, higher the discount.
• The discount is such that, the total of all the EMIs is equal to the listed price of the product.
To find out more about math behind the No-cost EMIs, refer to this post.
Let us consider an example. You are buying a mobile phone from an online retailer for Rs 12,999. If you opt for 6 month No-cost EMI, an upfront discount of Rs 479 shall be given and your credit card
will be billed Rs 12,520. Subsequently, this expense will be converted into a loan for 6 months at 13% p.a. (Had the interest rate been 15%, the discount would have been Rs 550 and the loan will be
given for Rs 12,449.)
By using the loan calculator, you can find the EMI for such a loan (12,520, 13%, 6 months) will be Rs 2,167. And Rs 2,167 x 6 = 12,999 (minor variation due to rounding off). From your perspective,
you don’t have to pay anything extra and you get the convenience of easy payment. By the way, you do pay a little extra because of the GST on the interest component of the EMI but that’s not much.
Here is an example.
Cost of the item ₹ 12,999
No Cost EMI Tenure (months) 6
Interest Rate (Assumed) 13%
No Cost EMI (Cost/No. of EMIs) ₹ 2,167
Net Loan (Amount charged to your credit card) ₹ 12,520
Upfront Discount to you ₹ 479
Loan Schedule
Month O/S at the start of the month EMI Interest Loan repaid during the month O/S at the end of the month GST Total Monthly outgo
1 12,520 2,167 136 2,031 10,489 24 2,191
2 10,489 2,167 114 2,053 8,436 20 2,187
3 8,436 2,167 91 2,075 6,361 16 2,183
4 6,361 2,167 69 2,098 4,264 12 2,179
5 4,264 2,167 46 2,120 2,143 8 2,175
6 2,143 2,167 23 2,143 0 4 2,171
How Moratorium Makes It a Costly Loan?
As a buyer, you must understand your loan is NOT a zero-interest loan. The banks can’t do that. It is the upfront discount (when you opt for No-cost EMI) that gives you the No-cost experience. Now,
if you avail EMI moratorium, the interest will continue to accrue. Why? Because the interest rate for your loan is not 0%. The interest rate is 13% (in the above illustration). It will continue to
get charged.
Let’s see how things will change. Continuing with the above example, let’s say you choose not to pay the EMI for such loan for 3 months. While your bank may not charge any penalty or report your
non-payment to the credit bureaus, your loan will still accrue interest. Rs 12,520 loan at 13% p.a. will grow to Rs 12,931 over 3 months.
(18% GST is applicable on the interest component. Had I considered the GST (to be added back to the principal), the outstanding principal at the end of 3 months would have been Rs 13,006. However, I
am not sure about the applicability of GST in this case. Therefore, I will ignore the impact of GST on accrued interest in the analysis.)
Now, by the time you start paying EMIs, the principal amount has gone up. Here is the revised schedule.
Month O/S at the start of the month EMI Interest Loan repaid during the month O/S at the end of the month GST Total Monthly outgo
1 12,931 2,238 140 2,098 10,834 25 2,263
2 10,834 2,238 117 2,120 8,713 21 2,259
3 8,713 2,238 94 2,143 6,570 17 2,255
4 6,570 2,238 71 2,166 4,404 13 2,250
5 4,404 2,238 48 2,190 2,214 9 2,246
6 2,214 2,238 24 2,214 0 4 2,242
You can see that the EMI has gone up from Rs 2,167 to Rs 2,238. Therefore, over the course of the loan, you will pay Rs 2,238 x 6 = Rs 13,426. If you include the GST impact (as shown in the above
table), the total payment will be Rs 13,515. Now, it is clearly not a No-cost EMI.
I have already discussed whether you should avail EMI moratorium. While the spreadsheet analysis will answer this question in the negative, there is no black or white answer. Everything depends on
your circumstances. I trust your judgement. | {"url":"https://emicalculator.net/emi-moratorium-makes-your-no-cost-emi-a-costly-emi/","timestamp":"2024-11-08T17:15:42Z","content_type":"text/html","content_length":"60468","record_id":"<urn:uuid:3aa44c3e-2b1b-4d7b-a6d8-064a5110a4b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00108.warc.gz"} |
Every AP Calculus AB Practice Test Available: Free and Official (2024)
One of the best ways to prepare for the AP Calculus AB exam, as well as stay on top of lessons in class throughout the year, is to take regular practice tests. Taking practice tests lets you estimate
how well you'll do on the AP exam, shows you the areas you need to focus your studies on, and helps you become more comfortable with the format of the AP exam.
There are a ton of AB Calc practice tests available; however, not all of them are created equally. Taking a poorly written practice test can give you a false idea of what the real AP exam will be
like and cause you to study the wrong things.
You can avoid those problems by reading this guide to AP Calculus AB practice tests. I'll go through every AP Calculus AB practice exam that's available, tell you which are highest quality, and
explain how you should use practice tests when preparing for the AP exam as well as throughout the year.
Official AP Calculus AB Practice Tests
Official practice exams (those developed by the College Board) are always the best to use because you can be sure they'll be an accurate representation of the real AP exam. There are three types of
official practice resources, and each is explained below.
Complete Practice Tests
The College Board has released two complete exams from prior administrations of the AP Calculus AB exam. The tests are from 1988 and 1998. The 1988 test has an answer key included; however, for some
reason, the 1998 exam does not. The College Board provided answers for the free-response questions in a separate document, but there is no official answer key available for the 1998 exam's
multiple-choice section. The answer key linked below is unofficial, but no one has publicly disagreed with any of the answers, so it's highly likely that it's correct.
Because these exams are from a while back, they both have some format differences compared to the current AP Calculus AB exam. But looking through these old exams can give you a sense of the test
format, and you can work some of the questions as practice, too.
The AP Calculus AB exam is 3 hours and 15 minutes long and has two sections. Both of these sections are divided into two parts. For reference, here's the current format of the exam:
Multiple-Choice Section
• 45 questions total
• 1 hour 45 minutes total
• Worth 50% of your total exam score
• Part A:
□ 30 questions
□ 60 minutes long
□ No calculator allowed
• Part B:
□ 15 questions
□ 45 minutes long
□ Calculator permitted
Free-Response Section
• Six questions total
• 1 hour 30 minutes total
• Worth 50% of your total exam score
• Part A:
□ Two questions
□ 30 minutes long
□ Calculator permitted
• Part B:
□ Four questions
□ 60 minutes long
□ No calculator allowed
You can only use a calculator for certain sections of the AP exam.
Both released exams have the same total number of multiple-choice and free-response questions as the current exam. However, the 1998 test does not have separate parts for the free-response section,
and students were allowed to use a calculator to answer all six questions.
Neither the multiple-choice nor the free-response sections of the 1988 exam were separated into different parts, and students were allowed to use their calculator for the entire exam. The
multiple-choice section was also only 90 minutes long, instead of 105 minutes.
When you take these exams for practice, it's not worth the time and effort needed to try and figure out which questions you wouldn't be allowed to solve with a calculator today. Instead, take the
tests with the calculator and timing rules that were in place when the tests were administered.
These variations between current and past exams do mean that these two complete released exams don't give quite as accurate a representation of the current AP exam as the complete released exams for
other AP subjects do.
However, they are still very useful because they cover the same content and are worded the same way as the current exam. Towards the end of this guide I'll explain exactly how to use these resources
and others.
AP Calculus AB Multiple-Choice Sample Questions
The College Board often reuses multiple-choice questions for multiple exams, so there are typically few official multiple-choice problems available for any AP exam, AP Calculus AB included.
Besides the complete practice tests discussed above, there are no full official multiple-choice sections available, but you can check out these official sample questions for Calculus AB. (The
questions start on page 5, and there are Calculus BC questions listed after the AB questions; be sure you're not accidentally looking at those.) This document contains 16 multiple-choice problems,
along with answers and the major skills each question tests. There are also two free-response questions.
AP Calculus AB Free-Response Sample Questions
Fortunately, there are more official free-response questions available and, since they are recent, they provide you with a very accurate idea of what to expect on the real exam.
The College Board has released free-response questions from 1998-2021 , along with scoring guidelines for each set of questions. These are a great resource, and you should definitely make use of them
during your review.
Khan Academy Resources
Khan Academy has partnered with the College Board to provide study resources for the PSAT, SAT, and some AP exams. This includes study resources for AB Calc.
On Khan Academy's website, there are explanation videos for several dozen previously administered questions, both multiple choice and free response. These videos can be particularly helpful if you've
gotten stuck on one of the official practice problems or if you just want to learn step-by-step how to solve a particular problem.
Unofficial AP Calculus AB Practice Tests and Quizzes
While not developed by the College Board, unofficial practice resources can still be very useful for your studying, particularly because there are so many resources available. For each resource
listed below, I explain what is offered as well as how you should make use of the resource. They are roughly listed from highest quality to lowest quality.
If you're looking for practice tests, this book has them: twelve of them, in fact! We love that they're all in one convenient resource, too. This book also breaks down the test concepts into study
units, so you can brush up on your weakest skills before you take the AP exam. The combination of high-quality instruction and excellent practice tests are why this book takes the top spot on our
unofficial AP Calculus AB practice tests list!
This study book is put out by The Princeton Review, which is a trustworthy test prep company. Even better: this book contains four practice tests that you can use to assess your current knowledge and
gauge how much you're improving as you study.
The other nice thing about this study guide is that it breaks down the key concepts of the exam as well. So if there are skills or ideas you've been struggling with, this book can help you get a
better handle on them before test day.
This exam was created by Patrick Cox, an AP Calculus teacher. The questions a good match for actual AP Calc questions. The answer key is available here. This exam is a good resource for students who
already have a good grasp of calculus concepts and can figure out pretty well on their own where they made mistakes for questions they got wrong.
Shmoop is the only resource listed in this guide that requires a fee to access any of its resources. Paying a monthly fee gets you access to a diagnostic exam, as well as eight complete practice
tests and additional practice questions. It also gets to access to Shmoop's study materials for other AP exams, as well as the SAT and ACT.
Varsity Tutors has a collection of three diagnostic tests and over 130 short practice quizzes you can use to study for the AP Calc AB exam. The practice quizzes are organized by topic, such as the
chain rule and finding the second derivative of a function. Difficulty levels are also given for each of the quizzes. The diagnostic tests are 40-45 questions long (all multiple-choice). They pretty
closely represent what questions from the actual AP exam are like, and, as a bonus, the score results show you how well you did in each topic area so you can focus your future studying on the areas
you need the most work in.
This site organizes quizzes into the three Big Ideas of Calculus AB, as well as more specific tags you can select (you don't need to worry about the Series quizzes, that's just for BC Calc). After
creating a free account, you can access hundreds of practice questions. Questions are ranked as easy, moderate, or difficult, they are not timed, and you see the correct answer (plus a detailed
explanation) after you answer each question. If that's not enough—or if you want to practice harder skills—there's a paid account option that gives you access to additional AP Calc questions.
This site has a 50-question multiple-choice test. The questions typically are easier and more basic than those you'd find on the actual AP exam, but if you're just starting your review or want to
brush up on the basics, this can be a good resource to use.
On this site there are 50+ quizzes, all multiple choice. It's a pretty standard example what most unofficial, free online practice AP resources are like. The questions are decent (though easier and
simpler to work through than actual AP Calc questions), and explanations are pretty barebones.
The ten quizzes and two full-length exams on this site are best for someone earlier on in the course/their AP studying. The quizzes are each about 15 questions long, and immediately after each
question, you'll be told if you answered correctly or incorrectly and be given an answer explanation. They're not the greatest match for actual AP Calculus questions, but because the quizzes are
organized by category, they can be helpful if you're looking to practice a specific topic.
This is a 20-question multiple-choice quiz. The questions are a bit overly simplistic, and it's not automatically graded, but if you're just looking for a quick study session, this fits the bill.
This resource also has practice questions or both the Calculus AB and BC exams, so you can get in a little additional practice, too.
This is a short quiz, and, unfortunately, it's not very high-quality. The questions are pretty basic and not nearly as complex or as in-depth as the ones you'll find on the AP exam. Additionally, the
format of this quiz is very poor, and it can be difficult to read. I wouldn't recommend using this quiz unless you're really desperate for review questions or you need a very basic quiz to get you
started with your review.
How to Use These AP Calculus AB Practice Tests
Knowing how to use each of these practice exams and quizzes will make your studying much more effective, as well as prepare you for what the real AP Calculus AB exam will involve. Below is a guide
for when and how to use the resources, organized by semester.
First Semester
During your first semester of Calc AB, you don't know enough material for it to be useful to take a complete practice exam. Therefore, you should spend this semester answering quizzes and
free-response questions on topics you've already covered. You'll probably want to begin answering practice questions about halfway through the semester.
Free-Response Practice
For free-response questions, use the official released free-response questions in the Official Resources section. Look through them to find questions you can answer based on what you've already
learned. It's best if you can take a group of them (up to six) together at a time in order to get the most realistic preparation for the real AP exam.
It also helps to time yourself when answering these questions, particularly as it gets later in the year. On the real AP exam, you'll have about 15 minutes to answer each free-response question, so
try to answer practice questions under those same time restrictions.
Multiple-Choice Practice
For multiple-choice practice, take unofficial quizzes that let you choose the subject(s) you want to be tested on. This will allow you to review content you've already learned and not have to answer
questions on material you haven't covered yet. The best resources for this are Albert and Varsity Tutors because their quizzes are clearly broken up by specific subject.
Sometimes the numbers can get overwhelming. Don't forget to take a break every now and then.
Second Semester
Second semester is when you can begin to take complete practice exams and continuing to review content you've learned throughout the year.
Step 1: Take and Score Your First Complete Practice Exam
Early on in this semester, when you have covered a majority of the content you need to know for the AP exam, take your first complete practice exam. This test should be taken in one sitting and with
official timing rules (see how the AP test is formatted above).
For this first practice test, I recommend using a test out of either the Barron's or Princeton Review's study guide and saving the official practice exams for down the line. After you take this
practice test, score the exam to see what you earned on the test.
Step 2: Analyze Your Score Results
After you've figured out your practice exam score, look over each problem you answered incorrectly and try to figure out why you got the question wrong.
As you're doing this, look for patterns in your results. Are you finding that you got a lot of questions on antiderivatives wrong? Did you do well on multiple choice but struggled with free response?
Did you get slowed down by questions you couldn't use a calculator to answer?
Figuring out which problems you got wrong and why is the best way to stop repeating your mistakes and begin to make significant improvements. Don't skip this step!
Now is also a good time to set a score goal if you haven't already. The minimum score you should be aiming for is a 3, since this is the lowest passing score. However, if you scored a 3 or higher on
this first practice exam, it's a good idea to set your goal score even higher. Getting a 4 or a 5 on the AP Calculus AB exam looks more impressive to colleges, and it can sometimes get you more
college credit.
Step 3: Focus Your Studying on Weak Areas
You should now have a good idea of what subject areas or skills you need to work on in order to raise your score. If there are specific content areas you need to work on, review them by going over
your notes, reading a review book, and/or answering multiple-choice and free-response questions that focus specifically on those topics.
If you're struggling with your test-taking techniques—like running out of time on the exam or misreading questions— the best way to combat these issues is to answer a lot of practice questions under
realistic testing conditions.
Take timed quizzes or time yourself for quizzes that aren't automatically timed. (On the real exam, you'll get about two minutes for multiple-choice questions you can't use a calculator to solve, a
little more than three minutes for multiple-choice questions where you can use a calculator, and 15 minutes per free-response question.) Taking multiple practice quizzes and tests will help you
become more familiar with the pacing needed for the AP exam.
Step 4: Take and Score Another Practice Exam
After you've identified your weak areas and spent time to improve them, it's time to see how all your hard work paid off.
Take and score another complete practice exam, timed and taken in one sitting. I'd recommend using either an official released practice exam or, if you want more recently-created questions, creating
your own practice test by combining a set of unofficial multiple-choice questions (such as the Varsity Tutors or 4Tests exam) with a set of official free-response questions. If you choose the second
option, you should have a total of 45 multiple-choice questions for the first part of the exam. As with the first test, this should be taken timed and in one sitting.
When you take this second practice exam, remember that it won't be formatted exactly the same way as the real AP test, where the multiple-choice and free-response sections will both be broken into
two parts, only one of which you can use a calculator on.
Step 5: Review Your Results to Determine Your Future Study Plan
Now you're able to see how much you've improved, and in which areas, since you took your first complete practice exam. If you've made improvements and have reached or are close to your target score,
you may only need to do some light studying from now until the AP exam.
However, if you haven't made much improvement, or you're still far from your score goal, you'll need to analyze the way you've been reviewing and think of ways to improve. The most common reason for
not improving is not actively studying, but only passively leafing through your notes or reviewing missed questions. Even though it may seem to take a while, in the long run, carefully analyzing why
you made the mistakes you did and devising ways to improve is really the only significant way to raise your score.
As you're studying, be sure to really understand your mistakes. If you don't understand why you got a question wrong, go back and review that particular skill! Also, when you're reviewing notes,
pause every few minutes and mentally go over what you just learned to make sure you're retaining the information.
You can repeat these steps as many times as you need to in order to make improvements and reach your target score.
Studying With AP Calculus AB Practice Exams: Key Tips
It would be difficult to score well on the AP Calculus AB exam without completing any practice exams. Official resources are the best to use, but there are plenty of high-quality unofficial quizzes
and tests out there as well.
During your first semester, you should focus on answering free-response and multiple-choice questions on topics you've already covered in class.
During your second semester, follow these steps:
□ Take and score your first complete practice exam
□ Analyze your score results
□ Focus your studying on weak areas
□ Take and score another complete practice exam
□ Review your results to determine your future study plan
What's Next?
Now that you have your practice tests, do you want to know more about the AP Calculus AB Exam? Our guide explains the complete format of the AP Calculus AB test, the question types you'll see, and
how to best prepare for the exam.
How many AP classes should you take? Get your answer based on your interests and your college goals.
Wondering how challenging other AP classes will be? Learn what the easiest AP classes are and what the hardest AP classes are so that you're prepared!
These recommendations are based solely on our knowledge and experience. If you purchase an item through one of our links, PrepScholar may receive a commission.
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About the Author
Christine graduated from Michigan State University with degrees in Environmental Biology and Geography and received her Master's from Duke University. In high school she scored in the 99th percentile
on the SAT and was named a National Merit Finalist. She has taught English and biology in several countries.
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Basic thing of Conic in Projective Geometry
• Thread starter wawar05
• Start date
In summary: P2 that has a homogenous quadratic equation and suitable coordinates. You can change the coordinates so that some terms can be dropped, and a degenerate conic is one that does not have a
nice smooth curve.
I am new about conic in projective geometry since it seems to be really different in euclidean plane.
A conic is a subset of P2 given by a homogenous quadratic equation:
aX^2 + bY^2 + cZ^2 + dXY + eXZ + fYZ = 0
why is it homogeneous?
meanwhile, it suitable coordinates we have aX^2 + bY^2 + cZ^2 = 0, with a, b, c element {0, 1, -1}.
why the part of dXY + eXZ + fYZ can be erased?
what is the difference of degenerate conic and non-degenerate conic?
Staff Emeritus
Science Advisor
Homework Helper
wawar05 said:
I am new about conic in projective geometry since it seems to be really different in euclidean plane.
A conic is a subset of P2 given by a homogenous quadratic equation:
aX^2 + bY^2 + cZ^2 + dXY + eXZ + fYZ = 0
why is it homogeneous?
An equation is homogeneous if all the terms have the same degree. A term of the form X
or XY all have degree 2.
meanwhile, it suitable coordinates we have aX^2 + bY^2 + cZ^2 = 0, with a, b, c element {0, 1, -1}.
why the part of dXY + eXZ + fYZ can be erased?
The point is that you can change coordinates in such a way such that the d, e and f can be dropped. See http://home.scarlet.be/~ping1339/reduc.htm for a reduction of a conic section to it's reduced
what is the difference of degenerate conic and non-degenerate conic?
A degenerate conic consists of lines and points, whilme a non-degenerate conic is a nice curve.
For example, the conic
has only (0,0) as a solution, thus the conic is just a point. The conic
is the same as
Thus the conic is two times the line X=-Y. Such a conics are degenerate because they are not the nice smooth curves we expect.
Science Advisor
Gold Member
I always find it helpful to go back to the basics when thinking about the conics. the geometry of it all is much simpler than equations and is SO easy to visualize. I've done up a drawing for you:
^^, thank you for the helps... I am now having good understanding of conic related to projective plane...
It is important to note that in projective geometry, the concept of a conic is defined differently than in Euclidean geometry. In projective geometry, a conic is a subset of the projective plane P2,
which is a space where points, lines, and planes are all treated equally. This is why we use homogeneous coordinates, where each point is represented by a triplet (X, Y, Z) rather than just (x, y) as
in Euclidean geometry.
The reason for the homogeneity of the conic equation is that it allows us to represent all points on the conic curve, including points at infinity. This is because when we use homogeneous
coordinates, the coordinates of a point are only determined up to a scalar multiple. Therefore, the equation aX^2 + bY^2 + cZ^2 + dXY + eXZ + fYZ = 0 represents all points on the conic, regardless of
their coordinates.
Regarding the part of dXY + eXZ + fYZ, this is known as the cross term and can be eliminated by choosing suitable coordinates. This is because in projective geometry, the cross term does not change
the shape of the conic, only its orientation. So, by choosing appropriate coordinates, we can simplify the equation and still represent the same conic curve.
The difference between a degenerate and non-degenerate conic lies in the number of points that satisfy the conic equation. A non-degenerate conic has exactly five points that satisfy the equation,
while a degenerate conic has fewer than five points. In Euclidean geometry, a degenerate conic would be a pair of intersecting lines or a single point, whereas in projective geometry, it can also be
a single line or a point at infinity.
In conclusion, the concept of a conic in projective geometry may seem different at first, but it allows us to study conic curves in a more general and inclusive way. Homogeneous coordinates and the
elimination of the cross term are essential tools in understanding and working with conics in projective geometry. The distinction between degenerate and non-degenerate conics is also important in
analyzing the properties of these curves.
FAQ: Basic thing of Conic in Projective Geometry
1. What is a conic in projective geometry?
A conic in projective geometry is a curve that can be defined as the intersection of a plane and a cone. It can take various forms such as a circle, ellipse, parabola, or hyperbola. In projective
geometry, conics are studied using the principles of projective transformations, which allow for the preservation of geometric properties.
2. What are the basic properties of a conic in projective geometry?
The basic properties of a conic in projective geometry include its center, axes, vertex, focus, and directrix. These properties vary depending on the type of conic. For example, a circle has a center
and a radius, while a parabola has a focus and a directrix. Understanding these properties is crucial in analyzing the behavior and characteristics of conics in projective geometry.
3. How is a conic represented in projective geometry?
In projective geometry, a conic can be represented by a general equation of the form Ax² + Bxy + Cy² + Dx + Ey + F = 0, where A, B, and C are not all equal to 0. This equation is known as the general
conic equation and can be used to represent all types of conics. By manipulating the coefficients, one can determine the type and orientation of the conic.
4. What are some real-world applications of conics in projective geometry?
Conics in projective geometry have various real-world applications, including in optics, astronomy, and engineering. For example, parabolic mirrors and lenses are used in telescopes and satellite
dishes, while hyperbolic shapes are used in radio antennas. Conics are also used in designing bridges, tunnels, and roads, as well as in the analysis of projectile motion.
5. How do conics in projective geometry differ from conics in Euclidean geometry?
In Euclidean geometry, conics are defined as the intersection of a plane and a cone in 3-dimensional space. However, in projective geometry, conics are studied in a higher-dimensional space, where
the principles of projective transformations apply. This allows for a more general and abstract approach to the study of conics, leading to new and unique properties and applications. | {"url":"https://www.physicsforums.com/threads/basic-thing-of-conic-in-projective-geometry.510576/","timestamp":"2024-11-03T09:10:17Z","content_type":"text/html","content_length":"96907","record_id":"<urn:uuid:cdbf8f6f-1c7c-4120-8e1d-2db8cac9ccb2>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00450.warc.gz"} |
What is the equation of the perpendicular bisector of a chord of a circle? | HIX Tutor
What is the equation of the perpendicular bisector of a chord of a circle?
Answer 1
For a chord AB, with $A \left({x}_{A} , {y}_{A}\right) \mathmr{and} B \left({x}_{B} , {y}_{B}\right)$. the equation of the perpendicular bisector is $y = - \frac{{x}_{B} - {x}_{A}}{{y}_{B} - {y}_{A}}
\cdot \left(x - \frac{{x}_{A} + {y}_{B}}{2}\right) + \frac{{y}_{A} + {y}_{B}}{2}$
Supposing a chord AB with #A(x_A,y_A)# #B(x_B,y_B)#
The midpoint is #M((x_A+x_B)/2,(y_A+y_B)/2)#
The slope of the segment defined by A and B (the chord) is #k=(Delta y)/(Delta x)=(y_B-y_A)/(x_B-x_A)# The slope of the line perpendicular to the segment AB is #p=-1/k# => #p=-(x_B-x_A)/(y_B-y_A)#
The necessary line equation is
#y-y_M=p(x-x_M)# #y-(y_A+y_B)/2=-(x_B-x_A)/(y_B-y_A)*(x-(x_A+x_B)/2)# Or #y=-(x_B-x_A)/(y_B-y_A)*(x-(x_A+x_B)/2)+(y_A+y_B)/2#
Note: the center of the circle, assummedly point C#(x_C,y_C)#, is in the line defined by the aforementioned equation, with radius #r=AC=BC#. Therefore the perpendicular bisector of the question is
also the locus of the possible centers of circles with the chord AB
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Answer 2
The equation of the perpendicular bisector of a chord of a circle is the equation of a diameter of the circle.
Let's assume that the circle's standard equation has radius r and a center at origin.
#color(blue)("The equation of the circle: "x^2+y^2=r^2)#
The chord AB's coordinates for its end points
#color(red)(A->(rcosa,rsina)" & " B->(rcosb,rsinb)#
The middle point C (x',y') coordinate for AB
#x'=r/2(cosa+cosb)=rcos((a+b)/2)cos((a-b)/2)# #y'=r/2(sina+sinb)=rsin((a+b)/2)cos((a-b)/2)#
The perpendicular bisector of AB has a slope of
The formula for AB's perpendicular bisector
Since this is, of course, the equation of a straight line through the circle's origin (0,0), or center, the perpendicular bisector of the chord equals the circle's diameter.
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Answer 3
The equation of the perpendicular bisector of a chord of a circle can be found using the midpoint formula and the negative reciprocal of the slope of the chord. Given the endpoints of the chord (x1,
y1) and (x2, y2), the midpoint (h, k) of the chord is found using: h = (x1 + x2) / 2 k = (y1 + y2) / 2
Then, the slope (m) of the chord is found using: m = (y2 - y1) / (x2 - x1)
The negative reciprocal of the slope is: m_perpendicular = -1 / m
Using the point-slope form of a line (y - k = m_perpendicular * (x - h)), substituting the midpoint (h, k) and the negative reciprocal slope (m_perpendicular), we get the equation of the
perpendicular bisector of the chord.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Aditya Hrdayam Again - Part 57 - Particles of Spinor field
In the last blog I explained how 'foam' is the Scalar field, 'conch' is the spinor field, 'Chakra' and other weapons are the 'Vector' fields of the Standard Model.
Let's see about the excitations that arise from these fields in this blog.
Conch - The Panchajanya - Pancha Jana
Conch is the Spinor, spin-½ field.
The excitation of Spinor field or conch gives rise to a class of particles called fermions. All the Universal matter are made of fermions. Thus it is said that the Universe arises out of the ‘sound’
(excitation) of the Conch.
The fermions are of five types or five races. They are called the ‘Pancha Jana’ in Vedas. The pancha jana are
1. quarks
2. antiquarks
3. electrons
4. positrons and
5. neutrinos.
These are five races because each race has number of particles based on flavor, color and handedness.
• Handedness is the Right-handed and left-handed (Chirality) that applies to all the five races. This is the basic aspect of ‘conch’, the fermions. All fermions have a left and right handed types.
• Flavor determines the ‘mass’ of the particle. It is the ‘coupling’ of these particles to ‘Scalar Higgs field’, the ‘foam’. The Scalar Higgs field, the foam, interacts with these fermions and
‘impedes’ their movement. This ‘impeding’ of their movement creates a higher energy density which we see as ‘mass’.
• There are three generations of each of these five races based on the flavor. Extremely massive, massive and light. Flavor applies to all five races.
Thus the fermionic ‘conch’, the spinor field, that gives rise to these five races is called ‘Pancha-Janya’, the Conch of Vishnu, the Higgs field.
Quarks and Antiquarks can also be classified further based on their ‘color charge’ which determines their interaction with ‘Strong force’ or Soma. Each of these quarks/antiquarks can be present in
any of three ‘colors’.
The pancha-jana, the five races are impacted by forces of ‘Vector fields’. They ‘suffer’ or ‘bear’ the effect of forces of vector fields and evolve due to the forces of these vector fields.
Chakra is the Vector, spin-1 field.
The excitation of spin-1 vector field or Chakra gives rise to a class of particles called Bosons. Bosons mediate ‘forces’ and are called ‘force-carriers’. They deliver the forces to fermions and
evolve the fermions.
Thus the various ‘Chakras’ or force-carriers are said to create/maintain/destroy and thereby progress the evolution. The sudarshan chakra that is most famous of the chakras is the ‘weak force
carrier’ which decays all five races of fermions (pancha-jana).
The visualization of spin-1 vector as a ‘Chakra’ (apart from the 360* rotation leading a particle to be in the same state) is to capture the point that they are force-carriers. Visualize a set of
wheels working together as in gears. Bosons mediate the forces similarly and deliver. (Remember its an analogy).
Pauli exclusion principle
Conch, the spinor field, give rise to fermions. They obey Pauli exclusion principle.
It means that no two fermions (excitation of conches) can occupy the same ‘quantum state’ at the same time. This can be visualized as a ‘Conch’ can produce only one sound at a time. Two different
excitations (sound) cannot arise from the same conch at the same time.
Chakra, the vector field, give rise to bosons. They don’t obey Pauli exclusion principle.
This means that two different bosons can occupy the same quantum state at the same time. This can be visualized like two wheels/chakras placed one over other, perfectly continuing to rotate 360*, due
to their ‘circular’ structure and transmitting force from one place to another.
Composite fermions
The excitations of the spinor field (fermions) combine together and form various composite particles. For eg. Quarks combine to form mesons and baryons. Baryons like protons and neutrons combine to
form nucleus of atoms. Nucleus of atoms combine with electrons to form atoms. Atoms combine to form molecules.
But these composite fermions can remain fermions with spin-½ (like Conch) or become the Chakra (Spin 1 Vector bosons) or become like foam (Spin 0 scalar bosons).
Quark based matter - Sura, Asura, Rakshasa, Yaksha
Of the five races, Quarks and anti-Quarks are subject to Soma the Strong force. From this Quark-strong force interaction, variety of matter particles arise. They are nucleons, Vector mesons,
pseudo-vector mesons, pseudo-scalar mesons and scalar mesons.
Sura are nucleons. Vector mesons and pseudo-scalar mesons are Rakshasas. Pseudo-vector mesons and Scalar mesons are the Yaksa.
Soma and Sura
Soma is strong force. Sura is what is ‘distilled’ from Soma, the strong force. Sura is the ‘residual strong force’ that is ‘distilled’ through the Quarks and antiquarks.
Surya are also those baryons that distill this strong force/soma into Sura, the residual strong force. They are the baryons and more specifically nucleons that use the Sura, the residual strong force
to make atomic nucleus. This process of making nucleus with residual strong force/sura is called ‘Surya’.
Asura are those that do not distil strong force. Hence they are not nucleons. Since asura, non-nucleons is a ‘wide name’, it would include leptons (electrons, positrons, neutrinos and their races)
and mesons as well.
Composite fermions as Vector bosons - Asuras - Rakshasas and Yaksas
Let’s say multiple conches (spinor structures) are connected in a way that their ‘spinor’ structure are aligned at the mouth. Then different excitations (sounds) can arise from the same ‘conch’.
Similarly multiple fermions can combine together and act like ‘bosons’ or excitations of vector fields, if their spins are aligned.
Examples of these kind of excitations are Vector / Pseudo-vector mesons which are quark-antiquark combinations, which act as vector fields with Spin 1. Nucleus of some atoms like deuterium are vector
bosons with Spin 1.
Such spin 1 bosons that arise from fermionic fields are called ‘Asuras’. These are called ‘Asuras’ as they do not use/drink ‘sura’. Sura are residual strong force. Asuras do not form nucleons.
Such Asuras which have Spin 1 are ‘force-carriers’. They can have either odd/negative intrinsic parity or even/positive intrinsic parity.
Odd parity means their ‘mirror rotation’ is different from the asura particle. Such asura particles with Spin 1 and odd parity are called vector mesons/particles. They are in pairs (a particle
distinct from its rotation). They are mithuna. Such mithuna, the Vector mesons with spin-1 or vector bosons are the rAkshasa.
Even parity means their ‘mirror rotation’ is same as the asura particle itself. Such Spin 1 particles arising out of fermions, but with even parity are called the pseudo-vector mesons/particles. They
are the Yakshas. Yakshas, the pseudo-vector mesons with even parity are ‘attendants’ of Gandharva or they remain around Gandharvas.
Composite fermions as Scalar bosons - Asuras - Rakshasa and Yaksas
Let’s say multiple conches (spinor structures) are connected in a way that their spinor structure are anti-aligned in such a way they totally close each other. Then such a conch can never produce any
excitation. It becomes like a ‘scalar’ field (foam, frozen) giving rise to a scalar boson with a ‘fixed’ value.
Examples of these kind of excitations are Scalar /Pseudo-scalar mesons which are quark-antiquark combinations, which act as scalar fields with Spin 0. Atoms like helium-4 or nucleus of carbon-12 also
act as scalar bosons with Spin 0.
Such spin 0 bosons that arise from fermionic fields are also called ‘Asuras’ as they do not use/drink Sura. Suras are nucleons. Asuras are non-nucleons.
Such Asuras which have Spin 0 can have either odd/negative intrinsic parity or even/positive intrinsic parity.
Odd parity means their ‘mirror rotation’ is different from the asura particle. Such asura particles with Spin 0 and odd parity are called pseudo-scalar mesons/particles. They are in pairs (a particle
distinct from its rotation). They are mithuna. Such mithuna, the pseudo-scalar mesons with spin-0 are the rAkshasa. They originate the rAksasa clans.
Even parity means their ‘mirror rotation’ is same as the asura particle itself. Such Spin 0 particles arising out of fermions, but with even parity are called the scalar mesons/particles. They are
the Yaksas.
Rakshasas and Yakshas
Summarizing, fermions which act as Vector or Scalar bosons with Odd parity are Rakshasas, while those with even parity are Yakshas.
Rakshasas are ‘mithuna’ (two). Their mirror-rotation (reflection in mirror) is not equal to them. Rakshasas are thus like ‘nara’ (human beings) and are said as ‘sent’ to earth.
Yakshas when rotated in mirror (mirror reflection) look the same. Thus they are compared to ‘ghosts’.
Composite fermions as nucleons - Suras
Let’s say multiple conches (spinor structures) are connected in a way that their spinor structure are neither totally aligned or anti-aligned. Then they produce an Unique vibration (sound) which is
different from a single conch, but also not multiple vibrations. These are called half-integer spin-fields (spin-3/2, 5/2 etc), which gives rise to composite fermions. They are called ‘Suras’.
Examples of these kind of excitations are Baryons with are three quark combinations, which act as spinor fields with spins of ½, 3/2, 5/2 etc. Nucleus of many atoms are made of these three quark
combinations, which retain their fermionic (Conch) nature. These are called Sura, those that use the nuclear force.
Protons and neutrons use this Sura, the nuclear force and form nucleus of atoms.
More to come | {"url":"https://www.vedabhasya.org/2018/02/aditya-hrdayam-again-part-57-particles.html","timestamp":"2024-11-11T18:30:03Z","content_type":"text/html","content_length":"130884","record_id":"<urn:uuid:1645d45a-1862-4ce0-9189-07fbafb1864b>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00479.warc.gz"} |
Law of Large NumbersUnderstanding the law of large numbers (without misconception)
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Generating synthetic gravitational fields
I was doing a quick study on “color from structure” and related concepts. The colors of butterfly wings are often not pigments in the traditional sense, but emergent colors because of the shapes of
the molecules.
What clicked in my mind is that the sub wavelength structures (the structures that produce the rather large “blue” wavelengths” are smaller than the wavelength) can be of atomic and nuclear
Back in the early 1980’s I wrote papers for the Gravity Research Foundation essay contest. In one of them I reflected on the gravitational energy density. Robert Forward who helped start
gravitational radiation detection gave an expression for the gravitational energy density and I found the relation to magnetic, electric, laser, thermal, and other forms of energy density. The
gravitational field at the earth’s surface has an energy density equal to a magnetic field of about 379 Tesla. This is larger than any static field we have been able to produce. I have tried to
visualize and understand and apply that for the last 40 years.
What I see is that ordinary matter, like molten iron at the earth’s core, is made of structures whose size and density determine the frequencies of gravitational potential waves they are immersed
in. And, since the gravitational field permeates all matter, including inside black holes, that means signals of all wavelengths are found in the gravitational field.
Specifically, at the surface of the earth, the gravitational energy density and the energy density of waves in the gravitational potential can be connected.
g^2/(8 pi G) = (4/c)*SB*T^4 where g is the acceleration, G is the gravitational constant, c is the speed of light and gravity, SB is the Stefan-Boltzmann constant, and T is the temperature of the
gravitational field in Kelvin.
T^4 = g^2 *c / (8 pi G * 4 SB)
T^4 = g^2 * (2.99792458E8)/(8*pi*6.674E-11 * 4 * 5.670374E-8)
For g = 9.8 meters/second^2, the temperature is about 2.95 Million Kelvin and the magnetic field is 379.37 Tesla
In radiation terms, TemperatureElectronVolts = BC * T / EC, where BC is Boltzmanns constant, EC is Electron Charge, and T is in Kelvin. For this example that is 254.17 electron volts, which is in
the soft x-ray region of the electromagnetic spectrum. What I am saying is that the earth’s color is primarily “soft x-ray” which is why gravity goes right through solids. They are immersed in it.
It flows through. and fills, the pores of matter, and its smallest particles move at the speed of light and gravity.
I have been trying for several years to design detectors that can see inside the earth and sun and moon or anything. To image in real time, the core of the earth and sun. This concept that the
color or spectrum of light is tied to its sources – to matter of unique shapes and sizes, not its periodicity, is powerful. It means that the hot mass of the core of the earth is visible from the
surface, by looking at the soft x-ray spectrum. And that is accessible, but forming structures and detectors that can image across all wavelengths. The wavelength of this light is about 4.878
nanometers. And pulses of that size can be made in many ways. This is freshman physics in college or something high school kids can study. And the tools for making the fields, you can buy off of
Amazon and Ebay.
Here are some of the kinds of searches I was doing
“nanodots” (“coloration” OR “coloring” OR “coloration” OR “pigment”) with 96,100 entries
“plasmon resonances” “colors” with 77,600 entries
“structural colors” OR “structural color” with 622,000 entries
“structural coloration” with 47,600 entries
(“coloring” OR “coloration”) (“photonic” OR “photonics”) with 643,000 entries
(“reflective coloring” OR “reflective coloration”) (“photonic” OR “interference” OR “diffraction” OR “Mie” Or “Bragg”) with 819
“nanorod” “color”
“tandem nanorods”
and many more | {"url":"https://theinternetfoundation.org/?p=2523","timestamp":"2024-11-07T22:51:01Z","content_type":"text/html","content_length":"48468","record_id":"<urn:uuid:cf226436-a267-46e4-b5a9-6a4590df46df>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00012.warc.gz"} |
Eastern New Mexico University - Roswell Catalog
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Introduction to Statistics
Dept of Mathematics Arts & Sciences Education
Course Title
Introduction to Statistics
Arts & Sciences Education
Course Description
Introduction to Statistics. Four credit hours. This course discusses the fundamentals of descriptive and inferential statistics. Students will gain introductions to topics such as descriptive
statistics, probability and basic probability models used in statistics, sampling and statistical inference, and techniques for the visual presentation of numerical data. These concepts will be
illustrated by examples from a variety of fields. Please consult your advisor before enrolling.
Course Attributes
General Education - Mathematics
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• (from the following course set: ) | {"url":"https://24-25-enmur.catalog.prod.coursedog.com/courses/MATH1350","timestamp":"2024-11-10T08:36:34Z","content_type":"text/html","content_length":"506182","record_id":"<urn:uuid:23c214a4-f367-4228-9e52-4018bcfb1f83>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00449.warc.gz"} |
(थ) द्विपद प्रमेय से (x+a)4 का प्रसार करो।... | Filo
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(थ) द्विपद प्रमेय से का प्रसार करो। हल-द्विपद प्रमेय से का प्रसार करने पर
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Question Text (थ) द्विपद प्रमेय से का प्रसार करो। हल-द्विपद प्रमेय से का प्रसार करने पर
Updated On Feb 5, 2023
Topic Calculus
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 73
Avg. Video Duration 13 min | {"url":"https://askfilo.com/user-question-answers-mathematics/begin-array-l-50-32-33-50-56-15-end-array-th-dvipd-prmey-se-34313232383136","timestamp":"2024-11-05T03:10:46Z","content_type":"text/html","content_length":"335623","record_id":"<urn:uuid:025e2949-2398-43f2-b00c-76b2a480bd19>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00245.warc.gz"} |
Statistical concepts to understand supervised learning | My Website
Algo AI based organization
Statistical Concepts You Should Learn To Understand Supervised Learning
Supervised machine learning is an interdisciplinary field that uses statistics, probability, algorithms to learn from provided data. These algorithms are used to build intelligent applications. Just
like Probability, a knowledge of Statistical concepts is invaluable when working on a machine learning project.
It would be fair to say that statistical concepts are required to effectively work through a supervised learning project. Here’s a list:
Data Analysis and Mining
To frame a problem- select its type and classify its structure. Once you have classified the types of inputs and outputs it involves, statistical methods like Exploratory Data Analysis and Data
Mining can be used to extract information. Exploratory Data analysis involves summarization and Visualisation to explore provisional views of the data. Data mining will help automatically discover
the structured relationships and patterns of the data.
Data Understanding
To understand the data you need to have a good grasp of both- the distributions of variables, and the relationships between them. Summary Statistics or Data Visualization is a method used to
summarise the distribution and relationships between variables using statistical quantities. It could be in the form of charts, plots or graphs.
Detection and Imputation
Records suggest that observations from a domain are not always uncorrupted. Although the process is automated and goes through security, it may have been subjected to processes that can damage the
fidelity of the data resulting in Data error or Data loss. The whole process of data cleaning requires identifying and repairing for any loss or error. Statistically, we can implement Outlier
Detection and Imputation for this process. Outlier Detection involves methods used for identifying observations that are far from expected value in a distribution. Imputation is filling up for the
missing or corrupted values in observations.
Data Sampling and Selection
Not all values and variables are relevant when modelling. Data sampling systematically creates smaller representatives from larger data sets, that can be used for predictions. You also need to
identify which variable is the closest to the expected outcome. The process is called Data Feature selection.
Data Preparation
Quite often you are required to tweak the shape or structure of data making it rather suitable for learning algorithms. Data preparation is executed using these statistical methods- Scaling:
Standardization and Normalization.
Encoding: Integer and one-hot encoding
Transforms: Power transforms like the Box-Cox methods.
Experimental Design
Routined evaluation of a learning method is significant in Supervised Learning. This often requires estimation of the skill of the model when making predictions. Experimental design is the process of
designing systematic experiments to compare the effect of independent variables on an outcome.
Statistical Hypothesis tests
Any given Supervised Learning algorithm often has a suite of parameters that allow the learning method to be tailored to a specific platform. The interpretation and comparison of results between
different parameters configurations is made using Statistical Hypothesis Tests. It involves methods that quantify the likelihood of observing the result based on an assumption regarding the result,
Once a final model is ready, it can be presented to stakeholders prior to being deployed on real data. However, a part of presenting the final product involves presenting the model first. Any
quantification in the skill of the final model is done with Estimation statistics.
Statistical concepts play a major role in Supervised Learning. To understand the data better, clean and prepare it for modelling, and finally in selection of the model and presenting the skill, and
predictions from final model- statistics helps you through the entire process. | {"url":"https://ulab.ai/statistical-concepts-to-understand-supervised-learning","timestamp":"2024-11-11T20:07:32Z","content_type":"text/html","content_length":"24954","record_id":"<urn:uuid:9a73970d-931d-4812-ad86-115897d9e808>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00467.warc.gz"} |
[EM] A 48% Group elects 60% of the Droop Members
DEMOREP1 at aol.com DEMOREP1 at aol.com
Mon Nov 15 12:51:32 PST 1999
The Droop ratio approaches the Hare ratio with an increasing number of
S= Number of Seats, D/H = Droop/Hare ratio (as a percentage)
S D/H (each is little more depending on the total number of votes)
1 50.00
2 66.67
3 75.00
4 80.00
9 90.00
19 95.00
49 98.00
99 99.00
999 99.90
Can a bare majority of Droop quotas in a single at large district produce
indirect minority rule ?
S= Number of seats, T = Total votes
Odd number of seats-
(S+1)/2 x (T/(S+1) +1) = (T + S + 1)/ 2 votes
Since the result is greater than T/2 the answer is NO.
Note that with a large T and a small S, the result will barely be over 50
percent of T.
Even number of seats-
(S/2 +1) x (T/(S+1) +1) = (S+2)/2 x (T + S + 1)/(S+1) votes
Not obvious.
S Result (votes)
2 2T/3 + 3
4 3T/5 + 5
6 4T/7 + 7
10 6T/11 +11
100 51T/101 +101
1000 501T/1001 +1001
With a large S the result nears T/2.
Since the result is greater than T/2 the answer is NO.
Note that with a large S, the result will barely be over 50 percent of T.
To avoid borderline apportionment problems in multi-party elections, there
would seem to be a need for a majority requirement (i.e. any 2 or more
parties with a majority of the votes must get a majority of the seats -- but
this may not be possible due to conflicting overlapping partial majorities
regarding the marginal seat(s) -- especially with 4 or more parties.)
Again I note that the various ratios of party votes/party seats for the
different parties will almost never be the same (especially with a low number
of seats in the legislative bodies of many private groups and local
I note again -- proxy p.r. avoids most, if not all, of the above math
More information about the Election-Methods mailing list | {"url":"http://lists.electorama.com/pipermail/election-methods-electorama.com/1999-November/101477.html","timestamp":"2024-11-13T15:22:28Z","content_type":"text/html","content_length":"4837","record_id":"<urn:uuid:99a5c088-2f77-4fcd-a9d9-c3075907cb2d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00334.warc.gz"} |
Chemistry Lab Calculations: Molarity and Stock Solutions
Question 1:
How many milliliters of a 2.50 M HCl stock solution are needed to prepare 250.0 mL of a 0.100 M HCl solution?
Question 2:
What is the molar concentration of glucose in a beaker after spilling 35.0 mL of a 3.00 M aqueous glucose solution into 150.0 mL of pure water?
Answer 1:
10.0 mL of the 2.50 M stock solution are required to make 250.0 mL of 0.100 M HCl.
Answer 2:
The molarity of glucose in the beaker after the spill is approximately 0.57 M.
In order to calculate the volume of the 2.50 M stock solution needed to prepare 250.0 mL of 0.100 M HCl, we can use the formula M1V1 = M2V2, where M1 is the molarity of the stock solution, V1 is the
volume of the stock solution needed, M2 is the desired molarity, and V2 is the desired volume.
Substituting the given values into the formula, we get V1 = (0.100 mol/L * 250.0 mL) / 2.50 mol/L, which simplifies to 10.0 mL of the stock solution needed.
After the spill of 35.0 mL of a 3.00 M glucose solution into a beaker containing 150.0 mL of water, the total volume of the solution becomes 185.0 mL. To find the new molarity of glucose in the
beaker, we calculate the amount of moles of glucose (3.00 mol/L * 0.035 L) and divide it by the total volume in liters (0.185 L), resulting in a new molarity of approximately 0.57 M. | {"url":"https://madlabcreamery.com/chemistry/chemistry-lab-calculations-molarity-and-stock-solutions.html","timestamp":"2024-11-08T02:58:13Z","content_type":"text/html","content_length":"21640","record_id":"<urn:uuid:96a2957e-e4cc-45f0-be94-c14faf09fb4e>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00848.warc.gz"} |
750V DC shunt motor takes 100A and runs at 3000rpm at full load.
The armature...
A 750V DC shunt motor takes 100A and runs at 3000rpm at full load. The armature...
A 750V DC shunt motor takes 100A and runs at 3000rpm at full load. The armature resistance is 1 ohm and the field winding resistance is 200 ohm. Assume magnetic linearity and flux is directly
proportional to the field current.
Compute its new speed if the developed torque remains the same after adjusting the field winding resistance to 250 ohm.
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Ask a Question | {"url":"https://justaaa.com/electrical-engineering/1299053-a-750v-dc-shunt-motor-takes-100a-and-runs-at","timestamp":"2024-11-11T08:17:14Z","content_type":"text/html","content_length":"40674","record_id":"<urn:uuid:826d8995-f2bb-4d71-8011-d960afb21b5c>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00584.warc.gz"} |
Z Score CalculatorZ Score Calculator
• Enter the Raw Score, Mean (μ), and Standard Deviation (σ) for your data.
• Click "Calculate Z-Score" to calculate the Z-Score and related values.
• Results, including the Z-Score, p-values, and confidence level, will be displayed below.
• Calculation steps will also be shown to explain how the Z-Score was computed.
• A chart visualizes the Z-Score in the context of the normal distribution.
• You can clear the entries, copy the results, and view calculation history.
A Z score calculator is a tool that allows users to calculate the Z score of a given value. The Z score is a measure of how many standard deviations a given value is away from the mean of a
population. It is calculated by subtracting the mean from the value and then dividing by the standard deviation.
The following are some of the key concepts that underlie Z score calculators:
• Mean: The mean is the average of a population. It is calculated by adding up all of the values in the population and dividing by the number of values.
• Standard deviation: The standard deviation is a measure of how spread out the values in a population are. It is calculated by taking the square root of the variance.
• Variance: The variance is a measure of how spread out the values in a population are. It is calculated by taking the average of the squared deviations from the mean.
The following formula is used to calculate the Z score of a given value:
Z score = (x - μ) / σ
• x is the value to calculate the Z score for
• μ is the mean of the population
• σ is the standard deviation of the population
Benefits of using a Z score calculator
There are several benefits to using a Z score calculator, including:
• Accuracy: Z score calculators are very accurate, as they use sophisticated mathematical algorithms to perform their calculations.
• Convenience: Z score calculators can save users a lot of time and effort, as they can perform complex calculations quickly and easily.
• Flexibility: Z score calculators can be used to calculate the Z score of any value, regardless of the distribution of the population.
• Versatility: Z score calculators can be used in a variety of fields, including statistics, psychology, and economics.
Interesting facts about Z scores
• Z scores are normally distributed, meaning that the majority of values have Z scores close to zero.
• A Z score of zero indicates that the value is equal to the mean of the population.
• A positive Z score indicates that the value is above the mean of the population.
• A negative Z score indicates that the value is below the mean of the population.
Scholarly references
• David G. Moore: The Basic Practice of Statistics, 7th Edition, W.H. Freeman & Company, 2018
• Richard J. Larsen and David A. Marx: An Introduction to Statistical Methods and Applications, 8th Edition, Pearson, 2016
• Gerald C. Mosteller and Frederick Mosteller: Data Analysis and Regression: A Second Course in Statistics, Addison-Wesley, 1988
Z score calculators are a valuable tool for anyone who needs to calculate the Z score of a given value. They are accurate, convenient, flexible, and versatile. Z score calculators can be used in a
variety of fields, including statistics, psychology, and economics.
Example of using a Z score calculator
Let’s say you are a student and you want to know how your score on a test compares to the rest of the class. You know that the mean score on the test was 75 and the standard deviation was 10.
To calculate your Z score, you would enter the following information into a Z score calculator:
• Value to calculate the Z score for: Your score on the test
• Mean of the population: 75
• Standard deviation of the population: 10
The calculator would then display the following result:
Z score = (x - μ) / σ = (Your score on the test - 75) / 10
For example, if you scored 85 on the test, your Z score would be:
Z score = (85 - 75) / 10 = 1
A Z score of 1 indicates that your score was one standard deviation above the mean.
Z score calculators can be used to compare your score on any test or assessment to the rest of the population. This can be helpful for determining how well you are doing and identifying areas where
you may need to improve. | {"url":"https://exactlyhowlong.com/z-score-calculator/","timestamp":"2024-11-12T07:06:45Z","content_type":"text/html","content_length":"151189","record_id":"<urn:uuid:ede536bb-7b49-472c-b4e7-dab866b02e49>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00471.warc.gz"} |
Is Zero an Even or an Odd Number
As a whole number that can be written without a remainder, 0 classifies as an integer. So to determine whether it is even or odd, we must ask the question: Is 0 divisible by 2?
A number is divisible by 2 if the result of its division by 2 has no remainder or fractional component—in other terms, if the result is an integer. Let’s break that down. When you go about dividing a
number, each part of an equation has a specific purpose and name based on what it does. For example, take a simple division by two: 10÷2=5. In this division statement, the number 10 is the dividend,
or the number that is being divided; the number 2 is the divisor, or the number by which the dividend is divided; and the number 5 is the quotient, or the result of the equation. Because the quotient
of this division by 2 is an integer, the number 10 is proved to be even. If you were to divide, say, 101 by 2, the quotient would be 50.5—not an integer, thereby classifying 101 as an odd number.
So, let’s tackle 0 the same way as any other integer. When 0 is divided by 2, the resulting quotient turns out to also be 0—an integer, thereby classifying it as an even number. Though many are quick
to denounce zero as not a number at all, some quick arithmetic clears up the confusion surrounding the number, an even number at that. | {"url":"https://www.thesoloreads.com/2019/01/is-zero-even-or-odd-number.html","timestamp":"2024-11-05T16:14:45Z","content_type":"application/xhtml+xml","content_length":"467087","record_id":"<urn:uuid:1460091d-ccb2-4cad-b312-4e0fcd492238>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00460.warc.gz"} |
Wave-Particle Duality, Uncertainty Relations, Etc.
Wave-particle duality was an early but important concept in standard quantum mechanics, and turns out to be a core feature of our models, independent even of the details of particles. The key idea is
to look at the correspondence between spacelike and branchlike projections of the multiway causal graph.
Let us consider some piece of “matter”, ultimately represented as features of our hypergraphs. A complete description of what the matter does must include what happens on every branch of the multiway
graph. But we can get a picture of this by looking at the multiway causal graph—which in effect has the most complete representation of all meaningful spatial and branchial features of our models.
Fundamentally what we will see is a bundle of geodesics that represent the matter, propagating through the multiway causal graph. Looked at in terms of spacelike coordinates, the bundle will seem to
be following a definite path—characteristic of particle-like behavior. But inevitably the bundle will also be extended in the branchlike direction—and this is what leads to wave-like behavior.
Recall that we identified energy in spacetime as corresponding to the flux of causal edges through spacelike hypersurfaces. But as mentioned above, whenever causal edges are present, they correspond
to events, which are associated with branching in the multiway graph and the multiway causal graph. And so when we look at geodesics in the bundle, the rate at which they turn in multiway space will
be proportional to the rate at which events happen, or in other words, to energy—yielding the standard E ∝ ω proportionality between particle energy and wave frequency.
Another fundamental phenomenon in quantum mechanics is the uncertainty principle. To understand this principle in our framework, we must think operationally about the process of, for example, first
measuring position, then measuring momentum. It is best to think in terms of the multiway causal graph. If we want to measure position to a certain precision Δ x we effectively need to set up our
detector (or arrange our quantum observation frame) so that there are O(1/Δ x) elements laid out in a spacelike array. But once we have made our position measurement, we must reconfigure our detector
(or rearrange our quantum observation frame) to measure momentum instead.
But now recall that we identified momentum as corresponding to the flux of causal edges across timelike hypersurfaces. So to do our momentum measurement we effectively need to have the elements of
our detector (or the pieces of our quantum observation frame) laid out on a timelike hypersurface. But inevitably it will take at least O(1/Δ x) updating events to rearrange the elements we need. But
each of these updating events will typically generate a branch in the multiway system (and thus the multiway causal graph). And the result of this will be to produce an O(1/Δ x) spread in the
multiway causal graph, which then leads to an O(1/Δ x) uncertainty in the measurement of momentum.
(Another ultimately equivalent approach is to consider different foliations, and to note for example that with a finer foliation in time, one is less able to determine the “true direction” of causal
edges in the multiway graph, and thus to determine how many of them will cross a spacelike hypersurface.)
To make our discussion of the uncertainty principle more precise, we should consider operators—represented by sequences of updating events. In the
And with this setup we can see why position and momentum, as well as energy and time, form canonically conjugate pairs for which uncertainty relations hold: it is because these quantities are
associated with features of the multiway causal graph that probe distinct (and effectively orthogonal) directions in multiway causal space. | {"url":"https://www.wolframphysics.org/technical-introduction/potential-relation-to-physics/wave-particle-duality-uncertainty-relations-etc/","timestamp":"2024-11-07T09:23:24Z","content_type":"text/html","content_length":"33569","record_id":"<urn:uuid:77fd71b6-a883-48e7-b61d-acceebd5f143>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00780.warc.gz"} |
Subject-based archive | Advanced Search | Total contents: 8 | Total contents visits: 148,087 |
January 19, 2011 -
Full Name: Dr. Jahed, Hamid Position: Associate ProfessorPhone: 98-21-77240540-50 Ex:2994Fax: 98-21-77240488Email: hjahedmo AT iust.ac.irAddress: Iran University of Science & Technology, Tehran,
IRAN University DegreesPHD, University of Waterloo, Canada MSc, University of Houston, USA BSc, University of Houston, USA Awards & HonorsDistinguished MSc Thesis supervisor in solid mechanics,
Awarded by Iranian Society of Mechanical Engineers, May 2005, Isfahan, Iran. Supervisor of the “Scaform” project, First prizewinner of the fifth Youth Kharazmi Award (The most prestige Iranian
national undergraduate award), December 2003, Tehran, Iran. Visiting Scholar, Department of Mechanical & Industrial Engineering, Ryerson University, July 2003-January 2004. Outstanding Researcher of
the year Award, in mechanical engineering, 2002, Iran University of Science & Technology, Tehran, Iran. Distinguished Chair of Department Award, Iran University of Science & Technology, 2001.
Supervisor of the "GREEN 2" hybrid HPV project, First prizewinner of the Second Youth Kharazmi Award (The most prestige Iranian national undergraduate award), 2000, Tehran, Iran. Outstanding
Achievement in Graduate Studies, October 25, 1997, the University of Waterloo. The 1996 Sanford Fleming Foundation Teaching Assistantship Award for the Mechanical Engineering Department, Waterloo
Chapter, November 1996. Mechanical Engineering Department Teaching Assistantship Award, University of Waterloo, Ontario, CANADA, for 4 academic terms (for TA in Machine Design and MODS II).
University of Waterloo Graduate Scholarship Award, University of Waterloo, Ontario, CANADA, for 8 academic terms from fall 1994 to winter 1997. Ministry of Culture and Higher Education’s PhD
Scholarship award, Tehran, Iran, 1993. Dean of Engineering honors list, University of Houston, Texas, USA, 1981. Current ResearchFailure analysis of turbo jet rotating parts. FEM code development
for finite deformation based on objective rates. Development of robust solution in Plasticity and Creeps. ExperiencesChair, Mechanical Engineering Department, IUST, Iran, 1999-2002 Head of planning
committee for development of a 2 year MSc program in automotive engineering, IUST, Iran, 2000-2001. (This led to the establishment of the department of automotive engineering at IUST in 2001) Head of
solid mechanics group, IUST, Iran, 2002- July 2004. Head of solid mechanics group, IUST, Iran, 1997-1999 Instructor, faculty of engineering, Ferdosi University of Mashad, Mashad, Iran 1984-1986
Instructor, Mechanical Engineering Department, IUST, Tehran, 1986-1993 Fields of InterestFailure Analysis. Robust solution in Plasticity Finite deformation FEM. Graduate Theses SupervisedEshraghi,
A, “Damage Model for Ductile Tearing in Pipe Line Steels,” (PhD), expected year of graduatin, 2008 Noban, M, “A Cyclic Plasticity Model for Non-Proportional Loading,” (PhD),expected year of
graduation 2006. Paryab, N, “Reliability and fatigue of a two Arm thermal actuator”, (MSc), Expected year of graduation, 2006 Khalalaji, E, “Energy-based fatigue properties for life prediction of
variableamplitude loading”,(MSc), Expected year of graduation, 2006. Ghilichi, R, “strength and fatigue analysis of CNG upgraded ‘Shahab Khodro’ urban bus frame”, (MSc), 2005. Karimi, M, “design and
testing of a laboratory size HIP equipment’s vessel”, (MSc), 2005. Eshraghi, A, “optimum wheel profile for Iranian commuter trains and railways”, (MSc), 2005. Ghafoori, R, “the effect of residual
stress induced by forming process on a vehicle rim fatigue life”, (MSc), 2004. Alizadeh, A, “Dynamic response of a tube under moving pressure”, (MSc), 2004. Rezae, S, “single and double swage
analysis using actual unloading behavior”, (MSc), 2004. Hosseini, M, “fatigue life prediction in autofrettage tubes”, (MSC), 2003. Bidabadi, J, “ A general axisymmetric method for creep analysis”,
(MSc), 2002. Faritus, M, “residual stress measurements of an autofrettage tube using hole drilling method”, (MSc), 2003. Mehrabian, M, “optimum design of rotating discs”, (MSc), co-supervisor:
professor B.Farshi, 2002. Khosnavaz, H, “flow induced high cycle fatigue analysis of J79 turbine blade”, (MSc), 2001. Ghanbari, G, “actual unloading behavior and its significance in residual stress
predictions of autofrettage tubes,” (MSc),2001. Ghasemi, A, “2-D large deformation analysis- A finite element code,” (MSc), 2001. Arjangian, M, “plasticity estimation ahead of crack tip“, (MSc),
2000. Noban, M, “strength and fatigue analysis of Toloo4 turbine blisk“, (MSc), 2000. Joulaee, N, “creep analyses of a rotating disc at elevated temperature”, (MSc), 1999. Sherkati, S,
“elastic-plastic analysis of rotating discs using VMP method”, (MSc),1999. Publications - Journal PapersJahed, H., Farshi, B. and Bidabadi, J., 2005, “Minimum Weight Design of inhomogeneous Rotating
Discs,” International Journal of Pressure Vessels and Piping, v 82, p 35; Farshi B, Jahed H, Mehrabian A, 2004, “Optimum design of inhomogeneous non-uniform rotating discs,” Computers and Structures,
v 82, n 9-10, p 773; Jahed H, Heydari Z, Hosseininia M, 2004, “Optimum pressure for Autofrettage with Bauschinger effect consideration,” 2004, International Journal of Engineering Sciences, v 17, n
3, p 123; Jahed H, Ahmadian H, Khoshnavaz H, 2004, "High Cycle Fatigue life Prediction of a Turbine Rotor Blade under Resonance Stresses," Accepted for publication in the International Journal of
Engineering Science. Jahed, H., Ghanbari G, 2003, "Actual Unloading Behaviour and its Significance on Residual Stress in Machined Autofrettage Tubes,” Journal of Pressure Vessels Technology,
Transaction of ASME, Vol. 125, pp 321-325; Jahed, Hamid, Bidabadi, Jalal, 2003, “An axisymmetric method of creep analysis for primary and secondary creep,” International Journal of Pressure Vessels
and Piping, v 80, n 9, p 597-606; Jahed H, 2002, "Mechanical Engineering Education in Iran: Present Situation," Iranian Journal of engineering education, Iran Science Academy, Vol. 14, pp 27-39;
Jahed, H., Shirazi, R, 2001, “Loading and unloading behaviour of a thermoplastic disc,” International Journal of Pressure Vessels and Piping, v 78, n 9, p 637; Jahed, H., Lambert, S.B., Dubey, R.N.,
2000, “Variable material property method in the analysis of cold-worked fastener holes,” Journal of Strain Analysis for Engineering Design, v 35, n 2, pp 137; Jahed H., Ramezani, M., 2000, “Residual
stress Field Prediction in Cold Worked Fastener Holes,” International Journal of Engineering Sciences, Vol. 11, No. 3, pp. 165-185; Jahed, H., Lambert, S. B. and Dubey R. N., 1998, “Total Deformation
Theory For Nonproportional Loading,” International Journal of Pressure Vessels and Piping, 75, pp. 633-642. Jahed, H., Sethuraman, R. and Dubey R. N., 1997, “A Variable Material Property Approach for
Solving Elastic-Plastic Problems,” International Journal Pressure Vessels & Piping, Vol 72, pp. 285-293; Jahed, H., and Dubey, R. N., 1997, “An Axisymmetric Method of Elastic-Plastic Analysis Capable
of Predicting Residual Stress Field,” Journal of Pressure Vessels Technology, Transaction of ASME, Vol 119, pp. 264-273; Casey J, Jahedmotlagh H, 1984 “S-D Effect in Plasticity,” International
Journal of Solids Structures, Vol. 20, No. 4, pp. 377-393. Publications - Conference ProceedingsJahed H, Farshi B and Karimi M, 2005, “Optimum Design of Multi layer Vessels,” to be presented at ASME
PVP 2005 conference in Denver, Colorado, July 2005. Jahed H, Ahmadi B and Shambouli M, 2005, “Re-autofrettage, A fatigue life enhancement method,” Proceedings of the 2nd GT conference, Oxford,
England. Jahed H, Farshi B and Karimi M, 2005, “Combined Autofrettage and Shrinkfit in Multi-layer Cylinders Assembly,” Proceedings of the 2nd GT conference, Oxford, England. Jahed H and Varvani A.,
2004, “Plasticity in Low Cycle Fatigue Analysis,” Proceedings of the 7th International Conference on Biaxial/Multiaxial Fatigue & Fracture, Berlin, pp 141-7. Jahed H, Tajik A, Shamae K and Rad S,
2004, “Rapid Prototyping of Starch- Based Bone Tissue Engineering Scaffolds,” International Conference on Biomedical Engineering in Kuala Lumpur, Malaysia, September 2004. IFBME Proc. 7: 369-372.
Jahed H, Rezae S, 2004, “Double Swage Autofrettage Analysis using Actual Material Unloading Behavior,” Proceedings of the 8th International Conference of ISME, Tarbeyat Modares University, Tehran. p
323. Jahed H, Tajik A, Shamae K and Rad S, 2004, “Scaffold Rapid Prototyping,” Proceedings of the 8th International Conference of ISME, Tarbeyat Modares University, Tehran, p 211. Jahed H, Noban M,
2004, “A robust method for analysis of non-proportional loading,” Proceedings of the 8th International Conference of ISME, Tarbeyat Modares University, Tehran, p 87. Jahed H, Hosseini M, 2004, “Gun
tubes fatigue life prediction,” Proceedings of the 8th International Conference of ISME, Tarbeyat Modares University, Tehran, p 306. Jahed H, Ghafouri R, 2004, “Residual stress induced by deep
drawing on fatigue life of rim,” Proceeding of the 2nd conference on material engineering, Sharif university, p 48. Jahed H, Nasr A and Eshraghi A, 2004, “Parameters influencing wear in Iranian state
railways,” proceedings of the 7th International conference of rail ways, Sharif university, p 57. Jahed H, Tajik A, Shamae K and Rad S, 2004, “Solid Freeform Fabrication of Bone Tissue Engineering
Scaffolds,” 6th International Symposium on Computer Methods in Biomechanics, Madrid, Spain. Jahed H, Ahmadian H, Khoshnavaz H, 2003, "HCF of a Turbine Rotor Blade," Proceedings of Fourth AERO
Conference in Iran, Aerospace Structures Vol., pp 601-611. Jahed H, Noban M, 2003, “Strain Field and History of Nonproportional Loadings,” Proceedings of Fatigue Damage 2003 conference, Toronto, pp
230. Jahed H, Bidabadi J, 2003, "First and Second Stage Creep Analysis of a Rotating Disc using Time and Strain Hardening Laws," Proceedings of Fourth AERO Conference in Iran, Aerospace Structures
Vol., pp 611-622. Jahed H, Rahmani O, 2003, "Hot Forging Simulation Based on the Anand Theory using ANSYS," Proceedings of SMEI 2003 Conference, pp. 545-553. Jahed H, Hamzee M, 2002, "Thermoelastic
and Modal Analysis of Paykan Valve," Proceedings of The Second International Conference on Internal Combustion Engines, Tehran Iran, pp. 122-130. Jahed H, Hamzee M, 2002, "Two Spring Valve Design for
Paykan," Proceedings of The Second International Conference on Internal Combustion Engines, Tehran Iran, pp. 130-139. Jahed H, Karimi M, 2002, "Deep drawing Simulation of a Vehicle Rim," Proceedings
of First Metal and Material Forming, Sharif University, Tehran, Iran, pp. 3-13. Jahed H, Eshraghi A, 2002, "Finite Elastoplastic Deformation Simulation of 2-D Problems," Proceedings of First Metal
and Material Forming, Sharif University, Tehran, Iran, pp. 25-39. Farshi B, Jahed H, Mehrabian A, 2002, "Optimum Design of Rotating Disc at High Temperatures," Proceeding of Sixth International
Conference of Iranian Society of Mechanical Engineers, ISME 2002, KN University, Tehran, Iran, Vol. 4, pp.2048-2056. Jahed H, Noban M, 2002, " A Fatigue Software Interface," Proceeding of Sixth
International Conference of Iranian Society of Mechanical Engineers, ISME 2002, KN University, Tehran, Iran, Vol. 4, pp.2305-2313. Jahed H, 2002, "Mechanical Engineering Educational Requirements in
Iran Immediate Future," Invited Keynote Speaker, Sixth International Conference of Iranian Society of Mechanical Engineers, ISME 2002, KN University, Tehran, Iran. Jahed H, Ghanbari G, 2002, "Actual
Unloading Curve Measurements and its Significant in Residual Stress Predictions, Proceedings of Gun Tube Conference, Guns and Pressure Vessels: Design, Mechanics and Materials, Keble College, Oxford,
England. Jahed H, Mehrabian A and Motlagh S, 2001, "Buckling of Cylindrical Shells with Opening and Reinforced Frames," Proceedings of First International Conference of Iranian Aerospace Engineers,
Sharif University, Tehran, Vol. 3, PP. 1045-1057. Jahed H, Shahidi Z and Shirian M, 2001, " Elasto-Plastic Analysis using Elastic Results of Aerospace Components," Proceedings of First International
Conference of Iranian Aerospace Engineers, Sharif University, Tehran, Vol. 3, PP. 985-995. Moalef H, Jahed H, and Arjomand N, 2001, " Static and Dynamic Analysis of the First Three Stages of Toloo 4
Compressors," Proceedings of First International Conference of Iranian Aerospace Engineers, Sharif University, Tehran, Vol. 3, PP. 1259-1269. Jahed H., Shamsaie H., 2001, “Residual Stress in Rotating
Disc with Variable Thickness at High Temperature,” Proceeding of fifth International Mechanical Engineering Conference of Iranian Society of Mechanical Engineers,ISME 2001, Guilan, Iran, Vol. 4, pp
Jahed H., Joulae N., 2001, “A Method for Creep Analysis of Rotating Discs,” Proceedings of 6th Biaxial/Multiaxial fatigue and Fracture, Lisbon, Portugal, Vol. 2, pp 425-431. Jahed H., Shirazi R.,
2001, “Loading and Unloading Behavior of a Thermoplastic Disc,” International journal of Pressure Vessel and Piping, Vol. 78, No. 9, pp. 637-645. Jahed H, Kazemkhani A, Asef A, Zamanisafa A, 2001, "
Design of a Human Hybrid Vehicle," Proceedings of First Hybrid Vehicles Conference, Iran University of Science & Technology, Tehran, Iran, pp.133-143. Jahed, H. & S. Sherkati, 2000, “Thermoplastic
Analysis of Inhomogeneous Rotating Disk with Variable Thickness,” Proceedings of Fatigue 2000 Conference, Fatigue & Durability Assessment of Materials, Components and Structures, Eds. Bache,
Blakemore, Draper, Jahed H, Arjomand N, 2000, "Residual Stresses in fitted Rotating Discs," Proceedings of 4th International Conference of Iranian Society of Mechanical Engineers, ISME 2000, Sharif
University, Tehran, Vol. 1, pp. 25-33. Jahed, H., Lambert, S. B. And Dubey R. N., 1998, “A Stress Based Deformation Theory for Linear Nonproportional Loading,” Proceedings of Six Annual Conference of
Iranian Society of Mechanical Engineers, Vol. 4, pp 287- 293. Jahed, H., 1998, “Autofrettage,” Proceedings of Third International Conference of Mechanical Engineering, IUST, Tehran, Vol. 4, pp.
361-369. R. N. Dubey, H. Jahed and A. Kumar, 1997, “A Technique for Solving Nonlinear Problems,” ANS Symposium, Trends in Structural Mechanics, Eds. J. Rorda & N. K. Srivastava, pp. 15-21, Waterloo,
CANADA. Jahed, H., Sethuraman, R., and Dubey R. N., 1996, “Variable Material Properties Approach in Elastic-Plastic Solution of Thick-Walled Cylinders,” Mechanics in Design, Ed. S. A. Meguid,
Toronto, pp. 187-197. Jahed, H., and Dubey R. N., 1996, “Residual Stresses Calculation in Autofrettage Using Variable Material Properties Method,” ASME PVP-Vol 327, Residual Stress in Design,
Fabrication, Assessment and repair, pp. 181-188 Jahed, H., and Dubey R. N., 1996, “A Consistent Inelastic Formulation Analogous to Elastic Problems with Variable Coefficients,” ASME PVP-Vol 343,
Development Validation, and Application of Inelastic methods for Structural Analysis and Design, pp. 215-223 Jahed, H., 1992, “A Way of Defining Overall Moduli in Orthotropic Materials,” Proceeding
of the First annual Conference of Iranian Society of Mechanical Engineers (ISME), p. 49. Publications - OthersJahed, H, Noban M, Proportional treatment of nonproportional loading, Chapter in
Advances on Fracture and Damage Assessment of Materials, Wessex University publishing Co, to appear in 2005. Jahed, H., Noban M., Eshraghi A., ANSYS Finite Elements, University of Tehran publishing
Co. 1000 pages, 2000 copies, First edition, August 2003; 3000 reprints, Nov 2004. Jahed, H., Noban M., Eshraghi A., ANSYS, how to get started, IUST publishing Co., 400 pages, 7000 copies (in three
editions), 2001.
January 19, 2011 -
Full Name: Dr. Gohari, Ali reza Position: Associate ProfessorPhone: 98-21-77240540-50 Ex:2953Fax: 98-21-77240488Email: gohari AT iust.ac.irAddress: Iran University of Science & Technology,
Tehran, IRAN University DegreesPHD, University of Wales, U.K. MSC, Nuclear Technology Survey University, England. BSC, Iran University of Science and Technology. Fields of InterestFracture Mechanics
Machine Design Finite Elements
January 19, 2011 -
Prof. Shokrieh
Full Name: Prof. Shokrieh, Mahmood M Position: ProfessorPhone: 98-21-7720-8127 & 98-21-77240540-50 Ex:2914Fax: 98-21-7749-1206 & 98-21-77240488Email: shokrieh AT iust.ac.irAddress: Iran University
of Science & Technology, Tehran, IRAN University DegreesPHD, McGill University, Canada. MSC, McGill University, canada. BSC, Iran University of Science & Technology Awards & HonorsSupervision of the
Best Bachelor Thesis of the year 2005, Iranian Society of Mechanical Engineers. Best Research Project, National Iranian Gas Company, (2004). Scientific Development Award, Khatam Festival, Iran
University of Science and Technology, (2004). Best Research Project, Iran Helicopter Support and Renewal Co (Panha Co), (2003). Supervision of the Best Masters Thesis of the year 2002, Iranian
Society of Mechanical Engineers. Best Masters Thesis of the year 2002, Kharazmi Festival Distinguished paper of Islamic Republic of Iran Broadcasting Conference (2001). Distinctive Researcher,
Mechanical Engineering Department, Iran University of Science and Technology, (2000). Selected Professor, Mechanical Engineering Department, Iran University of Science and Technology, (1999).
Selected Professor, Iran University of Science and Technology. (1998). Post Doctorate, National Science and Engineering Research Council of Canada, (1995-1996). Current ResearchImpact on Composite
Structures. Enviromental Effects on Composite Structures. Creep on Composite Materials. ExperiencesMember of Professors Promotion Committee of Iran University of Science and Technology, (2004-Now).
Research Deputy of Mechanical Engineering Dept., Iran University of Science and Technology (1999 - 2000). Chairman of Iran Composites Institute (1999 - Now). Scientific Consultant of Fajr Research
Center (1996 - 2002). Scientific Consultant of Presidency Technology Cooperation Office (1996 - Now). R& D Manager, Sap Institute, (1997 – Now). Chief Editor of Composites Bulletin, Iran Composites
Institute (2001 - Now). Saba Research Center, Tehran, Iran, (R&D Manager), (1988-1990). Saba Research Center, Tehran, Iran, (Solid Mechanics Researcher), (1986-1988). Saba Research Center, Tehran,
Iran, (Solid Mechanics Designer), (1985-1986). Fields of InterestMechanics of Composite Materials. Fatigue of Composite Materials. Finite Element Analysis. Funded Research ProjectsDesign and
manufacturing of a composite shell under external pressure, Satari Co., 2005. Design and manufacturing of composite pressure vessel, Eslami Co., 2005. Reinforcement of corroded gas pipes, National
Iranian Gas Co., 2003. Design and manufacturing of Composite belts of fiber optic manufacturing machine, Ghandi Co., 2004. Reinforcement of the heritage buildings, Iranian Ministry of Science,
Research and Technology, 2005. Reinforcement of the engine hood of Toyota car with composites, Fath Automaker Co., 2002. Design and manufacturing of the canopy of a helicopter, Havanirooz, 2002.
Reinforcement of the engine hood of Land Rover car with composites, Moratab Car Manufacturer Co., 2002. Technology transfer of composite wind turbine blade from Vesttas Co. to Sadid Sabaniroo, 2002.
Design and manufacturing of an electrical switch box, Tavanir Co., 2002. Design and manufacturing of wind shield of a helicopter, 2001. Design and manufacturing of a drop weight impact tester,
National Research Projects, 2001. Residual stresses in composite materials, Fajr Aviation and Composites Industry, 2001. Establishment of a design and standardization office, Sap Co., 1999. Design
and manufacturing of a composite helicopter nose, Iran Helicopter Support and Renewal Co (Panha Co), 2000. Design of a composite pressure vessel, Karimi Co., 1999. Sensitivity analysis of composite
materials, Karimi Co., 1999. Publications - Journal PapersShokrieh, M. M., and Kamali Shahri, S. M “Theoretical and Experimental Studies on Residual Stresses in Laminated Polymer Composites,”
Journal of Composite Materials, 2004. Accepted M. M. Shokrieh and Roham Rafiee, "Simulation of Fatigue Failure in a Full Composite Wind Turbine Blade," Composite Structures, Accepted 2005. Mahmood M.
Shokrieh and Meysam Rahmat, “On the Reinforcement of Concrete Sleepers by Composite Materials,” Composite Structures, Accepted 2005. Shokrieh, M. M., and Jamal Omidi, M., “Reinforcement of Metallic
Plates with Composite Materials,” Journal of Composite Materials, Vol. 39, No. 8, pp. 723-744, 2005. Hosseinzadeh, R. Shokrieh, M. M., and Lessard, L. B., “Parametric Study of Automotive Composite
Bumper Beams in Low-Velocity Impacts,” Composite Structures, 68 (2005) 419-425 Shokrieh, M. M., and Rezaee, D., “Replacement of a Metallic Leaf Spring With a Composite Materials,” Mechanical
Engineering, Vol. 12, No. 23, March 2004, pp. 10-13. Shokrieh, M. M., and Hasani, A., “Study of Shear Buckling of Composite Shafts under Torsion,” Composite Structures, Vol. 64/1 pp. 63-69, 2004.
Shokrieh, M. M., and Rezaee, D., “Analysis and Optimization of a Composites Leaf Spring,” Composite Structures, 60, (2003), pp. 317-325 Hassani, F., Shokrieh, M. M., and Lessard, L. B., “A Fully
Non-Linear 3-D Constitutive Relationship for the Stress Analysis of a Pin-Loaded Composite Laminate,” Composites Science and Technology, 62 (2002), pp. 429-439. Shokrieh, M. M., and Sokhanvar H. R.,
“Optimum Fiber Volume Fraction of Composite Materials, using Compressive Properties,” in Persian, International Journal of Engineering Science, .No. 3, Vol. 14, pp. 59-74, 2003. Shokrieh, M. M., and
Mozafari, M., “Study of Residual Compressive Strength of a Glass/Epoxy Plate under Low Velocity Impact,” Engineering Journal of Tehran University, Vol. 36, No. 3, pp. 409-416, 2002. Shokrieh, M. M.,
and Bohlool, A., “Sensitivity of Classical Laminated Plate Theory to Variation of Mechanical Properties of Unidirectional plies,” International Journal of Engineering Science, No. 5, Vol.13, Fall
2002, pp. 81-91. Shokrieh, M. M., and Haghiri, K., “Design and Failure Analysis of Composite Pressure Vessels using Effective Parameters,” International Journal of Engineering Science, In Persian,
Vol. 12, No. 4, pp.71-80, 2002 Shokrieh, M. M., and Taheri Behrooz, F., “Wing Instability of a Full Composite Aircraft,” Composite Structures, 54, pp. 335-340. 2001 Shokrieh, M. M., and Taheri
Behrooz, F., “Study on Bending-Torsion Flutter of wing of a Full Composite Aircraft,” International Journal of Engineering Science, In Persian, Vol. 12, No. 3, pp.11-22, 2001. Shokrieh, M. M., and
Taheri Behrooz, F., “Study on Wing Bending-Aileron Rotation Flutter of a Full Composite Aircraft,” International Journal of Engineering Science, In Persian, Vol. 12, No. 1, pp.153-162, 2001.
Shokrieh, M. M., and Bohlool, A., “Sensitivity of Finite Element Method to Variation of Mechanical Properties of Unidirectional plies,” in Persian, International Journal of Engineering Science, No.
3, Vol. 13, pp. 135-143, 2002. Shokrieh, M. M., and Lessard, L. B., “Progressive Fatigue Damage Modeling of Composite Materials, Part I: Modeling,” Journal of Shokrieh, M. M., and Lessard, L. B.,
“Progressive Fatigue Damage Modeling of Composite Materials, Part II: Material Characterization and Model Verification,” Journal of Composite Materials, Vol. 34, No. 13, pp. 1081-1116, (2000).
Shokrieh, M. M., and Mohamadi-Rad H., “The Effect of Sealing Liner on Mechanical Behavior of Composite Vessels under Internal Pressure,” International Journal of Engineering Science, In Persian. Vol.
10, No. 7, pp. 59-72, 1999. Diao, X., Lessard, L. B., and Shokrieh, M. M., “Statistical Model for Multiaxial fatigue Behavior of Unidirectional Plies,” Composites Science and Technology, 59, (1999),
pp. 2025-2035. Shokrieh, M. M., and Mohamadi-Rad H., “The Effect of Internal Liner on Mechanical Behavior of Composite Pressure Vessels,” Proceedings of ICCM-12, 12th International Conference on
Composite Materials, France, 1999. Shokrieh, M. M., and Lessard, L. B., “An Assessment of the Double-Notch Shear Test for Interlaminar Shear Characterization of Unidirectional Graphite/Epoxy Under
Static and Fatigue Loading,” Applied Composite Materials, Vol. 5, No. 1, 1998, pp. 46-63. Shokrieh, M. M., and Lessard, L. B., “Modification of the Three-Rail Shear Test for Composite Materials under
Static and Fatigue Loading,” 13th Symposium on Composite Materials: Testing and Design, ASTM STP 1242, pp. 217-233,1997. Shokrieh, M. M., Eilers, O., and Lessard, L. B., “Characterization of a
Graphite/Epoxy Composite under In-Plane Shear Fatigue Loading,” High Temperature & Environmental Effects on Polymeric Composites: Second Symposium, STP 1302, pp. 133-148, 1997. Shokrieh, M. M., and
Lessard, L. B., “Multiaxial Fatigue Behaviour of Unidirectional Plies Based on Uniaxial Fatigue Experiments I. Theory,” International Journal of Fatigue, Vol. 19, No. 3, pp. 201-207, 1997. Shokrieh,
M. M., and Lessard, L. B., “Multiaxial Fatigue Behaviour of Unidirectional Plies Based on Uniaxial Fatigue Experiments II. Experimental Evaluation,” International Journal of Fatigue, Vol. 19, No. 3,
pp. 209-217, 1997. Shokrieh, M. M., and Lessard, L. B., “Effects of Material Nonlinearity on the Three-Dimensional Stress State of Pin-Loaded Composite Laminates,” Journal of Composite Materials,
Vol. 30, No. 7, pp. 839-861, 1995 Lessard, L. B., Schmit, A. S., and Shokrieh, M. M., “Three-Dimensional Stress Analysis of Free-Edge Effects in a Simple Composite Cross-Ply Laminate,” Int. Journal
of Solids and Structures, Vol. 33, No. 15, pp. 2243-2259, 1996. Lessard, L. B., Shokrieh, M. M., “Two-Dimensional Modeling of Composite Pinned/Bolted Joint Failure,” Journal of Composite Materials,
Vol. 29, No. 5, 1995, pp. 671-697. Lessard, L. B., Eilers, O. P., and Shokrieh, M. M., “Testing of In-plane Shear Properties Under Fatigue Loading,” Journal of Reinforced Plastics and Composites,
Vol. 14, No. 9, 1995, pp. 965-987. Publications - Conference ProceedingsShokrieh, M. M., H. Nosrati and M. Akhbari, “Tensile Strength Studies of Woven Fabric Polyester/ Polyester Composites,” The
8th Asian Textile Conference, Textile and Modern Sciences, p. 94, Tehran, Iran, May 9-11, 205. Shokrieh, M. M., and R. Rafiee, “Extraction of Mechanical Properties of Multidirectional Composites
using Classical Lamination Theory”, the Fifth Canadian-International Composites Conference in Vancouver, BC, Canada. Submitted and accepted Shokrieh, M. M., and M. Rahmat, “Composite Material
Reinforcement of Concrete Sleepers under 3-point Bending Conditions,” the IV-th Moscow International Conference, Theory and Practice of technologies of Manufacturing Composite Materials and New Metal
All Shokrieh, M. M., and F. Taheri B. and R. Moslemian, “Reinforcement of Corroded Gas Pipes using Composite Materials,” in Persian, 13th annual (international) conference on Mechanical Engineering
in Isfahan, Iran, P. 308, 2005. Shokrieh, M. M., and Ghasemi, A. R., “Calibration Factors Determination for Central Hole Drlling Method to Measure Residual Stresses in Isotropic and Orthotropic
Materials,” In Persian, 13th annual (international) conference on Mechanical Engineering in I Shokrieh, M. M., and R. Rafiee, “Stochastic fatigue Analysis of a Horizontal Axis Wind Turbine Composite
Blade,” Fifth International Conference on Composite Science and Technology, American University of Sharjah, 1-3 February, 2005, pp. 197-202. Shokrieh, M. M., and A. Najafi, “Damping Characterization
of Polymer Matrix Composites Based on Micromechanics Approach, Theory and Experiment,” Fifth International Conference on Composite Science and Technology, American University of Sharjah, 1-3 Februa
Najafi A., Shokrieh, M. M., , Ghannadpour, S. A. M. and Mohammadi B., “The effect of Circular/Elliptical Cutouts on the Buckling Behavior of Laminated Composites Plates,” Fifth International
Conference on Composite Science and Technology, American Univers Mohammadi, B., Ghannadpour, S. A. M., Shokrieh, M. M., and A. Najafi, “The Post Buckling analysis of Laminated Composites Containig
Circular/Elliptical Holes,” Fifth International Conference on Composite Science and Technology, American University of Shar Shokrieh, M. M., and Mohammad R. Sareban, “A Method for Internal
reinforcement of Concrete with Composite Materials Waste,” Conference on Application of FRP Composites in Construction and Rehabilitation of Structures, pp. 19-29, May 4, 2004, In Persian. Shokrieh,
M. M., and Rafiee, R., “Life Prediction of a Composite Horizontal Axis Wind Turbine Blade,” Annual International Mechanical Engineering Conference, Tehran, in Persian, pp. 240, 2004. Shokrieh, M. M.,
and Kamali Shahri, S. M., “Residual stresses in Thermoset Polymer Composites” The Second International Conference on Structural Engineering Mechanics and Computation, 5-7 July 2004, Cape Town,. South
Africa, p. 47. Shokrieh, M. M., and A. Rezaee Azariani, “Effect of Flame on the Fiber Reinforced Polymers,” Proceedings of Fire Protections of Buildings, Building and Housing Research Center, Tehran,
Iran, pp. 55-66, 2004, In Persian. Shokrieh, M. M., and Jamal Omidi, M., “Reinforcement of Aluminum Cracked Plates Using Carbon/Epoxy Composites,” Second International and Fifth National Conference
of Iranian Aerospace Society, 2004, in Persian. Shokrieh, M. M., and Kamali Shahri, S. M., “Study of Classical Method in Determination of Residual stresses in Composite Materials” Aero 2003, The
Fourth Iranian Aerospace Society Conference, pp. 382-391, 2003, In Persian. Shokrieh, M. M., and Ali Babaee S., “Design and Manufacturing of Hybrid Composite Belt for a Weaving Machine,” In Persian,
Proceedings of the Sixth Conference of Manufacturing and Production Engineering, 2003. Shokrieh, M. M., and Chahardehi A. E., “Reinforcement of Spherical Vessels using Composite Materials,”
Proceedings of the Sixth Conference of Manufacturing and Production Engineering, 2003, In Persian Shokrieh M. M., Taheri Behrooz, F, and Fadaee Arasi, M., “Measurement of Hygro-thermal Properties of
Aramid/Epoxy Composite for Cylindrical Specimens and evaluation of Feek Equation,” Proceedings of the 11 th Annual Conference (International) of Mechanica Shokrieh M. M., and Seidi, A., “Design of
Skid of Bell 214 Helicopter Using Composites,” Proceedings of the 11 th Annual Conference (International) of Mechanical Engineering,” Iran, Mashhad, pp. 1264-1270, 2003, In Persian. Shokrieh M. M.,
and Zakeri, M., “Evaluation of Fatigue Life of Composite Materials using Progressive Damage Modeling and Stiffness Reduction,” Proceedings of the 11 th Annual Conference (International) of Mechanical
Engineering,” Iran, Mashhad, pp. 1256-1 Shokrieh M. M., and Paryab, N., “Repair and Reinforcement of Gas Pipes using Composite Materials,” Proceedings of the 11 th Annual Conference (International)
of Mechanical Engineering,” Iran, Mashhad, pp. 45-52, 2003, In Persian. Shokrieh, M. M., and Hasani, A., “Shear Buckling of Composite Drive Shafts under Torsion,” The Sixth International Conference
on Flow Process in Composite Materials (FPCM-6), Auckland, New Zealand, 15-17 July, 2002. Shokrieh, M. M., and Mahmoudi, J., “Shear Creep in Composite Materials,” Proceedings of the 10th Annual
International Mechanical Engineering Conference, Tehran, Vol. 4, pp. 2321-2328, May 25-27, 2002, in Persian. Shokrieh, M. M., and Ghasemi, A., “Aerodynamic Design and Static Analysis of a Composite
Wind Turbine Blade” Proceedings of the 10th Annual International Mechanical Engineering Conference, Tehran, Vol. 4, pp. 2335-2342, in Persian, May 25-27, 2002. Shokrieh, M. M., and Mousavi
Malevajerdi S. A., “Study of Bending Behaviour of Reinforced Concrete Beams, Strengthened with Glass/Epoxy and Polyester /Epoxy Composites,” The 1st Conference & Retrofitting of Structures, Submitted
and Accepted, in Persian, Shokrieh, M. M., and Mozafari, M., “Residual Compressive Strength of Composite Plates under Impact Loading,” Third Canadian International Composites Conference, 21-24 Aug.
2001, Montreal, Quebec, Canada. Shokrieh, M. M., and Rafiee, R., “Structural Design of Composite Transmission Poles,” International Conference on Materials for Advanced Technologies (ICMAT 2001),
Singapore, 1-6 July 2001 Shokrieh, M. M., and Rafiee, R., “Design and Analysis of a Composite Transmission Pole,” In Persian, Proceedings of 5th International & 9th Annual Mechanical Engineering
Conference, Iran, Rasht, Vol. 1, pp. 77-84, 2001. Shokrieh, M. M., Seyed A. Mousavi Malevajerdi, “Strengthening of Reinforced Concrete Beams Using Composite Laminates,” International Conference on
FRP Composites in Civil Engineering, Hong Kong, 12-14 Dec. 2001. Shokrieh, M. M., and Seyed A. Mousavi Malevajerdi, “Strengthening of Reinforced Concrete Structures Using Composite Plates,” In
Persian, Concrete and Development First International Conference, pp. 235-251, April-May, 2001. Shokrieh, M. M., and Bohlool, A., “Effects of Mechanical Properties of Unidirectional Plies on Stress
Analysis of Composite Structures,” ACCM-2000, Second Asian-Australasian Conference on Composite Materials, 18-20 Aug. 2000, Korea, Kyongju, pp.535-542. Shokrieh, M. M., and Haghiri, K., “Effective
Parameters on Optimum Design of Composite Pressure Vessels,” ECCM9, Composites - From Fundamentals to Exploitation, The Premier Composites Conference in Europe, 4-7 June 2000, Brighton, England.
Shokrieh, M. M., and Sadrabadi, B. B., “Design of a Composite Telescopic Mast,” First Seminar on Design, Manufacturing, Installation and Protection of Radio and Television Masts, IRIB of Islamic
Republic of Iran, pp. 1-6, 1999. Shokrieh, M. M., and Taheri Behrooz, F., “Wing Instability of a Full Composite Aircraft,” Third International Conference on Composite Science and Technology,
Sampe-2000, Moscow, Russia, pp. 730-743, 2000 Shokrieh, M. M., Maghsoudi, M., and S. M. Tajali Bakhsh, “ A Study on Adhesive Joints of Laminated Composites,” Sixth Annual Mechanical Engineering
Conference and Third International Mechanical Engineering Conference of the Iranian Society of Mechanical E Shokrieh, M. M., and Lessard, L. B., “Residual Fatigue Life Simulation of Laminated
Composites,” International Conference on Advanced Composites 98, Egypt,, 1998, pp. 79-86. Shokrieh, M. M., “Fatigue Damage Modeling of Mechanical Joints Made of Composite Materials,” Aviation 2000,
International Symposium, Zhukovsky, Moscow, Russia, August 19-24, 1997, pp. 343-351 Shokrieh, M. M., Lessard, L. B., and C. Poon, “Three-Dimensional Progressive Failure Analysis of Pin/Bolt Loaded
Composite Laminates,” AGARD Symposium, Meeting on Bolted/Bonded Joints in Polymer Composites Florence, Italy, pp. 7-1 - 7-10, 1996. Shokrieh, M. M, and Lessard, L. B., “Progressive Fatigue Modeling
of Laminated Composite Pinned/Bolted Connections,” Proceedings of ICCM-11, 11th International Conference on Composite Materials, Australia, Vol. II: Fatigue, Fracture and Ceramic Matrix Com Lessard,
L. B., and Shokrieh, M. M., “Overview of Failure in Composite Materials,” Applied Mechanics in Americas, Vol. III, pp. 255-260., Edited by L. A. Godoy, S. R. Idelsohn, and P. A. A. Laura, and D. T.
Mook, AAM, and AMCA, Santa Fe, Argentina, 1995, Shokrieh, M. M., Eilers, O. P., and Lessard, L. B., “Determination of interlaminar Shear Strength of Graphite/Epoxy Composite Materials in Static and
Fatigue Loading,” Proceedings of ICCM-10, 10th International Conference on Composite Materials, Whistler, Milette J. F., Shokrieh M. M. and Lessard L. B., “Static and Fatigue Behaviour of
Unidirectional Composites in Compression,” Proceedings of ICCM-10, 10th International Conference on Composite Materials, Whistler, B. C., Canada, 1995, pp. I-617 -- I-624. Shokrieh, M. M., and
Lessard, L. B., “Residual Strength and Fatigue Life of Unidirectional Composite Plies under Multiaxial Fatigue Loading,” SAMPE, Atlanta, 1994, pp. 303-314 Shokrieh, M. M., and Lessard, L. B.,
“Fatigue Behaviour of Composite Pinned / Bolted Joints - New Modeling Approach,” Cancom 93, Ottawa, Canada, (1993), pp. 871-877. Lessard, L. B., and Shokrieh, M. M., Esbensen, J. H., “Analysis of
Stress Singularities in Laminated Composite Pinned/Bolted Joints,” 38th SAMPE Symposium and Exhibition, May 10-13, (1993), Anaheim, California, pp. 53-60. Lessard, L. B., Shokrieh, M. M., and Schmit,
A. S., “Three Dimensional Stress Analysis of Composite Plates With or Without Stress Concentrations,” 9th International Conference on Composite Materials, 12-16 July, (1993) Madrid, Spain, pp.
243-250. Lessard, L. B., and Shokrieh, M. M., “Failure of Laminated Composite Materials,” International Conference on Engineering Applications of Mechanics, Tehran, Iran, (1992), pp. 475-483.
Lessard, L. B., and Shokrieh, M. M., “Pinned Joint Failure Mechanisms, Part I - Two Dimensional Modeling,” First Canadian International Composites Conference and Exhibition, CANCOM 91, Montreal,
Canada, (1991), pp. 15D1-15D8. Courses TaughtDesign of Aircraft Structures Engineering Mechanics, Statics Introduction to Mechanics of Composite Materials Mechanics of Materials Theory of Elasticity
January 19, 2011 -
Full Name: Dr. Hasheminejad, Seyed Mohammad Position: ProfessorPhone: 98-21-77240540-50 Ex:2936Fax: 98-21-77240488Email: hashemi AT iust.ac.irAddress: Iran University of Science & Technology,
Tehran, IRAN CAREER INTERESTS Preference for applied mechanics-related research and education; interested in development of analytical, computational and experimental techniques applicable to the
theory and practice of Mechanical Vibrations and Structural Acoustics focusing on problems in Multiple Wave Scattering, Fluid/Structure Interaction, Poroacoustics, Underground Sound, Atmospheric
Acoustics, Underwater Acoustics, Industrial Ultrasonics, Nondestructive Testing, Medical Ultrasonics, Transduction, Stress Wave Concentration, Dynamic Thermoelasticity, Environmental Acoustics,
Building Acoustics, Noise and Vibration Control, with special attention to acoustic wave propagation and (multi-)scattering in thermoviscous, poroelastic, thermoelastic, and viscoelastic medium.
University DegreesPHD, University of Colorado, U.S.A. MSC, University of Colorado, U.S.A. BSC, University of Colorado, U.S.A. ExperiencesLecturer, Dept. of Mech. Eng., Colorado School of Mines,
Golden, Colorado, USA, (1992-1993) Lecturer, Westminster Community College, Westminster, Colorado, USA (1992-1993) Lecturer, Dept. of Aerospace Eng., Research Div., Azad Univ., Poonak, Tehran,
1999-2002. Research Assistant, Dept. of Mech. Eng., U.C. Boulder, Colorado, 1988-1992. Engineering Technician, General Instruments, Palo Alto, California, 1983-1985. Fields of InterestAcoustic Fluid
– solid interaction Low Reynolds number flow Publications - Journal PapersS. M. Hasheminejad an d M. Komeili , “ Effect of imperfect bonding on axisymmetric elastodynamic response of a lined
circular tunnel in poroelastic soil due to a moving ring loa d , ” International Journal of Solids and Structures, 2009, Vol. 46, pp.398-411 . S. M. Hasheminejad and M. Rajabi, “ Effect of FGM core
on dynamic response of a buried sandwich cylindrical shell in poroelastic soil to harmonic body waves ,” International Journal of Pressure Vessels and Piping, 2008, Vol. 85, pp.762-771 . S. M.
Hasheminejad and A.K. Miri, “ Seismic isolation effect of lined circular tunnels with damping treatments ,” Earthquake Engineering and Engineering Vibration , 2008, Vol. 7, pp.305-319 . S. M.
Hasheminejad and A. Rafsanjani , “ Steady state vibration of an FGM plate strip under a moving line load ,” Mechanics of Advanced Materials and Structures , 2008, in press . S. M. Hasheminejad and M.
Rajabi, “ Scattering and active acoustic control from a submerged piezoelectric-coupled orthotropic hollow cylinder ,” Journal of Sound and Vibration, 2008, Vol. 318, pp.50-73 . S. M. Hasheminejad
and R. Avazmohammadi , “ Dynamic stress concentrations in lined twin tunnels within fluid-saturated soil,” ASCE Journal of Engineering Mechanics, 2008, Vol. 134, pp.542-554 . S. M. Hasheminejad and
Siavash Kazemirad , “Dynamic viscoelastic effects on sound wave scattering by an eccentric compound circular cylinder , ” Journal of Sound and Vibration, 2008, Vol. 318, pp.506-526. S. M.
Hasheminejad and M. Maleki, “ Acoustic wave interaction with a laminated transversely isotropic spherical shell with imperfect bonding ,” Archive of Applied Mechanics, 2008, in press . S. M.
Hasheminejad and M. Shabanimotlagh , “Sound insulation characteristics of functionally graded panels , ” Acustica/Acta Acustica, 2008, Vol. 94, pp. 290-300. S. M. Hasheminejad and R. Sanaei, “
Ultrasonic scattering by a fluid cylinder of elliptic cross section including viscous effects ” IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control, 2008, Vol. 55, pp. 391-404. S.
M. Hasheminejad, A. Shahsavarifard, and M. Shahsavarifard, “ Dynamic Viscoelastic Effects on Free Vibrations of a Submerged Fluid-filled Thin Cylindrical Shell ,” Journal of Vibration and Control,
2008, Vol. 14, pp.849-865 . S. M. Hasheminejad and Siavash Kazemirad , “Scattering and absorption of sound by a compound cylindrical porous absorber with an eccentric core , ” Acustica/Acta Acustica,
2008, Vol. 94, pp.79-90 . S. M. Hasheminejad and H. Hosseini, “ Dynamic interaction of a spherical radiator in a fluid-filled cylindrical borehole within a poroelastic formation, ” Mechanics Research
Communications , 2008, Vol. 35, pp.158-171 . S. M. Hasheminejad and H. Hosseini, “ Nonaxisymmetric interaction of a spherical radiator in a fluid-filled permeable borehole ,” International Journal of
Solids and Structures, 2008, Vol. 45, pp. 24-47 . S. M. Hasheminejad and Siavash Kazemirad , “ Dynamic response of an eccentrically lined circular tunnel in poroelastic soil under seismic excitation,
” Soil Dynamics and Earthquake Engineering, 2008, Vol. 28, pp. 277-292. S. M. Hasheminejad and M. A. Alibakhshi, “ Eccentricity and thermoviscous effects on ultrasonic scattering from a liquid-coated
fluid cylinder ,” Journal of Zhejiang University SCIENCE , 2008, Vol. 9, pp. 65-78. S. M. Hasheminejad and M. Maleki, “ Effect of interface anisotropy on elastic wave propagation in particulate
composites ,” Journal of Mechanics , 2008, Vol. 24, pp. 77-93. S. M. Hasheminejad and A.K. Miri, “ Ultrasonic energy transfer and stress concentrations in a single-fiber composite with absorbing
interface layer, ” Journal of Thermoplastic Composite Materials , 2008, Vol. 21, pp. 473-509 . S. M. Hasheminejad and M. Maleki, “ Acoustic resonance scattering from a submerged anisotropic sphere ,”
Acoustical Physics, 2008, Vol. 54, pp. 168-179 . S. M. Hasheminejad and A. H. Pasdar, “ Computation of acoustic field by a spherical source near a thermoviscous fluid sphere ,” Journal of
Computational Acoustics, 2007, Vol. 15, pp.159-1 80. S. M. Hasheminejad and M. Rajabi, “ Acoustic resonance scattering from a submerged functionally graded cylindrical shell ,” Journal of Sound and
Vibration, 2007, Vol. 302, pp. 208-228 . S. M. Hasheminejad and R. Sanaei, “ Acoustic radiation force and torque on a solid elliptic cylinder ” Journal of Computational Acoustics, 2007, Vol. 15, pp.
377-399 . S. M. Hasheminejad and R. Sanaei, “ Effects of fiber ellipticity and orientation on dynamic stress concentrations in porous fiber-reinforced composites ,” Computational Mechanics , 2007,
Vol. 40, pp. 1015-1036 . S. M. Hasheminejad and R. Sanaei, “ Ultrasonic scattering by a spheroidal suspension including dissipative effects ” Journal of Dispersion Science and Technology, 2007, Vol.
28, pp. 1093 – 1107. S. M. Hasheminejad and A.K. Miri, “ Dynamic interaction of an eccentric multipole cylindrical radiator suspended in a fluid-filled borehole within a poroelastic formation ,” Acta
Mechanica Sinica , 2007, Vol. 23, pp. 399- 408. S. M. Hasheminejad and M. Rajabi, “ Acoustic scattering characteristics of a thick-walled orthotropic cylindrical shell at oblique incidence ,”
Ultrasonics, 2007, Vol. 47 , pp. 32-48. S. M. Hasheminejad and M. Komeili,"Dynamic response of a thick FGM tube under a moving load," Proceedings of the IMECHE, Part C, Journal of Mechanical
Engineering Science, 2007, Vol. 221, pp.1545-1554 . S. M. Hasheminejad and R. Avazmohammadi , “ Elastic wave scattering in porous unidirectional fiber-reinforced composites, ” Journal of Reinforced
Plastics and Composites, 2007, Vol. 26, pp. 495-517 . S. M. Hasheminejad and R. Sanaei, “ Acoustic scattering by an elliptic cylindrical absorber ,” Acustica/Acta Acustica, 2007, Vol. 93, pp.
789-803. S. M. Hasheminejad and M. A. Alibakhshi, “Diffraction of sound by a poroelastic cylindrical absorber near an impedance plane,” International Journal of Mechanical Sciences, 2007, Vol. 49,
pp. 1-12. S. M. Hasheminejad and R. Avazmohammadi , “ Harmonic wave diffraction by two circular cavities in a poroelastic formation, ” Soil Dynamics and Earthquake Engineering, 2007, Vol. 27, pp.
29-41. S. M. Hasheminejad and M. Azarpeyvand, “ Radiation impedance loading of a spherical source in a two-dimensional perfect acoustic waveguide ,” Acoustical Physics, 2006, Vol. 52, pp.104-115. S.
M. Hasheminejad and M. A. Alibakhshi, “Dynamic viscoelastic and multiple scattering effects in fiber suspensions,” Journal of Dispersion Science and Technology, 2006, Vol. 27, pp. 219-234. S. M.
Hasheminejad and S. Mehdizadeh , “Acoustic performance of a multilayer close-fitting hemispherical enclosure,” Noise Control Engineering Journal, 2006, Vol. 54, pp. 86-100. S. M. Hasheminejad and R.
Avazmohammadi , “Acoustic diffraction by a pair of poroelastic cylinders,” Zeitschrift fur Angewandte Mathematik und Mechanik, 2006, Vol. 8, pp. 589-605. S. M. Hasheminejad and M. Azarpeyvand, “Sound
pressure attenuation in an acoustically lined parallel plate duct containing an off-center cylindrical radiator,” Acustica/Acta Acustica, 2006, Vol. 92, pp. 417-426. S. M. Hasheminejad and R. Sanaei,
“ Ultrasonic scattering by a viscoelastic fiber of elliptic cross section suspended in a viscous fluid medium ,” Journal of Dispersion Science and Technology, 2006, Vol. 27, pp. 1165-1179. S. M.
Hasheminejad and M. A. Alibakhshi, “ Ultrasonic scattering from compressible cylinders including multiple scattering and thermoviscous effects ,” Archives of Acoustics, 2006, Vol. 31, pp. 243-263. S.
M. Hasheminejad and M. A. Alibakhshi, “ Two-Dimensional Scattering from an Impenetrable Cylindrical Obstacle in an Acoustic Quarterspace ,” Forschung im Ingenieurwesen (Engineering Research) , 2006,
Vol. 70, pp. 179-186. S. M. Hasheminejad and M. Maleki, “ Diffraction of elastic waves by a spherical inclusion with an anisotropic graded interfacial layer and dynamic stress concentrations ,”
Journal of Nondestructive Evaluation, 2006, Vol. 25, pp. 67-81. S. M. Hasheminejad and M. Maleki, “ Interaction of a plane progressive sound wave with a functionally graded spherical shell ,”
Ultrasonics, 2006, Vol. 45, pp. 165-177 . S. M. Hasheminejad and A.K. Miri, “ Effect of inter-fiber distance on energy transfer in unidirectional composites containing transverse ultrasonic waves, ”
Advanced Composites Letters, 2006, Vol. 15, pp.157-166 . S. M. Hasheminejad, “Acoustic scattering by a fluid-encapsulating spherical viscoelastic membrane including thermoviscous effects,” Journal of
Mechanics, 2005, to appear. S. M. Hasheminejad and M. Azarpeyvand, “Radiation impedance loading of a spherical source in a two-dimensional perfect acoustic waveguide,” Acoustical Physics, 2005, to
appear. S. M. Hasheminejad and S. A. Badsar, “Elastic wave scattering by two spherical inclusions in a poroelastic medium,” ASCE Journal of Engineering Mechanics, 2005, to appear. S. M. Hasheminejad
and M. Azarpeyvand, “Acoustic radiation from a cylindrical source close to a rigid corner,” Zeitschrift fur Angewandte Mathematik und Mechanik, 2005, Vol. 85, pp. 66-74. S. M. Hasheminejad and N.
Safari, “Acoustic scattering from viscoelastically coated spheres and cylinders in viscous fluids,” Journal of Sound and Vibration, 2005, Vol. 280, pp. 101-125. S. M. Hasheminejad and M. Azarpeyvand,
“Acoustic radiation from a shell-encapsulated baffled cylindrical cap,” Acoustical Physics, 2005, Vol. 51, pp. 419-427. S. M. Hasheminejad and M. Azarpeyvand, “Sound radiation due to modal vibrations
of a spherical source in an acoustic quarterspace,” Shock and Vibration, 2004, Vol. 11, pp. 625-635. S. M. Hasheminejad and Saeed Mehdizadeh, “Acoustic radiation from a finite spherical source placed
in fluid near a poroelastic sphere,” Archive of Applied Mechanics, 2004, Vol. 74, pp. 59-74. S. M. Hasheminejad and M. Azarpeyvand, “Vibrations of a cylindrical radiator over an impedance plane,”
Journal of Sound and Vibration, 2004, Vol. 278, pp. 461-477. S. M. Hasheminejad and S. A. Badsar, “Acoustic scattering by a pair of poroelastic spheres,” Quarterly Journal of Mechanics and Applied
Mathematics, 2004, Vol. 57, pp. 95-113. S. M. Hasheminejad and M. Azarpeyvand, “Sound radiation from a liquid-filled underwater spherical acoustic lens with an internal eccentric baffled spherical
piston,” Ocean Engineering, 2004, Vol. 31, pp. 1129-1146. S. M. Hasheminejad and M. Azarpeyvand, “Harmonic radiation from a liquid-filled spherical acoustic lens with an internal eccentric spherical
source,” Mechanics Research Communications, 2004, Vol. 31, no. 1, pp. 493-506. S. M. Hasheminejad and S. A. Badsar, “Acoustic scattering by a poroelastic sphere near a flat boundary,” Japanese
Journal of Applied Physics, 2004, Vol. 43, no. 5A, pp. 2612-2623. S. M. Hasheminejad and M. Azarpeyvand, “Acoustic radiation from a pulsating spherical cap set on a spherical baffle near a hard/soft
flat surface,” IEEE Journal of Oceanic Engineering, 2004, Vol. 29, no. 1, pp. 110-117. S. M. Hasheminejad and M. Najafi, "Modeling and prediction of acoustic performance of reactive mufflers based on
transmission loss approach," Amirkabir Journal of Science and Technology, 2003, Vol. 14, pp. 755-764. S. M. Hasheminejad, “Modal acoustic impedance force on a spherical source near a rigid
interface,” Acta Mechanica Sinica, 2003, Vol. 19, no. 1, pp. 33-39. S. M. Hasheminejad and M. Azarpeyvand, “Non-axisymmetric acoustic radiation from a transversely oscillating rigid sphere above a
rigid/compliant planar boundary,” Acustica/Acta Acustica, 2003, Vol. 89, pp. 998-1007. S. M. Hasheminejad and M. Azarpeyvand, “Eccentricity effects on acoustic radiation from a spherical source
suspended within a thermoviscous fluid sphere,” IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control, 2003, Vol. 50, pp. 1444-1454 S. M. Hasheminejad and M. Azarpeyvand, “Energy
distribution and radiation loading of a cylindrical source suspended within a nonconcentric fluid cylinder,” Acta Mechanica, 2003, Vol. 164, pp. 15-30. S. M. Hasheminejad and M. Azarpeyvand, “Modal
vibrations of an infinite cylinder in an acoustic halfspace,” International Journal of Engineering Science, 2003, Vol. 41, pp. 2253-2271. S. M. Hasheminejad and N. Safari, “Dynamic viscoelastic
effects on sound wave diffraction by spherical and cylindrical shells submerged in and filled with viscous compressible fluids,” Shock and Vibration, 2003, Vol. 10, pp. 339-363. S. M. Hasheminejad
and B. Harsini, “Effects of dynamic viscoelastic properties on acoustic diffraction by a solid sphere,” Archive of Applied Mechanics, 2002, Vol. 72, pp. 697-712. S. M. Hasheminejad and H. Hosseini,
“Dynamic stress concentration near on a fluid-filled permeable borehole induced by general modal vibrations of an internal cylindrical radiator,” Soil Dynamics and Earthquake Engineering, 2002, Vol.
22, pp. 441-458. S. M. Hasheminejad and H. Hosseini, “Radiation loading of a cylindrical source in a fluid-filled cylindrical cavity embedded within a fluid saturated poroelastic medium,” ASME
Journal of Applied Mechanics, 2002, Vol. 69, pp. 675-683. S. M. Hasheminejad, “Modal acoustic force on a spherical radiator in an acoustic halfspace with locally reacting boundary,” Acustica/Acta
Acustica, 2001, Vol. 87, pp. 443-453. S. M. Hasheminejad, “Modal impedances for a spherical source in a fluid-filled spherical cavity embedded within a fluid-infiltrated elastic porous medium,”
International Journal of Solids and Structures, 1998, Vol. 35, pp. 129-148 S. M. Hasheminejad and T.L. Geers, “Modal impedances for two spherical surfaces in a thermoviscous fluid,” Journal of the
Acoustical Society of America, 1993, Vol. 94, pp. 2205-2214. S. M. Hasheminejad and T.L. Geers, “Doubly asymptotic approximations for an acoustic halfspace,” ASME Journal of Vibration and Acoustics,
1992, Vol. 114, pp. 555-563. S. M. Hasheminejad and T.L. Geers, “Linear vibration analysis of an ultrasonic cleaning problem,” Journal of the Acoustical Society of America, 1991, Vol. 90, no. 6, pp.
3238-3247 S. M. Hasheminejad and T.L. Geers, 1991, "Doubly asymptotic approximations for an acoustic halfspace," in Structural Acoustics, R F Keltie, et. al., eds., NCA-Vol. 12/AMD-Vol. 128, American
Society of Mechanical Engineers, New York , pp. 129-138.DownloadGuidelines on Scholarly Paper Publication Acoustics Research Laboratory
January 19, 2011 -
Full Name: Prof. Habibnejad Korayem, Moharam Position: ProfessorPhone: 98-21-77240540-50 Ex:2904Fax: 98-21-77240488Email: hkorayem AT iust.ac.irAddress: Iran University of Science & Technology,
Tehran, IRAN University DegreesPHD, University of Wollongong, Australia. MSC, Amirkabir University of Technology BSC, Amirkabir University of Technology. Awards & HonorsDistinguished Professor of
the year Award, in Iran, Ministry of Culture and Higher Education, Tehran, Iran, 2005. Distinguished Phd Thesis supervisor in solid mechanics, Awarded by Iranian Society of Mechanical Engineers,
Isfahan, Iran, May 2005. Supervisor of the "Jaro Khakandaz Robot" project, Second prizewinner of the Second Youth Kharazmi Award, Tehran, Iran, 2000. Distinguished Researcher of the year Award, in
Iran University of Science & Technology, Iran University of Science & Technology, Tehran, Iran, 2000. Distinguished Research project "Mobile Robot", Awarded by Iran University of Science &
Technology, Tehran, Iran, 2000. Distinguished Research project "Automated Load Carrying Using AtlasII Robot", Awarded by Iran University of Science & Technology, Tehran, Iran, 2000. Distinguished
Researcher of the year Award, in Iran University of Science & Technology, Iran University of Science & Technology, Tehran, Iran, 1999. Distinguished Researcher of the year Award, in Iran University
of Science & Technology, Iran University of Science & Technology, Tehran, Iran, 1998. Outstanding Lecturer of the year Award, in mechanical engineering, Iran University of Science & Technology,
Tehran, Iran, 1997. Ministry of Culture and Higher Education’s PhD Scholarship award, Tehran, Iran, 1989. Current ResearchAutonomous wheeled mobile robot. Robotics multimedia software. Modeling,
simulation and design of industrial robot. Fields of InterestElastic Robots. Trajectory Optimization. Symbolic Modeling and Control of Robot. Dynamic of Multi-body system.
Experiences6R-RobotPublications - Journal PapersMaximum Allowable Load of Flexible Manipulator for a Given Dynamic Trajectory Application of Symbolic Manipulation to Inverse Dynamics and Kinematics
of Elastic Robot Formulation and Numerical Solution of Elastic Robotic Dynamic Motions with Maximum Load Carrying Capacities Automated Symbolic Modeling of Flexible Manipulators with Application in
Trajectory Optimization Dynamic Load Carrying Capacity of Robotic Manipulators with Joint Elasticity Imposing Accuracy Constraints Optimal Load of Robotic Manipulator with Joint Elasticity Using
Accuracy and Actuator Constraints PC-Based Symbolic Modeling and Dynamic Simulation for Flexible Manipulators Automated Fast Symbolic Modeling of Robotic Manipulators With Compliant Links The
Kinematics of Closed Loop Robotic Manipulators Using Kane’s Method Optimal Trajectory of Flexible Manipulator with Maximum Load Carrying Capacity Soft Fruit Grading And Handling: A Robotic Approach
The Effect of Friction on the Formulation and Numerical Solution of Elastic Manipulators Dynamic Analysis of Robotic Manipulators with Flexible Link Experimental Investigation and Mathematical Models
for Gas Metal Arc Welding Process The Influence of Changing a Robot’s Geometrical Parameters on its Optimal Dynamic Performance Dynamic Behaviour Analysis of a Robot Subjected to different Velocity
Trajectories Simulation and Experimental Development of Atlas II Robot A Robotic Software Package for Autonomous Wheeled Mobile Robot Kinematics and Dynamic Modeling of the Human Shoulder An
Educational Autonomous Wheeled Mobile Robot: Measurement of Accuracy Mathematical Modeling and Simulation of Differentially Wheeled Mobile Robots Dynamic Equations A New Mechanical Design
Optimization of Servomotor with Respect to Performance An Algorithm for Design of Industrial Robotic Flexible, Manipulator Maximum Allowable Load on Wheeled Mobile Manipulators Maximum Allowable Load
on Wheeled Mobile Manipulators Imposing Redundancy Constraints Implementation of PLC Simulation for Robotics Multimedia Courseware Modeling of Muscle Activity of the Shoulder Mechanism The Effect of
Base Replacement on Load Carrying Capacity of Robotic Manipulators Optimal Load of Flexible Joint Mobile Manipulators: Considering Overturning Stability Constrains Analysis of Kinematic Redundancy
and Constraints on Robotic Mobile Manipulators Maximum Allowable Load of Mobile Manipulator for a Given Two End-points of the End-effector Dynamic Load Carrying Capacity of Mobile Base Flexible Joint
Manipulators Analysis of Wheeled Mobile Flexible Manipulator Dynamic Motions With Maximum Load Carrying Capacities The Roll of Base Mobility on DLCC of Robotic Manipulator Optimal Trajectory of
Mobile Manipulators with Maximum Load Carrying Capacity Optimal Load of Elastic Joint Mobile Manipulators Imposing Overturning Stability Constraint Effect of Dynamic Balancing on the Performance
Characteristics of an Articulated Robot Vision Based Robot Simulation and Experiment for Performance Tests of Robot Design, Manufacturing and Experimental Tests of Prismatic Robot for Assembly Line
Mathematical Analysis of Kinematic Redundancy and Constraints on Robotic Mobile Manipulators Load Carrying Capacity of Flexible Joint Manipulator with Feedback Linearization Wheeled Mobile Flexible
Manipulator Dynamic Motions With Maximum Load Carrying Capacities Optimal Actuator Sizing and End Point Deformation for Mobile Robotic Manipulators with Elastic Joint Based On Load Criteria
Trajectory Optimization of Flexible Mobile Manipulators Correlation Error Reduction of Images in Stereo Vision with Fuzzy Method and Its Application on Cartesian Robot Maximum Allowable Load of
Elastic Joint Robots: Sliding Mode Control Approach Simulation of Visual Servoing Control and Performance Tests of 6R Robot Using Image-Based and Position-Based Approaches Simulation and Experimental
Tests of Robot Using Feature-Based and Position-Based Visual Servoing Approaches Maximum Dynamic Load Carrying Capacity of 6UPS-Stewart Platform Manipulator Maximum Allowable Load of Flexible Mobile
Manipulators Using Finite Element Approach Improvement of 3P & 6R Mechanical Robots Reliability and Quality Maximum Payload for Flexible Joint Manipulators in Point-to-Point Task Using Optimal
Control Approach Maximum Payload for Flexible Joint Manipulators in Point-to-Point Task Using Optimal Control Approach Investigation on Effect of Different Parameters in Wheeled Mobile Robot
Error Publications - Conference ProceedingsInertial Measurement Unit Error Handling for Mobile Robot:Expert Algorithms Approach Determination of Maximum load of Flexible Manipulators Imposing
Residual Vibration Constraint Modeling, Simulation, and Experimental Analysis of 6R Robot Dynamic simulation of nanoparticle manipulation based on AFM nano-robot Maximum Payload of Flexible
Manipulators Simulation of Nan-Manipulation: Force Analysis A Vision System Based on Neural Network for a Mobile Robot and Statistical Analyses of Its Errors Location Estimation of a Mobile Robot by
Neural Network Infulence Payload on Residual Vibation of Flexible Manipulators Simulation of Nan-norobot (AFM) Maximum Dynamic Load Carrying Capacity in Flexible Joint Robots Using Sliding Mode
Control Maximum Dynamic Load Carrying Capacity of 6UPS-Stewart Platform Flexible Joint Manipulator Application of Painting Robot in Car Industry Metal Forming and Effect of Different Parameters on
Tesion Determination of Optimal Load for a Given Path Using Optimal Control Dynamic Modeling of Nan-norobot (AFM) Optimum Design Process of Robot Using FMEA Analysis Modeling and Experimental
Analysis of Flexible Mobile Robotic Manipulator Design, Manufacture and Experimental Analysis of 3R Robotic Manipulator Maximum dynamic payload carrying capacity of 6UPS-Stewart platform manipulator
Maximum Dynamic Load Carrying Capacity in Flexible Joint Robots Using Sliding Mode Control Design & Manufacturing of a Robot Wrist: Performance Analysis Design & Manufacturing of a Robot Wrist and
Experimental Analysis Modeling, Design and Simulation of Mechanical Manipulator and Experimental Analysis Feedback Linearization of Flexible Joint Manipulator Analysis and Design of Robot for
participating in Robo Contest Dynamic Load Carrying Capacity of flexible Joint Robot with Feedback Linearization Mechanical Design and Modeling of an Omni-directional Mobile Robot Performing
Laboratory Tests For 3p Robot Using Vision Design, Modeling and Experimental Analysis of Wheeled Mobile Robots Design of Mechanism for Nut Feeder Design and Simulation of Manipulator for Carrying of
Automobile Petrol Tank Modeling, Simulation and Odometry test of Differential Mobile Robot Dynamic Load Carrying Capacity of Flexible Joint Mechanical Manipulator Training of Medium Size Robot Using
Neural Network Training of Vision System of Soccer’s Robot by Using Neural Networks Modeling, Simulation and Control of Robot by Programmable Logic Controllers Dynamic Load Carrying Capacity of
Mobile Base Flexible Robots Applied Multimedia System for Analysis and Mechanical Design of Robot Hardware Control of Mobile Robot in Anonymous Environments Effect of Base Replacement on Maximum Load
of Mechanical Manipulator Manufacturing of 3P Sealer Robot Simulation and Experimental Test of Robot Using Vision Localization of Medium Size Robot Design and Manufacturing of Prismatic Robot for
Assembly Line Design and Manufacturing of Succor Robot Application of PLC in Robotic Work cell Application of Robot in Hospital Robotic Educational Software Using Multimedia System Distributed
Modeling of Flexible Mechanical Manipulator Modelling of Prismatic Robot on Sealing Station Robotics Intelligent Data Base Feasibility Study of Glass Making Robot Application of Standard in Research
Managements Flexible, Distributed Mechanical Arm Design Mathematical Modeling and Simulation of Differentially Wheeled Mobile Robots Using Lagrange’s Approach Robotics Multimedia Software for
Conceptual Understanding Multimedia Software for Engineering Mathematics Numerical and Experimental Analysis of Flow on Car Mathematical Analysis of Constraint and Redundancy of Mobile Robots
Mathematical Modeling and Simulation Studies of Mobile Robots Effect of Mathematica on Teaching and Learning Industrial Application of PLC Path Planning of Mobile Robot Modeling and Simulation of
Autonomous Mobile Robot Sequential Integration Method for Solution of Flexible Manipulator Dynamics Kinematics and Dynamics of Flexible Manipulator Using RLS Method Intelligent Mobile Robot For
Disables Kinematics and Dynamic Modeling of the Human Shoulder: A Robotic Approach A Novel Simulation and Experimental Setup for MoboLab Robot Numerical Control of Emco Turn 220 Optimization of
Mechanical Manipulator Subject to Load Carrying Capacity Multimedia Training Materials for Mechanical Manipulator Design of Non-Holonomic Wheeled Mobile Robot Motion Planning for Mobile Robots in the
Presence of Stationary Obstacles Dynamic Equation Analysis of Atlas II Robot Design and Modelling of Gas Turbine Blade Using Rapid Prototyping CAD/CAM/SLA Modeling and Simulation of Cement Kiln
Control of Gripper Position of Atlas II Robot A Novel Experimental Set up for Atlas II Robot Using Visual Basic Computer Animation of Mechanical Manipulators Using Mathematica Maximum Load of Elastic
Joint Robot for a Given Dynamic Trajectory Light Weight Robot Dynamic Using Equivalent Rigid Link Recursive Formulation of Robot and Its Application on Maximum Load Optimum Tool Path Generation for
Machining of Sculptured Surfaces Design and Modelling of Gas Turbine Blade Using CAD/CAM/SLA Experimental Design of AtlasII Robot Using Visual Basic Design and Modelling of Gas Turbine Blade Using
Rapid Prototyping CAD/CAM/SLA Optimal Time Control of Robot for a Given Trajectory Symbolic Derivation of Dynamic Equation of Motion of Motion for Flexible Manipulators Efficient Elastic Robot
Inverse Dynamics Algorithm Using Pc-Based Symbolic Manipulation Synthesis of Flexible Manipulator Dynamic Motions with Maximum Load Carrying Capacities Effect of Dynamic of Load and Joint Flexibility
on the Recursive Lagrangian Dynamics of Flexible Arm Dynamic Load Carrying Capacity for a Multi-Link Flexible Arm Symbolic Derivation and Dynamic Simulation of Flexible Manipulators Load Carrying
Capacity for a Two-Link Planer Flexible Arm Publications - OthersKorayem, M.H., Ghariblu H., Wheeled Mobile Flexible Manipulator Dynamic Motions with Maximum Load Carrying Capacities, Chapter in
Mobile Robots: New Research, Nova Science Publishers, Inc., 2005. Korayem, M.H., Mathmatica Software and Its Engineering Application, IUST publishing Co., First edition, 1998; 2000 reprints, 2002.
Korayem, M.H., Advanced Engineering Mathematics with Mathematica and Matlab, IUST publishing Co., 2002. Korayem, M.H., Numerical Dynamic of Robot, IUST publishing Co., 2000. Korayem, M.H., Proc. of
research Managements Conference, IUST publishing Co., 2000. Korayem, M.H., Proc. International Conference of ISME98, IUST publishing Co., (in four volumes), 1998. Korayem, M.H., Robot Control and
Analysis, IUST publishing Co., 1997 Courses TaughtAdv. Eng. Mathematics II Advanced Dynamics Advanced Robotics Advanced Engineering Mathematics Dynamics Sensors and Robot Calibration
January 19, 2011 -
Full Name: Prof. Daneshjo, Kamran Position: ProfessorPhone: 98-21-77240540-50 Ex:2906Fax: 98-21-77240488Email: kdaneshjo AT iust.ac.irAddress: Iran University of Science & Technology, Tehran, IRAN
University DegreesPhD,Studied for 3.5 years at Imperial College of London , U.K. (The Viva examination held at Amirkabir University of technology,Iran Jun 1989) MSC, Imperial College of London , U.K.
BSC, Queen Mary College , U.K. Current ResearchSatellite Structural Design and Analysis Long-Rod Penetration Investigation of Sound Transmission through Orthotropic Cylindrical Shells Structure of
Electromagnetic Launchers Random Vibration Analysis Fields of InterestStructural Dynamics Modal Analysis Composite Material Finite Element Publications - Journal Papers“Numerical Study of Long
Roads Penetration Depth in Semi-infinite Target in Electromagnetic Launcher”, IEEE Trans. Magn., (ISI), vol. 41, no. 1, pp. 375-379, Jan. 2005. “Thermal Stresses in Laminated Composite Plates Based
on non-classical Theories”, CSME Transactions (ISI), Vol.28, No.2B, 2004. “Neural Optimal Control of Flexible Spacecraft Slew Maneuver”, Acta Astronautica (ISI), Vol. 55, Issue 10, November 2004,
Pages 817-827. “Classical Coupled Thermoelasticity in Laminated Composite Plates Based on third-order Shear deformation Theory”, Composite Structures (ISI), 64, 2004, 369-378. “Coupled
Thermoelasticity In Laminated Composite Plates Based on Green-Lindsay Model”, Composite Structure (ISI), Vol. 55/4, Feb 2002. “Attitude Control System of Z-SAT”, Journal of Mechanical Eng.
Transaction of the ISME, Vol. 2. No. 1, Aug 2001. “Effect of Point Mass Damper on Satellite Stability Regions”, International Journal of Eng. Science, Vol. 2, No. 2, 2001. Publications - Conference
Proceedings“Analysis of Viscoelastic Boundary Conditions on Vibration of a Beam”, 13th International Mechanical Engineering Conference, May 2005. “Sound Transmission Analysis from Cylindrical
Isotropic Shells with Infinite Length”, 13th International Mechanical Engineering Conference, May 2005. “Calculation of RMS Von-Mises Stress in Structures Under Random Vibration Using SVD Method”,
13th International Mechanical Engineering Conference, May 2005. “An Efficient Algorithm for Failure Analysis of Mechanical Structures under Random Loading”, 12th International Mechanical Engineering
Conference, May 2004. “Estimation of Propulsion System Acoustic Loads”, 12th International Mechanical Engineering Conference, May 2004. “Computation of Von-Mises Stress Distribution on Structures
Under Random Loading Using Monte Carlo Simulation”, 12th International Mechanical Engineering Conference, May 2004. “Flexible Satellite Simulator Design”, 12th International Mechanical Engineering
Conference, May 2004. “Structural Design and Modeling of an Electromagnetic Launcher”, 11th International Mechanical Engineering Conference, May 2003. “Simulation and frequency Response Calculation
of a Space Structure due to Random Vibration”, 11th International Mechanical Engineering Conference, May 2003. “Conceptual Design of Attitude Control Simulators for Rigid and Flexible Satellites”,
Second Aero Space Industries Conference, Jan 2004. “Conceptual Design of Solar Array Mechanism”, Second Aero Space Industries Conference, Dec 2004. “Design of Satellite’s Acoustic Test Room”, Second
Aero Space Industries Conference, Dec 2004. “Random Vibration Analysis of a Small Satellite”, Second Aero Space Industries Conference, Dec 2004. “Design and Fabrication of E-Glass / Silicone
Composite With Elastic Properties, Using in Expandable and Retractable Space Structures”, 6th Iranian Metallurgy Conference, Nov 2002. “Study of Al 2024 Thermomechanical Aging and it’s affects on
Satellite Structural Weight”, First Metal Forming Conference, May 2002. “Mathematical Model for Natural Frequencies of a Satellite Structure”, Iranian Aero Space Conference, May 2002. “Physical -
Mechanical Property Optimization of Tungsten - Copper Composites for Using in the Structure of a Railgun E.M.L.”, 10th International Mechanical Engineering Conference, May 2002. “Design of Coil-able
control Boom with Simple Mechanical Structure”, 10th International Mechanical Engineering Conference, May 2002. “Dynamic and Shock Analysis of a Satellite Structure”, 10th International Mechanical
Engineering Conference, May 2002. “Study of Thermomechanical Aging in Tensile Properties of Al 2024”, 5th Iranian Metallurgy Conference, Nov 2001. “Study of Infiltration Process in Optimizing the
Tungsten - Copper Composite”, 5th Iranian Metallurgy Conference, Nov 2001. “Design and Optimization of a Satellite Stabilization”, 9th International Mechanical Engineering Conference, May 2001.
“Design and Static Analysis of an EML Structure with F.E. Method”, 9th International Mechanical Engineering Conference, May 2001. “Integration of Satellite Stabilization Control System”, First
Iranian International Aero Space Conference, Jan 2001. “Design of Control Boom in Gravity Gradient Stabilization”, 8th International Mechanical Engineering Conference, May 2001. “Analytical Model of
Sound Transmission through Orthotropic Cylindrical Shells”, 13th Annual (International) Mechanical Engineering Conference May 2005 Isfahan University of Technology. Isfahan, Iran. “Developing an
Efficient Method in Small Satellite’s System Integration in the Phase of Conceptual Design”, 12th Annual (International) Mechanical Engineering Conference, May 2004. “Design, Analysis and Modeling of
an Electromagnetic Launcher (HPEL)”, 12th Annual (International) Mechanical Engineering Conference, May 2004. “Reliability and Failure Analysis of a Small Satellite under Random Loading”, 12th Annual
(International) Mechanical Engineering Conference, May 2004. “Effect of Spin Stabilized Attitude Control System on Small Satellite’s Configuration in the Conceptual Design Phase”, International
Aerospace Conference, Feb 2004. “Structural analysis of Space Vehicle using Tresca Stress Criterion in Random Vibration Environment”, International Aerospace Conference, Feb 2004. “Tribological
Problems in Space Mechanisms”, International Aerospace Conference, Feb 2004. “Formulization of Satellite Structure Conceptual Design”, 11th International Mechanical Engineering conference, May 2003.
“Design, Fabrication and Testing of a Coil-able Control Boom in Scaled Laboratory Model for a Small Satellite”, International Aerospace Conference, Feb 2003. “Design and Analysis of Z-SAT Control
System”, 10th International Mechanical Engineering conference, May 2002. “FARAGAM Algorithm in Satellite Layout”, the 6th APC-MCSTA, Sep 2001. “Design and Analysis of ZS3-SAT Structure”, 9th
International Mechanical Engineering conference, May 2001. “Satellite Simultaneous Attitude Orbit Determination via Kalman Filtering of Magnetometer Data”, 9th International Mechanical Engineering
conference, May 2001. “Effects of Orbital Perturbation on Trajectory of the L.E.O Satellite”, International Aerospace conference, Dec 2000. “Dynamic Modeling and Stability Analysis of Z-SAT”, AIAA
Modeling and Simulation Conference, August 2000. “Control Boom Design of Z-SAT”, AIAA Modeling and Simulation Conference, August 2000. “Structural Design of Z Satellite”, 5th APC-MCSTA, 1999.
Publications - Others_____ Book (In Persian)___________ An Introduction to Satellite Design Dynamic of Satellite attitude Satellite Design: Structures and Mechanisms New Methods in Finite Element
Analysis Courses Taught______Under Graduate _________ Design of Airplane Structure Mechanical Vibrations Mechanics of Material Statics _______Graduate_______________ Design of Advanced Space
Structures Finite Element Method
January 19, 2011 -
Full Name: Prof. Ayatollahi, Majid Reza Position: Professor Phone: 98-21-77 240 201 Fax: 98-21-7724 0488 Email: m.ayat AT iust.ac.ir Address: Iran University of Science & Technology, Tehran,
IRAN University Degrees Ph.D, University of Bristol, UK (1999) M.Sc., Amirkabir University of Technology, Iran (1991) B.Sc., Iran University of Science and Technology( 1988) Current Research
Investigating the effects of geometry and constraint in fracture mechanics Rock Fracture Mechanics Experimental determination of crack parameters using Photoelasticity Mixed mode brittle fracture in
cracked and notched components Experiences Reviewer for International Journals: Engineering Fracture Mechanics, Intl. Journal of Fracture, Intl. Journal of Solids and Structures, Intl. Journal of
Pressure Vessels and Piping, Intl. Journal of Rock Mechanics and Minining Science, Materials and Design , Structural Engineering and Mechanics, Fatigue and Fracture of Engineering Materials and
Structures, Meccanica, Computational Materials Science, ASME-Journal of Applied Mechanics, ASCE-Journal of Engineering Mechanics. Reviewer for National Journals: Iranian Journal of Science and
Technology, ISME Journal, IUST International Journal of Engineering Science, Modarres Journal, Amirkabir Journal, Mechanical and Aerospace Journal of Engineering, JAST, Ferdowsi University, Esteghlal
Journal. Vice Chair in Research Affairs (2000- 2008), Department of Mechanical Engineering, IUST. Member of University research council (2000-2008), Iran University of Science and Technology. Member
of committee for promotion assessment (2004-present), Department of Mechanical Engineering, IUST. Head of the Solid Mechanics educational group (2008-present), Department of Mechanical Engineering,
IUST. Director of Fatigue and Fracture Research Laboratory (2001-present), Department of Mechanical Engineering, IUST. Director of the B.Sc. projects office (1989-1993), Department of Mechanical
Engineering, IUST. Professor Since May 2007, Department of Mechanical Engineering, IUST. Associate professor 2003-2007, Department of Mechanical Engineering, IUST. Assistant professor 1999-2003,
Department of Mechanical Engineering, IUST. Fields of Interest Fracture Mechanics. Computational Solid Mechanics. Experimental Solid Mechanics. Funded Research Projects “Failure analysis of
turbine shaft in a gas power plant”, Sponsored by Iran Power Plant Repairs Co, 2004-2005. “Damage tolerance analysis of a buried water pipeline” Sponsored by Tarvand Tadbir Co, 2003-2004. (In
collaboration with Dr. N.M. Nouri) “Crack tip parameters in mode I loading”, Sponsored by the Grant Department, Research Office, Iran University of Science and Technology, 2001-2003. “Development of
a strategic plan for the Structural Integrity Research Center”, Sponsored by the Grant Department, Research Office, Iran University of Science and Technology, 2001-2003. “Mixed mode fracture in
brittle materials”, Sponsored by the Grant Department, Research Office, Iran University of Science and Technology, 2000-2002. “ Brittle fracture in cracked components subjected to shear loading”,
Sponsored by the British Energy, UK, Feb. 1998- Jan. 1999. “Effects of crack tip constraint in tension-shear loading”, Sponsored by the British Energy, UK, Feb. 1995- Jan. 1998. Graduate Theses
Supervised PHD studentsZakeri, M., “Photoelastic analysis of cracks under mode I, mode II and mixed mode loading” (PhD), 2008. Aliha, M.R.M., “Mixed mode fracture in rock materials” (PhD), 2009.
Torabi, A.R. "Brittle fracture in U and V-notched components" (PhD), 2009. Shadlou, S. (PhD student), Expected year of graduation 2011. Akbardoost, J. (PhD student), Expected year of graduation 2012.
Pirmohammad S. (PhD student), Expected year of graduation 2012. Chamani H.R. (PhD student), Expected year of graduation 2013. MSc Students (not updated)Supervised 39 MSc theses 2000-2009. Shakeri,
A. “Brittle fracture in mixed mode II/III”, (MSc), 2006. Mostafavi, M. “Effects of loading mode on warm pre-stressed cracks”, (MSc), 2006. Hashemi, R. “Fracture analysis of an angled crack reinforced
by composite patching”, (MSc), 2005. Bagherifard, S. “Analysis of brittle fracture in a ceramic test specimen”, (MSc), 2005. Ashtari, R, “A statistical modeling for brittle fracture in marble
samples,” (MSc), 2005. Hasani, M, “Prediction of fracture trajectory in mode II crack specimens,”(MSc), 2005. Rokhi, H, “Free vibration of beams containing double edge cracks,”(MSc), 2005. Raghebi,
M, “Finite element analysis of curved Timoshenko beam using mode acceleration method,”(MSc),2004. Aliha, M.R.M., “Theoretical and experimental investigation of brittle fracture in the cracked
semi-circular bend specimen,”(MSc), 2004. Nikoobin, A, “Effect of lateral load on crack tip stresses using photoelasticity”(MSc), 2004. Khojeh, A, “Vibration analysis of a plate with special crack
elements,”(MSc),2004. Alinia, A, “The effects of warm pre-stressing on brittle fracture in mode I loading,”(MSc), 2004. Shokrani, A, “Crack tip parameters in an edge crack specimen under asymmetric
four-point bending,”(MSc), 2003. Nasr Esfahani, M, “Extraction of fracture curves using strain energy density criteria,”(MSc), 2003. Asadkarami, A, “Evaluation of fracture in mode II punch-type
specimens,”(MSc), 2002 Ardestani, S, “Mechanical and thermal shocks in multilayer circular plates”(MSc), 2002. Safari, G.H., “Evaluation crack tip stresses using photoelasticity method,”(MSc), 2002.
Zanbilbaf, A, “Crack growth path in pure shear loading”,(MSc), 2002. Abbasi, H, “Prediction of fracture initiation in angled cracked specimen”,(MSc), 2001. Publications - Journal Papers 73.
Ayatollahi, M.R., Aliha, M.R.M. (2010) "Fracture analysis of some ceramics under mixed mode loading", Journal of American Ceramics Society, In Press. 72. Ayatollahi, M.R., Aliha, M.R.M. (2010) "On
the use of Anti-symmetric Four-Point bend specimen for mode II fracture experiments" Fatigue and Fracture of Engineering Materials and Structures, In Press. 71. Ayatollahi, M.R., Sedighiani, K.
(2010) “Crack Tip Plastic Zone under Mode I, Mode II and Mixed Mode (I+II) Conditions” Structural Engineering and Mechanics, In Press. 70. Ayatollahi, M.R, Torabi, A.R. (2010) “ Investigation of
mixed mode brittle fracture in rounded-tip V-notched components ” Engineering Fracture Mechanics, In Press. 69. Ayatollahi, M.R, Nejati, M. (2010) “Experimental evaluation of stress field around the
sharp notches using photoelasticity” Materials and Design, In Press. 68. Ayatollahi, M.R. , Torabi, A.R., Azizi, P. (2010) “Experimental and Theoretical Assessment of Brittle Fracture in Engineering
Components Containing a Sharp V-Notch” , Experimental Mechanics, In Press 67. Ayatollahi, M.R, Nejati, M. (2010) “An overdeterministic method for calculation of coefficients of crack tip asymptotic
field from finite element analysis”, Fatigue and Fracture of Engineering Materials and Structures, In Press. 66. Ayatollahi, M.R., Dehghany, M.(2010) “On T-stresses near V-notches” International
Journal of Fracture, Vol. 165, No 1, PP. 121-126 65. Saghafi, H. Ayatollahi, M.R., Sistaninia, M. (2010) “ A modified MTS criterion (MMTS) for mixed-mode fracture toughness assessment of brittle
materials” Materials Science and Engineering: A, Vol. 527, No. 21-22, PP. 5624-5630 64. Aliha, M.R.M., Ayatollahi, M.R., Smith, D.J., Pavier, M.J. (2010) "Geometry and size effects on fracture
trajectory in a limestone rock under mixed mode loading"Engineering Fracture Mechanics, Vol. 77, No 11, PP. 2200-2212 63. Ayatollahi, M.R, Torabi, A.R. (2010) "Tensile fracture in notched
polycrystalline graphite specimens" Carbon, Vol. 48, No 8, PP. 2255-2265 62. Ayatollahi M.R., Khoramishad, H., (2010) "Stress intensity factors for a semi-elliptical internal crack embedded in a
buried pipe" International Journal of Pressure Vessels and Piping" Vol. 87, No. 4, PP. 165-169. 61. Zakeri, M., Ayatollahi M.R., Guagliano, M. (2010) " A Photoelastic study of T-stress in centrally
cracked Brazilian disc specimen under mode II loading" Strain, Article in Press, doi: 10.1111/j.1475-1305.2009.00680.x 60. Ayatollahi, M.R., Hashemi, R., Rokhi, H. (2010) "New formulation for
vibration analysis of Timoshenko beam with double-sided cracks" Structural Engineering and Mechanics, Vol. 34, No 4, PP. 475-490. 59. Ayatollahi M.R., Torabi A.R. (2010) "Determination of mode II
fracture toughness for U-shaped notches using Brazilian disc specimen" International Journal of Solids and Structures, Vol. 47, PP. 454-465. 58. Aliha, M.R.M., Ayatollahi, M.R. (2010) "Geometry
effects on fracture behaviour of polymethyl methacrylate" Materials Science and Engineering-A, Vol. 527, PP. 526-530. 57. Ayatollahi M.R., Torabi A.R. (2010) "Brittle fracture in rounded-tip V-shaped
notches" Materials and Design, Vol. 31, PP. 60-67 56. Khoramishad, H., Ayatollahi, M.R. (2009) "Finite element analysis of a semi-elliptical external crack in a buried pipe" Transactions of the
Canadian Society for Mechanical Engineering, Vol. 33, PP. 399-409. 55. Aliha, M.R.M., Ayatollahi, M.R., Kharazi, B. (2009) "Mode II brittle fracture assessment using ASFPB specimen" International
Journal of Fracture, Vol. 159, PP. 241-246. 54. Aliha, M.R.M., Ayatollahi, M.R. (2009) “Brittle Fracture Evaluation of a Fine Grain Cement Mortar in combined tensile-shear deformation” Fatigue and
Fracture of Engineering Materials and Structures, Vol. 32, PP. 987-994. 53. Ayatollahi M.R., Torabi A.R. (2009) "A criterion for brittle fracture in U-notched components under mixed mode loading"
Engineering Fracture Mechanics, Vol. 76, PP. 1883-1896. 52. Ayatollahi M.R., Torabi A.R. (2009) Investigation of Fracture in V-notched Brittle Polymers under Pure Shear Loading, Iranian Journal of
Polymer Science and Technology, Vol. 22, PP. 63-73. 51. Ayatollahi, M.R. and Bagherifard S., (2009) "Numerical analysis of an improved DCDC specimen for investigating mixed mode fracture in ceramic
materials", Computational Materials Science, Vol. 46, PP. 180-185. 50. Ayatollahi, M.R., Aliha, M.R.M., (2009) “Analysis of a new specimen for mixed mode fracture tests on brittle materials”
Engineering Fracture Mechanics, Vol. 76, PP. 1563-1573. 49. Ayatollahi, M.R. and Arian Nik, M. (2009) “Edge distance effects on residual stress distribution around a cold expanded hole in Al 2024
alloy” Computational Materials Science, Vol. 45, PP. 1134-1141. 48. Ayatollahi, M.R., Aliha, M.R.M.(2009) "Mixed-Mode Fracture in Soda-Lime glass Analyzed by Using the Generalized MTS Criterion"
International Journal of Solids and Structures, Vol. 46, PP. 311-321. 47. Ayatollahi, M.R. and Mousavi, M.M.S. (2008) "Confining pressure effects on mode I and mode II stress intensity factors of
cracked rocks" Iranian Journal of Mining Engineering, Vol. 3, No 5, PP. 21-32. 46. Aliha, M.R.M., Ayatollahi, M.R, Pakzad, R. (2008) "Brittle fracture analysis using a ring shape specimen containing
two angled cracks" International Journal of Fracture, Vol. 153 (1), PP. 63-68. 45. Aliha, M.R.M., Ayatollahi, M.R, Mousavi, M.M.S. (2008) "Confining pressure effects on stress intensity factors: A 3D
finite element analysis" Revue Synthese, Vol. 19, PP. 33-39. 44. Ayatollahi, M.R. and Aliha, M.R.M. (2008) "On the use of Brazilian disc specimen for calculating mixed mode I-II fracture toughness of
rock materials", Engineering Fracture Mechanics, Vol. 75, PP. 4631-4641. 43. Ayatollahi, M.R. and Aliha, M.R.M. (2008) "Mixed mode fracture analysis of polycrystalline graphite - A modified MTS
criterion", Carbon, Vol. 46(10), PP. 1302-1308. 42. Aliha, M.R.M. and Ayatollahi M.R. (2008) " On mixed mode I/II crack growth in dental resin materials" Scripta Materialia, Vol. 59, PP. 258-261. 41.
Ayatollahi, M.R. and Bagherifard, S. (2008) "Brittle fracture analysis of the offset-crack DCDC specimen" Structural Engineering and Mechanics, Vol. 29, No. 3, PP. 301-310. 40. Aliha, M.R.M. and
Ayatollahi M.R. (2008) " Fracture toughness evaluation for brittle polymers under combined tensile-shear loads" Iranian Journal of Polymer Science and Technology, Vol. 21, No 2, PP. 107-117. 39.
Ayatollahi, M.R., Rokhi , H. and Hashemi, R. (2008) "Free vibration of beam with double-sided edge cracks" International Journal of Science and Technology (Amirkabir), Vol. 18, PP. 51-48. 38.
Ayatollahi M.R. and Aliha M.R.M. and Rahmanian S. (2007) "Finite element analysis of an improved center crack specimen", Key Engineering Materials, Vol. 347, PP. 441-446. 37. Khojeh, A., Ayatollahi,
M.R. and Ahmadian H. (2007) "Vibrational analysis of cracked beams using a special crack element" IUST International Journal of Engineering Science, Vol.18, No 1, PP. 11-18. (in Persian) 36.
Guagliano M., Ayatollahi M.R., Zakeri M. and Colombo C. (2007) "Experimental investigation of T-stress effects on photoelastic fringes in Brazilian disk under mode II conditions" Key Engineering
Materials, Vol 348, PP. 969-972. 35. Zakeri, M., Ayatollahi, M.R. and Nikoobin, A. (2007) “Photoelastic Study of a Center-Cracked Plate - The Lateral Load Effects”, Computational Materials Science,
Accepted for publication. 34. Ayatollahi, M.R., and Zakeri, M. (2007) "T-Stress Effects on Isochromatic Fringe Patterns in Mode II", International Journal of Fracture, Vol.143, PP. 189-194. 33.
Ayatollahi, M.R., and Aliha, M.R.M. (2007) “Fracture toughness study for a brittle rock subjected to mixed mode I/II loading”, International Journal of Rock Mechanics and Mining Science, Volume 44,
No 4, PP. 617-624. 32. Ayatollahi M.R. and Hashemi, R. (2007) "Mixed mode fracture in an inclined center crack repaired by composite patching, Composite Structures, Vol. 81, PP. 264-273. 31.
Ayatollahi M.R. and Hashemi R. (2007) “Computation of stress intensity factors (KI, KII) and T-stress for cracks reinforced by composite patching” Composite Structures, Vol. 78, PP. 602-609. 30.
Ayatollahi M.R. and Mostafavi M. (2007) “Finite element analysis of a center crack specimen warm pre-stressed under different modes of loading” Computational Materials Science, Vol. 38, No. 4, PP.
847-856. 29. Ayatollahi M.R. and Aliha M.R.M. (2007) “Wide Range Data for Crack Tip Parameters in Two Disc-Type Specimens under Mixed Mode Loading” Computational Materials Science, Vol. 38, No. 4,
PP. 660-670. 28. Hashemi, R. and Ayatollahi M.R. (2006) “The effect of hygrothermal composite patch on fracture strength of a reinforced crack” Applied Mechanics and Materials, Vol. 5-6, PP. 189-196.
27. Aliha M.R.M. , Ashtari, R. and Ayatollahi M.R. (2006) “Mode I and Mode II fracture toughness testing for a coarse grain marble” Applied Mechanics and Materials, Vol. 5-6, PP. 181-188. 26. Smith,
D.J., Ayatollahi, M. R. and Pavier, M.J. (2006) “On the Consequences of T-stress in Elastic Brittle Fracture”, Proceedings of the Royal Society A, Vol. 462, PP. 2415-2437. 25. Ayatollahi M.R. and
Mostafavi M. (2006) “Effects of Crack Tip Blunting and Residual Stress on a Warm Pre-Stressed Crack Specimen” Computational Materials Science, Vol. 37, PP. 393-400. 24. Ayatollahi, M.R. and
Aliniaziazi, A. (2006) "Effects of lateral load on warm prestressing in a center crack plate, Materials Science and Engineering – A, Vol. 441, PP. 170-175. 23. Ayatollahi M.R., Aliha M.R.M. and
Hassani M.M. (2006) “Mixed mode brittle fracture in PMMA - An experimental study using SCB specimen” Materials Science and Engineering – A, Vol. 417, PP. 348-356. 22. Ayatollahi, M.R., and Aliha,
M.R.M. (2006) “On Determination of Mode II Fracture Toughness using Semi-circular Bend Specimen”, International Journal of Solids and Structures, Vol. 43, PP. 5217-5227. 21. Ayatollahi, M.R.,
Asadkarami, A. and Zakeri, M. (2005) “Finite element evaluation of punch-type crack specimens” International Journal of Pressure Vessels and Piping, Vol. 82, PP. 722-728 20. Ayatollahi, M.R., and
Aliha, M.R.M. (2005) “Cracked Brazilian disc specimen subjected to mode II deformation” Engineering Fracture Mechanics, Vol. 72, PP. 493-503. 19. Asadkarami, A. and Ayatollahi, M.R. (2005) “E-shape
specimen for determining fracture toughness in shear loading” IUST International Journal of Engineering Science, Vol. 16, No 4, PP. 11-20. 18. Ayatollahi, M.R. and Zanbilbaf, A. (2005) “Path of crack
growth under pure shear”, IUST International Journal of Engineering Science, Vol. 16, No 2, 75-83. 17. Ayatollahi, M.R., Faritus, M.R. and Asadkarami, A. (2004) “Three-dimensional stress analysis of
two standard crack specimens” IUST International Journal of Engineering Science, Vol. 15, No. 4, PP. 97-102. 16. Ayatollahi, M.R., Smith, D.J. and Pavier, M.J. (2004) “Effect of constraint on the
initiation of ductile fracture in shear loading” Key Engineering Materials, Vol. 261, PP. 183-188. 15. Ayatollahi, M.R., and Aliha, M.R.M. (2004) “Fracture Parameters for Cracked Semi-circular
Specimen”, International Journal of Rock Mechanics and Mining Science, Vol. 41, Supplement 1, PP. 20-25. 14. Ayatollahi, M.R., Abbasi, H. (2003) “Crack growth prediction based on the maximum hoop
strain criterion- Plane stress”, IUST International Journal of Engineering Science, Vol. 14, No. 4, PP. 123-13 13. Ayatollahi, M.R. and Safari, G.H. (2003) “Evaluation of crack tip constraint using
photoelasticity”, International Journal of Pressure Vessels and Piping, Vol. 80, No. 9, PP. 665-670. 12. Ayatollahi, M.R., Pavier, M.J. and Smith, D.J. (2002) "Mode I cracks subjected to large
T-stresses", International Journal of Fracture, Vol. 117, No 2, PP. 159-174. 11. Ayatollahi, M.R., Mohammad-Aliha, M.R., Aliniaziazi, A. (2002) “Variation of nonsingular stress parameter in terms of
crack length”, IUST International Journal of Engineering Science, Vol 13, No 3, PP 173-186. 10. Ayatollahi, M.R., Smith, D.J. and Pavier, M.J. (2002) “Crack tip constraint in mode II deformation”
International Journal of Fracture, Vol. 113, No 2, PP 153-173. 9. Ayatollahi, M.R. and Abbasi, H. (2001) “Prediction of fracture using a strain based mechanism of crack growth” Building Research
Journal, Vol. 49, No 3, PP 167-180. 8. Ayatollahi, M.R., Smith, D.J. and Pavier M.J. (2001) “Effect of higher order stresses in an internal crack problem” IUST International Journal of Engineering
Science, Vol. 12, No 2, PP 53-66. 7. Ayatollahi, M.R. (2001) “On biaxially loaded internal cracks” International Journal of Fracture, Vol. 109, PP. L9-L14. 6. Smith, D.J., Ayatollahi, M. R. and
Pavier, M.J.(2001) “The role of T-stress in brittle fracture for linear elastic materials under mixed mode loading”, Fatigue and Fracture of Engineering Materials and Structures, Vol. 24, No 2, PP
137-150. 5. Ayatollahi, M.R., Pavier, M.J. and Smith, D.J. (1998) "Determination of T-stress from finite element analysis for mode I and mixed mode I/II loading", International Journal of Fracture,
Vol. 91, PP 283-298. 4. Smith D.J., Ayatollahi, M.R., Davenport, J.C.W. and Swankie, T.D. (1998) "Mixed mode brittle and ductile fracture of a high strength rotor steel at room temperature",
International Journal of Fracture, Vol. 94, PP 235-250. 3. Ayatollahi, M.R., Smith D.J. and Pavier, M.J. (1998) "A method for calculating T-stress for mixed mode problems", International Journal of
Key Engineering Materials, Vol. 145, PP 83-88. 2. Ayatollahi, M.R., Pavier, M.J. and Smith, D.J. (1996) "On mixed mode loading of a single edge notched specimen", International Journal of Fracture,
Vol. 82, PP R61-R66. 1. Eslami, M.R. and Ayatollahi, M.R., (1993) "Modal Analysis of Shell of Revolution on Elastic Foundation ", International Journal of Pressure Vessels and Piping, Vol. 56, PP
351-368. Publications - Conference Proceedings (not updated) 64. Aliha M.R.M. and Ayatollahi M.R. (2007) "Combined mode fracture toughness assessment for a high strength cement mortar" Proceedings of
DAMAS conference, Turin, Italy. 63. Ayatollahi M.R., Azad H. and Hashemi R. (2007) "Analysis of cracked pipes reinforced by composite patching" First Iranian Pipe and Pipeline Conference, Tehran,
Iran. 62. Akbardoost J., Ayatollahi M.R. and Aliha M.R.M. (2007) "Strength evaluation of oil and gas pipes under external explosive loading" First Iranian Pipe and Pipeline Conference, Tehran, Iran.
61. Aliha M.R.M. and Ayatollahi M.R. (2007) "A fracture mechanics approach for analyzing spiral weld pipes containing crack" First Iranian Pipe and Pipeline Conference, Tehran, Iran. 60. Ayatollahi
M.R. and Khorramishad H. (2007) "Stress intensity factors for cracked buried pipes - 3D finite element analysis" Proceedings of 15th Annual International Conference on Mechanical Engineering, Tehran,
Iran. 59. Ayatollahi M.R. Aliha M.R.M. and Khademi S. (2007) "On determination of tensile strength of rock materials using disk-type specimens" Proceedings of 15th Annual International Conference on
Mechanical Engineering, Tehran, Iran. 58. Ayatollahi M.R. and Khorramishad H. (2007) "Two-dimensional analysis of cracks in buried oil and gas pipes" Proceedings of Third National Conference on Civil
Engineering, Tabriz, Iran. 57. Ayatollahi M.R. and Saravani M. (2007) "Finite element analysis of cracks existing in root of a V-notch" Proceedings of Third National Conference on Civil Engineering,
Tabriz, Iran. 56. Aliha M.R.M., Ayatollahi M.R. and Ashtari R. (2007) "Experimental investigation of brittle fracture in a cracked marble specimen" Proceedings of Third National Conference on Civil
Engineering, Tabriz, Iran. 55. Ayatollahi M.R. and Tiznobeik H. (2007) "A review of numerical techniques used for photoelastic analysis of cracks", Proceeding of 6th conference of Iranian Aerospace
Society, Tehran, Iran. 54. Ayatollahi M.R. and Saravani M. (2007) "Experimental determination of stress intensity factors in cracked V-notches using photoelasticity", Proceeding of 6th conference of
Iranian Aerospace Society, Tehran, Iran. 53. Ayatollahi M.R., Rokhy H. and Hashemi R. (2007) "Free vibration analysis of a cracked Timoshenko beam" Proceeding of 6th conference of Iranian Aerospace
Society, Tehran, Iran. 52. Ayatollahi M.R. A., Shakeri and M. Zakeri (2006) "A fracture criterion for strength analysis of cracked drills under drilling torsional moments" Proceedings of the First
Iranian Petroleum Engineering Congress, Tehran, Iran. 51. Ayatollahi M.R., Aliha M.R.M. and Takaffoli, M. (2006) "Application of fracture mechanics for determination of in-situ stresses during
hydraulic fracturing process" Proceedings of the First Iranian Petroleum Engineering Congress, Tehran, Iran. 50. Aliha M.R.M. and Ayatollahi M.R. (2006) "A theoretical investigation of hydraulic
fracturing in horizontal wells using fracture mechanics approach" Proceedings of the First Iranian Petroleum Engineering Congress, Tehran Iran. 49. Ayatollahi, M.R. and Mostafavi, M. (2006) “Mode I
preloading – mode II fracture in warm prestressing” Proceedings of the 16th European Conference on Fracture- ECF16, Alexandroupolis, Greece. 48. Ayatollahi, M.R. and Aliha, M.R.M. (2006) “Predictions
of mixed mode fracture toughness for a soft rock” Proceedings of the 16th European Conference on Fracture- ECF16, Alexandroupolis, Greece. 47. Ayatollahi, M.R. and Mostafavi, M. (2006) ”Warm
pre-stressing effects on mechanism of fracture in a central angled crack specimen” Proceedings of the 10th International ISME Conference, Isfahan, Iran. 46. Ayatollahi, M.R., Aliha, M.R.M and
Poursaeedi, E (2006) ”Failure evaluation of a fractured turbine shaft in a gas power plant using finite element analysis” Proceedings of the 10th International ISME Conference, Isfahan, Iran. 45.
Ayatollahi, M.R. and Hashemi, R. (2006) “Mixed mode brittle fracture in cracked components reinforced by composite patching” Proceedings of 7th ICCE International Conference, Tehran, Iran. (In
Persian) 44. Ayatollahi, M.R. and Hashemi, R. (2006) “Optimized fiber angle for composite reinforcement of a crack on a rivet hole” Proceedings of 7th ICCE International Conference, Tehran, Iran. (In
Persian) 43. Ayatollahi, M.R. and Takaffoli, M. (2006) “Finite element modeling for hydraulic fracturing in oil and gas wells” Proceedings of 7th ICCE International Conference, Tehran, Iran. (In
Persian) 42. Ayatollahi, M.R. and Hassani, M.M. (2005) “Fracture initiation angle for a cracked plate under pure shear” Proceedings of 2nd National Conference on Thin-Walled Structures, Kerman, Iran,
PP. 301-310. (in Persian) 41. Ayatollahi, M.R. and Hashemi, R. “Effects of composite reinforcement on a thin-walled plate containing a central crack” Proceedings of 2nd National Conference on
Thin-Walled Structures, Kerman, Iran, PP. 293-300. (in Persian) 40. Ayatollahi, M.R. and Aliniaziazi, A. (2005) “Finite element analysis of a warm pre-stressed center crack plate” Proceedings of 2nd
National Conference on Civil Engineering, Tehran, Iran. (in Persian) 39. Ayatollahi, M.R. and Raghebi, M. (2005) “In-plane vibrations of curved Timoshenko beams using finite element method”
Proceedings of 2nd National Conference on Civil Engineering, Tehran, Iran. (in Persian) 38. Ayatollahi, M.R. and Hassani, M.M. (2005) “Path of crack growth in a crack specimen subjected to pure mode
II” Proceedings of 2nd National Conference on Civil Engineering, Tehran, Iran. (in Persian) 37. Ayatollahi, M.R. and Ashtari, R. (2005) ”Statistical analysis of fracture toughness results for two
concrete materials using Weibull’s model” Proceedings of the 9th International ISME Conference, Isfahan, Iran. (in Persian) 36. Ayatollahi, M.R. Zakeri, M. (2005) ”Evaluation of iso-chromatic fringes
around mode II cracks using theory of photoelasticity” Proceedings of the 9th International ISME Conference, Isfahan, Iran. (in Persian) 35. Ayatollahi, M.R. Mostafavi, M. (2005) ”Effects of
preloading level on a warm pre-stressed crack specimen” Proceedings of the 9th International ISME Conference, Isfahan, Iran 34. Ayatollahi, M.R., Aliniaziazi, A. and Mostafavi, M. (2005) “Warm
pre-stress effects on cleavage fracture in center crack specimen” Proceedings of the 11th International Conference on Fracture- ICF11, Turin, Italy. 33. Ayatollahi, M.R. Zakeri, M. and Hassani, M.M.
(2005) “On the presence of t-stress in mode ii crack problems” Proceedings of the 11th International Conference on Fracture- ICF11, Turin, Italy. 32. Ayatollahi, M.R., and Aliha, M.R.M. (2005)
“Applications of fracture mechanics in rock cutting processes in drilling operation” Iranian Drilling Conference, Arak, Iran. (in Persian) 31. Ayatollahi, M.R., Ashtari, R. and Mohammadabadi, M.
(2005) “Finite element analysis of CNSR specimen for fracture toughness testing in rock materials” The Iranian Mining Engineering Conference, Tehran, Iran, Vol. 3, PP. 1439-1450. (in Persian) 30.
Ayatollahi, M.R. and Bagherifard, S. (2005) “A review of mixed mode test specimens for rock and ceramic materials” The Iranian Mining Engineering Conference, Tehran, Iran, Vol. 3, PP. 1451-1464. (in
Persian) 29. Ayatollahi, M.R., and Aliha, M.R.M. (2004) “Fracture Initiation angle in the centrally cracked circular disk specimen” Proceedings of the 15th European Conference on Fracture- ECF15,
Stockholm, Sweden. 28. Ayatollahi, M.R. (2004) “J-T Characterization of elastic-plastic stress fields in a mode II crack specimen” Proceedings of the 15th European Conference on Fracture- ECF15,
Stockholm, Sweden. 27. Ayatollahi, M.R. and Zakeri, M. (2004) “In-plane modes of deformation in crack problems” Proceedings of the 8th International ISME Conference, Tehran, Iran. 26. Ayatollahi,
M.R. and Khojeh, A. (2004) “Effect of traverse load on fractyure behavior of a cracked specimen” Proceedings of the 8th International ISME Conference, Tehran, Iran. 25. Ayatollahi, M. R. and
Nikoobin, A. (2004) “An experimental method for determination of T-stress” Proceedings of the 5th Iranian Aerospace Society Conference, PP. 379-386. 24. Ayatollahi, M.R. and Shokrani A.R. (2003)
“Shear geometry factor for a four-point-bend crack specimen” Proceedings of the 7th International ISME Conference, Mashhad, Iran, Vol. 3, PP 1159-1167. 23. Ayatollahi, M.R. and Nasr-Esfahani, M.
(2003) “A modification on the minimum distortional strain energy density criterion” Proceedings of the 7th International ISME Conference, Mashhad, Iran, Vol. 3, PP 1122-1128. 22. Ayatollahi, M.R.,
Aliniaziazi, A. and Aliha, M.R.M. (2003) “Stress intensity factors for a weld toe crack in a tubular joint” Proceedings of 6th International Conference on Civil Engineering (ICCE 2003), Isfahan,
Iran, Vol. 6, PP 361- 366. 21. Ayatollahi, M.R. and Abbasi, H. (2003) “Effects of higher order strains on brittle fracture based on LEFM” Proceedings of the 4th Iranian Aerospace Society Conference,
Tehran, Iran, Vol. 1, PP 87-94. 20. Ayatollahi, M.R. and Ashtari, R. (2003) “Shear deformation effects in blade vibrations” Proceedings of the 4th Iranian Aerospace Society Conference, Tehran, Iran,
Vol. 1, 495-504. 19. Ayatollahi, M.R. and Asadkarami, A., (2002) “Finite element study of a double edge crack specimen” Proceedings of 14th European Conference on Fracture, Krakow, Poland, Vol. 1, PP
153-160. 18. Ayatollahi, M.R., Pavier, M.J. and Smith, D.J. (2002) “A new specimen for mode II fracture tests” Proceedings of 14th European Conference on Fracture, Krakow, Poland, Vol. 1, PP 161-168.
17. Ayatollahi, M.R. (2002) “Evaluation of brittle fracture in spiral weld pipes” Proceedings of International Conference on Helical Seam Submerged-Arc Welded Pipe in Oil and Gas Industry, Tehran,
Iran. 16. Ayatollahi, M.R. and Zanbilbaf, A. (2002) “Fracture trajectory in shear loading” Proceedings of the 6th International ISME Conference, Tehran, Iran, Vol. 4, PP 2111-2117. 15. Ayatollahi,
M.R. and Ghandchi-Tehrani, M. (2002) “Dynamic analysis of a straight blade with asymmetrical cross-section” Proceedings of the 6th International ISME Conference, Tehran, Iran, Vol. 5, PP 64-71. 14.
Ayatollahi, M.R. and Smith D.J. (2001) “Lateral load effects in brittle fracture of internal cracks” Proceedings of the 6th International Conference on Biaxial and Multi-axial Fatigue and Fracture,
Lisbon, Vol. 2, PP 983-990. 13. Ayatollahi, M.R., Mohammad-Aliha, M.R., Aliniaziazi, A. (2001) “ The near-tip parameters for a few cracked specimens” Proceedings of the 5th International ISME
Conference, Rasht, Iran, Vol. 1, PP 441-448. 12. Ayatollahi, M.R. and Safari, G.H. (2001) “On the determination of the stress intensity factors using the method of photoelasticity” Proceedings of the
5th International ISME Conference, Rasht, Iran, Vol. 1, PP 239-245. 11. Ayatollahi, M.R. and Abbasi, H. (2000) “A modified criterion for mixed mode fracture” Proceedings of the 1st International
Aerospace Conference, Tehran, Iran, Vol. 3, PP. 975-983. 10. Ayatollahi, M.R. and Abbasi, H. (2000) “Crack growth prediction based on the maximum hoop strain criterion” Proceedings of the 8th
International NMCM Conference, Slovakia, paper No 94. 9. Ayatollahi, M.R. (2000) “Crack tip parameters for a tension-shear specimen” Proceedings of the 4th International ISME Conference, Tehran,
Iran, Vol. 3, PP. 61-66. 8. Smith D.J., Swankie, T.D., Ayatollahi, M.R. and Pavier, M.J. (1998) "Brittle and ductile failure under mixed mode loading", Proceedings of 12th European Conference on
Fracture (ECF12), Sheffield, UK, Vol. 2, PP 661-666. 7. Ayatollahi, M.R., Smith D.J. and Pavier, M.J. (1997) "Displacement based approach to obtain T-stress using finite element analysis",
Proceedings of ASME-Pressure Vessel and Piping Conference, Orlando, FL, Vol. 346, PP 149-154. 6. Ayatollahi, M.R. and Fuladi, M., (1993) "Nonsymmetrical Vibration of Thin Cylindrical Shells - An
Energy Approach", Proceedings of Iranian Society of Mechanical Engineers Annual Conference, Tehran, Iran. 5. Ayatollahi, M.R. and Shahani, A.R., (1993) "Thermally Induced Vibrations of Circular
Plates", Proceedings of ETCE`93, ASME, Houston, TX, PD-Vol. 52, PP. 45-49. 4. Ayatollahi, M.R. and Aghili, J., (1992) " Galerkin Approach in Finite Element Solution of Asymmetrically Loaded Circular
Plates ", Proceedings of the Third National Congress on Mechanics, Vol. 2, PP. 699-706, Athens, Greece. 3. Ayatollahi, M.R. (1991) " Static, Free and Forced Vibration Analysis of Shell of Revolution
on Elastic Foundation ", SMiRT 11 Transaction Vol. J, PP. 165- 70. 2. Eslami, M.R. and Ayatollahi, M.R., (1991) " Dynamic Analysis of Shell of Revolution by Modal Analysis", Proceedings of
ASME-Pressure Vessel and Piping Conference, 23-27 June, San Diego, CA., Vol. 218, PP. 151-156. 1. Eslami, M.R., Ahmadian, H. and Ayatollahi, M.R., (1990) " Static and Dynamic Finite Element Solution
of Shell and Plate of Revolution on Elastic Foundation", Proceedings of ASME-Pressure Vessel and Piping Conference , Nashville, TN., Vol. 194, PP. 21-2 Research Laboratory Fatigue and Fracture
Research Laboratory (Last update: Oct 2005) PDF Courses Taught Strength of Materials I (Undergraduate) Advanced Numerical Methods (Graduate) Engineering Mechanics - Statics (Undergraduate) Computer
Aided Machine Design (Undergraduate) Structural Integrity (Undergraduate) Strength of Materials Lab. (Undergraduate) Finite Element Methods (Graduate) Fracture Mechanics (Graduate)
January 19, 2011 -
Full Name: Prof. Ahmadian, Hamid Position: ProfessorPhone: 98-21-77240540-50 Ex:2907Fax: 98-21-77240488Email:Ahmadian AT iust.ac.irAddress: Iran University of Science & Technology, Tehran, IRAN
University DegreesPhD, University of Waterloo, Canada MSC, University of Amirkabir, IRAN BSc, Iran University of Science and Technology, IRAN Current ResearchNonlinear Mechanical Joints Modeling and
Identification Rotor Dynamics Investigating Chatter Vibrations in Machining Force Determination by Measuring the Structure Response Inverse Problems in Vibration Publications - Journal PapersHamid
Ahmadian, Hassan Jalali and Fatemeh Pourahmadian, Nonlinear model identification of a frictional contact support, Mechanical Systems and Signal Processing, Volume 24, Issue 8, November 2010, Pages
2844-2854 Hamid Ahmadian, and Mostafa Nourmohammadi, Tool point dynamics prediction by a three-component model utilizing distributed joint interfaces, International Journal of Machine Tools and
Manufacture, Article in Press Hassan Jalali, Hamid Ahmadian and Fatemeh Pourahmadian, Identification of micro-vibro-impacts at boundary condition of a nonlinear beam, Mechanical Systems and Signal
Processing, Article in Press H. Ahmadian, S. Faroughi, Superconvergent eigenvalue bending plate element , Inverse Problems in Science and Engineering, 28 July 2010 S Faroughi, H Ahmadian, Shape
functions associated with the inverse element formulations, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, Article in Press H Ahmadian and
H Azizi, Stability Analysis of a Nonlinear Jointed Beam under Distributed Follower Force , Journal of Vibration and Control , August 11, 2010 H. Jalali, B. T. Bonab and H. Ahmadian, Identification of
Weakly Nonlinear Systems Using Describing Function Inversion, Experimental Mechanics, Article in Press Hamid Ahmadian, and Arash Zamani, Identification of nonlinear boundary effects using nonlinear
normal modes, Mechanical Systems and Signal Processing, Volume 23, Issue 6, August 2009, Pages 2008-2018 M. Salahshoor, and Hamid Ahmadian, Continuous model for analytical prediction of chatter in
milling, International Journal of Machine Tools and Manufacture, Volume 49, Issue 14, November 2009, Pages 1136-114 Majid Mehrpouya, and Hamid Ahmadian, Estimation of Applied Forces on Railway
Vehicle Wheelsets from MeasuredVehicle Responses, Int. J. Vehicle Structures & Systems, 1(4), 2009 104-110 Hassan Jalali, Hamid Ahmadian, John E. Mottershead, "Identification of nonlinear bolted
lap-joint parameters by force-state mapping",International Journal of Solids and Structures 44 (2007) 8087–8105 K. Ahmadi, H. Ahmadian, "Modelling machine tool dynamics using a distributed parameter
tool–holder joint interface", International Journal of Machine Tools & Manufacture 47 (2007) 1916–1928 Ahmadian, H., Jalali, H.,"Generic Element Formulation for Modeling Bolted Lap Joints",Mechanical
Systems and Signal Processing, 21 (2007) 2318–2334 Ahmadian, H., and Jalali, H., ”Identification of Bolted Lap Joints Parameters in Assembled Structures", Mechanical Systems and Signal Processing, 21
(2007) 1041–1050 A. Khojeh, M. Ayat, and H. Ahmadian, “Investigating the crack effects on vibrational behaviour of a rotor using special element” , International Journal of Engineering Science,17
(5), 2007 H. Jahed, H. Ahmadian, and H. Khoshnavaz, “Estimating the fatigue life of an aero-blade under dynamical loading” , International Journal of Engineering Science, Vol. 17 (5), 2007 Ahmadian,
H., Mottershead, J.E., James, S., Friswell, M.I., and Reece, C.A., ”Modeling and Updating of Large Surface-to-Surface Joints in the AWE-MACE Structure ', Mechanical Systems and Signal Processing, 20,
868–880, 2006 Ahmadian, H., and Jalali, H., “Identification of Structural Joint Model From Modal Test Results', International Journal of Engineering Science, 15(3). 171-182,2004. Ahmadian, H.,
Mottershead, J.E., and Friswell, M.I, “Physical Realization of Generic Parameters in Updating', Journal of Vibration and Acoustics, 124, 628-633, 2002. Ahmadian, H., Mottershead, J.E., and Friswell,
M.I, “Boundary Condition Identification by Solution of Characteristic Equations', Journal of Sound and Vibration, 247(5), 755-763, 2001. Friswell, M.I., Mottershead, J.E., and Ahmadian, H., ”Finite
Element Model Updating Using Experimental Test Data: Parameterization and Regularization', Philosophical Transactions of the Royal Society of London, Series A, 359, pp. 169-186, 2001 Ahmadian, H.,
Mottershead, J.E., and Friswell, M.I, “Damage Detection Using Substructure Modes', Journal of Inverse Problems in Engineering. 8, pp. 309-323, 2000. Ahmadian, H., Mottershead, J.E., and Friswell,
M.I.,”Regularization Methods for Finite Element Model Updating', Mechanical Systems and Signal Processing, 12(1), 47-64, 1998. Friswell, M.I., Mottershead, J.E., and Ahmadian, H., ”Combining Subset
Selection and Parameter Constrains in Model Updating’’, Journal of Vibration and Acoustics, 120(4), 854-859, 1998. Ahmadian, H., Friswell, M.I., and Mottershead, J.E.,”Minimization of the
Discretization Error in Mass and Stiffness Formulation by an Inverse Method', International Journal for Numerical Methods in Engineering, 41,371-387, 1998. Ahmadian, H., Gladwell, G.M.L. and Ismail,
F., “Parameter Selection Strategies in Finite Element Model Updating", Journal of Vibration and Acoustics, 119(1). 37-45,1997. Gladwell, G.M.L., and Ahmadian, H.,”Generic Element Matrices Suitable
for Finite Element Model Updating', Mechanical Systems and Signal Processing, 9(6), 601-614, 1995. Ahmadian, H., Gladwell, G.M.L. and Ismail, F., “Finite Element Model Updating Using Modal Data",
Journal of Sound and Vibration, 172(5), 675-669, 1994. Courses TaughtAdvanced Vibrations Dynamics of Structures Measurement Sysytems Mechanical Vibrations Modal Testing | {"url":"https://idea.iust.ac.ir/page_arch.php?slc_pg_id=10791&slc_lang=en&sid=71","timestamp":"2024-11-07T13:45:17Z","content_type":"text/html","content_length":"159512","record_id":"<urn:uuid:b5f5423c-8e7f-4dc8-b153-8a041865a2f9>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00076.warc.gz"} |
Linear Relationships
Episode #8 of the course Everyday math by Jenn Schilling
Welcome to Lesson 8! Today, we will discuss linear relationships, which describes the behavior of two different variables when they vary in proportion to one another.
The Math
Let’s start with a few definitions. A variable in math is usually represented by a symbol, such as x or y. A variable represents something unknown or something that can change depending on different
factors. For example, variables are often used to represent lines or curves on a graph, and the equation of the line tells us how y changes based on x. Variables in an experiment or study represent
the different factors of the study that are either controlled or changed. There are three different types of variables in an experiment. The dependent variable is the observed variable that is
measured. The independent variable is the variable that is changed or manipulated. The controlled variable is held constant or kept the same.
Proportionality is another important term in linear relationships. Two variables are proportional if there is a constant ratio between them. Proportional relationships between two variables can be
represented by a straight line on a graph (the slope is constant).
So, linear relationships are interactions between variables that are proportional, meaning that as one variable changes, the other also changes at a constant rate. Linear relationships appear in
statistics when considering the correlation between variables, which tells us how close the relationship is between two different variables in a trial or experiment.
Everyday Applications
Linear relationships also occur in many everyday places. For example, any time you want to make a conversion between different types of measurement—say, inches to centimeters or cups to liters—you
use a linear relationship, because you multiple one measurement by a constant number to turn it into the other measurement.
Any time we consider rates, we are using linear relationships. For example, the time it takes to travel a certain distance depends on the speed at which you travel. Speed limit enforcement is often
done using radar, which measures the distance traveled over a certain period of time; the speed of the vehicle can then be determined by dividing distance by time.
Understanding proportionality is also useful when evaluating experiments and consuming news stories about scientific studies. Often, relationships between variables are presented without proper
explanations, which can make the conclusions challenging to understand. However, with this new knowledge of proportionality and variables (along with the statistics coming up in Lesson 10), you will
now be able to evaluate basic experiments for their validity. Did the experiment include control variables? Is the relationship between the independent variable and the dependent variable truly
linear and proportional? Now you can successfully interpret these questions!
Tomorrow, we will dive into probability before turning to statistics! See you then!
Share with friends | {"url":"https://gohighbrow.com/linear-relationships/","timestamp":"2024-11-04T10:58:31Z","content_type":"text/html","content_length":"65573","record_id":"<urn:uuid:52666c89-a08f-4cd4-abe3-4a21bfa64042>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00230.warc.gz"} |
Numbers | Documentation - Roblox Creator Hub
The number data type, or double, represents a double-precision (64-bit) floating-point number. Numbers can range from -1.7 * 10^308 to 1.7 * 10^308 (around 15 digits of precision, positive or
The sign of the number indicates whether it's positive or negative. For example, 1 is positive and -1 is negative. In Luau, the number -0 is equivalent to 0.
print(0 == -0) --> true
print(-0 > 1) --> false
print(-0 < 1) --> true
print(-0 > -1) --> true
print(-0 < -1) --> false
Luau doesn't distinguish between integers and numbers, but the API reference sometimes distinguishes between them to be more specific about how to use each API.
The float number type refers to a real number with a decimal place. In computer science terms, they are single-precision (32-bit) floating-point number, which isn't as precise as double-precision
floating-point numbers, but is sufficiently precise for most use cases and requires less memory and storage.
The integer number type, or int, refers to a 32-bit whole number, which ranges from -2^31 to 2^31 - 1. Properties and functions that expect integers may automatically round or raise errors when you
assign or pass non-integers to them.
The int64 number type refers to a signed 64-bit integer, which ranges from -2^63 to 2^63 - 1. This type of integer is common for methods that use ID numbers from the Roblox website. For example,
Player.UserId is an int64, and MarketplaceService:PromptPurchase() and TeleportService:Teleport() each expect int64 for the ID arguments.
Numbers are notated with the most significant digits first (big-endian). There are multiple ways to notate number literals in Roblox Lua:
To aid in the readability of long numbers, you can include underscores anywhere within a number literal without changing the value, except at the beginning where this would make it an identifier. For
example, 1_234_567 is the same as 1234567, both of which are equal to 1,234,567.
You can use logical and relational operators to manipulate and compare numbers. You can also use mathematical functions such as math.sqrt() and math.exp() in the math library and bitwise operations
in the bit32 library.
You can determine if a value x is a number by using type(x) or typeof(x). Both return the string number if x is a number.
local testInt = 5
local testDecimal = 9.12761656
local testString = "Hello"
print(type(testInt)) --> number
print(type(testDecimal)) --> number
print(type(testString)) --> string
print(typeof(testInt)) --> number
print(typeof(testDecimal)) --> number
print(typeof(testString)) --> string
You can round numbers using math.floor(), math.ceil(), or math.modf(). These functions return an integer result if Luau can represent it as an integer. If the number is too large, Luau returns it as
a float.
print(math.floor(3.3)) --> 3
print(math.floor(-3.3)) --> -4
print(math.ceil(3.3)) --> 4
print(math.ceil(-3.3)) --> -3
print(math.modf(3.3)) --> 3 0.2999999999999998
print(math.modf(-3.3)) --> -3 -0.2999999999999998 | {"url":"https://create.roblox.com/docs/luau/numbers","timestamp":"2024-11-08T23:43:22Z","content_type":"text/html","content_length":"293501","record_id":"<urn:uuid:f17dfae3-56fd-4642-bd96-f3416ff14529>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00010.warc.gz"} |
Avoiding Common Mathematical Errors: Tips and Tricks for Accurate Calculations - Finabliz
Avoiding Common Mathematical Errors: Tips and Tricks for Accurate Calculations
Ever find yourself frustrated by those pesky mistakes in your calculations? We’ve all been there. From simple addition blunders to complex formula mishaps, mathematical errors can leave anyone
feeling puzzled. But fear not! In this article, we’ll explore some of the most common mathematical errors and provide you with tips and tricks to avoid them. So, grab your calculator and join us as
we uncover the secrets to conquering those mathematical pitfalls.
Common Mathematical Errors
Misplacement of Decimal Points
One of the common errors in mathematics is the misplacement of decimal points. This error occurs when you incorrectly position the decimal point in a number, leading to incorrect calculations. For
example, if you mistakenly place the decimal point one place to the right or left of its correct position, it can significantly alter the value of the number. To avoid this error, it is crucial to
double-check your decimal placements when performing calculations or working with decimal numbers.
Forgetting to Carry or Borrow
Forgetting to carry or borrow is another common mathematical error that occurs during addition or subtraction. When adding or subtracting two numbers, you must correctly carry or borrow digits to
ensure accurate calculations. Failure to do so can lead to incorrect results. For instance, if you forget to carry a digit from the previous column or borrow a digit from the next column, it can
disrupt the overall calculation and provide an incorrect answer. Paying attention to carrying and borrowing is essential for precise arithmetic calculations.
Using the Wrong Operation
Using the wrong operation is a mathematical error that often occurs when solving problems. It happens when you apply the incorrect mathematical operation to find the solution. For example, if a
problem requires division, but you mistakenly use multiplication, the result will be inaccurate. To avoid this error, carefully read the problem and decide which operation is most appropriate.
Double-checking your calculations and ensuring that you are using the correct operation will help prevent this common mistake.
Order of Operations
Order of operations is a fundamental rule in mathematics that dictates the sequence in which mathematical operations should be performed. This rule, often represented by the acronym PEMDAS
(Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right), helps maintain consistency in calculations. Failing to follow the order of
operations can lead to incorrect answers. For instance, if you perform addition before multiplication, the result may be drastically different from the correct solution. It is essential to remember
and apply the order of operations to avoid this error.
Misinterpreting Word Problems
Word problems often present a challenge for many students, as misinterpreting the information given can lead to incorrect solutions. Misinterpreting word problems usually occurs when you misconstrue
the given information, fail to identify the relevant mathematical concepts, or misapply formulas or equations. To overcome this error, it is vital to carefully read and comprehend the problem before
attempting to solve it. Break down the problem into smaller parts, identify the key information, and translate it into mathematical expressions accurately. By paying attention to details and ensuring
a clear understanding of the problem, you can avoid misinterpreting word problems and effectively solve them.
Errors in Basic Arithmetic
Addition Errors
Addition errors are quite common in basic arithmetic. These errors can occur when adding two or more numbers and can arise from miscalculations, miscounting, or failing to correctly align the
numbers. For example, if you mistakenly add numbers from the wrong columns or overlook carrying digits, your final sum will be inaccurate. To avoid addition errors, it is crucial to take your time,
double-check your calculations, and ensure that you accurately align the numbers before adding.
Subtraction Errors
Similar to addition errors, subtraction errors can occur when subtracting two numbers. Common mistakes in subtraction include miscounting, incorrectly borrowing digits, or neglecting to align the
numbers correctly. If you skip or incorrectly perform the borrowing process or subtract numbers from the wrong columns, your result will be incorrect. To minimize subtraction errors, practice various
subtraction techniques, master the borrowing process, and pay close attention to aligning the numbers properly.
Multiplication Errors
Multiplication errors can happen when multiplying two or more numbers. They often stem from miscalculations, misalignments, or careless mistakes. For instance, overlooking a zero placeholder or
multiplying numbers from the wrong columns can lead to incorrect products. To avoid multiplication errors, ensure that you accurately align the numbers, properly handle zero placeholders, and recheck
your calculations. Practicing multiplication techniques and focusing on accuracy will help minimize errors.
Division Errors
Division errors are prevalent when dividing one number by another. These errors can occur from miscalculations, incorrect placement of the dividend and divisor, or misinterpreting the remainder.
Failing to consider the remainder or forgetting the decimal point placement can lead to inaccurate quotients. To reduce division errors, verify the placement of the dividend and divisor, correctly
handle remainders, and carefully position the decimal point. Thoroughly checking your division process will help you avoid these common mistakes.
Errors in Algebra
Misapplying the Distributive Property
Misapplying the distributive property is a common error in algebra. The distributive property allows you to simplify expressions by distributing a term across parentheses. However, misapplying this
property can result in incorrect simplifications. For example, if you incorrectly distribute a term or fail to apply the property to all terms within parentheses, your expression will be simplified
incorrectly. To avoid this error, carefully apply the distributive property to each term and pay attention to signs and operations.
Simplifying Expressions Incorrectly
Simplifying expressions is a crucial skill in algebra, but it is susceptible to errors. Mistakes in simplification can arise from miscalculations, mishandling negative signs, or neglecting to combine
like terms. If you fail to follow the order of operations, overlook the rules of combining like terms, or incorrectly perform arithmetic operations, your simplified expression will be incorrect. To
minimize simplification errors, double-check your calculations, carefully handle negative signs, and thoroughly combine like terms.
Getting the Signs Wrong
Algebra often involves manipulating and solving equations with variables and constants. Errors can occur when handling signs, such as positive (+) and negative (-). Misplacing or miscalculating signs
can lead to inaccurate equations and incorrect solutions. For example, if you incorrectly distribute a negative sign or neglect to change the sign when combining like terms, your equation will be
incorrect. To avoid sign errors, be cautious when distributing signs, pay attention to sign changes, and carefully simplify expressions.
Solving Equations Incorrectly
Solving equations is a fundamental skill in algebra, but errors can arise during the process. Common mistakes in solving equations include misapplying operations, neglecting to perform the same
operation on both sides of the equation, or mishandling fractions or decimals. If you incorrectly simplify an equation, fail to isolate the variable, or make an arithmetic error, your final solution
will be inaccurate. To prevent errors in solving equations, review the fundamental steps, perform the same operation on both sides of the equation, and carefully check your work at each step.
Errors in Geometry
Incorrectly Measuring Angles
Geometry involves the study of shapes, lines, and angles. One of the common errors in geometry is incorrectly measuring angles. Misreading a protractor, inaccurately placing the vertex, or
misinterpreting angle measurements can lead to incorrect angle values. To avoid this error, ensure that you correctly position the protractor, align the vertex with the protractor’s zero mark, and
accurately read the measurement. Taking your time and double-checking angle measurements will help minimize errors in geometry.
Using the Wrong Formula
Geometry problems often require the use of various formulas to find area, perimeter, volume, or other properties of shapes. Using the wrong formula is a common error that can result in incorrect
answers. For example, using the formula for the perimeter of a rectangle instead of the formula for the area will yield an incorrect value. To avoid this error, carefully read the problem, identify
the necessary formulas, and select the appropriate formula for each specific calculation. Checking your formulas before applying them will help ensure accurate results.
Making Mistakes in Proofs
Geometry proofs involve logical reasoning and step-by-step explanations to demonstrate the truth of a mathematical statement. Errors can occur when constructing proofs, misapplying theorems or
postulates, or making incorrect logical deductions. These mistakes can lead to flawed proofs and incorrect conclusions. To minimize errors in proofs, thoroughly understand the properties, theorems,
and postulates being utilized, carefully analyze the given information, and clearly articulate each logical step. Regular practice and attention to detail will enhance your proficiency in
constructing accurate proofs.
Misidentifying Geometric Shapes
Identifying geometric shapes correctly is essential in geometry. Misidentifying shapes can lead to errors in calculations, interpretations, or analysis of properties. For instance, mistaking a
parallelogram for a rectangle can result in inaccurate angle measurements. To avoid this error, familiarize yourself with the characteristics of different shapes, pay attention to specific
properties, and compare the given information with the defining attributes of each shape. By carefully identifying geometric shapes, you will ensure accurate deductions and calculations.
Errors in Calculus
Problems with Limits
Limits are a fundamental concept in calculus, but they can be challenging and prone to errors. Common mistakes with limits include misapplying limit laws, neglecting to simplify expressions before
taking limits, or overlooking the use of L’Hôpital’s rule when applicable. Failing to properly evaluate limits can lead to incorrect results and flawed calculus calculations. To avoid errors with
limits, review the limit laws, simplify expressions, and consider alternative techniques like L’Hôpital’s rule when necessary. Thoroughly reassessing your limit calculations will help ensure accurate
Integration Errors
Integration is a key component of calculus, but errors can occur during the integration process. Common integration mistakes include misapplying integration rules, neglecting to perform the
appropriate substitutions, or mishandling constants of integration. If you incorrectly integrate a term, overlook the need for a substitution, or fail to include the constant of integration, your
final integral will be incorrect. To minimize integration errors, master the integration rules, carefully identify substitution opportunities, and remember to add the constant of integration.
Double-checking your integration steps will help you achieve accurate results.
Differentiation Errors
Differentiation, or finding derivatives, is another crucial aspect of calculus. Errors in differentiation can arise from miscalculations, overlooking the need for the chain rule or product rule, or
mishandling exponential or logarithmic functions. If you incorrectly differentiate a term, fail to apply the appropriate rule, or neglect to simplify the derivative, your final result will be
inaccurate. To reduce differentiation errors, practice differentiation techniques, thoroughly understand the rules and formulas, and carefully analyze each term in the differentiation process.
Ensuring accuracy in differentiation will help you achieve precise results.
Inaccurate Graphing
Graphing enables visualization and analysis of functions and equations, but errors can occur during the graphing process. Common graphing mistakes include misidentifying key points, neglecting to
plot all relevant points, or incorrectly drawing the curve or line. If you incorrectly plot points, skip important points, or misinterpret the shape of the graph, your final graph will be inaccurate.
To avoid graphing errors, actively read and understand the given equation, identify crucial points (such as intercepts and critical points), and carefully plot them on the graph. Checking your graph
against the equation will help you create an accurate representation.
Errors in Statistics
Sampling Errors
In statistics, sampling errors refer to errors that arise from using a sample to estimate characteristics of a larger population. These errors can occur due to biased sampling methods, inadequate
sample sizes, or random variability. If a sample is not representative of the entire population or if the sample size is too small, the estimated statistics may differ significantly from the true
population values. To minimize sampling errors, use random sampling methods, ensure a sufficient sample size, and analyze the variability within the population. Careful consideration of these factors
will help improve the accuracy of statistical estimates.
Confusion with Mean, Median, and Mode
Mean, median, and mode are measures of central tendency commonly used in statistics. Confusion with these measures can lead to errors in data analysis and interpretation. For example, calculating the
mean instead of the median for a skewed dataset can produce a misleading representation of the average value. To avoid confusion, understand the characteristics and appropriate applications of each
measure. Consider the distribution of the data and choose the most suitable measure of central tendency accordingly. Clear understanding and thoughtful analysis will help prevent errors in
determining central tendencies.
Misunderstanding Probability
Probability is a fundamental concept in statistics, but it can be challenging to grasp. Errors in probability can arise from misinterpretation of probability concepts, incorrect use of probability
rules, or confusion with conditional probability. For instance, misunderstanding the concept of independence can lead to errors in calculating probabilities. To minimize errors, familiarize yourself
with the basic rules of probability, practice applying them in different scenarios, and seek clarification when unsure. Thoroughly understanding probability concepts will enhance your accuracy in
probability calculations.
Miscalculating Standard Deviation
Standard deviation is a measure of the spread or dispersion of a dataset in statistics. Miscalculating standard deviation can occur due to errors in arithmetic calculations, misinterpretation of the
formula, or omission of necessary steps. If you make a mistake in calculating the sum of squared deviations or fail to take the square root, your final standard deviation will be incorrect. To avoid
miscalculations, carefully follow each step of the standard deviation formula, recheck your arithmetic calculations, and verify the final result. Attention to detail and precision will help you
obtain accurate standard deviation values.
Errors in Probability
Confusion with Independent and Dependent Events
Understanding the distinction between independent and dependent events is crucial in probability. Errors in distinguishing the two can lead to incorrect probability calculations. For example,
assuming independence when events are dependent can substantially alter the probability estimate. To avoid this confusion, carefully analyze the relationship between events, consider conditional
probabilities when necessary, and verify independence or dependence before applying probability rules. Ensuring accurate identification of event dependencies will greatly enhance the reliability of
your probability calculations.
Applying the Wrong Probability Formula
Various probability formulas exist to calculate different types of probabilities, such as the probability of combinations, permutations, or conditional probabilities. Applying the wrong formula can
result in incorrect probability estimates. For instance, using a combination formula instead of a permutation formula when order matters can yield an inaccurate probability. To prevent this error,
carefully analyze the problem, identify the required probability calculation, and select the appropriate formula. Checking the problem conditions against the chosen formula will help you accurately
determine the probability.
Misunderstanding Conditional Probability
Conditional probability refers to the probability of an event occurring given that another event has already occurred. Misunderstanding conditional probability can lead to errors in probability
calculations. For example, miscalculating the likelihood of an event based on misleading conditional information can result in an incorrect probability estimate. To minimize errors, clearly
understand the given conditions, correctly assess the impact on event probabilities, and avoid common conditional probability fallacies. Thoroughly analyzing the conditional relationships will
enhance the accuracy of your probability calculations.
Forgetting to Account for Complementary Events
Complementary events in probability refer to events that are mutually exclusive and cover all possible outcomes. Forgetting to account for complementary events can result in errors in probability
calculations. For example, neglecting to consider the probability of the complement when calculating the probability of an event can lead to an inaccurate estimate. To avoid this error, remember to
account for both the event and its complement when determining probabilities. Including all possible outcomes will ensure accurate probability calculations.
Errors in Trigonometry
Using Incorrect Trig Identities
Trigonometric identities are crucial in trigonometry, but errors can occur when using them. Mistakes in applying trigonometric identities can result in incorrect simplifications or trigonometric
equations. For instance, using an incorrect Pythagorean identity or failing to recognize trigonometric ratios can lead to inaccurate solutions. To minimize errors, review the trigonometric
identities, practice their applications, and carefully analyze the given expressions or equations. Thoroughly assessing each step and verifying the identities used will ensure accurate results.
Solving Trig Equations Incorrectly
Solving trigonometric equations involves finding the values of angles or variables that satisfy the given equation. Errors in solving trig equations can arise from misapplying trigonometric
properties, overlooking possible solution sets, or neglecting to consider periodicity. If you skip potential solutions, incorrectly simplify expressions, or overlook the periodic nature of
trigonometric functions, your final solutions will be incorrect. To avoid this error, thoroughly analyze the equation, consider all possible solutions, and apply the appropriate trigonometric
properties or identities. Careful consideration and execution of each step will lead to accurate solutions.
Mistakes in Trig Function Graphs
Graphing trigonometric functions is a fundamental skill in trigonometry. Errors can occur when plotting points, misinterpreting the amplitude or period, or inaccurately sketching the graph. For
example, incorrectly identifying the maximum or minimum values or failing to account for phase shifts can lead to inaccurate graphs. To minimize mistakes in graphing trig functions, understand the
characteristics of each trigonometric function, identify key points, and correctly plot them. Paying attention to the amplitude, period, vertical shifts, and phase shifts will result in accurate trig
function graphs.
Confusion with Unit Circle
The unit circle plays a crucial role in trigonometry and provides a visualization of trigonometric ratios for specific angles. Mistakes can occur when interpreting or utilizing the unit circle.
Misinterpreting angles, misidentifying coordinates, or making errors in calculations can lead to inaccuracies in trigonometry problems. To avoid confusion, study and practice the unit circle,
memorize key angles and their corresponding coordinates, and carefully apply the trigonometric ratios. Regular exposure to the unit circle and its applications will enhance your understanding and
accuracy in trigonometry.
Errors in Number Theory
Factoring Errors
Factoring involves breaking down a number or expression into its prime factors. Errors can occur during the factoring process, resulting in incorrect prime factorizations or misunderstood
relationships between factors. For example, incorrectly identifying prime factors or neglecting remaining factors can lead to inaccurate breakdowns. To minimize factoring errors, practice prime
factorization techniques, cross-check your factorizations, and verify the multiplication of factors to ensure they yield the original number or expression. Careful attention to the factoring process
will help you obtain correct prime factorizations.
Misunderstanding Prime Numbers
Prime numbers are fundamental in number theory, but misunderstandings can lead to errors in mathematical reasoning. Mistakes can arise from misidentifying prime numbers, misapplying prime
factorization, or misinterpreting necessary conditions. For instance, incorrectly classifying a composite number as prime can lead to flawed conclusions. To avoid misunderstandings, study the
characteristics of prime numbers, practice prime factorization, and verify the primality of numbers through rigorous testing. Clear understanding and accurate identification of prime numbers will
enhance your reasoning and accuracy in number theory.
Wrongly Applying Theorems
Number theory involves the study of various theorems and properties related to numbers. Errors can arise when wrongly applying these theorems or misinterpreting their conditions. For example,
incorrectly assuming a number satisfies a theorem’s necessary condition when it does not can lead to erroneous conclusions. To prevent this error, thoroughly understand the assumptions and
requirements of each theorem, carefully evaluate the given number, and verify that the necessary conditions are met. Thoughtful application of theorems will help you draw accurate conclusions.
Inaccuracy in Number Patterns
Number patterns are prevalent in number theory and help identify relationships between numbers. Errors can occur when inaccurately analyzing or predicting number patterns, resulting in incorrect
conclusions or estimations. For example, misidentifying the pattern or jumping to unsupported conclusions can lead to false assumptions. To avoid inaccuracy in number patterns, carefully observe and
analyze the numbers, identify the underlying pattern or relationship, and validate your predictions through rigorous testing or proof. Vigilance in observing and interpreting number patterns will
improve the accuracy of your conclusions.
Errors in Mathematical Proofs
Invalid Logical Steps
Mathematical proofs involve logical reasoning and precise steps to demonstrate the truth of a mathematical statement. Errors can occur when invalid logical steps are taken, leading to flawed proofs.
For example, assuming a statement is true without proper justification or making unwarranted assumptions can undermine the validity of a proof. To minimize errors in proofs, carefully analyze each
step, verify the logical connections, and explicitly justify the reasoning behind each step. Rigorous scrutiny of each logical step will ensure the accuracy and validity of mathematical proofs.
Assuming What Needs to Be Proven
In mathematical proofs, it is crucial to prove the desired statement without assuming its truth. Errors can arise when assuming what actually needs to be proven or taking the desired conclusion for
granted. For instance, assuming that a statement holds true without providing substantial evidence or reasoning can render a proof incomplete or incorrect. To avoid this error, identify the specific
statement to be proven, start from the given information or known facts, and build a logical chain of reasoning to establish the desired conclusion. Avoiding unfounded assumptions will help ensure
comprehensive and accurate mathematical proofs.
Inadequate Communication of Ideas
Mathematical proofs require clear and concise communication of ideas and logical reasoning. Errors can occur when inadequately communicating the steps, explanations, or justifications in a proof.
Poor organization, ambiguous language, or missing details can undermine the clarity and believability of a proof. To improve the communication of ideas, structure your proof with appropriate
headings, clearly explain each step, use precise mathematical language, and provide sufficient details to support your arguments. Effective communication will enhance the clarity and credibility of
your mathematical proofs.
Missing or Incomplete Dependencies
Dependencies in mathematical proofs refer to the relationships between statements or theorems that are used to justify subsequent steps. Errors can occur when missing or incomplete dependencies lead
to unsupported assertions or flawed justifications. For example, referencing a theorem without proving it or neglecting a crucial step can result in gaps in the logical progression of a proof. To
avoid this error, explicitly identify and validate all necessary dependencies, provide proper justifications for each statement, and ensure a continuous flow of logical reasoning. Awareness of
dependencies will help establish the coherence and validity of your mathematical proofs. | {"url":"https://finabliz.com/avoiding-common-mathematical-errors-tips-and-tricks-for-accurate-calculations/","timestamp":"2024-11-04T20:31:46Z","content_type":"text/html","content_length":"237392","record_id":"<urn:uuid:afa22b68-b681-464f-87a3-126dca382864>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00496.warc.gz"} |
Particle number fluctuations, Rényi entropy, and symmetry-resolved entanglement entropy in a two-dimensional Fermi gas from multidimensional bosonization
We revisit the computation of particle number fluctuations and the Rényi entanglement entropy of a two-dimensional Fermi gas using multidimensional bosonization. In particular, we compute these
quantities for a circular Fermi surface and a circular entangling surface. Both quantities display a logarithmic violation of the area law, and the Rényi entropy agrees with the Widom conjecture.
Lastly, we compute the symmetry-resolved entanglement entropy for the two-dimensional circular Fermi surface and find that, while the total entanglement entropy scales as RlnR, the symmetry-resolved
entanglement scales as RlnR, where R is the radius of the subregion of our interest.
All Science Journal Classification (ASJC) codes
• Electronic, Optical and Magnetic Materials
• Condensed Matter Physics
Dive into the research topics of 'Particle number fluctuations, Rényi entropy, and symmetry-resolved entanglement entropy in a two-dimensional Fermi gas from multidimensional bosonization'. Together
they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/particle-number-fluctuations-r%C3%A9nyi-entropy-and-symmetry-resolved-","timestamp":"2024-11-11T04:21:33Z","content_type":"text/html","content_length":"46512","record_id":"<urn:uuid:69516f8f-74d7-4f80-b9b8-452014b1fe9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00657.warc.gz"} |
Introduction To Statistical Learning Stanford Pdf
Support Vector Machines Stanford Lagunita
Introduction to Statistical Learning and Personalized Medicine. 5/05/2015В В· Reference: (Book) (Chapter 2) An Introduction to Statistical Learning with Applications in R (Gareth James, Daniela
Witten, Trevor Hastie, Robert Tibshirani), ISLR: Data for an Introduction to Statistical Learning with Applications in R. We provide the collection of data-sets used in the book 'An Introduction to
Statistical Learning with Applications ….
Package †ISLR’ The Comprehensive R Archive Network
Statistics exploredegrees.stanford.edu. ISLR: Data for an Introduction to Statistical Learning with Applications in R. We provide the collection of data-sets used in the book 'An Introduction to
Statistical Learning with Applications …, Introduction to Statistical Learning Theory Olivier Bousquet1, St ephane Boucheron2, learning algorithms is thus to look for regularities (in a sense to be
de ned later) in the observed phenomenon (i.e. training data). These can then be generalized from the observed past to the future. Typically, one would look, in a collection of possible models, for
one which ts well the data, but at the.
An Introduction to Statistical Learning provides an accessible overview of the field of statistical learning, an essential toolset for making sense of the vast and complex data sets that have An
Introduction to Statistical Learning: (2013) (Springer Series in Statistics) by G. James, D. Witten, T. Hastie and R. Tibshirani Book Homepage pdf (9.4Mb, 6th corrected printing) The Science of
Bradley Efron (2008) Carl Morris and Robert Tibshirani (editors) The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer Series in Statistics) (2001 & 2009) by T
Stanford’s Department of Statistics, both renowned and near so many Internet and bioscience companies, is at the center of the boom. It received 800 résumés for next year’s 60 graduate
positions, twice the number of applications 24/06/2013В В· A great introduction book for statistical learning, a closely related field to machine learning. This is the accompany book for the course
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An Introduction to Statistical Learning Unofficial Solutions. Fork the solutions! Twitter me @princehonest Official book website. Check out Github issues and repo for the latest updates. Introduction
to Statistical Machine Learning - 2 - Marcus Hutter Abstract This course provides a broad introduction to the methods and practice of statistical machine learning, which is concerned with the
development of algorithms and techniques that learn from observed data by constructing stochastic models that can be used for making predictions and decisions. Topics covered include …
Notes and exercise attempts for "An Introduction to Statistical Learning" - asadoughi/stat-learning subject code STATS on the Stanford Bulletin's ExploreCourses web site. The department's goals are
to acquaint students with the role played in science and technology by probabilistic and statistical ideas and methods, to provide instruction in the theory and application of techniques that have
been found to be commonly useful, and to train research workers in probability and statistics. There
Introduction to Statistical Learning Theory Olivier Bousquet1, St ephane Boucheron2, learning algorithms is thus to look for regularities (in a sense to be de ned later) in the observed phenomenon
(i.e. training data). These can then be generalized from the observed past to the future. Typically, one would look, in a collection of possible models, for one which ts well the data, but at the
STAT7040 Statistical Learning This course provides an introduction to statistical learning and aims to develop skills in modern statistical data analysis. There has been a prevalence of “big
data” in many different areas such as finance, marketing, social networks and the scientific fields. As traditional statistical methods have become inadequate for analysing data of such size and
Respected Stanford professors Trevor Hastie and Robert Tibshirani, along with Martin Wainwright, not long ago released a new book titled "Statistical Learning with Sparsity: The Lasso and
Generalizations," which is available for purchase via its website, and has recently been made freely available as a PDF download. An Introduction to Statistical Learning: (2013) (Springer Series in
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Tibshirani (editors) The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer Series in Statistics) (2001 & 2009) by T
Class 1 Introduction to Statistical Learning Theory Carlo Ciliberto Department of Computer Science, UCL October 5, 2018 Introduction to Statistical Learning: With Applications in R Gareth James,
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(i.e. training data). These can then be generalized from the observed past to the future. Typically, one would look, in a collection of possible models, for one which ts well the data, but at the
GitHub asadoughi/stat-learning Notes and exercise
Book Review An Introduction to Statistical Learning. STAT7040 Statistical Learning This course provides an introduction to statistical learning and aims to develop skills in modern statistical data
analysis. There has been a prevalence of “big data” in many different areas such as finance, marketing, social networks and the scientific fields. As traditional statistical methods have
become inadequate for analysing data of such size and, 24/06/2013В В· A great introduction book for statistical learning, a closely related field to machine learning. This is the accompany book for
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subject code STATS on the Stanford Bulletin's ExploreCourses web site. The department's goals are to acquaint students with the role played in science and technology by probabilistic and statistical
ideas and methods, to provide instruction in the theory and application of techniques that have been found to be commonly useful, and to train research workers in probability and statistics. There.
5/05/2015 · Reference: (Book) (Chapter 2) An Introduction to Statistical Learning with Applications in R (Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirani) • Richard Sutton and
Andrew Barto, Reinforcement Learning: An introduction. MIT Press, 1998 MIT Press, 1998 • Trevor Hastie, Robert Tibshirani and Jerome Friedman, The Elements of Statistical Learning .
Introduction to Statistical Machine Learning - 2 - Marcus Hutter Abstract This course provides a broad introduction to the methods and practice of statistical machine learning, which is concerned
with the development of algorithms and techniques that learn from observed data by constructing stochastic models that can be used for making predictions and decisions. Topics covered include …
Class 1 Introduction to Statistical Learning Theory Carlo Ciliberto Department of Computer Science, UCL October 5, 2018
View Homework Help - stats216-summer18-homework1solutions-forstudents.pdf from STATS 216 at Stanford University. STATS216v Introduction to Statistical Learning Stanford University, Summer View An
Introduction to Statistical Learning.pdf from STA 380 at University of Texas. Gareth James Daniela Witten Trevor Hastie Robert Tibshirani An Introduction to Statistical Learning with
Support Vector Machines Here we approach the two-class classi cation problem in a direct way: We try and nd a plane that separates the classes in Support Vector Machines Here we approach the
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Respected Stanford professors Trevor Hastie and Robert Tibshirani, along with Martin Wainwright, not long ago released a new book titled "Statistical Learning with Sparsity: The Lasso and
Generalizations," which is available for purchase via its website, and has recently been made freely available as a PDF download. Introduction to Statistical Learning Theory Olivier Bousquet1, St
ephane Boucheron2, learning algorithms is thus to look for regularities (in a sense to be de ned later) in the observed phenomenon (i.e. training data). These can then be generalized from the
observed past to the future. Typically, one would look, in a collection of possible models, for one which ts well the data, but at the
An Introduction to Statistical Learning provides an accessible overview of the field of statistical learning, an essential toolset for making sense of the vast and complex data sets that have 24/06/
2013В В· A great introduction book for statistical learning, a closely related field to machine learning. This is the accompany book for the course with the same name by Stanford University online
MOOC platform. The content is intended for the beginners in machine learning therefore much less math than the other book - Element of Statistical Learning. The professors are actually the authors
An Introduction to Statistical Learning: (2013) (Springer Series in Statistics) by G. James, D. Witten, T. Hastie and R. Tibshirani Book Homepage pdf (9.4Mb, 6th corrected printing) The Science of
Bradley Efron (2008) Carl Morris and Robert Tibshirani (editors) The Elements of Statistical Learning: Data Mining, Inference, and Prediction (Springer Series in Statistics) (2001 & 2009) by T 5/05/
2015В В· Reference: (Book) (Chapter 2) An Introduction to Statistical Learning with Applications in R (Gareth James, Daniela Witten, Trevor Hastie, Robert Tibshirani)
statistical pdf - In January 2014, Stanford University professors Trevor Hastie and Rob Tibshirani (authors of the legendary Elements of Statistical Learning textbook) taught an online course based
on their newest textbook, An Introduction to Statistical Learning with Applications in R (ISLR). I found it to be Mon, 10 Dec 2018 05:27:00 GMT In-depth introduction to machine learning in 15 hours
Respected Stanford professors Trevor Hastie and Robert Tibshirani, along with Martin Wainwright, not long ago released a new book titled "Statistical Learning with Sparsity: The Lasso and
Generalizations," which is available for purchase via its website, and has recently been made freely available as a PDF download.
ISLR: Data for an Introduction to Statistical Learning with Applications in R. We provide the collection of data-sets used in the book 'An Introduction to Statistical Learning with Applications …
STATS216v Introduction to Statistical Learning Stanford University, Summer 2017 Problem Set 1 Due: Friday, July 7 Remember the universityhonor code.
Introduction to Statistical Learning and Personalized Medicine Spring, 2018 COURSE DESCRIPTION (3 credit hours) The rst part of the course gives an introduction to statistical learning … •
Richard Sutton and Andrew Barto, Reinforcement Learning: An introduction. MIT Press, 1998 MIT Press, 1998 • Trevor Hastie, Robert Tibshirani and Jerome Friedman, The Elements of Statistical
Learning .
An Introduction to Statistical Learning Unofficial Solutions. Fork the solutions! Twitter me @princehonest Official book website. Check out Github issues and repo for the latest updates. An
Introduction to Statistical Learning, with applications in R (with Gareth James and Daniela Witten, Springer-Verlag, 2013). Statistical Learning with Sparsity: the Lasso and Generalizations (with
Martin Wainwright, Chapman and Hall, 2015). | {"url":"https://hastingslearns.org/abitibi-70/introduction-to-statistical-learning-stanford-pdf.php","timestamp":"2024-11-04T18:50:19Z","content_type":"text/html","content_length":"66138","record_id":"<urn:uuid:dc28d475-1ca2-4155-b903-038a86514ff1>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00684.warc.gz"} |
Generating Odds
Most of the statistics in the scores table for each pool are based on simulations run using the Monte Carlo method.
The process:
1. Pull the latest values from kenpom.com for each team's adjusted efficiency margin and adjusted tempo.
2. For each actual or hypothetical matchup in the tournament, estimate team A's probability of defeating team B using the method described here. (As I understand it, there may be small discrepancies
if he decides that certain venues give teams partial home-court advantages, e.g. when UNC plays in Charlotte.)
3. Before the real tournament starts and each time a new result comes in, simulate a large number (several thousand) of hypothetical tournaments using the P(A,B) probabilities to determine random
outcomes for each of the games that hasn't yet been played. For each hypothetical tournament, score all of the entries in each pool and record how they performed (in total points and relative to
one another).
4. Aggregate the results from all of the thousands of runs to generate the statistics shown in the scores table.
Explanation of individual statistics:
• Best possible score (theoretical): This statistic doesn't require the Monte Carlo simulations. It's simply the score you could achieve if every remaining game went in your favor.
• Best possible score (observed): In all of the thousands of simulated tournaments, the maximum score that you achieved. At the beginning of the tournament this will typically be significantly
lower than the theoretical maximum; as the tournament progresses, the two numbers will converge. Caveat: you may occasionally see an observed best score that exceeds the corresponding potential
best score. This happens because the potential score updates almost immediately after a new result comes in, whereas the observed number will only update after the prediction model runs again
(which typically takes 10-20 minutes).
• Projected score: In all of the simulated tournaments, your average (mean) score.
• Best possible finish: In all of the simulated tournaments, the single best ranking you achieved.
• Worst possible finish: In all of the simulated tournaments, the single worst ranking you achieved.
• Odds of winning: The number of runs in which you finished first divided by the total number of runs. Note that the sum of these numbers will exceed 100% if there are possible scenarios in which
two or more entries tie for first place.
Known issues/limitations:
• Even if the prior probabilities are perfect, the Monte Carlo method is only an approximation. (In its defense, though, its results should converge to the true values if the number of runs is
sufficiently high.) But why not just calculate the exact numbers? Once the tournament is down to 16 or 8 teams, that's fine — but when 64 teams remain in the field, the number of possible
outcomes (2^63) is so large that even the fastest computer in the world couldn't iterate through all of them. I suspect you could make the computation tractable by taking clever shortcuts to
avoid considering most of the cases, but I'm not getting paid enough to work on that problem.
• If you have a tiny but non-zero probability of winning, it's possible that you won't finish first in a single run of the simulation — in which case the site will erroneously list your probability
of winning as being zero.
• As noted above, the probabilities I generate for individual matchups fail to account for partial home-court advantages.
Potential improvements in future years:
• Allow the user to view details of the simulation's runs so (s)he can identify the specific scenarios that will yield a victory in the pool.
• Provide an interface for the user to construct a hypothetical bracket (i.e. fill in winners for the remaining games) and then score a pool based on these results.
• (not related to the simulation) Show how many teams each entry has alive in upcoming rounds.
• Other ideas? Let me know. | {"url":"http://www.bracketastic.com/monteCarlo.jsp;jsessionid=282583F3108EB0CF19AD34DFBEC19AD2","timestamp":"2024-11-07T12:58:44Z","content_type":"text/html","content_length":"11591","record_id":"<urn:uuid:7668580a-14ef-40cc-adee-39a9a123578f>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00125.warc.gz"} |
How do I find the integral
int(x-9)/((x+5)(x-2))dx ? | HIX Tutor
How do I find the integral #int(x-9)/((x+5)(x-2))dx# ?
Answer 1
This integral can be solved by using the Partial Fractions approach, giving an answer of
#2ln(x+5)-ln(x-2) + C#
The partial fractions approach is useful for integrals which have a denominator that can be factored but not able to be solved by other methods, such as Substitution. This equation already has its
denominator factored, but note that if we were instead given the multiplied form:
we would need to factor the denominator to continue. We can now turn this function into its partial fraction equivalent:
#A/(x+5) + B/(x-2)# = #(x-9)/((x+5)(x-2))#
Multiplying by the common denominator: #(x+5)(x-2)#, we have:
#A(x-2) + B(x+5) = x-9#
Now we can choose any value of #x# to plug in on both sides, so the best solution is to choose values which will cancel out one of the terms on the left side. In this case, they will be #2# and #-5#.
Plugging in #-5#, we have:
#A(-5-2)+B(-5+5) = -5-9# #A(-7) = -14# #A = 2#
With #2#, we have:
#A(2-2) + B(2+5) = 2 - 9# #B(7) = -7# #B = -1#
Using these values in our original partial fractions representation from above, we have:
#A/(x+5) + B/(x-2)# = #2/(x+5) - 1/(x-2)#
Now you can integrate these terms separately using substitution. Both will end up being #u^(-1)#, resulting in #lnu + C#. Now you can arrive at your answer:
#2ln(x+5) - ln(x-2) + C#
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Answer 2
To find the integral of ( \frac{x-9}{(x+5)(x-2)} ) with respect to ( x ), you can use partial fraction decomposition. First, you need to decompose the rational function into partial fractions. Then,
you integrate each term separately.
The partial fraction decomposition of ( \frac{x-9}{(x+5)(x-2)} ) is:
( \frac{x-9}{(x+5)(x-2)} = \frac{A}{x+5} + \frac{B}{x-2} )
To find ( A ) and ( B ), multiply both sides by ( (x+5)(x-2) ) to clear the fractions, then equate coefficients of like terms.
Solving for ( A ) and ( B ), you'll find:
( A = -\frac{2}{7} ) and ( B = \frac{5}{7} )
Now, you integrate each term separately:
( \int \frac{-2}{x+5} ,dx + \int \frac{5}{x-2} ,dx )
This yields:
( -2\ln|x+5| + 5\ln|x-2| + C )
Where ( C ) is the constant of integration.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Information Content, Compressibility
Three years later I still think there is an inconsistency, but a Random Number Generator is not a good example to illustrate it. A Random Number Generator does not describe a particular random string
at all, and therefore cannot be the shortest description of a particular random string. It describes the set of all random strings. The RNG is not a reproducible description of a particular string,
because every generated string is unique. This is even truer for a true random sequence based on radioactive decay or any other natural source of true randomness (8).
However there are other systems like π (pi) and cellular automata (9) that reproducibly and uniquely describe an arbitrary long random sequence. The paradox is that the length of these simple
algorithms tells us nothing at all about the 'information content' or complexity of the sequence they produce/describe because the 'information content' can have any value depending on the length of
the output. Since π infinite length, its information content should be infinite.
Information and Meaning
The kind of 'information' produced by a mindless computer program or a natural, physical mindless process is cheap, worthless, meaningless information. Just let it run forever and it produces an
endless stream of 'information'! That is not what humans call information. There is the real paradox.
Another way of stating the paradox is: a long random string of letters has a higher Information Content then a book of the same length. This is so because a random string is hardly compressible at
all, while the content of a book can easily be compressed (as everyone knows who uses PC compression programs as pkzip, winzip, etc). In fact a random string has the maximum amount of information.
This definition of information was not invented by Paul Davies, but by Shannon, Chaitin and Kolmogorov.
The next 3 blocks, a grey block containing 'information', the next with some noise added, and one with the maximum amount of noise added, show an increasing information content according to the
mathematical definition:
│ increasing │ │ │ │
│ mathematical │ │ │ │
│ information │ 3.712 bytes │ 9.728 bytes │ 11.394 bytes │
│ content │ │ │ │
│ increasing │ │ │ │
│ meaningful │ no │ degraded │ information │
│ information │ information │ information │ │
│ Information content (same 3 jpg images are used in both rows) │
The 'information content' is reflected in the size of the jpeg files, because jpeg is a compression algorithm (all three have 50x75 pixels and have the same compression factor [14]).
So the ranking of human information (books) and random strings (noise) on the basis of the compressibility criterion yields a result clearly contradicting common sense knowledge. The point is that
the word 'information content' is misleading outside the mathematical context. I propose to call that kind of Information Content just what it is: compressibility and nothing more.
What is the difference between a random string and a book? The words in a book are carefully selected for their meaning. It should be no surprise that the compressibility definition of Information
Content yields such paradoxal results, because compressibility ignores the meaning of the symbols.
Another New Scientist reader Tony Castaldo wrote:
"It is only opposite if the book is going to be used by a human for human purposes, and of course everyday usage (in your world) is going to be about humans for human purposes. For a computer, the
book would have lower information content."
However a software program written in Fortran, C++, Visual Basic, etc would have a lower Information Content than a random string.
I now list a number of authors pointing out the difference between information and meaningful information.
Hubert Yockey
"I pointed out above in the discussion about finding a definition and a measure of 'information' that the mathematical definition does not capture all the meanings that are associated intuitively
with that word. By the same token, the Kolmogorov-Chaitin complexity may likewise not capture all that one thinks of in the word 'complexity', nevertheless, it will prove very useful in molecular
biology" [12].
John Maynard Smith
"Information theorists use the phrase 'information is data plus meaning'" [11]. This is exactly what I mean. But who are those 'information theorists'?
Radu Popa
"Shannon's information theory is purely mathematical and makes information oblivious to function, whence it cannot distinguish between meaningful (instructional) signals and noise." [13]
Richard Feynman
"How can a random string contain any information, let alone the maximum amount? Surely we must be using the wrong definition of "information"? But if you think about it, the N-bit strings could
each label a message ..." [3]
Labels! Those random strings are labels, not messages! Without label each random string is just a random string and nothing more. Labels create meaning. Those random strings could be labeled in many
ways. A code table must be created such as the ASCII code table or the morse code table or the genetic code table. But then there is information in the code table. This type of information must fall
outside the original set of random strings.
Ian Stewart
People talk an awful lot of nonsense about DNA as "information" for an organism - as if information has any meaning outside of a well-defined context. [...] However, what's really important about
a message is not the quantity of information, but its quality. "Two plus two makes seventeen" is longer than "2+2=4" but is also nonsense. [4]
Warren Weaver
"The word information, in Shannon's theory, is used in a special sense that must not be confused with its ordinary usage. In particular, information must not be confused with meaning..." [5]
Paul Davies
"All random sequences of the same length encode about the same amount of information, but the quality of that information is crucial..." [6]
The helpful insight of Davies is that meaningful information must be a very specific and small subset of all random sequences. However: how do we define 'quality'? How do we define the subset?
James Gleick
"The birth of information theory came with its ruthless sacrifice of meaning–the very quality that gives information its value and its purpose."
from The Information A History, a Theory, a Flood (2011) quoted by Andrew Robinson in Science 30 Sep 2011: "Introducing his The Mathematical Theory of Communication, the founder of information
theory, Claude Shannon, bluntly declared meaning to be "irrelevant to the engineering problem".
Jim Crutchfield
Later I discovered that Crutchfield noted the above-mentioned paradox: pure randomness produces a high Shannon information quantity and also a high Kolmogorov algorithmic complexity.
"Information theory as written down by Shannon is not a theory of information content. It's a quantitative theory of information."
Crutchfield designed a measure of meaningful information that escapes this paradox. He called it statistical complexity. Statistical complexity is low when randomness is very high, but is also low
when randomness is very low (regularities). But in the intermediate range, where there's randomness mixed in with regularity, the statistical complexity is high [7].
I am not sure if Crutchfield succeeded, but he tries to avoid the paradox.
As noted above I propose to call the mathematical Chaitin-Kolmogorov concept of information 'compressibility' in order to prevent confusion of 'mathematical information' with 'human information'.
Furthermore, inspired by William Dembski's idea of specification [2], I propose the following. The subset of random strings humans call information, is determined by 'a dictionary of words'. It is
really the dictionary that determines the interesting subset we call information. The amount of meaningful information in a string of symbols depends on the amount of matches it has with strings in
an established dictionary. This is valid for human language but also for DNA language. Geneticists determine the meaning of a piece of DNA by searching in an established 'gene dictionary'
(established from many other species). The rest of the DNA string can be assumed random until a match is found with a dictionary word. This method has recently been demonstrated by a team that used a
bigger human genome 'gene-dictionary' (11 in stead of 2 databases) and found 65.000 - 75.000 matches (in stead of 30.000).
Of course there is a relation with the presence of 'dictionary words' and compressibility. But there is no linear relation between compressibility and meaning. My proposal is Dictionary based
Information Content: information content based on a dictionary specified in advance. This is not the maximum compressibility that communication engineers and webgraphics designers are after, however
it is definitely not subjective or arbitrary, because it easily can be implemented and executed by a computer program and it will capture 'meaning' in a quantitative way. Additionally, it shows that
meaning is relative to a dictionary. A French text tested with a Dutch dictionary will result in very low values. The software itself can test which dictionary yields the best result.
The disadvantage of this proposal is that a text containing random but correctly spelled words or even a dictionary itself has the highest score, which is not what we want. A grammar-checker should
be added. The more grammar errors the less the information content. But then, a text without spelling or grammar errors can be gibberish because such a 'text' can be produced by a computer program!
[this has been done and submitted to a scientific journal]. Worse, a text with some spelling and grammar errors can still be very meaningful.
Conclusion: for detecting meaningful DNA (genes!) in a genome, the dictionary method works fine. The dictionary method works also for separating a string of random letters from a string of words, but
it is unable to detect meaningfull texts. Consolation: even humans can fail to detect meaning in correctly spelled texts.
July 2001
I am very honoured that Dr Huen YK, Cah Research Centre, Singapore, has referenced version 1 of this article in his Mathematical Properties of DNA: Algebraic Sequence Analysis, Online Journal of
Bioinformatics Volume 1: 42-59, 2001. [this paper has been removed].
See his Brief Comments on junk DNA: is it really junk? (Feb 2002)
Wolfram's rule 30 conflicts with algorithmic information theory
22 June 2002, updated 25 Jan 2004 (10)
Stephen Wolfram describes that simple algorithms (cellular automata) can produce complex behaviour ("A New Kind of Science", 9). One algorithm (cellular automata, rule 30) produces a perfect random
result. All statistical tests failed to find deviations from randomness (p.1085) and claims of others to find regularities are wrong, Wolfram says. The rule-30 algorithm is deterministic: "if one
always ran the cellular automation [rule 30] with the particular initial conditions, then one would always get exactly the same sequence of 0's and 1's" (p.317). Wolfram calls this result "intrinsic
randomness", because there is no random input to this system. Now, that is a most remarkable result: a deterministic algorithm produces a truly random sequence.
According to algorithmic information theory any sequence of data that cannot be produced by an algorithm that is shorter than the data itself is 'algorithmically random' (uncompressible).
But at the same time Wolfram states that "any data generated by a simple program can by definition never be algorithmically random" (p.1067). But this clearly conflicts with rule-30, because the
output of his rule-30 algorithm is truly random. (a) How does Wolfram solve this conflict? (b). I did not find any evidence that Wolfram rejects the notion of algorithmic randomness. On the contrary,
he even asserts that by the early 1990s the notion of algorithmic randomness has become generally accepted as the appropriate ultimate definition of randomness." (p.1068) (even though it turns out to
be undecidable in general whether any particular sequence is algorithmically random). The only thing he has to say is a very cryptic statement: 'algorithmic randomness cannot be directly relevant to
the kind of randomness of cellular automata"! (p.1067). But how could algorithmic information theory be irrelevant for any data set, if it is the ultimate and general theory of randomness? It is
amazing that Wolfram remains silent on this crucial point (c).
It seems to me, that rule-30 shows that the notion of algorithmic information content is flawed. Wolfram discusses several data compression methods applied to his cellular automata. Consider for
example the asymmetry of generating and compressing strings: on the one hand simple algorithms producing monotone and repetitive patterns and mixed repetitive/random patterns, and on the other hand
when data compression techniques are applied to the data sets, they give short descriptions of different lengths. Monotone and repetitive patterns are strongly compressible and random patterns not,
but very similar simple algorithms generate them all.
The more I think about the very idea of a cellular automata algorithm, the more it seems to invalidate the algorithmic theory of information and complexity. They can be run infinitely long, so the
output can grow to an infinite length. Especially rule 30 can produce a complex and non-compressible pattern of an arbitrary size (d). To measure the complexity of such an output by the length of its
shortest description is definitely meaningless, because no matter how long the output, the algorithm that produced it stays the same. The rule-30 cellular automata always produces the same output
with the same initial conditions (it is deterministic). So it really is a simple description/prescription of that particular output.
Finally, returning to William Dembski (2), his Law of Conservation of Information is refuted by rule-30, since rule-30 generates information for free! (e)
Comments of a mathematician who published about algorithmic information theory:
a. I would tend to agree that algorithmic information theory doesn't capture our intuitive notion of complexity. But I'd also point out that intuitive notions are often inchoate and inconsistent;
making these notions precise often shows how our intuition can be wrong. No one has yet succeeded in replacing algorithmic information theory with a better notion.
b. The conflict exists solely in Wolfram's mind. He is equivocating, using two different notions of "random". One is the notion from algorithmic information theory; the other is the vague, fuzzy
fact that a handful of statistical tests don't distinguish between the output of rule 30 and a random sequence. But there are MANY examples of deterministic systems that pass many statistical
tests, and they have been known for a while. (In fact, these systems are even better than Wolfram's, because you can prove that these systems pass all POLYNOMIAL-TIME statistical tests, if you
make a certain complexity-theoretic assumption. See the work of Andrew Yao and Blum-Blum-Shub, for example.) Rules 30 and 110 are only of interest because of their simplicity. No one has proved,
for example, that they pass all polynomial-time statistical tests.
c. It is only irrelevant in the sense that if one applies algorithmic information to rules like rule 30 and rule 110, one quickly finds they are NOT random in the algorithmic information sense. Thus
algorithmic information theory does not capture our "intuitive" notion that rule 110 is "complex". But perhaps this shows our intuition is wrong. Or perhaps it shows that "complex" is a vague and
uncapturable notion.
d. This is wrong. Rule 30's pattern IS NOT "non-compressible". You can compress it very simply by saying "Rule 30" + description of input + length of output.
e. It does not generate information in the algorithmic information sense.
This discussion shows the difficulty of an outsider to assess the value of theoretical claims outside one's own expertise. It seems that algorithmic information theory is a nearly unassailable truth
for professional mathematicians. [GK]
Letter Serial Correlation
Posted: October 20, 2009
In this section several articles are posted describing a novel method for a statistical analysis of texts. This method is based on analyzing the "letter serial correlation" (LCS) that has been
discovered to exist in semantically meaningful texts, regardless of language, authorship, style, etc. It is absent in meaningless strings of letters; applying the LSC text enables one to distinguish
between a meaningful message and gibberish even if the text's language is unknown and even if the text's alphabet is not familiar. Thus the statistics in question opens a way for an analysis of texts
beyond the limitations of the classic information theory.
Truly Random
20 feb 2020
"A truly random number is hard to find. The numbers chosen by people are not random at all, and the pseudorandom numbers created by computers are not very random either. Now scientists have
developed a way to generate random numbers from a genuine source of randomness – the unpredictable stage of exactly when, where and how a crystal starts to grow in a solution.
The researchers analysed the numbers generated from crystals grown in three solutions and found that they all passed statistical tests for the quality of their randomness."
Tom Metcalfe Really random numbers created from crystals, Chemistry World, 18 February 2020. (free access)
1. Letter to the New Scientist by Gert Korthof (9 Oct 1999) and a reply to my letter to the New Scientist (30 Oct 1999).
2. On the origin of information by means of intelligent design a review by Gert Korthof.
3. Richard Feynman (1999) Feyman Lectures on Computation, p120.
4. Ian Stewart (1998) Life's other secret, p239.
5. Claude Shannon and Warren Weaver (1949): The Mathematical Theory of Communication.
6. Paul Davies (1999) The Fifth Miracle, p119.
7. Tom Siegfried (2000), The Bit and the Pendulum, chapter 8, p169-171.
8. A pseudo-random string generated by a computerprogram. A true random string is generated by a natural process such as radioactive decay. See for explanation site of Mads Haahr and Genuine random
numbers, generated by radioactive decay.
9. Stephen Wolfram (2002) A New Kind of Science, p28, p1085. There is a critical letter to Nature 418, 17 (04 July 2002) by Michael Phillips Beautiful vistas, but is this really science?, explaining
that there is no link with reality. There is a (positive) review of the book by John L Casti "Science is a computer program", Nature, 417, 381-382 (23 May 2002). There is a news feature article
by Jim Giles (assistant news and features editor): "Stephen Wolfram: What kind of science is this?", Nature, 417, 216-218 26 May 2002.
10. I thank Otto B. Wiersma and Jeffrey Shallit for discussion.
11. John Maynard Smith, Eörs Szathmáry (1999) The Origins of Life. From the Birth of Life to the Origin of Language, p.11.
12. Hubert Yockey (1992) Information theory and molecular biology. See review on this site.
13. Radu Popa (2004) Between Necessity and Probability: Searching for the Definition and Origin of Life, p.95
14. The amount of noise added in both cases is 400% (maximum) and has a Gaussian distribution (Photoshop). Each resulting figure is unique and has different information content (file size). Only one
is given. The jpeg files do not have maximal compression, so file sizes are not equal to information content.
Further Reading:
• Christophe Menant Introduction to a Systemic Theory of Meaning. [ 14 Nov 2004 ]
• NewScientist.com news service 17 August 04: Prize draw uses hear for random numbers. "Intel's random number generating chip uses thermal noise from transistors as its source of randomness".
• Ricardo B.R. Azevedo et al (2005) "The simplicity of metazoan cell lineages", Nature, 433, 152-156 (13 Jan 2005) define the algorithmic complexity of a cell linage as the length of the shortest
description of the lineage based on its constituents sublineages. According to this definition the complexity the embryonic developmental process is simpler than would be expected by chance. This
is a very interesting example of the application of Kolmogorov complexity to biology. [ 13 Jan 2005 ]
• Thomas D. Schneider wrote to me: Resolution to your information paradoxes [14 May 2005]
Information Content, Compressibility and Meaning (this page)
You are right, there is a problem with the way most people present information measures. But if you go back to Shannon and read his work *carefully* you will see that *in the presence of noise*
one must take the information measure as a difference of uncertanties.
I have written up this pitfall at: http://www.lecb.ncifcrf.gov/~toms/pitfalls.html
My viewpoint is that 'algorithmic complexity' is a useless concept, especially since nobody can actually give you measures of the thing!
Shannon's measure works and can be used to evaluate information in living things. You can explore my entire web site but I think that you will find my paper on the evolution of information to be
the most interesting:
T. D. Schneider (2000), Evolution of Biological Information, Nucleic Acids Res, Second issue in the month, July 15, release date July 10, 28,14, pp2794-2799, 2000.
• See more about information and meaning "What is information?" (pp.92-104) in Computer Viruses, Artificial Life and Evolution by Mark A. Ludwig (review).
• Gregory Chaitin (2005) Meta Math! The Quest for Omega, Pantheon Books, 223 pages. Review: Nature, 439, 16 Feb 06 pp790-791.
• Gregory Chaitin (2006) "The Limits of Reason", Scientific American, Mar 2006. "Ideas on complexity and randomness originally suggested by Gottfired W. Leibniz in 1686, combined with modern
information theory, imply that there can never be a "theory of everything" for all of mathematics."
• Robert M. Hazen, Patrick L. Griffin, James M. Carothers, and Jack W. Szostak (2007) Functional information and the emergence of biocomplexity, PNAS, published online May 9, 2007; "It has been
difficult to define complexity in terms of a metric that applies to all complex systems." "Despite this diversity, a common thread is present: All complex systems alter their environments in one
or more ways, which we refer to as functions." "Function is thus the essence of complex systems". "Here we explore the functional information of randomly generated populations of Avida
organisms." Gene reductionism is prevented: an Avida genome consist of multiple lines of machine instructions. "None of these computational tasks can be performed by the execution of a single
instruction; indeed, the shortest functional program requires five instructions. The computational ability (function) of Avida organisms thus emerges from the interaction of instructions".
• Valerio Scarani (2010) 'Information science: Guaranteed randomness', News and Views. Nature 464, 988-989 (15 April 2010)
"You have received a device that is claimed to produce random numbers, but you don't trust it. Can you check it without opening it? In some cases, you can, thanks to the bizarre nature of quantum
"Starting with the comic strip, Dilbert's guide is uttering a scientific truth: the list 9, 9, 9, 9, 9, 9 is as valid an output of a generator of random numbers as is 1, 2, 3, 4, 5, 6 or 4, 6, 7,
1, 3, 8. In fact, in a long enough sequence of lists, any list of numbers should appear with the same frequency. Classic, 'black-box', tests of randomness exploit this idea: they check for the
relative frequencies of lists. But, in practice, no such test can distinguish a sequence generated by a truly random process from one generated by a suitable deterministic algorithm that repeats
itself after, for example, 10^23 numbers. Moreover, in this way, whether the numbers are private cannot be checked: even if the sequence had initially been generated by a truly random process, it
could have been copied several times, and the random generator may just be reading from a record."
• S. Pironio et al (2010) Random numbers certified by Bell's theorem', Nature, 15 Apr 2010.
"The characterization of true randomness is elusive. There exist statistical tests used to verify the absence of certain patterns in a stream of numbers, but no finite set of tests can ever be
considered complete, as there may be patterns not covered by such tests. For example, certain pseudo-random number generators are deterministic in nature, yet produce results that satisfy all the
randomness tests. At a more fundamental level, there is no such thing as true randomness in the classical world: any classical system admits in principle a deterministic description and thus
appears random to us as a consequence of a lack of knowledge about its fundamental description."
• Dual_EC_DRBG: Dual_EC_DRBG is a pseudorandom number generator that was promoted as a cryptographically secure pseudorandom number generator (CSPRNG) by the National Institute of Standards and
Technology. "the random-number generator, the 'Dual EC DRBG' standard, had been hacked by the NSA so that its output would not be as random as it should have been." Nature, 19 Dec 2013
• Commericial Random Number Generation: Quantis is a family of hardware random number generators (RNG) which exploit elementary quantum optical processes as a source of true randomness. [added: 20
sep 2017] | {"url":"https://www.wasdarwinwrong.com/kortho44a.htm","timestamp":"2024-11-07T08:43:03Z","content_type":"text/html","content_length":"35805","record_id":"<urn:uuid:1f053ac9-1298-431b-b193-f78482d9674e>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00615.warc.gz"} |
Computing an st-numbering
Lempel, Even and Cederbaum proved the following result: Given any edge {st} in a biconnected graph G with n vertices, the vertices of G can be numbered from 1 to n so that vertex s receives number 1,
vertex t receives number n, and any vertex except s and t is adjacent both to a lower-numbered and to a higher-numbered vertex (we call such a numbering an st-numbering for G). They used this result
in an efficient algorithm for planarity-testing. Here we provide a linear-time algorithm for computing an st-numbering for any biconnected graph. This algorithm can be combined with some new results
by Booth and Lueker to provide a linear-time implementation of the Lempel-Even-Cederbaum planarity-testing algorithm.
All Science Journal Classification (ASJC) codes
• Theoretical Computer Science
• General Computer Science
Dive into the research topics of 'Computing an st-numbering'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/computing-an-st-numbering","timestamp":"2024-11-03T10:21:22Z","content_type":"text/html","content_length":"45392","record_id":"<urn:uuid:16c7988b-d003-4dfa-bd42-5f336b55e1f2>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00069.warc.gz"} |
FiveMinute Check over Lesson 13 5 CCSS ThenNow
Download presentation
Five-Minute Check (over Lesson 13– 5) CCSS Then/Now New Vocabulary Example 1: Real-World Example: Identify Mutually Exclusive Events Key Concept: Probability of Mutually Exclusive Events Example 2:
Real-World Example: Mutually Exclusive Events Key Concept: Probability of Events That Are Not Mutually Exclusive Example 3: Real-World Example: Events That Are Not Mutually Exclusive Key Concept:
Probability of the Complement of an Events Example 4: Complementary Events Concept Summary: Probability Rules Example 5: Real-World Example: Identify and Use Probability Rules
Over Lesson 13– 5 Determine whether the event is independent or dependent. Samson ate a piece of fruit randomly from a basket that contained apples, bananas, and pears. Then Susan ate a second piece
from the basket. A. independent B. dependent
Over Lesson 13– 5 Determine whether the event is independent or dependent. Kimra received a passing score on the mathematics portion of her state graduation test. A week later, she received a passing
score on the reading portion of the test. A. independent B. dependent
Over Lesson 13– 5 A spinner with 4 congruent sectors labeled 1– 4 is spun. Then a die is rolled. What is the probability of getting even numbers on both events? A. 1 B. C. D.
Over Lesson 13– 5 Two representatives will be randomly chosen from a class of 20 students. What is the probability that Janet will be selected first and Erica will be selected second? A. B. C. D.
Content Standards S. CP. 1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of
other events (“or, ” “and, ” “not”). S. CP. 7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. Mathematical Practices 1 Make sense of
problems and persevere in solving them. 4 Model with mathematics.
You found probabilities of independent and dependent events. • Find probabilities of events that are mutually exclusive and events that are not mutually exclusive. • Find probabilities of
• mutually exclusive events • complement
Identify Mutually Exclusive Events A. CARDS Han draws one card from a standard deck. Determine whether drawing an ace or a 9 is mutually exclusive or not mutually exclusive. Explain your reasoning.
Answer: These events are mutually exclusive. There are no common outcomes. A card cannot be both an ace and a 9.
Identify Mutually Exclusive Events B. CARDS Han draws one card from a standard deck. Determine whether drawing a king or a club is mutually exclusive or not mutually exclusive. Explain your
reasoning. Answer: These events are not mutually exclusive. A king that is a club is an outcome that both events have in common.
A. For a Halloween grab bag, Mrs. Roth has thrown in 10 caramel candy bars, 15 peanut butter candy bars, and 5 apples to have a healthy option. Determine whether drawing a candy bar or an apple is
mutually exclusive or not mutually exclusive. A. The events are mutually exclusive. B. The events are not mutually exclusive.
B. For a Halloween grab bag, Mrs. Roth has thrown in 10 caramel candy bars, 15 peanut butter candy bars, and 5 apples to have a healthy option. Determine whether drawing a candy bar or something with
caramel is mutually exclusive or not mutually exclusive. A. The events are mutually exclusive. B. The events are not mutually exclusive.
Mutually Exclusive Events COINS Trevor reaches into a can that contains 30 quarters, 25 dimes, 40 nickels, and 15 pennies. What is the probability that the first coin he picks is a quarter or a
penny? These events are mutually exclusive, since the coin picked cannot be both a quarter or a penny. Let Q represent picking a quarter. Let P represent picking a penny. There a total of 30 + 25 +
40 + 15 or 110 coins.
Mutually Exclusive Events P(Q or P) = P(Q) + P(P) Probability of mutually exclusive events Simplify. Answer: ___ 9 22 or about 41%
MARBLES Hideki collects colored marbles so he can play with his friends. The local marble store has a grab bag that has 15 red marbles, 20 blue marbles, 3 yellow marbles and 5 mixed color marbles. If
he reaches into a grab bag and selects a marble, what is the probability that he selects a red or a mixed color marble? A. B. C. D.
Events That Are Not Mutually Exclusive ART Use the table below. What is the probability that Namiko selects a watercolor or a landscape? Since some of Namiko’s paintings are both watercolors and
landscapes, these events are not mutually exclusive. Use the rule for two events that are not mutually exclusive. The total number of paintings from which to choose is 30.
Events That Are Not Mutually Exclusive Let W represent watercolors and L represent landscapes. Substitution Simplify. Answer: The probability that Namiko selects a watercolor or a landscape is or
about 66%.
SPORTS Use the table. What is the probability that if a high school athlete is selected at random that the student will be a sophomore or a basketball player? A. B. C. D.
Complementary Events GAMES Miguel bought 15 chances to pick the one red marble from a container to win a gift certificate to the bookstore. If there is a total of 200 marbles in the container, what
is the probability Miguel will not win the gift certificate? Let event A represent selecting one of Miguel’s tickets. Then find the probability of the complement of A. Probability of a complement
Substitution Subtract and simplify.
Complementary Events Answer: The probability that one of Miguel’s tickets will not be selected is or about 93%.
RAFFLE At a carnival, Sergio bought 18 raffle tickets, in order to win a gift certificate to the local electronics store. If there is a total of 150 raffle tickets sold, what is the probability
Sergio will not win the gift certificate? A. B. C. D.
Identify and Use Probability Rules PETS A survey of Kingston High School students found that 63% of the students had a cat or a dog for a pet. If two students are chosen at random from a group of 100
students, what is the probability that at least one of them does not have a cat or a dog for a pet?
Identify and Use Probability Rules Understand You know that 63% of the students have a cat or a dog for a pet. The phrase at least one means one or more. So you need to find the probability that
either • the first student chosen does not have a cat or a dog for a pet or • the second student chosen does not have a cat or a dog for a pet or • both students chosen do not have a cat or a dog for
a pet.
Identify and Use Probability Rules Plan The complement of the event described is that both students have a cat or a dog for a pet. Find the probability of this event, and then find the probability of
its complement. Let event A represent choosing a student that does have a cat or a dog for a pet. Let event B represent choosing a student that does have a cat or a dog for a pet, after the first
student has already been chosen. These are two dependent events, since the outcome of the first event affects the probability of the outcome of the second event.
Identify and Use Probability Rules Solve Probability of dependent events Multiply.
Identify and Use Probability Rules Probability of complement Substitution Subtract. Answer: So, the probability that at least one of the students does not have a cat or a dog for a pet is or about
Identify and Use Probability Rules Check Use logical reasoning to check the reasonableness of your answer. The probability that one student chosen out of 100 does not have a cat or a dog for a pet is
(100 – 63)% or 37%. The probability that two people chosen out of 100 have a cat or a dog for a pet should be greater than 37%. Since 61% > 37%, the answer is reasonable.
PETS A survey of Lakewood High School students found that 78% of the students preferred riding a bicycle to riding in a car. If two students are chosen at random from a group of 100 students, what is
the probability that at least one of them does not prefer riding a bicycle to riding in a car? A. 32% B. 39% C. 43% D. 56% | {"url":"https://slidetodoc.com/fiveminute-check-over-lesson-13-5-ccss-thennow/","timestamp":"2024-11-09T20:31:53Z","content_type":"text/html","content_length":"155392","record_id":"<urn:uuid:87ab28b1-3360-4588-bcbe-0045c3cf914d>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00220.warc.gz"} |
Double Spring
Double Spring
This simulation shows two springs and masses connected to a wall. The graphs produced are called Lissajous curves and are generated by simple sine and cosine functions.
You can change parameters in the simulation such as mass or spring stiffness. You can drag either mass with your mouse to set the starting position.
The math behind the simulation is shown below. Also available: source code, documentation and how to customize.
Physics and Equations of Motion
The two springs act independently, so it is easy to figure out what are the forces acting on the two blocks. Label the springs and blocks as follows:
wall - spring[1] - block[1] - spring[2] - block[2]
We'll assume the origin is at the connection of the spring to the wall. Define the following variables (subscripts refer to block 1 or block 2):
• x[1], x[2] = position (left edge) of blocks
• v[1], v[2] = velocity of blocks
• F[1], F[2] = force experienced by blocks
• L[1], L[2] = how much spring is stretched
And define the following constants:
• m[1], m[2] = mass of blocks
• w[1], w[2] = width of blocks
• k[1], k[2] = spring constants
• R[1], R[2] = rest length of springs
The springs exert force based on their amount of stretch according to
F = −k × stretch
The forces on the blocks are therefore
F[1] = −k[1] L[1] + k[2] L[2]
F[2] = −k[2] L[2]
The stretch of the spring is calculated based on the position of the blocks.
L[1] = x[1] − R[1]
L[2] = x[2] − x[1] − w[1] − R[2]
Now using Newton's law F = m a and the definition of acceleration as a = x'' we can write two second order differential equations. These are the equations of motion for the double spring.
m[1] x[1]'' = −k[1] (x[1] − R[1]) + k[2] (x[2] − x[1] − w[1] − R[2])
m[2] x[2]'' = −k[2] (x[2] − x[1] − w[1] − R[2])
Numerical Solution
It is easy to convert the above second order equations to a set of first order equations. We define variables for the velocities v[1], v[2] . Then there are four variables x[1], x[2], v[1], v[2] and
a first-order differential equation for each:
x[1]' = v[1]
x[2]' = v[2]
v[1]' = −(k[1] ⁄ m[1]) (x[1] − R[1]) + (k[2] ⁄ m[1]) (x[2] − x[1] − w[1] − R[2])
v[2]' = −(k[2] ⁄ m[2]) (x[2] − x[1] − w[1] − R[2])
This is the form that we need in order to use the Runge-Kutta method for numerically solving the differential equation.
This web page was first published April 2001. | {"url":"https://www.myphysicslab.com/springs/double-spring-en.html?SHOW_ENERGY=true;SIM_CANVAS.BACKGROUND=;","timestamp":"2024-11-08T05:25:30Z","content_type":"text/html","content_length":"10178","record_id":"<urn:uuid:0d0a71b3-bbf1-45e6-972a-7b294811fe37>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00419.warc.gz"} |
What is the time complexity for accessing an element in an array - ITEagers
Data Structure - Question Details
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How counting by 10 helps children learn about the meaning of numbers - International Maths Challenge
How counting by 10 helps children learn about the meaning of numbers
When children start school, they learn how to recite their numbers (“one, two, three…”) and how to write them (1, 2, 3…). Learning about what those numbers mean is even more challenging, and this
becomes trickier yet when numbers have more than one digit — such as 42 and 608.
It turns out that the meaning of such “multidigit” numbers cannot be gleaned from simply looking at them or by performing calculations with them. Our number system has many hidden meanings that are
not transparent, making it difficult for children to comprehend it.
In collaboration with elementary teachers, the Mathematics Teaching and Learning Lab at Concordia University explores tools that can support young children’s understanding of multidigit numbers.
We investigate the impact of using concrete objects (like bundling straws into groups of 10). We also investigate the use of visual tools, such as number lines and charts, or words to represent
numbers (the word for 40 is “forty”) and written notation (for example, 42).
Our recent research examined whether the “hundreds chart” — 10 by 10 grids containing numbers from one to 100, with each row in the chart containing numbers in groups of 10 — could be useful for
teaching children about counting by 10, something foundational for understanding how numbers work.
When children start learning about numbers, they do not naturally see tens and ones in a number like 42. (Shutterstock)
What’s in a number?
Most adults know that the placement of the “4” and “2” in 42 means four tens and two ones, respectively.
But when young children start learning about numbers, they do not naturally see 10s and ones in a number like 42. They think the number represents 42 things counted from one to 42 without
distinguishing between the meaning of the digits “4” and “2.” Over time, through counting and other activities, children see the four as a collection of 40 ones.
This realization is not sufficient, however, for learning more advanced topics in math.
An important next step is to see that 42 is made up of four distinct groups of 10 and two ones, and that the four 10s can be counted as if they were ones (for example, 42 is one, two, three, four 10s
and one, two, “ones”).
Ultimately, one of the most challenging aspects of understanding numbers is that groups of ten and ones are different kinds of units.
Numbers can be arranged in different ways
The numbers in hundreds charts can be arranged in different ways. A top-down hundreds chart has the digit “1” in the top-left corner and 100 in the bottom-right corner.
A top-down hundreds chart. (Vera Wagner), Author provided (no reuse)
The numbers increase by 10 moving downward one row at a time, like going from 24 to 34 using one hop down, for instance. A second type of chart is the “bottom-up” chart, which has the numbers
increasing in the opposite direction.
A bottom-up hundreds chart. (Vera Wagner), Author provided (no reuse)
Counting by 10s
Children can move from one number to another in the chart to solve problems. Considering 24 + 20, for example, children could start on 24 and move 20 spaces to land on 44.
Another way would be to move up (or down, depending on the chart) two rows (for example, counting “one,” “two”) until they land on 44. This second method shows a developing understanding of
multidigit numbers being composed of distinct groups of 10, which is critical for an advanced knowledge of the number system.
For her master’s research at Concordia University, Vera Wagner, one of the authors of this story, thought children might find it more intuitive to solve problems with the bottom-up chart, where the
numbers get larger with upward movement.
After all, plants grow taller and liquid rises in a glass as it is filled. Because of such familiar experiences, she thought children would move by tens more frequently in the bottom-up chart than in
the top-down chart.
Study with kindergarteners, Grade 1 students
To examine this hypothesis, we worked with 47 kindergarten and first grade students in Canada and the United States. All the children but one spoke English at home. In addition to English, 14 also
spoke French, four spoke Spanish, one spoke Russian, one spoke Arabic, one spoke Mandarin and one communicated to some extent in ASL at home.
We assigned all child participants in the study an online version of either a top-down or bottom-up hundreds chart, programmed by research assistant André Loiselle, to solve arithmetic word problems.
What we found surprised us: children counted by tens more often with the top-down chart than the bottom-up one. This was the exact opposite of what we thought they might do!
This finding suggests that the top-down chart fosters children’s counting by tens as if they were ones (that is, up or down one row at a time), an important step in their mathematical development.
Children using the bottom-up chart were more likely to confuse the digits and move in the wrong direction.
Tools can impact learning
Tools used in the math classroom can impact children’s learning. (Shutterstock)
Our research suggests that the types of tools used in the math classroom can impact children’s learning in different ways.
One advantage of the top-down chart could be the corresponding
Our research suggests that the types of tools used in the math classroom can impact children’s learning in different ways.
One advantage of the top-down chart could be the corresponding left-to-right and downward movement that matches the direction in which children learn to read in English and French, the official
languages of instruction in the schools in our study. Children who learn to read in a different direction (for example, from right to left, as in Arabic) may interact with some math tools differently
from children whose first language is English or French.
The role of cultural experiences in math learning opens up questions about the design of teaching tools for the classroom, and the relevance of culturally responsive mathematics teaching. Future
research could seek to directly examine the relation between reading direction and the use of the hundreds chart.
For more such insights, log into our website https://international-maths-challenge.com
Credit of the article given to The Conversation | {"url":"https://international-maths-challenge.com/how-counting-by-10-helps-children-learn-about-the-meaning-of-numbers/","timestamp":"2024-11-08T14:10:08Z","content_type":"text/html","content_length":"152100","record_id":"<urn:uuid:e0c7ee81-7dc3-4f6d-a6a4-dbebb2e2fa02>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00266.warc.gz"} |
Firefly and PC GAMESS-related discussion club
Learn how to ask questions correctly
Re^3: Energy rises and oscillate at the end of optimization
Pavlo Solntsev
Dear Oleg.
It looks like you starting geometry very close to a transition state. I would recommend you to calculate hessian first and then perturb imaginary mode if it exists. Then try run optimization starting
from exact hessian. Algorithms for transition state search and minimum are similar.
Hope this helps.
On Thu Sep 11 '14 8:48pm, Oleg Levitskiy wrote
>Dear Pavlo,
>optimization with corrected parameters did 9 steps and unfortunately almost all of them with energy encrease. It seems that convergence to the predefined gradient value can't be achieved in case of
applying solvent modeling as you've supposed...
>Nonetheless I attached the resulting out in hope of appearing of any new ideas.
>Thanks again,
>On Thu Sep 11 '14 5:18am, Pavlo Solntsev wrote
>>I would recommend you try these parameters first:
>> $contrl scftyp=rhf dfttyp=pbe96
>> runtyp=OPTIMIZE icharg=-1 nprint=-5
>> maxit=100 ICUT=11 ITOL=30 NZVAR=168 $end
>> $system mwords=100 $end
>> $SCF DIRSCF=.true. maxdii=20 $end
>> $guess guess=moread $end
>> $ZMAT DLC=.T. AUTO=.T.$end
>> $p2p p2p=.t. dlb=.t. $end
>> $basis gbasis=TZV $end
>> $PCM PCMTYP=DPCM SOLVNT=input RSOLV=2.155 EPS=35.688 EPSINF=1.806874 $END
>> $PCMCAV RIN(58)=1.956 $END
>>Before apply a solvent correction, you need to optimize the geometry in a gas phase. Make sure you use a right functional. I am skeptical about pbe96. Do not expect very tight gradient for the
geometry with the solvent correction. Usually 10^-4 is ok.
>>On Tue Sep 9 '14 3:30pm, Oleg Levitskiy wrote
>>>Dear Firefly Users,
>>>I've tryed to perform an optimization of anionic complex of Ni (DFT/PBE96, PCM) and faced a problem with convergence. Energy has unexpectedly risen and fluctuate spontaneousely from step to step.
I've tryed to increase accuracy (ICUT, ITOL values and also DLCTOL and ORTTOL), to change optimization method (GDIIS, NR), to perform optimization in cartesian or delocalized coordinates, but the
result was similar in all cases. Could you advise me something else to solve this problem.
>>>The beginning of my input is following:
>>> $contrl scftyp=rhf dfttyp=pbe96
>>>runtyp=OPTIMIZE icharg=-1 nprint=-5
>>>maxit=100 ICUT=20 ITOL=30 NZVAR=168 $end
>>> $system kdiag=-1 mwords=100 $end
>>> $SCF DIRSCF=.true. maxdii=20 $end
>>> $guess guess=moread $end
>>> $STATPT method=NR $end
>>> $ZMAT DLC=.T. AUTO=.T.
>>>DLCTOL=1D-7 ORTTOL=1D-7 $end
>>> $INTGRL PACKAO=.true. $end
>>> $p2p p2p=.t. dlb=.t. $end
>>> $basis gbasis=TZV $end
>>> $PCM PCMTYP=DPCM SOLVNT=input RSOLV=2.155 EPS=35.688 EPSINF=1.806874 $END
>>> $PCMCAV RIN(58)=1.956 $END
>>>Typical out-file is attached.
>>>Thank you,
Fri Sep 12 '14 5:30am
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Holiday Printable Activities
Math Facts Printables
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tables in color 1 to 12 math worksheet. | {"url":"https://phpmyadmin.muycomputerpro.com/art/practice-exercises/math-facts-printables.html","timestamp":"2024-11-10T01:54:57Z","content_type":"text/html","content_length":"34328","record_id":"<urn:uuid:6749adcd-22b7-4b14-bc02-4a61da9fe76e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00720.warc.gz"} |
Find Volume Of Cylinder Worksheets [PDF]: Algebra 1 Math
How Will This Worksheet on "Find Volume of Cylinder" Benefit Your Student's Learning?
• Volume of a cylinder worksheets provide an interactive way for students to learn and visualize the concept.
• This allows students to clear their concepts as they progress through each section successfully.
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• This real-world application helps students understand the practical relevance of the concept.
How to Find Volume of Cylinder?
• Firstly identify the given measurements: Find the radius of the base and the height of the cylinder.
• Formula to be used: `V = π×r^2×h` where `π` is approximately equal to `3.14`, `r` is the radius of the base, and `h` is the height.
• Substitute the values of given radius and height into the formula then calculate the volume.
• Simplify the expression by multiplying the radius squared by the height and then multiplying by `π`. If necessary, use a calculator to find the value of `π`.
• Ensure final answer should be in cubic units, such as cubic centimeters or cubic meters.
Q. A cylinder has a base radius of $5 \mathrm{~cm}$ and a height of $16 \mathrm{~cm}$. What is its volume in cubic $\mathrm{cm}$, to the nearest tenths place? | {"url":"https://www.bytelearn.com/math-algebra-1/worksheet/find-volume-of-cylinder","timestamp":"2024-11-10T05:27:08Z","content_type":"text/html","content_length":"115797","record_id":"<urn:uuid:37bdb5a8-b3c6-4382-a55c-28e5dd2cc18e>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00283.warc.gz"} |
Informal Friday Seminar: Romanov -- The Lusztig-Vogan module of the Hecke algebra
SMS scnews item created by Anna Romanov at Mon 4 Nov 2019 1100
Type: Seminar
Modified: Mon 4 Nov 2019 1336
Distribution: World
Expiry: 3 Nov 2020
Calendar1: 8 Nov 2019 1430-1630
CalLoc1: Quad S225
CalTitle1: Romanov - The Lusztig-Vogan module of the Hecke algebra
Auth: romanova@10.17.127.166 (arom8272) in SMS-SAML
Informal Friday Seminar: Romanov -- The Lusztig-Vogan module of the Hecke algebra
Let G be a real reductive Lie group (think GL(n,R)). When studying the representation
theory of such a group, one quickly encounters a well-behaved class of representations
called admissible representations. The combinatorial behaviour of these representations
(e.g. composition series multiplicities of standard representations) is captured
by a certain geometrically-defined module over the associated Hecke algebra, the
Lusztig-Vogan module. In this talk, I will describe the construction of the
Lusztig-Vogan module, then we will see what it looks like explicitly in some SL2
examples. If we are lucky, we might see a glimpse of a mysterious feature called Vogan
duality. This talk is related to my previous IFS talks on unitary representation
theory, equivariant cohomology, and the admissible dual of SL(2,R), but I will assume
that the audience has no recollection of anything I have previously said.
Calendar (ICS file) download, for import into your favourite calendar application
UNCLUTTER for printing
AUTHENTICATE to mark the scnews item as read | {"url":"https://www.maths.usyd.edu.au/s/scnitm/romanova-InformalFridaySeminar-Rom-001","timestamp":"2024-11-11T14:49:33Z","content_type":"text/html","content_length":"3102","record_id":"<urn:uuid:d72a902c-3f5e-452e-a1a5-6381c9267836>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00024.warc.gz"} |
Linear Mixed Effects Models
Linear Mixed Effects Models¶
Linear Mixed Effects models are used for regression analyses involving dependent data. Such data arise when working with longitudinal and other study designs in which multiple observations are made
on each subject. Some specific linear mixed effects models are
• Random intercepts models, where all responses in a group are additively shifted by a value that is specific to the group.
• Random slopes models, where the responses in a group follow a (conditional) mean trajectory that is linear in the observed covariates, with the slopes (and possibly intercepts) varying by group.
• Variance components models, where the levels of one or more categorical covariates are associated with draws from distributions. These random terms additively determine the conditional mean of
each observation based on its covariate values.
The statsmodels implementation of LME is primarily group-based, meaning that random effects must be independently-realized for responses in different groups. There are two types of random effects in
our implementation of mixed models: (i) random coefficients (possibly vectors) that have an unknown covariance matrix, and (ii) random coefficients that are independent draws from a common univariate
distribution. For both (i) and (ii), the random effects influence the conditional mean of a group through their matrix/vector product with a group-specific design matrix.
A simple example of random coefficients, as in (i) above, is:
\[Y_{ij} = \beta_0 + \beta_1X_{ij} + \gamma_{0i} + \gamma_{1i}X_{ij} + \epsilon_{ij}\]
Here, \(Y_{ij}\) is the \(j^\rm{th}\) measured response for subject \(i\), and \(X_{ij}\) is a covariate for this response. The “fixed effects parameters” \(\beta_0\) and \(\beta_1\) are shared by
all subjects, and the errors \(\epsilon_{ij}\) are independent of everything else, and identically distributed (with mean zero). The “random effects parameters” \(\gamma_{0i}\) and \(\gamma_{1i}\)
follow a bivariate distribution with mean zero, described by three parameters: \({\rm var}(\gamma_{0i})\), \({\rm var}(\gamma_{1i})\), and \({\rm cov}(\gamma_{0i}, \gamma_{1i})\). There is also a
parameter for \({\rm var}(\epsilon_{ij})\).
A simple example of variance components, as in (ii) above, is:
\[Y_{ijk} = \beta_0 + \eta_{1i} + \eta_{2j} + \epsilon_{ijk}\]
Here, \(Y_{ijk}\) is the \(k^\rm{th}\) measured response under conditions \(i, j\). The only “mean structure parameter” is \(\beta_0\). The \(\eta_{1i}\) are independent and identically distributed
with zero mean, and variance \(\tau_1^2\), and the \(\eta_{2j}\) are independent and identically distributed with zero mean, and variance \(\tau_2^2\).
statsmodels MixedLM handles most non-crossed random effects models, and some crossed models. To include crossed random effects in a model, it is necessary to treat the entire dataset as a single
group. The variance components arguments to the model can then be used to define models with various combinations of crossed and non-crossed random effects.
The statsmodels LME framework currently supports post-estimation inference via Wald tests and confidence intervals on the coefficients, profile likelihood analysis, likelihood ratio testing, and AIC.
In [1]: import statsmodels.api as sm
In [2]: import statsmodels.formula.api as smf
In [3]: data = sm.datasets.get_rdataset("dietox", "geepack").data
In [4]: md = smf.mixedlm("Weight ~ Time", data, groups=data["Pig"])
In [5]: mdf = md.fit()
In [6]: print(mdf.summary())
Mixed Linear Model Regression Results
Model: MixedLM Dependent Variable: Weight
No. Observations: 861 Method: REML
No. Groups: 72 Scale: 11.3669
Min. group size: 11 Log-Likelihood: -2404.7753
Max. group size: 12 Converged: Yes
Mean group size: 12.0
Coef. Std.Err. z P>|z| [0.025 0.975]
Intercept 15.724 0.788 19.952 0.000 14.179 17.268
Time 6.943 0.033 207.939 0.000 6.877 7.008
Group Var 40.394 2.149
Detailed examples can be found here
There are some notebook examples on the Wiki: Wiki notebooks for MixedLM
Technical Documentation¶
The data are partitioned into disjoint groups. The probability model for group \(i\) is:
\[Y = X\beta + Z\gamma + Q_1\eta_1 + \cdots + Q_k\eta_k + \epsilon\]
• \(n_i\) is the number of observations in group \(i\)
• \(Y\) is a \(n_i\) dimensional response vector
• \(X\) is a \(n_i * k_{fe}\) dimensional matrix of fixed effects coefficients
• \(\beta\) is a \(k_{fe}\)-dimensional vector of fixed effects slopes
• \(Z\) is a \(n_i * k_{re}\) dimensional matrix of random effects coefficients
• \(\gamma\) is a \(k_{re}\)-dimensional random vector with mean 0 and covariance matrix \(\Psi\); note that each group gets its own independent realization of gamma.
• \(Q_j\) is a \(n_i \times q_j\) dimensional design matrix for the \(j^\rm{th}\) variance component.
• \(\eta_j\) is a \(q_j\)-dimensional random vector containing independent and identically distributed values with variance \(\tau_j^2\).
• \(\epsilon\) is a \(n_i\) dimensional vector of i.i.d normal errors with mean 0 and variance \(\sigma^2\); the \(\epsilon\) values are independent both within and between groups
\(Y, X, \{Q_j\}\) and \(Z\) must be entirely observed. \(\beta\), \(\Psi\), and \(\sigma^2\) are estimated using ML or REML estimation, and \(\gamma\), \(\{\eta_j\}\) and \(\epsilon\) are random so
define the probability model.
The marginal mean structure is \(E[Y|X,Z] = X*\beta\). If only the marginal mean structure is of interest, GEE is a good alternative to mixed models.
• \(cov_{re}\) is the random effects covariance matrix (referred to above as \(\Psi\)) and \(scale\) is the (scalar) error variance. There is also a single estimated variance parameter \(\tau_j^2\)
for each variance component. For a single group, the marginal covariance matrix of endog given exog is \(scale*I + Z * cov_{re} * Z\), where \(Z\) is the design matrix for the random effects in
one group.
The primary reference for the implementation details is:
• MJ Lindstrom, DM Bates (1988). Newton Raphson and EM algorithms for linear mixed effects models for repeated measures data. Journal of the American Statistical Association. Volume 83, Issue 404,
pages 1014-1022.
See also this more recent document:
All the likelihood, gradient, and Hessian calculations closely follow Lindstrom and Bates.
The following two documents are written more from the perspective of users:
Module Reference¶
The model class is:
MixedLM(endog, exog, groups[, exog_re, ...]) Linear Mixed Effects Model
The result class is:
MixedLMResults(model, params, cov_params) Class to contain results of fitting a linear mixed effects model.
Last update: Oct 03, 2024 | {"url":"https://www.statsmodels.org/stable/mixed_linear.html","timestamp":"2024-11-02T17:57:21Z","content_type":"text/html","content_length":"51308","record_id":"<urn:uuid:d391a7f1-0db5-4861-9fd6-029c1206f7a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00747.warc.gz"} |
Sizing a Gas-Fired Combi Boiler - GreenBuildingAdvisor
Sizing a Gas-Fired Combi Boiler
My house is 5 year old single story 2250 SQFT home with a Superior Wall insulated basement located in Warren Pa Zip Code 16365.
Currently my Heat/AC is provided with a Closed Loop Geothermal Heat pump (5 ton) connected to a mix of radiant tube and Forced Air Central Air system (See Attached Diagram). I have recently been
given access to Lease Gas and want to install a Gas fired Combi unit (into my existing system) to provide my DHW and home heating, which will significantly reduce my electric bill.
Initially I was told to just install a Weil McLain Aqua Balance 150 and it would do everything I will need, but after reading a lot of the posts on here about sizing boilers and combi units I think
an AB 150 is oversized and would have short cycle issues.
I do not have a Manual J for the home and have not been able to find anyone to perform one, but I did find a heat loss calculator on the web and was able to generate a Heat loss calculation (See
Attached Document). Unfortunately my blower door test did not include the garage so theACH50 infiltration rate does not include the garage so that skews the Infiltration data on the Heat Loss
Blower Door Test results for living area are: CFM50 1658, ACH@50 = 2.38, ACH Nat = .1575 (I know I need to install an HRV to provide ventilation, which I am currently researching)
As I sifted through all of the data provided and reviewed all of the boiler sizing posts I am confused on a couple of points:
When sizing a combi unit how do you select a the proper size? Is it the Heating load or the DHW requirement that dictates the size? (An AB150 seem oversized)
Given my need for 5 GPM of DHW and the BTUh of heating given in the Heat loss calculations there doesn’t seem to be a combi unit that fits the requirement.
How does a units Turn Down capability fit into the equation?
How does the outdoor reset effect the decision?
Would it make more sense to install a dual use hot water heater (like an HTP Versa Hydro)?
Thank You for any info and guidance.
1. Expert Member
Dana Dorsett | | #1
>" When sizing a combi unit how do you select a the proper size? Is it the Heating load or the DHW requirement that dictates the size?"
It's a function of both. The high-fire output needs to be able to cover the domestic hot water load, which usually several times the space heating load. The min-fire output has to be low enough
to not short-cycle on the available zone radiation. If the output is being buffered in a buffer tank of sufficient size (the Versa is inherently self buffered) the latter requirement goes away.
A temperature rise of 70F (mid-winter 35F incoming, 105F at the shower heads) and 5 gpm (= 2500lb/hr) flow takes 70F x 2500lb/hr= 175,000 BTU/hr of burner output. That is clearly outside the max
capacity of the W-M Aqua Balance 150, but within the range of the HTP UFTC-199 or Navien NCB-240E combi boilers.
Dlauffenburger | | #8
Thanks for the info. Attached is a copy of the heat loss calcs. I have been fighting with the system trying to get it add to my OP but I wouldn't allow 2 attachments.
joshdurston | | #2
Just out of curiosity, what temps are you maintaining your buffer tank at, it looks like everything is hydronic?
Fan coils and geothermal don't always make for an efficient pairing unless the coil was carefully selected.
But with properly sized and installed infloor heating, you should be able to get excellent efficiency.
I've seen some pretty poor performing geothermal system because they were set to maintain high temps all the time, but I've seen great performing systems with infloor running in the 75-95degF
Dlauffenburger | | #9
The buffer tank is a 40 gallon water heater, not connected to electric.
The Set Point on the Geo unit is 110F, but the system normally satisfies the heat call with the batch tank between 84 and 95F. I have only seen the batch tank get above 105 a couple of times.
The Air Handler is a MultiQqua 60CWA2.
1. Expert Member
Dana Dorsett | | #12
Running an EWT of ~90F water into a 5 ton air handler must deliver some pretty tepid air if it's used for heating. Can I assume that is only for cooling?
With a 4o gallon buffer configured as the hydraulic separator for a combi-boiler you can run the boiler fixed-temperature and still get there. If the buffer temp never exceeds 105F it
will always be running in condensing mode- no point to using outdoor reset. There is enough mass in the system to not really worry about modulating the burner with load too.
The Build-it-Solar load calculator isn't really very good as online load calculators go. Try Loadcalc.net if you really care about what the load number really are, not that it will change
the system by very much.
user-2890856 | | #3
Versa Hydro ( PHE130-55 ) would make the most sense considering your criteria . Low end modulation on most any combi will be too high for your loads for a VAST portion of the season . If you want
to avoid short cycling as stated you will require a buffer tank , probably ample sized , associated plumbing and cost .
REMEMBER : You will have to find that individual that can install either system properly . Your odds of that are much better with a package unit like the Versa Hydro
1. Expert Member
Dana Dorsett | | #5
The system currently has a 40 gallon buffer in the middle which could be put to good use here, and reasonable amounts of thermal mass in the zones to boot.
The performance issue would be the domestic hot water flow limitations, which are the same with any tankless. A Versa would fix that allowing MUCH higher flow rates, but it's an expensive fix
if not really needed. A 199KBTU tankless or combi can deliver about 5 gpm tub fills or showers even at the coldest PA incoming water temps.
4. Expert Member
Dana Dorsett | | #4
Looking at your zone radiation and system block schematic there is at least 75lbs of water in the smaller zone probably 80-90lbs interacting with every burn on that zone, which should be enough
to keep a ~17-20K min-fire combi from short cycling.
If the combi is set up to be heating the 40 gallon (333 lbs) buffer tank in the middle there is effectively zero possibility of short-cycling the combi on zone calls. Just about anybody's 199K
combi boiler would be fine.
user-2890856 | | #6
Have an idea that none of these zones are anywhere near low end modulation even at or below design Dana .
Dlauffenberger . Could you tell us what type buffer/ batch tank that is , who is the manufacturer ?
1. Expert Member
Dana Dorsett | | #7
True, neither zone will likely emit the min-fire output of a combi boiler, but there is also a reasonable amount of thermal mass to work with even in the zone radiation, enough to be able
to suppress truly egregious short cycling by opening up the differential swing.
Combi boilers are rarely a great fit, and that's true here too, but with some thermal mass to work with they don't have to be TERRIBLE (although evidence in the field might indicate
otherwise, given the weak grasp of hydronic design most installers installing them seem to have.)
To be sure, I wouldn't look at this system and immediatly think "combi boiler- great solution", quite the contrary! But could you make it work? Sure!
Dlauffenburger | | #11
Your hesitation on using a Combi unit for this configuration are enlightening, what would be a better and best option?
1. Expert Member
Dana Dorsett | | #13
A small modulating condensing boiler with a big turn-down ratio plus an indirect water heater operated as the "primary zone" is tried and true, easy to design for. There are
several mod-cons that can modulate down to almost 1/3 the minimum BTU rate of a 199K combi boiler.
With an indirect water heater the domestic hot water flow rates are not burner-limited- if the indirect is sized for the biggest tub you need to fill, the tub can be filled 10, 12
gpm or higher- whatever your water pressure and plumbing can deliver. And when showering multiple other high rate hot water draws can come on/off without major impact on "family
harmony" with the person singing / screeching in the shower.
Looking at your BIS load calculation output your 99% design heat load is about 46K, which could be covered by a Lochinvar KHB055 with some margin, with a max-fire output of about
52K, and a min-fire output of 7.9K (less than half that of a 199K combi). If the confidence isn't high on your load numbers and you want faster recovery on the indirect the KHB085
has a bigger, 10:1 turn down ratio, with about 81K out at high-fire, 8.1K out at low fire, which is STILL less than half the min-fire output of a combi boiler.
For less money the somewhat simpler HTP UFT-080W modulates between 7600 BTU/hr @ low fire to about 76K at high fire. The UFT boilers are even simpler to install, since they come
pre-plumbed with a secondary output port and controls to support an indirect water heater. (They are almost DIY-able, but it's best to study up on hydronic system design first.)
There are others. Find one that has reasonable local & regional support, call the distributor for contractor recommendations, since the distributor knows who is constantly turning
in bogus warranty claims on mis-installed systems or tying up the tech support with questions clearly answered in the manual, and who installs dozens per year with minimal need of
support. HTP's headquarters are less than a 90 minute drive from my house, which makes it an easier call for me, but I wouldn't turn down a Burnham ASPN-085 or Lochinvar KHB085,
for an application like this, both vendors have decent support in my area.
With a min-fire output less than 1/4 your design load it's possible to set it up under outdoor reset control where it will run nearly continuously through the entire heating
system with maximally stable room temperatures and high efficiency/low maintenance. Even with the 40 gallon buffer you can't quite do that with a combi boiler putting out nearly
half the design day load at min-fire.
Dlauffenburger | | #10
The batch tank is a Bradford White electric hot water heater that is not connected to electric. The boiler would be connected directly to the batch tank, and the hydronic circuits hook to
the batch tank via different connections.
5. Expert Member
Peter Engle | | #14
FWIW, I've got the HTP boiler that Dana mentions above, with a 30 gallon DHW indirect fired tank. Plenty of heat for a 2000+ sf 120 year old house in CZ5, and no hot water problems ever.
Dlauffenburger | | #15
Thank You for the reply and information. I think the indirect route will be a much better solution, just have to start the pricing comparison for the boiler and indirect tank setup.
Thank you for the info, it is always good to hear real world results.
rhl_ | | #16
I haven’t had a chance to read existing responses, but, let me warn you against the aquabalance. I own the AB 155c. I’m rather disappointed with it. It’s made quite cheaply, I’ve had it now for
two heat seasons, and I’ve had two fittings fail on it.
Also, more importantly, it doesn’t support external controls. A smart design is having the boiler be asked to output water at different _design_ temps based on both outside air and which zones
call for heat. So for example if it’s radiant floors only, don’t output more than 120 deg water.
I would look at Buderus as another option.
qofmiwok | | #17
This is so confusing. We will have 2 people in a fairly large, highly efficient house, and were thinking about doing some radiant heat loops in the garage, maybe a 5000 BTU max load. A Navien
combi-boiler keeps being recommended but it sure seems like overkill with the smallest one being 60kBTU heating and 160k DHW. Someone on another GBA post suggested an ATP Versa Hydro, and the
built in buffer tank handles the modulation, but the smallest one (55 gal) seems to be over $6k. Is that what I should expect to pay for what I need? Then I see other combi-boilers that are $2000
and claim to be just as efficient. Is the problem with these just that the turndown ratio is less? https://www.ecomfort.com/Noritz-NRCB199DV-NG/p82295.html
1. Expert Member
Dana Dorsett | | #18
Combi boilers basically suck even for regular code-min type houses. They're a better match for homes with big heat loads and small to moderate hot water needs. They suck even worse for highly
efficient houses.
That said, with high mass radiators such as concrete slabs they can be made to work without short-cycling.
BTW: Navien has some serious QC or design issues with heat exchangers on their tankless water heaters & combi-boilers leaking exhaust into the inside of the cabinet, which mixes the
combustion air causing lower efficiency & longevity. There may even be class-action suits coming. (Do an internet search on the terms [mikey pipes] [navien] [heat exchanger] ) Despite being
easier than most to size correctly and easy to install it's worth waiting until those issues get resolved before buying a water-tube heat exchanger type Navien. But the NFC series fire tube
versions are completely different, and simply cannot leak in the same way. https://www.navieninc.com/series/nfc
With it's 11: 1 turn down ratio and slab radiation an NFC-175 or NFC-199 should work just fine.
qofmiwok | | #19
Thanks for the heads up. I don't want to give myself a headache for something I don't need. And the NFC-175 is just crazy capacity for water for 2 people and an optional 5000 BTU of
garage heat. What would you recommend for a stand alone low load hot water system in a highly efficient house? Just a standard high efficiency NG water heater? | {"url":"https://www.greenbuildingadvisor.com/question/heat-loss-and-boiler-combi-proper-sizing","timestamp":"2024-11-05T21:44:03Z","content_type":"text/html","content_length":"129736","record_id":"<urn:uuid:7aed3205-71fe-4c22-b7ad-770f1f5543eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00538.warc.gz"} |
Table of Contents
The number of people voted in an Instagram poll
Instagram provides a polling feature, which allows the user to ask a question with two answers to the audiences. Once an audience pick one of the two answers, the percentage of users who pick each
answer is displayed.
So the questions are: how many people actually voted in the poll? How many people voted for each option?
I believe mathematical modelling of the voting process can provide an answer to these questions.
Update: there is a now a new solution to this problem.
Mathematical model of Instagram's polling feature
The Instagram poll can modelled using the following vector equation:
$$ poll(a,\ b) = \Big(nint(\frac{100a}{a+b}),\ nint(\frac{100b}{a+b})\Big), $$
where $a$ and $b$ are the number of audiences picked each of the option. $nint$ is the nearest integer function, such that $nint(x)$ is the nearest integer to $x$. The domain and the codomain of
$poll(a,\ b)$ is: $\mathbb{N}_0$. The range for $poll(a,\ b)$ is $0 \le x \le 100$.
Solution strategy
It is clear that $poll(a, b)$ has many-to-one mapping. This means that it cannot have an inverse function. The range of the function is quite small, and the actual input of the function tends to be
quite small. It is therefore a good strategy to build a look-up table by exhaustively enumerating all results produced by all possible combination of $a$ and $b$, up to a certain size limit.
Matlab implementation
I wrote the following Matlab functions to generate the look-up table and perform the search. The function IPRL takes two parameters - the number of audiences picked one option, and the maximum
potential number of participants in the poll. This function gives a list of potential number of participant in the poll.
function [x] = IPRL(yes, total)
%IPRL Instagram Poll Reverse Lookup
% Parameters :
% - yes : the number of audiences selected one of the options
% - total : the maximum potential number of audiences participated in the poll
tbl = GenTable(total);
[x, ~] = find(tbl == yes);
function [tbl] = GenTable(n)
%GENTABLE Generate the look-up table
tbl = zeros(n);
for i = 1:n
for j = 1:i
tbl(i,j) = round(j/i, 2) * 100;
Example scenario
I saw a poll with 27% of the users picked one option, and I suspect that maximum 30 users participated in the poll, so I run:
>> IPRL(27,30)
ans =
Then I get a friend of mine to vote in the option that had 27%, and now it changes to 33%.
>> IPRL(33,30)
ans =
It is clear that after I first voted, the number of participants could have been either 11 or 26. After my friend voted, the number of participants could have been 12 or 27. The idea is that the
second look-up should produce numbers that have increased by 1, compared to the first look-up. However from personal experiences, none of my Instagram poll ever broke the 20 participant barrier,
therefore 12 participants voted in it.
Other Strategies and Future Improvement
Perhaps I can record how the how the percentage number change over time, and automatically give a list of the most likely number. The number of participant monotonically increases over time. This
property can help the search. Perhaps I can write a function that automate this process.
Finally, it would be great to implement this whole idea using Javascript, so people can run it without Matlab.
Ethical Consideration
Well, if you do not want people to find out how many people have voted in your poll, do not do an Instagram poll. This work does not gather more information than what is already in the public.
I came up with this idea, after seeing an Instagram poll from the girl who gave me a toblerone. So, thank you for the inspiration.
public/the_number_of_people_voted_in_an_instagram_poll.txt · Last modified: 2019/04/16 10:46 by fangfufu | {"url":"https://www.fangfufu.co.uk/wiki/doku.php?id=public:the_number_of_people_voted_in_an_instagram_poll","timestamp":"2024-11-13T09:21:41Z","content_type":"text/html","content_length":"25054","record_id":"<urn:uuid:983d7573-0788-42b4-be7c-168e0a92a2e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00297.warc.gz"} |
How to Achieve a 90% Win Rate with the 10 Delta Put Credit Spread
Many tend to get into a slump, having no matter how much effort and research on how to stay consistent. This can be really demoralizing and even causes some to give up on trading just because they
think they are not good enough.
Seth Frey, head of the options trading desk at the New York City-based top proprietary trading firm, has developed one such strategy. They successfully develop methods to get their traders confident
and profitable again, one of them being options-leveraging—a versatile tool that, if used correctly, yields them high win rates.
This article introduces a simple yet effective options trading strategy that will offer a 90% win rate. The strategy, a 10 Delta Put Credit Spread, has been used by professionals and will help you
regain your trading mindset while keeping the promise of steady profits.
The 10 Delta Put Credit Spread
This trading vehicle comprised mostly an actively traded option on an exchange-traded fund: QQQ. QQQ tracks the NASDAQ 100 and is made up of a variety of big tech stocks such as Apple, Amazon, and
With a 10 Delta Put Credit Spread, traders can get an extremely high probability of winning. The Delta of the option is a measure of the chance that the option will expire in the money, and a 10
Delta only indicates a 10% chance of expiring in the money. Rather, this represents a 90% chance that the option will expire worthless, thus taking a win.
How the Strategy Works
Let’s break the steps of this high-probability strategy into segments:
Breaking this high-probability strategy into sections:
• Selection of the Ticker: The one used here is QQQ. On May 1st, 2023, QQQ was averaging $322.12
• Important to Choose the Choices:
□ Sell a put option with 10 Delta. The chance for that option to expire in-the-money is very low. In this scenario, the seller will sell the 295 put option. The chance to expire worthless is
relatively high.
□ Hedge the position by purchasing a put option with a lower strike price. For instance, the seller can purchase a 290 put to hedge against a more significant potential downside risk.
Risk vs Reward:
• Collecting the premium on the 295 put, the trader collects $1.26 a share, $1,260 for 10 contracts.
• Buying the 290 put at $0.93 a share ($930 for 10 contracts) leaves the trader with a net credit of $330.
The Expiration Day Outcome:
On the expiration day (June 23rd, 2023), if QQQ closes above $295, the 295 and 290 puts will both expire worthless, and the trader will be left to collect the $330 profit. This is what happens when
QQQ continues to close above the sold put’s strike price, which is highly likely given the Delta.
What are the Advantages of the 10 Delta Put Credit Spread?
• High Win Probability: Selling a put option that has a 10 Delta would be betting that there’s a 90% chance the stock stays above the strike price, thus ending up worthless.
• Tons of Maneuvering Room: Although QQQ might fall in price, it can easily hang above the strike of sold puts—in this case, $295. As long as the stock stays above that level, the trader still
• Continuous Cash Flow: The trader can establish these credit spreads month in and month out, earning regular premiums. In the example employed, this cycle would be repeated throughout the year,
earning the trader $3,850.
Calculation of Profit
For this case study, the trader was successful in all 12 trades over more than 12 months, peaking at almost $4,730. From the returns on investment after following this strategy over the year, the
return stands at 81.3%. Such a return based on high-probability trades is impressive and depicts a consistent manner through which wealth can be built over time.
Risk Management and Caveats
Even though this strategy has an extremely high win rate, it does pose some level of danger. Here are the most important points to take note of:
• Only Trade When Bullish: This strategy presumes a bullish or neutral bias with respect to the underlying asset. If the stock is not certain to stay high, such a trade should be avoided because a
dramatic drop in price can result in a loss.
• Managing Unfavourable Trades: In a very rare case where the stock ends up closing below the strike of the put sold, it would leave the trader with a choice either of taking delivery of the shares
or closing out the trade and foregoing some money.
• Realistic Expectations: Even though a string of 12 consecutive winning trades can and will occur in any one-year period, the practical outcome likely to be reached will probably be between 10 and
11 wins a year, with a winning percentage near 90 percent. Unpredictable market conditions mean that sometimes the losses will not be avoided throughout the year either
Why This Strategy Works
Professional Insight:
The 10 Delta Put Credit Spread is one of those strategies that professionals at firms such as SMB Capital employ because it strikes a good balance between risk and reward. The high-probability
success makes it a valuable tool in a professional’s toolkit.
Psychological Boost:
Complacency can be expected when no losses are ever experienced, as traders win 9 out of every 10 times; however, high-probability trades will make a trader more confident, especially for those that
have recently found themselves in losing situations.
This is a simple options strategy designed for traders to hit the 90% win rate — the benchmark of building long-term success. Selling 10 Delta puts and buying lower-strike puts to protect the
downside generates profits every month, as the QQQ example showed.
However, like any strategy, it must be used at the right time—that is, when you are bullish on the market—and then accompanied by good and meticulous management of risks. This method, when applied by
a good trader with proper execution, generates steady cash flow as well as high returns: it is an invaluable tool in any trader’s toolkit.
Add the 10 Delta Put Credit Spread to your trading plan as you move into learning more about high-probability options strategies.
Leave a Comment | {"url":"https://moneymatteronline.com/2024/10/24/how-to-achieve-a-90-win-rate-with-the-10-delta-put-credit-spread/","timestamp":"2024-11-06T08:48:48Z","content_type":"text/html","content_length":"123764","record_id":"<urn:uuid:9611bec0-d2ba-4ec4-aeab-9b2a854f4f92>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00509.warc.gz"} |
Section: New Results
Inverse Problems
Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains
Participants : Sébastien Imperiale, Philippe Moireau [correspondant] .
In [29] we are interested in reconstructing the initial condition of a wave equation in an unbounded domain configuration from measurements available in time on a subdomain. To solve this problem, we
adopt an iterative strategy of reconstruction based on observers and time reversal adjoint formulations. We prove the convergence of our reconstruction algorithm with perfect measurements and its
robustness to noise. Moreover, we develop a complete strategy to practically solve this problem on a bounded domain using artificial transparent boundary conditions to account for the exterior
domain. Our work then demonstrates that the consistency error introduced by the use of approximate transparent boundary conditions is compensated by the stabilisation properties obtained from the use
of the available measurements, hence allowing to still be able to reconstruct the unknown initial condition.
Analysis and numerical simulation of an inverse problem for a structured cell population dynamics model
Participants : Frédérique Clément, Frédérique Robin [correspondant] .
We have studied (with Béatrice Laroche, INRA) a multiscale inverse problem associated with a multi-type model for age structured cell populations [20] (see also [21] for another application). In the
single type case, the model is a McKendrick-VonFoerster like equation with a mitosis-dependent death rate and potential migration at birth. In the multi-type case, the migration term results in a
unidirectional motion from one type to the next, so that the boundary condition at age 0 contains an additional extrinsic contribution from the previous type. We consider the inverse problem of
retrieving microscopic information (the division rates and migration proportions) from the knowledge of macroscopic information (total number of cells per layer), given the initial condition. We have
first shown the well-posedness of the inverse problem in the single type case using a Fredholm integral equation derived from the characteristic curves, and we have used a constructive approach to
obtain the lattice division rate, considering either a synchronized or non-synchronized initial condition. We have taken advantage of the unidirectional motion to decompose the whole model into
nested submodels corresponding to self-renewal equations with an additional extrinsic contribution. We have again derived a Fredholm integral equation for each submodel and deduced the well-posedness
of the multi-type inverse problem. In each situation, we illustrate numerically our theoretical results.
Inverse problem based on data assimilation approaches for protein aggregation
Participants : Philippe Moireau [correspondant] , Cécile Della Valle [MAMBA] , Marie Doumic [MAMBA] .
Estimating reaction rates and size distributions of protein polymers is an important step for understanding the mechanisms of protein misfolding and aggregation. In a depolarization configuration, we
here extend some previous results obtained during the PhD Thesis of A. Armiento. Now, the depolarization rate is time-dependent or in the presence of an additional vanishing viscosity term. We
continue to develop our framework mixing inverse problems methodologies and optimal control approaches typically encountered in data assimilation, allowing to justify mathematically the methods but
also to adopt efficient numerical strategies. Publications of this work will be soon submitted.
Front shape similarity measure for data-driven simulations of wildland fire spread based on state estimation: Application to the RxCADRE field-scale experiment
Participants : Annabelle Collin [MONC] , Philippe Moireau [correspondant] .
Data-driven wildfire spread modeling is emerging as a cornerstone for forecasting real-time fire behavior using thermal-infrared imaging data. One key challenge in data assimilation lies in the
design of an adequate measure to represent the discrepancies between observed and simulated firelines (or “fronts”). A first approach consists in adopting a Lagrangian description of the flame front
and in computing a Euclidean distance between simulated and observed fronts by pairing each observed marker with its closest neighbor along the simulated front. However, this front marker
registration approach is difficult to generalize to complex front topology that can occur when fire propagation conditions are highly heterogeneous due to topography, biomass fuel and
micrometeorology. To overcome this issue, we investigate in this paper an object-oriented approach derived from the Chan–Vese contour fitting functional used in image processing. The burning area is
treated as a moving object that can undergo shape deformations and topological changes. We combine this non-Euclidean measure with a state estimation approach (a Luenberger observer) to perform
simulations of the time-evolving fire front location driven by discrete observations of the fireline. We apply this object-oriented data assimilation method to the three-hectare RxCADRE S5
field-scale experiment. This collaboration with CERFACS (M. Rochoux) and University of Maryland (C. Zhang and A. Trouvé) led to a publication [34] in the Proceedings of the Combustion Institute.
Model assessment through data assimilation of realistic data in cardiac electrophysiology
Participants : Antoine Gerard [CARMEN] , Annabelle Collin [MONC] , Gautier Bureau, Philippe Moireau [correspondant] , Yves Coudière [CARMEN] .
We consider a model-based estimation procedure – namely a data assimilation algorithm – of the atrial depolarization state of a sub- ject using data corresponding to electro-anatomical maps. Our
objective is to evaluate the sensitivity of such a model-based reconstruction with respect to model choices. The followed data assimilation approach is capable of using electrical activation times to
adapt a monodomain model simulation, thanks to an ingenious model-data fitting term inspired from image processing. The resulting simulation smoothes and completes the activation maps when they are
spatially incomplete. Moreover, conductivity parameters can also be inferred. The model sensitivity assessment is performed based on synthetic data generated with a validated realistic atria model
and then inverted using simpler modeling ingredients. In particular, the impact of the muscle fibers definition and corresponding anisotropic conductivity parameters is studied. Finally, an
application of the method to real data is presented, showing promising results. This collaborative work has been published, see [37]. | {"url":"https://radar.inria.fr/report/2019/m3disim/uid50.html","timestamp":"2024-11-10T06:20:13Z","content_type":"text/html","content_length":"43952","record_id":"<urn:uuid:6fd4c008-b7f2-446b-903b-3acf77f64c47>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00205.warc.gz"} |
Early life and education
William Thurston was born in Washington, D.C., to Margaret Thurston (née Martt), a seamstress, and Paul Thurston, an aeronautical engineer.^[1] William Thurston suffered from congenital strabismus as
a child, causing issues with depth perception.^[1] His mother worked with him as a toddler to reconstruct three-dimensional images from two-dimensional ones.^[1]
He received his bachelor's degree from New College in 1967 as part of its inaugural class.^[1]^[2] For his undergraduate thesis, he developed an intuitionist foundation for topology.^[3] Following
this, he received a doctorate in mathematics from the University of California, Berkeley under Morris Hirsch, with his thesis Foliations of Three-Manifolds which are Circle Bundles in 1972.^[1]^[4]
After completing his Ph.D., Thurston spent a year at the Institute for Advanced Study,^[1]^[5] then another year at the Massachusetts Institute of Technology as an assistant professor.^[1]
In 1974, Thurston was appointed a full professor at Princeton University.^[1]^[6] He returned to Berkeley in 1991 to be a professor (1991-1996) and was also director of the Mathematical Sciences
Research Institute (MSRI) from 1992 to 1997.^[1]^[7] He was on the faculty at UC Davis from 1996 until 2003, when he moved to Cornell University.^[1]
Thurston was an early adopter of computing in pure mathematics research.^[1] He inspired Jeffrey Weeks to develop the SnapPea computing program.^[1]
During Thurston's directorship at MSRI, the institute introduced several innovative educational programs that have since become standard for research institutes.^[1]
His Ph.D. students include Danny Calegari, Richard Canary, David Gabai, William Goldman, Benson Farb, Richard Kenyon, Steven Kerckhoff, Yair Minsky, Igor Rivin, Oded Schramm, Richard Schwartz,
William Floyd, and Jeffrey Weeks.^[8]
His early work, in the early 1970s, was mainly in foliation theory. His more significant results include:
• The proof that every Haefliger structure on a manifold can be integrated to a foliation (this implies, in particular, that every manifold with zero Euler characteristic admits a foliation of
codimension one).
• The construction of a continuous family of smooth, codimension-one foliations on the three-sphere whose Godbillon–Vey invariant (after Claude Godbillon and Jacques Vey) takes every real value.
• With John N. Mather, he gave a proof that the cohomology of the group of homeomorphisms of a manifold is the same whether the group is considered with its discrete topology or its compact-open
In fact, Thurston resolved so many outstanding problems in foliation theory in such a short period of time that it led to an exodus from the field, where advisors counselled students against going
into foliation theory,^[9] because Thurston was "cleaning out the subject" (see "On Proof and Progress in Mathematics", especially section 6^[10]).
The geometrization conjecture
His later work, starting around the mid-1970s, revealed that hyperbolic geometry played a far more important role in the general theory of 3-manifolds than was previously realised. Prior to Thurston,
there were only a handful of known examples of hyperbolic 3-manifolds of finite volume, such as the Seifert–Weber space. The independent and distinct approaches of Robert Riley and Troels Jørgensen
in the mid-to-late 1970s showed that such examples were less atypical than previously believed; in particular their work showed that the figure-eight knot complement was hyperbolic. This was the
first example of a hyperbolic knot.
Inspired by their work, Thurston took a different, more explicit means of exhibiting the hyperbolic structure of the figure-eight knot complement. He showed that the figure-eight knot complement
could be decomposed as the union of two regular ideal hyperbolic tetrahedra whose hyperbolic structures matched up correctly and gave the hyperbolic structure on the figure-eight knot complement. By
utilizing Haken's normal surface techniques, he classified the incompressible surfaces in the knot complement. Together with his analysis of deformations of hyperbolic structures, he concluded that
all but 10 Dehn surgeries on the figure-eight knot resulted in irreducible, non-Haken non-Seifert-fibered 3-manifolds. These were the first such examples; previously it had been believed that except
for certain Seifert fiber spaces, all irreducible 3-manifolds were Haken. These examples were actually hyperbolic and motivated his next theorem.
Thurston proved that in fact most Dehn fillings on a cusped hyperbolic 3-manifold resulted in hyperbolic 3-manifolds. This is his celebrated hyperbolic Dehn surgery theorem.
To complete the picture, Thurston proved a hyperbolization theorem for Haken manifolds. A particularly important corollary is that many knots and links are in fact hyperbolic. Together with his
hyperbolic Dehn surgery theorem, this showed that closed hyperbolic 3-manifolds existed in great abundance.
The hyperbolization theorem for Haken manifolds has been called Thurston's Monster Theorem, due to the length and difficulty of the proof. Complete proofs were not written up until almost 20 years
later. The proof involves a number of deep and original insights which have linked many apparently disparate fields to 3-manifolds.
Thurston was next led to formulate his geometrization conjecture. This gave a conjectural picture of 3-manifolds which indicated that all 3-manifolds admitted a certain kind of geometric
decomposition involving eight geometries, now called Thurston model geometries. Hyperbolic geometry is the most prevalent geometry in this picture and also the most complicated. The conjecture was
proved by Grigori Perelman in 2002–2003.^[11]^[12]
Density conjecture
Thurston and Dennis Sullivan generalized Lipman Bers' density conjecture from singly degenerate Kleinian surface groups to all finitely generated Kleinian groups in the late 1970s and early 1980s.^[
13]^[14] The conjecture states that every finitely generated Kleinian group is an algebraic limit of geometrically finite Kleinian groups, and was independently proven by Ohshika and Namazi–Souto in
2011 and 2012 respectively.^[13]^[14]
Orbifold theorem
In his work on hyperbolic Dehn surgery, Thurston realized that orbifold structures naturally arose. Such structures had been studied prior to Thurston, but his work, particularly the next theorem,
would bring them to prominence. In 1981, he announced the orbifold theorem, an extension of his geometrization theorem to the setting of 3-orbifolds.^[15] Two teams of mathematicians around 2000
finally finished their efforts to write down a complete proof, based mostly on Thurston's lectures given in the early 1980s in Princeton. His original proof relied partly on Richard S. Hamilton's
work on the Ricci flow.
Awards and honors
Personal life
Selected publications
• William Thurston, The geometry and topology of three-manifolds, Princeton lecture notes (1978–1981).
• William Thurston, Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, New Jersey, 1997. x+311 pp.
ISBN 0-691-08304-5
• William Thurston, Hyperbolic structures on 3-manifolds. I. Deformation of acylindrical manifolds. Ann. of Math. (2) 124 (1986), no. 2, 203–246.
• William Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), 357–381.
• William Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431
• Epstein, David B. A.; Cannon, James W.; Holt, Derek F.; Levy, Silvio V. F.; Paterson, Michael S.; Thurston, William P. Word Processing in Groups. Jones and Bartlett Publishers, Boston,
Massachusetts, 1992. xii+330 pp. ISBN 0-86720-244-0^[23]
• Eliashberg, Yakov M.; Thurston, William P. Confoliations. University Lecture Series, 13. American Mathematical Society, Providence, Rhode Island and Providence Plantations, 1998. x+66 pp. ISBN
• William Thurston, On proof and progress in mathematics. Bull. Amer. Math. Soc. (N.S.) 30 (1994) 161–177
• William P. Thurston, "Mathematical education". Notices of the AMS 37:7 (September 1990) pp 844–850
See also
Further reading
External links
• Media related to William Thurston at Wikimedia Commons
Wikiquote has quotations related to William Thurston.
• William Thurston at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "William Thurston", MacTutor History of Mathematics Archive, University of St Andrews
• Thurston's page at Cornell
• Tribute and remembrance page at Cornell
• Etienne Ghys : La géométrie et la mode
• "Landau Lectures | Prof. Thurston | Part 1 | 1995/6". YouTube. Hebrew University of Jerusalem. April 8, 2014.
• "Landau Lectures | Prof. Thurston | Part 2 | 1995/6". YouTube. Hebrew University of Jerusalem. April 8, 2014.
• "Landau Lectures | Prof. Thurston | Part 3 | 1995/6". YouTube. Hebrew University of Jerusalem. April 8, 2014.
• "The Mystery of 3-Manifolds - William Thurston". YouTube. PoincareDuality. November 27, 2011. 2010 Clay Research Conference
• Goldman, William (May 9, 2013). "William Thurston: A Mathematical Perspective". YouTube. UMD Mathematics. William Goldman (U. of Maryland), Collloquium, Department of Mathematics, Howard
University, 25 January 2013 | {"url":"https://www.knowpia.com/knowpedia/William_Thurston","timestamp":"2024-11-04T04:55:31Z","content_type":"text/html","content_length":"144060","record_id":"<urn:uuid:403cfd8c-a5d2-4f02-a6d4-626b4d712264>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00354.warc.gz"} |
How do the different measures of center compare? | Socratic
How do the different measures of center compare?
1 Answer
There are three main center measures.
The mean or average is calculated from the sum of all values divided by the number of cases. It gives - as the word says - an idea of the average. Disadvantage is, that it is susceptible to extreme
values. Example: Average income of a group of 1000 people is 500 a week. Add a person who gets 100000 a week and see what happens to the average.
$500000 / 1000 = 500$
$600000 / 1001 \approx 599$
The mode is the value or class value that happens most. It is the 'bump' in your frequency distribution. You have a problem if there are more bumps.
The median is the value, of which half the population has a lower value, and half a higher. It is the value of 'the middle man' if you order your cases by value. Greatest advantage of using median,
is that it is hardly susceptible to extremes. You might try the above example with the median.
In a perfect, or allmost perfect, normal distribution, these three are the same.
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statistics analysis - Academic Heroes
1. Randy Owen invested $6,700 at 12% annual interest, and left the money invested without withdrawing any of the interest for 10 years. At the end of the 10 years, Randy withdrew the accumulated
amount of money.
(a) What amount did Randy withdraw, assuming the investment earns simple interest?
The amount Randy withdrew
(b) What amount did Randy withdraw, assuming the investment earns interest compounded annually? (Round answer to 2 decimal places, e.g. 25.25.)
The amount Randy withdrew
For each of the following cases,
(1) In Table 1 (future value of 1):
Annual Rate
Number of
Years Invested
Case A
6 Annually
Case B 10% 6 Semiannually
(2) In Table 2 (future value of an annuity of 1):
Annual Rate
Number of
Years Invested
Case A
10 Annually
Case B 10% 6 Semiannually
Indicate to what interest rate columns you would refer in looking up the future value factor.
Table 1
Table 2
Case A
Case B
Indicate to what number of periods you would refer in looking up the future value factor.
Table 1
Table 2
Case A
Case B
3. Joyce Company signed a lease for an office building for a period of 10 years. Under the lease agreement, a security deposit of $7,400 is made. The deposit will be returned at the expiration of the
lease with interest compounded at 10% per year.
What amount will Joyce receive at the time the lease expires? (Round answer to 2 decimal places, e.g. 25.25.)
Amount at the time the lease expires
4. Bates Company issued $1,200,000, 10-year bonds and agreed to make annual sinking fund deposits of $79,000. The deposits are made at the end of each year into an account paying 8% annual interest.
What amount will be in the sinking fund at the end of 10 years? (Round answer to 2 decimal places, e.g. 25.25.)
Amount in the sinking fund
5. Frank and Maureen Fantazzi invested $5,800 in a savings account paying 8% annual interest when their daughter, Angela, was born. They also deposited $1,400 on each of her birthdays until she was
15 (including her 15th birthday).
How much was in the savings account on her 15th birthday (after the last deposit)? (Round answer to 2 decimal places, e.g. 25.25.)
Amount on 15th birthday
6. Hugh Curtin borrowed $33,100 on July 1, 2012. This amount plus accrued interest at 5% compounded annually is to be repaid on July 1, 2017.
How much will Hugh have to repay on July 1, 2017? (Round answer to 2 decimal places, e.g. 25.25.)
Amount to be repaid on July 1, 2017
7. For each of the following cases,
(1) In Table 3 (present value of 1):
Annual Rate
Number of
Years Involved
per Year
Case A
8% 5 Annually
Case B 11% 8 Annually
Case C 8% 9 Semiannually
(2) In Table 4 (present value of an annuity of 1):
Annual Rate
Number of
Years Involved
Number of
Payments Involved
Frequency of
Case A
11% 18 18 Annually
Case B 11% 6 6 Annually
Case C 8% 4 8 Semiannually
Indicate to what interest rate columns you would refer in looking up the discount rate.
Table 3
Table 4
Case A
Case B
Case C
Indicate to what number of periods you would refer in looking up the discount rate.
Table 3
Table 4
Case A
Case B
Case C
(a) What is the present value of $29,900 due 10 periods from now, discounted at 8%? (Round answer to 2 decimal places, e.g. 25.25.)
Present value
(b) What is the present value of $29,900 to be received at the end of each of 6 periods, discounted at 5%? (Round answer to 2 decimal places, e.g. 25.25.)
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Digital Logic Design Introduction | Download book PDF
Digital Logic Design and Lab
This note describes the following topics: Binary systems, Boolean algebra, logic gates, analysis or design of combinatorial circuits, synchronous sequential logic, registers, counters and memory,
laboratory experiments in digital circuits and logic Logic Design, Contemporary Logic Design, Switches: basic building block of digital computers, Relay networks, MOS transistors , CMOS network,
Combinational logic symbols, Implementation as a combinational digital system , Combinational Logic, Time behavior and waveforms, Product-of-sums canonical form, Karnaugh maps, Working with
Combinational Logic, Memory, Finite State Machines, Sequential Logic Technologies, Case Studies in Sequential Logic Design, Binary Number Systems and Arithmetic Circuits, Interfacing.
Author(s): Prof. Soo ik, Chae, Seoul National University
NA Pages | {"url":"https://www.freebookcentre.net/electronics-ebooks-download/Digital-Logic-Design-Introduction.html","timestamp":"2024-11-08T18:19:54Z","content_type":"text/html","content_length":"29736","record_id":"<urn:uuid:e3c1deef-dde7-4751-bc26-54b3516ef80f>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00114.warc.gz"} |
The length of the tangent of the curves x=acos^(3)theta,y=asi-Turito
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The length of the tangent of the curves
We have to find the length of the tangent. We are given the values of x and y as function of
The given values of x and y are as follows:
x = acos^3θ
y = asin^3
The formula for length of tangent is
Length =
We will first differentiate x and y w.r.t θ and then find
Differentiating x w.r.t θ
Differentiating y w.r.t θ
We will substitute this value in the formula of length of tangent.
So, length of the tangent is asin^2θ
For such questions, we should know the properties of trignometric functions and their different formulas.
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Computing bounds on
10.6 Computing bounds on poly_ints ¶
poly_int also provides routines for calculating lower and upper bounds:
‘constant_lower_bound (a)’
Assert that a is nonnegative and return the smallest value it can have.
‘constant_lower_bound_with_limit (a, b)’
Return the least value a can have, given that the context in which a appears guarantees that the answer is no less than b. In other words, the caller is asserting that a is greater than or equal
to b even if ‘known_ge (a, b)’ doesn’t hold.
‘constant_upper_bound_with_limit (a, b)’
Return the greatest value a can have, given that the context in which a appears guarantees that the answer is no greater than b. In other words, the caller is asserting that a is less than or
equal to b even if ‘known_le (a, b)’ doesn’t hold.
‘lower_bound (a, b)’
Return a value that is always less than or equal to both a and b. It will be the greatest such value for some indeterminate values but necessarily for all.
‘upper_bound (a, b)’
Return a value that is always greater than or equal to both a and b. It will be the least such value for some indeterminate values but necessarily for all. | {"url":"https://gcc.gnu.org/onlinedocs/gccint/Computing-bounds-on-poly_005fints.html","timestamp":"2024-11-07T22:57:25Z","content_type":"text/html","content_length":"5716","record_id":"<urn:uuid:e36ec1cf-1f22-400f-bfa0-c0c8b279673d>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00692.warc.gz"} |
[tlaplus] Proof by induction invoking the SequencesInductionTail theorem
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[tlaplus] Proof by induction invoking the SequencesInductionTail theorem
Hello all,
I have a theorem which depends on some lemmas. One of this lemmas is the fact that "the product of two sequence products is the sequence product of their concatenation". That is:
\A s1, s2 \in Seq(Nat) : SeqProduct[s1] * SeqProduct[s2] = SeqProduct[s1 \o s2]
SeqProduct[s \in Seq(Nat)] ==
IF s = <<>>
THEN 1
ELSE LET x == Head(s)
xs == Tail(s)
IN x * SeqProduct[xs]
For this one, im planning to reason inductively using the SequencesInductionTail theorem available on module SequenceTheorems (not saying this alone will suffice).
Now, having the following structure:
LEMMA SomeLemma == \A s1, s2 \in Seq(Nat) : SeqProduct[s1] * SeqProduct[s2] = SeqProduct[s1 \o s2]
<1> DEFINE Prop(s) == \A s2 \in Seq(Nat) : SeqProduct[s] * SeqProduct[s2] = SeqProduct[s \o s2]
<1> SUFFICES \A s1 \in Seq(Nat) : Prop(s1) OBVIOUS
<1>1. Prop(<<>>) PROOF OMITTED
<1>2. \A s \in Seq(Nat) : (s # << >>) /\ Prop(Tail(s)) => Prop(s) PROOF OMITTED
<1> HIDE DEF Prop
<1>3. QED
BY <1>1, <1>2, SequencesInductionTail
Ignoring now <1>1 and <1>2, for the high level part i was expecting TLAPS to prove the final step <1>3 and it couldn't.
The proof obligation generated is:
ASSUME Prop(<<>>) ,
\A s \in Seq(Nat) : s # <<>> /\ Prop(Tail(s)) => Prop(s) ,
ASSUME NEW CONSTANT S,
NEW CONSTANT P(_),
P(<<>>) ,
\A s \in Seq(S) : s # <<>> /\ P(Tail(s)) => P(s)
PROVE \A s \in Seq(S) : P(s)
PROVE \A s1 \in Seq(Nat) : Prop(s1)
which looks reasonable to me. Also tried the ASSUME ... PROVE form, but the result is the same.
Im missing something here to correctly invoke SequencesInductionTail ?
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Calculate the Perimeter of Irregular Shapes
The perimeter of a shape is simply the distance around the outside of the shape.
In this activity, we'll be working out the perimeter of a range of different 2D shapes.
When working out the perimeter we must make sure that we check the unit of measure - is it measured in mm, cm, m, km, or something else?
Sometimes we need to use a ruler to work out the perimeter.
In this activity, we will be given some of the measurements, so we won't need a ruler.
We'll use the measurements we are given to work out those we don't know.
Let's get started.
Let's have a look at a regular 2D shape.
This is a regular hexagon.
If a shape is regular, it has sides that are all the same length.
If the sides of this hexagon are 7 cm each, what is the total perimeter?
We know that the sides are all the same length because the shape is regular.
We know that a hexagon has 6 sides.
So, we simply multiply 7 cm by 6.
7 cm x 6 = 42 cm.
Let's have a look at another shape.
This time the shape is not regular.
This means that at least some of the sides will be different lengths.
We need to look at the information we have been given, in order to work out the missing information.
To work out the length on the right of the shape, we can add the two lengths on the left (vertical): 1 m and 3 m - this will give us 4 m.
To work out the length across the top of the shape, we can add the two lengths across the bottom (horizontal): 3 m and 2 m - this will give us 5 m.
Now, we have the measurements of all six sides.
The final job is to add all six lengths together.
Top tip: find a starting point and work around the shape.
1 m + 5 m + 4 m + 2 m + 3 m + 3 m = 18 m
The perimeter is 18 m in total.
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On-line version ISSN 1991-1696
Print version ISSN 0038-2221
SAIEE ARJ vol.109 n.1 Observatory, Johannesburg Mar. 2018
Energy efficient statistical cooperative spectrum sensing in cognitive radio networks
E. Kataka^I; T. Walingo^II
^IDiscipline of Electrical Electronic and Computer Engineering, University ofKwaZulu-Natal, Durban 4041, South Africa, E-mail: kataka.edwin@gmail.com
^IIDiscipline of Electrical Electronic and Computer Engineering, University ofKwaZulu-Natal, Durban 4041, South Africa, E-mail:walingo@ukzn.ac.za
Cooperative spectrum sensing (CSS) alleviates the problem of imperfect detection of primary users (PU)s in cognitive radio (CR) networks by exploiting spatial diversity of the different secondary
users (SUs). The efficiency of CSS depends on the accuracy of the SUs in detecting the PU and accurate decision making at the fusion center (FC). This work exploits the higher order statistical (HOS)
tests of the PU signal for blind detection by the SUs and combination of their decision statistics to make a global decision at the FC. To minimize energy, a two stage optimization paradigm is
carried out, firstly by optimal iterative selection of SUs in the network using Lagrange criterion and secondly optimized fusion techniques achieved by Neyman Pearson. The probability of detecting
the PU based on HOS and hard fusion schemes is investigated. The results indicate that the Omnibus HOS test based detection and optimized majority fusion rule greatly increases the probability of
detecting the PU and reduces the overall system energy consumption.
Key words: Cognitive radio, cooperative spectrum sensing, fusion techniques, higher order statistics, primary user, secondary user.
Cooperative spectrum sensing (CSS) utilizes multiple secondary users (SUs) to sense the vacant spectrum and send their decision to the fusion center (FC) for a final global decision to be made
regarding the presence of the primary user (PU) on the channel. CSS overcomes the challenges of wireless channel characteristics such as multipath fading, shadowing or hidden terminal problem
experienced when only one SU is employed to detect the PU. This is due to the spatial diversity of the different SUs cooperating to make the final decision on the status of the PU on the channel
[1,2]. A number of spectrum detection schemes have been proposed to detect the presence or absence of PU, among them include energy, matched filter and cyclostationary methods [3]. In most practical
systems the transmission channels are usually noisy hence causing tremendous reduction in signal to noise ratio (SNR) of the PU received signals. This has prompted the need for the higher order
statistical (HOS) detection techniques which have very high sensitivity at low SNR signal condition while maintaining reasonable circuit complexity [4]. CSS can generally be divided into two
detection stages; local update stage and global fusion stage. At the local update stage, the individual SUs detect the received PU's signals based on HOS. The SU then computes a local decision and
sends it to the FC for fusion. The commonly used metrics that utilize the HOS properties to detect the PU's received signals include Jarque-Bera, kurtosis, skewness and omnibus tests. These
statistical tests are utilized to determine the probability distribution function (PDF) of a group of data samples. This is crucial for benchmarking the distribution in order to make an informed
inference on a physical phenomena (existence of PU on the channel) [5].
In this paper, the performance analysis of the HOS tests on the PU signal is investigated with aim of selecting the best statistical technique in determining the status of the PU on the channel. This
has not been adequately addressed in literature.
The global fusion stage is performed at the fusion centre where either soft or hard combination schemes are employed to fuse the received signals from individual SUs [6]. Furthermore to reduce energy
consumption in the cooperative network not all the SU need to report their individual decisions. To optimize on the number of SUs selected to participate in the fusion process, this paper proposes a
two stage optimization strategy. The first stage is to select the SUs which qualify to transmit their individual decision data to the fusion center. To achieve this an iterative optimization
threshold algorithm is employed and determined based on the SUs' SNR. However, this is at the cost of minimizing on the error probability formulated by the Lagrange optimization criterion. The rest
of SUs that do not meet this threshold are rejected at this sensing point in time (they are not allowed to transmit). Those SUs selected during the first optimization stage are subjected to the
second stage optimization process, realized by a prudent and optimal choice of hard fusion criteria taken to fuse the SUs binary decisions. A strategic k out of n counting rule is adopted to
determine the optimal combinatorial order of the SUs to be considered for final global fusion. To realize this, Neyman-Pearson optimization criterion is employed through an iterative Bisection
numerical search algorithm formulated on k out of n rule. The cost function is to maximize the probability of detection subject to minimizing of the probability of false alarm. In summary, a hybrid
detection strategy of HOS local detection test and optimal global fusion technique was implemented. The simulated results show that an optimal k out ofn fusion rule based on omnibus test perform
better than other HOS tests in terms of detection probability. In this model, not all SUs participate in detection at any one sensing time frame hence great energy cost saving in the whole
cooperative spectrum sensing network.
The rest of the paper is organized as follows. Section II presents the related work, section III describes the system model, section IV is devoted on local spectrum sensing, section V focuses on the
fusion techniques, section VI presents the energy efficiency. Simulation results illustrating the effectiveness of the scheme are given in section VII and finally, section VIII, draws the
2. RELATED WORK
Cooperative spectrum sensing schemes have not exhaustively been studied in the current literature. In [7], authors investigated the performance of energy based CSS scheme where a group of SUs
cooperated to detect the presence or absence of primary user (PU) in fading channel environment. They also made comparative study on the three main hard fusion techniques i.e. OR-logic, AND-logic and
Majority-logic to make global decisions at the fusion center. In [8], authors proposed selection technique based on iteratively setting different thresholds for different signal to noise ratio (SNR)
of SUs in cooperative spectrum sensing with OR logic fusion technique done at the fusion centre. This scheme highly outperformed the traditional energy spectrum sensing with the same threshold in
terms of reduced probability of false alarm. Higher order test (HOS) have been utilized in literature to analyze data distribution and its degree of departure from the normal distribution. The
concept of separation is based on the maximization of the non-Gaussian property of separated signals to improve the robustness against noise uncertainty. The authors in [9], proposed kurtosis and
skewness (goodness-of-fit) test to check the non-Gaussianity of an averaged periodogram of received SUs signal. This is computed from the Fast Fourier transform (FFT) of the PU signal to justify its
existence and hence the availability or not of the spectrum for a cognitive radio transmission. Their findings showed improved detection of the PU signals especially under very low SNR conditions
i.e. the SUs are able to detect the primary channel with certainty even under very noisy environment. In [10], authors proposed Jarque-Bera tests based spectrum sensing algorithm and compared it to a
kurtosis & skewness combination test statistics. From their simulated results they concluded that Jarque-Bera showed better detection performance than the kurtosis & skewness in terms of the
reliability i.e. improved probability of detection for different values of SUs' SNR. In the emerging research on spectrum sensing schemes, researchers considered a number of modulation schemes on
multipath fading channel based on Jarque-Bera test in detection of the primary user. These schemes were considered to transcend the absence of a priori information of the spectrum occupancy under
additive white Gaussian noise channel [4]. In [11], authors showed Jarque-Bera as having rather poor small data sample properties, slow convergence of the test statistic to its limiting distribution.
In their findings the power of the statistical tests showed the same eccentric form, the reason being skewness and kurtosis are not independently distributed, and the sample kurtosis especially
attains normality very gradually. However, the JB test is simple to calculate and its power has proved to match other powerful statistical tests. A genuine omnibus tests should be consistent to any
departure from the null hypothesis. In [12], authors formulated omnibus test which is based on the standardized third and fourth moments. This was done to assess the normality of random variables by
calculating the transformed samples of kurtosis & skewness. In the computational economics these authors showed omnibus's simplicity provided by the chi-squared framework. In this work the omnibus
test is applied in CSS and compared to other well known Jarque-Bera, kurtosis and skewness tests. Fusion of the decisions received at the fusion center with a view to make the final global decision
on the status of the primary user is also another important challenge that has not been exhaustively studied. Fusion techniques are classified into soft and hard combination schemes. In hard decision
strategy the FC combines binary decisions using standard hard decision rules to achieve the global decision. Three hard combining decision rules used to arrive at the final decision are classified as
AND, OR and majority also called k out ofn counting rule [13]. In [14], authors made a comparative study of the performance of the three hard fusion techniques. In their findings they concluded that
AND rule was the most reliable fusion scheme followed by majority and the lastly the OR rule. Another comparative study on the performance of hard fusion schemes and soft decision schemes was done by
authors in [15]. In their study they confirmed earlier research done to justify that soft fusion decision reported better PU signal detection, albeit having significant data communication overheads.
Hard combination schemes however have attracted most attention from researchers since these fusion schemes are easy to implement by simple logics gates. The authors in [16], proposed strategies on
how the AND, majority and OR fusion rules are optimized based on the Neyman-Pearson criterion. Under this strategy the sensing objective was to maximize the probability of detection with the
constraint on the probability of false alarm of less than 10 percent. Their findings showed AND rule had higher detection performance than the other two. Spectrum sensing in the IEEE 802.22 standard,
for example requires stringent sensing of a false alarm probability of less than 0.1 for a signal as low as -20 dB (SNR) [17]. In [18], authors proposed an the iterative threshold cooperative
spectrum technique. Their objective was to optimize the thresholds of the cooperative spectrum sensing with different fusion rules including AND logic & OR logic. This was done in order to obtain the
optimal SUs in cooperative spectrum sensing and their optimal thresholds. Their algorithm achieved better detection performance for SUs' with different SNR. The optimal scheme also employed fewer SUs
in collaborative sensing at the fusion center. In [19], the authors proposed an optimized detection threshold in order to minimize both the error detection probabilities of single-channel and
multichannel cooperative spectrum sensing. In single-channel cooperative spectrum sensing, they performed an iterative optimal thresholds with AND logic, OR logic and k out of n rule respectively.
Their findings showed a great decrease in the error on detecting PU status on the channel. Energy efficiency in the cognitive radio network is defined as the ratio of throughput (average amount of
successfully delivered bits transmitted from SUs to the fusion centre) to the total average energy consumption in the system [20]. In order to reduce the energy consumed in spectrum sensing network,
not all SUs in each cluster send their sensed results to the fusion center of local cluster. In [21], authors optimized k out ofn by allowing those SUs with reliable sensing results to transmit to
the FC. This showed some reduction in energy consumption of the cognitive radio network. In this paper an optimal k out ofn is applied to improve on the probability of detection and reduce on the
energy system consumption by employing fewer SUs in the final detection on the presence or absence of the PU.
Notations: E[•] is the expectant operator, var is the variance, Im[•]and Re[•] are the imaginary and real parts of the signal X(•), erfc(•) is complementary error function and h is the circular
Gaussian channel.
3. SYSTEM MODEL
3.1 Practical cooperative sensing model
The system model in figure 1 shows a practical CSS network. In this scheme, a group of SUs sense the spectral band to determine the presence or absence of PU. They receive this information through
the control channel and independently analyze it by utilizing the statistical properties of the received PU's signal, and subsequently communicate their individual decisions through the reporting
channel to the FC. At the fusion center, the decisions from individual SUs are integrated together to finally make the global decision on whether the PU is transmitting on the channel or not. The SUs
can then opportunistically access and transmit on the channel if found idle.
3.2 Proposed Cooperative Spectrum Model
In the proposed lower level system model of figure 2, the secondary users (SU[1;]SU[2],...,SU[n]) collectively sense the PU channel based on HOS tests namely, kurtosis & skewness (kurt & skew),
omnibus (omnb) and Jarque-Bera ( JB) statistics tests. The hard binary local decisions made by SUs are transmitted over wireless Gaussian channel represented as (CH[1],CH[2], ...,CH[n]) to the data
FC. The binary data (b[1],b[2],...,b[n]) is fused to achieve the final global decision on the presence or absence of the primary user.
4.1 Spectrum sensing hypothesis
Generally the spectrum sensing problem can be formulated by the following two hypothesis [4,9]
where H[0] and H[1] are null and alternative hypothesis respectively, t is the digital samples numbering T , w(t) is the additive white Gaussian noise, s(t) is the PU's signal and x(t) is the signal
received at the fusion center. The received signal plus additive white Gaussian noise x(t) as function of SNR (y) is given as
where y is the PU signal to noise ratio (SNR). The probability of detection is formulated as hypothesis test P[d] = Prob(Signal Detected | H[1]), whereas the probability of false detection is
determined as Pf = Prob(Signal not Detected | H[1]). Another form of formulation is thresholding on the statistical test parameter. To detect the PU's spectrum effectively there is need to first
estimate and analyze the power spectral density (PSD) of the SUs received signal. A strategic periodogram PSD estimation technique can be used to accurately present the frequency-domain statistical
properties of a signal [9]. Based on the periodogram method and as formulated in algorithm 1, the received signal x(t) of T samples is firstly subdivided into L smaller segments. Then the i-th
segment signal can be formulated as [9]
where i = 0,..., T - 1 is the number of data samples, M = T/L is the length of each segment and t = 0,..., M - 1 are the Fast Fourier transforms (FFT) points in one segment. Performing FFT on signal
sample x[i](t), periodogram of the i-th SU, y[i](t) is given by
where i Є [t, T] is the number of samples, M is the length of each segment representing the elements of discrete Fourier transform (DFT) and y[i](t) is modeled as the PU signal and is utilized in the
next section to determine the skewness and kurtosis.
4.2 Spectrum sensing HOS techniques
Skewness and kurtosis: The estimated skewness (skew) is defined as third standard moment of a random variable x[i](t) of a Gaussian distribution. Estimated kurtosis (kurt) on the other hand is given
by fourth standard moment of a random distribution. The value tends to 3 as the sample size considered for the test increases [20]. For given sample set of yi(t) the estimated sample of skew is given
where y is the mean of a given signal data. Similarly, the estimated kurt of a random sample is formulated as
The test statistics ST(st) of the periodogram (power spectral density) is represented as the square root of the sum of squares of skew(y(t) and kurt(y(t)) as calculated in algorithm 1. When the value
of test statistics is larger than a set threshold Tλ, the distribution of the received signals averaged periodogram deviates from the AWGN's power spectral density, which is an indicator of the
presence of PUs signal. The test statistics of the periodogram estimate can be formulated as
where skew(y(t)) and kurt(y(t)) are the test statistics for skew and kurt respectively of the signal x(t). For a given probability of false alarm (Pf), the threshold (Tλ) for skew and kurt tests the
null hypothesis (H0). This is a chi-squared distribution defined as Pf = 1 - f (Tλ : H[0]) and hence is formulated as [9]
In order to derive the probability of detection (P[d]) and ( Pf) , the PDF for the test statistic is developed for both H[0] and H[1] as
Jarque-Bera (JB): The Jarque-Bera statistic has asymptotic chi-squared distribution with two degrees of freedom [10], formulated by considering the estimated skew and kurt on the transmitted PU
signal, defined as [11]
where M=M[FFT]is the number FFT points. In order to derive the P[d]and P[f]the hypothesis tests H[1] and H[0] are formulated as
For a given probability of false alarm (Pf), the threshold for JB test based on null hypothesis (H[0]), for an M[FFT ]points is expressed as [12]
For the null hypothesis to be accepted the test statistics must be smaller than a critical value that is positive and near zero. Higher values of JB indicate the sample do not follow the Gaussian
distribution. The probability of detection is iteratively determined as shown in the pseudo code for algorithm 2.
Omnibus (K^2) test: Omnibus is defined as the square root of a transformed skewness (skewT) and kurtosis (kurtT) test statistics. The asymptotic normal values for (skew) and (kurt) are used to
construct a chi-squared test involving the first two moments of the asymptotic distributions [12], mathematically expressed as
The hypothetical omnibus test is derived by comparing to defined threshold (K^2[λ]) formulated as
For a predetermined Pf the threshold for omnibus test is a fixed value determined by
where M[FFT]is the number of FFT points. The skewT on the estimated data sample is given as [11,12]
where Φ =W^2 = δ = (Y ) is the estimated skewness value of the random distributed data given as
where skew = skew(y(t)) is estimated skewness of the sampled signal data as given in equation (7), M is the number FFT data sample points. The skewness as a function of the variance ji[2](skew) is
formulated as
The transformed kurtosis (kurtT) on the random distributed received PU's signal is also formulated as [11,12]
where D is a constant that denotes the degrees of freedom for the chi-squared distribution. Solving for D to equate the third moment of theoretical and sampling distributions, it is possible then to
compute D as follows
where B[1] = μ[1](kurt) is the kurtosis as a function of the mean (μ1)given as
where kurt = kurt(y(t)) is the estimated kurtosis given in equation (7) and M is the number of samples. It is possible to standardize kurtosis by formulating the expression as
where the mean as a function of kurtosis is given as var [kurt] =, are all computed to determine transformed estimated kurtosis s shown in algorithm 2.
5. FUSION SCHEMES
5.1 Fusion strategy hypothesis tests
The null hypothesis (H[0]) for decision statistics of the omnibus test can be derived as
where X is the decision threshold which has to be optimized. The cost functions are formulated in terms of probability of misdetection and false alarm as conditioned on the channel, the probability
of misdetection is formulated as [22]
where w(t) is the AWGN. The probability of misdetection
Unlike in [22], this paper uses omnibus test (K^2) instead of kurtosis. λ[i]is the decision threshold, [00], a[10], a[20], a[21], a[30], a[31], a[40], a[41] & a[42] are given in table 1. The
conditional (on the channel) probability of false alarm is given as
where 0 is the phase angle, y is the SNR of the signal, O0 is the modulation constant and u0 is the mean of the data distribution as given in table 2.
5.2 First stage optimization on SU selection criteria
The aim of the first stage optimization is to iteratively select n SUs in SUs, in an r out of n counting rule where r is the number of SUs that form the combinatorial n fusion order and N is the
total number of SUs in CSS network. The criteria on selection is based on SUs decrementing SNR as formulated in algorithm 3. The error probability is further expressed as
where P(H[0]) is the null hypothesis, P(H[1]) is the alternative hypothesis, Qf is the global probability of false alarm and Q[m]is probability of misdetection. The sum of probability of false alarm
and misdetection is derived as a cost function to determine the global decremental error probability (Qe) in the detection of the primary user in CSS network. The minimization problem is formulated
as [15, 16, 18, 19]
where λ^optis the optimal decision threshold. Considering equation (25) and equation (26), the optimal threshold is formulated as
where P[f,i|][y,0] is the false alarm and P[m,i][|][y,0] is the misdetection of the ith SU. From equation (25), the probability of detection is similarly given as
Consequently from equation (29), the threshold is maximized as follows
By the Lagrange theorem, the maximum threshold is obtained by differentiating by parts as follows
where i = 1,...,n is the number of SUs selected to participate in fusion and λ[i] is the initial optimal threshold derived as
where σ^2[s ]is the noise variance, y[i] is the SNR of the i-th SU and M is the number of signal data samples. The global probability of detection in r out of n rule is derived as
where n ε {j = 1,..,N}, N is the total number of SUs, r is the actual number of SUs that form r out ofn counting rule and n is the total number of SUs selected to participate in decision making.
Similarly, the global probability of false alarm is formulated as
where n ε{j = 1,..,N}, algorithm 3. The minimization problem stated in equation (28) is formulated mathematically as
where Q[d] = 1 - Q[m]is the global probability of detection, the probability of false alarm is similarly derived as
The final iteration gives the optimal threshold n number of SUs, formulated as
where the optimal threshold is given in this scenario as
where B =[n] is the SNR for the n-th SU, σ^2[S] is the noise variance and M is the signal data samples. The decremented detection error is expressed as
where the P(H[0]) and P(H[1]) are the weights for probability of false and probability of detection n is the number of SUs participating in detection of the presence or absence of the PU on the
channel, y is the SNR and 0 is the uniformly distributed phase angle.
5.3 Second stage optimal strategy
At the FC, a specific k out of n strategy is employed to process the SUs' received decisions at the FC. Where k is number of SUs in the range of (1 < k < n) and n is the total number of SUs selected
from a total of N as realized in the first optimization stage. The idea behind this rule is to find the number of SUs whose local binary decisions is 1. If this number is larger than or equal k, then
the spectrum is said to be used otherwise the spectrum is unused. An iterative numerical search determined in algorithm 4 is carried out to find an optimal number of k SUs in k out ofn combinatorial
order is done at the FC. To achieve this an upper-threshold of global probability false alarm (Qf) of less than utilization level (ε) is set. The maximization problem can be formulated as [7,15,16]
The global probability of false alarm Qf based on k out ofn counting rule is formulated in algorithm 3 and mathematically derived as
where ε is the utilization level, k is number of SUs selected to participate in the k out of n fusion process, n is number of SUs iteratively found in the first optimization stage in section 5.2. The
derivative of global probability of false alarm (Q f) as function of (Pf) is derived as
From equation (43) it follows that cp is the binomial cumulative function give as
Subsequently the global probability of detection in k out of n case is given as
To optimize equation (45), we differentiate by parts the function as follows
From equations (25) and (26) the following probabilities must hold true.
Similarly the above equation can be further formulated as follows
From the above equation it is true to say Q[d] (k) is linearly increasing function of Qf (k). For all k Є [1,n] then the roots of Qf (k, Pf) are formulated in Bisection algorithm 3. The algorithm is
broken down as follows, for each P[d,i|y][,0] and Q[d](k, Pf), select the highest global probability, the value of k is the optimal number of SUs.
Energy efficiency is the ratio of throughput to average energy consumed during the cooperative spectrum sensing time. The throughput (THR) is formulated as [21,23]
where R is the data rate, t is the transmission time length, P(H[0]) is the probability that the spectrum is not being used, Q f is the global probability of false alarm. The average energy consumed
in the network by all SUs E[c]is derived as
where n is the total number of SUs selected from first optimization stage, e[su]is the energy consumed during CSS by all the SUs, e[st]is the energy consumed during data transmission, Pu is the
probability of identifying if the spectrum is idle, given as
where P(H[1]) = 1 - P(H[0]) is the probability of the spectrum being used, Q f is the global probability of false alarm and Qd is the probability of detection. Note that the energy consumption during
transmission occurs only ifthe spectrum is identified as unused. The efficiency (n) can be formulated as [20, 21]
where n is number of SUs in equation (52), computed as
where N is the total SUs in CSS network, k is the number of SUs in the k out ofn counting rule. A noisy channel is modeled as binary symmetric channel with error probability ( Pe) and it is the same
among all SUs.
In order to evaluate the HOS test for cooperative spectrum sensing capability, we considered a cognitive radio network with 15 SUs transmitting on 16 QAM constellation modulated signal built in
matlab software for simulation and analysis. It should be noted that any other modulation scheme can be used to model the PU signal. In all subsequent figures, the numerical results are plotted on
receiver operating characteristics curves (ROC). Simulation results are denoted with discrete marks on the curves. The simulation parameters are give in table 2.
In figure 3, the ROC curves show the performance of omnibus (omnb), Jarque-Bera (JB), kurtosis & skewness (kurt & skew) and kurtosis (kurt) test statistics as function of the SNR. In this scheme 2048
FFT sample points were considered. From the plot, as expected the probability of detection increased with increase in SNR starting from a low SNR. The omnb test displayed the highest probability of
detection progressively from a low SNR up to about -16 dB. The plot shows that omnb perform better at low SNR. This was followed by JB, then kurt & skew. The results of the other HOS tests are close
to those in [9,10,20]. In figure 4, the graph illustrates performance of the four HOS test considered under a smaller data sample of 512 FFT points. The plot shows omnb still has higher detection
probability for all ranges of SNR and even better under extremely low SNR (-30dB). The omnb test technique therefore tends to suppress the Gaussian noise showing an improved performance. From the two
results displayed in figures (3) and (4), it can be concluded that omnibus is a superior statistical test for both small and big data sample at low SNRs.
In figure 5, the performance of optimal k out ofn counting rule based on all HOS tests is displayed. The rules are for omnibus and majority rule (omnb and maj), Jarque-Bera and majority (JB and maj),
kurtosis & skewness and majority (kurt & skew and ma j). The optimal number of 8 out of 10 SUs was realized through a two stage optimization as given in algorithms (3) and (4). From ROC curves it can
deduced that a combination of omnb and ma j displayed a higher probability of detection for a false alarm of less than 0.1. This is as per the requirement of IEE 802.22 standards [17]. The
performance was then followed by JB and ma j and lastly kurt & skew and ma j. Figure 6, shows a comparative performance HOS based optimal majority rules; omnb and ma j, JB and ma j, kurt & skew and
maj and lastly kurt and maj. The optimal number of 8 out of 10 SUs was realized in the algorithm 3. From the plot, it can be deduced that omnb and ma j combination strategy displayed the lowest
probability of misdetection for all values of probability of false alarm as compared to the three other combinations. In conclusion, based on the figure (5) and (6), omnb and ma j rule showed the
highest probability of detection and the lowest misdetection as compared to all the other HOS based majority rule for all ranges of false alarm.
Figure 7, shows the performance of hybrid sensing scheme of k out of n counting rule based omnibus test for different numbers of SUs was investigated. The plot showed the comparative performance of
different numbers of SUs as selected in single stage compared to two stage optimization. Where n = 10, k = 5 and k = 8 respectively. From this plot, it can be deduced that omnibus a local detection
test based on a two stage optimization global detection scheme displayed higher probability of detection to that of single stage optimization for all ranges of false alarm.
Figure 8, shows the energy efficiency for the different k out of n counting rules representing three scenarios. The first case is when all the SUs in the cooperative spectrum sensing N = 15
participate in the detection of the PU. The second case is when an optimal number of SUs as found in the first optimization stage n = 10 and the third case is when n = 8 just for the purpose of
benchmarking. From this plot the optimal case showed the greatest energy efficiency of about 2 * 10^4 Bits per Joule. This was achieved when k = 8 SUs in the combinatorial order of 8 out o f10
counting rule. Note that due to the k out ofn rule the number of k can only go up to n number of SUs.
8. CONCULSION
In our proposed hybrid model, an optimal k out ofn based omnibus (K^2) statistics test was shown to be more superior to the other HOS tests. This model would be preferred to detect the PU in
cognitive radio networks operating under noisy conditions. Another advantage of this model is the overall reduction in energy consumption in the network due to the two stage optimization. Fewer SUs
make the final decision on the status of the PU on the channel but still maintain reliable decision outcomes.
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Networking , Vol. 1 No. 1, pp. 29-39, March 2017. [ Links ] | {"url":"https://scielo.org.za/scielo.php?script=sci_arttext&pid=S1991-16962018000100004&lng=en&nrm=iso&tlng=en","timestamp":"2024-11-05T09:08:13Z","content_type":"application/xhtml+xml","content_length":"82510","record_id":"<urn:uuid:bd047fc3-db19-4fd8-9283-a95c44ee8069>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00144.warc.gz"} |
Regression(y, b, I, K)
Given data points for a dependent variable «y» indexed by «I» and data for a "basis" (independent variables) «b» indexed by «I» and basis index «K», it returns coefficients C for a linear model:
Variable C := Regression(Y, B, I, K)
Variable Y_est := Sum(C*B, K)
where Y_est contains estimated values of Y.
Regression uses least-squares estimation, meaning that it minimizes the sum of squares of the residuals (estimation error) -- the difference between actual and estimated values:
Sum((y - Y_est)^2, K)
Values of the dependent variable, indexed by «I».
Values of the basis values (independent variables), indexed by «I and «K».
Index for the data points. Each element of «I» corresponds to a different data point.
(Optional)Basis index, or list of independent variables and, usually, a constant. This can be omitted when «b» is a scalar or has an implicit index (like a list) for the different basis terms.
See Regression analysis in Analytica User Guide for basic usage.
The form of a linear regression model is
[math]\displaystyle{ y = \sum_{k} c_k b_k(\bar x) }[/math]
where for any single data point, [math]\displaystyle{ \bar x }[/math] is a vector of values, know as the independent values, [math]\displaystyle{ b_k(\bar x) }[/math] are arbitrary functions of
[math]\displaystyle{ \bar x }[/math] known as the basis functions, and [math]\displaystyle{ y }[/math] is the dependent value. Sometimes the basis functions are trivial functions, such as [math]\
displaystyle{ b_0(x)=1 }[/math] and [math]\displaystyle{ b_1(x)=x }[/math], for [math]\displaystyle{ k=0,1 }[/math], which leads to the one-variable linear model:
[math]\displaystyle{ y = c_1 *x + c_0 }[/math]
When fitting an n^th-degree polynomial of one-variable, x, the basis functions are [math]\displaystyle{ b_k(x)=x^k }[/math] for [math]\displaystyle{ k=0..n }[/math]. Or when you have several
variables, [math]\displaystyle{ x_1, x_2, ..., x_n }[/math], you can fit a hyperplane using [math]\displaystyle{ b_0(\bar x) }[/math]=1 and [math]\displaystyle{ b_k(\bar x)=x_k }[/math] for [math]\
displaystyle{ k=1..n }[/math]. The hyperplane example could be encoded as
Index K := [0,1,2,3,4]
Variable B := Table(K)(1, x1, x2, x3, x4)
Most regressions include a constant in the basis, e.g., [math]\displaystyle{ b_0(\bar x)=1 }[/math], although this is not absolutely required. The coefficient associated with the constant term is
called the bias, offset or y-intercept, and the constant basis term is called the bias term. Without a constant basis term, the fitted model is forced to pass through the origin. This is appropriate
when you know in advance that y must be directly proportional to all your basis terms. But more commonly, leaving out a constant basis term is a mistake.
If you don't want to include a constant term in your basis, Regression() will return the bias as a second return value, provided that your expression captures the second return value. For example,
you can fit a single variable [math]\displaystyle{ y = m*x+b }[/math] model using simply
(m,b) := Regression( y, x, I )
The function knows whether the second return value is captured or not. When it isn't, the model must pass through the origin and the value for m will be different. When your independent variables x
are already indexed by K and you don't want to map these to a new index having one additional element for the constant basis term, it may be more convenient to capture the bias separately, for
Local (c, bias) := Regression( y, x, I, K );
Sum( c * x, K ) + bias
In this case, c is indexed by K, and bias is scalar. When you want to use this pattern, but you want c and bias to be global variables, a pattern that works nicely is
Variable bias := ComputedBy(c)
Variable c :=
Local a;
(a, bias) := Regression( y, x, I, K )
This captures the bias is variable bias, but passes the first return value through as the result for c.
Underconstrained Problems
When you do a regression fit, the number of data points, Size(I), should be greater than the number of basis terms, Size(K). When the number of data points is less than the number of basis terms, the
problem is under-constrained. Provided that there are no two data points having the same basis values but different «Y» values, the fit curve in an underconstrained problem will perfectly pass
through all data points, however, the co-efficients in that case are not unique. In the under-constrained case, Analytica will issue a warning, since this most likely indicates that the «I» and «K»
index parameters were inadvertently swapped. If you ignore the warning, embed the call within an IgnoreWarnings function call, or have the "Show Result Warnings" preference disabled, a set of
coefficients that passes through the existing data points is arbitrarily chosen and returned. The algorithm used is computational inefficient in the under-constrained case where Size(I) << Size(K) --
i.e., the number of basis terms is much larger than the number of data points. If you know your problem is highly underconstrained, then you probably do not intend to use a regression.
Bias term
Secondary Statistics
The Regression function computes the coefficients for the best-fit curve, but it does not compute secondary statistics such as parameter covariances, R-value correlation, or goodness-of-fit.
In what follows, we'll assume that Variable C is the computed regression coefficients, e.g.
Variable C := Regression(Y, B, I, K)
For each data point, the predicted expected value (from the regression) is given by Sum(C*B, K).
However, this prediction provides only the expected value. The RegressionDist function may be used to obtain a distribution over C, and hence a probabilistic estimate of «Y».
The R-squared value, also called the percentage of variance explained by the model, is given by:
Correlation(Y, Sum(C*B, K), I)^2
If your basis «B» might contain NaN or INF values, the corresponding coefficient in C will generally be zero. However, because 0*NaN and 0*INF are indeterminate, the expression Sum(C*B, K) will
return NaN in those cases. To avoid this, use the following expression instead:
Sum(If C = 0 Then 0 Else C*B, K)
If you know the measurement noise in advance, then S is given and may (optionally) be indexed by «I» if the measurement noise varied by data point. If you do not know s in advance, then S can be
obtained from the RegressionNoise function as RegressionNoise(Y, B, I, K, C)
Alternatively, S may be estimated as^†
Local y2 := Sum(C*B, K);
Sqrt( Sum((Y - Y2)^2, I)/(IndexLength(I) - IndexLength(K)))
Estimating S in either of these ways assumes that the noise level is the same for each data point.
In a generalized linear regression, the goodness of fit, or merit, is often characterized using a Chi-squared statistic, computed as:
Sum((Y-Sum(C*B, K))^2/S^2, I)
The S used in this metric must be obtained or estimated separately from the data used to computed the Chi-squared statistic. The resulting metric is a single number that follows a Chi-Squared
distribution with IndexLength(I)-IndexLength(K) degrees of freedom^† under the assumption that the underlying data was generated from a linear model of your basis with normally distributed noise with
a standard deviation of S. S is allowed to vary with I, although in most cases it does not.
Denoting the above as chi2, The probability that the data fit as poor as this would occur by chance is given as^†:
GammaI(IndexLength(I)/2 - 1, chi2/2)
This metric can be conveniently obtained using the RegressionFitProb function.
Another set of secondary statistics are the covariances of the fitted parameters. The covariance is an estimate of the amount of uncertainty in the parameter estimate given the available data. As the
number of data points increases (for a given basis), the variances and covariances tend to decrease. To compute the covariances, a copy of Index «K» is required (since the covariance matrix is square
in «K»); hence, you need to create a new index node defined as:
Index K2 := CopyIndex(K)
The co-variances are then computed as:
Variable CV_C := Invert(Sum(B * B[K = K2]/S^2, I), K, K2)
The diagonal elements of this matrix give the variance in each parameter. Since there is only a finite number of samples, the parameter estimate may be off a bit due to random chance, even if the
linear model assumption is correct; this variance indicates how much error exists from random chance at the given data set size.
With S and CV_C (the covariance of parameters C), a distribution on the expected value of «Y» can be obtained for a given input X (indexded by «I»), using:
Variable Coef := Gaussian(C, CV_C, K, K2)
Variable Y_predicted := Sum(Coef * X, K) + Normal(0, S)
The RegressionDist returns the uncertain coefficients directly, and is the more convenient function to use when you wish to estimate the uncertainty in your predicted value.
Numeric Limitations
The Regression function is highly robust to the presence of redundant basis terms. For example, if one of the basis functions is a linear combination of a subset of other basis functions, the
coefficients are not unique determined. For many other implementations of the Regression function (e.g., in other products), this can lead to numeric instabilities, with very large coefficients and
losses of precision from numeric round-off. Analytica uses an SVD-based method for Regression which is extremely robust to these effects, and guarantees good results even when basis terms are
Prior to build 4.0.0.59, this method that ensures robustness works for basis values up to about 100M. If basis values exceed 100M, then they should be scaled prior to using Regression. Starting with
build 4.0.0.59, Analytica automatically scales basis values so that large values are handled robustly as well.
Weighted Regression
Weighted regression assigns a non-negative weight to each data point. The weight, «w», therefore is indexed by «I». You may assign a weight of zero to points that you don't want to contribute at all
to the result.
One way to interpret these weights is to indicate the relative informativeness of each point -- or inversely the level of noise -- when that varies across the data points. Suppose each point comes
from a linear model with noise with distribution Normal(0, s/w_i), where the mean is zero, s is an unknown global noise level, and w_i is the noise for the ith data point. Compare this to a
non-weighted regression, where all points have the same amount of noise, according to Normal(0, s). A point with weight 0 has infinite standard deviation, and thus no usable information.
The coefficients for a weighted regression are given by
Regression(Y*w, B*w, I, K)
where «Y», «B», «I», and «K» are the customary parameters to regression, and «w» is the relative weighting which is indexed by «I».
Sometimes the raw data may include multiple observations with the same values, and the number of observations itself is included in the data. In this case, if n (indexed by «I») represents the number
of observations for each element of «Y» and «B», the weighting w should be Sqrt(n).
Using weights of 0 and 1 makes it possible to ignore certain points. However, to ignore points where a basis term or the «Y» value might be NaN, you need to test for w:
Local Y2 := If w = 0 Then 0 Else Y*w;
Local B2 := If w = 0 Then 0 Else B*w;
Regression(Y2, B2, I, K)
Plotting Regression lines Compared to Data
To overlay the regressed curves on the original data, a good method is to continue using a scatter plot for both data and curves. Create a variable as a list of identifiers -- the first identifier
being the original «Y»-value data, the second identifier being the fit-«Y» value (i.e., Sum(Basis*X, K)). Then plot the result as an XY plot, using your X value variable as an XY Comparison Source.
The model RegressionCompare.ana demonstrates this, with the plot shown here:
Comparison of Alternative Bases
Adding more basis terms to your regression model will improve the fit on your training data, but you may reach a point where the extra terms decreases the quality of predictions on new data. This
phenomena is referred to as over-fitting. As a general rule, as the number of data points increases, the more basis terms you can have before overfitting becomes a problem.
One approach that is often employed to evaluate whether the improvement justifies the additional basis terms is to use an F-Test. The output of this test is a p-value, which gives the probability
that the improvement obtained is just due to sample noise under the assumption that the extra basis terms do not actually contribute any additional information. A small p-value means that there is
statistically significant support that the extra basis terms improves the goodness of fit. One standard is to accept a model with more basis terms when the p-value is less than 5%.
To compute the F-test p-value, we'll use these variables:
• Basis1 indexed by K1 and «I» : The smaller basis (simpler model)
• Basis2 indexed by K2 and «I»: The larger basis (more complex model)
• «Y» indexed by «I» : The observed values for the dependent variable
Then the regression coefficients for each model are:
Variable c1 := Regression(Y, Basis1, I, K1)
Variable c2 := Regression(Y, Basis2, I, K2)
The forecasted values from each model are:
Variable y1 := Sum(c1*Basis1, K1)
Variable y2 := Sum(c2*Basis1, K2)
And the sum-of-square residuals are:
Variable Rss1 := Sum( (y1- y)^2, I)
Variable Rss2 := Sum( (y2 - y)^2, I)
The F-statistic is given by^†^:
Variable Fstat := (Rss1 - Rss2)/Rss2 * (IndexLength(I)-IndexLength(K2))/(IndexLength(K2) - IndexLength(K1))
And the p-value is^†^
Variable pValue := 1- CumFDist(Fstat, IndexLength(K2) - IndexLength(K1), IndexLength(I) - IndexLength(K2))
When a Null appears in «Y», the corresponding point is ignored.
When a Null appears in «B», the corresponding basis term along «K» is not used. The coefficient for that basis term in the final result is Null.
† Various expressions that appear on this page for the Secondary statistics use IndexLength(I) or IndexLength(K), which give the number of data points and number of basis terms respectively. For
example, these appear in the expressions for Fstat, pValue, S and chi2. However, when Nulls are present, this reduces the effective number of data point or basis terms, so if you have Nulls in «y» of
«b», you should substitute these
Starting in Analytica 5.0, if every term in «B» for a given point is Null, then the point is not used, and this does not cause any basis terms to be removed from consideration. However, if any
non-null appears for that point in «B», then the terms where the Null values appear are removed from the basis.
See Also | {"url":"https://docs.analytica.com/index.php?title=Regression&oldid=60945","timestamp":"2024-11-14T03:52:24Z","content_type":"text/html","content_length":"47515","record_id":"<urn:uuid:3ed81053-3999-432e-89f6-2a7ded8214c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00239.warc.gz"} |
[QSMS Seminar 2022-08-24] Vector bundles and representation theory on a curve
• Date : 2022-08-24 (Wed) 10:00 AM
• Place : 129-406 (SNU)
• Speaker : 문한봄 (Fordham University)
• Title : Vector bundles and representation theory on a curve
• Abstract :
Many classical constructions in representation theory can be extended to vector bundles on algebraic curves. The classical Borel-Weil theorem on the geometric description of irreducible
representations of simple algebraic groups has been extended to this setting by Teleman. I will introduce algebraic vector bundles and Borel-Weil-Bott-Teleman theory. Then I will describe its
consequences on the derived category of the moduli space of vector bundles on a curve and the Fano visitor problem of Jacobian varieties. This is ongoing joint work with Kyoung-Seog Lee. | {"url":"https://qsms.math.snu.ac.kr/index.php?mid=board_sjXR83&document_srl=2347&order_type=desc&listStyle=viewer&page=3","timestamp":"2024-11-13T17:51:49Z","content_type":"text/html","content_length":"20824","record_id":"<urn:uuid:0ab53ae1-36cb-46eb-97c3-5ff4738a9ad8>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00601.warc.gz"} |
In mathematics, the n-sphere is the generalization of the ordinary sphere to spaces of arbitrary dimension. It is an n-dimensional manifold that can be embedded in Euclidean (n + 1)-space.
For any natural number n, an n-sphere of radius r may be defined in terms of an embedding in (n + 1)-dimensional Euclidean space as the set of points that are at distance r from a central point,
where the radius r may be any positive real number. Thus, the n-sphere would be defined by:
In particular:
An n-sphere embedded in an (n + 1)-dimensional Euclidean space is called a hypersphere. The n-sphere of unit radius is called the unit n-sphere, denoted S^n. The unit n-sphere is often referred to as
the n-sphere.
When embedded as described, an n-sphere is the surface or boundary of an (n + 1)-dimensional ball. For n ≥ 2, the n-spheres are the simply connected n-dimensional manifolds of constant, positive
curvature. The n-spheres admit several other topological descriptions: for example, they can be constructed by gluing two n-dimensional Euclidean spaces together, by identifying the boundary of an n
-cube with a point, or (inductively) by forming the suspension of an (n − 1)-sphere.
For any natural number n, an n-sphere of radius r is defined as the set of points in (n + 1)-dimensional Euclidean space that are at distance r from some fixed point c, where r may be any positive
real number and where c may be any point in (n + 1)-dimensional space. In particular:
• a 0-sphere is a pair of points {c − r, c + r}, and is the boundary of a line segment (1-ball).
• a 1-sphere is a circle of radius r centered at c, and is the boundary of a disk (2-ball).
• a 2-sphere is an ordinary 2-dimensional sphere in 3-dimensional Euclidean space, and is the boundary of an ordinary ball (3-ball).
• a 3-sphere is a sphere in 4-dimensional Euclidean space.
Euclidean coordinates in (n + 1)-space
The set of points in (n + 1)-space: (x[1],x[2],…,x[n+1]) that define an n-sphere (S^n), is represented by the equation:
where c is a center point, and r is the radius.
The above n-sphere exists in (n + 1)-dimensional Euclidean space and is an example of an n-manifold. The ppl volume form ω of an n-sphere of radius r is given by
where * is the Hodge star operator; see Flanders (1989, §6.1) for a discussion and proof of this formula in the case r = 1. As a result,
The space enclosed by an n-sphere is called an (n + 1)-ball. An (n + 1)-ball is closed if it includes the n-sphere, and it is open if it does not include the n-sphere.
Topological description
Topologically, an n-sphere can be constructed as a one-point compactification of n-dimensional Euclidean space. Briefly, the n-sphere can be described as , which is n-dimensional Euclidean space plus
a single point representing infinity in all directions. In particular, if a single point is removed from an n-sphere, it becomes homeomorphic to . This forms the basis for stereographic projection.^
Volume and surface area
and are the n-dimensional volume and surface area of the n-ball and n-sphere of radius , respectively.
The constants and (for the unit ball and sphere) are related by the recurrences:
The surfaces and volumes can also be given in closed form:
where is the gamma function. Derivations of these equations are given in this section.
In general, the volumes of the n-ball in n-dimensional Euclidean space, and the n-sphere in (n + 1)-dimensional Euclidean, of radius R, are proportional to the nth power of the radius, R. We write
for the volume of the n-ball and for the surface of the n-sphere, both of radius .
Interestingly, given the radius R, the volume and the surface area of the n-sphere reaches a maximum and then decrease towards zero as the dimension n increases. In particular, the volume of the n
-sphere of constant radius R in n-dimensions reaches a maximum for dimension if <R< and if where for . Similarly, defining the sequence , the surface area of the n-sphere of constant radius R in n
dimensions reaches a maximum for dimension if and if .^[2]
The 0-ball consists of a single point. The 0-dimensional Hausdorff measure is the number of points in a set, so
The unit 1-ball is the interval of length 2. So,
The 0-sphere consists of its two end-points, . So,
The unit 1-sphere is the unit circle in the Euclidean plane, and this has circumference (1-dimensional measure)
The region enclosed by the unit 1-sphere is the 2-ball, or unit disc, and this has area (2-dimensional measure)
Analogously, in 3-dimensional Euclidean space, the surface area (2-dimensional measure) of the unit 2-sphere is given by
and the volume enclosed is the volume (3-dimensional measure) of the unit 3-ball, given by
The surface area, or properly the n-dimensional volume, of the n-sphere at the boundary of the (n + 1)-ball of radius is related to the volume of the ball by the differential equation
or, equivalently, representing the unit n-ball as a union of concentric (n − 1)-sphere shells,
We can also represent the unit (n + 2)-sphere as a union of tori, each the product of a circle (1-sphere) with an n-sphere. Let and , so that and . Then,
Since , the equation holds for all n.
This completes our derivation of the recurrences:
Closed forms
Combining the recurrences, we see that . So it is simple to show by induction on k that,
where denotes the double factorial, defined for odd integers 2k + 1 by (2k + 1)!! = 1 · 3 · 5 ··· (2k − 1) · (2k + 1).
In general, the volume, in n-dimensional Euclidean space, of the unit n-ball, is given by
where is the gamma function, which satisfies .
By multiplying by , differentiating with respect to , and then setting , we get the closed form
Other relations
The recurrences can be combined to give a "reverse-direction" recurrence relation for surface area, as depicted in the diagram:
Index-shifting n to n − 2 then yields the recurrence relations:
where S[0] = 2, V[1] = 2, S[1] = 2π and V[2] = π.
The recurrence relation for can also be proved via integration with 2-dimensional polar coordinates:
Spherical coordinates
We may define a coordinate system in an n-dimensional Euclidean space which is analogous to the spherical coordinate system defined for 3-dimensional Euclidean space, in which the coordinates consist
of a radial coordinate, and n − 1 angular coordinates where ranges over radians (or over [0, 360) degrees) and the other angles range over radians (or over [0, 180] degrees). If are the Cartesian
coordinates, then we may compute from with:
Except in the special cases described below, the inverse transformation is unique:
where if for some but all of are zero then when , and radians (180 degrees) when .
There are some special cases where the inverse transform is not unique; for any will be ambiguous whenever all of are zero; in this case may be chosen to be zero.
Spherical volume element
Expressing the angular measures in radians, the volume element in n-dimensional Euclidean space will be found from the Jacobian of the transformation:
and the above equation for the volume of the n-ball can be recovered by integrating:
The volume element of the (n-1)–sphere, which generalizes the area element of the 2-sphere, is given by
The natural choice of an orthogonal basis over the angular coordinates is a product of ultraspherical polynomials,
for j = 1, 2, ..., n − 2, and the e^ isφ[j] for the angle j = n − 1 in concordance with the spherical harmonics.
Stereographic projection
Just as a two-dimensional sphere embedded in three dimensions can be mapped onto a two-dimensional plane by a stereographic projection, an n-sphere can be mapped onto an n-dimensional hyperplane by
the n-dimensional version of the stereographic projection. For example, the point on a two-dimensional sphere of radius 1 maps to the point on the plane. In other words,
Likewise, the stereographic projection of an n-sphere of radius 1 will map to the dimensional hyperplane perpendicular to the axis as
Generating random points
Uniformly at random from the (n − 1)-sphere
To generate uniformly distributed random points on the (n − 1)-sphere (i.e., the surface of the n-ball), Marsaglia (1972) gives the following algorithm.
Generate an n-dimensional vector of normal deviates (it suffices to use N(0, 1), although in fact the choice of the variance is arbitrary), .
Now calculate the "radius" of this point,
The vector is uniformly distributed over the surface of the unit n-ball.
For example, when n = 2 the normal distribution exp(−x[1]^2) when expanded over another axis exp(−x[2]^2) after multiplication takes the form exp(−x[1]^2−x[2]^2) or exp(−r^2) and so is only dependent
on distance from the origin.
Another way to generate a random distribution on a hypersphere is to make a uniform distribution over a hypercube that includes the unit hyperball, exclude those points that are outside the
hyperball, then project the remaining interior points outward from the origin onto the surface. This will give a uniform distribution, but it is necessary to remove the exterior points. As the
relative volume of the hyperball to the hypercube decreases very rapidly with dimension, this procedure will succeed with high probability only for fairly small numbers of dimensions.
Wendel's theorem gives the probability that all of the points generated will lie in the same half of the hypersphere.
Uniformly at random from the n-ball
With a point selected from the surface of the n-ball uniformly at random, one needs only a radius to obtain a point uniformly at random within the n-ball. If u is a number generated uniformly at
random from the interval [0, 1] and x is a point selected uniformly at random from the surface of the n-ball then u^1/nx is uniformly distributed over the entire unit n-ball.
Specific spheres
The pair of points {±R} with the discrete topology for some R > 0. The only sphere that is not path-connected. Has a natural Lie group structure; isomorphic to O(1). Parallelizable.
Also known as the circle. Has a nontrivial fundamental group. Abelian Lie group structure U(1); the circle group. Topologically equivalent to the real projective line, RP^1. Parallelizable. SO(2)
= U(1).
Also known as the sphere. Complex structure; see Riemann sphere. Equivalent to the complex projective line, CP^1. SO(3)/SO(2).
Also known as the glome. Parallelizable, Principal U(1)-bundle over the 2-sphere, Lie group structure Sp(1), where also
Equivalent to the quaternionic projective line, HP^1. SO(5)/SO(4).
Principal U(1)-bundle over CP^2. SO(6)/SO(5) = SU(3)/SU(2).
Almost complex structure coming from the set of pure unit octonions. SO(7)/SO(6) = G[2]/SU(3).
Topological quasigroup structure as the set of unit octonions. Principal Sp(1)-bundle over S^4. Parallelizable. SO(8)/SO(7) = SU(4)/SU(3) = Sp(2)/Sp(1) = Spin(7)/G[2] = Spin(6)/SU(3). The
7-sphere is of particular interest since it was in this dimension that the first exotic spheres were discovered.
Equivalent to the octonionic projective line OP^1.
A highly dense sphere-packing is possible in 24-dimensional space, which is related to the unique qualities of the Leech lattice.
See also
1. ↑ James W. Vick (1994). Homology theory, p. 60. Springer
2. ↑ Loskot, Pavel (November 2007). "On Monotonicity of the Hypersphere Volume and Area". Journal of Geometry. 87 (1-2): 96–98. doi:10.1007/s00022-007-1891-1.
External links
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- version of the 12/3/2016. The text is available under the
Creative Commons Attribution/Share Alike
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Albert Einstein: Biography, Publications, Legacy
Born: Ulm, Germany. Died: Princeton, USA.
Albert Einstein was a very famous scientist; he was mostly famous for his theory of Relativity. In 1894 Einstein’s family moved to Milan and Einstein decided officially to relinquish his German
citizenship in favor of Swiss. In 1895 Einstein failed an examination that would have allowed him to study for a diploma as an electrical engineer at Zurich. After attending secondary school at
Aarau, Einstein returned (1896) to the Zurich Polytechnic, graduating (1900) as a secondary school teacher of mathematics and physics.
He worked at the patent office in Bern from 1902 to 1909 and while there he completed an astonishing range of theoretical physics publications, written in his spare time without the benefit of close
contact with scientific literature or colleagues. Einstein earned a doctorate from the University of Zurich in 1905. In 1908 he became a lecturer at the University of Bern, the following year
becoming professor of physics at the University of Zurich.
By 1909 Einstein was recognized as a leading scientific thinker. After holding chairs in Prague and Zurich he advanced (1914) to a prestigious post at the Kaiser-Wilhelm Gesellschaft in Berlin. From
this time he never taught a university courses. Einstein remained on the staff at Berlin until 1933, from which time until his death he held a research position at the Institute for Advanced Study in
In the first of three papers (1905) Einstein examined the phenomenon discovered by Max Planck, according to which electromagnetic energy seemed to be emitted from radiating objects in discrete
quantities. The energy of these quanta was directly proportional to the frequency of the radiation. This seemed at odds with the classical electromagnetic theory, based on Maxwell’s equations and the
laws of thermodynamics which assumed that electromagnetic energy consisted of waves which could contain any small amount of energy. Einstein used Planck’s quantum hypothesis to describe the
electromagnetic radiation of light.
Einstein’s second 1905 paper proposed what is today called the special theory of relativity. He based his new theory on a reinterpretation of the classical principle of relativity, namely that the
laws of physics had to have the same form in any frame of reference. As a second fundamental hypothesis, Einstein assumed that the speed of light remained constant in all frames of reference, as
required by Maxwell’s theory.
Later in 1905 Einstein showed how mass and energy were equivalent. Einstein was not the first to propose all the components of special theory of relativity. His contribution is unifying important
parts of classical mechanics and Maxwell’s electrodynamics. The third of Einstein’s papers of 1905 concerned statistical mechanics, a field of that had been studied by Ludwig Boltzmann and Josiah
After 1905 Einstein continued working in the areas described above. He made important contributions to quantum theory, but he sought to extend the special theory of relativity to phenomena involving
acceleration. The key appeared in 1907 with the principle of equivalence, in which gravitational acceleration was held to be indistinguishable from acceleration caused by mechanical forces.
Gravitational mass was therefore identical with inertial mass.
By 1911 Einstein was able to make preliminary predictions about how a ray of light from a distant star, passing near the Sun, would appear to be bent slightly, in the direction of the Sun. About
1912, Einstein began a new phase of his gravitational research, with the help of his mathematician friend Marcel Grossmann, by expressing his work in terms of the tensor calculus of Tullio
Levi-Civita and Gregorio Ricci-Curbastro. Einstein called his new work the general theory of relativity. After a number of false starts he published, late in 1915, the definitive version of general
When British eclipse expeditions in 1919 confirmed his predictions, Einstein was idolized by the popular press. Einstein returned to Germany in 1914 but did not reapply for German citizenship.
Einstein received the Nobel Prize in 1921 but not for relativity rather for his 1905 work on the photoelectric effect. He worked at Princeton on work which attempted to unify the laws of physics
until his death. | {"url":"https://schoolworkhelper.net/albert-einstein-biography-publications-legacy/","timestamp":"2024-11-14T10:23:03Z","content_type":"text/html","content_length":"611113","record_id":"<urn:uuid:9f5c1e32-9005-4639-9fed-41fb756a8a3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00256.warc.gz"} |
75 Hotel Statistics & Travel Trends in São Tomé, Sao Tome and Principe for 2024
75 Must-Know Hotel Statistics and Travel Trends in São Tomé, Sao Tome and Principe for 2024
Discover the most compelling hotel statistics and travel trends in São Tomé for 2024! Whether you're a traveler planning your next adventure or a professional in the hospitality industry, this
comprehensive guide of 75 key statistics and travel trends offers valuable insights into São Tomé's dynamic hotel scene. Backed by data from 16 hotels, 738 traveler reviews, and 2,076 price points,
we unveil the patterns and preferences shaping tourism in this remarkable destination.
Hotel and Travel Statistics for São Tomé
Top Hotel and Travel Statistics for São Tomé
• There are 16 hotels operating in São Tomé.
• The average hotel rating in São Tomé is 7.02, based on 801 reviews.
• Travelers can expect to pay around $104 per night for a hotel in São Tomé.
• If you're looking for the best month to visit São Tomé by rating, it's July with an average rating of 8.83.
• If you're looking for the cheapest month to visit São Tomé, it's November with an average price of $95.
• Prices range from $54 for the cheapest hotel to $187 for the most expensive hotel in São Tomé.
• The most expensive hotel in São Tomé is Hotel Praia with prices starting at $187.
• The cheapest hotel in São Tomé is Sweet Guest House with prices starting at $54.
• Friends are the most satisfied travelers when visiting São Tomé, rating their stays at 8.66 on average.
• Group Travelers are the least satisfied travelers when visiting São Tomé, rating their stays at 6.77 on average.
• Hotel prices in São Tomé peak in July, with an average price of $318.
Hotel Availability and Types in São Tomé
Number of Hotels
• There are 16 hotels in São Tomé.
Distribution by Star Rating
• São Tomé has 4 hotels with a 3-star rating, accounting for 25.0% of all hotels.
• São Tomé has 4 hotels with a 4-star rating, accounting for 25.0% of all hotels.
• São Tomé has 1 hotels with a 5-star rating, accounting for 6.3% of all hotels.
• We also have 7 hotels with an unknown star rating in São Tomé, accounting for 43.8% of all hotels.
Hotel Pricing Trends in São Tomé
Average Prices Over Time
• The average price of a hotel in São Tomé is $104 per night.
Average Price by Star Rating
• The average price of a 3-star hotel in São Tomé is $61 per night.
• The average price of a 4-star hotel in São Tomé is $151 per night.
• The average price of a 5-star hotel in São Tomé is $224 per night.
• The average price of a hotel in São Tomé with an unknown star rating is $59 per night.
Hotel Price Distribution
• There are 2 hotels in São Tomé priced in the $0-$50 range, accounting for 16.7% of all hotels.
• There are 5 hotels in São Tomé priced in the $50-$100 range, accounting for 41.7% of all hotels.
• There are 4 hotels in São Tomé priced in the $100-$200 range, accounting for 33.3% of all hotels.
• There are 1 hotels in São Tomé priced in the $200-$500 range, accounting for 8.3% of all hotels.
Best Month to Visit by Price
• The average price of a hotel in São Tomé in January is $112.
• The average price of a hotel in São Tomé in February is $111.
• The average price of a hotel in São Tomé in March is $110.
• The average price of a hotel in São Tomé in April is $119.
• The average price of a hotel in São Tomé in May is $317.
• The average price of a hotel in São Tomé in June is $317.
• The average price of a hotel in São Tomé in July is $318.
• The average price of a hotel in São Tomé in October is $102.
• The average price of a hotel in São Tomé in November is $95.
• The average price of a hotel in São Tomé in December is $97.
Hotel Ratings and Reviews in São Tomé
Number of Reviews
• We've collected 801 reviews for hotels in São Tomé.
Review Distribution by Traveler Type
• There are 160 reviews from business travelers in São Tomé, accounting for 20.0% of all reviews.
• There are 260 reviews from couples in São Tomé, accounting for 32.5% of all reviews.
• There are 140 reviews from families in São Tomé, accounting for 17.5% of all reviews.
• There are 50 reviews from friends in São Tomé, accounting for 6.2% of all reviews.
• There are 28 reviews from group travelers in São Tomé, accounting for 3.5% of all reviews.
• There are 107 reviews from solo travelers in São Tomé, accounting for 13.4% of all reviews.
• There are 56 reviews from travelers with an unknown type in São Tomé, accounting for 7.0% of all reviews.
Average Hotel Ratings Over Time
• The average rating for hotels in São Tomé in 2024 is 8.45, based on 194 reviews.
• The average rating for hotels in São Tomé in 2023 was 8.37, based on 141 reviews.
• The average rating for hotels in São Tomé in 2022 was 5.97, based on 87 reviews.
• The average rating for hotels in São Tomé in 2021 was 6.97, based on 28 reviews.
• The average rating for hotels in São Tomé in 2020 was 6.70, based on 13 reviews.
• The average rating for hotels in São Tomé in 2019 was 8.74, based on 43 reviews.
• The average rating for hotels in São Tomé in 2018 was 7.26, based on 59 reviews.
• The average rating for hotels in São Tomé in 2017 was 8.54, based on 70 reviews.
• The average rating for hotels in São Tomé in 2016 was 8.10, based on 57 reviews.
• The average rating for hotels in São Tomé in 2015 was 8.25, based on 33 reviews.
• The average rating for hotels in São Tomé in 2014 was 7.52, based on 24 reviews.
• The average rating for hotels in São Tomé in 2013 was 7.22, based on 21 reviews.
• The average rating for hotels in São Tomé in 2012 was 8.76, based on 14 reviews.
Average Ratings by Star Rating
• The average rating for 4-star hotels in São Tomé is 7.81.
• The average rating for 5-star hotels in São Tomé is 8.00.
• The average rating for hotels in São Tomé with an unknown star rating is 5.90.
Average Ratings by Traveler Type
• The average rating for business travelers in São Tomé is 7.08.
• The average rating for couples in São Tomé is 7.82.
• The average rating for families in São Tomé is 7.87.
• The average rating for friends in São Tomé is 8.66.
• The average rating for group travelers in São Tomé is 6.77.
• The average rating for solo travelers in São Tomé is 7.68.
• The average rating for travelers with an unknown type in São Tomé is 7.78.
Best Months to Visit by Ratings
• The average rating for hotels in São Tomé in January is 7.45.
• The average rating for hotels in São Tomé in February is 8.29.
• The average rating for hotels in São Tomé in March is 8.77.
• The average rating for hotels in São Tomé in April is 7.47.
• The average rating for hotels in São Tomé in May is 7.96.
• The average rating for hotels in São Tomé in June is 8.10.
• The average rating for hotels in São Tomé in July is 8.83.
• The average rating for hotels in São Tomé in August is 7.97.
• The average rating for hotels in São Tomé in September is 7.12.
• The average rating for hotels in São Tomé in October is 8.06.
• The average rating for hotels in São Tomé in November is 6.47.
• The average rating for hotels in São Tomé in December is 8.09.
Notable Hotels in São Tomé
Most Expensive Hotels in São Tomé
• The most expensive hotel in São Tomé is Hotel Praia with prices starting at $187.
Cheapest Hotels in São Tomé
• The cheapest hotel in São Tomé is Sweet Guest House with prices starting at $54. | {"url":"https://yoga-hotels.in/statistics/hotels/africa/sao-tome-and-principe/sao-tome","timestamp":"2024-11-03T16:18:44Z","content_type":"text/html","content_length":"898815","record_id":"<urn:uuid:0824a145-072b-4970-aeac-1a53979ea70b>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00049.warc.gz"} |
Next: MINIMUM COMMUNICATION COST SPANNING Up: Spanning Trees Previous: MAXIMUM MINIMUM METRIC K-SPANNING   Index
• INSTANCE: Graph B.
• SOLUTION: A spanning subgraph G such that the sum of the weights of the edges in E' does not exceed B.
• MEASURE: The diameter of the spanning subgraph.
• Bad News: Not approximable within 2405].
• Comment: Not approximable within 3/2 Not approximable within 5/4l(e)=1 for every edge e [405]. For results on minimum Steiner trees of bounded diameter, see MINIMUM STEINER TREE.
• Garey and Johnson: Similar to ND4
Viggo Kann | {"url":"https://www.csc.kth.se/~viggo/wwwcompendium/node76.html","timestamp":"2024-11-04T11:41:49Z","content_type":"text/html","content_length":"4580","record_id":"<urn:uuid:84973716-34ef-4ec9-808a-eb0dae6ecf84>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00312.warc.gz"} |
Mastering the table() Function in R » Data Science Tutorials
Mastering the table() Function in R, The table() function in R is a powerful tool for creating frequency tables, allowing you to quickly summarize the distribution of variables in your data.
In this article, we’ll explore the basics of table() and demonstrate its applications through practical examples.
Step-by-Step Data Science Coding Course
Syntax:Mastering the table() Function in R
The basic syntax of the table() function is:
Where x is a vector or a data frame.
Example 1: Frequency Table for One Variable
Let’s start with an example that demonstrates how to create a frequency table for the position variable in our data frame:
# Create data frame
df <- data.frame(player = c('AddJ', 'Bodkjgb', 'Chdgad', 'Dadgjdsn', 'dsjghdric', 'Frandgsk'),
position = c('A', 'B', 'B', 'B', 'B', 'A'),
points = c(51, 52, 52, 81, 70, 50))
# View data frame
# Calculate frequency table for position variable
The output will be a vector containing the frequency of each level of the position variable.
A B
Example 2: Frequency Table of Proportions for One Variable
In this example, we’ll use prop.table() to create a frequency table of proportions for the position variable:
# Calculate frequency table of proportions for position variable
The output will be a vector containing the proportion of each level of the position variable.
A B
0.3333333 0.6666667
Example 3: Frequency Table for Two Variables
Let’s create a frequency table for the position and points variable:
# Calculate frequency table for position and points variable
table(df$position, df$points)
The output will be a matrix containing the frequency of each combination of levels of the position and points variables.
A 1 1 0 0 0
B 0 0 2 1 1
Example 4: Frequency Table of Proportions for Two Variables
In this example, we’ll use prop.table() to create a frequency table of proportions for the position and points variable:
# Calculate frequency table of proportions for position and points variable
prop.table(table(df$position, df$points))
The output will be a matrix containing the proportion of each combination of levels of the position and points variables.
A 0.1666667 0.1666667 0.0000000 0.0000000 0.0000000
B 0.0000000 0.0000000 0.3333333 0.1666667 0.1666667
Tips and Variations
• You can use additional arguments with table() to specify specific levels or subsets of your data.
• You can use prop.table() to create frequency tables of proportions instead of frequencies.
• You can use options() to specify how many decimals to display in your proportion table.
• You can use table() with other types of data structures, such as lists or matrices.
In conclusion, the table() function is a powerful tool in R that allows you to quickly create frequency tables and summarize the distribution of variables in your data.
By mastering this function, you can gain valuable insights into your data and make informed decisions.
With its flexibility and versatility, table() is an essential tool for any R programmer. | {"url":"https://datasciencetut.com/mastering-the-table-function-in-r/","timestamp":"2024-11-06T22:08:31Z","content_type":"text/html","content_length":"111358","record_id":"<urn:uuid:35518fc9-704f-4d7f-85fb-545d5ca1877f>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00411.warc.gz"} |
Copyright (c) The University of Glasgow 2001
License BSD-style (see the file libraries/base/LICENSE)
Maintainer libraries@haskell.org
Stability stable
Portability portable
Safe Haskell Trustworthy
Language Haskell2010
Functions associated with the tuple data types.
data Solo a Source #
Solo is the canonical lifted 1-tuple, just like (,) is the canonical lifted 2-tuple (pair) and (,,) is the canonical lifted 3-tuple (triple).
The most important feature of Solo is that it is possible to force its "outside" (usually by pattern matching) without forcing its "inside", because it is defined as a datatype rather than a newtype.
One situation where this can be useful is when writing a function to extract a value from a data structure. Suppose you write an implementation of arrays and offer only this function to index into
index :: Array a -> Int -> a
Now imagine that someone wants to extract a value from an array and store it in a lazy-valued finite map/dictionary:
insert "hello" (arr index 12) m
This can actually lead to a space leak. The value is not actually extracted from the array until that value (now buried in a map) is forced. That means the entire array may be kept live by just that
value! Often, the solution is to use a strict map, or to force the value before storing it, but for some purposes that's undesirable.
One common solution is to include an indexing function that can produce its result in an arbitrary Applicative context:
indexA :: Applicative f => Array a -> Int -> f a
When using indexA in a pure context, Solo serves as a handy Applicative functor to hold the result. You could write a non-leaky version of the above example thus:
case arr indexA 12 of
Solo a -> insert "hello" a m
While such simple extraction functions are the most common uses for unary tuples, they can also be useful for fine-grained control of strict-spined data structure traversals, and for unifying the
implementations of lazy and strict mapping functions.
pattern Solo :: a -> Solo a
Applicative Solo Source # Since: base-4.15
Defined in GHC.Internal.Base
Functor Solo Source # Since: base-4.15
Defined in GHC.Internal.Base
Monad Solo Source # Since: base-4.15
Defined in GHC.Internal.Base
MonadFix Solo Source # Since: base-4.15
Defined in GHC.Internal.Control.Monad.Fix
Foldable Solo Source # Since: base-4.15
Defined in GHC.Internal.Data.Foldable
Traversable Solo Source # Since: base-4.15
Defined in GHC.Internal.Data.Traversable
Generic1 Solo Source #
Defined in GHC.Internal.Generics
type Rep1 Solo Since: base-4.15
Defined in GHC.Internal.Generics
Monoid a => Monoid (Solo a) Source # Since: base-4.15
Defined in GHC.Internal.Base
Semigroup a => Semigroup (Solo a) Source # Since: base-4.15
Defined in GHC.Internal.Base
Data a => Data (Solo a) Source # Since: base-4.15
Defined in GHC.Internal.Data.Data
Bounded a => Bounded (Solo a) Source #
Defined in GHC.Internal.Enum
Enum a => Enum (Solo a) Source #
Defined in GHC.Internal.Enum
Generic (Solo a) Source #
Defined in GHC.Internal.Generics
type Rep (Solo a) Since: base-4.15
Defined in GHC.Internal.Generics
Ix a => Ix (Solo a) Source #
Defined in GHC.Internal.Ix
Read a => Read (Solo a) Source # Since: base-4.15
Defined in GHC.Internal.Read
Show a => Show (Solo a) Source # Since: base-4.15
Defined in GHC.Internal.Show
Eq a => Eq (Solo a)
Defined in GHC.Classes
Ord a => Ord (Solo a)
Defined in GHC.Classes
type Rep1 Solo Source # Since: base-4.15
Defined in GHC.Internal.Generics
type Rep (Solo a) Source # Since: base-4.15
Defined in GHC.Internal.Generics
getSolo :: Solo a -> a Source #
Extract the value from a Solo. Very often, values should be extracted directly using pattern matching, to control just what gets evaluated when. getSolo is for convenience in situations where that is
not the case:
When the result is passed to a strict function, it makes no difference whether the pattern matching is done on the "outside" or on the "inside":
Data.Set.insert (getSolo sol) set === case sol of Solo v -> Data.Set.insert v set
A traversal may be performed in Solo in order to control evaluation internally, while using getSolo to extract the final result. A strict mapping function, for example, could be defined
map' :: Traversable t => (a -> b) -> t a -> t b
map' f = getSolo . traverse ((Solo $!) . f)
fst :: (a, b) -> a Source #
Extract the first component of a pair.
snd :: (a, b) -> b Source #
Extract the second component of a pair.
curry :: ((a, b) -> c) -> a -> b -> c Source #
Convert an uncurried function to a curried function.
>>> curry fst 1 2
uncurry :: (a -> b -> c) -> (a, b) -> c Source #
uncurry converts a curried function to a function on pairs.
>>> uncurry (+) (1,2)
>>> uncurry ($) (show, 1)
>>> map (uncurry max) [(1,2), (3,4), (6,8)]
swap :: (a, b) -> (b, a) Source #
Swap the components of a pair. | {"url":"http://hackage-origin.haskell.org/package/ghc-internal-9.1001.0/docs/GHC-Internal-Data-Tuple.html","timestamp":"2024-11-10T02:13:59Z","content_type":"application/xhtml+xml","content_length":"73126","record_id":"<urn:uuid:5525334c-e04c-40ef-88b3-c662fd5d85f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00686.warc.gz"} |
Schur Polynomials, Banded Toeplitz Matrices and Widom's Formula
Schur Polynomials, Banded Toeplitz Matrices and Widom's Formula
Keywords: Banded Toeplitz matrices, Schur polynomials, Widom's determinant formula, sequence insertion, Young tableaux, recurrence
We prove that for arbitrary partitions $\mathbf{\lambda} \subseteq \mathbf{\kappa},$ and integers $0\leq c<r\leq n,$ the sequence of Schur polynomials $S_{(\mathbf{\kappa} + k\cdot\mathbf{1}^c)/(\
mathbf{\lambda} + k\cdot\mathbf{1}^r)}(x_1,\dots,x_n)$ for $k$ sufficiently large, satisfy a linear recurrence. The roots of the characteristic equation are given explicitly. These recurrences are
also valid for certain sequences of minors of banded Toeplitz matrices.
In addition, we show that Widom's determinant formula from 1958 is a special case of a well-known identity for Schur polynomials. | {"url":"https://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i4p22","timestamp":"2024-11-03T13:40:12Z","content_type":"text/html","content_length":"14893","record_id":"<urn:uuid:64a799cf-0027-44a9-8d55-8a9c59b56ce5>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00166.warc.gz"} |
57. International Winter Meeting on Nuclear Physics
We derive the volume and surface components of the nuclear symmetry energy (NSE) and their ratio [1] within the coherent density fluctuation model [2, 3]. The estimations use the results of the model
for the NSE in finite nuclei based on the Brueckner and Skyrme energy-density functionals for nuclear matter. The obtained values of these quantities for the Ni, Sn, and Pb isotopic chains are
compared with estimations of other approaches which have used available experimental data on binding energies, neutron-skin thicknesses, and excitation energies to isobaric analog states. Apart from
the density dependence investigated in our previous works [4, 5, 6], we study also the temperature dependence of the symmetry energy in finite nuclei [7] in the framework of the local density
approximation combining it with the self-consistent Skyrme-HFB method using the cylindrical transformed deformed harmonic oscillator basis. The results for the thermal evolution of the NSE in the
interval T=0—4 MeV show that its values decrease with temperature. The same formalism is applied to obtain the values of the volume and surface contributions to the NSE and their ratio at finite
temperatures [8]. We confirm the existence of "kinks" of these quantities as functions of the mass number at T = 0 MeV for the double closed shell nuclei 78Ni and 132Sn and the lack of "kinks" for
the Pb isotopes, as well as the disappearance of these kinks as the temperature increases. References [1] A. N. Antonov, M. K. Gaidarov, P. Sarriguren, and E. Moya de Guerra, Phys. Rev. C 94, 014319
(2016). [2] A.N. Antonov, V.A. Nikolaev, and I.Zh. Petkov, Bulg. J. Phys. 6 (1979) 151; Z. Phys. A 297 (1980) 257; ibid 304 (1982) 239; Nuovo Cimento A 86 (1985) 23; A.N. Antonov et al., ibid 102
(1989) 1701; A.N. Antonov, D.N. Kadrev, and P.E. Hodgson, Phys. Rev. C 50 (1994) 164. [3] A.N. Antonov, P.E. Hodgson, and I.Zh. Petkov, Nucleon Momentum and Density Distributions in Nuclei, Clarendon
Press, Oxford (1988); Nucleon Correlations in Nuclei, Springer-Verlag, Berlin-Heidelberg-New York (1993). [4] M. K. Gaidarov, A. N. Antonov, P. Sarriguren, and E. Moya de Guerra, Phys. Rev. C 84,
034316 (2011). [5] M. K. Gaidarov, A. N. Antonov, P. Sarriguren, and E. Moya de Guerra, Phys. Rev. C 85, 064319 (2012). [6] M. K. Gaidarov, P. Sarriguren, A. N. Antonov, and E. Moya de Guerra, Phys.
Rev. C 89, 064301 (2014). [7] A. N. Antonov, D. N. Kadrev, M. K. Gaidarov, P. Sarriguren, and E. Moya de Guerra, Phys. Rev. C 95, 024314 (2017). [8] A. N. Antonov, D. N. Kadrev, M. K. Gaidarov, P.
Sarriguren, and E. Moya de Guerra, Phys. Rev. C 98, 054315 (2018). | {"url":"https://indico.mitp.uni-mainz.de/event/188/timetable/?view=standard_numbered","timestamp":"2024-11-05T15:22:23Z","content_type":"text/html","content_length":"365458","record_id":"<urn:uuid:2e893ac4-15d3-4062-8070-e3a52291b8a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00834.warc.gz"} |
The regional-scale surface mass balance of Pine Island Glacier, West Antarctica, over the period 2005–2014, derived from airborne radar soundings and neutron probe measurements
Articles | Volume 15, issue 3
© Author(s) 2021. This work is distributed under the Creative Commons Attribution 4.0 License.
The regional-scale surface mass balance of Pine Island Glacier, West Antarctica, over the period 2005–2014, derived from airborne radar soundings and neutron probe measurements
We derive recent surface mass balance (SMB) estimates from airborne radar observations along the iSTAR traverse (2013, 2014) at Pine Island Glacier (PIG), West Antarctica. Ground-based neutron probe
measurements provide information of snow and firn density with depth at 22 locations and were used to date internal annual reflection layers. The 2005 layer was traced for a total distance of 2367km
to determine annual mean SMB for the period 2005–2014. Using complementary SMB estimates from two regional climate models, RACMO2.3p2 and MAR, and a geostatistical kriging scheme, we determine a
regional-scale SMB distribution with similar main characteristics to that determined for the period 1985–2009 in previous studies. Local departures exist for the northern PIG slopes, where the
orographic precipitation shadow effect appears to be more pronounced in our observations, and the southward interior, where the SMB gradient is more pronounced in previous studies. We derive total
mass inputs of 79.9±19.2 and 82.1±19.2Gtyr^−1 to the PIG basin based on complementary ASIRAS–RACMO and ASIRAS–MAR SMB estimates, respectively. These are not significantly different to the value of
78.3±6.8Gtyr^−1 for the period 1985–2009. Thus, there is no evidence of a secular trend at decadal scales in total mass input to the PIG basin. We note, however, that our estimated uncertainty is
more than twice the uncertainty for the 1985–2009 estimate on total mass input. Our error analysis indicates that uncertainty estimates on total mass input are highly sensitive to the selected krige
methodology and assumptions made on the interpolation error, which we identify as the main cause for the increased uncertainty range compared to the 1985–2009 estimates.
Received: 09 Apr 2020 – Discussion started: 08 Jun 2020 – Revised: 09 Dec 2020 – Accepted: 17 Jan 2021 – Published: 11 Mar 2021
The stability of the West Antarctic Ice Sheet (WAIS) is a major concern for scientists seeking to predict global sea level rise. Transport of heat from upwelling circumpolar deep water has proved to
be a critical driver of Antarctic ice shelf thinning and grounding line retreat, thus initiating the acceleration of marine-terminating outlet glaciers (e.g. Hillenbrand et al., 2017). In particular
the Amundsen Sea sector has experienced an unprecedented acceleration in ice discharge since the beginning of satellite-based ice flow observations in the 1970s. Three-quarters of this ice discharge
stems from the Thwaites and Pine Island glaciers, with both showing evidence of rapid acceleration since the 1970s (Mouginot et al., 2014) and spreading of surface lowering along their tributaries
over the past two decades (Konrad et al., 2017). While spaceborne observations indicate that this acceleration has levelled off recently (Rignot et al., 2019), they also support model projections
suggesting modest changes in mass balance, i.e. the resulting net ice loss after accounting for all loss and gain processes, for the next decades to come (Bamber and Dawson, 2020). The dynamic ice
loss is mainly responsible for the negative mass balance of Pine Island Glacier (PIG). The net input is commonly referred to as the surface mass balance (SMB), i.e. snowfall minus sublimation,
meltwater runoff, and erosion/deposition of snow (Lenaerts et al., 2012; Medley et al., 2013). Various methods exist to measure the SMB on the ground (Eisen et al., 2008). The remoteness of WAIS
makes such measurements logistically challenging, in particular when extending these measurements to regional scales. Basin-wide total mass input estimates strongly depend on the coverage and quality
of SMB measurements. The study of Medley et al. (2014), hereinafter abbreviated as ME14, presents the first comprehensive survey of mean annual SMB between 1985 and 2009/2010 from airborne
radar-based observations of the Thwaites and Pine Island glaciers. The authors demonstrated that such airborne radar observations provide a critical means to overcome logistical challenges. However,
these measurements rely on assumptions about the dielectric properties of snow and firn, which include knowledge of their vertical density profiles. In this sense, ground-truthing measurements remain
an important tool for calibrating the radar soundings.
As part of the iSTAR Ice Sheet Stability Programme, a traverse across the Pine Island Glacier (PIG) was carried out in 2013/2014 (T1) and repeated the year after (T2). In total 22 sites were occupied
during both traverses. Boreholes of at least 13m depth were drilled at each site during traverse T1. Density–depth profiles were measured with a neutron probe (NP) device during both traverses (
Morris et al., 2017), and supplementary analysis of firn cores was performed for 10 sites during traverse T2 to determine additional independent proxies related to the annual snow accumulation (
Konrad et al., 2019).
The Alfred Wegener Institute (AWI) contributed to the iSTAR traverse T2 with radar soundings from the Airborne SAR/Interferometric Radar Altimeter System (ASIRAS) aboard the Polar 5 research plane.
Previous ASIRAS missions have demonstrated its capability to track annual snow accumulation layers of the upper firn column at regional scales over Greenland (Hawley et al., 2006; Overly et al., 2016
). The PIG flight track connects all iSTAR sites so that internal annual snow accumulation layers can be traced to make regional-scale SMB estimates. By comparison with earlier SMB measurements at
PIG, the vertical profiling based on the ASIRAS soundings achieves a resolution that is 1 order of magnitude higher (Table 1), which helps to trace narrow internal snow accumulation layers. In
addition, the ASIRAS flight track contains several crossovers, which we used to validate the same isochronal reflector from different directions.
In this study we first address local departures between SMB estimates from ASIRAS and NP measurements to evaluate the uncertainty of our regional-scale ASIRAS SMB estimates. We then compare our
results with those reported by ME14 and discuss differences between both data sets. Finally, we apply our new regional-scale SMB estimates to different PIG mass balance inventories to evaluate their
impact in light of the current stability of the study area. We include a list of abbreviations and notations in Appendix A.
2.1iSTAR traverse
The iSTAR traverse followed the PIG main trunk as well as its tributaries as shown in Fig. 1. A total flight track (black lines) of 2486km was covered by the ASIRAS measurements between 1 and
3 December 2014. Following ME14, the basin outlines (dashed lines) include the Wedge zone between PIG and Thwaites. The main emphasis of the iSTAR campaign was on the fast-flowing segments of PIG;
thus we lack measurements from the southward interior. Earlier observations from ME14 suggest that the SMB decreases towards the interior so the contribution from this area to the total mass input
will be less than that from the rest of the basin.
Additional SMB measurements were made with a ground-penetrating radar (GPR) during traverse T1 and published in Konrad et al. (2019). The authors selected the ∼1986 reflection layer, which
approximately coincides with the observed main reflector by ME14, and traced the layer along sections of the 900km traverse, amounting to a total of a 613km distance covered by GPR observations.
The route of these observations closely follows the ASIRAS flight track, and both are available at http://gis.istar.ac.uk/ (last access: 24 February 2021). Due to the limited maximum sampling depth
of the ASIRAS and NP measurements, the 1985/1986 reflection layer used by ME14 and Konrad et al. (2019) is not contained in most of our data. To benefit from the ASIRAS coverage while simultaneously
accounting for its limited depth range, we manually traced the continuous 2005 reflection layer over a distance of 2367km to derive mean annual SMB estimates for the 2005–2014 period. Due to the
reported consistency between the GPR and airborne SMB measurements in Konrad et al. (2019), we limit the comparison of our results to the basin-wide estimates by ME14. In addition, we assume that the
effect of strain history, which could affect our SMB estimates at the fast-flowing sections of PIG, is negligible. Konrad et al. (2019) conclude that the total effect over the whole catchment is
small, even though it can have a very significant effect at some sites. However, this effect is expected to be further reduced for the shallower reflection layer depths from the ASIRAS measurements.
(Konrad et al., 2019)
2.2Neutron probe measurements
NP measurements of snow and firn density were performed at all stations during both traverses as described in Morris et al. (2017). Further details on the calibration procedure, which is based on
theoretical considerations, can be found in Morris (2008). A comparison with gravimetric density measurements at existing core profiles did not indicate a systematic bias between both measurement
methods. To evaluate the effect of densification, the ground team repeated the density profiling in the same boreholes during traverse T2. Because the most recent accumulation is missing in these
profiles, they drilled an additional borehole of less than 6m depth and a nearby distance of about 1m to capture it during traverse T2. The only exception is site 2, where the ground team decided
to auger a completely new 14m borehole for the density profiling due to poor data from the T1 hole.
The deep firn cores (∼50m) shown in Fig. 1 were collected and analysed by the British Antarctic Survey. This analysis includes the annual variations with depth of the photochemical H[2]O[2] tracer
and density, which are phase shifted by about 6 months. According to Morris et al. (2017) the annual density variation is caused by the alternating late austral summer/autumn low-density hoar layer
with winter snow which has densified under the influence of warm summer temperatures. The different processes, which modulate the density and H[2]O[2] concentration with depth, allow for
an independent determination of annual snow accumulation at the 10 deep core sites. No volcanic reference horizon was detected in the cores (Robert Mulvaney, personal communication, 2020), which
therefore limits the annual markers to the H[2]O[2] and density profiles. Morris et al. (2017) applied an automatic annual layer identification routine to the vertical density profiles and used the
annual H[2]O[2] peak depths as an additional guidance for the annual layer dating. Thus, the depth–age scales from both annual markers are consistent.
We use a single regional density–depth profile we derive from the NP profiles of traverse T2 for the two-way-travel time (TWT)-to-depth conversion of the ASIRAS soundings. First, we merge the ∼13m
and nearby ∼6m density–depth profiles at each site (except at site 2) by linearly relaxing their overlapping segments. To reduce the effect of lateral noise convolution, we limit the relaxation
length to the overlapping segments that correlate well with each other. Then we align the intercepting depth–age scales to create a consistent depth–age scale for each compiled profile. The resulting
21 merged profiles and the single profile at site 2 are shown by the grey lines in Fig. 2. From these 22 profiles, we then determine a smoothed regional mean profile, which is denoted by the black
line. Morris et al. (2017) observed a two-stage Herron and Langway (1980) type densification at PIG, with the stages separated by an additional transition zone. We achieve a good fit to our regional
mean profile with a simple exponential function (red dashed line, Fig. 2), which we apply to the TWT-to-depth conversion. The blue dashed lines show the fitted standard deviation of the density as
a function of depth. Following Medley et al. (2013), we consider the fitted standard deviation to be representative of the spatial uncertainty of the regional-scale density–depth profile.
2.3ASIRAS soundings
ASIRAS is a Ku-band radar altimeter which operates at a carrier frequency of 13.5GHz and a bandwidth of 1GHz (Mavrocordatos et al., 2004). It was set to low-altitude mode (designed for heights less
than 1500m above ground) during its measurements at PIG. A synthetic aperture radar (SAR) processing of the collected data was performed, which yields the spatial resolution of the SAR level_1b data
shown in Table 1. The associated cross-track footprint is ∼15m. We use the electromagnetic wave speed $v=c/\sqrt{{\mathit{ϵ}}^{\prime }}$ to convert TWT to depth, where c is the vacuum speed of
light and ϵ^′ is the real part of the dielectric permittivity of the firn column. For the latter, we apply the commonly used empirical relation by Kovacs et al. (1995):
$\begin{array}{}\text{(1)}& {\mathit{ϵ}}_{\mathrm{kov}}^{\prime }=\left(\mathrm{1}+\mathrm{0.845}{\mathit{\rho }}_{\mathrm{s}}{\right)}^{\mathrm{2}},\end{array}$
where ${\mathit{\rho }}_{\mathrm{s}}=\mathit{\rho }/{\mathit{\rho }}_{\mathrm{w}}$ is the specific gravity of snow (or firn) at current depth with respect to the water density ρ[w]=1000kgm^−3.
An alternative model by Looyenga (1965) is
$\begin{array}{}\text{(2)}& {\mathit{ϵ}}_{\mathrm{loo}}^{\prime }={\left(\frac{\mathit{\rho }}{{\mathit{\rho }}_{\mathrm{ice}}}\left[\sqrt[\mathrm{3}]{{\mathit{ϵ}}_{\mathrm{ice}}^{\prime }}-\mathrm
with ${\mathit{ϵ}}_{\mathrm{ice}}^{\prime }=\mathrm{3.17}$ (Evans, 1965) and ${\mathit{\rho }}_{\mathrm{ice}}=\mathrm{917}\phantom{\rule{0.125em}{0ex}}\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm
{m}}^{-\mathrm{3}}$. Sinisalo et al. (2013), who consider a similar depth range to this study, conclude that the difference between wave speeds based on Eqs. (1) and (2) has a negligible impact on
their SMB estimates. This is also the case for our estimates (see Sect. 3). The maximum depth of the radargrams is ∼30m based on the TWT-to-depth conversion from the fitted regional mean profile of
density with depth and substituted Kovacs relation. The depth range of resolved internal stratigraphy varies along the flight track, but the layering remains visible for most of the upper 13m depth
covered by the NP measurements. Using the regional mean profile of density with depth, we determine the water equivalent (w.e.) depth value for each waveform bin and calculate the mass per unit area
between the selected reflection layer (magenta line in Fig. 3) and surface. We assume that internal reflection layers are generated by the dielectric contrast at embedded thin ice and hoar layers (
Arcone et al., 2004, 2005) and that these layers are formed at regional scales around summer/autumn (Medley et al., 2013). These layers may coincide with the annual density modulation, which we
observe with the NP measurements.
Before the layer tracing, we apply an automatic-gain-control filter to all waveforms and limit their dynamic range to twice the standard deviation centred around the mean amplitude of each waveform.
This improved the signal contrast of the radargram. Initially we tested a phase-following algorithm of the Paradigm EPOS geophysical processing software to trace the selected reflection layer
semi-automatically. However, this method became unstable for lower contrast and cases with close layer spacing. Furthermore, remaining SAR-processing artefacts were interfering with the
phase-following algorithm. Because of the complex nature of the observed stratigraphy, as has been also reported by Konrad et al. (2019), manual layer tracing was used. Following Richardson et al. (
1997), we attempted to bridge distorted or merged layer segments whenever distinct characteristics of a vertical layer sequence could be identified with confidence before and after the bridging.
Different processes can lead to distortion of the reflection layers, e.g. processes changing deposition of the annual snow layers or excessive rolling angles of the aeroplane, while merging layers
can result from low snow precipitation and ablative processes (e.g. wind scouring) or a combination of both. We checked the traced layer for possible mismatches which may have resulted from
systematic errors in the initial manual layer tracing at 34 crossover points and 8 nearby flight track segments. Such mismatches were particularly observed along challenging profile sections and
corrected by retracing the reflection layers, which yield the best match at the crossover points. In this sense, the layer tracing is performed independently from the annual layer dating at each
iSTAR site.
2.4Measurement error estimation
We attempt to trace the 2005 reflection layer, which is covered by all NP density–depth profiles. So far, we assumed that internal reflection layers form on an annual basis during summer/autumn, but
the potential formation of intra-annual reflection layers may challenge this assumption. For instance, Nicolas et al. (2017) found evidence of surface melt episodes over large parts of WAIS in
response to warm air intrusion events. Scott et al. (2010) observed a strong reflection layer, which coincides with an exceptional melt layer at 22m depth at one PIG ice core location. These
findings suggest that intra-annual reflection layers can form at the basin scale, even though the formation is less frequent, as it appears to be related to the complex coupling between different
atmospheric modes (e.g. Nicolas et al., 2017; Donat-Magnin et al., 2020). The frequency of intra-annual reflection layer formation may change towards the coast, where the snow accumulation is high.
For instance, Fig. 3 shows additional reflection layers with respect to the annual density markers from the NP measurements at site 21. Extreme solid precipitation events may also impact the density
modulation with depth (Turner et al., 2019), which is considered for the depth–age scale based on the NP measurements. Snow erosion may remove annual markers where accumulation rates are low. In
addition to annual layer counting errors, the timing between the reflection layer formation and snow densification may be offset during summer/autumn. All these factors challenge the tracing and
dating of the 2005 reflection layer, but combining the stratigraphic information from the ASIRAS and iSTAR observations helps reduce the risk of systematic errors from erroneous layer counting. To
account for the remaining risk in terms of isochronal accuracy, we assign an annual layer tracing uncertainty of $\stackrel{\mathrm{‾}}{\mathit{\delta }t}=±\mathrm{1}$ years.
Following Morris et al. (2017), we define mass balance years between the density peaks in the NP profiles (nominally 1 July). For instance, the mass balance year 2013 begins at the second annual
density peak below the surface (nominally 1 July 2013) and ends at the first peak (1 July 2014). Based on annual density markers, we can relate the snow and firn depth at each iSTAR site to its
associated age and determine the reflection layer age from its depth at each cross section. Here, we use an exponential fit of the local density–depth profile for the TWT-to-depth conversion. The
lateral displacement ΔD between the point of closest approach of the flight track and iSTAR site adds to the reflection layer dating uncertainty. We therefore consider all N points which lie within
a 2ΔD interval along the flight track. The interval is centred at the point of closest approach for the layer dating. Based on the local depth–age scale, we relate the estimated depths of N points to
their ages and assign the final layer date to their mean value. To account for the mass balance year definition above, we add 6 months to the mean layer date, which is listed for all iSTAR positions
in Table 2. In addition, we estimate the dating uncertainty from the N lateral estimates by their standard deviation σ[x]. In this sense, our error estimate is more conservative than the standard
error of the mean. Furthermore, we assume that the uncertainty due to local variation in the stratigraphy is isotropic, which does not generally need to be true. However, according to Table 2 the
overall impact of this effect is 1 order of magnitude smaller than the variability of layer age values among all iSTAR sites in most cases. As indicated in Table 2, we excluded dating estimates
around iSTAR sites 2 and 19. In both cases, our layer tracing revealed a large offset contrary to the neighbouring iSTAR sites. Possible reasons for these offsets could be systematic errors in the
layer dating from the NP profiles, the variability of internal stratigraphy between the ASIRAS measurements and their closest approach to both iSTAR sites, or systematic errors in the manual
reflection layer tracing. The remaining exclusion of layer age estimates at iSTAR sites 7, 12, and 18 is either due to high noise levels of the radargram or reflection layer depths significantly
exceeding the NP depth–age scales. Following Konrad et al. (2019) we estimate the final reflection layer year by the mean of dating values at each site with an uncertainty of $\mathrm{\Delta }t=\sqrt
{{\stackrel{\mathrm{‾}}{\mathit{\delta }t}}^{\mathrm{2}}+\mathit{\delta }{\stackrel{\mathrm{‾}}{t}}^{\mathrm{2}}+{\stackrel{\mathrm{‾}}{t}}_{x}^{\mathrm{2}}}$, with the standard deviation of dating
estimates $\mathit{\delta }\stackrel{\mathrm{‾}}{t}$ and the propagated error ${\stackrel{\mathrm{‾}}{t}}_{x}=\mathrm{1}/n\sqrt{{\sum }_{i}^{n}{\mathit{\sigma }}_{x}\left(i{\right)}^{\mathrm{2}}}$
from the n lateral error estimates around each iSTAR site (i=site number), which we introduced in addition. The resulting reflection layer dating estimate is $T=\mathrm{2004.8}±\mathrm{1.4}$, which
corresponds to a layer age of $a=\mathrm{10.1}±\mathrm{1.4}$. The associated average surface accumulation rate $\stackrel{\mathrm{˙}}{b}$ in terms of w.e. depth per year is
$\begin{array}{}\text{(3)}& \stackrel{\mathrm{˙}}{b}=\frac{\mathrm{1}}{a{\mathit{\rho }}_{\mathrm{w}}}\sum _{i=\mathrm{1}}^{m}\mathit{\delta }{z}_{i}{\mathit{\rho }}_{i},\end{array}$
where δz[i] is the ith depth increment of the radar waveform and ρ[i] is the associated density. Substitution of the wave propagation speed for δz[i] yields
$\begin{array}{}\text{(4)}& \stackrel{\mathrm{˙}}{b}=\frac{\mathrm{1}}{a{\mathit{\rho }}_{\mathrm{w}}}\sum _{i=\mathrm{1}}^{m}\frac{c{t}_{\mathrm{s}}}{\sqrt{{\mathit{ϵ}}_{i}^{\prime }}}{\mathit{\rho
where t[s]=0.37ns is the ASIRAS vertical bin sampling time (i.e. 0.5×TWT per bin), and ${\mathit{ϵ}}_{i}^{\prime }$ refers to the permittivity value at the ith bin. To avoid any confusion with
previous summations, the final index m refers to the traced waveform bin at the reflection layer depth. It is evident from Eq. (4) that the spatial uncertainty of the density profile affects both the
integration depth and incremental mass. Medley et al. (2013) and ME14 estimated the spatial uncertainty from the resulting SMB change by directly applying the standard deviation fits of their
regional density profiles to the TWT-to-SMB conversion. Instead, we may propagate the error in Eq. (4), assuming that errors are uncorrelated and normally distributed. Based on the Kovacs relation
according to Eq. (1) we account for the temporal, spatial, and digitization error components:
$\begin{array}{}\text{(5)}& \mathrm{\Delta }\stackrel{\mathrm{˙}}{b}=\frac{c{t}_{\mathrm{s}}}{a{\mathit{\rho }}_{\mathrm{w}}}\sqrt{\underset{\text{spatial}}{\underbrace{\sum _{i=\mathrm{1}}^{m}\left
(\frac{\mathrm{\Delta }{\mathit{\rho }}_{i}}{{\mathit{ϵ}}_{\mathrm{kov},i}^{\prime }}{\right)}^{\mathrm{2}}}}+\underset{\text{temporal}}{\underbrace{\left(\frac{\mathrm{\Delta }a}{a}\sum _{j=\mathrm
{1}}^{m}\frac{{\mathit{\rho }}_{j}}{\sqrt{{\mathit{ϵ}}_{\mathrm{kov},j}^{\prime }}}{\right)}^{\mathrm{2}}}}+\underset{\text{digitization}}{\underbrace{\left(\frac{\mathrm{1}}{\mathrm{3}}\sum _{k=m-\
mathrm{1}}^{m+\mathrm{1}}\frac{{\mathit{\rho }}_{k}}{\sqrt{{\mathit{ϵ}}_{\mathrm{kov},k}^{\prime }}}{\right)}^{\mathrm{2}}}}},\end{array}$
where $\mathrm{\Delta }a=±\mathrm{1.4}$ years is the temporal uncertainty, and Δρ[i] represents the standard deviation intervals according to Fig. 2. Due to the small incremental density change of
<0.7% along the entire profile, we approximate the digitization error by the mean SMB value of three consecutive bins centred at the final profile bin of the current integration depth. Figure 4a
displays the propagated individual measurement error components as well as the combined measurement error according to Eq. (5) as a function of geometric depth. In addition, we include the error
partitioning in Fig. 4b. The grey background shades highlight the distribution of layer depths to visualize the relevant error range of our SMB estimates, which peaks around (5, 8, and 10)m (darker
shades). In comparison with Medley et al. (2013) and ME14, we find that our spatial error estimate based on Eq. (5) is reduced by about 1 order of magnitude, while the standard deviation fits of
their regional density profiles cover a similar range compared to ours. We may ignore the spatial error compensation in Eq. (5) by replacing the root sum of squares (RSS) with absolute values:
$\sum _{i=\mathrm{1}}^{m}{\left(\mathrm{\Delta }{\mathit{\rho }}_{i}/{\mathit{ϵ}}_{\mathrm{kov},i}^{\prime }\right)}^{\mathrm{2}}\to {\left(\sum _{i=\mathrm{1}}^{m}\mathrm{|}\mathrm{\Delta }{\mathit
{\rho }}_{i}/{\mathit{ϵ}}_{\mathrm{kov},i}^{\prime }\mathrm{|}\right)}^{\mathrm{2}}.$
Hence, to comply with the studies above, we consider the more conservative spatial error propagation based on the sum of absolute values, but we keep the RSS of individual error components for the
combined measurement error estimate as shown in Fig. 4c–d. Following these assumptions, we find that our measurement error estimate is still dominated by the temporal layer dating uncertainty for
most of the traced layer depths, but the spatial error reaches a similar range to that reported in Medley et al. (2013). We consider the combined measurement error based on Fig. 4c–d for the SMB
estimates of this study.
2.5Kriging scheme
We focus on the regional-scale variability of the SMB distribution at PIG. Figure 5 shows our high-resolution (i.e. metre-scale) SMB estimates as well as smoothed SMB values with contour lines from
a digital elevation model (DEM) by Helm et al. (2014). We use the same 25km along-track smoothing window as ME14 and choose a sampling interval of half the smoothing window length. We initially
tested the same interpolation scheme as described in ME14 to estimate a regional-scale SMB field for the PIG basin from our smoothed SMB points. This scheme is based on the ordinary kriging (OK)
algorithm, a widely used geostatistical interpolation technique (e.g. Isaaks and Srivastava, 1991). Instead of a direct OK interpolation of smoothed SMB observations, ME14 consider the residual SMB
values with regard to an ordinary least squares linear regression model for the Thwaites–PIG basin area with northing, easting, and elevation as explanatory variables. This, in turn, yields a small
degree of skewness <0.5 with respect to the residual SMB distribution. However, we failed to reduce the skewness of residual SMB values from our estimates effectively using the same method, which may
be due to the different aerial coverage considered in our regression model. Examination of the DEM contour lines in Fig. 5 reveals that a simple relation between surface elevation and SMB is not
evident, which may hint that the prevailing synoptic-scale weather conditions at the Amundsen and Bellingshausen Sea sectors in combination with the precipitation shadowing effect of the mountain
ranges of Eights Coast (Fig. 1) require a more sophisticated model to capture the SMB at the PIG basin scale. We therefore searched for an alternative approach to generate krige estimates from the
SMB sample population of this study without the use of a regression model. Such alternative, which is also mentioned in ME14, is a logarithmic transformation of the SMB observations prior to the OK
$\begin{array}{}\text{(6)}& \stackrel{\mathrm{˙}}{B}\left({x}_{\mathrm{0}}\right)=\mathrm{ln}\left(\stackrel{\mathrm{˙}}{b}\left({x}_{\mathrm{0}}\right)+C\right),\end{array}$
where C is an arbitrary constant and x[0] represents the current interpolation location. After the OK interpolation of transformed SMB observations, the estimates must be transformed back into the
original measurement scale. This back transformation requires the addition of a correction term for each OK estimate to ensure that the expected value is equal to the sample mean and that the
smoothing effect is adequately compensated (i.e. resulting estimates reproduce the sample histogram and sample mean; Yamamoto, 2007). We implemented such an ordinary logarithmic kriging (OLK) method
in our analysis by adopting the four-step post-processing algorithm proposed by Yamamoto (2007) for the estimation of “nonbias terms”. According to Yamamoto (2008), OLK does not necessarily require
a log-normal sample distribution to produce improved estimates in terms of local accuracy. Furthermore, Yamamoto (2007) tested the impact of constant C according to Eq. (6) and found that a data
translation towards higher values yields an approximation from OLK to OK estimates, thus eliminating the advantage of improved sample mean reproduction and local accuracy of OLK estimates. Indeed, we
find that adding a negative constant C to all SMB values, such that the lowest SMB value reaches 0.1$\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\
mathrm{yr}}^{-\mathrm{1}}$, yields an improved reproduction of the observation data characteristics. Figure 6 shows the experimental isotropic semivariogram of our log-transformed SMB observations
from Fig. 5b together with a Gaussian model fit with a practical range of ∼190km, i.e. the range at which the spatial autocorrelation of sample points is vanishing. Following Yamamoto (2005, 2007),
we investigate the reproduction of observational data characteristics by means of PP plots (i.e. percentiles of cumulative distributions of observations and estimates against each other). Figure 7
shows the PP plots for our OLK and OK interpolation constrained to a maximum estimation range threshold R[max] with regard to the closest ASIRAS measurement locations of 100 and 190km and
nearest-neighbour locations. By comparison with Fig. 6, the 100 and 190km distances (dashed lines) approximately correspond to the lag distances at which the semivariogram has reached half the sill
and where it has levelled off, respectively. In addition, the average distance of PP points from the 1:1 line according to the definition in Yamamoto (2005) and the average SMB values for the OK and
OLK estimates are shown in the legend. Both the nearest-neighbour OK and OLK average SMB estimates are close to the average SMB observation value of 474$\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\
mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$. However, after increasing the range threshold R[max] to 100 and 190km, it is evident from Fig. 7 that the best
match exists between the observation and OLK estimation values. Hence, we limit our analysis to these values in the following.
Aside from the choice of the translational constant C and semivariogram model, we choose the method proposed by Deutsch (1996) to correct for negative kriging weights (Yamamoto, 2000) and constrain
all processing steps of the OLK estimation to the 16 nearest neighbours for each estimate according to the quadrant criterion. Depending on the neighbourhood considered, the effect of smoothing as
well as local stationarity of observation data is affected. As a guidance for our final setting, we aimed at generating an optimal PP relation according to Fig. 7 but also considered potential
artefacts which may arise from the OLK procedure.
In addition to each OLK estimate, we calculate the associated interpolation error. While ME14 choose the kriging standard deviation as a measure of interpolation error, our error estimation is based
on the interpolation standard deviation S[0] introduced by Yamamoto (2000) for two reasons. Firstly, as shown by the author, S[0] represents a more complete measure of local accuracy and has,
therefore, been implemented in the post-processing algorithm in Yamamoto (2007). Secondly, for the OLK method we need a corresponding back transformation of the interpolation error from the
logarithmic to the measurement scale, which has been investigated for S[0] in Yamamoto (2008). Thus, we adopted the proposed back transformation of S[0] in this study.
Following ME14, we estimate the total error of each SMB estimate by the RSS of the measurement error and back-transformed S[0]. The measurement error is estimated by generating 500 realizations of
OLK SMB estimates with added noise to the smoothed SMB observations, which follows a normal distribution with a mean of zero and standard deviation equal to the measurement error of the SMB
observation at x[0].
We have to keep in mind that the basin-wide SMB OLK estimation is limited in terms of the practical range according to Fig. 6. By comparison with the flight track shown in Fig. 1, even when
considering the practical range as a maximum threshold for the spatial SMB estimation, we do not cover the entire PIG basin (see Fig. 8). Hence, for the calculation of total mass input to the PIG
basin, we replace SMB OLK estimates with modelled SMB from a regional climate model at distances where the spatial autocorrelation of measurements is low. In the next section, we consider SMB
estimates from the RACMO2.3p2 (van Wessem et al., 2018) regional climate model (in the following abbreviated as RACMO) and the Modèle Atmosphéric Régional (MAR) according to Donat-Magnin et al. (2020
3.1Regional-scale SMB distribution
Based on the adopted OLK interpolation scheme, we produced the mean annual SMB map for the PIG basin from the ASIRAS observations in Fig. 8a. SMB observations and estimates are colour coded with the
same scale. Each estimate covers a pixel size of ∼5 by 5km^2 and refers to the averaging period between November 2004 and December 2014. The two surrounding dashed lines indicate the 100 and 190km
maximum distances from the ASIRAS measurement point cloud discussed earlier. The red triangle denotes an artificial interpolation cluster of 8 pixels with SMB values greater than 2000$\mathrm{kg}\
phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$, which we discuss in Sect. 4.4. Furthermore, some streak artefacts are visible from the
interpolation, which are mainly caused by the quadrant criterion of the OLK estimation. Increasing the number of nearest neighbours helps reduce these artefacts but at the cost of PP agreement in
terms of Fig. 7. We therefore kept the OLK settings according to Fig. 8a hereinafter.
Figure 8c and d show mean annual SMB estimates for the same period based on RACMO and MAR simulations, respectively. The horizontal resolution of simulated SMB is 27km for RACMO and 10km for MAR
runs. ASIRAS- and model-based estimates show similar main characteristics, i.e. increasing SMB rates towards the Amundsen Sea coastline, decreasing SMB rates towards the inland, and a region of low
SMB in response to the shadowing effect from the mountain ranges of Eights Coast. Similar characteristics also exist for the SMB map generated by M14. Furthermore, the ASIRAS observations start to
capture the transition to higher snow accumulation at the ice divide along the Eights Coast mountain range, as indicated by both regional climate models. This is best seen in the high-resolution
observations north of iSTAR site 10 according to Fig. 5.
Figure 8d and f show the relative difference between model and ASIRAS SMB estimates as defined in the caption. Local variations can be found in the agreement between ASIRAS- and model-based
estimates. Among others, the shadowing effect appears to be more pronounced in the MAR and ASIRAS estimates than in the RACMO estimates. Furthermore, the MAR estimates tend to be lower at the central
flight lines compared to the RACMO and ASIRAS estimates, whereas the agreement is the best between the MAR and ASIRAS estimates near coastal iSTAR sites. A common feature of the ASIRAS estimates is
the much less pronounced SMB gradient towards the southern interior compared to both model estimates but also to the M14 estimates. This can be explained by the missing observational constraints in
this region. We therefore generated hybrid SMB maps where ASIRAS estimates linearly relax into either MAR or RACMO estimates between the 100 to 190km range interval (dashed lines), as shown in Fig.
8b for complementary ASIRAS–RACMO estimates. The range interval was selected based on the spatial autocorrelation in terms of Fig. 6. It is evident from Fig. 8d and f that the SMB gradient towards
the southern interior is not the same for the MAR and RACMO simulations. Hence, the selection of complementary model data will impact total mass input estimates for the PIG basin.
(Mouginot et al., 2017)(Zwally et al., 2012)(Mouginot et al., 2017)(Zwally et al., 2012)(Mouginot et al., 2017)(Zwally et al., 2012)(Mouginot et al., 2017)(Zwally et al., 2012)
3.2Total mass input
Spatial integration of annual mean SMB from our generated hybrid maps yields the total mass input for the PIG basin, which we denote by Σ[+]. Table 3 summarizes Σ[+] and further statistical SMB
characteristics for different data sets and basin definitions according to Fig. 9d. Here, we replaced the interpolation artefact highlighted in Fig. 8a with averaged values from neighbouring pixels.
Σ[+] uncertainty estimates refer to the RSS of the interpolation and measurement error grids (Fig. 9c) in accordance with ME14. Because of a missing error grid for simulated SMB, we consider the
total combined error for the entire PIG basin a conservative error estimate. In this sense, we are augmenting the missing model error estimation. To quantify the relative contribution of ASIRAS to
the hybrid SMB estimates, OLK area and OLK Σ[+] denote the relative contribution in terms of covered land area and integrated SMB, respectively. For comparison with the hybrid-based estimates of this
study, we include results from RACMO, MAR, and ME14, which we converted from w.e. depth to SI units. Because of the different averaging periods between this study and ME14, we added model estimates
in brackets, which we extracted based on the same averaging period as for the ME14 results.
3.2.1Pine Island and Wedge zone
The Pine Island Σ[+] values are in agreement between all data sets within the estimated error margins. This is different for the Wedge area, where the RACMO Σ[+] estimates are between 35%–40% lower
compared to the estimates of this study and ME14. Increasing the averaging time of RACMO estimates to the 1985–2009/2010 period of the ME14 results yields an increase in Σ[+] by 2% for the Pine
Island and 8% for the Wedge area. However, the RACMO-based total mass input to the Wedge area remains below the observational error margins. In comparison with RACMO and MAR estimates, we find that
MAR-based Σ[+] values are about 5% higher for Pine Island and 38% higher for the Wedge area. The higher MAR SMB compared to RACMO towards the southern interior yields a 3% increase for hybrid SMB
estimates based on complementary MAR estimates. Considering the additional SMB properties according to Table 3, the hybrid-based SMB estimates of this study show the largest variability, except for
the Wedge area.
3.2.2Additional basin definitions
Table 3 includes results based on two additional basin definitions for PIG. Figure 9d shows a composite plot of all basin definitions used here. The surface areas range between 176.5, 178.6, and
208.8×10^3km^2 for the PIG basin (including Wedge) according to the definitions of Mouginot et al. (2017), Fretwell et al. (2013), and Zwally et al. (2012). With regard to the basin definition
according to Mouginot et al. (2017), Σ[+] increases by about 3% for the definition by Fretwell et al. (2013) and between 15% to 19% for the definition by Zwally et al. (2012), depending on which
data set is considered according to Table 3.
We discuss first the pronounced differences between annual layer dating from ASIRAS reflection and neutron probe density profiles at some sites and then secondly the systematic differences in SMB
distribution between the results of this study and those of ME14, RACMO, and MAR.
4.1Local SMB departures
Key to the evaluation of our selected internal reflection layer is its isochronic nature, which we assume based on matched depth–age relations from the iSTAR ground-truthing measurements. One may
argue that these measurements can be subject to local noise in the density profile, which would challenge any comparison with nearby radar observations. For instance, Laepple et al. (2016) observed
dominating stratigraphic noise at single pit density profiles near Kohnen station (East Antarctic plateau, Dronning Maud Land). Stacking of multiple profiles is one possibility to filter out noise.
While this is not possible for the single iSTAR sites, the estimated dating uncertainty of ±1.4 years according to this study suggests that iSTAR ground-truthing measurements at PIG are less prone to
stratigraphic noise, which is most likely to be related to the higher SMB compared to ∼70$\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}
^{-\mathrm{1}}$ near Kohnen station (Laepple et al., 2016). However, on a few occasions we identified larger departures in the annual layer dating, as is the case for iSTAR site 2 (Table 2). While
the layer tracing appears to be in agreement between site 1 and 3, the annual layer dating at site 2 would suggest an SMB of ∼290$\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\
phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ at the traced layer cross section rather than ∼150$\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}
{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ based on the 2004.8 layer dating of this study. Accordingly, local SMB results would increase by ∼100% if we used the uncorrected depth–age scale at site 2, which
most likely indicates a systematic error in the measurement scale. This is further corroborated by the measured SMB of 140$\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom
{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ at site 2 for the most recent 2014 layer, but also measured density and strain-rate profiles suggest a mean annual SMB of 200$\mathrm{kg}\phantom{\
rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ based on the Herron and Langway (1980) stage 1 equation; both observations are in a better
agreement with the collocated ASIRAS-based results than the SMB estimate based on the annual layer dating from the NP measurements at site 2. In this sense, the ASIRAS results allow us to be more
confident of the site 2 strain-rate measurements and therefore add to the densification analysis of Morris et al. (2017). The local SMB estimates near site 2 from ME14 and RACMO are within the 200 to
300$\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ range but lack the local precision of ASIRAS measurements and
therefore could not explain the measured density and strain-rate profiles at site 2. The bias to the ASIRAS observations also exists for the MAR estimates, which reach the 350$\mathrm{kg}\phantom{\
rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$ level near site 2 but experience a strong gradient along the PIG main trunk.
Nearby ASIRAS observations at site 18 and 19 in particular suggest higher SMB values compared to the dated NP profiles. Site 19 is directly located at the centre of a pronounced accumulation trough
of ∼2.5km width, which adds to the uncertainty in the layer matching because of the spatial displacement between the iSTAR site and point of closest approach. Because the traced reflection layer
significantly exceeds the depth range of the dated NP density profiles at site 18 and 19, we discarded both sites for the layer dating.
Additional local departures between our results and those from ME14 were identified for the northern slopes and southward interior of PIG. Because of difficulties in the layer tracing at the northern
slopes, the authors of M14 had to augment their SMB estimates with results from a different layer, which they dated back to 2002 and corrected for a temporal bias to the 1985 layer based on
overlapping segments. Thus, one possible explanation for the observed differences is that the true local temporal bias correction may be different from the regional-scale bias correction, which they
estimate from regression models. Other possible explanations are differences in the observational coverage and local accuracy from the different interpolation methods. With regard to the southward
interior, the spatial coverage is superior in the ME14 results. Despite the maximum range limit between 100 and 190km for the ASIRAS-based estimates, the missing observational constraints towards
the interior may still yield an underestimation of the southward SMB gradient. However, we also cannot rule out that the smaller gradient in our observations is due to a local increase in SMB between
the different observational periods in both studies. Additional observational constraints of the selected reflection layer may resolve the cause for the observed difference.
4.2Elevation-dependent model drift
The observational SMB estimates by M14 indicate an elevation-dependent drift of simulated SMB from RACMO. The authors find that RACMO underestimates the SMB at the high-elevation interior, which
would also impact our ASIRAS–RACMO-based estimates of total mass input. Indeed, this finding is also reflected in our data (see Supplement S1) and suggests that the ASIRAS–RACMO-based total mass
input estimates are biased by the underestimated SMB contribution from RACMO. According to Agosta et al. (2019), the opposite may apply for the ASIRAS–MAR-based estimates. The authors observe
a tendency for MAR to overestimate accumulation on Marie Byrd Land (Ross Ice Shelf) and conclude that differences between MAR and RACMO2 are very likely related to differences in the advection
inland. Similar to our elevation-dependent comparison between ASIRAS and RACMO SMB estimates, we find evidence of a drift in the MAR estimates with an opposite sign according to Supplement S1. We
conclude that the best estimate for total mass input lies between ASIRAS–RACMO and ASIRAS–MAR estimates.
Gardner et al. (2018)Zwally et al. (2012)Rignot et al. (2019)Mouginot et al. (2017)
4.3Impact on recent mass balance estimates
Despite the local differences in the SMB distribution, the difference between the Σ[+] estimates for the PIG catchment (including Wedge) between this study and ME14 is small; i.e. the ASIRAS–RACMO
hybrid Σ[+] is 1.7Gtyr^−1 larger, which corresponds to 2% of the ME14 value. Similarly, the ASIRAS–MAR hybrid estimates are 5% larger compared to ME14, which is still within the uncertainty range
estimated by the authors of M14. This indicates that the local differences in the SMB estimates between both studies cancel out. If we take into account that the temporal averaging time used by ME14
is about a factor of 2.7 larger than that used in this study, we cannot find evidence of a potential secular trend in SMB at decadal scales similar to that of the ice discharge at PIG. This provides
additional evidence to Medley et al. (2013) that the recent temporal evolution of the PIG mass balance is primarily driven by dynamic ice loss into the Amundsen Sea.
With regard to existing mass balance estimates for PIG, we have to take into account that basin outlines can differ significantly between studies as illustrated in Fig. 9d. To evaluate the impact of
our hybrid SMB estimates on recent mass balance inventories, we extracted results from the literature in Table 4 and added updated mass balance estimates ${\mathrm{\Sigma }}_{+}^{-}$ by replacing the
Σ[+] estimates from the literature with the Σ[+] estimates of this study. We assume that the SMB remains stationary for the mass balance calculation with regard to the shown periods. In addition, we
linearly interpolated the estimated ice discharge measurements in ME14 for the missing periods before 2007. Furthermore, we assume that the unspecified basin definitions in ME14 are in close
agreement with the basin definitions based on Fretwell et al. (2013).
The small difference between the Σ[+] estimates of this study and ME14 directly translates into the ${\mathrm{\Sigma }}_{+}^{-}$ mass balance estimates. The largest impact of our results is on the $
{\mathrm{\Sigma }}_{+}^{-}$ estimate by Gardner et al. (2018). After replacing their Σ[+] estimate from RACMO2.3 simulations with our ASIRAS–MAR hybrid Σ[+] estimate, the mass balance increases by
∼11 Gtyr^−1.
Medley et al. (2014)
4.4SMB uncertainty
While the agreement in Σ[+] estimates between this study and ME14 supports the hypothesis that the regional SMB of PIG is stationary at decadal scales, our uncertainty estimates are much larger. The
temporal error according to Fig. 4, which is ∼5% larger than in Medley et al. (2013) and ME14, cannot fully explain the difference between both uncertainty estimates. We also do not expect any major
differences with regard to the spatial uncertainty of the density profiles. According to the error-grid statistics of the ASIRAS–RACMO-based estimates in Table 5, we identify the back-transformed
interpolation standard deviation S[0] from the OLK scheme as the dominating error source of our results, while the combined error in ME14 is slightly above our measurement error. The dominating S[0]
uncertainty is also evident in Fig. 9a, b, and c, where the spatial features of the combined error grid are predominately determined by the S[0] grid. We find that the low accumulation zone at the
northern slopes of PIG, which is next to the main trunk between iSTAR site 1 to 6, shows combined S[0] patches that considerably exceed 100%. In contrast, combined error estimates in ME14 do not
exceed 20% at the same location.
Initial tests on our OLK setting revealed that the choice of the negative kriging weight correction method has a noticeable impact on the uncertainty estimates, a finding which according to our
knowledge has not been reported before. However, our applied method by Deutsch (1996) already yields the minimum uncertainty estimates for our results, whereas the additional methods cited in
Yamamoto (2000) yields an additional uncertainty increase between 20% (Froidevaux, 1993) and 50% (Journel and Rao, 1996).
Additional tests, where we used the kriging standard deviation based on non-transformed OK estimates, did not improve our interpolation uncertainty. Therefore the different choice of the
interpolation uncertainty measure is not the source of the larger uncertainty range of this study. We hypothesize that despite the homoscedastic (i.e. data-value-independent) nature of the krige
standard deviation, the reduction of data variance after subtracting the regression surface according to ME14 is most likely the cause of their significantly lower uncertainty estimates.
In addition to the larger uncertainty range of this study, we note that the choice between cell-by-cell summation and RSS of grid errors has a quite substantial impact on the Σ[+] uncertainty
estimates. If we make the optimistic assumption that gridded errors are independent and choose the calculation of RSS instead, Σ[+] uncertainty estimates would reduce to ±0.5Gtyr^−1 (i.e. ∼97%
less) for the combined Pine Island and Wedge basin.
4.5Systematic retrieval impacts
In addition to the uncertainty assessment in Sect. 4.4, we evaluated the impact of artificial cluster removal, the choice of permittivity model, and the non-transformed OK scheme.
4.5.1Artificial cluster removal
Inspection of the artificial cluster highlighted in Fig. 8 revealed that it is centred around the location with the lowest observed SMB and is essentially generated by the local nonbias terms of the
OLK procedure. Owing to its steep contrast with the surroundings, it appears to be plausible to replace this cluster by averaged values of its nearest neighbours. However, due to the limited extent
of this cluster, its additional contribution to the Σ[+] estimates would be less than 0.8%. Similarly, the impact on the PP plot is negligible. Increasing the translational constant C helps remove
this cluster but at the cost of statistical agreement between observations and estimates.
4.5.2Looyenga-based results
Defining ϵ^′ by Eq. (2) instead of Eq. (1) yields a minor reduction of Σ[+] for the PIG catchment of 0.6%, which we expect from Sinisalo et al. (2013). However, despite the minor impact of the
alternative definition for ϵ^′, we noticed an additional small impact on the layer dating, which shifted our estimated layer formation from November to September 2004. Thus, we had to adjust the time
range in the RACMO SMB extraction for the calculation of hybrid SMB estimates. While the choice of the ϵ^′ model only has a minor impact on our total mass input estimates, it is worth noting that the
effect on our annual layer dating is detectable.
4.5.3Non-transformed kriging results
If we choose the OK procedure instead, Σ[+] increases by 4% for the Pine Island and 12% for the Wedge area, which would further increase the offset between this study and ME14. However, inspection
of the SMB distribution (not shown) indicates that estimates tend to overshoot near the coastline of the Amundsen Sea, which becomes particularly evident for the Wedge area. Hence, the OK procedure
appears to be more sensitive to the limited observational constraints near the Wedge area. In addition, S[0]-based uncertainty estimates increase by 27% and 88% for the Pine Island and Wedge area,
which highlights the improved performance of the OLK procedure.
Our analysis provides updated mean annual SMB estimates for the PIG basin and 2005–2014 averaging period based on a comprehensive airborne radar and ground-truthing survey and complementary model
simulations. Based on these estimates, we calculated a total mass input of 79.9±19.2 and 82.1±19.2Gtyr^−1 for the PIG basin area when using complementary RACMO and MAR SMB estimates, respectively.
In comparison with earlier estimates from airborne radar observations, which consider the 1985–2009 averaging period, our results show a greater total mass input between 2% and 5%. This increase is
still within the uncertainty range of both studies. Hence, no distinct trend is visible for the total mass input between both averaging periods. We conclude that our results provide further evidence
that the recent total mass input can be considered stationary at decadal scales. This implies that the increased dynamic ice loss over past decades remains the driving force in the recent mass
balance evolution of PIG. However, departures between both observations at the northern slopes and southward interior of PIG, which cancel out for the estimates on total mass input, may indicate
temporal changes in the local SMB distribution. Furthermore, our radar-based observations can resolve a discrepancy between strain-rate and SMB measurements at iSTAR site 2, which highlights the
benefit of such complementary SMB measurements for future missions.
Despite the minor changes in total mass input between both studies, the more than 2-fold uncertainty range of our results remains striking. Neither the applied model for the wave propagation speed of
radar soundings nor the uncertainty related to the regional density profile can explain the larger uncertainty of this study. The same also applies for the reduced temporal averaging time.
A comprehensive evaluation of our uncertainty estimation revealed that assumptions on the geostatistical interpolation error as well as grid-error dependences can have a substantial impact on the
uncertainty estimation. In terms of the error partitioning, our interpolation error is the dominating source of combined grid errors. Moreover, varying basin definitions have an impact on our total
mass input estimate by up to 19%. This highlights the importance of a thorough documentation of uncertainty estimates and basin definitions to improve future intercomparisons between different SMB
and mass balance inventories.
Appendix A:List of abbreviations and notations
ASIRAS Airborne SAR/Interferometric Radar Altimeter System
GPR ground-penetrating radar
M14 Medley et al. (2014)
Combined error root sum of squares of measurement and interpolation standard deviation
Measurement error root sum of squares of spatial, temporal, and digitization error components
NP neutron probe
OK ordinary krige procedure
OLK ordinary logarithmic kriging procedure
RACMO RACMO2.3p2 regional climate model
PP plot percentile–percentile plot
RSS root sum of squares
SMB surface mass balance, $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$
T1 iSTAR traverse 2013/2014
T2 iSTAR traverse 2014/2015
TWT two-way-travel time of radar soundings
PIG Pine Island Glacier
w.e. water equivalent
WAIS West Antarctic Ice Sheet
a layer age, years
$\stackrel{\mathrm{˙}}{b}$ annual mean SMB, $\mathrm{kg}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{-\mathrm{1}}$
ΔD closest distance between ASIRAS track and iSTAR site
ϵ^′ real part of the dielectric permittivity
N number of reflection layer points considered for annual dating
ρ density, kgm^−3
R[max] maximum range threshold to ASIRAS measurements, km
σ[x] standard deviation of N layer dating estimates
Σ[+] total mass input, Gtyr^−1
${\mathrm{\Sigma }}_{+}^{-}$ total mass balance, Gtyr^−1
S[0] interpolation standard deviation
SK conceived of the presented idea, designed the computational framework, adapted and tested the geostatistical krige methods, accomplished the reflection layer tracing in large parts, reprocessed
the neutron probe density profiles for the data calibration, and wrote the manuscript with input from all authors; VH performed the SAR level_1b ASIRAS data processing, provided the digital elevation
model, and established access to the RACMO2.3p2 data; EM delivered the neutron probe density profiles; and OE contributed to layer analysis and interpretation. All authors discussed the results and
contributed to revising the manuscript.
Olaf Eisen is co-editor in chief of The Cryosphere.
The authors gratefully acknowledge the excellent logistical support provided by British Antarctic Survey's (BAS) Rothera Research Station and members of the iSTAR traverse and Alfred Wegener
Institute and Polar 5 flight crew during the field campaign, which has been funded by the UK Natural Environment Research Council (NERC, grant no. NE/J005681/1). The authors express their gratitude
towards Andrew Shepherd, PI of the iSTAR-D project, for his general support and data contribution to this study. The authors thank Robert Mulvaney (BAS, UK) and Hannes Konrad (CPOM, University of
Leeds, UK; and DWD, Germany) for provision of data sets and discussions in an early stage. The authors gratefully acknowledge the provision of RACMO2.3p2 model output by
Stefan Roderick Martijn Ligtenberg and Melchior van Wessem (IMAU, Utrecht University, NL). The authors would like to thank Emerson E&P Software, Emerson Automation Solutions, for providing licenses
in the scope of the Emerson Academic Program. The authors sincerely appreciate the valuable comments and suggestions by the referees and editor. The authors acknowledge support by the Open Access
Publication Funds of Alfred-Wegener-Institut Helmholtz-Zentrum für Polar- und Meeresforschung.
This research has been supported by the German Ministry of Economics and Technology (grant no. 50EE1331 to Veit Helm).
The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.
This paper was edited by Michiel van den Broeke and reviewed by Brooke Medley and two anonymous referees.
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mass balance on Fimbul ice shelf, East Antarctica: Comparison of field measurements and large-scale studies, J. Geophys. Res.-Atmos., 118, 11625–11635, https://doi.org/10.1002/jgrd.50875, 2013.a, b
Turner, J., Phillips, T., Thamban, M., Rahaman, W., Marshall, G. J., Wille, J. D., Favier, V., Winton, V. H. L., Thomas, E., Wang, Z., van den Broeke, M., Hosking, J. S., and Lachlan-Cope, T.: The
Dominant Role of Extreme Precipitation Events in Antarctic Snowfall Variability, Geophys. Res. Lett., 46, 3502–3511, https://doi.org/10.1029/2018GL081517, 2019.a
U.S. Geological Survey: Landsat Image Mosaic of Antarctica (LIMA), Tech. rep., available at: http://pubs.er.usgs.gov/publication/fs20073116 (last access: 25 April 2019), 2007. a, b, c, d
van Wessem, J. M., van de Berg, W. J., Noël, B. P. Y., van Meijgaard, E., Amory, C., Birnbaum, G., Jakobs, C. L., Krüger, K., Lenaerts, J. T. M., Lhermitte, S., Ligtenberg, S. R. M., Medley, B.,
Reijmer, C. H., van Tricht, K., Trusel, L. D., van Ulft, L. H., Wouters, B., Wuite, J., and van den Broeke, M. R.: Modelling the climate and surface mass balance of polar ice sheets using RACMO2 –
Part 2: Antarctica (1979–2016), The Cryosphere, 12, 1479–1498, https://doi.org/10.5194/tc-12-1479-2018, 2018.a
Yamamoto, J. K.: An Alternative Measure of the Reliability of Ordinary Kriging Estimates, Math. Geol., 32, 489–509, https://doi.org/10.1023/A:1007577916868, 2000.a, b, c
Yamamoto, J. K.: Correcting the Smoothing Effect of Ordinary Kriging Estimates, Math. Geol., 37, 69–94, https://doi.org/10.1007/s11004-005-8748-7, 2005.a, b
Yamamoto, J. K.: On unbiased backtransform of lognormal kriging estimates, Comput. Geosci., 11, 219–234, https://doi.org/10.1007/s10596-007-9046-x, 2007.a, b, c, d, e
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ant_grn_drainage_systems.php (last access: 1 March 2021), 2012.a, b, c, d, e, f, g, h | {"url":"https://tc.copernicus.org/articles/15/1285/2021/","timestamp":"2024-11-09T13:24:13Z","content_type":"text/html","content_length":"386882","record_id":"<urn:uuid:616167cc-fb93-4b1c-a79f-4829460e481a>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00411.warc.gz"} |
Truth Tables (… you can handle the truth)
About eight years ago I worked on my first major commercial software project. Before I worked as a freelancer and had mostly rather small and dull projects. This project however had a team of eight
people plus management and it was surprisingly complex at times.
One day we had to implement a feature which depended on four independent conditions. Independent in this case means that there were no exclusive branches of logic. To implement it multiple branches
with similar but not identical logic were necessary. I was a bit scared because I knew that this would be one big ugly hairball of a nested if / case statement which is impossible to get right on the
first attempt and will probably be hard to test.
Then our head of engineering came along, an experienced old school hacker, who told me: »Lets implement it with a truth table!«
At first I didn’t know what he was talking about and then I felt like this would be some kind of arcane low level hack solution to this complex problem but it turned out to make everything clean and
easy to reason about.
Since then I had maybe three or four other occasions where I used truth tables to solve similar complex problems and I was always grateful for this advice. Every time I proposed to use a truth table
in these situations there were always some colleagues who either didn’t know about them, just like me seven years ago, or knew them theoretically but never used them in practice.
The last occurrence was just recently in my current project at Wooga which is why I thought it might be a good idea to share the concept with everyone who does not already know what I’m talking
Show Me The Tables
If you have studied computer science and/or philosophy you should have seen truth tables on paper. They are often used to describe the behavior of logic gates for example. This is how a truth table
for an AND gate would look like:
A | B | Output
0 | 0 | 0
1 | 0 | 0
0 | 1 | 0
1 | 1 | 1
This table tells you that the output of an AND gate will be 1 only if input A AND B are 1.
In philosophy the same tables are used for reasoning about logic statements in arguments
(philosophers, please correct me here if thats not accurate enough, I know you care :).
Practical Application of Truth Tables
For practical applications truth tables have a slightly different form. Imagine we have a website where users generate a lot of content and they have the option to download their entire history which
is so massive that we have to process and compress it. Now the app can be in a couple of different states and in only one of the states the user can actually download the archive.
Lets start simple and say we have two boolean variables which are important to determine the state of our application.
requested_archive: true | false
processing_finished: true | false
For each of the variables we would now assign a unique binary flag:
requested_archive: 01 (decimal: 1)
processing_finished: 10 (decimal: 2)
Now the truth table would look like this:
requested | processing |
archive | finished | result | meaning
00 | 00 | 00 | not requested, not processed
01 | 00 | 01 | requested but not processed
00 | 10 | 10 | impossible
01 | 10 | 11 | requested and processed
The basic idea here is to evaluate the conditions separately and use a binary OR to compute the resulting bit flag.
Like I said, it is a simple example which does not strictly require implementing a truth table but even in this situation a truth table in form of a comment above the nested conditional statement can
have a benefit for the next person reading your code.
Also it provides an overview of all the possible combinations and what cases are actually relevant for you. You know what unit tests to write and you can be certain that you’ve covered everything.
Ok now lets implement this truth table. We already know what cases we have and what they mean so the first thing is to assign them symbolic names in your code. In this example I will use ruby but it
should work in a similar way in your favorite language.
For this simple example it might not seem obvious why truth tables are great but it illustrates some aspects already. The evaluation of conditions is isolated into separate functions which you can
unit test very easily.
When you unit test the archive_state function and you get an unexpected state, the bit flags will tell you exactly which of the conditions was failing your expectations.
You give each state a symbolic name and you avoid a nested conditional.
I know, I know this looks crazy complex but its just an example and as soon as you add one or more conditions it will be much more clear why I prefer a truth table over nested conditions any time.
The next code example is from a real world erlang application and its again about determining the state of an application. Based on the day of the season, the state of the user data and the state of
the processing we have to do different things and there is no exclusive branch for one of the conditions which means the implementation would be painful. (Note: there is a hidden fourth condition in
the first one, checking that the season is really over as in all matches have been simulated on the last day of the season)
With truth tables we know we have 2^3 possible combinations, of which two are impossible, we know exactly which cases mean what and what we want to do for each of them, we also can implement each
conditions in neat and tidy functions which are easily unit testable, we have symbolic names and the final case statement is actually readable.
Without the comment which shows the actual truth table and without all the other comments, this whole code snippet would be a lot less intuitive of course but with them it is self sufficient
The best part is, even if you have four or five conditions the complexity of the code does not increase, it is just more work to write down the initial truth table of 16 or 32 possibilities but the
implementation would follow the same pattern and the evaluating case statement would still be easy to read.
Personally I went as far as four conditions and when I would encounter a situation where five conditions are evaluated I would first spent a few hours trying to reduce the amount of conditions maybe
even refactoring code but if you really have this kind of complex system, truth tables are your best chance to implement it without losing your sanity.
Let me know in the comments if that was helpful.
15 thoughts on "Truth Tables (… you can handle the truth)"
1. Hej hukl, this is a good advice indeed. When implementing such conditions I often write down truth tables, but most of the time I just translate them to boolean expressions and if/else
statements, while using constants for both conditions and results is indeed way more readable and easier to apply changes on. Thanks for sharing! Regards, mechko
2. Excellent Article! Thank you for that!
Your CSS-Declaration of pre-Tags lacks the generic Font-Name “monospace”, thus your Code-Examples look ugly on Linux.
1. is it better now?
3. The sample code is missing in your RSS Feed 🙁
1. I had to embed it from github – thats why I guess
4. I heard about truth tables, but never used them so far (I did not study computer science or philosophy). But, I think I’ll consider using them in future projects as I think they are a very useful
tool. Thanks for sharing your insights.
5. Kind of looks like a half-assed deterministic finite automaton, only with non-obvious, externally handled state transitions. You may want to check out DFAs, or “Deterministische endliche
Automaten” (DEA).
1. Hmm sample implementations / examples would be nice. Because from what I read on wikipedia I cannot directly map that to our use case in the example
6. Hi hukl,
Is it intended that you overwrite FINISHED_PROCESSING in your first example? Or am I wrong and should read it one more time? 🙂
Greets, Marcus
1. ah good catch – fixed it! thanks!
1. You’re welcome. 🙂
1. Nevermind, it’s a great post. I will use it more often. Thanks!
7. Karnaugh maps have been used in games (Bards Tale, Maze) to simplify the logic of a 2D map and tiles.
8. In Erlang (and of course Elixir), you can directly use pattern matching to implement the truth table, no? E.g., assuming the test functions are reimplemented to return booleans:
case {is_current_season_over(), is_finished_processing(), is_state_reset()} of
{false, true, false} -> noop;
{true, false, true} -> noop;
{true, true, false} -> start_processing();
{true, false, false} -> resume_processing();
{false, false, true} -> fc_league_state:reset_state();
Unexpected -> throw({broken_state, Unexpected})
This gets rid of most of the boilerplate and in fact looks like a literal truth table.
1. Sure that works too. I prefer the described way but this is just as valid. | {"url":"https://smyck.net/2015/12/10/truth_tables/","timestamp":"2024-11-09T20:07:10Z","content_type":"text/html","content_length":"37041","record_id":"<urn:uuid:75cfbc06-3834-4165-997e-5805451ffe38>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00776.warc.gz"} |
SSC CHSL Topic Wise Study Material - General Intelligence - Order and Sequence | Recruitment Topper
SSC CHSL Topic Wise Study Material – General Intelligence – Order and Sequence
SSC CHSL Study MaterialSSC CHSL Previous PapersSSC CHSL Practice Workbook
The process of determining the position or place of a person or a thing on the basis of comparison or relative position of other person or thing is called ranking. In ranking test, relative position
or ranking of different groups of persons/objects are given. Students are required to establish the ranking or position of other individual in the same group with respected to one another.
Two types of questions covered in this chapter are as follows
1. Sequential order of arrangement
2. Position test
1. Sequential Order of Arrangement
This type involves determining the sequential order of two or more persons/objects based on the comparison of parameters such as age, height, mark, salary, weight etc. Questions based on ranking are
generally given with a set of information in jumbled form and based on that it is required to systematically arrange the given information and answer the questions asked.
Example In a group of five districts, Akbarpur is smaller than Fatehpur, Dhanbad is bigger than Palamu and Bara Banki is bigger than Fatehpur but not as big as Palamu. Which district is the biggest?
ssc (10+2) 2011
(a) Akbarpur
(b) Fatehpur
(c) Dhanbad
(d) Palamu
(c) Dhanbad > Palamu > Bara Banki > Fatehpur > Akbarpur
Hence, Dhanbad district is biggest amongst them.
2. Position Test
In this type, the rank or position of persons/objects from either of the two ends of a row/queue is given and it is required to determine the total number of persons in the group or the position of
the persons from left or right side in a row/queue.
Different types of position tests will give you a better idea
(i) Rank of a Person/Object from Top or Bottom from Left or Right
In this type of questions, it is asked to determine the rank or position of a person/object in a group of persons/objects either from the left (or right) or from the top (or bottom) depending upon
the arrangement.
The position or rank can be calculated with the help of following formulae
• Position (or rank) from the left end (or top) = (Total number of persons/students – Rank from the right end (or bottom) + 1
• Position (or rank) from the right end (or bottom)=(Total number of persons./students – Rank from the left end (or top) + 1
Example In class of 45 students rank of Ayush is 15 from top, then rank of Ayush from bottom is
(a) 30
(b) 32
(c) 31
(d) 35
(ii) Total Number of Objects/Persons in Queue
In this type of questions, it is asked to Calculate the total number of persons when the rank of a person from both the ends is given.
Following formula is helpful in the calculation of total number of objects/persons in a queue
Total number of persons in a row or queue = (Position (or rank) of a person from the left end (or top or front) + Position (or rank) of the person from the right end (or bottom or last) -1
Example In a queue, Soham is 10th from left and 28th from right. How many people in the queue?
(a) 38
(b) 35
(c) 37
(d) 42
Reference Corner
1.P, Q, R, S and T are sitting together. T is at one extreme end. P is the neighbour of T and is third to the left of Q. Who is fourth to the right of T? SSC (10 + 2) 2017
(a) P
(b) T
(c) Q
(d) S
2.Barun is taller than Sanjay. Bipul is taller than Barun. Krishna is also not as tall as Bipul, but is taller than Barun. Who is the tallest? SSC (10+2) 2013
(a) Barun
(b) Bipul
(c) Krishna
(d) Sanjay
(b) The information given in the question may be interpreted systematically as follow
Bipul > Krishna > Barun > Sanjay
So, Bipul is the tallest.
3.Sunita is the 11th from either end of a row of girls. How many girls are there in that row? SSC (10+2) 2013
(a) 19
(b) 20
(c) 21
(d) 22
(c) The total number of girls = 11 + 11-1 = 22-1
= 21
4.At a birthday party, 5 friends are sitting in a row. ‘M’ is to the left of ‘O’ and to the right of ‘P’. ‘S’ is sitting to the right of T’, but to the left of ‘P’. Who is sitting in the middle? SSC
(10+2) 2013
(a) M
(b) O
5.In a row of letters, a letter is 5th from left end and 12th from the right end. How many letters are there in a row? SSC (10+2) 2013
(a) 15
(b) 16
(c) 17
(d) 18
(b) Number of letters in the row = 5 + 12-1=16
6.Lakshmi is elder than Meenu. Leela is elder than Meenu but younger than Lakshmi. Latha is younger than both Meenu and Hari but Hari is younger than Meenu. Who is the youngest? SSC (10+2) 2013
(a) Lakshmi
(b) Meenu
(c) Leela
(d) Latha
(d) According to the question.
Lakshmi > Meenu
Lakshmi > Leela > Meenu
Meenu > Hari > Latha
Lakshmi > Leela > Meenu
> Hari > Latha
Hence, Latha is the youngest.
Practice Exercise
1.Nitin ranks eighteenth in a class of 49 students. What is his rank from the last?
(a) 18
(b) 19
(c) 31
(d) 32
2.Ram ranked ninth from the top and thirty-eighth from the bottom in a class. How many students are there in the class?
(a) 45
(b) 46
(c) 47
(d) 48
3.A class of boys stands in a single line. One boy is nineteenth in order from both the ends. How many boys are there in the class?
(a) 27
(b) 37
(c) 38
(d) 39
4.In a row of boys, A is thirteenth from the left and D is seventeenth from the right. If in this row A is eleventh from the right, then what is the position of D from the left?
(a) 6th
(b) 7th
(c) 10th
(d) 12th
5.In a class of 60, where girls are twice that of boys, Kamal ranked seventeenth from the top. If there are 9 girls ahead of Kamal, how many boys are after him in rank?
(a) 3
(b) 7
(c) 12
(d) 23
6.In a row of boys, Jeevan is seventh from the start and eleventh from the end. In another row of boys, Vikas is tenth from the start and twelfth from the end. How many boys are there in both the
rows together?
(a) 36
(b) 37
(c) 39
(d) 38
7.Manoj and Sachin are ranked seventh and eleventh respectively from the top in a class of 31 students. What will be their respective ranks from the bottom in the class?
(a) 20th and 24th
(b) 24th and 20th
(c) 25th and 21st
(d) 26th and 22nd
8.Ravi is 7th ranks ahead of Sumit in a class of 39. If Sumit’s rank is seventeenth from the last, what is Ravi’s rank from the start?
(a) 14th
(b) 15th
(c) 16th
(d) 17th
9.In a class, among the passed students, Amisha is twenty-second from the top and Sajal, who is 5 ranks below Amisha, is thirty-fourth from the bottom. All the students from the class have appeared
for the exam. If the ratio of the students who passed in the exam to those who failed is 4 : 1 in that class, how many students are there in the class?
(a) 60
(b) 75
(c) 90
(d) Data inadequate
10. Richard is fifteenth from the front in a column of boys. There were thrice as many behind him as there were in front. How many boys are there between Richard and the seventh boy from the end of
the column?
(a) 33
(b) 34
(c) 35
(d) Data inadequate
11. Forty boys are standing in a row facing the North. Amit is eleventh from the left and Deepak is thirty-first from the right end of the row. How far will Shreya, who is third to the right of Amit
in the row, be from Deepak?
(a) 2nd
(b) 3rd
(c) 4th
(d) 5th
12. In a queue, A is eighteenth from the front while B is sixteenth from the back. If C is twenty-fifth from the front and is exactly in the middle of A and B, then how many persons are there in the
(a) 45
(b) 46
(c) 47
(d) 48
13. N ranks fifth in a class. S is eighth from the last. If T is sixth after N and just in the middle of N and S, then how many students are there in the class?
(a) 23
(b) 24
(c) 25
(d) 26
14. In a row of forty children, P is thirteenth from the left end and Q is ninth from the right end. How many children are there between P and R if R is fourth to the left of Q?
(a) 12
(b) 13
(c) 14
(d) 15
15. In a class of 35 students, Kunal is placed seventh from the bottom whereas Sonali is placed ninth from the top. Pulkit is placed exactly in between the two. What is Kunal’s position from pulkit?
(a) 9
(b) 10
(c) 11
(d) 13
16. Statement Rakesh is senior to Rajesh and Rajesh is senior to Rahul.
I. Rakesh is senior to Rahul.
II. Rahul is not senior to Rakesh. SSC (10 + 2) 2012
(a) Only Conclusion I follows
(b) Only Conclusion II follows
(c) Both Conclusions I and II follow
(d) Neither Conclusion I nor II follows
17. There are five friends Suresh, Kaushal, Madhur, Amit and Ramesh. Suresh is shorter than Kaushal . but taller than Ramesh. Madhur is the tallest. Amit is a little shorter than Kaushal but little
taller then Suresh. If they stand in the order of their heights, who will be the shortest? SSC (10 + 2) 2010
(a) Amit
(b) Madhur
(c) Ramesh
(d) Kaushal
Hints & Solutions | {"url":"http://www.recruitmenttopper.com/ssc-chsl-topic-wise-study-material-general-intelligence-order-sequence/12483/","timestamp":"2024-11-10T04:38:32Z","content_type":"text/html","content_length":"81996","record_id":"<urn:uuid:3ac629d6-1f37-47f3-a53f-69311ce5c23c>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00211.warc.gz"} |
Lesson 13
Decomposing Bases for Area
Let’s look at how some people use volume.
13.1: Are These Prisms?
1. Which of these solids are prisms? Explain how you know.
2. For each of the prisms, what does the base look like?
1. Shade one base in the picture.
2. Draw a cross section of the prism parallel to the base.
13.2: A Box of Chocolates
A box of chocolates is a prism with a base in the shape of a heart and a height of 2 inches. Here are the measurements of the base.
To calculate the volume of the box, three different students have each drawn line segments showing how they plan on finding the area of the heart-shaped base.
1. For each student’s plan, describe the shapes the student must find the area of and the operations they must use to calculate the total area.
2. Although all three methods could work, one of them requires measurements that are not provided. Which one is it?
3. Between you and your partner, decide which of you will use which of the remaining two methods.
4. Using the quadrilaterals and triangles drawn in your selected plan, find the area of the base.
5. Trade with a partner and check each other’s work. If you disagree, work to reach an agreement.
6. Return their work. Calculate the volume of the box of chocolates.
The box has 30 pieces of chocolate in it, each with a volume of 1 in^3. If all the chocolates melt into a solid layer across the bottom of the box, what will be the height of the layer?
13.3: Another Prism
A house-shaped prism is created by attaching a triangular prism on top of a rectangular prism.
1. Draw the base of this prism and label its dimensions.
2. What is the area of the base? Explain or show your reasoning.
3. What is the volume of the prism?
To find the area of any polygon, you can decompose it into rectangles and triangles. There are always many ways to decompose a polygon.
Sometimes it is easier to enclose a polygon in a rectangle and subtract the area of the extra pieces.
To find the volume of a prism with a polygon for a base, you find the area of the base, \(B\), and multiply by the height, \(h\).
\(\displaystyle V = Bh\)
• base (of a prism or pyramid)
The word base can also refer to a face of a polyhedron.
A prism has two identical bases that are parallel. A pyramid has one base.
A prism or pyramid is named for the shape of its base.
• cross section
A cross section is the new face you see when you slice through a three-dimensional figure.
For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.
• prism
A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.
Here are some drawings of prisms.
• pyramid
A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.
Here are some drawings of pyramids.
• volume
Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.
For example, the volume of this rectangular prism is 60 units^3, because it is composed of 3 layers that are each 20 units^3. | {"url":"https://im-beta.kendallhunt.com/MS/students/2/7/13/index.html","timestamp":"2024-11-04T11:57:53Z","content_type":"text/html","content_length":"95177","record_id":"<urn:uuid:e4bccc5a-92c6-4832-812b-d0b5897149af>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00644.warc.gz"} |
CS2810 OOAIA: A5
● To learn about templates in C++ and implement algorithms to evaluate an arithmetic expression.
Data Structure
Implemented a templated Stack class. It should support the standard push, pop and top
operations. You are not allowed to use the std::stack class that C++ STL provides.
Use the templated class to implement the algorithm to evaluate arithmetic expressions in
infix form. These expressions consist of terms, operators and parentheses. Terms can be
either integers or Polynomials. You are free to use code from your previous assignments
in your submission.
Input Format
Each testcase will consist of n expressions. Each expression has either integer terms or
Polynomial terms. The first line of the expression is “int” or “poly” accordingly.
If it is “int”, the expression will be given in the next line in a single line with space
separated tokens, that is, all parentheses, operators and numbers will be space separated.
If the expression is “poly”, it is followed by an integer on the next line which denotes the
number of lines to follow. Each following line is either a polynomial, an operator, or a
parenthesis. A polynomial having m terms is represented by 2m numbers in one line, where
each pair of numbers represents one term. Each term has the exponent followed by the
Please note the following:
– All coefficients are integers. No doubles are given as input.
– The operators we are testing are + (addition), – (subtraction) and * (multiplication).
We have provided an input.cpp file on Moodle which provides appropriate logic to take
both “int” and “poly” expressions as input. Feel free to use this in your submissions.
Output Format
The output for an “int” expression will be an integer which has to be printed irrespective of
whether it is zero or not. The output for a “poly” expression will be a Polynomial which is
printed similar to A3. However as each coefficient is an integer and not a double, there is
no need of printing the decimal point or the digits after that. All other rules as in A3 are still
followed such as only printing non-zero terms and printing a blank line for a zero
– There will be a maximum of 100 operations per testcase.
– The maximum degree of any input Polynomial will be 10.
Sample Testcase
2 → Number of operations
int → Int expression to follow
( ( 2 + 3 ) – 100 ) * 5 – ( 1 – 2 )
poly → Poly expression to follow
11 → Next 11 lines define the poly expression
1 2 3 4 → 2x^1 + 4x^3
2 3 4 5 → 3x^2 + 5x^4
1 5 → 5x^1
2 3 → 3x^2
-9x^3 + 9x^4 + 12x^5 + 15x^6
Design Submission Format
For the design submission on Moodle, please submit a .tar.gz file named as your roll number. | {"url":"https://codingprolab.com/answer/cs2810-ooaia-a5/","timestamp":"2024-11-02T13:46:49Z","content_type":"text/html","content_length":"107257","record_id":"<urn:uuid:6875f7b6-c8eb-4aff-9d38-6735f30aae29>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00618.warc.gz"} |
HARMEAN Function: Definition, Formula Examples and Usage
HARMEAN Function
Are you familiar with the HARMEAN function in Google Sheets? If not, let me introduce you to this handy tool. The HARMEAN function allows you to calculate the harmonic mean of a set of numbers in a
Google Sheet. The harmonic mean is a type of average that is used when working with rates or ratios, rather than the more common arithmetic mean which is used for most other types of data. In other
words, the HARMEAN function is especially useful when you want to find the average of a set of numbers that are all reciprocals of each other (e.g. speeds, rates, etc.).
So, how does the HARMEAN function work? It’s actually quite simple. All you need to do is provide the function with a range of cells containing the numbers you want to include in the harmonic mean
calculation, and the function will return the harmonic mean as a result. For example, if you have a range of cells containing the numbers 1, 2, 3, and 4, you can use the HARMEAN function to calculate
their harmonic mean by entering “=HARMEAN(A1:A4)” into a cell. And that’s all there is to it!
Definition of HARMEAN Function
The HARMEAN function in Google Sheets is a built-in function that calculates the harmonic mean of a set of numbers. The harmonic mean is a type of average that is used to calculate the mean of a set
of numbers that are all reciprocals of each other, such as rates or speeds. To use the function, you simply need to provide it with a range of cells containing the numbers you want to include in the
calculation, and it will return the harmonic mean as a result. For example, to calculate the harmonic mean of the numbers 1, 2, 3, and 4, you would enter “HARMEAN(A1:A4)” into a cell. The HARMEAN
function is useful for finding the average of rates or ratios, and can be a useful tool in data analysis and statistical calculations.
Syntax of HARMEAN Function
The syntax for the HARMEAN function in Google Sheets is as follows:
=HARMEAN(number1, [number2], ...)
The function requires at least one argument, which is the range of cells containing the numbers you want to include in the harmonic mean calculation. Additional arguments can be included to specify
additional ranges of cells or individual numbers to be included in the calculation. For example, you could use the following syntax to calculate the harmonic mean of the numbers in cells A1, A2, and
A3, as well as the number 5:
=HARMEAN(A1:A3, 5)
Note that the HARMEAN function only accepts numerical values as arguments, and will return an error if any non-numeric values are included in the range or as individual arguments.
It’s also worth noting that the HARMEAN function will return an error if any of the numbers included in the calculation are zero or negative, as the harmonic mean is not defined for these values.
Examples of HARMEAN Function
Here are three examples of how to use the HARMEAN function in Google Sheets:
1. Calculating the harmonic mean of a range of cells:
Let’s say you have a range of cells in your Google Sheet containing the numbers 1, 2, 3, and 4, and you want to calculate their harmonic mean. To do this, you could use the following formula:
This will return the harmonic mean of the numbers 1, 2, 3, and 4 as a result.
2. Calculating the harmonic mean of a range of cells with additional numbers:
Now let’s say you want to calculate the harmonic mean of the same range of cells (A1:A4), but you also want to include the number 5 in the calculation. To do this, you can use the following
=HARMEAN(A1:A4, 5)
This will return the harmonic mean of the numbers 1, 2, 3, 4, and 5 as a result.
3. Calculating the harmonic mean of multiple ranges of cells:
Finally, let’s say you have two ranges of cells in your Google Sheet, A1:A4 and B1:B3, and you want to calculate the harmonic mean of all the numbers in both ranges. To do this, you can use the
following formula:
=HARMEAN(A1:A4, B1:B3)
This will return the harmonic mean of all the numbers in both ranges as a result.
Use Case of HARMEAN Function
Here are a few real-life examples of how the HARMEAN function could be used in Google Sheets:
1. Calculating the average speed of a vehicle:
Imagine you are tracking the speed of a vehicle over a period of time and want to find the average speed. You could use the HARMEAN function to calculate the harmonic mean of the speeds recorded.
For example, if you have a range of cells containing the speeds in miles per hour (mph), you could use the following formula:
This would return the average speed in mph as a result.
2. Calculating the average conversion rate for an e-commerce website:
If you are running an e-commerce website, you might want to track the conversion rate of visitors to customers. The conversion rate is the percentage of visitors who make a purchase on your
website. You could use the HARMEAN function to calculate the harmonic mean of the conversion rates for a given period of time. For example, if you have a range of cells containing the conversion
rates in percentage form (e.g. 0.5 for 50%), you could use the following formula:
This would return the average conversion rate as a result.
3. Calculating the average exchange rate for a currency:
If you are tracking the exchange rate of a currency, such as the US dollar to the euro, you could use the HARMEAN function to calculate the harmonic mean of the exchange rates over a period of
time. For example, if you have a range of cells containing the exchange rates in the form of the number of euros per dollar (e.g. 0.8 for 0.8 euros per dollar), you could use the following
This would return the average exchange rate as a result.
Limitations of HARMEAN Function
There are a few limitations to keep in mind when using the HARMEAN function in Google Sheets:
1. The HARMEAN function only works with numerical values:
The HARMEAN function only accepts numerical values as arguments, and will return an error if any non-numeric values are included in the range or as individual arguments.
2. The HARMEAN function is not defined for zero or negative numbers:
The harmonic mean is not defined for zero or negative numbers, so the HARMEAN function will return an error if any of the numbers included in the calculation are zero or negative.
3. The HARMEAN function does not work with empty cells:
If any of the cells included in the range or as individual arguments are empty, the HARMEAN function will return an error.
4. The HARMEAN function can only handle a maximum of 255 arguments:
The HARMEAN function can only handle a maximum of 255 arguments, so if you have more than 255 numbers to include in the calculation, you will need to use a different method.
Commonly Used Functions Along With HARMEAN
Here are a few commonly used functions that are often used along with the HARMEAN function in Google Sheets:
1. SUM: The SUM function allows you to add up a range of cells or a list of numbers. For example, you could use the SUM function to add up the numbers in a range of cells that you want to include in
a harmonic mean calculation with the HARMEAN function. For example:
=HARMEAN(SUM(A1:A4), SUM(B1:B3))
This would calculate the harmonic mean of the sum of the numbers in cells A1:A4 and the sum of the numbers in cells B1:B3.
2. AVERAGE: The AVERAGE function allows you to calculate the arithmetic mean (or average) of a range of cells or a list of numbers. You can use the AVERAGE function along with the HARMEAN function
to compare the harmonic mean and arithmetic mean of a set of numbers. For example:
This would calculate the harmonic mean and arithmetic mean of the numbers in cells A1:A4 and display the results in two separate cells.
3. COUNT: The COUNT function allows you to count the number of cells in a range that contain numerical values. You can use the COUNT function along with the HARMEAN function to ensure that you are
including the correct number of values in your harmonic mean calculation. For example:
This would calculate the harmonic mean of the numbers in cells A1:A4 and display the number of values included in the calculation in a separate cell.
The HARMEAN function in Google Sheets is a built-in function that allows you to calculate the harmonic mean of a set of numbers. The harmonic mean is a type of average that is used to calculate the
mean of a set of numbers that are all reciprocals of each other, such as rates or speeds. To use the function, you simply need to provide it with a range of cells containing the numbers you want to
include in the calculation, and it will return the harmonic mean as a result.
There are a few limitations to keep in mind when using the HARMEAN function, such as the fact that it only works with numerical values and is not defined for zero or negative numbers. Additionally,
the function does not work with empty cells and can only handle a maximum of 255 arguments.
In summary, the HARMEAN function is a useful tool for calculating the average of rates or ratios in Google Sheets. If you have a set of numbers that are reciprocals of each other and want to find
their average, give the HARMEAN function a try! You might be surprised at how easy it is to use and how useful it can be in your data analysis and statistical calculations.
Video: HARMEAN Function
In this video, you will see how to use HARMEAN function. We suggest you to watch the video to understand the usage of HARMEAN formula.
Related Posts Worth Your Attention
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Power BI Blog: LINEST and LINESTX Now Added to Power BI
Welcome back to
this week’s edition of the Power BI blog series. This week, we look at two new DAX functions
just added to the Power BI DAX library.
Two new statistical DAX functions have just been added to the Power BI repertoire: LINEST and LINESTX. These two functions
perform linear regression, leveraging the Least Squares method, to calculate a
straight line that best fits the given data and return a table describing that
line. These functions are especially
useful in predicting unknown dependent values (y) given known independent
values (x).
Both functions
return a single-row table describing the line and additional statistics. The
resulting table includes columns such as slopes, intercepts, standard errors
and the coefficient of determination. The
equation of the fitted line can be constructed as follows
y = Slope1 * x[1] + Slope2 * x[2] + … + Intercept.
The difference
between LINEST and LINESTX is that LINEST expects columns
of known x and y values to be provided, whereas LINESTX expects a table and expressions to be evaluated for each row of the table to
obtain the x and y values.
For the following
examples, consider the following data, which includes Sales Amount and Gross
National Product, GNP_Per_Capita:
In the example
below, we will use LINESTX to predict total sales based upon GNP per
LinestX_example =
VAR CountryGNP =
Sales”, SUM(Sales[Sales Amount])
VAR SalesPrediction
= LINESTX(
[Total Sales],
Example_GNP_Per_Capita = 50000
) *
Example_GNP_Per_Capita +
This expression not
only leverages LINESTX but also leverages the result to perform a
prediction for a fictitious country with gross national product per capita of
$50,000. The result is a predicted total
sales of $17,426,123.29. Of course, this
is a fabricated scenario and it’s rare to have a fixed value such as the
$50,000 above as part of the expression.
We may do the same
using LINEST assuming the required tables are all in the model, e.g. as calculated tables. In this example, we’ve
added the following calculated tables:
• CountryDetails, defined as:
Sales”, SUM(Sales[Sales Amount]))
• SalesPredictionLINEST, defined as:
= LINEST(‘CountryDetails'[Total Sales], ‘CountryDetails'[GNP_Per_Capita]).
Now we may use
following measure expression to obtain the same result as above:
Linest_example =
Example_GNP_Per_Capita = 50000
MAX (
SalesPredictionLINEST[Slope1] ) * Example_GNP_Per_Capita
+ MAX ( SalesPredictionLINEST[Intercept]
In the meantime, please remember we offer training in Power
BI which you can find out more about here. If you wish to catch up on past articles,
you can find all of our past Power BI blogs here.
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Cross electromagnetic nanofluid flow examination with infinite shear rate viscosity and melting heat through Skan-Falkner wedge
This demonstration of study focalizes the melting transport and inclined magnetizing effect of cross fluid with infinite shear rate viscosity along the Skan-Falkner wedge. Transport of energy
analysis is brought through the melting process and velocity distribution is numerically achieved under the influence of the inclined magnetic dipole effect. Moreover, this study brings out the
numerical effect of the process of thermophoresis diffusion and Brownian motion. The infinite shear rate of viscosity model of cross fluid reveals the set of partial differential equations (PDEs).
Similarity transformation of variables converts the PDEs system into nonlinear ordinary differential equations (ODEs). Furthermore, a numerical bvp4c process is imposed on these resultant ODEs for
the pursuit of a numerical solution. From the debate, it is concluded that melting process cases boost the velocity of fluid and velocity ratio parameter. The augmentation of the minimum value of
energy needed to activate or energize the molecules or atoms to activate the chemical reaction boosts the concentricity.
Bibliographical note
Publisher Copyright:
© 2022 the author(s), published by De Gruyter.
• 2-D cross fluid
• Brownian motion
• inclined magnetized flow
• infinite shear rate viscosity
• melting process of energy
• thermophoresis diffusion
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A conceptual look at Bellman operator - Intuitive Tutorials
A conceptual look at Bellman operator
Bellman operators come in Reinforcement Learning (RL). When I first encountered it, I had many questions regarding it. I often feel it is interesting to observe that what feels intriguing to someone.
Many questions surface out in our minds.
Why is it called an operator?. What are the inputs to it and output from it?. What properties does it hold?. Who is Bellman anyway?. Regarding Bellman operator, I spent some time answering such
questions. This blog post is a documented version of the answers I found.
The Mathematical Operator
From the mathematical point of view, an operator is a mathematical object that does some transformation to its input.
A simple example could be $y=x+1$, in this mathematical expression whatever be the quantity in $x$ will be incremented by 1 and will be represented by the variable $y$. The ‘$+$’ here is a
mathematical operator.
Graphical illustration of mathematical operator
Wikipedia defines a mathematical operator as a function or mapping that acts on elements in one space and produce elements of another space or even the same space. Here the space in this example is a
set of real numbers. Adding 1 to them maps them to the same space itself.
It is through one or more combinations of such operators we describe transformation happening to mathematical quantities. Operators have many properties like linearity which often makes the analysis
That being said, what operation does a Bellman operator do?. On what quantities does it operate?. What are the properties does it hold?. Before trying to answer all these, let’s see who is Bellman.
Some Historical Notes
Richard E. Bellman (Source: wikipedia.org)
Richard Ernest Bellman was an American applied mathematician. If you are from the tech industry chances are high that you probably heard about him before.
Perhaps the most popular thing from his inventory in today’s scenario is the ‘curse of dimensionality‘. It relates to the problem of an exponential increase in computational difficulty while adding
extra dimensions to the mathematical space.
The driving technologies of today’s world are Artificial Intelligence (AI) and Machine Learning (ML). But we can say they were born realizing the challenges posed by the curse of dimensionality. AI
and ML use approximate solution methods to tackle this problem.
He is also known for Dynamic Programming (DP) which again tries to tackle the exponential dimensionality problem. We can say that DP follows the idea of “don’t reinvent the wheel” and make use of
intermediate results for computations.
Bellman Operator in Reinforcement Learning
Bellman’s ideas are best applied in problems of recursive nature. In RL we have the value function whose present value depends upon the value in the previous state. The value function assigns a
numerical value for being in a state. The higher the value will be the better the state we are in.
In simple terms, my bank balance today will be depending upon my balance from the past. A student’s final grades will depend upon all the hard work she/he will put in many days before the final exam.
The recursive nature of returns in RL and some visual intuitions are documented here.
In the form of an equation, the Bellman equation says the value of being in the current state will be the immediate reward plus the discounted version of all the future reward an agent is going to
get. Lets us take a look at a simplified version of the Bellman equation.
$v(s) = \cal{R}_s+ \gamma\sum \cal{P}_{ss’} v(s’) $
Here $v$ represents the value function, $s$ current state, $s’$ next state, $R_s$ is the immediate reward, $\gamma$ is the discount factor, and $\cal{P}_{ss’}$ is the state transition probability.
This equation concisely tells that the value of being in the current state will be the immediate reward plus the discounted value of the next state value. Bellman equation holds this relation for all
The above equation ties up all such relations into a general equation using the properties of linear algebra. When expanded, the equation will look like the following.
$\left. \begin{bmatrix} v_\pi(1) \\ \vdots \\ v_\pi(n) \end{bmatrix}= \begin{bmatrix} \cal{R}_1^\pi \\
\vdots \\ \cal{R}_n^\pi \end{bmatrix} +\gamma \begin{bmatrix} \cal{P}_{11}^\pi & \dots & \cal{P}_{1n}^\pi\\
\vdots & \ddots & \vdots\\ \cal{P}_{n1}^\pi & \dots & \cal{P}_{nn}^\pi \end{bmatrix} \begin{bmatrix}
v_\pi(1) \\ \vdots \\ v_\pi(n) \end{bmatrix} \right.$
Here the additional $\pi$ represents the policy an agent is following.
As you can see here, according to the equation, each state value is equated as a sum of respective immediate rewards plus discounted future values. This means the Bellman operator operates on the
value function.
Mathematically this can be defined as an operator as shown below
$\cal{T^\pi}(v) = \cal{R}^\pi + \gamma \cal{P}^\pi v$
Hence Bellman operator maps state values to its next state values. The significance lies in the fact that Bellman operators have the property called contraction mapping. In simple terms, repeated
application of the Bellman operator converges the value function to a fixed value.
Closing thoughts
We now have at least some first answers to the questions we had. As of now, today’s leading technologies rely heavily on approximation methods rather than analytical solutions. These concepts from
Bellman have a different viewpoint on problem-solving by leveraging the recursive nature of problems. Who knows whether time will show us the rebirth of analytical methods!.
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Condense Logarithms Using The Quotient Property Worksheets [PDF]: Algebra 2 Math
How Will This Worksheet on "Condense Logarithms Using the Quotient Property" Benefit Your Student's Learning?
• Condensing logarithms using the quotient property simplifies math problems by combining separate logarithmic terms that are being subtracted into a single logarithm representing their division.
• This makes calculations faster because we are dealing with fewer steps.
• It also helps develop stronger math skills by teaching how to handle complex expressions effectively.
• Mastering this technique prepares students for more advanced math subjects by building a solid understanding of logarithmic principles.
• Additionally, it reduces the chances of errors in calculations by presenting clearer and more straightforward forms of logarithmic expressions.
How to Condense Logarithms Using the Quotient Property?
• Begin with separate logarithms that are subtracted, such as \(\log_b(x) - \log_b(y)\).
• Use the quotient property of logarithms, which states `\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)`.
• Condense the expression by converting the subtraction of logarithms into a single logarithm of their quotient.
• Ensure that the condensed logarithm accurately represents the original subtraction of logarithms, maintaining clarity and correctness in mathematical operations.
Q. Condense the logarithm. Assume all expressions exist and are well-defined.$ewline$$\log_2 9 - \log_2 4$ | {"url":"https://www.bytelearn.com/math-algebra-2/worksheet/condense-logarithms-using-the-quotient-property","timestamp":"2024-11-10T04:29:33Z","content_type":"text/html","content_length":"119081","record_id":"<urn:uuid:30aef6ae-8a55-47d1-bb97-0ada4e686643>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00042.warc.gz"} |
How to calculate effect sizes for predictors in lavaan regression model?
When conducting regression analyses in R, particularly with the lavaan package, one critical aspect researchers need to consider is effect sizes for predictors. Understanding effect sizes provides
insights into the strength of relationships between variables, enhancing the interpretability of your results.
Problem Scenario
Original Problem: "create me article about: How to calculate effect sizes for predictors in lavaan regression model?"
To put it simply, you're looking to learn how to calculate effect sizes for predictors within a regression model utilizing the lavaan package in R.
What is Lavaan?
The lavaan package in R (short for "latent variable analysis") is designed for structural equation modeling (SEM). However, it is also used to perform traditional regression analyses, allowing users
to evaluate direct and indirect effects among variables in complex models.
Calculating Effect Sizes
To calculate effect sizes in a lavaan regression model, one can use the R-squared values and standardized coefficients (often referred to as Beta coefficients). The following steps outline the
Step 1: Install and Load the Lavaan Package
Make sure the lavaan package is installed and loaded into your R session:
Step 2: Specify Your Model
Assume you have a dataset and you want to build a regression model. Here's an example model:
model <- '
# Regression
outcome_variable ~ predictor_variable1 + predictor_variable2
Step 3: Fit the Model
You can fit the model using your data. Here is an example with fictitious data:
data <- data.frame(
outcome_variable = rnorm(100),
predictor_variable1 = rnorm(100),
predictor_variable2 = rnorm(100)
fit <- sem(model, data = data)
Step 4: Extract the Summary for Effect Sizes
You can extract the summary of the fitted model, which includes coefficients, R-squared values, and fit indices:
summary(fit, standardized = TRUE)
In the summary output, pay attention to the standardized coefficients (under Std.lv or Std.all) to assess the effect sizes of your predictors. The R-squared values will indicate how much variance in
the outcome variable is explained by the predictors.
Step 5: Interpreting Effect Sizes
• Standardized coefficients: A standardized coefficient of 0.3, for instance, indicates a small effect size, while 0.5 indicates a medium effect size, and 0.8 or higher indicates a large effect
• R-squared: Values closer to 1 indicate a greater proportion of variance explained. For example, an R-squared value of 0.25 indicates that 25% of the variance in the outcome variable is explained
by the predictors.
Practical Example
Let's look at a practical example of calculating effect sizes.
Assuming you built a regression model predicting academic performance from study hours and attendance:
model <- '
AcademicPerformance ~ StudyHours + Attendance
fit <- sem(model, data = your_data)
summary(fit, standardized = TRUE)
The output might show standardized coefficients like this:
• StudyHours: 0.45 (medium effect)
• Attendance: 0.30 (small effect)
This output suggests that for every one standard deviation increase in Study Hours, Academic Performance increases by 0.45 standard deviations.
Calculating effect sizes for predictors in a lavaan regression model is essential for understanding the impact of independent variables on dependent outcomes. By using standardized coefficients and
R-squared values, researchers can present more informative and interpretable results.
Useful Resources
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MIDAS - Microwave Inductors Design Automation on Silicon
MIDAS: The "golden touch" for
Microwave Inductors Design
Automation on Silicon
MIDAS can cover the range of inductances from about 0.1 to 5nH according to the following approximated formula [1]:
Where N is the number of turns, w is the track width, s is the spacing between adjacent tracks dout, din and davg refer to the outer, inner and average diameter of the spiral to be drawn. The
coefficients c1, c2, c3 and c4 given in Table 1 allow the formula to be adapted to both octagonal and square spiral inductors.
Table 1. Values of the coefficients.
Figure 2. Example of an octagonal spiral inductor with PGS created by the EM Structure Simulator.
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Zariski's Main Theorem - (Commutative Algebra) - Vocab, Definition, Explanations | Fiveable
Zariski's Main Theorem
from class:
Commutative Algebra
Zariski's Main Theorem states that every ideal in a Noetherian ring can be expressed as a finite intersection of primary ideals, and the primary components can be associated with the prime ideals of
the ring. This theorem is fundamental in understanding the structure of ideals and their relationships with algebraic varieties, connecting primary decomposition to geometric properties such as Krull
dimension and the behavior of varieties under morphisms.
congrats on reading the definition of Zariski's Main Theorem. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Zariski's Main Theorem provides a way to decompose ideals into primary ideals, which helps to understand their structure more clearly.
2. The primary components associated with an ideal reveal important geometric information about the variety defined by that ideal.
3. Every primary ideal corresponds to a unique prime ideal, which helps link algebraic concepts to geometric structures.
4. In Noetherian rings, Zariski's Main Theorem assures that this decomposition is finite, simplifying many algebraic computations.
5. The theorem plays a crucial role in algebraic geometry by connecting the algebraic properties of rings with the topological properties of varieties.
Review Questions
• How does Zariski's Main Theorem facilitate the understanding of the structure of ideals in a Noetherian ring?
□ Zariski's Main Theorem shows that any ideal in a Noetherian ring can be expressed as a finite intersection of primary ideals. This decomposition into primary components allows for clearer
insight into how these ideals interact and relate to each other. It emphasizes the importance of prime ideals and their role in determining the geometric properties of algebraic varieties.
• Discuss the implications of Zariski's Main Theorem on the relationship between algebraic geometry and commutative algebra.
□ Zariski's Main Theorem bridges commutative algebra and algebraic geometry by linking ideals to geometric objects. The theorem allows us to interpret primary decompositions as geometric
intersections of varieties. Consequently, it provides tools for studying how algebraic structures influence geometric properties, such as dimension and singularity, deepening our
understanding of both fields.
• Evaluate how Zariski's Main Theorem influences the study of Krull dimension and its properties in commutative algebra.
□ Zariski's Main Theorem impacts the study of Krull dimension by highlighting how the decomposition of ideals relates to chains of prime ideals. Since primary components correspond to prime
ideals, analyzing these decompositions allows for insights into the lengths of chains that define Krull dimension. This interconnection reinforces the significance of dimensionality in both
algebraic and geometric contexts, demonstrating how structural properties of rings can influence their dimensional characteristics.
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Gamers solve problems faster than a computer
Humans routinely find better solutions to difficult quantum mechanical problems than computers, shows new research.
Computers might be able to beat us at chess but we are still better at recognising patterns, and apparently, we are also better at finding solutions to quantum mechanical problems.
In an article published in Nature, scientists explain how gamers can beat computers, when it comes to figuring out how to move atoms quickly.
”We were very surprised to find that gamers were able to come up with results on the other side of what we thought was the quantum speed limit-- that is, the maximum speed at which the computer can
operate,” says Jacob Sherson, associate professor and leader of Centre for Community Driven Research (CODER), Aarhus University, Denmark.
The scientists can use this new knowledge to help design quantum computers.
The basic unit of information in computing is a bit, which has a value of either '0' or '1'.
In quantum computing, the basic unit of information is the quantum bit (sometimes qubit or qbit).
A quantum bit can simultaneously have a value of ‘0’ or ‘1’ at any one time.
Quantum bits provide many new possibilities for calculations and a quantum computer will be able to solve certain tasks much faster than a normal computer.
Watch the video at the bottom of the article to learn more about how gamers are helping scientists develop a quantum computer.
Quantum speed limit holds back the quantum computer
In the new study, scientists simulated a quantum computer with quantum bits that the gamers could manipulate. But this is not a straightforward task, says Sherson.
”The challenge is that the individual quantum bits connect with the surroundings and then the quantum information flows out of them. We have to make calculations before the information is lost and
that is a big challenge,” he says.
The physicists know that there is a fundamental limit to how quickly things can happen at quantum level as set out by the laws of quantum mechanics. But it is impossible to calculate this quantum
speed limit for complex systems.
To move an atom in a quantum computer is comparable to moving a glass of water, which is full to the brim. It is easy if you take it nice and slow, but if you run, the water will slosh around the
glass and spill. The challenge is finding the quickest way to move the water without spilling it. And that, is what the gamers have helped to solve.
10,000 players help beat the quantum limit
The gamers helped Sherson and his colleagues to calculate the quantum limit. Computers simulated a process, such as moving an atom, multiple times to see how quickly they could finish the task.
Sherson and his colleagues thought they knew approximately where the limit would be for their quantum computer, but they quickly revised this estimate after almost 10,000 gamers played the game
Quantum Moves half a million times.
It turned out that some of the players were able to finish one part of the game much quicker than the computer. And the computer tried it 100 million times.
”The computer was unable to find any solutions in under 0.26 seconds,” says Sherson.
“We would normally trust that the computer worked near the quantum speed limit. So it’s very exciting, that the gamers were able to come up with solutions on the other side of that limit,” says
Computers can learn from humans
The scientists assumption, that the computer would find the limit turned out to be wrong. Some of the players were intuitively better at finding solutions to the complex problem.
Based on the results of the repeated games, the scientists were able to improve the computer algorithm and essentially taught the computer to use the players' methods. This led to even better results
and the quantum speed limit increased again.
The result is good news for the development of quantum computers. It proves that it is possible to perform processes more quickly and efficiently than scientists previously thought.
Video: Aarhus University
Read the Danish version of this story on Videnskab.dk
Translated by: Stephanie Lammers-Clark
Scientific links
External links
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Logarithms do not only involve the use of logarithm tables; it also works with the use of the logarithms laws which are interrelated to the laws of indices. Moreover, the laws of indices have
equivalent laws of logarithms as we will see in this article.
In this article, we are going to look at the laws of logarithms and examples of all laws with step-by-step solutions. We will likewise drop the worksheet. This will contain quizzes similar to the
examples in the article.
The following are the different rules that can be used when solving problems on logarithms:
Product Law
In some clans, this can be called the ‘addition law‘. It is the same, this simply implies that when two logarithms of the same base are added, the result is the product of the two logarithms to their
common base. That is, ${\color{Emerald}&space;loga^{x}+&space;loga^{y}=loga^{xy}}$On the other hand, this can be the logarithm of products of two numbers with a common base. Which will result in the
addition of the two numbers with the same base. This can be represented by ${\color{Emerald}&space;log&space;a^{MN}=log&space;a^{M}+loga^{N}}$.
For example, $log2^{8}=log2^{(2\times&space;4)}$ which will give $log2^{2}&space;+&space;log&space;2^{4}$.
Quotient Law
This is also referred to as the ‘subtraction law‘. It shows that when logarithms are subtracted, the one being subtracted is used to divide the other, all expressed in their common base. That is,
On the other hand, these can be written as ${\color{Emerald}&space;loga^{\frac{M}{N}}=loga^{M}-loga^{N}}$.
The questions on these two laws can come in any of the mentioned forms. The proper application of the laws is what is required.
For example, $log10^{\frac{1000}{100}}=log10^{1000}-log10^{100}$.
Power Law
When a logarithm is raised to a certain power, the power is used to multiply the logarithm itself. This law is expressed in the form: ${\color{Emerald}&space;loga^{M^{p}}=&space;Ploga^{M}}$
For example, $log2^{2^{3}}=3log2^{2}$.
Apart from the above laws and more that will be discussed later, there are principles that can be applied when solving questions on the laws of logarithms. We can call them the key logarithm rules.
These rules must be known since they will most probably be applied in some of these laws.
Recommended: Standard form of a number
There are two of these rules:
1. Logarithms to its own Base: The logarithm of any value to its own base is equal to 1. That is, $loga^{a}=1$. For example, $log10^{10}=1,&space;log8^{8}=1,&space;log100^{100}=1$ etc.
2. Logarithm of 1: The logarithm of $1$ to any base is equal to zero. Tha is, $loga^{1}=0$, where $a$ ≠ $1$. For example, $log10^{1}$.
To prove that $log10^{1}$ is equal to zero, Let $log10^{1}=x$.
According to a logarithm law, the above becomes $10^{x}=1$.
Recall that $10^{0}=1$.
So $10^{x}=10^{0}$ (the 10 will clear the 10),
∴ $x=0$
So you see, $log10^{1}=0$.
We will see the application of these various laws in complex examples, later in this article. Before then, let’s look at other laws of logarithms. Consider the fractional power law.
Fractional Power law
In some cases, a logarithm might have fractions as its powers. These are fractional power logarithms, and it can be solved like the power law. It is represented by,
Which can also be expressed as ${\color{Emerald}&space;\frac{xloga^{M}}{y}}$
For example, $log10^{100^{\frac{3}{2}}}=\frac{3}{2}log10^{100}=\frac{3log10^{100}}{2}$
Root Law
This is a special case of fractional power law. In this case, $x=1$ and the root law is applied. Recall from the laws of indices that, $a^{\frac{1}{2}}=\sqrt{a},&space;a^{\frac{1}{4}}=\sqrt[4]{a}$
etc. So root law can be expressed as
Note, this can also be written as $\frac{1}{y}loga^{M}=\frac{loga^{M}}{y}$
Whatever method you use in solving questions like this, you will arrive at the same result.
For example, $log2^{16^{\frac{1}{4}}}=log2^{\sqrt[4]{16}}$.
Reciprocal Law
When the reciprocal of a logarithm is required, the base and the number interchange their positions. That is, ${\color{Emerald}&space;loga^{M}=\frac{1}{logM^{a}}}$
For example, $log2^{8}=\frac{1}{log8^{2}}$
Change of Base Law
This law shows that when the base of a logarithm is changed, the initial base is used as a separate logarithm to divide the initial logarithm. All to the new base. That is, ${\color{Emerald}&space;
From the above, the initial base divides the initial logarithms which is ‘$M$‘, all to the same new base ‘$x$‘.
For example, $log100^{1000}=\frac{log10^{1000}}{log10^{100}}$.
Finally, let’s consider some examples, explaining the above formulas. As you go through these examples, see how the laws are applied, that will help you assimilate them properly and also apply them
when necessary.
We are going to consider a series of examples that will explain the practical application of the above laws.
Example 1: Simplify $log8^{16^{-2}}$
From the above question, $log8^{16^{-2}}=&space;-2log8^{16}$ (applying the Power law)
$-2log8^{16}=-2(\frac{log2^{16}}{log2^{8}})$ (change of base law)
Simplify further by reducing $16$ and $8$,
$-2(\frac{log2^{2^{4}}}{log2^{2^{3}}})$$=-2(\frac{4log2^{2}}{3log2^{2}})$, Recall one of the rules we discussed above, rule 1, $(log2^{2}=1)$
Example 2: Simplify $\frac{1}{2}log5^{25}-&space;log5^{0.2}$
Firstly, let’s convert the decimal to fractions, $0.2=\frac{2}{10}=\frac{1}{5}$,
Secondly, lets apply the power law and this law of indices $\frac{1}{5}=5^{-1}$,
Applying the rule, $log5^{5}=1$ and remembering that $-\times&space;-=+$ gives us
Example 3: Simplify $2log3^{6}+log3^{12}-log3^{16}$ (WAEC)
Use the product and quotient laws, since all the values have the same base,
Applying the Power law,
$3log3^{3}$ (recall that $log3^{3}=1$)
∴ $3\times&space;1=3$
Example 4: If $log2=0.3010,$ evaluate $log&space;32$.
Using the product law,
$=log10^{2^{3}}+log10^{2^{2}}$ (Applying the power-law)
since $log2=log10^{2}=0.3010$
In conclusion, let us learn how to relate logarithms and indicial equations. An indicial equation is an equation involving one or more unknowns as the exponent (or index). The example below will
explain this further.
Example 5: Solve the logarithm equation $log9^{(\frac{1}{3})^{x}}-log9^{(\frac{1}{9})^{x}}=2$
Applying the logarithm power law,
Using the division rule, $\frac{1}{3}$ ÷ $\frac{1}{9}=3$
So $x(log9^{\frac{1}{3}}-log9^{\frac{1}{9}})=2$ gives $xlog9^{3}=2$
Reversing the power law, $log9^{3^{x}}=2$
Recall that if $log2^{x}=6$, $x=2^{6}$. Applying that to the above, we get
$3^{x}=3^{2^{2}}$ (clear the common base, 3)
∴ $x=4$
Example 6: Solve for x in the logarithm equation $log_{3}{x}+log_{3}{(x-8)}=2$. Watch the video below for the solution.
For questions on this topic, “Logarithms Laws and Example” download the law of logarithms work Sheet. Click HERE to download.
For insight on this topic and more, get a copy of the textbook “New Track Mathematics for JAMB UTME SERIES 1” It contains comprehensive mathematics step-by-step solutions with JAMB Past Questions and
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Electric Drive for Automation and Robotics
Code Completion Credits Range Language
B3B14EPR Z,ZK 4 2P+2L Czech
In order to register for the course B3B14EPR, the student must have registered for the required number of courses in the group BEZBM no later than in the same semester.
Course guarantor:
The course gives a brief overview of basic types of electric drives. It deals with drives with DC, asynchronous, synchronous and special motors including power electronic converters.Another
topics include control strategies such as scalar, vector, direct, sensorless control of AC drives, pulse width modulation strategies and various load types. It is focused on understanding the
physical nature of a given type of drive, general derivation of basic differential equations describing transient and steady states, and creating corresponding mathematical models of analyzed
systems suitable for both off-line simulation and online-adapted dynamic and real-time control using the basis of modern microprocessor technology. Problems of operating states, sensors and
diagnostics of electric drives are also discussed. Basic knowledge of mathematics, mechanics, kinematics, dynamics, theory of electromagnetic field, circuit theory and control theory are assumed.
Syllabus of lectures:
1. Energy sources, batteries and accumulators, solar batteries, fuel cells, materials used in electric drives
2. DC drives for automation - DC - commutator with self-supporting winding
3. Electronically commutated drives for automation - EC
4. Asynchronous multiphase cage motor - ASM
5. Universal serial engine - mass use
6. Stepper, reluctance motor, Linear electric drives in automation - principle and control
7. Mathematical model of BLDC motor
8. Drive kinematics
9. Drive control
10. Drive control
11. Drive control
12. Drive control
13. Drive control
14. Drive control
Syllabus of tutorials:
1. Measurement on MAXON drives - setting of current, speed and position controller - digital control
2. Measurement of basic parameters of DC motor - mathematical model Ra, La …….
3. Measurement of DC motor start-up - comparison of real measurement with model in MatLab
4. Measurement of asynchronous motor - type test - automated system NI - PXI
5. Linear drive - motion trajectory setting, stepper motor - working characteristics
6. Drive control using PLC system - DC motor, AM asynchronous motor
7. Real application STM32 Nucleo - BLDC servo drive control
8. Modeling of servomotors in MATLAB -
9. Assignment of individual work
10. individual work
11. individual work
12. individual work
13. individual work
14. Homework control - credit
Study Objective:
Study materials:
1. Dr. Urs Kafader - Selection of high-precision microdrives CH-6072 Sachsen / Switzerland 2006 ISBN 3-9520143
2. Maxon motor ag - Magnetism - Basics, Forces, Applications CH-6072 Sachsen / Switzerland 2008 ISBN 978-3-9520143-5- 6
3. Dr. Otto Stemme, Peter Wolf, - Principles and Properties of Highly Dynamic DC Miniature Motors - Interelectric AG, CH- 6072 Sachsen / Switzerland 1994
4. Formulae Handbook, Jan Braun, maxon Academy, Sachseln 2012
5. Francis H. Raven - Automatic Control Engineering, McGraw-Hill, Inc. ISBN 0-07-051341-4, 1995
6. Elektrické stroje – Teoria a príklady – V.Hrabovcová, P.Rafajdus – Žilina 2009 ISBN 978-80-554-0101-0
7. Moderné alaktrické stroje – V.Hrabovcová, P.Rafajdus – Žilina 2009 ISBN 978-80-554-0101-0
Time-table for winter semester 2024/2025:
Time-table is not available yet
Time-table for summer semester 2024/2025:
Time-table is not available yet
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Back to Papers Home Back to Papers of School of Physics
Paper IPM / P / 15975
School of Physics
Title: Black hole subregion action and complexity
Author(s): 1. M. Alishahiha
2. K. Babaei Velni
3. M.R. Mohammadi Mozaffar
Status: Published
Journal: Phys. Rev. D
No.: 12
Vol.: 99
Year: 2019
Pages: 126016
Supported by: IPM
We evaluate finite part of the on-shell action for black brane solutions of Einstein gravity on different subregions of spacetime enclosed by null boundaries. These subregions include the
intersection of the Wheeler-DeWitt patch with past/future interior and left/right exterior for a two-sided black brane. Identifying the on-shell action on the exterior regions with subregion
complexity, one finds that it obeys the subadditivity condition. This gives an insight to define a new quantity named mutual complexity. We will also consider a certain subregion that is a part of
spacetime, which could be causally connected to an operator localized behind/outside the horizon. Taking into account all terms needed to have a diffeomorphism-invariant action with a well-defined
variational principle, one observes that the main contribution that results in a nontrivial behavior of the on-shell action comes from joint points where two lightlike boundaries (including the
horizon) intersect. A spacelike boundary gives rise to a linear time growth, while we have a classical contribution due to a timelike boundary that is given by the free energy.
Download TeX format
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